Cardiac perfusion and pumping engineering

Cardiac Perfusion and Pumping Engineering This page intentionally left blank Cli n i c a l l y - O rien ted B iomed...

Author: N. Ghista Dhanjoo | Eddie Yin-kwee Ng

155 downloads 1243 Views 13MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Cardiac Perfusion and Pumping Engineering

This page intentionally left blank

Cli n i c a l l y - O rien ted B iomedical Engineer ing — Vol. 1

Cardiac Perfusion and Pumping Engineering

Dhanjoo N. Ghista Eddie Yin-Kwee Ng Nanyang Technological University, Singapore

World Scientific NEW JERSEY

•

LONDON

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Clinically-Oriented Biomedical Engineering — Vol. 1 CARDIAC PERFUSION AND PUMPING ENGINEERING Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-696-6 ISBN-10 981-270-696-8

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

Steven - Cardiac Perfusion.pmd

1

8/29/2007, 3:47 PM

July 4, 2007 21:17 WSPC/SPI-B421 Cardiac Perfusion and Pumping Engineering Fm

REVIEWS “Cardiac Perfusion and Pumping are inter-related, and hence together constitute an intriguing phenomenon that governs cardiac performance. This fascinating book is unique in addressing the clinical features and bioengineering characteristics of cardiac perfusion and pumping. The book also addresses assisted pumping (in the form of left ventricular assist devices) and cardiac tissue engineering (to replace and regenerate myocardial infarcts). The bioengineering formulations of the various chapters are not only sufficiently rigorous to be instructive for bioengineering courses, but are also clinically oriented. The book should be very useful to biomedical engineers as well as cardiologists and cardiac surgeons.” Ghassan S. Kassab Thomas J. Linnemeier Guidant Endowed Chair and Professor of Biomedical Engineering Professor of Surgery, Cellular and Integrative Physiology Indiana-Purdue University, USA “This book is unique! It describes the multiple aspects of cardiac ventricular contraction (provided by cardiac perfusion and pumping characteristics), by observing the specific features associated with contraction with an amplified attention. The result is an intermediate step between the well-accepted biomechanical interpretations and fully new and original scientific descriptions. This book is not just a book of a collection of accepted and validated concepts. Nor, is it a special issue of a scientific journal (devoted to perfusion and pumping), with proposals of new concepts and descriptions. Rather, it is an intermediate status of the matter. If you are searching for concrete aspects, you will find therein the relevant models and rules currently accepted by the scientific community. However, if your mind wants to fly out of the actual constraints, you have so many opportunities to compare your most original ideas with those described by the authors, that you will be engaged in a fascinating game. You will find in the book an acceptable and agreeable scent of science, which impregnates every page of the book and drags a bright mind into a knowledge paradise!” Romano Zannoli Professsor of Medical Physics University of Bologna, Italy v




PREFACE Cardiac pumping is dependent on cardiac perfusion. Hence, it is only natural that we address both cardiac perfusion and pumping in this book. We have gone one step further in also considering assisted perfusion by coronary bypass surgery and myocardial regeneration by means of stem cells transformed into implantable cardiomyocytes. The book is hence divided into three sections: (1) Cardiac Perfusion, (2) Cardiac Pumping Characteristics, (3) Assisted Perfusion and Pumping, and Myocardial Repair. Section I on Cardiac perfusion starts out with the chapter on physiomics of coronary microcirculatory perfusion, which supplies nutrients to the heart myocardium for its contraction. The following chapter deals with the phenomenon of myocardial inhomogeneity, and provides an answer to this enigma as the basis of providing cardiac functional reserve. The next chapter is on quantification of cardiac perfusion and function, using nuclear cardiac imaging. The final chapter in this section is a synthesis of cardiac perfusion and pumping. It analyses left-ventricular (LV) pumping in terms of intra-ventricular blood flow velocity and pressure distributions, and also computationally depicts the distribution of pressure and flow velocities in discrete regions of the heart. Section II is on how cardiac pumping is initiated by myocardial contraction, causing stresses and strains in the myocardium. More importantly, the mechanism of how myocardial contraction causes LV torsion is also discussed. The LV myocardial fibers are spirally wound inside its wall. Thereby, when they contract, the LV twists and then unwinds during relaxation. While this LV torsion is an end-product of LV myocardial fibers geometry and contraction, it can also serve as an index of contractility. Finally, in order for LV to contract or to depict its inability to contract adequately for adequate blood outflow, we have developed indices to assess its contractility, in the form of some intrinsic indices that correlate well with the traditional contractility index of (dP /dt)max . In doing so, we have also addressed a hitherto unexplained phenomenon of LV suction during its early filling state. The LV sarcomere is still contracting (albeit decreasing vii


viii

Preface

its contractile force and shortening velocity) during early filling. It is this mechanism that causes LV suction (and resulting LV pressure decrease), before the left atrium starts to contract and cause LV filling. Cardiovascular disease results in in vivo bioelectrical abnormalities, blocked coronary vessels and consequently myocardial infarcts. Hence, it is but natural that this book also deals (in Section III) with methods of augmented myocardial perfusion by coronary bypass surgery, mechanical circulatory support in the form of cardiac-assist devices, and myocardial regeneration by means of implantable three-dimensional cardiomyocyte constructs converted from embryonic stem cells. In this regard, this section discusses in detail the reasons for blocked distal coronary graft-occluded vessel anastomosis, the mechanical analysis of circulatory-support ventricularassist systems, and in particular of an axial blood pump. The last two chapters deal with the basis and prospects of tissue engineering and novel approaches to cell transplantation following the conversion of embryonic stem cells into cardiomyocyte scaffolds for implantation. We hope that this book can serve as a major reference resource in cardiology and cardiac surgery, as well be employable as a course text for a course on this topic in a biomedical engineering program. With these aims in mind, the individual chapters deal in adequate rigor with both theory and clinical applications. Thank you. Dhanjoo N. Ghista and Eddie Yin-Kwee Ng Nanyang Technological University Singapore 639798 October 2006


CONTENTS Reviews Preface Book Summary

v vii xiii

Section I. Cardiac Perfusion Chapter 1. Physiomics of Coronary Perfusion and Cardiac Pumping Fumihiko Kajiya, Masahito Kajiya, Taro Morimoto, Tatsuo Iwasaki, Yousuke Inai, Masanori Hirota, Takahiko Kiyooka, Yuki Morizane, Takehiro Miyasaka, Satoshi Mohri and Juichiro Shimizu Chapter 2. Left Ventricular Inhomogeneity and the Heart’s Functional Reserve Felix Blyakhman Chapter 3. Quantification of Cardiac Perfusion and Function Using Nuclear Cardiac Imaging Ru-San Tan, Liang Zhong, Terrance Chua and Dhanjoo N. Ghista Chapter 4. Regional Mechanics of the Beating Heart Martyn P. Nash and Peter J. Hunter Chapter 5. Left Ventricular (LV) Pumping–Perfusion Analysis: Myocardial Properties, Intra-LV Velocity and Pressure, Detection of Myocardial Ischemic and Infarcted Segments, Perfusion Depiction by SPECT Imaging, Computation of Blood Flow Pressure and Velocity Patterns Within Myocardial Regions Eddie Y. K. Ng, Dhanjoo N. Ghista, Jian Jun Shu, Reginald C. Jegathese and Meena Sankaranarayanan ix

1

17

57

83

129


x

Contents

Section II.

Cardiac Pumping Characteristics

Chapter 6.

Left Ventricular (LV) Pressure Increase Mechanism During Isovolumic Contraction, and Determination of the Equivalent LV Myocardial Fibers Orientation Dhanjoo N. Ghista, Li Liu, Liang Zhong, Si Yong Yeo, Leok Poh Chua, Ru-San Tan, and Yong Seng Tan

Chapter 7.

Chapter 8.

Chapter 9.

Left Ventricular Filling Performance Characteristics Dhanjoo N. Ghista, Liang Zhong, Ru-San Tan and Eddie Y. K. Ng New Clinically Relevant Left Ventricular Contractility Index (Based on Normalized Wall Stress) Dhanjoo N. Ghista, Liang Zhong, Ru-San Tan, Eddie Y. K. Ng and Leok Poh Chua Characterization of Cardiac Dysfunction During Systolic Ejection Dhanjoo N. Ghista, Liang Zhong, Eddie Y. K. Ng and Ru-San Tan

Chapter 10. Strain Analysis and Visualization of LV Deformation During a Cardiac Cycle, As an Index of Contractility Jinah Park and Sang I. L. Park

165

191

203

215

229

Section III. Assisted Perfusion and Pumping, and Myocardial Repair Chapter 11. Augmented Myocardial Perfusion by Coronary Bypass Surgical Procedure: Emphasizing Flow and Shear Stress Analysis at Proximal and Distal Anastomotic Sites Providing the Basis of Better Graft Patency Rates

255


Contents

xi

Dhanjoo N. Ghista, Meena Sankaranarayanan, Leok Poh Chua, Yong Seng Tan and Eddie Y. K. Ng Chapter 12.

Numerical Simulation and PIV Measurement of Two Proximal Anastomosis Models Leok Poh Chua, Jun-Mei Zhang and Dhanjoo N. Ghista

313

Chapter 13.

Mechanical Circulatory Support Systems Mustafa Akdis and Helmut Reul

351

Chapter 14.

Development of an Axial Blood Pump Weng Kong Chan and Yew Wah wong

383

Chapter 15.

Mathematical Modeling of Ventricular-Assist Devices S. Vandenberghe, P. Segers and P. Verdonck

Chapter 16.

Chapter 17.

Index

Tissue Engineering for the Infarcted Heart: Cell Transplantation Therapy Genevieve M. Y. Tan, Lei Ye, Winston S. N. Shim, Husnain Kh. Haider, Alexis B. C. Heng, Terrance Chua, Tian Hai Koh and Eugene K. W. Sim Tissue Engineering of Artificial Heart Tissue Genevieve M. Y. Tan, Lay Poh Tan, N. N. Quang, Winston S. N. Shim, Alfred Chia, Subbu V. Venkatramen and Philip E. H. Wong

419

477

541

579




BOOK SUMMARY Section I.

Cardiac Perfusion

Chapter 1. Physiomics of Coronary Perfusion and Cardiac Pumping by Fumihiko Kajiya, Masahito Kajiya, Taro Morimoto, Tatsuo Iwasaki, Yousuke Inai, Masanori Hirota, Takahiko Kiyooka, Yuki Morizane, Takehiro Miyasaka, Satoshi Mohri and Juichiro Shimizu Physiome is considered to be a powerful successor to the genome. Physiome refers to a quantitative description of the physiologic dynamics or functions of the intact organism. It includes integration of knowledge through functional modules and modelling of hierarchic system elements of biologic systems. Biomechanics offers potent tools to promote the physiome concept. By using modern microvisualization technology with physiomic model of coronary circulatory network, this chapter introduces our physiomic approach to coronary microcirculation, which supplies oxygen and nutrients to heart muscles. The heart is unique, among other organs, in that coronary arterial flow is exclusively diastolic while venous flow is systolic. That is, blood pooled in coronary microvessels (during diastole) is squeezed out to the coronary vein by myocardial contraction. In this chapter, we first describe the biomechanical interaction between coronary blood flow and cardiac contraction. Then, the physiome of coronary capillary network and its functions are discussed. Chapter 2. Left Ventricular Inhomogeneity and the Heart’s Functional Reserve by Felix Blyakhman This chapter concerns with the study of myocardial inhomogeneity in the left ventricular wall. Inhomogeneity is an attribute of both the normal heart and the pathologically compromised heart. In the course of the last couple of decades, this phenomenon has revealed that myocardial inhomogeneity is a modulator of cardiac contractility and/or pump function, although the significance of inhomogeneity for the normal heart has not yet been clarified. Why has nature created such an inhomogeneous device? In this chapter, we seek an answer to this question. We present evidence that the possible role of inhomogeneity in the normal heart is to provide functional reserve xiii


xiv

Book Summary

for the left ventricle, which is tapped (as needed) to maintain stability of cardiac pumping function throughout the course of life. Chapter 3. Quantification of Cardiac Perfusion and Function Using Nuclear Cardiac Imaging by Ru-San Tan, Liang Zhong, Terrance Chua and Dhanjoo N. Ghista Myocardial ischemia occurs when tissue metabolic needs outstrips coronary blood flow or perfusion. The deficiency of the latter is commonly caused by atherosclerotic coronary artery disease, and is diagnosed by various myocardial perfusion imaging techniques that track blood flow heterogeneity between myocardium supplied by normal versus narrowed arteries. Nuclear myocardial perfusion imaging, the most established and ubiquitous of these methods, uses radioactive isotopes (commonly thallium-201or technetium-99m-based tracer agents) combined with stress and rest imaging protocols, to evaluate regional relative coronary flow reserves, and hence diagnose areas of myocardial ischemia and infarction. Modern image acquisition, using single photon emission computed tomography, allows the reconstruction of a three-dimensional image dataset that facilitates visual analysis as well as quantitation of perfusion. This increases reproducibility of interpretation, and is especially useful in the assessment of myocardial viability. Further, electrocardiographic gating, during the scan acquisition, allows assessment of left ventricular function, which has great prognostic significance. Quantitated perfusion and functional data can be displayed as polar maps that are amenable to comparison with normal databases, thus enhancing the clinical applicability of the technique. Chapter 4. Regional Mechanics of the Beating Heart by Martyn P. Nash and Peter J. Hunter Mathematical modeling provides a useful tool to understand the normal and abnormal mechanical function of the heart. The large deformations that take place during the cardiac cycle require that finite deformation elasticity must be used with the governing laws of physics. In addition, the complex geometry and microstructural arrangement of cardiac muscle requires that numerical and computational methods need to be used to solve the resulting nonlinear equations. This chapter summarizes a continuum mechanics approach to analyzing myocardial soft tissues, and details how the orthotropic nature of the ventricular myocardium may be efficiently represented. Based on this framework, a finite element analysis of canine ventricles is presented, and


Book Summary

xv

the distributions of deformation and regional mechanical stress throughout the heart cycle are quantified. Chapter 5. Left Ventricular (LV) Pumping–Perfusion Analysis: Myocardial Properties, Intra-LV Velocity and Pressure, Detection of Myocardial Ischemic and Infarcted Segments, Perfusion Depiction by SPECT Imaging, Computation of Blood Flow Pressure and Velocity Patterns Within Myocardial Regions by Eddie Y. K. Ng, Dhanjoo N. Ghista, Jian Jun Shu, Reginald C. Jegathese and Meena Sankaranarayanan This chapter essentially provides an overview of LV perfusion and pumping (and connects these two LV functions), by employing different methods for characterizing: (i) pumping in terms of LV passive and active myocardial properties, as well as intra-LV flow velocity and pressure distributions; (ii) perfusion based on detection of LV myocardial ischemic and infarcted segments (by means of echocardiographic texture analysis), SPECT imaging, and computational analysis of intra-myocardial regional perfusion. Section II. Cardiac Pumping Characteristics Chapter 6. Left Ventricular (LV) Pressure Increase Mechanism During Isovolumic Contraction, and Determination of the Equivalent LV Myocardial Fibers Orientation by Dhanjoo N. Ghista, Li Liu, Liang Zhong, Si Yong Yeo, Leok Poh Chua, Ru-San Tan, and Yong Seng Tan Herein, a biomechanical thick-walled cylindrical model of the left ventricle (LV) is developed to demonstrate that the mechanisms of LV internal pressure increase during isovolumic contraction is due to the contraction of the LV myocardial fibers helically wrapped inside the LV wall. The contraction of these fibers deforms and twists the LV. Hence, we can indirectly associate LV twisting with LV contractility. Associated with the LV pressure increase, we have determined the LV (radial, longitudinal, and twist) deformation state. We then determine the LV wall stresse`s associated with the deformations, and thereby the principal stresses in the LV wall, along with the axial shortening force and the torque experienced by the LV. We now hypothesize that the LV principal stresse`s orientation corresponds to the orientation of the LV cylindrical model myocardial fibers. This is how we are able to postulate that the contraction of these LV myocardial fibers causes LV deformations, inducing torsion of the LV and associated LV twist angle. Further, the derived orientation of the LV myocardial fibers


xvi

Book Summary

may be deemed to be an intrinsic property of the LV, and determine its capacity for adequate blood outflow into the aorta. Chapter 7. Left Ventricular Filling Performance Characteristics by Dhanjoo N. Ghista, Liang Zhong, Ru-San Tan and Eddie Y. K. Ng Left ventricular (LV) filling in turn influences LV contraction because inadequate filling will cause inadequate contraction. Hence, it is important to develop an appropriate LV filling index in terms of monitorable LV pressure and volume. Hence, the prime objective of this chapter is to develop an index to assess the filling functional performance of the LV in terms of monitorable LV pressure and volume. For this purpose, the LV volume response to the driving pressure term is formulated in terms of a differential equation. This equation is solved for LV volume expression (in terms of the model parameters) for two filling phases: (i) the early filling phase due to LV suction, during which the LV myocardial sarcomere contractile element is relaxing and the driving pressure term (in the governing differential equation) is zero, and (ii) the second phase of filling, due to left atrial contraction. Chapter 8. New Clinically Relevant Left Ventricular Contractility Index (Based on Normalized Wall Stress) by Dhanjoo N. Ghista, Liang Zhong, Ru-San Tan, Eddie Y. K. Ng and Leok Poh Chua It may be said that any comparative analysis of contractility indices in the intact heart is somewhat arbitrary, due to lack of an ideal descriptor of the contractile state. Hitherto, left ventricular (LV) (dP/dt)max has been employed as a measure of LV contractility, and has been shown to be a relatively load-dependent index. It is however an extrinsic measure of LV contractility, because the LV pressure itself is developed by LV wall stress caused by LV myocardial sarcomeric contraction. It is hence natural to represent LV contractility by means of this intrinsic property of the LV (namely its circumferential wall stress normalized with respect to LV pressure), which is independent of the preload and afterload. For this purpose, our LV spherical model’s wall stress is normalized with respect to LV pressure, and its maximum value is adopted as an index of LV contractility. Our new index is an intrinsic property of the LV, as well as easily and noninvasively obtainable in terms of measurable LV volume and myocardial volume. For the formulation of this contractility index, the LV is modeled


Book Summary

xvii

as a pressurized thick-walled sphere; to reduce the mathematical complexity and for clinical application convenience. The high degree of correlation between our new and simple contractility index and (dP/dt)max shows that it is well capable of separating normal LVs from LVs with impaired LV contractility. Chapter 9. Characterization of Cardiac Dysfunction During Systolic Ejection by Dhanjoo N. Ghista, Liang Zhong, Eddie Y. K. Ng and R. S. Tan It can be said that it is the left ventricular (LV) sarcomere contractility that develops the LV pressure in response to its volume during filling and ejection phases. In this chapter, we develop the governing equations of dynamics of the LV sarcomere contained within the wall of the LV cylindrical model. We then relate the sarcomere stress and displacement to the monitored LV pressure and volume, in terms of the sarcomere elements’ parameters (namely the sarcomere contractile element (CE) force and shortening velocity), and evaluate them. We next determine the power generated by the sarcomere (CE) element. All of these indices are deemed to be important LV functional determinants. Chapter 10. Strain Analysis and Visualization of LV Deformation During a Cardiac Cycle, As an Index of Contractility by Jinah Park and Sang I. L. Park The three-dimensional (3D) deformation fields of the LV models were initially estimated from tagged MRI data sets, which provide in-plane temporal correspondence of material points. While there is some initial experience in the use of tagged MRI and related techniques to study the 3D motion of a heart, there is still no generally accepted method for analysis and display of the 3D heart motion. Our parameter-function model captures the 3D deformation field in terms of its model parameters, and the volumetric model can then be regenerated based on the estimated parameters, allowing us to select any desired volume element within the myocardium for a conventional strain analysis. We have verified that the results of a conventional strain analysis performed on the parameter-function model are in agreement with those from a conventional finite element model. Furthermore, we have proposed a new methodology in visualizing multi-dimensional rendition of the LV overall myocardial strain variation. This will help gain a thorough and localized


xviii

Book Summary

understanding of the LV motion, and should significantly increase the clinical utility of LV motion analyses.

Section III.

Assisted Perfusion and Pumping, and Myocardial Repair

Chapter 11. Augmented Myocardial Perfusion by Coronary Bypass Surgical Procedure: Emphasizing Flow and Shear Stress Analysis at Proximal and Distal Anastomotic Sites Providing the Basis of Better Graft Patency Rates by Dhanjoo N. Ghista, Meena Sankaranarayanan, Leok Poh Chua, Yong Seng Tan and Eddie Y. K. Ng Coronary artery bypass grafting (CABG) surgery is an effective treatment modality for patients with severe coronary artery disease. CABG is a routine surgical treatment for ischemic heart disease. A large number of CABG cases fail postoperatively due to intimal hyperplasia within months or years, due to deleterious blood-flow velocity and shear–stress distributions at the graft–artery junctions. The conduits used during the surgery include both the arterial and venous conduits. Long-term graft patency rate for the internal mammary arterial graft is superior, but the same is not true for the saphenous vein grafts. At 10 years, more than 50% of the vein grafts would have occluded, and many of them are diseased. This chapter presents the fluid-dynamics of blood flow in (i) the aorto-right in-plane CABG model (the centerline of the aorta, graft, and the host artery all lie in a plane) and (ii) an out-of-plane CABG model (the centerline of the aorta, graft, and the host artery do not lie in a plane), wherein the left anterior descending artery is bypassed using the sapheneous vein. In our model, the dimensions of the aorta, saphenous vein, and the coronary artery simulate the actual dimensions at surgery, and we employ three-dimensional computational fluid-dynamics, to analyze the blood flow at both proximal and distal anastomoses. Our results have revealed that (i) maximum perfusion of the occluded artery occurs during mid-diastole, (ii) the maximum wall shear–stress variation is observed around the distal anastomotic region, and (iii) there is a decrease in the magnitude of the peak wall shear–stress at the bed of the anastomosis in the non-planar CABG model as compared to the planar geometry, supporting the view that non-planarity of the blood vessel may lead to better graft patency.


Book Summary

xix

Chapter 12. Numerical Simulation and PIV Measurement of Two Proximal Anastomosis Models by Leok Poh Chua, Jun-Mei Zhang and Dhanjoo N. Ghista Hemodynamics is widely believed to influence the stenosis of coronary artery bypass graft (CABG). Although distal anastomosis has been extensively investigated, further studies on proximal anastomosis are still necessary, as the extent and initiation of stenosis process may be influenced by the flow at proximal anastomosis per se. Therefore, in this study, firstly, two models (namely 90◦ and 135◦ anastomotic models) were designed and constructed to mimic the proximal anastomosis of CABG for left and right coronary arteries, respectively. Flow characteristics of these models were studied by both numerical simulation and particle image velocimetry (PIV) measurement, so as to acquire physical insight of hemodynamics in proximal anastomosis and to validate the simulation result simultaneously. The simulation results showed disturbed flow (such as flow separation, stagnation point, etc.) as well as abnormal hemodynamic parameters (HPs) distributions (including the low and high time-averaged wall shear stress (WSS), oscillation shear index, and time-averaged wall shear stress gradient regions in both the models). In contrast to the 90◦ model studied, the 135◦ model is proposed to provide better patency rate, as it has reduced disturbed flow and abnormal HPs. A fair agreement between numerical and experimental data has been observed in terms of flow characteristics, velocity profiles, and WSS distributions. The discrepancy could be due to the difference in detail geometry of the physical and computational models because of manufacturing limitations to have the exact shape of the computational model when making the Pyrex glass model. Chapter 13. Mechanical Circulatory Support Systems by Mustafa Akdis and Helmut Reul Mechanical blood pumps for temporary or permanent support of cardiac function are classified into the traditional engineering categories of displacement and rotary pumps. Herein, the clinical use and indications of the various pump categories are outlined, and a detailed description of currently available systems is given. The first part deals with extracorporeal as well as implantable ventricular-assist devices (VADs) of the displacement type, and is followed by a section on current developments in the field of total artificial hearts (TAHs). The second part covers rotary


xx

Book Summary

blood pumps (RBPs) that are intended as VADs. A brief summary of each RBP system (including specific design aspects of axial, radial, and diagonal pumps) is outlined. This chapter concludes with recommendations on major challenges and future trends in the field of mechanical circulatory support. Chapter 14. Development of an Axial Blood Pump by Weng Kong Chan and Yew Wah Wong This chapter describes some early development efforts on an axial blood pump as a left-ventricle-assist device (LVAD). The performance of the preliminary design model (comprising a flow straightener, inducer/impeller, and a diffuser) is first refined, followed by drive system integration in the prototype. The impeller is first studied in isolation, using numerical flow simulations to improve the impeller geometry in relation to blade thickness, blade angle distribution, wrap angle and hub contour, followed by studies on impeller and diffuser interactions. The objective is to minimize flow reversal and reduction of scalar stresses in the impeller passage at the designed impeller speed of 10,000 rpm, delivering 5 L/min flow against 100 mmHg pressure. Numerical results predict scalar stresses in the impeller passage near the hub surface to be ranging from 5 ×10−4 to 8 Pa, while those near the impeller tip region to be ranging from 1 to 150 Pa. The residence time of particles through the impeller is typically less than 0.1 s, suggesting the unlikelihood of hemolysis. The electric drive is integrated to the pump by incorporating rotor magnets in the inducer–impeller hub and packing stator coils around the pump casing. A brushless rotating field, with feedback from Hall-effect sensors, regulates motor operation. The first prototype has been bench tested, and its results are presented. The pump developed holds promise as a viable LVAD. Chapter 15. Mathematical Modeling of Ventricular-Assist Devices by S. Vandenberghe, P. Segers and P. Verdonck This chapter describes the development of a bio-mathematical model used to assess the interaction between an assist device and the native cardiovascular system. First, an overview of mathematical cardiovascular modeling is presented, based on the time-varying elastance concept, an afterload model, and an appropriate assist device model (either a rotary pump or


Book Summary

xxi

a pneumatic or electromechanical displacement pump). In the second part of the chapter, the mathematical model of the rotary pump is presented, followed by a case study that investigates its unloading capacities of the rotary blood pump in different operating modes. Chapter 16. Tissue Engineering of the Infarcted Heart: Cell Transplantation Therapy by Genevieve M. Y. Tan, Lei Ye, Winston S. N. Shim, Husnain Kh. Haider, Alexis B. C. Heng, Terrance Chua, Tian Hai Koh and Eugene K. W. Sim Cardiovascular Disease (CVD) is one of the world’s most prevalent causes of death. Transcending geographical boundaries and population demographics, CVD accounted for 38.5 percent of all deaths or 1 of every 2.6 deaths in the United States in 2001, of which CHD was responsible for more than 1 of every 5 deaths. CHD was thus identified as the single largest killer of Americans (All data resourced from the American Heart Association journal report; 2004 update). There is currently no effective remedy for cardiovascular disease. There is a compelling need to arrest (if not reverse) the incidence progression of heart disease. Cell transplantation is a promising alternative therapy for myocardial repair. This novel approach seeks to compensate for the extensive loss in cardiomyocytes, which is in turn a consequence of left ventricular dysfunction. The rationale for engrafting cardiomyocytes onto a compromised myocardium is intuitive, and has prompted the initiative for subsequent “proof-of-concept” experimentation. To date, its application in various animal models has been reported with success. In this chapter, we discuss systematically, novel approaches to cell transplantation, specifically, applications in (i) (ii) (iii) (iv)

myoblast transplantation, adult mesenchymal, embryonic stem cell engineering, and phase I human clinical trials

We also report accomplished milestones in the respective fields of study, and outline the principles underlying each approach. At the same time, we highlight the limitations in each technique, and review current progress in the afore-listed strategies for cell transplantation in the infarcted heart.


xxii

Book Summary

Chapter 17. Tissue Engineering of Artificial Heart Tissue by Genevieve M. Y. Tan, Lay Poh Tan, N. N. Quang, Winston S. N. Shim, Alfred Chia, Subbu V. Venkatramen and Philip E. H. Wong Cardiovascular disease is widespread in the developed world, and many suffer from damaged heart tissue resulting in heart failure. Tissue engineering is a field combining the interests of clinicians, molecular scientists, and materials engineers. With the rapid advancement of “materials” science and an understanding of how it interacts with living cells, creating artificial tissues for organ repair seems more of a reality today than ever before. Recent reports on creating spontaneously contracting artificial heart tissue has created excitement, and has spurred on research in this area. This chapter offers a current perspective on strategies employed in engineering artificial heart tissues for reparative therapy of the damaged heart.

June 6, 2007 6:7 WSPC/SPI-B421 Cardiac Perfusion and Pumping Engineering ch01

SECTION I.

CARDIAC PERFUSION




PHYSIOMICS OF CORONARY PERFUSION AND CARDIAC PUMPING FUMIHIKO KAJIYA∗,†,‡ , MASAHITO KAJIYA‡ , TARO MORIMOTO‡ , TATSUO IWASAKI‡ , YOUSUKE INAI‡ , MASANORI HIROTA‡ , TAKAHIKO KIYOOKA‡ , YUKI MORIZANE‡ , TAKEHIRO MIYASAKA‡ , SATOSHI MOHRI‡ and JUICHIRO SHIMIZU‡ † Kawasaki

Medical School University Graduate School of Medicine Dentistry and Pharmaceutical Sciences 577 Matsushima, Kurashiki, Okayama, Japan ∗[email protected]

‡ Okayama

1. Introduction Arterial blood inflow into the myocardium is almost exclusively diastolic, whereas venous outflow is predominantly systolic.1,2 This finding suggests a substantial phasic volume change in the intramyocardial vessels by myocardial contraction and relaxation during a cardiac cycle. Thus, observation of the dynamics of intramyocardial microvessels during cardiac cycles is crucial for understanding the basic mechanism of intramyocardial blood perfusion. Mechanical stresses acting on intramyocardial blood vessels are different between endocardium and epicardium, hence studies of coronary hemodynamics at different transmural depths provide important information about intramyocardial influence on blood distributions in the myocardial wall. Such comparative studies may reveal why the deeper portion of myocardium is relatively vulnerable to ischemia, a principal question in coronary pathophysiology.

2. Coronary Blood Flow and Its Mechanical Interaction with Cardiac Contraction and Relaxation Coronary vessels in the myocardium are subjected to the phasic mechanical influences of cardiac contraction and relaxation, resulting in a unique instantaneous blood flow pattern: unlike blood flow in other organs, coronary arterial flow exhibits a predominantly diastolic pattern, while venous 1


2

Fumihiko Kajiya et al.

Fig. 1. Coronary arterial and venous blood velocity waveforms measured by an optical fiber sensor of laser Doppler velocimeter. The fiber sensor was fixed by cyanoacrylate adhesive on distal small artery and vein to evaluate myocardial inflow and outflow. Note the phase difference between arterial inflow and venous outflow; that is, diastolic versus systolic preponderance.

flow a systolic pattern (Fig. 1). Thus, arterial inflow into the myocardium during diastole should be stored in intramyocardial capacitance vessels (unstressed volume (UV) and ordinary capacitance), and an almost equal amount of blood should be squeezed out into epicardial veins in the next systole (Fig. 2). UV is defined as a capacitance to accommodate blood during diastole without significant increase in pressure exceeding outflow venous pressure. To investigate the intramyocardial mechanical interaction between myocardium and coronary vessels and flow directly, we introduced a portable needle-probe video microscope with a charge-coupled device (CCD) camera to observe the subendocardial and intramural vessels.3–5 The phasic diameter of the intramural and subendocardial arterioles decreased 10–20% by cardiac contraction (Fig. 3). In contrast, the diameter of the subepicardial arterioles changes little during a cardiac cycle. Mori et al. observed similar systolic narrowing of perforating arterial branches with length by using synchrotron radiation angiography.6 The degree of systolic compression was high in deeper myocardium. More recently, using the portable needle-probe CCD video microscope with a high-speed camera (200 frame/s), the movement of visible blood flow markers (blood velocities) exhibited exclusively diastolic flow in subendocardial arterioles with two-phased systolic reverse flow, while a remarkable forward flow was recognized in subepicardial arterioles although the diastolic predominant flow pattern was common throughout myocardial layers.7–9


Physiomics of Myocardial Perfusion and Pumping

3

Fig. 2. Model of intramyocardial microcirculation based on Spaan et al.’s model. The model consists of variable arterial and venous resistances, and capacitance (UV and ordinary capacitance). UV accommodates diastolic arterial inflow without increase of UV pressure above outflow venous pressure. During systole, myocardial pump (pressure) propels pooled blood in UV to coronary vein.

Fig. 3. Subepicardial and subendocardial arteriolar images with flow markers. Although diastolic predominant pattern is similar between subepicardium and subendocardium, remarkable systolic forward flow (↓) is observed in subepicardial arterioles, while negligibly small forward flow with two-phased reverse flows ( ) is noticed in subendocardium. The diameter of subendocardial arterioles decreased in systole by 10–20%, while that of subepicardium almost remained unchanged (revised from Ref. 9).


4


3. Functional Role of Capillaries As Capacitance and UV Through Myocardial Wall To clarify the functional characteristics of intramyocardial capillaries, as well as possible transmural difference, we visualized the 3D capillary architecture by intracoronary injection of custom-made contrast medium consisting of BaSO4 , India ink, gelatin, and distilled water in diastolic- and systolic-arrest rat hearts.10 A confocal laser-scanning microscope (CLSM) was used for the 3D microvisualization, and 200-µm-thick block sample was used after slicing by a microtrom. Figure 4 shows epimyocardial (left) and endomyocardial (right) capillary images during diastole (top) and systole (bottom). Capillary volume decreased by 32% on average throughout myocardium. The volume fraction of capillaries per unit myocardial mass is 10 times as large as that of arterioles and venules, indicating that capillaries function as major capacitance and UV (see Fig. 2). Transmurally, the reduction in volume fraction of capillary from systole to diastole was by 37% in endocardium, by 34% in midcardium, and by 19% in epicardium (Fig. 5). Thus, the functions of capillaries as capacitance may be more remarkable in the deeper layers.

Fig. 4. Epimyocardial (left) and endomyocardial (right) capillary images in diastolic (top) and systolic (bottom) arrested rat hearts. Note that the capillary diameters decreased in both epi- and endomyocardial layers during systole, but the degree is greater in endomyocardium. The capillary sinus is also compressed during systole, especially in endomyocardial layer.



5

Fig. 5. Volume reduction of transmural capillaries from diastole to systole. Volume reduction by cardiac contraction is most remarkable in endomyocardium (ENDO), while that of epimyocardium (EPI) is half of endomyocardium. The midmyocardial (MID) volume change is little less than endomyocardium (revised from Ref. 10).

4. Capillary Flow Dynamics and Regional Myocardial Perfusion Recently, we visualized the epicardial capillary network of the beatings in dog hearts in vivo, using high-resolution needle-lens probe microscope (Fig. 6).11 A capillary originating from a point in an arteriolar zone carries blood running parallel to the muscle fibers and ends in a point in the venular zone. The capillary length from the smallest arteriole and venule is several hundred µm.7 This path length is 8–10 times larger than a single capillary length, indicating 8–10 interconnection along the path length. The flow through Y-, T-, H-, and hairpin-type interconnection seen in the CLSM image (Fig. 4) was recognized by in vivo visualization. The flow of neighboring capillaries exhibited co- and counter-corrents, and cross-connecting flow occurred abundantly through interconnection, facilitating oxygen supply to myocytes. The transit time of blood in the capillary path length was about 1.5 s, again indicating that the capillary works as UV, which stores the arterial inflow blood for this period without emergence of venous flow.12 The other characteristics of capillary hemodynamics were coexistence of systolic and diastolic preponderant flows. This coexistence of temporal flow preponderance implies that the watershed between diastolic arterial and systolic venous flows is located within capillaries. The diastolic preponderance may be arteriolar capillary, while the systolic is venular one.


6


Fig. 6. Visualization of coronary capillary network in dogs with our high-magnification needle-lens CCD microscope (A). Microblood flow distribution by tracer autoradiography with 3 H-desmethyl-imipramine in rabbit subendocardial and subepicardial layers (B, C). Note the dense capillary network with H-, Y-, T-, and hairpin-type interconnection. There is coexistence of systolic and diastolic preponderant flow through capillary network (A). Marked global microheterogeneity (high flow in one region, but low flow in other region) of blood flow is recognized in subendocardial layer, while local (point-topoint) heterogeneity is more remarkable in subepicardial layer (revised from Refs. 12–14).

In Fig. 6, flow distributions within subendocardial (B) and subepicardial (C) layers of a rabbit left ventricular wall are shown. These images were obtained by tracer digital radiography with 100-µm resolution.13,14 Both show a marked spatial heterogeneity as quantitated by the coefficient of variation of flows (CV = standard deviation/mean) and the correlation coefficient of adjacent regional flows (CA). Both CA and CV values are higher in subendocardium than in subepicardium, indicating the existence of more clearly distinct aggregates of high or low regional flows in subendocardium. The greater anatomic myocardial flow heterogeneity may explain the greater CV in subendocardium, which is inferred from the higher flow



7

heterogeneity in subendocardium observed in the fully vasodilated, arrested heart.15 In addition, the extravascular compressible force may also contribute to the transmural difference of CV and CA, because its enormous effect on subendocardium3,5 appears to vary spatially to the larger extent within subendocardium than subepicardium due to the existence of the papillary muscle.

5. Simulation of Intramyocardial Microcirculation We modified the intramyocardial pump model originally proposed by Spaan et al.16 In our model, the myocardium was divided into three layers.17 The vascular compartments of each layer were consisted of arterioles, capillaries, and venules. A lumped epicardial artery was placed proximal to all of these layers, and a lumped vein was placed distally (Fig. 7). Intramyocardial vascular diameter is determined by a transmural pressure (PTR = P − Pim ) and a vascular elasticity, wherein P is intravascular pressure and Pim intramyocardial pressure. To calculate instantaneous blood volume in each compartment, Pim was assumed to be 5/6 of left ventricular pressure (LVP) at subendocardial layer, 3/6 at midmyocardial layer, and 1/6 at subepicardial layer assuming endo-, mid-, and epimyocardial compartments located at the depth of 1/6, 3/6, and 5/6 from endocardium, respectively, because it is well documented that the intramyocardial hydrostatic pressure decreased almost linearly from endocardium to epicardium. The relation between vascular volume (V ) in each compartment and transmural pressure (PTR ) can be expressed by PTR = A exp(Ke(t) V ) + B log(Ke(t) V ) − P0 .

(1)

Here, the elastance of a vessel consists of two components: (i) The first component is the time-dependent elastance relating to local myocardial stiffness. This implies that vascular elastance increases with increasing cardiac stiffness during systole (by crossbridge formation) and decreases with decreasing of stiffness during diastole. This is expressed by the timedependent parameter Ke(t), which is based on Krams et al.’s article,18 extended from Suga’s original work.19 Since cardiac contractility is considered to be uniform transmurally, Ke(t) value is set to be equal in all three layers. (ii) The second component involves vascular stiffness changes following the state of contraction and relaxation of vascular smooth muscle itself. This is modeled by change in the time-independent parameter A, B,


8


Fig. 7. Three layers (epi-, mid-, and endomyocardium) coronary circulation model consisting of nine vascular compartments based on Spaan et al.’s model. Each compartment is basically same, but intravascular and intramyocardial pressures, and vascular compliance C of each compartment are different transmurally and temporalily. C relates to parameters k, a, b and P0 in Eq. (1). The resistance (r) was calculated from the volume V stored in C by the Poisuille law, and is located in proximal and distal portions of each compartment on a fifty-fifty basis.

and P0 in Eq. (1). The values of these constants of each vascular compartment are determined based on published data17,20 and our own animal experimental data. Resistance is represented to be inversely proportional to square of vascular volume,16 R = Kr /V 2 ,

(2)

where Kr is a constant, the value of which is also given as in our earlier report.21 We input a typical measured data of aortic pressure and LVP obtained from our laboratory.



9

Fig. 8. Simulated blood flow waveforms in subepicardial, midmyocardial, and subendocardial arterioles. Note that there is systolic reverse flow in deeper myocardium, while systolic forward flow in superficial portion. The amount of diastolic flow component is larger in deeper portion.

Figure 8 shows an example of our simulation results of arteriolar flows simulation in three different layers.17 In subendocardium, two-peaked systolic retrograde flow with predominant diastolic flow was simulated, as seen in our in vivo visualization experiments (see Fig. 3). The subepicardial arteriolar flow exhibited two peaked forward flows, one during systole and the other during diastole, consistent with our experimental data except a sharp and short dip during isovolumic contraction phase, probably due to local muscle strain. The flow pattern in the arteriole in midlayer exhibited an in-between pattern, but we have no available experimental data at present. The diameters of subendocardial and midmyocardial arterioles decreased during systole as our physiologic measurements, and the degree of diameter change was greater in the deeper myocardium (data not shown). The subepicardial arterioles changed very little, which is also good agreement with our experiments. Figure 9 shows a simulation study result of transmural autoregulation of coronary circulation. When the coronary perfusion pressure decreased from normal physiologic pressure to about 70 mmHg, coronary flow in all layers did not change; that is, autoregulatory mechanism operated well. However, when the coronary perfusion pressure further decreased, the subendocardial flow began to decrease, while autoregulation still operated in mid- and subepicardial layers. Further pressure decrease less than 60 mmHg caused


10


Fig. 9. Computer simulation of autoregulation of subepicardial, midcardial, and subendocardial microvessels. Note that autoregulation is working more robustly in subepicardial layer than subendocardial layer. The autoregulation in midcardium lies in-between.

disappearance of autoregulation of midlayer, and finally subepicardial regulation waned for lower pressure range below 50 mmHg. Collectively, this simulation study confirmed that epimyocardial layers are robust against ischemia, while endomyocardial layers are weakest and midcardial layers in-between.

6. Capillary As Capacitance and Its Effect on Coronary Arterial Inflow and Venous Outflow Figure 10 shows the relation between coronary arterial inflow and bloodpooled conditions in UV.22 The protocol of this experiment in an AVblocked dog was as follows: the blood pooled in capacitance vessels (mainly UV) was squeezed out until coronary venous flow became nearly zero by cardiac contractions after coronary artery (left anterior descending artery, LAD) occlusion and then long diastole was induced by pacing-off. During long diastole, LAD perfusion pressure was increased stepwise (two levels). After application of LAD perfusion pressure, the great cardiac vein (GCV) flow was absent for 1 or 2 s, indicating that UV was not filled yet. Then, the GCV flow resumed and gradually increased, indicating filling of UV. It is noticeable that the LAD flow was significantly higher in UV-unfilled conditions than in UV-filled conditions. This is physiologically important as a biomechanical feedback mechanism that pooled blood volume in the capacitance vessels exceeding UV reduces coronary arterial inflow.



11

Fig. 10. Coronary arterial (LAD) flow and GCV flow responses to stepwise increase in LAD perfusion pressure during artificially induced long diastole. After perfusion pressure increase, GCV flow was absent for 1 or 2 s, indicating that UV is unfilled, and then GCV flow increased gradually; that is, UV filled. Note that LAD flow in UV unfilled is greater than that in filled condition. This implies that the blood volume pooled in UV has a mechanical feedback control against coronary arterial inflow (revised from Ref. 22).

Fig. 11. A typical example of the decaying process of the coronary venous flow velocity after coronary inflow occlusion. The GCV flow decreased exponentially.

Figure 11 shows arterial and venous flows before coronary inflow occlusion and the decay phenomenon process of the GCV flow velocity after occlusion. It should be noted that the adjusted mean GCV flow averaged for each beat decreased exponentially.23 Thus, the process can be


12


expressed as VGCV (t) = VGCV (0) e−t/τ ,

(3)

where VGCV (t) is the decaying flow after the coronary artery occlusion, VGCV (0) is the mean flow just before occlusion, and τ is the decaying time constant. Integration of Eq. (3) with respect to time t gives ∞ VGCV (0)e−t/τ dt = VGCV (0) τ. (4) 0

The value of VGCV (0)τ represents the total displaceable volume V0 (mainly UV) stored in the intramyocardial capacitance vessels before occlusion: V0 = τ VGCV (0).

(5)

VGCV (0) = V0 /τ.

(6)

Alternatively,

Figure 12 depicts this relationship between coronary venous outflow VGCV (0) and the total displaceable volume V0 . The correlation coefficient between the total displaceable volume V0 and the coronary vein flow VGCV (0) at one beat before occlusion was significantly high (r = 0.86, p < 0.01), whereas the time constant is almost constant irrespective of various coronary venous flows (τ .=· 1.8 s). This result implies that the greater

Fig. 12. Relationship between venous outflow and total displaceable blood volume in intramyocardial capacitance vessels (left) and positive feedback from total displaceable blood volume to venous outflow (right). Coronary perfusion pressure (CPP) or coronary vascular tonus were changed to obtain the scatter diagram (left). In spite of wide variation of CPP and vascular tone, the correlation coefficient is significantly high. The observation that increase in total displaceable volume in UV augments venous outflow indicates the mechanical feedback from stored blood in UV to venous flow (right).



13

the blood volume stored in the capacitance vessels, the more the venous blood flow out (Fig. 12 (left)). This can draw an analogy to the Starling’s law of the heart, namely: more the blood stored in the left ventricle during diastole, more the blood pumped out into aorta in the next systole (Fig. 12 (right)). In other words, there is a positive feedback from the stored blood volume in capacitance vessels to venous outflow. We also found that there is a negative feedback from the stored blood in diastole to diastolic arterial inflow; that is, the more the blood stored in diastole, the less the blood inflow in the same diastole.

7. Conclusion In conclusion, a physiomic understanding of mechanical interaction between coronary microcirculation and cardiac pumping is fundamentally important to the understanding of intramural coronary hemodynamics. The transmural morphology and function of capillaries are the key issues for the mechanical control of coronary circulation together with arteriolar and venular mechanics.

Acknowledgments The authors thank Drs. M. Goto, T. Yada, E. Toyota, T. Matsumoto, O. Hiramatsu, Y. Ogasawara, K. Tsujioka and T. Okamoto for their collaboration of studies. Thanks are also given to Ms. H. Izushi, E. Nawachi and K. Yoshioka for their help in the preparation of this manuscript. This study was partly supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Technology, Sports and Culture (S and A), and the Research Grant for Cardiovascular Diseases (14-1) from the Ministry of Health, Labor and Welfare. We are grateful to World Scientific, for permission to reproduce in part our paper published in JMMB Vol. 5 No. 1 (2005) 1–924 . References 1. W. M. Chilian, and M. L. Marcus, Phasic coronary flow velocity in intramural and epicardial coronary arteries, Circulation Research 50, 6 (1982) 775–781. 2. F. Kajiya, G. Tomonaga, K. Tsujioka, Y. Ogasawara, and H. Nishihara, Evaluation of local blood flow velocity in proximal and distal coronary arteries by laser Doppler Method, Journal of Biomechanical Engineering 107, 1 (1985) 10–15.


14


3. T. Yada, O. Hiramatsu, A. Kimura, M. Goto, Y. Ogasawara, K. Tsujioka, S. Yamamori, K. Ohno, H. Hosaka, and F. Kajiya, In vivo observation of subendocardial microvessels of the beating procine heart using a needle-probe videomicroscope with a CCD camera, Circulation Research 72, 5 (1993) 939–946. 4. F. Kajiya, T. Yada, A. Kimura, O. Hiramatsu, M. Goto, Y. Ogasawara, and K. Tsujioka, Endocardial coronary microcirculation of the beating heart, Advances in Experimental Medicine and Biology 346 (1993) 173–180. 5. O. Hiramatsu, M. Goto, T. Yada, A. Kimura, Y. Chiba, H. Tachibana, Y. Ogasawara, K. Tsujioka, and F. Kajiya, In vivo observations of the intramural arterioles and venules in beating canine hearts, Journal of Physiology 509, 2 (1998) 619–628. 6. H. Mori, E. Tanaka, K. Hyodo, M. Uddin Mohammed, T. Sekka, K. Ito, Y. Shinozaki, A. Tanaka, H. Nakazawa, S. Abe, S. Handa, M. Kubota, K. Tanioka, K. Umetani, and M. Ando, Synchrotron microangiography reveals configurational changes and to-and-for flow in intramyocardial vessels, American Journal of Physiology 276, 2 (1999) H429–H437. 7. F. Kajiya, and M. Goto, Integrative physiology of coronary microcirculation, Japanese Journal of Physiology 49, 3 (Review) (1999) 229–241. 8. F. Kajiya, T. Yada, T. Matsumoto, M. Goto, and Y. Ogasawara, Intramyocardial influences on blood flow distributions in the myocardial wall, Ann Biomed Eng 28, 8 (Review) (2000) 897–902. 9. E. Toyota, Y. Ogasawara, O. Hiramatsu, H. Tachibana, F. Kajiya, S. Yamamori, and W. M. Chilian, The dynamics of flow velocities in endocardial and epicardial coronary arterioles, American Journal of Physiology 288, 4 (2005) H1598–1603. 10. E. Toyota, K. Fujimoto, Y. Ogasawara, T. Kajita, F. Shigeto, T. Matsumoto, M. Goto, and F. Kajiya, Dynamic changes in three-dimensional architecture and vascular volume of transmural coronary microvasculature between diastolic- and systolic-arrested rat hearts, Circulation 105, 5 (2002) 621–626. 11. T. Kiyooka, O. Hiramatsu, F. Shigeto, T. Yamamoto, H. Nakamoto, T. Yada, Y. Ogasawara, T. Morimoto, Y. Morizane, H. Minami, S. Mohri, J. Shimizu, T. Ohe, and F. Kajiya, Functional role of capillaries in reactive hyperemia by direct observation with a pencil-lens intravital videomicroscope, Microcirculation Annual 2003, eds. M. Asano and S. Miura (Nihon-Igakukan, Tokyo, 2003) 63–64. 12. T. Kiyooka, O. Hiramatsu, F. Shigeto, H. Nakamoto, H. Tachibana, T. Yada, Y. Ogasawara, M. Kajiya, T. Morimoto, Y. Morizane, S. Mohri, J. Shimizu, T. Ohe, and F. Kajiya, Direct observation of epicardial coronary capillary hemodynamics during reactive hyperemia and during adenosine administration by intravital videomicroscopy, American Journal of Physiology 288, 3 (2005) H1437–1443. 13. T. Matsumoto, M. Goto, H. Tachibana, Y. Ogasawara, K. Tsujioka, and F. Kajiya, Microheterogeneity of myocardial blood flow in rabbit hearts during normoxic and hypoxic states, American Journal of Physiology 270 (1996) H435–441.



15

14. T. Matsumoto, J. Ebata, H. Tachibana, M. Goto, and F. Kajiya, Transmural microcirculatory blood flow distribution in right and left ventricular free walls of rabbits, American Journal of Physiology 277 (1999) H183–191. 15. R. E. Austin, Jr, N. G. Smedira, T. M. Squiers, and J. I. Hoffman, Influence of cardiac contraction and coronary vasomotor tone on regional myocardial blood flow, American Journal of Physiology 266, 6 Pt 2 (1994) H2542–2553. 16. P. Bruinsma, T. Arts, J. Dankelman, and J. A. Spaan, Model of coronary circulation based on pressure dependence of coronary resistance and compliance, Basic Research in Cardiology 83, 5 (1988) 510–524. 17. S. Ohta, Y. Ogasawara, T. Matsumoto, O. Hiramatsu, H. Nakamoto, T. Okamoto, and F. Kajiya, Analysis of coronary haemodynamics by coronary circulation model with three layered myocardium, IEEE Computers in Cardiology (1995) 673–677. 18. R. Krams, P. Sipkema, and N. Westerhof, Varying elastance concept may explain coronary systolic flow impediment, American Journal of Physiology 257, 26 (1989) H1471–1479. 19. H. Suga, K. Sagawa, and A. A. Shoukos, Load independence of the instantaneous pressure-volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio, Circulation Research 32, 3 (1973) 314–322. 20. J. Dankelman, H. G. Stassen, and J. A. Spaan, Coronary circulation mechanics, Coronary Circulation-Basic Mechanism and Clinical Relevance eds. Kajiya et al. (Springer-Verlag, Tokyo, 1990), 75–87. 21. F. Kajiya, T. Yada, T. Matsumoto, M. Goto, and Y. Ogasawara, Intramyocardial Influences on blood flow distributions in the myocardial wall, Annals of Biomedical Engineering 28, 8 (2000) 897–902. 22. M. Goto, K. Tsujioka, Y. Ogasawara, Y. Wada, S. Tadaoka, O. Hiramatsu, M. Yanaka, and F. Kajiya, Effect of blood filling in intramyocardial vessels on coronary arterial inflow, American Journal of Physiology 258 (1990) H1042– 1048. 23. F. Kajiya, O. Hiramatsu, M. Goto, and Y. Ogasawara, Mechanical characteristics of coronary circulation, Journal of Mechanics in Medicine and Biology 1, 2 (2001) 67–77. 24. M. Kajiya, O. Hiramatsu, T. Yada, E. Toyota, T. Kiyooka, S. Mohri, J. Shimizu, Y. Ogasawara, and F. Kajiya, Physiomic approach to biomechanics of coronary microcirculation, Journal of Mechanics in Medicine and Biology 5, 1 (2005) 1–9.




LEFT VENTRICULAR INHOMOGENEITY AND THE HEART’S FUNCTIONAL RESERVE FELIX BLYAKHMAN Department of Physics, Ural State University, 51 Lenin Ave Ekaterinburg 620083, Russia Tel.: +7(343)375-6548 [email protected]; [email protected]

1. Introduction The normal left ventricle (LV) is a structurally inhomogeneous pump according to the features such as its geometry,1 the intricacy of the muscular fiber architecture,2,3 regional differences in wall thickness,4,5 and peculiarities of myocardial blood supply and conduction systems.6,7 LV structural complexity results in a spatial and temporal non-uniformity of chamber wall dynamics,8–11 which, in turn, is manifested as differences in regional stresses, shortening, thickening, and lengthening.12–16 Thus, inhomogeneity is known to be an attribute of the normal LV.5,17 Meanwhile, the significance of inhomogeneity for the normal heart has not yet been clarified. It is implicitly assumed that inhomogeneity provides a certain optimization of pump efficiency and/or muscle function.9,18–20 In physical terms, however, it is difficult to imagine that the performance of an inhomogeneous muscular pump could be higher than that of a homogeneous one, such as a sphere. To understand the possible role of inhomogeneity in the normal LV wall, we first try to elucidate the performance of a homogeneous muscular pump shaped like a thin-walled sphere. For this purpose, we look into the pump function of the pathologically transformed LV and experimental model of spherical ventricle, which is represented by isolated cardiac muscle. Clinical and experimental data have pushed us toward the conclusion that at a given level of myocardial contractility a homogeneous spherical LV should have no functional reserve; that is, no ability to maintain pump function in the case of additional load. Then, we turn our attention to the performance of the inhomogeneous LV with different extents of myocardial damage. We consider evidence to 17


18

F. Blyakhman

support the possible dependence of LV functional reserve on the contractility and on the extent of inhomogeneity. For this purpose we provide clinical data from patients with different extents of coronary artery stenotic lesion. On the basis of regional analysis with three-dimensional (3D) reconstruction, we show that the LV functional reserve degradation is accompanied by progressive decrease of both the contractility and the inhomogeneity. According to these experimental results and the concept of myocardial remodeling, we conclude that the inhomogeneity is a possible contributor to the LV functional reserve. Using the experimental model of spherical ventricle, we simulate different extents of inhomogeneity through “implantation” of a second muscle into the model ventricular wall, in addition to the first. The results push us toward the conclusion that at a certain degree of myocardial contractility, LV functional reserve is only possible in the case of inhomogeneity. In conclusion, we compare pump function of a homogeneous and an inhomogeneous LV, formulating the hypothesis that inhomogeneity is present in the normal ventricle to create functional reserve, which is utilized as needed to maintain pump-function stability throughout the course of life.

2. General Considerations To avoid misunderstanding, we first define “inhomogeneity”. Typically, to describe the differences in LV wall structure or function, we use different terminologies such as “non-uniformity”, “non-homogeneity”, “heterogeneity”, or “inhomogeneity”. Commonly, different terms are used to describe the same phenomenon, and the same term is used to describe different phenomena. Perhaps the main source of confusion lies in the anticipated causal relation between differences of LV structure and function. In fact, the performance of LV is the result of interaction between LV architecture, with its attendant peculiarities, and myocardial properties. In the pathologic state, disease impairs structure and function dynamically and unequally. Hence, the consequences are difficult to interpret. Meanwhile in the normal heart, the LV function must strongly correspond to LV structure. At the microscopic level, myocardial function is based on muscular contractility. Commonly, contractility is taken as the ability to generate mechanical tension and the ability to shorten. The basis of contractility lies in the molecular structure and regulatory mechanisms of the cardiac cell. Generally, at a given level of contractility, muscular function depends on loading conditions; that is, sarcomere pre- and after-load.


LV Inhomogeneity and the Heart’s Functional Reserve

19

At the macroscopic level, regional loading of myocytes’ depends on local wall structure which results in regional differences of function. At end diastole, the variations in thickness, and circumferential and longitudinal radii of curvature create regional differences in strain and stress.21,22 In other words, at the beginning of systole, myocytes in different wall regions have different load and initial length. For example, at the end of diastole, wall stress increases toward the endocardium,21 and initial sarcomere lengths are longer in the endocardium region than the epicardium.23 During systole, regional shortening and wall thickening are controlled mainly by myocyte after-load and instantaneous length. Those values depend on regional morphology and initial loading conditions.5,21,24,25 At the same time, differences in fiber orientation and thickness, as well as excitation delays, may play important roles in the regional loading profile, giving rise to load and length redistributions among fiber layers and regions as the cardiac cycle progresses. These features are amply demonstrated in experiments using the interaction model of two cardiac muscles connected in parallel or in series.26–28 Variations of loading conditions throughout the LV wall determine the regional contributions to the ventricular ejection fraction (EF). For example, with the use of magnetic resonance myocardial tagging in the normal adult LV, the different contributions of circumferential and longitudinal shortening and thickening to total EF was demonstrated.5 Thus, the normal pattern of LV regional function is complex and closely corresponds to LV structure. At the same time, loading conditions at the macroscopic level may influence the cardiac structure and function at the molecular level.29–32 Such long-term feedback control may explain variations of the contractile protein enzymes33,34 or cardiac cell electrophysiologic properties35,36 between normal epicardium and endocardium. Meanwhile, the possible role of these differences in determining normal LV function is largely unknown. At the same time, the feedback control plays an important role for myocardial adaptation in the pathologic state.37,38 Thus, the main functional differences within the normal LV wall are based on differences of LV structure. Structural inhomogeneity may include variations of curvature radii, thickness, transmural fiber orientation, conduction system, and coronary arterial morphology. As opposed to the word inhomogeneity, we use the term non-uniformity to describe the resulting functional variations. Thus, we postulate that inhomogeneity of normal LV structure results in non-uniformity of myocardial function.


20

F. Blyakhman

3. Performance of the Homogeneous LV We now seek to answer the following question: “Why isn’t the LV a homogeneous spherical shell?” We first consider the performance of such a shell. As is well known, an LV close in shape to a sphere is seen only in cases of severe myocardial damage (e.g. in dilated or ischemic cardiomyopathy).37,38 It is not seen in the normal heart. Figure 1 presents a 3D-reconstructed LV of a cardiomyopathy patient 2 months before heart transplant. For comparison, the LV of a relatively healthy man suffering from paroxysmal tachycardia examined during a paroxysm-free period is shown on the right. Reconstruction of LV endocardial and epicardial surfaces was carried out by means of transesophageal ultrasound examination (for details, see Appendix and Ref. 39).

3.1. Performance of the pathologic muscular sphere As seen in Fig. 1, the pathologic LV closely resembles a sphere. The ratio between short and long axes of LV is about 0.9. On the contrary, for a relatively normal LV this ratio is around 0.55. On average in the pathologic sample, LV end-diastolic volume is two times greater than normal, and LV wall thickness is two times smaller. According to the regional thickness map shown in the figure, the spherical LV appears quite homogeneous compared to the relatively normal LV.

Fig. 1. Example of 3D-reconstructed cardiomyopathic LV (left) and relatively healthy ventricle (right). The distribution of LV regional thickness is represented as colored map. Color-coding scheme is shown in the center.



21

The specimen in the figure was damaged homogenously due to diffusive microcircular endothelial dysfunction, ischemia, and reparative interstitial fibrosis.40–42 The basis of this dysfunction is a genetic mutation and involvement of the organ-specific autoantibodies to proteins of the cardiac sarcomere and cytoskeleton, as well as a mitochondrial disorder.43–45 At the same time, notwithstanding the impaired wall properties, the LV stroke work (product of end-diastolic pressure and stroke volume) in the steady state is close to normal.46–48 Meanwhile, during such cardiomyopathy the patients’ quality of life is extremely low. Usually, they are unable to perform any physical work.49 Hence, maximum work occurs in the basal state. Thus, at a given level of myocardial contractility, the impaired spherically transformed LV has no functional reserve with which to maintain pump function in the case of additional load.

3.2. Performance of the normal muscular sphere To estimate the performance of a spherical LV with normal myocardial contractility, a hybrid model of the ventricular chamber was developed.27,50 LV wall properties are represented by an isolated cardiac muscle, and chamber architecture by a mathematical model of thin-walled sphere. The muscle interacts with the spherical model by means of computer feedback control in real time. The experimental system components are outlined in Fig. 2. One end of the muscle is attached to the force transducer, and the other end to the linear motor. The muscle is pre-stretched to some initial length by the motor, and stimulated with rectangular pulses of 2–3-ms duration. At any initial length, the muscle produces both passive and active forces. Passive force depends on the muscle’s elastic properties and active force is a result of muscle contraction. The electrical signal from the force transducer (F ) corresponds to the muscle’s tension (σ) by σ = F/s, where s is the muscle’s cross section. According to Laplace equation, the pressure (P ) of spherical LV was calculated as: P =

2hF F 2hσ = = kf , R Rs R

(1)

where R is the radius of the model sphere, h is muscle thickness, and kf is the scale coefficient. The LV chamber radius was considered as: R = kl l, where l is the actual muscle length and kl is the coefficient for converting


22

F. Blyakhman

Fig. 2. Schematic diagram of experimental apparatus for ventricular mechanics modeling. The system contains a force transducer (1), a linear motor for applying mechanical deformations (2), an optical transducer to measure the muscle length (3), and a sustenance system (4) with the bath for muscle perfusion and platinum electrodes connected to the electronic generator (5). Measurement and control devices are interfaced with a computer (6).

the actual length into chamber radius. The chamber volume (V ) of the ventricle was calculated as: V =

4 3 πR . 2

(2)

The scheme demonstrating feedback control is presented in Fig. 3. At the initial experimental conditions, the ventricle has end-diastolic volume (Ved ) and pressure (Ped or pre-load) which depend on the actual length and passive tension of the pre-stretched muscle. When an electric pulse is applied (this moment of time labeled “1” in Fig. 3), LV pressure increases, and the increase depends on the muscle’s contractile properties. At constant end-diastolic volume (traces labeled as dotted lines) and given extent of myocardial contractility, the LV produces maximum possible isovolumic pressure (P0 ), while external work is equal to zero. In this case, feedback control maintains constant muscle length; that is, constant radius of the spherical chamber. To simulate LV ejection, a pressure value less than P0 must be set. Then, beginning from the moment of time “2”, the LV ejects blood with respect to a chosen aortic pressure (after-load). In this case, feedback control maintains constant pressure by means of dynamic muscle length changes. At the end of the ejection phase (moment of time “3”), the LV has its minimum value of volume (end-systolic volume, Ves ) and end-systolic pressure (Pes ).



23

Fig. 3. Schematic presentation of the main hemodynamic parameters of LV modeled by means of the apparatus feedback control. The traces with dotted lines correspond to LV isovolumic contraction, and those with smooth line correspond to LV contraction with physiologic loading regime. P0 , isovolumic pressure; Ped and Ved , end-diastolic pressure and volume, respectively; Pes and Ves , end-systolic pressure and volume, respectively. 1–5 are moments of time (explanation in text).

Since the feedback control maintained constant pressure, the values of afterload and Pes are the same. The period of time from “3” to “4” correspond to the LV isovolumic relaxation phase, when LV pressure falls at constant end-systolic volume. In this case, feedback control maintains constant muscle length. At time “4”, LV pressure is close to Ped , and the servomotor applies a deformation to stretch the muscle to its initial length; that is, to expand the LV to its end-diastolic volume. This process simulates LV filling as blood moves through the mitral valve.


24

F. Blyakhman

Thus, the feedback control system realizes the main phases of the LV cardiac cycle: 1–2: isovolumic contraction; 2–3: ejection; 3–4: isovolumic relaxation; and 4–5: diastolic filling. Figure 4 demonstrates an example of the LV model behavior during the course of the whole cardiac cycle. Feedback control realizes LV pump function in accordance with the main phases of cardiac cycle. Cardiac muscle from the frog ventricle, around 4 mm in length and about 1–2 mm2 cross section, was chosen for examination. Values of LV pre-load pressure and end-diastolic volume were specified initially. The traces labeled “1” correspond to LV contraction when the after-load pressure is 30% of the maximum possible isovolumic pressure (P0 ) at a given LV end-diastolic volume. The traces labeled “2” correspond to the experimental condition when the LV after-load pressure was set to ∼60% of P0 . According to the modeling data of Fig. 4, the increase of LV after-load at a given level of end-diastolic volume is accompanied by a decrease of LV stroke work, defined as the area inside the loop of the “pressure–volume”

Fig. 4. Example of the LV work modeling during the cardiac cycle. The traces labeled “1” correspond to LV contraction with after-load pressure 30% of maximal isovolumic pressure (P0 ) at the given LV end-diastolic volume. The traces labeled “2” correspond to the experimental condition when LV after-load pressure was specified to be ∼60% of P0 . On the right, pressure–volume loops obtained with the feedback control system are shown. x-axis, LV end-diastolic volume (Ved ); y-axis, LV pressure normalized to its maximal (isovolumic) value (P0 ). Cardiac cycle phases: (a) isovolumic contraction; (b) ejection; (c) isovolumic relaxation; and (d) diastolic filling.



25

curve. Thus, the modeled spherical LV has no opportunity to maintain pump function in the case of additional load, and therefore has no functional reserve. In summary, we have shown that neither the modeled spherical ventricle, nor the pathologic LV (whose shape is close to a sphere) has any functional reserve. 3.3. LV contractility reserve Commonly, LV functional reserve is considered as a reserve of myocardial contractility, and is based on regulatory mechanisms such as the Frank–Starling mechanism, heart rate, inotropic agents, etc. In other words, functional reserve is based on the properties of LV material itself. At constant sarcomere length, the myocardial fiber is able to realize a three- to fivefold tension increase or velocity increase at constant sarcomere after-load. This increase results from an increase of calcium activation and/or myosin ATPase activity.51 Thus, contractility reserve represents the capacity of the myocardium to increase its contractile function. This reserve reflects the maximal myocardial function mediated by all possible regulatory mechanisms. Contractility reserve of the cardiomyopathic ventricle is extremely low due to pathologic processes, and its functional reserve is close to zero. A minor increase of physical load inevitably diminishes LV pump function and may cause patient death.46,52 According to experimental data (Fig. 4), the increase of LV after-load results in a diminution of ventricular stroke work. This is caused by the absence of regulatory mechanisms in the modeled pump to support functional reserve, in contrast to the intact normal heart. For comparison, in the normal heart physical exercise results in heart rate increase and endogenous catecholamine release to increase the blood volume ejected by LV per minute and myocardial contractile function. At the same time, LV stroke work is close to the value seen during steady-state conditions. In the experimental model, however, contractility reserve could be introduced to come from outside. One could add, for example, a positive inotropic agent (say, calcium or adrenaline) to the perfusion bath that surrounds the muscle. In that case, the higher the after-load level that is set, the larger the quantity of inotropic agent needed. In other words, we recruit myocardial contractility reserve as needed to maintain LV stroke work stability. In contrast, the pathologic LV has the potential to recruit positive inotropic agents but the myocardium is not able to realize this possibility due to impaired contractility.


26

F. Blyakhman

Meanwhile, at a given level of myocardial contractility (say, normal), a homogeneous spherical LV performs maximal stroke work, and therefore has no functional reserve because a slight increase of after-load inevitably results in a decrease of LV pump function. This result is in good agreement with the data obtained in the intact spherical LV in the pathologic state where an existent level of myocardial contractility is extremely low. Thus, in the absence of myocardial contractility reserve, any homogeneous spherical ventricle has no functional reserve to maintain stability of ventricular pump function in the face of additional load.

4. Performance of the Inhomogeneous LV To maintain pump-function stability in extreme situations, be it an extra load or a pathologic process, LV functional reserve is needed. Indeed, despite the fact that cardiac pathology impairs myocardial contractility, in the steady-state LV pump-function parameters (such as end-systolic pressure, stroke volume, and stroke work) remain close to normal; and with respect to relative range of physical loads (stress test), LV stroke work is characterized by high stability.28,46,53–57 Therefore, the functional reserve of the heart is spent to keep LV pump function close to normal values. If we assume that the normal LV has maximal functional reserve, then this reserve must correspond to maximal contractility reserve, with a certain degree of wall structure inhomogeneity as well. Then, a contractility reserve decrease due to cardiovascular pathology development must be accompanied by a progressive decrease of LV functional reserve with changes in wall structure. On the one hand, this implies the existence of a causal relationship between the extent of LV functional reserve and the extent of LV contractility reserve; and on the other, it implies a possible relationship between the extent of LV functional reserve and the extent of LV inhomogeneity. We now seek evidence for the proposed relations. As is well known, the development of coronary atherosclerosis in ischemic heart disease (IHD) results in a decrease of myocardial contractility, with LV structural transformation.37 Thus, IHD at different levels of myocardial damage is a suitable pathology for looking for correlation between LV functional reserve and inhomogeneity, or LV functional reserve and myocardial contractility reserve. We present data from clinical observations on 23 patients with IHD based on LV 3D reconstruction by means of transesophageal ultrasound examination. With this technology we provide regional analysis of



27

structure and function of the LV wall. Additionally, all patients underwent angiographic examination to verify coronary lesions. Preliminary results on some of these studies were published earlier.39,46,54–56 The extent of LV functional reserve was estimated as functional class (FC) of angina pectoris according to the patient tolerance to stress test (treadmill). FC was defined according to the standard method, taking into account the maximal power of stress and the product of heart rate and arterial pressure.57 In some cases FC had intermediate values (0.5; 1.5). These correspond to cases in which the functional class was not clearly defined; for example, cases in which the stress test was stopped because of the increase in heart rate and/or arterial pressure reached the highest allowable limit. However, this limit value was not accompanied by pain or ECG changes characteristic of angina pectoris. The patients’ FC was in the range from 0.5 to 3.0 (mean value: 2.1 ± 0.2).

4.1. LV functional reserve and contractility reserve relation 4.1.1. Definition of LV contractility reserve In IHD, contractility reserve mainly depends on myocardial contractility and coronary artery reserve; that is, the ability of arteries to increase coronary flow to provide the myocardium with maximal functional realization. Since patients with different extents of coronary artery stenotic lesions were chosen for this investigation, LV coronary reserve was initially impaired by different extents. Eight patients had single-vessel (left anterior descending coronary artery, LAD), nine had double-vessel (LAD and left circumflex coronary artery, or LAD and right coronary artery), and six had multi-vessel coronary artery disease. Seven patients had a history of myocardial infarction. According to New York Heart Association (NYHA), three patients were classified as functional class I, eight as class II, and twelve as class III. All 23 patients had stable heart rate on average 73 ± 3 beats per minute (bpm). It is reasonable to assume that at a given level of coronary reserve (i.e. at the moment of patient examination), steady-state myocardial function reflects contractility reserve. Usually, ejection fraction (EF) is used as a measure of myocardial function. The EF was calculated as LV stroke volume divided by end-diastolic volume, and multiplied by 100%. LV endocardial shapes at the end of systole and diastole were chosen by means


28

F. Blyakhman

of synchronization of echo- and electrocardiographic signals. The LV enddiastolic volume varied from 135 to 285 ml with mean value of 187 ± 31. On average patients had 0.71 ± 0.02 ratio between the short and long axes of LV, and a depressed EF with mean value of 44.6 ± 2.8%. To estimate the contribution of LV regions to EF, segmental ejection fractions (s-EF) for 24 LV wall regions were determined. The LV volume was divided into 24 pyramids. The apex of each pyramid was set against the LV center of mass, and the LV surface region coincided with the bottom of each pyramid. Figure 5 shows an example of s-EF distribution in the LV wall for an IHD patient compared to a patient with relatively normal LV. As is seen in Fig. 5, the contribution of s-EF to the LV total EF for IHD patients is highly depressed and non-uniform relative to the normal LV. Some LV regions show paradoxical motion in the direction opposite to most regions. This implies that the LV wall is expanding during systole in the impaired regions. To estimate the extent of functional non-uniformity of LV regions’, the variation coefficient (CV) of s-EF was chosen. CV was calculated as RMS deviation of the parameter used, divided by its average value, and multiplied by 100%. For example, the CV of s-EF for pathologic and relatively healthy heart presented in Fig. 5 are 55% and 17%, respectively. The functional pattern in Fig. 5 (left) corresponds to the changes in LV myocardial contractility associated with the pathologic state. This conclusion is based on existence of a close correlation between EF as the measure of LV total myocardial contractility and CV s-EF as the measure of LV wall regions’ functional non-uniformity.46,54,55 In particular, for the data

Fig. 5. s-EF distribution for the pathologic (left) and relatively healthy (right) heart. IVS: interventricular septum; AW: anterior wall; FW: free wall; and PW: posterior wall.



29

presented in this section the correlation coefficient between EF and CV s-EF was −0.57 (n = 23; p < 0.01). It means that the decrease of total myocardial contractility due to IHD development results in an increase of LV wall functional non-uniformity. Thus, the CV s-EF is a sensitive measure of myocardial contractile properties. To estimate the extent of LV contractility reserve, we also analyzed LV diastolic elastic properties. LV wall elasticity is a very sensitive measure of myocardial contractility and cellular degeneration. The low values of myocardial elasticity reflect the reparative fibrosis and/or myocyte contracture induced by coronary artery occlusion.37,58,59 In fact, under the influence of an external force, the thickness of heart wall regions with different elastic properties changes differently during diastolic filling. While filling, the LV can be considered a passive structure, and the wall thickness change depends only on the chamber blood pressure and myocardium’s elastic properties. Because the force of pressure is equal for all myocardial regions, a relative change of wall thickness (∆H) may be used as an indirect measure of LV regional elastic properties. The advantage of using ∆H for diagnostic purposes has been confirmed in numerous studies.60,61 LV wall elasticity was assessed by means of ∆H, mapped during the LV filling phase. To define ∆H, a uniform surface mesh with cell size approximately 4 mm2 was placed on the LV endocardium at the beginning of the LV filling phase. On average (n = 23), the individual LV wall contained 1321 ± 27 surface regions. To determine the direction of every cell’s displacement from the beginning to the end of filling, we developed new method (for details, see Appendix and Ref. 39). The tracking of the cell node’s direction of motion for every moment (frame) of LV stretching enabled us to locate every region at the end of the diastole, and to assess the ∆H, correctly. ∆H was calculated as: ∆H = (H1 − H2 )/H1 × 100%, where H1 and H2 are the values of LV wall region thickness corresponding to the beginning and the end of diastole. On average patients had ∆H with mean value: 19.77 ± 1.89%, which is less than normal (∼40%39 ). Figure 6 shows the distribution of ∆H in the LV wall for both IHD and the relatively normal heart. The distribution of regional ∆H to total LV wall elasticity seems fairly uniform as opposed to the pathologic LV, where it is not. For this example, the CVs of ∆H for the pathologic and relatively healthy heart are 61% and 24%, respectively. Probably, a slight variation of the elastic properties of the normal LV depends on the differences of wall thickness, which in turn depend on the amount of cytoskeletal proteins per myocardial cross section. In the pathologic state, a high extent of ∆H


30

F. Blyakhman

Fig. 6. Relative thickness change (∆H) distribution for the pathologic (left) and relatively healthy (right) heart. IVS: interventricular septum; AW: anterior wall; FW: free wall; and PW: posterior wall.

variation reflects differences in elasticity between the damaged myocardial regions and those of adjacent ones. Thus, total EF, CV s-EF, and mean value of diastolic elasticity (∆H) are introduced as contributors, for estimation of LV contractility reserve.

4.1.2. Effect of the contractility reserve on LV functional reserve Figure 7 shows the relationship between LV EF and the patients’ FC of angina pectoris, defined on the basis of stress test (treadmill). The inverse correlation implies that the myocardial functional decrease results in FC increase. The higher value of FC implies the decrease of patients’ tolerance to physical load; that is, degradation of LV functional reserve. Figures 8 and 9 present the relationships between the patients’ FC and the CV s-EF, and the mean value of LV diastolic elasticity, respectively. The obtained data show that a CV s-EF increase results in an FC increase, and a ∆H decrease also leads to an FC increase. In other words, the decrease of myocardial contractility that arose from the increase of LV wall nonuniformity and stiffness resulted in the decrease of patient tolerance to physical load. Thus, the decrease of LV contractility reserve is accompanied by degradation of the functional reserve of heart: the lower the contractility, the smaller the LV functional reserve.



Fig. 7.

31

Dependence of LV EF on FC of angina pectoris.

Fig. 8. Dependence of the LV functional non-uniformity (CV s-EF) on FC of angina pectoris.

Fig. 9.

Dependence of mean LV elasticity (∆H) on FC of angina pectoris.


32

F. Blyakhman

4.2. LV functional reserve and inhomogeneity relation 4.2.1. Definition of LV inhomogeneity To characterize the extent of LV inhomogeneity, regional thickness (H) was mapped at the end of diastole. Figure 10 illustrates the distribution of H in the LV wall for IHD-modified and relatively normal LV. Additionally, the values of 40 radii at a single LV cross section were analyzed. The cross section was chosen at the level of LV’s half long-axis, and the center of mass of this section was defined. The radii were calculated as a distance between the center of mass and a point at the endocardial contour of chosen LV cross section. An example of the radii distribution for pathologic and relatively normal LV is presented in Fig. 11. Visually, the distribution of LV thickness and radii look fairly inhomogeneous in the relatively normal LV compared to the pathologic one. Quantitatively, the extent of LV inhomogeneity was characterized by a variation coefficient of thickness (CV-H) and radii (CV-R). The values of CV-H and CV-R are plotted in Figs. 10 and 11. On average CV-H amounted to 27.43 ± 1.70% and CV-R to 13.10 ± 0.68%. The results of LV regional diastolic elasticity analysis were used for LV inhomogeneity definition as well. According to angiographic data, values of ∆H less than 5% mainly correspond to the areas of postinfarction aneurysm

Fig. 10. End-diastolic thickness (H) distribution for the pathologic (left) and relatively healthy (right) heart. H normalized by the mean value of whole LV thickness (Hm ). CV: variation coefficient of H/Hm ; IVS: interventricular septum; AW: anterior wall; FW: free wall; and PW: posterior wall.



33

Fig. 11. Radii (R) distribution at LV cross-sectional level of half long axis, for the pathologic (left) and relatively healthy (right) heart. CV: variation coefficient; IVS: interventricular septum; AW: anterior wall; FW: free wall; and PW: posterior wall.

or scar. In other words, this area of the LV wall contains mainly noncontractile material. To estimate the contribution of fibrotic regions to the LV wall, the size of squares with ∆H ≤ 5% was normalized to the whole LV surface. On average, patients had a scale fibrosis factor (SFF) with mean value of 26.73 ± 3.01%, which implies loss of about a quarter of the contractile material. Thus, the CV-H, CV-R, and SFF in the ventricular wall are introduced as contributors, for estimation of LV inhomogeneity.

4.2.2. Dependence of LV functional reserve on inhomogeneity Figures 12 and 13 illustrate the relationships between the LV inhomogeneity parameters and functional reserve. As is seen in Fig. 12, CV-R varied from 8% to 18%. As far as we used only a single LV cross section for chamber curvature description, CV-R provided limited information on LV geometric complexity. Meanwhile, even with using such limited inhomogeneity parameter, the relationship between CV-R and FC is significant. The FC increase (i.e. LV functional reserve degradation) was accompanied by a CV-R decrease (i.e. the decrease of radii distribution inhomogeneity). The correlation between the CV-H and FC (Fig. 13) was stronger than the “CV-R–FC” relationship. A thickness variation decrease results in the FC increase. In other words, the smaller the extent of thickness inhomogeneity, the less the LV functional reserve (i.e. higher FC). The same conclusion appears to follow from the relationship between SFF and FC, presented in Fig. 14. If we assume that the fibrosis structure is highly homogeneous over


34

F. Blyakhman

Fig. 12.

Dependence of CV-R in LV cross section on the FC of angina pectoris.

Fig. 13.

Dependence of CV-H (CVH) on FC of angina pectoris.

the myocardium, then the LV functional reserve degradation results from an inhomogeneity decrease by means of the fibrosis area increase. Thus, the LV functional reserve is closely correlated with the extent of inhomogeneity: the lower the inhomogeneity, the less the reserve. In other words, the more the inhomogeneity, the higher the LV functional reserve. In summary, the performance of the inhomogeneous LV depends on contractility reserve and inhomogeneity. LV functional reserve degradation is accompanied by progressive decrease of both the contractility reserve and the inhomogeneity. Remarkably, in spite of the limited number of patients used for analysis (n = 23), most parameters of LV contractile reserve and inhomogeneity were correlated significantly. For example, the correlation coefficient between EF and SFF is −0.42 (p < 0.05); CV s-EF and CV-H is −0.58 (p < 0.01); ∆H and CV-R is 0.52 (p < 0.01). All relations show



Fig. 14.

35

Dependence of the LV scale fibrosis factor (SFF) on FC of angina pectoris.

that the decrease of LV contractility reserve accompanied by decrease in inhomogeneity. Thus, the contractility reserve decrease due to IHD impairs LV performance, and involves LV structural transformation toward lower inhomogeneity. Generally, the transition from the normal structure of LV to the pathologic one is called myocardial or cardiac remodeling. This is the point of the next section.

5. Myocardial Remodeling Myocardial remodeling implies the rearrangement of normally existing structures, and is triggered by mechanical stretch.37 Remodeling is a universal adaptation mechanism, and to a lesser or greater extent is characteristic of most cardiovascular diseases.62 The adaptation process is associated with changes in genetic expression. Generally, remodeling allows both the myocytes and the collagen network to adapt to new working conditions. This is a complex process, associated with structural changes from the molecular to the macroscopic level. The macroscopic structural changes in the LV wall due to IHD are the subject of our interest here. LV remodeling in IHD is a multi-step process, which has been investigated in numerous studies.37,62–65 The loss of contractile material following coronary occlusion is accompanied by acute dilation of the infarction area, LV volume increase, lengthening of the LV perimeter, and a blunting of the normal curvature. The changes of myocyte loading trigger fibrosis, which results in an increase of diastolic pressure and wall stress,


36

F. Blyakhman

and a progressive thinning of the damaged area. Simultaneously, elevation of end-diastolic stress causes volume-overload compensatory hypertrophy in non-infarcted regions. Progressive expansion of ischemia and fibrosis is associated with additional volume enlargement, decrease of ventricular thickness, and LV sphericity increase. The general remodeling strategy is the optimization of energy efficiency. At the microscopic level, this strategy is realized by means of rate reduction of the main mechanisms responsible for myocardial contractility.37 At the LV level, energy economy is provided by means of cardiac hypertrophy and chamber dilation. The hypertrophy compensatory role appears as an increase of the number of contractile units and reduction of wall stress according to Laplace law. LV volume and sphericity increase results in an increase of radii of curvature. Hence, according to Laplace law, at a given value of LV after-load, a lower tension is necessary to eject the same stroke volume. The myocardial remodeling due to coronary occlusion is asymmetric and is associated with infarct expansion. In the development of dilated cardiomyopathy, remodeling is symmetric and is associated with diffusive myocardial damage.37 Generally, remodeling causes transition from a nonspherical shape to a near-spherical one; it is of compensatory character and aims at maintaining adequate blood circulation. Thus, remodeling is a strongly tuned mechanism of interaction between different levels of myocardial structure. This adaptation process is a biologic response to cardiac pathology, required to support to LV performance in the extreme case. Therefore, we can assume that the same as contractility reserve, myocardial remodeling, is a possible source of LV functional reserve. In general, the data presented in the previous section are in a good agreement with findings by other authors. The development of IHD (the increase of patient FC by NYHA, of angina pectoris) is accompanied by LV dilation, an increase of index sphericity (the ratio between LV short and long axes), compensatory hypertrophy, and fibrosis expansion. At the same time, LV dilation is associated with an inhomogeneity decrease, so that all inhomogeneity parameters were closely correlated with LV end-diastolic volume (Figs. 15–17). On average, LV dilation and even local hypertrophy eliminates the differences in the radii and thickness distributions (see Figs. 10 and 11), and the fibrosis expansion replaces the structurally complex contractile myocardium for relatively simple collagen. As we have seen above, the development of IHD is accompanied by a decrease of LV myocardial contractility on the one hand, and by a



Fig. 15.

Fig. 16.

37

Dependence of LV end-diastolic volume (Ved ) on CV-H.

Dependence of LV end-diastolic volume (Ved ) on CV-R in LV cross section.

progressive decrease of LV wall structure inhomogeneity on the other. Hence, as the contractility reserve is exhausted, the LV resorts to the help of myocardial remodeling, which tends to decrease the inhomogeneity. However strange it may seem, nature has apparently not made the normal LV a simple sphere, reserving it for extreme cases. In the next section we seek to answer the following question: “Is inhomogeneity a possible contributor to LV functional reserve?”

6. On the Role of Inhomogeneity in Normal LV To understand the contribution of inhomogeneity to LV functional reserve, we need somehow to adjust a stationary level of the contractility reserve


38

F. Blyakhman

Fig. 17.

Dependence of LV end-diastolic volume (Ved ) on the LV SFF.

contribution to the ventricular ability of maintain pump function stability. As we demonstrated above (see Section 3), the LV spherical model is suitable for this purpose. What happens with modeled ventricular pump function if we introduce inhomogeneity in the ventricular wall structure? To do this, we represent the wall of the spherical ventricle by two strips of cardiac muscle effectively in series. With this model, we can simulate different degrees of inhomogeneity, and investigate the impact of inhomogeneity on stroke work. The results of these studies were published earlier.66 6.1. Methodologic approach The experimental model for simulation of particular extents of inhomogeneity in a spherical LV wall is based on a two-muscle interaction principle proposed many years ago,26 and modified later.55,67 A servo system for the registration and control of the mechanical activity of two isolated cardiac muscles was created. The system contains two independent apparatuses as presented in Fig. 2 (see Section 3.2), each consisting of identical devices. Measurement and control devices of the two apparatuses are interfaced with a common computer. The two muscles were connected serially (as tandem) by means of computer feedback control. To do this, the muscles’ force has to be maintained equal, and this was done by dynamic redistribution of the muscles’ lengths by means of servo motors. The force produced by the cardiac muscles was converted into pressure by Eq. (1) (see Section 3.2). LV chamber radius and



39

its change were calculated using the actual length of the muscles and their changes in length during the experiment: R = kl1 l1 + kl2 l2 , ∆R = kl1 ∆l1 + kl2 ∆l2 ,

(3) (4)

where l1 and l2 are the actual lengths of the muscles, and kl1 and kl2 are the coefficients for converting the actual lengths into the chamber radius. The chamber volume (V ) of the ventricle was calculated by Eqs. (2) and (3), and the volume change (∆V ) during a small time interval was calculated as: ∆V = 4πR2 ∆R.

(5)

The variation of kl1 and kl2 coefficients enabled us to change the size of the sphere-wall segments so that either one muscle dominated or neither muscle dominated. Hence, we had an opportunity to simulate different extents of inhomogeneity. Thus, the computer feedback control included a geometric model of the LV in the form of a sphere, and in contradistinction to the homogeneous ventricle described in Section 3.2, LV wall properties were represented by two muscles. Two assumptions for the spherical LV model were used: (1) the thickness of the sphere’s shell is much less than its radius and (2) the sphere does not change its shape during contraction. Experiments (n = 16) were performed on the cardiac muscles of the frogs (Rana radibunda), which had been sacrificed by decapitation. Two longitudinal samples from the ventricle wall were dissected. Typically, samples were 3–5 mm in length and had a cross section of about of 1–2 mm2 . The muscles were stimulated with rectangular pulses of 2–3-ms duration, applied by platinum electrodes. Stimulation rate was 20 bpm. Figure 18 demonstrates an example of LV work modeling during the course of the whole cardiac cycle. Two muscles with different cross sections were chosen for examination. The coefficients kl1 and kl2 were set equal. The values of LV pre-load pressure and end-diastolic volume were specified initially by muscle pre-stretching. The traces labeled “1” correspond to LV isovolumic contraction when the after-load chamber pressure is set equal to P0 at the given LV end-diastolic volume (see Section 3.2 for details). In spite of the fact that LV volume is constant, length redistribution takes place. The reason is a difference in cross section between muscles. The muscle labeled L1 had larger cross section than second muscle (L2 ), and, therefore, produced higher force at a given initial length. At LV isovolumic contraction, the feedback control maintains constant length of the tandem muscle pair and the same force in both. As far as the time course


40

F. Blyakhman

Fig. 18. Experimental records of LV mechanical parameters during cardiac cycle.66 P : pressure inside the spherical LV, obtained with the feedback control system; V : volume of the LV chamber, obtained with the feedback control system; L1 and L2 : changes in length of the muscles (muscle shortening upward). 1: Traces for the isovolumic LV and 2: traces obtained in course of the whole cardiac cycle modeling. Cardiac cycle phases: (a) isovolumic contraction; (b) ejection; (c) isovolumic relaxation; and (d) diastolic filling.

of force development in different muscles are concerned, individual lengths change to eliminate differences in instantaneous force. As a result, a sphere radius remains constant during LV isovolumic contraction. The traces labeled “2” correspond to the experimental condition in which the LV post-load pressure was specified to be ∼70% of P0 . In this case, feedback control realizes LV pump function in accordance with the main phases of the cardiac cycle. The contribution of each muscle to the ejection process (phase “b” in Fig. 18) is seen to be much different.



41

It is necessary to consider our assumption that the ventricular chamber does not change its shape during contraction. According to Laplace equation, differences in LV wall thickness must be accompanied by changes in LV regional curvature. This applies at the beginning and during contraction as well. Thus, the results of inhomogeneity modeling contain some error source, which minimizes the real effect of inhomogeneity on LV stroke work.

6.2. Modeling of inhomogeneity in ventricular wall structure Two different types of structural inhomogeneity were simulated. Regional thickness difference was modeled when two samples with different cross sections were used. Temporal excitation difference between regions was simulated by using muscles with almost equal mechanical characteristics, but with variable delays between the electrical stimuli. The dependence of stroke work of the spherical ventricle on the extent of regional thickness inhomogeneity is shown in Fig. 19. The data were obtained with 0.4 P0 LV after-load. The area size of thinner-wall segments as a percentage of the total area of the modeled ventricle surface is plotted on the abscissa. LV stroke work is plotted on the ordinate. The stroke work value is normalized to that of a homogeneous sphere; that is, when only a single muscle represented the model ventricle. The data in Fig. 19 show that the increase of relative inhomogeneity results in a decrease of LV stroke work. So, the larger the

Fig. 19. Dependence of spherical LV stroke work on extent of regional wall thickness difference.66


42

F. Blyakhman

Fig. 20. Dependence of stroke work of spherical LV on the extent of temporal excitation difference between wall surface regions.66

contribution of the thin region (weak muscle) to the total area of LV wall, the less the pump’s efficacy. The dependence of LV stroke work on the extent of temporal excitation difference between regions is shown in Fig. 20. The data were obtained with both segments of the modeled LV surface having the same size, and 0.4 P0 LV after-load. The delays between electric stimuli as a percentage of the contraction cycle’s total duration are plotted on the abscissa. LV stroke work values are plotted on the ordinate. The stroke work is normalized to that of synchronous LV; in other words, the excitation delay was equal to zero. According to the data (Fig. 20), the temporal excitation difference between regions results in a decrease of the pump’s stroke work. Similar to the data in Fig. 19, the increase in inhomogeneity leads to a reduction of LV efficacy. Thus, independently of the type of modeled inhomogeneity in the LV wall structure, an increase of inhomogeneity results in a decrease of LV pump function. At first sight, this experimental result does not agree with clinical observations, presented in Section 4 (see Figs. 12–14). In spite of great differences in LV wall inhomogeneity, the patients with varying extents of myocardial contractility damage were able to pass the stress test. In other words, the LV provided adequate pump function with a higher or lesser range, estimated as the FC of angina pectoris. The advantage of the model ventricle is the possibility of excluding the contractility reserve influence on LV pump function (i.e. to characterize the effect of inhomogeneity on LV stroke work at constant level of myocardial contractility). As we have demonstrated above (see Fig. 4 in Section 3, and



43

Figs. 19 and 20 in this section), at a given level of myocardial contractility a homogeneous spherical LV performs maximal stroke work, and therefore has no functional reserve. Now, we reasonably assume that the normal LV has some extent of thickness and excitation inhomogeneity, say by the values indicated by the dotted line in Figs. 19 and 20. In this case, the LV carries out work less than maximal. Therefore, the LV functional reserve emerges. For example, Fig. 20 shows that for a 10% ventricular functional reserve, 5% excitation delay is necessary. Then, the decrease of LV functional reserve will be accompanied by a decrease of inhomogeneity. Thus, the conclusions following from the clinical and experimental data are qualitatively the same: the more the inhomogeneity, the higher the LV functional reserve. In summary, through use of inhomogeneity modeling we have shown that at a certain degree of myocardial contractility, LV functional reserve is only possible with a certain degree of inhomogeneity in the ventricular structure. As far as pump function is concerned, inhomogeneity diminishes LV performance relative to a homogeneous sphere. Indeed, at a given value of surface area, a sphere has a maximal volume in contrast to any geometric figure. This implies that at a given value of surface area change, the sphere ejects maximum volume. In other words, at constant after-load pressure, the sphere carries out maximal stroke work. With respect to pump-function efficiency, the normal LV is not an optimal pump, but it has high functional reserve for adaptation to extreme situations that might occur over time. At the same time, nature has designed many special “devices” to eliminate fully or partly the negative influence of LV non-optimal construction on the pump efficiency. For example, there are: the trabecular apparatus to improve hemodynamic68 ; fiber orientation to reduce functional differences between epicardium and endocardium69 ; a massive fibrotic mitral ring to facilitate ventricular filling during the LV systolic period,70 etc. Probably, the reason for LV conduction system is the same. Thus, inhomogeneity in the normal LV wall constitutes functional reserve, which is based on the peculiarities of LV structure.

7. Conclusion Figure 21 summarizes the concepts described above. A hypothetic dependence of LV functional reserve on LV contractility reserve and inhomogeneity are presented. In a 3D coordinate system the normal LV is positioned at a point whose coordinates correspond to the maximum value of LV


44

F. Blyakhman

functional reserve (ymax ), arbitrarily chosen values of LV inhomogeneity (zmax ), and LV contractility reserve (xmax ). In this concept the spherical ventricle in a pathologic state occupies a position extremely close to the origin of the coordinates, and the modeled spherical ventricle is positioned at the point with coordinates (xmax , 0, 0). According to the clinical observations, cardiovascular pathology development goes by a vector marked with the dotted line in Fig. 21. This vector would appear to demonstrate, on the one hand, the existence of a causal relationship between the extent of LV inhomogeneity and the extent of LV functional reserve (Figs. 12–14); and on the other, a relationship between the extent of LV functional reserve and the extent of LV contractility reserve (Figs. 7–9). Thus, at a given pathologic state, LV functional reserve depends on corresponding extents of contractility reserve and inhomogeneity extent. This conclusion has an important prognostic significance in clinical practice. According to the inhomogeneity modeling data (Figs. 19 and 20), a vector marked with the zipper line implies that the decrease of inhomogeneity in the normal LV wall inevitably resulted in the decrease of LV functional reserve even at the normal (maximal) extent of LV contractility reserve. To find evidence for this assumption in the normal living heart seems difficult, but not impossible. Probably, the future LV modeling will confirm the existence of the predicted relationship.

Fig. 21. Schematic dependence of functional reserve on extent of inhomogeneity and contractility reserve (see text for explanation).



45

The modeling data imply that LV inhomogeneity determines functional reserve. From a technical point of view, inhomogeneity diminishes LV performance relative to a homogeneous pump. At the same time, the thickness and curvature differences, and fiber orientation create a safety margin for the extreme situation, be it an extra load or local myocardial pathology: for example, acute coronary artery stenosis. It is easy to imagine what happens with spherical ventricle in case of regional cell death. From a biologic point of view, inhomogeneity constitutes a structural resource for LV changes in the course of life. If we look to the myocardial remodeling process, the changes in genetic expression are adapting to a “homogeneity program”. This program ranges from local structural homogeneity (cell death) to the whole LV (as was presented in Fig. 1), including fibrosis and hypertrophy. Therefore, LV inhomogeneity is the long-term structural basis of the remodeling adaptation process. Thus, much the same as the myocardial contractility reserve, inhomogeneity appears as an independent source of LV functional reserve, which is maximal in the normal heart. And the same as myocardial contractility reserve, inhomogeneity reserve, is exhausted in extreme situations, be it an extra load or a pathologic process. Acknowledgments I thank my colleagues and students from the Biophysics Laboratory of Ural State University and the Diagnostics Department of Moscow Transplantology Institute who contributed much to the methods development, experimental and clinical data, and discussions. Their names are cited in Refs. 39, 46, 54–56, 66, 71, 72. I thank Prof. Gerald H. Pollack from the University of Washington for useful discussion and critical reading of the manuscript, and Prof. Frits W. Prinzen from University Maastricht for useful discussion as well. The results presented in this chapter were obtained in course of investigations supported by Russian Foundation for Basic Research. Appendix A: The Method for Regional Elastic Properties Determining on the Basic of LV 3D Reconstruction (for More Details, see Refs. 39, 71) A.1. LV 3D reconstruction Initial data were obtained by inspection of the patients with stable heart rate at the Diagnostics Department of the Institute of Transplantology


46

F. Blyakhman

and Artificial Organs (Moscow). LV two-dimensional (2D) long-axis sections were obtained with a “Powervision-380” (Toshiba, Tochigiken, Japan) transesophageal echocardiographic imager, at a scan rate of 25 Hz. Echocardiographic data were captured at least for two cycles to detect the LV filling phase precisely. The data for only one cycle were used for regional elastic properties evaluation. Subsequently, the transducer was rotated by 15◦ until a complete volume scan was obtained, Fig. 22. The angle of 15◦ corresponds to the minimal step between adjacent scanning planes for the used transducer. Typically, the complete volume scan takes around 1 min, and contains 12 scanning planes. The LV endocardial and epicardial borders for all sections were manually outlined frame by frame with “DICOR”c software.72 The software provides: the real-time video capture from different sources with frame rate up to 30 fps; captured image pre-edition (cutting and video format conversion); input video image quality improvement (noise smoothing, brightness/contrast adjustment, object edges emphasizing, subtraction, etc.); frame-by-frame manual heart chamber contour tracing. Device FlyVideo98 was used for video capture. It provides spatial resolution of about 0.3 mm in both vertical and horizontal dimensions. The LV filling phase was determined by means synchronization of the ultrasonic apparatus to ECG signal. The beginning of LV filling phase

Fig. 22. LV 2D sections for 3D reconstruction, obtained by means of the transesophageal ultrasonic examination.39 The probe was precisely rotated with angle steps of 15◦ . X, Y, Z and x, y correspond to global and local coordinate systems, respectively.



47

corresponded to the end of “T” wave of ECG (i.e. to the end of myocardium relaxation). The end of LV filling phase conformed to the end of “P” wave. Only the data with the same number of LV filling frames in each section were used for reconstruction. 3D reconstruction of the LV was carried out by a rotation of the plane of each cross section around the y-axis by the angle corresponding to the transducer plane. In a global coordinate system, see Fig. 22, X-axis is directed along the ultrasonic probe axis, Z-axis is directed along the transducer’s axis of rotation, and the Y -axis completes the system. The local coordinate system (x, y) was aligned with the transducer-scanning plane (see Fig. 22); y-axis was directed along the rotation Z-axis of the transducer-scanning plane, x-axis was orthogonal to y-axis. Cross-section rotational was carried out within the Cartesian coordinate system. Coordinate conversion from the local coordinate system (xi , yi ) into the global one (Xi , Yi , Zi ) was done by means of rotating every point around y-axis by angle γ. Xi = xi cos γ,

(A1)

Yi = −xi sin γ,

(A2)

Zi = yi .

(A3)

To compensate for the LV spatial displacement, at every moment of diastolic filling time, the superposition of LV cross sections was carried out by its centers of mass.

A.2. LV surface approximation As the number of 2D heart cross sections was rather limited, LV surface approximation was carried out using fitted spherical functions. The procedure resulted in as minimal as possible mean square deviance between the LV-approximated surface and the initial data. 3D description of the endocardial and epicardial surfaces was converted from Cartesian coordinates (X, Y, Z) into spherical ones (r, θ, ϕ). The center of mass of the endocardial surface was placed at the origin of spherical coordinates to provide the ranges of θ and ϕ from 0◦ to 180◦ and 0◦ to 360◦ , respectively. The function F (θ, ϕ) was introduced to locate any arbitrary surface point for a given angular coordinates θ and ϕ (in a spherical coordinate system). The function F (θ, ϕ) was presented by means of a set of spherical


48

F. Blyakhman

harmonics fnk (θ, ϕ) having the following equation: F (θ, ϕ) = =

n m n=0 k=0 n m

fnk (θ, ϕ) Pn(k) (cos θ)[Ank cos kϕ + Bnk sin kϕ],

(A4)

n=0 k=0 (k)

where only the coefficients Ank and Bnk are unknown; Pn (cos θ) is the attached Legendre polynomial. To achieve the best approximation of the LV geometry, the unknown coefficients Ank and Bnk of F (θ, ϕ) expansion were selected, so that the relative mean square deviance (∆m ) would be minimal: N N 2 ∆m = [r(θi , ϕi ) − F (θi , ϕi )] r(θi , ϕi )2 × 100%. (A5) i=1

i=1

A suitable approximation order was selected so that the value ∆m would not change significantly in the case of the increase of spherical function order. The plot in Fig. 23 demonstrates the relationship of the approximation deviance (∆m ) on the order of spherical functions (m). According to the presented data, ∆m changes quite insignificantly (less 0.1%) beyond the fourth order of m. The higher spherical harmonic orders are necessary for the approximation of abrupt curvature surfaces. As for

Fig. 23. Dependence of approximation error (∆m ) on the spherical function of order (m).39



49

our case, the LV chamber surface has no sharp bends, so the fourth order of spherical functions was chosen.

A.3. The analysis of LV regional motion at the filling phase To calculate the regional ∆H, the right position of a region at LV endocardial end-diastolic surface must be determined. At LV filling phase, the myocardium resembles a passive structure. The velocity of LV pressure increase is sufficiently less (at least for two orders) than the tension propagation rate in myocardium. For every moment of time, it implies some equilibrium between the blood pressure force and stretching forces. According to experimental findings,5 the normal deformations in LV wall are much more significant than the shear ones. As far as the pressure force effects to LV endocardial surface, perpendicularly, is concerned, the region displacement is mostly directed the normal. The assumption above is correct for regions with similar elastic properties and there is no certainty for a case when regions have sufficiently different elasticities (e.g. a normal and impaired myocardium). To test this, a simple physical model was considered, Fig. 24. Two LV regions with different elastic properties were represented by two elements AC and CB, moreover χ1 (AC) χ2 (BC), where χ is the stiffness coefficient. The ends of the elements are fixed at points A and B. An external force (F ) is applied perpendicularly to the point C. For a small time interval the point C moves to the point C along some trajectory close to circle with radius approximately equal to distance AC. So the element −−→ BC is stretching. The resulting displacement of the point C (vector CC ) −−→ −→ may be represented as a vector sum of normal (CD) and tangential (DC )

Fig. 24. Physical model for explaining LV wall regional displacement principle.39 Details in text.


50

F. Blyakhman

−→ −→ displacements. For small value of the angle α, the ratio of |CD| to |DC| is given by: −→ α cos α |CD| ∼ 1 (A6) −−→ = sin α = {α → 0} = α − 2 α→0 1. |DC | The expression shows that the less α, the more the difference between −−→ −→ −→ normal and tangent components of vector CC, so that CC |DC |. For −→ −→ a small time interval of the point C motion (when |CC| |AC|(|CB|)), −−→ −−→ the angle deviation of vector CC from CD is also small (the second-order infinitesimal). It implies the small displacement of the point C will take place into the normal direction. If the element ends A and B are free (see Fig. 24) the conclusion above is correct too for a short time interval. For this case, the ration between normal and tangential components of the point C displacement will change insignificantly; a displacement of the point B will be compensated by the CB element stretching, and the position of point A will be almost stable. According to the physiologic value of heart wall motion velocity (about of 0.02 m/s), we assumed that the displacement direction of any LV wall point is normal also for the interframe interval with 40 ms. A.4. LV regional ∆H calculation The approach of the region normal displacement was realized for an intact LV. At the beginning of the diastolic filling phase, LV endocardial surface was divided into the equal angular segments with step ∆θ = ∆ϕ = 5◦ in a spherical coordinate system. It resulted in the linear size of segments of about 4 × 4 mm. The assumption of the myocardium homogeneity within each LV surface element was used. The node points motion directions for every element of LV surface were calculated for the current frame as a normal to this point: → → → → → (A7) s1 × s2 , N = s1 × s2 →

→ →

where s1 , s2 are tangent vectors to the surface at a given node point; N is the normal vector to the surface at the same point. Tangent vectors to the point with coordinates (θ, ϕ) were calculated as: ∂x → ∂y → ∂z → j + i + k, ∂ϕ ∂ϕ ∂ϕ ∂x → ∂y → ∂z → → j + s2 = i + k. ∂θ ∂θ ∂θ

→

s1 =

(A8) (A9)



51

Figure 25 illustrates the building of normals at node points during the −−− → −→ j LV filling phase. Normal vectors Nij and Ni+1 were constructed for node points i and i + 1 on the endocardial surface at the beginning of LV filling (time t1). Then, points of intersection of these vectors with the endocardial surface for the next frame were determined. As a result of this procedure of frame-by-frame normal building, a location of each node point was calculated. Figure 25 demonstrates the complexity of a node-point trajectory, which is different from the radial direction. The wall thickness at the beginning (H1i ) and at the end (H2i ) of LV filling phase was calculated as the length of perpendicular dropped from

Fig. 25. Example of normal building at the node points i and i + 1 during LV filling phase.39 Direction of point displacement during the filling phase is determined by the −−−−→ frame-by-frame construction of the normal. The normal ( Nij+1 ) is constructed from the point i on the internal LV surface from time (tj ) to the internal one at (tj+1 ). Beginning (H1i ) and end (H2i ) wall thickness are calculated as the length of perpendicular dropped from node point i at the internal LV surface onto the external surface.


52

F. Blyakhman

node point i at the internal LV surface on the external one:

H i = (xi2 − xi1 )2 + (y2i − y1i )2 (z2i − z1i )2 ,

(A10)

where (xi1 , y1i , z1i ) are coordinates of the node point at the endocardial LV surface; and (xi2 , y2i , z2i ) are coordinates of the node point at the epicardial LV surface. Thus, the movement tracking of every wall point at LV stretching enables us to assess the myocardial relative thickness change: ∆H i = (H1i − H2i )/H1i × 100%. Possible error sources were considered to evaluate the method quantitatively. According to the different tests, the error of manual LV wall contours tracing, including the apparatus visualization error, is 2.3%; the errors of the approximation algorithm are 4.6%. The RMS error of the method as a whole is about 5.4%.

References 1. J. Lessick, Y. Fisher et al., Regional three-dimensional geometry of the normal human left ventricle using cine computed tomography, Annals of Biomedical Engineering 24, 5 (1996) 583–594. 2. D. D. Streeter, H. M. Spotnitz et al., Fiber orientation in the canine left ventricle during diastole and systole, Circulation Research 24 (1969) 339–347. 3. D. D. Streeter, and W. T. Hanna, Engineering mechanics for successive states in canine left ventricular myocardium: I. Cavity and wall geometry, Circulation Research 33 (1973) 639–655. 4. D. D. Streeter, Gross morphology and fiber geometry of the heart (Chapter 4), Topics in Handbook of Physiology. The Cardiovascular System. Heart (American Physiological Society, Bethesda, 1979) 61–112. 5. J. Bogaert, and F. Rademarkers, Reginal nonuniformity of normal adult human left ventricular, American Journal of Physiology — Heart and Circulatory Physiology 280 (2001) H610–H620. 6. A. M. Scher, and A. C. Young et al., Spread of electrical activity through the wall of ventricle, Circulation Research 1 (1953) 539. 7. G. Arisi, E. Macchi et al., Potential fields on the ventricular surface of the exposed dog heart during normal excitation, Circulation Research 52 (1983) 706–715. 8. C. J. Wiggers, The interpretation of intraventricular pressure curve on the basis of rapidly summated fractionate contractions, American Journal of Physiology 80, 1 1–11. 9. W. Y. Lew, and M. M. LeWinter, Regional comparison of midwall segment and area shortening in the canine left ventricle, Circulation Research 58, 5 (1986) 678–691.



53

10. F. E. Rademakers, W. J. Rogers et al., Relation of regional cross-fiber shortening to wall thickening in the intact heart. Three-dimensional strain analysis by NMR tagging, Circulation 89, 3 (1994) 1174–1182. 11. Y. Takayama, K. D. Costa, and J. W. Covell, Contribution of laminar myofiber architecture to load-dependent changes in mechanics of LV myocardium, American Journal of Physiology — Heart and Circulatory Physiology 282, 4 (2002) H1510–H1520. 12. R. A. Greenbaum, and D. G. Gibson, Regional non-uniformity of left ventricular wall movement in man, Br Heart J 45, 1 (1981) 29–34. 13. F. J. Villarreal, and W. Y. Lew, Finite strains in anterior and posterior wall of canine left ventricle, American Journal of Physiology 259 (1990) H1409–H1418. 14. P. Balzer, A. Furber et al., Regional assessment of wall curvature and wall stress in left ventricle with magnetic resonance imaging, American Journal of Physiology 277 (1999) H901–H910. 15. S. Delepine, A. Furber et al., MRI quantification of regional variations of left ventricular parietal stress in normal subjects, Archives des Maladies du Coeur et des Vaisseaux 92, 9 (1999) 1189–1196. 16. W. Krahwinkel, G. Haltern, and H. Gulker, Echocardiographic quantification of regional left ventricular wall motion with color kinesis, American Journal of Cardiology 85, 2 (2000) 245–250. 17. G. Barletta, and M. Baroni, Regional and temporal nonuniformity of shape and wall movement in the normal left ventricle, Cardiology 90, 3 (1998) 195–201. 18. D. L. Brutsaert, Nonuniformity: a physiologic modulator of contraction and relaxation of the normal heart, Journal of American College of Cardiology 9, 2 (1987) 341–348. 19. E. Hexeberg, K. Matre, and J. Lekven, Transmural fibre direction in the anterior wall of the feline left ventricle: theoretical considerations with regard to uniformity of construction, Acta Physiological Scandinavica 141, 4 (1991) 497–505. 20. J. Rijcken, P. H. Bovendeerd et al., Optimization of cardiac fiber orientation for homogeneous fiber strain during ejection, Annals of Biomedical Engineering 27, 3 (1999) 289–297. 21. M. B¨ uchi, O. M. Hess et al., Left ventricular wall stress distribution in chronic pressure and volume overload: effect of normal and depressed contractility on regional stress-velocity relations, Basic Research in Cardiology 85 (1990) 367–383. 22. L. K. Waldman, Y. C. Fung, and J. W. Covell, Transmural myocardial deformation in the canine left ventricle. Normal in vivo three-dimensional finite strains, Circulation Research 57 (1985) 152–163. 23. C. Yoran, J. W. Covell, and J. J. Ross, Structural basis for the ascending limb of left ventricular function, Circulation Research 48 (1973) 297–303. 24. P. Stein, and D. M. Marzilli et al., Systolic and diastolic pressure gradients within the left ventricular wall, American Journal of Physiology — Heart and Circulatory Physiology 238 (1980) H625–H630.


54

F. Blyakhman

25. L. K. Waldman, D. Nosan et al., Relation between transmural deformation and local myofiber direction in canine left ventricle, Circulation Research 63 (1988) 550–562. 26. J. V. Tyberg, W. W. Parmley, and E. H. Sonnenblick, In-vitro studies of myocardial asynchrony and regional hypoxia, Circulation Research 25 (1969) 569–578. 27. F. A. Blyakhman, V. S. Markhasin et al., The effect of asynchronous contraction of the myocardium on its mechanical function, Fiziol Zh SSSR Im IM Sechenova 75, 7 (1989) 923–930. 28. F. A. Blyakhman, Do we know much about the heart? Journal of Priroda 11 (1999) 23–30. 29. V. Ya. Izakov, L. B. Katsnelson et al., Cooperative effects due to calcium binding by troponin and their consequences for contraction and relaxation of cardiac muscle under various conditions of mechanical loading, Circulation Research 69, 5 (1991) 1171–1184. 30. R. L. Chen, D. J. Penny et al., Stretch-induced regional mechanoelectric dispersion and arrhythmia in the right ventricle of anesthetized lambs, American Journal of Physiology — Heart and Circulatory Physiology 286, 3 (2004) H1008–H1014. 31. A. J. Brady, Time and displacement dependence of cardiac contractility: problems in defining the active state and force–velocity relations, Fed. Proc. 24 (1965) 1410–1420. 32. Y. Yazaki, H. Tsuchimochi et al., Distribution of myosin isozymes in human atrial and ventricular myocardium: comparison in normal and overloaded heart, European Heart Journal 5 (1984) F103–F110. 33. L. Gorza, J. J. Mercadier et al., Myosin types in the human heart. An immunofluorescence study of normal and hypertrophied atrial and ventricular myocardium, Circulation Research 54, 6 (1984) 694–702. 34. J. Leger, C. Dechesne et al., Characterization and distribution of myosin variants in normal and pathological human hearts, European Heart Journal 5 (1984) F111–F117. 35. K. A. Yamada, E. M. Kanter et al., Transmural distribution of connexins in rodent hearts, Journal of Cardiovascular Electrophysiology 15, 6 (2004) 710–715. 36. C. Antzelevitch, and J. Fish, Electrical heterogeneity within the ventricular wall, Basic Res Cardiol 96, 6 (2001) 517–527. 37. B. Swynghedauw, Molecular mechanisms of myocardial remodeling, Physiological Reviews 79, 1 (1999) 215–262. 38. K. L. Schwartz, P. Carrier et al., Molecular basis of familial cardiomyopathies, Circulation 91 (1995) 532–540. 39. S. G. Kolchanova, A. A. Grinko et al., The regional elastic properties analysis of myocardium based on echocardiographic 3D reconstruction of the left ventricle, Journal of Ultrasound in Medicine and Biology 30, 3 (2004) 311–320. 40. V. Marti, R. Aymat et al., Coronary endothelial dysfunction and myocardial cell damage in chronic stable idiopathic dilated cardiomyopathy, International Journal of Cardiology 82, 3 (2002) 237–245.



55

41. A. F. van den Heuvel, D.J. van Veldhuisen et al., Regional myocardial blood flow reserve impairment and metabolic changes suggesting myocardial ischemia in patients with idiopathic dilated cardiomyopathy, Journal of American College Cardiology 35, 1 (2000) 19–28. 42. H. Mollnau, B. Munkel, and J. Schaper, Collagen VI in the extracellular matrix of normal and failing human myocardium, Herz 20, 2 (1995) 89–94. 43. A. L. Caforio, J. H. Goldman et al., Organ-specific cardiac autoantibodies in dilated cardiomyopathy — an update, European Heart Journal 16 (1995) O68–O70. 44. A. de Roos, J. Doornbos et al., Cardiac metabolism in patients with dilated and hypertrophic cardiomyopathy: assessment with proton-decoupled P31 MR spectroscopy, Journal of Magnetic Resonance Imaging 2, 6 (1972) 711–719. 45. A. L. Caforio, J. H. Goldman et al., Evidence for autoimmunity to myosin and other heart-specific autoantigens in patients with dilated cardiomyopathy and their relatives, International Journal of Cardiology 54, 2 (1996) 157–163. 46. V. V. Chestukhin, B. L. Mironkov et al., The functional inhomogeneity of myocardium at IHD is a factor determining indications to heart transplantation, Transplantologiya 2 (1995) 3–8. 47. P. Zardini, P. Marino et al., Ventricular remodeling and infarct expansion, American Journal of Cardiology 72, 19 (1993) 98G–106G. 48. D. E. Wallis, J. B. O’Connel, and R. E. Henkin, Segmental wall motion abnormalities in dilated cardiomyopathy, Journal of American College of Cardiology 4 (1984) 674–682. 49. R. M. de Jong, J. H. Cornel et al., Abnormal contractile responses during dobutamine stress echocardiography in patients with idiopathic dilated cardiomyopathy, European Journal of Heart Failure 3, 4 (2001) 429–436. 50. F. A. Blyakhman, V. Ia. Izakov, and A. D. Mikh, Method of representing the process of mechanical loading in isolated strips of myocardium (imitation of the cardiac cycle), Fiziol Zh SSSR Im IM Sechenova 70, 11 (1984) 1582–1584. 51. D. L. Brutsaert, W. W. Parmley, and E. H. Sonnenblick, Effects of various inotropic interventions on the dynamic properties of the contractile elements in heart muscle of the cat, Circulation Research 27, 4 (1970) 513–522. 52. J. N. Cohn, Critical review of heart failure: the role of left ventricular remodeling in the therapeutic response, Clinical Cardiology 18 (1995) IV4–IV12. 53. F. Fantini, G. Barletta et al., Alterations in left ventricular shape in patients with angina and single-vessel coronary disease, Coronary Artery Disease 5, 11 (1994) 901–908. 54. E. M. Hurs, S. Yu. Sokolov et al., The mechanical remodeling as the criterion of a choice sugrical treatment at a post infarction aneurysms of the left heart ventricular, The Bulletin Transplantology 2 (1999) 26–31. 55. F. A. Blyakhman, V. V. Chestukhin, and V. I. Shumakov, Asynchrony as modulator of the myocardial contractility and heart pump function (Chapter 1.7), Topics in Physiological Essays, ed. V. I. Shumakov (Repronics, Moscow, 1999) 180–210.


56

F. Blyakhman

56. F. A. Blyakhman, E. Marchenko, et al., Effect of the myocardium nonuniformity on the heart functional reserve, Journal of Mechanics in Medicine and Biology 5(1) (2005) 29–37. 57. G. F. Fletcher, G. J. Balady, and E. A. Amsterdam, Exercise standards for testing and training, Circulation 104 (2001) 1694–1740. 58. L. M. King, F. Boucher, and L. H. Opie, Coronary flow and glucose delivery as determinants of contracture in the ischemic myocardium, Journal of Molecular and Cellular Cardiology 27 (1995) 701–720. 59. S. F. Nagueh, I. Mikati, and D. Weilbaecher, Relation of the contractile reserve of hibernating myocardium to myocardial structure in humans, Circulation 100 (1999) 490–496. 60. H. Azhari, S. Sideman et al., Three-dimensional mapping of acute ischemic regions using MRI: wall thickening versus motion analysis, American Journal of Physiology 259, 5 Pt 2 (1990) H1492–H1503. 61. S. U. Sys, and D. L. Brutsaert, Is stiffness increased during ischemia? American Journal of Cardiology 63, 10 (1989) 83E–86E. 62. J. N. Cohn, R. Ferrari, and N. Sharpe, Cardiac remodeling-concepts and clinical implications: a consensus paper from an international forum on cardiac remodeling, Journal of American College of Cardiology 35, 3 (2000) 569–582. 63. M. A. Pfeffer, and E. Braunwald, Ventricular remodeling after myocardial infarction, Circulation 81 (1990) 1161–1172. 64. G. M. Hutchins, and B. H. Bulkley, Infarct expansion versus extension: two different complications of acute myocardial infarction, American Journal of Cardiology 41 (1978) 1127–1132. 65. G. Olivetti, J. M. Capasso et al., Side-to-side slippage of myocytes participates in ventricular wall remodeling acutely after myocardial infarction in rats, Circulation Research 67 (1990) 23–34. 66. F. A. Blyakhman, T. F. Shklyar et al., Why the left ventricle is not a sphere, Applied Bionics Biomechanics 2 (2004) 101–105. 67. F. A. Blyakhman, and Kh. M. Nafikov et al., An experimental model of the mechanical heterogeneity of the myocardium, Fiziol Zh SSSR Im I M Sechenova 74, 8 (1988) 1191–1195. 68. H. J. Appell, and C. Stang-Voss, A peculiar fibrillar pattern in the myocardial cells of trabeculae carneae in the right ventricle of the rat, mouse, and rabbit, Celland Tissue Research 208, 1 (1980) 165–168. 69. J. Rijcken, P. H. Bovendeerd et al., Optimization of cardiac fiber orientation for homogeneous fiber strain during ejection, Annals of Biomedical Engineering 27, 3 (1999) 289–297. 70. G. Keren, E. Sonnenblick et al., Mitral annulus motion. Relation to pulmonary venous and transmitral flows in normal subjects and in patients with dilated cardiomyopathy, Circulation 71, 3 (1988) 621–629. 71. S. G. Kolchanova, M. L. Shur, and F. A. Blyakhman, The method of definition of regional elastic properties for a hollow organ wall, Patent RU (1) No 2217042 33 (2002). 72. S. Yu. Sokolov, S. G. Kolchanova et al., A program DICOR for the determination of the regional myocardial contractility, Program for PC, database, IC topology Patent RU RU No 2002610607, 3 (2002) 79.


QUANTIFICATION OF CARDIAC PERFUSION AND FUNCTION USING NUCLEAR CARDIAC IMAGING RU-SAN TAN∗ , LIANG ZHONG∗,† , TERRANCE CHUA∗,‡ and DHANJOO N. GHISTA§ ∗Department

of Cardiology, National Heart Centre, Singapore †[email protected] ‡Terrance [email protected]

§School of Mechanical and Aerospace Engineering, College of Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 [email protected]

1. Physiology of Myocardial Ischemia 1.1. Definition The coronary arteries transport oxygen- and nutrient-rich blood to the myocardium (heart muscle) to sustain the heart’s normal contractile pumping function. The pumping heart in turn generates pressure and flow within the coronary circulation to supply blood and oxygen to the heart. In coronary artery disease (CAD), atherosclerotic fatty deposits in the coronary artery wall narrow the lumen (opening) of the arteries, compromising blood flow and oxygen delivery to the heart. Myocardial ischemia is said to occur when coronary blood flow (CBF) or perfusion is insufficient for myocardial metabolic needs. This can occur when there is a reduction in blood flow due to a severe obstruction of the coronary arteries, or when increased metabolic demands are not met by an increased supply of oxygen, such as during physical exercise. During exercise, the workload of the heart increases so as to increase cardiac output to meet the increased requirements of the skeletal muscle. This increased workload requires an increase in coronary blood supply, which is met by vasodilation of the arteriolar bed. However, in patients with CAD, this normal increase in CBF with exercise may be limited due to narrowing (stenosis) of one or more coronary arteries. Thus in a patient with a coronary stenosis, ischemia may arise when myocardial metabolic requirements and oxygen demand increase but supply is limited by coronary disease, such as when the heart is stressed during exercise. Hence, a decrease in 57


58

Ru-San Tan et al.

Heart rate Contractility Systolic wall stress

Blood oxygenation level Hemoglobin level Coronary blood flow

Myocardial oxygen demand

Demand/supply imbalance

Myocardial oxygen supply

Myocardial ischemia

Fig. 1.

Factors influencing myocardial ischemia.

myocardial perfusion and/or an increase in myocardial metabolism impose an imbalance in supply–demand that results in ischemia (Fig. 1).

1.2. Ischemic cascade The onset and manifestations of myocardial ischemia usually occur in a stepwise sequence, which has been termed the ischemic cascade. The initial abnormality is that of insufficient CBF, either a reduction in CBF or an increase in metabolic demand that is not matched by an increase in blood supply. There is heterogeneity of CBF with lower blood flow in ischemic territories (supplied by diseased coronary arteries) compared to non-ischemic territories, and also from the subendocardium to the subepicardium. Impairment of myocardial myofiber-active relaxation and contraction ensue, resulting in diastolic (relaxation) and systolic (contraction) dysfunction, respectively. Depending on the extent of myocardial involvement and adequacy of compensatory hyperfunction of non-ischemic myocardium, global myocardial function, left ventricular ejection fraction (LVEF) and stroke volume deteriorate variably. Ischemic changes on surface electrocardiograms (ECG) appear late, often heralding the onset of anginal symptoms. The evaluation of ischemia relies on the detection and measurement of these event parameters. Using various imaging and diagnostic technologies, alterations in coronary perfusion, myocardial contractile function, and ECG may be identified and quantified (Fig. 2).


Quantification of Cardiac Perfusion and Function

Ischemic cascade

59

Methods to detect ischemia

Onset of ischemia

Flow heterogeneity

Myocardial perfusion imaging (nuclear perfusion , myocardial contrast echo )

Diastolic dysfunction

Systolic dysfunction

Stress echocardiography

ECG changes

Stress ECG

Anginal symptoms

Fig. 2. Diagnosis of ischemia by detection of abnormalities associated with the ischemic cascade of events.

1.3. Coronary autoregulation Coronary arteries arborize into a network of fine arterioles. The walls of these arterioles contain smooth muscle cells that contract in the basal state, contributing to coronary tone and resistance. When a proximal epicardial coronary artery develops stenosis, the corresponding distal arterioles dilate in response, thus lowering coronary resistance and effectively maintaining unchanged CBF. With exercise stress, tissue accumulation of metabolites occurs, which relaxes arteriolar wall smooth muscle cell contraction. Arterioles further dilate (coronary hyperemia) by varying amounts depending on the basal coronary tone. CBF in non-stenosed arteries expand more than in stenosed arteries because arterioles supplied by the latter are already semi-dilated even before exercise begins. Hence, the coronary flow reserve (CFR), defined as the ratio of stress to rest CBF, is lower in stenosed arteries compared to normal arteries (Fig. 3).


60

Ru-San Tan et al.

Rest

No stenosis

Stress Physiologically significant stenosis

Epicardial arteries

A

B

C

D

1×

1×

3×

1.5×

3

1.5

Arteriole lumen Regional coronary blood flow

Coronary flow reserve

Fig. 3. CFRs in normal and stenotic coronary arteries. In normal arteries, arteriolar wall contraction maintains coronary tone at rest (A). With proximal epicardial coronary artery stenosis, the corresponding distal arterioles dilate, lowering resistance and maintaining normal CBF (B). With exercise stress, arterioles dilate (coronary hyperemia). The quanta of increase, thence CFR, are greater in normal (C) versus stenotic (D) arteries because of the former’s greater basal coronary tone. With pharmacologic vasodilatory stress, there is maximal arteriolar dilation and even greater CBF enhancement.

2. General Principles of Myocardial Perfusion Imaging 2.1. Myocardial perfusion tracer agents Blood flow to the myocardium may be tracked using parenterally administered compounds, such as radioisotope-labeled tracers, echocardiographic microbubbles, magnetic resonance relaxivity, and X-ray contrast agents. Besides CBF, other factors determine myocardial blood flow tracer uptake characteristics, including the relative distribution of the tracer between intravascular (within blood vessels) and extravascular compartments, the degree of trapping within versus redistribution outside myocardial cells, and the radionuclide half-life in the case of radiopharmaceuticals. The latter two factors determine persistence of the tracer in the myocardium. If persistent, myocardial perfusion imaging (MPI) may be performed at any convenient time after the tracer has been administered. For blood flow tracers that transit briefly and rapidly clear from the myocardium, MPI



61

must be performed during their initial passage through the myocardium and is technically more challenging. 2.2. Assessing coronary flow and CFR It is possible to quantify absolute CBF (expressed as ml/min/gr of myocardial tissue) in human beings using pure intravascular flow agents, such as radiolabeled 15 O-water, 13 N-ammonia, or rubidium-82, all positron emission tomography (PET) tracer agents. However its expense, the limited availability of PET facilities, and expertise impede widespread application of the technology. More commonly, the relative CFR of areas of myocardium supplied by normal versus stenosed coronary arteries are assessed by performing MPI at rest and under conditions of exercise, pharmacologically induced cardioexcitatory, or vasodilatory stress. While CBF in all myocardial territories may be equal at rest (yielding homogeneous tracer uptake in all myocardial regions), attenuated amplification of regional CBF during stress implies presence of physiologically significant coronary stenosis in the corresponding proximal epicardial arterial territory (Fig. 3). Alternatively, MPI during acute ischemic episodes, for example, during anginal chest pain at the coronary care unit or emergency department, may similarly uncover heterogeneity of myocardial tracer uptake. 2.3. MPI techniques Various MPI techniques are employed to visualize the distribution and concentration of perfusion tracers within the heart. All exploit the heterogeneity of blood flow between myocardial regions that are perfused by normal coronary arteries and regions that are supplied by coronary vessels with physiologically significant coronary artery stenoses. Table 1 lists these techniques, their strengths, and limitations. Of these, nuclear MPI is arguably the most established and commonly used technique.

3. Nuclear MPI Nuclear cardiology studies use small doses of radioactive material to assess myocardial blood flow, evaluate the pumping function of the heart, as well as visualize the size and location of a heart attack.

Technique

PET

Tracer

15 O–H

2 O, 13 N–NH , 3

Comparison of MPI techniques.

Parameter

Quantitative analysis

Spatial resolution (mm)

Renal risk

Ionizing radiation

CBF, CRF

Yes

5–6

−

++

Rb-82 Yes

10–12

−

++

CBF, CRF

Yes

2–3

+∗

−

MCE

Microbubble

CBF, CRF

Yes

1–2

−

−

Angiography

X-ray contrast

No

0.1

+++

++

CT scan

X-ray contrast

Myocardial blush grade CRF

No

0.5

+++

+++

High cost, limited availability Attenuation, long scan protocols Pacemakers, ICD contraindicated Little consensus on echo technique Invasive method, qualitative test Early development

CBF, coronary blood flow; CRF, coronary flow reserve; CT, computed tomography; ICD, implantable cardioverter–defibrillator; MCE, myocardial contrast echocardiography; MRI, magnetic resonance imaging; PET, positron emission tomography; SPECT, single photon emission computed tomography. ∗ Rare cases of a serious generalized skin condition, nephrogenic, fibrosing dermopathy, have been reported in subjects with impaired kidney function.

Ru-San Tan et al.

CRF

MRI

Th-201, Tc-99 m agent Gadolinium

SPECT

Limitations


62

Table 1.



63

3.1. Nuclear myocardial perfusion tracer agents MPI is performed with intravenously injected radionuclides, thallium-201or technetium-99 m-based agents (sestamibi, tetrofosmin, and teboroxime compounds). These tracers are trapped in the myocardium in proportion to the regional CBF. Further, intact cell membrane and mitochondrial function are obligatory for radioactive thallium-201- or technetium-99 m-based tracers, respectively, to enter myocardial cells. Hence, myocardial uptake of these tracers reflects both myocardial perfusion and myocardial viability. Thallium-201 was one of the earliest tracers to be used for MPI, with an excellent tissue extraction fraction.1 The rate of tracer uptake by the myocardium is almost linearly correlated to CBF, except at very high levels of CBF, when it starts to plateau. Thallium-201 has however, some limitations. Its relatively long half-life (73 h) exposes the patient to prolonged radioactivity, limiting the dose that can be administered. Second, its main photopeak is suboptimal for gamma camera imaging (see below). These disadvantages adversely impacted the technical quality of the images, providing incentive for the development of the newer technetium99 m-based tracers. The tissue extraction fractions of the commonly used technetium-99 m-based tracers (sestamibi and tetrofosmin) are less than that of thallium-201. In spite of this, the latter’s image quality is superior due to a more favorable photon energy (140 keV) as well as the higher amounts of radioactive technetium-99 m that can be administered due to its shorter half-life (6 h) and the persistence of the tracer in the myocardium. Unlike thallium-201, which slowly redistributes from perfused into nonperfused myocardial tissue, technetium-99 m (sestamibi and tetrofosmin) persists in the perfused myocardial tissue. This enables MPI to be conveniently performed long after initial injection of the tracer.2 It also allows for repeat scanning without concern that the tracer has washed out or redistributed from its initial pattern of uptake, if there is a technical problem such as patient motion that requires a repeat scan. 3.2. Stress agents and scan protocols Graded exercise stress is performed either on a treadmill or a bicycle ergometer, using various standardized protocols. The ubiquitous symptom-limited Bruce treadmill exercise protocol consists of consecutive 3-min exercise stages at successively faster speeds and steeper gradients. Exercise stress induces coronary hyperemia, resulting in heterogeneous


64

Ru-San Tan et al.

increase in regional CBF in non-stenosed arteries (with high basal coronary tone) versus stenosed arteries (with lower basal coronary tone) due to their different CFR. For patients who are unable to exercise adequately, the physiologic effects of exercise may be simulated by infusion of drugs. The adrenergic drug dobutamine, administered at a dose of 5 µg/kg/min with increment every 3 min to 40 µg/kg/min, increases heart rate and myocardial contractility, thereby growing myocardial oxygen demand that leads to coronary hyperemia. Other pharmacologic agents may also directly induce coronary hyperemia. Intravenous adenosine or dipyridamole, administered at a dose of 140 µg/kg/min, maximally dilates coronary arterioles, augmenting flow heterogeneity in myocardial territories supplied by normal versus stenosed epicardial arteries. This is known as vasodilator stress imaging, and is the most commonly applied radionuclide pharmacologic stress technique. All stress procedures are performed with continuous monitoring of patients’ vital signs (symptoms, blood pressure, and pulse rate) and ECG. Radionuclides tracers are administered during maximal exercise, pharmacologic cardioexcitatory or vasodilatory stress. As the radionuclides tracers persist in the myocardium for some time, MPI can be delayed allowing sufficient time for patient to be transferred to the scanner after the stress. Technetium-99m-based agents, provide more counts than thallium-201 and offer the possibility of imaging of the tracer as it passes through the heart after rapid bolus infusion of tracer. This is known as first-pass imaging, and allows additional information to be obtained such as an estimation of left ventricular (LV) function during exercise. To take advantage of this possibility, a gamma camera capable of rapid counts, such as a multi-crystal gamma camera, is recommended. First-pass imaging during rapid bolus infusion of radioactive tracer is less commonly performed, as it requires the patient to be in the scanner during the stress procedure (which limits the choice of exercise stress possibilities). Different scan protocols may be employed. Separate day acquisitions for stress and rest scans allow sufficient time for the radioactive tracer to be washed out between scans through radioactive decay and bodily excretion. For technetium-99 m-based agents with short radioactive halflife, same day stress-rest or rest-stress acquisitions is feasible, with the two radionuclide boluses administered a few hours apart. As thallium-201 redistributes from non-ischemic into ischemic tissues, a second scan may be



65

performed a few hours (or even 24 h) after tracer administration to assess rest (redistribution) perfusion. In addition, a dual-tracer protocol combining rest thallium-201 followed shortly by stress technetium-99m (sestamibi) injections may be employed. This exploits the distinct photopeaks associated with each tracer to enable rest and stress perfusion images to be separately acquired at one sitting.3

3.3. Image acquisition and processing Radionuclides administered into patients course through the blood circulation and enter perfused viable myocardial cells, from where they emit gamma radiation, a form of electromagnetic energy. The gamma rays are detected using scintillation counters mounted onto photomultipliers. Gamma particles interact with matter within the scintillator (usually made of sodium iodide crystal) to produce photoelectrons with characteristic energy quanta, photopeaks, specific to the radionuclide. These generate light flashes that are in turn converted to electric pulses by a system of photomultiplier tubes. The electric pulses, with voltages proportional to the energy of the original electrons, are sorted and analyzed by computers to form two-dimensional spatial images of the scintillation count densities, which reflect the distribution and relative concentration of radioactive tracer present in the heart. Imaging the heart from fixed standard external positions yields simple projection perfusion images (planar imaging). This has now been superseded by single photon emission computed tomography (SPECT) acquisition technique. In SPECT, multiple projection images at equally spaced angular displacement intervals are acquired of the chest by one or more gantry-mounted scintillation detectors, positioned near the chest wall, that rotate around the patient’s body axis as the patient lies supine on the SPECT table. Three-dimensional image reconstruction using filtered backprojection4 enables good separation of different heart chambers in space, which surmounts the problem of structure overlap inherent in planar imaging (Fig. 4). Further, SPECT offers improved perfusion defect detection (because of enhanced contrast resolution), localization, and sizing compared to planar imaging.5 In general, myocardial perfusion SPECT acquisition takes from 10 to 30 min, depending on the number of detectors and the type and dose of the radiopharmaceutical. ECG gating can be performed using electrodes and a gating device. This yields temporally resolved SPECT datasets


66

Ru-San Tan et al.

Fig. 4.

Stress myocardial perfusion SPECT.

(usually eight per heart cycle) that when lined up sequentially and displayed as a cine loop enables visualization and quantification of cardiac contractile function.

4. Interpretation of Myocardial Perfusion SPECT The acquisition, computer processing, and display of scintigraphic (radionuclide imaging) data by high-speed computers constitute a fundamental step in the process of nuclear imaging. Quantitative analysis of the digitized information reduces the subjectivity of visual assessment and augments the reproducibility of evaluation and interpretation of perfusion SPECT data.6 4.1. Inspection of raw projection data The rotating raw projection images are first reviewed to ensure that they have been acquired over the appropriate acquisition arc and are devoid of artifacts such as excessive motion between projections. Motion artifacts may result from either excessive patient movement within the scanner or a postexercise gradual shift in the position of the diaphragm from a lower (more inflated lung) to a higher position (“upward creep”) when the patient is



67

lying supine during the course of image acquisition. As technetium-99 m tracers, sestamibi and tetrofosmin, are excreted through the intestines, there may occasionally be areas of increased tracer uptake in gut tissue near the inferior surface of the heart. These potentially cause error during image reconstruction and must be eliminated either by digital image postprocessing or repeat image acquisition. In addition, abnormal extracardiac loci of high radionuclide tracer uptake may rarely be found in the lungs or other organs within the imaging field. These warrant further investigation to determine their nature and exclude malignant lesions. One common problem encountered in ECG-gated SPECT is inaccurate gating due to arrhythmia or poor ECG signal. This may manifest as a “flashing” appearance. Rejected beats and low counts in some projections result in line artifacts in the back-projected reconstructed volume. Where there is concern that gated acquisition could have introduced error into perfusion data, the acquisition is repeated without ECG gating.

4.2. Visual assessment of SPECT myocardial perfusion SPECT allows evaluation of perfusion patterns in reconstructed thin slices of myocardium. While the three-dimensional SPECT perfusion image dataset may be freely reformatted and manipulated at will, a uniform approach to SPECT image display has been universally adopted. The reconstructed tomographic images are reoriented relative to the major axes of the LV yielding short-axis, vertical long-axis, and horizontal long-axis slices. Imaging slices from different acquisitions (i.e., stress–rest, stress– redistribution) are displayed in an interleaved fashion, and aligned to ensure that each slice of the stress MPI correctly matches its corresponding rest slice. Comparison of stress and rest images help in differentiating ischemia from infarct scar. A perfusion abnormality present in the stress study that resolved in the rest study represents ischemia, whereas a perfusion abnormality on both the stress and rest studies represents scar (Fig. 5). An alternative explanation for a persistent perfusion defect is an attenuation artifact. This is caused by the non-uniform decrease in gamma radiation as the rays traverse soft tissue of varying thicknesses before hitting the scintillation detector surface. These attenuation artifacts typically produce non-reversible perfusion defects present both on the stress and rest studies and are associated with characteristic locations (e.g., the basal inferior wall in men, anterior wall in women). The absence of wall motion abnormality on gated SPECT helps to distinguish these non-reversible defects from


68

Ru-San Tan et al.

Fig. 5. Myocardial perfusion SPECT. Alternating rows of stress and rest images are aligned. Contiguous thin slices of myocardium are displayed, from left to right: apical to basal short-axis slices (top 2 rows), septal to lateral vertical long-axis slices (middle 2 rows), inferior to superior horizontal long-axis slices (bottom 2 rows). On the color image, maximal tracer uptake (which implies high CFR) is depicted in bright orange, and areas of perfusion defect are in graduated darker hues of blue-green. Note the large area of stress perfusion defect at the apex, lateral and inferior walls that is largely reversible on the rest images. This patient exhibits severe coronary ischemia that correspond to multiple epicardial coronary arterial territories.

scarring due to prior myocardial infarction and damage. Special hardware and software may be used to correct for soft tissue attenuation. The LV may be divided into 17 or 20 segments for semiquantitative analysis. Our laboratory uses the latter system (Fig. 6). Each segment is scored according to a 5-point scheme: 0, normal; 1, slight reduction of tracer uptake; 2, moderate reduction of tracer uptake; 3, severe reduction of tracer uptake; and 4, absence of tracer uptake. Post-processing of SPECT images to yield a stepped color display facilitates visual assessment of regional relative CFR (Fig. 7). This systemic approach to SPECT myocardial perfusion interpretation is conceivably more reproducible than simple qualitative evaluation.



69

Fig. 6. The 20-segment model of the LV. The entire LV is represented by three shortaxis slices (apical, mid and basal) that are each subdivided into six segments (clockwise when viewed from the apex: anterior, anterolateral, inferolateral, inferior, inferoseptal, and anteroseptal), as well as two apical segments (anteroapical, inferoapical) visualized in the mid-vertical long-axis imaging slice. Each segment constitutes approximately 5% of the LV volume. The extent of perfusion defect can be estimated by counting the number of segments involved.

Fig. 7. Myocardial perfusion SPECT in a patient with a large anterior myocardial infarct. The regional tracer uptake is normalized with respect to the maximum tracer activity detected count the myocardium, and then displayed according to deciles of percentage maximal counts using a stepped color scale (right). Note the large area of stress perfusion defect at the inferior, septal, distal anterior and apical walls that is nonreversible at rest. In addition, the tracer activity in these areas is less than 50–60% of maximum counts, implying that they are nonviable and are unlikely to improve with revascularization of the diseased coronary arteries.


70

Ru-San Tan et al.

4.3. Quantitative assessment of SPECT myocardial perfusion The quantitative approach is generally recognized as a desirable tool to standardize analysis and improve the reproducibility of the nuclear cardiac assessment. When quantitative methods are used, the reported sensitivities and specificities are 90% and 70% for thallium-201 SPECT imaging,6–13 versus 83% and 82% for technetium-99 m SPECT images analyzed using one commonly used commercial automated software (AutoQuant, CedarsSinai)14,15 The apparent decline in specificity may be due partially to the effect of referral bias16 increasing over time. Once a diagnostic test becomes widely accepted, patients with abnormal test results are more likely to be referred for angiography than patients with normal results. Although this is appropriate clinically, it has a profound effect on test specificity since patients with normal coronary arteries are unlikely to undergo angiography unless they have falsely positive test results. Various software algorithms are employed to isolate and automatically segment the LV for analysis (Table 2).14,15,17–21 Nevertheless, all quantitative output for myocardial perfusion should be checked visually for accuracy of the program in defining the myocardial borders. It may be necessary to adjust the gain settings for patients with severe perfusion defects to do this. The potential for failure of automated edge-detection programs exists, particularly in patients with extra-cardiac activity close to the heart or with large and severe defects such as those observed in LV aneurysms. Most quantitative analyses display the tomographic slices in a polar map format. This provides a comprehensive representation of the extent, severity, and reversibility of regions of perfusion abnormality expressed as percentages of the entire LV, on a single compact two-dimensional image. The individual patient’s polar map perfusion can then be compared with a parametric reference polar map derived from disease-free normal patients, with low likelihood of coronary artery disease, and visually normal images (Figs. 8 and 9). The criteria for abnormality are typically derived from the mean and standard deviation of the count uptake in each region of the LV. It is important to note that normal databases are usually specific to the scan protocol, gender, and tracer (thallium-201, technetium-99 m sestamibi or tetrofosmin, the dual isotope) used. Acquisition and image processing parameters (filters) need also be specified.


Quantification of Cardiac Perfusion and Function Table 2.

71

Available software for automated SPECT quantitation of mycardial perfusion.

Name CEqual17

CardioMatch18

Emory Toolbox18

3D-MSPECT19

Yale-CQ20,21

AutoQuant14,15

Software function and components Automated sampling of LV volume (40 points per circumferential profile) to yield two-dimensional polar maps, which are compared to normal database of patients with normal perfusion, low likelihood of CAD and visually normal images. Criteria for abnormality are derived from the mean and standard deviation of the count uptake in each LV region. The patient’s stress profiles are normalized to the normal database, by dividing the heart is divided into 10 regions and scaling the most normal patient sector (highest average counts) to the corresponding sector of the normal database. Automated registration of three-dimensional stress and rest images, comparison with normal database to generate measurements of perfusion defect size and extent. Database is stratified by gender, age, weight, and heart size. CEqual software, databases for different protocols, automated calculation of transient ischemic dilation ratio, gated SPECT display, three-dimensional display of perfusion with coronary artery tree overlay (PerSPECTive), expert systems analysis of results (PERFEX). Automated processing and comprehensive quantitative analysis of perfusion and function, three-dimensional SPECT, gated SPECT and attenuation-corrected SPECT with normal databases. Automated sampling of LV circumferential count profiles using an automatic algorithm to localize the center and define the edges of short-axis SPECT slices. Uniformly distributed three-dimensional sampling of LV employing an ellipsoid model. Counts are averaged across the entire myocardial thickness. Integrated program for simultaneous assessment of perfusion (QPS) and function (QGS, using gated SPECT). Automated perfusion quantification, calculation of LVEF, volumes, lung–heart ratio, transient ischemic dilation ratio, regional wall thickening, wall motion. Standard normal databases, plus optional user-defined database generation.

4.4. Quantitative assessment of myocardial viability from perfusion SPECT The likelihood of functional recovery subsequent to successful revascularization has been shown to be proportional to the amount of tracer uptake within each myocardial region. A cutoff value of 50–60% of maximal counts appears to best identify LV segments that will recover functionally following coronary revascularization.22,23 Normal myocardial perfusion tracer uptake at rest or redistribution implies the presence of myocardial viability.


72

Ru-San Tan et al.

Fig. 8. Polar map quantitation. Tracer activity at the LV apex is depicted in the center. Count activities in the short-axis slices are displayed around the center as successive flat annular rings of increasing radii as the slices progress toward the base of the LV. In this diagram, the subject exhibits a large perfusion defect in the left anterior descending artery (LAD) territory that largely spares the left circumflex (LCX) and right coronary artery (RCA) territories.

Sublingual nitrate (which dilates the coronary vasculature, hence improving coronary flow) administered prior to rest tracer injection augments the detection of viable myocardium.24 If a region has severely reduced or no uptake of radioactivity in these settings, then it is considered to be nonviable (Fig. 7). Areas with moderate reduction of counts are usually partially viable, and patients in this group have a variable response in terms of recovery of myocardial contractile functional improvement after operation. Dichotomous classification of perfusion defects on thallium-201 imaging into reversible or non-reversible using qualitative assessment fails to detect many visually non-reversible defects that exhibit evidence of viability on PET imaging (considered the gold standard for viability assessment) or functional recovery after revascularization. However, when defects are quantitated, correlation between thallium-201 and PET imaging25 improves substantially to about 80%; and agreement between thallium-201 and technetium-99 m sestamibi imaging improves to 96%.26 Thus quantification of defect severity is highly recommended for the assessment of myocardial viability.



73

Fig. 9. Quantitative myocardial perfusion assessment using AutoQuant (QPS) program. Algorithm-derived myocardial borders and segmentation are superimposed on reconstructed myocardial tomographic slices, facilitating visual check of the accuracy of myocardial border definition (left column). Stress and rest perfusion polar maps, as well as three-dimensional displays, show a moderate-sized partially reversible defect at the apex, distal anterior, and anteroseptal walls (middle two columns). Calculated numerical data on perfusion defect size and severity are also depicted (right column). Automated quantitative regional perfusion scores at stress and rest (bottom right corner) yield sum stress score (SSS) 14, sum rest score (SRS) 2, and sum difference score (SDS) 12, implying significant ischemia (top right corner). The quantitative algorithm compares acquired perfusion datasets with normal databases that are gender, radionuclide, and protocol dependent.

4.5. Assessment of LV function from gated perfusion SPECT LVEF assessment from gated perfusion SPECT (especially of technetium99m tracer) can be achieved at very little additional cost. In general, gated perfusion SPECT measurements of LVEF are highly reproducible and correlate well with other imaging modalities.19,27–34 However, in patients with small LV volumes (end-diastolic volume 20%), prior infarction and large infarction, in whom the capability of an automated approach would be most severely tested. Further, we demonstrated good agreement between gated SPECT and echocardiographic assessment of regional function. We concluded that gated SPECT LVEF measurements are generally accurate and reliable, even in the presence of large defects.50

5. Clinical Applications of Quantitative Myocardial Perfusion SPECT In general, quantification has previously not had a major impact on the diagnostic accuracy of perfusion SPECT, although the advent of more



77

sophisticated computer programs is changing this. Indeed, interpretation of studies using polar map quantitation alone without visual assessment of the tomographic slices is not advisable. Nevertheless, quantitation provides a means of describing the extent and severity of perfusion abnormalities, and may call attention to subtle decreases in tracer uptake below the normal limits that might otherwise go unnoticed on visual inspection. In contrast, quantification has been an important factor in the success of gated perfusion SPECT because of its ability to provide reliable reproducible LVEF measurements. Beyond the simple detection of CAD, nuclear cardiology has enormous value as a tool for risk-stratification and selection of patients for medical management or interventional therapy because of the large database of studies demonstrating its ability to prognosticate. Semiquantitative scoring systems of visually interpreted images, in particular, has been shown to be an effective method. The development of automated programs for obtaining semiquantitative perfusion scores may improve their reproducibility and reduce variability in interpretation between different laboratories. Berman et al. showed that the automatic scoring program (AutoQuant/QPS) has equivalent prognostic accuracy compared to expert visual assessment. Stress defects >10% and reversibility >5% of LV are associated with worse outcome.51 Similarly, in patients undergoing risk stratification following myocardial infarction, stress perfusion defects and defect reversibility quantified by polar map programs appear to be correlated with prognosis.52 In addition to the widely accepted value of LVEF, volume measurements by gated SPECT have been shown to have prognostic value in a large study involving dual isotope rest thallium stress sestamibi gated SPECT. Sharir et al. showed that LVEF < 45%, end-diastolic volume >120 ml and end-systolic volume >70 ml were associated with adverse prognosis.53 The advent of quantitative gated SPECT has made automated measurement of volumes routinely feasible and highly reproducible. The ability of gated SPECT to provide additional information on LV function contributes further to the prognostic power of nuclear cardiology.

6. Conclusion One of the major advantages that nuclear cardiology has over other imaging modalities is the ease by which nuclear cardiac data can be quantified and analyzed. In principle, quantification especially when coupled with


78

Ru-San Tan et al.

automation of computer processing offers the potential to reduce or eliminate the subjectivity of visual assessment and to increase the reproducibility of evaluation. Quantification is also required to allow for comparison of perfusion data with a normal database thereby assisting in the detection of disease and the objective assessment of extent and severity. In patients with CAD, the presence and extent of inducible myocardial ischemia, the extent of myocardial infarction, and LV function are among the most powerful predictors of subsequent cardiac events. These powerful prognostic indicators are provided by nuclear cardiac studies and their value is greatly enhanced when they are provided in objective, quantitative terms. References 1. H. F. Weich, H. W. Strauss, and B. Pitt, The extraction of thallium-201 by the myocardium, Circulation 56 (1977) 188–191. 2. P. Kailasnath, and A. J. Sinusas, Technetium-99 m-labeled myocardial perfusion agents: Are they better than thallium-201? Cardiology Review 9 (2001) 160–172. 3. D. S. Berman, H. Kiat, J. D. Friedman et al., Separate acquisition rest thallium-201/stress technetium-99 m sestamibi dual-isotope myocardial perfusion single-photon emission computed tomography: a clinical validation study, Journal of American College of Cardiology 22 (1993) 1455–1464. 4. R. A. Brooks, and G. Di Chiro, Theory of image reconstruction in computed tomography, Radiology, 117 (1975) 561–572. 5. D. J. Fintel, J. M. Links, J. A. Brinker, T. L. Frank, M. Parker, and L. C. Becker, Improved diagnostic performance of exercise thallium-201 single photon emission computed tomography over planar imaging in the diagnosis of coronary artery disease: a receiver operating characteristic analysis, Journal of American College of Cardiology 13 (1989) 600–612. 6. N. Tamaki, Y. Yonekura, T. Mukai, S. Kodama, K. Kadota, H. Kambara, and C. Kawai, K. Torizuka, Stress thallium-201 transaxial emission tomography: quantitative versus qualitative analysis for evaluation of coronary artery disease, Journal of American College of Cardiology 4 (1984) 1213–1221. 7. E. V. Garcia, K. Van Train, J. Maddahi et al., Quantification of rotational thallium-201 myocardial tomography, Journal of Nuclear Medicine 26 (1985) 17–26. 8. J. Maddahi, K. Van Train, F. Prigent, E. V. Garcia, J. Friedman, E. Ostrezega, and D. Berman, Quantitative single photon emission computed tomography for detection and localization of coronary artery disease: optimization and prospective validation of a new technique, Journal of American College of Cardiology 14 (1989) 1689–1699. 9. K. F. Van Train, J. Maddahi, D. S. Berman et al., Quantitative analysis of tomographic stress thallium-201 myocardial scintigrams: a multicenter trial, Journal of Nuclear Medicine 31 (1990) 1168–1179.



79

10. E. DePasquale, A. Nody, G. DePuey et al., Quantitative rotational thallium201 tomography for identifying and localizing coronary artery disease, Circulation 72 (1988) 316–327. 11. J. J. Mahmarian, T. M. Boyce, R. K. Goldberg, M. K. Cocanougher, R. Roberts, and M. S. Verani. Quantitative exercise thallium-201 single photon emission computed tomography for the enhanced diagnosis of ischemic heart disease, Journal of American College of Cardiology 15 (1990) 318–329. 12. S. Borges-Neto, J. J. Mahmarian, A. Jain et al., Quantitative thallium-201 single-photon emission computed tomography after oral dipyridamole for assessing the presence, anatomic location and severity of coronary artery disease, Journal of American College of Cardiology 11 (1988) 962. 13. A. S. Iskandrian, J. Heo, B. Kong et al., Effect of exercise level on the ability of thallium-201 tomographic imaging in detecting coronary artery disease: analysis of 461 patients, Journals of American College of Cardiology 14 (1989) 1477. 14. G. Germano, P. B. Kavanagh, P. B. Waechter, J. S. Areeda, J. Gerlach, S. Van Kriekinge, T. Sharir, H. C. Lewin, and D. S. Berman, A new algorithm for the quantitation of myocardial perfusion SPECT. I. Technical principles and reproducibility, Journal of Nuclear Medicine 41 (2000) 720–727. 15. T. Sharir, G. Germano, P. B. Waechter, P. B. Kavanagh, J. S. Areeda, J. Gerlach, X. Kang, H. C. Lewin, and D. S. Berman. A new algorithm for the quantitation of myocardial perfusion SPECT. II. Validation and diagnostic yield, Journal of Nuclear Medicine 41 (2000) 720–727. 16. A. Rozanski, G. A. Diamond, D. S. Berman et al., The declining specificity of exercise radionuclide ventriculography, New England Journal of Medicine 309 (1983) 518. 17. K. F. Van Train, J. Areeda, E. V. Garcia et al., Quantitative same-day rest– stress technetium-99 m sestamibi SPECT: definition and validation of stress normal limits and criteria for abnormality, Journals of Nuclear Medicine 34 (1993) 1494–1502. 18. P. J. Slomka, G. A. Hurwitz, J. Stephenson et al., Automated alignment and sizing of myocardial stress and rest scans to three-dimensional normal templates using an image registration algorithm, Journal of Nuclear Medicine 36 (1995) 1115–1122. 19. T. L. Faber, S. D. Cooke, R. D. Folks, J. Vansant, K. J. and Nichols, E. G. DePuey, Left ventricular function and perfusion from gated SPECT perfusion images: an integrated method, Journal of Nuclear Medicine 40 (1999) 650– 659. 20. Y. H. Liu, A. J. Sinusas, P. DeMan, B. L. Zaret, and F. J. Wackers, Quantification of SPECT myocardial perfusion images: methodology and validation of the Yale-CQ method, Journal of Nuclear Cardiology 6 (1999) 190–204. 21. S. Kirac, F. J. Wackers, and Y. H. Liu, Validation of the Yale circumferential quantification method using 201 Tl and 99 mTc: a phantom study, Journal of Nuclear Medicine 41(8) (2000) 1436–1441.


80

Ru-San Tan et al.

22. J. E. Udelson, P. S. Coleman, J. Metherall et al., Predicting recovery of severe regional ventricular dysfunction: comparison of resting scintigraphy with 201 Tl and 99 mTc-sestamibi, Circulation 89 (1994) 2552–2561. 23. G. J. Kauffman, T. S. Boyne, D. D. Watson, W. H. Smith, and G. A. Beller, Comparison of rest thallium-201 imaging and rest technetium-99m sestamibi imaging for assessment of myocardial viability in patients with coronary artery disease and severe left ventricular dysfunction, Journal of American College of Cardiology 27 (1996) 1592–1597. 24. R. Sciagra, G. Bisi, G. M. Santoro et al., Comparison of baseline technetium99m sestamibi with rest–redistribution thallium-201 tomography in detecting viable hibernating myocardium and predicting post-revascularization recovery, Journal of American College of Cardiology 30 (1997) 384–391. 25. R. O. Bonow, V. Dilsizian, A. Cuocolo, and S. L. Bacharach, Identification of viable myocardium in patients with chronic coronary artery disease and left ventricular dysfunction. Comparison of thallium scintigraphy with reinjection and PET imaging with 18 fluorodeoxyglucose, Circulation 83 (1991) 26–37. 26. V. Dilsizian, J. A. Arrighi, J. G. Diodati et al., Myocardial viability in patients with chronic coronary artery disease. Comparison of 99 mTcsestamibi with thallium reinjection and 18 fluorodeoxyglucose, Circulation 89 (1994) 578. 27. T. L. Faber, M. S. Akers, R. M. Peshock, and J. R. Corbett, Threedimensional motion and perfusion quantification in gated single photon emission computed tomograms, Journal of Nuclear Medicine 32 (1991) 2311–2317. 28. E. G. DePuey, K. Nichols, and C. Dobrinsky, Left ventricular ejection fraction assessment from gated technetium-99m sestamibi SPECT, Journal of Nuclear Medicine 34 (1993) 1871–1876. 29. G. Germano, H. Kiat, P. B. Kavanagh et al., Automatic quantification of ejection fraction from gated sestamibi SPECT, Journal of Nuclear Medicine 36 (1995) 2138–2147. 30. K. Nichols, E. G. DePuey, and A. Rozanski, Automation of gated tomographic left ventricular ejection fraction, Journal of Nuclear of Cardiology 3 (1996) 475–482. 31. K. A. Williams, and L. A. Taillon, Left ventricular function in patients with coronary artery disease assessed by gated tomographic myocardial perfusion images. Comparison with assessment by contrast ventriculography and firstpass radionuclide angiography, Journal of American College of Cardiology 27(1) (1996) 173–181. 32. W. H. Smith, R. J. Kaster, D. A. Calnon, D. Segalla, G. A. Beller, and D. D. Watson, Quantitative gated single photon emission computed tomography imaging: a counts-based method for display and measurement of regional and global ventricular systolic function, Journal of Nuclear Cardiology 4 (1997) 451–463. 33. D. A. Calnon, R. J. Kastner, W. H. Smith, D. L. Segalla, G. A. Beller, and D. D. Watson, Validation of a new counts-based gated SPECT method for quantifying left ventricular systolic function: comparison with equilibrium radionuclide angiography, Journal of Nuclear Cardiology 4 (1997) 464–471.



81

34. M. Y. Shen, Y. H. Liu, A. J. Sinusas, R. Fetterman, W. Bruni, O. E. Drozhinin, B. L. Zaret, and F. J. Wackers, Quantification of regional myocardial wall thickening on electrocardiogram-gated SPECT imaging, Journal of Nuclear Cardiology 6(6) (1999) 583–595. 35. E. Vallejo, D. P. Dione, W. L. Bruni, R. T. Constable, P. P. Borek, J. P. Soares, J. G. Carr, S. G. Condos, F. J. Wackers, and A. J. Sinusas, Reproducibility and accuracy of gated SPECT for determination of left ventricular volumes and ejection fraction: experimental validation using MRI, Journal of Nuclear Medicine 41 (2000) 874–882. 36. K. Nakajima, T. Higuchi, J. Taki, M. Kawano, and N. Tonami, Accuracy of ventricular volume and ejection fraction measurement by gated myocardial SPECT: comparison of 4 software programs, Journal of Nuclear Medicine 42 (2001) 1571–1578. 37. P. V. Ford, S. N. Chatziioannou, W. H. Moore, and R. D. Dhekne, Overestimation of the LVEF by quantitative gated SPECT in simulated left ventricles, Journal of Nuclear Medicine 42 (2001) 454–459. 38. T. Sharir, C. Bacher-Stier, S. Dhar, H. C. Lewin, R. Miranda, J. D. Friedman, G. Germano, and D. S. Berman, Identification of severe and extensive coronary artery disease by postexercise regional wall motion abnormalities in Tc-99 m sestamibi gated SPECT, American Journal of Cardiology 86 (2000) 1171–1175. 39. T. Chua, H. Kiat, G. Germano, M. Maurer, K. Van Train, J. Friedman, and D. Berman, Gated technetium-99m sestamibi for simultaneous assessment of stress myocardial perfusion, postexercise regional ventricular function and myocardial viability. Correlation with echocardiography and rest thallium201 scintigraphy, Journal of American College of Cardiology 23 (1994) 1107–1114. 40. C. Anagnostopoulos, M. G. Gunning, D. J. Pennell, R. Laney, H. Proukakis, and S. R. Underwood, Regional myocardial motion and thickening assessed at rest by ECG-gated Tc-99 m-MIBI emission tomography and by magnetic resonance imaging, European Journal of Nuclear Medicine 23 (1996) 909–1016. 41. M. G. Gunning, C. Anagnostopoulos, G. Davies, S. M. Forbat, P. J. Ell, and S. R. Underwood, Gated technetium-99 m tetrofosmin SPECT and cine MRI to assess left ventricular contraction, Journal of Nuclear Medicine 38 (1997) 438–442. 42. J. C. Stollfuss, F. Haas, I. Matsunari, J. Neverve, S. Nekolla, J. Schneider, U. Schricke, S. Ziegler, and M. Schwaiger, Regional myocardial wall thickening and global ejection fraction in patients with low angiographic left ventricular ejection fraction assessed by visual and quantitative resting ECG-gated 99 mTc-tetrofosmin single-photon emission tomography and magnetic resonance imaging, European Journal of Nuclear Medicine 25(5) (1998) 522–530. 43. K. Nichols, E. G. DePuey, A. Rozanski, H. Salensky, and M. I. Friedman, Image enhancement of severely hypoperfused myocardia for computation of tomographic ejection fraction, Journal of Nuclear Medicine 38(9) (1997) 1411–1417.


82

Ru-San Tan et al.

44. K. Nichols, E. G. DePuey, N. Krasnow, D. Lefkowitz, and A. Rozanski, Reliability of enhanced gated SPECT in assessing wall motion of severely hypoperfused myocardium: an echocardiograpic validation, Journal of Nuclear Cardiology 5 (1998) 387–394. 45. DePuey, and A. Rozanski, Using technetium-99 m sestamibi SPECT to characterize fixed perfusion defects as infarct or artifact, Journal of Nuclear Medicine 36 (1995) 952–955. 46. R. Taillefer, E. G DePuey, J. E. Udelson, G. A. Beller, Y. Latour, and F. Reeves, Comparative diagnostic accuracy of Tl-201 and Tc-99 m sestamibi SPECT imaging (perfusion and ECG-gated SPECT) in detecting coronary artery disease in women, Journal of American College of Cardiology 29 (1997) 69–77. 47. P. E. Smanio, D. D. Watson, D. L. Segalla, E. L. Vinson, W. H. Smith, and G. A. Beller, Value of gating of technetium-99 m sestamibi single photon emission computed tomographic imaging, Journal of American College of Cardiology 30 (1997) 1687–1692. 48. T. Sharir, D. S. Berman, P. B. Waehter, J. Areeda, P. B. Kavanaugh, J. Gerlach, X. Kang, and G. Germano, Quantitative analysis of regional motion and thickening by gated myocardial perfusion SPECT: normal heterogeneity and criteria for abnormality, Journal of Nuclear Medicine 42 (2001) 1630–1638. 49. A. Manrique, M. Faraggi, P. Vera, D. Vilain, R. Lebtahi, A. Cribier, and D. Guludec, 201 Tl and 99 m Tc-MIBI gated SPECT in patients with large perfusion defects and left ventricular dysfunction: comparison with equilibrium radionuclide angiography, Journal of Nuclear Medicine 40 (1999) 805–809. 50. T. Chua, C. Y. Lee, H. T. Tan, B. C. Tai, Z. P. Ding, and Y. L. Lim, Accuracy of automated assessment of left ventricular function using gated perfusion SPECT in the presence of perfusion defects and left ventricular dysfunction: correlation with equilibrium radionuclide ventriculography and echocardiography, Journal of Nuclear Cardilogy 7 (2000) 301–311. 51. D. S. Berman, X. Kang, K. F. Van Train et al., Comparative prognostic value of automatic quantitative analysis vs semiquantitative visual analysis of exercise myocardial perfusion SPECT, Journal of American College of Cardiology 32 (1998) 1987–1995. 52. H. A. Dakik, N. S Kleinman, J. A. Farmer, J. A. He, Z. X. Wendt, C. M. Pratt, M. S. Verani, and J. J. Mahmarian, Intensive medical therapy versus coronary angioplasty for suppression of myocardial ischemia in survivors of acute myocardial infarction: a prospective, randomized pilot study, Circulation 10 (1998) 2017–2023. 53. T. Sharir, G. Germano, X. Kang, H. C. Lewin, R. Miranda, I. Cohen, R. D. Agafitei, J. D. Friedman, and D. S. Berman, Prediction of myocardial infarction versus cardiac death by gated myocardial perfusion SPECT: risk stratification by the amount of stress-induced ischemia and the post-stress ejection fraction, Journal of Nuclear Medicine 42 (2001) 831–837.


REGIONAL MECHANICS OF THE BEATING HEART MARTYN P. NASH and PETER J. HUNTER Bioengineering Institute and Engineering Science The University of Auckland, New Zealand [email protected]

1. Introduction Heart failure is a leading cause of death worldwide, and a significant proportion of these deaths is due to disturbance of the normal cardiac rhythm. To gain insight into the mechanisms underpinning cardiac arrhythmias, a knowledge of the structure and electro-mechanical function of heart muscle is important. For many years, cardiologists have used simple clinical measurements such as heart rate, blood pressure and the electrocardiogram to diagnose diseases and pathologies of the heart. Mathematical modeling provides a useful framework to interpret the wide variety of experimental clinical observations and physiologic measurements of cardiac function in a rational, systematic and quantitative manner. With advances in computing hardware and software, large-scale numerical representations of the heart that simulate its function based on fundamental laws of physics have been made possible. Such analyses are not only useful for reproducing and understanding various aspects of ventricular function, but also provide quantitative estimates of performance indicators that may not be directly measureable, such as the dynamic changes in mechanical stress experienced by the myocytes during the heart cycle. In this regard, continuum mechanics deals with the movement of materials when subjected to applied forces. The motion of a deformable solid can be described by a continuous displacement field resulting from a set of forces acting on the solid body. In general, the displacements and forces may vary continuously with time, but for the present purpose a two-state quasistatic analysis will be discussed. The initial unloaded state of the material is referred to as the reference or undeformed state, since the displacements are zero everywhere. The material then reconfigures due to applied loads and reaches an equilibrium state referred to as the deformed state. The mechanics concepts of strain (a measure of length change or displacement gradient) and stress (the force per unit area acting on an infinitesimally 83


84

M. P. Nash and P. J. Hunter

small plane surface within the material) are of fundamental importance for finite deformation elasticity theory. The equations that govern the motion of deformable materials can be grouped into the following four categories: 1. Kinematic relations, which define (i) the components of the strain tensor in terms of displacement gradients and (ii) the incompressibility constraint for incompressible materials. 2. Stress equilibrium, or equations of motion derived from the laws of conservation of linear momentum and conservation of angular momentum. 3. Boundary conditions, which specify the external loads or displacement constraints acting on the deforming body. 4. Constitutive relations, which express the relationship between stress and strain for a particular material, and are typically empirical equations based on experimental observations. The first two steps define relationships that hold for all materials and is outlined in Sections 2 and 3, respectively. The application of boundary constraints is specific to the model and loading conditions under consideration, and is also covered in Section 3. The fourth step is concerned with relations determined experimentally for a particular material, and is explained in Section 4. Brief summaries are provided here, but further information on solid mechanics can be found in Refs. 1–4. The application of continuum mechanics to the analysis of soft biologic tissues such as myocardium typically requires a numerical representation of the mathematical model. Finite element modeling provides an efficient computational technique to integrate the nonlinear equations that govern the mechanics of the heart. Section 5 describes a computational finite element model of the canine ventricles, and presents an anlysis of the deformation and stress distributions during the heart cycle. 2. Kinematic Relations The key to analyzing strain in a material undergoing large displacements and deformation is to establish two coordinate systems and the relationship between them. The first is a material coordinate system to effectively tag individual particles in the body. The second is a fixed spatial coordinate system. Deformation is quantified by expressing the spatial coordinates of a material particle in the deformed state, as a function of the coordinates of the same particle in the undeformed state. Length changes of material segments can then be determined from the known deformation fields, and thus strain tensors may be calculated.


Regional Mechanics of the Beating Heart

85

2.1. Material versus spatial coordinates Deformation is defined by the movement of material particles, which can be thought of as small non-overlapping quantities of material that occupy unique points within the undeformed body. For this reason a method of labeling the particles is required. One convenient method is to define each material particle X by a set of rectangular cartesian coordinates (X1 , X2 , X3 ) in the undeformed body. As the body deforms, the coordinate axes deform with it and hence orthogonal coordinate axes in the undeformed state will not, in general, be orthogonal in the deformed configuration. These coordinates are referred to as material (or Lagrangian) coordinates because as the body deforms, a unique material particle is always identified by the same coordinate values. Each point in space may be defined by a set of spatial (or Eulerian) coordinates relative to a fixed reference cartesian coordinate system. A particular spatial point x, with coordinates (x1 , x2 , x3 ), may identify different material particles as they pass through the point x during the deformation. Conversely, a fixed material particle X may move to several spatial positions during the deformation. Note that it is often convenient to choose (without loss of generality) the material coordinates X to coincide with the rectangular cartesian spatial coordinates in the undeformed state. 2.2. Deformation gradients To quantify the deformation of a material it is necessary to consider the change in length of material segments, or sets of adjacent material particles within the body. In Fig. 1, an infinitesimal material line segment dX in the undeformed body B0 has components dX 1 , dX 2 and dX 3 with respect to global rectangular cartesian coordinates (Y1 , Y2 , Y3 ). In the deformed body B, the same material particles that constituted dX have reconfigured (due to applied loads) into dx, which has components dx1 , dx2 and dx3 with respect to (Y1 , Y2 , Y3 ). The deformation is quantified by the deformation gradient tensor, which carries the line segment dX into dx = FdX, or in component i dX M . The deformation gradients are defined in Eq. (1) as: form, dxi = FM i = FM

∂xi . ∂XM

(1)

As well as mapping line elements from the undeformed to the deformed configurations, the deformation gradient tensor also maps differential volumes and areas. The scalar triple product of three vectors u·(v × w) results


86


Fig. 1.

The deformation gradient tensor F relates line segment dX to dx.

in the volume spanned by those vectors.2 Consider the three vectors dxi (i = 1, 2, 3) in the deformed configuration and the three vectors dXi (i = 1, 2, 3) in the reference configuration. The undeformed reference volume, dV , maps to the volume in the deformed configuration, dv, as follows: = ijk dxi1 dxj2 dxk3 dv = dx1 · (dx2 × dx3 ) i = F · dX1 · (F · dX2 × F · dX3 ) note: dxi1 = FM dX1M , etc. i = det(F)dX1 · (dX2 × dX3 ) since ijk FM FNj FPk = det(F)MN P = det(F)dV = JdV,

(2)

where J is the determinant of the deformation gradient tensor and is a Jacobian. In the case of differential area elements, consider two area elements dA and da in the reference and deformed configurations, respectively, each having the normal vectors N and n, respectively. Consider a volume element with volume dV and edges dX1 , dX2 , dX3 in the undeformed state and with volume dv and edges dx1 , dx2 , dx3 in the deformed state. Then, N dA = dX2 × dX3 and n dA = dx2 × dx3 , wherein N and n represent the unit normal of the parallelogram formed by the two edges of the undeformed and deformed states, respectively. If dA and da are the respective areas



87

of the parallelograms in the undeformed and deformed states, respectively, then the volume in the deformed state can be calculated as follows: dv = dx1 · (n da) = (F · dX1 ) · (n da) = dX1 · FT · (n da).

(3)

Expanding Eq. (2): dv = J dV = J(dX1 · (N dA)).

(4)

Combining Eqs. (3) and (4) yields Nanson’s relations,2 defined in Eq. (5) as follows: 1 (5) n da = JN dA · F −1 , N dA = n da · F. J 2.3. Polar decomposition of deformation Any deformation can be split into two parts: (a) a rigid body rotation and (b) a stretch. This polar decomposition can be represented mathematically by considering the deformation gradient tensor to be a product, F = RU, of an orthogonal rotation tensor, R, and a symmetric positive definite stretch tensor, U. Thus the undeformed line segment components dX M L i are stretched into dy L = UM dX M before being rotated into dxi = RL dy L . Equivalently, the line segment could be rotated first and then stretched. However, for present purposes it is more convenient to interpret the stretch in terms of material coordinates and then relate the stretched material lines to the spatial coordinates through the rotation tensor, R. For further details on polar decomposition refer to Ref. 1 (Section 1.4) or Ref. 4 (Section. 2.5). It is important to note here that the stretch tensor, U, contains a complete description of the material strain, independent of any rigid body motion. 2.4. Strain tensors Strain in a deforming body is determined by measuring segment length changes. Equation 6 uses Pythagoras’ theorem to determine the arc length of the deformed segment dx: ds2 = dxi dxi = dxT dx = (F dX)T F dX = dXT FT F dX = dXT CdX, (6) where C = FT F or CMN =

∂xk ∂xk . ∂XM ∂XN

(7)


88


Equation (7) defines Green’s deformation tensor or the right Cauchy– Green deformation tensor∗ (Ref. 1, p. 12), which indicates how each component of the undeformed line segment dX contributes to the squared length of the deformed line segment dx. The deformation tensor C is related to the stretch tensor U in Eq. (8) using the polar decomposition theorem: T

C = FT F = (RU) RU = UT RT RU = UT U = U2 ,

(8)

since R is orthogonal (RT = R−1 ) and U is symmetric. Note that like U, C is symmetric and positive definite, and that both U and C are expressed in terms of material coordinates. One method for computing the stretch tensor U from the deformation gradient tensor F is to first calculate C = FT F, then calculate the eigenvalues (λ1 )2 , (λ2 )2 and (λ3 )2 , and eigenvectors s1 , s2 and s3 of C using the similarity transformation in Eq. (9) (Ref. 5, p. 239). U may then be constructed using Eq. (10):   0 (λ1 )2 0 (9) C = ΩΛ2 ΩT = Ω  0 (λ2 )2 0  ΩT , 0 0 (λ3 )2   λ1 0 0 (10) U = ΩΛΩT = Ω  0 λ2 0  ΩT , 0 0 λ3 where (i) the columns of Ω are the eigenvectors of C and are the principal axes of stretch and (ii) λi are the principal stretches (there is no shear when the deformation is referred to the principal axes). Note that since C is a real symmetric matrix, the eigenvectors are orthogonal and therefore Ω is an orthogonal matrix, ΩT Ω = I . In essence, the similarity transformation diagonalizes C, and the positive square root of the resulting diagonal matrix is used to compute the stretch tensor U. The two orthogonal tensors R and Ω, derived from F, have quite different physical interpretations. R describes the rigid body rotation component of the deformation with no information about the material stretching. On the other hand, the columns of Ω are the orientations of the principal stretch axes relative to the material coordinates. In 3D, the deformation tensor is a 3 × 3 matrix. There are three invariants (scalar combinations of the components of C), which remain ∗ The

left Cauchy–Green deformation tensor B = FFT is also defined, but is not useful here since it is not independent of rigid body rotation.



89

unchanged under coordinate rotations at a given state of deformation. These principal invariants are given by (Ref. 1, Section. 1.4): 1 [(tr C)2 − tr C2 ], I3 = det C, (11) 2 where the trace of C, denoted by tr C, is the sum of the diagonal terms, CMM , and the determinant of C, det C, is a measure of volume change. The similarity transformation of Eq. (9) may be used to express the invariants of C in terms of the principal stretch ratios as in I1 = tr C,

I2 =

I1 = (λ1 )2 + (λ2 )2 + (λ3 )2 , I2 = (λ1 )2 (λ2 )2 + (λ2 )2 (λ3 )2 + (λ3 )2 (λ1 )2 , 2

2

(12)

2

I3 = (λ1 ) (λ2 ) (λ3 ) . Equation (13) is an additional kinematic constraint that must be imposed on the deformation field for incompressible materials. This is discussed further in Section 4: det C = I3 = (λ1 λ2 λ3 )2 = 1.

(13)

Equation (14) shows how the Lagrangian Green’s strain tensor is related to the right Cauchy–Green deformation tensor (Ref. 4, p. 72): E=

1 (C − I), 2

cf. small strain approximation: SS Eij

ui = xi − Xi ,

1 = 2

(14)

∂ui ∂uj + ∂Xj ∂Xi

.

The Eulerian equivalent description is the Almansi strain tensor defined with respect to the deformed configuration (hence in terms of the left Cauchy–Green deformation tensor B) using: e=

1 (I − B−1 ). 2

(15)

2.5. Exercises 1. Compare Green’s and small strain tensors for the biaxial, 3D isochoric extension given below: x1 = λ1 X1 ,

x2 = λ2 X2 ,

x3 = X3 /(λ1 λ2 ).

Evaluate both tensors for λ1 = 1.1 and λ2 = 1.2.


90


2. Compare Green’s and small strain tensors for the simple shear deformation given below: x1 = X1 + γX2 ,

x2 = X2 ,

x3 = X3 .

Evaluate both tensors for γ = 0.3.

3. Stress Equilibrium Having established the kinematic framework for finite deformation analysis, the next step is to consider the governing force and momentum balances which follow from Newton’s laws of motion. To apply these equations to materials that undergo large deformations, it is necessary to define stress tensors and the way they enter into the governing equations. 3.1. Stress tensors Stress is defined as the force per unit area acting on an infinitesimally small plane surface. If the line of action of the force is normal to the plane, then a normal or axial stress results. On the other hand, a shear stress arises when the line of action of the force is tangential to the plane. The quantities of force and area can be referred either to the reference (undeformed) or deformed configurations, which leads to three important ways of representing stress in a deforming body, namely using the Cauchy, first or second Piola–Kirchhoff stress tensors. Refer to Ref. 3 (p. 220) for a more complete explanation. 1. The Cauchy stress tensor, denoted as σ ij , represents the force measured per unit deformed area acting on an element of surface in the deformed configuration. The first index indicates the direction of the normal to the surface on which σ ij acts and the second index indicates the direction of the stress component. It should be noted that the Cauchy stress tensor is symmetric for nonpolar materials (see Section 3.3) and that in rectangular cartesian coordinates, σ ij are also the physical components of stress. 2. The first Piola–Kirchhoff stress tensor, denoted as sMj , represents the force acting on an element of surface in the deformed configuration but measured per unit undeformed area. The first index is written in uppercase, since it refers to the normal of the surface in the undeformed state, and is thus a material coordinate index. The second index denotes the direction of the force acting on the deformed material, and is a spatial



91

coordinate index. For this reason, the first Piola–Kirchhoff stress tensor is generally not symmetric. It is sometimes referred to as the Lagrangian stress tensor and is often used in experimental testing, where force is measured in the deformed tissue but the area over which it acts is measured in the undeformed tissue. 3. The second Piola–Kirchhoff stress tensor, denoted as T MN , represents the force measured per unit undeformed area, P, acting on an element of surface in the undeformed configuration. This force may be determined from the actual force, p, in the same way that the undeformed material vector, dX, is determined from the deformed material vector, dx. Specifically P = F−1 p, just as dX = F−1 dx (Ref. 3, p. 222). The primary use of the second Piola–Kirchhoff stress tensor is for representing material behavior at a point, independent of rigid-body motion. This is discussed further in Section 4, which describes relationships between stress and strain tensors at a point. The main idea here is that the second Piola– Kirchhoff stress tensor is defined solely in terms of material coordinates, just as for Green’s strain tensor. Note that the second Piola–Kirchhoff stresses must be transformed into first Piola–Kirchhoff stresses for use in the equilibrium equations, which require a global spatial frame of reference. Equations (16) and (17) define the relationships between the second Piola–Kirchhoff (T), first Piola–Kirchhoff (S) and Cauchy ( ) stress tensors as follows: T = S(FT )−1 = JF−1 (FT )−1 ; S = JF−1 , sMj = J

∂XM ij σ , ∂xi

T MN = sMj

∂XN ∂XM ij ∂XN =J σ ; ∂xj ∂xi ∂xj

(16)

or inversely S = TFT , sMj = T MN

∂xj , ∂XN

=

σ ij =

1 J FS

1 ∂xi Mj s J ∂XM

1 FTFT ; J ∂xi MN ∂xj = J1 T ; ∂XM ∂XN

=

(17)

where J is the Jacobian of the transformation from reference to deformed coordinates, defined in Eq. (18). Note from Eq. (16) that the second Piola– Kirchhoff stress tensor is symmetric whenever the Cauchy stress tensor is symmetric: (18) J = det F = I3 = λ1 λ2 λ3 .


92


3.2. Conservation of mass The conservation of mass principle relates the mass densities in the undeformed and deformed bodies (denoted by ρ0 and ρ, respectively) using Eq. (19) (Ref. 6, p. 15): ρ0 dV0 = ρ dV = ρJ dV0 . (19) V0

V

V0

Thus for arbitrary volumes, the mass densities for the undeformed and deformed bodies are related as follows: (20) ρ0 = Jρ = I3 ρ. Thus for incompressible materials I3 = 1. 3.3. Conservation of angular momentum The conservation of angular momentum equates the time rate of change of the total angular momentum for a set of particles to the vector sum of the moments of the external forces acting on the system. For stress equilibrium of nonpolar materials, this principle is equivalent to the symmetry condition on the Cauchy stress tensor, namely σ ij = σ ji (see Ref. 3, (Section 5.3) or Ref. 4, (Section 7.5) or for a full derivation). Note that if the Cauchy stress tensor is symmetric (as is the case for the nonpolar materials being considered here), the second Piola–Kirchhoff stress tensor is also symmetric as a direct consequence of Eq. (16). This implies that there are only six independent components of stress — three normal components and three shear components. 3.4. Conservation of linear momentum For a given set of particles, the time rate of change of the total linear momentum equates to the vector sum of all external forces acting on the particles of the set (Ref. 3, Section 5.3). This is expressed mathematically in Eq. (21), where t is the traction vector (external surface forces per unit area), b represents the body forces (per unit mass) and the rate of change of momentum is written in terms of the material derivative (d/dt) and the velocity vector v: d t dS + ρb dV = ρv dV. (21) dt S

V

V



93

Cauchy’s formula, defined in the following equation, t dS = σ ij n ˆ i ij dS

(22)

(where σ ij are components of the Cauchy stress tensor and are physical stresses, since ij are unit vectors) projects the components of a stress vector t (the force per unit area acting on some deformed surface dS, with unit normal n ˆ=n ˆ j ij ) onto the set of orthogonal base vectors for the rectangular cartesian reference coordinate system, ij . Cauchy’s formula is substituted into Eq. (21) to form dv j σ ij n ˆ i dS + ρ bj − dV = 0, (23) dt S

V

which is appropriate for a material with constant density. Note that Eq. (23) is written in component form, where the body force and velocity vectors have components b = bj ij and v = v j ij , respectively. Applying the divergence theorem to Eq. (23) yields the following: ij ∂σ + ρbj − ρf j dV = 0, (24) ∂xi V

j

j

where f = dv /dt are components of the acceleration vector. If Eq. (24) is to be valid for arbitrary volumes, then the integrand must vanish (it is assumed here that the integrand is continuous). This results in ∂σ ij + ρbj = ρf j , (25) ∂xi which is the component form of Cauchy’s first law of motion for rectangular cartesian coordinates. It is often convenient to express Cauchy’s first law of motion in terms of the second Piola–Kirchhoff stress components to be independent of the spatial frame of reference. The derivation is similar to that of Eq. (25), except that the momentum balance Eq. (21) is first expressed in terms of integrals over the undeformed area and volume, and that Cauchy’s formula Eq. (22) for the traction vector is expanded in terms of the second Piola– Kirchhoff stress components. The result is a different form of the stress equilibrium equations, as follows: ∂ MN j T FN + ρ0 bj = ρ0 f j , (26) ∂XM where ρ0 is the density of the undeformed material. Note that Eq. (26) can also be determined from Eq. (25) by substituting Eqs. (17) and (20) into


94


Eq. (25). Also note that the term in parenthesis is the first Piola–Kirchhoff stress, sMj . For static equilibrium of the material, important in solid mechanics, the right-hand-side acceleration term in Eq. (25) vanishes. In the absence of body forces, this relation reduces to the statement of stress equilibrium in Eq. (27) for rectangular cartesian coordinates: ∂ MN j ∂σ ij T (27) = 0 or FN = 0. ∂xi ∂XM 3.5. Principle of virtual work Now consider a body of volume V and surface S loaded by a surface traction s which is in equilibrium with the internal stress vector t. If the body is subjected to an arbitrarily small displacement δv, which satisfies compatibility and any displacement boundary conditions specified on S (where δv must be zero), then the principle of virtual work can be expressed in the form of (see Ref. 3 (Section 5.5) or Ref. 7 (p. 168)): s · δv dS = t · δv dS, (28) S2

S

where S2 is the portion of the boundary that is not subjected to displacement boundary conditions. The virtual displacements may be resolved into components δv = δvj ij . Cauchy’s formula (Eq. (22)) is then substituted into the virtual work Eq. (28) to yield the following: s · δv dS = σ ij n ˆ i δvj dS. (29) S2

S

The right-hand-side surface integral in Eq. (29) is transformed into a volume integral (using Gauss’ theorem Ref. 8 (p. 117) to give: ij ∂σ ij ∂δvj dV. (30) s · δv dS = δvj + σ ∂xi ∂xi S2

V

Cauchy’s first law of motion in Eq. (25) can be substituted into the volume integral in Eq. (30) to give ∂δvj σ ij dV = ρ(bj − f j ) δvj dV + s · δv dS. (31) ∂xi V

V

S2



95

Moreover, Eq. (17) is used to express Eq. (31) in terms of the second Piola– Kirchhoff stress tensor, as follows: MN 1 ∂xj ∂δvj j j T dV = ρ(b − f )δvj dV + s · δv dS. (32) J ∂XM ∂XN V

V

S2

To solve the stress equilibrium equations, it is necessary to evaluate the surface integral on the right-hand-side of Eqs. (31) and (32) as outlined in Section 3.6. The final step is to express the stress components in terms of the deformation to characterize the specific material behavior. This is achieved using appropriate constitutive relations, as discussed in Section 4. 3.6. Boundary constraints and surface tractions To solve Eq. (32), the right-hand-side integral involving the surface traction vector s must be evaluated. If external surface pressures are applied, this integral must be evaluated for those portions of the boundary that sustain the loads. In the absence of boundary pressures this term vanishes. Consider a deforming surface, with unit normal n ˆ=n ˆ j ij . If the surface is loaded by a pressure, p(appl) (a physical stress), then the surface traction ˆ j ij and the right-hand-side surface intevector has components s = p(appl) n gral of Eq. (32) is evaluated using Eq. (33). Note that point forces can also be applied through the use of an appropriately located Dirac delta in the surface integral: s · δv dS = p(appl) n ˆ j δvj dS. (33) S2

S2

This surface integral is then substituted into Eq. (32) to yield the governing equations for finite deformation elasticity with respect to rectangular cartesian coordinates, as given below: 1 ∂xj ∂δvj T MN dV = ρ bj − f j δvj dV + p(appl) n ˆ j δvj dS. J ∂XM ∂XN V

V

S2

(34) It then remains to solve Eq. (34) in terms of the unknown virtual displacements δvj , subject to any displacement boundary constraints, which reduce the number of equations in the global system. For geometrically simple bodies with straightforward material behavior, the stress equilibrium equations of continuum mechanics can be solved analytically in their present form. However, for curvilinear shapes and/or more


96


complex constitutive relations, it may be convenient to take advantage of appropriate coordinate systems. In this case, general closed-form solutions to the equilibrium equations are rarely possible, thus the equations need to be solved using an appropriate numerical technique, such as the finite element method. Numerical methods are outside the scope of this chapter, but the application of the finite element method for cardiac mechanics has been described previously.9,10 Section 5 demonstrates the application of these modeling techniques with an analysis of regional mechanics of the cardiac ventricles. But first the mechanical response of ventricular myocardium must be characterized using an appropriate consitutive relation, as discussed in Section 4.

4. Constitutive Relations for Soft Tissue Mechanics Unlike the previously described kinematic relations and stress equilibrium equations that hold for most materials, constitutive relations are required to characterize individual materials and their response to external loads. In the context of finite deformation elasticity, constitutive equations are used to represent the behavior of a material through empirical relationships between experimentally observed stress and strain tensors. This section will only treat constitutive equations concerned with the mechanical behavior of materials. There are several important considerations which should be addressed when formulating constitutive relations. Perhaps the most important is that they are robust enough to predict behavior in various experimental situations using different samples of the same type of material. It is unreasonable, however, to expect to simulate all aspects of a material’s behavior with one set of constitutive equations. For this reason the most important and relevant behavioral features should be identified for the particular application, and it is these features that the constitutive relations should approximate. The resulting equations will be more concise, numerically efficient and thus more suitable for use in large-scale computer models. The form of the constitutive relation is not universal, because a constitutive relation is material behavior specific. It describes the gross behavior of a material under the specific conditions of interest, since it is impractical to attempt the modeling of the behavior under all conditions due to the difficulty of the problem. Therefore, it may be necessary to develop more than one constitutive relation for a material, if a variety of different conditions are of interest. For the development of a constitutive model, the following



97

five steps have been proposed2 : 1. Delineate general characteristics of the material (e.g. measure stress versus strain or loading force versus displacement). 2. Establish an appropriate theoretic framework (e.g. hyperelastic, viscoelastic, poroelastic, plastic). 3. Identify a specific funtional form of the relation (e.g. polynomial, exponential). 4. Estimate appropriate values of the material parameters. 5. Evaluate the predictive capability of the final relation. The first two steps above are discussed in Sections 4.1 and 4.2, respectively. Steps 3 and 4 are demonstrated for cardiac muscles in Sections (4.3)– (4.5), and finally step 5 is illustrated in Section 5.

4.1. General material characteristics The material must be classified as a solid, fluid or a mixture of the two to determine the fundamental approach of the modeling process. For a solid, it must be determined if the material is elastic/inelastic, isotropic/anisotropic, linear/nonlinear, homogenous/heterogenous (nonhomogenous), and compressible/incompressible. These classifications are defined as follows: • Isotropy: The response of a material to an applied load, relative to a prescribed configuration, is independent of the direction of loading. Examples: adipose (fat) and brain tissues. • Transversely isotropic: The response is independent of all directions transverse (perpendicular) to a single preferred direction. Examples: skeletal muscles and tendons. • Orthotropic: There are three preferred orthogonal directions. Application of equivalent loads in each of the three directions will result in different responses, as for example in the case of cardiac muscle. • Homogenous: The mechanical properties are the same at all points in the body. • Heterogenous: Nonhomogenous material, for which the mechanical response depends on the position within the body. • Incompressibility: The material preserves its volume under arbitrary loading conditions. Most soft tissues behave in this manner, since they primarily consist of immobile water.


98


Note that a material may be homogenous in the reference configuration and heterogenous once subjected to external loads, thus homogeneity must be specified with respect to a specific configuration.2 Similarly, isotropic and anisotropic behaviors are the results of internal symmetries in the microstructure of a material, which may change with applied loads. Incompressibility is a kinematic constraint that is independent of the loading conditions. Most biologic tissues consist largely of water, which is nearly incompressible. Poroelasticity or mixture theories can be used to account for solid–fluid interactions; examples include intramyocardial blood perfusion, and the extruding or imbibing of fluid by a tissue as it deforms.2 Examples of the implementation of these type of throries can be found in Refs. 11 and 12. A constitutive relation can be postulated to define the stress as a function of the deformation gradient tensor1 : σ ij = f ij (FNm ),

(35)

where FNm are the components of the deformation gradient tensor, σ ij are components of a stress tensor and f ij are general functions. To conserve angular momentum, f ij = f ji . This formulation represents a behavior known as elasticity. Hyperelasticity is the behavior of a material when it is possible to define a single-valued function called a strain energy function, which is only dependent on the deformation gradient tensor.1 Inelastic materials may be viscoelastic by exhibiting behaviors of elasticity and viscosity. Unlike hyperelastic or elastic materials, viscoelastic materials undergo stresses that are dependent on the strain history experienced by the material.13 Alternatively, inelastic materials may be plastic, and once a limiting strain (called the yield strain) is exceeded, they experience irreversible strains that are not dependent on stress. In the range of deformations that biologic tissues experience under normal physiologic conditions, plasticity does not usually play a role. For this reason, the ventricular mechanics model presented in this chapter has treated the myocardium as a hyperelastic material. 4.2. Theoretic constitutive framework It is essential that constitutive relations are based on experiments using real materials, but once the general characteristics of the material have been identified, theoretic arguments regarding the form of the constitutive



99

equations aid in the development of a relation for a specific material. There are four principles in constitutive theory that must be satisfied during the development process23 : • Principle of determinism: This principle implies that the material behavior can only be influenced by state variables of the present and past configurations. Future configurations of the material do not influence its present behavior. • Principle of local action: The behavior of a particular material particle will not be influenced by state variables which are far away from the neighborhood of the particle; this is also known as St. Venant’s Principle. • Principle of material frame indifference (objectivity): This principle states that the constitutive relation must be independent of the frame of reference. The same material law must apply from all views that an observer may take. In other words, the constitutive equations must be independent of the choice of coordinate system, since they characterize the constitution of an individual material and not the frame of reference from which the material is observed. Thus rigid-body motions should play no role in the constitutive relation. This is also known as the axiom of objectivity (see Ref. 14 (p. 163)) • Principle of equipresence: This states that an independent variable assumed to be present in one constitutive equation of a material should be assumed to be present in all constitutive equations of the same material, unless its presence contradicts an assumed symmetry of the same material, or contradicts the principle of material frame indifference or some other fundamental principle. Mathematically, material frame indifference is satisfied by postulating the existence of a single-valued strain energy function, W , to be a scalar potential that depends on the components of either the right Cauchy–Green deformation tensor or Green’s strain tensor: that is, W = W (C) or W = W (E), respectively. Components of the second Piola–Kirchhoff stress tensor are then given by the derivatives of W with respect to the components of C or E, as defined below (Ref. 15, p. 6): T

MN

1 = 2

∂W ∂W + ∂EMN ∂EN M

,

(36)

where the components of the Green’s strain tensor, EMN , are referred to XM material coordinates.


100


Material symmetry imposes further theoretic restrictions on the form of the constitutive relation. Isotropic materials possess no preferred direction, and exhibit rotational symmetry about all directions and reflectional symmetry with respect to all planes. For isotropic materials, the strain energy is constant for all orientations of the coordinate axes, or mathematically W (C) = W (QCQT ), where Q is any constant orthogonal tensor. Thus the strain energy is an invariant function of C. It can be shown that any invariant function of C can be expressed as a function of the three principal invariants of C, which are defined in Eq. (11) (see Ref. 4, Section. 10.2). This reduces the functional form of the strain energy function to W = W (I1 , I2 , I3 ). For incompressible materials, a further restriction on the form of the constitutive relation can be imposed. In this case, the additional kinematic constraint I3 = 1 (Eq. (13)) is applied. Note that it is not sufficient to set I3 = 1 in the constitutive equation, since certain derivatives of W tend to infinity in the limiting case of an incompressible material (Ref. 4, p. 141). This problem is overcome by introducing an arbitrary Lagrange multiplier λ into the constitutive equation. Then the unspecified strain energy term W is limited to be a function of I1 , I2 only. Thus for isotropic, incompressible materials, W = W (I1 , I2 ) + λ(I3 − 1),

(37)

where the first and second terms are referred to as the distortional and dilatational components, respectively. Equation (37) shows the functional form of the strain energy function. The mechanical effect of the incompressibility condition is to give rise to a reaction stress referred to as the hydrostatic pressure (denoted by p), which does not contribute to the deformation of the body. In other words, the addition of a hydrostatic pressure to an incompressible elastic body alters the state of stress, but does not directly change the shape of the material. Substituting Eq. (37) into (36) results in, T MN =

1 2

∂W ∂W + ∂EMN ∂EN M

− pC MN ,

(38)

which defines the components of the second Piola–Kirchhoff stress tensor with respect to XM material coordinates (Ref. 4, Section 10.2).



101

For isotropic, incompressible materials: ∂W ∂W ∂I1 ∂W ∂I2 = + . ∂EMN ∂I1 ∂EMN ∂I2 ∂EMN

(39)

Note that ∂I3 /∂EMN = 2δ MN , where the Kronecker delta δ MN = 1 if M = N (zero otherwise). In addition, the arbitrary Lagrange multiplier is chosen to be λ = − 12 pC MN . The multiplier C MN arises via the transformation from a Cauchy (physical) stress component to a second Piola– Kirchhoff stress (noting that J = 1), and is calculated by inverting the right Cauchy–Green deformation tensor: that is, {C MN } = {CMN }−1 (since the XM coordinates are orthogonal). These details ensure that the additional diagonal component of the stress tensor is a true hydrostatic stress. A suitable form of W (I1 , I2 ) must then be chosen, based on experimental observations of the material. Certain types of rubber exhibit almost isotropic behavior, and are referred to as Mooney–Rivlin materials: W (I1 , I2 ) = c1 (I1 − 3) + c2 (I2 − 3)

(40)

characterizes this type of material using material constants (mechanical properties) c1 and c2 , which must be estimated experimentally. A subset of the Mooney–Rivlin materials are the neo-Hookean materials, which are characterized by setting c2 = 0 in Eq. (40). Note that the use of (I1 − 3) and (I2 − 3) ensures that the strain energy is zero when the strain, E, is zero. This may be demonstrated by using Eq. (14) to show that C = I for zero strain, in which case Eq. (11) reduce to I1 = I2 = 3 and I3 = 1. However, note that the distortional stress is non-zero when the strain is zero. This (physically odd) feature of these relations may be offset using the dilational stress term in the stress–strain relation. Specifically, by setting an appropriate hydrostatic pressure value to produce a dilational stress that is equal and opposite to the above nonzero distortional stress, a zero total stress state can be achieved when the strain is zero. Isotropic biologic tissues (such as fat, lung and brain tissues) have been modeled using a similar type of constitutive relation. Nonlinear stress–strain behavior can been accounted for using higher order powers of the terms in Eq. (40), or by using nonlinear functions. For example, the mechanical response of passive lung tissue has been modeled using16 : W = c1 (I1 − 3) + c2 (I1 − 3)2 + c3 (I1 − 3)3 + c4 (I1 − 3)4 ,

(41)


102


whereas adipose (fat) tissue has been represented using: W = c1 (ec2 (I1 −3) − 1).

(42)

Transversely isotropic materials possesses a single preferred direction at every point. These materials exhibit rotational symmetry about the preferred axis and reflectional symmetry with respect to all planes containing this axis. To account for this, the framework for isotropy above may be extended by allowing W to depend on two further strain invariants associated with the plane of isotropy (Ref. 15, p. 28). A major objection to the above approaches concerning the formulation of constitutive equations is that the parameters bear no obvious relation to the underlying structure of the material. An approach that incorporates parameters, which directly reflect mechanical or structural properties of the material would potentially yield a more robust and biophysically meaningful constitutive relation. In addition, variations in material properties could be more directly interpreted in terms of their effect on the mechanical behavior of the material. With regard to this specific issue, Section 4.4.2 details the development of a microstructurally based constitutive relation for passive cardiac tissue. But to begin to develop biophysically based constitutive relations for heart mechanics, one must first understand the microstructural organization of myocardial tissue, as described in Section 4.3. 4.3. Microstructural architecture of the heart To develop constitutive equations that are relevant to the myocardial tissue structure, we first need to describe the tissue organization of myocardium. Cardiac muscle cells or myocytes are typically cylindrical with lengths that range from 80 to 100-µm and diameters ranging from 10 to 20-µm. The fundamental contractile unit is the sarcomere, which is about 2-µm long. The sarcomere spans between adjacent Z-lines along the longitudinal axis of the cell and contains the contractile apparatus. Cells consist of about 40–50 sarcomeres in series, and branch and interconnect end to end through intercalated disk junctions. The branching angle is usually acute, so that adjacent cells run almost parallel with one another. In this way, the contractile apparatus between cells is aligned for efficient mechanical function. Intercalated disks contain nexi or gap junctions, which provide electric continuity between cells. Consequently, electric impulses propagate more rapidly along the axis rather than across the axis of the constituent fibers.



103

Studies of cardiac tissue architecture date back to the early 1900s, when the heart was viewed as an assembly of discrete fiber bundles originating at the base of the ventricles and spiralling toward the ventricular apices.17,18 This notion was generally accepted for the first half of the 20th century, but descriptions were largely qualitative. Subsequent quantitative studies showed that there is a smooth transmural variation of fiber orientation, which led to the predominant view that myocardium is a single muscle mass, which is more appropriately described as a continuum than as discrete muscle bundles.19,20 Further detailed studies supported this view across different species, 21,22 including human hearts. 23,24 However, of these studies, none appear to have measured the muscle fiber orientation at more than eight sites on a single heart, and just one study seemed to have have sampled the right ventricle.24 Furthermore, the muscle fiber architecture was not quantitatively referred to ventricular geometry, and therefore the data obtained provided a limited and essentially qualitative description of ventricular fiber orientation. More recently, a systematic approach was adopted to characterize the muscle fiber orientation throughout all regions of the ventricular myocardium.25,26 This technique involved cutting the right and left ventricles into thick serial slices transverse to the base–apex axis, which were then subdivided into wedges. The distribution of muscle fiber orientations across each wedge was determined by sectioning them in the transmural plane and employing standard histologic techniques. Problems with this procedure include difficulties with spatial registration between slices, and loss of detail through averaging. To date, the most thorough quantitative study of cardiac muscle fiber orientation is that of Le Grice,27 who progressively removed fine layers of myocardium from a mounted intact preparation. Muscle fiber orientation was measured together with the absolute coordinates at a large number of sites over successive myocardial surfaces (see Ref. 28 for further details). Using this procedure, spatial registration was implicitly preserved, and local muscle fiber orientation could be referred to the surrounding myocardium. Measurements obtained from this work confirm the more selective quantitative findings reported previously.21

4.3.1. Laminar-sheet structure of the heart Ventricular sections, particularly near the midwall, reveal extensive extracellular gaps and thick longitudinal ventricular sections that have a


104


layered appearance due to radial cleavage planes that run across the wall.29,30 Moreover, in an extensive review of fiber studies, Streeter Jr.31 acknowledges that there is a substantial discontinuity in the muscular architecture of the ventricles at both the microscopic and macroscopic level. These findings remained essentially qualitative until recent quantitative anatomic studies reported that the ventricular myocardium should not be viewed as a uniformly continuous structure.32 These detailed studies revealed that cardiac tissue is a composite of discrete layers of myocardial muscle fibers tightly bound by endomysial collagen, as illustrated in Fig. 2. These myocardial laminae or sheets are loosely coupled by perimysial collagen and have the ability to slide over each other with relative ease. Myocardial laminae are on average four to six cells thick, and continuously branch in each direction throughout the ventricular walls. The sheet plane is typically oriented perpendicularly to the ventricular surfaces,

Fig. 2. Schematic of cardiac microstructure showing branching sheet structures, with a convenient set of microstructural material axes.



105

except in the subendocardial and subepicardial regions, where the sheets appear to become almost tangential to the wall surfaces. To quantify the distribution of sheet orientations, Le Grice et al.32 systematically recorded measurements from thick longitudinal transverse sections of ventricular myocardium. As in their previous muscle fiber study,27 regional distributions of sheet orientations were quantified with reference to the ventricular geometry, so that the 3D orthotropic structure of the entire ventricles could be completely characterized and referred to the anatomy. For modeling purposes, it is convenient to define a natural set of material directions to characterize the structure of myocardial tissue at an arbitrary point in the heart wall (see Fig. 2). The first of these directions is referred to as the fiber axis and coincides with the muscle fiber direction at each point. The sheet axis is defined to lie in the plane of the muscle layer and is perpendicular to the fiber direction. The third axis is defined to be orthogonal to the first two and is referred to as the sheet-normal axis, since it is perpendicular to the muscle layer. Using quantitative data on the orientation of the myocardial sheets, an efficient mathematic representation of ventricular geometry and microstructure was formulated.33 The model was developed using finite element fitting techniques, and forms the anatomic basis for the ventricular mechanics model presented in this chapter. The next step is to characterize the mechanical properties of ventricular myocardium using appropriate constitutive relations, as described in Sections 4.4 and 4.5. 4.4. Mechanical properties of passive myocardium Biaxial tension tests on thin sections of ventricular myocardium34–37 have revealed highly nonlinear, anisotropic stress–strain behavior (typical of most soft biologic tissues). The stress–strain properties along each of the microstructurally relevant directions are quite different, reflecting in part the organization of collagen relative to these three axes. Figure 3 schematically summarizes typical stress–strain behavior of myocardium when stretched along each of the three microstructural axes.38 The most striking difference between each of the three axes is the limiting strain for an elastic response. When the tissue is stretched along the myocyte (fiber) axis the limiting extension ratio∗ is about 1.3, whereas the limiting extension ratio for the sheet axis is approximately 1.5. Below an extension ratio of ∗ The

muscle fiber extension ratio is defined as the sarcomere length relative to a resting sarcomere length of approximately 1.95 µm for the unloaded muscle.


106


Fig. 3. Typical nonlinear stress–strain properties of ventricular myocardium. The parameters a1 , a2 and a3 represent the limiting strains for elastic deformations along the fiber, sheet and sheet-normal axes, respectively. Note the highly nonlinear behavior as the elastic limits are approached.

1.5, very little tension is developed in the direction of the sheet-normal, but tension increases rapidly above this and irreversible damage occurs when the sheet-normal extension ratio exceeds about 1.7.38 Variations in the axial limiting strains can be explained by the organization of the extracellular connective tissue matrix. The high fiber stiffness is most likely to be due to intracellular titin protein coupled with the tightly bound endomysial collagen coils that surround individual myocytes.39 As the tissue is stretched along the fiber axis, these coils straighten and it is the taut length of the collagen that determines the limiting fiber strain.40,41 In contrast, the relatively low sheet-normal stiffness is most likely to be due to the sparse density of perimysial collagen links that cross the cleavage planes to connect myocardial sheets.32 Based on this information, fully 3D orthotropic constitutive relations for myocardium are described below. 4.4.1. An orthotropic exponential constitutive relation Following the pioneering work of Fung,8 many exponential constitutive relations have been proposed to model the mechanics of a wide variety of soft biologic tissues. Along these lines, an empirical exponential relation to represent the orthotropic mechanical function of ventricular tissue42,43 proposed: W =

c1 Q (e − 1), 2

(43)



107

where 2 2 Q = cff Eff2 + css Ess + cnn Enn + 2cfs ( 12 (Efs + Esf ))2

+ 2cfn( 12 (Efn + Enf ))2 + 2cns ( 12 (Ens + Esn ))2 .

(44)

where Eαβ are the Green’s strain components referred to fiber–sheet coordinates and cαβ are the constitutive parameters for this relation. Note that the distortional components of stress may be determined by substituting derivatives of this relation into Eq. (36). For example: Tff =

∂W ∂W ∂Q = = c1 cff Eff eQ . ∂Eff ∂Q ∂Eff

(45)

All of the parameters of this relation are entirely empirical and their values bear no direct relation to the underlying structure of the tissue. However, good fits to experimental data have been obtained using this equation. For example, a series of studies44–46 used finite element methods with nonlinear optimization to estimate the constitutive parameters of the orthotropic exponential relation using experimental observations from small cubes of myocardial tissue subjected to 3D simple shear loading conditions.47 Mean and variances of the fitted parameter values taken across the six experiments are listed in Table 1. 4.4.2. An orthotropic pole–zero constitutive relation Stress–strain behavior along one axis is very nearly independent of the degree of lateral stretch.34 This means that the contribution of the deformation along one of the material axes to the total strain energy is nearly independent of the contribution from the other two axes — there is a small degree of cross-axis coupling via the hydrostatic pressure, but this coupling does not occur in a biaxial tension test experiment, since the hydrostatic pressure is zero (because the out-of-plane stress is zero). It thus follows that a strain energy function for myocardium should be separated into individual expressions in terms of the stretch along each of the material axes. It is also evident from biaxial tests that the axial stress is very low for small Table 1. Parameters of the orthotropic exponential constitutive relation42 fitted to data from 3D shear experiments on ventricular myocardium.47 Fitted parameters summarized from Schmid et al.44

Mean Variance

c1 (kPa)

cff

css

cnn

cfs

cfn

cns

0.26 0.09

37.2 13.8

18.9 9.5

9.1 6.5

12.0 1.2

12.0 1.3

10.9 3.3


108


axial strains, but increases rapidly as the strain approaches the limiting strain for that axis. These characteristics, microstructural observations and biaxial test results have been encapsulated in the pole–zero strain energy function for ventricular myocardium10,38,48 given by: 2 2 Eff2 Ess Enn + k + k ss nn |aff − |Eff ||bff |ass − |Ess ||bss |ann − |Enn ||bnn 2 2 2 Efs Efn Esn + kfs + k + k , fn sn |afs − |Efs ||bfs |afn − |Efn ||bfn |asn − |Esn ||bsn

W = kff

(46) where the components of Green’s strain tensor have been referred to the fiber coordinates and the constitutive parameters (“a”s, “b”s and “k”s) have the following interpretations: • The limiting strains or poles, denoted by aαβ , are physical properties of the tissue that may be estimated directly from microstructural observations. In particular, elastica theory was applied to the collagen helices surrounding myofibers40 to determine the fiber axis pole (yield strain) aff = 0.523. Alternatively, these yield strains may be estimated by fitting the model directly to experimental stress–strain data (e.g. see Ref. 34). Note that the strain components are assumed to be less than the limiting strains for an elastic response. If they exceed the pole, a different relation must be used to model the plastic strain. • The second group of parameters, denoted by bαβ , are related to the curvature of the uniaxial stress–strain relationships for each mode of deformation and have been estimated using the biaxial tension test results.34 • Lastly, the kαβ parameters weigh the contribution of the corresponding mode of deformation to the total strain energy of the material. Note that the distortional components of the second Piola–Kirchhoff stress tensor can be determined by substituting the derivatives Eq. (46) into (36) to yield: ∂W kff Eff bff Eff = (47) 2 + , 0 < Eff < aff . Tff = ∂Eff (aff − Eff )bff (aff − Eff ) The constitutive parameters of Eq. (46) are naturally split into six groups, one for each mode of deformation. These groups correspond to the six independent components of Green’s strain tensor. The first three terms in Eq. (46) refer to the three axial modes of deformation (fiber, sheet and sheet-normal, denoted ff, ss and nn, respectively). The parameters



109

Table 2. Parameters of the pole–zero constitutive relation for ventricular myocardium.9 Type

Axial parameters

Shear parameters

Coefficients

kff kss knn

1.937 0.028 0.310

kfs kfn ksn

1.0 1.0 1.0

Poles

aff ass ann

0.523 0.681 1.037

afs afn asn

0.731 0.731 0.886

Exponents

bff bss bnn

1.351 5.991 0.398

bfs bfn bsn

2.0 2.0 2.0

associated with these terms have been estimated using a combination of microstructural observations, biaxial tension test results and noninvasive magnetic resonance imaging data, and are listed in Table 2. The remaining terms relate to modes of shear deformation between the microstructural axes (fiber/sheet, fiber/sheet-normal and sheet/sheet-normal, denoted by subscripts fs, fn, and sn, respectively). It is reasonable to expect that the cellular structures responsible for resisting the shearing deformations are exactly those structures responsible for limiting axial deformations. These load-bearing connections are simply the collagen struts that link individual cardiac fibers and sheets. The important implication of this assumption is that the parameters of the shear terms in Eq. (46) are strongly correlated to the parameters of the axial terms. A fiber distribution model has been used to help understand and quantify some of these parameter correlations.10 Experiments involving shear deformations of cardiac tissue.47,49 could be used to verify these correlations. Equation (46) can be regarded as a first-order approximation of a power series in the pole–zero terms. A more general description would include crossproduct terms reflecting coupling between different modes of axial and shear deformation and may be warranted following further experimental testing. However, this cannot be justified on the basis of the biaxial experiments. 4.5. Active ventricular contraction During the cardiac cycle, the wave of excitation that propagates throughout the ventricular myocardium stimulates myofiber contraction in synchrony. Following stimulation, myocardial cells generate contractile forces that are primarily aligned with their longitudinal axes (note that there is some


110


evidence for the development of active tension along the other microstructural axes,50 however this has been neglected in this study). To this end, just one extra term needs to be added to Eq. (38) to simulate active muscle contraction, as given below: ∂W 1 ∂W T MN = + (48) − pC MN + Ta C 11 δ1M δ1N , 2 ∂EMN ∂EN M M is the Kronecker delta, and Ta is the active contractile stress where δN generated along the fiber axis. Detailed biophysical models of the contractile stresses generated my myocytes have been previously proposed.51 However, for the purposes of this study, it was sufficient to use the following steady-state relationship between the active contractile stress Ta , the intracellular calcium concentration [Ca2+ ]i and the fiber extension ratio λ9 : (Caactn · Ca2+ max )h · T · [1 + β(λ − 1)], Ta (λ, Caactn ) = (Ca · Ca2+ )h + (c )h ref actn

2+

max

50

(49)

where Ca max = 2.5 mM and c50 = 0.5 mM are the intracellular calcium concentrations for maximal and 50% force generation, respectively; h = 3 is the Hill coefficient controlling the shape of the [Ca2+ ]i saturation curve; Tref = 100 kPa is the isometric active stress at the resting sarcomere length and with saturating Ca2+ i ; and β = 1.45 is a nondimensional parameter associated with the length dependence of force generation. The control variables in this relation are the fiber extension ratio λ, which was determined from the local deformation metrics of the model, and a nondimensional parameter Caactn that was prespecified to control the level of activation for each loading state. Note that the effects of the electric impulse propagation were neglected here, since excitation typically occurs on a much faster time scale than muscle shortening. Thus, all points in the ventricles were assumed to be activated simultaneously. The soft tissue mechanics framework outlined above was used together with finite element modeling techniques, to analyze the regional myocardial deformation and stress distributions during the heart cycle, as described in Section 5. 5. Modeling the ventricular mechanics cycle An anatomically realistic finite element model of the ventricles33 was used to analyze deformation and stress in the beating heart. The geometry



111

and microstructural architecture of the ventricles were efficiently described by 120 finite elements that used high-order (cubic-hermite) interpolations of nodal parameters.9,10 The analysis is based on the theory of finite deformation elasticity, developed in Sections 2 and 3. The model incorporates the microstructurally based pole–zero constitutive law for the passive ventricular mechanics properties (Section 4.4.2), together with a simple quasi-static approximation of the active stresses generated by myofibers during active contraction (Section 4.5). The resulting nonlinear equations were solved using the Galerkin finite element method subject to physiologically realistic boundary constraints, including the ventricular pressures acting on the endocardial surfaces, and the limiting effects of the pericardial sac. The CMISS software package∗ was used to perform the simulations. This section compares model predictions with experimental observations of regional deformation in the intact heart, and presents predicted regional distributions of myofiber stress during the heart cycle. 5.1. Pressure versus volume relations The mechanical performance of the heart is often characterized by the left ventricular (LV) cavity pressure–volume relation. Many studies have quantified this relationship using a variety techniques in isolated, supported and in situ hearts. The ventricular mechanics model predictions are compared to the data from three studies of canine hearts in Fig. 4. For comparison purposes, LV cavity volume has been transformed into a percentage change relative to that at the zero pressure reference state. McCulloch et al. (Ref. 52, Fig. 2) measured the passive LV volume change for a range of cavity pressures averaged over eleven static loading cycles in six isolated potassium-arrested canine hearts. The normalized pressure–volume relationship is represented by asterisks (∗) in Fig. 4, with volume changes (expressed as mean ± SD) referred to the mean LV cavity volume of 40 ml at zero pressure. In seven isolated potassium-arrested canine LVs, Omens et al. (Ref. 53, Fig. 3) obtained stable passive pressure–volume curves up to LV pressures of 2.7–4 kPa (20–30 mmHg). Mean changes in LV cavity volume were reported for each of four pressure loads. The mean unloaded LV cavity volume was not quantified in this study, but an unloaded volume of approximately 20 ml ∗ Information

on the CMISS computational physiology software can be found at http://www.cmiss.org/.


112


Fig. 4. Diastolic pressure–volume relations for the canine LV. The ventricular mechanics model () predicted realistic diastolic stiffening of the LV compared to the experimental studies (see text for details). Error bars show standard deviations.

was reported for one heart. This volume was used to normalize the reported volume changes. The normalized passive pressure–volume relationship is represented by triangles () in Fig. 4, with volume changes expressed as mean ± SD. McCulloch et al. (Ref. 54, Table 3, Fig. 2) reported LV pressure versus percentage volume change for five isolated potassium-arrested dog hearts subjected to static pressure loading. These observations are represented by box symbols () in Fig. 4, with volume changes expressed as mean ± SD. The reported mean unloaded LV volume was 36 ± 8 ml (SD). The ventricular mechanics model was passively inflated to LV and RV pressures of 3.0 kPa (22.5 mmHg) and 0.6 kPa (4.5 mmHg), respectively, using 30 equal load steps. The LV volume was computed at each step, for which the model had converged to an error tolerance of 10−3 . Volume changes were normalized by the unloaded LV volume of 32 ml. The predicted



113

Fig. 5. Ventricular mechanics model predictions of the LV pressure–volume relations during the cardiac cycle. IVC: isovolumic contraction.

normalized passive pressure–volume relationship is represented by diamonds () in Fig. 4. It is clear from this comparison that the ventricular mechanics model predicted sufficiently realistic diastolic pressure–volume characteristics. Following diastolic filling, there is a short phase of contraction during which the ventricular valves are all closed, and hence the volumes of the ventricular cavities remain constant. This so-called isovolumic contraction phase was modeled using a simple constraint that controlled the ventricular cavity volume changes. Following end-diastole, contraction was simulated by successively incrementing the activation parameter in Eq. (49), until the resulting LV pressure exceeded 15 kPa. The subsequent ejection phase was simulated by allowing the volumes of the cavities to decrease in a controled manner as a result of the contractile stresses (see Ref. 9 for full details). Figure 5 illustrates the pressure–volume relationship during the ventricular cycle. The isovolumic relaxation (passive recoil) phase of diastole was not considered in this study.

5.2. End-diastolic strains referred to cardiac coordinates Many experimental studies have quantified various aspects of in vivo diastolic ventricular deformation using imaging techniques such as angiography, echocardiography, computed tomography, sonomicrometry,


114


Fig. 6. The cardiac coordinate system (wc , wl , wr ) denoting the circumferential, longitudinal and radial base vectors, repectively.

bi-plane cineradiography or magnetic resonance imaging. To quantify myocardial deformations in terms of a topologically relevant system of local coordinates, the cardiac coordinate system was introduced.55 The orthonormal base vectors of the cardiac coordinate system (wc , wl , wr ) are illustrated in Fig. 6 and may be constructed by first considering a small surface tangential to the heart wall. The circumferential base vector, wc , is defined by the intersection of the wall surface and the short-axis (y, z) plane, and is oriented clockwise when viewed from the base toward the apex. The longitudinal base vector, wl , is also defined to lie in the wall surface, and is perpendicular to wc . By convention, wl is oriented in the opposite hemisphere to the x-axis. The transmural or radial base vector is defined as wr = wc × wl and represents the outward normal to the heart wall. In the cardiac coordinate system, the six independent components of the strain tensor are the circumferential (Ecc ), longitudinal (Ell ) and radial (Err ) normal strains representing deformations along the three axes, and the wall-plane shear (Ecl ) and transverse shears (Ecr and Elr ) representing changes in the angles between the corresponding pairs of coordinate axes. Cardiac coordinates have been used by many researchers to characterize ventricular deformation in terms of components relative to the epicardial or endocardial wall surfaces. Omens et al. (Ref. 53, Fig. 6) computed end-diastolic strains referred to the cardiac coordinate system for the equatorial region of the anterior midwall. Transmural strain distributions were reported for seven isolated



115

potassium-arrested canine hearts, except for at the most endocardial depth, where data from only five hearts were available. Mean LV end-diastolic pressure (LVEDP) was 1.1 ± 0.5 kPa (8 ± 4 mmHg). Generally, normal strain components decreased in magnitude from the endocardium to the epicardium. Ecc and Ell were consistently positive, whereas Err was negative, representing wall thinning. The mean wall-plane shear was small and negative and the transverse shears were negligible, with very small transmural gradients. Observations from this study are represented by box symbols () in Fig. 7, where the standard deviations for the shear strain components were estimated from the ranges reported in Fig. 4 legend of Ref. 53. Using similar methods, May-Newman et al. (Ref. 56, Fig. 4) quantified transmural end-diastolic cardiac coordinate strains from the midanterior LV free wall of eight isolated potassium-arrested dog hearts. LV pressure was 1.3 kPa (10 mmHg) and deformations were referred to the unloaded, residually stressed ventricular state. Deformation trends were qualitatively similar to the previous study and are represented by crosses (×) in Fig. 7. The error bars represent the standard deviations and were computed by √ multiplying the reported standard error of the mean (SEM) by n, where n = 8. To assess the accuracy of the ventricular mechanics model, the LV and RV cavities were inflated to 1.0 kPa (7.5 mmHg) and 0.2 kPa (1.5 mmHg), respectively. Predictions of the normal strain components for the equatorial anterior wall (represented by diamonds [] in Fig. 7) were typically within 1SD of the experimental observations. Shear strain components with respect to cardiac coordinates achieved similar accuracy, however it is likely that the subendocardial distribution of the transverse shear strain, Ecr , was inaccurate. It is plausible that this discrepancy was due to the choice of constitutive parameters associated with the shearing modes of deformation (see Section 4.4.2). It is interesting to note that the axisymmetric model of the canine LV proposed by Guccione et al. (Ref. 57, Fig. 6), which incorporated a transversely isotropic constitutive law into a geometrically accurate description of the anterior LV free wall, also failed to reproduce the experimentally observed transmural distribution of Ecr . The predominantly negative Ecr distributions during diastolic filling indicate a greater anticlockwise twist of the endocardium relative to the epicardium as viewed from the apex. This mode of deformation may be partially responsible for ventricular wall thinning during diastole.


116


Fig. 7. Transmural distributions of end-diastolic strain referred to cardiac coordinates for the equatorial region of the anterior ventricular wall. Model predictions () are compared to the experimental observations from Ref. 53 (, n=7) and Ref. 56 (×, n = 8). Error bars show standard deviations.

The predicted transmural distribution of the wall-plane component of shear strain Ecl was predominantly negative. This is consistent with a left-handed torsional deformation about the longitudinal axis during diastolic filling, for which the apex rotates clockwise with respect to a fixed



117

base, as viewed from the apex. Predicted ventricular torsion increased monotonically with wall depth and was maximal at the endocardial surface. This agreed well with the experimental observations, as illustrated in Fig. 7.

5.3. End-systolic strains referred to cardiac coordinates Many experimental studies have reported end-systolic strains referred to cardiac coordinates. The majority of these studies have referred strains to the end-diastolic configuration. Four studies have been selected to compare to ventricular mechanics model predictions of end-systolic cardiac coordinate strain distributions. From bead studies for the anterior equatorial LV free wall of seven open-chest dogs, Waldman et al. (Ref. 58, Tables 1 and 2) computed endsystolic 3D cardiac coordinate strains with respect to the end-diastolic reference state. Observations from this study are represented by crosses (×) in Fig. 8. Villarreal et al. (Ref. 59, Table 1) reported transmural distributions of end-systolic cardiac coordinate strains with respect to the end-diastolic reference state, for the anterior LV wall in seven open-chest dogs. Results from this study are represented by triangles () in Fig. 8. McCulloch et al. (Ref. 60, Fig. 3) computed end-systolic cardiac coordinate strains with respect to the end-diastolic reference state for the anterior LV free wall of six open-chest dogs, from the bead study data reported by Waldman et al.61 Observations from McCulloch et al. (Ref. 60, Fig. 3) are represented by plus symbols (+) in Fig. 8. Le Grice et al. (Ref. 62, Table 1) reported end-systolic cardiac coordinate strains referred to the end-diastolic state for the anterior LV free wall of eight open-chest dog hearts. Observed strains are represented by box symbols () in Fig. 8. For this study, the peak systolic LV and RV pressures were 14.7 ± 1.1 kPa (110 ± 8 mmHg) and 4.4 ± 0.8 kPa (33 ± 6 mmHg), respectively, and the LVEDP was 0.9 ± 0.1 kPa (7 ± 2 mmHg). Ventricular mechanics model predictions of the end-systolic cardiac coordinate strains (represented by diamonds [] in Fig. 8) were computed for the anterior equatorial LV wall and were referred to the predicted end-diastolic configuration. Although the longitudinal strain Ell was reasonably realistic, Fig. 8 illustrates differences between experimental and predicted strain distributions. In particular, the oscillatory nature of the radial strain distributions, and the large subendocardial wall


118


Fig. 8. Transmural distributions of end-systolic strain (with respect to the end-diastolic state) referred to cardiac coordinates for the equatorial region of the anterior ventricular wall. Model predictions () are compared to the observations from four experimental studies (see text for details). Error bars show standard deviations.

thickening strain Err were likely to be unrealistic. These discrepancies were possibly due to the homogenous myocardial material property distributions used for this analysis. Spatially heterogenous material property estimates based on experimental tests of myocardial tissue would



119

most likely improve the accuracy of the ventricular mechanics model predictions.

5.4. Fiber strain during the cardiac cycle When referred to the fiber-sheet microstructural material coordinate system (described in Section 4.3), fiber strain components represent deformations experienced by the myocardial cells. More specifically, the fiber strain (Eff ) is directly related to sarcomere length changes during the cardiac cycle. Direct measurement of fiber strain is made difficult by the complex 3D branching arrangement of the myocytes. To quantify fiber strains, researchers have typically measured segment length changes with respect to some topologically based coordinate system (such as the cardiac coordinate system) and then have transformed strain components into the fiber coordinate system using post-mortem measurements of the fiber orientations. Several such indirect studies are summarized below to compare the experimental fiber strain estimates with ventricular mechanics model predictions for the end-diastolic and end-systolic configurations. Omens et al. (Ref. 53, Fig. 8) reported the mean transmural wall-plane fiber angle distribution for the equatorial region of the anterior midwall in six isolated potassium-arrested dog hearts. Fiber angles varied linearly from approximately −40◦ at the epicardium, to 70◦ at the endocardium (errors were estimated to be up to 10◦ ), where a positive angle represented an anticlockwise rotation from the circumferential direction in the wall plane. Cardiac components of strain at a LV pressure of 1.1 ± 0.5 kPa (8 ± 4 mmHg) were rotated to wall-plane fiber strains using interpolated fiber angles at the various wall depths (see Ref. 53, Fig. 9). The transmural distribution of fiber axis strain (Eff ) at a typical end-diastolic state is represented by box symbols () in Fig. 9 (left). Using similar techniques, May-Newman et al. (Ref. 56, Fig. 5) plotted mean transmural fiber strains from the anterior LV free wall of eight isolated potassium-arrested dog hearts inflated to a LV pressure of 1.3 kPa (10 mmHg). Fiber angles varied from −37 ± 18◦ at the epicardium to 70 ± 24◦ at the endocardium. The resulting fiber strain distribution for the configuration closest to an end-diastolic state is represented by crosses (×) in Fig. 9 (left). LeWinter et al. (Ref. 63, Tables 1 and 2) used pairs of ultrasonic crystals to estimate fiber shortening in the anterior LV of open-chest dogs relative to their end-diastolic lengths. Crystal pairs were implanted into the


120


Fig. 9. Transmural distributions of end-diastolic (left) and end-systolic (right) fiber strain for the equatorial region of the anterior ventricular wall. Model predictions () are compared to data from a variety of experimental studies (see text for details).

anterior LV midwall of six hearts and oriented circumferentially to be parallel to the midwall fiber direction. For these studies, the peak systolic LV pressure was 14.9 ± 0.6 kPa (112 ± 4.7 mmHg (SEM)) and LVEDP was 0.6 ± 0.1 kPa (4.8 ± 1.1 mmHg). Additional pairs of crystals were sewn to the anterior LV epicardium of four hearts and oriented 70◦ clockwise from the circumferential direction to approximate the fiber direction reported by Streeter et al.21 For these epicardial studies, the peak systolic LV pressure was 16.4 ± 1.3 kPa (123 ± 9.8 mmHg (SEM)). Observations from this study are represented by triangles () in Fig. 9 (right). From bead studies for the anterior equatorial LV free wall of seven openchest dogs, Waldman et al. (Ref. 58, Tables 1 and 2) computed end-systolic fiber strains with respect to the end-diastolic reference state. They found that in the outer portions of the wall, greatest shortening occurred in the fiber direction. Observations from this study are represented by crosses (×) in Fig. 9 (right). In ten closed-chest dogs, Rademakers et al. (Ref. 64, Table 1) used MRI tissue tagging to compute end-systolic fiber strains with respect to the end-diastolic configuration. In the plane of the wall, the average fiber angles with respect to the circumferential axis (anticlockwise positive) were −68.6 ± 12.7◦ , 10.5 ± 3.6◦ and 74.5 ± 3.2◦ for the epicardial, midwall and endocardial regions, respectively. The peak systolic blood pressures was 19.1 ± 0.9 kPa (143 ± 7 mmHg). The end-systolic wall-plane fiber strain reported for the equatorial region of the anterior free wall (referred to as location 4 in the study) is represented by box symbols () in Fig. 9 (right).



121

Ventricular mechanics model predictions of the fiber strain in the anterior equatorial wall are represented by diamonds () in Fig. 9. During diastole, the LV and RV were inflated to 1.0 kPa (7.5 mmHg) and 0.2 kPa (1.5 mmHg), respectively. The end-diastolic fiber strain (Eff ) was relatively uniform through the wall, and typically within 1SD of the mean experimental estimates. Model predictions of the transmural end-systolic fiber strain distribution were also consistent with the relevent experimental studies, and predict that sub-endocardial cells undergo greatest shortening during systole.

5.5. Spatial distributions of myofiber stress The ability of the ventricular mechanics model to reproduce realistic distributions of cardiac and fiber strain distributions implies that the predicted fiber stress distributions were also physiologically reliable. During diastole, fiber stresses typically dominated all other predicted stress components throughout the ventricles. Figure 10(a) illustrates an anterior view of the predicted epicardial and endocardial end-diastolic fiber stress distributions superimposed on the inflated ventricles. At end-diastole, the predicted tensile fiber stress was greatest near the endocardial region of the apex, while small tensile stresses were predicted for epicardial fibers at apical and equatorial regions. This is consistent with sarcomere length increases that have been observed in these regions during diastole.65 On the other hand, small compressive fiber stresses were predicted for epicardial regions near the base. This was likely to be due to the stiff constraining effects of the basal ring on the mechanics of the ventricles. Figure 10 also illustrates the predicted epicardial and endocardial fiber stress distributions superimposed on the deformed ventricles at the end of the isovolumic contraction and ejection phases of systole. At the end of isovolumic contraction, small compressive stresses were predicted for the most of the LV endocardium. However, the large tensile stresses predicted for regions near the apex are likely to be somewhat misleading. RV endocardial and epicardial fiber stresses were typically tensile. At end-systole, small compressive stresses were also predicted for the majority of the LV endocardium. It is likely that the large predicted tensile fiber stress near the basal portions of the LV septal endocardium was due to the stiff basal constraints. Interestingly, the large tensile stress near the endocardial apex at the end of isovolumic contraction was not as pronounced at end-systole.


122


Fig. 10. Distributions of fiber stress superimposed on the deformed ventricles. Lines depict element boundaries of the finite element mesh.



123

Large tensile fiber stresses were predicted across most of the epicardium during the systolic contraction phase of the cardiac cycle. 6. Summary An anatomically realistic mathematical model of the ventricles has been developed to predict regional distributions of myocardial deformation and stress throughout the heart cycle. The ventricular mechanics model is based on the nonlinear finite element method for finite deformation elasticity, and incorporated the biophysically based orthotropic pole–zero constitutive law to represent the passive mechanical properties of ventricular tissue. Ventricular contraction was simulated using a steady-state calcium–length–tension relation to approximate the active stresses generated by the myofibers during ventricular systole. Mechanical indices of cardiac function and regional distributions of ventricular deformation were analyzed for the various phases of the heart cycle, and showed good agreement with relevant experimental observations (where available). Moreover the nature of the model enabled prediction of the regional myocardial stress distributions, known to be related to oxygen consumption and remodeling of the myocardium. Computational modeling of this kind provides the necessary framework to quantitatively interpret the wide variety of experimental and clinical observations on cardiac function. Quantitative models may be used to better understand the mechanisms underpinning cardiac dysfunction, and provide useful tools to evaluate the efficacy of interventions and treatment for the management and prevention of cardiac arrhythmias. References 1. R. J. Atkin, and N. Fox, An Introduction to the Theory of Elasticity (Longman Group Limited, London, 1980). 2. J. D. Humphrey, Cardiovascular Solid Mechanics — Cells, Tissues, and Organs (Springer, 2002). 3. L. E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, Inc, Englewood Cliffs, New Jersey, 1969). 4. A. J. M. Spencer, Continuum Mechanics (Longman Group Limited, London, 1980). 5. L. Fox, An Introduction to Numerical Linear Algebra (Oxford University Press, London, 1967). 6. J. T. Oden, Finite Elements of Nonlinear Continua (McGraw-Hill Book Company Inc, New York, 1972).


124


7. J. E. Marsden, and T. J. R. Hughes, Mathematical Foundations of Elasticity (Prentice-Hall, Inc, New Jersey, 1983). 8. Y. C. Fung, Foundations of Solid Mechanics (Prentice-Hall, Inc: Englewood Cliffs, New Jersey, 1965). 9. M. P. Nash, Mechanics and Material Properties of the Heart Using an Anatomically Accurate Mathematical Model. Ph.D. thesis, The University of Auckland, New Zealand (1998). 10. M. P. Nash, and P. J. Hunter, Computational mechanics of the heart: from tissue structure to ventricular function, Journal Elasticity 61, 1/3 (2000) 113–141. 11. J. M. Huyghe, D. H. van Campen, T. Arts, and R. M. Heethaar, A two-phase finite element model of the diastolic left ventricle, Journal of Biomechanics 24, 7 (1991) 527–538. 12. J. M. Huyghe, D. H. van Campen, T. Arts, and R. M. Heethaar, The constitutive behaviour of passive heart muscle tissue: a quasi-linear viscoelastic formulation, Journal of Biomechanics 24 (1991) 841–849. 13. F. Lockett, Nonlinear Viscoelastic Solids (Academic Press, 1972). 14. A. C. Eringen, Mechanics of Continua (Krieger, New York, 1980). 15. A. E. Green, and J. E. Adkins, Large Elastic Deformations, 2nd edn. (Clarendon Press, Oxford, 1970). 16. G. C. Lee, N. T. Tseng, and Y. M. Yuan, Finite element modelling of lungs including interlobar fissures and the heart cavity, Journal of Biomechanics 16, 9 (1983) 679–690. 17. J. B. MacCallum, On the muscular architecture and growth of the ventricles of the heart, Johns Hopkins Hospital Report 9 (1900) 307–335. 18. F. P. Mall, On the muscular architecture of the ventricles of the human heart, American Journal of Anatomy 11, 3 (1911) 211–266. 19. W. Hort, Mikrometrische untersuchungen an verschieden weiten meerschweinchenherzen, Verhandl Deut Ges Kreislaufforsch 23 (1957) 343–346. 20. D. D. Streeter, and D. L. Bassett, An engineering analysis of myocardial fiber orientation in pig’s left ventricle in systole, 155 (1966) 503–511. 21. D. D. Streeter, H. M. Spotnitz, D. P. Patel, J. Ross Jr., and E. H. Sonnenblick, Fibre orientation in the canine left ventricle during diastole and systole, Circulation Research 24 (1969) 339–347. 22. J. A. Armour, and W. C. Randall, Structural basis for cardiac function, American Journal of Physiology 218, 6 (1970) 1517–1523. 23. C. C. Fox, and G. M. Hutchins, The architecture of the human ventricular myocardium, Johns Hopkins Medical Journal 130 (1972) 289–299. 24. R. A. Greenbaum, S. Y. Ho, D. G. Gibson, A. E. Becker, and R. H. Anderson, Left ventricular fibre architecture in man, British Heart Journal 45 (1981) 248–263. 25. P. M. F. Nielsen, The Anatomy of the Heart: A Finite Element Model. Ph.D. thesis, The University of Auckland, New Zealand (1987). 26. M. R. McLean, and J. Prothero, Coordinated three-dimensional reconstruction from serial sections at macroscopic and microscopic levels of resolution; the human heart, Anat Rec 219 (1987) 434–439.



125

27. I. J. Le Grice, A Finite Element Model of Myocardial Structure: Implications for Electrical Activation in the Heart. Ph.D. thesis, The University of Auckland, New Zealand (1992). 28. P. M. F. Nielsen, I. J. Le Grice, B. H. Smaill, and P. J. Hunter, Mathematical model of geometry and fibrous structure of the heart. American Journal of Physiology, 260 (Heart and Circulation Physiology 29) (1991) H1365–H1378. 29. J. S. Robb, and R. C. Robb, The normal heart. Anatomy and physiology of the structural units, American Heart Journal 23 (1942) 455–467. 30. H. M. Spotnitz, W. D. Spotnitz, T. S. Cottrell, D. Spiro, and E. H. Sonnenblick, Cellular basis for volume related wall thickness changes in the rat left ventricle, Journal of Molecular Cellular Cardiology 6 (1974) 317–331. 31. D. D. Streeter Jr., Gross morphology and fibre geometry of the heart. Handbook of Physiology (Section 2: The Cardiovascular System. Vol. 1: The Heart), eds. R.M. Berne, N. Sperelakis, and S. R. Geigert, (American Physiological Society, Williams and Wilkins Company, Baltimore, 1979) Chapter 4, 61–112. 32. I. J. Le Grice, B. H. Smaill, L. Z. Chai, S. G. Edgar, J. B. Gavin, and P. J. Hunter, Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog, American Journal of Physiology 269, 38 (1995) H571–H582. 33. I. J. Le Grice, P. J. Hunter, and B. H. Smaill, Laminar structure of the heart: a mathematical model, American Journal of Physiology 272 (Heart and Circulation Physiology 41 (1997) H2466–H2476. 34. B. H. Smaill, and P. J. Hunter, Structure and function of the diastolic heart: material properties of passive myocardium. Theory of Heart: Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac Function, eds. L. Glass, P.J. Hunter, and A.D. McCulloch, (Springer-Verlag: New York, 1991), 1–29. 35. L. L. Demer, and F. C. P. Yin, Passive biaxial mechanical properties of isolated canine myocardium, Journal of Physiology (London) 339 (1983) 615–630. 36. J. D. Humphrey, R. K. Strumpf, and F. C. P. Yin, Determination of a constitutive relation for passive myocardium: II. Parameter estimation, ASME Journal of Biomechanical Engineering, 112 (1990) 340–346. 37. F. C. P. Yin, R. K. Strumpf, P. H. Chew, and S. L. Zeger, Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading, Journal of Biomechanics 20 (1987) 577–589. 38. P. J. Hunter, M. P. Nash, and G. B. Sands, Computational electromechanics of the heart. Computational Biology of the Heart, eds. A. V. Panfilov and A. V. Holden, (John Wiley and Sons Ltd, West Sussex, England, 1997), Chapter 12, (1997) 345–407. 39. T. F. Robinson, M. A. Geraci, E. H. Sonnenblick, and S. M. Factor, Coiled perimysial fibers of papillary muscle in rat heart: morphology, distribution and changes in configuration, Circulation Research 63 (1988) 577–592. 40. D. A. MacKenna, J. H. Omens, A. D. McCulloch, and J. W. Covell, Contributions of collagen matrix to passive left ventricular mechanics in isolated rat hearts, American Journal of Physiology 266 (1994) H1007–H1018.


126


41. D. A. MacKenna, J. H. Omens, and J. W. Covell, Left ventricular perimysial collagen fibers uncoil rather than stretch during diastolic filling, Basic Research in Cardiology 91, 2 (1996) 111–122. 42. K. D. Costa, J. W. Holmes, and A. D. McCulloch, Modelling cardiac mechanical properties in three dimensions, Philosophical Transactions of Royal Society of London A 359 (2001) 1233–1250. 43. T. P. Usyk, R. Mazhari, and A. D. McCulloch, Effect of laminar orthotropic myofiber architecture on regional stress strain in the canine left ventricle, Journal of Elasticity 61, 1/3 (2000) 143–164. 44. H. Schmid, P. O’Callaghan, M. P. Nash, W. L. Lin, I. J. Le Grice, B. H. Smaill, A. A. Young, and P. J. Hunter, Myocardial material parameter estimation from simple shear tests — a non-homogeneous finite element study, Biomechanics and Modeling in Mechanobiology (in press), 2007. 45. W. L. Lin, Heart tissue mechanics: a comparative finite element study of orthotropic constitutive laws for simple shear (a part 4 project). Technical Report, Department of Engineering Science, The University of Auckland, New Zealand, (2005). 46. T. Zhao, Heart mechanics using orthotropic constitutive laws (a part 4 project). Technical Report, Department of Engineering Science, The University of Auckland, New Zealand, (2004). 47. S. Dokos, B. H. Smaill, A. A. Young, and I. J. Le Grice, Shear properties of passive ventricular myocardium, American Journal of Physiology 283, 6 (2002) H2650–H2659. 48. P. J. Hunter, Myocardial constitutive laws for continuum mechanics models of the heart. Molecular and Subcellular Cardiology: Effects of Structure and Function, eds. S. Sideman and R. Beyar, (Plenum Press, 1995), Chapter 30, 303–318. 49. S. Dokos, I. J. Le Grice, B. H. Smaill, J. Kar, and A. A. Young, A triaxialmeasurement shear-test device for soft biological tissues, ASME Journal of Biomechanical Engineering 122, 65 (2000) 471–478. 50. D. H. S. Lin, and F. C. P. Yin, A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus. ASME Journal of Biomechanical Engineering 120 (1998) 504–517. 51. P. J. Hunter, A. D. McCulloch, and H. E. D. J. ter Keurs, Modelling the mechanical properties of cardiac muscle, Progresses in Biophysics and Molecular Biology 69 (1998) 289–331. 52. A. D. McCulloch, B. H. Smaill, and P. J. Hunter, Regional left ventricular epicardial deformation in the passive dog heart, Circulation Research 64 (1989) 721–723. 53. J. H. Omens, D. A. MacKenna, and A. D. McCulloch, Measurement of twodimensional strain and analysis of stress in the arrested rat left ventricle, Advances in Bioengineering 20 (1991) 635–638. 54. A. D. McCulloch, P. J. Hunter, and B. H. Smaill, Mechanical effects of coronary perfusion in the passive canine left ventricle, Am J Physiol 262 (Heart and Circulation Physiology 31), (1992) H523–H530.



127

55. G. D. Meier, M. C. Ziskin, W. P. Santamore, and A. A. Bove, Kinematics of the beating heart, IEEE Transactions Biomedical Engineering, 27, 6 (1980) 319–329. 56. K. May-Newman, J. H. Omens, R. S. Pavelec, and A. D. McCulloch, Threedimensional transmural mechanical interaction between the coronary vasculature and passive myocardium in the dog, Circulation Research 74, 6 (1994) 1166–1178. 57. J. M. Guccione, K. D. Costa, and A. D. McCulloch, Finite element stress analysis of left ventricular mechanics in the beating dog heart, Journal of Biomechanics, 28, 10 (1995) 1167–1177. 58. L. K. Waldman, D. Nossan, F. Villareal, and J. W. Covell, Relation between transmural deformation and local myofiber direction in the canine left ventricle, Circulation Research 63 (1988) 550–562. 59. F. J. Villarreal, W. Y. W. Lew, L. K. Waldman, and J. W. Covell, Transmural myocardial deformation in the ischemic canine left ventricle, Circulation Research 68, 2 (1991) 368–381. 60. A. D. McCulloch, and J. H. Omens, Non-homogeneous analysis of threedimensional transmural finite deformation in canine ventricular myocardium, Journal of Biomechanics, 24, 7 (1991) 539–548. 61. L. K. Waldman, Y. C. Fung, and J. W. Covell, Transmural myocardial deformation in the canine left ventricle: normal in vivo three-dimensional finite strains, Circulation Research 57, 1 (1985) 152–163. 62. I. J. Le Grice, Y. Takayama, and J. W. Covell, Transverse shear along myocardial cleavage planes provides a mechanism for normal systolic wall thickening, Circulation Research 77 (1995) 182–193. 63. M. M. LeWinter, R. S. Kent, J. M. Kroener, T. E. Carew, and J. W. Covell, Regional differences in myocardial performance in the left ventricle of the dog, Circulation Research 37 (1975) 191–199. 64. F. E. Rademakers, W. J. Rogers, and W. H. Guier, Relation of regional cross-fiber shortening to wall thickening in the intact heart: three-dimensional strain analysis by NMR tagging, Circulation 89, 3 (1994) 1174–1182. 65. E. K. Rodriguez, W. C. Hunter, M. J. Royce, M. K. Leppo, A. S. Douglas, and H. F. Weisman, A method to reconstruct myocardial sarcomere lengths and orientations at transmural sites in beating canine hearts. American Journal of Physiology 263 (Heart and Circulation Physiology) (1992) H293–H306.




LEFT VENTRICULAR (LV) PUMPING–PERFUSION ANALYSIS: MYOCARDIAL PROPERTIES, INTRA-LV VELOCITY AND PRESSURE, DETECTION OF MYOCARDIAL ISCHEMIC AND INFARCTED SEGMENTS, PERFUSION DEPICTION BY SPECT IMAGING, COMPUTATION OF BLOOD FLOW PRESSURE AND VELOCITY PATTERNS WITHIN MYOCARDIAL REGIONS EDDIE Y. K. NG∗ , DHANJOO N. GHISTA† , JIAN JUN SHU, REGINALD C. JEGATHESE and MEENA SANKARANARAYANAN School of Mechanical and Aerospace Engineering College of Engineering, Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 ∗[email protected] † [email protected]

1. LV Pumping 1.1. Introduction The severity of myocardial disease (ischemic versus infarcted myocardial segments) and its extent will influence the function and performance of the left ventricle (LV) as a perfusion pump, based on which we can assess the patient’s treatment and (if necessary) candidacy for coronary bypass surgery. However, currently, we have vague means of assessing LV pumping function, for instance in terms of global indices such as ejection fraction, rather than in terms of intrinsic (and regional, instead of global) indices of “resistance-to-filling” and “contractility”. If such intrinsic indices were available, it would also be possible to a priori simulate the reperfusion effects of coronary bypass surgery (by administering a myocardial perfusing agent), and study whether and how significantly the values of these indices are altered, so as to provide guidelines for bypass surgery candidacy. In our earlier articles,1–4 we have proposed the use of diastolic and systolic modulus of the LV wall, as measures of LV stiffness (LV-STIFF) 129


130

Eddie Y. K. Ng et al.

during filling and LV contractility (LV-CONT). Herein, we have depicted LV-STIFF and LV-CONT in terms of LV diastolic-modulus versus LV wall stress, and LV stress (σ) versus strain rate (ε). ˙ We have also depicted these indices of LV resistance-to-filling (LV-RTF) and LV contractility (LVCONT) in the form of intra-LV blood-flow velocity and pressure gradients, during LV-filling and ejection phases. 1.2. LV pumping-function indices of resistance-to-filling and contractility, and candidacy for bypass surgery 1.2.1. State-of-the-art and its shortcomings Traditionally, the major determinants of LV pump function have been enddiastolic pressure, stroke volume and stroke power. However, these determinants do not intrinsically assess LV pump function or dysfunction, in the form of LV-RTF and LV-CONT. Further, for deciding candidacy for coronary bypass surgery, improved LV wall motion and ejection fraction (EF) following administration of nitroglycerin are employed as indicators. However, because LV wall motion and EF are not sensitive and intrinsic indicators of LV-RTF and LV- CONT, they do not indicate if, following coronary bypass surgery, the myocardium will contribute to improved ventricular functional efficiency by: • decreased LV-RTF, in terms of decreased mitral-valve-to-apex pressure gradient and • more favorable LV-CONT, in terms of better intra-ventricular apex-toaortic valve pressure gradient during ejection. 1.2.2. Our approach For assessment of the LV as a blood pump, instead of the traditional global measures of stroke volume and stroke power, we have carried out analyses of: • The stiffness of the pressurized spherical model of LV during the filling phase as an index of LV-RTF, and of its wall stress (σ) strain rate (ε) ˙ as a measure of LV-CONT. • Intra-LV blood-flow analysis, to determine the intra-LV distributions of blood-flow velocity and differential pressure (by means of finite-element flow analysis), to characterize LV-RTF and LV-CONT. Further, by comparing the values of intra-myocardial blood flow and pressure-gradient distributions, before and after administration of a myocardial perfusing agent (such as nitroglycerin), we can assess how the


Left Ventricular Pumping–Perfusion Analysis

131

myocardium is going to respond after coronary bypass surgery, and thereby develop the basis for recommending surgery. 1.3. Analysis of LV-filling-phase stiffness and ejection-phase wall σ versus strain rate ε˙ Figure 1 depicts cineangiographically derived LV pressure versus time, and spherical LV model dimensions, equivalent radius R [= (3 ∗ LV volume/4π)1/3 ] and wall thickness t, during a cardiac cycle. This data is employed in the following analysis: Instantaneous LV myocardial incremental stress (using Laplace law) stress: ∆σi = ∆Pi Ri /2ti

(1)

Due to incompressibility of LV muscle, 2 4πRi2 ti = 4πRi−1 ti−1 .

(2)

Therefore, from (1) and (2), 2 ∆σi = ∆Pi Ri3 /2Ri−1 ti−1 .

Total instantaneous wall stress: σi =

∆σi .

(3)

(4)

Instantaneous wall strain: ∆εi = (Ri − Ri−1 )/Ri−1 .

Fig. 1.

(5)

Cineangiographically derived data on LV pressure and geometric parameters.


132


Total instantaneous strain: εi =

εi .

(6)

Instantaneous strain rate: ε˙i = ∆εi /∆t.

(7)

Instantaneous myocardial-wall elasticity: E = ∆σi /∆εi

(8)

where ρ ∼ = intra-LV pressure, R ∼ = equivalent LV spherical model radius, t = LV wall thickness, σ = circumferential wall stress; Pi , σi , εi , Ri , and ti , respectively, refer to pressure, wall stress, wall strain, radius, and wall thickness at frame (or instant) i. From the above analysis, we compute (i) elastic modulus of LV (E) versus wall stress (σ) during the filling phase and (ii) LV wall stress (σ) versus strain rate (ε) ˙ during the ejection phase. The results of the analysis are depicted later on in Fig. 2. Figures 2a and 2b depict LV E versus σ during the filling phase, and σ versus ε˙ (strain rate) during the ejection phase for the patient whose data is depicted in Fig. 1. Note that we are obtaining an in vivo inverse relationship between σ and ε, ˙ corresponding to the one-dimensional (1D) in vitro experimental observation of “force versus velocity” of isolated muscle preparation.

Fig. 2a.

Computed LV-filling-phase stiffness E versus wall stress.



Fig. 2b.

133

∼ LV contractility). Measure of LV myocardial stress versus strain rate (=

1.4. Finite-element analysis of blood flow in the LV (using monitored LV wall-motion data) The data required for the FEA consists of • LV two-dimensional long-axis frames during LV diastolic and systolic phases. • LV pressure versus time, associated with these LV frames (optional). • Computation of LV instaneous wall velocities as well as instaneous velocity of blood entering the LV during the filling phase and leaving the LV during the ejection phase. From this FEA, we determine the instantaneous distributions of intraLV blood-flow velocity and differential pressure during filling and ejection phases, to intrinsically characterize LV-RTF and LV-CONT, respectively. Additionally, if we know the catheterized pressure at a point in the LV, we can also determine the absolute pressure distribution in the LV. The analysis is detailed in an earlier work with Dr. Fallen and Dr. Subbaraj,5 and summarized in the Appendix. Further, by comparing intra-LV pressure gradients before and after administration of nitroglycerin (a myocardial perfusing agent, and hence a quasi-simulator of coronary bypass surgery), we can infer how the myocardium is going to respond and how these LV functional indices will improve after coronary bypass surgery. Figures 3a, 4a, and 5a display the superimposed sequential diastolic and systolic endocardial frames for three patients PETE, HOWS and FRAN. From these images, we can compute the instantaneous wall displacements


134


as well as the input and exit velocities during the four time-intervals of diastole and systole,5,6 as shown in Figs. 3(b, c), 4(b, c) and 5(b, c). From the endocardial wall velocities, the intra-LV blood-flow velocities are computed by finite-element analysis (FEA), and shown in Figs. 3(d, e), 4(d, e)

Patient PETE: Superimposed sequential diastolic and systolic endocardial frames, whose aortic valves centres and the long axis are matched.

Patient PETE: Instantaneous endocardial wall displacement distortions during the periods of diastole.

Fig. 3. LV segmental wall motions and intra-LV flow-velocity distributions, for patient PETE, as reproduced from our work.5



Patient PETE: Instantaneous endocardial wall displacement distrotions during the periods of systole.

Patient PETE: Instantaneous intra-LV velocity distributions during the periods of diastole.

Fig. 3.

(Continued)

135


136


Patient PETE: Instantaneous intra-LV velocity distributions during the periods of systole.

Fig. 3.

(Continued)

and 5(d, e). Thereafter, intra-ventricular pressure distributions are computed by employing Bernoulli equation. The analysis (employed for computing intra-LV flow velocity and pressure distributions) is detailed in the Appendix. 1.4.1. Intra-LV flow during diastolic filling (Figs. 3d, 4d, and 5d) How well and how easily the LV fills is depicted by the instantaneous intraventricular flow distributions, which are governed by the segmental stiffness of the LV. In general, the flow is highest during the first half of the diastole (20–30 cm/s) in all patients, and the relative flow during all phases of diastole is maximum in the inflow segment of the LV (just below the mitral valve). The results suggest that the early-filling phase could possibly be due to the actively relaxing LV wall setting up a pressure-gradient conducive to filling (denoted by LV suction). Later on, the LV filling is due to the LV



137

Patient HOWS: Superimposed sequential diastolic and systolic endocardial frames, whose aortic valves centres and the long axis are matched.

Patient HOWS: Instantaneous endocardial wall displacement distortions during the periods of diastole.

Fig. 4. LV segmental wall motions and intra-LV flow-velocity distributions, for patient HOWS, as reproduced from our work.5


138


Patient HOWS: Instantaneous endocardial wall displacement distortions during the periods of systole.

Patient HOWS: Instantaneous intra-LV velocity distributions during the periods of diastole.

Fig. 4.

(Continued)



139

Patient HOWS: Instantaneous intra-LV velocity distributions during the periods of systole.

Fig. 4.

(Continued)

wall motion responding passively to blood flow due to atrial contraction. Subsequently, during the late-filling phases, the increasing stiffness of the LV wall (due to increasing LV volume) provides increased resistance to LV filling in the form of reduced flow. In the LV of patient PETE, without segmental asynergy, there is an alternating sequence in the direction of flow (Fig. 2d), with the major velocity vector direction being toward the anterior wall in the first quarter of diastole, then toward the posterior wall in the second quarter, with the sequence repeating itself during the second half of diastole. This alternating clockwise and counter-clockwise rotation of the velocity vectors is not seen in the other two patients, HOWS and FRAN, with segmental asynergy (Figs. 4d and 5d). In these cases, the direction of flow is for the most part toward the wall opposite the relatively stiff akinetic or hypokinetic segments.


140


Patient FRAN: Superimposed sequential diastolic and systolic endocardial frames, whose aortic valves centres and the long axis are matched.

Patient FRAN: Instantaneous endocardial wall displacement distortions during diastole.

Fig. 5. LV segmental wall motions and intra-LV flow-velocity distributions, for patient FRAN, as reproduced from our work.5



Patient FRAN: Instantaneous endocardial wall displacement distortions during systole.

Patient FRAN: Instantaneous intra-LV velocity distributions during diastole.

Fig. 5.

(Continued)

141


142


Patient FRAN: Instantaneous intra-LV velocity distributions during systole. Fig. 5.

(Continued)

1.4.2. Intra-LV flow during systolic ejection (Figs. 3e, 4e, and 5e) During systolic contraction, the overall direction of the velocity vector shifts toward the aortic valve. In the synchronously contracting ventricle of PETE, the maximum velocity of systolic ejection occurs during the latter half of systole (Fig. 3e); on the other hand, in the two ventricles with asynergy, the maximum velocity occurs during the first half of systole (Figs. 4e and 5e). During the systolic phase of maximum velocity, the velocity vectors are more or less uniform in magnitude throughout the ventricular cavity in the synchronous ventricle of PETE (Fig. 3e). In the case of the two patients (HOWS and FRAN) with asynergy, the normal LV wall segments appear to contribute more to the magnitude of the velocity vector (Figs. 4e and 5e); here, the maximum velocity is occurring in the upper half of the ventricular cavity, just below the aortic valve (Figs. 4e and 5e), and is in of the order of 40–50 cm/s.



143

Also, there is an alternating sequence in the direction of flow systole in the case of patient PETE. The direction of the velocity profile is primarily from the posterior wall during the first half of systole, and from the anterior wall in the latter half of systole (Fig. 4e). However, in both the asynergic ventricles, the direction of velocity flow is from the contralateral wall throughout systole (Figs. 4e and 5e).

1.4.3. Candidacy for coronary-bypass surgery The ideal situation is for the wall contraction to be so graded that adequate flow is generated in the apical region and a near-uniform flow is maintained throughout the LV chamber. The factors contributing to adequate intra-LV flow and cardiac output, with a smooth washout, are strong LV wall contraction and uniformly accelerating wall motion. If following administration of nitroglycerin, the LV wall can contract more uniformly and thereby set up a more favorable intra-LV velocity field, instead of a pattern of compensatory regional hypercontractility (and associated high wall tension and oxygen demand) to make up for a region of hypocontractility, then such a patient would be a good candidate for coronary bypass surgery. Referring to Fig. 6, for the patient (with a myocardial infarct) TURN, • a1 and a2 depict superimposed LV outlines during diastole and systole, before (a1 ) and after nitroglycerin administration (a2 ); • b1 and c1 depict intra-LV blood-flow velocity distributions during diastole and systole, before nitroglycerin administration; • b2 and c2 depict intra-LV blood-flow velocity distributions during diastole and systole, after nitroglycerin administration; • d1 and e1 depict intra-LV blood-flow pressure distributions during diastole and systole, before nitroglycerin administration; • d2 and e2 depict intra-LV blood-flow pressure distribution during diastole and systole, after nitroglycerin administration. From a computational viewpoint, the intra-LV flow is determined from the LV wall-motion boundary condition to the potential-flow equation (A-1). The intra-LV pressure gradient can in turn be computed from the flow by employing Bernoulli equation (A-4). However, we could interpret the phenomenon as if the LV wall stiffness were providing the resistance to wall motion for filling during diastole, and that the contracting LV is


144


facilitating emptying of the LV during systole, thereby setting up the requisite intra-LV pressure gradients and velocity distributions. In the case of the patient TURN, Fig. (6d1 , e1 ) demonstrate poor LV-RTF and LV-CONT in terms of adverse intra-LV blood pressure gradients during filling and ejection phases, respectively. However, following

a1

a2

Patient TURN: Superimposed sequential diastolic and systolic endocardial frames (whose aortic valves centres and the long axis are matched) before (1) and after (2) administration of nitroglycerin. Fig. 6. Patient TURN (as reproduced from our work5 ). Parts a1 and a2 depict LV wall motions, before and after administration of nitroglycerin; parts b1 and b2 depict intraLV flow-velocity distributions during the filling phase, before and after nitroglycerin administration; parts c1 and c2 depict intra-LV flow-velocity distributions during the ejection phase, before and after administration of nitroglycerin; parts d1 and d2 depict intra-LV pressure distributions during the filling and ejection phases, before and after nitroglycerin administration.



b1

b2

Patient TURN: Instantaneous intra-LV distributions of velocity during diastole, before (1) and after (2) administration of nitroglycerin.

c1

Fig. 6.

(Continued)

145


146


c2

Patient TURN: Instantaneous LV distributions of velocity during ejection phase, before (1) and after (2) administration of nitroglycerin.

d1

Fig. 6.

(Continued)



147

d2

Patient TURN: Instantaneous intra-LV distributions of pressuredifferentials during diastole, before (1) and after (2) administation of nitroglycerin Fig. 6.

(Continued)

administration of nitroglycerin, these filling and ejection phases’ pressuregradients (and hence LV-RTF and LV-CONT) are improved (Fig. 6d2 , e2 ), thereby providing the basis for advocating coronary bypass surgery for this patient. 2. Detection of Myocardial Ischemic and Infarcted Segments The intra-myocardial flow-velocity profiles provide some idea of how intra-myocardial stress and decreased permeability of ischemic myocardial segments influence the pressure gradients during filling and ejection. Aside from determining intra-ventricular blood-flow velocity and pressure distributions, we also need to know the distribution of normal, ischemic, and infarcted myocardial segment in the LV, resulting from myocardial


148


e1

Fig. 6.

(Continued)

perfusion distribution. Whether or not a myocardial segment has become ischemic can be determined from the echocardiographic texture analysis carried out by us,7 to determine the initial stages of myocardial ischemia, even before myocardial infarct occurrence. In Fig. 7, the middle (dark-green colored) region is infarcted, the surrounding salvageable (brown-shaded) region is ischemic, and the outside (white colored) region is normal. The highly stressed ischemic (brown-shaded) region has a high oxygen demand, and is prone to becoming infarcted. This myocardial texture analysis enables noninvasive tracking of the spread of the (green-shaded) myocardial infarcted segment, and to assess the efficacy of pharmacologic treatment. In this way, we can (i) develop the total myocardial texture profile of the LV of a patient, (ii) track the alteration in the texture with pharmologic



149

e2

Patient TURN: Instantaneous intra-LV distributions (pressuredifferential) during ejection phase, before (1) and after (2) administation of nitroglycerin. Fig. 6.

(Continued)

treatment, and (iii) study the alteration in the ischemic segmental sites with nitroglycerin administration, to determine candidacy for coronary bypass surgery.

3. Myocardial Perfusion 3.1. Perfusion based on SPECT imaging Myocardial perfusion imaging is a noninvasive method for assessing the regional myocardial flow by means of radioactive tracers (thallium-201 and


150


Fig. 7. (a) Long-axis 2D ultrasonic view of a pediatric patient’s heart showing highly echo-reflectile regions 1 and 2 and a healthy region 3. (b) Echocardiographic texture analysis showing echo-intensity levels from myocardial region 1. This figure is partially reproduced from our work in Ref. 7.

technicium-99) injected intravenously (IV). Ischemic areas pertaining to less blood flow do not take up these tracers. These tracers get trapped in the myocardium in proportion to the regional blood flow. This trapping allows external imaging using gamma cameras to assess myocardial perfusion. Myocardial perfusion provides added information about the hemodynamic viability of myocardial segments. Assessment of myocardial perfusion is performed, using single photon emission computed tomography (SPECT)



151

or positron emission tomography (PET). However, even though SPECT imaging is widely used, this method suffers from attenuation artifacts and also exposes the patient to radiation. SPECT allows evaluation of perfusion patterns in thin slices of myocardium. Comparative studies of stress-to-rest images help in differentiating ischemia from scar. A perfusion abnormality present on both the stress and rest studies represents scar tissue, while a perfusion abnormality that is present in stress study alone represents ischemia. The images shown below (in Fig. 8) were obtained during exercise in a 35-year-old man, with a low probability of coronary disease. Figure 8 describes a SPECT-computed 3D perfusion profile of the heart containing serial slices of the LV in the short axis (beginning at the apex and ending at the base), vertical long axis (beginning at the septum and ending at the lateral wall) and horizontal long axis (beginning at the inferior wall

Fig. 8. Perfusion pattern images of a 35-year-old man with a low probability of coronary artery disease.


152


Stress

Rest

Stress

Rest

Stress

Rest

Fig. 9. Perfusion images of a 52-year-old patient suffering from 85% stenosis of the right coronary artery. The abnormal stress images are depicted in the figure.

and ending at the anterior wall). For each tomographic plane, the stress images are shown at the top followed by the rest images. Figure 9 describes perfusion images obtained by SPECT of a 52-yearold patient, who suffers from 85% stenosis of the right coronary artery. The stress images are the upper set of each pair of horizontal row of images, and the rest images are the corresponding lower set of each pair. The images reveal that the stress images are abnormal, while the rest images are normal. The stress images show severely reduced tracer uptake in the inferior wall of the heart indicating the presence of a significant stenosis in the right coronary artery. At rest, the tracer uptake is normal. This implies that there is a large potentially reversible perfusion defect of the inferior wall.



153

3.2. Computational perfusion analysis To gain further insight into the intra-myocardial flow in the LV section, we have developed a biomechanics model of intra-myocardial blood flow through a porous myocardium medium, using a modified form of Darcy’s law in which blood velocity is dependent on intra-myocardial pressure gradient (∇p), myocardial permeability (k), as well as myocardial-stressdependent hydrostatic pressure (H). The equation governing blood flow through the porous but stressed myocardium is the Darcy law,8 for a porous media given by k V = − (∇p − H), µ

(9)

where V is velocity of blood flow, k is permeability of the tissue, µ is dynamic viscosity of blood, p is the pressure, and H is the hydrostatic pressure. The hydrostatic pressure is calculated by H=

1 (σr + σθ + σφ ), 3

(10)

and for a thick-walled spherical shell, we have:7,8 σθ = σφ =

p(a3 /b3 + a3 /2r3 ) (1 − a3 /b3 )

and

σr =

p(a3 /b3 − a3 /r3 ) . (1 − a3 /b3 )

(11)

To analyze this model, we first solve the Laplace equation of ∇2 p = 0 to obtain the p variation in the LV myocardial sector subject to input and output pressure conditions depicted in Figs. 10 and 11. We then determine the H distribution for the LV using Eq. (10), corresponding to its geometry and the computed p distribution. Thereafter, we employ Eq. (9) to compute the V distribution. The pressure and velocity distribution for a specific case in the sector of the LV are shown in Figs. 10 and 11. Now if we wish to analytically simulate the SPECT data, we would need to vary the permeability parameter k. Then for the patient, we would also need to know the inlet and outlet pressures for the LV chamber sector. This data is difficult to obtain. On the other hand, if we primarily want to analytically study the perfusion flow patterns in the LV myocardial sectors (to understand how intra-myocardial stress also affects intra-myocardial perfusion), then we could adopt (i) Pin (inlet pressure at the sector) to be equal to the mean LV pressure and Pout (the exit pressure at the sector) to be equal to the right atrial pressure. Thus, we could employ catheterized


154


Fig. 10. LV annular myocardial wall model (with inlet, pin = 100 mmHg, and outlet, pout = 30 mmHg, at a distance) depicting pressure variation (kPa) inside the segment.

LV pressures to obtain the detailed V distribution profiles in the various sectors. On the other hand, if we intend to have some idea of intra-myocardial flow distributions, we can adopt Pin (= mean ausculatory pressure) and



155

Fig. 11. LV annular myocardial wall segment model having flow configuration and conditions as in Fig. 10: (a) velocity (maximum vector of 0.9 m/s) and (b) streamline.

Pout (= zero), and assume LV-filling pressure = 10 mmHg. The steps entail (i) determing Pin (= mean ausculatory pressure); (ii) from LV geometry for the sector, to solve ∇2 p = 0 to obtain p distribution in the sector corresponding to Pin ; (iii) computing H variation corresponding to the LV-filling pressure; and (iv) solving Eq. (9) to obtain the velocity distribution.


156


3.2.1. Theory and governing equations The governing equations required to compute intra-myocardial pressure and velocity distributions are given as follows:9,10 Continuity equation: ∇ • (φs us + φf uf ) = 0,

(12)

φs + φf = 1.

(13)

Momentum equations: ∇ • σ s + bs = 0, ∇ • σ f + bf = 0.

(14)

σ s = −φs pI + σ ˜ s,

(16)

σ f = −φf pI.

(17)

(15)

Constitutive equations:

Effective solid stress: σ ˜ s = Bs tr (Ts ) I + 2µs es .

(18)

Darcy’s law: bs − bf = µf (φf uf − φs us ) , (19) κ where the subscripts s and f refer to the incompressible, elastic solid phase and incompressible, viscous fluid (blood) phases, respectively; ∇: gradient operator; φ: volumetric concentration; u: velocity vector of solid; σ: partial stress; b: diffusive body force vector; p: hydrostatic pressure; I: identity tensor; σ ˜ s : effective solid stress tensor; Bs : elastic bulk modulus; tr(•): trace operator that yields first invariant of its tensorial argument; Ts : solid strain tensor; µs : elastic shear modulus; es = Ts − 13 tr (Ts ) I: deviatoric component of solid strain tensor; µf : fluid dynamic viscosity; κ: permeability. In our situation, we assume that φf >> φs (i.e., φf → 1, φs → 0). Equations (11)–(19) may now be rewritten as: Continuity equation: ∇ • uf = 0.

(20)

Momentum equations: ∇ • σ s + bs = 0, ∇ • σ f + bf = 0.

(21) (22)



157

Constitutive equations: σs = σ ˜s,

(23)

σ f = −pI.

(24)

Effective solid stress: σ ˜ s = Bs tr (Ts ) I + 2µs es . Darcy’s law: uf =

κ bs − bf . µf

Inserting Eqs. (21)–(24) into (26), the Darcy’s law becomes: κ ∇p − bs uf = − µf or κ uf = − (∇p + ∇ • σ ˜ s) . µf

(25)

(26)

(27)

(28)

In Eq. (27), since the body force bs is a field force (∇ × bs = 0 in Eq. (21)), bs can be rewritten as bs = ρf ∇Ω, where ρf is the fluid density and Ω is the potential. In Eq. (28), if we put H = −∇ • σ ˜ s for the ideal myocardium perfusion modeling case with a thick-walled spherical shell, Eqs. (9)–(11) are recovered. Implementation. The above set of equations enable us to analyze the behavior of (i) incompressible, elastic solid phase and (ii) incompressible, viscous fluid phase. For the analysis [10], the bulk blood velocity is given by Darcy’s law, under the assumption that the pores are small and interconnected, as: κ (29) V = − (∇p − ρ∇Ω) , µ where Ω is the potential for body forces bi : bi =

∂Ω . ∂x i

(30)

Rewriting V gives,

p κρg κ 1 ∇φ. V = − ρg ∇ − ∇Ω = − µ ρg g µ

(31)


158


The pressure head is given as, φ=

Ω p − . ρg g

(32)

Darcy’s law is expressed in terms of hydraulic conductivity, −κρg C = , µ

(33)

which gives (from Eqs. (21) to (23)): ∂φ . (34) ∂x i For an incompressible blood, ∇ · V = 0, and the pressure head φ satisfies V i = −C

∇ • (C ∇φ) = 0.

(35)

For an anisotropic medium, V i = −C ij

∂φ . ∂xj

(36)

For this analysis, the last estimated value for Vi gives the velocity. Application. Table 1 summarizes the parameters used for the analysis of myocardial tissue,10 to gain some idea of pressure and flow-velocity patterns in certain regions of the LV myocardium, for reasonable input and boundary conditions, depicted in Figs. 10 and 11. Table 2 outlines the steps involved in the analysis. Figure 10a shows the annular model and the boundary conditions (BCs) with inlet and outlet at a distance. Figure 10b shows the computed pressure distribution. Figures 11a and 11b depict the velocity distribution and streamline pattern. It is interesting to observe how the blood enters the region, flows almost parallel to the walls, and then exits the region. Table 1.

Parameters used for analysis of LV myocardium tissue11 .

Parameter description

Value

Density (ρ) Acceleration due to gravity (g) Dynamic viscosity (µ) Permeability (k) LV wall thickness (t) Mean inlet pressure (pin ) Mean outlet pressure (pout ) Mean LV chamber pressure (p1 )

1000 kg/m3 9.81 m/s2 0.00015 N/s/m2 1 × 10−15 m2 10 mm 100.0 mmHg (13.3 kPa) 30.0 mmHg (3.9 kPa) 10.0 mmHg (1.3 kPa)


Left Ventricular Pumping–Perfusion Analysis Table 2.

159

A typical flow chart of poroelastic fluid-structure analysis.

STRUCTURE

FLUID

Initial displacement

Read Pin , Pout and given boundary conditions

Calculate strain tensor, Ts

update displacement Iterate until no change

Calculate effective solid stress, σ~s using Eq (25)

Calculate diffusive body force, bs using Eq (21)

Calculate potential Ω using Eq (30)

Calculate boundary conditions for Φ using Eq (32)

Solve fluid flow Φ using Eq (35)

Calculate velocity u f using Eq (34)

Calculate pressure P using Eq (32)

Figures 12 and 13 depict regional distributions of pressure and flow velocity, for the case when the inlet and outlet are adjacent to each other. The geometry and boundary conditions for this case are depicted in Fig. 12a. Figure 12b shows the pressure distribution. Figure 13a shows the associated velocity distribution and Fig. 13b gives the velocity streamline plot. The pressure is uniform away from the entry/exit region, but non-uniform near inlet and outlet, which are consistent with the velocity vector plots. This intra-myocardial analysis can provide some insight into how perfusion occurs in different segments of the myocardium.

4. Myocardial Reperfusion Using Coronary Bypass Surgery It is obvious, from the previous illustrations in Section 3, that there is less amount of blood flow in some ischemic and infarcted areas of the


160


Fig. 12. LV annular myocardial model (with inlet, pin = 100 mmHg, and outlet, pout = 30 mmHg, adjacent to each other) depicting pressure variation (kPa) inside the segment.

myocardium, which suggests the severity of stenosis present in the corresponding regional coronary vessels. To overcome this defect, coronary artery bypass grafting surgery can be performed. The aim of coronary artery bypass grafting is thus to increase blood flow to malperfused myocardium that results in maximal perfusion through the bypass conduits. The details of this topic can be found in Ref. 11.



161

Fig. 13. LV annular myocardial model having flow configuration and conditions as in Fig. 12: (a) velocity (maximum vector of 0.88 m/s) and (b) streamline.

5. Conclusion This chapter provides an overview of LV pumping and perfusion results, by employing different methods for characterizing (i) pumping (in terms of, for instance, intra-myocardial flow velocity and pressure distributions as for instance displayed by Fig. 6, (ii) perfusion based on SPECT imaging in Fig. 8 and 9, and (iii) computation analysis of intra-myocardial perfusion in Figs. 10–13.


162


References 1. D. N. Ghista, and H. Sandler, Elastic modulus of the human intact left ventricular determination of physiological interpretation, Med. Bio. Eng. 13, 2 (1975). 2. L. Zhong, N. Dhanjoo, D. N. Ghista, E. Y.-K. Ng, S. T. Lim, and T. S. J. Chua, Determination of aortic pressure–time profile, along with aortic stiffness and peripheral resistance, Journal of Mechanics in Medicine and Biology 4, 4 (2004) 499–509. 3. L. Zhong, N. Dhanjoo, D. N. Ghista, E. Y.-K. Ng, S. T. Lim, T. S. J. Chua, and C. N. Lee, Left ventricular shape-based on contractility index, Journal of Biomechanics 39, 13 (2006) 2397—2409. 4. D. N. Ghista, L. Zhong, L. P. Chua, E. Y.-K. Ng, S. T. Lim, R. S. Tan, and S. J. Chua, Systolic modeling of the left ventricle as a mechatronic system: determination of myocardial fiber’s sarcomere contractile characteristics and new performance indices, Molecular and Cellular Biomechanics 2, 4 (2005) 217–233. 5. K. Subbaraj, D. N. Ghista, and E. L. Fallen, Intrinsive indices of the left ventricle as a blood pump in normal and infarcted left ventricles, Biomedical Engineering (1987) 215. 6. K. Subbaraj, D. N. Ghista, and E. L. Fallen, Determination of intra-cardiac pressure and flow distributions from cineangiograms, Automedica 6, 3 (1985) 137–141. 7. M. V. Kamath, R. C. Way, D. N. Ghista, T. M. Srinivasan, C. Wu, S. Smeenk, C. Manning, and J. Cannon, Detection of myocardial scars in neonatal infants from computerized echocardiographic texture analysis, Engineering in Medicine 15, 3 (1986) 137–141. 8. C. R. Jegathese, E. Y.-K. Ng, and D. N. Ghista, Analysis of left ventricular myocardial properties, Journal of Mechanics in Medicine and Biology 4, 2 (2004) 173–185. 9. E. Y.-K. Ng, D. N. Ghista, and C. R. Jegathese, Numerical approach to fluidstructure analysis of soft biological tissue, Selected Paper From ICMMB-13, Journal of Mechanics in Medicine and Biology 5, 1 (2005) 11–28. 10. E. Y.-K. Ng, C. R. Jegathese, and D. N. Ghista, Perfusion studies of steady flow in poroelastic myocardium tissue, International Journal of Computer Methods in Biomechanics and Biomedical Engineering 8, 6 (2005) 349–358. 11. D. N. Ghista, S. Meena, L. P. Chua, and Y. S. Tan, Augmented myocardial perfusion by coronary bypass surgical procedure: emphasising flow and shear stress analysis at proximal and distal anastomotic sites providing the basis of better graft patency rates, Chapter 12 in this book, (2006).

Bibliography 1. F. Kajiya, and M. Goto, Integrative physiology of coronary microcirculation, Japanese Journal of Physiology 49, 3 (1999) 229–241.



163

2. O. C. Zienkiewicz, The Finite Element Method, 3rd edn. (McGraw Hill Book Co., New York, 1977) 423–449. 3. J. E. Akin, Applications and Implementation of Finite Element Methods (Academic Press, London, 1982). 4. A. G. Hansen, Fluid Mechanics, (John Wiley, New York, 1967) 349 pp. 5. L. M. Milne-Thompson, Theoretical Hydrodynamics (McMillan, New York 1965).

Appendix A5 A1. Solution for velocity potential (φ) A two-dimensional (2D) inviscid fluid-flow finite-element analysis (FEA) is carried out in the antero-posterior projection plane, by assuming a quasisteady flow in the time interval between successive frames. The intraventricular flow is governed by the potential equation: ∇2 φ = 0,

(A1)

where ∇2 is the Laplacian operator, φ is the velocity potential, and ∇φ is the velocity vector. For FEA, the governing differential equation (1) ∂2φ ∂2φ + 2 =0 ∂x2 ∂y

(A2)

for a 2D planar flow domain, is transformed to a finite element equation form, by making use of the Galerkin-weighted residual procedure. The resulting stiffness-matrix system of equations [K]{φ} = {F }

(A3)

can be solved for φ at those point(s) in the flow domain, by specifying ∂φ/∂n(Vn ) along the endocardial boundary, and φ at those point(s) on the boundary where Vn is not specified. By specifying φ to be constant along the open boundaries, the flow can be constrained to be normal to that boundary; this constraint also allows the solution to obtain a flow balance. The value of the constant φ is arbitrary, as only derivatives of the shape functions Ni are used in the computational process. In the present analysis, φ = 0 is specified along the open boundary. The matrix system [K], in Eq. (A3), is symmetrical and banded. Equation (A3) is solved for φ using a Gaussian elimination method, which transforms the matrix system [K] into an equivalent triangular system whose solution can be obtained by back substitution. From the computed values of φ at each internal point, we determine the velocity components


164


at each internal point of the LV chamber and hence obtain instantaneous maps of intra-LV blood-flow velocity patterns. A2. Analysis for intra-LV pressure distribution5,6 The intra-cardiac pressure distribution at any point inside the LV chamber can be obtained, from the Bernoulli equation for unsteady potential flow as: 1 P + ρV 2 + ρ(∂φ/∂t) = C(t), 2

(A4)

where P is the pressure, 12 ρV 2 is the dynamic pressure term, ρ is the density of blood, and V is the velocity of blood; ρ(∂φ/∂t) is the effect due to acceleration; and C(t) represents the total pressure as sensed by a pressure probe facing the oncoming fluid, C(t) is a constant, and the gravitational or hydrostatic effects are neglected. The partial derivative (∂φ/∂t) is computed from the value of φ at the same point at successive instants, using the finite difference scheme. Since we want the procedure to not utilize the catheter-pressure data, we can obtain the pressure distribution relative to a reference point in the chamber, say at the center of the aortic or mitral orifice. Hence the differential pressure field at a point s, in terms of the pressure po at the inlet (during diastole) or outlet (during the ejection phase) of the ventricle, is given by 1 ρ(Vo2 − Vs2 ) + ρ(∂φ/∂t|o − ∂φ/∂t|s ), (A5) 2 where Vo and Vs are, respectively, the velocity of blood flow at the center of the orifice (i.e. at the aortic or mitral orifice during systolic or diastolic phase) and at a point s inside the LV chamber. The differential pressure (ps − po ) can be expressed and displayed in non-dimensional form, as 1 2 ρV , (A6) Cp = (ps − po ) 2 o ps − po =

where Cp is the non-dimensional pressure coefficient. This instantaneous graphic display of the relative pressure distribution in the LV chamber can provide an indication of the resistance-to-filling as well as of the effectiveness of the LV contraction in setting up the appropriate pressure distribution in the chamber, so as to promote adequate emptying.


SECTION II.

CARDIAC PUMPING CHARACTERISTICS




LEFT VENTRICULAR (LV) PRESSURE INCREASE MECHANISM DURING ISOVOLUMIC CONTRACTION, AND DETERMINATION OF THE EQUIVALENT LV MYOCARDIAL FIBERS ORIENTATION DHANJOO N. GHISTA∗ , LI LIU† , LIANG ZHONG‡ , SI YONG YEO‡ , LEOK POH CHUA† , RU-SAN TAN‡ , and YONG SENG TAN‡ †,∗School

of Mechanical and Aerospace Engineering, College of Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 ∗[email protected] ‡Department

of Cardiology, National Heart Centre, Singapore 168752

1. Introduction and Scope 1.1. What this chapter is about? Background: The left ventricular (LV) myocardial fibers are adroitly oriented such that their contraction causes compression and twisting, resulting in intra-LV pressure increase during isovolumic contraction. Although there has been some work on mechanics of active contraction of the heart muscle, it has not been shown that the development of principal stresses (corresponding to the orientation of the myocardial fibers) causes adroit compression and twisting of the LV, which in turn cause internal pressure increase. In this chapter, a biomechanical thick-walled cylindrical model of the LV is developed to demonstrate that the mechanisms of LV internal pressure increase during isovolumic contraction is due to the contraction of the LV myocardial fibers helically wrapped inside the LV wall. The contraction of these fibers deforms and twists the LV. Hence, we can indirectly associate LV twisting with LV contractility. Associated with the LV pressure increase, we have determined the LV (radial, longitudinal, and twist) deformation state. We then determine the LV wall stresses associated with the deformations, and therefore the principal stresses in the LV wall, along with the shortening force and the torque experienced by the LV. We then hypothesize that the LV principal stresses 165


166

D. N. Ghista et al.

orientation corresponds to the orientation of the LV myocardial fibers. Hence we postulate that the contraction of these LV myocardial fibers causes LV deformations inducing twisting of the LV and the associated LV twist angle.

1.2. Earlier works For many years, researchers have employed the famous Laplace law for the determination of LV wall stress.1–3 In particular, Sandler and Dodge1 approximated the LV as an axisymmetric ellipsoidal thick-walled shell and used the Laplace law to determine the stress and tension in the wall. However, the Laplace law is not suitable for determining the wall stress variation in thick-walled structures as the LV.4,5 The main limitation of this model was their thin-walled representation, which assumed that the ventricular wall thickness is small relative to the radii of curvature. With this constraint, variations of material properties across the wall thickness were ignored and the stresses were assumed to be constant across the wall thickness. Wong and Rautaharju6 were among the first to come up with a thickwalled ventricular model. The wall stress was formulated with ellipsoidal LV shells, assumed to be isotropic, linearly elastic, and compressible. Despite further simplification of the analysis using small strain theory, they discovered nonlinear stress distributions through the wall thickness, an outcome that cannot be predicted by the Laplace law. It was Mirsky7 who first applied large deformation elasticity for the evaluation of LV stress. He discovered that the stresses at the endocardium using large strain theory were about ten times as high as the values predicted by linear small strain models. Several authors including Arts et al.,8 Tözeren,9 and Guccoine et al.10 have developed various cylindrical models to investigate the mechanics and material properties of LV. Azhari et al.,11 recognizing that a cylindrical model is not adequate to describe the LV geometry and the non-uniformity of LV wall motion, constructed a conical LV model. Chaudhry et al.12 formulated a mathematical model based on a thick-walled truncated conical shell and incorporated the theory of large deformation elasticity. It was found that conical models lead to more realistic results than the spherical models, and can better evaluate stresses and strains from base to apex. Noting the irregular LV geometry, Gould et al.13 formulated one of the first LV finite element (FE) models, with axisymmetric geometry based


LV Pressure Increase Mechanism and Determination of EMFO

167

on the longitudinal cross section of the LV. In this analysis, they found that the irregularity of the LV geometry had considerable effects on wall stresses. Following this, many FE models14–18 of the LV have been constructed for the three-dimensional analysis of stress and strain distribution in the LV wall. In addition, the effects of anisotropy due to myofiber orientation on the ventricular wall stresses have also been examined. Several other studies19–24 have also shown that the architecture and orientation of myocardial fiber have significant effects on the mechanical properties of myocardium. Recently, Nash and Hunter25 formulated an FE model of the LV based on finite deformation elasticity. This anatomically representative model took into account the effects of muscle fiber orientation, to analyze the deformations of the heart during different phases of the cardiac cycle.

1.3. Our work: LV pressure generation mechanism during isovolumic contraction While all of these works have been oriented to LV wall stress and deformation, we believe that the mechanism of LV pressure generation during isovolumic contraction is of great significance to LV systolic function, LV contractility, and LV ejection. It is this phase of early systole that sees the fastest rise in LV pressure. However, few studies are dedicated to the analysis of LV mechanism during isovolumic contraction. In this chapter, we develop the analysis to determine the stresses developed in the LV myocardial fibers during isovolumic contraction from LV deformation, using large deformation theory. Although simple geometries cannot accurately represent the LV shape, we have employed a thick-walled cylindrical LV model to reduce the mathematical intricacy and focus mainly on the effects of myofiber stresses on LV pressure development. In this process, we also obtain the constitutive properties of the myocardium and the relevant LV wall stresses, and, most importantly, the equivalent myofiber orientation responsible for the LV deformation. Our model can lend insight into how the myofiber orientation and the stress developed in the myofibers can cause the observed rapid LV pressure rise and the LV deformation during isovolumic contraction. At this stage, this is intended to be a theoretic mechanistic model. Later on, when we acquire the technological capability to monitor the complex LV deformation (radii, axial, and twist in the cylindrical coordinate frame) along with LV pressure, this model can be employed to determine the equivalent myocardial fiber orientation (EMFO), along with the LV myofiber stress state. We


168

D. N. Ghista et al.

deem this LV EMFO to be an intrinsic property of the LV, which governs how effectively the LV can develop pressure to effect adequate stroke volume ejection. In this regard, EMFO could well come to be regarded as an LV diagnostic property characterizing LV contraction. 1.4. Method of analysis The LV is modeled as a fluid-filled thick-walled cylindrical shell subjected to internal pressure increase during isovolumic contraction. Our objective is to elucidate that the tremendous internal pressure build-up during isovolumic contraction is caused by stress development in the spirally wound myocardiac fibers (due to their contraction). The chamber pressure increase is accompanied by chamber blood volume decrease (associated with the bulk modulus of blood), and corresponding radial and longitudinal deformations. The LV wall stresses associated with these deformations and the accompanying LV twist are determined by a finite elasticity analysis of the internally pressurized LV shell. The LV shell model wall stresses are made to satisfy the boundary conditions of internal pressure, by using a nonlinear optimization method, to determine (i) the LV myocardial constitutive parameters and (ii) the LV wall stress distribution. From this stress state in the LV wall, the principal stresses are determined. We then adopt the principal compression stresses to be the stresses developed in the myocardial fibers (by their contraction), and the angle of the principal stress to be the orientation of the myocardial fibers. In this way, we are able to determine the stresses within the myocardial fibers as well as their orientation. 1.5. Significance The results show that both the myocardial fiber orientation and the principal stresses change continuously during the isovolumic phase. The stress generation in the LV cylindrical model (to cause the pressure build-up within it) is equivalent to the development of active compression force and torsion within the model, as a mechanism for the high intra-LV cavity pressure build-up during isovolumic contraction. An important determinant of our analysis is the orientation of the myocardial fibers. Conversely, it can be said that the fibers are so optimally oriented, that their contraction causes LV deformation, which in turn causes appropriate increase in intra-LV pressure. This is an exemplification of anatomic optimality for optimal physiologic functionality.



169

2. Analysis Approach 2.1. Scope The LV myocardial wall is made up of helically oriented fibers. As the bioelectric wave propagates along these fibers, it causes concomitant contraction wave propagation. The contraction of the helically oriented myocardial fibers causes compression and twisting of the LV, resulting in increase of its internal pressure. Herein, we simulate this phenomenon of LV isovolumic contraction (which causes the intra-LV pressure to rise so fast during 0.04– 0.06 s of isovolumic contraction) by means of a finite elasticity analysis of the thick-walled LV cylindrical shell under incremental pressure increase. We have analyzed that the pressure generated during isovolumic contraction is caused by stress state in the LV wall equivalent to LV twisting and compression, due to contractile stress in the helically wound myocardial fibers (Fig. 1). Based on experimental studies, myocardial fibers are found to be helically wound, at angle varying from −60◦ to 60◦ across the LV wall.4–7 However, there is no way that this fiber orientation can be measured; nor can the LV deformation (for a cylindrical model of the LV) be precisely measured. Hence, in this chapter, we determine the EMFO averaged across the LV wall as well as the stress in the LV wall, from LV deformation and pressure increase. The stress in the myofibers will then be taken to be the derived principal stresses in LV wall. Conversely, it is intended thereby to indicate that the stress developed in myofibers of the specific equivalent orientation (average across the wall)

Pi

Ro

r oj

Pi

Ri

r ij

lj

L

Θ

Fig. 1.

Schematics of fiber orientation and twisting of an LV cylindrical model.


170

D. N. Ghista et al.

causes LV pressure increase and LV deformation. Although there has been some work on mechanics of active contraction of the heart muscle,8,9 it has not been hitherto shown that the development of principal stresses and the principal directions cause adroit compression and twisting of the LV, which in turn cause internal pressure increase. In this context, this work is pioneering in providing analytic demonstration of LV twisting and shortening (by the contraction of the helically wound myocardial fibers) being associated with and causing the rise of LV pressure during isovolumic contraction. It has also been shown that the fiber angle changes during isovolumic contraction. 3. Methodology 3.1. Analysis approach We approximate LV during isovolumic contraction as a fluid-filled, thickwalled cylindrical shell that is closed at both ends and constrained in the longitudinal direction at the base (Fig. 2). We further consider that LV wall behaves as an incompressible, transversely isotropic, hyperelastic material. Step 1. Starting with known LV pressure increase, we determine the corresponding LV internal volume change, based on the bulk modulus of blood.

ri Compressible fluid Bulk modulus (K )

L

ξ ro

∆P

∆F

T

Hyperelastic material strain energy function Ψ = C ( exp (Q ) − 1) 2

Fig. 2. The fluid-filled LV cylindrical shell model: (i) geometry and (ii) material property; equivalent compression (∆F ) and torsion (∆T ) associated with its internal stress state due to internal pressure rise within it.



171

From this volume change, we characterize the radial and longitudinal deformations of the LV cylindrical model. We also adopt a reasonable value of LV twist deformation (because the current technology does not permit us to determine this deformation, along with the LV deformation due to change in its internal volume). Step 2. From these deformations, we determine the corresponding stretches, and therefrom the strains in the LV wall, using finite elasticity formulations. Step 3. Next, we express the LV wall stresses in terms of the LV wall strains, in terms of the constitutive properties of LV wall material. Step 4. We now impose the boundary conditions involving equilibrating the LV stress in the inner wall to the LV internal blood pressure. In these equations, the LV constitutive properties are unknown. By computationally solving these equations, we determine the best values of these myocardial constitutive properties. Step 5. Once the LV constitutive properties are determined, we can go back to the expressions for the LV wall stresses and evaluate the LV wall stress state. We then continue on to determine the principal stresses from the above-obtained stress state. Step 6. We designate the relevant principal stresses and accompanying principal stress directions to correspond to the LV myocardial fiber stresses and orientations, respectively. By doing so, we can estimate the equivalent myocardial fiber angle. The computed principal stress magnitude is found to be in the same range as the typical isometric tension value, which lends credibility to our analysis. Step 7. From the LV wall stress state, we can also determine the equivalent torsion (T ) and compression (F ) applied at the apical end of the cylindrical model (which is taken to be fixed at the base of upper end) by the active contraction of the myocardium. 3.2. Data obtained for modeling Although this chapter presents a theoretic model for the mechanism of development of LV pressure, we have employed some relevant clinical data of a subject, in order to elucidate the model. The subject in this study was studied in a resting recumbent (baseline) state, after premedication with 100◦ C 500 mg of sodium pentobarbital by retrograde aortic catheterization.


172

D. N. Ghista et al.

The LV chamber pressure was measured by a pigtail catheter and Statham P23Eb pressure transducer; the pressure was recorded during ventriculography. Angiography was performed by injecting 30◦ C 36 ml of 75% sodium diatrizoate into the LV at 10–12 ml/s. These findings are used to justify the use of single-plane cine techniques, which allow for beat-to-beat analysis of the chamber dimensions. For our study, monoplane cineangiocardiograms were recorded in an RAO 30◦ projection from a 9-in. image intensifier using 35-mm film at 50 frame/s using INTEGRIS Allura 9 system at the National Heart Centre (NHC), Singapore. Automated LV analysis was carried out to calculate LV volume and myocardial wall thickness. The LV data, derived from the cineangiographic films consist of measured volume and myocardial thickness of the chamber as well as the corresponding pressure. All measurements are corrected for geometric distortion due to the respective recordings systems. 3.3. LV cylindrical biomechanical model (a) We model the LV as an incompressible thick-walled cylindrical shell, which is constrained in the long-axis direction at one end to represent the suspension of LV by the aorta at its base. Considering the LV at end diastole (ED) to be in the unloaded reference configuration, the cylindrical model in its undeformed state is represented geometrically in terms of cylindrical coordinates (R, Θ, Z) by29 Ri ≤ R ≤ Ro ,

0 ≤ Θ ≤ 2π,

0 ≤ Z ≤ L,

(1)

where Ri , Ro and L denote the inner and outer radii, and the length of the undeformed LV model, respectively. The LV model in its deformed state can then be defined in terms of cylindrical coordinates (r, θ, z) as ri ≤ r ≤ ro ,

0 ≤ θ ≤ 2π,

0 ≤ z ≤ l,

(2)

where ri , ro , and l denote the inner and outer radii, and the length of the deformed cylindrical model, respectively. (b) We further consider the incompressible LV model in its reference state to be subjected to twisting, radial, and axial deformations in the radial and long-axis directions during isovolumic contraction, such that (i) based on incompressibility criterion of π(R2 − Ri2 )L = π(r2 − ri2 )λz l, where R2 − Ri2 r= + ri2 . λz



(ii)

173

Φ , z = λz Z, (3) L where λz is an axial stretch per unit unloaded length and Φ is the angle of twist measured at the apex (relative to the base). θ =Θ+Z

(c) For the myocardium material, we assume a Fung-type exponential strain energy function of the form10 : Ψ = C(exp(Q) − 1)/2

(4)

where Q characterizes the material’s transverse isotropy in the cylindrical polar coordinate system, and is given by11 2 2 2 Q = b1 Eθθ + b2 EZZ + b3 ERR + 2b4 Eθθ EZZ + 2b5 ERR EZZ 2 2 2 + 2b6 Eθθ ERR + 2b7 EθZ + 2b8 ERZ + 2b9 EθR .

(5)

Herein, bi , (i = 1, 2, . . . , 9) are non-dimensional material parameters, and EIJ , (I, J = R, Θ, Z) are components of the modified Green–Lagrange strain tensor referred to cylindrical polar coordinates (R, Θ, Z). To reduce the mathematical complexity of the problem, we assume negligible transverse shear during isovolumic contraction. Thus ERZ and EθR in Eq. (5) and their corresponding stress components (i.e., σRZ and σθR ) are neglected in the subsequent analysis. (d) The stress equilibrium relation (in the cylindrical coordinate system) is (σrr − σθθ ) dσrr + = 0, dr r

(6)

where σrr and σθθ denote the radial and circumferential stresses, respectively. The other two equilibrium equations (in the cylindrical coordinate system) are satisfied trivially, because the transverse shear stresses (σRZ and σθR ) are treated as zero. We further consider the epicardial surface to be free of external pressure, while the endocardial is subjected to the intra-cavity blood pressure. Therefore, the boundary conditions on the outer and inner surfaces of the LV cylindrical model can be designated to be σrr (r = ro ) = 0;

σrr (r = ri ) = −pi ,

(7)

where pi denotes the internal or cavity blood pressure acting on the inner surface of the LV model.


174

D. N. Ghista et al.

By integrating Eq. (6), we obtain the Cauchy radial stress σrr as ro dr σrr (ξ) = (σrr − σθθ ) , ri ≤ ξ ≤ ro . r ξ

(8)

Substituting this into the boundary condition equation (7), the internal pressure pi can be denoted as ro dr (9) pi = − (σrr − σθθ ) . r ri Further, since the valves are closed during isovolumic contraction, we impose a set of boundary conditions at both the top and bottom of the internal LV surface, giving σzz π(ro2 − ri2 ) = p(πri2 ),

(10)

where σzz denotes the axial component of the Cauchy stresses. 3.4. LV geometry, pressure, volume data and deformation (a) The geometric parameters of the LV cylindrical model are defined in Fig. 1. The volumes of myocardial wall (MV) and of the LV (V ) are given as M V = π(Ro2 − Ri2 )L = π(2Ri + h)hL, V =

πRi2 L,

(11) (12)

wherein Ro is outer radius, Ri is inner radius, h is the wall thickness, and L is the length. Herein, the LV volume (V ), wall thickness (h), and myocardial volume (M V ) are obtained by cineventriculography. Using Eqs. (1) and (2), we can calculate the instantaneous radii Ri (t) and length L(t) (or any time instant t) in terms of measured MV, V , and h as 2V h/M V + (2V h/M V )2 + 4V h2 /M V , (13) Ri = 2 L = V /πRi2 . (14) Then, Ro = Ri + h. (b) We consider blood in the LV cavity to be slightly compressible, based on the value of its bulk modulus (K = 2.0 × 109 Pa). By doing so, we are allowing a small change in cavity volume as a result of the substantial pressure change during isovolumic contraction. The instantaneous change



175

in cavity volume (∆V ) can thus be expressed in terms of the instantaneous pressure change (∆P ) and the bulk modulus (K) of blood as ∆V /V = ∆p/K.

(15)

From Eq. (15), we can calculate the sequence ∆V , and hence V (= Vj − ∆Vj ). For the change ∆Vj in the LV volume, by assuming the ratios of our LV cylindrical-model length and radius changes (∆rj and ∆lj ) to be equal during isovolumic contraction, we obtain their expressions as (16) ∆lj = (1 − 3 1 − ∆p/K)L, ∆rij = (1 − 3 1 − ∆p/K)Ri . It is to be noted that this assumption may not be clinically valid; however, we are employing it in order to obtain the change in LV model length and radius. From Eq. (16), the ∆lj and ∆rij can be evaluated, and hence lj (= lj − ∆rij ) and rij (= rij − ∆rij ). So the wall thickness h can be obtained from 2 M V /lj + πrij − rij . (17) hj = π (c) Now, let ∆φ denote the relative angle of twist measured at the apex, at each of the three stages of isovolumic contraction phase, obtained by magnetic resonance imaging (MRI). At this stage as we are unable to simultaneously measure pressure and twist angle at the different stages of isovolumic contraction phase, we will adopt reasonable values of ∆φ from literature26,27 to be the twist angle. Table 1 shows a sample data including pressure, volume, and model parameters for the cylindrical LV model. The pressure was obtained by catheterization for different instants. The volume (V ), myocardial volume (MV ), and wall thickness (h) were measured by ventriculography, from the beginning of the isovolumic contraction (t = 0). The cylindrical model dimensions Ri , Ro , and L are calculated using Eqs. (13) and (14). Then the volume, radius, and length change ∆Vj , ∆rij , and ∆lj are calculated using Eqs. (15) and (16) for the instants (t = 0.02, 0.04, and 0.06); hence the instantaneous values of V, rij , and lj are obtained. The wall thickness for the cylindrical model is calculated using Eq. (17). The outer radius roj is then obtained (= rij + hj ).


176

D. N. Ghista et al.

3.5. Determination of LV wall strains, stresses, and constitutive properties (a) We can now determine the stretches in the different directions as follows28 : R r l ∂r r∂θ = = , (18) λz (R) = , λr (R) = , λθ (R) = L ∂R rλz R∂Θ R where l = L − ∆l and ri = R − ∆ri , noting that λz λr λθ = 1. On the other hand, the twist stretch due to torsion can be expressed as λφ (r) =

r∆Φ r∂θ = . ∂z l

(19)

(b) The components of the Green–Lagrange strain tensor (Eij ) can thus be expressed in terms of the stretches and deformations, obtained from Eqs. (13) and (14), as: 1 2 1 (λr − 1), Eθθ = (λ2θ − 1), 2 2 1 2 γλz λθ 2 . (20) Ezz = (λz (1 + γ ) − 1), Eθz = 2 2 By expressing the Cauchy stress tensor in terms of the strain energy density function (Ψ) described in Eq. (4), we get29 : Err =

∂Ψ ∂Ψ ∂Ψ + 2γλz λθ + γ 2 λ2z − p¯, ∂Eθθ ∂Eθz ∂Ezz ∂Ψ ∂Ψ ∂Ψ ∂Ψ = λ2r − p¯, σzz = λ2z − p¯, σθz = λz λθ + γλ2z , ∂Err ∂Ezz ∂Eθz ∂Ezz (21)

σθθ = λ2θ σrr

where p¯ denotes the hydrostatic pressure and Ψ is derived from Eqs. (4) and (5). (c) By employing these stress expressions, we are now in a position to satisfy the boundary conditions of the stress along the inner surface of the LV wall to be equal to the internal pressure in the LV. These boundary conditions are expressed by Eqs. (9) and (10). By integrating Eq. (9) along with the boundary conditions described in Eq. (10), we can determine the material parameters bi , (i = 1, 2, . . . , 9) and p¯, using a nonlinear least squares method. For the sample case shown in Table 1, the material parameters are determined, and shown in Table 2 (b1 = 5946.2278, b2 = 15690.58158, b3 = 422.514993, b4 = 16157.10454, b5 = 16360.53744, b6 = 33299.28998, b7 = 680.7385218, b8 = 0, b9 = 0).

0 0.02 0.04 0.06

0 0.02 0.04 0.06

p (mmHg)

∆p (mmHg)

V (ml)

∆V (ml)

rij (cm)

18 43 63 81

25 45 63

1.36700000E+02 1.36699773E+02 1.36699591E+02 1.36699427E+02

2.27263750E−04 4.09074750E−04 5.72704650E−04

2.03208400E+00(= Ri ) 2.032083992E+00 2.032083985E+00 2.032083979E+00

∆rij (cm) 8.46701669E−09 1.52406301E−08 2.13368822E−08 ∆Φ (◦ )

lj (cm)

∆lj (cm)

hj (cm)

roj (cm)

1.053745000E+01(= L) 1.053744996E+01 1.053744992E+01 1.053744989E+01

4.39060418E−08 7.90308757E−08 1.10643226E−07

1.085247000E+00 1.085246679E+00 1.085246684E+00 1.085246689E+00

3.117331000E+00(Ro ) 3.117330670E+00 3.117330669E+00 3.117330667E+00

Table 2.

0 0.667 1.333 2.00

The parameters of the strain energy function for the sample case shown in Table 1.

Parameter

b1

b2

b3

b4

b5

b6

b7

b8

b9

Value

5946.2278

15690.58158

422.514993

16157.10454

16360.53744

33299.28998

680.7385218

0

0

177


t (s)

Pressure–volume and model parameters for a sample subject with M V = 185 ml.


Table 1.


178

D. N. Ghista et al.

Table 3. The stretches calculated from Eqs. (21) and (22) for the sample case shown in Table 1. t

λz

λr

λθ

λφ

0 0.02 0.04 0.06

0.9999999958 0.9999999925 0.9999999895

1.000000008E+00 1.000000015E+00 1.000000021E+00

9.99999996E−01 9.99999993E−01 9.99999990E−01

1.28626948E−01 2.56482520E−01 3.85688000E−01

(d) We next determine (i) the stretches from Eqs. (21) and (22) shown in Table 3, and therefrom (ii) the strains Eij from Eq. (23). By substituting the computed strain components (Eij ) and the strain energy function (Ψ) into Eq. (24), we can then determine the LV wall stress components for the cylindrical LV model (as shown in Tables 4 and 5). 3.6. Determining the principal stresses We further consider that the active contraction force developed by the myocardial fibers can be approximated by an equivalent torsion ∆T and axial compression ∆F applied at the apical end, such that ro σzz r dr, (22) ∆F = 2π ri ro ∆T = 2π σθz r2 dr. (23) ri

The computed values of the axial compression and torque are depicted in Table 5. 3.7. Determining the equivalent axial compression ∆F and torque ∆T acting on the LV cylindrical model, induced by LV contraction Given the stress components σθz , σθθ , and σzz , we can compute the principal stresses σ1 and σ2 , and the principal angle φ as 2 σzz − σθθ σzz + σθθ 2 σ1,2 = ± + σθz , (24) 2 2 tan 2φ =

2σθz . σθθ − σzz

The computed principal stresses are also shown in Table 5.

(25)

0.02 0.04 0.06

Endocardium 0 −24.0809 −44.2917 −6.21E+01

Epicardium 1.99E−02 −23.4615 −43.1341 −6.05E+01

1.02E−01 −17.2801 −31.6543 −5.43E+01

Table 5.

t (s) 0 0.02 0.04 0.06

2.37E−01 −12.9839 −23.7482 −4.48E+01

4.08E−01 −8.73378 −15.9694 −3.39E+01

0.591715 −4.9888 −9.13029 −2.31E+01

0.762765 −2.11726 −3.88098 −1.34E+01

0.89833 −0.41173 −0.75569 −5.75E+00

0.98014 −0.2144 −0.3542 −1.13E+00

1 0 0 0

Results for the sample subject at different instants.

Circumferential stress (Pa)

Radial stress (Pa)

Axial force (N)

Torque (N m)

Axis stress (Pa)

Shear stress (Pa)

Principal stress tension (Pa)

Principal stress compression (Pa)

Principal direction (◦ )

0.00 3358.33 5318.61 4481.83

0.00 −2844.53 −5956.38 −8530.94

0.00 −13.51 −46.06 −102.38

0.00 0.38 0.89 1.30

0.00E+00 −1.35E+04 −4.70E+04 −1.04E+05

0.00E+00 3.29E+04 8.21E+04 1.22E+05

0.00 28221.68 63836.28 81673.18

0.00 −38395.38 −105547.96 −180897.35

37.82 36.18 33.02

179


t (s)

Radial stresses distributions along LV wall from endocardium to epicardium.


Table 4.


180

D. N. Ghista et al.

4. Results (a) Compression force ∆F and torque ∆T Using Tables 1–5, the variation of the induced equivalent active torque ∆T and axial compressive force ∆F during the isovolumic phase is calculated, and shown in Fig. 3. We hence inversely demonstrate that the big increment of internal pressure in LV cavity during the isovolumic phase is

0 Axial force -20

F = Σ∆Fi (N)

-40

-60

-80

-100

-120 0

0.01

0.02

0.03 Time (s)

0.04

1.4

0.05

0.06

Torque

1.2

T = Σ∆Ti (Nm)

1

0.8

0.6

0.4

0.2

0 0

0.01

0.02

0.03 Time (s)

0.04

0.05

0.06

Fig. 3. Variations of axial force and torque induced within in the LV as a function of time during isovolumic phase.



181

caused by the corresponding torque and axial force developed within the LV, due to contraction of the LV myocardial fibers. (b) LV wall stresses The variations of the LV wall radial and circumferential stresses (σrr , σθθ ), as well as axial and shear stresses (i.e. σzz and σθz ) are shown in Table 5 and Fig. 4. It can be seen that axial and shear stresses are much greater than the circumferential stress and the radial stress in the LV wall. 5

1.5

x 10

Axial stress (σ zz ) Shear stress (σ z ) θ

1

Stress (Pa)

0.5

0

-0.5

-1

-1.5

0

0.01

0.02

0.03 Time (s)

0.04

0.05

0.06

0.04

0.05

0.06

6000 4000 2000

Stress (Pa)

0 -2000 -4000 -6000 Radial stress (σ rr )

-8000

Circumferential st ress (σ ) θθ

-10000 0

Fig. 4.

0.01

0.02

0.03 Time (s)

Variations of stresses as a function of time during isovolumic phase.


182

D. N. Ghista et al. 0

-10

Radial stress (mmHg)

-20

-30

-40

-50

-60

-70

t=0.02 s t=0.04 s t=0.06 s 0

0.1

0.2

Endocardium

Fig. 5.

0.3

0.4 0.5 0.6 Wall thickness (%)

0.7

0.8

0.9

1

Epicardium

Radial stresses distributions along LV wall.

(c) Radial stress The radial stress distribution along the thickness of LV wall at different times is shown in Table 4 and Fig. 5. It can be seen that the radial stress distribution along the LV wall becomes linear with time, during the isovolumic phase. The radial stress has maximum magnitude at the endocardium (equal to the LV internal pressure) and is zero at the epicardium. The notable result from Fig. 6 is that both the principal stresses and their orientation angle keep changing during the isovolumic phase. At the end of the isovolumic phase, the magnitude of the compression principal stress is around 1.75×105 Pa, which is in good agreement with the isometric tension value of 1.40 × 105 Pa achieved under maximal activation.8 4.1. Equivalent myocardial fiber angle It is seen from Fig. 6 that the EMFO is 38◦ at the start of isovolumic contraction, and becomes 33◦ at the end of the isovolumic phase. In other words, the monitored internal pressure increase during isovolumic phase from 25 to 45 to 63 mmHg is attributed to the active contraction of the helically woven myocardial fibers from 38◦ to 33◦ . It is noteworthy that the computed EMFO is in the range of the value determined experimentally.5 The principal stresses and the corresponding angle along the LV wall thickness during isovolumic phase are shown in Fig. 7. The principal stresses



183

5

1

x 10

Principal stress (tension) Principal stress (compression)

0.5

Stress (Pa)

0

-0.5

-1

-1.5

-2

0

0.01

0.02

0.03 Time (s)

0.04

0.05

0.06

0

0.01

0.02

0.03 Time (s)

0.04

0.05

0.06

38 37.5 37

Principle angle (°)

36.5 36 35.5 35 34.5 34 33.5 33

Fig. 6. Variations of the principal stresses and the principal angle (of the principal compressive stress) as functions of time, during the isovolumic phase.

(tension and compression) do not vary much from endocardium to epicardium. However, the principal angle (Fig. 8) increases slightly from endocardium to epicardium during the isovolumic contraction phase. The relationships between the torque induced in the LV and the twist angle as well as between the principal stress angle and the twist angle are shown in Fig. 9. Table 5 provides a summary of all the computations.


184

D. N. Ghista et al.

9

x 10

4

Principle stress (tension) (Pa)

8

7

6

t=0.02 s t=0.04 s t=0.06 s

5

4

3

2

0 0.1 0.2 Endocardium x10

-0.2

0.3


0.7

0.8

0.9 1 Epicardium

5

Principle stress (compression) (Pa)

-0.4 t=0.02 s t=0.04 s t=0.06 s

-0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2

0 0.1 Endocardium

0.2

0.3


0.7

0.8

0.9

1

Epicardium

Fig. 7. Variation of the principal stresses along the wall thickness during the isovolumic contraction phase.

5. Conclusion This chapter is meant to theoretically demonstrate that the LV pressure build-up is due to contraction of the LV spirally wound myocardial fibers. It also enables us to provide a measure of the LV EMFO. In other words, what is implied is that if we can determine the LV deformation state, we can



185

38

37

Angle (°)

36 t=0.02 s t=0.04 s t=0.06 s

35

34

33

32

0 0.1 0.2 Endocardium

0.3


0.7

0.8

0.9

1

Epicardium

Fig. 8. Variations of the principal angle (for the principal compressive stress) along wall thickness during isovolumic phase.

determine the LV myocardial fiber orientation and stresses that cause this deformed state. From the wall stress state, we can determine the principal stresses and their orientations. The flow chart in Fig. 10 summarizes the calculation of the principal stresses and their orientation. We can therefore postulate that this EMFO of the LV can be employed as an intrinsic property of the LV, which governs its contractility of the LV. Taking into consideration that it is not possible to determine the LV wall fiber orientation, the determination of this equivalent fiber orientation can provide an important clue to why some persons are not able to effectively raise their LV pressure and are more prone to impaired LV contractility. We have shown that the incremental intra-LV pressures (∆pi ) and LV model deformations during isovolumic contraction are associated with and result in the development within the LV of incremental torsion and axial compression of the LV, and incremental principal stresses. The compressive principal stress corresponds to the active contractile stress generated in the myocardial fibers, while the angle of the compressive principal stress corresponds to the myocardial fiber’s equivalent helical angle, which is in agreement with the experimental data on the fiber angle. Conversely, we


186

D. N. Ghista et al. 3

2.5

T (Nm)

2

(A)

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotational angle (°)

42 40

Principal angle (°)

38 36

(B)

34 32 30 28 26 1.5

2

2.5

3

3.5

4

4.5

5

Rotational angle (°)

Fig. 9. Relation between (A) torque induced in the LV and rotational twist angle and (B) between the compressive principal stress angle and the rotational twist angle during the isovolumic phase.

have been able to analytically demonstrate how active twist and shortening, caused by the contraction of the helically wound and specifically oriented myocardial fibers, result in the substantial increase in LV pressure during isovolumic contraction.



187

Cineventriculography & catheter pressure

End diastolic volume, myocardial volume,wall thickness (h) & pressure during isovolumic contraction (Table 1)

Expresse stress using Eq. (24) Boundary condition using Eqs. (9 &10)

Calculate volume change using Eq. (18) (Table 1)

Calculate constitutive parameters using Eqs. (9 & 10) (Table 2)

Calculate deformation using Eqs. (19 & 20) (Table 1)

Calculate stress using Eq. (24) (Table 4)

Calculate stretch using Eqs. (21 & 22) (Table 3)

Calculate strain using Eq. (23)

Calculate principle stress using Eq. (27) (Table 5)

Calculate principle angle using Eq.(28) (Table 5)

Calculate equivalent torsion and axial compression force using Eqs. (25 & 26) (Table 5)

Postulate principal stress orientation to be the EMFO

Fig. 10. Flow chart for computation of principal stresses and the principal angle during the isovolumic contraction phase.

References 1. H. Sandler, and H. T. Dodge, Left ventricular tension and stress in a man, Circulation Research 13 (1963) 91–104. 2. I. Mirsky, Left ventricular stresses in the intact human heart, Biophysical Journal 9 (1969) 189. 3. R. R. Martin, and H. Haines, Application of Laplace’s law to mammalian hearts, Comparative Biochemistry and Physiology 34 (1970) 959–962. 4. I. Mirsky, Basic terminology and formulae for left ventricular wall stress, Cardiac Mechanics: Physiological, Clinical, and Mathematical Considerations, eds. I. Mirsky, D. N. Ghista, and H. Sandler (John Wiley & Sons, Inc., New York, 1974) 3–10. 5. T. F. Moriarty, The law of Laplace, its limitations as a relation for diastolic pressure, volume or wall stress of the left ventricle, Circulation Research 46 (1980) 321–331. 6. A. Y. K. Wong, and P. M. Rautaharju, Stress distribution within the left ventricular wall approximated as a thick ellipsoidal shell, American Heart Journal 75 (1968) 649–662. 7. I. Mirsky, Ventricular and arterial wall stresses based on large deformations analyses, Biophysical Journal 13 (1973) 1141–1159.


188

D. N. Ghista et al.

8. T. Arts, R. S. Reneman, and P. C. Veenstra, A model of the mechanics of the left ventricle, Annals of Biomedical Engineering 7 (1979) 299–318. 9. A. T¨ ozeren, Static analysis of the left ventricle, ASME Journal Biomechanical 105 (1983) 39–76. 10. J. M. Guccoine, A. D. McCulloch, and L. K. Waldman, Passive material properties of the intact ventricular myocardium determined from a cylindrical model, ASME Journal of Biomechanical Engineering 113 42–55. 11. H. Azhari, M. Buchalter, S. Sideman, E. Shapiro, and R. Beyar, A conical model to describe the non-uniformity of the left ventricular twisting motion, Annals of Biomedical Engineering 20 (1992) 149–165. 12. H. R. Chaudhry, B. Bukiet, and A. M. Davis, Stresses and strains in the left ventricular wall approximated as a thick conical shell using large deformation theory, Journal of Biological Systems 4, 3 (1996) 353–372. 13. P. Gould, D. Ghista, L. Brombolich, and I. Mirsky, In-vivo stresses in the human left ventricular wall: analysis accounting for the irregular 3-dimensional geometry and comparison with idealized geometry analyses, Journal of Biomechanics 5, 521–539. 14. D. D. McPherson, K. J. Skorton, S. Kodiyalam, L. Petree, M. P. Noel, R. Kieso, R. E. Kerber, S. M. Collins, and K. B. Chandran, Finite element analysis of myocardial diastolic function using three dimensional echocardiographic reconstructions: application of a new method for study of acute ischemia in dogs, Circulation Research 60 (1987) 647–682. 15. P. J. Hunter, A. D. McCulloch, P. M. F. Nielsen, and D. H. Smail, A finite element model of passive ventricular mechanics. Computational Methods in Bioengineering, eds. R. L. Spilker, and B. R. Simon, BED-9 (ASME, Chicago, 1988) 387–397. 16. A. D. McCulloch, and P. J. Hunter, Finite element modeling of left ventricular mechanics, Proceedings of the First World Congress of Biomechanics, La Jolla, California, 1990. 17. A. D. McCulloch, J. M. Guccione, J. M. Rogers, and P. J. Hunter, Threedimensional finite element analysis of stress and activation in the heart, Mechanics Computing in 1990’s and Beyond: Proceedings of the ASCE Engineering Mechanics Specialty Conference (ASCE, New York, 1991) 514–581. 18. J. M. Guccoine, and A. D. McCulloch, Finite element modeling of ventricular mechanics, Theory of Heart: Biomechanics, Biophysics and Nonlinear Dynamics of Cardiac Function, eds. L. Glass P. J. Hunter and A. D. McCulloch (Springer, New York, 1991) 121–144. 19. D. D. Streeter Jr, and D. L. Basset, An engineering analysis of myocardial fiber orientation in pig’s left ventricle in systole, Anatomical Record 155 503–511. 20. D. D. Streeter Jr, and W. T. Hanna, Engineering mechanics for successive states in canine left ventricular myocardium: I. Cavity and wall geometry, Circulation Research 33 (1973) 639–655. 21. D. D. Streeter Jr, and W. T. Hanna, Engineering mechanics for successive states in canine left ventricular myocardium: II, Fiber angle and sarcomere length, Circulation Research 33 (1973) 656–664.



189

22. F. C. P. Yin, R. K. Strumpf, P. H. Chew, and S. L. Zeger, Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading, Journal of Biomechanics 20 (1987) 577–589. 23. P. M. F. Nielsen, I. J. LeGrice, B. H. Smail, and P. J. Hunter, A mathematical model of the geometry and fibrous structure of the heart, American Journal of Physiology 260 (1991) H1365–H1378. 24. K. D. Costa, J. W. Holmes, and A. D. McCulloch, Modeling cardiac mechanical properties in three dimensions, Philosophical Transaction of the Royal Society of London 359 (June 2001) 1233–1250. 25. M. P. Nash, and P. J. Hunter, Heart mechanics using mathematical modeling, Second NZ Postgraduate Conference for Engineering and Technical Students, Auckland, NZ, Sep. 1995. 26. E. Nagel, M. Stuber, B. Burkhard, S. E. Fischer, M. B. Scheidegger, P. Boesiger, and O. M. Hess, Cardiac rotation and relaxation in patients with aortic valve stenosis, European Heart Journal 21 (2000) 582–589. 27. R. E. Henson, S. K. Song, J. S. Pastorek, J. J. H. Ackerman, and C. H. Lorenz, Left ventricular torsion is equal in mice and humans, American Journal of Physiology — Heart and Circulatory Physiology 278 (2000) 1117–1123. 28. G. A. Holzapfel, and T. Gasser, A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of Elasticity 61 (2000) 1–48.




LEFT VENTRICULAR FILLING PERFORMANCE CHARACTERISTICS DHANJOO N. GHISTA†,∗ , LIANG ZHONG‡ , RU-SAN TAN‡ and EDDIE Y. K. NG†,§ †School

of Mechanical and Aerospace Engineering, College of Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 ∗[email protected] § [email protected] ‡Department


1. Introduction The heart may be conceived as a pump that receives blood from a lowpressure system and raises it to a high-pressure system. By periodic stimulation of its muscles, it contracts and pumps blood throughout the body. In diastole, the ventricular wall must be compliant to allow rapid and complete filling at low filling pressure. Therefore, diastolic dysfunction of the left ventricle (LV) is defined as an impaired capacity to fill the LV by LV suction due to decreasing sarcomere contraction and primarily by left atrial (LA) contraction. The clinical importance of the diastolic function of the LV is associated with the cardiac pathophysiology of many forms,1−4 and the recognition that impairment of LV diastolic function is associated with diseases such as myocardial infarction and hypertrophy.5 This underscores the need for biomechanical characterization of LV diastolic function. The prime objective of this chapter is to develop an index to assess the filling functional performance of the LV. This index will be denoted by LVFPI, the LV filling performance index. 1.1. LV filling phase At the end of isovolumic relaxation, the LV enlarges its volume, while its pressure continues to decrease after transmitral flow has commenced. In other words, the model must account for LV suction. A simple explanation of this LV suction mechanism is the storage of elastic “strai” energy in the ventricular wall and surrounding structures during systole, which is recovered and manifested as elastic recoil during diastole.6,7 The LV pressure 191


192

D. N. Ghista et al.

profile of the ventricular suction phase has been experimentally demonstrated by Sabbah.6 After the occurrence of the “e” wave, the LA starts to contract, and this phenomenon is manifested as the “a” wave, as illustrated in Fig. 1. Thus the model should also incorporate these phenomena. The diastolic filling can be considered to comprise two temporal phases: (i) Phase I (tDV , measured in seconds), corresponds to ventricular passive filling; it comprises the interval from the opening of the mitral valve until the initiation of atrial systole (phase I). This filling is initiated by somewhat explosive LV wall recoil. (ii) Phase II (tDA ) corresponds to the LA contraction phase of diastole, as shown in Fig. 1. The total duration of diastole tD is given by tD = tDV + tDA .

(1)

Volume (ml)

The contour of an LV volume, obtained from cineventriculography, is shown schematically in Fig. 1. The flow rate is derived by differentiating

tDV

tDA

0

Flow (ml/s)

e wave

a wave

0 tDV

tDA

Fig. 1. Schematic of LV volume versus time and flow rate versus time, via cineventriculography. Phase I of diastolic filling occurs for 0 ≤ t ≤ tDV . Phase II encompasses interval tDV ≤ t ≤ tDA .


LV Filling Performance Characteristics

193

the volume with respect to time. Ventricular diastolic biomechanical model (incorporating inertial, resistive, and compliant properties) needs to account for both “e” and “a” waves, associated with LV suction and LA contraction.

2. Biomechanical Model From Appendix A, the associated governing differential equation of LV volume (V ) response (which includes inertial, filling-resistance and compliance terms) is given by V = F (t), (2) M V¨ + Re V˙ + C where M represents inertial term, Re represents resistance to filling, C represents myocardial wall compliance, and F (t) represents the driving term due to the atrial pressure generated by LA contraction. As mentioned earlier, the diastolic filling encompasses two phases. Phase I is passive diastolic filling or LV suction caused by LV recoil, for which F (t) = 0. The LV suction is caused by the decreasing active elastance (Ea ) of the LV, and maintained by the passive elastance (Ep ) of the LV. This phenomenon of LV suction caused by decreasing Ea and increasing Ep is described in the next section. The total filling mechanism can hence be represented by two phases: phase I, for which F (t) = 0; and phase II in which the active elastance is increasing and the passive elastance is decreasing. In phase II, the LA actively contracts and forces blood into the ventricle; this phase is represented by the time period tDA . For this process, we assume F (t) = F0 sin(ωt). The biomechanics model represents LV volume response to F (t), which can be viewed as the driving term due to the atrial pressure generated by atrial contraction (the “a” wave). Hence the model parameters F0 and ω characterize LA contraction. 2.1. Myocardial sarcomere model Based on the conventional Hill three-element model8 and Huxley crossbridge theory,9,10 we have developed a new myocardial sarcomere model (for the LV filling phase) involving the LV myocardial mass, series elastic element (SE), viscous elastic element (VE), and the contractile element (CE), as shown in Fig. 2. We now link the anatomical associations of these myocardial model elements with the microscopic structure of the heart muscle.


194

D. N. Ghista et al.

σCE

σ CE

k

σ

k

SE

m

CE B VE

m

CE B

σ SE

VE

myosin filament

σCE actin filament

Fig. 2. Sarcomere element contractile model, involving: the effective mass (m) of the myocardial tissue that is accelerated; elastic parameter k of the series element stress σSE (k is the elastic modulus of the sarcomere element); viscous damping parameter B of the stress σVE in the parallel viscous element, the generated contractile stress σCE between myosin (thick) and actin (thin) filaments.

In Fig. 2, the sarcomere represents the fundamental structural and functional unit of contraction. Herein, the CE corresponds to the actin–myosin filaments, as presented in Fig. 2. It is this unit that makes the myocardial muscle element contract, distort its shape, and perform work. The active elastance Ea arises from this sarcomere model. The connective tissue between the fibers of the myocardium and the sarcolemma constitute the SE and VE. In Fig. 2, m is the LV myocardial mass, B is the viscosity parameter, k is the elastic-modulus parameter, xT is the displacement of sarcomere relative to the centerline, x2 is the displacement of the LV sarcolemma, σCE is the stress generated by the CE, σV E is the stress by the VE, and σ is the resulting total active stress that is related to the chamber pressure of LV. Now, the LV active elastance Ea is expressed as11 zC z t t−d R Ea = Ea0 1 − exp − exp − , (3) τC τR where t represents time measured from start of isovolumic contraction; Ea0 represents the active elastance coefficient; τC is the time coefficient of the rate of elastance raise during LV contraction; τR is the time coefficient of



195

the rate of elastance fall during LV relaxation, and d is the time constant. The exponents ZC and ZR have been introduced to smoothen out the Ea curve during isovolumic contraction and relaxation. The active elastance Ea is caused by the contraction of the LV myocardial sarcomere. The rapid decrease of Ea during isovolumic relations causes LV suction, as explained by us earlier.12 Now, simultaneously with Ea , we also have passive elastance Ep , expressed as12 Ep = (dP/dV ) = Ep0 ezp V ,

(4)

where Ep0 is the passive elastance coefficient, zp is the passive elastance exponent, and V is the LV volume. This passive elastance represents how LV pressure develops as its volume increases. Now, as Ea decreases and LV volume increases (due to Ep ), the LV pressure increases due to the passive elastance Ep . Hence in Fig. 1, this period tDV comprises the LV suction phase. The period tDA is due to atrial contraction.

3. Determination of Solution to the Governing Eq. (2), and Formation of LV Filling Performance Characteristics 3.1. Phase I: early diastolic filling (refer to Appendix B, Eq. (B-8)) During this phase, there is no atrial contraction. However, the decreasing Ea and increasing Ep are both characterized by the LV compliance term (C). We adopt an over-damped solution for Eq. (2) as −αt α sinh(βt) + cosh(βt) , (5) V (t) = V0 e β at the start of the filling phase, where V0 is initial volume of LV, α = Re /2M , and β = α2 − 1/(CM ). When t = 0, V = V0 . 3.2. Phase II: late diastolic filling (refer to Appendix B) The solution to Eq. (2) (due to LA contraction) is given by V (ta ) = C3 e−(α−β)ta + C4 e−(α+β)ta + A cos(ωta ) + B sin(ωta ),

(6)


196

D. N. Ghista et al.

where A = F0

−ωR , ω 2 )2 + ω 2 R2

M 2 (ω02 − 1/2

ω0 = (1/CM )

B = F0

M (ω02 − ω 2 ) , M 2 (ω02 − ω 2 )2 + ω 2 R2

,

(α + β)[V (tDV ) − A] + V˙ (tDV ) − Bω , 2β (β − α)[V (tDV ) − A] − V˙ (tDV ) + Bω . C4 = 2β

C3 =

and

3.3. Simulating LV volume data by the solution of our governing Eq. (2) By simulating the LV volume during filling phase with our V (t) solution expressions (Eqs. (5) and (6)), Vactu = V (t) + (Ves − V0 ),

(7)

where Vactu is actual volume of LV and Ves is end-systolic volume, we can evaluate the parameters V0 , α, and β as listed in Table 1. The modelcomputed volume matches very well with the data measured (Fig. 3). Now we will proceed to solve for the remaining parameters: F0 and ω. During phase II, the solution of V (t) must satisfy the following two Table 1.

Parameters related to case as shown in Fig. 1 during diastolic filling.

Parameters

Values

Fitting

Remarks

Volume fit during phase I V0 (ml)

−57.93 ± 2.33

α(s−1 ) β(s−1 ) M (mmHg/(ml/s2 )) Re (mmHg/(ml/s)) C (ml/mmHg) F0 (mmHg) ω(s−1 )

11.11 ± 0.92 0 3.66e−6 0.81e−4 2212.73 0.36 44.13

RMSE R-square

1.646 0.99

This case is classified as the critical mode (because damping β = 0).



197

(a)

e wave

a wave

(b)

Fig. 3.

Volume and flow rate for case study as shown.

boundary conditions: Vactu (t = tDA ) = Ved ,

V˙ actu (t = tDA ) = 0,

(8)

where Ved is end-diastolic volume. We can hence determine the values of F0 and ω, as listed in Table 1. Hence all model parameters (V0 , α, β, F0 and ω) can be evaluated. Also, M equals the mean value of [ρh/(4πRe2 )] during phase I. Therefore, Re and C can also be determined from the following equations: Re = 2αM,

C = 1/[(α2 − β 2 )M ],

for over-damped case.

(9)

Hence, after determining the parameters α, β, and M (associated with phase I), we match the phase II solution (given by Eq. (9)) to the LV volume data, and evaluate the parameters F0 and ω, which characterizes LA contraction.

4. Results: Model Parameters and Interpretation, and Performance Characteristics We apply the above procedure to the LV data shown in Fig. 1, and present the results in Fig. 3a and Table 1. It is noted that from the volume–time


198

D. N. Ghista et al.

profile, we compute the flow-rate–time curve in Fig. 3b, in which we can identify the “e” and “a” waves. Let us now interpret these parameters physiologically. High M implies enlarged LV; high R represents mitral stenosis; low C represents a stiff LV myocardium; high F0 means a well-contracting LA; and low α means normal damping due to normal LV myocardial elastance.

Appendix A: Derivation of Ventricular Diastolic Phase Biomechanical Model Dynamic equilibrium of the LV myocardial element gives me u¨ + Pel Ae − PLV Ae = 0

or ms u¨ + Pel − PLV = 0,

(A1)

where the myocardial element mass, me = ms Ae , ms (the myocardial surface density or mass per unit surface area) = ρh, ρ, is the myocardial density, and u is the radial displacement (as depicted in Fig. 4). Now since, ms =

me (element mass) = ρh, Ae (element surface area)

(A2)

∆V = 4πR2 ∆u = V˙ ∆t, ¨, ∴ V˙ = 4πR2 u˙ and V¨ = 4πR2 u

(A3)

LA LAP(t) d1 P1 d2

LV LVP(t)=P3 Volume V

Ae (=dxdy) is the surface area of this element

dPLV

(incremental LV pressure) dx

dσ dσ

(incremental wall stress, contributory to elastic Compliance (C) recoil pressure dPel)

dy

h u (displacement)

Fig. 4. Dynamic equilibrium of a myocardial element. Element mass me = ρdxdyh = (ρh)dxdy = ρhAe = ms Ae ; σ = Pel R/(2h), where Pel is elastic recoil pressure. For ¨ = 0 or PLV − 2σh/R + dynamic equilibrium of the myocardial element, PLV − Pel + me u ¨ = 0. me u



we can write: ms u¨ =

(ρh)V¨ = 4πR2

ρ hV¨ = ρs hV¨ = M V¨ , 4πR2

199

(A4)

where ρs is the surface density, M = ρs h, and V is the LV volume. Now, refer to Fig. 4, PLA − P2 = P1 − P2 = Re V˙

(A5)

and ρf v 2 , (A6) 2 where Re is the resistance to LV filling (through the open mitral valve), ρf is the blood density, and v is the blood velocity at site 2. Hence, P2 = PLV −

ρf v 2 . PLV = PLA − Re V˙ + 2

(A7)

Then, V , (A8) C where Pel,0 is the elastic recoil pressure at the start of filling phase. Hence, from Eqs. (A1), (A4), and (A8), we have Pel (elastic recoil pressure) = Pel,0 +

M V¨ +

ρf v 2 V = PLA − Re V˙ + − Pel,0 C 2

(A9)

or V ρf v 2 = PLA + − Pel,0 , M V¨ + Re V˙ + C 2 so that we can write: V = F (t). M V¨ + Re V˙ + C

(A10)

(A11)

Appendix B: Solution of the Governing Equation (Eq. (2) or (A11)) Based on Appendix A, the associated governing differential equation (which includes inertial, filling-resistance, and compliance terms) is given as V = F (t). M V¨ + Re V˙ + C Rewriting Eq. (B1), we have V F (t) Re ˙ V + = V¨ + M MC M

(B1)


200

D. N. Ghista et al.

or F (t) V¨ + 2αV˙ + ω02 V = , (B2) M where α = Re /2M, ω0 = 1/M C is the natural frequency of the system, α is the attenuation or damping constant of the system, and β = α2 − ω02 is the angular frequency of damped oscillation of the system. B.1. Phase I: early diastolic filling (corresponding to LV suction) Equation (B2) becomes V¨ + 2αV˙ + ω02 V = 0.

(B3)

The relative sizes of the parameters define three regimes of motion, which generate a family of solutions to Eq. (B3). In keeping with the nature of LV volume–time data, we adopt an overdamped solution for LV volume response. The definition of an over-damped volume response is that Re2 > 4M/C, so that β is real. Therefore, the solution of Eq. (B3) is a product of two real exponentials. If Re2 = 4M/C, the solution is referred as critically damped. If Re2 < 4M/C, the solution is referred as an under-damped. Here, as indicated earlier, we adopt the over-damped solution, as it resembles the LV volume represented by the tDV time interval in Fig. 1. This solution is given by V = C1 e−(α−β)t + C2 e−(α+β)t .

(B4)

For this phase of diastole, the initial conditions that we shall impose are dictated by V (0) = V0 = C1 + C2 , V˙ (0) = −(α − β)C1 − (α + β)C2 = 0.

(B5) (B6)

Solving Eqs. (B5) and (B6) for C1 and C2 we get C1 = V0

(α + β) 2β

and C2 = V0

(β − α) . 2β

Hence, the complete expression for Eq. (B4) is given by α V (t) = V0 e−αt sinh(βt) + cosh(βt) . β

(B7)

(B8)



201

B.2. Phase II: late diastolic filling For mathematical convenience, we make a shift in the time variable and redefine it as ta = t − tDV . This changes the limits of our temporal region of interest to (as per Eq. (1)) 0 ≤ ta ≤ tD − tDV .

(B9)

The governing equation (B2) can then be written (by adopting F (t) = F0 sin ωt) as V¨ + 2αV˙ + ω02 V = F0 sin(ωta )/M.

(B10)

Note that M, Re , and C may have different values from their values during phase I of diastole; ω is estimated by the duration of atrial filling. The general solution to Eq. (B10) is the sum of the homogeneous and particular solutions; that is, Vg (ta ) = Vh (ta ) + Vp (ta ).

(B11)

For over-damped case, we obtain from Eq. (B11) Vh (ta ) = C3 e−(α−β)ta + C4 e−(α+β)ta

(B12)

Vp (ta ) = A cos(ωta ) + B sin(ωta ),

(B13)

and

where A = F0

−ωR , 2 2 M (ω0 − ω 2 )2 + ω 2 R2

B = F0

M (ω02 − ω 2 ) , M 2 (ω02 − ω 2 )2 + ω 2 R2

ω0 = (1/CM )1/2 .

(B14)

As the LV volume between phases I and II is continuous, this determines the initial condition for phase II. Hence, Vh (0) + Vp (0) = C3 + C4 + A = V (t = tDV )

(B15)

V˙ h (0) + V˙ p (0) = −(α − β)C3 − (α + β)C4 + Bω = V˙ (tDV ).

(B16)

and

Solving Eqs. (B12) and (B13), we have C3 =

(α + β)[V (tDV ) − A] + V˙ (tDV ) − Bω , 2β

(B17)

C4 =

(β + α)[V (tDV ) − A] − V˙ (tDV ) + Bω . 2β

(B18)


202

D. N. Ghista et al.

Hence, the complete expression for Eq. (B11) is given by Vg (ta ) = C3 e−(α−β)ta + C4 e−(α+β)ta + A cos(ωta ) + B sin(ωta ).

(B19)

References 1. I. Mirsky, Assessment of passive elastic stiffness of cardiac muscle: mathematical concepts, physiologic and clinical considerations, directions of future research, Progress in Cardiovascular Diseases XVIII, 4, (1976) 277–308. 2. J. C. Gilbert, and S. A. Glantz, Determinants of left ventricular filling and the diastolic pressure–volume relation, Circulation Research 64, 5, (1989) 827–852. 3. W. C. Little, and T. R. Downes, Clinical evaluation of left ventricular diastolic performance, Progress in Cardiovascular Disease 2 (1990) 273–290. 4. E. L. Yellin, S. Nikolic, and R. W. M. Frater, Left ventricular filling dynamics and diastolic function, Progress in Cardiovascular Disease VXXXII, 4, (1990) 247–271. 5. A. H. Dougherty, Congestive heart failure with normal systolic function, American Journal of Cardiology 54 (1984) 778–788. 6. H. N. Sabbah, and P. D. Stein, Negative diastolic pressure in the intact canine right ventricle: evidence of diastolic suction, Circulation Research 49 (1981) 108–113. 7. F. R. Robinson, S. M. Factor, and E. H. Sonnenblick, The heart as a suction pump, Scientific American 254, 6, (1986) 84–91. 8. A. V. Hill, The heat of shortening and dynamic constants of muscle, Proceedings of the Royal Society of London Series B 126, (1938) 136–195. 9. A. F. Huxley, Muscle structure and theories of contraction, Progress in Biophysics 7 (1957) 255–318. 10. A. F. Huxley, Muscle contraction, Journal of Physiology 243 (1974) 1–43. 11. L. Zhong, D. N. Ghista, E. Y. K. Ng, and S. T. Lim, Passive and active ventricular elastances of the left ventricle, Biomedical Engineering online 4 (2005) 10. 12. L. Zhong, D. N. Ghista, and E. Y. K. Ng, Explaining the left ventricular pressure dynamics in terms of ventricular passive and active elastances, Journal of Engineering in Medicine 220 (2006) 647–655.


NEW CLINICALLY RELEVANT LEFT VENTRICULAR CONTRACTILITY INDEX (BASED ON NORMALIZED WALL STRESS) DHANJOO N. GHISTA†,∗ , LIANG ZHONG‡ , RU-SAN TAN‡ , EDDIE Y. K. NG†,§ , and LEOK POH CHUA† †School

of Mechanical and Aerospace Engineering, College of Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 ∗[email protected] §[email protected] ‡Department


1. Introduction The formulation of left ventricular (LV) contractility index has been an enigma ever since cardiology has matured to characterize an impaired LV. Multiple studies have been undertaken to find indices of ventricular contractility that are independent of other major factors influencing the performance of the LV.1–7 Most studies have indicated the peak of the first time derivative of the ventricular pressure dP/dtmax to be the most sensitive cardiac index of inotropic changes.4 The maximum value of dP/dt is reached just before aortic valve opening, after the myocardial fibers have started to shorten and developed a high actin–myosin contraction force. However, the intraventricular LV pressure is obtainable only by cardiac catheterization. In keeping with the tradition of LV elastance property,8–11 a more intrinsic measure of contractility has been proposed by us in the form of active elastance index Ea,max , due to activation of the LV myocardial sarcomere. The active elastance Ea,max can be obtained from LV pressure–volume data. Despite this rationale of Ea,max as an intrinsic property of the LV, its use can be hampered by the need to measure pressure in clinical settings. Additionally, the calculation of Ea,max is somewhat complex and time-consuming. We have also developed a LV contractility model in terms of its sarcomere dynamics of contractile element stress and shortening velocity.7 The traditional LV contractility index dP/dtmax is based on LV pressure, which is an outcome of LV myocardial stress developed by myocardial sarcomere contraction. Hence, we deem it be more appropriate to employ LV 203


204

D. N. Ghista et al.

wall stress, normalized with respect to LV pressure, to formulate a new LV contractility index. Our new contractility index is based on the maximum rate of development of LV wall stress (σ) with respect to LV pressure (P ). Herein, we will define LV contractility to represent the capacity of the LV to develop necessary and sufficient intra-myocardial stress (with respect to LV pressure) to eject blood volume as rapidly as possible. Based on this definition and in keeping with the traditional LV contractility index of dP/dtmax , we have formulated another LV contractility index in terms of the maximum value of d(σ/P )/dtmax . The advantage of this index over Ea,max as well as over the more rigorous LV contractility model involving sarcomere dynamics7 is that its computation is more simple and does not require LV pressure data. 2. Analysis for LV Contractility Index Based on LV Wall Stress 2.1. LV passive and active elastances Earlier, we have defined LV passive elastance as Ep = (dP/dV) = Ep0 ezp V ,

(1)

wherein Ep0 is the passive elastance coefficient, zp is the passive elastance exponent, and V is the LV volume and LV active elastance as Z Zc t t−d R exp− , (2) Ea = Ea0 1 − exp − τC τR wherein t represents time measured from start of isovolumic contraction, Ea0 represents the active elastance coefficient, τC is the time coefficient of the rate of elastance rise during LV contraction, τR is the time coefficient of the rate of elastance fall during LV relaxation, and d is the time constant. The exponents ZC and ZR have been introduced to smoothen out the Ea curve during isovolumic contraction and relaxation. Ep denotes the LV pressure response to LV volume changes (such as during LV filling and ejection). On the other hand, Ea denotes LV elastance (= dP/dV ), produced by LV myocardial sarcomere. The determination of the parameters of Ep and Ea from clinical data is explained in our earlier work.7 Hence, we can regard Ea as a contractility index. However, the determination of Ea requires LV pressure–volume data, obtained from cineventriculography, and requires LV pressure measurement by cardiac catheterization, which we would want to avoid. This has led us to the development of another index of LV contractility, which is based on


New Clinically Relevant Left Ventricular Contractility Index

205

normalized LV wall stress and hence obviates the need for cardiac catheterization. This new index is now developed in the following sections. 2.2. Determination of LV wall stress We approximate the LV as a thick-walled spherical shell made of incompressible, homogenous, isotropic, elastic material. The assumption of a spherical shape has been made to reduce the mathematical complexity and make it convenient for clinical usage. The inner and outer radii of the shell are denoted by ri and re , respectively. The epicardial surface is considered to be free of applied pressure, and the endocardial surface is subjected to blood pressure P (t), where t is time. We can put down for wall stress (σθ ) in the circumferential direction, 3 ri ri3 re3 ri3 + 1 + r3 r3 2r3 re3 r3 2r3 = P e , σθ (r) = P 3 i 3 1 + e3 = P (3) 3 re − ri 2r ri ri3 1− 3 1− 3 re re where P is ventricular pressure (the relevant analysis is provided in Appendix A). The wall stress in the endocardium is maximum, and given by 3 3 ri /re + 1/2 . (4) σθ (ri ) = P 1 − ri3 /re3 We want to express σθ (ri ) in terms of non-invasively measurable ventricular myocardial volume (MV ) and chamber volume (V ), given by M V = 4π(re3 − ri3 )/3,

V = 4πri3 /3,

(5)

and measured by cineangiography. From Eqs. (4) and (5), the wall stress in the endocardium can be expressed in terms of MV and V as: (V /(M V + V ) + 1/2) σθ (r = ri ) = P 1 − V /(M V + V ) 3V + M V 3V 1 + =P =P . (6) 2M V 2M V 2 Now, because we want to formulate a non-invasively determinable contractility index, we normalized the wall stress with respect to LV pressure P , as 3V 1 + . (7) σθ /P = 2M V 2


206

D. N. Ghista et al.

2.3. Defining a new LV contractility index Of importance in assessing cardiac function is an understanding and knowledge of the rate of development of stress within the wall of the LV during a cardiac cycle. The generated wall stress (GWS) is a measure of the effectiveness of the contractile machinery of the LV myocardium, for carrying out its ejection function. In keeping with the traditional contractility index of dP/dtmax , we can define a new non-invasively determinable contractility index (based on Eq. (7)) as d(σθ∗ ) d(σθ /P ) 3 dV = = . (8) CONT = dt max dt max 2M V dt max Moreover, because we want an entirely non-dimensional index, we define d(σθ∗ ) LVCT2 = × ejection period dt max 3 dV = 2M V dt

× ejection period.

(9)

max

3. Clinical Application 3.1. LV pressure and volume measurement (obtained by cineventriculography) The subjects in this study were studied in a resting recumbent state, after premedication with 100–500 mg of sodium pentobarbital by retrograde aortic catheterization. LV chamber pressure was measured by a pigtail catheter and Statham P23Eb pressure transducer; the pressure was recorded during ventriculography. Angiography was performed by injecting 30–36 ml of 75% sodium diatrizoate into the LV at 10–12 ml/s. It has been found by using biplane angiocardiograms that calculated orthogonal chamber diameters are nearly identical.12 These findings are used to justify the use of single-plane cine techniques, which allow for beat-to-beat analysis of the chamber dimensions. For this study, monoplane cineangiocardiograms were recorded in an RAO 30◦ projection from a 9-in. image intensifier using 35-mm film at 50 frame/s using INTEGRIS Allura 9 system at the National Heart Centre (NHC), Singapore. Automated LV analysis was carried out to calculate LV



207

Fig. 1. An example of a patient’s measured LV pressure and volume during a cardiac cycle: t = 0–0.08 s is the isovolumic contraction phase, t = 0.08–0.32 s is the ejection phase, t = 0.32–0.40 s is the isovolumic relaxation phase, and t = 0.40–0.72 s is the filling phase.

volume and myocardial volume. The LV chamber pressure was obtained by LV catheterization, and with a pigtail catheter and Statham P23Eb pressure transducer. Typical LV pressure–volume patient data are shown in Fig. 1. 3.2. Results The above analysis is now applied to clinically obtained data consisting of the subject’s LV (instant-to-instant) dimensions (obtained by cineangiocardiography) and chamber pressure (obtained by cardiac catheterization). For each subject, passive and active elastances (Ep and Ea ) are determined from the LV data.7 Also, the new contractility index LVCT1 (Ea,max ) is determined. Table 1 provides the measured data and the model-derived parameters for three clinically representative subjects (subject HEL, DDM, and ML). Subject HEL serves as a sample patient with myocardial infarction, subject DDM with double-vessel disease (DVD) and hypertension, treated with PTCA; subject ML with idiopathic myocardial hypertrophy (IMH). Figure 2 shows the correlation between LVCT2 and the invasive measure of LV contractility dP/dtmax at an r-value of 0.9014, as LVCT2 = 0.0025 dP/dtmax − 1.8(r = 0.9014). Figure 3 shows the relationships between LVCT2 and LVCT1 (Ea,max ) (LV CT 2 = 0.39Ea,max + 0.3, r = 0.9393). Statistical analysis: To test the hypothesis that LVCT2 can be employed as noninvasive quantitative indices of contractility, the LCVT2 value was


208

D. N. Ghista et al. Table 1. Clinical history, measured hemodynamic data, and computed active elastance and contractility index LVCT2. Subject Disease LVP (mmHg) AOP (mmHg) EDV/ESV (ml) EF Ea,max (mmHg/ml) dP/dtmax (mmHg/s) LVCT2

HEL

DDM

ML

MI, DVD 122/18 125/75 132.5/84.3 0.36 1.37 1200 0.80

DVD, HTN 170/24 169/99 121.7/41.3 0.66 3.58 1475 2.18

IMH 109/12 115/70 368/284 0.23 0.57 1125 0.74

Clinical history, measured hemodynamic data and calculated active elastance Ea for subjects (HEL, DDM, and ML). Herein LVP: left ventricle chamber pressure, AOP: aortic pressure, EDV: end-diastolic volume, ESV: end-systolic volume, EF: ejection fraction, MI: myocardial infarction.

3

2.5

y = 0.0025*x- 1.8 r=0.9014

LVCT2

2

1.5

1

0.5

0

-0.5 700

800

900 1000 1100 1200 1300 1400 1500 1600 1700 1800 dP/dtmax (mmHg/s)

Fig. 2. Relating our contractility index LVCT2 to dP/dtmax factor, with r being the correlation coefficient.

compared with the values of LV traditional contractility indices dP/dtmax , ejection fraction (EF), and Ea,max by linear regression analysis, with the correlation coefficients. The MedCalc and Matlab 6.5 softwares were used for that purpose and for computing regression equations. It is seen that the new non-invasively determinable contractility index LVCT2 relates well to the traditional and invasively determinable contractility index of dP/dtmax as well as to LVCT1 (Ea,max ). The advantage of the



209

3 y = 0.39*x +0.3 r=0.9393

2.5

LVCT2

2

1.5

1

0.5

0

0

0.5

1

1.5

2 2.5 3 3.5 LVCI1, Ea,max (mmHg/ml)

4

4.5

5

Fig. 3. Relating our contractility index LVCT2 to Ea,max factor, with r being the correlation coefficient.

contractility index LVCT2 is that it enables very convenient computation, and can be obtained without the use of LV pressure data.

4. Discussion It may be said that any comparative analysis of contractility indices in the intact heart is somewhat arbitrary, due to lack of an ideal descriptor of the contractile state. Hitherto, LV dP/dtmax has been employed as a means of LV contractility, and has also been shown to be a relatively load-dependent index. It is however an extrinsic measure of LV contractility, because the LV pressure is developed by LV wall stress, caused by LV myocardial sarcomeric contraction. It is hence important to represent LV contractility by means of intrinsic property of the LV, which is independent of the preload and afterload. In that regard, Ea,max is an effective contractility index. However, its determination requires measurement of LV pressure. Our new index LVCT2 is an intrinsic property of the LV, as well as easily and non-invasively obtainable in terms of LV volume and myocardial volume, which can be determined from ventriculography.13 For the formulation of LVCT2, the LV is modeled as a pressurized thick-walled sphere; spherical geometry is assumed to reduce the mathematical complexity and for clinical application convenience. The high degree of correlation between


210

D. N. Ghista et al.

LVCT2 and dP/dtmax shows that LVCT2 is well capable of separating normal from reduced LV contractility.

Appendix A: Wall Stress in LV Spherical Thick-Wall Model A.1. Strain state Consider the LV as a spherical shell (under interpressure P ). If ur represents the radial displacement of a spherical surface of radius r, then ur + (∂ur /∂r)dr is the radial displacement of the adjacent surface of radius r + dr. Hence, the strain εr in the radial direction is: ∂ur dr − ur ur + ∂ur ∂r εr = = . (A1) dr ∂r The strain εφ in the circumferential direction is εφ =

ur 2π(r + ur ) − 2πr = = εθ . 2πr r

(A2)

The strains along the three axes are hence εr =

∂ur , ∂r

εθ =

ur , r

εφ =

ur . r

(A3)

A.2. Equilibrium state Consider a spherical wall element of thickness dr at distance r, subtending a small angle 2θ at the center (Fig. 4). Because of spherical symmetry σθ = σφ , hence for equilibrium in the radial direction (by ignoring the body force), we have −σr (2θr)(2θr) + (σr + dσr )(r + dr)2θ(r + dr)2θ dr dr −2 r + 2θdrσφ sin θ − 2 r + 2θdrσθ sin θ = 0. 2 2

(A4)

Putting σθ = σφ , dσr = (∂σr /∂r)dr, sin θ ≈ θ (θ ∼ = 0), and ∆r2 ≈ 0 (∆r ≈ 0), the above equation reduces in the limit to 2 1 d 2 (r σr ) = σφ . r2 dr r

(A5)



σφ φ (ε φ )

σθ

2θ

r (ε r )

σr

2θ r

σθ

θ (εθ ) (b)

211

σ r+dσ r σφ

(a)

r

dr

θ

ur [(r +dr)+(ur +dur)]-[r +ur]=d(r +ur)

u r + dur

(c) Fig. 4. (a) Strain in (r, θ, φ) coordinates. (b) Stresses acting on an LV spherical-model wall element. (c) Schematic of LV wall segment before and after deformations.

A.3. Constitutive equations Based on Hooke’s law, we can put down (from Eq. (A1)): εr =

1 ∂ur [σr − v(σθ + σφ )] = ; E ∂r

or

dur 1 = [σr − 2vσφ ], dr E

(A6)

where E is the elastic modulus, and v is the Poisson’s ratio. Likewise, from Eq. (A2) εφ =

1 ur [σφ − v(σθ − σr )] = E r

or

ur 1 = [(1 − v)σφ − vσr ]. r E

(A7)

A.4. Combining the equations to obtain a differential equation is σr Differentiating Eq. (A7) with respect to r, we get dur 1 ∂(rσr ) ∂(rσφ ) = −v (1 − v) . dr E ∂r ∂r

(A8)


212

D. N. Ghista et al.

For Eqs. (A6) and (A8), we get d(rσr ) d(rσφ ) −v − (σr − 2vσφ ) = 0. dr dr Upon substituting σφ from Eq. (A5), we get (1 − v)

d(rσr ) v d(r2 σr ) 1 − v d2 (r2 σr ) − σr + = 0. −v 2 2 dr dr r dr Now, upon putting r2 σr = y, Eq. (A10) becomes d2 y v dy vy y v dy 1 (1 − v) 2 − + 2 − 2+ = 0; 2 dr r dr r r r dr

or

(A9)

(A10)

d2 y y − 2 2 = 0. 2 dr r (A11)

A.5. Solution of the σr differential equation, and imposing the boundary conditions This is a homogeneous linear equation, with the solution: B y B y = Ar2 + , and σr = 2 = A + 3 , r r r where A and B are constants. From Eqs. (A5) and (A12), we get

(A12)

B . (A13) 2r3 For a spherical thick-walled shell with radii ri and re , subjected to internal pressure Pi , the boundary conditions are σr = −Pi when r = ri , and σr = 0 when r = re . Hence from Eq. (A12), σφ = A −

A+

B = −Pi ri3

A=

ri3 Pi re3 − ri3

B = 0. re3

(A14)

re3 ri3 Pi . re3 − ri3

(A15)

and A +

Hence, we get and B =

A.6. Final expression for σr and σθ (= σφ ) Then, from Eqs. (A12)–(A14), the expressions for radial and circumferential stress are ri3 re3 σr = Pi 3 1− 3 . (A16) re − ri3 r r3 r3 σφ = σθ = Pi 3 i 3 1 + e3 . (A17) re − ri 2r



213

References 1. D. T. Mason, E. Braunwald, J. W. Covell, E. H. Sonnenblick, and J. Ross, Assessment of cardiac contractility, Circulation 44 (1971) 47–58. 2. K. L. Peterson, D. Skloven, P. Ludbrook, J. B. Uther, and J. Ross, Comparison of isovolumic and ejection phase indices of myocardial performance in man, Circulation 49 (1974) 88–101. 3. C. R. Lambert, W. W. Nichols, and C. J. Pepine, Indices of ventricular contractile state: comparative sensitivity and specificity, American Heart Journal 106 (1983) 136–144. 4. D. A. Kass, W. L. Maughan, Z. M. Guo, A. Kono, K. Sunagawa, and K. Sagawa, Comparative influence of load versus inotropic states on indices of ventricular contractility: experimental and theoretical analysis based on pressure–volume relationships, Circulation 6 (1987) 1422–1436. 5. D. A. Kass, and R. Beyer, Evaluation of contractile state by maximal ventricular power divided by the square of end-diastolic volume, Circulation 84 (1993) 1698–1708. 6. L. Zhong, D. N. Ghista, E. Y.-K. Ng, and S. T. Lim, Passive and active ventricular elastances of the left ventricle, Biomedical Engineering Online 4, 10 (2005) 13. 7. L. Zhong, D. N. Ghista, E. Y.-K. Ng, S. T. Lim, and T. S.-J. Chua, Measures and indices for intrinsic characterization of cardiac dysfunction during filling & systolic ejection, Journal of Mechanics in Medicine and Biology 5, 2 (2005) 307–332. 8. H. Suga, Time course of left ventricular pressure–volume relationships under various end-diastolic volume, Japanese Heart Journal 10 (1969) 509–515. 9. H. Suga, K. Sagawa, and A. A. Shoukas, Load independence of the instantaneous pressure–volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio, Circulation Research 32 (1973) 314–322. 10. H. Suga, and K. Sagawa, Instantaneous pressure volume relationships and their ratio in the excised, supported canine left ventricle, Circulation Research 35 (1974) 117–126. 11. K. Sagawa, The ventricular pressure–volume diagram revisited, Circulation Research 43 (1978) 677–687. 12. H. Sandler, and H. T. Dodge, The use of single plane angiocardiograms for the calculation of left ventricle volume in man, American Heart Journal 75 (1968) 325–334. 13. C. Masquet, M. S. Slama, A. Dibie, F. H. Sheehan, and J. Lienard, Normal left ventricular volume and ejection fraction: assessment with quantitative digital cardioangiography, International Journal of Cardiac Imaging 14 (1998) 27–36.




CHARACTERIZATION OF CARDIAC DYSFUNCTION DURING SYSTOLIC EJECTION DHANJOO N. GHISTA†,∗ , LIANG ZHONG‡ , EDDIE Y. K. NG†,§ and RU-SAN TAN‡ of Mechanical and Aerospace Engineering, College of Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 ∗[email protected] §[email protected] †School

‡Department


1. Introduction 1.1. Dilemma “How to intrinsically assess the systolic function of the LV” has long been the major problem in physiology and cardiology, and this problem remains yet unsolved. The systolic function of the LV has been viewed at different levels.1 Way back in 1895, Frank characterized ventricular function in terms of the pressure-volume diagram. In 1918, Starling first viewed the LV as a pump to generate cardiac output (proportional to its filling) against an afterload. Hill2 and Huxley3,4 investigated muscle contraction by means of muscle force–velocity relationship at the micro-structural level (cross-bridge theory). Then, more recently, Suga5 proposed that the systolic function of the LV is a process of converting the cross-bridge chemical energy into the mechanical energy during contraction. In systole, the ventricular wall myocardium must generate adequate force to build sufficient pressure to open the aortic valve, and shorten (and thicken) the LV wall to pump an appropriate stroke volume. Therefore, systolic dysfunction is the inability of the myocardium to shorten against the aortic pressure load, characterizing LV disability to eject blood into a high-pressure aorta.

215


216

D. N. Ghista et al.

1.2. Scope In this chapter, we develop the governing equations of dynamics of the LV sarcomere, contained within wall of the LV cylindrical model. We then relate the sarcomere stress and displacement to the LV pressure and volume, in terms of the sarcomere elements parameters; and hence evaluate them. We next determine the sarcomere contractile-element (the actin-myosin unit) “stress versus shortening-velocity” characteristics as well as the power generated by the sarcomere (CE) element. There are termed to be important LV functional indices. Finally, we express LV pressure in terms of sarcomere displacement and hence LV volume, and demonstrate that we can thereby determine LV pressure noninvasively.

1.3. Data, model simulation and parameter determination The methodology delineated in this chapter comprises the following steps: 1. We monitor the patients’ LV pressure and volume, by LV catheterization and cineventriculographic analysis using the Pie Medical Imaging’s CAAS II QCA/LVA software.6 The subject in this study was studied in a resting recumbent (baseline) state, after premedication with 100–500 mg of sodium pentobarbital by retrograde aortic catheterization. The LV chamber pressure was measured by a pigtail catheter and Statham P23Eb pressure transducer; the pressure was recorded immediately during angiocardiography. Angiography was performed by injecting 30–36 ml of 75% sodium diatrizoate into the LV at 10–12 ml/s. Monoplane cineangiocardiograms were recorded in a RAO 30◦ projection from a 9 image intensifier using 35 mm film at 50 frames/s using INTEGRIS Allura 9 system. 2. We then carry out the analysis of volume response to generated LV wall stress during the systolic phase, by developing a model of myocardial sarcomere contraction for the systolic phase. 3. We next match the differential-equation’s model solutions with the clinical data, and evaluate the model parameters representing ejection phase. 4. Finally, we formulate and calculate indices for systolic-ejection phase in terms of the model parameters.


Characterization of Cardiac Dysfunction During Systolic Ejection

217

Fig. 1. Pressure and dimension during a cardiac cycle with thick-wall cylindrical model for a typical subject.

A typical LV clinical pressure (P ) versus time (t) and volume (V ) versus time (t) data are given in Fig. 1. Also depicted is volume rate versus time variation.

2. LV Contractility Index (LVCI) 2.1. Myocardial muscle model Based on the conventional Hill three-element model2 and Huxley crossbridge theory,3,4 we have developed a new myocardial model involving the LV myocardial mass, series-elastic element (SE), viscous-elastic element (VE) and the contractile-element (CE), as shown in Fig. 2. Therein, the sarcomere represents the fundamental structural and functional unit of contractions, which makes the muscle element contract and performs work. In this myocardial sarcomere model, the contractile element (CE) corresponds to the actin-myosin filaments, as indicated in Fig. 2. The connective tissue between the fibers of the myocardium and the sarcolemma constitute the SE and VE.


218

D. N. Ghista et al.

x2

xT

σCE

σ CE

k

σ

k

m

SE x1=x2-xT

CE B σ VE VE

m

CE B

σ VE

σ SE

VE

myosin filament

σCE actin filament

Fig. 2. Sarcomere element contractile model, involving: the effective mass (m) of the muscle tissue that is accelerated; elastic parameter k of the series element stress σSE (k is the elastic modulus of the sarcomere element); viscous damping parameter B of the stress σVE in the parallel viscous element, the generated contractile stress σCE between myosin (thick) and actin (thin) filaments.

In Fig. 2, m denotes the LV myocardial mass, B is the viscosity parameter, k is the elastic-modulus parameter, xT is the displacement of sarcomere relative to the center line, x2 is the displacement of the LV sarcolemma, σCE denotes the stress generated by the CE, σVE denotes the stress by the VE, and σ denotes the resulting total active stress which is related to the chamber pressure of LV. In this study, the LV is represented as a thick-walled cylinder contracting symmetrically. Transverse isotropy is assumed with respect to the axis of cylinder. The lumped myocardial fiber is depicted in the LV cross-section (Fig. 3). For the sake of convenient, it comprises of four sarcomere units, arranged circumferentially. The integration of these four sarcomere units causes LV myocardial contraction, as a result of which circumferential active force is generated per unit area of the myocardium. The resulting active circumferential stress is denoted by σ. 2.2. Myocardial sarcomere model dynamics and governing equation The organization of the sarcomere units within the LV cylindrical model is shown in Fig. 3. Each of these sarcomere units is further depicted in Fig. 4.



Ro

h

Ri

l/h

h P

σ =P

P0

P 2 2 ( Ro + Ri )

( Ro − Ri ) 2

2

T

t x2 en m ce pla σ VE B dis

en tx

σ CE

m

di sp la ce nt x 1 m

ceme di sp la

k

219

σ SE Unit cross-section 1/h

h

h

R

Contractile element

Series element Mass unit Viscous element

Fig. 3. Section (of unit cross-sectional area) of a thick-wall cylinder representing the LV. Herein, four sarcomere units are arranged circumferentially. Each of these sarcomere units is further depicted in Fig. 4. Their contraction integration causes the development of active circumferential stress within the LV wall, which in turn develops LV pressure. x1ed is the displacement of x1 at the end-diastolic phase.


220

D. N. Ghista et al.

Centre-line

x2

xT

σ CE myosin

σ SE σ

m

k x1= x2-x T

actin

σ VE B

Fig. 4. Typical sarcomere unit model: effective mass (m) of the myocardial sarcomere tissue is mass per unit area; k is the elastic modulus of series element, B the viscousdamping parameter of parallel viscous element, σ denotes the total generated stress caused by the contractile stress σCE , σSE the stress in the series element, and σVE is the stress in the viscous element.

From Fig. 4, the governing differential equation for sarcomere, due to the generated contractile stress, can be expressed as: m¨ x2 + B x˙ 2 − σCE + kx1 = 0 or

xT , m¨ x1 + B x˙ 1 + kx1 = σCE − B x˙ T − m¨

(1)

where σCE is the applied stress exerted by the contractile-element, m the muscle mass of per unit cross-section area = πRρm /2, B the viscous damping parameter of parallel viscous element, k the elastic stiffness (or modulus) of SE, x2 (the displacement of muscle mass m relative to centreline) = xT + x1 , and xT is the displacement of the muscle half-unit relative to its centre-line σVE = B x˙ 2

and σSE = k(x1 + x1ed ).

Because the term m¨ x1 and m¨ xT can be neglected due to their small values as compared to other terms, Eq. (1) can be rewritten as B x˙ 1 + kx1 = σCE − B x˙ T .

(2)

The systolic contraction includes two temporal phases. Phase I, denoted by tiso (and measured in seconds), corresponds to the duration of isovolumic contraction; it comprises the interval from the closing of the mitral valve



221

until the aortic valve open. Phase II, (te ), corresponds to the ejection phase of systole. The total duration of contraction ts is given by ts = tiso + te .

(3)

During the phase of systolic contraction, the stress of CE, σCE , can be considered as (analogous to the LV pressure wave shape) σCE = A3 sin(ω2 t)e−A4 t ,

(4)

where ω2 = π/te , A3 and A4 are coefficients.

2.2.1. Phase I: Isovolumic contraction phase (refer to Appendix A) The solution of Eq. (2) is given by x1 = xc1 = C1 e−kt/B + (a sin ω2 t + b cos ω2 t)e−A4 t

(5)

where a=

A3 (k − A4 B) , (k − A4 B)2 + (Bω2 )2

b=

BA3 ω2 , (k − A4 B)2 + (Bω2 )2

C1 = −b.

2.2.2. Phase II: Ejection phase (see Appendix A) The solution to Eq. (2) is given by x1 = xe1 = C2 e−kta /B + (a sin ω2 (ta + tiso ) + b cos ω2 (ta + tiso ))e−A4 (ta +tiso ) + (c sin ωta + d cos ωta )e−A2 ta where c=−

BA1 (k − BA2 ) , (k − BA2 )2 + (Bω)2

C2 = C1 e−ktiso /B − d.

d=

B 2 A1 ω , (k − BA2 )2 + (Bω)2

(6)


222

D. N. Ghista et al.

3. Determining the Parameters of the Governing Myocardial Sarcomere Dynamics Equation (2) The expression for the LV myocardial stress σ, generated (by sarcomere σCE ) is then given by σSE = σ = k(xe1 + x1ed ),

(7)

where xe1 is given by Eq. (6), and x1ed is end-diastolic displacement of SE. The resulting LV intra-cavity pressure P developed (by σ) can then be expressed as P =

(Ro2 − Ri2 )k(xe1 + x1ed ) σ(Ro2 − Ri2 ) = , 2 2 (Ro + Ri ) (Ro2 + Ri2 )

(8)

where Ro and Ri are the outside and inside radii of the LV cylindrical model. By matching the model generated P with the actual LV pressure data of a patient (Fig. 1) and carrying out parameter-identification accordingly, we can determine the parameters k, B, A3 , and A4 (as summarized in Table 1). Figure 5a suggests that the model-computed pressure match the actual pressure very well.

4. Power Generated by the Sarcomere Contractile-Element Now, we define the contractile-element power as Power = FCE × x˙ 2 ,

(9)

where (i) FCE is the contractile force per unit height or length of the LV cylindrical model, (ii) both FCE and x2 are functions of time, (iii) FCE , the Table 1. phase.

LV sarcomere model parameters, related to case shown in Fig. 1 during ejection

Parameters

Values

A1 (cm) A2 (s−1 ) A3 (g/s2 ) A4 (s−1 ) B (g/s) k (g/s2 )

5.726 ± 0.624 6.673 ± 0.708 1.184 ± 0.243 −1.002 ± 0.778 2.596 ± 0.598 3.954 ± 0.867

Fitting xT fit during ejection phase Pressure fit during ejection phase

RMSE R-square RMSE

0.0421 0.9902 2.2690

R-square

0.9605

The parameters A1 , A2 , A3 , A4 and B are defined in Eqs. (5) and (6); the parameter k is depicted in Fig. 4.



223

Fig. 5. Results for sample case as shown in Fig. 1, showing (a) pressure versus time; (b) displacement x1 , x2 , xT versus time; (c) shortening velocity of contractile element versus time; (d) contractile stress versus time; (e) contractile stress versus shortening velocity; (f) power generated by CE versus time.

contractile force generated by its contractile element, is given (based in Fig. 3) by FCE = σCE (h × 1/h) = σCE

(10)

and (iv) x˙ 2 is the shortening velocity of the CE, expressed by x˙ 1 + x˙ T .

5. Results of Left Ventricular Contractility Index (LVCI) Using the values of parameters as presented in Table 1, we have determined and plotted x1 , x2 , and xT versus time respectively in Fig. 5b. The velocity and CE stress are included in Fig. 5c and d. The power generated by CE is shown in Fig. 5f. The higher the power input, the better will be the contractility. Herein, in quantifying the performance of the LV, contractility is defined as the ability of the LV myocardium to produce a contractile force with a high shortening-velocity capability, so as to exert maximum contractile


224

D. N. Ghista et al.

power.7 Accordingly, we define LV contractility index (LVCI) as Powermax , which in the present sample case is 0.54 Nm/s.

6. Conclusion This chapter demonstrates the theory, methodology and computation of clinical-diagnostic measures represented by (i) LV contractile element “σCE versus x˙ 2 ”, and (ii) power generated. These indices can be viewed as important LV contractility indices. We have recently published an even more elaborate model of LV contractility in which the myocardial fibers are spirally oriented inside the wall of the LV cylindrical model.8

Appendix A: Solution of Governing Equation (2) A.1. Phase I: Isovolumic contraction (solution of Eq. (2)) Since both the mitral and aortic valves are closed, the volume of blood in the ventricle is constant and pressure inside the ventricle increases. So xT = x˙ T = x ¨T . Equation (2) now becomes B x˙ 1 + kx1 = A3 sin(ω2 t)e−A4 t .

(A1)

The solution of Eq. (A1) is given by x1 = xc1 = C1 e−kt/B + (a sin ω2 t + b cos ω2 t)e−A4 t ,

(A2)

where A3 (k − A4 B) , (k − A4 B)2 + (Bω2 )2 C1 = −b. a=

b=

BA3 ω2 , (k − A4 B)2 + (Bω2 )2

For this phase of contraction, the initial conditions that we shall impose are dictated by xc1 (0) = C1 + b = 0.

(A3)

Solving Eq. (A3), we have C1 = −b.

(A4)

Hence the solution for this phase is given by x1 = xc1 = −be−kt/B + (a sin ω2 t + b cos ω2 t)e−A4 t .

(A5)



225

A.2. Phase II: Ejection phase (solution of Eq. (2)) For mathematical convenience we make a shift in the time variable and redefine it as ta = t − tiso . This changes the limits of our temporal region of interest to 0 ≤ ta ≤ te − tiso .

(A6)

In this phase, xT is no longer zero, and hence we need to relate it to the LV volume. In this analysis, we assume that the LV is a thick-wall cylindrical chamber of radius R and wall thickness h. Now we can express xT in terms of LV model radius R, as (refer Fig. 2): xT =

π π ∆R = (Red − R(t)), 2 2

(A7)

where Red denotes LV radius at end-diastole, and R(t) represents the LV radius at time t during the systolic phase. In order to fit an expression to the [π(Red − R(t))/2] clinical data, we adopt x˙ T = A1 sin(ωta )e−A2 ta

(A8)

to correspond to the shape of the LV model-change radius (obtained from LV volume change curve) shown in Fig. 6. Upon integrating, we get xT =

A1 A1 ω (A2 sin(ωta ) + ω cos(ωta ))e−A2 ta + 2 . A22 + ω 2 A2 + ω 2

(A9)

Hence, the governing Eq. (2) now becomes B x˙ + kx1 = A3 sin(ω2 (ta + tiso ))e−A4 (ta +tiso ) − BA1 sin(ωta )eA2 ta . (A10) The solution of Eq. (A10) is given by x1 = xe1 = C2 e−kta /B + (a sin ω2 (ta + tiso ) + b cos ω2 (ta + tiso ))e−A4 (ta +tiso ) + (c sin ωta + d cos ωta )e−A2 ta ,

(A11)

where c=−

BA1 (k − BA2 ) , (k − BA2 )2 + (Bω)2

C2 = C1 e−ktiso /B − d.

d=

B 2 A1 ω , (k − BA2 )2 + (Bω)2


226

D. N. Ghista et al.

V (ml)

150 100 50 0 0

0.2

0.4

0.6

0.8

0.6

0.8

0.6

0.8

0.6

0.8

Time (second)

dV/dt (ml/s)

600 400 200 0 -200 0

0.2

0.4

-400 Time (second)

2.5

R (cm)

2 1.5 1 0.5 0 0

0.2

0.4 Time (second)

dR/dt (cm/s)

4 3 2 1 0 -1 0 -2

0.2

0.4

-3 Time (second)

Fig. 6. LV volume (V ) versus time, LV volume rate (dV /dt) versus time, LV radius (R) versus time and radius rate (dR/dt) versus time, for a typical subject shown in Fig. 1.

Because the xT changes between phases I and II are continuous, this determines the initial condition for phase II. Hence xe1 (0) = C2 + (a sin(ω2 tiso ) + b cos(ω2 tiso ))e−A4 tiso + d = C1 e−k/Btiso + (a sin(ω2 tiso ) + b cos(ω2 tiso ))e−A4 tiso .

(A12)



227

Solving Eq. (A12), we have C2 = C1 e−k/Btiso − d.

(A13)

Hence the solution for this phase is given by x1 = xe1 = (−be−ktiso /B − d)e−kta /B + (a sin ω2 (ta + tiso ) + b cos ω2 (ta + tiso ))e−A4 (ta +tiso ) + (c sin ωta + d cos ωta )e−A2 ta . (A14)

References 1. O. Frank, Zur Dynamik des Herzmuskels, Z. Biology 32 (1895) 370–447, Translated into English by C. B. Chapman and E. Wasserman. On the dynamics of cardiac muscle. American Heart Journal 58 (1959) 282–317 and 467–478. 2. A. V. Hill, The heat of shortening and dynamic constants of muscle, Proceedings of the Royal Society of London Series B 126 (1938) 136–195. 3. A. F. Huxley, Muscle structure and theories of contraction, Progress in Biophysics 7 (1957) 255–318. 4. A. F. Huxley, Muscle contraction, Journal of Physiology 243 (1974) 1–43. 5. Pie Medical Imaging BV http://www.piecaas.com, accessed 28 July 04, 2004. 6. H. Suga, Paul Dudley White International Lecture: Cardiac performance as viewed through the pressure-volume window, Japanese Heart Journal 35 (1993) 263–279. 7. L. Zhong, D. N. Ghista, E. Y.-K. Ng, S. T. Lim, and S. J. Chua, Dynamic of contraction of the left ventricle: modeling and application, Third Scientific Meeting of the Biomedical Engineering Society, Singapore, 2004, pp. 114–116. 8. D. N. Ghista, L. Zhong, L. P. Chua, E. Y.-K. Ng, S. T. Lim, R. S. Tan, and S. J. Chua, Systolic modeling of the left ventricle as a mechatronic system: determination of myocardial fibers sarcomere contractile characteristics and new performance indices, Molecular and Cellular Biomechanics 2 (2005) 217–233.




STRAIN ANALYSIS AND VISUALIZATION OF LV DEFORMATION DURING A CARDIAC CYCLE, AS AN INDEX OF CONTRACTILITY ∗School

JINAH PARK∗ and SANG IL PARK† of Engineering, Information and Communications University 119 Moonjee-ro Yusong-ku Daejeon 305-714, Korea [email protected]

†Robotics

Institute, Carnegie Mellon University, Pittsburgh, PA, USA

1. Introduction An effective presentation of collected data is as important as collecting the data to draw conclusions from the presented information. Especially when the data set is multi-dimensional, analyzing the data requires an effective visualization tool which can display a certain mode of information dynamically. Also, when it is not yet determined which mode gives rise to the critical piece of information, a tool that allows a user to test out a wide variety of information in a user-friendly manner becomes extremely valuable. For several decades many clinicians and researchers have strived to gain a better understanding of the left ventricular (LV) functional status related to heart diseases, because the LV of a heart plays a major role in the cardiovascular system and is the very place where most symptoms of heart diseases appear. Clinical observations have shown that the earliest indicator of an ischemic heart disease is alteration in myocardial wall motion. If it can be detected at all, many lives will be saved from, for example, myocardial infarction. With the advances in imaging technologies of magnetic resonance (MR) tagging,2,27 it has become possible to obtain suitable data sets for the extensive cardiac motion estimation.13,19 MR tagging is a non-invasive technique to locally perturb the magnetization in the tissue with the following characteristics: (1) Magnetization tags are seen as dark regions in subsequent images, within a certain relaxation time T1 . (2) As magnetization moves with tissue, the magnetization tags will move in the corresponding image sequence, directly reflecting the motion of the underlying tissue as shown in Fig. 1. Therefore, the technique allows us to directly image motion patterns within otherwise featureless structures such as the heart wall. 229


230

Jinah Park and Sang Il Park

Fig. 1. MRI-SPAMM images of an LV (short-axis view) during systole (courtesy of Dr. Leon Axel).

Unfortunately, since a heart undergoes complex motion during its cycle, there is no established way of describing its motion. Moreover, the variability among the healthy individuals needs to be accounted for to discriminate an “abnormal” motion from a “normal” motion. A conventional way of describing the LV motion is to show the local strain patterns. The strain tensor at each location throughout the myocardium has six components: three normal strains and three shear strains. Which components are useful in diagnosing a heart disease are still under study, and a dynamic and interactive visualization tool will greatly help in analyzing the extracted motion parameters. To study cardio-mechanics, it is necessary to construct a model representing the LV. Simple analytical shapes like spheres, ellipsoids, or cylinders have been employed in the past.1,4 However, these geometric models may limit the analyses because of the simplified assumptions about the shape and the governing equations of motion. With deformable three-dimensional (3D) models, one can certainly improve the shape representing an LV more closely at any phase during its cycle. Recently, with an increased popularity of deformable models, their associated computer vision or graphics techniques have been developed to capture the shape and motion of the LV from medical image data.3,5,8,14,15,17,25 Readers are referred to Ref. 7 for a comprehensive list of the techniques. Although these previous techniques have shown good results for qualitative visualization, they have not been suitable for further quantitative analyses for clinical diagnosis. Traditionally, a finite element method (FEM)10,11,25 has been the choice for constructing volumetric LV models among the researchers with clinical backgrounds, because it provides a means for an extensive strain and stress analysis.17 Although its representation does not directly lend itself to an intuitive understanding of the underlying kinematics and therefore requires a non-trivial post-processing, many reported clinical findings related to LV functions are based on the FEM.


Strain Analysis and Visualization of LV Deformation

231

Taking the physically-based approach,15 where flexible materials are incorporated into purely geometric models, we have previously developed the volumetric deformable models with parameter functions (or parameter-function models) and successfully applied them to characterize 3D LV motion in terms of a few physical parameters that offer sufficient accuracy.19,20 Although our previous work21,22 has demonstrated that the model parameter-functions capture and describe the deformation field in an intuitive and clinically useful manner, we also have wanted the ability to compute the local strain tensors since previous studies with finite elements are based on strain calculations. In this way not only we can easily compare our findings with previous research results, but also provide alternative ways to seek out meaningful parameters for a clinical use. We also address the problem of an effective presentation for thousands of local parameters derived by FEM. The six strain components at each location are reduced to three principal strains, and further down to one normalized strain — the Von Mises’ yield criterion. This new way of visualizing the LV strains allows the new clinical interpretation of a cardiac function. By employing a smoothing filter, we have been able to visualize the overall trend of each strain variation. Our visualization system is integrated in a PC environment with user-friendly interfaces including viewing the volume elements at each level along longitudinal, circumferential and transmural directions. This chapter is based on our original technical paper (Ref. 23) on strain analysis and visualization of the left ventricle. In Section 2, we present the underpinning for deformation and strain as a tutorial. Section 3 introduces the parameter-function model. Section 4 relates traditional strain components with parameter-function model parameters by providing interpretations for the deformation degree of freedom using a cylindrical model as an example. Finally, our visualization system is described along with the LV strain analysis results in Section 5.

2. Basic Concepts In this section, we briefly introduce the basic concepts on deformation gradient and strain of a continuous medium for those who are not familiar with mechanics. For a more comprehensive discussion, we refer the readers to a classical textbooks on continuum mechanics such as Ref. 9. Our presentation focuses on six strain components, principal strains and Von Mises strain for analyzing and visualizing the LV deformation.


232


2.1. Deformation and strain When physical forces act on a body, the body experiences deformation. Strain is a commonly used concept for describing the deformation of the body in continuum mechanics. One of the simplest cases of the deformation can be found in a uniaxial tension test where a specimen is stretched along one direction. In this case, the deformation along the loading axis is intuitively described as a change in length per unit of the initial length, which is a typical definition of strain. If a specimen of 10 cm length is stretched to 10.1 cm, for example, the strain is 0.01. As illustrated in Fig. 2, there are three simple cases of strain in two dimension (2D): two for extension and one for shear. These strains are defined as follows: 1 ∆ux ∆uy ∆ux ∆uy 1 π , εy = , and εxy = −ψ ≈ + . εx = ∆X ∆Y 2 2 2 ∆Y ∆X (1) The above definition will hold when the body is deformed uniformly over the space. However, deformations can occur differently from point to point on the body. Let us generalize the concept of strain. We have defined the strain in Eq. (1) as the change in length. Thus, we can also say that the strain is closely related to the change of the distance between any two points on the body. As shown in Fig. 3, on an object, there are two points P and Q in the undeformed (i.e., original) configuration, and they go to p and q, respectively, after deformation. The distances between two points are denoted as dX = Q − P and dx = q − p, respectively. Then, we can set up a transformation between those two displacements written as dx = F · dX,

∆ ux

∆u y ∆Y ∆X

ψ ∆ uy

∆ux

(a) Uniaxial extension in x direction Fig. 2.

(b) Uniaxial extension in y direction

(c) Pure shear without rotation

Three kinds of the strain in a 2D case.


Strain Analysis and Visualization of LV Deformation z

233

deformed configuration

undeformed configuration

q p Q

P y

x

Fig. 3.

The undeformed and deformed configurations.

where F is called deformation gradient that maps the undeformed PQ segment vector to its deformed state pq. Note that dX and dx are vectors and that F is a second-order tensor (i.e., matrix). The change of the displacement can be measured by comparing the length of the displacement in the undeformed configuration with that in the deformed configuration. The length dS of P Q in the original configuration is dS 2 =

3

dXi dXi ,

i=1

where dXi is the ith component of vector dX. Similarly, the length ds of pq in the new configuration is ds2 =

3

dxi dxi .

i=1

Note that in these and the following derivations we assume that both the coordinate systems for the undeformed and deformed state are the same rectangular Cartesian (rectilinear and orthogonal) coordinate system for the sake of simplicity in derivation. If we assume that the change of the configuration of the body is continuous (i.e., neighborhoods change to neighborhoods), we can describe the length as ! 3 3 " 3 ∂Xi ∂Xi 2 dxl dxm dS = ∂xl ∂xm m i=1 l=1


234

and


! 3 3 " 3 ∂xi ∂xi dXl dXm . ds2 = ∂Xl ∂Xm m i=1 l=1

Then, the difference between the squared lengths is ! 3 3 " 3 ∂xi ∂xi 2 2 ds − dS = − δlm dXl dXm , ∂Xl ∂Xm m i=1 l=1

where δlm is the Dirac-delta function which returns 1 when l = m and otherwise zero. We define the strain tensors 3 ∂xi ∂xi − δlm , elm = ∂Xl ∂Xm i=1 so that ds2 − dS 2 = 2elm dXl dXm , where elm is the (i, m) component of the strain tensor. This strain tensor E was introduced by Green and St. Venant, and is called the Green’s strain tensor. Now, we introduce the displacement vector u = x − X. Then, ∂xi ∂ui = + δil . ∂Xl ∂Xl Finally, by plugging this result into the definition of E, we obtain 3 1 ∂ul ∂um ∂ui ∂ui + + . elm = 2 ∂Xm ∂Xl ∂Xm ∂Xl i=1 If we apply this equation to the 2D case, we obtain ∂ux ∂uy 1 ∂ux ∂uy , εy = , and εxy = + εx = , ∂X ∂Y 2 ∂Y ∂X which becomes identical with Eq. (1) as expected if we take limit for dX and dY to zero. Here is an alternative definition of E based on the deformation gradient F , which contains a rigid-body rotation that does not contribute to the deformation. We first separate F into two tensors: one is related to the deformation or strain and the other to the rigid-body rotation. It is well



235

known when we divide F into a symmetric matrix and a skew-symmetric matrix, the rigid-body rotation is included in the skew-symmetric matrix. The symmetric matrix C called the Cauchy deformation tensor is obtained as follows: C = F T F. Based on C, the strain e is defined as e=

1 (C − I) . 2

2.2. Six components of strain In a 3D case, the strain tensor e can be represented with six components e11 , e22 , e33 , e12 , e23 , and e31 due to its symmetry (i.e., eij = eji ). Just as in the 2D case, the three components with the same indices, e11 , e22 , and e33 , are related to contraction or extension and the other three components are related to shearing of the body. Since we want to compute the strain tensor with respect to the physical circumferential and longitudinal coordinates of the LV, we convert the strain components in the Cartesian frame of reference into the strain components in the cylindrical coordinate according to the tensor transformation law9 , which results in er , eθ , ez , erθ , ezr , and eθz . The cylindrical coordinate system is set up to have its orgin at the apex of the LV model and its z-axis to be aligned with the long axis of the LV. Figure 4 shows the characteristics of six components of strain tensor in the cylindrical coordinate system. From the six strain components in the Cartesian coordinate system, the six components in the cylindrical coordinate system are obtained

z

z eθ ez

erθ erz

eθz

er

Fig. 4.

Six components of a cylindrical strain tensor.


236


as follows: er = e1 cos2 θ + 2e12 sin θ cos θ + ee sin2 θ, eθ = e1 sin2 θ − 2e12 sin θ cos θ + ee cos2 θ, ez = e3 , erθ = (e2 − e1 ) sin θ cos θ + e12 (cos2 θ − sin2 θ), eθz = e31 cos θ + e23 sin θ, ezr = −e31 sin θ + e23 cos θ. 2.3. Principal strains As described in the previous subsection, there are six components in a strain tensor, where each component represents its specific characteristics of deformation. To analyze the overall deformation easily, it is desirable to reduce the number of components. Due to the symmetry of the strain tensor, these six strain components are often reduced down to three mutually perpendicular strains without the shear strains. These strains are called the principal strains and the directions of them are called the principal directions — where the maximum and minimum principal directions have some meaningful information related to LV deformation.a The principal directions are found by solving the eigen vector problem. Stated another way, the principal strain does not change its value for the infinitesimal perturbations along its principal direction. Let ni be the ith principal direction. Then, the eigen vector problem is 3

eij nj − eni = 0 i = 1, 2, 3.

j=1

The resulting three eigen values are corresponding to the values of the principal strains, which are conventionally denoted as eI , eII , and eIII ordered algebraically from the largest to the smallest. 2.4. Von Mises strain We reduce the strain components further down to one scalar value so that we can visualize the overall effect of the deformation easily. The Von Mises strain (or stress) is one of the most widely used concepts for providing a The maximum (most positive-valued) and the minimum (most negative-valued) principal strains are oriented approximately in the radial direction and within the circumferential–longitudinal plane, respectively.



237

practical guidelines in estimating the mechanical safety of the structure in the field of computational mechanics. The Von Mises strain is originally introduced to predict failure of a given material based on the experimental data. The basic assumption of this kind of the yield (failure) criteria is that failure will occur when the value of a certain component of the strain or the stress reaches a critical value. The critical value is usually obtained by a uniaxial context. However, we are dealing with the strains in a multiaxial state, which complicates the prediction; shear strains contribute to the other normal strains. We can alleviate this complexity by using the principal strains making the shear strains zero. Almost every yield criterion is based on this principal theory. The simplest way of approximately predicting the failure is to assume that failure occurs when the absolute value of the maximum or minimum principal strain equals to or exceeds the uniaxial failure strain of the given material. Thus, max(|eI |, |eIII |) can be a single measure for failure in this case. On the other hand, failure can also occur when the absolute shear strain exceeds a certain critical shear strain according to the maximum shear theory. For the given strain in a multiaxial state, the maximum strain is represented as the maximum difference between the principal strains based on the coordinate transformation theory; that is (eI −eIII ). Integrating those two concepts directly leads to the well-known Tresca yield criterion. The Von Mises yield criterion is the improved version of the Tresca. The Von Mises yield criterion predicts failure based on the distortion energy. According to the distortion theory, the energy is dependent on the shear strains, which leads to the definition of the Von Mises strain εvm : εvm ≡ 2 [(eI − eII )2 + (eII − eIII )2 + (eIII − eI )2 ]. Equivalently, the Von Mises strain εvm is represented with the six Cartesian components of the strain as follows: 1 (e11 − e22 )2 + (e22 − e33 )2 + (e33 − e11 )2 εvm 2 = 3 + 2 (e12 )2 + (e23 )2 + (e31 )2 . The display of this value will give some initial guidelines as to which area of the LV is strained most considering all six components.23

3. Parameter-Function Model The physics-based deformable modeling framework16 is utilized for representing the shape and motion of the LV of a heart. In particular, the


238


parameter-function model20 captures the deformation of the LV during its cardiac cycle with its explicit modeling parameters that are intuitive for describing the non-rigid motion. It is a novel way to analyze the strain of the LV. This section introduces the physics-based deformable modeling framework, the parameter-function model description, and its utility in the quantification of the deformation of the LV. 3.1. Deformable model kinematics and dynamics The class of deformable models is based on the use of a small number of parameters that are functions. The fact that these parameters are functions and not fixed allows the localization and the detailed estimation of the LV shape and motion. These parameters are a convenient grouping of the many local parameters needed to estimate the LV shape and motion. We first define an inertial frame of reference Φ and we express the positions on our model by a vector-valued, time-varying function x(u, t) = (x(u, t), y(u, t), z(u, t))T , where T denotes transposition and u = (u, v, w) is the material coordinates of a model. We also set up a non-inertial, model-centered reference frame φ and express the position of a point in the model as x = c + Rs, where the center of the model c(t) is the origin φ of and the rotation matrix R(t) gives the orientation of φ relative to Φ with a reference shape s. Thus, s(u, t) gives the positions of points in the model relative to the model frame. The referential shape of an LV model s is defined with its parameters (a0 , a1 , a2 , a3 , τ, eo1 , eo2 ) as follows:   e1 cos(τ (u)) − e2 sin(τ (u)) + e01 (u)   s = e1 sin(τ (u)) − e2 cos(τ (u)) + e02 (u), e3 where

   a1 (u) cos u cos v e1   e =  e2  a0 w  a2 (u) cos u cos v . e3 a3 (u) sin u 

The parameter a0 is the overall scaling factor that is used to normalize the size of the LV. The parameters a1 , a2 and a3 are the aspect



239

ratio along the x-, y- and z-axis, respectively. These capture the contractions of the LV along each axis independently. As they are indicated as functions of u, the values may change throughout the LV material. The parameter τ is for the twist about the z-axis, capturing the wringing motion of the LV along its long axis. Finally, the parameters eo1 and eo2 are the axis offsets along the x-axis and y-axis, respectively. They are also functions of u, allowing local translation of the long axis to capture the bending motion of the long axis. In the above equation, e = (e1 , e2 , e3 )T is the generalized ellipsoid which is represented as a parametric form in u = (u, v, w). We orient our model in such a way that the center of the model is at the centroid of the LV with y-axis pointing toward the right ventricle. It will allow us to interpret intuitively the parameters of the model without any further complex processing. For example, since the shortand the long-axis views coincide with xy- and yz-planes, respectively, the parameter functions a1 (u) and a2 (u) (i.e., the aspect ratio along x- and y-axis) will capture the radial contraction, and the parameter function a3 (u) (i.e., the aspect ratio along z-axis) will capture the longitudinal contraction. For the material coordinates u = (u, v, w), u runs from the apex to the base of the LV, v starts and ends at the point where the septum is located, and w is used for the definition of model points between the inner and outer walls of the deformable model. The resolution of discretization of the volume depends on how densely the data points (i.e., MR tagging data) are distributed throughout the volume. By incorporating the geometric definition of the models into the physicsbased framework, we create dynamic models that deform due to forces exerted by data points and thus conform to the given data set, Dq˙ = LT f , where q is the vector of the model’s degrees of freedom, D is the damping matrix used as a stabilizing factor, L is the Jacobian matrix, and f is the 3D force applied to the model. The detailed algorithm for extracting and distributing forces from the given MR tagging data to material points within the volumetric model is described in Ref. 20. In computing the correct forces from the data the algorithm exploits the geometry of the motion of the MR tagging data points over time. Once these forces are computed, we use a Lagrangian dynamics formulation16 to estimate


240


the model parameters. The parameters of the LV model that are estimated during the fitting process directly reflect the shape changes during systole. 3.2. LV model description The deforming LV model is a series of fitted parameter-function models at specified time phases of the cardiac cycle from end-diastole to end-systole. It is defined by six model parameter functions of u = (u, v, w), the material coordinates, where u spans from apex to base of the LV; v covers from septum, anterior, lateral, and posterior of the LV; and w goes out from the endocardial surface to epicardial surface (see Fig. 5). Since the model parameter values are estimated over the cardiac cycle using the technique described in the previous section, the dynamic behavior of the model s can be expressed in the definition as follows: s = s(u, t; a1 (u, t), a2 (u, t), a3 (u, t), τ (u, t), e01 (u, t), e02 (u, t)) . Note that the time t here refers to the cardiac cycle, rather than the iterations of the “fitting” time. Therefore, given the translation and the orientation of a model frame of reference, the positions of all points in the model at a specific cardiac cycle can be computed from the estimated parameter functions. That is, we can track the time-varying positions of any material points in the volumetric model. We can re-tessellate the model, if desired, to create meshes of a finer resolution in all u, v, and w directions, since we can extrapolate the values of the model parameters at any given material position. Then, the volume elements (see Fig. 9) are formed by connecting eight nodes of coordinates (ui , vi , wi ), (ui+1 , vi , wi ), (ui , vi+1 , wi ), (ui+1 , vi+1 , wi ), (ui , vi , wi+1 ),

w

v

u

u=(u,v,w) Fig. 5.

Material coordinates u for a parameter-function of LV model.



241

(ui+1 , vi , wi+1 ), (ui , vi+1 , wi+1 ), and (ui+1 , vi+1 , wi+1 ), where i and i + 1 are indices of immediate neighbor nodes.b Since the volume elements are distorted, especially toward the apex, we use a cuboid master element and define Jacobian matrix for the coordinate transformation. Within each element, the position is given as a function of element local coordinates as a weighted average of positions at the nodes with a linear-element-basis function. Since we do not know the material properties of the heart, we assume the linear variation within the element. 3.3. Quantification and visualization The main limitation of most of the other techniques for LV modeling is that they do not provide intuitive motion parameters to model the motion of the LV. Rather, they represent the LV motion as a set of local displacement vectors which requires a non-trivial post-processing to provide meaningful geometric information on the nature of the deformation. Since the semantics of each of our model parameters are intuitive, we can directly plot each parameter function for comparison purposes and visualization. For example, since u varies from the apex to the base of an LV, by plotting extracted parameters as functions of u we can compare and quantify the motion variation from the apex to the base. Figure 6a shows plots of the a2 parameter (which captures radial contraction along the constrained direction) at the endocardium win for each time phase t = 1, . . . , 5), where t = 5 represents end-systole. For example, we can determine that the overall percent changes of the magnitude of radial contraction of the inner wall during systole is approximately 30%, and the contractions at the apex and at the base are approximately 35% and 22%, respectively. By plotting the parameter for each time phase, we can observe the progression of the deformation during systole. Figure 6b shows the same parameter function a2 at the endocardium, at end-systole. We can observe that the abnormal LV does not contract as much as the normal LV does, especially toward the apex. The abnormal LV contracts only approximately 60% of the amount found for the normal LV (−20% versus −35%). For a complete set of graphs of other parameters, the readers are referred to Ref. 20. b We

did not compute the strains at the very apex of the LV where we can only create wedge-shaped volume elements. Since the parameter-function model does not provide an accurate deformation at the apex due to the lack of proper MRI-SPAMM data points to begin with, it is not unreasonable to exclude the region for the analyses.


242


Fig. 6. Parameter-function model fitting results: parameter graphs for a2 parameter. ED: end-diastole, ES: end-systole, and HTCM: hypertropic Cardiomyopathy.

For each parameter-function, its value is evaluated at each node of the model. Then this value is used to compute a triplet of RGB color for the corresponding hue of the predetermined color-coding scheme. Figure 7 shows such renderings of the model for parameter τ estimated for a normal (left) and an abnormal (right) LV at the end systole. The connected red line segments in the figure are the traces of the sequential positions of nodes from

Fig. 7.

Parameter-function model fitting results: color mapping for τ parameter.



243

end diastole to end systole. The parameter values at epicardium are colored onto the epicardial surfaces. White, red, and blue correspond to zero twist, +10◦ twist and −10 twist, respectively. We can clearly observe that both LVs have undergone a wringing motion during systole, and that the abnormal LV has twisted more than the normal LV. For more visualization techniques employed for the parameter functions and their results, readers are referred to Ref. 21.

4. Strain Analysis In this section, we present the strain analysis conducted from the nodal displacements of the parameter-function LV models. As described in the previous section, the parameter-function model is a new class of physicsbased deformable models where the deformation fields are captured into their global parameters that are functions. The fact that these parameters are functions allows the localization and the detailed estimation of the LV shape and motion. This method provides a new way to view the strains for the LV motion based on intuitive parameters (i.e., radial contraction, longitudinal contraction, twisting about and bending of its long-axis). The deformation fields are derived from the MRtagging image data sets, in particular MRI-SPAMM2 data sets. Figure 8 shows the overview of the LV motion estimation and analysis based on parameter-function models in conjunction with MRI-SPAMM data. From the sets of two orthogonal view images (i.e., short axis and long axis) obtained by MRI-SPAMM techniques, volumetric sets of time-varying 2D data points are extracted by utilizing active contours.11 These data points become the inputs to the parameter-function modeling framework that estimates the 3D (rigid and non-rigid) motion of the heart. The model fitting process is essentially the process of recovering the model parameter-functions at each time phase. The output of the process is 3D models deforming over time as defined by their parameter functions. The node positions of the recovered 3D model at each time phase are the inputs to our strain analysis program presented in this chapter. We first compute three normal and three shear strain tensors from nodal displacements and convert them into strain components in an LVspecific coordinate system for a better interpretation. We then compute three principal strains from the strain tensor and visualize the maximum


244


Sampled 2D data over time Long-axis view image

MRISPAMM

t=1

t=N

Parameter function modeling framework Model Fitting Over time

Short-axis view image

t=1

Strain analysis and Visualization

t=N

Quantify and visualize 3D motion from model parameters

Fig. 8. Flowchart of our work from MRI-SPAMM data to LV strain analysis and visualization.

and minimum principal directions. We also compute normalized strains, Von Mises’ yield criteria, from the principal strains. All of the abovecomputed components are visualized in a user-friendly PC environment for an effective analysis of the LV function. 4.1. Interpretations of strain components: a cylinder model We have created a test parameter-function model of a thick-walled cylindrical shell so that we can understand better all six strain components in conjunction with parameters in parameter-function models. The cylinder model has three parameter functions (α, a3 , and τ ) over the material coordinates u. We define the model as follows: ec = ec (u = (u, v, w); α(u), a3 (u), τ (u))   α(u) cos v cos τ (u) − α(u) sin v sin τ (u)   = w  α(u) cos v sin τ (u) − α(u) sin v cos τ (u) , a3 (u) where −π/5 ≤ u ≤ π/5, −π ≤ v ≤ π, and win = 0.5 ≤ w ≤ 1.0 = wout .


Strain Analysis and Visualization of LV Deformation Table 1.

245

Parameters of parameter-function models.

LV model

Representation

Cylinder model

a1 , a2 a3 τ e01 , e02

Radial contraction Longitudinal contraction Twisting about the long axis Bending of the long axis

α a3 τ —

w

v

u

v

u w

in top out base Fig. 9.

Subscripted labels for the test model.

For the original LV model proposed in Ref. 20, six parameter-functions were employed to describe four types of LV wall motion (see Table 1). It turned out that the changes in values of e01 and e02 over the cardiac cycle were not significant and therefore these parameters were only used initially to recover the initial shape of the LV. Also note that, as pointed out in Ref. 20,a1 and a2 parameters may be combined into one parameter, namely α = a21 + a22 , to capture the radial contractile motion. In this case, the long-axis bending parameters e01 and e02 should not be used in order for all parameters to be independent. Initial parameter values for the undeformed state are set to the following: α(u) = 0.8, a3 (u), and τ (u) = 0.0. The radii of inner and outer walls at the undeformed state are 0.4 and 0.8 unit, respectively. The model was decomposed into 32 linear volume elements. For convenience we use the subscripted labels of “in”, “out”, “top”, and “base” as shown in Fig. 9. We examine the following seven cases of deformation as illustrated in Fig. 10.

4.1.1. Case 1: Radial contraction I The radii of inner and outer walls become 0.3 and 0.6 unit, respectively. In this case, the changes in parameter values are α(−, −, win ) = α(−, −, wout ) = 0.6, and therefore the radial contraction for both inner and outer walls is 25% (= (0.8 − 0.6)/0.8). The associated strain tensor


246


Fig. 10.

Deformation of the test model.

components are er = −0.25 and eθ = −0.25. Then the interpretation is as follows: The radial displacements for inner and outer walls are 0.1 and 0.2, respectively. Therefore, the wall thickness has changed from 0.4 to 0.3 (25% thinning). Radial contractile motion causes circumferential shortening of 25%. 4.1.2. Case 2: Radial contraction II The radii of inner and outer become 0.4 and 0.7 unit, respectively. In this case, the changes in parameter values are α(−, −, win ) = 0.6 and α(−, −, wout ) = 0.7. It indicates that there is more radial contractile motion at the inner wall (25%) than at the outer wall (12.5% = (0.8 − 0.7)/0.8). The associated strain tensor components are er = 0.0 and eθ = −0.17. The interpretation is as follows: The radial displacements for both inner and outer walls are 0.1 unit. Therefore, there was no change in wall thickness, yet there was circumferential shortening of 17% due to radial contraction. 4.1.3. Case 3: Regional rotation I Inner wall rotates 20◦ and outer wall rotates 10◦ in the circumferential direction. In this case, the changes in parameter values are τ (−, −, win ) = 0.175 rad = 10◦ , indicating that there were 10 more degrees of rotation at the inner wall. The corresponding strain tensor component is erθ = −0.12. It tells us that there were no length changes along radial, circumferential,



247

and longitudinal direction. However, there was the shearing in radial– circumferential direction. 4.1.4. Case 4: Regional rotation II The model undergoes 10◦ of clockwise rotation at the base and 10◦ of counterclockwise rotation at the top. In this case, the changes in parameter values are τ (ubase , −, −) = 10◦ , τ (utop , −, −) = −10◦ . It shows that there is 10◦ of twisting about the long and axis (10◦ clockwise at the base and 10◦ counterclockwise at the top). The corresponding strain tensor component is ezθ = −0.11. It shows another example for representing shearing, where the shearing direction is in the longitudinal–circumferential direction. 4.1.5. Case 5: Radial contraction III The model is gradually contracted toward the base in the radial direction, with no change in wall thickness. In this case, the changes in parameter values are functions of u: α(−, −, win ) = 0.8 − 0.2u and α(−, −, wout ) = 0.8 − 0.16u. They indicate linear contractions (because of negative slope) along u (from apex to base), where the inner wall contracts more than the outer wall (steeper slope for inner wall). The corresponding strain tensor components are erz = +0.10 and eθ = −0.25 ∼ 0.0. The interpretation is as follows: The main characteristic in this motion is the shearing in the radial–longitudinal direction. But, due to the radial contraction at the base, we see an increase in circumferential along the longitudinal direction. 4.1.6. Case 6: Longitudinal contraction I Only inner wall contracts longitudinally, and the change in parameter values is a3 (−, −, win ) = 0.8, indicating that there was 20% (= (1.0 − 0.8)/1.0) longitudinal contraction at the inner wall. The corresponding strain tensor components are erz = ±0.06 and ez = −0.2 ∼ 0.0, and the interpretation is as follows: Upper elements experience negative shearing in the radial– longitudinal direction, and lower elements experience positive shearing in the same direction. We also see an increase in longitudinal shortening along the radial direction. 4.1.7. Case 7: Longitudinal contraction II There is a uniform contraction along longitudinal direction, with the change in parameter values is a3 (−, −, −) = 0.8 . It indicates that there was 20%


248


(= (1.0 − 0.8)/1.0) longitudinal contraction everywhere. In this case, the corresponding strain tensor component is ez = −0.2. It tells us that there was only longitudinal shortening, which is reflected in the ez component. 5. Experiment 5.1. Implementation Our strain analysis and visualization tool is implemented using visual C++ in the PC environment. The computer program takes an input file, which has a set of lists of node positions of the volumetric model at each time phase from the undeformed state to the end of a deformed state. Prior to computing strain tensors, a user may choose an option to perform the smoothing operations on the model. For a better display the variation of the strains, each volume element is divided into 4 × 4 × 4 uniform sub-elements, and the strain is computed at the center of these sub-elements. Figure 11 (top) shows the snapshot of the program display. The program allows a user to select either to view the model at any time phase or to

Fig. 11.

Screenshot of our system for strain visualization.



249

animate the deforming model. In either mode, the user can choose dynamically which component among the six strain tensor components and the Von Mises strain to be color coded onto the model being displayed. Since the model is volumetric, for an effective visualization, the program also allows the user to visualize a set of elements forming a column or a circular ring at each level individually. If the model consists of multiple layers in the radial direction, each layer can be displayed individually. An example is shown in Figs. 11a–c. 5.2. Strain analysis We have applied the strain analysis to the parameter-function LV models. These models are generated from the estimated parameter functions for LV motion. The mesh dimensions for the model are 14 × 24 × 2, and, therefore, the model is decomposed into 672 linear volume elements. The systolic motions of the LV are sampled at 5 time frames from end diastole to end systole. We have conducted experiments with both normal and diseased hearts. 5.2.1. Evaluation of the parameter-function model Young and Alex26 have reported the 3D LV strain analysis results estimated by a (traditional) FEM from MRI-SPAMM data. Since we have used similar data sets of MRI-SPAMM to estimate 3D deformation fields in the parameter-function modeling framework, we have compared our strain analysis results to the results reported in Ref. 26. Table 2 shows the averages and the ranges of each strain component computed at the mid-wall layer. Our results are in good agreement with Young’s results. Therefore, though we need to study further against a larger database, we have verified that Table 2. Strain components

Results from our model Average value

er eθ ez erθ eθz erz

Strain analysis results (normal LV).

0.10 −0.17 −0.15 0.02 0.00 0.00

Range [min to max] [0.06 [−0.24 [−0.16 [−0.04 [−0.04 [−0.06

to to to to to to

0.15] −0.04] −0.14] 0.06] 0.04] 0.04]

Results from Young’s model26 Average value 0.1 −0.2 −0.15 0.01 0.03 0.010

Range [min to max] [0.02 [−0.28 [−0.2 [−0.02 [0.01 [−0.04

to to to to to to

0.20] −0.15] −0.1] 0.06] 0.05] 0.04]


250


the strain information gathered indirectly from parameter-function models is consistent with previously reported work. 5.2.2. Normal versus abnormal LV The results of strain analyses from two LV models (normal and abnormal) are shown in Figs. 12 and 13. Originally, the normal LV model was recovered from a data set of a healthy volunteer, and the abnormal LV model was recovered from a data set of a patient with hypertropic cardiomyopathy (HTCM). Figure 10 displays the Von Mises strain patterns of near epicardium from end diastole (i.e., undeformed state) to end systole. The color ranges from blue to red depending on the computed normal strain value at each location. Blue represents the minimum (in this case, 0.0) and red represents the maximum (in this case, 0.42). We can observe that the Von Mises strain values can be used to find the regions of large deformation for each model, without having to examine each component of a strain tensor, and that the distribution and the range of the Von Mises strain values can be utilized to differentiate these two LVs. Figure 13 shows the distribution patterns of six components of strain tensors near the epicardium at end systole. Green, blue, and red colors are used for zero, most-negative, and most-positive values, respectively, of each

Fig. 12. systole.

Von Mises strain distribution of normal and abnormal LV models during



Fig. 13.

Table 3.

Normal HTCM

251

Six components of strain tensors at the end systole.

Average values of strain components: normal versus abnormal LV (HTCM). er

eθ

ez

erθ

eθz

erz

0.10 0.11

−0.17 −0.11

−0.15 −0.06

−0.018 0.024

−0.003 0.022

−0.003 −0.029

strain component. For each strain component, the same color scale was used for both models. The ranges of values used in Fig. 11 for each component are: eθ = [−0.40, 0.40], er = [−0.43, 0.43], ez = [−0.38, 0.38], eθz = [−0.43, 0.43], erz = [−0.33, 0.33], and erθ = [−0.30, 0.30]. We can easily observe that the abnormal LV has different strain values in each component compared with the normal LV: (1) The distributions of ez -component for two models are different. Blue color dominates the region of the normal LV compared with the abnormal one. It means that the contraction along z-axis is dominant motion for the normal LV, but the heart with a disease has relatively small contraction along z-axis (longitudinal contraction). (2) The ranges of eθz and erθ for the abnormal LV are much bigger than those for the normal one. It means that the abnormal LV twists along θ-axis (circumferential direction) more than the normal one does. (3) The ranges of er and erz for the abnormal LV are bigger than those for the normal one, and the high values are locally concentrated in the abnormal case compared to the normal case. The average values of each component are tabulated in Table 3 for both the normal and the abnormal cases.


252


6. Conclusion The 3D deformation fields of the LV models were initially estimated from tagged MRI data sets,20 which provide in-plane temporal correspondence of material points. While there is some initial experience in the use of tagged MRI and related techniques to study the 3D motion of a heart, there is still no generally accepted method for analysis and display of the 3D heart motion. The parameter-function model captures the 3D deformation field in its model parameters, and the volumetric model can be regenerated based on the estimated parameter, allowing us to select any desired volume element within the myocardium for a conventional strain analysis. We have verified that the results of a conventional strain analysis performed on the parameter-function model are in agreement with those from a conventional FEM model.26 Furthermore, we have proposed a new methodology in visualizing multi-dimensional local quantities for a better rendition of overall strain variation.23 It will help gain a thorough and localized understanding of the LV motion, and should significantly increase the clinical utility of such heart motion analyses.

References 1. T. Arts, W. C. Hunter, A. S. Douglas, M. M. Muijtjens, J. W. Corsel, and R. S. Raneman, Macroscopic three-dimensional motion patterns of the left ventricle, Advances in Experimental Medicine and Biology 346 (1993) 383–392. 2. L. Axel, and L. Dougherty, MR imaging of motion with spatial modulation of magnetization, Radiology 17, 3 (1989) 841–845. 3. E. Bardinet, N. Ayache, and L. C. Cohen, Fitting of iso-surfaces using superquadrics and free-form deformations, Proceedings of IEEE Workshop on Biomedical Image Analysis, Seattle, WA, 1994, pp. 184–193. 4. R. Beyar, and S. Sideman, Effect of the twisting motion on the nonuniformities of transmural fiber mechanics and energy demands — a theoretical study, IEEE Transaction on Biomedical Engineering 32 (1985) 764–769. 5. C. W. Chen, T. W. Huang, and M. Arrott, Modeling, analysis and visualization of left ventricle shape and motion by hierarchical decomposition, IEEE Transaction on Pattern Analysis and Machine Intelligence 16, 4 (1994) 342–356. 6. J. Declerck, J. Feldmar, and N. Ayache, Definition of a four-dimensional continuous planispheric transformation for the tracking and analysis of leftventricle motion, Medical Image Analysis 2, 2 (1988) 197–213. 7. A. F. Frangi, W. J. Niessen, and M. A. Viergever, Three-dimensional modeling for functional analysis of cardiac images: a review, IEEE Transaction on Medical Imaging 20, 1 (2001) 2–25.



253

8. D. Friboulet, I. E. Magnim, and D. Revel, Assessment of a model for overall left ventricular three-dimensional motion from MRI data, International Journal of Cardiac Imaging 8 (1992) 175–190. 9. Y. F. Fung, Foundations of Solid Mechanics (Prentice-Hall, New Jersey, 1976). 10. J. M. Guccione, K. D. Costa, and A. D. McCulloch, Finite element stress analysis of left ventricular mechanics on the beating dog heart, Journal of Biomechanics 28, 10 (1995) 1167–1177. 11. P. J. Hunter, and B. H. Smaill, The analysis of cardiac function: a continuum approach, Progress in Biophysics and Molecular Biology 52 (1998) 101–164. 12. M. Kass, A. Witkin, and D. Terzopoulos, Snakes: active contour models, International Journal of Computer Vision 1, 4 (1988) 321–331. 13. S. Masood, G. Z. Yang, D. J. Pennel, and D. N. Firmin, Investigating intrinsic myocardial mechanics: the role of MR tagging, velocity phase mapping and diffusion imaging, Journal of Magnetic Resonance Imaging 12 (2000) 873–883. 14. A. Matheny, and D. B. Goldgof, The use of three- and four-dimensional surface harmonics for rigid and nonrigid shape recovery and representation, IEEE Transaction on Pattern Analysis and Machine Intelligence 17 (1995) 967–981. 15. T. McInerney, and D. Terzopoulos, A dynamic finite element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image analysis, Computerized Medical Imaging and Graphics 19, 1 (1995) 69–83. 16. D. Metaxas, and D. Terzopoulos, Shape and nonrigid motion estimation through physics-based synthesis, IEEE Transaction on Pattern Analysis and Machine Intelligence 15, 6 (1993) 569–579. 17. C. C. Moore, C. H. Lugo-Olivieri, E. R. McVeigh, and E. A. Zerhouni, Threedimensional systolic strain patterns in the normal human left ventricle: characterization with tagged MR imaging, Radiology 214 (2000) 453–466. 18. C. Nastar, and N. Ayache, Frequency-based nonrigid motion analysis application to four dimensional medical images, IEEE Transaction on Pattern Analysis and Machine Intelligence 18 (1996) 1069–1079. 19. J. Park, D. Metaxas, A. A. Young, and L. Axel, Deformable models with parameter functions for cardiac motion analysis form tagged MRI data, IEEE Transaction on Medical Imaging 15, 3 (1996) 278–289. 20. J. Park, D. Metaxas, and L. Axel, Analysis of left ventricular wall motion based on volumetric deformable models and MRI-SPAMM, Medical Image Analysis 1, 1 (1996) 53–71. 21. J. Park, D. C. Bloomgarden, Z. A. Fayad, and L. Axel, 3D model of deformation: validation studies using MR tagging, Proceedings of International Society for Magnetic Resonance in Medicine, Vancouver, B.C., Canada, 1997, pp. 147. 22. J. Park, D. Metaxas, and L. Axel, Quantification and visualization of the 3D nonrigid motion of the left ventricle, Proceedings of SPIE International


254

23. 24.

25.

26.

27.


Symposium on Medical Imaging, Physiology and Function from Multidimensional Images, Newport Beach, CA, 1997, pp. 298–308. J. Park, and S. Park, Strain analysis and visualization: left ventricle of a heart, Computers and Graphics 24 (2000) 701–714. K. Shoemake, ARCBALL: a user interface for specifying three-dimensional orientation using a mouse, Proceedings of Graphics Interface, Vancouver, B.C., 1992, pp. 151–156. P. Shi, G. Robinson, C. T. Constable, A. Sinusas, and J. Duncan, A modelbased integrated approach to track myocardial deformation using displacement and velocity constraints, Proceedings of the 5th International Conference on Computer Vision (ICCV95), Cambridge, Massachusetts, 1995, pp. 687–692. A. A. Young, and L. Axel, Three-dimensional motion and deformation of the heart wall: estimation with spatial modulation of magnetization — a model-based approach, Radiology 185 (1992) 241–247. E. A. Zerhouni, D. M. Parish, W. J. Rogers, A. Yang, and E. P. Shapiro, Human heart: tagging with MR imaging — a method for noninvasive assessment of myocardial motion, Radiology 169, 1 (1988) 59–63.


SECTION III.

ASSISTED PERFUSION AND PUMPING, AND MYOCARDIAL REPAIR




AUGMENTED MYOCARDIAL PERFUSION BY CORONARY BYPASS SURGICAL PROCEDURE: EMPHASIZING FLOW AND SHEAR STRESS ANALYSIS AT PROXIMAL AND DISTAL ANASTOMOTIC SITES PROVIDING THE BASIS OF BETTER GRAFT PATENCY RATES DHANJOO N. GHISTA School of Mechanical and Aerospace Engineering College of Engineering, Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 [email protected] MEENA SANKARANARAYANAN∗ Media Division, Institute for Infocomm Research 21 Heng Mui Keng Terrace Singapore 119613 [email protected] LEOK POH CHUA School of Mechanical and Aerospace Engineering Nanyang Technological University, Singapore 639798 [email protected] YONG SENG TAN Department of Cardiothoracic Surgery National Heart Centre, Singapore 168752 EDDIE Y. K. NG School of Mechanical and Aerospace Engineering College of Engineering, Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 [email protected]

∗ This work was done while Meena was at Nanyang Technological University, Singapore 639798.

255


256

D. N. Ghista et al.

1. Introduction The complex anatomy of the coronary vessels has made the investigation of coronary flow and hemodynamics one of the most difficult and challenging studies even today. We owe a lot to all those who have contributed to the understanding of the anatomy and the function of the cardiovascular system. In this regard, we can mention Leonardo da Vinci (generally called as “man ahead of time”), who developed insightful sketches (some 550 years ago) of the anatomy of the heart and its blood vessels (Fig. 1). The major cause of death in both the developed and developing countries is cardiovascular disease. Arteries can become clogged over time by the buildup of fatty plaque. The coronary arteries are a common site of plaque formation. This blockage causes a decrease in the coronary flow, which in turn results in poor heart perfusion. One method to overcome this problem of depleted perfusion of the affected myocardium is by performing cardiac bypass surgery (generally referred to as coronary artery bypass grafting (CABG)), whereby new routes around narrowed and blocked arteries are

Fig. 1.

Leonardo da Vinci’s sketches showing the distribution of heart and blood vessels.


Augmented Myocardial Perfusion by CABG Procedure

257

created (by means of grafts anastomosed to the coronary vessels at distal to the occluded sites) to deliver oxygen and nutrients to the myocardium. Coronary bypass surgery has been proven to be a reliable and effective long-term treatment for coronary atherosclerosis over the past 35 years. Randomized prospective trials have shown that bypass surgery prolongs the life expectancy of high-risk patients, consistently improves or eliminates symptoms of angina, and improves left ventricular (LV) function and symptoms of congestive heart failure for selected patients. While, the Achilles heel of bypass surgery (namely the vein graft failure) has not been eliminated, recent data indicate a rate of vein graft failure of 30–35% by 10 postoperative years. Furthermore, strong evidence suggests that the consistent use of platelet inhibitors and coenzyme A inhibitors (statins) may further decrease vein graft failure rates. In contrast, the LITA–LAD graft has a failure rate of only 10% by 20 years. Despite CABG being an effective surgical technique to reperfuse the myocardium, 20–50% of bypass grafts fail due to the formation of intimal hyperplasia.1–3 The mechanism of this disease is unclear, but it is believed that many interrelated issues involved include local hemodynamic factors,4 compliance mismatch between graft material and host artery,5 and biomaterials incompatibility.6 These diseases are prone to develop near the bends and junctions of graft.7 Among all these factors, it is noted that hemodynamic factors like wall shear stress (WSS) play a vital role in atherogenesis.

1.1. Relation between WSS and intimal hyperplasia Since lesion development takes place close to the blood–vessel wall interface, the nature and distribution of WSS has been of specific interest. The role of WSS related to arterial disease has been the subject of debate over decades. Fry8 reported that if the endothelial cells are exposed to a WSS magnitude of over 37 N/m2 , the endothelial surface would then be subject to denudation. Endothelial cell degeneration can be produced by turbulence or increased shear stress on the cell surface. This is due to the fact that where the wall shear rate is high, an increased amount of mechanical energy is stored in the wall, which causes damage to the endothelium. On the other hand, Caro et al.9 have reported that early atherosclerotic lesions are present in low shear stress areas due to shear-dependent mass transport for atherogenesis. It is observed that low shear rates retard the transport of circulating particles away from the wall, thereby increasing intimal accumulation of lipids in turn to cause early atheroma. Furthermore,


258

D. N. Ghista et al.

regions of low WSS and flow recirculation have been shown to correlate with locations of atheroma in coronary arteries.10 With regard to end-to-side anastomosis, some interesting results have been reported on the relationship between WSS gradient and intimal hyperplasia thickening. A photochromic technique, in a Plexiglas model with 45◦ graft–host vessel anastomosis angle, was adopted by Ojha et al.11 to determine the velocity profiles and WSS. The proximal limb of the artery was totally occluded. A sinusoidal pulsatile flow waveform was adopted as the inlet condition. The results showed that during the systolic acceleration phase, a jet from the graft was directed toward the artery floor and deflected toward the distal artery. A stagnation point on the artery floor was found to fluctuate during the flow cycle, which in turn causes fluctuations in the direction of shear stress resulting in high spatial gradient of shear stress. This shear stress variation on the floor of the artery could stimulate the endothelial cells to cause arterial disease. During the deceleration phase, strong secondary flow patterns were also seen in the distal artery. Thus, their studies reveal a relation between fluctuating WSS gradients and intimal hyperplasia. To understand the overall flow patterns at the distal anastomosis of steady and pulsatile flow, the flow visualization technique was adopted by Hughes and How.12 Transparent polyurethane models of the distal anastomosis were fabricated (using a dip-coating process) with anastomosis angles of 15–45◦. During the systolic acceleration phase, the flow in all the models was laminar with the particle pathlines dominated by the axial velocity component. As the flow began to decelerate, the flow pattern in the distal artery changed and became dominated by secondary components. A vortex was observed at the heel of the anastomosis; as the deceleration phase progressed, this vortex was seen to move toward the center of the anastomosis and then into the graft. Observations reveal that the strength of secondary flow increases with the angle of grafting and with Reynolds number in the graft. These flow patterns may be one cause that enhances platelet activation and eventually leads to intimal hyperplasia. Additionally, animal model studies have been employed to identify the conditions for the initiation of two different types of anastomotic intimal thickening (suture line and arterial floor intimal thickening). Bassiouny et al.13 have shown that a stagnation zone exists along the floor of the artery, and that separation region and secondary flows exist at both the heel and the toe of the graft (Fig. 2). These regions of low WSS could cause pathophysiologic responses in endothelial cells leading to intimal hyperplasia.



259

Fig. 2. Schematic of the model flow visualization showing (i) a stagnation point at the floor of the artery and (ii) secondary flow patterns at the heel and the toe of the anastomosis.

1.2. Anastomosis geometry and artery flow conditions Literature clearly reveals that two major characteristics that determine the flow structure and the hemodynamic factors causing intimal hyperplasia are the geometry of the graft–vessel junction and its input flow waveform. Unfortunately, even though a few studies that have been conducted to replicate the in vivo arterial and graft geometry in the computational fluid dynamics (CFD) model (to study its hemodynamics), most of the modeling work in graft flow is limited to only a part of the bypass conduit geometry, namely the anastomosis site. As regards to anastomosis geometry, Moore et al.14 have discussed the extent to which an arterial flow model needs to be anatomically accurate, so as to capture the essential hemodynamics in CABG. In connection with the influence of the anastomosis geometry, Min-Ho Song et al.15 developed a Y-figure anastomotic model for proximal arterial stenosis (at angles ranging from 10◦ to 30◦ ), to analyze the three-dimensional (3D) simulation of CABG. In their work, on the endto-side anastomosis model, all the vessels were adopted to be of the same diameter. The boundary conditions of velocity and flow in the model were based on the average velocity and flow in the coronary artery, measured intraoperatively. They found the WSS (a determinant of intimal hyperplasia) was smallest at the heel position in 10◦ anastomosis and largest at 30◦ anastomosis, thereby concluding that a more acute anastomosis angle may lead to better graft patency. Even 3D numerical simulations have been performed (using CFD) by Bertolotti and Deplano16 on end-to-side anastomosis of a stenosed coronary bypass. Herein, the anastomosis geometry was based on the assumption that the graft and the host vessel are of the same diameter, with the graft


260

D. N. Ghista et al.

inclined at 45◦ to the host vessel. The boundary conditions in the model were (i) fully developed velocity profiles, at both (the graft and the host coronary) inlets, (ii) a traction-free surface condition at the outflow, and (iii) the no-slip condition at the vessel walls. The flow features were compared for different flow rates and distance of the anastomosis from the site of occlusion. It was concluded that the risk of intimal hyperplasia could be minimized if the anastomosis was sutured at a sufficient distance of length. Kute and Vorp17 using a CFD model for steady flow through the idealized geometry model consisting of equal diameter vessels with the graft inclined at 30◦ . The boundary conditions for the model involved blunt velocity profiles in the graft and in the proximal artery, with a fixed pressure at the distal artery outlet. The velocity vectors for all the flow conditions (prograde, zero, retrograde) exhibited skewing toward the floor of the artery, with the prograde flow in the proximal artery providing more perfusion of the distal artery. Ethier et al.18 studied the role of inlet flow on the WSS distribution in an end-to-side anastomosis geometry, consisting of two cylindrical conducts of equal diameter intersecting at 45◦ . With a study of the effect of different flow waveforms on the anastomotic WSS patterns, it was seen that the temporal and spatial gradients of WSS on the host artery bed were larger for femoral waveform than the coronary artery waveform. As regards the influence of the graft non-planarity on graft flows, it was found that non-planarity resulted in a 10% reduction in the peak WSS magnitude at the bed of the anastomosis.19 However, data on in vivo physiologic velocity at the aortic inlet are mostly not available, and thus fully developed velocity profiles (Womersley profiles) are assumed as inlet conditions. Hence, to reduce the complexity, CFD studies of the flow in the CABG models have mainly been on distal anastomotic flows, without taking into account the proximal anastomosis region. Also, most studies consider the intersection of the graft–host vessel to be circular. But in reality as the diameter of the graft is larger than that of the host vessel, the intersection between the graft and the host vessel results in an elliptic-shaped anastomosis conduit.

1.3. Our work in light of previous studies Most of the blood flow analyses have been essentially simulating specific bypass conduits geometries and flow conditions. Notwithstanding the limitations in replicating the realistic geometry and the in vivo conditions from which the quantitative data can be extracted, the determination



261

and adoption of appropriate geometry and flow boundary conditions are important for studying hemodynamics in the bypass conduits. Unfortunately, most of these works have not considered (i) the complete bypass flow domain, (ii) realistic physiologic flow conditions, and even (iii) the non-planarity of the bypass graft vessel. Therefore in our work, we have addressed these drawbacks to some extent by studying the flow characteristics, by means of a 3D CABG model involving the complete domain under representative physiologic conditions.

2. Formulation 2.1. Model assumptions The blood is assumed to be incompressible, with a Newtonian behavior having dynamic viscosity (µ) of 0.00408 Pa and a density (ρ) of 1050 kg/m3 . The blood vessel walls are assumed to be rigid and impermeable. For a quasi-steady, 3D laminar flow, the Navier–Stokes equations (for mass and momentum conservation) governing fluid motion are written as follows: equation of continuity: ∇q = 0, ∂q equation of momentum: ρ + q · ∇q = −∇p + µ∇2 q, ∂t

(1) (2)

where q denotes the velocity vector in 3D. The distributions of velocity and WSS are obtained by computationally solving the above equations subject to the boundary conditions. 2.2. Fluid-dynamic simulation setup The fluid-dynamics simulations are performed by using a control-volumebased technique, implemented in the CFD flow solver, Fluent.20 The computation procedure of the commercial code consists of (i) construction of the geometry using a pre-processor (Gambit21 ), (ii) meshing the computation domain, (iii) assigning boundary conditions based on the flow waveforms discussed earlier, (iv) assigning fluid properties, and (v) the solution algorithm. In the solution algorithm used by Fluent, the governing equations are solved sequentially. As the governing equations are non-linear (and coupled), several iterations of the solution loop need to be performed before a converged solution is obtained. Using this approach, the resultant algebraic equations for the dependent variables in each control volume are solved


262

D. N. Ghista et al.

sequentially by a point implicit (Gauss Seidel) linear equation solver, in conjunction with an algebraic multi-grid (AMG) method. The governing equations are solved iteratively until convergence of all flow variables is achieved. The solutions of all the flow variables are deemed to have converged once their residuals computed from two successive iterations are below the set desired convergence criteria of 10−4 .

3. Flow Studies in 3D Aorto-Right CABG System This section presents the fluid dynamics of blood flow in the aorto-right CABG model. 3D CFD simulations are developed for the blood flow in CABG systems, using the CFD software (FLUENT 6.0.1). We have calculated the flow-field distributions of the velocity and the WSS at four typical instants of the cardiac cycle: two during systole and the other two during the diastole phase. We infer from our work that sites favoring the development of arterial diseases are the toe region and the floor of the artery, where the spatial WSS variations are high. 3.1. Geometrical model The bypass model simulating the flow field of the anastomosis in the right CABG is shown in Fig. 3a. For the sake of clarity, the geometry indicating the specific sites is depicted in Fig. 3b. The ascending aorta (A–B) is 80 mm in length and 25 mm in diameter. The left coronary artery (C) is of circular cross section with 4 mm in diameter. The occluded right coronary artery (F–G) is a straight tube, 45 mm in length and 2 mm in diameter. Its proximal part (E–F) is taken to be completely occluded. The CABG (D–E) is anastomosed with the aorta at D and with the occluded coronary artery at E. The venous graft is taken to have a non-uniform circular cross section, larger than that of the coronary artery. The intersection between the graft and the right coronary vessel has an elliptic shape, which is due to the deformation of the larger diameter graft while suturing it to the smaller diameter coronary vessel surface. 3.2. Data input and boundary conditions 3.2.1. Systolic phase During the systolic phase, the aortic valve opens allowing blood to flow into the aorta A–B. This flow is obtained by prescribing a uniform velocity



Fig. 3a. system.

263

Geometry (plane view) and dimensions (in mm) of the aorto-right CABG

100 % Occluded Region Vein Graft Toe G

Right Coronary Artery

E

Heel F

Aorta A

D

C

B

Left Coronary Artery Fig. 3b. Geometry specifying the aortic inlet A, ascending aorta B, left coronary inlet C, aorta–graft junction D, artery–graft junction E, bypassed right coronary inlet F, and the right coronary artery exit G.


264

D. N. Ghista et al.

(across the cross section) at the inlet to the aorta A (Fig. 3b), from the LV. The velocity magnitude is computed from the physiologic representative stroke volume over the ejection period, based on the flow waveform22 shown in Fig. 4a. Throughout this period, the LV contracts and squeezes the coronaries, which are embedded in the cardiac muscle. This allows very little amount of blood to flow into the coronary arteries, the left coronary vessel C, and the bypassed right coronary vessel F–G. Hence, the inputs to the model consist of monitored time-varying flowrate waveform (i) at the inlet of the aorta A (Fig. 4a, adopted from Ganong22 ); (ii) at the left coronary artery C (Fig. 4c), obtained from in vivo intravascular Doppler ultrasonic technique23 ; and (iii) imposed at the exit of the right coronary artery G (Fig. 4d), obtained using a Doppler flow-meter catheter.24 The flow into the graft at D is determined from the monitored physiologic flow rates in the ascending aorta B (Fig. 4b), by taking into account the run-off conditions in the grafted right coronary vessel F–G. 3.2.2. Diastolic phase

Flow Rate (l/min)

At the start of diastole, there exists a small amount of back-flow into the LV through the aortic inlet A (Fig. 4a). At all other instants during the 30 25 20 15 10 5 0 -5

0

0.2

0.4

0.6

0.8

1

Time (s)

Flow Rate (l/min)

Fig. 4a.

Flow-rate waveform at the inlet of the aorta. 30 25 20 15 10 5 0 -5

0

0.2

0.4

0.6

0.8

1

Time (s)

Fig. 4b.

Flow-rate waveform at the ascending aorta.



265

Flow Rate (l/min)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05

0

0.2

0.4

0.6

0.8

1

Time (s)

Fig. 4c.

Flow-rate waveform at the left coronary artery.

Flow Rate (l/min)

0.25 0.2 0.15 0.1 0.05 0

0

0.2

0.4

0.6

0.8

1

Time (s)

Fig. 4d.

Flow-rate waveform at the right coronary artery.

deceleration phase, the aortic valve remains closed. In all these cases, blood flows from the ascending aorta B to the left coronary artery at C and to the right coronary vessel F–G. The data input to the model consists of the uniform velocity prescribed at the ascending aorta B (calculated from the monitored time-varying flow-rate waveform), as shown in Fig. 4b. The flow conditions at the inlet of the left coronary vessel C and at the exit of the bypassed right coronary artery G are calculated from the monitored time-varying input flow-rate waveforms (Figs. 4c and d). To prescribe the velocities at the entry of the left coronary artery C and at the exit of the bypassed right coronary artery G from the aorta, we take into account that a major portion (two-thirds) of the back-flow into the ascending aorta B (flow waveform is displayed in Fig. 4b) goes into the left coronary artery C (Fig. 4c), allowing the remainder to flow into the right coronary vessel F–G (Fig. 4d). This yields the blood volume flowing into the left coronary vessel (during a cardiac cycle) to be 2.67% of the stroke volume, with the remaining 1.33% of the stroke volume going into the right coronary artery. Based on this, the input data of the flow rates are specified at the inlet of the left coronary artery C and at the exit of the bypassed right coronary artery G.


266

D. N. Ghista et al.

3.3. Results and discussion For our numeric flow simulations, we have incorporated the unsteady flow character by (i) dividing the cycle into a number of small time intervals, (ii) adopting the time-dependent input conditions of the flow-rate waveforms, (Figs. 4a–d), and (iii) analyzing for steady flow within these small time intervals. To observe the velocity distribution features of the entire flow field, the computed velocity vectors are illustrated in the plane of symmetry at four different instants of the cardiac cycle, two during the acceleration phase and the other two during the deceleration phase. The simulated flows for the aorto-right CABG model are shown in Figs. 5–8. 3.3.1. Flow field in the aorto-right CABG model at onset of ejection, t = 0.0 s Figure 5a shows the distribution of velocity vectors in the flow field at the onset of ejection at t = 0.0 s. The flow-velocity distributions in different sections of the flow field are depicted in Figs. 5b–e. At the start of systole, blood flows from the aortic inlet A (with a uniform velocity across the cross section) into the ascending aorta B. The uniform velocity magnitude at the

flow rate (l/min)

30

m/s

G

20 10 0

0

0.2

-10

E

0.4 0.6 0.8 time (sec)

1

F

A

D

C

B

Fig. 5a. Illustration of the velocity vectors in the center plane of the aorto-right coronary bypass model at the start of ejection, t = 0.0 s.



A

267

D

C

Fig. 5b. At the start of systole, t = 0.0 s, blood flows from the aortic inlet A to the ascending aorta B with a uniform velocity 0.035 m/s. The fluid at a distance of 10 mm from the entrance of the aorta behaves like an inviscid flow. A small amount of flow gets diverted into the graft D. There is a small amount of blood flow entering into the aorta through the left coronary artery inlet C, that may be due to the high pressure in the myocardium.

m/s

A D

Fig. 5c. Parabolic profiles of the velocity vectors are seen inside the graft at the start of systole. Slight skewing is observed due to the curvature effects and low flow rate at the graft entrance D. The peak velocity magnitude half-way into the graft is 0.09 m/s. As the flow progresses into the graft, the maximum velocity shifts toward the centerline of the graft vessel.


268

D. N. Ghista et al.

Fig. 5d. Maximum velocity magnitude of blood flow exiting the graft is around 0.18 m/s. Major portion of the flow exiting from the graft E moves toward the distal portion of the right coronary vessel G. As the proximal portion F of the right coronary vessel is 100% occluded, the flow in this region hits and reverses its direction thereby creating a region of recirculation at the heel region.

m/s

E

Fig. 5e. The flow pattern shows slight skewing toward the floor of the artery. With increasing distance skewing disappears thus shifting the maximum velocity magnitude 0.371 m/s to the axially centered position in the host artery. The velocity profile remains the same at any cross section of the vessel, thereby exhibiting the fully developed nature of the flow.



269

Pa

D B

A

C

Fig. 5f. The uniform WSS distribution in the proximal anastomosis region of the graft is negligible because the spatial variation in the flow velocity is very small.

Pa

F

E

Fig. 5g. A wide variation of WSS is seen in the distal anastomotic region. Maximum WSS magnitude, 5.9067 Pa, is located at the toe region E and the floor of the bypassed right coronary artery. The weak recirculation zone results in negligible WSS around the heel region F.


270

D. N. Ghista et al.

aortic inlet A, obtained from the flow-rate waveform adopted from Fig. 5a, is 0.035 m/s. At a distance of about 10 mm from the entrance to the aorta, the computed flow field appears to be like an inviscid flow. Note that there is hardly any flow into the graft entrance D, at the start of systole (see Fig. 5b). This is due to the contraction of the LV, which in turn affects the coronaries which are embedded in the cardiac muscle. A very small amount of blood flow (with average velocity magnitude of 0.01 m/s) is determined to return to the aortic domain through the left coronary artery C; this could be due to the high stress in the LV myocardium during systolic contraction. There is a low flow rate into the graft at its entrance D; the flow pattern in the graft (anastomed to the aorta at a distance of around 30 mm from the aorta entrance A) is parabolic (as observed in Fig. 5c), with a slight skewing due to the curvature effect. As the flow progresses into the graft, the maximum velocity shifts toward the centerline of the graft vessel. The peak velocity magnitude half-way into the graft (Fig. 5c) is 0.09 m/s; however, the maximum further downstream in the graft is around 0.18 m/s. The forward flow exiting through the graft outlet E enters the host artery F–G, exhibiting a small region of recirculation (with negligible velocity magnitude) at the heel of the distal anastomosis, at the entrance to the occluded bypassed segment of the coronary artery F (Fig. 5d). Lesions tend to develop in regions where there is a large spatial variation in the blood velocity. This is observed around the distal anastomosis region, specifically at the toe and on the floor of the artery just below the anastomosis E. As shown in Fig. 5e, as the flow progresses into the distal coronary artery from E to G, the maximum velocity shifts to the axially centered position in the host vessel and thereafter remains along the centerline indicating that a fully developed flow profile is exiting from the right coronary vessel G (with a maximum velocity magnitude of 0.371 m/s). The computed WSS (product of velocity gradient at the wall and the viscosity of fluid) is then calculated. Figure 5f shows the uniform WSS distribution in the proximal anastomosis region of the graft, as the spatial variation in blood flow velocity is very small. However, unlike the proximal region, the spatial WSS variation in the distal anastomosis is very high (as seen in Fig. 5g), with the magnitude of peak WSS being 5.9067 Pa. Figure 5g depicts (i) a low WSS region near flow-stagnation region along the floor of the artery and (ii) a high WSS around the toe region of the anastomosis. These adjacent regions of highest and lowest wall shear are areas susceptible to arterial diseases.18



271

3.3.2. Flow field in the aorto-right CABG model close to mid-ejection, t = 0.12 s The distribution of the velocity vectors at t = 0.12 s is shown in Fig. 6a. Herein, it is shown that the input flow rate (also depicted in Fig. 4a) at t = 0.12 s is maximum, thus allowing maximum flow to enter the aorta A–B through the aortic inlet A. The velocity at the entrance to the aorta is 0.8912 m/s (Fig. 6b). The high flow rate and the small pressure drop in the aorta (from A to B) allow most of the blood to flow into the ascending aorta at B. The amount of flow entering the coronaries increases during this acceleration phase from t = 0.0 s to t = 0.12 s. Secondary flow patterns are observed near the entrance to the graft at D, resulting in skewed velocity profiles, with higher velocity magnitudes present near the outer wall of the curved graft vessel (Fig. 6c). Further downstream, the flow profile becomes more symmetric. Here, the maximum velocity magnitude in the graft is around 0.5 m/s, compared to 0.18 m/s at the start of ejection. In Fig. 6d, it is seen that the flow exiting from the graft outlet at E is diverted into the distal portion of the bypassed right coronary vessel, with a strong recirculation zone and a velocity magnitude of around 0.175 m/s

flow rate (l/min)

30

m/s

20 10 0 0

0.2

-10

G

0.4 0.6

0.8

1

time (sec)

E F

D

A C

B

Fig. 6a. Velocity-vector plots displayed in the center plane of the aorto-right CABG model close to mid-ejection, t = 0.12 s.


272

D. N. Ghista et al.

m/s

A

D

C

Fig. 6b. At t = 0.12 s, maximum flow enters the aorta through the aortic inlet A. Flow enters the aorta with a uniform velocity of 0.8912 m/s. The high flow rate and pressure drop allow most of the fluid to go straight toward the ascending aorta B. During systole LV contracts, thereby squeezing the coronaries that are embedded in the cardiac muscle. Hence, there is only minimal flow entering the left coronary inlet C.

m/s

A D Fig. 6c. Secondary flow patterns are observed at t = 0.12 s near the entrance of the graft D causing skewness of the velocity profiles with larger velocity magnitudes present close to the outer wall of the curvature of the graft vessel. Further downstream the flow becomes smooth exhibiting symmetry in the flow pattern. The maximum velocity magnitude in the graft is around 0.5 m/s.



273

m/s

E

Fig. 6d. The flow exiting from graft outlet E moves toward the distal portion of the right coronary vessel, with a minimal recirculation region at the heel region and extends upto the proximal portion of the host vessel F. A slight flow reversal is also seen near the toe region. These flow disturbances cause skewing of the profiles toward the inner wall of the bypassed vessel.

m/s

E

Fig. 6e. The maximum velocity of blood in the bypassed right coronary vessel is around 1.12 m/s. With increasing time, it is noted that the magnitude of peak velocity increases contributing to better perfusion of the bypassed vessel.


274

D. N. Ghista et al.

Pa

D B

A

C

Fig. 6f. Variation in velocity near the entrance of the aorta A and the graft inlet D. At the graft inlet D, the WSS is around 7.94 Pa, due to the secondary flow patterns seen in Fig. 6c.

Pa

F

E

Fig. 6g. The region close to the toe and around the heel region, F exhibits zero WSS. Regions of maximum WSS are at the toe and the floor of the artery as expected due to the presence of strong velocity gradients.



275

at the heel of the distal anastomosis. A small flow reversal is also seen near the toe region. These distorted flow patterns cause pronounced skewness of the velocity profile in the host artery. In Fig. 6e, it is shown that the flow in the bypassed right coronary vessel attains a maximum velocity of 1.12 m/s (compared to 0.371 m/s at the start of ejection), resulting in better perfusion of the host artery. The high secondary flow pattern close to the entrance of the graft at D (Fig. 6c) contributes to high WSS at the aorta– graft junction of magnitude 7.94 Pa (Fig. 6f). In the distal anastomosis, the WSS is even higher, with a maximum magnitude of 26.473 Pa. This maximum WSS at the toe and the floor of the artery (Fig. 6g) is expected due to the presence of high velocity gradients (Fig. 6d). 3.3.3. Flow field in the aorto-right CABG model at the start of diastole, t = 0.32 s Similar analysis has been carried out at the start of diastole for the aortoright CABG model (Fig. 7). The distribution of velocity vectors in the flow

flow rate (l/min)

30

m/s

20 10 0 0

0.2

-10

G

E

0.4 0.6

0.8

1

time (sec)

F

A C

D

B

Fig. 7a. Velocity-vector plots at the start of diastole, t = 0.32 s displayed in the center plane of the aorto-right CABG model.


276

D. N. Ghista et al.

m/s

A

D

C

Fig. 7b. At the start of diastole, the leaflets are in the process of closing, and allow only a small amount of the blood to flow into the LV through the aortic inlet A. During diastole the LV relaxes, thereby allowing more blood to flow into the coronaries. Velocity vectors at t = 0.32 s depict the back-flow from the ascending aorta B.

field at t = 0.32 s is shown in Fig. 7a. The velocity vectors (shown in Fig. 7b) depict a back-flow from the ascending aorta B. At this instant, the aortic leaflets are in the process of closing, and a small amount of the blood flows into the LV through the aortic inlet A. Now, as the LV relaxes and the myocardial stress is low, more blood flows into the coronaries. The major portion of flow is shown to enter the left coronary inlet at C (Fig. 7b), with a high velocity magnitude of around 0.285 m/s. The flow at the entrance to the graft is subject to two forces: (i) the pressure gradient, which allows blood to flow along the curvature of the graft; and (ii) the centrifugal force, which makes the fluid to move away from the center of curvature resulting in skewed velocity profiles (see Fig. 7c). Half-way into the graft, the maximum velocity attained at the center of the graft is 0.22 m/s; at the graft exit E, the maximum velocity increases to around 0.475 m/s. The flow exiting from the graft outlet at E is pushed away from the proximal portion of the host artery due to the presence of the recirculation zone located at the heel of the anastomosis (Fig. 7d). Skewing



277

m/s

A D Fig. 7c. Back-flow from the ascending aorta B enters the graft D subjecting the fluid to two forces: the pressure gradient induced in the flow allows blood to flow along the curvature of the graft; the centrifugal force forces the fluid to move away from the center of curvature resulting in skewed profiles. Half-way into the graft, the maximum velocity magnitude attained at the center of the graft is 0.22 m/s; this maximum increases as the flow progresses further downstream.

m/s

E

Fig. 7d. The maximum velocity magnitude of blood flow exiting the graft is around 0.475 m/s. This high velocity flow exiting from the graft outlet E interacts with the region of recirculation seen at the heel which in turn directs the flow toward the distal portion of the bypassed right coronary vessel.


278

D. N. Ghista et al.

m/s

E

Fig. 7e. Skewing of the profile seen toward the floor of the host artery. The flow moves downstream to right coronary artery exit G, with a peak velocity magnitude of 1.0072 m/s.

Pa

D B

A

C

Fig. 7f. Velocity gradients are negligible in the aorta thus producing negligible WSS. The WSS magnitude is around 4 Pa at the aorta–graft junction.



279

Pa

F

E

Fig. 7g. The location of the recirculation region is mirrored in the WSS plot. This plot indicates the presence of elevated shear stresses whose magnitude is 22.8 Pa at the toe of the graft–host junction and on the floor of the artery.

of the flow profile is seen toward the floor of the host artery. The maximum velocity in the host artery is 1.0072 m/s at G (see Fig. 7e). A WSS magnitude of around 4 Pa is seen at the aorta–graft junction (Fig. 7f). The WSS is minimum at the heel (with the presence of the recirculation region), while a high WSS of maximum magnitude (22.808 Pa) is seen at the toe and the floor of the artery (Fig. 7g).

3.3.4. Flow field in the aorto-right CABG model at the mid-diastolic instant, t = 0.57 s At the mid-diastolic instant of t = 0.57 s, Fig. 8a illustrates the velocityvector plots. Now, the major amount of the back-flow coming from the ascending aorta B enters into the graft D, while the remainder goes into the left coronary vessel C (Fig. 8b). Along the bypass graft, the peak velocity is skewed toward the outer wall. Initially, the flow is seen to follow the outer wall, and then this slowly turns to the center (Fig. 8c).The high flow rate at the graft inlet D, along with the curvature effect, results in the highly skewed velocity pattern in the graft. The peak velocity magnitude half-way


280

D. N. Ghista et al.

flow rate (l/min)

30

m/s

20 10 0 0

0.2

-10

0.4 0.6 0.8 time (sec)

1

E

G

F

D

A C

B

Fig. 8a. Velocity-vector plot at mid-diastole instant, t = 0.57 s, in the center plane of the aorto-right CABG model.

m/s

A

D

C

Fig. 8b. At t = 0.57 s, the mid-diastolic instant, the aortic valve is fully closed thus allowing the back-flow from the ascending aorta B to enter the two outlets: the left coronary inlet C and the graft entrance D. Major portion of the back-flow enters into the graft D.



281

m/s

A D

Fig. 8c. The flow entering the graft D follows the same trend as shown in Fig. 8b, but with a difference in the magnitude of velocity. At the mid-diastolic instant, t = 0.57 s, the magnitude of peak velocity half-way into the graft is 0.45 m/s. Maximum perfusion of the graft occurs at mid-diastole.

m/s

E

Fig. 8d. Maximum flow velocity approaching the graft exit E is around 0.95 m/s. The flow exiting the graft E with a high flow velocity results in a stronger impingement on the floor of the artery. The strong recirculation region seen at the heel of the distal anastomosis drives the flow toward the right coronary exit G.


282

D. N. Ghista et al.

m/s

E

Fig. 8e. Significant skewing of the velocity profiles is seen toward the inner wall of the host artery due to the high flow rate. The maximum flow velocity magnitude, 2.0458 m/s, seen close to the floor tends to shift to the center of the vessel only on reaching the right coronary exit G.

Pa

D B

A

C

Fig. 8f. The WSS distribution is negligible in the entire domain of the aorta due to minimal variation in the flow velocity. At the graft entrance, the WSS is minimal.



283

Pa

F

E

Fig. 8g. Maximum perfusion occurs at the mid-diastolic instant, t = 0.57 s. The large difference in velocity magnitude around the toe region E and the floor of the artery results in maximum WSS, whose magnitude is 81.8438 Pa.

into the graft (Fig. 8c) is 0.45 m/s; however, the maximum velocity further downstream in the graft is around 0.95 m/s. Indeed, maximum perfusion of the graft occurs during the mid-diastolic instant. The high flow rate in the graft results in a stronger impingement of the flow on the floor of the artery (Fig. 8d). At the heel (near the occluded end of the artery F) a strong recirculation zone is observed, whereas a small flow reversal is noted at the toe. We observe that because of the high strength of recirculation at the heel, the flow is accelerated toward the distal portion of the bypassed right coronary artery, with a significant skewing of the velocity profile. Now, the peak velocity magnitude in the host artery is 2.0458 m/s (compared with 1.0072 m/s at the start of diastolic phase), depicting maximum perfusion of the bypassed vessel (Fig. 8e). At the graft entrance, the WSS is minimal (Fig. 8f). However, the wall WSS at the toe and the floor of the artery is very high of magnitude 81.8438 Pa. From the heel to the proximal portion of the host artery (Fig. 8g), the WSS is minimal. This strong spatial variation in WSS may make this region prone to development of arterial wall disease, thereby affecting graft patency.


284

D. N. Ghista et al.

4. Blood Flow in an Out-of-plane Aorto-Left CABG Model This section models blood flow in a 3D aorto-left CABG surgery, taking into account the influence of the out-of-plane geometry of the graft on the blood flow features. The finite volume technique has been employed to model 3D blood flow, and to determine the velocity and WSS distributions. We have calculated the distributions of the velocity and the WSS at four typical instants of the cardiac cycle: two during systole (start of ejection, t = 0.0 s; mid-ejection, t = 0.15 s) and the other two during the diastole phase (early diastole, t = 0.32 s; mid-diastole, t = 0.57 s). Plots of velocity vector and the WSS are displayed in the aorto-graft-coronary arterial flowfield domain, providing an insight into the link between fluid dynamics and arterial diseases. 4.1. Geometrical model The model simulation of the flow field of the anastomosis in aorto-left CABG is illustrated in Fig. 9a. The ascending aorta (A–B) is considered to have a length of 80 mm with a diameter of 25 mm. The right coronary artery, RCA (inlet C, in Fig. 9a) has a circular cross section of 3.2-mm diameter and the left main coronary artery, LCA (inlet D, appearing diametrically

C Graft outer wall

LAD

D

G Toe F Heel Artery junction

Junction of the graft with the aorta

A E

Bed of the distal anastomosis H

RCA

LCA Graft inner wall

B Aorta

Fig. 9a. A schematic of out-of-plane aorto-left CABG model which includes (i) the ascending aorta A–B; (ii) the right coronary artery (RCA) inlet C; (iii) the left coronary artery (LCA) inlet D; (iv) the non-planar bypass graft vessel E–F; and (v) the 100% proximally occluded left anterior descending (LAD) artery G–H.



285

opposite to the RCA) has a circular cross section with 4-mm diameter. The LAD is assumed to be a straight tube with length and diameter of 45 and 3.2 mm, respectively. The proximal portion of the LAD (F–G) is assumed to be fully occluded. The venous graft is simulated as a non-planar vessel of circular cross section with diameter of 4 mm. It originates from the aorta at E and anastomosis with the LAD at F. The intersection between the graft and the LAD artery has an elliptic shape which is caused by the deformation of the larger diameter graft due to its sutured attachment to the smaller LAD vessel. The overall dimensions of the total aorto-left CABG model are based on surgical observations by our cardiac surgeon co-author (Dr. Tan Yong Seng). Figure 9b illustrates the different cross sections of (i) the aorta (A, (Ai , i = 1, . . . , 4) and B) with A corresponding to the aortic inlet and B to that of the ascending aorta exit; (ii) CABG (Gi , i = 1, . . . , 6), with G1 corresponding to the first section close to the proximal anastomotic junction and G6 close to the distal anastomotic junction; and (iii) the LAD artery (Ci , i = 1, . . . , 4) with C1 just after the distal anastomotic junction and C4 closest to the exit of the LAD. The flow characteristics that include the axial velocity profiles and in-plane velocity vectors (depicting

A

G1

G2 H

C4 C3 C2 C1

A1 G6

G3 G5

G4

A2 A3 A4

B

Fig. 9b. The CABG model illustrates the different cross sections of (i) the aorta A (Ai , i = 1, . . . , 4), and B, (ii) the graft Gi (i = 1, . . . , 6), and (iii) the LAD artery Ci , (i = 1, . . . , 4).


286

D. N. Ghista et al.

the secondary flow patterns) are displayed on the above-mentioned sections of the CABG model. 4.2. Data input and boundary conditions 4.2.1. Systolic phase The systolic flow is obtained by prescribing a blunt velocity profile (along the cross section) at the inlet to the aorta, from the LV. The velocity magnitude is computed from the physiologically measured stroke volume over the ejection period based on the flow waveform22 as shown in Fig. 10a. During this period, the extravascular myocardial compression of the coronary circulation reduces the blood flow through the coronary arteries significantly. During systole, the inputs to the model consist of: (i) a blunt velocity derived from the time-varying flow-rate waveform (QA ) at the inlet A (shown in Fig. 9a) of the aorta (depicted in Fig. 10a) as adopted from Ganong22 ; (ii) the calculated time-varying flow-rate waveform (QB ) at the exit (B) from the ascending aorta (Fig. 10b); (iii) the time-varying flowrate waveform (QH ) at the distal end H of the LAD artery (Fig. 10c), as derived from the mean velocity waveform obtained from phase-contrast MRI scanning25 ; (iv) the time-varying flow-rate waveform (QD ) (Fig. 10d) at the LCA entrance (D, shown in Fig. 9a) is equal to the flow rate in LCX obtained by phase-contrast MRI scan25 ; and (v) the time-varying flow-rate waveform (QC ) imposed at the entrance C of the RCA (Fig. 10e), as obtained by means of a Doppler flow-meter catheter.26 4.2.2. Diastolic phase At the start of diastole, there exists a small amount of back-flow in the LV through the aortic inlet (see Fig. 10a). During the diastolic phase, the aortic valve remains closed, and it is the back-flow from the ascending aorta (Fig. 10b) that perfuses the coronary arteries. Figures 10c–f reveals that the majority of the flow enters the LCA, which is in agreement with the earlier works.27,28 It can be noted that the perfusion to the coronary arteries is much greater during diastolic phase as compared to the systolic phase as shown in Fig. 10a–f. This implies that the coronary vascular resistance in the left coronary circulation is significantly lower than that of the right. In summary, the data input to the model consists of (1) the calculated blunt velocity profile at the ascending aorta (Fig. 10b) and (2) the flow


25

25

20

20

Flow Rate(l/min)

Flow Rate (l/min)


15 10 5 0

15 10 5 0

0

0.1

0.2

-5

0.3

0.4

0.5

0.6

0.7

0.8

0

0.1

0.2

-5

Time (sec)

0.3

Flow Rate (l/min)

Flow Rate (l/min)

0.2 0.15 0.1 0.05 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.6

0.7

0.8

(b)

0.25

0

0.4

Time (sec)

(a)

0.5

0.6

0.7

0.8

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Time (sec)

(c)

(d)

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0.4 Flow Rate(l/min)

Flow Rate (l/min)

287

0.35 0.3 0.25 0.2 0.15 0.1 0.05

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

(e)

0.6

0.7

0.8

0 0

0.1

0.2

0.3 0.4 0.5 0.6 Time (sec)

0.7 0.8

(f)

Fig. 10. (a) Flow-rate waveform at the inlet of the aorta A, obtained from Ref. 22. (b) The calculated flow-rate waveform at the ascending aorta B. (c) Flow-rate waveform at the distal end of the LAD H, derived from velocity measurements performed using phase-contrast MRI scanning.25 (d) Flow-rate waveform at the LCX derived from velocity measurements performed using phase-contrast MRI scanning.25 (e) Flow-rate waveform at the entrance of the right coronary artery C, measured using a Doppler flowmeter catheter.26 (f) Flow-rate waveform at the left coronary artery D, obtained as a sum of the flow-rate waveforms of the LAD and the LCX.


288

D. N. Ghista et al.

conditions at the LAD exit, LCA and RCA entrances, obtained from the time-varying input flow-rate waveforms (Figs. 10c–e). 4.3. Results Simulations were carried out at various time instants of the cardiac cycle. Herein, we depict the simulation results at the (i) start of ejection, t = 0.0 s; (ii) mid-ejection, t = 0.15 s; (iii) early diastole, t = 0.32 s; and (iv) middiastole, t = 0.57 s. At these time instants, the flow velocity, secondary flow motion, and WSS distributions were determined. The axial velocity profiles in the vessels (aorta, graft, and LAD) were computed along the vertical and horizontal centerlines at all cross sections (Fig. 9b). The in-plane velocity vectors were computed at specific sections of (1) the aorta, at A1 and A3 located at distances of 10 and 40 mm, respectively, from the aortic inlet; (2) the graft, at G1 (close to the proximal anastomosis junction) and G5 (close to the distal anastomosis junction); and (3) the LAD artery, at C1 and C4 located at 20 and 40 mm, respectively, from the proximal end of the artery G.H. 4.3a. Systole phase During systole, blood flows in the forward direction from the LV into the ascending aorta and the coronary vessels. This is termed as forward flow. 4.3a.1. Start of ejection, t = 0.0 s In order to capture the flow features, the computed axial velocity profiles (in the aorta, graft, and LAD) along both the horizontal and vertical centerlines (at all cross sections of the flow domain) are illustrated in Figs. 11a–f. In Figs. 11a and b, it can be seen that the blunt velocity profile assigned at the aortic inlet slowly changes to a parabolic profile as blood flows downstream in the ascending aorta. In the aorta, the axial velocity distribution along the horizontal centerline results in an M-shaped profile due to the influence of the flow entering the RCA. This double-peaked velocity profile gradually changes to a somewhat parabolic profile with increasing distance from the aorta entrance (section A). In Fig. 11a, it can be noted that as the flow progresses further downstream in the aorta, the effect of the proximal anastomotic junction is not significantly felt on the velocity profile, since there is only a very small amount of flow entering the graft. In Fig. 11b, the computed axial velocity


Augmented Myocardial Perfusion by CABG Procedure 0.025

0.02

Aorta Diameter (m)

Aorta Diameter (m)

0.025

Aorta Inlet(A) 10mm from A (A1)

0.015

30mm from A (A2) 0.01

40mm from A (A3) 60mm from A (A4)

0.02

Aorta Inlet (A) 10 mm from A (A1)

0.015

30 mm from A (A2) 40 mm from A (A3)

0.01

60 mm from A (A4) 0.005

0.005

0

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

(a)

0.05

0.06

(b) 0.004

0.0040 0.0035

0.0035

0.0030

Graft Graft Graft Graft Graft Graft

0.0025 0.0020 0.0015

Section Section Section Section Section Section

Graft Diameter(m)

Graft Diameter (m)

0.04

Axial Velocity (m/s)


(G1) (G2) (G3) (G4) (G5) (G6)

0.0010

0.003 Graft Section (G1) Graft Section (G2) Graft Section (G3) Graft Section (G4) Graft Section (G5) Graft Section (G6)

0.0025 0.002 0.0015 0.001

0.0005

0.0005

0.0000 0.000

0.010

0.020

0.030

0

0.040

0

0.01


0.02 0.03 Axial Velocity (m/s)

0.0032

0.0028

0.0028

0.0024 0.0020 20 25 35 40

0.0012

mm mm mm mm

from from from from

the proximal end (G) of LAD (C1) G (C2) G (C3) G (C4)

0.0008 0.0004

Artery Diameter (m)

0.0032

0.0016

0.04

(d)

(c)

Artery Diameter (m)

289

0.0024 0.0020 0.0016 0.0012 0.0008

20 mm fromthe proximalend (G) ofLAD(C1) 25 mm fromG(C2) 35 mm fromG(C3) 40 mm fromG(C4)

0.0004

0.0000 0

0.01

0.02

0.03

0.04

0.05

0.0000 0.00

0.01

0.02

0.03



(e)

(f)

0.04

0.05

Fig. 11. At t = 0.0 s, (a) axial velocity profiles (m/s) extracted along the horizontal centerline of all the cross sections of the aorta. The blunt velocity profile changes to a parabolic profile with increasing distance from the aortic inlet. (b) Axial velocity profiles computed along the vertical centerline of the aorta sections exhibit a slight skewing toward the top wall of the aorta as a result of the junction effect. (c) Axial velocity profiles in the graft sections are parabolic in nature with slight variations in velocity magnitude along different sections. (d) Vertical centerline axial velocity profiles in the graft exhibit slight skewing due to the effect of curvature of the graft vessel. (e) Axial velocity profiles in the different LAD sections are parabolic in nature. (f) Vertical center-line axial velocity plot shows indicates that blood flows in the LAD vessel with a consistent velocity except with a slight difference seen on C1 due to the effect of the distal anastomotic junction.


290

D. N. Ghista et al.

Top

Top

Bottom

Bottom (g)

O

(h)

I

O

(j)

(i)

I

O

(k)

I

I

O

(l)

Fig. 11. (g) In-plane velocity vector components in A1 exhibit symmetric vortices close to the top wall. (h) Similar behavior in the in-plane velocity-vector pattern is observed in A3 . (i) Vector components are all directed toward the inner vessel wall of the graft section G1 . (j) In-plane velocity-vector components shifts from the inner to the outer wall as seen in G5 caused due to the curvature effects of the graft. (k) Small recirculation region observed in the center of C1 of the LAD vessel. (l) Bicellular patterns of negligible velocity magnitude observed in section C4 reflecting the symmetric nature of flow. O: outer wall and I: inner wall.



291

profiles along the vertical line exhibit a slight skewing toward the top wall of the aorta, due to the junction effects. In the graft, it is noted (in Figs. 11c and d) that as blood flow progresses from the proximal junction of the graft to the distal end, the axial velocity exhibits a parabolic profile with slight skewing. Figures 11e and f shows the velocity profiles at different sections in the LAD. Figures 11g–l depict the in-plane flow field, in terms of the in-plane velocity vectors at specific sections of the vessels mentioned earlier. Figures 11g and h show the computed secondary flow vector fields in the cross-sectional plane of the aorta at A1 and A3 . In Fig. 11g, symmetric vortices are observed close to the top wall, while the rest of the core flow is directed upward; this may be due to some flow entering into the RCA junction. On section A3 (after the proximal anastomotic junction), a similar pattern is observed in Fig. 11h with a slightly decreased peak velocity magnitude. In the graft section G1 , the in-plane velocity vectors (shown in Fig. 11i) are all directed toward the inner wall. As flow progresses downstream, the flow direction at section G5 (as in Fig. 11j) changes from the inner to the outer wall of the graft, due to the curvature of the vessel. The flow distribution at section C1 of the LAD immediately after the distal anastomotic junction (in Fig. 11k) shows a small recirculation region at the center of the cross section; herein, the flow is skewed toward the bed of the artery. On reaching the distal end of the LAD (section C4 ), the secondary flow weakens in strength and the flow is purely in the axial direction (see Fig. 11l). The flow-velocity distributions result in WSS distributions (see Figs. 12a and b). The flow disturbances due to the junction effects result in steep velocity gradients around the RCA entrance C and the proximal anastomotic junction E. Hence, the maximum WSS (∼ 0.58 Pa) occurs at the proximal anastomotic junction (Fig. 12a). The magnitude of WSS around the distal anastomotic junction ranges from 0.3 to 0.41 Pa, while at toe magnitude of WSS is 0.27 Pa. As the proximal portion (F–G) of the LAD vessel is completely occluded, the flow is stagnant with negligible WSS (Fig. 12b).

4.3a.2. Mid-ejection, t = 0.15 s During mid-acceleration, the blood enters the aorta with a very high velocity magnitude of around 0.85 m/s (compared to 0.04 m/s at the start of


292

D. N. Ghista et al.

E C

Fig. 12a. WSS distribution (Pa) at t = 0.0 s. Peak magnitude of WSS (0.58 Pa) is present at the proximal anastomotic junction. Through the aortic domain the WSS is negligible with an exception around the RCA entrance (∼ 0.35 Pa) caused due to the flow disturbance.

G

Bed of the distal anastomosis

Heel Toe

H

Fig. 12b. The toe region experiences an appreciable WSS of magnitude 0.27 Pa compared to the heel (∼ 0 Pa) due to the stagnation of flow. WSS is uniformly distributed along the bed of the LAD with magnitude 0.24 Pa.



293

ejection). This instant corresponds to the maximum flow in the ascending aorta during the cardiac cycle. In Figs. 13a and b, the computed axial velocity profiles at different sections of the aorta exhibit a blunt profile, with a slight variation in the magnitude of peak velocity. As expected, the magnitude of the peak velocity increases as the flow proceeds toward the ascending aorta. The flow in the core region has a blunt profile throughout the flow domain (Figs. 13a and b), similar to an inviscid flow. In the graft, significant skewing of the axial velocity profiles is seen at the different sections of the graft (Figs. 13c and d). The curvature of the graft vessel results in skewing of the flow toward the outer wall. In Fig. 13c, it is seen that as blood flows toward the distal anastomotic junction, the peak velocity gets skewed toward the inner graft wall. The distal junction effects along with the non-planarity of the graft are reflected in the axial velocity profiles. In the LAD artery, it is observed (in Fig. 13e and f) that initially the velocity profile is skewed toward the bed of the artery and becomes more uniform downstream. Figures 13g and h depict that the in-plane velocity-vector components in the sections of the aorta are all directed toward the top wall (caused by the flow diversion into the RCA and the graft). The in-plane velocityvector components in the graft sections G1 and G5 are shown in Figs. 13i and j. Counter-rotating vortices in the LAD section C1 , with the peak velocity being on the floor of the artery are shown in Fig. 13k. On reaching the LAD exit, the bicellular secondary flow pattern gradually changes to a single vortex pattern (as seen in section C4 ) of negligible magnitude as depicted in Fig. 13l; that is, it tends toward developed flow. The corresponding WSS distributions at the proximal and distal junctions of the graft are shown in Figs. 14a and b. The blunt axial velocity profiles result in a uniform WSS magnitude of around 2.16 Pa in the aorta. Sites of peak WSS (6.185 Pa) are at the RCA and graft junctions (Fig. 14a) due to high velocity gradients at these sites. At the distal anastomotic region, the WSS at the toe is around 0.62 Pa. The WSS along the bed of the LAD artery is almost uniformly distributed (at 0.45 Pa), as seen in Fig. 14b.

4.3b. Diastole phase During diastole, the back-flow enters the ascending aorta at B (Fig. 9b) due to the elastic recoil of the aortic wall. The flow profiles are hence depicted with a negative sign to indicate the reverse in the flow direction.


294

D. N. Ghista et al. 0.025

0.020

Vertical Distance (m)

Horizaontal Distance (m)

0.025

Aorta Inlet (A) 10 mm from A (A1)

0.015

30 mm from A (A2) 40 mm from A (A3)

0.010

60 mm from A (A4) 0.005

0.020 Aorta Inlet (A) 10 mm from A 30 mm from A 40 mm from A 60 mm from A

0.015

0.010

(A1) (A2) (A3) (A4)

0.005

0.000

0.000 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4


(a)

1

(b) 0.0035

0.0030


0.0025 0.0020 0.0015 0.0010


Graft Diameter (m)

0.0035 Graft Diameter (m)

0.8

0.0040

0.0040

(G1) (G2) (G3) (G4) (G5) (G6)

0.0030


0.0025 0.0020 0.0015 0.0010


(G1) (G2) (G3) (G4) (G5) (G6)

0.0005

0.0005

0.0000

0.0000 0

0.02

0.04

0.06

0

0.08

0.02

0.04

(c)

0.0028

0.0028 Artery Diameter (m)

0.0032

0.0024 20 mm fromthe proximal end (G) of LAD (C1) 25 mm fromG (C2)

0.0016 0.0012

35 mm fromG (C3)

0.0008

40 mm fromG (C4)

0.08

(d) 0.0032

0.0020

0.06



Artery Diameter (m)

0.6


0.0024 0.0020 0.0016 0.0012 0.0008

20 mm from the proximal end (G) of LAD(C1) 25 mm from G (C2) 35 mm from G (C3) 40 mm from G (C4)

0.0004

0.0004 0.0000 0

0.02

0.04

0.06

0.08

0.1

0.0000 0.000

0.020

0040.

0.060



(e)

(f)

0.080

0.100

Fig. 13. At t = 0.15 s, (a) axial velocity profiles at sections of the aorta are all similar in shape to the blunt velocity profile assigned at the aortic inlet A. (b) The vertical centerline plots exhibit a similar profile except with a slight variation in velocity magnitude. (c) Horizontal centerline axial velocity profiles in the graft sections exhibit skewing initially toward the outer graft wall and then on proceeding downstream as the flow gets skewed toward the inner graft wall due to the presence of curvature. (d) Similar pattern is observed in the vertical centerline profiles except with a change in the magnitude of the axial velocity in different vessel sections. (e) The presence of the distal anastomotic junction causes a shift in the peak velocity in the axial velocity profile in C1 . (f) Axial velocity profiles extracted along the vertical centerline of the LAD sections.



Top

Top

Bottom

Bottom

(g)

O

(h)

I

O

(i)

I

I

(j)

O

(k)

295

I

O

(l)

Fig. 13. (g) In-plane velocity vectors on A1 are directed toward the top wall. (h) The magnitude of the in-plane velocity vectors on A3 are similar to that seen on A1 . (i) Inplane velocity-vector components in G1 rarely directed toward the inner graft wall. (j) There is a change in the flow direction on reaching G5 . (k) The LAD section C1 exhibits a bicellular flow pattern with one vortex of negligible strength compared to the other reflecting the skewing of the velocity profiles. (l) The bicellular pattern changes to a unicellular pattern as we move from C1 to C4 . O: outer wall and I: inner wall.


296

D. N. Ghista et al.

E C

(a)

G


Heel

Toe

H (b) Fig. 14. (a) WSS distribution (Pa) at mid-ejection, t = 0.15 s. The steep-velocity gradients observed near the RCA entrance and the proximal anastomotic junction result in maximum WSS (6.18 Pa). At all other regions in the aortic domain, the WSS is almost uniform (∼ 2.16 Pa). (b) The WSS magnitude in the bed of the LAD vessel (∼ 0.43 Pa) is, lesser compared to the toe region (0.62 Pa). The non-planarity of the graft vessel results in a uniform WSS distribution along the bed of the LAD artery.



297

4.3b.1. Start of diastole, t = 0.32 s At this point, there is only a small amount of back-flow from the ascending aorta to the LV (through the aortic valve). Thereafter, the reverse pressure gradient on the aortic valve leaflets (protruding into the coronary sinus) makes the valve close quickly and allows only a small amount of blood to flow back to the LV. A blunt velocity profile is prescribed at the ascending aorta exit B. As the flow progresses from B (shown in Fig. 9b) into the ascending aorta, the axial velocity profile gradually increases in magnitude (see Figs. 15a and b).

20 mm from B (A4) 0.015

40 mm from B (A3) 50 mm from B (A2)

0.010

70 mm from B (A1)

0.020

Aorta Diameter (m)

0.020

Ascending Aorta Exit(B)

0.005

Ascending Aorta Exit (B) 20 mm from B (A4) 40 mm from B (A3) 50 mm from B (A2) 70 mm from B (A1)

0.015 0.010 0.005

0.000 -0.125

-0.1

-0.075

-0.05

-0.025

0.000

0

-0.15

-0.1


-0.05

0


(a)

(b)


(G1) (G2) (G3) (G4) (G5) (G6)

0.0040

0.0035

0.0035

0.0030 0.0025 0.0020 0.0015 0.0010

Graft Diameter (m)


0.0040

Graft Section (G1) Graft Section (G2) Graft Section (G3) Graft Section (G4) Graft Section (G5) Graft Seciton (G6)

-0.125

-0.025

0.0020 0.0015

0.0005

0.0000 -0.225

0.0025

0.0010

0.0005 -0.325

0.0030

Graft Diameter (m)

-0.15

Aorta Diameter (m)

0.025

0.025

0.0000 -0.325

-0.275

-0.225

-0.175

-0.125



(c)

(d)

-0.075

-0.025

Fig. 15. At t = 0.32 s, (a) blood flows from the ascending aorta to the coronaries and as the aortic valve is fully not closed there is a small amount entering the LV. The flow profiles are depicted with a negative sign to indicate change in flow direction. The blunt profile gradually begins to develop as flow progresses toward the aortic inlet. (b) The different sections of the aorta exhibit a similar trend in the axial velocity profile when extracted along the horizontal centerline. Most of the flow enters the coronaries, in specific the LAD, through the graft. (c) Blood enters the graft with a high flow rate. This along with the non-planarity of the graft vessel result in considerable skewing. (d) Pronounced movement of blood from the outer vessel to the inner graft vessel is observed.


20 mm from the proximal end (G) of LAD (C1) 25 mm from G (C2)

0.003

0.0032

0.003

0.0028

0.002 0.002 0.002

35 mm from G (C3)

0.001

40 mm from G (C4)

0.001

0.0024 20 mm fromthe proximal end (G) of LAD (C1) 25 mmfrom G (C2) 35 mmfrom G (C3) 40 mmfrom G(C4)

0.000 -0.35 -0.3

-0.25 -0.2

-0.15 -0.1


(e)

-0.05

0

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

0.0016 0.0012 0.0008 0.0004

0.000 -0.4

0.0020

Artery Diameter (m)

D. N. Ghista et al.

Artery Diameter(m)

298

-0.05

0.0000 0.00


(f)

Fig. 15. At t = 0.32 s, (e) the presence of M-shaped profiles in the LAD sections in C1 and C2 are due to the high flow coming out of the distal anastomotic junction. (f) Predominant skewing of the axial velocity profiles is observed along the bed of the LAD as seen in C1 and C2 . As the flow progresses downstream toward the LAD exit the velocity maximum shifts and the flow pattern exhibits symmetry.

The velocity profiles in the graft (Figs. 15c and d) exhibit considerable skewing. The double curvature of the graft vessel results in change in the skewness of the velocity profile from the outer to inner wall as shown in Figs. 15c and d. Figure 15e shows that the M-shaped velocity profiles at sections C1 and C2 changes to quasi-parabolic profiles at downstream sections C3 and C4 . Prominent skewing of the axial velocity profiles toward the floor of the artery is observed along the vertical centerline in Fig. 15f. Because the flow advances in the ascending aorta from B toward the aortic inlet, the in-plane velocity vector components are first displayed at section A3 followed by A1 (between the RCA and the proximal anastomotic junction) in Figs. 15g and h. On approaching the proximal anastomotic– graft junction E, the suction effect is felt on the in-plane velocity-vector components thereby pulling the velocity vectors toward the top wall as seen in Fig. 15g. It is evident from Fig. 15h that the amount of flow entering the LCA is relatively higher than the flow entering the RCA, thereby directing more flow toward the bottom wall. The significant skewing of the axial velocity profiles (Figs. 15c and d) is reflected in the in-plane velocity vectors (Figs. 15i and j). In the graft, the velocity vectors (Fig. 15i) are directed toward the inner wall and the top portion of the graft vessel. Figure 15j shows a small region of recirculation in the graft section G5 . In the LAD sections (C1 and C4 ), the secondary flow patterns (Figs. 15k and l) are qualitatively similar to those observed at the earlier time instant t = 0.15 s. The corresponding WSS distributions are shown in Figs. 16a and b. Negligible WSS is seen in the aortic domain, with the aorta–graft junction



299

Top

Top

Bottom

Bottom

(g)

(h)

O

I

O

(i)

I

(j)

O

(k)

I

I

O

(l)

Fig. 15. (g) The in-plane velocity components in A3 are seen directed toward the top wall as a result of the suction effect. (h) The presence of the RCA and LCA result in inplane velocity-vector components to be pulled on either side. (i) The high flow rate in the graft along with the curvature of the vessel wall motivates skewing in G1 . (j) Movement of flow from the top to bottom is observed reflecting the presence of the second bend in the graft vessel. Skewing of the in-plane velocities toward the bottom portion of the outer wall results in a small region of recirculation near the top wall. (k) C1 exhibits counterrotating vortices of comparable magnitude reflecting the M-shaped profile of the axial velocity. (l) A small region of recirculation is seen in section C4 at a distance of 5 mm from the LAD exit. This indicates that flow is almost parabolic in nature as observed in the axial velocity plot. O: outer wall and I: inner wall.


300

D. N. Ghista et al.

E C

Fig. 16a. WSS distribution (Pa) at early diastole, t = 0.32 s. During diastole most of the flow enters the coronaries, in specific the LCA. WSS is negligible in the aortic domain. Blood entering the non-planar graft vessel with a high flow rate experiences steep velocity gradients that result in high WSS (5.5 Pa).

G


Heel Toe

H

Fig. 16b. The bed of the artery just opposite to the distal anastomotic junction experience a higher WSS of around 6.0 Pa compared to that at the toe (around 1.5 Pa).



301

depicting a high WSS of around 5.0 Pa. (Fig. 16a). The WSS at the bed of the artery opposite to the distal junction (in Fig. 16b) is relatively higher (6.0 Pa) as compared to that at the toe region (∼ 1.5 Pa). This is expected due to the impingement of blood on the floor of the artery. 4.3b.2. Mid-diastole, t = 0.57 s At this instant, the aortic valve remains closed and the majority of the backflow comes into the ascending aorta and enters into the graft vessel, with the remainder perfusing the LCA and RCA. Figures 17a and b show the axial velocity distributions at different cross sections of the aorta along the horizontal and vertical centerlines, respectively. With increasing distance from B, it is noted that the blunt profile (Fig. 17a) becomes parabolic. It is interesting to observe (in Fig. 17b) that at section A3 (50 mm from B), the axial velocity distribution exhibits a “bidirectional behavior”. This bidirectional behavior may be due to the strong suction of the branching flow into the graft. The non-planarity of the graft vessel breaks the flow symmetry, resulting in pronounced movement of the fluid from the outer wall toward the inner wall (shown in Figs. 17c and d) caused by the pressure gradient induced by the curvature of the graft vessel. The flow (in the LAD exiting from the graft at high speed) results in pronounced skewing of the axial velocity profiles toward the bed of LAD (Figs. 17e and f). The in-plane velocity components in Figs. 17g–l, illustrate that the flow features are found to be qualitatively similar to those observed at the earlier diastolic times except with an increase in the velocity magnitude. The WSS in the aorta is uniform and low (∼ 1.0 Pa), except at the proximal anastomotic junction exhibiting a WSS of magnitude around 17 Pa, (Fig. 18a). The WSS at the toe (around 3.0 Pa) is much lower as compared to that observed on the floor of the artery (14.0 Pa) opposite to the distal anastomotic junction, Fig. 18b. 4.4. Discussion The major objective of this study was to determine the effect of the outof-plane geometry on the flow pattern in a 3D aorto-left CABG model. Quasi-steady-flow simulations were carried out using CFD. Our analysis has focused on the blood flow pattern and WSS, as they are intimately related to the development of arterial diseases. The present model predicts that the non-planarity of the graft vessel geometry is responsible for low WSS observed at the toe and the bed of the LAD artery opposite to the


D. N. Ghista et al.

0.020 Ascending Aorta Exit (B)

0.015

20 mm from B (A4) 40 mm from B (A3) 50 mm from B (A2)

0.010

70 mm from B (A1)

0.025 Aorta Diameter (m)

0.025

0.020

Ascending Aorta Exit (B) 20 mm from B (A4)

0.015

40 mm from B (A3) 50 mm from B (A2)

0.010

70 mm from B (A1)

0.005

0.005

0.000

0.000

-0.0325 -0.0275 -0.0225 -0.0175 -0.0125 -0.0075 -0.0025

-0.035 -0.03 -0.025 -0.02


-0.015 -0.01 -0.005

0


(a)

(b) 0.0040

0.0040

0.0035

0.0025

Graft Section (G3)

0.0020

Graft Section (G4) Graft Section (G5)

0.0015

Graft Section (G6)

0.0030

Graft Section (G1) Graft Section (G2) Graft Section (G3) Graft Section (G4) Graft Section (G5) Graft Section (G6)

0.0025 0.0020 0.0015

0.0010

0.0010

0.0005

0.0005 0.0000

0.0000 -0.4

-0.3

-0.2

-0.1

-0.65

0

-0.55

-0.45

-0.35

-0.25

-0.15

-0.05

Axial Velocity(m/s)

Axial Velocity(m/s)

(c)

(d) 0.0032

0.0028

0.0028

0.0024 0.0020

35 mm from G (C3)

0.0016

40 mm from G (C4)

0.0012 0.0008

Artery Diameter (m)


0.0032

0.0024 0.002

0.0004


0.0016

35 mm from G (C3)

0.0008

40 mm from G (C4)

0.0004

0.0012

0.0000 -0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Artery Diameter (m)

-0.5

Graft Diameter(m)

Graft Section (G2)

0.0035 Graft Diameter (m)

0.0030

Graft Section (G1)

-0.6

Aorta Diamete r(m)

302

0 -0.7

-0.6

-0.5

-0.4

-0.3

-0.2



(e)

(f)

-0.1

0

Fig. 17. At t = 0.57 s, (a) the axial velocity profiles along the horizontal centerline of different sections of aorta exhibit a parabolic profile. (b) The junction effects are strongly reflected in the axial velocity profiles. (c) Pronounced movement of the fluid from the outer wall toward the inner wall is seen. (d) Predominant skewing of the velocity profile with movement from the outer to the inner graft vessel is motivated by the curvature of the graft vessel. (e) The high flow rate from the graft results in double-peaked and asymmetric velocity profiles. (f) With increasing distance along the LAD vessel, the skewed velocity profile seen in C1 and C2 gradually shifts to the centerline of the host vessel.



Top

Top

Bottom

Bottom

(g)

O

(h)

I

O

(i)

I

I

( j)

O

(k)

303

I

O

(l)

Fig. 17. (g) Similar trend as seen during early diastole is observed in the in-plane velocities in A3 . (h) Suction effect due to the presence of the LCA and RCA junctions is seen in A1 of the in-plane velocity plot. (i) A c-shaped profile is seen in G1 of the graft vessel. The high flow in the graft results in an increase in magnitude of the in-plane velocities as compared to early ejection. (j) The pattern of in-plane velocities is qualitatively similar to that seen at t = 0.32 s except with an increase in magnitude. (k) Bicellular in-plane velocity vector pattern in C1 , with one vortex of higher strength (seen close to O) reveal the effect of the distal anastomotic junction. (l) The unicellular vortex pattern in C 4 of reasonable velocity magnitude reflects the asymmetric distribution in the axial velocity profile. O: outer wall and I: inner wall.


304

D. N. Ghista et al.

E C

Fig. 18a. WSS distribution (Pa) at mid-diastole, t = 0.57 s. The distribution of WSS is qualitatively similar as observed during early diastole. The WSS in the aorta is only around 1.0 Pa. The high velocity gradients are responsible for high WSS at the proximal anastomotic junction (around 17.0 Pa).

G


Heel

Toe

H

Fig. 18b. The bed of the artery just opposite to the distal anastomotic junction experience a high WSS of around 14.0 Pa due to the strong impingement of blood on the floor of the artery while the WSS at the toe is around 3.0 Pa.



305

distal anastomotic junction. Our results on WSS distribution are in agreement with Sherwin et al.19 who carried out the flow studies within a distal end-to-side anastomosis model that was fully occluded proximal to the anastomotic junction taking into account the non-planarity of the bypass vessel. Their results showed that non-planarity resulted in a 10% reduction in the peak WSS magnitude on the bed of the anastomosis. It was noted that the WSS is uniformly distributed along the bed of the artery, resulting in small spatial WSS gradients. This may be again due to the non-planarity of the vessel geometry as seen in Caro et al.29 who reconstructed the geometry of the aorta at sites of curvature and bends using casts, and employed MRI techniques to measure the flow patterns in the arteries. Their findings revealed that non-planarity enhanced flow mixing, and resulted in a more uniform WSS distribution. These observations underscore the role of graft geometry on hemodynamics. The value of WSS in the toe region observed in this study is close to the physiologic range of WSS (1–2 Pa), with an exception at mid-diastole where the toe experiences a higher WSS of about 3.0 Pa. Hence, it may be that the in vivo non-planar graft geometry enhances the graft patency.

5. Conclusions and Future Works 5.1. Conclusions The study was undertaken with the aim of understanding the flow dynamics in both in-plane and out-of-plane aorto-CABG models using CFD. The dimensions of the CABG used to construct the models were based on surgical observations (obtained from our cardiac surgeon Dr. Tan Yong Seng). Our work is pioneering in (i) depicting the entire flow field, from the aorta through the entire graft and into the distal coronary vessel; (ii) incorporating monitored physiologic input data at the aortic entrance A, the ascending aorta B, the left coronary inlet C, and the right coronary artery exit G; and (iii) thereby simulating dynamic flow in the entire graft from the proximal to the distal anastomosis. Our results of flow patterns of the aorto-right CABG model at four different instants of the cardiac cycle reveal the following features: (i) At the onset of ejection (at t = 0.0 s), very little flow enters the graft, with a maximum velocity of around 0.18 m/s. A maximum WSS of 5.9067 Pa is seen at the toe and the floor of the artery, with a negligible


306

D. N. Ghista et al.

WSS at the heel and the proximal portion of the right coronary artery. The maximum flow velocity in the distal artery is 0.3710 m/s. (ii) At close to mid-ejection (at t = 0.12 s) due to the secondary flow patterns, the flow profile in the graft is highly skewed toward the outer wall, with a maximum velocity of around 0.5 m/s (compared to 0.18 m/s at t = 0.0 s). In comparison to the start of systole, the maximum WSS magnitude at the toe and the floor of the distal anastomosis region is found to increase fourfold (from 5.9067 to 26.473 Pa. The maximum flow velocity in the distal artery is 1.12 m/s. (iii) At the start of diastole (at t = 0.32 s), there is some flow entering the LV, and more blood flows into the left coronary vessel. This decreases the amount of flow into the graft. The value of maximum flow velocity in the graft is around 0.475 m/s. The maximum WSS at the top and the floor of the distal anastomosis region is found to be 22.808 Pa. The maximum flow velocity in the distal artery is 1.0072 m/s. (iv) At the mid-diastolic instant (at t = 0.57 s), the graft perfusion is maximum, and the maximum flow velocity is around 0.95 m/s. The WSS at the top and the floor of the distal anastomosis region is also maximum of 81.8438 Pa. The maximum flow velocity in the distal artery is 2.0458 m/s. This indicates that the myocardial perfusion is maximum at mid-diastole. We have thus provided a comprehensive picture of the flow and shear stress fields in the grafted region, from the proximal anastomotic region to the distal anastomotic region. The high spatial velocity variations with corresponding high WSS at artery are sites susceptible to arterial disease. Our work on the out-of-plane aorto-left CABG model has shed some light on the fluid-dynamic changes in this simulated real-life bypass model. We are reporting the following findings, deemed to be of clinical relevance: 1. Shear stresses are present at the proximal (aorto-saphenous vein) anastomosis. We may need to review our current way of performing proximal anastomosis to reduce or eliminate the WSS at this location by altering the angle or making use of connector devices. 2. The twisted graft vessel geometry has resulted in swirling of blood between the inner and outer wall of the graft. This may be deemed to be advantageous, as it would scour the arteries of deposits. Our current method of constructing CABG is to construct a relatively straight vascular tube using the saphenous vein between the aorta and the obstructed



307

coronary artery. With this new information, we hope to improve coronary bypass surgery by taking the helical geometry into consideration. Non-planar helical bypass could theoretically prevent graft failure in two ways. Firstly, they would eliminate the dead spots where the graft meets the artery, therefore the blood could scour out the inside of the bend, preventing plaque formation. Secondly, it can eliminate WSS fluctuation which is the potent stimulus for smooth muscle cell over-proliferation in intimal hyperplasia. 3. The out-of-plane (twisted) geometry of the aorto-left CABG is seen to reduce the spatial variation of WSS distribution along the bed of the distal anastomosis, thereby reducing the damage to the endothelium. However, high shear stress is still present at the toe region of the distal anastomosis. To eliminate this, a connector device may be useful to change the fluid-dynamic profile at this location.

5.1.1. Clinical implications of study The role of WSS in arterial disease has been the subject of debate for over three decades. Fry8 reported that if the endothelial cells are exposed 2 to a WSS magnitude of over 37 N/m , the endothelial surface would be 9 subject to denudation. Caro et al. on the other hand, have reported that early atherosclerotic lesions are present in low WSS areas due to sheardependent mass transport for atherogenesis. Furthermore, regions of low WSS and flow recirculation have been shown to correlate with locations of atheroma in coronary arteries.10 In summary, the hemodynamic culprits for intimal hyperplasis and atherogenesis are disturbances to streamlined flow, including WSS magnitude (too low or too high relative to a homeostatic value) and direction (forward versus reverse flow), WSS gradients (temporal and spatial), high oscillatory shear index (OSI), flow separation and secondary flow, and long particle residence time (see review in Kassab and Navia30 ). The present paradigm is that these hemodynamic features influence endothelial cell response by inducing platelet activation, cell migration, and vascular smooth muscle cell (VSMC) proliferation, and release of mitogenic factors and proteinases.30 In the present study, we determined the WSS distribution from the velocity profiles based on realistic out-of-plane 3D geometry of CABG. Our results show that the WSS is uniformly distributed on the bed of the anastomosis (opposite to the distal anastomotic junction) and the magnitude of WSS at the toe region is close to the physiologic range. The section of the


308

D. N. Ghista et al.

host artery close to the heel is continually exposed to low WSS, due to the proximity of relatively stagnant fluid in this vessel segment. Our non-planar WSS model results are consistent with those obtained by Sherwin et al.19 for a distal end-to-side anastomosis junction. The non-planar graft vessel geometry has resulted in swirling of blood between the outer and inner walls of the graft. This may be deemed advantageous as it might wash out the deposits of the arteries that cause arterial disease.31

5.2. Future works The simulations studies in the CABG models considered have clearly shown that how a slight modification in the geometry contributes significantly to the flow dynamics in the CABG. Thus in future we would like to work on (i) optimization of the graft geometry that may improve the graft patency; (ii) to make some in vivo measurements of the flow velocity at different inlets of the vessels (by our team) so as to simulate the flow dynamics based on physiologic measurements; and (iii) to compare the hemodynamic patterns and WSS values obtained through CFD with experimental results (done using PIV) so as to enable a necessary validation of our work. Finally, having acquired more insight into the fluid-dynamic changes in the coronary bypass graft, the challenges for the team ahead are how to incorporate all these findings in designing the ideal proximal and distal coronary connectors and to validate the concept of non-planar helical graft in practice. This may lead to a new revolution in our thinking with regards to the concept of the shape of the bypass graft in that a “kinky” graft could be more advantageous compared to a straight graft.

References 1. A. M. Imparato, A. Braccco, G. F. E. Kim, and R. Z. Zeff, Intimal and neointimal fibrous proliferation causing failure of arterial reconstruction, Surgery 2 (1972) 1007–1017. 2. S. T. Nikkari and A. W. Clowes, Restenosis after vascular reconstruction, Annals of Medicine 26 (1994) 95–100. 3. A. W. Clowe, Pathologic intimal hyperplasia as a response to vascular injury and reconstruction, Vascular Surgery, 4th edn., ed., R. B. Rutherford (WB Saunders Company, Philadelphia, 1995) 285–295. 4. F. W. Lo Gerfo, T. Soncrant, T. Teel, and C. F. Dewey Jr., Boundary layer separation in models of side-to-end arterial anastomoses, Archives Surgery 114 (1979) 1369–1373.



309

5. R. E. Clark, S. Apostolou, and J. L. Kardos, Mismatch of mechanical properties as a cause of arterial prosthesis thrombosis, Surgery Forum 27 (1976) 208–210. 6. F. W. Lo Gerfo, W. C. Quist, M. D. Nowak, H. M. Crawshaw, and C. C. Haudenschild, Downstream anastomotic hyperplasia. A mechanism of failure in Dacron arterial grafts, Annals of Surgery 197 (1983) 479–483. 7. D. P. Giddens, C. K. Zarins, and S. Glagov, The role of fluid mechanics in localization and detection of atherosclerosis, ASME Journal of Biomechanical Engineering 115 (1993) 588–594. 8. D. L. Fry, Acute vascular endothelial changes associated with increased blood velocity gradients, Circulation Research 22 (1968) 165–197. 9. C. G. Caro, J. M. Fitzgerald, and R. C. Schroter, Atheroma and arterial wall shear stress. Observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis, Proceedings of the Royal Society of London Series B 177 (1971) 109–159. 10. T. Asakura, and T. Karino, Flow patterns and spatial distribution of atherosclerotic lesions in human coronary arteries, Circulation Research 66 (1990) 1045–1066. 11. M. Ojha, C. Ethier, K. Johnston, and R. Cobbold, Steady and pulsatile flow fields in an end-to-side arterial anastomoses model, Journal of Vascular Surgery 12 (1990) 747–753. 12. P. E. Hughes, and T. V. How, Effects of geometry and flow division on flow structure in models of the distal end-to-side anastomosis, Journal of Biomechanics 29 (1996) 855–872. 13. H. S. Bassiouny, S. White, S. Glasgov, E. Choi, D. P. Giddens, and C. K. Zarins, Anastomotic intimal hyperplasia: mechanical injury or flow induced, Journal of Vascular Surgery 15 (1992) 708–717. 14. J. A. Moore, D. A. Steinmann, S. Prakash, K. W. Johnson, and C. Ethier, A numerical study of blood flow patterns in anatomically realistic and simplified end-to-side anastomoses, Journal of Biomechanical Engineering 121 (1999) 265–272. 15. Min-Ho Song, S. Masaru, and U. Yuichi, Three-dimensional simulation of coronary artery bypass grafting with the use of computational fluid dynamics, Surgery Today 30 (2000) 993–998. 16. C. Bertolotti, and V. Deplano, Three-dimensional numerical simulations of flow through a stenosed coronary bypass. Journal of Biomechanics 33 (2000) 1011–1022. 17. S. M. Kute, and D. A. Vorp, The effect of proximal artery flow on the hemodynamics at the distal anastomosis of a vascular bypass graft: computational study, Journal of Biomechanical Engineering 123 (2001) 277–283. 18. C. R. Ethier, S. Prakash, D. A. Steinman, R. L. Leask, G. G. Couch, and M. Ojha, Steady flow separation patterns in a 45◦ junction, Journal of Fluid Mechanics 411 (2000) 1–38. 19. S. J. Sherwin, O. Shah, D. J. Doorly, J. Peiro, Pappaharilaou, N. Watkins, C. G. Caro, and C. L. Dumoulin, The influence of out-of-plane geometry


310

20. 21. 22. 23.

24.

25.

26.

27. 28.

29.

30.

31.

D. N. Ghista et al.

on the flow within a distal end-to-side anastomosis, ASME Journal of Biomechanical Engineering 122 (2000) 86–95. Fluent: Fluent User’s Guide. New Hamisphere; 2001. Gambit: Gambit User’s Guide. New Hamisphere; 2001. W. F. Ganong, Review of Medical Physiology (Appleton & Lange, 1999). B. K. Lee, H. M. Kwon, D. Kim, Y. W. Yoon, J. K. Seo, and I. J. Kim, Computed numerical analysis of the biomechanical effects on coronary atherogenesis using human hemodynamic and dimensional variables, Yonsei Medical Journal 39, 2 (1998) 166–174. D. Zeng, Z. Ding, M. H. Friedman, and C. R. Ethier, Effect of cardiac motion on right coronary artery hemodynamics, Annals of Biomedical Engineering 31 (2003) 420–429. J. Suo, Y. Yang, J. Oshinski, A. Tannenbaum, J. Gruden, and D. Giddens, Flow patterns and wall shear stress distributions at atherosclerotic-prone sites in a human left coronary artery — an exploration using combined methods of CT and computational fluid dynamics, Proceedings of the 26th Annual International Confernece of the IEEE EMBS San Francisco, CA, USA, September 1–5, 2004, 3789–3791. S. Matsuo, M. Tsuruta, M. Hayano, Y. Imamura, Y. Eguchi, T. Tokushima, and S. Tsuji, Phasic coronary artery flow velocity determined by Doppler flowmeter catheter in aortic stenosis and aortic regurgitation, American Journal of Cardiology 62 (1988) 917–922. W. A. Mc Alpine, Heart and Coronary Arteries (Springer-Verlag, New York, 1975). M. Zamir, and M. D. Silver, Morpho-functional anatomy of the human coronary arteries with reference to myocardial ischemia, Canadian Journal of Cardiology 1 (1985) 363–372. C. G. Caro, D. J. Doorly, M. Tarnawski, K. T. Scott, Q. Long, and C. L. Dumoulin, Non-planar curvature and branching of arteries and nonplanar-type flow, Proceedings of the Royal Society of London Series A 452 (1996) 185–197. G. S. Kassab, and J. A. Navia, Biomechanical considerations in the design of CABG: the mechanical homeostatis hypothesis, Annals of Thoracic Surgery (in press). Jonathan Knight, Cunning plumbing, New Scientist (1999) 32–37.

List of Publications Journal papers 1. S. Meena, L. P. Chua, D. N. Ghista, and Y. S. Tan, Computational model of blood flow in the aorto-coronary bypass graft, Biomedical Engineering OnLine 4 (2005) 14. 2. S. Meena, L. P. Chua, D. N. Ghista, and Y. S. Tan, Flow studies in three dimensional aorto-right coronary bypass graft system, Journal of



311

Medical Engineering and Technology, 30, 5 (September/October 2006) 269–282. 3. S. Meena, D. N. Ghista, L. P. Chua, Y. S. Tan, and G. Kassab, Analysis of blood flow in an out-of-plane CABG model, Am. J. Physiol. Heart Circ Physiol, July 2006, 291: H283–H295. Papers in refereed conference proceedings 1. S. Meena, D. N. Ghista, L. P. Chua, and Y. S. Tan, Blood flow velocity and pressure profile in the coronary graft, Singapore Cardiac Society 16th Annual Scientific Meeting, Singapore, March 2004, pp. 58–59. 2. S. Meena, L. P. Chua, D. N. Ghista, and Y. S. Tan, Flow studies in aorto-right coronary bypass graft system, 1st International Bioengineering Conference 2004, Singapore, December 2004, pp. 274–275. 3. S. Meena, D. N. Ghista, L. P. Chua, and Y. S. Tan, Effect of out-ofplane geometry on the flow in an aorto-left coronary artery bypass-graft model, ASME 2005 Summer Bioengineering Conference, Vail, Colorado, June 22–26, 2005. 4. S. Meena, D. N. Ghista, L. P. Chua, and Y. S. Tan, Influence of aorto-left coronary bypass graft geometry on wall shear stress distribution, 27th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS 2005, Shangai China, September 1–4, 2005. 5. S. Meena, D. N. Ghista, Chua Leok Poh, and Tan Yong Seng, Comparison of wall shear stress distribution in the normal left coronary artery tree and bypassed left coronary artery tree, ICBME, Singapore, December 7–10, 2005.




NUMERICAL SIMULATION AND PIV MEASUREMENT OF TWO PROXIMAL ANASTOMOSIS MODELS LEOK POH CHUA∗ , JUN-MEI ZHANG and DHANJOO N. GHISTA† School of Mechanical and Aerospace Engineering College of Engineering, Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 ∗[email protected] †[email protected]

Nomenclature Bn DG f i J0 J1 N OSI

Fourier coefficients Diameter of graft (6 mm) Frequency (Hz) √ −1, unit imaginary number First kind of Bessel function of order 0 First kind of Bessel function of order 1 Total number of Fourier transform terms Oscillatory shear index, defined as  1 1− OSI = 2

p QG Q(t) r R Re ReG t T

)  T 0 τw dt  )T |τ | dt w 0

Static pressure (Pa) Mean flow rate in the graft during the pulsatile flow cycle (m3 /s) Flow rate at time t(m3 /s) Radial location (m) Radius of the aorta (10 mm) Reynolds number of aorta, defined as Re = 2ρuR/µ Reynolds number of graft, defined as ReG = ρuDG /µ Time (s) Time period (s) 313


314

L. P. Chua et al.

ui

Velocity in i direction (m/s), i = 1, 2, 3 for x, y, z directions in Cartesian coordinate, respectively u(r, t) The distribution of axial velocity (m/s) at different radial location (r) and time (t) WSS Time-averaged wall shear stress (Pa), defined as )T WSS = 1/T 0 |τw |dt WSSG Normalized time-averaged wall shear stress gradient defined as 2 2 2 T ∂τy ∂τz ∂τx 1 DG WSSG = + + dt T τG 0 ∂x ∂y ∂z xi

α µ ρ τG

Location in Cartesian coordinate (m), i = 1, 2, 3 for x, y, z directions, respectively Womersley number, defined as α = R ω/υ Dynamic viscosity of the working fluid (Pa s) Density of the working fluid (kg/m3 ) Poiseuille-type wall shear stress at the graft corresponding to the mean flow rate in the graft (Pa), defined as τG =

τi τw υ ω

32µQG 3 πDG

Wall shear stress in Cartesian coordinate (Pa), i = x, y, z for x, y, z directions, respectively Wall shear stress (Pa), defined as τw = µ(∂u/∂n)|wall , where ∂u/∂n|wall is the normal velocity gradient at the wall Kinematic viscosity of the working fluid (m2 /s) Angular frequency (rad/s) of the oscillatory motion, defined as ω = 2πf

1. Introduction Since the introduction of saphenous vein grafting in the late 1960s, coronary artery bypass grafting (CABG) remains the standard treatment for the management of intractable angina due to coronary artery occlusive disease. However, CABG is not without complications.1 The saphenous vein graft provides only palliation of the ongoing process. Approximately 15 to 20% vein grafts occlude in the first year and 50% occlude within the first 2 years. After 10 years, approximately 60% of vein grafts are patent; only


Numerical Simulation and PIV Measurement

315

50% of these vein grafts remain free of significant stenosis.2 The pathology in the grafts, documented by angiography and histological examinations, includes acute thrombosis and intimal hyperplasia (IH) during the first postoperative year and the onset of progressive atherosclerosis beyond 3–5 years. It is therefore not unexpected that numerous research works have been carried out to investigate the occlusion symptom or commonly known as stenosis. Previous studies have shown that hemodynamics is the most important factor.1,3–5 Hemodynamic parameters (HPs) were thus suggested to encapsulate “disturbed flow” that might trigger a cascade of abnormal biological process leading to intimal thickening and/or thrombi formation.6 These HPs included low wall shear,7,8 low and oscillating shear,9 high wall shear,10 “safe-bandwidth” of wall shear stress (WSS),11 wall shear stress gradient (WSSG),12,13 and oscillation shear index (OSI).14 Utilizing these parameters as “indicator” functions, internal branching blood vessel geometries were analyzed, and altered for early detection of possibly highly stenosed vessel segments, prediction of future disease progression, and vessel redesign to potentially improve long-term patency rates.6 In terms of anastomoses, only a few investigations have employed the HPs concept to generate a new-optimal distal anastomosis;13,15 also, proximal anastomosis has received relatively little attention although it could provide the condition to form mitogens and activated platelets, which are convected down to the distal part.16 Therefore, one objective of this study is to investigate the flow structure and distributions of HPs in different proximal anastomosis models, to obtain an optimal model. So far, the comparison of numerical results with experimental data has either not been conducted or has been conducted in a simplified model using a flow visualization method5,17,18 or laser Doppler anemometer (LDA) measurements.19,20 Although particle image velocimetry (PIV) method was recently implemented for hemodynamics investigation21,22 as a wholeflow-field technique in contrast to the traditional point-wise velocity measurement techniques (such as hot-wire and LDA), comparisons between the results of numerical simulation and PIV measurements have not been reported to date. In this study, PIV measurements were also carried out to complement the computational investigations of the hemodynamics for proximal anastomosis. Subsequent comparisons between the numerical and experimental results were performed to verify the validity of numerical simulation.


316

L. P. Chua et al.

2. Methodology and Theory 2.1. Physical models and physiological flow conditions The Pyrex glass models used in PIV experiments had the same geometry as the models of the numerical simulation, which were designed to represent the proximal anastomosis based on clinical data of Asian patients from the National Heart Centre of Singapore. The internal diameters of the aorta and graft were 20 and 6 mm, respectively. As the results of Moore et al.23 have shown that simplified models provided sufficient information for comparing hemodynamics with qualitative or average disease locations, this study focuses on the influence of grafting angle. Shown in Fig. 1 are the schematic views of 90◦ and 135◦ proximal anastomosis models, which typically represent the CABG for left and right coronary arteries, respectively.

Aorta

90º Toe

Heel

Bypass graft

Inner Wall

Outer Wall

(a)

135º Toe

Heel

Aorta

Inner Wall

Bypass graft Outer Wall

(b) Fig. 1.

Schematic design of the (a) 90◦ and (b) 135◦ proximal anastomotic models.



317

100

Mean Velocity(cm/sec)

80 60 40 20 0 0

40

80

120

160

200

240

280

320

360

-20 -40

Phase Angle (Degree)

Fig. 2. Comparison of pulsatile waveform for simulation and PIV experiment (with error bars, — o ) and Nichols and O’Rourke, (.......)24

Figure 2 shows the inlet flow waveform used for pulsatile flow study, to be close to the in vivo measurements of Nichols and O’Rourke,24 it is characterized by a first peak Reynolds number (Re) of 5,430, a second peak Re of 667, a mean Re of 1,293, and a Womersley number (α) of 12.2. The Re throughout the anastomosis was lower than the critical value of 5,500, indicated by Peacock et al.25 This suggests that the flow was laminar, and this was further validated by PIV measurements, as turbulence was not observed in the flow field during the experiments.

2.2. Computational simulation method As CABG is usually performed on the elderly and associated with vessels > 2 mm in diameter, their blood in general has low hematocrit with high shear rate, and thus can be approximated as a Newtonian fluid. The study of Xu et al.26 has also demonstrated that the predicted velocity profile did not differ much, for either treating blood as Newtonian or non-Newtonian fluid. In addition, Steinman and Ethier27 concluded that effects of wall distensibility were less pronounced than those associated with changes in arterial geometry and flow conditions. Hence, this justified our assumptions of incompressible Newtonian fluid and rigid arterial and graft walls in this study. To be consistent with the working fluid used in PIV measurements, blood was assumed to have a dynamic viscosity of 4.08 × 10−3 Pa s and a


318

L. P. Chua et al.

density of 1,055 kg/m3 . The corresponding unsteady laminar blood flow is governed by the following mass and momentum conservation equations: ∂ (ui ) = 0. ∂xi ∂ui ∂ui ∂τij ∂p ρ + uj + . =− ∂t ∂xj ∂xi ∂xj

(1) (2)

The commercial software Fluent (version 5.5), based on finite-volume methodology, was applied to solve mass and momentum conservation equations, with user-supplied C programs for calculating input-velocity profile. Since the flow was fully developed at the inlet of model within the circulation loop of PIV experiment, a transient Womersley solution28 was implemented as the time-varying inlet-velocity profile as per the flow waveform shown in Fig. 2. In addition, the outlets were assumed to have zero gauge pressure so as to keep consistent with that of PIV experiment. In this study, all the equations were discretized by the second-order upwind scheme, which provided similar results with the Quick scheme. The resulting coupled nonlinear equations were solved by a segregated solver. Fully implicit scheme was adopted for temporal discretization. Convergence of iterations was controlled by the scaled residual, for both velocity and pressure and the limit used was 5 × 10−5 . All calculations were performed on a workstation (SGI Origin 2000) with the operating system SGI IRIX. After some trials, 1/25 of time period (T = 1.25 s) was adopted to be the preferential time step. To eliminate the startup effect of transient flow, the computation was carried out over at least two and a half periods (more than 48 h of CPU time). The pulsatile flow results presented here are solutions obtained after one cycle, with the variation of velocity within 0.002% compared to the third period. To reduce computational time and memory, only one symmetric-half of models were built and meshed. A total of 45,451 nodes and 77,736 elements were used for each model. Mesh density near the wall was larger than elsewhere, to get more accurate WSS. Grid independency has been demonstrated by comparing the velocity profiles with different meshes. The lengths of grafts and aorta were also checked to fulfill the fully developed flow. There is strong biological evidence that HPs encapsulate “disturbed flow”, which may trigger a cascade of abnormal biological processes leading to intimal thickening and/or thrombi formation.6 Hence, the following HPs were computed, including WSS, time-averaged WSS, time-averaged WSSG,



319

and OSI, for these two models, by using the following equations: τw = µ(∂u/∂n)|wall . 1 T |τw | dt. WSS = T 0 2 2 2 ∂τy ∂τz ∂τx 1 DG T WSSG = + + dt. T τG 0 ∂x ∂y ∂z  )  T τ dt 0 w 1 . OSI = 1− )T 2 |τw | dt

(3) (4)

(5)

(6)

0

Segmental averages of time-averaged WSS, WSSG, and OSI were calculated by Ref. 29 HP dAsurface segment , (7) HP = dAsurface segment

because the segmental average of HPs (especially WSSG) was found to show a statistically significant correlation with the intimal white blood cell (WBC) densities.30 2.3. Experimental arrangement and methodology Figure 3 shows a schematic drawing of the constructed experimental test rig.31 The fluid from the reservoir tank (also acting as head tank) was initially allowed to fill up the entire system, so as to remove any trapped air bubble in the piston, pipes, etc. After the system was fully filled, the control valve (5) was closed and the motor attached with a cam was turned on. The specially designed cam could activate the piston for creating the required waveform, and a control circuit provided the external triggering signal to synchronize the PIV system for capturing the images of wholeflow-field with respect to the phase of pulsatile flow. The error bars for flow waveform adopted in the PIV measurements were shown in Fig. 2, with average error of 2% and the experimental curve was exactly the same as that of numerical simulation. At the visualization section, the Nd:YAG laser illuminated the cross-sectional plane of test model, which was immersed in the transparent tank filled with the same liquid as the working fluid. The CCD camera, mounted vertically above test model, instantaneously


320

L. P. Chua et al. Main outlet flow to Graft outlet flow to Valve (3) Flexible tubing Valve (2) Adapter

Reservoir Tank

Overflow Drainage Inlet flow from Reservoir

Link to PIV Synchronization unit

(6)

Laser Sheet

Test Model

Plan view of the transparent tank Backpressure Tank

CCD Camera (2) (3)

(4)

(5) Sump Tank

EMF. (2)

Centrifugal Pump Cam

(1) EMF. (1) Contraction Cone Transparent Tank

Flow Direction

Piston Motor

Laser

Electromagnetic flow meter Control Valve Check Valve

Fig. 3.

Schematic drawing of the experimental arrangement.

captured two digital images at a time interval of a few mini-seconds. These images can be processed to obtain the velocity vectors of the flow fields. To match the refractive index of the Pyrex glass model and mimic the fluid properties of blood, the working fluid was a mixture of ammonium thiocyanate, glycerin, and water, which had the refractive index and dynamic viscosity of 1.47 and 4.08 × 10−3 Pa s at 22◦ C, respectively. Polyamid seeding particles (PSP-20, Dantec Measurement Technology) with mean diameter of 20 µm were selected, based on a compromise between an adequate tracer response of the particles in the fluid and a high signalto-noise ratio (SNR) of the scattered light signal. The experimental parameters (such as the time between pulses, pulse duration, and separation) were carefully determined and verified during measurements. The experimental WSS distribution was based on the detection of flow velocities in the vicinity of wall within the viscous boundary layer. A curve was then fitted to these data, for obtaining a near-wall velocity profile, and the slope of the curve at the wall was determined as a measure of velocity gradient or shear rate.32 In this study, a linear approximation with three experimental points was implemented to calculate WSS, for a straight cylindrical tube as elucidated in the next section.



321

3. Pre-validation of Numerical Simulation and PIV Measurement 3.1. Pre-validation of numerical simulation Before carrying out numerical simulation of the proximal anastomosis models, a validation of simulation method was carried out by comparing with the Womersley solution for fully developed pulsatile flow in a straight circular tube, to assess the accuracy of numerical methods and examine such important issues as spatial and temporal discretization errors. Accordingly, a circular artery with a length of 3 cm and radius of 0.2 cm, which was subject to uniform inlet flow, was used for computation validation. The inlet velocity was uniform in space and periodic in time. The time variation was described by a sinusoidal function u(t) = u[1 + sin(2πt/T )] with mean velocity u = 13.5 cm/s and period, T of 0.2 s. The fluid had a kinematics viscosity of 0.04 poise, which resulted in a mean Reynolds number of 135 and Womersley number of 5.6. Axis-symmetry was exploited, and hence a half-model was constructed. The model was discretized into 65,098 nodes and 59,640 hexahedral elements. The simulated results show that the flow fully developed in a length equivalent to approximately 10 times of radius from the inlet (as evidenced by the axial velocity distribution of symmetry plane) does not vary with axial position thereafter. In addition, the outflow volume was monitored to compare the inflow volume, to verify that incompressibility and continuity were satisfied. As observed in Fig. 4(a) and (b), the outlet-volume flow rate did match the inflow-volume flow rate. Further, comparison of the computed axial velocity results at the outlet of the artery with the Womersley solution valid at the corresponding time interval is shown in Fig. 4(c). The simulation solution was found to match the theoretical velocity profile well, with differences observed primarily along the centerline of the vessel in the first half of the cardiac cycle, where the numerical results deviate from the exact solution a little. Although there were minor differences of velocity distributions in Fig. 4(c), it is worth to note that the volume flow rate was exactly the same for both analytical and numerical solutions. The level of discretization used for this example resulted in approximately 20 nodal points across the diameter, and a maximum error of less than 4.07%. As the exact solution is known for this problem, it should be possible to further improve the agreement between the numerical simulation results and the Womersley solution with a finer mesh.


322

L. P. Chua et al.

Fig. 4.

Pre-validation of numerical simulation.

3.2. Pre-validation of PIV measurement Before measurements, it is necessary to quantify the accuracy of velocity profiles obtained from PIV. In this application, one straight glass tube with 20 mm inner diameter was taken for verification. As the measurement



323

under pulsatile flow condition is very tedious and time consuming, the validation was carried out under steady flow condition. Firstly, the flow rate obtained from electromagnetic flowmeter (0.06 m3 /h) was compared with that derived by integrating the velocity profile captured from PIV measurements. With the aid of small apparatus (such as the prolonged tube circuit between the test section and the reservoir tank, the honeycomb contraction structure and flow straighteners just before the test section, etc.), the flow into test section became fully developed. This was demonstrated by the nearly constant measured axial velocity contours and vectors in symmetry plane, as shown in Fig. 5(a). ) ¯ was calculated from ¯ = r u(r)2πr dr, the average flow rate, Q, Using Q 0 the velocity profile. It is to be noted that u(r) is the local velocity with respect to radius (r) of the tube, and dr is the small increment in length of the cross-sectional annulus along the radial direction. From measured velocity, we derived the average flow rate at plane 1 (x/D = 1) and plane 2 (x/D = 2) (where x represents the axial distance from the inlet to the measured plane and D is the inner diameter of the tube) to be 0.0562 and 0.0560 m3 /h, respectively. The discrepancies between the values attained by PIV measurements and flowmeter recording (0.06 m3 /h) were within 6.7%. This small discrepancy could be due to the disturbance of background noise, wall reflection, and the accuracy of positioning the illuminating laser sheet at the measuring plane as well as the uncertainty of the flowmeter. Additionally, the velocity profile was observed to be parabolic in shape, as shown in Fig. 5(a). The quantitative comparisons between normalized velocities calculated at both planes 1 and 2 as well as those of theoretical solution are shown in Fig. 5(b). From the figure, it is found that the experimental results of velocity distributions agree well with the theoretical Hagen–Poiseuille solution for circular pipe flow, with a maximum error of 4.1%; thus the accuracy of the velocity profiles obtained from PIV measurements is acceptable. Since the accuracy of PIV ultimately depends on the uncertainty of the measurements of particle displacement, the uncertainty of velocity measurement in this study can be elaborated as follows. If a standard cross-correlation-based sub-pixel PIV-processing technique is used and valid correlation peaks are detected, a commonly accepted uncertainty in the displacement measurement is in the order of 0.1 pixel.33 With the double-pulse separation of 600 µm, 0.1 pixel translated to an error of 0.0029 m/s, which was 5.47% of the pipe mean velocity. As regards the approximation error for wall shear rate (WSR) estimation, Fatemi and Rittgers32 found that it depends not only on the number of points and order of the curve-fitting, but also on the distance of the


324

L. P. Chua et al.

u (m/s) 0.093 0.084 0.075 0.065 0.056 0.047 0.038 0.029 0.020 0.011 0.002

(a) Velocity contour and vectors for symmetry plane of the straight tube 1.0

U/Umax

0.8

0.6

0.4

Velocity Ratio at plane 1 Velocity Ratio at plane 2 Theoretical Results

0.2

0.0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

r/R

(b) Comparison of normalized velocity profiles between measured and theoretical results Fig. 5.

Pre-validation of PIV measurements.

first experimental point from the wall and the increment between the subsequent points in the radial direction of LDA measurements. To obtain better results, the later criterion was achieved, using small size seeding particle (20 µm in diameter) and interrogation area (32 × 32 pixels) as well as high-resolution CCD camera (pixel size 6.45 µm × 6.45 µm). Table 1 shows the calculated WSR at planes 1 and 2 (x/D = 1 and x/D = 2) of the previously illustrated straight tube, to justify the preferential number of points (N ) and order of the curve-fitting (n). As the increment between subsequent experimental points was 0.25 mm, and under


Numerical Simulation and PIV Measurement Table 1.

325

Calculated WSR at planes 1 and 2 with different n and N .

Plane 1 (x/D = 1)

N n a1

2 1 32.5

1 18.75

Plane 2 (x/D = 2)

N n a1

2 1 25

1 20

3

4 2 46.25

1 17

2 28.25

3 59.58

1 17.25

2 30

1 18

4 2 25.5

3 33.33

1 18

3

5 3 42.32

4 72.08

5 2 3 20.86 31.61

4 34.58

2 20.82

steady flow condition the boundary layer thickness was estimated around 1.24 mm, the maximum number of points (N ) used for curve-fitting was thus 5. Because the theoretical WSR was 20 s−1 in this case, a linear approximation with three experimental points and a second-order polynomial curve-fit with five experimental points were shown to produce a more accurate estimation of WSR than others. However, the fifth experimental point was outside the viscous boundary layer under pulsatile flow condition. So a linear approximation (n = 1) with three experimental points (N = 3) was implemented to calculate the WSS in this study.

4. Computational Simulation Results for Proximal Graft Anastomosis Figure 6 shows the velocity vector maps obtained for the mid-plane of 90◦ proximal anastomosis model at different phases. Note that a full period was 1.25 s and the timing shown was after one complete cycle. At t1 = 1.35 s as observed in Fig. 6(a), the main flow from the aorta maintained its flow path on reaching the heel; only part of the flow diverted into the graft. Along the graft outer wall, the deviated main flow approached the curved surface, and then bifurcated into two streams, with the formation of a stagnation point at the toe. The velocity magnitude was generally small as the inlet flow has just began to accelerate. Note that only flow field at the joint region is shown here as it has a high tendency to form mitogens, activated platelets, etc., which lead to eventual graft failure. At peak flow rate (t2 = 1.6 s), a recirculation region with 10 mm (about 1.67 DG ) length was observed along the graft inner wall, as shown in Fig. 6(b). This was similar to flow structures noted by Hughes and How16 under steady flow conditions, where the Reynolds number is 306 and 76% of the inlet fluid flowed downstream through the mother tube. The recirculation region was three-dimensional (3D) in nature. To study the 3D flow in detail, two cross-sectional planes of the graft (indicated as A and B)


326

L. P. Chua et al.

Fig. 6. Velocity vectors at center plane in 90◦ proximal anastomosis model at different time phases: (a) t1 = 1.35 s; (b) t2 = 1.6 s; (c) t3 = 1.7 s; (d) t4 = 1.85 s; (e) t5 = 2.4 s.



Fig. 6.

327

(Continued).

in Fig. 6(b) were extracted at 13 and 15 mm from the centerline of aorta, respectively. The flows in planes A and B are shown in Fig. 7(a), and will be discussed later. When the flow just began to decelerate, such as at t3 = 1.7 s shown in Fig. 6(c), the flow separation region near the inner wall distended into the graft center region and occupied more space than those of previous time phase. At t4 = 1.85 s, the vortex center moved upward into the aorta and shifted toward the heel, which induced more fluid in the graft to flow back to the aorta and form a vortex at the downstream of aorta as shown in Fig. 6(d). For t5 = 2.4 s as demonstrated in Fig. 6(e), backflow in the graft had disappeared, and thus the fluid had moved toward the downstream of the aorta when some fluids are channeled into the graft with small velocity magnitude.


328

L. P. Chua et al.

Fig. 7. Distributions of velocity vector and streamlines in symmetry and horizontal sectional planes A and B at peak flow rate t2 = 1.6 s.

Since the velocity field at peak flow rate was rather complex than other time intervals, Figs. 7(a) and (b) demonstrates the velocity vectors (and the corresponding streamlines) in both the symmetry plane and planes A and B of 90◦ and 135◦ proximal anastomosis models, respectively. Note that the origin of x, y, and z Cartesian coordinate was set at the center of the inlet cross-sectional plane. Only half of the x–z planes were shown, as the models are symmetrical with respect to the x-axis. From the figures, it was concluded that at peak flow rate, the flow separation region could



329

be found along the graft inner wall immediately downstream of the heel, which was represented by the diverged streamlines from a point for the 90◦ and 135◦ proximal anastomosis models. The nodal points viewed as sources in Figs. 7(a) and (b) were the attachment points for 3D flow separation.34 In addition, the angle of anastomosis was observed to affect the separation region near the heel. With the increase of anastomosis angle from 90◦ to 135◦ , the flow separation region reduced in size. Similar effect has also been reported by Hughes and How,16 when the downstream of mother tube is partially occluded. A pair of vortices was found at both planes A and B of the 90◦ (and 135◦ ) grafts, as shown in Figs. 7(a) (and (b)). For the 90◦ anastomotic model, the extension of vortex increased from 3.5 mm (0.58 DG ) at plane A to 5.0 mm (0.82 DG ) at plane B in x-axis, when the cross-sectional area decreased from plane A to B. However, for the 135◦ proximal anastomotic model, the vortex reduced in length when moving planes A to B due to an increase in cross-sectional area. Since the presence of vortex in the graft would increase contact between blood and the thrombogenic graft surface, which may result in platelet activation,16 increasing the cross-sectional area may be in favor of graft patency. In addition, it should be noted that the vortex center for the 90◦ graft was closer to the wall than that of the 135◦ graft. The vortex center of the 135◦ graft even moved far away from the wall when shifting the cross-sectional plane from A to B, which may reduce the probability of pathological changes. As the development of IH at the suture joint is believed to closely related with WSS,1,13,35 the simulation results of WSS distributions at the joint of these two models were also investigated; this will be further explained in the next section. Besides the instantaneous WSS distributions along the graft inner and outer walls, Fig. 8 shows the surface contour plots of the timeaveraged WSS, WSSG, and OSI calculated by Eqs. (4), (5), and (6), respectively, for the 90◦ proximal anastomosis model. To facilitate the comparison with the results of Buchanan et al.30 for a rabbit aorto-celiac junction, the 90◦ model was inverted in the figure and white arrows represented flow directions. The surface contour plots of each HP are presented in two separate figures to allow for a clear comprehensive viewing, namely Figs. 8(a) and (b) for time-averaged WSS, (c) and (d) for time-averaged WSSG, and (e) and (f) for OSI. The highest time-averaged WSS values were seen on both heel and toe, and the region downstream of the toe, with a locally decreasing at the stagnation region. These high time-averaged WSS regions experienced


330

L. P. Chua et al.

Fig. 8. Contours of HPs on the surface of 90◦ facing model: (a) and (b) time-averaged WSS; (c) and (d) time-averaged WSSG; and (e) and (f) OSI (L: low and H: high).

less reverse flow, as shown by the low OSI values in Fig. 8(e). There were also two significant areas with low time-averaged WSS, near the middle of anastomosis at the aorta wall, as indicated in Fig. 8(a) and along the graft inner wall as demonstrated in Fig. 8(b). These low time-averaged



331

WSS regions were regions of flow separation through the whole cycle, as indicated by the elevated OSI values at the same locations as shown in Figs. 8(e) and (f). In addition, high time-averaged WSSG regions were associated with the high time-averaged WSS regions near the toe and heel as observed in Fig. 8(c). Comparing Fig. 8 with Fig. 4 of Buchanan et al.,30 a good agreement can be found, except that the low time-averaged WSS region at the lateral to the celiac opening of Buchanan et al.30 had a much larger size than the present result, and their OSI distributions along the aorta wall had more obvious variations. This is due to the large diameter ratio (3.33) between the aorta and the graft in the present study, if comparing to their < 2 (1.64) diameter ratio, which should induce more flow from the aorta to the branch. In the study of Buchanan et al.,30 comparisons between HPs with the animal experimental intimal WBC density, low-density lipoprotein (LDL) permeability, and lesion growth data were explored. The results show that the segmental averages of time-averaged WSSG significantly correlated with WBC densities and LDL permeability positively. Early and continued atherosclerotic lesion growth is associated with regions of low time-averaged WSS and an elevated oscillatory component (i.e., low-WSS–high-OSI) and regions of high time-averaged WSS and low OSI (i.e., high-WSS–low-OSI). Using this as a reference, the comparisons of HP distributions between different models could reveal more physiological information, as shown in Fig. 9, and Tables 2 and 3. Figures 9(a)–(c) shows the surface contour plots of the time-averaged WSS, WSSG, and OSI for a complete cycle, obtained by using Eqs. (4), (5), and (6), respectively, for the two models. The first and second columns of Fig. 9 represent the results for the 90◦ and 135◦ proximal anastomosis models, respectively. From the first row, it can be observed that the timeaveraged WSS at the heel, toe, and the region downstream of the toe was higher than at other places for both the models. In addition, the high timeaveraged WSS at the heel of the 135◦ proximal anastomosis model was smaller in size than that of the 90◦ model. Note that for ease of comparison, the contour plots of time-averaged WSS, WSSG, and OSI for both the models adopt the same darkness code within the same range of limits. Three low time-averaged WSS regions could be found for 90◦ model at the region near the middle of anastomosis at the aortic wall, downstream of the heel along the graft inner wall and the region further downstream of the graft inner wall. On the other hand, only two low time-averaged


332

L. P. Chua et al.

Fig. 9. Contours of HPs (a) time-averaged WSS, (b) time-averaged WSSG, and (c) OSI on the surface of 90◦ and 135◦ proximal anastomosis models (L: low and H: high).

WSS regions were found for 135◦ model at the region near the middle of anastomosis at the aortic wall and the region further downstream of the graft inner wall, respectively. In contrast to the 90◦ model, the 135◦ proximal anastomosis model has a larger size of the low time-averaged WSS region just downstream of the heel. The reason is that the 135◦ model experienced the highest reverse flow at the peak flow and deceleration phases, as indicated by elevated OSI shown in Fig. 9(c).


Numerical Simulation and PIV Measurement Table 2. models.

333

Comparison of HP range between different 90◦

135◦

Time-averaged WSS (Pa) Minimum Maximum

0.968 5.678

1.17 4.94

Time-averaged WSSG Minimum Maximum

5.53 78.49

5.58 71.52

OSI Minimum Maximum

0.013 0.498

0.021 0.498

Note: 90◦ and 135◦ represent 90◦ and 135◦ proximal anastomosis models, respectively.

Table 3. Sketch maps 90◦

135◦

Comparison of HP between different models. Location Area (mm2 ) WSS (Pa) WSSG OSI

SP

A

872.68

2.02

31.47

0.23

0.56

B

87.50

2.76

56.56

0.14

0.87

A

1034.48

2.07

28.37

0.24

0.54

B

67.60

2.50

38.73

0.17

0.69

The distributions of time-averaged WSSG as demonstrated in Fig. 9(b) were quite similar for the two models. Three high time-averaged WSSG regions were distributed at the heel and the toe as well as downstream of the toe, respectively. The magnitude of time-averaged WSSG for 90◦ model was larger than the 135◦ model, as shown by its darkness in color. In addition, the spatial distribution of OSI for the 135◦ model demonstrated a high OSI region near the middle of anastomosis at the aortic wall, which corresponded to the low time-averaged WSS region. For both the models, there were three low OSI regions at the heel, toe, and downstream of the toe separately and one high OSI region downstream of the heel. Detailed comparisons of the quantitative results are given in Tables 2 and 3.


334

L. P. Chua et al.

As listed in Table 2, the maximum OSI indicated relatively high values for both the models because the flow was highly dependent on the present input pulse with reverse flow. However, the time-averaged WSS variation range of the 135◦ model was smaller than that of 90◦ model, which has the higher minimum time-averaged WSS (1.17 Pa) and the lower maximum time-averaged WSS (4.94 Pa). Bearing in mind that early and continued atherosclerotic lesion growth is associated with regions of high-WSS–lowOSI and low-WSS-high–OSI, respectively, the smaller variation range of time-averaged WSS should have reduced the potential of initializing the intimal hyperplasia at the anastomosis and thus prevented the graft conduit from stenosis. This is consistent with the “safe-bandwidth theory”,11 which postulates that both low and high shear stresses contribute to the growth of plaque formation and assumes a “safe-bandwidth” of the WSS (within the low and high limits). As the “safe-bandwidth” theory has been proven useful in determining some susceptible sites and growth patterns of atherosclerotic lesions, the 135◦ model may merit higher graft patency rate than the 90◦ model, due to its smaller variation range of time-averaged WSS. In addition, the maximum time-averaged WSSG for the 135◦ model is smaller than the 90◦ model, although the minimum time-averaged WSSG for 135◦ model is marginally larger than the 90◦ model. Because the high time-averaged WSSG is believed to be associated with elevated LDL permeability30 and intimal hyperplisia,36 the 135◦ model is postulated to be better than the 90◦ model in this aspect. To have a more complete picture of the time-averaged WSS, WSSG, and OSI distributions, besides the extreme values of these HPs presented in Table 2, Table 3 shows the segmental average of HPs distribution calculated by Eq. (7) on two surfaces at the joint, which is above and below the anastomosis line, namely A and B , respectively. From the table, it could be observed that the segmental average of time-averaged WSS for surface A has not much discrepancy with that of 90◦ model, while the 135◦ model had lower segmental average of time-averaged WSS for surface B and the lower segmental average of time-averaged WSSG for both the surfaces A and B . As the segmental averages of time-averaged WSSG significantly correlate with WBC densities,30 these findings further support that 135◦ proximal anastomosis model merits a high patency of graft. To synthesize the effect of different HPs, Kleinstreuer et al.6 proposed a composite severity parameter (SP), which was defined as N HPnorm , SP = average(HPnorm ) = n=1 N



335

where HPnorm is the HP normalized by its maximum value and N is the number of HPs used. As the segmental average of OSI was not much different between two model as listed in Table 3, the SP in this study synthesized the effects of time-averaged WSS and WSSG (N = 2). The resulting values showed that significant difference in SP values was found especially for surface B . The SP for 135◦ model was lower than the other. As a result, it is expected that the 135◦ proximal anastomosis model will have less potential for stenosis and may result in improving the graft patency rate as mentioned earlier.

5. Comparison Between Simulation and Experimental Results To qualitatively compare the experimental and numerical results, Figs. 10(a)–(e) show the measured velocity vector maps within the center plane of 90◦ proximal anastomosis model, at different time phases, respectively. From Fig. 10(a), it can be observed that at t1 = 1.35 s, part of the fluids has diverted into the graft from the main flow in the aorta. Upon hitting the graft outer wall, the diverted flow bifurcated into two streams with the formation of a stagnation point at the toe. The velocity magnitude was generally small, as the inlet flow just began to accelerate and the similarity between experimental and simulation results can be observed in Figs. 10(a) and 6(a). At peak flow rate (t2 = 1.6 s), as shown in Fig. 10(b), a flow separation region was observed along the graft inner wall. Similar flow structures of numerical simulation were shown in Fig. 6(b) at corresponding time phase. When flow just began to decelerate, at t3 = 1.7 s, the flow separation region near the graft inner wall evolved into a vortex, with its center moved toward the middle of graft as observed in Fig. 10(c), which is similar to the simulation result shown earlier in Fig. 6(c). At the late deceleration phase (t4 = 1.85 s), two vortices were observed at aorta upstream near the heel and downstream near the toe separately in the PIV measurements as shown in Fig. 10(d). The fluid in the graft flowed back into the aorta. This is in agreement with numerical simulation results observed in Fig. 6(d), except that the vortex observed at aorta downstream in PIV measurements is smaller in size and is occupied a smaller region than that of numerical simulation. At t5 = 2.4 s, backflow in graft disappeared, and the fluid thus moved toward the downstream of the aorta, while some amount of fluid was channeled into the graft with a small velocity magnitude, just like that observed in


336

L. P. Chua et al.

Fig. 10. Experimental results of velocity vectors at center plane in 90◦ proximal anastomosis model at different time phases: t1 = 1.35 s; t2 = 1.6 s; t3 = 1.7 s; t4 = 1.85 s; and t5 = 2.4 s.



Fig. 10.

337

(Continued).

Figs. 10(e) and 6(e). Comparison between PIV measurements (Fig. 10) and numerical results (Fig. 6) showed reasonably qualitative agreement. Similar observations and comparisons can be made for experimental and numerical simulations of 135◦ proximal anastomosis model as illustrated in Figs. 11(a) and (b), respectively. Further quantitative comparisons of velocity profiles are demonstrated in Fig. 12. Shown in Figs. 12(a) and (b) are experimental and computational velocity profiles at two planes (namely AA and BB) within the graft of 90◦ and 135◦ proximal anastomosis models, respectively, at different time phases. Note that, the velocity profile represents the velocity in the x–y plane (i.e.,


338

L. P. Chua et al.

√ velocity magnitude is u2 + v 2 ). The one with positive v velocity of ycoordinate will be above the cutting plane (AA or BB), while negative v velocity is below the cutting plane. The negative v velocity would be preferred in the graft as the blood is proceeding forward to perfuse the coronary artery.

Fig. 11. Comparison of (a) experimental and (b) numerical velocity vectors at center plane in 135◦ proximal anastomosis models at different time phases: t1 = 1.35 s; t2 = 1.6 s; t3 = 1.7 s; t4 = 1.85 s; and t5 = 2.4 s.



Fig. 11.

339

(Continued).

A similar trend can be observed between numerical and experimental results of 90◦ proximal anastomosis model as shown in Fig. 12(a), although some differences can be found. For example, at t2 = 1.6 s, the experimental and simulated velocity near the inner wall at BB location had different signs there. The reason was that the flow separation region observed in PIV experiment as illustrated in Fig. 10(c) was smaller in size than that of simulation shown in Fig. 6(c), this had resulted in measured velocity, which became not only negative in sign but also higher in magnitude than those of simulation throughout nearly the whole BB plane, except the region near the outer wall. With the same reason, the negative WSS region was shorter (0.57 ≤ x1 /DG ≤ 0.81) in the measurements along the graft inner wall than that of numerical simulation (0.67 ≤ x1 /DG ≤ 1.37) at peak flow phase (t2 = 1.6 s) as observed in Fig. 13(a). In addition, the measured velocity near the outer wall of location AA at t4 = 1.85 s, as demonstrated in Fig. 12(a), had different sign with that of simulation on account of smaller


340

L. P. Chua et al.

vortex size observed in PIV experiment at the aorta downstream. Correspondingly, experimental WSS as observed in Fig. 14(a) changed sign from positive to negative earlier than that of simulation at t4 = 1.85 s. To sum up, the simulated velocity profiles were in fair agreement with those of PIV

Fig. 12. Comparison of velocity profiles between simulation (Sim.) and experimental (Exp.) results at two locations for (a) 90◦ and (b) 135◦ proximal anastomosis models at different time phases: t1 = 1.35 s; t2 = 1.6 s; t3 = 1.7 s; t4 = 1.85 s; and t5 = 2.4 s.



Fig. 12.

341

(Continued).

experiment qualitatively; the percentage of differences ranged from 9% to 46% for 90◦ model. Note that this variation range maintained about the same order of magnitude for different time phases. Similar comparisons of experimental and computational velocity profiles for 135◦ proximal anastomosis model were demonstrated in Fig. 12(b). Some discrepancies can be observed although the shapes of the velocity profiles were qualitatively similar. At t2 = 1.6 s, the experimental velocity near the inner and outer walls of location AA were different from those of simulation. The larger flow separation region observed in PIV experiment resulted in positive velocity at the inner wall and the stagnation point had also been shifted further downstream along the outer wall as demonstrated by the positive velocity there. Consequently, the experimental WSS delayed the changing of sign from negative to positive along the graft outer wall (at x2 /DG = 1.1) in contrast with those of simulation (at x2 /DG = 0.88), which could be observed in Fig. 14(b). Similarly at t3 = 1.7 s, as the vortex


342

L. P. Chua et al.

induced in the graft of PIV experiment occupied larger space than that of numerical simulation, the experimental velocity near the graft inner wall at location BB as demonstrated in Fig. 12(b) was still positive whereas computational velocity was negative. In addition, the measured velocity near the outer wall of location AA at t4 = 1.85 s as seen in Fig. 12(b) had different sign with that of numerical simulation on account of smaller vortex size observed in PIV experiment at the aorta downstream. The difference of the velocity near the outer wall of location AA at t5 = 2.4 s was due to the shifting of stagnation point. The experimental stagnation point shifted upstream along the outer wall, which can be observed in Fig. 14(b) at t5 = 2.4 s. In general, the computational velocity profiles were qualitatively similar with those of PIV experiment, and the range of difference varied between 8% and 55% for 135◦ proximal anastomosis model. This variation range maintained at almost the same level for different time phases. Figures 13 and 14 show the comparisons between computational and experimental WSS distributions along graft inner and outer walls, respectively, for both 90◦ and 135◦ proximal anastomoses. Along the graft inner wall of 90◦ proximal anastomotic joint as shown in Fig. 13(a), the experimental WSS had the following characteristics. At t1 = 1.35 s, the experimental WSS increased when accessing the heel, then it decreased once entering the low velocity region (0.37 ≤ x1 /DG ≤ 0.65) and increased gradually at further downstream of the graft. When inlet flow rate increased to peak from t1 to t2 , the WSS level reached its maximum to 14 Pa. Corresponding to the flow separation region formed near the heel at t2 = 1.6 s, WSS kept negative in the region (0.57 ≤ x1 /DG ≤ 0.82). When flow just began to decelerate at t3 = 1.7 s, the flow characteristics and WSS distributions had the similar pattern as those of peak flow. However, the maximum WSS decreased to 8 Pa and the flow separation region near the heel was increased in size (0.34 ≤ x1 /DG ≤ 1.38) as indicated by the negative WSS obtained experimentally. At t4 = 1.85 s, WSS was almost all negative along the inner wall on account of the backflow in graft and the vortex formed upstream of the aorta. The WSS came back to positive when t5 = 2.4 s, and varied slightly along the graft inner wall but maintained positive value. When comparing the numerical simulation results with those of measurements, it was found that they had similar distribution trends except at t4 = 1.85 s, although in general the simulation results mostly had a little higher magnitude. The reason for the different trend at t4 = 1.85 s was due to the slight difference of curvature along the graft inner wall between the Pyrex glass model and numerical model, which lead to different size of



343

Fig. 13. Comparison of simulation (Sim.) and experimental (Exp.) WSS distributions along the graft inner wall of (a) 90◦ and (b) 135◦ proximal anastomosis models at different time phases.

vortex observed at the upstream of aorta. The vortex size found by PIV measurements was smaller than that of simulation, especially along the graft inner wall as shown in Figs. 10(d) and 6(d). Thereby positive WSS existed for the experiment, while numerical WSS was all negative along the graft inner wall. In addition, the different size of flow separation regions at t2 = 1.6 s and t3 = 1.7 s were due to the differences of vortex size observed at the downstream of heel as seen in Figs. 10(b) and (c) and Figs. 6(b) and (c). Similar observations can be found for 135◦ proximal anastomotic joint as demonstrated in Fig. 13(b).


344

L. P. Chua et al.

Fig. 14. Comparison of simulation (Sim.) and experimental (Exp.) WSS distributions along the graft outer wall for (a) 90◦ and (b) 135◦ proximal anastomosis models at different time phases.

Along the graft outer wall of 90◦ proximal anastomosis, the characteristics of experimental WSS distributions can be described as follows. At the accelerating stage (t1 = 1.35 s) as shown in Fig. 14(a), WSS varied from −0.5 to 1 Pa and changed sign at stagnation point (x2 /DG = 0.72). At peak flow phase (t2 = 1.6 s), the WSS magnitude obtained from PIV measurements attained its maximum of 8.8 Pa at x2 /DG = 1.4 as shown in Fig. 14(a). When flow just began to decelerate (t3 = 1.7 s), the WSS maintained the same pattern as that of peak flow. However the stagnation point moved upward a little toward the aorta. At late deceleration phase (t4 = 1.85 s), because of the vortex formed at the downstream of



345

aorta, WSS became positive at the beginning of x2 and the separation point was different from that of simulation and occurred at much earlier (x2 /DG = 0.35) location. Then it changed to negative downstream, which was unlike those of early time intervals in virtue of flow reverses from graft back to the aorta. At t5 = 2.4 s, the variations of WSS were similar to those of t1 = 1.35 s. Meanwhile, similar trend can be found for numerical WSS distributions, although the shifting of stagnation points can be observed at t2 = 1.6 s, t3 = 1.7 s, and t5 = 2.4 s. Similar observations can be found for 135◦ proximal anastomosis model as demonstrated in Fig. 14(b). In general, the WSS distribution patterns derived from numerical simulation were observed to have similar trend with those of PIV experiments along the graft inner and outer walls, although differences in magnitude still existed. The shifting of stagnation points can be found at t2 = 1.6 s, t3 = 1.7 s, and t4 = 1.85 s, these were due to the slight differences of velocity profiles near the wall. In addition, most experimental WSS along graft outer wall was observed to be smaller than those of numerical simulation. As the velocity along graft outer wall varied tremendously both in direction and magnitude, the 0.25 mm spatial resolution of PIV experiments would not be fine enough to capture the small variation of velocity distribution, especially in the region with tremendous curvature variations. Whereas numerical simulation can provide more reliable results as the nearest distance of grids to the wall was within 0.15 mm and more meshes were put near the wall to capture the small variations of velocity distribution. To sum up, the WSS distributions of PIV measurement and numerical simulation were qualitatively similar, although discrepancies can be found quantitatively. Higher fluctuations (range from –22 to 21 Pa) of instantaneous WSS were found in the simulation results.

6. Conclusions In this study, hemodynamics of 90◦ and 135◦ proximal anastomosis models were comprehensively studied by both numerical simulation and PIV measurements. The existence of flow separation, vortex and, the small shifting of stagnating locations were observed, which may lead to the formation of IH. Meanwhile the simulation results were validated by PIV experimental data. A fair match between numerical and experimental data was observed for both velocity vectors and WSS distributions. The difference between them varied from 8% to 55%, which was comparable as Lei et al.19 and Bertolotti et al.20 also reported the same order of error ranged from 1% to


346

L. P. Chua et al.

40% by means of LDA measurements. In addition, the error range maintained almost the same level for different time phases. This was different from the study of Lei et al.,19 who observed that the relative error between numerical and LDA experimental data was smaller (typically around 1– 8%) during the forward flow phases and much larger (ranged from 1% to 40%) during the reverse flow phase. The discrepancy may be due to the different measuring instrument used. PIV measurement was capable of capturing the instantaneous whole-flow-field in cross-sectional plane, while for LDA measurement, the obtained data at a location was the average value of several cycles at the same location, which relied heavily on the repeatability of experiment and required more time to analyze the entire flow field. PIV measurement was therefore capable of providing accurate whole-flow-field information in relatively shorter amount of time than that of LDA as demonstrated also by Browne et al.37 The discrepancy between the simulation and experimental results could be due to disturbance of background noise, wall reflection, and the accuracy of positioning the measuring plane in PIV techniques as well as the accuracy of flowmeter and cam mechanism used to mimic the pulsatile flow. Above all, the discrepancy should be in virtue of the differences in geometrical detail of the Pyrex glass model and the numerical model. If silicone rubber model was manufactured instead of the Pyrex glass model with the aid of computeraided design/computer-aided manufacture (CAD/CAM), and rapid prototyping for the PIV experiment, which would faithfully reproduce the numerical model, the difference between the numerical and experimental results should be greatly reduced. In contrast with PIV measurements, the numerical simulation could produce more versatile results such as the HPs distributions with enough mesh density, which provides more comprehensive information of hemodynamics. Analysis based on HPs such as time-averaged WSS, WSSG, OSI, and segmental average of time-averaged WSS and WSSG, etc. proposed that the 135◦ proximal anastomosis model (which represents the CABG for right coronary artery) would alleviate the potential of intimal thickening and/or atherosclerosis than that of 90◦ proximal anastomosis model (which represents the CABG for left coronary artery). However, CABG for right coronary artery clinically has lower patency rate than that of left coronary artery, which may be due to other reasons such as the out-of-plane characteristics of left coronary artery and in general artery (such as internal mammary artery) is used as graft for left coronary artery while saphenous vein is used for right coronary artery. Therefore, the investigation of the



347

whole anastomosis model, including both proximal and distal anastomoses was necessary and would be reported in due course.

Acknowledgement The financial support of A*STAR Project 0221010023 is gratefully acknowledged.

References 1. D. A. Vorp, Fluid mechanical considerations in vascular grafts, American Society for Artificial Internal Organs Journal 43, 3 (1997) 237–238. 2. P. J. Shah, I. Gordon, J. Fuller, S. Seevanayagam, A. Rosalion, J. Tatoulis, J. S. Raman and B. F. Buxton, Factors affecting saphenous vein graft patency: clinical and angiographic study in 1402 symptomatic patients operated on between 1977 and 1999, The Journal of Thoracic and Cardiovascular Surgery 126, 6 (2003) 1972–1977. 3. M. H. Friedman, C. B. Bargeron, D. D. Duncan, G. M. Hutchins and F. F. Mark, Effects of arterial compliance and non-Newtonian rheology on correlations between intimal thickness and wall, Journal of Biomechanical Engineering 114 (1992) 317–320. 4. S. F. C. Stewart and D. J. Lyman, Effects of a vascular graft/natural artery compliance mismatch on pulsatile flow, Journal of Biomechanics 25 (1992) 297–310. 5. M. Hofer, G. Rappitsch, K. Perktold, W. Trubel and H. Schima, Numerical study of wall mechanics and fluid dynamics in end-to-side anastomoses and correlation to intimal hyperplasia, Journal of Biomechanics 29 (1996) 1297– 1308. 6. C. Kleinstreuer, S. Hyun, J. R. Buchanan, Jr., P. W. Longest, J. P. Archie, Jr. and G. A. Truskey, Hemodynamic parameters and early intimal thickening in branching blood vessels, Critical Reviews in Biomedical Engineering 29, 1 (2001) 1–64. 7. S. E. Rittgers, P. E. Karayannacos, J. F. Guy, R. M. Nerem, G. M. Shaw, J. R. Hostetler and J. S. Vasko, Velocity distribution and intimal proliferation in autologous vein grafts in dogs, Circulation Research 42 (1978) 792–801. 8. R. L. Binns, D. N. Ku, M. T. Stewart, J. P. Ansley and K. A. Coyle, Optimal graft diameter: effects of wall shear stress on vascular healing, Journal of Vascular Surgering 10 (1989) 326–337. 9. H. S. Bassiouny, S. White, S. Glagov, E. Choi, D. P. Giddens and C. K. Zarins, Anastomotic intimal hyperplasia: mechanical injury or flow induced, Journal of Vascular Surgering 15 (1992) 708–717. 10. D. L. Fry, Certain histological and chemical responses of the vascular interface to acutely induced mechanical stress in the aorta of the dog, Circulation Research 24 (1969) 93–108.


348

L. P. Chua et al.

11. C. Kleinstreuer, M. Nazemi and J. P. Archie, Jr., Hemodynamics analyses of a stenosed carotid bifurcation and its plaque-mitigating design, Journal of Biomechanical Engineering 113, 3 (1991) 330–335. 12. F. S. Henry, M. W. Collins, P. E. Hughes and T. V. How, Numerical investigation of steady flow in proximal and distal end-to-side anastomoses, Journal of Biomechanical Engineering 118 (1996) 302–310. 13. C. Kleinstreuer, M. Lei and J. P. Archie, Jr., Flow input waveform effects on the temporal and spatial wall shear stress gradients in a femoral graft-artery connector, Journal of Biomechanical Engineering 118 (1996) 506–510. 14. X. He and D. N. Ku, Pulsatile flow in the human left coronary artery bifurcation: average conditions, Journal of Biomechanical Engineering 118 (1996) 74–82. 15. M. Lei, C. Kleinstreuer and J. P. Archie, Jr., Hemodynamic simulations and computer-aided designs of graft-artery junctions, Journal of Biomechanical Engineering 119 (1997) 343–348. 16. P. E. Hughes and T. V. How, Flow structures at the proximal side-to-end anastomosis. Influence of geometry and flow division, Journal of Biomechanical Engineering 117 (1995) 224–236. 17. D. A. Steinman, B. Vinh, C. R. Ethier, M. Ojha, R. S. C. Cobbold and K. W. Johnston, A numerical simulation of flow in a two-dimensional endto-side anastomosis model, Journal of Biomechanical Engineering 115 (1993) 112–118. 18. J. S. Cole, L. D. Wijesinghe, J. K. Watterson and D. J. A. Scott, Computational and experimental simulations of the haemodynamics at cuffed arterial bypass graft anastomoses, Proceedings of the Institution of Mechanical Engineering (Part H) 216 (2002) 135–143. 19. M. Lei, D. P. Giddens, S. A. Jones, F. Loth and H. Bassiouny, Pulsatile flow in an end-to-side vascular graft model: comparison of computations with experimental data, Journal of Biomechanical Engineering 123 (2001) 80–87. 20. C. Bertolotti, V. Deplano, J. Fuseri and P. Dupouy, Numerical and experimental models of post-operative realistic flows in stenosed coronary bypass, Journal of Biomechanics 34 (2001) 1049–1064. 21. C. J. Bates, D. M. O’Doherty and D. Williams, Flow instabilities in a graft anastomosis: a study of the instantaneous velocity fields, Proceedings of the Institution of Mechanical Engineering (Part H) 215 (2001) 579–587. 22. M. Heise, S. Schmidt, U. Kr¨ uger, R. R¨ uckert, S. R¨ osler, P. Neuhaus and U. Settmacher, Flow pattern and shear stress distribution of distal end-toside anastomoses. A comparison of the instantaneous velocity fields obtained by particle image velocimetry, Journal of Biomechanics 37 (2004) 1043–1051. 23. J. A. Moore, D. A. Steinman, S. Prakash, K. W. Johnston and C. R. Ethier, A numerical study of blood flow patterns in anatomically realistic and simplified end-to-side anastomoses, Journal of Biomechanical Engineering 121 (1999) 265–272. 24. W. W. Nichols and M. F. O’Rourke, McDonald’s Blood Flow in Arteries: Theoretic, Experimental, and Clinical Principles (Lea & Febiger, Philadelphia, 1990).



349

25. J. Peacock, T. Jones, C. Tock and R. Lutz, The onset of turbulence in physiological pulsatile flow in a straight tube, Experiments in Fluids 24 (1998) 1–9. 26. X. Y. Xu, M. W. Collins and C. J. H. Jones, Flow studies in canine artery bifurcations using a numerical simulation method, Journal of Biomechanical Engineering 114 (1992) 504–511. 27. D. A. Steinman and C. R. Ethier, The effect of wall distensibility on flow in a two-dimensional end-to-side anastomosis, Journal of Biomechanical Engineering 116 (1994) 294–301. 28. J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, Journal of Physiology 127 (1955) 553–563. 29. L. P. Chua, J. M. Zhang, S. C. M. Yu, D. N. Ghista and Y. S. Tan, Numerical study on the pulsatile flow characteristics of proximal anastomotic model, Proceedings of the Institute Mechnical Engineering (Part H) 219, 5 (2005) 361–379. 30. J. R. Buchanan, Jr., C. Kleinstreuer, G. A. Truskey and M. Lei, Relation between non-uniform hemodynamics and sites of altered permeability and lesion growth at the rabbit aorto-celiac junction, Atherosclerosis 143 (1999) 27–40. 31. L. P. Chua, W. F. Ji and T. M. Zhou, In vitro study on the steady flow characteristics of proximal anastomosis models, Internation Communication in Heat and Mass Transfer 32 (2005) 464–470. 32. R. S. Fatemi and S. E. Rittgers, Derivation of shear rates from near-wall LDA measurements under steady and pulsatile flow conditions, Journal of Biomechanical Engineering 116 (1994) 361–368. 33. R. D. Keane and R. J. Adrian, Theory of cross-correlation analysis of PIV images, Applied Scientific Research 49 (1992) 191–215. 34. A. Filippone, 3-D flow separation, Advanced Topics in Aerodynamics, 2004 [online]. Available: http://aerodyn.org/Wings/3dsepar.html. 35. M. Ojha, Spatial and temporal variations of wall shear stress within an endto-side arterial anastomosis model, Journal of Biomechanics 26 (1993) 1377– 1388. 36. M. Lei, C. Kleinstreuer and G. A. Truskey, Numerical investigation and prediction of atherogenic sites in branching arteries, Journal of Biomechanical Engineering 117 (1995) 350–356. 37. P. Browne, A. Ramuzat, R. Saxena and A. P. Yoganathan, Experimental investiongation of the steady flow downstream of the St. Jude bileaflet heart valve: a comparison between laser Doppler velocimetry and particle image velocimetry techniques, Annals of Biomedical Engineering 28 (2000) 39–47.




MECHANICAL CIRCULATORY SUPPORT SYSTEMS MUSTAFA AKDIS∗ and HELMUT REUL† ∗MAQUET Cardiopulmonary AG Neue Rottenburger Str. 37 D-72379 Hechingen, Germany [email protected]

1. Introduction Mechanical blood pumps are gaining widespread acceptance as therapeutic instruments for the treatment of cardiac insufficiency. Their primary task is to maintain blood circulation and, thus, to provide sufficient oxygen supply to organs and body tissue, if the natural heart is failing. Modern development in mechanical circulatory support (MCS) has resulted in a large number of different pumping principles reaching from intra-aortic balloon pumps (IABPs) to the total artificial heart (TAH) according to the specific clinical indication. Application times may vary between a few hours over days and weeks to chronic. Indication, timing of introduction, and the selection of the appropriate device are, therefore, as important as the specific pump design itself. Other issues for consideration are quality of life, reliability, and cost-effectiveness. 2. Blood Pump Classification Mechanical pumps can be classified into two main categories: displacement pumps and rotary pumps (Fig. 1). The energy transfer in displacement pumps is characterized by periodic changes of a working space. In rotary pumps, the energy transfer to the fluid is established by velocity changes within the impeller vanes. Depending on impeller geometry, rotary blood pumps (RBPs) can be classified into three main categories: axial, radial (centrifugal), and diagonal (mixed flow) pumps. ∗ Corresponding

† Passed

author. away on November 4, 2004. 351

Mustafa Akdis and Helmut Reul


352

Classification of blood pumps in the world. Fig. 1.


Mechanical Circulatory Support Systems

353

Generally speaking, rotary pumps are best suited for high flows up to 20 l/min at differential pressures lower than 500 mmHg. The radial design is most capable of producing high pressures and low flows, whereas axial pumps generate high flows at low pressure differences. Diagonal pumps, often referred to as mixed flow systems, tend to have the capability of high generated pressures and high flows. This simple classification of potential pump designs is normalized with respect to pump size, taking into consideration that a 60-mm-diameter centrifugal pump can naturally pump more fluid at significantly higher pressures than a 6-mm-diameter axial pump. RBPs have a number of potential physiologic and technological advantages, yet they have their own hazards. They are characterized by low blood trauma, lower anticoagulation levels, and thus, less hemorrhage. Depending on pump design, priming volume may be low and a surface modification like heparin coating is possible. Due to their small size, rotary pumps also allow implantability as well as transportability and integration into more complex devices. Figure 2 represents an inventory of RBPs. It is based on an extensive literature review and is hopefully fairly complete.

3. Clinical Use and Indications In the medical arena, displacement-type pumps (e.g., roller pumps) have been established over decades for cardiopulmonary bypass in heart–lung machines or in dialysis machines. Roller pumps are still predominant and are commonly used for up to several hours. Their main advantages are simplicity of operation, low cost of disposable tubing, and reliability, whereas disadvantages can be seen in blood damage and spallation. RBPs have several theoretic and practical advantages in extracorporeal circulation (ECC) in terms of lower blood damage, smaller size, lower filling volume, better transportability, absence of spallation, less bleeding, less neurologic complications, and shorter stay in intensive care, among others. Also, the application time may be extended up to several days, if necessary. Therefore, for ECC applications, centrifugal pumps are gaining market share, especially in the USA. Main cause of death after open-heart operation is an insufficiently low cardiac output, known as “low output syndrome (LOS)”. Due to its relatively high blood damage (hemolysis) and the high doses of anticoagulation drugs, the use of roller pumps is usually limited to several hours. Therefore, LOS is subsequently treated by positive inotropic and vasoactive drugs as well as by intra-aortic balloon pumps (IABPs) in extreme cases.


354


HeartMate III Thoratec Corp / USA

RotaFlow MAQUET AG / GERMANY

Hemobear Vienna Univ / AUSTRIA

CentriMag Levitronix Gmbh / CH

EvaHeart Sun Medical Corp / JAPAN Incor Berlin Heart AG / GERMANY

Jarvik 2000 Jarvik Heart Inc / USA

Tandem Heart Cardiac Assist Inc / USA

Ablomed CF Ablomed Inc / USA

Impella ABIOMED Inc. / USA

CorAide Arrow International Inc / USA

MicroDiagonal HIA - Medos AG / GERMANY

VentrAssist HeartMate II Thoratec Corp / USA

DeltaStream

Ventracor Ltd / AUSTRALIA HIA - Medos AG - GERMANY

MicroVAD Circulite Inc / USA

DuraHeart Terumo Corp / JAPAN

HeartQuest WorldHeart Corp / USA

HiFlow HIA-Medos AG / GERMANY

FLOW STRAIGHTENER STATOR SHELL

MagneVAD BLOOD Gold Medical Inc / USA INFOLW

Valvo Pump Keio Univ / JAPAN MITI Heart MiTi Heart Inc / USA

FLOW TUBE INDUCER/IMPELLER DIFFUSER

DeBakey VAD Micromed Inc / USA BioPump Medtronic Inc / USA Isoflow Bard Inc / USA Delphin Pump Sams - 3M Inc / USA

Caplox Terumo Corp / JAPAN Gyro Pump Baylor College / USA

Fig. 2.

Overview of RBPs in the world.

Cardiogenic shock or cardiac fibrillation require more efficient means of cardiac support. For this purpose, continuous flow centrifugal, micro-axial, or pulsatile pneumatic displacement blood pumps are used (see Fig. 2). The micro-axial pumps have the advantages of quick applicability and minimal invasiveness. They are introduced via the femoral artery and the inflow cannula is placed through the aortic valve into the left ventricle. The first axial-type blood pump, the so-called HEMOPUMPTM , was an intra-arterial miniature circulatory support system. It is no longer available, but the newly developed ImpellaTM pumps are an excellent alternative. End-stage heart failure (ESHF) is today the leading cause of death in the Western hemisphere. In Europe, it affects an estimated number of 3.5 million people (US: 5 million), who suffer from terminal cardiac failure.



355

The natural history of these patients is comparable to patients with intestinal or breast cancer: 5 years after diagnosis only 50% of the patients are still alive. Every year 250,000 patients (EU) die with this primary diagnosis, whereas in 350,000 patients (EU) this disease is diagnosed de novo. Conservative treatment includes angiotensin-converting enzyme (ACE) inhibitors, angiotensin II receptor blockers, β-receptor blockers, and diuretics. However, even under this modern neurohumoral therapy, mortality rates are still high: 34% mortality after 4 years. Until recently cardiac transplantation represented the only treatment with acceptable long-term survival in ESHF. However, donor shortage limits this therapeutic modality to less than 1,000 recipients per year overall Europe (US: 2,500), compared to an estimate of at least 25,000 (EU: 10% of patients) candidates requiring hemodynamic assist. In this context, mechanical cardiac support systems have recently reached the threshold of long-term applicability. European centers were among the first to apply these devices in clinical studies, and achieved excellent results. Nevertheless, the success varied widely between different centers with regards to selected patients and devices used, indicating the still non-standardized and complex interactions between the interdisciplinary contributors to this new therapeutic approach. Today, these pumps can provide a very acceptable quality of life for several months to a couple of years. However, the inherent risks of thromboembolism, infection, and pump-related complications require still a lot of concomitant medical and technical support. Till now, most of these pumps are produced in the USA, but currently some European companies catch up with these developments and achieve important advantages such as miniaturization down to catheter implantability and magnetic levitation. Furthermore, necessary monitoring and therapeutic interventions remain to be clarified and standardized. Particularly rotary pumps (which are smaller in size, yield fewer infections, complement activation, and potentially fewer thromboembolic complications than previous systems) may be eligible for extended use. For medium-term bridging periods of a few days up to a few months, pneumatic displacement blood pumps are successfully used for left-, right-, or biventricular support. The pneumatically activated displacement pumps require thoracotomy and are connected between left atrium or left ventricular apex and aorta, or right atrium and pulmonary artery, respectively. Disadvantages include the percutaneous blood cannulae that lead from the heart to the extracorporeal pumps. They present an inherent risk


356


of infection at their skin penetration sites. Additionally, the patients have restricted mobility due to the bulky driving consoles. If recovery of the heart cannot be established by means of the above systems, the emphasis of MCS shifts from recovery to bridging until a donor heart is available. For bridging periods ranging from several months up to 2 years, which may either be caused by the lack of a suitable donor heart or by a longer than anticipated recovery period, fully implantable electromechanical left-ventricular-assist devices (LVADs) are in clinical use since 1984 in the USA and since 1993 in Europe. Several systems are currently available: the r r 3 from World Heart Inc., USA; the HeartMate I from Thoratec Novacor1,2 4 Corporation, USA; and the LionHeart from Arrow International, USA. About 20–30% of the patients requiring long-term left ventricular support, also have right ventricular complications, which may lead to complete cardiac failure.5 Also, in the presence of additional complications such as septum or valvular defects, singular left ventricular support is not sufficient.6,7 In these cases, either transplantation or the use of a TAH is indicated. An orthotopic TAH is defined as a blood pump which replaces the explanted natural heart in terms of anatomic placement and function. Until recently, only pneumatic TAHs with extracorporeal driving systems have been clinically used, the most prominent examples being the Jarvik 7 and its successor, the Cardiowest TAH. They have been used as bridge-totransplant or, in a few cases, as permanent replacement systems.8 A novel electrohydraulic TAH, the Abiomed AbioCorTM received approval of its IDE application to begin clinical trials in January 2001. The first patient was implanted on July 2, 2001 and the trial is ongoing.9 A second category of blood pumps, namely rotary pumps, has entered the clinical arena with growing numbers and success rates. The most prominent examples are currently the Micromed DeBakey,10 the Jarvik r II,12 and the Berlin Heart Incor.13 Other sys2000,11 the HeartMate tems like the Terumo DuraHeartTM ,14 the Australian VentrAssistTM ,15 the CorAideTM ,16 and the HeartQuestTM 17 are currently being prepared for clinical studies in the near future. More recently, MCS with extracorporeal membrane oxygenation (ECMO) or ventricular-assist devices (VADs) has become an increasingly important tool for the treatment of infants and children with congenital and acquired cardiovascular diseases who experience cardiopulmonary failure and circulatory collapse.18 Most clinical experience to date has involved the short-term support of patients weighing 6 kg and more. Currently, ECMO remains the most commonly used form of MCS for these patients.



357

In the field of VAD systems, several pulsatile and continuous flow devices exist that are appropriate for pediatric use in terms of size and hemodynamic performance. The Medos/HIA VAD, Berlin Heart Excor, Medtronic r have proven successful for Biomedicus Pump, and Abiomed BVS 5000 short- and mid-term pediatric circulatory support. The increasing need for long-term pediatric circulatory support systems has led to an NIH grant in April 2004 in the total amount of $22 million that have been distributed for R&D activities to the following institutions: Penn State University, Cleveland Clinic, Ension Inc., Jarvik Heart Inc., and University of Pittsburgh. As identified in the literature,19,20 the toughest challenges of designing a pediatric VAD include concerns about patients anatomy and device size, incidence of thrombosis and hemolysis, and the availability of different pump sizes corresponding to desired stroke volumes. The shortterm success of extracorporeal pediatric VADs and the evolving progress in VAD design encourages the belief that implantable VADs for long-term circulatory support to the pediatric population will become possible in the near future.

4. Existing Blood Pump Systems and Current Developments Due to the large number of different blood pumps, which are already in clinical use or under development, the following section includes only an assortment of pumps selected under the criteria of clinical importance, technology, novelty, and partly of personal bias. The collection of technical data for each of the listed pump systems was extremely difficult, and may be subjected to actualization due to the continuous progress and change related to each device in question. Therefore, the presented data are neither complete nor consistent. 4.1. Displacement pumps 4.1.1. Extracorporeal VAD systems Thoratec VAD The Thoratec VAD21 (Fig. 3) is approved for use in patients, both adults and children, who are eligible for heart transplantation, as well as for patients whose failing ventricle(s) require temporary assistance after heart surgery. It is an external heart-assist device that is capable of supporting either the right and/or left ventricle, but does not replace the heart. More than 4,300 of these devices have been


358


Fig. 3.

Thoratec VAD.

used in the treatment of over 2,800 patients worldwide. With the introduction of the implantable ventricular-assist device (IVADTM ), Thoratec delivers the first and only IVAD for left-, right-, and biventricular support for bridge-to-transplantation and for post-cardiotomy recovery. In addition, r Portable VAD Driver provides hosthe accompanying Thoratec TLC-II pitals with the first mobile system that allows these uni- or biventricular VAD patients to be discharged home to await cardiac transplantation or myocardial recovery. r r VAD The Abiomed BVS 500022 (Fig. 4) is the most Abiomed BVS 5000 widely used mechanical cardiac-assist device for patients with a potentially recoverable heart. It was the first extracorporeal VAD on the market and is the most widely used bridge-to-recovery device with systems located in more than 600 institutions throughout the world. The BVS was the first FDA-approved device for the short-term support of all patients with reversible heart failure. Since its US approval in 1992, the system has supported thousands of patients ranging from 8 to 84 years old in over 500 medical centers. r can support one or both sides of the failing heart and The BVS 5000 can be operated with one of the three Abiomed pneumatic consoles: the BVS 5000i, BVS 5000t, and the new AB5000 console. The typical length of support on the BVS 5000 is approximately 7–10 days.

Medos/HIA VAD The Medos/HIA VAD System23 (Fig. 5) is a heart-assist device for short- and medium-term applications. Main indications for this system are cardiogenic shock, post-cardiotomy as well as cardiomyopathy. The systems have been used for uni- and biventricular support in over 80 European centers; it has been efficiently used in more than 500 applications



Fig. 4.

Fig. 5.

359

r. Abiomed BVS 5000

Medos/HIA VAD.

and was able to help patients in extreme life-threatening situations. The pumps are equipped with three leaflet valves made of polyurethane (PU), which have been especially designed for this purpose. To assure an optimal support from infant up to heavyweight adults, the Medos VAD ventricles are provided in following sizes: • • • •

Adult HiFlux (80/72 ml, left/right stroke volume) Adult (60/54 ml) Pediatric (25/22.5 ml) Infant (10/9 ml).


360


To match the anatomic conditions, Medos offers venous cannulae in various sizes (14–58F) and angular positions, whereas the angular position can still be adjusted during surgery. The Apex-cannulae, as an alternative to the venous cannula, can be applied directly to the apex of the heart and is also provided in various sizes (6, 8, 12, and 15 mm). The driving unit of the Medos VAD system has been designed especially for the uni- and biventricular operation of the VAD ventricles. Currently, a new portable driving console is being prepared for clinical certification, that will enhance patient mobility. Berlin Heart Excor VAD The extracorporeal VAD system Berlin Heart Excor (Fig. 6) is primarily a bridge-to-transplant device.24 In individual cases, the Berlin Heart Excor has been used for ventricular recovery with subsequent device removal. Blood pumps are available in the sizes of: 10, 25, 30, 50, 60, and 80 ml. The pediatric pumps (10, 25, and 30 ml) are provided with PU valves for optimal functionality. All other pump sizes are available either with PU valves or tilting disc valves. The capabilities of this system have been proven in over 850 applications worldwide (status June 2004). The system permits uni- or biventricular support. For the past 4 years, the device has been used in special cases as destination therapy (6 patients, over 1350 days).

Fig. 6.

Berlin Heart Excor.



361

With a small portable driving unit introduced 4 years ago, most of the patients are fully mobile and living at home waiting for a suitable transplant organ.

4.1.2. Implantable VAD systems r r LVAS The Novacor left-ventricular assist system (LVAS)1,2 Novacor (Fig. 7) is an electromechanically driven pump, which is implanted within the abdominal wall. The system is monitored by an electronic controller and powered by primary and reserve battery packs, worn on a belt around the waist or carried in a shoulder bag, or by a small bedside monitor. The controller is connected to the implanted pump by a percutaneous lead. The system is self-regulating, responding instantaneously to the recipient’s r LVAS was designed primarily for longcirculatory demands. The Novacor term use. In Europe, the device has unrestricted approval for use as a bridge-totransplantation, an alternative to transplantation, and to support patients who may have an ability to recover the use of their natural heart. It is commercially approved as a bridge-to-transplantation in the USA and Canada. r LVAS recipients worldwide. A There have been over 1350 Novacor total of 288 recipients were supported on their device for more than 6 months; 106 recipients for more than 1 year; 23 recipients for more than 2 years; 9 recipients for more than 3 years; and 3 for more than 4 years,

Fig. 7.

r LVAS. Novacor


362


Fig. 8.

r I LVAD. HeartMate

a record unmatched by any other implanted cardiac-assist device. A clinical trial (INTrEPID) is underway in the USA and Canada to evaluate long-term use of the device as an alternative to medical therapy. r r I LVAD The HeartMate I LVAD3 (Fig. 8) is indicated for HeartMate use in adult patients both awaiting heart transplantation and for permanent application for those not eligible for transplantation. It is manufactured by Thoratec Corporation, and is utilized in over 100 medical centers in the USA. It is designed to assist the heart in pumping blood to the body in r is an implantable patients with a failing left ventricle. The HeartMate blood pump that is about 2-in. thickness, 4 in. in diameter, and weighs about 2.5 lb. It contains an internal motor and a drive line, enabling electric wires to exit through the patient’s upper abdominal wall. r LVAD is indicated for use in patients both awaiting The HeartMate heart transplantation and those not eligible for transplantation. This device r is used in adults only (body surface area, BSA >1.5 m2 ). The HeartMate SNAP-VE LVAS is now FDA approved as a long-term permanent implant, called destination therapy, for ESHF patients who are not eligible for heart transplants. r LVAS has been approved as a bridge-toWhile the HeartMate transplant since 1994 and used in more than 3,300 patients worldwide, FDA r SNAP-VE marks the first time in history that approval of the HeartMate an LVAD has been approved for permanent implantation.

LionHeart T M LVAS The Arrow LionHeartTM 4 (Fig. 9) LVAS manufactured by Arrow International Inc., is designed to be used as a “destination therapy” for patients with progressive, irreversible, end-stage (Class IV)



Fig. 9.

363

LionHeartTM LVAS.

congestive heart failure, for which heart transplantation is not an option. It is not intended as a bridge-to-transplant or as a bridge-to-recovery of ventricular function. It is a therapeutic option for patients who are ineligible for heart transplantation. Its fully implantable components eliminate the currently required lines (drive and venting) through the skin of the commercially available VADs. This is expected to result in significant reduction in the risk of infection, improved mobility, and enhanced quality of life. These lines and tethers are eliminated through the use of a transcutaneous energy transmission system. This represents a significant advance in mechanical circulatory-assist technology. The combined weight of the implanted components is 3.2 lb or 1.3 kg. The blood pump is electrically powered and is implanted in the preperitoneal space, beneath the left costal margin. The motor controller and internal coil control the operation of the blood pump. The motor controller is placed under the anterior abdominal wall in the pre-peritoneal space, beneath the right costal margin. The internal coil is placed in the subcutaneous tissue of the chest wall. The compliance chamber and access port serve as a variable gas-volume accumulator. The compliance chamber is placed in the left pleural space. The access port is passed through the intercostal space and located in the subcutaneous tissue over the left anterior chest wall.


364


Fig. 10.

HIA VERSUS LVAD.

HIA VERSUS LVAD The HIA VERSUS25 (Fig. 10) LVAD is an electromechanically driven pulsatile displacement pump, which is currently being developed at the Helmholtz Institute, Aachen to support the heart of patients for recovery from ESHF. To enable the implantation within the hemi-thorax an extremely flat energy converter has been designed. Its main mechanism is a gear system, which converts the rotational movement of a brushless DC motor into the translational movement of the pusher plate. In the diastolic phase of the pump, blood coming from the left atrium enters the pump chamber. When the pusher plate moves upward, it expels the blood out of the pump into the aorta (systolic phase). The pump unit delivers an output of 1.9–3.6 l/min at pump rates between 80 and 140 bpm. Power consumption averages 4.1 W. First animal and cadaver studies have been completed successfully. The tests showed stable pump operation without technical failures and very good fitting of the flat pump. Further animal tests are ongoing. On the technical site, the next generation VERSUS with smoother surfaces, easier assembly, and reduced weight is currently being tested. Further animal tests with the new generation device are planned within 2005. 4.1.3. Total artificial heart AbioCor T M TAH The AbioCorTM 9 (Fig. 11) implantable replacement heart is a fully implantable prosthetic system, intended as a substitute for severely diseased human hearts in patients suffering from coronary heart disease or some form of end-stage congestive heart failure. After



Fig. 11.

365

AbioCorTM TAH.

implantation, the device does not require any tubes or wires to pass through the skin. Power to drive the prosthetic heart is transmitted across the intact skin, avoiding skin penetration that may provide opportunities for infection. The AbioCorTM consists of two blood-pumping chambers. Each of the two pumps is capable of delivering more than 8 l of blood every minute. An internal controller regulates power delivered to the prosthetic heart. A rechargeable internal battery allows the patient to be completely free of the external power transmission unit for some period of time monitored by the internal system. The AbioCorTM system is designed to increase or decrease its pump rate in response to the body’s needs. It also includes an active monitoring system that provides detailed performance feedback and alarms in the event of irregularities. The AbioCorTM is currently in an initial clinical trial. Of the seven patients enrolled in the study so far, five were successfully supported. Four of the patients met the first survival milestone of 60 days. Three of these patients lived respectively for 142, 151, and 292 days; the remaining patient is alive at close to 500 days and has been living at home for the past 9 months. AbioCorTM was able to maintain normal hemodynamics in all supported patients. The majority were able to walk, shower, carry out normal interactions with family and friends and normal activities outside the hospital environment. ACcor TAH The development of the ACcor TAH26 (Fig. 12) was initiated at the Helmholtz Institute for Biomedical Engineering in 1990. The final


366


Fig. 12.

HIA ACcor TAH.

goal is an orthotopic pulsatile electromechanical TAH for use as a longterm bridge-to-transplantation, and finally for use as a permanent heart replacement system. The ACcor TAH is a double-chamber diaphragm pump, which replaces the explanted ventricles functionally and anatomically. It consists of three main components: two diaphragms pump chambers with inlet and outlet valves, and the electromechanical energy converter. The inlets of the pump chambers are connected to the natural atria while the outlets are connected to the aorta and the pulmonary artery, respectively. The energy converter consists of a brushless electronically commutated motor and a gear unit, which transforms the rotational movement of the motor into translation of the pusher plates. The pump delivers flows between 4 and 8 l/min at aortic pressures of 80 and 140 mmHg at different pump rates. Long-term animal studies using the ACcor TAH in calves are ongoing. 4.2. Rotary pumps 4.2.1. Axial pumps r (Fig. 13) was developed by DeBakey VAD The DeBakey VAD10 MicroMed Technology Inc. in conjunction with Drs. Michael DeBakey, r is a miniaturized George Noon, and NASA engineers. The DeBakey VAD axial flow LVAD capable of fully supporting the left ventricle. Its mechanical blood-immersed bearings showed no significant wear or failures after 2 years of continuous operation and demonstrated an expected durability



Fig. 13.

367

r. DeBakey VAD

in excess of 5 years. The titanium inflow cannula is inserted through a core in the left ventricular apex and secured with a sewing ring. The outflow graft is anastamosed to the ascending aorta. The pump is attached to the controller, which monitors its functions, for example speed, current, r is the only VAD power, battery status, and flow. The DeBakey VAD with an integral flow meter. The entire blood contacting surface is coated r . Patients are anticoagulated initially with heparin, then with Carmeda converted to anti-platelet and anti-vitamin K maintenance. r is CE marked and is in its pivotal bridge-toThe DeBakey VAD transplant trial in the USA. By mid-2004, over 70 DeBakey VADs have been implanted worldwide, representing an overall experience of over 40 patient-years. Forty-two patients have been discharged home to date, with a total out of hospital experience of over 8 years. Over 20 patients have had their implants over 6 months, and five over a year. Technical data: Weight 95 g, flow rate 5–10 l/min, rotational speed 7,500–12,000 rpm, power requirements 6 W, blood-immersed ceramic bearings. Jarvik 2000 LVAD The Jarvik 200011 (Fig. 14) is an axial flow blood pump that uses electric power to rotate a vaned impeller. The pump is connected between the left ventricular apex and the descending aorta. The impeller is a neodymium–iron–boron magnet, which is housed inside a welded titanium shell. The impeller is supported by ceramic bearings. A small cable, which exits the body through the abdominal wall, delivers power to the impeller. All of the blood-contacting surfaces are made of highly polished titanium. The normal operating range for the control system is 8,000–12,000 rpm, which will generate an average pump flow rate of 5 l/min. The pump speed is


368


Fig. 14.

Jarvik 2000.

controlled by an analog system controller. The pump speed can be manually adjusted in increments of 1,000 rpm. The control unit monitors the pump function and the remaining power in the batteries. Audible and visual alerts notify the user of any problems. Technical data: Diameter 25 mm, length 55 mm, weight 85 g, flow rate 5–6 l/min, rotational speed 8,000–12,000 rpm, power requirements 7–10 W, blood-immersed ceramic bearings. r r II LVAD The HeartMate II LVAD12 (Fig. 15) is manufacHeartMate tured by Thoratec Corporation and is presently in the first stages of implantation in Europe. This device is indicated for use in critically ill patients

Fig. 15.

r II LVAD. HeartMate



369

eligible for heart transplantation, but is not currently listed on a cardiac transplant list and for those not eligible for heart transplantation. r II LVAD is a high-speed, electric, axial flow, rotaryThe HeartMate r LVAD, assist device that does not replace the heart. Like the HeartMate this pump is designed to assist the heart in pumping blood to the body in r patients with a failing left ventricle. Like its predecessor, the HeartMate II LVAD has a drive line enabling electric wires to exit through the patient’s upper abdominal wall. The system circuitry, called the controller, interfaces between the patient and the power supply to operate the LVAD. There are two power source options for the LVAD: a power base unit, which plugs into r II LVAD a standard AC outlet and two 12-V batteries. The HeartMate is indicated for use in patients eligible for heart transplantation, but not currently listed on a cardiac transplant list and for patients not eligible for r II LVAD heart transplantation. Patients implanted with the HeartMate may live at home. Technical data: Diameter 43 mm, length 81 mm, weight 370 g, flow up to 10 l/min, rotational speed 8,000–15,000 rpm, power requirement 10 W, blood-immersed ceramic bearings. Impella microaxial pump The technology developed by Impella (Aachen, Germany) follows and goes beyond the tradition of the famous Hemopump and consists of a family of intracorporeal miniature axial pumps (Fig. 16). The world’s first biventricular intracardiac pump system, namely

Fig. 16.

r pump. Impella


370


r the Impella elect,27 represents a technological development which facilitates biventricular support during beating heart CABG procedures. The r recover was developed left ventricular intracardiac pump system Impella on the basis of the Impella platform technology. Only 6.4 mm in diameter, this first Recover system for cardiac surgery is pumping up to 4.5 l/min of blood. It consists of a left ventricular pump, a portable console including a docking station for controlling, and an infusion syringe pump for purging r recover is placed directly within of the intracardiac pump. The Impella the left ventricle and supports the patient’s circulatory system for up to 7 days. Possible indications include postoperative use for wean from the heart–lung machine, use in the event of minimal cardiac output, use in the event of cardiac shock or acute myocardial infarction to mention just a few. Technical data: Diameter 6.4 mm, length 30 mm, weight 5 g, flow 4.5 l/min, rotational speed up to 30,000 rpm, power requirement 10 W, polymeric shaft seal.

HIA MicroVAD The MicroVAD28 (Fig. 17) has been under development at the Helmholtz Institute Aachen and is currently being manufactured by Circulite Inc. (New Jersey, USA) as a miniature size axial blood pump, that features the smallest VAD system worldwide. It consists of a singlestage impeller and is powered by an integrated brushless electric motor. The torque is transmitted to the impeller via magnetic coupling. A hybrid bearing solution was chosen which combines three different bearing concepts: mechanical, magnetic, and hydrodynamic. The system underwent long-term durability tests, in vitro hemolysis tests, and thrombogenicity

Fig. 17.

HIA MicroVAD.



371

studies. Furthermore, animal tests have been performed in adult sheep. The MicroVAD is free of thrombi for up to 3 months of implantation without anticoagulation treatment. Further development activities focus on specific aspects of long-term stability and design optimization. Technical data: Diameter 30 mm, length 80 mm, operating point 4 l/min against 80 mmHg, rotational speed 20,000 rpm, priming volume 1.5 ml, hybrid bearing system. Berlin Heart Incor LVAD The implantable LVAD Berlin Heart Incor13 (Fig. 18) consists of electrically powered axial pump with a percutaneous pump cable, the inflow cannula, the outflow angle section and the outflow cannula, the extracorporeal control unit for power supply and communication, and two rechargeable batteries. The control unit and the batteries are carried in a shoulder bag. The axial pump is placed intrathoracally. The pump impeller is suspended by wear-free magnetic bearings. The Incor system maintains a constant mean circulation flow, the pulsatility of this flow depending on the remaining spontaneous activity of the heart. Coating of r BioActive Surface. blood contact surfaces is by Carmeda Currently the system is being evaluated in the INIT Incor Initial Trial. As of January 2003 the system has been implanted in 15 patients (13 males, 2 females, average age 45 years) in five different centers. The longest implantation period is >224 days and the average implantation period is 83 days. The cumulative experience with the device is >1, 250 days. One patient was successfully weaned from the device after improvement of cardiac function. Two patients died from bleeding-associated complications (one intracerebral hemorrhage and another multiorgan failure after mass-transfusion).

Fig. 18.

Berline Heart Incor.


372


Clinical results are very encouraging. There are no signs of significant hemolysis. Technical data: External diameter 30 mm, length 114 mm, mass 200 g, volume 60 cm3 , design point: 5 l/min at 8,500 rpm against 100 mmHg, power requirements: 1 W. magnetic bearings, 2–4 W motor, 5 W electronics, magnetic bearings. 4.2.2. Radial pumps Terumo DuraHeart T M LVAD The DuraHeartTM 14 (Fig. 19) incorporates a 3-dimensional magnetically levitated centrifugal pump. The primary system is comprised of the implantable Mag-lev centrifugal pump and drive unit, a wearable controller, wearable Li ion battery packs, AC/DC converter, battery charger, and hospital console. In the past year, several key clinical milestones have been successfully met: The final cadaver-fit studies (conducted at the Cleveland Clinic Foundation) verified a BSA ≥ 1.1 m2 relative to device placement; and multiple animal tests verifying component integrity. At the Utah Artificial Heart Institute, final calf studies have been conducted encompassing biocompatibility and final system validation. After successfully passing clinical studies at the Heart and Diabetes Center (Bad Oeynhausen, Germany) the system is currently being implanted in patients awaiting heart transplantation.

Fig. 19.

DuraHeartTM LVAD.



373

Technical data: Weight 400 g, volume 96 cm3 , design point: 5 l/min at 2,000 rpm, power requirements: 15 W (9 W for magnetic suspension), magnetic bearing. VentrAssist T M LVAD The VentrAssistTM 15 (Fig. 20) device consists of a small diameter diagonal flow impeller with an integrated rare-earth-magnet motor. The rotor is suspended by contact-free hydrodynamic forces. The pump is encased in a biocompatible titanium alloy shell. An externally worn battery and controller provides power and regulate operation according to physiologic demands. Based on current battery technology, operation for periods of over 8 h is possible without battery changes. The pump is implanted below the diaphragm and is connected to the left ventricular apex and the ascending aorta using standard grafts. Animal studies have been successfully completed. On March 10, 2004 the fifth VentrAssist human implantation has been successfully performed in a pilot trial at the Alfred Hospital in Melbourne, Australia. Technical data: Weight 400 g, diameter 60 mm, hydrodynamic bearing, passively suspended rotor, 7–8 W, rotational speed 1,800–3,000 rpm, sensorless motor control. CorAide T M LVAD The Arrow CorAideTM 16 (Fig. 21) LVAD is an electrically powered circulatory assist system. It is comprised of an implanted blood pump with inlet and outlet cannulae, a percutaneous cable, an external portable electronic module with electric cables, and Ni-metal hybrid battery packs. A patented apical cuff allows for positioning and fixation of the apical cannula to optimize flow into the pump. The pump

Fig. 20.

VentrAssistTM LVAD.


374


Fig. 21.

CorAide LVAD.

and tether weigh 303 g, the pump being fabricated from titanium with a biocompatible coating being applied to much of the blood-contacting surfaces. The rotating assembly does not require mechanical bearings, but utilizes a combination of magnetic and hydrodynamic forces for its suspension and function. An automatic control algorithm has been designed to calculate and control the flow through the pump. An algorithmic watchdog adjusts pump speed to guard against inadvertent suck down of the ventricle and to adjust to the variable physiologic needs. One R&D pump operated for 2 years of continuous operation at a pressure head of 95 mmHg and flows from 4.8 to 5.2 l/min over the duration of the study. Additional tests with pulsatile input are under way. Using this LVAD, 18 chronic in vivo experiments were completed previously in the calf model: 14 systems operated for approximately 30 days, and 4 operated for approximately 90 days. Mean pump flow was 5.9 l/min, average pump speed was 2,700 rpm, average MAP was 95 mmHg. There were no occurrences of bleeding or organ dysfunction. The last t6 implants were done without anticoagulation. Technical data: Weight 210 g, volume 62 cm3 , design point: 5 l/min at 3,000 rpm, power requirement 6–7 W, combination of magnetic and hydrodynamic bearing.



Fig. 22.

375

HeartQuestTM LVAD.

HeartQuest T M LVAD MedQuest Inc. has completed the design, assembly, bench testing, and animal experimentation of an implantable, magnetically suspended HeartQuestTM VAD17 (Fig. 22) clinical prototype. The device’s only moving part, a rotating impeller, is completely suspended in the housing using electromagnetic fields. HeartQuestTM clinical systems have been assembled and extensive animal and bench testing has begun. It is anticipated that the HeartQuestTM VAD will be available for clinical trials in patients in 2002. The first HeartQuestTM VAD to be introduced clinically, the HQ-I, will have a small diameter flexible cable through the skin to transmit power and data to the implanted blood pump from an external patient controller. Technical data: Weight 600 g, volume 275 cm3 , flow rate up to 10 l/min, nominal rotational speed 2,500 rpm, power requirements 6.5 W for magnetic bearing, hybrid magnetic bearing, combination of permanent magnets and electromagnets.

4.2.3. Diagonal pumps r r The DeltaStream29 (Fig. 23) blood pump, Medos DeltaStream developed by the Helmholtz Institute, Aachen and manufactured by Medos AG (Stolberg, Germany), is a rotary pump with a diagonal flow impeller. It has been developed for ECC with the main focus on simplified bypass systems. Small size and an embedded electric motor are key pump features. Due to its compact construction, the pump can be placed extremely flexible


376


Fig. 23.

Medos DeltaStream.

and very near to the patient. The option of a pulsatile flow mode for VAD applications has also been demonstrated in vitro. Durability studies have been performed to assess the lifetime of the pump especially with regard to the incorporated polymeric seal. Seal lifetime of up to 28 days have been achieved, using different blood substitutes. In animal tests, biocompatibility, low hemolysis, and non-thrombogenicity have been demonstrated. Besides heart lung machine and simplified bypass system applications, ventricular assist and ECMO up to several days appear also as promising potential applications. The pump is in successful clinical use since end of 2001. Technical data: Diameter 41 mm, length 150 mm, flow 0–10 l/min, rotational speed 1,000–10,000 rpm, pressure head 0–600 mmHg, priming volume 30 cm3 , polymeric shaft seal. r HIA Microdiagonal Pump As a follow-up version of the DeltaStream the Microdiagonal pump (MDP) (Fig. 24) is a mixed-flow-type RBP, which is currently being developed at the Helmholtz Institute, Aachen as a long-term VAD for clinical use.30 The compact rotary pump consists of a brushless DC motor, which drives the mixed flow impeller by means of a magnetic coupling system. Inside the pump, the blood stream simultaneously serves to cool the integrated electric motor. Two pump designs have been worked out for mid-term and long-term use, respectively. The mid-term MDP comprises a hybrid bearing structure based on a combination of a mechanical pivot bearing and a fluid film bearing. This pump will be used as a paracorporeal VAD system. The long-term MDP has a completely contact-less impeller suspension without friction and wear. The unique feature of this rotor bearing lies in its sole use of passive elements based on a permanent



Fig. 24.

377

Microdiagonal pump.

magnetic bearing and a fluid film bearing, without the need for any additional sensors and actuators. Both pump configurations are currently in the animal testing phase. Technical data: Diameter 30 mm, length 80 mm, flow 0–10 l/min, rotational speed 1,000–10,000 rpm, pressure head 0–300 mmHg, priming volume 15 cm3 , hybrid bearing system. r r : The MAQUET RotaFlow (Fig. 25) is an extracorporeal RotaFlow centrifugal blood pump which has been developed in cooperation between the German Heart Centre (Munich, Germany) and the MAQUET Cardiopulmonary AG (Hirrlingen, Germany), formerly Jostra Medizintechnik AG.31,32 It has a spinning rotor driven by an external DC brushless motor and a permanent magnetic radial coupling. The disposable pump head itself consists of a polycarbonate pump housing including a spiral volute, a polycarbonate shrouded impeller that incorporates a set of NdFeB magnets as the driven component of the magnetic coupling and a magneto-mechanical bearing system. The bearing system to stabilize the rotor against axial and radial displacements within the pump housing comprises a blood immersed single pivot bearing (ceramic/plastic material combination) for axial stabilization and a passive magnetic bearing implemented within the magnetic coupling to provide additional stabilization in radial and axial direction. r is mainly used for short-term cardiopulmonary bypass The RotaFlow in continuous and pulsatile flow mode. In combination with a hemocompatr r the RotaFlow recently obtained CE mark approval ible Bioline Coating for midterm applications up to 14 days.


378


Fig. 25.

r MAQUET RotaFlow

r The RotaFlow is in clinical use since 1998 and has successfully pasted more than 100,000 clinical applications. Efforts on the long-term version of r are currently ongoing. the RotaFlow Technical data: Priming volume 32 ml, diameter 50 mm, max. speed 5000 rpm, max. pressure rise 750 mmHg, max. flow 10 l/min, flow sensor integrated in drive unit, biocompatible coating, pulsatile flow option.

5. Conclusions The various pump configurations presented and their clinical applications, from short-term use in cardiopulmonary bypass to long-term implantation, demonstrate that the field of blood pumps is extremely broad and challenging. Also, basic physiologic questions, such as the need for pulsatile flow in long-term applications, have not yet been fully answered. Therefore, both pump categories (i.e., displacement and rotary pumps) have their specific value in terms of cardiac and circulatory support. Future R&D activities on blood pumps will focus on long-term implantable systems. The successful use of short- and mid-term blood pumps strengthens the belief that long-term circulatory support toward destination therapy (permanent use) will become possible.



379

RBPs will be more appropriate for miniaturization than pulsatile displacement pumps and therefore more preferable for pediatric circulatory support. However, continuous flow RBPs must still be validated clinically in the long-term run and additional pulsatile pump operation may be needed. To enhance device safety and durability magnetic rotor bearings will be favored over mechanical bearings in the pump design process. The most challenging issues in future blood pump development will concentrate on miniaturization, antithrombogenicity, low anticoagulation levels, patient mobility, and device cost-efficiency. It is hoped that interdisciplinary research efforts among engineers, clinicians, and regulatory institutes will lead to new technical solutions that will improve patient care together with quality of life in a wide variety of applications. Suggested Reading 1. Artificial Heart, Vols. 1–5, eds. T. Akutsu and H. Koyanagi (Springer Verlag, Tokyo, Berlin, Heidelberg, New York, 1986–1996). 2. Assisted Circulation, Vols. 1–4, ed. F. Unger (Springer Verlag, Berlin, Heidelberg, 1979–1995). 3. Rotary Blood Pumps (Special Issues 2001, 2002); Artificial Organs, Vol. 26, 11; Artificial Organs, Vol. 25, 5. References 1. P. M. Portner, A totally implantable ventricular assist device for end-stage heart disease, Assisted Circulation, Vol. 2, ed. F. Unger (Springer Verlag, Berlin, 1984), 115–141. 2. P. M. Portner, A totally implantable heart assist system: the Novacor program, Heart Replacement, eds. T. Akutsu, and H. Koyanagi (Springer Verlag, Tokyo, 1993), 71–80. 3. T. J. Myers, K. A. Dasse, O. H. Frazier et al., Use of a left ventricular assist device in an outpatient setting, ASAIO Journal 40, 3 (1994) 471–475. 4. A. El-Banayosy, L. Arusoglu, R. Korfer et al., Preliminary experience with the LionHeart left ventricular assist device in patients with end-stage heart failure, Annals of Thoracic Surgery 75, 5 (2003) 1469–1475. 5. W. J. Kolff, Total artificial hearts, LVADs or nothing? And muscle and airpowered LVADs, Heart Replacement (Springer Verlag, 1993), 3–11. 6. C. Cabrol, A. Gandjbakhch, A. Pavie et al., Heart transplantation — status in Europe, Artificial Heart Vol. 3, eds. T. Akutsu, and H. Koyanagi (Springer Verlag, Tokyo, 1990), 263–267. 7. W. S. Pierce, The artificial heart: paths, progress and patience, Artificial Organs 16, 3 (1992) 314–326.


380


8. J. G. Copeland, R. Smith, T. Icenogle et al., Orthotopic total artificial heart as a bridge to transplantation, preliminary results, Journal of Heart Transplantation 8, 2 (1989) 123–128. 9. R. D. Dowling, L. A. Gray, O. H. Frazier et al., Initial experience with the AbioCor implantable replacement heart system, Journal of Thoracic and Cardiovascular Surgery 127, 1 (2004) 131–141. 10. M. E. DeBakey, Development of a ventricular assist device, Artificial Organs 21, 11 (1997) 1149–1153. 11. S. Westaby, O. H. Frazier, R. K. Jarvik et al., Implant technique for the Jarvik 2000 Heart, Annals of Thoracic Surgery 73, 4 (2002) 1337–1340. r II left ven12. B. P. Griffith, R. L. Kormos, K. C. Butler et al., HeartMate tricular assist system: from concept to first clinical use, Annals of Thoracic Surgery 71, 3 (2001) 116–120. 13. R. Hetzer, Y. Weng, J. Mueller et al., First experiences with a novel magnetically suspended axial flow left ventricular assist device, Artificial Organs 28, 2 (2004) 218–225. 14. C. Nojiir, T. Kijima, J. Maekawa et al., Terumo implantable left ventricular assist system: results of long-term animal study, ASAIO Journal 46, 1 (2000) 117–122. 15. P. A. Watterson, J. C. Woodard, J. A. Reizes et al., VentrAssist hydrodynamically suspended, open, centrifugal blood pump, Artificial Organs 24, 6 (2000) 475–477. 16. R. L. Gerhart, D. J. Horvath, L. A. Golding et al., The effects of impact on the CorAide ventricular assist device, ASAIO Journal 48, 4 (2002) 449–452. 17. E. A. Allaire, H. G. Wood, D. B. Olsen et al., Blood flow in a continuous flow ventricular assist device, Artificial Organs 23, 8 (1999) 769–773. 18. A. L. Throckmorton, P. E. Allaire, H. P. Gutgesell, D. B. Olsen et al., Pediatric circulatory support systems, ASAIO Journal 48 (2002) 216–221. 19. G. D. Pennington, and M. T. Swartz, Circulatory support in infants and children, Annals of Thoracic Surgery 35 (1993) 233–237. 20. A. Sidiropoulos, H. Hotz, and W. Kronertz, Pediatric circulatory support, Journal of Heart Lung Transplant 17 (1998) 1172–1176. 21. D. J. Farrar, and J. D. Hill, Univentricular and biventricular Thoratec VAD support as a bridge to transplantation, Annals of Thoracic Surgery 55, 1 (1993): 276–282. 22. P. A. Wassenberg, The Abiomed BVS 5000 biventricular support system, Perfusion 15, 4 (2000) 369–371. 23. W. Konertz, H. Hotz, M. Schneider, M. Redlin, and H. Reul, Clinical experience with the MEDOS HIA-VAD system in infants and children: a preliminary report, Annals of Thoracic Surgery 63, 4 (1997) 1138–1144. 24. T. Drews, M. Loebe, E. Hennig, F. Kaufmann, J. Muller, and R. Hetzer, The ‘Berlin Heart’ assist device, Perfusion 15, 4 (2000) 387–396. 25. F. Foerster, R. Kaufmann, H. Reul, and G. Rau, A small pulsatile blood pump for ventricular support during end-stage heart failure, Artificial Organs 24, 5 (2000) 373–376.



381

26. R. Kaufmann, H. Reul, and G. Rau, The Helmholtz total artificial heart Labtype, Artificial Organs 18, 7 (1994) 537–542. 27. T. Schmidt, J. Siefker, S. Spiliopoulos, and O. Dapunt, New experience with the paracardial right ventricular axial flow micropump impella elect, European Journal of Cardio-Thoracic Surgery 24, 2 (2003) 307–308. 28. B. Meyns, W. Kerkhoffs, H. Reul et al., Animal testing of Microvad: from extracorporeal model to implantation without anticoagulation, Cardiovascular Engineering 8, 3 (2003) 25. 29. C. Goebel, R. Eilers, H. Reul et al., Development of the MEDOS/HIA Delta Stream extracorporeal rotary blood pump, Artificial Organs 25, 5 (2001) 358–365. 30. M. Akdis, M. Martin, C. G¨ obel, H. Reul et al., Development and testing of a microdiagonal blood pump, International Journal of Artificial Organs 25 (2002) 666. 31. N. Mendler, F. Podechtl, G. Feil, P. Hiltmann and F. Sebening, Seal-less centrifugal blood pump with magnetically suspended rotor: Rot-A-Flot, Artificial Organs 19, 7 (1995) 620–624. 32. Y. Orime, M. Shiono, S. Yagi et al., Jostra Rota Flow RF-30 pump system: a new centrifugal blood pump for cardiopulmonary bypass, Artificial Organs 24, 6 (2000) 437–441.




DEVELOPMENT OF AN AXIAL BLOOD PUMP WENG KONG CHAN∗ and YEW WAH WONG College of Engineering, School of Mechanical and Aerospace Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 ∗[email protected]

1. Introduction In the United States, about 800,000 new cases of heart disease are reported each year.1 Of these, about 50,000 Patients are in need of a new heart but there are only about 2,500 donor hearts available. Similar situation can be drawn worldwide. In Singapore, heart-failure-related disease is the second biggest killer after cancer.2 Each year, of the 16,000 cases or so of reported deaths, nearly one-quarter is due to heart disease. The Ministry of Health has also predicted that in 20 years’ time heart disease could cause up to 40% of all deaths. Unfortunately, the number of donors is largely insufficient to cater for the demand needed. This makes cardiac-assist devices an attractive alternative as a bridging device to sustain the life of the patient to enable a weakened heart to recover or until a donor heart can be found. Recent technology advances in cardiac-assist devices have paved the way to a better quality of life for end-stage heart patients. Rotary blood pumps have proven to be a viable choice as compared to the pusher-platetype diaphragm blood pumps.3–5 The engineering problems related to the development of rotary blood pumps are complex. Reliability problems with mechanical parts, valves, and thrombus formation (clotting of blood) have hampered efforts in the development of ventricular-assist devices (VADs). Recently, magnetic bearings have been used to completely suspend the rotor. This approach eliminates mechanical contact and avoids the need for a drive shaft. Another major difficulty encountered in the development of VADs is hemolysis, the destruction of red blood cells (RBCs). When blood flows through a blood pump, high level of energy is transmitted from the impeller to blood cells. Blood cells, when in direct contact with any mechanical devices, will be damaged if they are exposed sufficiently long 383


384

W. K. Chan and Y. W. Wong

to high shear forces. This is critical especially when VADs are designed to run at high rotational speed. The design of the blade profile becomes important since the blade geometry plays an important role in determining the nature of the flow field in the pump. Vortices or flow separations are highly undesirable as they could lead to hemolysis and thrombus formation. The investigation of the trajectories of RBCs during its passage through the pump is therefore important to minimize hemolysis, thrombosis, and shear-related issues. In this chapter, we will present our work on the design, computational fluid dynamics (CFD) simulations, novel manufacturing techniques used in the fabrication of the impellers and diffusers, and hydraulic performance characterization of a miniature axial blood pump being developed in our laboratory. Our preliminary work on the motor design is also described.

2. Background The left ventricle (LV) has the heaviest workload as compared to the other three chambers in the heart, because it pumps blood to the rest of the body. As such, if the LV is weakened, vital organs like the brain and kidney will not receive sufficient blood, which could lead to severe consequences. As most patients die due to the failure of the ability of LV to supply blood to the rest of the body, attention is focused on the development of the left-ventricular-assist devices (LVADs), which can either be pulsatile and nonpulsatile (rotary blood pumps). Though the working mechanism of the pulsatile pump resembles the pumping action of the heart, its design is relatively more complex and expensive. Coupled with the complexity and high cost of pulsatile which VADs, doctors and patients have sought alternatives such as rotary blood pumps. Rotary blood pumps have several potential advantages over pulsatile pumps. Basically, they are smaller in size and could be used in patients of smaller body frame; they also have fewer moving parts compared to the pulsatile pumps. They require a lower energy input and do not require a compliance chamber in the system. Currently, available pulsatile flow support devices are too large and expensive to be used for the general population. Rotary blood pumps such as axial and centrifugal blood pumps are small and inexpensive. However, current pump designs can be used only for a limited duration because of problems such as hemolysis and thrombus formation. Initial concerns on the feasibility of using rotary blood pumps as LVADs were proven unfounded when clinical trials demonstrated that rotary blood


Development of an Axial Blood Pump

385

pumps have higher efficiencies and are more reliable.7 In addition, when compared to a pulsatile blood pump, rotary blood pump has a simpler design and can be manufactured easily. The only major concern for the use of a rotary blood pump is the physiologic and circulatory abnormalities caused by its nonpulsatility. In 1979, studies by Golding and colleagues8 established that hemodynamic parameters displayed by nonpulsatile blood are not different to those of pulsatile total body perfusion. Hence, rotary blood pumps are recognized to be potential candidates as cardiac-assist devices. One of the primary requirements of such a pump is that it should be able to generate 5 l/min of blood flow at a pressure of 100 mmHg, which is the nominal output of the human heart. Also, for the pump to be implantable, it should be compact and efficient so that it can be operated using battery packs. Rotary blood pump technology has been applied successfully to circulation outside the body, especially in cardiopulmonary bypass (CPB). Several types of centrifugal pumps have been developed and used successfully for open-heart surgery. Clinical experience illustrated that rotary blood pumps have many substantial advantages over conventional roller pumps. In this chapter, our studies on the development of an axial blood pump are presented. The overall pump development strategy is shown in Fig. 1. The focus of this chapter is to present our work on the pump geometry and the drive system. The integration of these subsystems will provide the prototype for in vitro testing. When deciding on the choice of the pump, the size of the patient is an important factor to be considered. Centrifugal pumps may be simpler compared to axial pumps, but they are larger and may not be suitable for smaller patients.6 To make VADs more practical and implantable, a smaller pump is necessary. However, due to its smaller size, an axial blood pump has to operate at a much higher rotational speed. This calls for a need to study closely and analyze the flow pattern in axial flow blood pumps. The axial flow pump contains an impeller that spins at 10,000–20,000 rpm and can pump about 5–7 l of blood/min. The centrifugal pump can achieve the same flow rate at 2,000 rpm due to its larger size. The smaller-size axial flow pump is ideal for Asian and smaller-size patients. Once the pump type is selected, the drive system and the materials to be used are the other factors to be considered. In this chapter, we first describe the preliminary design specifications of the pump. A test rig was developed to test its performance. Numerical modeling is also carried out to examine the flow characteristics of the initial design. The results of the numerical modeling were validated against the experimental results. Based on these


386


Conceptual Design of Blood Pump * Pump Type * Drive system

Flow Analysis

Drive System

Material

CFD Modelling Prototype Fabrication Analysis/ Measurement

Motor

Bearings

Refinement

Integration

Prototype Characterization

Fig. 1.

Overall development chart for a blood pump.

results, important information on the position and gap clearance between the inducer and the impeller were obtained. An iterative approach using CFD was adopted to improve the geometry of the impeller. We examine the flow pattern, pressure distribution, and shear stresses in the blade passage as these are contributory factors that affect both hemolysis and thrombosis. After a satisfactory design is obtained, the next step is to manufacture the impellers. Different RP processes were tested and it was concluded that models built using selective laser sintering (SLS) yield the best results. Then, experimental studies on the pressure rise along the impeller were conducted and these were used to validate the CFD results. Blood trauma predictions were also conducted to estimate the shear stresses on the fluid particles passing through the pump. Finally, the design of the motor was



387

described. To avoid clotting at the shaft, the impeller has to be driven magnetically. Different designs were explored and a model was fabricated to test its feasibility. 3. Preliminary Design Specifications The function of an axial blood pump is to assist a diseased heart when it is not operating in the manner the body requires. Therefore, the pump, like the heart, must be capable of producing a pressure of 100 mmHg at a flow rate of 5 l/min. This forms the design specifications of the pump developed in our laboratory. The pump is designed to operate at 10,000 rpm. Since the pump has to be inserted into the human chest, the physical dimensions of the pump are limited. The pump consists of four parts, namely straightener, inducer, impeller, and diffuser. Both the straightener and the diffuser are designed to be stationary and the only rotating components are the inducer and impeller manufactured as an integrated part. The main function of the straightener is to prevent pre-rotation of fluid before it enters the inlet. The airfoil profile of the blades and the hub extension enable a more streamlined flow. Due to size constraints, the diameter of the straightener is 16 mm and its length is 18 mm. Fluid leaving the straightener enters the inducer. The inducer has an inlet angle of 5◦ and outlet angle of 30◦ . Two sets of inducers–impellers were fabricated: one with two blades while the other has three blades. Figure 2 shows the two inducers–impellers. The main function of the inducer is to impart kinetic energy gradually to the fluid. This is important since cavitation must be avoided at the pump inlet. Fluid leaves the inducer

Fig. 2.

Double- and triple-start inducer–impeller.


388


and enters the impeller where they will receive most of the kinetic energy. The inlet blade angle of the impeller is kept at 30◦ while the outlet angle is fixed at 90◦ . The current impeller–inducer measure 40 mm in length, and the diameter of the hub is 9 mm. Fluid leaving the impeller enters the diffuser. The function of the diffuser is to convert swirl energy from the inducer–impeller to potential energy. The diffuser inlet angle must be close to the direction of fluid particles leaving the impeller. Initial calculations placed the outlet angle of fluid stream at approximately 81◦ . The wrap angle for the diffuser is 100◦ . The trailing edge angle of the diffuser is 90◦ . The trailing edge is extended 3 mm to provide a straightening effect on the flow leaving the pump. Preliminary experimental studies will be conducted based on this design to determine the hydraulic performance of the pump designed. It is expected that the flow within the inducer, impeller, and diffuser to be non-optimal and further refinement is necessary. Details of this work can be found in Ref. 9.

4. Experimental Setup A closed-loop test rig consisting of a primary drive system, an electromagnetic flow meter, a pressure transducer, and the pump elements is designed and fabricated. The test rig is shown in Fig. 3. The straightener, inducer, impeller, and diffuser are housed in transparent pump housing. The inducer

Fig. 3.

Closed loop axial pump test rig.



389

and impeller, are fabricated from a single aluminium rod and it is shaft driven. The drive system is made up of a DC motor, a tachometer, and a motor controller linked to a computer system. The test facility runs in a closed-loop mode. The output angular speed of the DC motor is measured by the tachometer, which is mounted at the opposite end of the motor shaft. The tachometer provides feedback to the controller to maintain motor speed at a constant value. The DC motor drives the inducer–impeller in the test section. This DC motor speed is user selectable through the computer system, which also displays the actual operating speed. The software can also control the start– stop functions of the motor. Two valves along the piping system are used to control the flow rate. An open reservoir upstream of the pump inlet serves as an accumulator for the test fluid. Blood analog is used as the test fluid. The volume flow rate is measured through a measuring probe that is connected to an electromagnetic blood flow meter MFV-3200 (Nihon Kohden Corporation), which converts electromagnetic signals to digital signals. These digital signals are displayed as numerical values, which can then be read off by the user. The last component in the system is the pressure transducer (Druck 0–300 mmHg), which measures the head generated across the test section.

4.1. Comparison of pump performance at different operating speeds The pump characteristics for the double-start inducer–impeller are plotted for different operating speeds in Fig. 4. It can be noted, in general, the characteristics curves for the four speeds are similar to each other. This indicates that the pump characteristics at one speed can be used to derive the characteristics at other speeds. In addition, it is observed that at a nominal flow rate of 5 l/min, the pump needs to be operated at a speed higher than 8,000 rpm to provide a head of at least 100 mmHg which is the physiologic demand of the human body. Indeed, at an operating speed of 10,000 rpm, a head of 100 mmHg is attainable for flow rates less than 7 l/min.

4.2. Comparison of pump performance with different inducers The pump characteristics for the two different inducers operating at a speed of 10,000 rpm are shown in Fig. 5. The difference between the two inducers


390


Comparison of pump performance at different operational speeds 300.00 Pump pressure (mmHg)

6000 rpm

250.00

8000 rpm 10,000 rpm 11,000 rpm

200.00 150.00 100.00 50.00 0.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

Flow rate (l/min) Hydraulic performance of double-start inducer at different rotational speeds.

Comparison of pump performance with different Inducer– Impellers and 100 degree Diffuser (blood analog) at rotational speed of 10,000 rpm 240.00 220.00

Double start inducer-impeller Triple start inducer-impeller

200.00

Pump pressure (mmHg)

Fig. 4.

180.00 160.00 140.00 120.00 100.00 80.00 60.00 40.00 20.00 0.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

Flow rate (l/min) Fig. 5.

Hydraulic performance of double- and triple-start inducers.



391

lie in the number of blades they have at the inducer. One is a doublestart inducer–impeller, while the other is a triple-start inducer–impeller. It can be observed that the former configuration generate a higher pressure across the pump for a wider range of flow rates in comparison to the latter configuration. The same observation was also noted when the two inducers were used with different diffusers. This may be due to a higher blockage at the inducer inlet for the triple-start configurations. Further analysis also revealed that at a flow rate of 5 l/min, the doublestart inducer–impeller configuration gives a pressure rise of approximately 120 mmHg and is able to maintain this at above 100 mmHg for up to a flow rate of 7 l/min. However, for the triple-start inducer–impeller configuration, it can only give a pressure rise of approximately 100 mmHg at 5 l/min and beyond that, the pressure generated drops below 100 mmHg. This suggests that the latter is not an appropriate design for blood pump, as it does not provide sufficient pressure head at a higher flow rate, which may be a result of changes in physiologic conditions. The inlet of the triple-start inducer serves, as a blockage to incoming flow, thus operating it at a higher flow rate will result in greater pressure losses in the inducer. This explains the steeper pump performance curve for the triple-start inducer–impeller configuration. Comparatively, for the double-start inducer–impeller, it allows operation of the pump over a range of flow rates while maintaining pressures above 100 mmHg, which is more suitable as cardiac-assist device. The experimental studies demonstrated that the pump can meet the requirements of the human physiologic system. However, it is important to examine the flow pattern within the pump assembly since regions of high shear and flow separations are undesirable as these may cause hemolysis and thrombus formation. To have a quick and cost-effective means of determining the flow field, a numerical approach is adopted.

5. Numerical Flow Simulation CFD is a tool that enables researchers to study the fluid motion in or around devices of interest. In CFD, a computational model that represents a system is first constructed. By applying fluid flow physics to the system, the software computes the flow pattern. It predicts flow behavior, transfer of heat, mass, phase change, chemical reaction, mechanical movement, stress, and solid deformation. When properly formulated and applied, CFD can help to reduce blood pump development time to an order of months. However,


392


CFD results must be applied with cautions, and experimental verifications are needed to ensure that the results predicted are reliable and representative of the actual flow pattern in the system. Pump designs are usually studied computationally in the preliminary stages. Promising designs will then be selected for experimental studies and validations. This approach is usually adopted as it is both time- and cost-effective. In this study, CFX-TASCflow,10 a commercial software, was employed to model the flow in the entire pump. The CFX-TASCflow system, developed by AEA Technology Engineering Software Limited, is an integrated Navier–Stokes’ fluid flow prediction software. One important feature which makes it suitable for this project is its capability to model flows in rotating frames of reference. The meshed grids are created using CFX-Turbogrid. All grid files were then imported into TASCflow, followed by setting the appropriate boundary conditions. The typical CPU time for a problem is about 12–36 h depending on the number of grids and capability of the work station. Figure 6 shows the computational blood pump model created using CFX software. The CFD model includes the flow straightener, inducer, impeller, diffuser, and two tubes with axial length 10 and 20 times the diameter representing the inlet and outlet, respectively. Skew angles of not less than 20◦ were ensured during the construction of the grids. Fluid properties were assumed to follow that of blood, having a viscosity of 3.5 × 10−3 Pa s and a density of 1055 kg/m3 . The inducer and impeller were set to rotate at

Diffuser

Flow straightener

Inducer–impeller

Fig. 6.

Meshed grids of the blood pump.



393

10,000 rpm. Walls of the channel, straightener, and diffuser were set to stationary in the absolute (stationary) frame, while the walls of the inducer and impeller were stationary in the relative (rotating) frame; that is, rotating together with frame at 10,000 rpm. At the interfaces between the rotor and stator (straightener–inducer and impeller–diffuser), a frozen rotor interface is adopted. At a frozen rotor interface, information such as velocities are passed from the impeller side to the diffuser as the flow crosses the interface without imposing any circumferential averaging. The relative position of the two components remains fixed for the entire simulation. At the outlet, a zero pressure over the cross section is prescribed and a mass flow rate of 0.0879 kg/s (corresponding to a flow rate of 5 l/min) is set at the inlet. At the inlet, the turbulent kinetic energy k and its dissipation ε are set according to the following relations: k=

3 (T u · v)2 , 2

(1)

ε=

k 1.5 , L

(2)

where T u is the turbulence intensity, which is assumed to be 2% and L is the eddy length scale, assumed to be 10% of the inlet tube diameter. Typically for axial blood pumps, the Reynolds number is small (in the order of several tens of thousand) and the upstream flow is not fully turbulent. Hsu11 has shown in his studies that the flow patterns and turbulent stresses within the pump are in general independent of the assumed turbulent stresses at the inlet. Hence, we assume that magnitude of the turbulence intensity at the inlet will not play a significant part in the solution. In general, it is believed that the turbulence generated by the rotating element will dominate the flow pattern. In our study, the Reynolds number based on the diameter of the impeller is about 71,000. This is fairly low and to better approximate the flow field, the k–ω based SST turbulence model instead of the commonly used k–ω or the k–ε model is assumed in the present study. Here k, ε, and ω represent the turbulent kinetic energy, turbulent dissipation rate, and turbulent frequency, respectively. The latter two models use the gradient diffusion hypothesis to relate the Reynolds stresses to the mean velocity gradients and the turbulent viscosity. The turbulent viscosity is modeled as the product of a turbulent velocity and turbulent length scale. The k–ε model has been known for its robustness, economy, and reasonable accuracy for a wide range of turbulent flow. However, it tends to predict the onset of separation too late and under-predict the amount of


394


separation. The k–ω model which had shown more accurate prediction of separation in a number of test cases and in industrial applications, on the other hand, does not predict as well as the k–ε model in region outside the boundary layers. The k–ω-based SST turbulence model was chosen for this study as it combines the advantages of the k–ω and k–ε models in the nearwall regions and outside the boundary layers, respectively. One of the major problems of standard two-equation turbulence models is that they often fail to predict the onset and the amount of flow separation under adverse pressure gradient conditions. The k–ω-based SST model was designed to give a highly accurate representation of the separation phenomena by the inclusion of transport effects into the formulation of the eddy viscosity. It accounts for the transport of the turbulent shear stress and gives highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients. Studies12–15 have demonstrated that the SST turbulence model provides better accuracy than both k–ω and k–ε models in pressure-driven flows. Grid dependency and validation of the numerical simulation were carried out and reasonably good agreement was obtained when the predicted pump performance is compared with available experimental data as shown in Fig. 7. The number of cells used is about 350,000. We have compared these results with two flow rates (5 and 6 l/min) ran using 400,000 cells and the

35000

200

20000

150

15000 100 10000

Pressure, mmHg

25000 Pressure, Pa

250

Experimental Computational (120 k grids) Computational (250 k grids) Computational (350 k grids) Computational (400 k grids)

30000

50

5000 0

0 0

2

4

6

8

10

Flow rate (l/min) Fig. 7.

Pump characteristics at 10,000 rpm (CFD versus experimental data).



395

agreements are reasonably good. In view of this, we have used 350,000 cells in all our simulations to save time and computing resources. Since both the inducer and impeller are rotating, the velocity vectors shown in all the figures are relative to the rotating frame. The flow pattern in the mid-circumferential plane of the inducer is illustrated in Fig. 8, where it is observed that fluid are, in general, guided smoothly through the inducer. As the inducer is rotating at 10,000 rpm, it is essential that fluid particles entering the inducer be guided gently. This is the reason for the inducer having a wrap angle larger than 180◦ . No flow separations or backflow were observed. Investigations on the effects of the inducer angular

Fig. 8.

Velocity vectors of inducer at mid-circumferential plane.


396


Flow direction

Flow direction

Fig. 9. Axial velocity contours on hub surface of impeller (blue regions represent flow reversals).

alignment with respect to the impeller as well as the axial gap between the two were also investigated and are described in Ref. 16. However, results from CFD simulations17 conducted on the pump, as shown in Fig. 9, reveal large regions of flow reversal at the blade trailing edge near to the hub surface. This is unacceptable since this may lead to hemolysis and thrombus formation. Hence, further modification of the impeller geometry is necessary. 5.1. Modifications of impeller design Initial numerical studies have shown reasonably good flow throughout the inducer and it was decided that the inducer geometry is to be retained.



397

Attention was instead focused on the impeller and diffuser as severe flow reversal was found in hub regions near the impeller–diffuser junction. The impeller and diffuser have to be redesigned to minimize flow reversal. After analyzing the initial design, modifications are made to the geometry and studied computationally. The design is revised if CFD results are unsatisfactory. This process is done iteratively until a satisfactory design is obtained. During the design process, it was noted that backflow is a consequence of the non-ideal work distribution that is imparted on fluid particles as they traverse the impeller passage. The blade angle as well as the wrap angle are related to the swirl distribution along the blade profile. From Euler equation for turbomachinery, it is a well-known fact that the swirl distribution along the blade profile indicates the rate at which work or energy is imparted from the impeller blade to the fluid particles. The result of the evolution of the impeller geometry after successive iteration is shown in Fig. 10. It is noted that the hub diameter was increased from the leading edge to the trailing edge. The wrap angle as well as the blade angle vary continuously along the blade profile. The final design shown in Fig. 10 is obtained through this process and will be built and tested experimentally. Further details of this work are described in Ref. 18. Figure 11 shows the comparison of axial velocity component at the hub regions. After modifications, the negative axial velocity (blue regions of I) has been totally eliminated in both impellers II and III. Figure 12 shows the pressure contour plots of the three designs. The pressure contour plot reveals the poor pressure build-up in impeller I. Design II showed parallel pressure contours toward the downstream of the

I Fig. 10.

II

III

CAD drawing of the modified impellers I, II, and III.


398


Flow direction

I

II Fig. 11.

I

III

Progressive improvement of the axial velocity.

II Fig. 12.

III

Pressure contours plots of impellers I, II, and III.

impeller. This indicates an evenly distributed work done between the pressure and suction side as pressure increased gradually along the flow. Pressure distribution was further improved in design III, shown by the gradual increase in pressure contours throughout the flow passage.



399

By varying the blade angle distribution, wrap angles, and hub contours of the impeller, flow reversal in the original design (design I) was totally eliminated in design III impeller. Though the pressure of 9.3 kPa generated is lower than what the heart requires, an addition of a diffuser will increase the overall pump performance.

6. Manufacturing of Impellers In this developmental process, the impeller was designed and its performance was evaluated with the aid of CFD. Prototypes of those designs, where the CFD results are promising, were needed in sufficient quantities at a low cost for experimental validation of the CFD results. As the impeller is less than 16 mm in diameter with a maximum blade thickness of about 1.5 mm, innovative manufacturing techniques are explored to determine the best process for quick fabrication of prototypes that are dimensionally accurate, structurally robust, and at low cost. Four RP techniques were explored. Notwithstanding CFD being recognized as a good design tool in the initial design stage, the computational results must be validated with experimental data. It is therefore critical to have access to rapid manufacturing methods that can produce dimensionally accurate, small, and complex parts. In the case of axial blood pumps where the diameter is typically less than 20 mm and the length of the rotating element is less than 80 mm, the prototypes are normally machined using computer numerical control (CNC) machining techniques. The complexity of the blade profile has necessitated the use of CNC machines with five-axis machining capabilities resulting in a substantial increase in the cost and manufacturing time required. The ability to create prototypes rapidly for use in in vitro experiments during the preliminary evaluation process gives a significant boost to the progress of the project. The four RP systems used are: selective laser sintering (SLS), stereolithography apparatus (SLA), model maker II (MMII), and fused deposition modeling (FDM), respectively. The details are described in Ref. 19. 6.1. SLS process The SLS method spreads a thin layer of powder onto a working platform. The laser energy is directly scanned onto the powder causing the powder to melt and then solidify. The working platform is next lowered to allow


400


a new covering of powder layer to be spread and the scanning is repeated until the object is complete. The object is removed from the part-build chamber, and any powder particles are brushed or blown away. The object may undergo further treatment, such as sanding or annealing, before final use.20 There are a number of materials that can be used by this process. In this chapter, the material used is DuraFormTM , which is a nylon-based material. The SLS specimen of heat pump is shown in Fig. 13. 6.2. SLA process SLA process uses ultraviolet (UV) light-sensitive liquid polymer as base material. When a laser beam strikes the liquid surface, the liquid quickly solidifies. Once one layer is completely traced, the platform in the liquid vat is lowered a layer-thick distance and a second layer is traced right on top of the first, and so on. The self-adhesive property of the material causes the layers to bond to one another and eventually form a complete, threedimensional (3D) object after many such layers are formed. Figure 14 shows the SLA specimen of heat pump impeller. 6.3. MMII process MMII process is similar to 3D printing. The system has two print heads: one for jetting build material and another for jetting support material. The print head jets out micro-material droplets and deposits them on the foam plate or the previously built layer. From bottom-up, layer by layer, a 3D specimen with support can be printed out. The support material can be then dissolved in an organic liquid and the final green wax specimen is left behind. Figure 15 shows the MMII specimen of heat pump impeller.

Fig. 13.

Impeller made by SLS method.



Fig. 14.

Fig. 15.

401

Impeller made by SLA method.

Impeller made by MMII method.

6.4. FDM process The FDM machine builds the specimen by extruding a semi-molten filament through a heated nozzle in a prescribed pattern onto a platform. A second nozzle may extrude a second material if required. When the first layer is complete, the platform lowers by one-layer thickness and the process begins again. This part is easily removed from the platform, supports are removed, and the specimen is ready. Thakker et al.21 used the FDM process to manufacture impulse blades and directly used them for a turbine motor. Figure 16 shows the FDM specimen of heat pump impeller. 6.5. Merits and demerits of the four RP processes From the appearance of the parts, it can be seen that not all features are formed completely on the SLA prototype of the heat pump impeller. SLS


402


Fig. 16.

Impeller made by FDM method.

can create all features well while FDM can create most of the features reasonably well. As for MMII all features are created, but the specimen was so fragile that the thin blade of MMII prototype easily break off. Both the SLS and FDM specimens attained good strength and they can be used directly for some fluid test. The SLS specimen being built from powder is weaker in strength. MMII specimen is very fragile. SLA and MMII specimens cannot be directly used for functional test. However, all the four kinds of RP parts are suitable as sacrificial pattern in investment casting to derive a metal part. The surfaces of all specimens display clearly a step phenomenon resulting from the layer-by-layer forming method. This is clearly seen on the spiral blades which has the appearance of a spiral stair. The FDM specimen has the worst surface quality among the four types of RP specimens, followed by SLS, while MMII produced the best surface quality. The latter can be used for investment casting without any further surface polishing. The accuracy and surface quality depend on the size of the material element and density of deposition. For both SLS and SLA processes, the laser point determines the size of deposited material element. Nevertheless, SLA has a better building quality than SLS because its base material is originally a liquid while SLS base material is a powder, and liquid will deposit more densely than powder. For FDM process, the material element is a two-dimensional line. The thinnest filament extruded out is 1.8 mm (0.07 in.). The material element in MMII is a micro-droplet, which is in the order of several microns, and the layer thickness is as small as 0.0125 mm. Among the four RP processes, MMII has the best building accuracy and can produce some fine features, but the strength of MMII is very weak.



403

Even though the fine features are completely formed, they can easily break off. The FDM specimen has the only demerit of surface quality, which can be compensated by brushing with an organic liquid solvent. This liquid can partially dissolve ABS material and thus eliminate the layered steps on the part’s surface. Generally speaking, FDM is the most economical in material- and timecost, and can also assure a good strength and building quality. But FDM is not suitable for small features whose dimensions are less than 2-mm thickness. On the other hand, MMII is a good choice for fabricating small and complicated parts rapidly, but requires a further step of producing the investment cast object. Based on the above evaluation, it was decided to manufacture the prototypes using SLS and the build material used is nylon powder. As the resolution of SLS is not as precise as conventional machining, the surface finish of the prototypes produced by SLS is not smooth, which could result in losses due to higher frictional shear along the surfaces of the impellers. It is believed that with improved surface finish of the prototypes by external polishing and with a more compatible diffuser; the pressure head generated across the pump with these new impellers could be higher than that of the existing pump.

7. Experimental Validation of CFD Results For validation purposes, experimental data on the pressure gain along the axial length of the inducer–impeller was tabulated and compared with CFD predictions. Five pressure taps were made along the pump as shown in Fig. 17. Pressure differences from points 1 to 5 were measured with respect to point 1. The pump characteristic has not been carried out in this test, as this configuration does not include a diffuser. This will be carried out when the diffuser is designed to couple with this impeller. The impeller obtained through the CFD analysis was selected for this study. The CFD pressure curve is obtained by averaging the pressure difference (with respect to inlet) on the circumferential wall of the inducer–impeller casing wall. The comparison between the experimental and computational results shown on Fig. 18 showed relatively good agreement between the two. It is observed that the measured pressure rise in the initial section of the inducer is higher than that predicted by the CFD results. One possible reason is that in reality, pre-rotation exists and fluid particles entering the inducer are subjected to a certain degree


404


Fig. 17.

Schematic diagram of the pressure taps along the inducer–impeller.

P (Pa)

PRESSURE DISTRIBUTION ALONG THE PUMP 16000 14000 12000 10000 8000 6000 4000 2000 0 0

10

20

30

40

z (mm)

0

inducer

19 computational

Fig. 18.

22

impeller

38

experimental

Pressure head gain along the axial length of the inducer–impeller.



405

of pre-rotation. This may explain the discrepancy between the computed results and those measured in the inducer section. Pressure losses can be seen along the 3-mm gap between the inducer and impeller shown by the dip in pressure gain in the CFD curve. In addition to this pressure drop, pressure gain along the pump was quite consistent. Most of the pressure is generated within the inducer; at point 3, the ∆p is around 10,000 Pa accounting for more than 60% of the pressure generated across the inducer– impeller. The ∆p at the trailing edge of the impeller is about 14 kPa.

8. Blood Trauma Prediction In the application of blood pumps, the axial pump has the advantages of simpler design and smaller size over the pulsatile pump. However, its high rotating speed could be a threat to RBCs as hemolysis occurs in regions of high shear stress. We adopt the scalar stress method as proposed by the approach of Bludszuweit.22 This method attempts to represent the overall stress tensor by a scalar stress value, which can be expressed as 1/2 2 σij . σ = 1/6 (σii − σjj )2 +

(3)

The scalar stresses on the hub and tip surfaces for the prototype based on the effective viscosity were computed. On the impeller hub surface, the scalar stress ranges from 5 × 10−4 to 8 Pa while at the impeller tip surface, it ranges from 1 to 150 Pa. The scalar stress contours at the tip surfaces are shown in Fig. 19. The magnitude of scalar stresses demonstrates that the stresses subjected on RBCs are not excessive and hence, hemolysis is quite unlikely.23 Particle streaklines showed that the residence time of particles in the impeller is typically less than 0.1 s. However, it is also noted that while the scalar stress plot provides indication of the likelihood of hemolysis, it does not shed any information on the occurrence of thrombus formation. This can only be obtained by analyzing the velocity profiles; any flow reversals are indications that thrombus formation may be possible. In general, geometries of pump elements should be optimized to reduce flow separations while generating sufficient pressure. Good flow conditions are important for low shear stresses and shorter residence time, thus reducing blood damage.


406


Fig. 19.

Contour plot of scalar stresses at the tip surface of design III.

9. Preliminary Design of Motor In vitro tests have shown that the performance of the existing flow test pump model is capable of meeting the requirements of a circulatory-assist device. In the present form, an external motor drives the pump impeller via a drive shaft. A shaft seal is used to prevent fluid from seeping out of the flow conduit during pump operation. The main use of the pump is to aid in the refinement of the flow in the pump. Its shortcomings in regard to its size and reliance on seals are secondary to the study. However, in an implantable device, its physical size and the reliance on seals do matter. Seals tend to wear and succumb to leakage after prolonged use, thus the next development phase would focus on the compact, seal-less drive system. A compact motor-integrated pump suggests an impeller assembly embedded with permanent magnets forming the motor-rotor suspended on bearings. The bearings are housed in the straightener and diffuser which are fixed onto an outer tube to form the pump flow section. The stator coils surround the tube to provide the motive flux (Fig. 20).



Fig. 20.

407

Initial embodiment design of the pump.

It is ideal to have as small a gap as possible in the magnetic flux path between the motor-rotor and stator. A smaller gap would reduce magnetic flux leakage and thus provides higher torque and efficiency. Keeping all other factors constant, a longer motor increases the flux in the motor thereby increasing the motor torque. Installing stator coils having higher number of turns increases the magnetic strength of the stator and thus a higher torque is produced due to stronger attraction and repulsion force between the rotor and stator. The primary purpose of the initial design is to package the pump into a more manageable form for implantation and also to achieve seal-less operation. Despite knowing the limitation that a large gap has on motor performance, placing the motor magnets in the impeller hub was still preferred as it allows the existing geometry of the new inducer–impeller to be maintained. This is a compromise to avoid unnecessary re-engineering of the inducer–impeller. As the impeller is made using RP, a minimum thickness of 1 mm in the hub is needed for strength. The diameter of the 30-mm-long embedded rotor magnet core is restricted to 6 mm. The outermost diameter of the motor yoke is set to 60 mm in this first motor-integrated prototype. The motor coils were arranged in three-phase connections so as to minimize switching losses. Having a higher phase count in the motor would mean that the stator has to switch from one state to the next more frequently as the effect of each phase is active for a lower percentage for each turn of the rotor. In this initial design (Fig. 21), there are six coils, with two coils per phase. The number of coils was kept small, to allow more turns of wire to each coil as increasing the number of coils would require more teeth in the core, leaving less space for the coils. The size of this motor indirectly limits the number of coils in the design. Due to this consideration for space, the lower number of


408


Fig. 21.

Three-phase motor with six stator coils.

coils would be preferred. However, six coils were selected, as it would give a more balanced torque production. Concentrated coil windings were selected in favor of distributed windings primarily because the end turns are smaller; thus reducing the length of the motor. Having concentrated windings will also simplify the manufacturing process and reduce the likelihood of interphase faults. For a six-coil three-phase motor it is only feasible to have two pole-pairs in the rotor. As a larger number of poles will increase the produced torque, four rotor magnets will be used. The number of turns per phase and method of commutation has to be decided before the theoretic motor torque constant can be determined. Since the motor coils in the prototype are hand wound, the actual number of turns can only be determined during fabrication. The motor stator core consists of the yoke and teeth. The yoke is split into six 60◦ segments each with a tooth at its center. This is to facilitate convenient winding of the stator coils. The yoke and base of the teeth are made longer than the center support to act as a guard for the end turns of the stator windings. The iron core has a maximum flux density of 1.5 Wb/m2 . Magnetostatic simulation results indicate that for every 100 amp-turns the coil will cause 6 × 10−6 Wb of magnetic flux to pass through the stator tooth. With an all round thickness of 2 mm, the stator tooth has an area of 60 mm2 at the center support. It is able to accommodate the flux linkages from a coil with 1450 amp-turns and the permanent magnets in the rotor. With the fluid passage around the pump rotor, coil commutation cannot be achieved by conventional brushes. Instead, commutation is done



409

externally using transistors activated in sequence to achieve smooth and continuous rotation of the motor rotor. A signal generator consisting of a combination of permanent magnets and Hall-effect sensors was used to produce the sequential pulse signal for activation of the respective transistors. A diagram of the pulse signal generator is shown in Fig. 22. The signal generator has permanent magnets with an arc angle of 180◦ mounted to the shaft of a separate motor (pulse signal motor). Three Hall-effect sensors are placed around the magnets at equal intervals of 120◦. The Hall-effect sensors are powered by a 5-Vregulated DC power supply. The rate of switching is controlled by the speed of the pulse signal motor. Increasing the signal motor operating speed will increase the switching frequency of the transistors. One revolution of the pulse signal motor will generate six commutation states (one commutation cycle). As the motor rotor has two pole-pairs, the pulse signal motor rotates through two complete cycles to produce one revolution of the rotor. As a result, operating speed of the motor-integrated prototype is exactly half that of the pulse signal motor. The operating speed of the motor-integrated prototype is measured by a digital tachometer reading off Hall-effect sensor. As suitable Hall-effect sensors that are able to read signals off magnets mounted onto the pump rotor could be sourced in time for the tests, a direct drive shaft affixed with a set of permanent magnets is extended through the pump diffuser. As the

Fig. 22.

The pulse signal generator.


410


magnets rotate together with the shaft, the Hall-effect sensor that is placed outside the pump casing will be activated, thus enabling the tachometer to measure the pump speed. The motor-rotor magnets are housed within the impeller hub as shown in Fig. 23. The center portion of the shaft has a square cross section to seat the magnets. The total length of the magnetic core is 30 mm with an effective diameter of 6 mm. The pump casing is machined from aluminium and inserted through the motor stator. The pump casing has an internal diameter of 16 mm and external diameter of 18 mm. The flow straightener, inducer–impeller, and diffuser are housed within the pump casing. An end-cap is inserted into the upstream and downstream ends of the pump-casing keeping the pump components in place and also serves as connectors between the pump and flow tubes. 9.1. Performance testing of the motor-integrated prototype The motor-integrated prototype was tested at its maximum operating speed at different voltage settings for the motor. The test setup is similar to the one used for pump performance test as shown in Fig. 24. The pump characteristics curves are obtained by measuring the pump pressure and flow rate for the highest operating speeds. The motor-integrated prototype achieved a maximum speed of 4,550 rpm at a voltage setting of 14 V. The maximum operating speed increase along with the voltage setting. The gradient of the curve decreases as voltage supply and operating speed increase. This is likely to be caused

(a) With end-caps removed Fig. 23.

(b) Complete assembly

The motor-integrated prototype.



411

Maximum operating speed (RPM)

Maximum operating speed vs Voltage supply 5000 4000 3000 2000 1000 0 0

5

10

15

Voltage supply (V) Fig. 24.

Graph of prototype maximum operating speed against voltage supply.

by greater pump and bearing frictional losses and rate of work done by the pump when operating at a higher speed. This shows that the increase in torque requirement of the motor-integrated prototype is greater than that produced by the motor. Pump performance testing conducted on the motor-integrated prototype shows that the results correlate well with the pump performance tests. This can be seen from the dimensionless pump characteristics curve shown in Fig. 25. Furthermore, no leakages or malfunction was observed during operation, thus proving that the design concept of the motor is feasible. Figure 26 shows the graph of motor speed and brake torque at different supply voltages. The results show that the brake torque and motor CP vs CQ 0.12 Motor integration prototype 4550 rpm

0.10

Pump performance test 8000 rpm

CP

0.08 0.06 0.04 0.02 0.00 0.00

0.01

0.02

0.03

CQ Fig. 25.

Dimensionless pump characteristics curve.

0.04


412


Fig. 26.

Graph of motor speed against brake torque.

speed the brake torque tends to decrease linearly with an increase in motor speed. With a voltage supply of 14 V, the pump is able to operate at 4,550 rpm; this corresponds to a motor brake torque of 0.68 N mm and motor brake power is 0.325 W. From experimental results, the maximum pump power coefficient is 0.009, thus the maximum power delivered to the fluid is 0.08 W. The pump efficiency at this operating condition is 24%. The test results shown in Fig. 27 illustrates that the motor torque and motor current have a linear relationship. Theoretically, the y-axis intercept should be zero, since no torque will be produced if no current flows in the stator coils. However, tests results show that the y-axis intercept is in the negative region. This difference accounts for the frictional torque of the motor, which is the torque required to overcome friction. As the voltage supply increases, the frictional torque also shows a corresponding increase due to the higher running speed when the motor is operating at a higher voltage setting. The frictional torque for the motor-integrated prototype is significant when compared to the useful torque produced by the motor. At 8, 11, and 14 V, the frictional torque is 1.36, 2.02, and 2.82 N mm respectively. The frictional torque is significant when compared to the 1.2–1.3 N mm of useful torque produced by the motor at 8–14 V. In fact, the frictional torque is generally much larger than the useful torque at higher voltage ratings. This indicates that performance of the end bearing imposes too much drag



Fig. 27.

413

Graph of brake torque against motor current.

on the torque to overcome the friction torque rather than produce useful work, especially at high operating speeds. When the voltage supply is kept constant, an increase in operating speed will decrease the average current flow into the coils. This is due to the transistors having to switch at an increased frequency and a higher induced back-EMF. As the motor torque is directly proportional to motor current, the motor torque also decreases along with the motor current as speed is increased. The gradient of the graph or motor torque against motor current gives the value of the motor torque constant. The experimental results yielded a motor torque constant of between 2.9 and 3.0 N mm/A for voltage ratings of 8, 11, and 14 V. This experimental value of motor torque constant is near to the theoretical value of 3.44 N mm/A for an ideal motor. The graph of back-EMF versus motor speed is shown in Fig. 28. When functioning as a generator, the motor-integrated prototype can produce a back-EMF value of 2.7 mV s/rad. At an operating speed of 4,550 rpm, the induced back-EMF is about 1.23 V. The theoretical motor back-EMF constant of 3.44 mV s/rad is larger than the measured value. However, this is within expectation as the theoretic value is calculated based on an ideal DC motor. Furthermore, numerical value of the motor torque constant is slightly larger than the motor back-EMF constant. This is acceptable, as only an ideal motor will have its motor torque constant equal to its motor back-EMF constant.


414


Fig. 28.

Graph of back-EMF against motor speed.

The seal-less pump in a motor-integrated prototype was designed and fabricated. During the performance evaluation test of the motor-integrated prototype, it was noted that the increase in maximum pump operating speed at higher supply voltages is not as significant as at lower voltages. The maximum operating speed achieved by this prototype was 4,550 rpm when a voltage of 14 V was supplied. Pump evaluation tests showed that the performance curve for the motor-integrated prototype is the same as that obtained in earlier pump performance tests. The motor torque constant is 3.0 N mm/A and the motor back-EMF constant is 2.8 mV s/rad. It was also observed that the frictional torque present within the motor-integrated prototype is very high. With voltage supply of 14 V, the frictional torque present during pump operation was 2.82 N mm. This is significantly larger than the useful torque of 0.7 N mm produced at 4,550 rpm. At present, the maximum operating speed of the motor-integrated prototype is 4,550 rpm. At this speed, the motor-integrated prototype is only able to provide partial cardiac assist. To be capable for full assist, the pump has to operate in the region of 10,000 rpm. Some improvements can be made to the existing design so as to increase the maximum operating speed of the pump prototype. The gap between the motor stator and rotor magnets is 6.5 mm. Even in large brushless motors, this gap is typically less than 1 mm. A large gap will cause large amount of magnetic flux leakage, resulting in weaker magnetic interaction between the motor stator and rotor. Increasing the rotor magnet size and reducing the air gap can improve the motor torque production.



415

The gap can be reduced to 5 mm by increasing the rotor magnet diameter to 7 mm and decreasing the pump-casing outer diameter to 17 mm. Wall thickness of the proposed rotor hub and pump-casing is 0.5 mm; thinner than the existing thickness of 1 mm and 1.5 mm, respectively. Furthermore, the rotor magnets can be contoured to fit the increasing hub diameter at the impeller end. This will not only increase the size of magnet but also reduce the motor air gap, increasing the magnetic flux and reducing flux leakages in the motor. In the motor torque evaluation, it was noted that the friction torque in the prototype was much greater than the useful torque. Currently, the rotor shaft is inserted 5 mm into the bush bearings. By reducing contact area between the rotor shaft and bush bearings, frictional losses can be reduced. It is proposed that the diameter to be reduced to 2 mm and the insertion depth to be reduced to 2 mm. By doing so, there is a 75% reduction in the contact area. In this prototype, as a precaution from overheating, larger wires were used for winding the stator coils to cater for a continuous operating current of up to 2 A. At present, the motor current is about 1.15 A when running at 4,550 rpm with a supply voltage of 14 V. With a larger rotor motor and smaller air gap, the motor torque production is expected to improve. After modifying the bearing system, the motor-integrated prototype is expected to operate at a higher speed with the same voltage supply, as less torque will be used to overcome friction. In view of this, it is possible that the stator coils can be rewound with thinner wires to increase the number of coil turns. This will result in stronger magnetic properties for the stator coil. In addition, the existing coils were hand wound and the slot fill factor was only 51%. Using thinner wires will also increase the slot fill factor. For concentrated windings, it is possible to have a slot fill factor of 60–70%.

10. Conclusion This chapter describes the development of an axial blood pump. It covers the design, numerical simulations, manufacturing processes, experimental studies, and preliminary motor design of an axial blood pump prototype. The pump consisting of a flow straightener, inducer–impeller, and diffuser was developed for application as an LVAD. Preliminary studies on the original pump design revealed severe flow reversals near the hub regions of the impeller and diffuser. The poor design of the impeller and diffuser was found to be the cause of this poor flow condition. The geometries of the


416


impeller were altered to minimize flow reversals. The impeller was first studied without the diffuser by varying various parameters. The redesigning of the impeller relies heavily on numerical simulation as it is a cost-effective and time-saving tool. Simulations are made at operating conditions with a flow rate of 5 l/min and rotational speed of 10,000 rpm. Modifications made to the impeller geometry include blade thickness, blade angle distribution, wrap angle, and hub contour. These modifications proved useful as flow reversal was eliminated in the final design. The final design of the impeller presented a balance between relatively good flow and pressure generation. Blade angle distribution, wrap angles, and hub contours had been shown to play an important role in the pump performance in terms of flow and pressure generation. RP techniques were employed to manufacture the impellers. Experimental tests were conducted to validate the CFD predicted results and comparison showed good agreement between the two. Scalar stresses in the impeller passage also revealed that on the impeller hub surface, it ranges from 5×10−4 to 8 Pa, while at the impeller tip surface, it ranges from 1 to 150 Pa. This is unlikely to cause hemolysis because particle tracking also reveals that the residence time of particles within the impeller is typically less than 0.1 s. The preliminary design and testing of a motor design were also conducted. Currently, the maximum operating speed of the prototype is 4,550 rpm which is insufficient to meet the requirements. To increase the operating speed, the air gap, the wire used in the winding as well as the wall thickness of the rotor hub, and pump-casing could be reduced. In addition, the size of the magnet in the rotor could be increased. This would help to increase the operating speed of the prototype. Thus, the pump developed holds promise as a viable LVAD. However, further investigations need to be conducted on the ancillaries of the overall pump system as they are important components of the system.

Acknowledgments The authors acknowledge the contributions of the following persons: C. K. Chua, C. W. Lee, and C. Feng on the rapid prototyping work, V. Chong on the motor integration work and S. Y. Koh, W. Ong, A. Aw, H. Y. Lau, E. Tan, H. T. Wong and M. F. Lum on the model refinement and experimental work.



417

References 1. M. E. DeBakey, The Odyssey of the artificial heart, Artificial Organs 24 (2000) 405–411. 2. Singapore National Heart Association. Website: http://www.snha.org/heart matters/hearthealth singapore.htm 3. Y. Nosè, The need for a nonpulsating pumping system, Artificial Organs 12 (1998) 133–135. 4. K. Yamazaki, R. Kormos, T. Mori, M. Umezu, M. Kameneva, J. Antaki, E. Outa, P. Litwak, J. Kerrigan, and J. Tomczak, An intraventricular axial flow blood pump integrated with a bearing purge system, ASAIO Journal 41, 3 (1995) M333–M336. 5. C. H. Park, K. Nishimura, T. Yamada, H. Mizuhara, T. Akamastsu, T. Tsukiya, K. Matsuka, and T. Ban, A magnetically suspended centrifugal pump. In vitro and in vivo assessment, ASAIO Journal 41, 3 (1995) M345–M350. 6. G. P. Noon, Bio-medicus ventricular assistance, Ann Throac Surg 52 (1991) 80–81. 7. Y. Nosè, K., Kawahito, and T. Nakazaki, Can we develop a permanent rotary blood pump? Yes, we can, Artificial Organs 20 (1996) 467–474. 8. L. R. Golding, G. Jacobs, T. Murakami, S. Takatani, F. Vadès, H. Harasaki, and Y. Nosè Y, Chronic nonpulsatile blood flow in an alive, awake animal: 34 day’s survival, Trans Am Soc Artif Intern Organs 26 (1980) 251–255. 9. W. K. Chan, Y. W. Wong, M. F. Lum, V. Chong, and S. Y. Koh, Preliminary hydraulic performance characterization of an axial blood pump, Journal of Mechanics in Medicine and Biology 4, 3 (2004) 227–236. 10. CFX-TASCflow version 2.11 Users Manual, AEA-Advanced Scientific Computing, Waterloo, Ontario, Canada, 1995. 11. C. H. Hsu, Flow study on a newly developed impeller for a left ventricular assist device, Journal of Artificial Organs 6 (2003) 92–100. 12. J. E. Bardina, P. G. Huang, and T. J. Coakley, Turbulence Modeling Validation. NASA Technical Memorandum 110446, 1997. 13. J. E. Bardina, P. G. Huang, and T. J. Coakley, Turbulence Modeling Validation, 28th Fluid Dynamics Conference, AIAA paper 97-2121, June 1997. 14. T J. Coakley, Development of Turbulence Models for Aerodynamic, Applications, 28th Fluid Dynamics Conference, AIAA paper 97-2009, June 1997. 15. M. A. Leschziner, P. Batten, and H. Loyau, Modelling of shock-affected near-wall flows with anisotropy-resolving turbulence closures, Proceedings of the 4th International Symposium on Engineering Turbulence Modelling and Measurements Ajaccio, Corsica, 1999. 16. W. K. Chan, Y. W. Wong, W. Ong, S. Y. Koh, and V. Chong, Numerical investigation of the effects of the gap clearance between the inducer and impeller of an axial blood pump, Artificial Organs (in press). 17. S. Y. Koh, V. Chong, Y. W. Wong, and W. K. Chan, Development and flow investigation of an axial blood pump, 21st Southern Biomedical Engineering Conference, Washington DC, USA, September 2002.


418


18. W. K. Chan, Y. W. Wong, S. Y. Koh, and V. Chong, Performance Characterisation and CFD analysis of an axial blood pump, Journal of Mechanics in Medicine and Biology (in press). 19. W. K. Chan, Y. W. Wong, C. K. Chua, C. W. Lee, and C. Feng, Rapid manufacturing techniques in the development of an axial blood pump impeller, Journal of Engineering in Medicine 217, H6 (2003) 469–475. 20. J. J. Beaman and J. R. Laser, Sintering for desktop manufacturing, Machine Design, March 26 (1990) 65–66. 21. A. Thakker, C. Sheahan, P. Frawley, and H. B. Khaleeq, Innovative manufacture of impulse turbine blades for wave energy power conversion, Journal of Engineering Manufacture 216, Pt B (2002) 1053–1059. 22. C. Bludszuweit, Model for a general mechanical blood damage prediction, Artificial Organs 19, 7 (1995) 583–589. 23. R. Paul, J. Apel, S. Klaus, F. Schjugner, P. Schwindke, and H. Reul. Shear stress related blood damage in laminar couette flow, Artificial Organs 27, 6 (2003) 517–529.


MATHEMATICAL MODELING OF VENTRICULAR-ASSIST DEVICES S. VANDENBERGHE∗ , P. SEGERS and P. VERDONCK† Institute Biomedical Technology Ghent University, Ghent, Belgium ∗[email protected] †[email protected]

A. Overview of Mathematical Cardiovascular Modeling 1. Introduction The history of mathematical models is closely related to the evolution of mathematics. A model can be as simple as one equation, but the contemporary available computing power has opened the doors for much more complicated models. The main advantage of mathematical models over, for instance, in vitro models is that they are cheap and flexible. In addition, parameter values in a mathematical model can be set exactly and kept constant, while in vitro models are more difficult to control. When focusing on the cardiovascular system, models of different functional levels can be created: for example electrophysiologic models that simulate exchange of substances between cells, models that simulate the electrical conductivity of the heart, or models that simulate the mechanical behavior of valve leaflets. Models related to heart failure and ventricular unloading are mainly created in two forms: (i) finite element models and (ii) lumped models. In this chapter, the focus is on lumped parameter models in a mathematical form. The principle of such a model is to condense the properties of the cardiovascular system that are normally geometrically distributed, in a number of discrete components. Models of cardiovascular function are often linked to electronic circuit theory and the depiction of such a model as an electric network facilitates composition of the necessary equations. These equations can then be solved in the same way as electronic circuits are solved, and several options are available: the time domain, the frequency domain, a state-space approach, or even commercially available 419


420

S. Vandenberghe et al.

design software for electronic circuits. It is, however, not the focus of this chapter to explain electric network theory, but rather to demonstrate how such networks can be linked to the cardiovascular system and assist devices, and what the underlying rationale of the components is. There are various uses for lumped mathematical models of the assisted cardiovascular system. If the model includes a detailed simulation of the device, design optimization may be performed while obviating the need for prototypes for each optimization step. The model can for instance estimate the required motor torque and power to obtain the desired flow characteristics, and this can be tested over a wide range of operating conditions by varying heart rate, peripheral resistance, blood volume, etc. Also cannula specifications or pneumatic drive pressure requirements can be defined based on simulations. Control algorithms for a device can be included in a model and thus various algorithms can be simulated and evaluated. Simulations can also focus on the impact of the device on the perfusion of a patient or the unloading of his/her heart. In the past, mathematical models have also been used as a diagnostic tool: by simulating various fault conditions and comparing the results to data acquired from a patient with an actual implant, a problem with a device can be detected and corrected. Lumped models focus on pressure (∼voltage), volume (∼charge), and flow (∼current) that are generated in the cardiovascular system. To simulate the interaction between the heart and an assist device, the total model needs to combine the model of the heart, the assist device, and the afterload experienced by those two “pumps”. These three components of a total model will be discussed in detail in the following sections.

2. Cardiac Models 2.1. Time-varying elastance The original time-varying elastance theory of Suga and Sagawa forms the basis of the most cardiac models and is extensively discussed in literature. In short, ventricular elastance is defined as the ratio of instantaneous left ventricular pressure to left ventricular volume, corrected for a “dead volume” (V0 ), the volume in the ventricle at zero pressure1 : E(t) =

pLV (t) [VLV (t) − V0 ]

or pLV (t) = E(t)[VLV (t) − V0 ],

(1)


Mathematical Modeling of Ventricular-Assist Devices

421

with E(t), time-varying elastance (mmHg/ml); pLV (t), left ventricular (LV) pressure (mmHg): VLV (t), left ventricular volume (ml); and V0 , volume at zero pressure (ml). In a pressure–volume (PV) plane, instantaneous elastance can be presented as the slope of a line connecting the V0 intercept on the volume axis with the considered PV data point at that instant. Over the course of a heart cycle, the left ventricular PV variations will create a counterclockwise loop in the PV plane. Consequently, the slope of the line will rise and fall, resulting, respectively, in an increase and decrease of the elastance. This elastance varies according to a certain pattern that is typical for a given heart, independent of preload or afterload condition. Therefore the time-varying elastance theory is suitable for use in models of the heart where different loading conditions will be simulated. Moreover, if the elastance function is normalized with respect to time and amplitude, a curve is obtained that is typical for the species.2 This means that a model only has to contain one normalized elastance curve that can be used to simulate any individual by simply rescaling it in time and amplitude to the actual maximal elastance and heart rate of that individual. It is assumed in the original time-varying elastance theory that the end-systolic and end-diastolic PV relations (ESPVR and EDPVR, respectively, see Fig. 2) are linear. It has meanwhile been shown, however, that the EDPVR is best presented by an exponential or a logarithmic relation. The EDPVR in fact characterizes the passive compliance of a relaxed ventricle. It even extends to negative pressures, suggesting that the ventricle is able to suck blood during filling within a certain volume range.3 It was found that the ESPVR is also curvilinear but to a much lesser extent.4 Several approaches are possible to implement the elastance theory in a mathematical model. The equations used are normally set for one cardiac cycle and are solved repeatedly, where a scaling in time is applied to obtain results at the desired heart rate. Here we discuss about three approaches: (i) The first approach is based on the original elastance theory (see Eq. (1)), where all relations are assumed linear, including the EDPVR. If the elastance curve is plot as a function of time, then its maximum is the slope of the ESPVR in the PV plane and its minimum relates to the slope of the EDPVR. Since diastolic pressure is always positive and non-zero (with the exception of extreme low volume conditions), the EDPVR will have


422


a positive value and the elastance curve as a function of time will consequently hover completely above the time axis. This first approach is the most simple one, but it has proven to be sufficiently accurate in many cases.5−7 (ii) Another approach is to recognize the EDPVR as the passive compliance behavior of the left ventricle, which is always present regardless of whether the heart actively contracts or not. The pressure generated in the left ventricle can then be described as: PLV (t) = pactive (t) + ppassive (t) = E(t)[VLV (t) − V0 ] + EDPVR,

(2)

where the active pressure (pactive ) is generated by the contraction of the myocardium and the passive pressure (ppassive ) is the result of the compliance of the myocardium. This latter part can be modeled as an exponential or logarithmic term (the mathematical description of the EDPVR) that is added to the active part. For the active part, the linear elastance theory can then be used, where minimum elastance equals zero. Thus in contrast to the previous approach, the diastolic part of the elastance curve would coincide with the time axis. In other words, the active part alone simulates a PV loop where the diastolic filling phase coincides with the volume axis, while the passive part lifts the loop onto the EDPVR. Both approaches (i) and (ii) can simulate an increase in contractility by rescaling the elastance curve and increasing its maximum, resulting in a steeper ESPVR. (iii) The final approach to implement the time-varying elastance theory is to define the ESPVR and EDPVR as (time-invariant) boundaries by a mathematical equation with a level of complexity chosen by the developer. This may be linear, but implementation of curved boundaries is easy with this approach. The third equation then describes a time-varying activation function (α(t)) that determines in what proportion each boundary contributes to the instantaneous PV. In other words, it is a transition function between 0 and 1 that describes how an instantaneous data point migrates from the EDPVR to the ESPVR. Consequently, the equation that defines the left ventricular pressure as a function of volume and time has a term for ESPVR and a term for EDPVR: p(V, t) = α(t)ESPVR + [1 − α(t)]EDPVR.

(3)

When the activation α(t) = 0, the instantaneous PV data point will be on the EDPVR, while when the activation equals 1 it will be on the ESPVR.



423

To increase the contractility of the left ventricle in this approach, it is necessary to redefine the ESPVR, while the activation function can stay the same. A simplification adopted by all models is that the volume intercept V0 remains constant throughout the cardiac cycle and thus acts in the PV diagram as the pivot point for all the instantaneous PV relations (Fig. 1). In reality, the volume intercept varies over the cardiac cycle, although within a limited range.8 Especially during systole the variations are very small, and modeling it as a constant value is acceptable. A more realistic approach would be to use a volume value at a slightly negative pressure as a pivot point. This would reconcile the idea of Sagawa10 that ESPVR and nonlinear EDPVR cross below the volume axis (Fig. 2). 2.2. Examples of elastance/activation curves For modeling purposes, it is desirable to use relatively simple mathematical functions that can be created within the model, based on the time

Fig. 1. Three PV loops from the same subject at different loading conditions. The solid lines connect data points measured at the same time reference in the cardiac cycle (isochronic lines). The extrapolated dashed lines illustrate the variation of V0 on the volume axis. Reproduced from McKay et al.9


424


Fig. 2. Schematic diagram to indicate the difference between V0 (volume intercept of ESPVR) and Vd (volume at which no pressure variations can be generated). Reproduced from Sagawa.10

scale and a minimum number of parameters. A commonly used activation curve consists of the combination of a raised cosine function for activation with an exponential decaying function for deactivation.11 A similar approach can be applied for elastance curves: mathematical functions can be combined to obtain an elastance curve that mimics patient or animal elastance functions. The simplest way is to combine linear segments, but this has the disadvantage that many parameters are required to define the total “curve”, and that a lot of steps are necessary to position the line segments relatively to one another.12 Polynomials may be used instead, but the elastance curve is also then best split up in to several parts. Splitting the curves adds one extra difficulty, especially when they have to be rescaled; the connecting points need to be continuous in value and first derivative. Below are some examples of elastance curves used in left ventricular mathematical models. The mathematical expressions were obtained by curve fitting to elastance data that were calculated from PV loops acquired in live subjects. All example expressions contain parameters that have some physiologic meaning as they separately control the steepness of the rising and falling legs of the elastance curves, relating to contraction and relaxation, respectively.



425

2.2.1. Sinusoidal approximation In the normalized elastance curve that is used as an example here,13 the cardiac cycle is split into three parts: a rising part for contraction, a falling part for relaxation, and a part where the elastance is zero for ventricular filling (Fig. 3). This curve can be used in models according to approach (ii) as described above: Rising part : En (t) = sin

tπ 2Trise

,

(4)

(t − Trise )π , Falling part : En (t) = 1 − sin 2Tfall

(5)

elastance (mmHg/ml)

LV pressure (mmHg)

with En (t), normalized time-varying elastance (−); t, time (s); Trise , 0.833 × duration of systole (s); and Tfall, 0.167 × duration of systole (s).

160 140 120 100 80 60 40 20 Tfall

Trise 1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

time (s) Fig. 3. Sinusoidal model of a normalized human elastance curve, referenced to left verticular (LV) pressure obtained with a model simulation. Trise = 0.333 and Tfall = 0.066.


426


2.2.2. Exponential approximation This method allows to model the elastance curve with only two parts: a rising exponential and a decaying exponential with an asymptote equal to the minimum elastance. Four constant parameters plus the time indication of the maximum elastance are necessary to define the elastance curve in this way14 : −t (6) Rising part : E(t) = Emax 1 − exp + Emin , τ1 Falling part : E(t) = Emax exp

−(t − tmax ) + Emin , τ2

(7)

with E(t), time-varying elastance (mmHg/ml); t, time (s); Emax , Emin : scaling factors (mmHg/ml); and τ1 , τ2 : time constants (s) (Fig. 4).

elastance (mmHg/ml)

LV pressure (mmHg)

160 140 120 100 80 60 40 20 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

time (s) Fig. 4. Exponential model of a human elastance curve, referenced to left vertricular (LV) pressure obtained with a model simulation. Emax = 2.6 mmHg/ml; Emin = 0.3 mmHg/ml; τ1 = 0.15 s; and τ2 = 0.03 s.



427

2.2.3. Double-Hill function In contrast to the previous methods, the “double-Hill” function describes the elastance curve as a whole, with five parameters6: n1     t α1 T 1 n1   n2  , En (t) = a (8) t 1 + α1 T 1 + α2tT

elastance (mmHg/ml)

LV pressure (mmHg)

where En (t), normalized time-varying elastance (mmHg/ml); t, time (s); T , period of a heart cycle (s); a, amplitude scaling factor (mmHg/ml); α1 , α2 , shape factors (−); and n1 , n2 : exponential shape factors (−) (Fig. 5). The first part of the function is an actual Hill function that determines the rising leg of the curve, while the second part is an inverted Hill function that determines the falling leg. The alpha parameters define the timing of the separate parts relative to the period T , and thus α1 needs to be smaller than α2 . The exponents control the steepness of the rising and 160 140 120

100 80 60 40 20

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

time (s) Fig. 5. “Double-Hill” model of a normalized human elastance curve, referenced to left vertricular (LV) pressure obtained with a model simulation. a = 1.66 mmHg/ml; α1 = 0.303; α2 = 0.508; n1 = 1.32; and n2 = 21.9.


428


falling leg, and there too n1 will be smaller than n2 , resulting in a much steeper falling leg. 2.3. Right ventricular and atrial modeling Most of the studies and models are focusing on the left ventricle, as it is the most powerful heart chamber that provides flow to multiple organs. PV measurements on patients have mostly been performed in the left ventricle, and data are readily available. The situation is different for the right ventricle, where only limited patient data are available. Compared to the left ventricle, right ventricle PV loops exhibit a more triangular shape. This is caused by a combination of a distinct right ventricular geometry and a markedly different afterload. As a consequence of the low pulmonary afterload pressures, the isovolumic contraction and relaxation phases of the right ventricle are shorter than in the left ventricle. From geometric point of view, the left ventricle can be considered as one symmetric cavity (often approximated by an ellipsoid or cone), but the right ventricle consists of two distinct parts: (i) the sinus and (ii) the conus. These appear not to work perfectly synchronous, resulting in a twophased contraction.15,16 Therefore, it is questionable whether the whole right ventricle should be modeled as one part and whether its function should be determined from one complete loop. The situation is even more confusing for models of the atria, where PV loops show a figure of 8-shape. Atria obviously have a completely different geometry compared to the left ventricle and their function and timing are also incomparable. In summary, experience and good data are available for left vertricular models, but to model the right ventricle and the atria, several assumptions need to be made, or conversions from animal data have to be used. 2.4. Alternative cardiac modeling methods 2.4.1. Elastance-resistance model Although the original time-varying elastance concept of Suga and Sagawa as described above is still widely used in mathematical models, it was revised by the same authors in 1980. They suggested the following extension: t dV (t) dV (t) + C3 V (t) dt, + C2 p(t) = E(t)[V (t) − V0 ] + C1 dt dt 0 max (9) where C1 , C2 , C3 : weighing factors; and other variables as described above.



429

Since ventricular volume decreases during ejection, the three additional right-hand terms are negative and result in an extraventricular pressure drop. The second term of Eq. (9) represents the visco-eleastic behavior of the myocardium, which originates from the extracellular matrix, of which collagen is an important component. The two following terms are dependent on the deactivation of the muscle fibers. Basically, the second term of Eq. (9) contains the ejection flow of the ventricle. It appears that higher ejection rates result in a lower pressure generation. The third term is the peak ejection flow rate, while the last term is in fact the total volume that is ejected from the ventricle since the start of the beat. The term with the highest impact proved to be the ejection flow and as a consequence, the elastance-resistance model was created and used in different forms and variations by various researchers.17−19 Ignoring the last two terms in Eq. (9) and reformulating the remaining extra term results in a common way of incorporating ventricular resistance in the model: p(t) = E(t)[V (t) − V0 ] − Rv Q(t),

(10)

where Q(t) = dV (t)/dt, ejection flow (ml/s); and Rv , ventricular resistance (mmHg s/ml). The ventricular resistance is an intrinsic property of the heart and it can be quantified by creating a situation where p(t) equals zero; that is, a beat with no afterload. The generated flow will then be the maximum flow (Qmax ) that can be generated by the specific ventricle that is modeled: Rv =

E(t)[V (t) − V0 ] . Qmax

(11)

Consequently, the instantaneous left ventricular pressure can be modeled by substituting Eq. (11) in Eq. (10): Q(t) . (12) p(t) = E(t)[V (t) − V0 ] 1 − Qmax This model contains three parameters that characterize a specific ventricle in a specific contractile state: E(t), V0 , and Qmax . The first parameter, E(t), is a waveform similar to the one discussed before while Qmax and V0 are constants. Since it is not feasible to create a no-load condition in an in vivo experiment without dissecting the aorta, a method based on the combination of an ejecting beat and an isovolumic beat is usually applied to estimate the three parameters.


430


In short, the combination of these beats allows the model to be rewritten so that only two parameters need to be determined. These can be estimated with curve fitting techniques from the pressure data of the two beats and from the flow data of the ejecting beat. An isovolumic beat has the advantage that Q(t) is zero and V (t) is a constant with end-diastolic volume, Ved . Consequently, the isovolumic pressure waveform, piso (t), will reveal the shape of the elastance curve waveform: piso (t) = E(t)[Ved − V0 ] = E(t)/A.

(13)

The eventual equation needed for the estimation procedure is given as: t t p(t) Q(t) 1− =A Q(t) dt + Q(t) dt . (14) 1−A piso (t) Qmax 0 0 Only A and Qmax need to be estimated and E(t) can be obtained afterward from Ref. 13. Depending on the available equipment, volume measurements can be performed instead of flow measurements and Q(t) can be replaced with dV (t)/dt. This latter approach is more exact, especially in hearts where mitral regurgitation is expected. After all, the generated pressure depends on muscle fiber action and matrix interaction that results in volume variations, but the myocardium does not sense that this results in an effective ejection. A disadvantage of this method is the need for an isovolumic beat, thereby excluding human validation. 2.4.2. Isovolumic description The elastance function obtained from ejecting beats is in fact characteristic for the combination of the ventricle with the particular arterial load that was present at the time of measurement. Grignola and Ginés stated that the peculiar shape of the right ventricular PV loops, for instance, is not related to different contracting mechanisms, but is due to the different load imposed by the pulmonary circulation.20 Nevertheless, elastance is used in models to describe the function of the ventricle alone, regardless of the load. A more realistic model of the ventricle would be based on data acquired with the pump separated from the load. In practice, this can be achieved with isovolumic beats, where the aorta is clamped, and hence there is no ejection and no ventricular volume variation. In an isovolumic beat, the heart always works against the same (infinite) afterload, independent of the arterial system. Consequently, isovolumic beats are directly related to the ventricle’s contractile state and therefore a model based on isovolumic



431

pressure measurements was proposed by Palladino et al.21 In an isovolumic contracting ventricle, the instantaneously generated pressure is dependent on the volume contained in the ventricle and on the time point in the heart cycle. The volume stretches out the myocardium and thereby creates a passive pressure, independent of time. Active contracting mechanisms add an active pressure that is dependent on time and the contained volume, as was already discovered by Frank in the 19th century. Isolated dog heart experiments revealed the following relationship for the passive (diastolic) pressure: ppassive = a(V − b)2 ,

(15)

where ppassive , passive left ventricular pressure (mmHg); V : left ventricular volume (ml); and a, b: constants. This is in fact a quadratic description of the EDPVR, in contrast to the exponential or logarithmic definitions that were mentioned before. The constant “b” can be perceived as the volume intercept at zero pressure, while the constant “a” relates to the end-diastolic elastance. A description of the active pressure generation during an isovolumic beat, however, differs more from the instantaneous elastance concept. The active pressure is divided in to a contraction and relaxation phase, taking the form: α α t t − tb exp − , (16) pactive = A 1 − exp − τ1 τ2 where pactive , active left ventricular pressure (mmHg); A, volume-dependent factor (mmHg); α, exponential shape factor (−); τ1 , τ2 , shape factors for contraction and relaxation (s); and tb : time delay for onset of relaxation (s). The total description of the left ventricular pressure is given as follows: α  α  b exp − t−t 1 − exp − τt1 τ2 α α  . p(V, t) = a(V − b)2 + (cV − d) tp tp −tb 1 − exp − τ1 exp − τ2 (17) Eight parameters are required for this equation, which can all be derived from experimental data with curve fitting procedures. Isovolumic beats at different volumes are therefore required and the resulting equation then characterizes a particular ventricle. The time tp is the point at which the maximum ventricular pressure is reached, and c and d are constants that are proportional to the active pressure. The pressure in this model is instantaneous and dependent on V , which represents here the time-varying left ventricular volume, and not the


432


constant (isovolumic) volume at which the measurements were performed. Consequently, this model can be used for both ejecting and isovolumic beats. In the traditional time-varying elastance model (which is derived from ejecting beats), the pressure waveform of an isovolumic beat would have the same shape as the elastance curve. The factor (V − V0 ) will be no more than a constant, and thus a scaling factor that relates elastance to pressure. This means that that pressure waveform would be skewed to the right, while experimentally measured isovolumic pressure waveforms are symmetrical. Therefore, this alternative model performs better on isovolumic beats than the traditional elastance approach (see Fig. 6). Moreover, it can also be used on ejecting beats and it describes the function of the heart independent of the arterial load. However, this model is difficult to use because of the many required parameters and the necessary experimental data that is difficult to obtain from in situ hearts.

Fig. 6. Fitting (solid lines) of Eq. (17) to pressure waveforms (crosses) acquired in dogs with induced isovolumic (non-ejecting) beats at different volumes.21



433

3. Afterload Models 3.1. Windkessel models Simulating the vascular system with a mathematical model is similar to in vitro setups or the so-called mock circulatory systems, where the systemic and pulmonary circulations are modeled with combinations of hydraulic valves, resistors, and compliance chambers all connected with tubing. Mathematical modeling of such systems, however, is more flexible, and the components are ideal and really discrete. Various models with a wide range of complexity have been developed, some of which are shown in Fig. 7. It may be desirable that the lumped afterload model is set up with components that have a specific physiologic meaning and translate to the actual properties of the vascular system that they model. The addition of one component that does not relate to a specific property will result in a “meaningless model” that may fit the vascular system input impedance well, though with meaningless values for the parameters. Lumped afterload models that are currently in use for cardiovascular studies contain either two, three, four, or five components and they are all based on the classic windkessel model as it was used by Frank.22 The basic components are a compliance (electric: capacitance, abbreviated as C or C2 in Fig. 7) and a resistor (abbreviated as R, Rp , or R2 in Fig. 7) that are connected in parallel. The former simulates the elasticity of the large arteries and the latter the resistance formed by the arterioles. This model, also referred to as an RC model, summarizes the basic functionality of the arterial system, but its input impedance spectrum differs notably from measured mammalian data. Impedance is defined in electric network theory as the ratio of voltage to current, and can be interpreted here as the ratio of aortic pressure over aortic flow. The modulus of the RC model impedance tends to 0 for higher frequencies and the phase angle has an asymptote at −90◦ , while it should go to 0◦ to simulate in vivo impedance. At high frequencies the ventricle sees an afterload that is free of reflection and is thus equal to the local high-frequency behavior of the proximal aorta, as if there were no further arterial system. This observation suggested that an extra component was necessary to model the proximal aorta: the characteristic impedance (Zc or Rc or R1 in Fig. 7), which is mainly a resistive component. This characteristic component was added in series to the windkessel circuit by Westerhof and it was termed the modified windkessel model (RCR model).23 At high frequencies, the capacitor will behave like a bypass over R and thus Rc will be the ruling component. Hence the phase


434


Fig. 7. Left: Electric analogs of various windkessel models with their impedance derived in the frequency domain. Right: The impedance characteristics of the different models. Adapted from Ref. 24.



435

of this model in the impedance spectrum tends to 0◦ at high frequencies and the modulus has Rc as asymptote, which is a major improvement over the classic windkessel model. The third model that is very popular is the four-element (RLRC) windkessel model, where an inductance (L) is added either in series or in parallel with the characteristic impedance.25 This inductance represents the total inertance resulting from the mass of fluid in the arterial system. Pulsatile flow requires an acceleration (and deceleration) of this mass and thus an extra force or pressure buildup. An inductance component blocks all high frequencies and thus if it is placed in series with the characteristic impedance, it will be the only ruling component at high frequencies. Therefore it only relates to reality if it is used in parallel with the characteristic impedance, although the series composition may yield better fitting results.26,27 The value of the inductance in the series model will be a magnitude lower than in the parallel model, and the value of Rc will be higher, which is out of the physiologic range.25 This four-element model can further be extended with another compliance element that represents the compliance of the proximal aorta. It is split off from the total arterial compliance and separated from it with an inductor. In the impedance spectrum, this fifth element seems to result only in a small improvement that does not justify the addition of the extra complexity. Comparison of the flow waveforms generated with this type of model, conversely demonstrates a superior match with human data.24 The key is that the presence of two compliance elements introduces a small resonance into the model, which is not present in any of the other models presented here.28 To simulate the systemic vasculature of an individual (e.g., for use in a more extended model), one needs to estimate the parameter values for the different components used in the selected windkessel model. To obtain these values, a fitting procedure can be applied that requires aortic pressure and flow waveforms measured at the aortic root of the individual. The pressure waveform can be fed as input to the windkessel model (either in the time domain or frequency domain), and the resulting output will be a flow waveform that is compared to the measured flow waveform. The values of the afterload model components are then adjusted (with iterative optimization procedures) to minimize the error between measured and calculated flow and the best fit yields parameter values that represent the real arterial system in which the data were acquired. Similarly, a flow waveform can be used as input while matching a generated pressure waveform to a measured


436


pressure waveform. It is important that the in vivo data are measured synchronously and at the same site, or that time delays are corrected. It is also important to realize that the resulting parameters characterize the considered arterial system in a specific condition. Interventions (e.g., induced vasodilation) may alter the system, and thus the parameters need to be adjusted accordingly or recalculated from another data set. Not only is estimation of the windkessel parameters an excellent method for simulating an individual’s vasculature, but it can also be useful diagnostically because accurate estimates of arterial compliance and peripheral resistance are obtained. Knowledge of these properties is important for diagnosing hypertension and its underlying causes, and as such the windkessel model can also be a useful monitoring tool for vascular changes during disease or pregnancy.

3.2. Combinations of windkessel models Windkessel models are mainly used to mimic the systemic arterial system. The whole pulmonary circulation can in fact be neglected, when it is assumed that the blood volume ratio between pulmonary and systemic circulation stays constant (no storage).29 Nevertheless, windkessel models contain the passive properties of a branched system of elastic vessels and they even incorporate to some extent the sequence of different types of vessels with different properties (elastic vessels before resistive). The same models can also be used to simulate the pulmonary arterial system with adjustments of the resistance, inductance, and capacitance values. A meaningful model for the arterial system, however, may be meaningless for the venous circulation because of the inverted anatomic buildup; venous diameters increase downstream and junctions are present instead of bifurcations. The only model that is useful for both circulations (with adapted component values) is the classic windkessel (RC) model because it only simulates basic compliance and resistance. The modified windkessel (RCR) model considers specific behavior at higher frequencies, which is partly due to wave propagation and reflection. This type of behavior is completely different in the venous system and consequently the meaning of Rc in the arterial system cannot be translated to the venous system. The same reasoning can be followed for the four- and five-element windkessel models. When ventricular assist is simulated with an outflow cannulation to the ascending aorta, the afterload for both heart and pump is the same: the whole arterial system. However, it may be of interest to study the impact of



437

an assist device and the altered flow and pressure patterns on the circulation toward, for instance, the brain. In that case, the arterial system can be split up in an upper and a lower body part, where the former originates from the aortic arch and combines the carotid and subclavian arteries and their branches. Since this part consists of the same structure of vessel types as the whole arterial system, it can be meaningfully modeled with one of the windkessel models discussed before. This model should be used in parallel with an alike model that simulates the lower body part, with the inclusion of the proper component values. In some cases, however, the outflow graft of a ventricular-assist device is anastomosed to the descending aorta because it offers an easier surgical access and for some devices also a more direct flow path (e.g., the “Jarvik 2000” axial rotary blood pump). To simulate this, it is necessary to split up the arterial system into three: a part downstream of the cannulation site, a second part for the upper body, and a third part between the aortic root and the cannulation site. In this latter part, the flow generated by the pump will be a retrogradefloar, while the heart tries to force an antegrade flow. Since this part is only a piece of large artery, it needs to be modeled with an adapted windkessel, while the other parts can use traditional models.

3.3. Transmission line and T-tube model Although windkessel models (except for the classic one) try to mimic the high-frequency behavior that results from pressure wave propagation and reflection, they do not actually model pressure waves. Taylor was the first to approach arterial hemodynamics with a transmission line model that effectively simulates these pressure waves.30 Taylor’s first model was a single transmission line that ended in a high resistance, which can be compared to an elastic tube with a uniform diameter and a closed end. A transmission line is derived from electric network theory and it allows to calculate the pressure and flow as a function of time and location in the tube. Consequently, the input impedance can be derived from the pressure and flow wave patterns at the inlet of the tube and it can be compared to human data. Such a model has the advantage that there is a spatial distribution of the vascular properties. Further studies with transmission line models revealed that a better approximation could be obtained by modeling the tube as geometrically


438


and elastically tapered, and by using a modified windkessel as a terminal load.31,32 Experimental data also revealed that there appear to be two major reflection sites in the arterial circulation: (i) a location near the iliac bifurcation and (ii) other the head.33 The latter results in a faster reflection due to its location closer to the heart. To cope with this dual reflection, the concept of the T-tube model was developed, where two parallel elastic tubes of different lengths are used that each end in their own terminal load. The short tube represents the upper body part while the long end represents the lower body. This model can accurately simulate mammalian input impedance spectra.34 The basic asymmetric T-tube model is not much more complex than windkessel models: it has only six independent parameters if a three-element windkessel model is used as a terminal load. Such a model may be useful in studying the impact of altered input frequency spectra as a result of the interference of the heart and a pulsatile pump. It further offers the advantage that the impact of an assist device can be studied separately for the upper and lower body part. 3.4. Arterial network models Since the models above do not consider the actual arterial geometry, they do not allow to predict pressure and flow at a specific location in the body. This requires mathematical arterial network models that simulate the entire geometry, just as Westerhof did in his analog model with resistors, coils, and capacitors.35 Avolio composed a linear transmission line model of 128 arterial segments,36 which is very complicated and difficult to use in simulations. Nevertheless, such complex network models are not very meaningful for ventricular-assist device studies.

4. Assist Device Models 4.1. Cannulas To simulate the energy losses in the inflow and outflow cannulas of an assist device, several aspects need to be considered. Basically, the pressure drop in the cannulas is the result of viscous losses and a resistive component is necessary to simulate this behavior with a cannula model. When pulsatile devices are used, an additional pressure drop exists due to the acceleration and deceleration of the fluid, and thus an inductance should also be added to the cannula model. This in fact also applies to “continuous flow devices”, as the interaction with the heart and the varying pressure head



439

may also result in a pulsatile pump flow. Consequently, the typical cannula model consists of a series combination of a resistor and an inductor. Additionally, a compliance component can be added to the cannula model, but this is often superfluous because the used cannulas are very stiff compared to blood vessels. Compliance elements may, however, be useful to simulate ventricular suction as a result of rotary blood pumps running at high speed. Vollkron et al. mimicked ventricular suction with a pressuredependent variable resistance in the inflow cannula.13 4.2. Pumps 4.2.1. Rotary pumps Various rotary pump models are found in literature. Some only model the hydraulic part, while others include the motor and thus the energy conversion of electric current to fluid flow. This latter approach is especially of interest to relate a pump model to parameters that can actually be measured when the device is implanted: consumed power (or current) and rotational pump speed. Such pump models can, for instance, be created as a tool to develop parameter estimation methods. If the correlation between measurable pump parameters (current and speed) and circulatory parameters (ventricular contractility, resistance, compliance, etc.) can be obtained in a mathematical model, then current and speed can be used to estimate the patients’ circulatory parameters once an actual pump is implanted in a patient. Thus parameter estimation methods can turn the assist device in a diagnostic tool to obtain parameters that otherwise require invasive measurement techniques. The simulation of the motor is also necessary for development and validation of physiologic control algorithms for rotary blood pumps. These are necessary to provide higher flows when a patient becomes more active, and also to limit the risk of ventricular suction. The flow generated by a rotary pump is dependent on the pressure head Eq. (18), which is in contrast to displacement pumps where preload and afterload are independent determinants: Q2 = ±|3.571 × 10−5 ω 2 − 2.838 × ∆p − 12.66|,

(18)

where Q, pump flow (l/min); ω, impeller rotational speed (rpm); and ∆p: pressure head over the pump (mmHg). Consequently, it can be expected that the pre- and afterload have an inverse effect on the rotary pump performance and that the flow sensitivity to either variable is equal for a certain rotational speed. An increase in


440


preload has the same effect on the pressure head (a decrease) as a decrease in afterload with the same increment. To characterize a rotary pump, a pump graph presenting the generated flow as a function of pressure head can be composed from in vitro data, and this for multiple rotational speeds. An example of a purely hydraulic model of a rotary pump is the model used for numerical simulations of the current CorAide centrifugal pump (Arrow Int., Reading, PA) while it was still in a development phase.37 Below is an example of a model including the electromechanical relations, where the current can be used as an input or controllable parameter.38 dQ = −b0 Q − b1 ω 2 + b2 ∆p dt J

3 dω = KI − Bω − a0 ω 3 + a1 Qω 2 , dt 2

(19) (20)

where Q, pump flow (m3 /s); ω, impeller rotational speed (rad/s); ∆p, pressure head over the pump (N/m2 ); J, impeller inertia (kg m2 ); K, constant of back electromagnetic force (V s); I, motor current (A); B, damping coefficient (kg m2 s−1 ); and a0 , a1 , b0 , b1 , b2 , constants. Equation (19) defines the hydraulic characteristics of the device (an axial rotary pump) while Eq. (20) defines the function of the brushless DC electromotor and relates the current to the electromagnetic force, the resulting torque, and the torque of the load in function of the flow (last term).

4.2.2. Pneumatic displacement pumps With the modeling of displacement pumps, more parameters need to be considered. Of importance are the stroke volume and the filling and ejection timing. These pumps can again be modeled as a hydraulic component on its own, or with inclusion of the actuation mechanism. The latter option allows simulating control algorithms, which can be based on feedback from specific sensors. The Thoratec Pneumatic VAD (Thoratec Corp. Pleasanton, CA), for instance, has a switch to indicate when the blood chamber is completely filled. Pneumatically actuated pumps often also incorporate a means to monitor the air flow rate to and from the pump to estimate the blood flow generated by the pump. The derivative of the air flow from the pump can also be an indicator for the contractility of the ventricle.



441

The older driving consoles of pneumatic-assist devices are based on alternate switching between pressure reservoirs (vacuum and positive pressure) with an electronically controlled valve. Simple mathematical models of the hydraulic part of pneumatic displacement pumps are based on this principle, where a positive and a negative constant voltage source in the electric analog represent the pressure reservoirs. More sophisticated models that integrate control need to use time-varying voltage sources or combine the sources with controlled resistances. The tubes between the driver and the air chamber of the pump are often modeled similar to cannulas, but a compliance component (capacitor) is mandatory in this case to simulate the compressibility of the air. The pump chamber of a pneumatic actuated device contains a sac or a membrane to separate the air from the blood. Often, the pump housing is slightly larger than the membrane in unstressed state, leaving a gap between the two. When the pump is then actuated, the elastic properties of the membrane come into play at the end of ejection and filling. At that time, the membrane cannot rest against the housing and will be stretched out by the pressure difference between blood and air chamber. At other times in the pump cycle, there will be a balance between the pressure in both chambers, and the membrane will be unstressed, and in reality even wrinkled. A volume-dependent compliance component can be used to model this behavior of the membrane, where compliance is only present below and above certain volume thresholds. The elasticity of the membrane is not necessarily equal in both the directions: stretching toward the blood side may show a different compliance than that toward the air side. This is due to the multi-layered composition of certain membranes and to the unstressed shape of the membrane. The compliance of the membrane can be combined with the compliance of the gas and the values can be determined experimentally (Fig. 8). An example of a constitutive equation for a pneumatic displacement pump can therefore be presented with the equation below, which refers to Fig. 9: dpp (t) aps + (1 − a)pd − pp (t) . = Qi (t) − Q0 (t) + C(V ) aRs + (1 − a)Rd dt

(21)

The constant “a” is a binary term that determines the activation of the pump: the pump ejects when “a” is 1, while it fills when “a” equals 0. Hence “a” simulates the control of the electronic valves and it determines the pump rate.


442


Pump Pressure (mmHg)

300 200 100 Vmin 0 Vmax -100 -200 -300 0

20

40

60

Pump Volume (ml) Fig. 8. PV relation of a pneumatically actuated assist device with a stroke volume of 70 ml. Vmin : 15 ml, Vmax : 55 ml. Suction compliance: 0.06 ml/mmHg, Ejection compliance: 0.05 ml/mmHg.

Li

Ri

Lo

Ro

pp

Qo

Qi

Rs ps

Rd

C

pd

Fig. 9. Electric analog of a pneumatic displacement pump. Qi , Q0 : blood inflow and outflow, respectively; Li , L0 : inertance of cannulas; Ri , R0 : resistance of cannulas and valves; pp : pump pressure; Rs , Rd : resistance of the pneumatic tubing for pump during systole and diastole, respectively; ps , pd : positive pressure and vacuum in the pressure reservoirs; and C: pump compliance.

4.2.3. Electromechanical displacement pumps Simulation of a mechanically actuated displacement pump can again be performed by either modeling only the hydraulic part or the hydraulic part in combination with the actuator. However, these devices are more complex than the ones described in the previous sections, and they usually



443

incorporate more or better sensors to feed back information to the controller. Each actuation mechanism results in a specific pressure or flow waveform, and therefore this type of device should be modeled with the inclusion of the actuation mechanism. If desired, the control algorithms used by the controller can also be implemented, or algorithm prototypes can be simulated. As a result of the combination of control algorithm and actuation mechanism, it is not possible to model these devices with a time-varying compliance or elastance, as is the case for the heart. More complicated experiments and fitting techniques are usually required to obtain a model that eventually gives pressure and volume as a function of time, preload, and afterload. The model discussed below simulates the function of the Novacor N100 assist device (see also Fig. 10), which is in fact a magnetically actuated device. In summary, the device contains a valved blood sac with a pusher plate on either side. Each pusher plate is connected to one end of a beam spring, while the other ends of the springs are fixed in a frame with a central pivot. Thus the springs can rotate toward each other and compress the sac. To actuate the springs, each frame-half contains an electromagnet, and powering the magnets results in a collapse of the frame (latching) which flexes the springs and loads them. The springs will flex back to their unloaded geometry by moving the pusher plates and squeezing the blood out of the sac. To model the pump function of the Novacor N100, the actuator is simulated in two parts: the spring and the latching system.39 A combination

Fig. 10. Electric analog of a mathematical model of the Novacor N100 device.39 See text for description of the parameters.


444


of a capacitance (Cso ) and a resistor (Rso ) is used to model the dynamic behavior of the latching (solenoid) system, where Rso and Cso form the time constants of solenoid closure. A time-varying capacitance (CVAD ) simulates the spring behavior. Consequently, the P (V ) point in the model describes the static PV relation that is only dependent on the springs. The solenoid parameters and the CVAD function were derived from isovolumic beats at different volumes. Mock loop experiment data allowed to refine the model by adding a resistor (Rp ) and a flow-dependent inductance (Lp (Q)) to simulate the dynamic fluid behavior; that is, the viscosity and inertia effects. All together, these parameters and functions allow to compute the pump chamber pressure (Pcp ) as a function of the instantaneous pump volume V (t), and it can therefore be implemented in a model of the cardiovascular system. Yu et al. modeled the cannulas of the device as described in Section 4.1, with the exception of the outflow cannula resistor: a flowdependent resistor was used to better match the experimental data. Presumably this correction is necessary because of the turbulent flow pattern at the outflow.

B. The Hydraulics Laboratory Mathematical Model 1. Introduction The mathematical model developed at the Hydraulics Laboratory of Ghent University is intended for studies of heart–arterial load interaction. It is developed in Matlab 5.0 (The MathWorks Inc., Natick, MA) and it makes use of the ODE45 solver for the computation of variables that are embedded in differential equations. The mathematical model has previously been validated in several forms. In one study it was used to predict the stroke volume and aortic blood pressure in sheep, where the model generated data proved to accord well with the experimental data.7 In another study, normotensive and hypertensive patients were simulated to assess the concept of wall stress normalization as an initiator for left ventricular hypertrophy.40 The maximum discrepancy between modeled and simulated diastolic blood pressure was 7% and for systolic pressure it measured 4.8%. The basic heart–arterial model was further extended to allow simulation of rotary blood pumps. In particular pulsatile control of these devices was



445

studied. It has not been the intention to simulate or predict specific clinical situations, but the aim is to indicate the impact of this new concept on left ventricular load. In its current form, the model does not work with a fixed volume of blood and cannot simulate effects of bleeding or infusion. All simulations were performed with a cardiovascular model matched to sheep data to allow validation with available animal data. Yet, the model composition was chosen as basic as possible to minimize the number of assumptions and estimations, and to eliminate the need for parameters from literature data. All necessary parameters for this basic model can be derived from an animal experiment where left ventricular PV, and aortic pressure and flow are acquired. Even though sheep with induced heart failure were used as a reference for some studies, it should be considered that clinical heart failure is usually more severe than it can be modeled in animals. 2. Cardiac Model 2.1. Elastance The cardiac model is restricted to the left heart, with the option to include or exclude the atrium. Both the heart chambers are modeled according to the original time-varying elastance theory, where the EDPVR is assumed linear. The elastance curve can be chosen from a normalized curve that is implemented as a Fourier series, or a double-Hill curve. The choice of elastance curve only determines the shape, since this curve has been normalized in time and amplitude. At the start of a simulation, a rescaling of the normalized elastance curve is performed based on the parameters that can be set for each heart chamber. These parameters are the maximum and minimum elastance (Emax , Emin , respectively), the heart rate, and the time of maximal elastance (Tmax ). Additionally, the volume intercept (V0 ) needs to be set for the cardiac model, but this does not affect the elastance curve. A disadvantage of the linear time-varying elastance concept is that the diastolic filling in the PV plane is linear instead of curved. This is, however, only the case if a predefined elastance curve with a flat diastolic phase (e.g., the double-Hill curve) is used, as the filling of the heart chamber will then be prescribed completely by the linear EDPVR. This can be prevented by creating a custom elastance curve where the diastolic part is not flat. In this case only the minimum of the elastance curve will actually fall on the EDPVR line. The nonlinearity of the real EDPVR is then in fact incorporated in the elastance curve. The remaining discrepancy with reality will be the effect of large volume shifts, where the PV loop in this


446


model will be trapped between linear boundaries, while in reality trapped between curvilinear boundaries. This will result in an underestimation of filling pressure when dilation of the heart is modeled. Conversely, the linear approach of the ESPVR may result in an overestimation of systolic left vertricular pressure in the simulation of hearts with extremely large volumes: the ESPVR is believed to curve down toward the volume axis as of a certain volume. Different in vivo ESPVR characteristics can be found depending on the type of contraction (e.g., isovolumic, isotonic13 ) and both convex and concave curvature have been reported for the lower volume range.41,42 Therefore, the choice for a curved ESPVR is not obvious. In summary, the cardiac model is an approximation that is not valid for a seriously decompensated heart as seen in terminal patients, but it can be used to model heart failure as it is observed in animal models by decreasing Emax , and increasing Emin and heart rate. One way to implement a custom (normalized) elastance curve is the Fourier series description (Table 1). This is available for human from literature,2 and for sheep from own experimental data. An example of both

Table 1. Fourier series data for human and sheep normalized elastance curves. Harmonic

Human Modulus M (mmHg/ml)

0 1 2 3 4 5 6 7 8 9 10 11 12

Phase ϕ (rad)

Sheep Modulus M (mmHg/ml)

0.2839 0 0.2251 0.3759 0.0837 0.2771 0.2102 −1.4868 0.2227 0.0767 2.8657 0.1491 0.0481 0.1677 0.0767 0.0418 4.6302 0.0326 0.0194 3.0884 0.0395 0.0059 −0.3054 0.0450 0.0118 4.4107 0.0355 0.0084 3.1815 0.0187 0.0002 1.2429 0.0067 0.0031 4.1568 0.0108 0.0032 2.9462 0.0125 P12 En = M (0) + h=1 [M (h) sin(h2πf t + ϕ(h))]

Phase ϕ (rad) 0 0.6298 −0.3252 −1.3229 3.8192 2.0308 0.0551 −1.1606 4.0587 2.9214 0.9517 −1.0987 4.0776

En : normalized elastance function; h: harmonic number; f : harmonic frequency: 0.2890 Hz for human and 0.2 Hz for sheep.



447

elastance (mmHg/ml)

3.0 human sheep

2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

time (s) Fig. 11. Reconstructed left ventricular elastance curves for sheep and human at a heart rate of 60 BPM. Emax = 2.6 mmHg/ml. Emin = 0.3 mmHg/ml.

the reconstructed curves is shown in Fig. 11, scaled for an identical contractility and heart rate. The normalized sheep elastance was constructed with PV data of four healthy sheep and two sheep with induced chronic heart failure by injection of microspheres.43 After determination of V0 from caval vein occlusion data, the real elastance curve for each sheep was calculated from an averaged beat. The data were then normalized with respect to time and amplitude, therfore the maximal elastance becomes 1 and occurs at time 1, where time and elastance are both non-dimensional measures. Next, the data were consecutively resampled to obtain the same time step, averaged over all the animals, and decomposed in Fourier components. A subprogram in the mathematical model recomposes the Fourier series up to the 12th harmonic with the time step that is also used in the solver of the differential equations, so no further interpolation is required. The obtained normalized curve is then rescaled according to Emax , Emin , heart rate, and Tmax given by the user. Tmax is set at 30% of the heart period by default, but it can and should be adjusted by the user. Since no control of the heart rate according to autonomous nervous activity was incorporated, an algorithm for determination of the systolic-to-diastolic time ratio was superfluous. In most cases, the simulations will be based on actual experimental data, and thus the actual Tmax can be determined without the use of an estimation algorithm.


448


2.2. Preload Since the right heart is completely neglected in this mathematical model, the preload of the left heart needs to be decoupled from the afterload in another way. A common approach is the use of a constant pressure (voltage) source between the afterload and the atrium.44−46 This method simulates a constant pulmonary venous filling pressure which roughly approximates reality because the pulmonary circulation is a very compliant circuit with a low pressure source. The venous filling pressure can be set by the model user and it can be derived from animal experiments as the end-diastolic atrial pressure or pulmonary venous pressure, although these data are rarely available. 2.3. Valves The mitral and aortic valves are basically modeled as a resistor in series with an inductor. Leakage is also included and consequently the electric analog of the valves can be seen as a parallel circuit of two resistor–inductor combinations with a diode in each branch (see Fig. 17). The diodes are opposed to each other and their purpose is to determine the flow direction through the valve in function of the pressure (voltage) difference. The practical method in the model is the comparison of downstream and upstream pressure. If upstream pressure is highest, then forward flow will open the valve and the resistor–inductor combination with the lowest values will be used; therefore, a high flow and small pressure drop will be present over the valve. In the opposite case, the combination with the highest resistance value is used which will result in a small backward (regurgitant) flow. The resistance and inductance values of the valves can be set manually for forward and backward flow. The default values are taken from literature and from fitting procedures on experimental data. 2.4. Wall stress To quantify the load on the heart, or the degree of unloading that an assist device can provide, an estimate of the stress in the myocardium can be used. According to Arts et al., wall stress in the left ventricle is homogenous so it can be represented by one indicative number.47 The cardiac model, therefore, allows estimating the wall stress in the left ventricle as a single parameter over the time course of the cardiac cycle. For this, the ventricle is modeled as an isotropic, homogeneous thick-walled ellipsoid; and the



449

meridional wall stress at the equator is calculated according to the following equation: σm =

pLV D2 , 4h(D + h)

(22)

where σm , meridional wall stress at the equator (mmHg); pLV , left ventricular pressure (mmHg); D, internal ventricular diameter (cm); and h, wall thickness (cm). The internal diameter of the left ventricle is required for this calculation, just as myocardial wall thickness (h). The diameter is computed from a formula used in echocardiography,48 that relates the ventricular volume (VLV (ml)), which is a variable in the mathematical model, to the internal diameter: VLV =

7D3 . 2.4 + D

(23)

As the wall thickness is rarely measured and recorded in animal experiments, an estimation of the variations in wall thickness throughout the cardiac cycle is used. It is based on the knowledge that the myocardial wall volume (Vwall ) is constant. So, if wall volume and instantaneous internal volume are known, the outer dimensions and wall thickness of the ellipsoid can be computed. To calculate the wall volume, an arbitrary value of endsystolic wall thickness is assumed (e.g., 1) and the resulting wall stress data should be interpreted as normalized with respect to end systole: σc =

pLV D(2L2 − D2 ) , 4h(L2 + Dh)

(24)

where σc , circumferential wall stress at the equator (mmHg) and L, longaxis dimension (cm). Circumferential stress can be calculated according to Eq. (24) , which additionally requires the internal long-axislength (L) of the ellipsoidal model. As the model is assumed axisymmetric, and the diameter (D) and internal volume are already known, its internal length can be calculated from the standard formula for volume calculation: 4 (25) Vellipsoid = πD2 L. 3 It was found that this method yields only very little variation of the long-axis dimension L and, consequently, the circumferential wall stress calculated with this method was deemed unreliable. Furthermore, according to Arts et al., circumferential stress is very much dependent on the geometry,


450


and an ellipsoidal approximation is not a good starting point for calculation of that type of stress.47 The result depends on the curvature, and thus higher stresses are found at the equator compared to the top of the ellipsoid. The circumferential stress calculated according to Eq. (24) is in fact the equatorial stress, which is therefore an overestimation of mean stress. The three variables necessary for fiber stress calculation according to Eq. (26) are already computed for the meridional wall stress, and consequently a calculation of the fiber stress (σf ) is easily implemented in the cardiac model. σf represents the stress in the fibers of the extracellular matrix, which are presumed to form the mechanical strength of the myocardium and absorb the majority of wall stress. For the derivation of this formula, it was assumed that the fibers are arranged in parallel with the isobars of tissue pressure in the myocardial wall: VLV (26) pLV . σf = 1 + 3 Vwall The three possible stress calculations were compared on a set of in vivo data, with special attention to the waveform morphology over a heart cycle. In most cases the end-systolic wall thickness will be set at 1 cm and “normalized stress” will be compared between simulations, so that the amplitude of the different types of wall stress is of lesser importance. For comparison, the circumferential and meridional wall stresses were scaled to match the maximum of fiber stress. It can be noted that the variation in meridional wall stress and fiber stress fit closely, while circumferential stress deviates from those two (Fig. 12). The relatively simple approach of the thick-walled ellipsoid and the resulting meridional stress calculation of Falsetti et al. was recommended previously as a parameter that suffices for clinical evaluation of myocardial wall stress.49

3. Afterload Model Two options for the afterload model are programmed and can be selected: a three-element and a four-element windkessel model, both have been discussed before (see section A, 3.1). The three-element model is the modified windkessel model (RCR) consisting of a characteristic resistance in series with a parallel combination of a resistance and compliance element.23 The four-element model (RLRC) has an additional inductance in parallel with the characteristic resistance.25 Even though the series combination of the inductance and characteristic resistance tends to give better results for data



wall stress (mmHg)

300

LVP (mmHg)

BASELINE

100

451

UNLOADED

σf σm σc

250 200 150 100 50

80 60 40 20

LVV (ml)

80 70 60 50 40 0.0

0.1

0.2

0.3

time (s)

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

time (s)

Fig. 12. Example of different wall stress calculation methods applied on sheep data at baseline and with support of a rotary pump (unloaded) that produced 3.5 l/min. The wall stress curves are referenced to left ventricular pressure (LVP) and volume (LVV) curves. σf : muscle fiber stress; σm : meridional wall stress; σc : circumferential wall stress; σm and σc are rescaled to match the amplitude of σf . The original maxima of σc and σm were, respectively, 310.0 and 81.7 mmHg at baseline and 183.4 and 61.4 mmHg when unloaded.

fitting, the parallel combination was implemented because it relates more closely to the actual physiologic afterload. This model, just as all the others, is solved in the time domain.

4. Assist Device Model The mathematical model studies discussed in this chapter focus on a specific rotary blood pump, the Medos-HIA microdiagonal pump (Medos Medizintechnik GmbH, Stolberg, Germany). Since this device was still in


452


a development phase at the time this mathematical model was completed, all necessary experiments were performed with a prototype of the pump. This prototype had a diameter of 30 mm, a length of 80 mm, and weighed 120 g. The microdiagonal pump is intended to be implanted via a left thoracotomy (between the ribs), which is less traumatic than the traditional sternotomy where the whole sternum is split. Moreover, the less traumatic implantation and small size justify the use of this device in an earlier stage of heart failure, where the device should only deliver part of the total blood flow. Consequently, the device is intended for use as a bridge to recovery or to transplantation, and it aims at a use of 2 years, where apical or atrial cannulation can be applied (Fig. 13). The pump and the cannulas were modeled separately because the pump is only available in one size, while the cannulas can vary in size and are cut at length by the surgeon during the implantation. Only the hydraulic part of the pump is modeled and the only pump parameter that can be set by the user is the rotational speed of the impeller. The device is modeled as to simulate apical to ascending aorta cannulation, and the cannulas are assumed to be cylindrical. Consequently, the length and diameter of the inflow and the outflow cannula are the remaining four parameters that can be set. The microdiagonal pump model (see Eq. (27) and Fig. 14) was obtained from fitting procedures on experimental data acquired in an in vitro flow loop at the Helmholtz Institute in Aachen (Germany). The basic experimental data consist of standard pressure head-flow measurements of the pump when running at different speeds. A broad range of pressure heads

Fig. 13. Picture of the Medos-H/A microdiagonal pump prototype that was used in the here discussed studies (left) and a schematic drawing of its intended placement (right).



453

Fig. 14. Example of the fit of Eq. (27) on a data set of the microdiagonal pump (r 2 = 0.99). The rotational speed varied from 3,000 (most left plot) to 9,000 rpm (most right plot) with a 1,000 rpm increment. The final parameters of the fit are: A = 2.956 × 10−6 mmHg/rpm2 and Rp = −2.415 mmHg min2 /l2 .

and according flows were obtained by varying the resistance in the flow loop. Apart from the different speeds, data were also acquired when the pump was switched off as to simulate pump failure. These data characterize the hydraulic viscous losses over the pump and were performed with both forward and backward flow. The different flow levels through the switched off pump were achieved with an extra pump. Note that backward flow can also occur in vivo, even when the pump is running: in diastole aortic pressure is higher than left ventricular pressure, and thus a negative pressure head will be imposed on the pump. If it is running at a low rotational speed, the pump may not generate sufficient power to overcome this pressure head, and blood will regurgitate from the aorta to the ventricle. This situation was not mimicked in the experimental setup. To attain pump characteristics of backward flow through a running pump, the forward flow characteristics were mirrored by switching the sign of the pump resistance (see below). In electric network theory, a realistic voltage source can be considered as a series combination of an ideal voltage source and a resistor that models the energy losses. Analogous, a rotary pump can be considered as the combination of an ideal pressure source and a resistive element. Consequently, the pump can be modeled as the sum of a term in function of the rotational


454


speed (generated energy) and a term in function of the flow (energy losses). Since the flow pattern in the pump can be expected to be turbulent, the pressure loss in the pump will correlate to the square of the flow, rather than to the flow itself. The resulting pump equation, that was used to fit to the data, is given below: ∆p = Aω 2 + Rp Q2p ,

(27)

where ∆p, pressure head over the pump (mmHg); A, pump constant (mmHg/rpm2 ); ω, rotational speed (rpm); Rp , pump resistance (mmHgmin2 /l2 ); and Qp : pump flow (l/min). To offer flexibility for future implementation of other devices, the pump equation is in fact rewritten in the following form: α3 1 + α4 ∆pα5 + α6 . Qα p = α2 ω

(28)

The α parameters in this equation are constants that are stored in a separate file, containing other possible pump models in the same format. In the model of the microdiagonal pump, α2 and α4 represent A and Rp from Eq. (27), respectively, while α3 and α5 are 2. These parameters can be retrieved from the pump model file for combination with the cannula model. The cannula model consists of a series combination of an inductor and a resistor for each cannula. The values of these components are calculated from the cannula dimensions as given by the user. The extra pressure drop generated by the cannulas is then: ∆pc = ∆pL + ∆pR ,

(29)

dQp + (Ri + R0 )Q2p , (30) dt where ∆pL , pressure drop due to inertance effects (mmHg); ∆pR , pressure drop due to viscous effects (mmHg); ∆pc , pressure drop over the cannulas (mmHg); Li , inlet cannula inertance (mmHg min2 /l); Lo , outlet cannula inertance (mmHg min2 /l) Ri , inlet cannula resistance (mmHg min2 /l2 ) Ro inlet cannula resistance (mmHg min2 /l2 ). The inertance L of a cylindrical tube is given by: ∆pc = (Li + L0 ) ·

4ρl . (31) πD2 The flow through the cannulas is expected to be laminar, given the dominantly low Reynolds numbers (Re < 3,000). Therefore, the resistance is characterized by a linear relation between pressure drop and flow (∆p ∼ Q). However, to conform with Eq. (27) and to allow an easier implementation L=



455

in the model, the relation was inserted in Eq. (30) as being quadratic (∆p ∼ Q2 ). To transform this back to a linear relationship, the pump flow (Qp ) was included in the denominator of the resistance calculation. The linear pressure drop due to resistance (∆pR ) is thus calculated from the Poiseulle equation for laminar flow: 64 1 u2 1 ρ Re D 2 133.33 32vlρ Q2p 60 × 106 ⇒ ∆pR = uD2 A 133.33 32vlρ ρ 60 × 103 2 Q ⇒ ∆pR = Qp D2 A 133.33 p ∆pR =

⇒ ∆pR =

128µl 60 × 103 2 Q = Ri,o Q2p · Qp πD4 133.33 p

(32) (33) (34) (35)

where Re = uD/v, Reynolds number (−); u, fluid velocity (m/s); D, cannula diameter (m); l, cannula length (m); ρ, fluid density (kg/m3 ); v, dynamic fluid viscosity (m2 /s); µ = vρ, kinematic viscosity (N s/m2 ). In these equations, flow and pressure are, respectively, expressed in l/min and mmHg, and the other variables in SI units. As a consequence of expressing the resistive behavior as a quadratic relationship, the pump flow is in the denominator of the cannula resistance (Ri,o ) which makes it a flow-dependent variable. The alternative is to express the resistive behavior as a linear relationship, which would give a non-flow-dependent resistance. However, the dimensions of Ri and Ro would differ from the dimensions of Rp and as a consequence they would make two separate terms in the resulting differential equation. A combination of the formulas above leads to a differential equation that describes the hydraulic part of the assist device in the following form: dQp , (36) dt where pao , aortic pressure (mmHg); pLV , left ventricular pressure (mmHg); Rtot = Rp +Ri +Ro , flow-dependent resistance (mmHg min2 /l2 ); and Ltot = Li + Lo , cannula inertance (mmHg min2 /l). Additionally, the assist device model was extended with the option to modulate the rotational speed throughout the cardiac cycle. This is a simulation of synchronous pulsatile use of rotary blood pumps. Two additional parameters need to be set: (i) the type of modulation and (ii) the phase shift between the pump cycle and the heart cycle. This latter is set in percentage of the total cycle (see Fig. 15). There are three types of modulations, pao − pLV = Aω 2 + Rtot (Qp )Q2p + Ltot


456


30%

100%

4000

2000 1.5 1.0 0.5 0.0 0.0

0.5

E(t) (mmHg/ml)

pump rate (rpm)

6000

1.0

time (s) Fig. 15. Illustration of the pulsatile pump speed pattern, here indicating a time shift of 30% of the full cycle. The reference point is the onset of systole assessed from the time-varying elastance curve.

besides the option to have a constant speed. Each type was scaled to have a peak-to-peak difference of 4,000 rpm and set off to obtain a mean rotational speed equal to the speed value set by the user. The implemented modulation waveforms are: an inverted elastance curve, a negative part of a sine wave, and an inverted complete sine wave (Fig. 16).

type 1

type 2

type 3

rot. speed (rpm)

7000 6000 5000 4000 3000 2000 1000 0.0

0.5

1.0

1.5

2.0

2.5

3.0

time (s)

Fig. 16. Illustration of the three rpm modulation types that are implemented in the mathematical model. All modulations have a peak-to-peak value of 4,000 rpm and an average of 4,000 rpm.



457

5. Program Setup 5.1. Overview of all the model components An overview of all the model components is depicted in Fig.17. 5.2. Program sequence When all the required parameter values are inserted via the user interface (Fig. 19) and a simulation is started (with the “run” button), the program determines the appropriate solver. Because valves can be simulated as a simple resistor or a resistor–inductor combination, and because there is a choice of pump and afterload models, several solvers were implemented. The following discussion assumes the presence of an atrium and a rotary pump in the chosen solver, relating to the complete model (Fig. 18). The selected solver starts with calculation of the inductance and resistance values of the pump cannulas and with determining the A and Rp values of the pump model (see Eq. (27)). The solver will then create an empty vector for each variable, where the length is determined by the time step and the heart rate as given by the user. The rotational speed of the pump will also be calculated as a vector, as a function of the type of modulation and the phase shift that is chosen. Initial Rmvb

Lmvb

Raovb

Laovb pao

Rv

Lmv

Laov

Rmv

Raov

Lo

Li

Pv

PLA

PLV

Ro

Rpump

Rc

L

Ri

pc

Ppump C

Rp

Components: L: inductor (inertance). C: capacitor (compliance). R: resistor (resistance). P: voltage (pressure) source. Indices: v: venous. LA: left atrium. LV: left ventricle. mv: mitral valve. mvb: mitral valve backward. aov: aortic valve. aovb: aortic valve backward. i: in. o: out. c: characteristic. p: peripheral Variables: pao: aortic pressure. pc: pressure before the peripheral resistor

Fig. 17. Electric analog of the Hydraulics Laboratory mathematical model. Since it is possible to include or exclude certain parts, the model should be visualized with the addition of switched bypasses.


458

S. Vandenberghe et al. insert / load parameters save parameters START

determine solver load default initial values

based on: heart model arterial model pump model

calculate cycle determine new initial values

plot / append waveforms calculate RMSE value

NO

RMSE < precision YES calculate output from last cycle plot graphs fill textboxes

Fig. 18. Simplified flow chart of the mathematical model simulation sequence. RMSE: root mean squared error.

estimates for the first time point (default initial values) of each variable are also set by the solver before it actually solves the differential equations with Matlab’s ODE45 function for every time point of one cardiac cycle. There are seven state variables in the complete model:

1. 2. 3. 4. 5.

Flow through the mitral valve (Qmv ). Flow through the aortic valve (Qao ). Flow through the pump (Qp ). The pressure proximal to the arterial compliance (pc ). The aortic pressure (pao ).



459

Fig. 19. Overview of the graphical user interface of the mathematical model after a simulation where the microdiagonal pump was run at 4,000 rpm, the heart was modeled only by the left ventricle, and the arterial system by a four-element windkessel model.


460


6. Left ventricular volume (VLV ). 7. Left atrial volume (VLA ). Once these are solved over the heart period, further output variable vectors, left ventricular pressure and left atrial pressure, are derived from them. Since the volumes of these heart chambers are known from the calculation, the pressures can be derived with the original elastance formula from Suga and Sagawa (Eq. (1)). A temporary output is then plotted on the user interface, hence the user can follow the progress of the simulation. Next, the values of the state variables at the end of the cardiac cycle are used as the initial values for the following cycle. The differential equations are then solved for a second time, the output variables are calculated once more and this second cycle is plotted adjacent to the first cycle on the user interface. Before continuing the simulation, a convergence criterion is tested: the root mean squared error (RMSE) value of the aortic pressure between the last beat and the previous beat is compared to a preset precision that is entered by the user at the start of the simulation (default: 2 mmHg): The RMSE value is defined as: n 1 2 [pao (n)c − pao (n)c−1 ] , (37) RMSE = n 1 where n, number of time points in a heart cycle (−); pao , aortic pressure (mmHg); C, last calculated cycle; and C − 1, previous cycle. If the RMSE value is lower than this precision, convergence is assumed and the simulation is halted. Otherwise a following cycle is calculated until convergence is achieved. In case divergence is detected, the initial values for the following beat will be taken as the average of the end values of the previous two beats, instead of the values of only the last beat. Once convergence is obtained and the simulation is stopped, additional parameter calculations are performed and the output is displayed on the user interface, as explained in Section 5.4. 5.3. Differential equations The complete set of differential equations is given below: for the model as depicted in Fig. 17. The equations are given in matrix notation and the parameter and variable abbreviations relate to the ones in Fig. 17 and above. The matrix notation should be interpreted as x˙ = Ax + Bu, where x is a vector with the unknown (state) variables and u a vector with the



461

inputs. The matrix A is usually referred to as the system matrix and B is the input matrix. Additional parameters that are not indicated in Fig. 17 or the text are: E: left ventricular elastance (known vector). Ea : left atrial elastance (known vector). V0 : volume intercept of the left ventricular ESPVR (known constant). V0a : volume intercept of the left atrial ESPVR (known constant). Qv : venous flow from the lungs (variable used to compose equations). QRc : flow through the characteristic resistor (used to compose equations). 2 dQmv 3 2−Rmv dt 6 dQ ao 6 6 dt 6 dQ 6 p 6 dt 6 6 dpc 6 dt 6 6 dpao 6 dt 6 6 dV 6 LV 4 dt dVLA dt

76 76 76 76 76 76 76 76 7=6 76 76 76 76 76 76 54

0

0

0

0

−Raov Laov

0

0

0

0

−Rtot Qp Ltot

0

1 C

1 C

Lmv

0

1 C

aov − RcL·R aov

1

−1

−1

0

2Q

3

2

6 6 Qao 7 6 6 7 6 6 Qp 7 6 6 7 6 6 7 6 × 6 Qc 7 + 6 6 7 6 6 Qao 7 6 6 7 6 4 QLV 5 6 6 4 QLA mv

1 C

−

Rc Rtot Ltot

0

Qp RLc − RL pC

−1

0

0

0 1 Lmv

0

0

+

Rc Ltot

0 0 0

0

0

0

1 Rv

0

0

E Lmv −E Laov

−Ea Lmv

3

7 0 7 7 7 7 −E 1 0 7 Ltot Ltot 7 7 0 0 0 7 7 “ ” 7 Rc Rc Rc Rc Rc 0 7 Laov + Ltot − L −E Laov + Ltot 7 7 0 0 0 7 5 −Ea 0 0 Rv 1 Laov

−1 Rp C

−1 Lmv 1 Laov 1 Ltot Rc Laov

0

3

7 0 72 3 7 E · V0 07 76 7 6 Ea · V0a 7 7 0 07 7. 76 Pv 5 74 0 A7 2 7 ω 0 07 5 1 Rv 0 0 0

(38)

As mentioned in the introduction, the equations can be derived by applying traditional electronic circuit theory on the schematic of Fig. 17. It can be seen here that the equations are solved in the time domain. Most important for the derivations are Kirchow’s node and loop laws, and the following relations between pressure (p) and flow (Q) over electric components (remember that pressure–voltage and flow–current): (39) Resistor : ∆p = RQ (Ohm’s law). 1 dp Capacitor : dp = Q dt or C . (40) C dt dQ . (41) Inductor : dp = L dt As for the heart valves, it is tested within each iteration of the solver whether the pressure proximal to the valve is higher than the distal pressure


462


(see also Section B, 2.3). If so, the valve is open and the normal values for valve resistances and inertances are used (Rmv , Raov , Lmv , Raov ). Else, the higher backflow values Rmvb , Raovb , Lmvb , and Laovb are used to simulate a closed (potentially leaking) valve. This can be achieved according to the schematic in Fig. 17 by seeing the diode elements as a multiplier that can have a value of either 0 or 1, thereby creating a broken link or a closed link in the circuit, respectively. Below are the initial equations that can be derived directly from the schematic: dQmv . (42) PLV − PLA = Qmv + Lmv dt dQaov Pao − PLV = Qaov + Raov + Laov . (43) dt The above equations give the relationship between the pressure drop over a valve and the flow through the components that make the valve: a resistor and an inductor: dQp . (44) Pao − PLV = Rtot Q2p − Aω 2 + Ltot dt This is the pump equation including the cannulas (as discussed before): Qao + QP = C

Pc dpc + . dt Rp

(45)

The node law applied on pc : the total flow goes in that node and must equal the sum of the flows through C and Rp : Pv − PLA dVLA = Qv − Qmv = − Qmv . dt Rv

(46)

The difference between what flows into the left atrium (venous flow) and what flows out (mitral flow) is the volume change of the atrium. dVLV = Qmv − Qao − Qp . dt

(47)

The difference between what flows into the left ventricle (mitral flow) and what flows out (aortic flow and pump flow) is the volume change of the ventricle: dQL = Pao − Pc = QRc Rc . (48) L dt The pressure drop over the afterload inductor equals the pressure drop over the characteristic resistor because the parallel model is used.



463

With Eqs. (42)–(48) as a starting point, it is left to the reader to perform the appropriate substitutions and reformat the equations to the matrix notation of Eq. (38). 5.4. Model input and output The possible settings and inputs for the model were described in the previous sections. Parameters are inserted in textboxes, while a listbox allows to make choices; for example, for the different elastance curves or the pump modulation. Check boxes are further used to include or exclude components such as the inertia of the heart valves. After a simulation has ended, output data is presented on the user interface and it can be saved to a data file. It concerns time-varying variables that are given in vector format over the whole last heartbeat and averaged and derived parameters that are given as a single value. These separate output parameters are displayed on the user interface in textboxes (see Fig. 19): • Left ventricular stroke volume and end-diastolic volume (SV, LVEDV). • Mean, systolic, and diastolic aortic pressure (Psys, Pdia, Pmean); • External work (EW), potential energy (Pot En), pressure volume area (PVA), and mechanical efficiency (Mech Eff) of the left ventricle. These are all parameters that relate to the energetics and oxygen consumption of the heart.a • Systolic and diastolic wall stress (Tsys, Tdia). • Average mitral and aortic valve flow, and assist device flow (Qmv , Qao , Qpump). • Forward and regurgitant flow for each heart valve plus the leakage duration (Forward, Leak, T Leak). • The pressures at which leakage occurs (PLV, PLA, Pao (leak)). The vector data contains the following variables: • Time • Left atrial pressure and volume • Left ventricular pressure and volume a These

parameters are derived from PV loops. External work is calculated as the area within the PV loop and quantifies the energy that the LV delivers to the blood. Potential energy is calculated as the triangular area between ESPVR, EDPVR, and diastolic portion of the PV loop. This area quantifies the energy that is consumed by the left ventricle for a single muscle contraction.The sum of these areas is known as the PV area and is linearly related to myocardial oxygen consumption.57


464

• • • • •


Aortic flow and pressure Mitral flow Pulmonary venous flow Left ventricular wall stress Assist device flow

These data are displayed on the user interface in graphs. Three similar graphs (pressures, flows, and volumes) show data in function of time for all the calculated beats. Other graphs display data only obtained from the last calculated heartbeat: a PV loop with indication of ESPVR and EDPVR, a plot of wall stress over time and a plot of aortic and left ventricular pressure over time.

C. Case Study: Effect of a Rotary Blood Pump on Left Ventricular Energeticsb 1. Introduction This study investigates the capacity of the Medos-HIA microdiagonal pump to unload the left ventricle. This is investigated through simulations with the mathematical model described above. The unloading is evaluated with energetic parameters and wall stress, and it is important for potential recovery of a failing heart. In certain types of heart failure (cardiomyopathy) the myocardium remodels itself as a response to the disease. Presumably, partial or complete recovery of select patients can be achieved if this remodeling can be reversed so the myocardium returns to its original state. Barbone et al. recently demonstrated that the load imposed on the heart is one of the major determinants of the remodeling process,52 but the question remains how much unloading is required in relation to the patient’s condition, to obtain reversed remodeling. Not only is the degree of unloading important, but it is also important that the unloading level can be controlled. Currently, there is general belief that a severely failing heart needs initial maximal support to optimally perfuse the patient and to prevent further deterioration of cardiac function. However, it is also believed that maximal support for a too long a period will render the heart atrophic and dysfunctional. Hypothetically, there is a subset of patients that can be recovered b More

details and extended results of this study can be found in: S. Vandenberghe, P. Segers, B. Meyns, and P. Verdonck, Unloading effect of a rotary blood pump assessed by mathematical modeling, Artificial Organs 27, 12 (2003) 1094–1101.



465

by gradually reducing the support (and increasing the load on the heart) and thus training the heart and stimulating reversed remodeling. For those patients, accurate determination of the loading level is necessary.

2. Methods 2.1. Setup of the mathematical model The study discussed below was based on experimental data acquired in a healthy sheep (56 kg) that had a Medos-HIA microdiagonal pump implanted. The data included waveforms of left ventricular pressure and volume, arterial (carotid) pressure, and mean aortic flow and pump flow. Apart from steady-state measurements, vena cava occlusions were performed for determination of V0 . Thus the experiment yielded the following LV reference values for the simulations: Ees = 1.55 mmHg/ml; Eed = 0.12 mmHg/ml, heart rate = 104 bpm, tEes = 0.275 s and V0 = 14.8 ml. Since the sheep was already heavily instrumented, no atrial data could be obtained and thus for the mathematical model a solver was chosen that did not include an atrium. A four-element (RLCR) windkessel model was used as a simulation of the sheep’s systemic circulation. Since there was no aortic flow waveform available, the fitting procedure as described in Section A, 3 could not be applied. An alternative method was used to determine the systemic load of our reference animal: mean carotid pressure (MCP), pulse pressure (PP), stroke volume (SV), and heart rate (HR) were used in the following formulas: MCP (mmHg s/ml). SV × HR SV (ml/mmHg). C= PP × 1.4

Rp =

(49) (50)

Rc and L were adjusted within limited range to improve the agreement between measured and simulated PV loops at baseline (pump turned off and clamped). This yielded the following arterial reference parameter set: Rp = 1.13 mmHg s/ml, C = 1.27 ml/mmHg, Rc = 0.15 mmHg s/ml, and L = 0.05 mmHg s2 /ml. The microdiagonal pump model as described above was used to simulate LV to aorta cannulation, where the cannula dimensions were copied from the experiment; the inlet cannula was 50F × 290 mm (diameter × length) and the outlet cannula measured 8 × 395 mm.


466


2.2. Simulations Unless otherwise stated, all simulations were performed adopting the reference model parameters as given above. The reference pump speed was chosen as 4,000 rpm since preliminary simulations pointed out that this is the optimal setting at which there is still cyclic flow through the aortic valve (systole), while the backflow through the pump (diastole) is minimal. In a first series of simulations, the unloading of the left ventricle by a rotary blood pump running at constant speed was studied for a range of pump speeds from 2,000 to 10,000 rpm in 2,000 rpm increments. In a second series of simulations, the rotary blood pump was simulated in a synchronous pulsatile mode, wherein the pump speed varies throughout the cardiac cycle from 2,000 to 6,000 rpm according to a type 3 modulation (see Figs. 15 and 16). A range of phase shifts from 0% to 100% was explored in 10% increments. A 0% shift can be seen as counterpulsation, while a 50% shift is similar to co-pulsation.

3. Results and Discussion 3.1. Continuous mode simulations To illustrate the effect of pump speed on the calculated variables, Fig. 20 shows a calculated PV loop, left ventricular wall stress, left ventricular and aortic pressure, and aortic flow at 2,000 and 10,000 rpm. The increase in speed results in a reduction of both left ventricular volume and pressure. The combination of these effects results in a considerable decrease in maximal wall stress (31%). The peak wall stress is always reached before the end of systole. At 2,000 rpm, the aortic valve opens and closes cyclically as evidenced by the partial overlap of aortic and left ventricular pressure, and the aortic flow pattern. This is in contrast to the data at 10,000 rpm, where there is no flow through the aortic valve (0 1/min). This is related to the pressure head over the valve, where the aortic pressure is high and nearly constant (150 mmHg) while the left ventricular pressure stays below 65 mmHg. Typical for unloading at high constant speeds is the triangular shape of the left ventricular PV loop,13 (Fig. 20). Even though isovolumic contraction and relaxation phases are very short in the normal heart, the pump evacuates blood rapidly from the left ventricular resulting in volume variations during these phases. Figure 21 further summarizes the effects of pump speed on the different parameters under study. Aortic flow decreases steeply with increased pump



LV & Aortic pressure

467

Aortic flow

160

Qao (l/min)

pressure (mmHg)

15

120 80

10

5

40 0 0.0

0

0.1

0.2

0.3

0.4

0.5

0.0

0.6

0.1

0.2

time (s)

LV PV-loop

0.5

0.6

0.5

0.6

LV wall stress

10000 rpm 2000 rpm

80

(mmHg)

80

0.4

100

60 40

m

pressure (mmHg)

100

0.3

time (s)

60 40 20

20 0 0

20

40

60

80

volume (ml)

100

120

0 0.0

0.1

0.2

0.3

0.4

time (s)

Fig. 20. Comparison of results obtained with a constant pump speed of 2,000 (solid lines) and 10,000 (dotted lines) rpm. The gray lines represent left ventricular (LV) pressure, while the black lines in the same graph show aortic pressure. Aortic flow is calculated at the aortic valve.

rate and becomes zero from 6,000 rpm onward. This finding, combined with the fact that as of 6,000 rpm the systolic aortic pressure is higher than maximum LV pressure, indicates that the aortic valve stays closed throughout the cardiac cycle, and thus that the total flow in the circulation is produced by the pump only. A phenomenon noticed at low pump speeds, when there is minimal unloading, is backflow through the pump due to the negative pressure head during diastole. Overall mean pump flow increases linearly with increasing pump speed, but it starts at a backflow of −1.4 1/min at 2,000 rpm and resulting in a maximum flow of 8 1/min at 10,000 rpm. The contributions of the pump to total flow are −49% and 33% for 2,000 and 4,000 rpm,



120 80 40

100 80 60

Pump Flow Aortic Flow

6 4 2 0 -2 2

4

6

8

RPM (x103)

10

PVA (x103 mmHg.ml)

40

8

Flow (l/min)

Stress (mmHg)

max LV Press Aortic Press

160

EW, PE (x103 mmHg.ml)

Pressure (mmHg)

468

EW PE

3 2 1

2

4

6

8

RPM (x103)

10

5 4 3 2 1 2

4

6

8

10

RPM (x103)

Fig. 21. Summary of the effect of various (constant) pump speeds. Aortic pressure was calculated at systole. Stress is the maximal meridional wall stress; EW is external work; PE is potential energy, and PVA is the PV area of the calculated PV loop.

respectively, and 100% for the higher pump speeds. The increasing pump speed also results in improved unloading of the LV, which can be concluded from the decreasing wall stress and PV area, which indicates a lower oxygen consumption of the myocardium. 3.2. Pulsatile mode simulations The simulations with the time-varying pump speed show that timing has an important effect on the hemodynamics and ventricular energetics (see Fig. 22). The thin horizontal lines in Fig. 22 represent the results from the constant speed simulations at 4,000 rpm, which is our point of comparison. It can be deduced from Fig. 22 that pump flow and aortic flow respond oppositely to the phase shift, resulting in a total mean systemic flow that is fluctuating around the mean total flow that would be obtained if the pump were running at a constant speed of 4,000 rpm. For a phase shift range of 20–50%, systolic and mean aortic pressure for pulsatile operation are decreased from the case of constant speed, while diastolic aortic pressure is reduced in a slightly higher range. Consequently, the pulse pressure in the pulsatile simulations is only higher than in the constant speed simulations for shifts of 50% and above. For the wall stress,



Flows aortic flow

pump flow total flow

4

2

Aortic pressures

100

pressure (mmHg)

flow (l/min)

6

469

75

50

systolic pressure diastolic pressure

0 0

20

40

60

80

0

100

phase shift (% of full cycle)

PVA

(mmHg) m,max

2000

0 0

20

40

60

80


100

60

80

100

max. LV wall stress

100

external work

4000

40


Energetic parameters

6000

20

75 50 25 0 0

20

40

60

80

100


Fig. 22. Results of the pulsatile mode simulations as a function of the phase shift (thick lines). The thin horizontal lines represent the values calculated with a constant pump speed of 4,000 rpm. LV: left ventricular; PVA: PV area.

40% phase shift is the limit up to where the values obtained with the pulsatile pump speed stay noticeably lower than the constant speed value. Phase shifts below 30% yield lower PVA values and less left ventricular external work than at the constant speed of 4,000 rpm. Higher shifts result in external work that is markedly higher than the constant speed value. The maximal decrease in PVA that can be obtained with a well-timed pulsatile pattern is 14% of the constant speed value. The maximum wall stress σm,max can be reduced by 16%. Consequently, these data demonstrate that by modulating an average pump speed and selecting the proper phase shift, the left ventricle can be unloaded without increasing the aortic pressure. This is in contrast to continuous speed operation, where unloading can only be achieved at


470


high pump speeds that also cause an increase in arterial pressure. The use of modulated pump speed only causes limited changes in total flow and therefore these simulations suggest that a pulsatile pattern as used in this study can maintain perfusion while providing superior preload and afterload reduction to the LV. The results presented in Fig. 22 indicate that a phase shift of approximately 20% yields most favorable results for the energetic parameters, which supports the counterpulsation idea. Figure 23, shows the results of a 20% phase shift in more detail. The influence of the pulsatile pattern is reflected in the pump flow, where it has to be considered that the pressure head over the pump varies due to the cardiac pumping action.

Pump flow

LV PV-loop 6

continuous pulsatile

80

Qpump (l/min)

pressure (mmHg)

100

60 40 20

20

40

60

80

100

2 0 -2 0.0

0 0

4

120

0.1

0.2

LV & Aortic pressure

0.5

0.6

LV wall stress 100

80

80

(mmHg)

100

60 40

m

pressure (mmHg)

0.4

time (s)

volume (ml)

20 0 0.0

0.3

60 40 20

0.1

0.2

0.3

0.4

time (s)

0.5

0.6

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

time (s)

Fig. 23. Comparison of results obtained with a constant pump speed of 4,000 rpm (dotted lines) versus a pulsatile pump pattern with 20% phase shift and an average speed of 4,000 rpm (solid lines). The gray lines represent left ventricular (LV) pressure, while the black lines in the same graph show aortic pressure.



471

Since we considered sheep data, the simulations cannot simply be translated to clinical cases and as such, our results have to be seen as indicative, rather than absolute. This is particularly true for the calculated wall stress, where it is assumed that the wall thickness is always 10 mm at end systole. Nevertheless, this model gives a good view on the influence of unloading and the variations in effective wall stress.

4. Conclusions We used a validated mathematical model to investigate the unloading effect of a rotary blood pump (Medos-HIA microdiagonal pump). This effect was evaluated by hemodynamic data and mechano-energetic parameters derived from left ventricular PV data. Our results indicate that for a rotary blood pump running at constant speed, complete unloading of the LV can only be achieved at high pump speeds. A more efficient unloading is achieved by running the pump in a pulsatile mode in a counterpulsation pattern. This operation mode can decrease LV external work, wall stress, and oxygen consumption without impeding the flow. It is thus beneficial to have a low pump speed during ventricular systole and high pump speed during ventricular diastole.

References 1. H. Suga, and K. Sagawa, Instantaneous pressure–volume relationships and their ratio in the excised, supported canine left ventricle, Circulation, Research 35, 1 (1974) 117–126. 2. H. Senzaki, C. H. Chen, and D. A. Kass, Single-beat estimation of end-systolic pressure–volume relation in humans. A new method with the potential for noninvasive application, Circulation 94, 10 (1996) 2497–2506. 3. N. B. Ingels, Jr, G. T. Daughters, S. D. Nikolic, A. DeAnda, M. R. Moon, A. F. Bolger et al., Left ventricular diastolic suction with zero left atrial pressure in open-chest dogs, American Journal of Physiology 270, 4, Pt 2 (1996) H1217–H1224. 4. D. Burkhoff, S. Sugiura, D. T. Yue, and K. Sagawa, Contractility-dependent curvilinearity of end-systolic pressure–volume relations, American Journal of Physiology 252, 6, Pt 2 (1987) H1218–H1227. 5. P. Segers, N. Stergiopulos, and N. Westerhof, Quantification of the contribution of cardiac and arterial remodeling to hypertension, Hypertension 36, 5 (2000) 760–765. 6. N. Stergiopulos, J. J. Meister, and N. Westerhof, Determinants of stroke volume and systolic and diastolic aortic pressure, American Journal of Physiology 270 6, Pt 2 (1996) H2050–H2059.


472


7. P. Segers, P. Steendijk, N. Stergiopulos, and N. Westerhof, Predicting systolic and diastolic aortic pressure and stroke volume in the intact sheep, Journal of Biomechanics 34, 1 (2001) 41–50. 8. W. L. Maughan, A. A. Shoukas, K. Sagawa, and M. L. Weisfeldt, Instantaneous pressure–volume relationship of the canine right ventricle, Circulation Research 44, 3 (1979) 309–315. 9. R. G. McKay, J. M. Aroesty, G. V. Heller, H. D. Royal, S. E. Warren, and W. Grossman, Assessment of the end-systolic pressure–volume relationship in human beings with the use of a time-varying elastance model, Circulation 74, 1 (1986) 97–104. 10. K. Sagawa, The end-systolic pressure–volume relation of the ventricle: definition, modifications and clinical use, Circulation 63, 6 (1981) 1223–1227. 11. J. D. Thomas, J. Zhou, N. Greenberg, G. Bibawy, P. M. McCarthy, and P. M. Vandervoort, Physical and physiological determinants of pulmonary venous flow: numerical analysis, American Journal of Physiology 272, 5, Pt 2 (1997) H2453–H2465. 12. J. W. Clark, R. C. Pruett, D. L. Baldridge, R. Srinivasan, H. M. Bourland, J. S. Cole et al., Functional model for the characterisation of the ventricular mechanics of the human subject, Medical and Biological Engineering and Computing 15, 4 (1977) 335–348. 13. M. Vollkron, H. Schima, L. Huber, and G. Wieselthaler, Interaction of the cardiovascular system with an implanted rotary assist device: simulation study with a refined computer model, Artificial Organs 26, 4 (2002) 349–359. 14. Y. Sun, Modeling the dynamic interaction between left ventricle and intraaortic balloon pump, American Journal of Physiology 261, 4, Pt 2 (1991) H1300–H1311. 15. W. P. Santamore, G. D. Meier, and A. A. Bove, Effects of hemodynamic alterations on wall motion in the canine right ventricle, American Journal of Physiology 236, 2 (1979) H254–H262. 16. J. C. Grignola, J. Pontet, M. Vallarino, and F. Gines, The characteristics proper of the cardiac cycle phases of the right ventricle, Rev Esp Cardiol 52, 1 (1999) 37–42. 17. W. C. Hunter, J. S. Janicki, K. T. Weber, and A. Noordergraaf, Systolic mechanical properties of the left ventricle. Effects of volume and contractile state, Circulation Research 52, 3 (1983) 319–327. 18. K. B. Campbell, J. A. Ringo, G. G. Knowlen, R. D. Kirkpatrick, and S. L. Schmidt, Validation of optional elastance-resistance left ventricle pump models, American Journal of Physiology 251 2, Pt 2 (1986) H382–H397. 19. S. G. Shroff, J. S. Janicki, and K. T. Weber, Left ventricular systolic dynamics in terms of its chamber mechanical properties, American Journal of Physiology 245, 1 (1983) H110–H124. 20. J. C. Grignola, and F. Ginés, Left ventricle-like mechanical properties of the right ventricle due to an acute afterload increase, 1st virtual congress of cardiology, Facultad de Medicina, Universidad de la Rep´ ublica MontevideoUruguay, 2003.



473

21. J. L. Palladino, S. Y. Rabbany, J. P. Mulier, and A. Noordergraaf, A perspective on myocardial contractility, Technology and Health Care 5, 1–2 (1997) 135–144. 22. O. Frank, Die Grundform des arterielen Pulses erste Abhandlung: mathematische Analyse, Z Biol 37 (1899) 483–526. 23. N. Westerhof, Analog studies of human systemic arterial hemodynamics, Thesis, Philadelphia, PA: University of Pennsylvania (1968). 24. M. K. Sharp, and R. K. Dharmalingham, Development of a hydraulic model of the human systemic circulation, ASAIO Journal 45, 6 (1999) 535–540. 25. N. Stergiopulos, B. E. Westerhof, and N. Westerhof, Total arterial inertance as the fourth element of the windkessel model, American Journal of Physiology 276 (1999) H81–H88. 26. M. K. Sharp, G. M. Pantalos, L. Minich, L. Y. Tani, E. C. McGough, and J. A. Hawkins, Aortic input impedance in infants and children, Journal of Applied Physiology 88, 6 (2000) 2227–2239. 27. B. Lambermont, P. Gerard, O. Detry, P. Kolh, P. Potty, and J. O. Defraigne et al., Comparison between three- and four-element Windkessel models to characterize vascular properties of pulmonary circulation, Archives of Physiology and Biochemistry 105, 7 (1997) 625–632. 28. S. Toy, J. Melbin, and A. Noordergraaf, Reduced models of arterial systems, IEEE Transactions on Biomedical Engineering 32, 2 (1985) 174–176. 29. T. Matsumoto, Study of parameters for evaluation of the effect of assist pump on the natural heart, PhD Thesis, Sapporo: Hokkaido University, 1987. 30. M. G. Taylor, An approach to an analysis of the arterial pulse wave. I. Oscillations in an attenuating line, Physics in Medicine and Biology 1, 3 (1957), 258–269. 31. R. Fogliardi, R. Burattini, and K. B. Campbell, Identification and physiological relevance of an exponentially tapered tube model of canine descending aortic circulation, Medical Engineering and Physics 19, 3 (1957) 201–211. 32. R. Burattini, and G. Gnudi, Computer identification of models for the arterial tree input impedance: comparison between two new simple models and first experimental results, Medical and Biological Engineering and Computing 20, 2 (1982) 134–144. 33. R. Burattini, G. G. Knowlen, and K. B. Campbell, Two arterial effective reflecting sites may appear as one to the heart, Circulation Research 68, 1 (1991) 85–99. 34. S. G. Shroff, D. S. Berger, C. Korcarz, R. M. Lang, R. H. Marcus, and D. E. Miller, Physiological relevance of T-tube model parameters with emphasis on arterial compliances, American Journal of Physiology 269, 1, Pt 2 (1995) H365–H374. 35. N. Westerhof, F. Bosman, C. J. De Vries, and A. Noordergraaf, Analog studies of the human systemic arterial tree, Journal of Biomechanics 2 (1969) 121–143. 36. A. P. Avolio, Multi-branched model of the human arterial system, Medical and Biological Engineering and Computing 18, 6 (1980) 709–718.


474


37. J. Zhou, G. P. Armstrong, A. L. Medvedev, W. A. Smith, L. A. Golding, and J. D. Thomas, Numeric modeling of the cardiovascular system with a left ventricular assist device, ASAIO Journal 45, 1 (1999) 83–89. 38. G. A. Giridharan, M. Skliar, D. B. Olsen, and G. M. Pantalos, Modeling and control of a brushless DC axial flow ventricular assist device, ASAIO Journal 48, 3 (2002) 272–289. 39. Y. C. Yu, J. R. Boston, M. A. Simaan, P. J. Miller, and J. F. Antaki, Pressure–volume relationship of a pulsatile blood pump for ventricular assist device development, ASAIO Journal 47, 3 (2001) 293–301. 40. P. Segers, N. Stergiopulos, J. J. Schreuder, B. E. Westerhof, and N. Westerhof, Left ventricular wall stress normalization in chronic pressureoverloaded heart: a mathematical model study, American Journal of Physiology — Heart and Circulatory Physiology 279, 3 (2000) H1120–H1127. 41. M. R. Wolff, P. P. de Tombe, Y. Harasawa, D. Burkhoff, S. Bier, and W. C. Hunter, et al., Alterations in left ventricular mechanics, energetics, and contractile reserve in experimental heart failure, Circulation Research 70, 3 (1992) 516–529. 42. D. A. Kass, and W. L. Maughan, From “Emax” to pressure–volume relations: a broader view, Circulation 77, 6 (1988) 1203–1212. 43. T. Nishida, B. Meyns, M. Zietkiewicz, B. Perek, Z. Xia, C. Goebel et al., The effect of sudden failure of a rotary blood pump on left ventricular performance in normal and failing hearts, Artificial Organs 24, 11 (2000) 893–898. 44. Y. Ishida, J. S. Meisner, K. Tsujioka, J. I. Gallo, C. Yoran, R. W. Frater et al., Left ventricular filling dynamics: influence of left ventricular relaxation and left atrial pressure, Circulation 74, 1 (1986) 187–196. 45. Y. Sun, B. J. Sjoberg, P. Ask, D. Loyd, and B. Wranne, Mathematical model that characterizes transmitral and pulmonary venous flow velocity patterns, American Journal of Physiology 268 1, Pt 2 (1995) H476–H489. 46. J. Alexander, Jr, K. Sunagawa, N. Chang, K. Sagawa, Instantaneous pressure–volume relation of the ejecting canine left atrium, Circulation Research 61, 2 (1987) 209–219. 47. T. Arts, P. H. Bovendeerd, F. W. Prinzen, and R.S. Reneman, Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall, Biophysics Journal 59, 1 (1991) 93–102. 48. L. Teichholz, T. Kreulen, M. Herman, and R. Gorlin, Problems in echocardiographic volume determinations: echocardiographic–angiographic correlations in the presence of absence of asynergy American Journal of Cardiology 37, 1 (1976) 7–11. 49. F. C. Yin, Ventricular wall stress, Circulation Research 49, 4 (1981) 829–842. 50. O. H. Frazier, First use of an untethered, vented electric left ventricular assist device for long-term support, Circulation 89, 6 (1994) 2908–2914. 51. R. Hetzer, J. H. Muller, Y. Weng, R. Meyer, and M. Dandel, Bridging to recovery, Annals of Thoracic Surgery 71, 3 Suppl (2001) S109–S113; discussion S114–S115.



475

52. A. Barbone, J. W. Holmes, P. M. Heerdt, A. H. The, Y. Naka, and N. Joshi, et al., Comparison of right and left ventricular responses to left ventricular assist device support in patients with severe heart failure: a primary role of mechanical unloading underlying reverse remodeling, Circulation 104, 6 (2001) 670–675. 53. S. Vandenberghe, P. Segers, B. Meyns, and P. R. Verdonck, Effect of rotary blood pump failure on left ventricular energetics assessed by mathematical modeling, Artificial Organs 26, 12 (2002) 1032–1039. 54. H. Suga, K. Sagawa, and A. A. Shoukas, Load independence of the instantaneous pressure–volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio, Circulation Research 32, 3 (1973) 314–322. 55. S. Vandenberghe, P. Segers, B. Meyns, and P. Verdonck, Hydrodynamic characterisation of ventricular assist devices, International Journal of Artificial Organs 24, 7 (2001) 470–477. 56. H. Ishihara, M. Yokota, T. Sobue, and H. Saito, Relation between ventriculoarterial coupling and myocardial energetics in patients with idiopathic dilated cardiomyopathy, Journal of American College of Cardiology 23, 2 (1994) 406–416. 57. H. Takaoka, M. Takeuchi, M. Odake, and M. Yokoyama, Assessment of myocardial oxygen consumption (Vo2) and systolic pressure–volume area (PVA) in human hearts, European Heart Journal 13, Suppl E ( 1992) 85–90. 58. H. L. Falsetti, R. E. Mates, C. Grant, D. G. Greene, and I. L. Bunnell, Left ventricular wall stress calculated from one-plane cineangiography, Circulation Research 26, 1 (1970) 71–83. 59. J. Linneweber, K. Nonaka, T. Takano, S. Kawahito, S. Schulte-Eistrup, T. Motomura et al., Hemodynamic exercise response in calves with an implantable biventricular centrifugal blood pump, Artificial Organs 25, 12 (2001) 1018–1021. 60. E. Tayama, Y. Niimi, Y. Takami, Y. Ohashi, G. Ohtsuka, J. A. Glueck et al., Hemolysis test of a centrifugal pump in a pulsatile mode: the effect of pulse rate and RPM variance, Artificial Organs 21, 12 (1997) 1284–1287. 61. S. S. Wang, S. H. Chu, N. K. Chou, and K. X. Qian, The pulsatile impeller pump for left ventricular assist, Artificial Organs 20, 12 (1996) 1310–1313.




TISSUE ENGINEERING FOR THE INFARCTED HEART: CELL TRANSPLANTATION THERAPY GENEVIEVE M. Y. TAN∗,‡,‡‡ , LEI YE# , WINSTON S. N. SHIM∗,‡ , HUSNAIN KH. HAIDER∗∗ , ALEXIS B. C. HENG† , TERRANCE CHUA§ , TIAN HAI KOH , and EUGENE K. W. SIM∗,¶,‡,††,## ∗Department of Surgery, Faculty of Medicine National University of Singapore, Singapore 117597 ‡‡genevieve

tan [email protected]

##[email protected] †Dean’s ‡Research

Office, Faculty of Dentistry, National University of Singapore Singapore 117597 and Development Unit, National Heart Centre, Singapore 169609 §Department

¶Department Medical

of Cardiology, National Heart Centre Singapore 168752

of Cardiothoracic Surgery, National Heart Centre Singapore 168752

Director’s Office, National Heart Centre, Singapore 168752 ∗∗Department

of Pathology, Faculty of Medicine University of Cincinnati, Ohio, USA

†† Gleneagles

JPMC Sdn Bhd, Brunei Darussalam

# National

University Medical Institutes Faculty of Medicine, National University of Singapore, Singapore 117597

1. Myoblast Transplantation 1.1. Myoblast transplantation: substitute of cardiomyocytes Skeletal muscle, unlike heart muscle, has the ability to regenerate and repair injury due to the presence of myoblasts as skeletal muscle cells can proliferate and differentiate when activated in response to muscle injury. Skeletal myoblasts are mononucleated unipotent progenitor cells that can be expanded in vitro. Skeletal myoblasts can be genetically modified in vitro to deliver angiogenic cytokines and growth factors to encourage angiomyogenesis. Animal studies have shown that grafted myoblasts form myotubes in the myocardium, and eventually mature to become well-formed myofibers with contractile apparatus. 477


478

Genevieve M. Y. Tan et al.

1.2. Animal study of myoblast transplantation for cardiac repair It was in 1989 that Kao et al.1 first demonstrated the feasibility of skeletal myoblast transplantation for cardiac repair. Myoblasts were injected into the center of the scar using cryo-injured myocardial model. Histological studies demonstrated viable myofibers in the scar the tissue. Marelli and colleagues2 further developed this strategy. In the subsequent 5 years, many researches confirmed this finding.3−10 1.3. Efficacy of myoblast transplantation for cardiac repair A landmark study by Taylor et al.11 first demonstrated the effectiveness of skeletal myoblast transplantation for the improvement of cardiac function in a rabbit heart model of cryo-injury. Furthermore, skeletal myoblast transplantation improved diastolic compliance prior to systolic performance.12 The improved diastolic function was related to increased regional strain, decreased dynamic stiffness, unaffected static stiffness, and reversed diastolic creep. Both regional13−20 and global17,21,22 ventricular systolic function can be improved by myoblast transplantation. The systolic function improvement was shown as improved developing pressure21,23 and limited left ventricular ejection fraction (LVEF) deterioration.18 Myoblast transplantation delimits or inhibits ventricular remodeling. Myoblast transplantation decreases left ventricular (LV) diastolic dimension,17,19,20,24 increases myocardial wall thickness,19,20,22 and minimizes global ventricular dilation.18,22 The efficacy of skeletal myoblast transplantation is at least as efficient as neonatal cardiomyocytes or bone marrow cells for cardiac repair.25 Scorsin et al.15 showed that LVEF markedly increased in animals groups receiving fetal cardiomyocytes or neonatal myoblasts at 1 month after treatment. Thompson et al.20 had documented similar findings. Autologous myoblast transplantation improved regional heart function to a level similar to that of bone marrow transplantation. Both the cell transplantation procedures resulted in a similar degree of improvement in a derivative of stroke work. 1.4. Skeletal myoblasts in human studies 1.4.1. The European experience The first clinical application was performed by Menasche and colleagues as an adjunct to coronary artery bypass grafting (CABG) surgery. In this


Tissue Engineering for the Infarcted Heart

479

procedure, they used cultured autologous skeletal myoblasts in a 72-year old patient. This patient was in New York Heart Association (NYHA) class III and presented with a mean LVEF of 21±2% by echocardiography. Follow-up at 5 months showed that the patient had an improved LVEF (∼ 30%)was in NYHA class II. The procedure was performed without any complications. The promising results of this study paved the way for phase I clinical trials of myoblast transplantation. Since the first landmark study, Menasche and colleagues have reported nine more patients as a part of phase I trials.27 The patients (mean age 60±3 years) were diagnosed with severe LV dysfunction (LVEF = 35%). An average of 8.74 × 108 autologous myoblasts were injected into the akinetic, nonrevascularizable, and nonviable scar as assessed by dobutamine echocardiography and positron emission tomography (PET). The cell transplantation procedure was uneventfull, and without any perioperative complications. The nine operative survivors were subjected to follow-up assessments for up to 8 months. The results showed an improvement in NYHA class for all the patients, from NYHA class 2.7 ± 0.2 to 1.6 ± 0.1 (p < 0.02), in parallel with documented improvement in LVEF from 24 ± 1% to 34 ± 1% (p = 0.02). The improvement in cardiac function restoration was observed during the first 3 months post-transplantation. This improvement has not deteriorated since. These positive findings have resulted in a randomized multicenter trial. Siminiak and colleagues28 in Poland documented autologous myoblast transplantation for the treatment of post-myocardial infarction (post-MI) depressed myocardial function. The procedure was carried out as an adjunct to CABG in a 55-year-old female patient who suffered a transmural anterior wall MI. Functional echocardiographic functional assessment revealed extensive akinesia in the apex and apical segments of the anterior wall and the septum. The absence of the viability in the akinetic region was shown by low-dose dobutamine echocardiography. Echocardiography at 4 weeks after myoblast transplantation revealed significant improvement in segmental contractility, especially in the apical region. Chachques and colleagues29 initiated a phase I study for autologous myoblast transplantation as an adjunct to route CABG in four male and one female patient in NYHA functional class III. The study has been extended to include 18 patients (90% male) with an average NYHA class 2.6. The patients were diagnosed with impaired LV function (LVEF = 32 ± 5%) and LV posterior wall postischemic scars (akinetic and absence of metabolic viability). The propagation of the cells was carried out in complete human


480

Genevieve M. Y. Tan et al.

medium, using the patients’ own sera. There were no complications related to the cell transplantation procedure. The patients showed uneventful recovery and were discharged from the intensive care unit (ICU) 2 days after surgery. No cardiac arrhythmias were observed at follow-up (mean 9 ± 3 months). Echocardiographic studies showed an improvement in regional wall motion (from akinetic cardiomyopathy to hypokinetic ventricular wall). The infarct scar size appeared to be significantly reduced from 21 ± 5 to 8 ± 3 cm2 (p < 0.05). Myocardial viability tests showed regenerating nodes, with patients moving from heart failure class 2.6 to 1.3. In addition to the above-mentioned phase I studies, clinical myoblast transplantation studies are under way in Spain (14 cases), the Netherlands (13 cases), Germany (D¨ usseldorf and Rostock), and Italy (5 patients).

1.4.2. The American experience In addition to myoblast transplantation as an adjunct to CABG, the procedure has also been carried out in association with left-ventricular-assist device (LVAD) implantation as the first part of a multicenter trial sponsored by Diacrin, Inc., Massachusetts.30 Five patients (median age 60 years) with a history of ischemic cardiomyopathy were selected for the study. These patients were on the waiting list for heart transplantation and were to receive an LVAD implantation as a bridge-to-transplantation. They were on maximal inotropic support with a median LVEF of 15%. A total of 300 × 106 cells were injected into each patient, with the exception of one patient who required urgent LVAD implantation before sufficient cells could be cultured. Dib and colleagues31 at the Arizona Heart Institute presented a study involving skeletal myoblast transplantation into the scarred region of the heart in 16 patients as an adjunct to CABG (n = 11) or LVAD (n = 5). The study was a part of multicenter industry-sponsored trial from Diacrin and involved a dose–escalation study. Autologous myoblasts were purified and proliferated in vitro from each patient’s thigh muscle biopsy samples. A total of 10 million to 300 million cells were injected into the scar region. There were no intraoperative or postoperative complications related to the procedure. Follow-up evaluation by magnetic resonance imaging (MRI), echocardiography, and PET showed successful survival of the transplanted cells within the heart at the site of graft, which registered improvement in LVEF from 21% to 29% at 3 weeks follow-up.


Tissue Engineering for the Infarcted Heart

481

Bioheart has initiated the multicenter Food and Drug Administration (FDA) approved MYOHEART Clinical Trial to be carried out at Mount Sinai Hospital (New York), Duke University and the American Cardiovascular Research Institute (Atlanta). Similarly, Genzyme has announced a MAGIC Multicenter Trial in Europe and USA. Other institutes involved in myoblast transplantation include the University of California at Los Angles, Temple University at Philadelphia, the University of Michigan at Ann Arbor, and the Cleveland Clinic.

1.4.3. The Asian experience Our group in Singapore carried out the first autologous myoblast transplantation on a beating heart as a part of a phase I clinical study. A 55-year old male patient presented with acute MI. The apex, anterior wall, and septum of the LV were akinetic and the LVEF was 31%. After informed consent and as part of an institutional review board-cleared clinical trial, the patient received 3.78 × 108 autologous myoblasts at 20 different sites in and around the infarct region during CABG. The cells were >98% pure for desmin expression, with >99% viability at the time of injection. A 6-month followup revealed a perfusion defect involving the anterior wall and the apex with partial reversibility on 99m Tc-tetrafosamine nuclear scan. These findings were in agreement with the previously published reports and demonstrated the safety, viability, and benefit of autologous myoblast transplantation as an adjunct to off-pump CABG. Although more patients need to be evaluated, the benefits of reduced risks associated with off-pump CABG cardiopulmonary bypass deem this technique attractive for the delivery of cells for transplantation. More recently, Zhang and colleagues32 from Nanjing Medical University, People’s Republic of China, have reported a phase I study including three patients with a history of coronary heart disease. The patients underwent CABG and implantation of autologous myoblasts. Myoblasts were isolated from muscle biopsies of the right vastus lateralis muscle after enzymatic treatment. The heart was placed under hypothermic cardioplegic arrest, and 4 ml of cell suspension divided into approximately 40 doses was injected into the ventricular wall of the ischemic area in

Cardiac perfusion and pumping engineering

On Bypass: Advanced Perfusion Techniques (Current Cardiac Surgery)

Pumping Iron

Pumping Nylon

Practical Introduction to Pumping Technology

Capture Pumping Technology

The Practical Pumping Handbook

The Practical Pumping Handbook

The Practical Pumping Handbook

Clinical MR Neuroimaging: Diffusion, Perfusion and Spectroscopy

Cardiac Arrest

Cardiac Arrest

Cardiac Transplantation

Cardiac Mapping

Cardiac Markers

Cardiac Markers

Cardiac Cell and Gene Transfer

Cardiac Arrest

Cardiac Electrophysiology Methods and Models

Practical Introduction to Pumping Technology

Practical Introduction to Pumping Technology

Cardiac Catheterization and Percutaenous Interventions

Management of Cardiac Arrhythmias

Myocardial Perfusion Scintigraphy: From Request to Report

Cardiac Drug Therapy Khan

Cardiac Transplantation (Contemporary Cardiology)

Quantitative Cardiac Electrophysiology

Cardiac perfusion and pumping engineering

On Bypass: Advanced Perfusion Techniques (Current Cardiac Surgery)

Pumping Iron

Pumping Nylon

Practical Introduction to Pumping Technology

Capture Pumping Technology

The Practical Pumping Handbook

The Practical Pumping Handbook

The Practical Pumping Handbook

Clinical MR Neuroimaging: Diffusion, Perfusion and Spectroscopy

Cardiac Arrest

Cardiac Arrest

Cardiac Transplantation

Cardiac Mapping

Cardiac Markers

Cardiac Markers

Cardiac Cell and Gene Transfer

Cardiac Arrest

Cardiac Electrophysiology Methods and Models

Practical Introduction to Pumping Technology

Practical Introduction to Pumping Technology

Cardiac Catheterization and Percutaenous Interventions

Management of Cardiac Arrhythmias

Myocardial Perfusion Scintigraphy: From Request to Report

Cardiac Drug Therapy Khan

Cardiac Transplantation (Contemporary Cardiology)

Quantitative Cardiac Electrophysiology

Recommend Documents