Standard Handbook of Biomedical Engineering and Design

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Editor in Chief

MCGRAW-HILL

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Library of Congress Cataloging-in-Publication Data Standard handbook of biomedical engineering and design / Myer Kutz, editor in chief. p. cm. Includes index. ISBN 0-07-135637-1 1. Biomedical engineering—Handbooks, manuals, etc. 2. Medical instruments and apparatus—Design and construction—Handbooks, manuals, etc. I. Kutz, Myer. R856.15.S73 610'.28—dc21

2003 2002029356

Copyright © 2003 by The McGraw-Hill Companies, Inc. AU rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 6 7 8 9 0

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ISBN 0-07-135637-1 The sponsoring editor for this book was Kenneth P. McCombs, the editing supervisor was David E. Fogarty, and the production supervisor was Sherri Souffrance. It was set in Times Roman by Progressive Information Technologies, Inc. Printed and bound by R. R. Donnelley & Sons Company. This book is printed on acid-free paper. McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please write to the Director of Special Sales, Professional Publishing, McGraw-Hill, Two Penn Plaza, New York, NY 10121-2298. Or contact your local bookstore.

Information contained in this work has been obtained by The McGraw-Hill Companies, Inc. ("McGraw-Hill") from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

For Arlene, forever

C O N T R I B U T O R S

Ronald S. Adrezin

College of Engineering, University of Hartford, West Hartford, Conn, (CHAPS. 7, 31)

Ronald E. Barr Department of Mechanical Engineering, University of Texas, Austin, Texas (CHAP. 6) Christopher Batich

Materials Science and Engineering Department, University of Florida, Gainesville, FIa.

(CHAP. 11)

Ravi V. Bellamkonda

Department ofBiomedical Engineering, Case Western Reserve University, Cleveland,

Ohio (CHAP. 16)

Anthony J. Brammer National Research Council, Ottawa, Ontario, Canada and University of Connecticut Health Center, Farmington, Conn. (CHAP. 10) Thomas S. Buchanan Albert M. Cook

Center for Biomedical Research, University of Delaware, Newark, Del. (CHAP. 5)

Faculty of Rehabilitation Medicine, University of Alberta, Edmonton, Alberta, Canada

(CHAP. 30)

John Michael Currie Alfred M. Dolan

SmithGroup, Inc., Washington, D.C. (CHAP. 38)

Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto,

Canada (CHAP. 36)

David Elad

Department of Biomedical Engineering, Tel Aviv University, Tel Aviv, Israel (CHAP. 3)

Shmuel Einav

Department of Biomedical Engineering, Tel Aviv University, Tel Aviv, Israel (CHAP. 3)

Laurie L. Fajardo The Russell Morgan Department of Radiology and Radiological Science, Johns Hopkins Medical Institutions, Baltimore, Md. (CHAP. 28) Kenneth L. Gage

Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, Penn.

(CHAP. 20)

Michele J. Grimm James B. Grotberg

Bioengineering Center, Wayne State University, Detroit, Mich. (CHAP. 15) Department of Biomedical Engineering, University of Michigan, Ann Arbor, Mich.

(CHAP. 4)

Michael N. Helmus

Boston Scientific Corporation, Boston, Mass. (CHAP. 14)

Arif lftekhar Department ofBiomedical Engineering and Gleason Laboratory, University of Minnesota, Minneapolis, Minn. (CHAP. 12) Bhaskara R. Jasti Department of Pharmaceutics and Medicinal Chemistry, University of the Pacific, Stockton, Calif. (CHAP. 22) Arthur T. Johnson

Department of Biological Resources Engineering, University of Maryland, College Park,

Md. (CHAP. 21)

Leo Joskowicz Computer-Aided Surgery and Medical Image Processing Laboratory, The Hebrew University of Jerusalem, Israel (CHAP. 29) Tony M. Keaveny Departments of Mechanical Engineering and Bioengineering, University of California, Berkeley, Calif, and Department of Orthopaedic Surgery, University of California, San Francisco, Calif. (CHAP. 8)

David H. Kohn

Department of Biomedical Engineering, University of Michigan, Ann Arbor, Mich.

(CHAP. 13)

Patrick Leamy

Materials Science and Engineering Department, University of Florida, Gainesville, FIa.

(CHAP. 11)

Xiaoling Li Department of Pharmaceutics and Medicinal Chemistry, University of the Pacific, Stockton, Calif. (CHAP. 22)

Mark T. Madsen Kurt. T. Manal

Department of Radiology, University of Iowa, Iowa City, Iowa (CHAP. 27) Center for Biomedical Research, University of Delaware, Newark, Del. (CHAP. 5)

Nancy J. Meilander

Department of Biomedical Engineering, Case Western Reserve University, Cleveland,

Ohio (CHAP. 16)

Elise F. Morgan

Department of Mechanical Engineering, University of California, Berkeley, Calif.

(CHAP. 8)

Jit Muthuswamy

Department of Bioengineering, Arizona State University, Tempe, Ariz. (CHAP. 18)

Patrick J. Nolan

DDL, Inc., Eden Prairie, Minn. (CHAP. 23)

Michael D. Nowak Department of Civil and Environmental Engineering, University of Hartford, West Hartford, Conn. (CHAP. 9) James P. O'Leary Marcus Pandy

Department of Mechanical Engineering, Tufts University, Medford, Mass. (CHAP. 19)

Department of Biomedical Engineering, University of Texas, Austin, Texas (CHAP. 6)

Donald R. Peterson Narender P. Reddy

University of Connecticut Health Center, Farmington, Conn. (CHAPS. 7, 10) Department of Biomedical Engineering, University of Akron, Akron, Ohio (CHAP. 1)

David J. Reinkensmeyer Irvine, Calif (CHAP. 35) Peter Rockett

Department of Mechanical and Aerospace Engineering, University of California,

Department of Engineering Science, Oxford University, Oxford, U.K. (CHAP. 26)

Blair A. Rowley

Biomedical, Industrial and Human Factors Engineering, Wright State University, Dayton,

Ohio (CHAP. 34)

Daniel J. Schaefer David M. Shade

GE Medical Systems, Waukesha, Wise. (CHAP. 24)

Johns Hopkins Pulmonary Laboratory, Johns Hopkins School of Medicine, Baltimore, Md.

(CHAP. 21)

Steve I. Shen

Department of Pharmaceutics and Medicinal Chemistry, University of the Pacific, Stockton,

Calif. (CHAP. 22)

M. Barbara Silver-Thorn

Department of Biomedical Engineering, Marquette University, Milwaukee, Wise.

(CHAP. 33)

John Smith

Hospital for Sick Children, Toronto, Ontario, Canada (CHAP. 37)

Roger W. Snyder Ira Tackel

LVAD Technology, Inc., Detroit, Mich. (CHAP. 14)

Department of Biomedical Instrumentation, Thomas Jefferson University Hospital, Philadelphia,

Penn. (CHAP. 39)

Russell Taylor

Computer Science Department, Johns Hopkins University, Baltimore, Md. (CHAP. 29)

Kai E. Thomenius

Ultrasound Program, GE Corporate Research and Development, Schenectady, NY.

(CHAP. 25)

Bruce C. Towe

College of Engineering, Arizona State University, Tempe, Ariz. (CHAP. 17)

William R. Wagner Departments of Surgery, Bioengineering & Chemical Engineering, University of Pittsburgh, Pittsburgh, Penn. (CHAP. 20)

Ge Wang

Department of Radiology, University of Iowa, Iowa City, Iowa (CHAP. 26)

Richard F. ff. Weir Prosthetics Research Laboratory and Rehabilitation Engineering Research Center, Northwestern University, Chicago, III. (CHAP. 32) Mark B. Williams Oscar C. Yeh Liang Zhu

Department of Radiology, University of Virginia, Charlottesville, Va. (CHAP. 28)

Department of Mechanical Engineering, University of California, Berkeley, Calif, (CHAP. 8)

Department Mechanical Engineering, University of Maryland, Baltimore County, Md. (CHAP. 2)

P R E F A C E

How do important medical advances that change the quality of life come about? Sometimes, to be sure, they can result from the inspiration and effort of physicians or biologists working in remote, exotic locations or organic chemists working in the well-appointed laboratories of pharmaceutical companies with enormous research budgets. Occasionally, however, a medical breakthrough happens when someone with an engineering background gets a brilliant idea in less glamorous circumstances. One afternoon in the late 1950s, the story goes, when an electrical engineer named Wilson Greatbatch was building a small oscillator to record heart sounds, he accidentally installed the wrong resistor, and the device began to give off a steady electrical pulse. Greatbatch realized that a small device could regulate the human heart, and in two years he had developed the first implantable cardiac pacemaker, followed later by a corrosion-free lithium battery to power it. In the mid-1980s, Dominick M. Wiktor, a Cranford, New Jersey, engineer, invented the coronary stent after undergoing open heart surgery. You often find that it is someone with an engineer's sensibility—someone who may or may not have engineering training, but does have an engineer's way of looking at, thinking about, and doing things—who not only facilitates medical breakthroughs, but also improves existing healthcare practice. This sensibility, which, I dare say, is associated in people's consciousness more with industrial machines than with the human body, manifests itself in a number of ways. It has a descriptive component, which comes into play, for example, when someone uses the language of mechanical engineering to describe blood flow, how the lungs function, or how the musculoskeletal system moves or reacts to shocks, or when someone uses the language of other traditional engineering disciplines to describe bioelectric phenomena or how an imaging machine works. Medically directed engineer's sensibility also has a design component, which can come into play in a wide variety of medical situations, indeed whenever an individual, or a team, designs a new healthcare application, such as a new cardiovascular or respiratory device, a new imaging machine, a new artificial arm or lower limb, or a new environment for someone with a disability. The engineer's sensibility also comes into play when an individual or team makes an application that already exists work better—when, for example, the unit determines which materials would improve the performance of a prosthetic device, improves a diagnostic or therapeutic technique, reduces the cost of manufacturing a medical device or machine, improves methods for packaging and shipping medical supplies, guides tiny surgical tools into the body, improves the plans for a medical facility, or increases the effectiveness of an organization installing, calibrating, and maintaining equipment in a hospital. Even the improved design of time-released drug capsules can involve an engineer's sensibility. The field that encompasses medically directed engineer's sensibility is, of course, called biomedical engineering. Compared to the traditional engineering disciplines, whose fundamentals and language it employs, this field is new and rather small. Although there are now over 80 academic programs in biomedical engineering in the United States, only 6500 undergraduates were enrolled in the year 2000. Graduate enrollment was just 2500. The U.S. Bureau of Labor Statistics reports total biomedical engineering employment in all industries in the year 2000 at 7221. The bureau estimates this number to rise by 31 percent to 9478 in 2010. The effect this relatively young and small field has on the health and well being of people everywhere, but especially in the industrialized parts of the world that have the wherewithal to fund the field's development and take advantage of its advances, is, in my view, out of proportion to its age and size. Moreover, as the examples provided earlier indicate, the concerns of biomedical en-

gineers are very wide-ranging. In one way or another, they deal with virtually every system and part in the human body. They are involved in all phases of healthcare—measurement and diagnosis, therapy and repair, and patient management and rehabilitation. While the work that biomedical engineers do involves the human body, their work is engineering work. Biomedical engineers, like other engineers in the more traditional disciplines, design, develop, make, and manage. Some work in traditional engineering settings—in laboratories, design departments, on the floors of manufacturing plants—while others deal directly with healthcare clients or are responsible for facilities in hospitals or clinics. Of course, the field of biomedical engineering is not the sole province of practitioners and educators who call themselves biomedical engineers. The field includes people who call themselves mechanical engineers, materials engineers, electrical engineers, optical engineers, or medical physicists, among other names. The entire range of subjects that can be included in biomedical engineering is very broad. Some curricula offer two main tracks: biomechanics and bioinstrumentation. To some degree, then, there is always a need in any publication dealing with the full scope of biomedical engineering to bridge gaps, whether actually existing or merely perceived, such as the gap between the application of mechanical engineering knowledge, skills, and principles from conception to the design, development, analysis, and operation of biomechanical systems and the application of electrical engineering knowledge, skills, and principles to biosensors and bioinstrumentation. The focus in the Standard Handbook of Biomedical Engineering and Design is on engineering design informed by description in engineering language and methodology. For example, the Handbook not only provides engineers with a detailed understanding of how physiological systems function and how body parts—muscle, tissue, bone—are constituted, it also discusses how engineering methodology can be used to deal with systems and parts that need to be assisted, repaired, or replaced. I have sought to produce a practical manual for the biomedical engineer who is seeking to solve a problem, improve a technique, reduce cost, or increase the effectiveness of an organization. The Handbook is not a research monograph, although contributors have properly included lists of applicable references at the ends of their chapters. I want this Handbook to serve as a source of practical advice to the reader, whether he or she is an experienced professional, a newly minted graduate, or even a student at an advanced level. I intend the Handbook to be the first information resource a practicing engineer reaches for when faced with a new problem or opportunity—a place to turn to even before turning to other print sources or to sites on the Internet. (The Handbook is planned to be the core of an Internet-based update or current-awareness service, in which the Handbook chapters would be linked to news items, a bibliographic index of articles in the biomedical engineering research literature, professional societies, academic departments, hospital departments, commercial and government organizations, and a database of technical information useful to biomedical engineers.) So the Handbook is more than a voluminous reference or collection of background readings. In each chapter, the reader should feel that he or she is in the hands of an experienced consultant who is providing sensible advice that can lead to beneficial action and results. I have divided the Handbook into eight parts. Part 1, which contains only a single chapter, is an introductory chapter on applying analytical techniques to biomedical systems. Part 2, which contains nine chapters, is a mechanical engineering domain. It begins with a chapter on the body's thermal behavior, then moves on to two chapters that discuss the mechanical functioning of the cardiovascular and respiratory systems. Six chapters of this part of the Handbook are devoted to analysis of bone and the musculoskeletal system, an area that I have been associated with from a publishing standpoint for a quarter-century, ever since I published David Winter's book on human movement. Part 3 of the Handbook, the domain of materials engineering, contains six chapters. Three deal with classes of biomaterials—biopolymers, composite biomaterials, and bioceramics—and three deal with using biomaterials, in cardiovascular and orthopedic applications, and to promote tissue regeneration. The two chapters in Part 4 of the Handbook are in the electrical engineering domain. They deal with measuring bioelectricity and analyzing biomedical signals, and they serve, in part, as an introduction to Part 5, which contains ten chapters that treat the design of therapeutic devices and diagnostic imaging instrumentation, as well as the design of drug delivery systems and the development

of sterile packaging for medical devices, a deceptively robust and complex subject that can fill entire books on its own. Imaging also plays a role in the single-chapter Part 6 of the Handbook, which covers computer-integrated surgery. The last two parts of the Handbook deal with interactions between biomedical engineering practitioners and both patients and medical institutions. Part 7, which covers rehabilitation engineering, includes chapters that treat not only the design and implementation of artificial limbs, but also ways in which engineers provide environments and assistive devices that improve a person's quality of life. Part 8, the last part of the Handbook, deals with clinical engineering, which can be considered the facilities-planning and management component of biomedical engineering.

Acknowledgments The contributors to this Handbook work mainly in academia and hospitals. Several work in commercial organizations. Most work in the United States and Canada; a few work in Israel. What they all have in common is that what they do is useful and important: they make our lives better. That these busy people were able to find the time to write chapters for this Handbook is nothing short of miraculous. I am indebted to all of them. I am additionally indebted to multiple-chapter contributors Ron Adrezin of the University of Hartford and Don Peterson of the University of Connecticut School of Medicine for helping me organize the biomechanics chapters in the handbook, and for recruiting other contributors, Mike Nowak, a colleague at the University of Hartford and Anthony Brammer, now a colleague at the University of Connecticut Health Center. Also, contributor AIf Dolan of the University of Toronto was especially helpful in recommending contributors for the clinical engineering chapters. Thanks to both of my editors at McGraw-Hill—Linda Ludwig, who signed the Handbook, and Ken McCombs, who saw the project to its completion. Thanks also to Dave Fogarty, who managed McGraw-Hill's editing process smoothly and expeditiously. I want to give the final word to my wife Arlene, the family medical researcher and expert, in recognition of her patience and support throughout the life of this project, from development of the idea, to selection and recruiting of contributors, to receipt and editing of manuscripts: "It is our hope that this Handbook will not only inform and enlighten biomedical engineering students and practitioners in their present pursuits, but also provide a broad and sturdy staircase to facilitate their ascent to heights not yet scaled." MYER KUTZ

Albany, New York

A B O U T T H E EDITOR MYER KUTZ is president of Myer Kutz Associates, Inc., a publishing and information services consulting firm. Formerly, he was vice president for professional scientific and technical publishing at John Wiley and Sons. Mr. Kutz has been a member of the board of trustees of the Online Computer Library Center and was chair of the ASME publications committee. He has a B.S. degree in mechanical engineering from MIT and an M.S. from RPI. Mr. Kutz resides with his wife in Delmar, N. Y.

Contents

Contributors ................................................................................................

ix

Preface .......................................................................................................

xv

About the Editor ..........................................................................................

xviii

Part I. Biomedical Systems Analysis 1.

Modeling and Simulation of Biomedical Systems ...........................................

1.3

1.1

Mathematical Modeling ..................................................................

1.3

1.2

Compartmental Models ..................................................................

1.4

1.3

Electrical Analog Models ................................................................

1.9

1.4

Models with Memory and Models with Time Delay .........................

1.14

1.5

Artificial Neural Network Models ....................................................

1.20

1.6

Mechanical Models ........................................................................

1.23

1.7

Model Validation ............................................................................

1.24

References ..............................................................................................

1.24

Part II. Mechanics of the Human Body 2.

Bioheat Transfer ..............................................................................................

2.3

2.1

Introduction ....................................................................................

2.3

2.2

Fundamental Aspects of Bioheat Transfer .....................................

2.3

2.3

Bioheat Transfer Modeling .............................................................

2.6

2.4

Temperature, Thermal Property, and Blood Flow Measurements ...............................................................................

2.12

Hyperthermia Treatment for Cancers and Tumors .........................

2.19

References ..............................................................................................

2.26

2.5

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vii

viii 3.

4.

5.

6.

Contents Physical and Flow Properties of Blood ...........................................................

3.1

3.1

Physiology of the Circulatory System .............................................

3.1

3.2

Physical Properties of Blood ..........................................................

3.4

3.3

Blood Flow in Arteries ....................................................................

3.5

3.4

Blood Flow in Veins .......................................................................

3.14

3.5

Blood Flow in the Microcirculation ..................................................

3.16

3.6

Blood Flow in the Heart .................................................................

3.18

3.7

Analog Models of Blood Flow .........................................................

3.21

References ..............................................................................................

3.23

Respiratory Mechanics and Gas Exchange ...................................................

4.1

4.1

Anatomy ........................................................................................

4.1

4.2

Mechanics of Breathing .................................................................

4.3

4.3

Ventilation ......................................................................................

4.4

4.4

Elasticity ........................................................................................

4.7

4.5

Ventilation, Perfusion, and Limits ...................................................

4.9

4.6

Airway Flow, Dynamics, and Stability .............................................

4.11

References ..............................................................................................

4.12

Bibliography .............................................................................................

4.14

Biomechanics of Human Movement ...............................................................

5.1

5.1

Why Study Human Movement? ......................................................

5.1

5.2

Forward Versus Inverse Dynamics ................................................

5.2

5.3

Tools for Measuring Human Movement .........................................

5.5

5.4

Analysis of Human Motion: an Inverse Dynamics Approach ..........

5.12

5.5

Concluding Remarks ......................................................................

5.24

References ..............................................................................................

5.25

Biomechanics of the Musculoskeletal System ...............................................

6.1

6.1

Introduction ....................................................................................

6.1

6.2

Mechanical Properties of Soft Tissue .............................................

6.3

6.3

Body-segmental Dynamics ............................................................

6.8

6.4

Musculoskeletal Geometry .............................................................

6.11

6.5

Muscle Activation and Contraction Dynamics ................................

6.16

6.6

Determining Muscle Force .............................................................

6.23

6.7

Muscle, Ligament, and Joint-contact Forces ..................................

6.27

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Contents

ix

References ..............................................................................................

6.32

Biodynamics: a Lagrangian Approach ............................................................

7.1

7.1

Motivation ......................................................................................

7.1

7.2

The Significance of Dynamics ........................................................

7.3

7.3

The Biodynamic Significance of the Equations of Motion ...............

7.4

7.4

The Lagrangian (an Energy Method) Approach .............................

7.4

7.5

Introduction to the Kinematics Table Method .................................

7.16

7.6

Brief Discussion .............................................................................

7.24

7.7

In Closing ......................................................................................

7.25

References ..............................................................................................

7.25

Bone Mechanics ..............................................................................................

8.1

8.1

Introduction ....................................................................................

8.1

8.2

Composition of Bone .....................................................................

8.2

8.3

Bone as a Hierarchical Composite Material ...................................

8.2

8.4

Mechanical Properties of Cortical Bone .........................................

8.6

8.5

Mechanical Properties of Trabecular Bone ....................................

8.11

8.6

Mechanical Properties of Trabecular Tissue Material .....................

8.16

8.7

Concluding Remarks ......................................................................

8.17

References ..............................................................................................

8.17

Finite-element Analysis ...................................................................................

9.1

9.1

Introduction ....................................................................................

9.1

9.2

Geometric Concerns ......................................................................

9.2

9.3

Material Properties .........................................................................

9.3

9.4

Boundary Conditions .....................................................................

9.5

9.5

Case Studies .................................................................................

9.6

9.6

Conclusions ...................................................................................

9.9

References ..............................................................................................

9.9

10. Vibration, Mechanical Shock, and Impact ......................................................

10.1

10.1 Introduction ....................................................................................

10.1

10.2 Physical Measurements .................................................................

10.6

7.

8.

9.

10.3 Models and Human Surrogates ..................................................... 10.12 10.4 Countermeasures .......................................................................... 10.20 References .............................................................................................. 10.25 This page has been reformatted by Knovel to provide easier navigation.

x

Contents

Part III. Biomaterials 11. Biopolymers .....................................................................................................

11.3

11.1 Introduction ....................................................................................

11.3

11.2 Polymer Science ............................................................................

11.4

11.3 Specific Polymers .......................................................................... 11.13 References .............................................................................................. 11.29 12. Biomedical Composites ...................................................................................

12.1

12.1 Introduction ....................................................................................

12.1

12.2 Classification .................................................................................

12.3

12.3 Constituents ...................................................................................

12.3

12.4 Processing .....................................................................................

12.7

12.5 Physical Properties ........................................................................

12.7

12.6 Fracture and Fatigue Failure .......................................................... 12.10 12.7 Biologic Response ......................................................................... 12.12 12.8 Biomedical Applications ................................................................. 12.13 References .............................................................................................. 12.16 13. Bioceramics .....................................................................................................

13.1

13.1 Introduction ....................................................................................

13.1

13.2 Bioinert Ceramics ..........................................................................

13.3

13.3 Bioactive Ceramics ........................................................................

13.8

13.4 Ceramics for Tissue Engineering/Biological Therapy ..................... 13.16 13.5 Biomimetic Ceramics ..................................................................... 13.17 References .............................................................................................. 13.22 14. Cardiovascular Biomaterials ...........................................................................

14.1

14.1 Introduction ....................................................................................

14.1

14.2 Materials ........................................................................................

14.2

14.3 Testing ...........................................................................................

14.6

14.4 Material Processing and Device Design ......................................... 14.10 14.5 Material Replacement .................................................................... 14.10 References .............................................................................................. 14.11 15. Orthopedic Biomaterials ..................................................................................

15.1

15.1 Introduction ....................................................................................

15.1

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Contents

xi

15.2 Natural Materials ............................................................................

15.2

15.3 Engineered Materials .....................................................................

15.8

15.4 Conclusion ..................................................................................... 15.18 References .............................................................................................. 15.19 16. Biomaterials to Promote Tissue Regeneration ...............................................

16.1

16.1 Background ...................................................................................

16.1

16.2 Structural Component ....................................................................

16.2

16.3 Biochemical Component ................................................................ 16.13 16.4 Conclusions ................................................................................... 16.22 References .............................................................................................. 16.22

Part IV. Bioelectricity 17. Bioelectricity and Its Measurement .................................................................

17.3

17.1 Introduction ....................................................................................

17.3

17.2 The Nature of Bioelectricity ............................................................

17.3

17.3 Action Events of Nerve ..................................................................

17.9

17.4 Volume Conductor Propagation ..................................................... 17.15 17.5 Detection of Bioelectric Events ...................................................... 17.18 17.6 Electrical Interference Problems in Biopotential Measurement ................................................................................. 17.26 17.7 Biopotential Interpretation .............................................................. 17.39 References .............................................................................................. 17.50 18. Biomedical Signal Analysis .............................................................................

18.1

18.1 Introduction ....................................................................................

18.1

18.2 Classifications of Signals and Noise ..............................................

18.2

18.3 Spectral Analysis of Deterministic and Stationary Random Signals ...........................................................................................

18.5

18.4 Spectral Analysis of Nonstationary Signals ....................................

18.8

18.5 Principal Components Analysis ...................................................... 18.13 18.6 Cross-correlation and Coherence Analysis .................................... 18.19 18.7 Chaotic Signals and Fractal Processes .......................................... 18.23 References .............................................................................................. 18.27

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xii

Contents

Part V. Design of Medical Devices and Diagnostic Instrumentation 19. Medical Product Design ..................................................................................

19.3

19.1 Introduction/Overview ....................................................................

19.3

19.2 Scope ............................................................................................

19.5

19.3 Qualities of Successful Product Design .........................................

19.5

19.4 Concurrent Engineering .................................................................

19.5

19.5 Goals .............................................................................................

19.6

19.6 Team/Talent ..................................................................................

19.6

19.7 Planning/Resources .......................................................................

19.7

19.8 Developing User Needs .................................................................

19.7

19.9 Product Specifications ...................................................................

19.9

19.10 Concept Development ................................................................... 19.10 19.11 Concept Evaluation ........................................................................ 19.11 19.12 Architecture/System Design ........................................................... 19.13 19.13 Detail Design ................................................................................. 19.14 19.14 Design for Manufacture .................................................................. 19.14 19.15 Rollout ........................................................................................... 19.14 19.16 Process Review ............................................................................. 19.15 19.17 Prototyping .................................................................................... 19.15 19.18 Testing ........................................................................................... 19.16 19.19 Documentation ............................................................................... 19.16 19.20 Tools .............................................................................................. 19.17 19.21 Regulatory Issues .......................................................................... 19.17 19.22 Closure .......................................................................................... 19.18 References .............................................................................................. 19.18 20. Cardiovascular Devices ..................................................................................

20.1

20.1 Introduction ....................................................................................

20.1

20.2 Artificial Heart Valves .....................................................................

20.1

20.3 Stents and Stent-grafts: Percutaneous Vascular Therapies ...........

20.6

20.4 Pacemakers and Implantable Defibrillators .................................... 20.11 20.5 Artificial Vascular Grafts ................................................................ 20.17 This page has been reformatted by Knovel to provide easier navigation.

Contents

xiii

20.6 Artificial Kidneys ............................................................................ 20.20 20.7 Indwelling Vascular Catheters and Ports ....................................... 20.24 20.8 Circulatory Support Devices .......................................................... 20.28 20.9 Artificial Lungs ............................................................................... 20.35 References .............................................................................................. 20.39 21. Design of Respiratory Devices ........................................................................

21.1

21.1 Introduction ....................................................................................

21.1

21.2 Pulmonary Physiology ...................................................................

21.1

21.3 Important Principles of Gas Physics ..............................................

21.4

21.4 Device Components .......................................................................

21.7

21.5 Common Respiratory Measurements ............................................. 21.15 21.6 Other Devices ................................................................................ 21.23 21.7 Design of Respiratory Devices ....................................................... 21.23 References .............................................................................................. 21.28 22. Design of Controlled-release Drug Delivery Systems ....................................

22.1

22.1 Physicochemical Properties of Drug ..............................................

22.2

22.2 Routes of Drug Administration .......................................................

22.3

22.3 Pharmacological and Biological Effects .........................................

22.4

22.4 Prodrug ..........................................................................................

22.4

22.5 Diffusion-controlled Delivery Systems ............................................

22.5

22.6 Dissolution/Coating-controlled Delivery Systems ...........................

22.9

22.7 Biodegradable/Erodible Delivery Systems .....................................

22.9

22.8 Osmotic Pump ............................................................................... 22.10 22.9 Ion Exchange Resins ..................................................................... 22.11 22.10 New Macromolecular Delivery Approaches .................................... 22.12 22.11 Conclusion ..................................................................................... 22.14 References .............................................................................................. 22.14 23. Sterile Medical Device Package Development ...............................................

23.1

23.1 Regulatory History .........................................................................

23.2

23.2 Functions of a Package .................................................................

23.5

23.3 Package Types ..............................................................................

23.6

23.4 Packaging Materials ......................................................................

23.9

23.5 Common Testing Methods ............................................................. 23.13 This page has been reformatted by Knovel to provide easier navigation.

xiv

Contents 23.6 Package Process Validation .......................................................... 23.19 23.7 Shelf Life Studies ........................................................................... 23.25 23.8 Final Package Validation ............................................................... 23.30 References .............................................................................................. 23.33

24. Design of Magnetic Resonance Systems .......................................................

24.1

24.1 Introduction ....................................................................................

24.1

24.2 MR Magnet Characteristics ............................................................

24.3

24.3 Gradient Characteristics ................................................................

24.5

24.4 Radio-frequency Magnetic Field and Coils .....................................

24.8

24.5 Other MR Systems ........................................................................ 24.12 24.6 Safety Standards ........................................................................... 24.13 24.7 NEMA MR Measurement Standards .............................................. 24.14 References .............................................................................................. 24.15 25. Instrumentation Design for Ultrasonic Imaging ...............................................

25.1

25.1 Introduction ....................................................................................

25.1

25.2 Basic Concepts ..............................................................................

25.1

25.3 Typical System Block Diagram ......................................................

25.4

25.4 Beam Formation ............................................................................

25.7

25.5 Signal Processing and Scan Conversion ....................................... 25.16 25.6 Summary ....................................................................................... 25.16 26. The Principles of X-ray Computed Tomography ............................................

26.1

26.1 Introduction ....................................................................................

26.1

26.2 The Interaction of X-rays with Matter .............................................

26.3

26.3 The Mathematical Model ................................................................

26.9

26.4 The Microtomography System ....................................................... 26.27 References .............................................................................................. 26.51 27. Nuclear Medicine Imaging Instrumentation ....................................................

27.1

27.1 Introduction ....................................................................................

27.1

27.2 Scintillation Cameras .....................................................................

27.2

27.3 Spect Systems ............................................................................... 27.14 27.4 Summary ....................................................................................... 27.20 References .............................................................................................. 27.20

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Contents

xv

28. Breast Imaging Systems: Design Challenges for Engineers .........................

28.1

28.1 Introduction ....................................................................................

28.1

28.2 Breast Anatomy .............................................................................

28.2

28.3 Current Clinical Breast Imaging .....................................................

28.3

28.4 New and Developing Breast Imaging Modalities ............................

28.6

28.5 Future Directions – Multimodality Imaging ..................................... 28.11 References .............................................................................................. 28.12

Part VI. Engineering Aspects of Surgery 29. Computer-integrated Surgery and Medical Robotics .....................................

29.3

29.1 Introduction: Coupling Information to Surgical Action .....................

29.3

29.2 An Overview of CIS Systems .........................................................

29.8

29.3 The Technology of CIS Systems .................................................... 29.10 29.4 Examples of CIS Systems ............................................................. 29.24 29.5 Perspectives .................................................................................. 29.36 References .............................................................................................. 29.36

Part VII. Rehabilitation Engineering 30. Technology and Disabilities ............................................................................

30.3

30.1 Introduction ....................................................................................

30.3

30.2 The Context for Rehabilitation Engineering ....................................

30.3

30.3 A Working Definition of Assistive Technologies .............................

30.4

30.4 Control Interfaces for Electronic Assistive Technologies ................

30.4

30.5 Computer Access by Persons with Disabilities ...............................

30.7

30.6 Augmentative and Alternative Communication ............................... 30.11 30.7 Aids for Manipulation ..................................................................... 30.13 30.8 Future Trends ................................................................................ 30.14 30.9 Summary ....................................................................................... 30.16 References .............................................................................................. 30.16 31. Applied Universal Design ................................................................................

31.1

31.1 Motivation ......................................................................................

31.1

31.2 Design Methodology ......................................................................

31.2

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xvi

Contents 31.3 Special Topics ............................................................................... 31.16 31.4 Web Sites ...................................................................................... 31.19 31.5 What's Next? ................................................................................. 31.19 References .............................................................................................. 31.19

32. Design of Artificial Arms and Hands for Prosthetic Applications ....................

32.1

32.1 Introduction ....................................................................................

32.1

32.2 The Nature of the Problem .............................................................

32.3

32.3 General Design Considerations .....................................................

32.6

32.4 Anatomical Design Considerations ................................................ 32.28 32.5 Multifunction Mechanisms .............................................................. 32.33 32.6 Safety ............................................................................................ 32.36 32.7 Control ........................................................................................... 32.36 32.8 In Conclusion ................................................................................. 32.54 References .............................................................................................. 32.56 33. Design of Artificial Limbs for Lower Extremity Amputees ...............................

33.1

33.1 Overview ........................................................................................

33.1

33.2 History of Limb Prosthetics ............................................................

33.1

33.3 Amputation Surgery .......................................................................

33.3

33.4 Prosthetic Clinic Team ...................................................................

33.5

33.5 Prosthesis Design ..........................................................................

33.8

33.6 Amputee Gait ................................................................................. 33.24 33.7 Recent Developments .................................................................... 33.25 References .............................................................................................. 33.28 34. Home Modification Design ..............................................................................

34.1

34.1 General Considerations .................................................................

34.1

34.2 The Kitchen ...................................................................................

34.2

34.3 The Bathroom ................................................................................ 34.14 Bibliography ............................................................................................. 34.20 35. Rehabilitators ..................................................................................................

35.1

35.1 Introduction ....................................................................................

35.1

35.2 Rationale for Rehabilitators ............................................................

35.1

35.3 Design of Rehabilitators .................................................................

35.7

35.4 The Future of Rehabilitators .......................................................... 35.11 This page has been reformatted by Knovel to provide easier navigation.

Contents

xvii

35.5 Conclusion ..................................................................................... 35.13 References .............................................................................................. 35.13

Part VIII. Clinical Engineering 36. Clinical Engineering Overview ........................................................................

36.3

36.1 Biomedical Engineering .................................................................

36.3

36.2 Clinical Engineering .......................................................................

36.5

36.3 Role of Clinical Engineering ...........................................................

36.7

36.4 Future Development of Clinical Engineering .................................. 36.10 References .............................................................................................. 36.14 37. Technology Planning for Health Care Institutions ..........................................

37.1

37.1 Introduction ....................................................................................

37.1

37.2 Strategic Planning ..........................................................................

37.2

37.3 Planning Principles ........................................................................

37.3

37.4 Components of Technology Planning ............................................

37.5

37.5 Equipment Planning for New Facilities ........................................... 37.13 37.6 Project Management ...................................................................... 37.14 37.7 Facility Components ...................................................................... 37.14 37.8 Business Cases ............................................................................. 37.15 37.9 Priorities and Funding .................................................................... 37.16 37.10 Outsourcing ................................................................................... 37.17 37.11 Corporate Relationships ................................................................ 37.17 37.12 Summary ....................................................................................... 37.18 References .............................................................................................. 37.19 38. An Overview of Health Care Facilities Planning .............................................

38.1

38.1 Introduction ....................................................................................

38.1

38.2 Health Care Facilities Planning and Design ...................................

38.6

38.3 Key Points of Influence: Checklist for Medical Technology Input into the Health Care Facilities Planning Process ................... 38.19 References .............................................................................................. 38.20 39. Department/Program Management ................................................................

39.1

Index ..........................................................................................................

I.1

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P - A - R - T

-

1

BIOMEDICAL SYSTEMS ANALYSIS

CHAPTER 1

MODELING AND SIMULATION OF BIOMEDICAL S Y S T E M S Narender P. Reddy University of Akron, Akron, Ohio

1.1 MATHEMATICAL MODELING 1.3 1.2 COMPARTMENTAL MODELS 1.4 1.3 ELECTRICAL ANALOG MODELS 1.9 1.4 MODELS WITH MEMORY AND MODELS WITH TIME DELAY 1.14

7.7

MATHEMATICAL

1.5 ARTIFICIAL NEURAL NETWORK MODELS 1.20 1.6 MECHANICAL MODELS 1.23 1.7 MODELVALIDATION 1.24 REFERENCES 1.24

MODELING

Models are conceptual constructions that allow formulation and testing of hypotheses. A mathematical model attempts to duplicate the quantitative behavior of the system. Mathematical models are used in today's scientific and technological world because of the ease with which they can be used to analyze real systems. The most prominent value of a model is its ability to predict as yet unknown properties of the system. The major advantage of a mathematical or computer model is that the model parameters can be easily altered and the system performance can be simulated. Mathematical models allow the study of subsystems in isolation from the parent system. Model studies are often inexpensive and less time consuming than corresponding experimental studies. A model can also be used as a powerful educational tool, since it permits idealization of processes. Models of physiological systems often aid in the specification of design criteria for the design of procedures aimed at alleviating pathological conditions. Mathematical models are useful in the design of medical devices. Mathematical model simulations are first conducted in the evaluation of the medical devices before expensive animal testing and clinical trials. Models are often useful in the prescription of patient protocols for the use of medical devices. Pharmacokinetic models have been extensively used in the design of drugs and drug therapies. There are two types of modeling approach: the black box approach and the building block approach. In the black box approach, a mathematical model is formulated on the basis of the inputoutput characteristic of the system without consideration of the internal functioning of the system. Neural network models and autoregressive models are some examples of the black box approach. In the building block approach, models are derived by applying the fundamental laws (governing physical laws) and constitutive relations to the subsystems. These laws together with physical constraints are used to integrate the models of subsystems into an overall mathematical model of the system. The building block approach is used when the processes of the system are understood. However, if the system processes are unknown or too complex, then the black box approach is used. With the building block approach, models can be derived at the microscopic or at the macroscopic

levels. Microscopic models are spatially distributed and macroscopic models are spatially lumped and are rather global. The microscopic modeling often leads to partial differential equations, whereas the macroscopic or global modeling leads to a set of ordinary differential equations. For example, the microscopic approach can be used to derive the velocity profile for blood flow in an artery, but the global or macroscopic approach is needed to study the overall behavior of the circulatory system including the flow through arteries, capillaries, and the heart. Models can also be classified into continuous time models and models lumped in the time domain. While the continuous time modeling leads to a set of differential equations, the models lumped in time are based on the analysis of discrete events in time and may lead to difference equations or sometimes into difference-differential equations. Random walk models and queuing theory models are some examples of discrete time models. Nerve firing in the central nervous system can be modeled by using such discrete time event theories. Models can be classified into deterministic and stochastic models. For example, in deterministic modeling, we could describe the rate of change of volume of an arterial compartment to be equal to the rate of flow in minus the rate of flow out of the compartment. However, in the stochastic approach, we regard the probability of increase in the volume of the compartment in an interval to be dependent on the probability of transition of a volume of fluid from the previous compartment and the probability of transition of a volume of fluid from the compartment to the next compartment. While the deterministic approach gives the mean or average values, the stochastic approach yields means, variances, and covariances. The stochastic approach may be useful in describing the cellular dynamics, cell proliferations, etc. However, in this chapter, we will consider only the deterministic modeling at the macroscopic level. The real world is complex, nonlinear, nonhomogeneous, often discontinuous, anisotropic, multilayered, multidimensional, etc. The system of interest is isolated from the rest of the world by using a boundary. The system is then conceptually reduced to that of a mathematical model by using a set of simplifying assumptions. Therefore, the model results have significant limitations and are valid only in the regimes where the assumptions are valid.

1.2

COMPARTMENTAL

MODELS

Compartmental models are lumped models. The concept of a compartmental model assumes that the system can be divided into a number of homogeneous well-mixed components called compartments. Various characteristics of the system are determined by the movement of material from one compartment to the other. Compartmental models have been used to describe blood flow distribution to various organs, population dynamics, cellular dynamics, distribution of chemical species (hormones and metabolites) in various organs, temperature distribution, etc. As an example, let us consider a simple one-compartment model for the prescription of treatment protocols for dialysis by an artificial kidney device (Fig. 1.1). While the blood urea concentration (BUN) in the normal individual is usually 15 mg% (mg% = milligrams of the substance per 100 mL of blood), the BUN in uremic patients could reach 50 mg%. The purpose of the dialysis is to bring the BUN level closer to the normal. In the artificial kidney, blood flows on one side of the dialyzer membrane and dialysate fluid flows on the other side. Mass transfer across the dialyzer membrane occurs by diffusion due to concentration difference across the membrane. Dilysate fluid is a makeup solution consisting of saline, ions, and the essential nutrients that maintains zero concentration difference for these essential materials across the membrane. However, during the dialysis, some hormones also diffuse out of the dialyzer membrane along with the urea molecule. Too-rapid dialysis often leads to depression in the individual because of the rapid loss of hormones. On the other hand, too-slow dialysis may lead to unreasonable time required at the hospital. Simple modeling can be used to calculate the treatment protocols. Let us consider a one-compartment model of the tissue where we assume that the blood and tissue are well mixed and that the concentration of urea is uniform throughout the body. Let C0 be the concentration of urea at the outlet of the body, i.e., at the inlet of the dialyzer in the arterial line that takes blood into the dialyzer, and let C1- be the concentration of urea at the inlet of the compartment, i.e., at the exit of the dialyzer

BODY

C Q

C0 BLOOD DIALYZATE FLUID DIALYZER

FIGURE 1.1 A one-compartment model of the human body to analyze the patient-dialyzer interactions.

in the venous line that brings the blood back to the body. Mass balance demands that the rate of change of mass in the body be equal to the net rate of mass coming into the body from the dialyzer, plus the metabolic production rate. (1.1) where V = tissue volume plus the blood volume Q = blood flow rate to the kidney m = metabolic production rate of urea in the body It should be noted that mass is equal to volume V times the concentration C. Usually, the dialysate flow rate in the artificial kidney is much larger than that of the blood flow rate. Regardless of the type of the dialyzer (cocurrent, countercurrent, or mixed flow), the extraction ratio E can be expressed as (Cooney, 1980): (1.2) where A is the interfacial membrane surface area for mass transfer and k is the permeability of the membrane for that particular solute (urea in the present context). Since Q does not change during dialysis, and since k and A are design parameters, extraction ratio E remains a constant. Extraction can be further expressed in terms of the concentrations as follows: (1.3) It should be pointed out that C0 is the concentration at the outlet of the body and therefore at the

inlet of the dialyzer, and C1 is the concentration in the blood coming into the body and therefore going out of the dialyzer. Also, it should be noted that the concentration C0 in the blood going out of the body is the same as the concentration in the body, since we assumed that the entire body (tissue and blood) constitutes a homogeneous, well-mixed compartment. Therefore, Eq. (1) can be rewritten as follows on substitution of Eq. (3) (1.4) When the dialyzer is turned on, metabolic production rate (m) can be neglected when compared to the other term in the equation, and upon integration will result in (1.5) where C 0 is the initial concentration of urea in the tissue. When the patient is not on dialysis, then the blood flow to the dialyzer Q is zero, and therefore (1.6) When the patient is not on dialysis, the concentration of urea will increase linearly if the metabolic production rate is constant or will increase exponentially if the metabolic production rate is a linear function of the concentration (first-order reaction). When the patient is on dialysis, the concentration would decrease exponentially. This way, the treatment protocol can be prescribed after simulating different on and off times (e.g., turn on the dialyzer for 4 hours every 3 days) to bring the BUN under control. Now, let us examine the limitations of the one-compartment model. First, the entire blood and tissue are assumed to be in equilibrium. However, it is well known that intracellular urea concentration may be significantly different from the extracellular compartment. Moreover, urea may be preferentially produced in certain organs like the brain, heart, muscle, etc. An accurate treatment requires a multicompartment model. Let us consider a two-compartment model (Fig. 1.2) consisting of an intracellular pool (compartment 1) and an extracellular pool (compartment 2). Urea is produced by the intracellular pool and is transported across the cell membrane into the interstitial fluids and then into the blood stream. Mass balance for these two compartments can be expressed as (1.7) where B (C1 — C2) = the interfacial transfer from compartment 1 to compartment 2 (from intracellular pool to extracellular pool) C1 and C2 = concentrations of urea in compartments 1 and 2 B = a constant The constant B is a product of permeability of the cellular membrane for urea and the interfacial surface area. (1.8) Blood flow to the dialyzer (Q) is zero when the patient is not on the dialysis machine. Babb et al. (1967) simulated the two-compartmental model and the model results were in agreement with the experimental data (Fig. 1.3). However, the two compartmental model may not be sufficient if one wants to find the concentration of urea in the brain tissue. A multicompartment model involving

Ci

Intracellular

C2 Extracellular Ci

C0

DIALYZER FIGURE 1.2 A two-compartmental model of the body used to simulate the patientdialyzer interactions.

separate compartments for brain, heart, kidney, lean tissue, etc., may be needed to accurately determine the concentration of urea in critical organs. Similar compartmental models have been used in the development of metabolic modeling. Now, let us consider an example of cellular dynamics in bone. Cellular dynamics plays an important role in natural and stress-induced bone remodeling. There are four kinds of functionally distinct cells: mesenchymal cells, osteoclasts, osteoblasts, and osteocytes. Mesenchymal cells have an outstanding capacity for proliferation and are capable of further differentiating into an osteoclast or an osteoblast. While the mesenchymal cell undergoes cell division, osteoblasts, osteoclasts, or the osteocytes do not undergo cell division. The osteoclast is responsible for bone resorption. The osteoblast is responsible for new bone formation. The osteoclast can transform into an osteoblast and vice versa. The osteocyte is a resident cell of the bone. When the osteoblast has surrounded itself with a matrix (after new bone formation), it becomes an osteocyte and it loses the characteristics of an actively secreting cell. Let us assume that these four types of cells can be divided into four different, well-mixed homogeneous compartments (Fig. 1.4). Let W be the number of mesenchymal cells, X be the number of osteoclasts, Y be the number of osteoblasts, and Z be the number of osteocytes at any given time. Let compartments 1 through 4, respectively, represent these four types of cells. The birth rate and death rate of cells in a compartment depend on the number of cells in that compartment. Let B be the birth rate of mesenchymal cells per individual cell per unit time. Let D1 and D2 represent the death rates of osteoclasts and osteocytes per cell per unit time. Equations representing the cell population dynamics can be expressed as follows by integrating the notions of bone cell physiology with the concepts of compartmental modeling. The rate of change of the number of mesenchymal cells (W) can be expressed as (1.9)

(mg%) BUN

DAYS FIGURE 1.3 Model simulation results of the patient BUN (blood urea nitrogen levels) in mg% (mg% = mg of the substance per 100 mL of blood) plotted as a function of time. [From Babb etal (1967).] Closed circles are experimental observations.

FIGURE 1.4 A four-compartment model of bone cells: Compartment 1 represents mesenchymal cells, compartment 2 represents osteoclasts, compartment 3 represents osteoblasts, and compartment 4 represents osteocytes. Mesenchymal cells reproduce and transform either into osteoclasts or into osteoblasts. The osteoclast can transform into an osteoblast and an osteoblast can transform into an osteoclast. The osteoblast can transform into an osteocyte.

The first term on the right-hand side represents the birth rate of mesenchymal cells, the second term represents the rate of number of mesenchymal cells transforming into an osteoclast, and the last term represents the rate of transformation of mesenchymal cells into osteoblasts. The rate of change of the number of osteoclasts (X) as a function of time can be expressed as (1.10) The first term on the right-hand side represents the rate of mesenchymal cells transforming into an osteoclast, the second term represents the rate of transformation of osteolasts into osteoblasts, the third term represents the rate of transformation of osteoblast into osteoclasts, and the last term represents the rate of death of osteoclasts. The rate of change of the number of osteoblasts (Y) as a function of time can be expressed as (1.11) The first term on the right-hand side represents the rate of transformation of mesenchymal cells into osteoblasts, the second term represents the rate of transformation of osteoclasts into osteoblasts, the third term represents the rate of transformation of osteoblasts into osteoclasts, and the last term represents the rate of transformation of osteoblasts into osteocytes. The rate of change of the number of osteocytes (Z) as a function of time can be expressed as (1.12) The first term in the above equation represents the rate of osteoblast transformation into osteocytes, and the last term represents the rate of death of osteocytes. Reddy and Joshi (1987) simulated the stochastic compartmental model of bone cells in which the equation for the population means are the same as the above equations. In addition, the stochastic analysis provides information about the variations and covariences of cellular populations. Figure 1.5 shows the normalized number of osteoblasts plotted as a function of age of the individual when C1 and C2 are assumed to be sinusoidal functions in time. These simulation results of Reddy and Joshi (1987) are consistent with experimental observations of Frost (1963). Compartmental models are used in the analysis of thermal interactions. Simon and Reddy (1992) formulated a mathematical model of the infant-incubator dynamics. Neonates who are born preterm often do not have the maturity for thermal regulation and do not have enough metabolic heat production. Moreover, these infants have a large surface area to volume ratio. Since these preterm babies can not regulate heat, they are often kept in an incubator until they reach thermal maturity. The incubator is usually a forced-convection heating system with hot air flowing over the infant. Incubators are usually designed to provide a choice of air control or skin control. In air control, the temperature probe is place in the incubator air space and the incubator air temperature is controlled. In the skin control operation, the temperature sensor is placed on the skin and the infant's skin temperature is controlled. Simon and Reddy (1992) used a four-compartment model (Fig. 1.6) to compare the adequacy of air control and skin control on the core temperature of the infant. They considered the infant, air, mattress, and the wall to be four separate, well-mixed compartments.

1.3

ELECTRICAL ANALOG

MODELS

Electric analog models are a class of lumped models and are often used to simulate flow through the network of blood vessels. These models are useful in assessing the overall performance of a system or a subsystem. Integration of the fluid momentum equation (longitudinal direction, in cy-

TIME IN YEARS FIGURE 1.5 Simulation results of the normalized number of osteoblasts plotted as a function of age in years. [From Reddy andJoshi (1987).]

lindrical coordinates) across the cross section results in the following expression (Reddy, 1986; Reddy and Kesavan, 1989):

where p is the fluid density, Q is the flow rate, a is the wall radius, P is the pressure, / is the length, and rw is the fluid shear stress at the wall. If we assume that the wall shear stress can be expressed by using quasi-steady analysis, then the wall shear stress can be estimated by rw = 4 /xQ/a3. Substituting for the wall stress and rearranging results in

(JL\^ \m2/

dt

= AP.(M)Q W4/

(U4)

= LP-RQ

(1.15)

V

}

which can be rewritten as L^ dt where L = pl/iira2) and R = 8^//(raz4).

CORE

AIR SPACE

SKIN

WALL

MATTRESS

FIGURE 1.6 A lumped parameter model of the infant-incubator dynamics used by Simon and Reddy (1994) to simulate the effect of various control modes in a convectively heated infant incubator. Infants core, and skin are modeled as two separate compartments. The incubator air space, the incubator wall, and the mattress are treated as three compartments. Heat interactions occur between the core (infant's lungs) and the incubator air space through breathing. Skin-core heat interactions are predominantly due to blood flow to the skin. Heat transfer between the infant's skin and the incubator air are due to conduction and convection. Heat transfer from the skin to the mattress is via conduction, and heat transfer to the wall is via radiation from skin and convection from the air.

It can be easily observed that flow rate Q is analogous to electrical current /, and AP is analogous to the electrical potential drop (voltage) AE. In Eq. (1.15), L is the inductance (inertance) and R is the resistance to flow. Therefore, Eq. (1.15) can be rewritten as L^- = AE-Ri at

(1.16)

The fluid continuity equation when integrated across the cross section can be expressed as

^ = AS = e i n - e out

an)

where V is the volume. However, volume is a function of pressure (from the momentum balance for the vessel wall): P = P^ + №o)