The Artificial Kidney: Physiological Modeling and Tissue Engineering (Tissue Engineering Intelligence Unit 3)

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c-Myc Function in Neoplasia Chi V. Dang and Linda A. Lee, Johns Hopkins University

Functional Heterogeneity of Liver Tissue: From Cell Lineage Diversity to Sublobular Compartment-Specific Pathogenesis Fernando Vidal-Vanaclocha, Universidad del Pais Vasco

Endothelins David J. Webb and Gillian Gray, University of Edinburgh

Host Response to Intracellular Pathogens Stefan H.E. Kaufmann, Institute für Mikrobiologie und Immunologie der Universität Ulm

Cellular Inter-Relationships in the Pancreas: Implications for Islet Transplantation Lawrence Rosenberg and William P. Duguid, McGill University Anti-HIV Nucleosides: Past, Present and Future Hiroaki Mitsuya, National Cancer Institute Heat Shock Response and Organ Preservation George Perdrizet, University of Connecticut Glycoproteins and Human Disease Inka Brockhausen, Hospital for Sick Children—Toronto Exercise Immunology Bente Klarlund Pedersen, Rigshospitalet—Copenhagen Chromosomes and Genes in Acute Lymphoblastic Leukemia Lorna M. Secker-Walker, Royal Free Hospital-London Surfactant in Lung Injury and Lung Transplantation James F. Lewis, Lawson Research Institute Richard J. Novick, Roberts Research Institute Ruud A.W. Veldhuizen, Lawson Research Institute

Management of Post-Open Heart Bleeding Rephael Mohr, Jacob Lavee and Daniel A. Goor, The Chaim Sheba Medical Center

Premalignancy and Tumor Dormancy Eitan Yefenof, Hebrew University - Hadassah Medical School Richard H. Scheuerman, University of Texas Southwestern Myocardial Preconditioning Cherry L. Wainwright and James R. Parratt, University of Strathclyde Cytokines and Inflammatory Bowel Disease Claudio Fiocchi, Case Western Reserve Bone Metastasis F. William Orr and Gurmit Singh, University of Manitoba Cancer Cell Adhesion and Tumor Invasion Pnina Brodt, McGill University Cutaneous Leishmaniasis Felix J. Tapia, Instituto de Medicina-Caracas

Estrogen and Breast Cancer W.R. Miller, University of Edinburgh Molecular Mechanisms of Hypercoagulable States Andrew I. Schafer, University of Texas-Houston Organ Procurement and Preservation for Transplantation Luis Toledo-Pereyra, Michigan State University Liver Stem Cells Stewart Sell and Zoran Ilic, Albany Medical College

Cytokines in Reproduction: Molecular Mechanisms of Fetal Allograft Survival Gary W. Wood, University of Kansas

Skin Substitute Production by Tissue Engineering Mahmoud Rouabhia, Laval University

Computers in Clinical Medicine Eta Berner, University of Alabama

HIV and Membrane Receptors Dimitri Dimitrov, National Institutes of Health Christopher C. Broder, Uniformed Servives University of the Health Sciences

Immunology of Pregnancy Maintenance in the First Trimester Joseph Hill, Harvard University Peter Johnson, University of Liverpool

TEIU

The Artificial Kidney: Physiological Modeling and Tissue Engineering

3

Molecular Basis of Autoimmune Hepatitis Ian G. McFarlane and Roger Williams, King’s College Hospital

The Glycation Hypothesis of Atherosclerosis Camilo A.L.S. Colaco, Qandrant Research Institute

Cellular & Molecular Biology of Airway Chemoreceptors Ernest Cutz, University of Toronto

3

John K. Leypoldt

Hyperacute Xenograft Rejection Jeffrey Platt, Duke University Transplantation Tolerance J. Wesley Alexander, University of Cincinnati

TISSUE ENGINEERING I N T E L L I G E N C E U N I T

Interferon-Inducible Genes Ganes Sen and Richard Ransohoff, Case Western Reserve University

Inherited basement Membrane Disorders Karl Tryggvaso, Karolinska Institute

Artificial Neural Networks in Medicine Vanya Gant and R. Dybowski, St. Thomas Medical School— London

Cartoid Body Chemoreceptors Constancio Gonzalez, Universidad de Madrid

von Willebrand Factor Zaverio M. Ruggeri, Scripps Research Institute

Molecular Biology of Leukocyte Chemostasis Antal Rot, Sandoz Forschungsinstitut—Vienna

Immune Mechanisms in Atherogenesis Ming K. Heng, UCLA

Breast Cancer Screening Ismail Jatoi, Brook Army Medical Center

The Biology of Germinal Centers in Lymphoral Tissue G.J. Thorbecke and V.K. Tsiagbe, New York University


Gamma Interferon in Antiviral Defense Gunasegaran Karupiah, The John Curtin School of Medical Research—The Australian National University

p53 B Cells and Autoimmunity Christian Boitard, Hôpital Necker-Paris

LEYPOLDT

Myocardial Injury: Laboratory Diagnosis Johannes Mair and Bernd Puschendorf, Universität Innsbruck

R.G. LANDES

Genetic Mechanisms in Multiple Endocrine Neoplasia Type 2 Barry D. Nelkin, Johns Hopkins University

C O M PA N Y

MEDICAL INTELLIGENCE UNIT

R.G. LANDES C OM PA N Y

R.G. LANDES C OM PA N Y

TISSUE ENGINEERING INTELLIGENCE UNIT 3

The Artificial Kidney: Physiological Modeling and Tissue Engineering John K. Leypoldt, Ph.D. Research and Service Veteran Affairs Medical Center and Departments of Internal Medicine and Bioengineering University of Utah Salt Lake City, Utah, USA

R.G. LANDES COMPANY AUSTIN, TEXAS U.S.A.

TISSUE ENGINEERING INTELLIGENCE UNIT The Artificial Kidney: Physiological Modeling and Tissue Engineering R.G. LANDES COMPANY Austin, Texas, U.S.A. Copyright ©1999 R.G. Landes Company All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in the U.S.A. Please address all inquiries to the Publishers: R.G. Landes Company, 810 South Church Street, Georgetown, Texas, U.S.A. 78626 Phone: 512/ 863 7762; FAX: 512/ 863 0081

ISBN: 1-57059-602-6

While the authors, editors and publisher believe that drug selection and dosage and the specifications and usage of equipment and devices, as set forth in this book, are in accord with current recommendations and practice at the time of publication, they make no warranty, expressed or implied, with respect to material described in this book. In view of the ongoing research, equipment development, changes in governmental regulations and the rapid accumulation of information relating to the biomedical sciences, the reader is urged to carefully review and evaluate the information provided herein.

Library of Congress Cataloging-in-Publication Data

The Artificial Kidney: Physiological Modeling and Tissue Engineering / John K. Leypoldt. p. cm. -- (Tissue engineering intelligence unit) Includes bibliographical references and index. ISBN 1-57059-602-6 (alk. paper) 1. Artificial kidney. 2. Hemodialysis. 3. Biomedical engineering. I Title. II. Series. [DNLM: 1. Hemodialysis--methods. 2. Biomedical Materials-therapeutic use. 3. Biomedical Engineering--methods. 4. Kidney, Artificial. 5. Models, Biological. 6. Tissue Culture--methods. WJ 378 L684p 1999] RC901.7.A7L49 1999 617.4'61059--dc21 99-33005 for Library of Congress CIP

PUBLISHER’S NOTE Landes Bioscience produces books in six Intelligence Unit series: Medical, Molecular Biology, Neuroscience, Tissue Engineering, Biotechnology and Environmental. The authors of our books are acknowledged leaders in their fields. Topics are unique; almost without exception, no similar books exist on these topics. Our goal is to publish books in important and rapidly changing areas of bioscience for sophisticated researchers and clinicians. To achieve this goal, we have accelerated our publishing program to conform to the fast pace at which information grows in bioscience. Most of our books are published within 90 to 120 days of receipt of the manuscript. We would like to thank our readers for their continuing interest and welcome any comments or suggestions they may have for future books. Michelle Wamsley Production Manager R.G. Landes Company

CONTENTS 1. Fluid Removal During Hemodialysis ....................................................... 1 John K. Leypoldt and Alfred K. Cheung Compartmentalization of Body Fluids and Ions ................................... 1 Determinants of Changes in Fluid Volumes During Hemodialysis .......................................................................... 4 Mathematical Modes of Sodium Kinetics and Fluid Distribution ........................................................................ 8 Modeling Changes in Blood Volume During Hemodialysis ............... 16 Conclusion ............................................................................................. 19 Notation ................................................................................................. 20 2. Urea Removal During Hemodialysis ..................................................... 25 Daniel Schneditz Urea Removal During Hemodialysis .................................................... 25 Transport and Elimination of Urea ...................................................... 25 Compartment Modeling ....................................................................... 34 The Inverse Problem ............................................................................. 45 Conclusion ............................................................................................. 51 Notation ................................................................................................. 54 3. Transport Kinetics During Peritoneal Dialysis ..................................... 59 Michael F. Flessner Peritoneal Anatomy ............................................................................... 60 Physiology of the Dialysis Transport Process ...................................... 60 Mathematical Models of Peritoneal Transport .................................... 76 Notation ................................................................................................. 86 4. The Bioartificial Renal Tubule ............................................................... 89 H. David Humes, Sherrill MacKay and Janeta Nikolovski In Vitro Development and Characterization of the RAD ................... 90 Ex Vivo Performance of the RAD ......................................................... 93 Conclusion ............................................................................................. 96 5. Percutaneous Access for Peritoneal Dialysis: A Tissue Engineering Approach ............................................................................ 99 Jennifer A. LaIuppa and Clifford J. Holmes PD Catheters ........................................................................................ 100 Tissue Response to Current PD Catheters ......................................... 100 Complications Associated with PD Catheters ................................... 102 Previous Attempts to Solve Access Complications ............................ 103 Tissue Engineering Approach to Reduce PD Catheter Complications ............................................................ 104 Surface Architecture ............................................................................ 105 Surface Treatments .............................................................................. 109 Growth Factors .................................................................................... 114 Conclusion ........................................................................................... 116 Notation ............................................................................................... 116

6. Tissue Engineering in the Peritoneal Cavity: Genetic Modification of the Peritoneal Membrane .......................................... 123 Catherine M. Hoff and Ty R. Shockley Gene Therapy ....................................................................................... 124 Peritoneal Membrane .......................................................................... 126 Genetic Modification of the Peritoneal Membrane ........................... 132 Potential for Improving Peritoneal Dialysis Through Genetic Modification .................................................................................... 136 Potential Limitations of Gene Therapy for Peritoneal Dialysis ..................................................................... 140 Formulation of a Successful Mesothelial Cell-Mediated Gene Therapy Strategy .................................................................... 141 Conclusion ........................................................................................... 141 Notation ............................................................................................... 141

EDITORS John K. Leypoldt, Ph.D. Research and Service Veteran Affairs Medical Center and Departments of Internal Medicine and Bioengineering University of Utah Salt Lake City, Utah, USA Chapter 1

CONTRIBUTORS Alfred K. Cheung, M.D. Medical Service Veteran Affairs Medical Center and Department of Internal Medicine University of Utah Salt Lake City, Utah, USA Chapter 1 Michael F. Flessner, M.D., Ph.D. Nephrology Unit University of Rochester Rochester, New York, USA Chapter 3 Catherine M. Hoff, Ph.D. Baxter Healthcare Corporation Renal Division, Scientific Affairs McGaw Park, Illinois, USA Chapter 6 Clifford J. Holmes, Ph.D. Baxter Healthcare Corporation Renal Division, Scientific Affairs McGaw Park, Illinois, USA Chapter 5 H. David Humes, M.D. Department of Internal Medicine University of Michigan Ann Arbor, Michigan, USA Chapter 4

Jennifer A. LaIuppa, Ph.D. Baxter Healthcare Corporation Renal Division, Scientific Affairs McGaw Park, Illinois, USA Chapter 5 Sherrill MacKay, B.S. Department of Internal Medicine University of Michigan Ann Arbor, Michigan, USA Chapter 4 Janeta Nikolovski, M.S. Department of Internal Medicine University of Michigan Ann Arbor, Michigan, USA Chapter 4 Daniel Schneditz, Ph.D. Karl-Franzens University Graz, Austria Chapter 2 Ty R. Shockley, Sc.D. Baxter Healthcare Corporation Renal Division, Scientific Affairs McGaw Park, Illinois, USA Chapter 6

PREFACE

P

rofessional engineering activities towards improving the artificial kidney continue to evolve and expand. Early research focused on machine and hemodialyzer development, with emphasis on producing dialysis delivery systems that were simple and practical, and on improving hemodialyzer efficacy for fluid and solute removal. These efforts were very successful; indeed, fluid and uremic toxins can now be removed during hemodialysis treatments at rates that are so rapid that the primary limitation to the removal process is the patient, not the artificial kidney. Biomedical engineers or physiologists (and nephrologists) with an engineering or mathematical bent are well-positioned to play a significant role in understanding these physiological limitations to fluid and solute removal. Mathematical modeling of fluid and solute removal during treatment by the artificial kidney has long been an important area of research. Early models led to a better understanding of the removal of uremic toxins by the dialyzer or the dialysis membrane; this endeavor could be called kinetic modeling of the dialysis membrane. Recent models have instead emphasized limitations to fluid and solute removal within the patient, an effort that can be called physiological modeling. The first three Chapters of this book detail the importance of understanding the physiological limitations to fluid and solute removal during treatment by the artificial kidney. The first Chapter describes fluid removal during hemodialysis and the importance of sodium kinetics and fluid distribution within body compartments to this removal process. Fluid removal remains a significant limitation to shortening hemodialysis treatment time. Too rapid fluid removal can lead to hypotension and other intradialytic symptoms that are uncomfortable to the patient. On the other hand, insufficient fluid and sodium removal may lead to chronic hypertension in these patients. The optimal fluid removal method for treating chronic hemodialysis patients remains to be described. Chapter 2 is an introduction to and a thorough description of the most recent concepts regarding the use of urea kinetic modeling during hemodialysis. The author of this Chapter, Daniel Schneditz, has recently made several meaningful contributions to this research area and has also made them practical for the clinical nephrologist. The importance of urea kinetic modeling to the practice of clinical dialysis cannot be underestimated. Today, all dialysis staff and patients know the meaning of the urea removal index, Kt/V, and its importance to the outcome of hemodialysis and peritoneal dialysis patients. It should be emphasized that urea Kt/V is derived directly from the engineering concept of a dimensionless parameter. Peritoneal dialysis is an alternative therapy for treating end stage renal failure patients, and Chapter 3 describes the physiological basis for solute removal during peritoneal dialysis. Over the past two decades, Michael F. Flessner and his colleagues have continued to remind the dialysis community that the peritoneum is not simply a hemodialysis membrane placed inside the patient. Indeed, it is a physiological transport barrier that has unique characteristics. It is

worth mentioning that the physiological modeling approach employed in both Chapters 2 and 3 was inspired by the early pioneering research of Robert L. Dedrick and Kenneth B. Bischoff. Their seminal paper (ref. 39 of Chapter 2) first introduced the concept of physiological modeling to the dialysis community and was the inspiration for the first part of this volume’s title. Chapters 4-6 describe the application of tissue engineering principles to further improve artificial kidney therapy. Chapter 4 describes the construction and initial testing of a novel device, a bioartificial tubule, that mimics the reabsorptive and metabolic functions of the native kidney. The ultimate goal of this research is to develop a bioengineered kidney consisting of both a bioartificial glomerulus and the described bioartificial tubule. In a sense, this work can be considered an extension of work by Lee W. Henderson, Clark K. Colton and others in the 1970s who developed a membrane device that mimicked glomerular ultrafiltration of blood, an artificial glomerulus. This work is in its infancy, but these investigators are creating a unique and potentially more natural way to remove uremic toxins. Chapters 5 and 6 describe two research programs that apply tissue engineering principles to improve peritoneal dialysis therapy. In contrast to the other Chapters in this book, these last two Chapters describe truly pioneering efforts into uncharted territory. Chapter 5 reviews the potential for developing improved peritoneal catheters that can modify the tissue response and promote normal tissue attachment to this implanted device. Such developments are crucial to further advance peritoneal dialysis as a long term renal replacement therapy. Chapter 6 describes the potential for genetically altering cells within the peritoneal cavity, specifically peritoneal mesothelial cells, to enhance peritoneal membrane function and extend its longevity as a dialyzing membrane. This area is particularly exciting and the potential for this research in the near future appears to be limited only by the imagination. This book was designed for use by bioengineering graduate students or professional bioengineers interested in undertaking research related to the artificial kidney. It is hoped that nephrologists will also find the topics and information in this book helpful to their interests. John K. Leypoldt January 1999

CHAPTER 1

Fluid Removal During Hemodialysis John K. Leypoldt and Alfred K. Cheung

T

he total volume and distribution of body fluids are highly regulated in normal individuals because of precise excretion of water and ions by the kidney.1,2 In end stage renal disease, control of fluid volume becomes progressively abnormal; at a certain stage, water and ions must be removed by dialysis therapy to maintain normal physiologic functions of the body, especially those of the heart and lungs. While the importance of fluid removal during dialysis therapy has long been recognized,3-5 we are entering a new era because of recent technical developments that permit more accurate evaluation of volume status in hemodialysis patients.6,7 This chapter reviews current understanding of the changes in body fluid distribution that occur during maintenance hemodialysis therapy. Emphasis is placed on the physiology of fluid distribution in chronic hemodialysis patients and on physiological models that describe fluid and sodium kinetics during hemodialysis. Physiological models can be defined as mathematical models that attempt to capture the underlying transport physiology and therefore aim to understand transport mechanisms; the extent of detail needed for a given model depends on the study goals or objectives.8 Kinetic modeling of body fluid distribution in hemodialysis patients focuses on describing the time dependence of body fluid compartment volumes and certain plasma solute concentrations, since many of these variables can be experimentally measured, are directly related to transport processes that limit fluid and solute removal and directly affect patient outcome. Two fundamental physiological transport processes that can limit fluid and solute removal are slow intercompartmental diffusion and blood flow. The current chapter will consider physiological limitations to fluid and sodium removal using models without flow limitations, since body fluids and ions are known to be highly compartmentalized into distinct anatomical entities.1,2 In contrast, the next chapter will emphasize compartmental, flow-limited models to describe urea removal during hemodialysis.

Compartmentalization of Body Fluids and Ions Body Fluids In the normal individual approximately 60% of body weight is water. Two-thirds of body water is intracellular, and one-third is extracellular. The extracellular compartment can be further subdivided into the interstitial and plasma compartments. The interstitial fluid volume is approximately three-fourths and plasma volume is approximately one-fourth of extracellular fluid volume. These relationships are illustrated schematically in Figure 1.1. There is significant interindividual variability in body composition even among normal individuals; the amount of total body water has been previously shown to depend on age,

The Artificial Kidney: Physiological Modeling and Tissue Engineering , edited by John K. Leypoldt. ©1999 R.G. Landes Company.

2


Fig.1.1. Approximate distribution of fluid within the human body. The extracellular compartment is the sum of the intersitital and plasma compartments.

gender and body fat content. Altered physiological states, such as end stage renal disease, can also alter body fluid composition.9 Routine determination of body fluid composition is difficult. The volume of total body water, the extracellular compartment and plasma can be measured by determining the dilution of marker (usually RADiolabeled) solutes that are confined to the respective spaces. The accuracy of determinations of total body water by these methods is approximately ± 3% or ± 1.5 L. Extracellular fluid volume cannot be determined as precisely as total body water, because there does not exist an ideal marker solute that is rapidly distributed throughout this volume but completely excluded from cells. For example, estimates of extracellular fluid volume in normal male individuals as a percentage of body weight range from approximately 16% (using inulin or mannitol as marker) to 28% (using bromide as marker).9 Assuming total body water is 60% of body weight, these estimates of extracellular fluid volume correspond to 27-47% of total body water. Estimates of extracellular fluid volume using bioelectrical impedance analysis (Model 4000B, Xitron Laboratories, San Diego, CA) closely approximate the distribution volume of bromide10,11 and are therefore at the high end of these estimates. Intracellular fluid volume can only be determined by subtraction of extracellular fluid volume from total body water volume. The determination of total body water and extracellular fluid volumes in chronic hemodialysis patients may become increasingly routine with the improved development of techniques based on bioelectrical impedance analysis; however, these techniques are only currently employed in research settings. Most of current knowledge about body fluid distribution derives from studies on normal individuals or in patients without renal disease. In one of the few studies on renal failure patients, Bauer and Brooks12 found that body fluid composition of 10 well-nourished patients maintained on chronic hemodialysis therapy had blood, extracellular and total body water volumes that were similar to those of normal individuals. The only difference

Fluid Removal During Hemodialysis

3

observed was a high plasma volume during renal failure, presumably because of a change reciprocal to that of red cell mass in order to maintain normal blood volume. It is unclear from these data, however, if these fluid volumes are optimal for chronic hemodialysis patients.

Ions

The ionic composition of intracellular and extracellular fluids differs markedly.1,2 The intracellular compartment contains predominantly potassium as the cationic constituent and proteins and organic phosphates as anion constituents. The extracellular compartment contains predominantly sodium as the cationic constituent and chloride as the anionic constituent. Plasma differs from interstitial fluid in that it contains substantially more protein. These compositional differences are maintained by barriers or membranes, and the characteristics of the membrane separating the intracellular and interstitial compartments are materially different from that separating the interstitial and plasma compartments. The former barrier is the cell membrane and can be considered readily permeable to water but impermeable to all ions. The latter barrier is the capillary wall and can be considered readily permeable to water and small solutes but almost completely impermeable to large proteins such as albumin. For the purposes of this chapter, we will consider the cell membrane to be a perfect osmometer such that it is impermeable to all ions. In reality, these barriers are considerably more complex. The ionic composition of body fluid compartments is a major determinant of the volume of each compartment since these volumes are determined passively by maintaining osmotic equilibrium between the compartments. The volume of the extracellular compartment is determined largely by its sodium content (and its attendant anions). In the normal individual the kidney, largely under the influence of circulating antidiuretic hormone or vasopressin, maintains serum or plasma sodium concentration in a narrow range, 138-142 mEq/L,13 despite great variations in sodium and fluid intake. By regulating sodium and water excretion, the kidney maintains a constant extracellular fluid volume. Small differences in ionic concentration can be critical in ion and fluid kinetics during hemodialysis; therefore, the methods for measuring ionic concentrations, and a detailed understanding of the factors that influence these concentrations, are important.14-16 Methods for measuring ionic concentration have changed over the past decade. Previously, flame photometry was commonly used to measure ionic concentrations in routine clinical practice; this method measures the total concentration of an ion in a given volume. Most laboratories currently use ion selective electrodes for measuring concentration; this method measures ionic activity within the water content of the solution. Solute or ionic activity is a complex parameter; it generally assesses the ability of the ion to take part in electrochemical or physicochemical processes. For example, the difference in ionic activity across a barrier or membrane determines its diffusion rate or its osmotic effectiveness. Changes in ionic activity (as determined electrochemically) during hemodialysis may not, however, accurately reflect total ion removal.14 These issues can be illustrated by considering concentrations of sodium in plasma and dialysate as measured by flame photometry and ion sensitive electrodes.15 These alternative techniques give very similar values (within 1%) for sodium concentration in plasma because of two different counteracting factors. First, the concentration of sodium in plasma water is approximately 6% higher than that in plasma because of the excluded volume occupied by plasma proteins. Accounting for this factor alone, one might expect ion selective electrodes to yield values significantly higher than those determined by flame photometry. On the other hand, sodium is complexed with proteins and other small anions by approximately 5%; this lowers sodium activity by this same factor. Thus, these

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two factors approximately cancel each other in plasma. In dialysate, however, the absence of proteins requires that the dialysate activity of sodium, measured by ion selective electrodes, is 3-5% lower than that measured by flame photometry. In this review, we will neglect any further distinction between concentration and activity except in the numerical simulations (described below).

Determinants of Changes in Fluid Volumes During Hemodialysis In the end stage renal failure patient, the normal mechanisms for regulatory body fluid volumes and ionic composition are no longer present. In the absence of residual renal function, body fluids are largely removed only intermittently during hemodialysis therapy. One of the primary goals of hemodialysis therapy is therefore to alter patient volume status and body fluid distribution; the target is the so-called dry weight, which can be defined as the postdialysis body weight where the patient is in a state of normohydration. Because the dry weight of chronic hemodialysis patients cannot be directly evaluated, however, it is often defined simply to minimize the occurrence of clinical signs and symptoms indicating either overhydration or underhydration. It should be emphasized that rigorous evaluation of dry weight by these clinical criteria is difficult; nevertheless, we will assume in this chapter that this practice truly achieves a state of overall normohydration. In the ideal case, therefore, maintenance of dry weight during hemodialysis controls the volume of total body water. Fluid gain during the interdialytic interval is distributed into the intracellular and extracellular compartments based on the ratio of dietary sodium to fluid intake.17 For example, if fluid ingested during the intRADialytic interval contains sodium at the same concentration as the postdialysis extracellular fluid compartment, then the ingested fluid will remain entirely within the extracellular compartment. In this case the plasma sodium concentration will remain constant. If the fluid ingested between treatments has a sodium content higher than that contained within the postdialytic extracellular fluid compartment, then the ingested fluid will remain extracellular and will be accompanied by the movement of intracellular water into the extracellular compartment to dilute the excess sodium. If the fluid ingested between dialysis treatments has a low sodium content (the most typical case), then the ingested fluid will be distributed between both the intracellular and extracellular compartments. In the latter case, the plasma sodium concentration decreases during the interdialytic interval; for a fixed sodium intake, the decrease is greater the larger the amount of fluid ingested.17 Another goal of dialysis therapy is therefore to remove sufficient sodium to maintain a volume of extracellular fluid that normalizes blood pressure. Because sodium concentrations in plasma and dialysate are very similar, it is often difficult to determine sodium removal during routine therapy. In practice, the sodium concentration in dialysate is usually fixed to a value that is approximately the desired concentration in plasma after therapy, since equilibrium between plasma and dialysate sodium concentrations is assumed to be achieved by the end of the dialysis treatment. Simply removing a quantity of sodium equal to that ingested between dialysis treatments does not, however, guarantee maintenance of extracellular volume at normal values (see below).

Changes in Intracellular and Extracellular Fluid Volumes

Van Stone et al18,19 determined total body water volume, extracellular fluid volume and plasma volume using RADiolabeled isotopes of H2O, sulfate, and albumin and demonstrated that volume changes in these compartments during hemodialysis were a function of the dialysate sodium concentration. These investigators studied chronic hemodialysis patients at three different dialysate concentrations of sodium: one dialysate sodium concen-


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tration was equal to that in the patient’s plasma, one was 7% less than that in patient plasma and one was 7% more than that in patient plasma. When patients were dialyzed at an ultrafiltration rate of 0.5 L/h using a dialysate sodium concentration equal to that in plasma, fluid was removed during hemodialysis predominantly from the extracellular compartment. When the dialysate sodium concentration was lower than that in patient plasma, fluid was also removed largely from the extracellular compartment; in addition, there was a significant shift of fluid from the extracellular to the intracellular compartment. When the dialysate sodium concentration was higher than that in patient plasma, fluid was removed from both the extracellular and intracellular compartments. This study also demonstrated that decreases in plasma volume due to either a high ultrafiltration rate or a slow vascular refilling rate were lessened when using a high dialysate sodium concentration. Since the use of a high dialysate sodium concentration had been previously shown to reduce hypotension and other symptoms during hemodialysis,20-22 these observations were consistent with the concept that these symptomatic events are volume-related. Subsequent studies showed that these observations are consistent with those predicted by mathematical models of sodium and fluid kinetics during hemodialysis. Kimura et al23 developed a model assuming that sodium and its accompanying anions were the only important osmotic substances in extracellular fluids and that the absolute amount of intracellular osmotic solutes remained constant throughout hemodialysis (see below for more details). This model accurately predicted changes in plasma sodium concentration during hemodialysis when patients were treated with low, normal and high dialysate sodium concentrations.23 Changes in intracellular and extracellular fluid volumes were also estimated by this model with reasonable accuracy, although the measured fluid shifts induced by low dialysate sodium concentrations were significantly smaller than predicted. While the above studies confirm that changes in intracellular and extracellular fluid volumes can be induced by altering dialysate sodium concentration, the conditions used in these experiments do not necessarily apply during maintenance hemodialysis where the dialysate sodium concentration for a given patient is often fixed. The body compartments from which fluid is removed under these latter conditions is instead determined by fluid and sodium intake during the interdialytic interval. This can be explained by assuming that the plasma sodium concentration is always reduced to approximately the same value at the end of each treatment regardless of the predialysis plasma sodium concentration. Consider first the patient who ingests fluid with a sodium concentration equal to that in the extracellular compartment. All ingested fluid will be confined to the extracellular compartment and the plasma sodium concentration will be equal to the dialysate sodium concentration. In this case, fluid will be removed only from the extracellular compartment; this case was experimentally verified by Bauer and Brooks.12 Consider next the same patient who ingests fluids with a sodium concentration less than that of the extracellular compartment. This fluid will be distributed in both the intracellular and extracellular compartments and the predialysis plasma sodium concentration will be less than the dialysate sodium concentration.17 Since the dialysate sodium concentration will be higher than that in plasma, fluid will be removed from both intracellular and extracellular compartments during each hemodialysis treatment. A corresponding scenario applies when this patient ingests fluid with a sodium concentration higher than that in the extracellular compartment. During routine hemodialysis, therefore, the compartment from which fluid is removed is determined by the composition of the fluid the patient ingests in the interdialytic interval.

Changes in Plasma Volume The amount of fluid removed during hemodialysis therapy is roughly equal to the total volume of plasma. If there were no mechanisms for refilling the plasma compartment, plasma

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and blood volume during hemodialysis would drop to fatal levels. The rate of plasma or vascular refilling has been measured in a few studies and was reported to be between 0.25 and 0.56 L/h,24-26 values substantially less than the typical rate of extracorporeal fluid removal during hemodialysis. Therefore, plasma volume can decrease to a variable extent during hemodialysis depending on the vascular refilling rate. The factors that affect the plasma refilling rate are only partially understood. Keshaviah et al27 determined the decrease in plasma volume (calculated from increases in total plasma protein concentration) in uremic dogs and chronic hemodialysis patients undergoing isolated ultrafiltration. For a given ultrafiltration volume, the decrease in plasma volume was greater for higher rates of ultrafiltration. In chronic hemodialysis patients, there was considerable interpatient variability in the degree of intravascular volume depletion at a given ultrafiltration rate. These observations were further analyzed by a mathematical model based on Starling’s hypothesis of transcapillary fluid exchange where the rate of vascular refilling is governed by perturbations of Starling forces induced by the ultrafiltration process. The capillary ultrafiltration coefficient (Kv) was estimated by fitting the observed changes in total plasma protein concentration with predictions from the mathematical model. The Kv value calculated in 14 hemodialysis patients was 4.3 ± 0.7 (SEM) ml/min/mm Hg, a value comparable to that previously reported in the physiology literature. Low values of Kv indicated low vascular refilling rates and were associated with a greater degree of hypovolemia. Schneditz et al28 subsequently described a similar mathematical model, but included a small but finite concentration of protein in the fluid refilling the vascular compartment. These investigators estimated two separate parameters, the initial blood volume and Kv, by fitting their model predictions with observed changes in blood volume during hemodialysis after a short pulse of rapid ultrafiltration. The value of Kv estimated in the 13 patients was 5.3 ± 2.2 (SD) ml/min/mm Hg when normalized to a lean body mass of 50 kg, a value similar to that reported by Keshaviah et al. These investigators also showed that the estimated initial blood volume was 21% higher on average than that predicted from an anthropometric formula, suggesting the presence of fluid overload at the beginning of hemodialysis treatment. Furthermore, they showed that hypovolemia was less for patients with high values of Kv and for patients who were overhydrated. This latter result is consistent with observations by others29,30 that vascular refilling is enhanced in states of overhydration. The extracorporeal ultrafiltration rates employed in the studies by Keshaviah et al and Schneditz et al were greater (>1.2 L/hr) and the duration of ultrafiltration (30-120 min) was shorter than those typically observed during routine hemodialysis. These studies showed an exponential increase in vascular refilling during ultrafiltration because of the rapid increase in plasma oncotic pressure. Changes in blood or plasma volume during routine hemodialysis31-35 do not, however, follow the pattern expected from these experiments. The decrease in plasma volume during maintenance hemodialysis can be linear is some patients; in others, there is no significant decrease in plasma volume for the first 1 or 2 hours of treatment, after which plasma volume can decrease precipitously. The disparity between changes in plasma or blood volume during routine hemodialysis and those predicted by the models of Keshaviah et al and Schneditz et al are illustrated in Figure 1.2. This disparity between model predictions and measured changes in plasma volume during routine hemodialysis was described quantitatively by Tabei et al36 using a mathematical model based on Starling’s hypothesis. These investigators reported that Kv estimated in 14 patients was not constant during the treatment despite a constant extracorporeal ultrafiltration rate of 0.71 ± 0.05 (SEM) L/hr. Instead, Kv decreased dramatically, by a factor of approximately four, during the last three hours of the dialysis session. After one hour of hemodialysis Kv was 6.8 ± 1.3 (SEM) ml/min/mm Hg, and at the end of the session Kv was only 1.6 ± 0.2 (SEM) ml/min/mm Hg; these values bracket those reported by Keshaviah et


7

Fig. 1.2. A schematic illustration of intRADialytic changes in blood volume. Shown in the solid line with squares is the profile expected if intRADialytic increases in plasma oncotic pressure largely govern changes in blood volume as suggested by the studies of Keshaviah et al27 and Schneditz et al.28 Shown in the solid line and in the long and short dashed lines are three typical profiles of intRADialytic changes in blood volume observed in chronic hemodialysis patients.35

al and Schneditz et al. A significant concern with these mathematical models is that quantitative values for several assumed parameters cannot be directly measured in chronic hemodialysis patients. Sensitivity analysis of the assumed parameters by Tabei et al36 showed, however, that the decrease in Kv with time during the treatment was unlikely due to invalid estimates of the initial blood volume or arterial, venous or capillary blood pressure. In a separate preliminary report,37 these investigators showed that the decrease in Kv was not correlated with norepinephrine concentrations since continuos norepinephine infusions during hemodialysis only resulted in a relatively small decrease in Kv. Further analysis of these same data38 showed that the decrease in Kv closely correlated with plasma concentrations of both atrial natriuretic peptide (ANP) and cyclic guanosine 3',5'-monophosphate (cGMP). This observation led these investigators to postulate that the decrease in Kv during hemodialysis was due to a decrease in the water permeability of the capillary wall by ANP. An alternative explanation relates to the relationship between plasma concentrations of both ANP and cGMP and patient volume status or the hydration status of interstitial fluids.39-42 The intRADialytic decrease in ANP and cGMP likely reflects a reduction in hydration status of the interstitial compartment, a factor that has been previously shown to influence the vascular refilling rate.29,30 In these mathematical models such mechanisms would be observed as a decrease in calculated Kv values, since none of the above mathematical models accounts for changes in the hydration status or pressure in the interstitial space. Indeed, more complex mathematical models43 have reported that the capacitance of interstitial fluids can play an important role in altering intRADialytic changes in plasma volume. Alternatively, Lopot and Kotyk44 have argued that Kv only minimally influences intRADialytic

8


changes in plasma volume; instead, they suggest that the compliance of the cardiovascular system is the main determinant of intRADialytic changes in plasma volume. Nevertheless, these changes in vascular compliance must act by altering Starling forces to effect vascular refilling and intRADialytic changes in plasma volume. The above mathematical models have also neglected the effect of simultaneous changes in intracellular and extracellular volumes on intRADialytic changes in plasma volume. Since the plasma is part of the extracellular compartment, one should expect there to be a relationship, albeit complex, between intRADialytic changes in these compartmental volumes. For example, Van Stone et al19 measured changes in plasma, extracellular and total body water volumes using RADioisotopes in six patients during hemodialysis in the absence of ultrafiltration using three different dialysate sodium concentrations. Extracellular and plasma volumes decreased when using the low sodium dialysate (131 ± 1 (SEM) mEq/L); they remained relatively constant when using the normal sodium (141 ± 2 (SEM) mEq/L); and they increased when using the high sodium dialysate (153 ± 2 (SEM) mEq/L). The intRADialytic decrease in plasma volume when using the low sodium dialysate concentration in the absence of ultrafiltration was greater than 6%. These results show that changes in extracellular and intracellular fluid volumes may significantly impact on intRADialytic changes in plasma volume, especially if the latter are less than 10%.

Mathematical Modes of Sodium Kinetics and Fluid Distribution The experiments and models describing changes in fluid volumes during hemodialysis described in the above section have had a major impact on the treatment of chronic hemodialysis patients. Dialysate sodium concentrations have been increased over the past two decades to reduce intRADialytic hypotension and other volume-related symptoms; however, the importance of increasing dialysate sodium concentration on the volume of the extracellular fluid compartment has not been emphasized. We now examine this relationship using a mathematical model.

Models During Hemodialysis Mathematical models of sodium and fluid removal during hemodialysis were originally developed to determine whether changes in dialysate sodium concentration would increase the retention of sodium during hemodialysis therapy. Gotch et al45,46 first proposed a model of sodium and fluid removal during hemodialysis; this model is fundamental and is shown schematically in Figure 1.3. Body fluids are assumed to be distributed between the intracellular and extracellular compartments based on the concentrations of two different osmotic solutes. The extracellular osmotic solute is assumed to be sodium, and sodium is assumed to be distributed uniformly throughout and confined exclusively to the extracellular compartment. Another solute (e.g., potassium) is assumed to be the intracellular osmotic solute; this solute is assumed to be impermeable to cell membranes and therefore confined exclusively to the intracellular compartment. Fluid and solute balance equations for the intracellular and extracellular compartments are the following:45,46

dVi = K f (Ci − Ce ) dt

[Eq. 1.1]


9

Fig. 1.3. A schematic model of fluid and sodium kinetics during hemodialysis. Fluid is distributed between the intracellular and extracellular compartments with volumes Vi and Ve, respectively. The intracellular osmotic solute is confined to the intracellular compartment with a concentration Ci, and sodium is confined to the extracellular compartment with a concentration Ce. The extracorporeal ultrafiltration rate is denoted by Qf, Qs denotes the sodium removal rate and Kf denotes the whole body ultrafiltration coefficient.

dV e = −Q f − K f (Ci − Ce ) dt

[Eq. 1.2]

d(ViCi ) =0 dt

[Eq. 1.3]

d(V eCe ) = −Qs dt

[Eq. 1.4]

where: Vi and Ve = intracellular and extracellular volumes; Ci and Ce = intracellular osmotic solute and plasma sodium concentrations, respectively; Kf = the whole body (or cell membrane) ultrafiltration coefficient, a product of the permeability of cell membranes to water and the effective cell membrane surface area; Qf and Qs = the extracorporeal ultrafiltration rate and the extracorporeal sodium removal rate, respectively. The latter parameter can be calculated from the dialysance of sodium (D) by the following equation:45,46


10

Qs = [D(1− Q f / Qb ) + Q f ]Ce − D(1− Q f / Qb )Cd

[Eq. 1.5]

where: Qb and Cd denote the blood water flow rate and dialysate concentration of sodium. Eqs. [1.1]-[1.5] can only be solved numerically when appropriate initial conditions are specified. Certain algebraic modifications or simplifications to these equations were made by Gotch et al.45 First, Eqs. [1.1] and [1.2] can be added together to yield

dV e dVi dV T + = = −Q f dt dt dt

[Eq. 1.6]

where: VT denotes the sum of intracellular and extracellular volumes or total body water volume. This equation indicates that changes in total body water volume resulting from hemodialysis can be calculated as the amount of total ultrafiltered water or the change in body weight during the treatment. The second simplification made by Gotch et al was to replace extracellular volume with total body water volume in Eq. [1.4]. This was assumed valid because measurements of the apparent or osmotic volume of distribution for sodium approximate total body water, not extracellular volume.47 With this assumption, Eq. [1.4] becomes

d(V TCe ) = −Qs dt

[Eq. 1.7]

The two compartment model for sodium and fluid kinetics was therefore transformed into an apparent single compartment model. Eqs. [1.5]-[1.7] were then solved analytically to yield the following equation describing the dependence of plasma sodium concentration on time during hemodialysis:

(C (t) − C ) / (C (0) − C ) = [V e

d

e

d

D(1/ Q f −1/ Q b )

T

(t) / V T (0)]

[Eq. 1.8]

This equation was shown to accurately predict the postdialysis plasma sodium concentration from the initial plasma sodium concentration, the dialysate sodium concentration, the initial volume of total body water (assumed to be equal to the urea distribution volume), the hemodialysis operating conditions and the change in body weight


11

during hemodialysis. Petitclerc et al48 derived Eq. [1.8] using a similar approach; however, these investigators assumed that VT could be substituted for V e in Eq. [1.4] because the intracellular osmotic solute was impermeable to the cell membrane. While this latter approach simplifies the model, it does not identify the key assumption in the derivation of Eq. [1.8]. Kimura et al23 derived a two compartment sodium kinetic model in finite difference, instead of differential, form that is essentially equivalent to that derived by Gotch et al. Instead of assuming that the apparent sodium distribution volume could be substituted for the true extracellular distribution volume for sodium, these investigators assumed that osmotic equilibrium was maintained at all times between intracellular and extracellular fluids. These investigators then solved the finite difference equations using a computer and showed that the change in plasma sodium concentration during hemodialysis using low, normal and high dialysate sodium concentrations could also be well predicted by their model. This result is not very surprising since the models of Gotch et al, Petitclerc et al and Kimura et al are all equivalent, as we now show. The solution to Eqs. [1.1]-[1.5] depends significantly on the parameter ε, which is defined as

ε = D / K fCd

[Eq. 1.9]

This parameter is a ratio of the time scale for osmotic equilibration between the intracellular and extracellular compartments (τoe = VT(0)/KfCd) and the time scale for sodium diffusional equilibrium within the extracellular compartment (τde = VT(0)/D) during hemodialysis. When ε is small, then osmotic equilibrium is rapid, as assumed by Kimura et al. We have shown elsewhere using perturbation analysis49 that when ε is very small or zero, Eq. [1.1] can be approximated by

Ci = Ce

[Eq. 1.10]

Substituting Eq. [1.10] into [1.3] and adding together Eqs. [1.3] and [1.4], the result is Eq. [1.7]. This analysis shows that the assumption that the apparent distribution volume for sodium can be used to evaluate changes in plasma sodium concentration45 is equivalent to that assuming rapid osmotic equilibration between the intracellular and extracellular compartments.23 The amount of sodium removal during a treatment session can be simply calculated using the above model from the change in plasma sodium concentration and the change in total body water (i.e., the apparent sodium distribution volume).50 It should be noted, however, that this model cannot be used to determine total body content of sodium, VeCe, since it does not provide a unique value for extracellular fluid volume. The inability of the above model to uniquely predict intracellular and extracellular fluid volumes in absolute terms limits, to some extent, its routine application. More complex models of sodium kinetics that attempt to more accurately determine changes in plasma sodium concentration during hemodialysis have been recently formulated. Heineken et al51 measured changes in plasma sodium concentration during hemodialysis when the dialysate sodium concentration was abruptly changed to very high (170 mEq/L) and to very low (110 mEq/L) values. They showed that the changes in plasma sodium concentration could not be explained using a single compartment sodium kinetic model assuming rapid osmotic equilibrium between the intracellular and extracellular fluid compartments. To fit the experimental data, these investigators developed a more complex

12


model assuming that both sodium and urea were osmotically active across the cell membrane and reported excellent agreement between model predictions and experimental measurements of plasma sodium concentration. A surprising result was that the estimated value of the whole body ultrafiltration coefficient or Kf was exceedingly low (0.22 ml/min/ mm Hg); indeed, this value is even less than previous estimates of the capillary ultrafiltration coefficient of approximately 5 ml/min/mm Hg (see above). Calculation of such a low value was likely due to their assumption that urea is equivalent to sodium as an osmotic solute across cell membranes. This is not a valid assumption, since urea is highly permeable across cell membranes and its osmotic reflection coefficient across the cell membrane is much less than one, even though integral membrane transport proteins are usually required for movement of urea across cell membranes. On the other hand, the osmotic reflection coefficient for sodium should be approximately equal to one. It should be emphasized that direct experimental evidence for a significant osmotic effect of urea on body fluid distribution during maintenance hemodialysis is lacking. Indeed, Van Stone et al19 found virtually no change in extracellular volume when chronic renal failure patients were dialyzed (causing a substantial and rapid decrease in plasma urea concentration) using an approximately isonatremic dialysis solution in the absence of ultrafiltration. While urea may play a significant osmotic role when its concentration is very high, as in the disequilibrium syndrome,52 the osmotic role of urea is likely small during routine hemodialysis. Ursino et al53 have used a similar kinetic model to describe sodium kinetics during hemodialysis using either a constant or variable dialysate sodium concentration, except they have assumed osmotic equilibrium across the cell membrane. A similar concern over this model exists regarding the validity of assuming that urea is an effective osmotic solute across cell membranes. Ahrenholz et al54 observed that intRADialytic changes in plasma volume were dependent on the dialysate sodium concentration. These investigators proposed a two compartment sodium kinetic model comprising the plasma and the rest of total body water that provided a good fit between theoretical predictions and their experimental observations. Their model assumed that differences in sodium concentration across the capillary wall induced an osmotic flow into plasma. This assumption is physiologically untenable, however, since the osmotic reflection coefficient for sodium across the capillary wall (except perhaps in the brain) is very small.55 Further experimental and modeling efforts are needed to clarify additional factors which alter fluid exchange between the intracellular and extracellular compartments during hemodialysis.

Models in the Interdialytic Interval In order to produce a model for illustrating the role of important dialysis and dietary parameters on plasma sodium concentration and body fluid distribution, a mathematical model of sodium and fluid kinetics in the interdialytic interval is necessary. Eqs. [1.1]-[1.4] and [1.6]-[1.7] remain valid during the interdialytic interval except that fluid intake, denoted by Nf, and sodium intake, denoted by Ns, replace and are opposite in sign to Qf and Qs, respectively. We also assume that Nf and Ns are constants and that changes in body sodium and fluid from sweat, stool and urinary losses are negligible. With these assumptions, changes in plasma sodium concentration and extracellular volume during the interdialytic interval can be described by the following equations:


13

Ce (θ ) =

V T (0)Ce (0) + Nsθ V T (0) + Nfθ

[Eq. 1.11]

V e (θ ) =

V e (0)Ce (0) + Nsθ Ce (θ )

[Eq. 1.12]

The variables in Eqs. [1.11] and [1.12] are functions of time during the interdialytic interval (θ). To calculate a time-averaged value of Ve, we have used the arithmetic mean of the postdialysis and predialysis values. While an analytical expression for time-averaged Ve can be derived by integrating Eq. [1.12] over the interdialytic interval, the result is algebraically complex, and we have not found that the algebraic expression differs by more than a small fraction of that calculated by simply averaging the predialysis and postdialysis values. This general approach for modeling sodium and fluid kinetics during the interdialytic interval is similar to that first described by Kimura and Gotch.17

Model of a Chronic Hemodialysis Patient A mathematical model describing sodium and fluid kinetics for a hemodialysis patient with constant sodium and fluid intake can be constructed by combining the above models during the intRADialytic and interdialytic intervals. For illustrative purposes, we assume that the patient was always dialyzed to the same postdialysis total body water volume of 42 L and that conventional hemodialysis was performed for 4 hours, 3 times per week. The blood water flow rate was assumed to be 350 ml/min, and the dialyzer and the dialysate flow rate were such that the sodium dialysance was 270 ml/min. These values are characteristic of a typical hemodialysis session using a dialyzer containing a membrane with surface area of 1.5-2.0 m2. In these simulations the plasma sodium concentration was not altered, but the dialysate sodium concentration was divided by 1.0323 to account for differences between sodium activity and concentration (see above). These treatment sessions were assumed to be symmetrically positioned throughout the week with equal interdialytic intervals, and the patient was simulated to steady state conditions over 15 consecutive treatment sessions using the above equations. Eq. [1.12] shows that this model is not truly a single compartment model, since the initial or target postdialysis extracellular fluid volume and plasma sodium concentration must be assumed. We have assumed these values as 14 L and 140 mEq/L, respectively. Because of these required assumptions, the calculated values of extracellular volume are not absolute in magnitude. Nevertheless, relative differences in extracellular volume effectively represent the effect of altering dialysis and dietary parameters on plasma sodium concentration and body fluid distribution. Several general conclusions regarding sodium and fluid kinetics can be made from experience with this model. First, sodium and fluid removal during hemodialysis equals sodium and fluid intake during the interdialytic interval. This relationship must hold because the patient is assumed to be at steady state. This does not imply, however, that the patient has a fixed postdialysis extracellular volume equal to the target value. Second, the compartment from which fluid is removed during hemodialysis depends on the relationship between the dialysate sodium concentration and the sodium concentration of fluids ingested during the interdialytic interval. If the sodium concentration of fluid ingested during the interdialytic interval is hypotonic compared with the dialysate sodium concentration, then fluid removal during hemodialysis will be derived from both the intracellular and extracellular compartments. If, however, the sodium concentration of fluid

14


ingested during the interdialytic interval is hypertonic compared with the dialysate sodium concentration, then fluid removal during hemodialysis will be derived exclusively from the extracellular compartment and there will be a concurrent flux of fluid from the extracellular to the intracellular compartment. These results concur with predictions stated above from general principles. Third, the value of the assumed urea removal index, Kt/V (see chapter 2), has only a minor effect on the calculated results. Figures 1.4-1.7 show examples of results of these simulations for fluid intake rates of 1-3 L/day, sodium intake rates of 2-8 g/day and dialysate sodium concentrations of 135-145 mEq/L. Figure 1.4 shows the intRADialytic decrease in extracellular volume predicted for a constant fluid intake of 2 L/day for several sodium intake levels and dialysate sodium concentrations. The intRADialytic decrease in extracellular volume is greater for higher sodium intake and lower dialysate sodium concentrations. This effect of dialysate sodium concentration on the intRADialytic decrease in extracellular volume is identical to that shown empirically by Van Stone et al. 18,19 Interestingly, the level of sodium intake has only a moderate influence on the intRADialytic decrease in extracellular volume until high levels (i.e., 8 g/day) are reached. Figure 1.5 shows the effect of these same dietary and dialysis parameters on time-averaged values of extracellular volume. Time-averaged extracellular volume is greater with a higher sodium intake and a higher dialysate sodium concentration. This example suggests that the effect of an increase in sodium intake by 3 g/day on timeaveraged extracellular volume is approximately equal to that for an increase in the dialysate sodium concentration by 5 mEq/L. Figures 1.6 and 1.7 show the effect of changing sodium and fluid intake on the intRADialytic decrease in and time-averaged extracellular volume at a fixed dialysate sodium concentration of 140 mEq/L. Figure 1.6 shows that the intRADialytic decrease in extracellular volume is greater for both higher sodium and higher fluid intake. These results are not unexpected for a steady state patient, since sodium and fluid removal rates increase in parallel with increases in sodium and fluid intake. The effect of fluid intake on timeaveraged extracellular volume shown in Figure 1.7 is minimal; actually, time-averaged extracellular volume decreases with higher fluid intake in this example. This is because excess body fluid remains predominantly intracellular and the higher fluid removal rate results in enhanced sodium removal. Similar results have been observed using other parameter combinations. These simulations show that treatment by hemodialysis leads to a decrease in extracellular volume and that this decrease is a complex function of sodium intake, fluid intake, and the dialysate sodium concentration. Although not demonstrated here, the intRADialytic decrease in extracellular volume is also a strong function of the extracorporeal ultrafiltration rate. These conclusions are consistent with clinical experience in the dialysis unit and reflect the necessity to remove the sodium and fluid ingested between hemodialysis sessions. The importance of low levels of sodium and fluid intake in ameliorating some of the adverse effects of the treatment, including large shifts in fluid volume, have long been advocated. The effect of these dietary and dialysis parameters on time-averaged extracellular volume contrast with those used to treat the routine hemodialysis patient. It is generally considered that the dialysate sodium concentration should be set to the target postdialysis plasma sodium concentration. If equilibration between the dialysate and plasma sodium concentrations and the prescribed postdialysis body weight were always achieved, then each and every dialysis treatment would lower extracellular volume to a fixed value that would be independent of sodium and fluid intake. Our simulations show, however, that true equilibration between the plasma and dialysate sodium concentrations is never truly achieved, even though the initial difference in concentration is never very large. Since plasma sodium concentration approaches the dialysate sodium concentration similarly to the manner


15

in which plasma urea concentration approaches zero (compare Eq. [1.8] with Eq. [2.25]), it is apparent that equilibration is only 60-70% complete. Further, a small difference between plasma and dialysate sodium concentrations can produce a significant change in extracellular volume. Because of this disequilibration between plasma and dialysate sodium concentrations at the end of treatment, this analysis shows that both sodium and fluid intake rates can influence the volume of extracellular fluids immediately postdialysis and during the interdialytic interval, even though a constant postdialysis body weight is achieved.

Control of Extracellular Volume and Blood Pressure The above simulations show that time-averaged extracellular volume is a complex function of sodium intake, fluid intake, dialysate sodium concentration and postdialysis body weight, yet the importance of these fundamental relationships is not well appreciated. This is rather surprising because of the known relationship between time-averaged extracellular volume and blood pressure in hemodialysis patients.56 Although the studies by Van Stone et al18,19 showed that the use of high dialysate sodium concentration resulted in increased extracellular fluid volume at the end of hemodialysis, the importance of such a high postdialysis extracellular volume in contributing to hypertension is generally not acknowledged. Indeed, the high dialysate sodium concentrations routinely used to maintain hemodynamic stability during hemodialysis treatments can lead to increased thirst57 and higher blood pressure, especially in hypertensive patients.58 The latter observation is likely a direct result of the increase in time-averaged extracellular volume when increasing dialysate sodium concentration as shown in the above simulations (Fig. 1.5). Further evidence for such a relationship has recently been reported in studies reporting an effect of dialysate sodium concentration on the blood pressure of chronic hemodialysis patients.59,60 It is important to note that fluid intake per se has little effect on time-averaged extracellular volume. Although fluid and sodium intake rates are often correlated in chronic hemodialysis patients, this model shows that sodium intake, not fluid intake, largely determines time-averaged extracellular volume and presumably blood pressure. Recognizing that rigorous control of extracellular volume is a mainstay of hemodialysis therapy, the above simulations suggest that dialysate sodium concentrations should ideally be individualized for each patient, depending on sodium intake and fluid intake. One approach for individualizing dialysate sodium concentration would be to achieve a target postdialysis plasma sodium concentration, perhaps by noninvasive monitoring of plasma water conductivity.61,62 This approach would only produce a target postdialysis extracellular fluid volume, not a time-averaged value. It would be possible using a mathematical model to extend this approach to target time-averaged extracellular fluid volume if sodium intake could be routinely determined by measuring predialysis and postdialysis plasma sodium concentrations.50 A difficulty with any such approach is that the target extracellular fluid volume for each patient would not be known a priori and would need to be determined by incrementally decreasing the dialysate sodium concentration. A decrease in dialysate sodium concentration to lower extracellular volume may, however, lead to poor hemodynamic stability during treatment; thus, lowering dialysate sodium concentration would have to be performed cautiously. Further, a strategy for decreasing dialysate sodium concentration would need to be empirical, perhaps until a practical mathematical model for predicting intRADialytic changes in blood volume can be developed. An alternative guide for lowering dialysate sodium concentration is direct measurement of extracellular fluid volume using multifrequency bioimpedance spectroscopy in these patients. This latter approach for optimizing dry weight (or more precisely, extracellular volume) determinations is currently being evaluated.63

16


Fig. 1.4. IntRADialytic decrease in extracellular volume (ECV) for a hypothetical patient with a postdialysis total body water volume of 42 L after a 4 hour hemodialysis treatment using a dialyzer and dialysate flow rate such that sodium dialysance was 270 ml/min and the blood water flow rate was 350 ml/min. The patient was at steady state, and fluid intake was assumed to be 2 L/day. Results are shown for various dialysate sodium concentrations and for sodium intakes of 2 (black bars), 5 (white bars), and 8 (gray bars) g/day.

Modeling Changes in Blood Volume During Hemodialysis Mathematical modeling of changes in blood volume during hemodialysis is of high interest because this parameter is related to symptomatic hypotension and other intRADialytic morbid events7 and because of recent technical developments that permit continuous, real-time measurement of intRADialytic changes in blood or plasma volume.64 As discussed above, previous mathematical models do not accurately describe intRADialytic changes in blood volume during hemodialysis because they only account for changes in Starling forces in plasma during the treatment. Besides the Starling forces in plasma, changes in intracellular and extracellular fluid compartment volumes and changes in Starling forces in interstitial tissues are also important. Several additional factors described below have also been proposed as important in evaluating intRADialytic changes in blood volume. The details of modeling intRADialytic changes in blood volume are beyond the scope of this review. The most advanced mathematical models to date have not completely accounted for all of the above factors;43,65,66 this is an active and fruitful area for future research. To determine intRADialytic changes in blood volume from continuous measurements of hematocrit or plasma protein concentration, it must be assumed that there is no loss from or addition of red blood cells or plasma proteins within the circulation. For example, Yu et al67 showed that intRADialytic hypovolemia induced a release of red blood cells from the splanchnic, and perhaps the splenic, vascular beds that increased the total number of circulating red blood cells during treatment. This concern might be heightened in hemodialysis patients who are performing exercise during treatments, since exercise in normal individuals has been shown to induce a significant splenic release of red blood cells.68 A corresponding concern exists when monitoring plasma protein concentrations since it is


17

Fig. 1.5. Time-averaged extracellular volume (ECV) during the interdialytic interval for the patient and treatment schedule described in the legend to Figure 1.4.

known that plasma proteins, in particular albumin, are partially permeable across the capillary wall. Indeed, Schneditz et al28 suggested that there is a net increase in protein content within the circulation as the result of extracorporeal ultrafiltration due to a small, but finite, protein content in the fluid entering the circulation during vascular refilling. Incorporation of these effects into a mathematical model is difficult. A second, and perhaps more significant, concern when modeling intRADialytic changes in blood volume is the assumption that circulating red blood cells are evenly distributed throughout the vasculature. This assumption is clearly false, but the extent to which changes in blood flow distribution during hemodialysis alter hematocrit in different parts of the body is unknown. It has long been known that the hematocrit in the microcirculation is lower than that in large blood vessels and that the measured hematocrit varies among different organs.69 Any change in blood flow distribution, therefore, could produce a change in hematocrit in a peripheral blood vessel without any change in total blood volume. For example, Lundvall and Lindgren70 have recently shown that changes in hematocrit in a peripheral artery upon standing from the prone position can be analyzed as if the body behaves as a two compartment system. Changes in arterial hematocrit did not reflect overall changes in plasma volume since dependent tissues have higher blood flows than other tissues; thus, the true change in plasma volume was underestimated when calculated from the change in arterial hematocrit. While previous work has suggested that blood flow distribution to various organs is altered during hemodialysis,71 the effect of this phenomenon on intRADialytic changes in hematocrit measured in a peripheral blood vessel is unknown. Evidence suggesting these effects to be unimportant has recently been reported in preliminary form,72 where intRADialytic changes in blood volume measured from changes in total plasma protein concentration using ultrasound technology were not different from those measured from changes in hematocrit.

18


Fig. 1.6. IntRADialytic decrease in extracellular volume (ECV) for the patient and treatment schedule described in the legend to Figure 1.4. In this case, the dialysate sodium concentration was fixed at 140 mEq/L. Results are shown for various rates of fluid intake and for sodium intakes of 2 (black bars), 5 (white bars), and 8 (gray bars) g/day.

Despite the theoretical possibility, evidence to date does not support an important role for the addition of red blood cells or protein to the circulation nor for the importance of changes in blood flow distribution in governing intRADialytic changes in hematocrit or plasma protein concentration. As recently stated by Lundvall and Lindgren,70 “this problem (the validity of hemogloblin or hematocrit as a marker of changes in plasma volume) has resisted a solution for almost a century.” It appears that further discussion of this issue without additional data would be nonproductive. The present discussion suggests that two features are essential in the development of a future mathematical model of blood volume changes during hemodialysis. First, changes in Starling forces, both intravascularly and extravascularly, need to be included. Inclusion of the latter phenomena have only been incorporated into such models recently43,66 and need further development. Second, changes in extracellular volume need to be taken into account. The models of Kimura et al65 and Ursino et al53 attempt to account for this phenomenon, but the latter model likely does not accurately describe intRADialytic changes in extracellular volume for reasons described above. Only after these factors are added to the existing models of Keshaviah et al27 and Schneditz et al28 can the development of more meaningful and practical mathematical models occur.

Conclusion One of the main indications for hemodialysis therapy is the removal of excess body fluid and the accompanying extracellular sodium ions; yet, optimal methods for removing fluid and sodium during routine hemodialysis remain elusive. Mathematical models of fluid removal are coupled to those of sodium kinetics and require knowledge of the distinct compartmentalization of total body fluid and sodium content. This review shows that


19

Fig. 1.7. Time-averaged extracellular volume (ECV) during the interdialytic interval for the patient and treatment schedule described in the legend to Figure 1.6.

routine use of high dialysate sodium concentrations during maintenance hemodialysis is a double-edged sword. While high dialysate sodium concentrations improve hemodynamic stability during the treatment, they can also increase time-averaged extracellular volume and presumably blood pressure. This important role of dialysate sodium concentration in regulating time-averaged extracellular volume, however, is largely ignored in algorithms that attempt to optimize intRADialytic changes in blood volume.73,74 Further developments in modeling fluid removal and sodium kinetics are needed to show how to individualize dialysate sodium concentration and optimize intRADialytic hemodynamic stability without increasing time-averaged extracellular fluid volume.

Notation Cd Ce Ci D Kf Kv Qb Qf Qs SD SEM t Ve Vi

sodium concentration in dialysate sodium concentration in extracellular fluid and plasma concentration of the intracellular osmotic solute dialysance of sodium whole body ultrafiltration coefficient governing fluid movement between the intracellular and the extracellular compartments capillary ultrafiltration coefficient blood water flow rate in the hemodialysis circuit ultrafiltration rate in the hemodialysis circuit solute removal rate in the hemodialysis circuit standard deviation standard error of the mean time during the dialysis treatment extracellular fluid volume intracellular fluid volume


20

VT total body water volume Greek: ε dimensionless parameter defined in Eq. [1.9] θ time during the interdialytic interval τoe time scale for osmotic equilibrium between intracellular and extracellular compartments τde time scale for sodium diffusional equilibrium within the extracellular compartment

References 1. Pitts RF. Physiology of the Kidney and Body Fluids. 3rd ed. Chicago: Year Book Medical, 1974. 2. Valtin H, Schafer JA. Renal Function. 3rd ed. Boston: Little Brown and Company, 1995. 3. Hegstrom RM, Murray JS, Pendras JP et al. Two year’s experience with periodic hemodialysis in the treatment of chronic uremia. Trans Am Soc Artif Intern Organs 1962; 8:266-280. 4. Thomson GE, Waterhouse K, McDonald HP Jr et al. Hemodialysis for chronic renal failure. Arch Int Med 1967; 120:153-167. 5. Vertes V, Cangiano JL, Berman LB et al. Hypertension in end-stage renal disease. N Engl J Med 1969; 280:978-981. 6. Cheung AK. Stages of future technological developments in haemodialysis. Nephrol Dial Transplant 1996; 11[Suppl 8]:52-58. 7. Leypoldt JK, Cheung AK. Evaluating volume status in hemodialysis patients. Adv Ren Repl Ther 1998; 5:64-74. 8. Aris R. Mathematical Modelling Techniques. London: Pitman, 1978. 9. Fannestil DD. Compartmentation of body water. In: Narins RG, ed. Clinical disorders of fluid and electrolyte metabolism. 5th ed. New York: McGraw-Hill, 1994:3-20. 10. van Marken Lichtenbelt WD, Snel YEM et al. Deuterium and bromide dilution, and bioimpedance spectrometry independently show that growth hormone-deficient adults have an enlarged extracellular water compartment related to intracellular water. J Clin Endcrinol Metab 1997; 82:907-911. 11. de Lorenzo A, Andreoli A, Matthie J et al. Predicting body cell mass with bioimpedance by using theoretical methods: A technological review. J Appl Physiol 1997; 82:1542-1558. 12. Bauer JH, Brooks CS. Body fluid composition in chronic renal failure. Clin Nephrol 1981; 16:114-118. 13. Kumar S, Berl T. Sodium. Lancet 1998; 352:220-228. 14. Gotch FA, Evans MC, Keen ML. Measurement of the effective dialyzer Na diffusion gRADient in vitro and in vivo. Trans Am Soc Artif Intern Organs 1985; 31:354-357. 15. Locatelli F, Ponti R, Pedrini L et al. sodium kinetics and dialysis performances. Contrib Nephrol 1989; 70:260-266. 16. Flannigan MJ. Sodium flux and dialysate sodium in hemodialysis. Semin Dial 1998; 11:298-304. 17. Kimura G, Gotch FA. Serum sodium concentration and body fluid distribution during interdialysis: Importance of sodium to fluid intake ratio in hemodialysis patients. Int J Artif Organs 1984; 7:331-336. 18. Van Stone JC, Bauer J, Carey J. The effect of dialysate sodium concentration on body fluid distribution during hemodialysis. Trans Am Soc Artif Intern Organs 1980; 26:383-386. 19. Van Stone JC, Bauer J, Carey J. The effect of dialysate sodium concentration on body fluid compartment volume, plasma renin activity and plasma aldosterone concentration in chronic hemodialysis patients. Am J Kidney Dis 1982; 2:58-64. 20. Stewart WK, Fleming LW, Manuel MA. Muscle cramps during maintenance haemodialysis. Lancet 1972; i:1049-1051. 21. Port FK, Johnson WJ, Klass DW. Prevention of dialysis disequilibrium syndrome by use of high sodium concentration in the dialysate. Kidney Int 1973; 3:327-333.


21

22. Wehle B, Asaba H, Castenfors J et al. The influence of dialysis fluid composition on the blood pressure response during dialysis. Clin Nephrol 1978; 10:62-66. 23. Kimura G, Van Stone JC, Bauer JH et al. A simulation study on transcellular fluid shifts induced by hemodialysis. Kidney Int 1983; 24:542-548. 24. Kim KE, Neff M, Cohen B et al. Blood volume changes and hypotension during hemodialysis. Trans Am Soc Artif Intern Organs 1970; 16:508-514. 25. Rouby JJ, Rottembourg J, Durande JP et al. Importance of plasma refilling rate in the genesis of hypovolaemic hypotension during regular dialysis and controlled sequential ultrafiltration-hemodialysis. Proc Eur Dial Transplant Assoc 1978; 15:239-244. 26. Swartz RD, Somermeyer MG, Hsu CH. Preservation of plasma volume during hemodialysis depends on dialysate osmolality. Am J Nephrol 1982; 2:189-194. 27. Keshaviah PR, Ilstrup KM, Shapiro FL. Dynamics of vascular refilling. In: Atsumi K, Maekawa M, Ota K, eds. Progress in Artificial Organs-1983. Cleveland: ISAO Press, 1984:506-510. 28. Schneditz D, Roob J, Oswald M et al. Nature and rate of vascular refilling during hemodiaysis and ultrafiltration. Kidney Int 1992; 42:1425-1433. 29. Koomans HA, Geers AB, Mees EJD. Plasma volume recovery after ultrafiltration in patients with chronic renal failure. Kidney Int 1984; 26:848-854. 30. Wizemann V, Leibinger A, Mueller K et al. Influence of hydration state on plasma volume changes during ultrafiltration. Artif Organs 1995; 19:416-419. 31. de Vries J-PPM, Olthof CG, Visser V et al. Continuous measuremnt of blood volume during hemodialysis by an optical method. ASAIO J 1992; 38:M181-M185. 32. de Vries JPPM, Donker AJM, de Vries PMJM. Prevention of hypovolemia-induced hypotension during hemodialysis by means of an optical reflection method. Int J Artif Organs 1994; 17:209-214. 33. Bogaard HJ, de Vries JPPM, de Vries PMJM. Assessment of refill and hypovolaemia by continuous surveillance of blood volume and extracellular fluid volume. Nephrol Dial Transplant 1994; 9:1283-1287. 34. Leypoldt JK, Cheung AK, Steuer RR et al. Determination of circulating blood volume by continuously monitoring hematocrit during hemodialysis. J Am Soc Nephrol 1995; 6:214-219. 35. Lopot F, Kotyk P, Bláha J et al. Use of continuous blood volume monitoring to detect inadequately high dry weight. Int J Artif Organs 1996; 19:411-414. 36. Tabei K, Nagashima H, Imura O et al. An index of plasma refilling in hemodialysis patients. Nephron 1996; 74:266-274. 37. Tabei K, Sakurai T, Iimura O et al. Effect of noRADrenaline (NA) on water permeability coefficient (Lpp) in hemodialysis (HD) patients. [Abstract]. J Am Soc Nephrol 1994; 5:529. 38. Iimura O, Tabei K, Nagashima H et al. A study of regulating factors on plasma refilling during hemodialysis. Nephron 1996; 74:19-25. 39. Saxenhofer H, Gnädinger MP, Weidmann P et al. Plasma levels and dialysance of atrial natriuretic peptide in terminal renal failure. Kidney Int 1987; 32:554-561. 40. Lauster F, Gerzer R, Weil J et al. Assessment of dry body-weight in haemodialysis patients by the biochemical marker cGMP. Nephrol Dial Transplant 1990; 5:356-361. 41. Lauster F, Fülle H-J, Gerzer R et al. The postdialytic plasma cyclic guanosine 3':5'-monophosphate level as a measure of fluid overload in chronic hemodialysis. J Am Soc Nephrol 1992; 2:1451-1454. 42. Fishbane S, Natke E, Maesaka JK. Role of volume overload in dialysis-refractory hypertension. Am J Kidney Dis 1996; 28:257-261. 43. Ursino M, Innocenti M. Mathematical investigation of some physiological factors involved in hemodialysis hypotension. Artif Organs 1997; 21:891-902. 44. Lopot F, Kotyk P. Computational analysis of blood volume dynamics during hemodialysis. Int J Artif Organs 1997; 20:91-95. 45. Gotch FA, Lam MA, Prowitt M et al. Preliminary clinical results with a sodium-volume modeling of hemodialysis therapy. Proc Dial Transplant Forum 1980; 10:12-16.

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46. Sargent JA, Gotch FA. Principles and biophysics of dialysis. In: Jacobs C, Kjellstrand CM, Koch KM et al, eds. Replacement of Renal Function by Dialysis. 4th ed. Dordrecht: Kluwer Academic, 1996:34-102. 47. Wolf AV, McDowell ME. Apparent and osmotic volumes of distribution of sodium, chloride, sulfate and urea. Am J Physiol 1954; 176:207-212. 48. Petitclerc T, Man N-K, Funck-Brentano J-L. Sodium modeling during hemodialysis: A new approach. Artif Organs 1984; 8:418-422. 49. Leypoldt JK, Cheung AK. Extracellular volume in nocturnal hemodialysis. Semin Dia 1999; 12[Suppl 1]:S50-S54. 50. Kimura G, Van Stone JC, Bauer JH. The amount of sodium removed by hemodialysis. Am J Kidney Dis 1986; 8:253-256. 51. Heineken FG, Evans MC, Keen ML et al. Intercompartmental fluid shifts in hemodialysis patients. Biotechnol Prog 1987; 3:69-73. 52. Ross EA, Barri YMH. Hemodialysis. In: Tisher CC, Wilcox CS, eds. Nephrology. 3rd ed. Baltimore: Williams & Wilkins, 1995:228-240. 53. Ursino M, Coli L, La Manna G et al. A simple mathematical model of intRADialytic sodium kinetics: “In vivo” validation during hemodialysis with constant or variable sodium. Int J Artif Organs 1996; 19:393-403. 54. Ahrenholz P, Falkenhagen D, Hähling D et al. Measurements of plasma colloid osmotic pressure, total protein and sodium concentration during haemodialysis: Can single-pool sodium modelling explain the results? Blood Purif 1990; 8:199-207. 55. Chrone C, Levitt DG. Capillary permeability to small solutes. In: Renkin EM, Michel CC, eds. Handbook of Physiology, Section 2: The Cardiovascular System, Volume IV. Microcirculation, part 1. Bethesda: American Physiology Society, 1984:411-466. 56. Charra B, Bergström J, Scribner BH. Blood pressure control in dialysis patients: Importance of the lag phenomenon. Am J Kidney Dis 1998; 32:720-724. 57. Henrich WL, Woodard TD, McPhaul JJ Jr. The chronic efficacy and safety of high sodium dialysate: Double-blind, crossover study. Am J Kidney Dis 1982; 2:349-353. 58. Cybulsky AVE, Matni A, Hollomby DJ. Effects of high sodium dialysate during maintenance hemodialysis. Nephron 1985; 41:57-61. 59. Flanigan MJ, Khairullah QT, Lim VS. Dialysate sodium delivery can alter chronic blood pressure management. Am J Kidney Dis 1997; 29:383-391. 60. Sang GLS, Kovithavongs C, Ulan R et al. Sodium ramping in hemodialysis: A study of beneficial and adverse effects. Am J Kidney Dis 1997; 29:669-677. 61. Pedrini LA, Ponti R, Faranna P, et al. Sodium modeling in hemodiafiltration. Kidney Int 1991; 40:525-532. 62. Locatelli F, Di Filippo S, Manzoni C et al. Monitoring sodium removal and delivered dialysis by conductivity. Int J Artif Organs 1995; 18:716-721. 63. Katzarski KS, Divino-Filho JC, Bergstrom J. Importance of the removal of fluid excess on blood pressure control in hemodialysis patients. [Abstract]. J Am Soc Nephrol 1998; 9:254A. 64. Schneditz D, Levin NW. Non-invasive blood volume montoring during hemodialysis: Technical and physiological aspects. Semin Dial 1997; 10:166-169. 65. Kimura G, Van Stone JC, Bauer JH. Model prediction of plasma volume change induced by hemodialysis. J Lab Clin Med 1984; 104:932-938. 66. Ursino M, Innocenti M. Modeling arterial hypotension during hemodialysis. Artif Organs 1997; 21:873-890. 67. Yu AW, Nawab ZM, Barnes WE et al. Splanchnic erythrocyte content decreases during hemodialysis: A new compensatory mechanism for hypovolemia. Kidney Int 1997; 51:1986-1990. 68. Laub M, Hvid-Jacobsen K, Hovind P et al. Splen emptying and venous hematocrit in humans during exercise. J Appl Physiol 1993; 74:1024-1026. 69. Albert SN, Jain SC, Shibuya J et al. The Hematocrit in Clinical Practice. Springfield: Charles A. Thomas, 1965. 70. Lundvall J, Lindgren P. F-cell shift and protein loss strongly affect validity of PV reductions indicated by Hb/Hct and plasma proteins. J Appl Physiol 1998; 84:822-829.


23

71. Chaignon M, Chen WT, Tarazi RC et al. Effect of hemodialysis on blood volume distribution and cardiac output. Hypertension 1981; 3:327-332. 72. Schneditz D, Chamney PW, Greenwood RN et al. Relative blood volume changes during hemodialysis (HD) measured by optical and ultrasonic techniques. [Abstract]. J Am Soc Nephrol 1997; 8:172A. 73. Bonomini V, Coli L, Scolari MP. Profiling dialysis: A new approach to dialysis intolerance. Nephron 1997; 75:1-6. 74. Santoro A, Mancini E, Paolini F et al. Blood volume regulation during hemodialysis. Am J Kidney Dis 1998; 32:739-748.

CHAPTER 2

Urea Removal During Hemodialysis Daniel Schneditz

U

rea evolved as the carrier for the excretion of nitrogen in most mammals, including man, probably because it is relatively inert, highly soluble in water, and highly permeable across membranes. These features are important for transport in biological systems. Urea appears in body water in significant amounts as a result of the synthesis and degRADation of proteins and enzymes which represent the dominant structural and functional entities of the organism. Urea is relatively stable and easily analyzed by different laboratory techniques.1 These characteristics make urea a unique marker for hemodialysis. In order to monitor hemodialysis using urea as a marker, the effects of hemodialysis on urea concentration in the blood, in the tissues, and in the whole body have to be known, and they will be discussed in this chapter. Solute removal from the body during dialysis is determined by flow and diffusion. The term dialysis implies that diffusion is the dominant mode of solute transport across the membrane within the artificial kidney. However, the transport of solutes from the tissues into dialysate, crossing a characteristic distance of meters, is governed largely by forced convection. In a multistep process, the overall rate of the process is limited by the slowest step in the sequence. Thus, in high efficiency dialysis the rate of small solute elimination is controlled by convective transport. The common cause of flow limitation is recirculation, which is extensively discussed in this chapter. The transport of substances added to the body, their distribution, transformation, and elimination are studied by pharmacokinetics. The tools to study and to quantitate elimination of substances such as urea during hemodialysis are the same as those used in pharmacokinetic modeling. Modeling of urea elimination is based on physical principles such as the conservation of mass and physical relations such as the law of diffusion. Analysis of the dynamic problem results in ordinary differential equations with initial conditions, usually in first-order form, but with associated algebraic equations. An initial discussion of physical relations and equations involved in compartment modeling is followed by a presentation of compartment models and their evolution, leading to current concepts of how to determine the amount of urea removal and the dose of delivered dialysis in everyday practice.

Transport and Elimination of Urea Basic Mechanisms Flow and transport of mass is a feature of life. The study of uptake and elimination of a wide range of substances lies at the very heart of physiology. The concept of transport is also linked to containment and selective transport either to obstruct or to facilitate transport of specific substances. It is important to recognize the scale of distances to be covered in a complex organism.2 The length scale ranges from nanometers to meters, a linear ratio of The Artificial Kidney: Physiological Modeling and Tissue Engineering, edited by John K. Leypoldt. ©1999 R.G. Landes Company.


26

109, and a volume ratio of 1027. The large range of dimensions involved in transport translates into a range of time constants which serve as estimates of time required for a given transient process to be effectively complete. Once characteristic time scales have been established, one can restrict attention to individual processes with response times of the same order. Those an order of magnitude faster can be treated as instantaneous, and those ten times longer can be assumed not to happen at all. Effective transport from the molecular to the cellular level, and from one organ to the other, requires a combination of transport mechanisms. Transport on the small scale is determined by diffusion.3,4 Diffusion The Brownian motion of molecules and particles in solution, which results from intermolecular collisions with the surrounding fluid, is the basis of transport at the molecular scale. This motion does not take any predictable direction and the movement is random, but particles under observation tend to move farther from their origin with increasing time and with increasing coefficient of diffusion (D, in cm2/s). If the molecules in solution are uniformly distributed, the net movement is zero. But, if the solute is not evenly distributed in the solution, there is a net movement of substance from regions of higher concentration to regions of lower concentration. Since diffusion is based on random movements of solutes, it is more probable in a statistical sense that solutes diffuse from regions of higher concentration to regions of lower concentration than in the opposite direction. The same process will be observed if a region of high solute concentration adjacent to a region of low solute concentration is separated by a membrane permeable to the solute. For a thin membrane (∆x), the diffusive flow (Jd, in g/s) of uncharged molecules across this membrane is determined by:

Jd = −

DA ⋅ ∆c ∆x

[Eq. 2.1]

where: ∆c = the difference in concentration of the solute on both sides of the membrane; A = the membrane surface area; D = the diffusion coefficient of the solute in the membrane. Convection As diffusion becomes exceedingly inefficient with increasing distance, nature has developed other mechanisms such as fluid flow for the transport of solutes to places where they are needed and where diffusion can take over the process of distribution.5 The treatment with the artificial kidney takes advantage of both diffusion and convection. Blood flow is used to move solutes from the microcirculation of the tissues to central parts of the circulation, to the extracorporeal circulation, and to the artificial kidney. Dialysate flow removes the solutes from the artificial kidney. Intracorporeal blood flow and blood flow distribution is controlled by physiologic mechanisms, whereas extracorporeal blood and dialysate flow is controlled by the dialysis machine. The distinction is important because a major limitation to increasing hemodialysis efficiency is determined by physiologic blood flow regulation and blood flow limitation. Even though the profile of particle velocities at steady flow in a rigid tube such as the extracorporeal blood line is inhomogeneous, changes in input concentration tend to even out because of lateral and longitudinal diffusion at points farther downstream from the

Urea Removal During Hemodialysis

27

inflow. Turbulence and pulsatility will increase cross-stream mixing so that the concentration sampled from the bloodstream at sufficient distance from the inflow can be assumed to be homogeneous. The convective solute flux (Jv, in g/s) can be approximated by

Jv = Qc

[Eq. 2.2]

where: c = solute concentration; Q = volume flow. Transport by convection not only occurs with blood flow but is also important when fluid is filtered through a membrane and solutes are carried with the fluid by solvent drag.

Aspects in Dialysis Extraction and Clearance Blood serves as a carrier for solutes to be extracted from the body by the artificial kidney (Fig. 2.1). As blood is brought into close contact with the tissues in the microcirculation and with dialysate in the artificial kidney by convection, solutes such as urea will tend to equilibrate across permeable membranes by diffusion. If urea concentrations are high in the tissues and low in the dialysate, there is net convective and diffusive flow of urea from the tissues to the dialysate. Blood is saturated in the microcirculation and cleared in the artificial kidney. The recirculating closed loop nature of convective transport between locations with diffusive transport is a characteristic for cardiovascular transport in general, and for hemodialysis in particular. The extraction (E) of solute from blood in the dialyzer is defined as the fraction of solute removed from the dialyzer relative to the amount of solute delivered to the dialyzer6

E=

Qin cin − Qout cout Qin cin

[Eq. 2.3]

where: cin and cout = urea concentrations; Qin and Qout = blood flows entering and leaving the system, respectively. The subscripts (in, out) refer to a view from the dialyzer. It follows from mass balance that extraction into the dialyzer (Ed) is equal to negative extraction from the tissue (Etis). A major component of the hemodialysis treatment refers to fluid removal from the overhydrated patient. Body fluid accumulating in the interdialysis interval must be removed from the patient during hemodialysis by ultrafiltration. The problems associated with fluid overload and fluid removal are discussed in chapter 1 of this book. Since the dialyzer membrane is permeable both to solute and to solvent, a certain amount of fluid will be filtered through the membrane when a given pressure gRADient is applied across the membrane. Dialysis membranes are manufactured to retain cellular components and large molecules such as proteins, and to allow the passage of small molecules such as urea. Except for components larger than the cutoff, the composition of ultrafiltrate is comparable to the composition of blood. A significant amount of solute is removed from the body by ultrafiltration without changes in solute concentration. The solute flux by ultrafiltration is a convective flux. Ultrafiltration is an integral part of hemodialysis. The difference between dialyzer blood inflow (Qin) and outflow (Qout) is given by the ultrafiltration rate (UFR):


28

Fig. 2.1. Extraction, diffusion, and convection during hemodialysis.

Qin − Qout = UFR

[Eq. 2.4]

and Eqn [2.4] can be rewritten as a sum of two expressions

E=

cin − cout UFR cout + ⋅ cin Qin cin

[Eq. 2.5]

where the first expression on the right side of Eq. [2.5] refers to extraction without ultrafiltration; the second expression refers to extraction due to ultrafiltration. Extraction of solute without ultrafiltration is often referred to as diffusive extraction, even though blood and dialysate flows determine overall solute transport over a wide range of dialyzer performance. Also, in different sections of the hollow fiber membrane, solute is transported with filtration and backfiltration because of changing hydrostatic and colloid osmotic pressure gRADients between the blood and the dialyzer compartment.7 Extraction by ultrafiltration is referred to as convective extraction. The contribution of convective to overall extraction is small, especially with high blood flows, but significant in quantitating the amount of solute removed. With constant ultrafiltration, UFR rarely exceeds 30 ml/min and with high efficiency hemodialysis where high ultrafiltration rates are accompanied by high blood flows, the filtration fraction UFR/Qb will be less than 10% in most cases. The extracted flow is defined as clearance (Cl):


29

[Eq. 2.6]

Cl = Qin ⋅ E

Clearance is the equivalent flow of a reference fluid from which the solute is completely extracted. The choice of reference fluid is arbitrary, but may be important for the interpretation of clearance. In applications with the artificial kidney, the reference fluid may be whole blood, plasma, plasma water or blood water. However, if urea clearance is used to determine the urea distribution volume in a hemodialysis patient, the reference fluid must be blood water. Therefore, it is good practice to transform plasma urea concentrations to plasma water concentrations (correct for the plasma protein content) and to relate clearances to blood water flow.8 It follows from Eq. [2.5] and Eq. [2.6] that clearance consists of a diffusive and a convective component according to:

Cl = Qin

cin − cout c + UFR out cin cin

[Eq. 2.7]

Tissue The flow of solute between tissue and blood in the microcirculation depends on both convection and diffusion of solute. The system of flowing blood equilibrating with stationary tissue was studied by Renkin:9,10

Cl = Qb ⋅

cart − cven = Qb ⋅ 1 − e − PS Qb cart

(

)

[Eq. 2.8]

where: Qb = the blood (or blood water) flow; cart and cven = arterial and venous concentrations, respectively; Cl = the diffusive clearance according to Eq. [2.7]. The clearance in Eq. [2.8] can be interpreted as the hypothetical volume which comes into complete equilibration with the tissue fluid compartment per unit time. The dimension of the permeability ⫻ surface area product (PS) is that of a flow and represents the maximum capillary clearance possible for a given substance in a capillary bed of given permeability and surface area at infinite blood flow. At finite blood flows, a clearance smaller than this maximum will be realized (Fig. 2.2). Renkin’s formula was derived from experiments using 42K+ as a tracer added to the bloodstream and the tissue concentration (ctis) could be assumed as negligible in this special case. However, most of the time tissue concentration is not negligible (ctis ≠ 0) and the original relationship developed by Kety is a more general description of solute flow between the blood and the tissue:5,11

Qb ⋅

cart − cven = Qb ⋅ 1 − e − PS Qb cart − ctis

(

)

[Eq. 2.9]

30


Fig. 2.2. Tissue and dialyzer clearance. Larger increase in tissue clearance (Cl, Eq. [2.8]) than dialyzer clearance (Kd, Eq. [2.10]) with increasing blood flow at the same permeability ⫻ surface area product (PS = K0A = 100, 300, or 700 ml/min, Qd=1.5 ⫻ Qb).

Renkin’s formula represents a particular solution of Kety’s formula when ctis = 0. Eq [2.9] is helpful to understand the relationship between arteriovenous concentration difference (cart - cven) and the tissue to blood concentration difference (cart - ctis). Venous outflow concentration readily equilibrates with tissue concentration with small blood flows and high PS. In this case, transport of solute is flow-controlled. When blood flow increases relative to PS, the arteriovenous difference decreases and the transport of solute is diffusion-controlled. Artificial Kidney Compared to the microcirculation, where the stationary tissue is continuously exposed to a moving phase, both blood and dialysate are continuously replaced by fresh phase in the artificial kidney. Blood flow (Qb) and dialysate flow (Qd) continuously deliver and remove solutes from the blood and dialysate compartments separated by the dialyzer membrane in order to maintain a high concentration gRADient (∆c/∆x) to drive the diffusive solute flux. The constant exchange considerably increases solute flux across the membrane. If diffusion is fast compared to blood flow, solutes will equilibrate and the process becomes flowcontrolled. If diffusion is slow compared to blood flow, the process is diffusion-controlled (Fig. 2.2).


31

Solute transport is determined by flow and permeability, with the special effect that both compartments are continuously exchanged in modern dialyzers. With countercurrent flow, dialyzer clearance (K) is given as:

K = Qb

1 − e W (1− Z ) Z − e W (1− Z )

[Eq. 2.10]

where: Z = Qb/Qd; W = K0A/Qb.12 Some common abbreviations have been adapted for hemodialysis where clearance and permeability ⫻ surface area product are usually abbreviated by K and K0A, respectively. The K0A defines the maximum clearance to be attained at infinite blood and dialysate flows. A comparison of Eq. [2.8] and Eq. [2.10] shows that with the same permeability ⫻ surface area product (K0A = PS), Renkin’s formula yields higher values than the clearance calculated for dialyzers at finite dialysate flows (Fig. 2.2). The difference can be explained by Renkin’s assumption of a zero tissue concentration (ctis=0), and of a finite solute concentration in the dialysate compartment in the derivation of Eq. [2.10]. The solute concentration in the dialysate compartment reduces the concentration gRADient between the blood and the dialysate and reduces clearance. Only when Qd is much greater than Qb, solute concentration in the dialysate compartment becomes very small, and Eq. [2.10] reduces to Renkin’s formula. The permeability ⫻ surface area product for urea is available for most commercial dialyzers, so that actual urea clearance can be calculated from blood water and dialysate flows.13 Typically, K0A values range from 350-1100 ml/min. Dialyzers with K0A values greater than 700 ml/min are considered high efficiency dialyzers. Listed K0A usually refer to tests done with aqueous solutions and in vivo blood water clearances tend to be lower than calculated from manufacturer data using Eq. [2.10]. Recirculation Recirculation is basic to hemodialysis. It is intrinsic to the process insofar as only a fraction of the volume is removed from the body per unit of time, cleared from urea, and returned to the body. If all the volume (V) were cleared in a single pass using a blood flow (Qb) and a clearance (K) the process would be complete at t = V/Qb. The extraction of solute from the volume using a single pass process (Es) is given by the extraction of the dialyzer (E, Eq. [2.3]):

Es = E =

K ⋅t V

[Eq. 2.11]

However, if blood flow is returned to the volume and perfect mixing is assumed, the extraction obtained with recirculating blood flow (Er) for the same duration is given as:

Er = 1 − e − Es

[Eq. 2.12]


32

The difference (Es - Er) increases with increasing dialyzer extraction (E). Even with a perfect dialyzer (Es = 1) the extraction of the mixed compartment (Er) only reaches 63% if the process is maintained for the same time t = V/Qb. Access and Systemic Clearance Access clearance (Kac) is the equivalent flow passing the access that is cleared from solute. Under normal conditions and without access recirculation (Rac) the concentration of solute such as urea is the same in blood entering the access (cac,in) and the dialyzer (cd,in), and access clearance is equal to dialyzer clearance. In the simplest analysis, access recirculation occurs when access blood flow is insufficient to meet the demands of the blood pump (Fig. 2.3). With local access recirculation, access clearance becomes smaller than dialyzer clearance, because a fraction of cleared blood returns to the inlet of the arterial line and dilutes the concentration of solute entering the extracorporeal circulation. A flow dependent concentration gRADient (fac) develops between the blood entering the access (cac,in) and blood entering the dialyzer (cd,in). It follows from mass balance that dialyzer clearance has to be corrected for the concentration gRADient between the access and the arterial line inflow which develops because of access recirculation. Access clearance is defined as a fraction of dialyzer clearance determined by the intra-to-extracorporeal solute gRADient (fac):

Kac = fac Kd

[Eq. 2.13]

The intra-to-extracorporeal solute gRADient (fac) depends on the degree of access recirculation (Rac), dialyzer clearance (Kd) and extracorporeal blood flow (Qb) according to:14

fac =

cd , in 1 − Rac = cac, in 1 − Rac ⋅ (1 − Kd Qb )

[Eq. 2.14]

Without access recirculation, fac = 1 and cd,in = cac,in. With a given recirculation (Rac) and high efficiency dialyzers, fac approaches values of (1 - Rac). A recirculation of 30% is not uncommon with reversed position of blood lines, and dialyzer inflow concentrations may only reach 70% of access inflow concentrations in this case.15 An important practical consequence derived from Eq. [2.14] relates to the effect of blood flow on access clearance in the presence of access recirculation. An increase in blood flow will increase dialyzer clearance, but may decrease access clearance in the presence of access recirculation.16 Systemic clearance (Ksys) is the equivalent flow cleared from systemic tissue compartments. The concentration of solute in blood entering the access (cac,in) is equal to arterial concentration (cart) in the peripheral and to mixed venous concentration (cven,mix) in the central venous access, respectively. Arteriovenous Access The peripheral arteriovenous access is established for chronic hemodialysis. However, access blood flow bypasses systemic tissue compartments, which causes a small but significant reduction in extracorporeal clearance. A fraction of access flow returns to the peripheral access without systemic equilibration because of compartment recirculation


33

Fig. 2.3. Access and compartment recirculation. Concentration of solute entering the dialyzer is smaller than concentration entering the access with access recirculation (cd,in130 mm Hg) have large amounts of fluid filtered, but the fluid filtration decreases significantly when the hypertension is controlled. These patients may be better hydrated (see below), in which case rates of convection may increase, or perhaps the Starling forces are shifted toward filtration from the capillaries to a greater degree than under normal pressures.

Tissue-Level Mechanisms The examination of transport mechanisms at the level of the tissue requires a working concept. Figure 3.7 illustrates a hypothetical tissue structure which lies adjacent to the peritoneum. Solutes circulate in the blood capillaries and transport passively across the capillary endothelium (cell layer and its basement membrane which make up the blood capillaries) into the tissue interstitium (space outside of blood vessels which surrounds the cells of any tissue). Once in the interstitium, the solute will continue passive diffusion in accordance with the concentration gradient and will be subject to convection by the solvent

66


Fig. 3.6. Effect of i.p. hydrostatic pressure on the net ultrafiltration in 34 dialysis patients after two hours of dialysis with a 3.86% solution. Data replotted from ref. 22.

flow present. The presence of fresh dialysis fluid in the cavity in contact with the peritoneum overlying the tissue sets up the blood-to-cavity concentration gradient for endogenous substances such as urea and creatinine. In contrast to endogenous substances, the typical dialysis solution has a high concentration of glucose, which sets up a cavityto-blood concentration gradient. As each substance moves through the interstitium, uptake and metabolism by cells may occur. In addition, solutes may be taken up by other blood capillaries or by lymphatic capillaries and returned to the venous system. Although the blood and lymph capillaries are illustrated to be uniformly distributed in the tissue space, there is considerable variation among the different tissues. Lymphatics, for example, tend to be located in the tissue planes between layers of muscle; in contrast, they are more diffusely located in the wall of the gut.23 The major barriers to transport are the blood capillary endothelium and the interstitium. The peritoneum itself does not present any more barrier than the equivalent cellular and interstitial layer which underlies it. A recent review has detailed several studies of transmesenteric permeability. 24 The mesentery is essentially a double-walled fold of peritoneum with a small amount of connective tissue and vessels in between the layers, and it has been used as a surrogate for the peritoneum, which is difficult to dissect from the surrounding tissues. Unfortunately, in vitro permeabilities tend to be unreliable because mesothelial cells degRADe quickly in vitro and detach from the remainder of the peritoneum.25 However, the permeability of the peritoneum can be assessed indirectly. As noted above in the discussion of absorption, solutions containing 8% serum albumin are readily absorbed without change in protein concentration in the cavity12 (see Fig. 3.3). Absorption of solutions

Transport Kinetics During Peritoneal Dialysis

67

Fig. 3.7. Tissue-level model of the peritoneal transport system. Solutes and water must cross the blood capillary endothelium, the interstitial space, and the mesothelial cell layer, which together with several layers of connective tissue makes up the peritoneum (From ref. 83).

containing immunoglobulin G occurs the same way: without sieving of the protein at the peritoneum.26 In the past there have been proposals27 that protein is transported across via vesicles. However, other histologic studies have demonstrated intermesothelial gaps of approximately 50 nm,28 which would offer little resistance to the passage of macromolecules. Functional studies29 have shown that labeled iron coupled to transferrin transports across the mesothelium without disassociation. Since dissociation of the iron from the apoprotein would occur if the protein were taken up in an acidic compartment of the vesicle, it is unlikely that this transport occurs via endocytic vesicles. Because of the large intercellular gaps, the mesothelium also does not present a barrier to the osmotic agent glucose. Therefore, the same osmotically-induced transport mechanism which occurs across renal epithelium cannot be invoked, since there is no retardation of the transmesothelial transport of the glucose. This is further supported by the fact that interstitial hydrostatic pressures adjacent to the peritoneum are not negative, as one would anticipate if water were extracted by osmosis across a membrane.30

Blood Capillary The capillary barrier depends on the tissue type which determines the capillary density (surface area per unit volume of tissue) and the capillary permeability. For example, the liver has been reported to have a capillary permeability x area density of nearly 40 times that of skeletal muscle.31 Capillaries of the gut mucosa are known to be much more permeable than those of muscle.31 However, most of the exchange is with the outer layers of the gut,

68


which are made up of muscle, or with capillaries in skeletal muscle attached to parietal peritoneum. Therefore, muscle capillaries appear to be the dominant capillary exchange element, since most of the parietal tissue is equivalent to skeletal muscle (retroperitoneum, abdominal wall, and diaphragm) and most of the visceral tissue is made up of the gastrointestinal tract with smooth muscle in the outer layers adjacent to the peritoneum. Starling’s Law is the basis of transport across a homogeneous membrane and has been modified by irreversible thermodynamics.32 Briefly, the volume flux can be described as:

J v = L p ( ∆ P - Σni=1σ i ∆ π i )

[Eq. 3.1]

where: Jv = volume flux; P = hydrostatic pressure; σi = the reflection coefficient for substance ‘i’ (fraction of solute ‘i’ which would be retained or reflected during convection of a solute through the membrane and accounts for the fact that most biological membranes are not truly impermeable to small solutes); πi = osmotic pressure of substance ‘i’. The general equation for solute transfer is

[Eq. 3.2] where: J s,i = flux of solute ‘i’; p i = diffusional permeability of membrane to ‘i’; Cm,i = mean concentration of ‘i’ in membrane (typically calculated as the arithmetic mean or log-mean). The major paRADigm which has been used to describe transcapillary transport is pore theory, originated by Pappenheimer and colleagues.33 In this theory, transport occurs through holes or “pores” in the membrane which provide a sieving action or retardation of the solute according to its molecular size. Rippe and colleagues have most eloquently applied this theory to peritoneal transport in the so-called Three-Pore Theory.34,35 Figure 3.8 depicts the essence of the theory, which attempts to account for the observed size discrimination and water flux of the peritoneal barrier. Let us imagine that we are looking at the capillary endothelium with the lumen to the right and the tissue interstitium on the left. The protein content (large circles) within the capillary is usually greater than in the interstitium while in peritoneal interstitium the concentration of dextrose or other small solutes (small circles) can be higher than in the capillary lumen. If we focus on the very small pore at the top of the diagram, termed “transcellular pore,” we see that no solutes, but only water, can pass through. The value for σ is one, and all solutes exert the full potential osmotic pressure across the pore. The RADius of this pore is on the order of 0.2-0.4 nm. Even if only 1-2% of the pores are of this type, as much as 40% of the filtration induced by hypertonic glucose solution in the peritoneal cavity can occur across these pores because they function as a true semipermeable membrane. Since this 40% is solute free, the introduction of this pore resolved the dilemma of the great difference between capillary membrane transmittance coefficients (equal to 1 - σ, where σ = capillary reflection coefficient, which is typically 0.05 for small solutes) and the sieving coefficient of the lumped structures (often called the “peritoneal membrane”) of 0.6-0.7 for small solutes. The morphologic equivalent of the “water-only” pore is a new class of “aquaporin.” The existence of this structure was discovered by Agre and his colleagues in the red cell membrane and was termed the CHIP-28 molecule.36 CHIP-28 is also located in the mammalian kidney (proximal tubule and the descending thin limb of the nephron), in the eye (ciliary and lens epithelium, corneal endothelium), in the gastrointestinal tract (hepatic bile duct, gall


69

Fig. 3.8, top. Illustration of the three-pore theory of transendothelial transport. See text for details. Fig. 3.9, above. Solute clearance rates calculated from the triple-pore theory illustrated in Figure 3.8. The curves have been replotted from ref. 75. In calculating these rates, Rippe and colleagues (see Rippe43) assumed that the interstitium and the peritoneum offered negligible resistance to mass transfer.

70


bladder epithelium), in eccrine sweat glands, lymphatic endothelium and in nonfenestrated endothelia.36 The next pore to discuss is the “small” pore which allows passage of small substances (molecular RADius 20 nm) at the bottom. Gunnar Grotte, a Swedish physiologist, was the first to propose in 1956 that the capillary endothelia had to possess large leaks or pores which would allow passage of large proteins and dextrans.38 Others proposed that this transport occurred by vesicular transport. Jens Frokhar-Jensen39 subsequently carried out rigorous electron microscopic studies in which he sequentially sectioned capillary endothelia in 7.5 nm sections from the interstitium to the lumen. He found that so-called “vesicular transporters” were in fact blind invaginations into the cell, which appeared to be circular “vesicles” on cross-section. However, a few of these (1/50,000) could be observed to fuse together to form a large pore. These pores offer no resistance to the passage of small solutes and minimal resistance to large solutes, and they account for 3-5% of the total pore area. There is essentially no osmotic or oncotic force across the pore and therefore hydrostatic pressure dominates in the Starling relationship. Typical capillary hydrostatic pressures vary from 9.5 mmHg in the rat40 to 18 mmHg in the human.41 The interstitial pressures are typically -0.4 to -0.5 mmHg41,42 for both species. Therefore, transport is one-way out of the capillary because there is almost always a 10 mmHg hydrostatic pressure gradient out of the capillary. Since their transport is dominated by convection under normal conditions, large proteins will pass from the circulation to the interstitium but will not return. The chief role for lymphatics within the tissue is the return of this protein and fluid from the interstitium to the general circulation. Although fluid and small solutes are carried out with the protein in the convection through the large pore, the total solute or fluid flow is small compared with that through the other two pores.43 The mathematical approach to this theory modifies Eqs. [3.1] and [3.2] as follows. The volume flux across the capillary wall is as follows:

J total = J TC + J Sv + J vL = L p ( ∆P - Σ 3j=1α j ( Σi σ i, j ∆ π i )) v v [Eq. 3.3] where: Jvtotal = net total volume flux across the capillary; JvTC = volume flux across the transcellular pores; JvS = volume flux across the small pores; JvL = volume flux across the large pores. αj = fraction of Lp accounted for by pore ‘j’; Σj αj = 1; σi,j = reflection coefficient of pore ‘j’ for solute ‘i’. An analogous equation for the solute flux sums the contributions of both the small and the large pores. (By definition, no solute flux occurs across the transcellular pore.)

J total = J iS + J iL = Σ 2j=1( pi, j ∆ C i + J vj C m,i (1 - σ i,j )) i

[Eq. 3.4]


71

With this theory Rippe, Stelin, and Haraldsson have applied fundamental microcirculatory physiology to describe the sieving characteristics of the peritoneal capillary membrane. Figure 3.9 is a plot of the transperitoneal solute clearances for a variety of solute sizes, based on their theory and plotted with calculations from their own data. These clearances are estimated by assuming that the interstitial barrier is negligible and that transfer occurs directly between dialysate and blood across the capillary barrier. The pore RADius curve of 4.7 nm fits solutes from urea to albumin. However, the transition to larger molecules such as IgG or IgM requires introduction of a larger pore of RADius 30 nm. Finally, to calculate the total solute or water transport for a particular tissue element, the flux must be multiplied by the capillary area-density, which has been estimated to be 70 cm2/g tissue.31 While the model can be reasonably fitted to peritoneal solute clearance data, the parameters which are often obtained appear to be unrealistic with respect to the tissue-level mechanisms. Because of the lack of the interstitial barrier, the efficiency of transfer increases substantially, and the capillary surface area in the model of Rippe and colleagues required for the total cavity transfer would be contained in 50 g of tissue; from tissue concentration gradients of small and large solutes, the actual amount of tissue involved in transfer between the blood and peritoneal cavity likely approaches ten times this quantity (see below).8,44 In addition, the simultaneous hydrostatic pressure-driven fluid loss from the cavity and the osmotic flow into the cavity cannot be explained by a single membrane system.

Interstitium After transporting across the blood capillary endothelium, the solute enters the tissue interstitium, through which it must move toward the peritoneal cavity (see Fig. 3.7). Studies in rats which employ quantitative autoRADiography to measure tissue concentration profiles, have demonstrated that the tissue concentration of 14C-EDTA (approximately the molecular size of sucrose), which is injected i.p. and allowed to diffuse into the surrounding tissue for 60 min, does not approach the plasma concentration until a distance of 500-800 mm from the peritoneum (see Fig. 3.10).44 Since the rat peritoneum is 25 µm thick, the finding that the concentration profile extends to hundreds of microns suggests that the majority of the capillaries actively involved with the transport are contained within the underlying tissue. Since individual capillaries are located at variable distances from the cavity, the barrier presented by interstitium will vary depending on the location of the capillary with respect to the peritoneum. That the interstitial barrier is significant is supported by studies of gas transfer from the cavity. If inert gases equilibrate with the blood flowing in capillaries, their rate of transfer should be the same and proportional to the rate of local blood flow, if the interstitium is an insignificant part of the barrier. When Collins45 measured the simultaneous clearance of several inert gases from the peritoneal cavity of piglets, he found a threefold range of gas transfer rate. That the clearance rate of each gas was proportional to its diffusivity in water implied that gas transport was limited by diffusional barriers of the interstitium in conjunction with the blood capillary wall. The interstitial barrier varies among peritoneal tissues due to the tissue type and the pressure forces on the tissue. During its transit through the interstitium, the solute will be excluded from much of the tissue space by cells, collagen fibers, and large molecular weight interstitial matrix proteins, called glycosaminoglycans. Small solutes such as sucrose (MW = 360 daltons) are restricted to as little as 20% of the extravascular space.46,47 Large solutes such as albumin (MW = 58,000 daltons) are excluded from approximately 90% of the extravascular space.47 This tissue exclusion results in proportionate decreases in the rate of diffusion. Added to this is the tortuous path of the solute, which has been estimated to be two to three times the linear distance between two points.48,49 This causes further reduction

72


of rates of transport. Depending on the solute charge, it may be further retarded in its progress through the matrix maze. The result of the exclusion phenomena, the tortuousity, and the charge effects is a decrease in the rate of diffusion by 1-2 orders of magnitude. The effective diffusivity or Deff is defined by

Deff = DT θ s

[Eq. 3.5]

where: DT = the diffusivity within the interstitium and incorporates the effects of tortuousity and charge; θs = nonexcluded fraction of tissue which is available to solute. Figure 3.11 displays the estimated diffusivities in tissue (DT) versus molecular weight. The diffusivities in water are plotted for comparison. In order to estimate the relative contribution of the capillary wall and the interstitium to the total peritoneal transport resistance during steady-state conditions, we define the following relationship for small solutes (MW 1 or

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