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Transport of fluid and solutes in the body : a compartmental model approach Gyenge, Cristina C. 2000

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T R A N S P O R T OF F L U I D A N D S O L U T E S IN T H E B O D Y A Compartmental Model Approach By Cristina C. Gyenge B. A. Sc. (Chemical Engineering) University 'Babes-Bolyai', Cluj, Romania, 1990 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES C H E M I C A L and BIOLOGICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A September, 2000 . © Cristina C. Gyenge, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract A mathematical model is formulated to study the transport and redistribution of fluid, proteins and small ions between the circulating blood, interstitium and cells. To achieve this task, the human/animal body was schematically divided into two distinct compartments, namely the plasma and interstitium. Two additional cellular compartments representing the red blood cells and generalized tissue cells were introduced as sub-compartments embedded in the two extracellular compartments. Two major sites of exchange are accounted for to characterize the movement of materials between these four fluid compartments. The microvascular exchange system (MVES) involves the movements of fluid, proteins and small ions from plasma into the interstitium across the capillary membrane as well as the return of these materials from the interstitial space back into plasma via the lymphatic system. Across the cellular membrane separating the intra- and extracellular compartments, there are dynamic exchanges of fluid and the three important ions, Na + , K + and CI". These exchanges are assumed to occur by both passive and active mechanisms. The general model consists of a large set of time-dependent differential-algebraic equations that must be solved simultaneously to predict both clinically measurable data (e.g., plasma and blood volumes, plasma solute concentrations, and osmotic pressures) and experimentally difficult or impossible to measure variables (e.g., intracellular volumes and small ion concentrations, cellular transmembrane potentials, and transmembrane fluid shifts). The solution of these equations is carried out by the use of numerical methods. To describe mass exchange within the M V E S and across the cell membranes, the transport characteristics of the principle resistances encountered by the exchanging materials must be known. The set of transport parameters needed to describe fluid and protein exchanges across the capillary membrane and within the lymphatic system were estimated previously from human data by other researchers in our group. As part of the present work, the transport parameters related to the movement of small ions across the capillary membrane (i.e., the reflection coefficient, O ION , and the permeability-surface area product, PS ION) were estimated using data from studies in which animals were successively infused with iso-osmolar saline (NS) i i and hyperosmolar saline (HS) solutions. Also, the transport parameters associated with cellular exchange (i.e., the cell membrane permeabilities for sodium, potassium and chloride, pNa, PK and pci, as well as the rate of the sodium-potassium pump, RP, were determined from the steady-state equations that describe cell volume regulation, together with the known normal distribution of ions between the intra- and extracellular fluids. Additional transport parameters required to accommodate external infusions of macromolecular species such as dextran were obtained from the literature. The validation of the model with these newly introduced parameters was carried out by comparing model-predicted results with experimental data from animals and humans that had undergone different resuscitation protocols (i.e., different rates and volumes of fluid administration using different types of infusates (NS, Ringer's solution (RS), HS or hyperosmotic saline/dextran solution (HSD)). Considering the physiological complexity of the body, the model-predicted results compared very well with the experimental data in the majority of cases simulated. As a subset of this study, mathematical expressions are developed to describe the excretion of fluid and small ions by the kidney. The formulation of this renal model is based on the physiological role of the kidney in maintaining the plasma volume and plasma sodium concentration at their normal values. Thus, it is assumed that the kidney responds via a negative feedback to any changes in these two values from their normal set-points. The generalized four-compartment model that includes the 'kidney module' was tested using experimental animal and human data involving infusions of NS or HS solutions, or non-treated hemorrhages. The model predictions were generally in very good agreement with the measured results for all the cases simulated. Finally, the applicability of the model to the study of hemorrhagic shock was exemplified through a series of simulations that describe the distinct stages in the progression of shock. Empirical equations were proposed to characterize the release of glucose and other solutes that occur during the compensatory (hemodilution) phase of hemorrhage, as well as the disturbed cellular transport that takes place during the decompensatory (hemoconcentration) stage of shock. The weaknesses and strengths of the model to clarify certain mechanisms related to hemorrhagic shock were underlined. iii Table of Contents Abstract ii List of Figures ix List of Tables xv Acknowledgments xviii Chapter 1: INTRODUCTION 1 Chapter 2: PHYSIOLOGICAL OVERVIEW OF THE COMPARTMENTS OF 7 STUDY 2.1 Introduction 7 2.2 Water Distribution in Fluid Compartments 7 2.3 Distribution of Solutes between Different Fluid Compartments 9 2.4 Characterization of the Vascular Compartment 11 2.4.1 Cardiovascular system and vascular segments 11 2.4.2 Composition of blood 15 2.4.2.1 Cellular fractions of blood 16 2.4.2.2 Plasma 16 2.5 Characterization of the Interstitial Compartment 17 2.5.1 Structure of the interstitium 17 2.5.1.1 Interstitial fluid 18 2.5.1.2 Structural molecules present in interstitium 18 2.5.2 Physical characteristics of the interstitium 19 2.5.2.1 Volume exclusion 19 2.5.2.2 Interstitial compliance 20 2.6 Characterization of the Lymphatic System 21 Chapter 3: TRANSPORT ACROSS THE C A P I L L A R Y AND C E L L U L A R 24 M E M B R A N E S 3.1 Introduction 24 PART I: T R A N S C A P I L L A R Y E X C H A N G E 24 3.2 Structure and Function of the Microcirculation 24 3.3 The Capillary Wall and its Transport Pathways 25 3.4 Transcapillary Transport Equations 27 3.4.1 Fluid transport across the capillary wall 27 3.4.2 Protein transport across the capillary wall 31 3.4.3 Transport of small ions across the capillary 32 PART II: C E L L U L A R E X C H A N G E 33 iv 3.5 Theoretical Considerations 33 3.5.1 Cell membrane 34 3.5.2 General characteristics of transport through the cellular membrane 34 3.5.3 Cell membrane permeability 35 3.6 Exchanges of Ions and Water across the Cellular Membrane 38 3.6.1 Mechanistic description of a typical cell 38 3.6.2 Transport of ions due to electrical and diffiisional driving forces ... 39 3.6.3 The resting membrane potential 43 3.6.4 Electroneutrality condition 45 3.6.5 Osmotic pressure, osmolarity and isotonicity condition 46 Chapter 4. M O D E L F O R M U L A T I O N . 4 9 4.1 Introduction 49 4.2 Literature Review 50 4.2.1 Compartmental M V E S models 50 4.2.2 Cellular volume regulation (CVR) models 51 4.2.3 Mathematical models of whole-body fluid and solute exchange 54 4.3 Model Description 54 4.3.1 General considerations 55 4.3.2 Model compartments 55 4.4 Model Assumptions 57 4.5 Model Equations : 62 4.5.1 Mass balance equations for extracellular compartments 62 4.5.2 Mass balance equations for intracellular compartments 63 4.5.3 Transport equations 66 4.5.3.1 Transport across the capillary membrane 66 4.5.3.2 Transport through the lymphatics 68 4.5.4 Constitutive relationships 70 4.5.4.1 Capillary hydrostatic pressure and circulatory compliance 70 4.5.4.2 Interstitial pressure and interstitial compliance 71 4.5.4.3 Compartmental concentrations and colloid osmotic pressure relationships 74 4.5.4.4 Mathematical representation of the Donnan distribution ... 76 4.5.5 Fluid and solute inputs and outputs 78 4.5.6 Characterization of Dextrans, Dextran 70 80 4.6 Steady state condition 80 4.7 Initial conditions for subjects other than the 'Reference man' 87 Chapter 5: N U M E R I C A L METHODS A N D COMPUTATIONAL PROCEDURES 89 5.1 General Aspects 89 5.2 Conceptual Design of the Program 89 5.3 Numerical Methods 91 v Chapter 6: V A L I D A T I O N OF THE M O D E L BASED ON A N I M A L STUDIES 96 6.1 Introduction 96 6.2 Solutions for Infusion 96 6.2.1 Iso-osmolar saline (Ringer's) solutions 97 6.2.2 Hyperosmolar/hypertonic solutions 98 6.3 Estimation of the Permeability-Surface Area Product and the Reflection Coefficient for the Transcapillary Transport of Small Ions 100 6.3.1 Experimental information 101 6.3.2 Initial conditions 101 6.3.3 Estimation of parameters 105 6.4 Model Validation Based on Additional Animal Studies 110 6.4.1 Experimental information 110 6.4.2 Comparisons with Onarheim's data 113 6.4.3 Comparisons with Manning and Guyton's data 116 6.4.4 Summary of validation study 121 6.5 Implications of the Model 121 6.5.1 Iso-osmotic solution (NS or RS) infusions 121 6.5.2 Hyperosmotic solution (HS) infusions 125 6.6 Conclusions from Validation Studies 129 Chapter 7: M O D E L VALIDATION BASED O N H U M A N STUDIES 130 7.1 Introduction 130 7.2 Model Validation for Human Studies 131 7.2.1 Model description 131 7.2.2 Available experimental data 131 7.3 Comparison of Model Predictions with Experimental Data 135 7.4 Implications of the Model 143 7.5 Application of the Model to Fluid Resuscitation 148 7.6 Conclusions from the Validation Studies 151 Chapter 8: M O D E L OF THE R E N A L FUNCTION 152 8.1 Introduction 152 8.2 Some Aspects of Kidney Anatomy and Physiology 152 8.2.1 The nephron 153 8.2.2 Nephron capillary system 155 8.3 Quantitative Analysis of the Kidney Function 156 8.3.1 Urine formation 157 8.3 2 Renal clearance 159 8.3.3 Average values for urine and solute excretions in humans 160 8.4 Formulation of Model for Renal Excretion of Fluid and Solutes 161 8.4.1 Modeling background 161 8.4.2 Renal response to perturbed extra-renal conditions 163 8.4.2.1 Renal response to isosmolar Ringer's (RS) or normal saline (NS) infusions 163 vi 8.4.2.2 Renal response to hyperosmolar (HS) or hyperosmolar/ hyperoncotic (HSD) infusions 166 8.4.2.3 The renal response to hypovolemia caused by external hemorrhage (HEM) 167 8.5 Model description 169 8.5.1 General assumptions 169 8.5.2 Model equations 171 8.5.3 Initial conditions for renal excretion in humans 177 8.5.4 Parameter estimation 178 8.5.4.1 Estimation of k„ using data from Watenpaugh et al 178 8.5.4.2 Estimation of k ° using data from Lucas and Ledgerwood 179 8.5.4.3 Estimation of k N a , k K and k c l using data from 182 Tollofsrud et al. 8.5.4.4 Estimation of k c based on the data of Cannon et al 184 8.5.5 Model validation and discussion 191 8.5.5.1 Validation based on NS or HSD infusions in normal swine 192 8.5.5.2 Validation based on data from mild hemorrhage followed by HSD administration 201 8.5.5.3 Validation based on data from graded hemorrhages in pigs 206 8.6 Conclusions 207 Chapter 9: APPLICATION OF THE M O D E L TO THE STUDY OF H E M O R R H A G E 212 9.1 Introduction 212 9.2 Hemorrhagic Shock 213 9.2.1 Classification of hemorrhagic shock 213 9.2.2 Progression of hemorrhagic shock 215 9.2.3 Hemodilution phase 217 9.2.4 Hemoconcentration phase 222 9.2.4.1 Cellular damage during shock 223 9.3 Application of the model to the study of hemorrhage 228 9.3.1 Proposed time-course of events for modeling hemorrhage 228 9.3.2 Model description 230 9.3.3 Simulated hemorrhage scenarios 235 9.3.3.1 Simulation of hemorrhage in the absence of glucose 235 release 9.3.3.2 Simulations of glucose release and osmolarity increase after blood removal 246 9.3.3.3 Cellular defect in hemorrhagic shock 257 9.3.3.4 Application of the model in identifying the hemodilution and hemoconcentration stages of hemorrhage 273 Chapter 10: CONCLUSIONS A N D RECOMMENDATIONS 276 vii Nomenclature 282 References 286 Appendix A: Derivation of the Electrodiffusional and Membrane Potential Equations... 302 Appendix B: The Jakobsson Model of Cell Volume Regulation 305 Appendix C: Scaling procedure for Determination of the Initial Compartmental Values 307 Appendix D: Initial distribution of proteins, small ions and glucose between plasma, 313 interstitium and cells Appendix E: Compartmental values for swine 315 Appendix F: Compartmental values for dogs 318 Appendix G: Compartmental values for rats 319 Appendix H : Program Listing 321 viii List of Figures 2-1 Distribution of body fluids in compartments as a percentage of body 8 weight for a 70 kg man. 2-2 Distribution of blood volume in different segments of the circulatory 13 system. 2-3 Distribution of cardiac output to several organs in normal human at rest. 15 2-4 The change from normal of interstitial fluid volume AVrr (ml/g dry 21 weight) vs. the change from normal of interstitial pressure, A P I X (mmHg), for experiments with dogs. 2- 5 Schematic diagram of the initial lymphatic and collecting vessels. 23 3- 1 Transport of different types of solutes (see bellow) through the multiple 26 pathways identified for transcapillary exchange. 3-2 The Starling forces that contribute to fluid exchange across the capillary 29 membrane. 3-3 Schematic representation of the Donnan partition of small ions across the 31 capillary. 3- 4 Distribution of ionic species across the cellular membrane as a 41 consequence of the different types of transport mechanisms. 4- 1 Schematic of a general model of fluid, protein, small ion and additional 56 solute (introduced through infusion) exchange between plasma interstitium and cells. 4-2 The 'most-likely' human compliance curve with its three distinct segments 73 for dehydration, moderate hydration and overhydration. 6-1 Combined sum-of-squares-of-differences between the model predictions 107 and Wolfs [1982] plasma volume and plasma osmolarity measurements as a function of P S I O N / P S and CJION. 6-2 Comparison of the model predictions for plasma volume changes vs. time 108 with experimental data from Wolf [1982]. ix 6-3 Comparison of the model predictions for changes in plasma osmolarity vs. 109 time with experimental data from Wolf [1982]. 6-4 Comparison of the model predictions for blood volume change vs. time 118 with experimental data from Manning and Guyton [1980]. 6-5 Comparison of the model predictions for plasma protein concentration 119 change vs. time with experimental data from Manning and Guyton [1980]. 6-6 Comparison of the model predictions for extracellular volume change vs. 120 time with experimental data from Manning and Guyton [1980]. 6-7 Starling forces: Model predictions performed according to the 123 experimental protocol described by Wolf [1982] for the NS infusion. 6-8 Model predictions for the relative transcapillary flow (i.e., JIT /JIT, NL) VS . 124 time, forRS infusions [Manning and Guyton, 1980]. 6-9 Model predictions for changes in cell volume vs. time following HS 127 infusion [Onarheim, 1995]. 6- 10 Model predictions for the relative transcapillary fluid flow vs. time after 128 HS infusion based on the studies of Onarheim [1995] and Wolf [1982]. 7- 1 A schematic diagram depicting the compartments and flows which 132 comprise the human model. 7-2 Model predictions and experimental data [Watenpaugh et al. 1992] for 136 changes in plasma volume ( V P L ) , the colloid osmotic pressure of plasma (TCP.PL) interstitial fluid volume (V IT ) , hematocrit (Hct) and the net transcapillary fluid flow, after NS infusion. 7-3 Model predictions and experimental data involving HSD infusion in 140 normovolemic humans by Tollofsrud et al. [1997]. 7-4 Model predictions and experimental data involving HSD infusion in 142 hypovolemic humans by Tollofsrud et al. [1997]. 7-5 Model predictions for the hypovolemic condition described in Tollofsrud 144 et al. [1997] for interstitial fluid volume (V IT ) , red blood cell volume ( V R B C ) and interstitial cell volume ( V T C ) . 7-6 Model predictions for the hypovolemic condition described in Tollofsrud 147 et al. [1997] for transcapillary fluid exchange (JIT) and lymph flow ( J L ) . X 7-7 Model predictions of blood volume dynamics for three HSD infusion 150 times. 8-1 Differences between a cortical nephron and a juxtamedullary nephron. 154 8-2 Schematic of a functional nephron indicating the renal corpuscle 154 composed of the glomerulus and Bowman's capsule and a series of tubules. 8-3 The three main mechanisms of urine formation. 157 8-4 Schematic representation of kidney part of the overall model of whole- 172 body fluid and solute exchange. 8-5 Average experimental urinary output reported by Watenpaugh et al. [1992] 180 vs. average model predicted urinary outputs obtained by using the estimated parameter kj}. 8-6 Calculated urinary output rate vs. change in plasma volume obtained by 181 using the estimated k ° and k^ constants. 8-7 Comparison between the computed averages urinary rates and the 183 experimentally measured values for the normovolemic HSD case investigated by Tollofsrud et al. [1997]. 8-8 Comparison between the computed averages for sodium excretion rates 185 and the corresponding experimentally measured values for the normovolemic HSD case investigated by Tollofsrud et al. [1997]. 8-9 Comparison between the computed averages for sodium concentration in 186 urine and the experimentally measured values for the normovolemic HSD case investigated by Tollofsrud et al. [1997]. 8-10 Comparison between the computed averages for potassium excretion rates 187 and the corresponding experimentally measured values for the normovolemic HSD case investigated by Tollofsrud et al. [1997]. 8-11 Comparison between the computed averages for potassium concentration 188 in urine and the experimentally measured values for the normovolemic HSD case investigated by Tollofsrud et al. [1997]. 8-12 Comparison between the computed averages for chloride excretion rates 189 and the corresponding experimentally measured values for the normovolemic HSD case investigated by Tollofsrud et al. [1997]. xi 8-13 Comparison between the computed averages for chloride concentration in 190 urine and the experimentally measured values for the normovolemic HSD case investigated by T0ll0fsrud et al. [1997]. 8-14 Least-squares fitting of the experimental data for the change in C 2 + 191 excretion rate vs. the change in sodium excretion rate at peak natriuresis after HS fluid infusions. 8-15 Comparison between model predictions and experimental data for Hct 193 values from the study of Sondeen et al. involving NS infusion in swine. 8-16 Model-predicted for continuous changes in plasma volume and the 194 comparison between model predictions and experimental data for the urinary output values from the study of Sondeen et al. [1990(a)] involving NS infusion in swine. 8-17 Experimental data from Sondeen et al. [1990(a)] and model-predicted 195 changes in the concentration of sodium in plasma and the relative change in the sodium excretion rate after NS infusion in swine. 8-18 Comparison between model predictions and experimental data for Hct 196 values from the study of Sondeen et al. [1990(a)] involving HSD infusion in swine. 8-19 Model-predicted for continuous changes in plasma volume and the 197 comparison between model predictions and experimental data for the urinary output values from the study of Sondeen et al. [1990(a)] involving HSD infusion in swine. 8-20 Experimental data from Sondeen et al. and model-predicted changes in 198 the concentration of sodium in plasma and the relative change in the sodium excretion rate after HSD infusion in swine. 8-21 Comparison between the computed average urinary rates and 203 experimentally measured values for the hypovolemic HSD case investigated by T0llofsrud et al. 8-22 Model predicted sodium renal clearances of sodium following HSD 205 infusions for normovolemic humans (solid line) and mildly hemorrhaged humans (dashed line) based on the hypovolemic HSD case investigated by T0ll0fsrud et al. [1997]. 8-23 Urinary output profile predicted by the model for increasing degrees of 208 hemorrhage and the comparison between the computed average urinary rates with the experimentally measured values based on the work of Sondeen etal. [1990(b)] xii 8-24 Sodium excretion rate profile predicted by the model for increasing degrees of 209 hemorrhage and the comparison between the computed average sodium excretion rates with the experimentally measured values reported by Sondeen etal. [1990(b)] 8- 25 Comparison between the model-predicted renal clearance of sodium and the 210 values calculated using measured data from Sondeen et al. [1990(b)]. 9- 1 The change in arterial pressure after different degrees of hemorrhage. 214 9-2 Changes in blood pressure, transcellular membrane potential and plasma 225 and interstitial potassium concentrations as a function of time post-hemmorhage. 9-3 A schematic diagram depicting the compartments and flows, which 234 comprise the hemorrhage model, is presented. 9-4 Predicted changes in the compartmental fluid volumes after a 10% and a 237 30% hemorrhage of long duration. 9-5 Model predictions for normalized transcapillary and lymphatic fluid flow 238 rates after a 10% and a 30% hemorrhage of long duration. 9-6 Model predictions for the capillary and interstitial hydrostatic pressures 239 after a 10% or a 30% hemorrhage of long duration. 9-7 Model predictions for the colloid osmotic pressures of plasma and 240 interstitium after a 10% and a 30% hemorrhage of long duration. 9-8 Model predictions for Hct from studies involving 10% and 30% 243 hemorrhages of short or long duration. 9-9 Model predictions for the compartmental fluid volumes from studies 244 involving 10% and 30% hemorrhages of short or long duration. 9-10 Model predictions for fluid excretion rates after 10% and 30% 245 hemorrhages of short or long duration. 9-11 Comparison between model-predicted and experimental results of Gann et 251 al. for a 10% hemorrhage. 9-12 Comparison between model-predicted and experimental results of Gann et 252 al. for a 20% hemorrhage. 9-13 Comparison between model-predicted and experimental results of Gann et 253 al. for a 30% hemorrhage. xiii 9-14 Model predictions of B V R in the presence and absence of solute release to 255 plasma after 10, 20 and 30% hemorrhages. 9-15 Simulation protocol for a severe 34% hemorrhage based on the 258 experiments of Illner and Shires [1980]. 9-16 Changes in the membrane potential of interstitial cells after an increase in 260 PNa,xc or a decrease in R P T C 9-17 Model-predicted changes in the compartmental fluid volumes after a 261 shock-producing hemorrhage which corresponds to the experiment of Illner and Shires [1980]. 9-18 The normalized transcapillary and lymph flows after a hemorrhage which 262 corresponds to the experiment of Illner and Shires [1980]. 9-19 Comparison of the model predictions with the experimental results of 265 Illner and Shires [1980] for changes in tissue cell concentrations of sodium, potassium and chloride. 9-20 Comparison of the model predictions with the experimental results of 268 Illner and Shires [1980] for changes in plasma concentrations of sodium, potassium and chloride. 9-21 Model-predicted changes of the cell membrane potential of RBCs after a 270 severe hemorrhage followed by an increase in pNa,Tc or a decrease in R P T C 9-22 Model predictions for the Donnan distribution across the capillary after a 271 severe hemorrhage. 9-23 Model predictions for a hypothetical scenario where a severe external loss 274 of blood is followed by an initial 2 h of compensation and a subsequent 2 h of continuous poisoning of the Na +-K +-pump. xiv List of Tables 2-1 Normal physiological concentration ranges and average values for small ions 10 and proteins showing their partition between the intra- and extracellular compartments. 2-2 Characteristics and function of different vessels of the circulation. 13 2- 3 Average geometric properties of normal red blood cells in humans. 16 3- 1 The permeabilities for red blood cells and skeletal muscle cells. 37 4- 1 Initial steady-state compartmental values for the 70-kg "Reference" man. 81 4-2 Steady-state values for small ions and proteins showing their partition between 84 intra- and extracellular compartments. 4- 3 Calculated and literature values for steady-state cellular membrane parameters. 87 5- 1 Cash-Karp coefficients for the Runge-Kutta-Fehlberg method. 95 6- 1 Examples of compositions for isotonic and hypertonic fluids used most 99 commonly for experimental infusion studies. 6-2 Resuscitation protocols used by Wolf [ 1982 ]. 101 6-3 Initial steady-state compartmental values for Wolfs [1982] dogs. 103 6-4 Initial steady-state compartmental values common to humans and all 105 experimental animals. 6-5 Initial steady-state compartmental values used in animal validation 111 studies. 6-6 Interstitial compliance relationships for 'scaled rat' [Onarheim, 1995] and 112 'Scaled dog' [Guyton and Manning, 1980]. 6-7 Resuscitations protocols for model validation based on dog studies by [Guyton 112 and Manning, 1980] and rat studies by [Onarheim, 1995]. 6-8 Comparison between model predictions and Onarheim's [1995] experimental 113 data at 60 minutes after fluid infusion. 7-1 Inputs and outputs used for the model for the infusion of NS solution in human 134 subjects [Watenpaugh et al., 1992]. 7-2 Inputs and outputs used for the model for the infusion of HSD solution in 136 human subjects. Data corresponds to the normovolemic case, N V , as reported in the original study of Tollofsrud et al. [1997]. 7-3 Inputs and outputs used for the model for the infusion of HSD solution in 137 human subjects. Data corresponds to the hypovolemic case, HV, as reported in the original study of Tollofsrud et al. [1997]. 7- 4 Comparison between the model predictions and experimental results for the 141 changes in plasma osmolarity for the experiments of Tollofsrud et al. [1997]. 8- 1 Average concentrations of sodium, potassium, chloride and glucose in the 161 protein-free filtrate and in the urine. 8-2 Diagrams of renal and vascular responses to fluid infusion and hemorrhage. 164 8-3 Vascular and renal responses to different degrees of hemorrhage. 168 8-4 Initial values for renal excretion in a 70-kg normal human 177 8-5 Set of estimated parameters of the kidney model obtained by least-square fitting 179 technique. 8- 6 Renal excretion of fluid, sodium, potassium and chloride after a 10% 204 hemorrhage followed by 4 ml/kg HSD infusion: comparison between model predictions and experimental data from the study of Tollofsrud et al. [1997]. 9- 1 Summary of changes in plasma osmolarity, plasma glucose and other plasma 220 solutes for a severe, non-resuscitated hemorrhage of 35% in cats [Jarhult, 1975]. 9-2 The rate of refill of plasma volume at different times after a non-resuscitated 221 35% hemorrhage in cats Jarhult [1975] and humans (estimated based on Jarhult's data). 9-3 Experimental studies showing the direct correlation between the decrease in 226 arterial pressure and cell membrane depolarization for different animal species. 9-4 Factors hypothesized to be responsible for: cellular swelling, cellular sodium 228 and chloride content increase, and cellular potassium decrease during hemorrhagic shock. xvi 9-5 Literature information regarding the percentage of blood loss where 230 hypovolemic shock was first detected. 9-6 Initial compartmental values for glucose amounts and the transcapillary 231 transport parameters. 9-7 Conditions used for simulating mild and severe hemorrhages. 237 9-8 Model predictions for small ion concentration in plasma, interstitial fluid and 264 interstitial cells in control (t = 0 h) and during shock characterized by a 23% V m depolarization (t = 2.95 h). xvii Acknowledgements It would be very difficult to adequately thank everyone who either has helped make this work possible or had been an important part of my life during my stay in this university. First of all, I would like to thank to my two supervisors, Professors J. L . Bert and B. D. Bowen, for giving me the opportunity to pursue this work and for their continuous guidance. Dr. Bert was instrumental in making this thesis possible. He was responsible for keeping things on track and for being an important driving force in surpassing many obstacles encountered during this project. On both professional and personal levels, I have learned a lot from him, but like any other student most probably not enough. If I were to single out an important piece of advice I received from him it would be ... keep it simple. Keeping it simple was the most complex task I was called upon to master and, unfortunately, it is still a work in progress. I take this opportunity, therefore, to thank him for all the useful advice, time, understanding and continuous effort put into my development during these past few years. Gertrude Stein once said, 'a difference, to be a difference, must make a difference'. Among the many people whom I had the fortune to meet during my work here, Dr. Bowen was the person who made that difference. I would like to extend my sincere thanks to him for his guidance, tireless patience and effort put into making this work better. Several friends and colleagues brought a lot of enjoyment to our life. Although they are too many to be enumerated, I am very grateful for their presence. M y kind thoughts are especially extended to a dear friend, Antonio. Our lively conversations, exchange of ideas and good times are dearly missed. Also, the staff in the Chemical and Biological Engineering Department was always gracious and prompt in helping me in times of need. I thank them for this and for creating a pleasant environment in our department. I would also like to thank my parents for their support and love. It must be very difficult for a parent to let an offspring fly when the time comes. It must be also very difficult to visualize your child through occasional photographs and scribbled lines. I thank them for understanding that all this effort was necessary in order for me to become who I am. Last but not least, I am greatly indebted to Elod for making everything possible. His kind presence, unconditional love, continuous support and encouragement gave me the unique opportunity to understand the unbelievable depth and force of still waters. Thank you Elod for being here. xviii Chapter 1: Introduction 1 Chapter 1: INTRODUCTION To gain an improved understanding of how the body responds to different perturbations, scientists have attempted to simulate the behaviour of the real system through the use of models. In fact, experimental measurements are not ultimately directed toward simply acquiring an extensive collection of data; rather, their purpose is to achieve a 'model image' of how and why the body functions the way it does under normal conditions, or how it can be assisted in resuming its normal function under perturbed circumstances. A variety of models have been developed to serve this purpose. Such models can be qualitative (e.g., a conceptual description of the cellular membrane), experimental (e.g., use of animal models for studying hemorrhage) or mathematical (e.g., mass balances to investigate the redistribution of materials within the body). Each of these examples can, in turn, give rise to more and more complex representations of the system depending, ultimately, on the purpose of the model. A l l models are idealized speculative representations of reality and, hence, have to be constantly questioned and tested against the information obtained from the real system. As science progresses, and as the collection of experimental data provides more and more useful information, any model needs to be subjected to continuous changes and improvements or, if proven unsatisfactory, discarded in favor of more realistic descriptions. In the present work, a mathematical model that aims to approximate and interpret the complex dynamics of material exchange within an animal or human body is presented. Specifically, this work focuses on the distribution and transport of fluid, proteins and small ions between the plasma, interstitium and cells, based on the two important sites of exchange within the body, namely the microvascular exchange system (MVES) and the transport across the cell membranes. When the body is at steady state, there is an unchanging balance of fluids, proteins and small ions, characterized by no net movement of materials across the capillary membrane, that separates the vascular and interstitial compartments, as well as across the cellular membrane that separates the intra- and extracellular spaces. Over a reasonable (e.g., 24 h) averaging period, the external inputs to the body in the form of food, fluids, etc., exactly balance the outputs from the body in the form of renal excretions, sensible/insensible losses, etc. However, extrinsic disturbances to the system (such as hemorrhage or burns) or intrinsic ones (such as venous Chapter 1: Introduction 2 constriction, venous dilation, hypoproteinemia, or cellular damage) compromise the normal physiological fluid distribution between the vascular, interstitial and cellular compartments, and furthermore, influence the renal elimination of fluid and solutes from the body. A l l of these disturbances have a direct effect on the health and well-being of the body. In order to better understand the interplay of different mechanisms and forces acting at the capillary and cellular levels under both normal and perturbed conditions, mathematical models have been used as an important complement to experimental studies. These models, which are now commonly-used tools for describing physiological systems, can be categorized into two general classes: 1) compartmental models and 2) distributed models. In the former case, the system is defined by a series of compartments and sub-compartments that are assumed to be homogenous and well-mixed. Thus, fluid and solutes entering the compartment are instantaneously dispersed throughout its entire volume. Fluid and solutes are exchanged between compartments at rates that are directly proportional to a driving force (mechanical, chemical or electrochemical) and inversely proportional to the resistances encountered in crossing a given boundary. Mass balance equations for fluid and each of the solutes (e.g., proteins, ions) involved, combined with auxiliary transport equations describing mass-exchange rates or the properties of the individual compartments (colloid osmotic pressure, pressure-volume relationship, etc.) are written to characterize the system. The mathematical formulation of a compartmental model can be described by a set of time-dependent ordinary differential equations. The degree of complexity of the model depends on the number of compartments considered and the number of transported species accounted for in these models. One major disadvantage of these types of models is their inability to provide any information about local gradients and mass transport behavior within an individual compartment. Distributed models, on the other hand, try to account for spatial heterogeneity within a compartment. In these types of models, the compartment is usually a specific organ or even a small part of an organ. Volume averaging procedures are applied and a continuum representation of the system is obtained. Usually this leads to a set of non-linear, coupled, partial differential equations which necessitate a far more complex mathematical description than the compartmental models and a more demanding computational effort. For these reasons, distributed models are difficult to employ when dealing with a large number of transporting species, especially when the information available with respect to the properties of a given compartment is scarce. Chapter 1: Introduction 3 Previous studies by our group have approached the representation of the M V E S system through both compartmental [e.g., Bert et al., 1988] and distributed [e.g., Taylor, 1996] models. Because of our interest in developing a model that can be used to simulate the behaviour of the whole body in the presence of externally-applied perturbations, a compartmental approach was adopted here. Much of the previous effort in developing compartmental models has been directed towards mathematically describing the transport properties of the capillary wall [Curry, 1986; Haraldson, 1986], simulating whole body fluid and plasma protein exchange under both normal and pathological states [Guyton et al., 1973; Wiederhielm, 1979; Arturson et al., 1984; Bert et al., 1988; Carlson et al., 1996] and predicting the effect of resuscitation with iso- and hyperosmolar solutions [Wolf, 1982; Mazzoni et al., 1988]. Other models, fewer in number, have also focussed on the topic of transport across the cellular membrane of an isolated cell [Jakobsson, 1980; Gordon and MacKnight, 1991]. Each of the models cited above provides insight into various aspects of either the M V E S or transport between the intra- and extracellular media. However, in all the cases cited, the mathematical models are limited in scope. Common weaknesses of many of these models are incomplete and/or out-of-date descriptions of both the M V E S and cellular transports. For example, although Wolf [1982] attempted to emphasize the importance of cells during resuscitation with hyperosmolar solutions, his model did not consider a detailed description of solute transport through the cellular membrane. The goal of the present work, therefore, is to develop a more general mathematical model which describes the combined effect of cellular and transcapillary transport during non-traumatic perturbations such as infusions with varied resuscitants, and following traumatic perturbations such as hemorrhage. This work extends the previous compartmental M V E S models developed by our group [e.g., Bert and Pinder, 1982; Bert et al., 1988; Bowen et al., 1989; Chappie et al., 1993; Xie et al., 1995]. Initially, the interpretation of the role of the M V E S in the dynamics of whole body fluid and solute distribution was approached by providing a realistic characterization of the interstitial and plasma compartments and by establishing appropriate relationships for the interstitial and plasma compliances together with the capillary and lymphatic transports. These early M V E S models were concerned with fluid and protein redistribution under normal conditions, and incorporated transport parameters that were estimated based on statistical comparison of the model predictions with experimental data that corresponded to this case. The Chapter 1: Introduction 4 good predictions offered by these M V E S models under normal circumstances made it possible to further increase their complexity to allow the investigation of traumatic conditions. For example, a model developed to simulate burn injuries [e.g., Ampratwum et al., 1995; Bert et al., 1997] considered the transport of fluid and proteins between three fluid compartments, namely, plasma, a normal interstitium and an injured (burned) interstitium. In the present work, the complexity of these previous M V E S models is taken a step further to account for more fluid compartments (i.e., intracellular compartments), more species participating in transport (i.e., small ions), an independent description of the renal function, and new traumatic scenarios such as hemorrhagic shock. The work undertaken here can be divided into three major sections that can be summarized as follows: 1) First, a generalized compartmental model that includes additional cellular sub-compartments is formulated and validated against experimental data. The newly introduced cellular components become important sites of exchange, for example, when attempting to describe the response of the body to external infusions of hypertonic saline solutions with or without added colloidal compounds. The specific objectives of this section are: • to introduce two intracellular compartments, namely the red blood cells and a generalized tissue cell compartment, and to establish their physical and transport characteristics, • to elucidate the governing principles of cell volume regulation, • to account for the presence of all small ions involved in exchanges across the cellular membranes and to describe their transport between the intra- and extracellular compartments as well as between the plasma and interstitium, and • to validate the model by comparing its predictions with animal and human data that involve infusions of different resuscitants and different resuscitation schemes in an otherwise non-traumatized body. 2) Second, a new 'kidney module', that is linked to and used in conjunction with the model of whole-body fluid and solute transport, is formulated. The two specific objectives targeted in this section are: • to propose a model formulation of kidney function that accounts for the renal excretion of fluid and all the small ions present in the system, Chapter 1: Introduction 5 • to validate the whole-body model with the newly introduced kidney module against data involving infusions in normal animals and/or humans, and against data involving external blood losses through hemorrhage. 3) Third, the applicability of the model to the study of hemorrhage and the evolution of hemorrhagic shock is demonstrated. Hemorrhagic shock is a traumatic condition of considerable interest to physicians. However the succession of events leading to shock is not completely understood and, in some respects, is controversial. The highest priority of this section, therefore, is to assemble a summary of the available literature information on hemorrhagic shock and to clarify the succession and importance of all the pathophysiological mechanisms that can bring about an altered mass exchange within the body. As part of this section the model is used to provide insights into the dynamics of mass exchange that follows the hemorrhage for the following cases: • in the absence of a hyperosmolar compensatory response, • in the presence of an increased plasma osmolarity due to the release of glucose and/or other solutes, and • during a severe shock state, in the presence of an impaired cellular function that causes disturbances in the transport of water and small ions across the cell membranes. Additionally, based on the model-predicted results, this section attempts to identify if the cellular defect that occurs after a massive hemorrhage is the pathophysiological mechanism responsible for vascular collapse observed in the final stages of shock. The rest of the thesis is organized as follows. Chapter 2 briefly introduces the physiological background required for understanding the mass exchange behaviour of the compartments under study. Chapter 3 provides both a qualitative and quantitative description of the various exchanges that take place across the two important barriers within the body, namely, the capillary and cellular membranes. Chapter 4 presents the general model of microvascular and cellular exchange, while Chapter 5 describes the important numerical methods used in solving the model equations. Chapters 6 and 7 provide separate validations of the general model using experimental data obtained for animals and humans, respectively. Data obtained for the administration of resuscitant fluids that are of ongoing interest in clinical research (e.g., hypertonic saline solutions with or without added colloids) were included in these validation studies. Due to its central role in controlling the body's fluid and solute reserves, a kidney module is another important feature that was lacking in previous models. A new approach to Chapter 1: Introduction 6 modeling the renal function, based on clinical concepts, is presented in Chapter 8. In Chapter 9, the complete model, which includes cellular compartments and a kidney module, is applied in a preliminary study of hemorrhagic shock. Finally, Chapter 10 summarizes the important conclusions resulting from this work and recommends several avenues for extending and further testing this model. Chapter 2: Physiological Overview 7 Chapter 2: PHYSIOLOGICAL OVERVIEW OF THE COMPARTMENTS OF STUDY 2.1 Introduction On average the total body water (TBW) accounts for approximately 55 to 60% of the male body weight (BW) and around 50 to 55% of the female B W [e.g., Guyton, 1991]. This is the medium in which solutes (organic and inorganic) found in the human body are dissolved. One of the most important functional features of the body is to maintain constancy of the volume of water and its dissolved solutes. Under normal circumstances, over a reasonable period (e.g., during the course of a 24-h day), the amount of water and solutes taken into the body (by ingestion through the gastrointestinal tract and by metabolic production) is equal to the amount of water and solutes lost from the body (mainly through the kidneys but to some extent through lungs, skin, etc.). Beyond this simple mass balance that imposes equality between the inputs and outputs of the body, are complex physiological processes such as capillary and cellular exchanges that act synergistically to bring about this task. These exchanges normally provide a sufficient turnover of water and solutes between the different compartments of the body that homeostasis is assured. In abnormal situations (e.g., overhydration, hemorrhage, burns, etc.), when these exchanges are altered, the volume and composition of all the fluid compartments that participate in exchange are compromised. In order to understand the exchange processes mentioned above, it is essential to first understand what compartments are involved in exchange and which solutes are exchanged between them. The objective of this chapter, therefore, is to give a general overview of the compartments that participate significantly in fluid and solute exchanges within the human body. The following discussion is primarily concerned with normal physiological conditions with only brief references made to traumatic conditions, where necessary. 2.2 Water Distribution in Fluid Compartments The total body water of about 42 1 in the average human can be divided into two compartments. The fluid found inside the cells is called the intracellular fluid. This is the medium in which all the biochemical reactions associated with cell metabolism occur. About 65 to 75% of the total body water (or about 28 to 301) is contained within the cellular compartment [Guyton, 1991; Berne and Levy, 1988]. A l l of the fluid found outside the cells is known as Chapter 2: Physiological Overview 8 extracellular fluid. The intracellular volume (ICV) and the extracellular volume (ECV) are separated by a semi-permeable cellular membrane. In normal humans, the E C V accounts for about 27 to 35% of the TBW (or about 12-15 1) [Guyton, 1991; Kleinman and Lorenz, 1984] and includes the plasma volume ( V P L ) and the interstitial volume (V IT ) . The interstitial fluid comprises by far the highest proportion of the E C V , i.e., about 67 to 74%, or the equivalent of 20 to 25% of TBW. The rest of the E C V (or about 7 to 9% of TBW) is plasma. The percentages presented above can change significantly from one individual to another depending on age, weight, percentage of body fat, and status of hydration. Normally, for modeling studies, it is convenient to express the fluid distribution in the body as a percentage of body weight corresponding to the archetypal 'Reference man'. This hypothetical reference man is defined as a well-hydrated, healthy young man, in supine position who has a height of 170 cm and a weight of 70 kg [Reference Man ICRP 23, 1975]. A summary of the fluid distribution amongst the various compartments discussed above for this 70 kg man is presented in Fig. 2-1. Total body water in a 70 kg man (TBW) 60% B W (42 I) Extracellular water (ECV) -16.6% BW (11.61) Intracellular water (ICV) -43.4% BW (30.4 I) Cell membrane Plasma (VPL) ; Interstitial volume (VIT) -4.6% BW i 12% BW (84!) (3.2 1) ] Capillary membrane Figure 2-1. Distribution of body fluids in compartments as a percentage of body weight for a 70 kg man [Reference Man ICRP 23, 1975]. For discussion see main text. Adapted from Arfors and Buckley [1989]. Chapter 2: Physiological Overview 9 The E C V represents the real 'internal environment' of the body and can be envisaged as a transport system that circulates throughout the body between the vascular compartment and the tissue spaces (i.e., the interstitium). The partition of the E C V between plasma, as part of the vascular compartment, and the interstitial fluid, as part of the interstitium, is determined by the dynamic exchange of fluid and solutes across the capillary membrane and by the physical characteristics of the two fluid compartments. The partition of total body fluids between the intra- and extracellular spaces is governed by exchange across the cellular membranes. Both of these exchanges are very important in the model developed in this study and will be detailed separately in the following chapters. 2.3 Distribution of Solutes between Different Fluid Compartments Table 2-1 presents a compilation of the literature data [Guyton, 1991; Greger and Windhorst, 1996; Ganong, 1991; Kleinman and Lorenz, 1984] on the distribution of small ions between the E C V and ICV in the human body. As exemplified in this table, although similar organic and inorganic solutes can be found in both compartments, the partition of these solutes between the two is quite different. Each pair of columns in Table 2-1 shows the normal concentration range as well as the normal average value for each solute. As can be seen from Table 2-1, the principal cation that resides in the ICV is potassium. In normal humans, 98 to 99% of the total body potassium of about 60 mEq/(kg-BW), resides inside the cells and has an average concentration of about 150 mEq/1 [Kleinman and Lorenz, 1984]. The remaining 1-2% can be found in the combined plasma and interstitial fluids. By contrast, of the total 27 mEq/(kg-BW) sodium found in the body fluids, approximately 85 to 87% is located in the E C V while the remainder yields an average ICV concentration of about 10 mEq/1 [Kleinman and Lorenz, 1984]. The remaining positive species present in the body fluids are represented as a single entity in Table 2-1. These species consist mainly of calcium and magnesium, each having a positive charge of +2. The negatively-charged chloride ion is the major anion in the E C V . The total chloride content normally found in the body fluids is about 20 mEq/(kg-BW) [Guyton, 1991], of which only about 8 to 9% is located in the ICV. Inside the cells, the other negatively-charged species are the proteins and smaller anions such as sulfates, phosphates and acidic amino-acids. Chapter 2: Physiological Overview 10 ECV (Plasma and Interstitium) ICV Species Plasma Interstitium (mEq/L) (mEq/L) (mEq/L) Sodium, N a + (139-145) -142 (135-142) -138 (8-15) -10 Potassium, K + (3.5-5) -4.2 (4-6) -4 (135-165) -150 Other Positive Charges (10-20) -11.2 (10-20) -11 (10-40) -30 Chloride, CI" (101-108) -105 (100-115) -108 (2-5) ~4 Other Negative Charges (20-50) -34.7 (20-50) -35 (30-100) -70 Proteins (15-35) -17.7 (8-12) -10 (80-120) -116 Table 2-1: Normal physiological concentration ranges and average values (in bold) for small ions and proteins showing their partition between the intra- and extracellular compartments [Guyton, 1991; Greger and Windhorst, 1996; Ganong, 1991; Kleinman and Lorenz, 1984]. The plasma and interstitial fluids are presented as distinct compartments of the E C V in order to provide clear evidence for the Donnan distribution between the two compartments. The values for the intracellular compartment (ICV) are representative average values of skeletal muscle cells. Table 2-1 presents two separate entries for all anions other than chloride, one for proteins and one for the remaining negative species. The different distribution of proteins (mainly albumin), between the plasma and interstitial compartments, leads to an additional partition of small ions between these compartments due to the Donnan effect. This redistribution of small ions caused by the Donnan effect will be discussed in detail in later chapters. Transport across the cellular membranes, due to both active and passive mechanisms, brings about the differences in small ion concentrations between the ICV and ECV. Of all the solutes present, Na + , K + and CI" are the most important ions involved in transport across the cellular membrane. These species also play an important role in governing the electrical properties of cells. However, all the solutes presented in Table 2-1 contribute to the osmotic properties of the cells. Despite the differences in distribution of the various solutes between the E C V and ICV, the total chemical activity of the cations and anions within each compartment must be equal, i.e., Chapter 2: Physiological Overview 11 electrical neutrality must be respected. Additionally, the ICV is in osmotic equilibrium with the ECV, i.e., the osmolarity of the two compartments must be the same. More about the specific solute compositions of the various fluid compartments will be presented and discussed in later chapters. The remainder of this chapter presents a brief physiological background describing some of the physical characteristics associated with the vascular and interstitial compartments. 2.4 Characterization of the Vascular Compartment Blood is the medium of the vascular compartment. The circulation of blood throughout the vasculature is carried out by the circulatory system. A description of both the system for transport and the transported material is essential for understanding the role of the vascular compartment in promoting fluid and solute exchanges within the body. 2.4.1 Cardiovascular system and vascular segments The circulatory system fulfils multiple functions, among which the delivery of nutrients (e.g., O2, glucose, amino-acids, etc.) to the tissue cells, the removal of metabolic waste products from these cells and the regulation of the body temperature, are just a few. These tasks are achieved through the movement of blood in a close circuit from the heart to the different organs of the body and then, back to the heart. The circulatory system can be divided into the systemic circulation (or peripheral circulation) and the pulmonary circulation. As depicted schematically in Fig. 2-2, the cardiovascular system is comprised of two components: the heart which acts as a variable pump for blood, and a system of vessels which are branching elastic tubes. The contraction of the right or left ventricles of the heart pumps blood to either the pulmonary or systemic circulation, respectively. In healthy humans at rest, the blood pumped by the heart through the vessels per unit time, i.e., the cardiac output, is around 5-6 1/min. After exiting the heart, the blood enters the aorta which, in turn, branches into arteries, arterioles and ultimately capillaries. The blood from the capillaries is returned to the right side of the heart via the venous segment of the circulation. Except for the capillaries and venules, all the vessels of the circulation act as passive conduits for blood. The capillaries and to some extent the venules are the major sites of exchange between the blood and interstitium and are referred to as the microcirculation. A summary of the types, characteristics and functions of the various vessels that make up the cardiovascular system Chapter 2: Physiological Overview 12 is given in Table 2-2. Except for the large arteries and veins, in order to exemplify the relative sizes of different vessels, all values presented in Table 2-2 correspond to an arbitrary human tissue. It should be noted, however, that these values do differ from tissue to tissue and also with age, gender and species. The volumetric flow rate of blood that is distributed to specific tissues varies greatly according to the need of the tissue to perform its specific function as well as with the level of metabolic activity within the tissue. The normal distribution of the cardiac output to different organs in the human body, under resting conditions, is depicted in Fig. 2-3. The values indicated are basal values adequate for survival. In both normal and abnormal physiological situations, the increase in the metabolic needs of an organ is paralleled to some extent by an increase in the cardiac output and, therefore, the flow to the organ is augmented. However, the maximal cardiac output in a normal human reaches a limiting value of about 25-28 1/min [Greger and Windhorst, 1996] (i.e., about 4-5 times the baseline value) which represents about half the amount that would be theoretically required by all the combined tissues, to maintain their maximal levels of activity. Therefore, when the blood flow to an organ is augmented, the remaining so-called 'inactive organs', that are not directly involved in a specific process at the time, are collaterally vasoconstricted and maintain blood flows at low basal levels [Greger and Windhorst, 1996]. Examples for different patterns of blood flow can be given for normal as well as traumatic scenarios. For instance, during normal physical exercise skeletal muscles require an increased amount of blood flow which is obtained at the expense of other organs. Alternatively, in a pathological scenario such as hemorrhage, a reduction in blood volume can cause a powerful vasoconstriction of the skeletal muscle, kidney and the splanchnic bed. This will result in a redistribution of blood, by as much as 50%, toward the vital organs (heart and brain). It would appear from the above discussion (see also the values presented in Fig. 2-3) that, at the tissue level, the circulatory system displays specific blood flow characteristics. However, the functional features of the circulatory system can be generalized and averaged such that a common description can be provided for all parts of the circulatory system. Chapter 2: Physiological Overview 13 Pulmonary circulation—9% Superior vena cava Inferior vena cava Aorta Heart—7% Arteries—13% Arterioles and capillaries —7% Veins, venules, and venous sinuses—64% Figure 2-2. Distribution of blood volume in different segments of the circulatory system. The figure illustrates the two components of the circulatory system, i.e., the pulmonary and the systemic circulations. Although only four generic capillary beds are shown in this simplified diagram, it should be understood that each organ possesses its own microcirculatory site of exchange. From Guyton [1991], Chapter 2: Physiological Overview fl o •a •n I* 2 ~S , , — o P H S3 • a 13 • a a <L> ( A G =3 fl g T3 P O 3 s eo ' 3 a T 3 o fl w u £p ca o r o 60 •c Hi 60 S3 S a g S fl 5 o VO o o o r-x o o ti "3 '8 c/J w 1 ~ s 1) 5 o ( A 1> _ 9-13 fl <L> 8 * r-C N • C O C N O O o o o CN C N O O • • f l —J C O T3 T 3 o o 5 : fl <L> in 1 -6 8 eg f t o .2 o i—I I o o m • © CN CN O CD 3 s ca .3 3 CO C O c O T3 a o o C to C D a 3 o c3 C O C D O T3 c CD s a> 3 cn cd C D s C O -a e o CL, C O CD l i V - c o O a> j 3 > 43 <D +-> cd C O C D & 2 c0 t-. CD > cd tjfj 2 8 C O S .S CD 3 13 o |1 a .S y C D u, Jd 3 3 C O O ° > ON C + H O 2P K f 2 13 C d ffi C O +-> ON o oi a •5 > 3 § 1 .§ P? CD 43 C •o cd co ;=i cd -f5 CD ° c £ T3 «4-l O •rj Cd 3 a o 43 3 ' ^ O CC o t>0 CD . „ s •? '-' 5 co ON O i co o co o rt o i_ S cd S O 43 u C N « J ^ •si 3 H -5 < Chapter 2: Physiological Overview 15 5500 ml/min pulmonary bed 100% right heart left heart 1045 ml/min cerebral and coronary circulation 1650 ml/min liver and splanchnic bed 1100 ml/min m skeletal muscle 1100 ml/min 4 renal circulation 605 ml/min skin and skeleton circulation 19% 30% 20% 20% 11% Figure 2-3: Distribution of cardiac output to several organs in normal human at rest. Adapted from Holtz [1996]. 2.4.2 Composition of blood As previously stated, blood is the medium transported in the vasculature. It is a fluid with a viscosity three times higher than that of water [Guyton, 1991] and it accounts for about 5 to 8% of the total body weight [Berne and Levy, 1988; Guyton, 1991]. This fluid consists of a suspension of cellular elements, that occupies approximately 40-45% of the total volume, in an aqueous solution containing various solutes such as small ions, proteins, glucose, etc. [Berne and Levy, 1988]. Centrifugation of blood causes a separation of blood into two fractions: plasma and cells. Clotting of blood allows for a separation of serum from the blood. With the exception of one macromolecular species, fibrinogen, which is removed in the clot, plasma and serum have similar compositions and are both cell-free fluids. Chapter 2: Physiological Overview 16 2.4.2.1 Cellular fractions of blood The cellular fractions of blood include red blood cells (or erythrocytes) (RBCs), white blood cells (leukocytes) and a variety of platelets or cell fragments. RBCs constitute by far the highest proportion of blood cells. The remaining cells contribute only to about 10"3 of the total cell fraction. Therefore, although they are functionally important, quantitatively they can be safely ignored from a modeling perspective. By virtue of their accessibility and size, the RBCs have been the subject of numerous investigations. Among the metabolic functions performed by the RBCs, by far the most important is their role in the exchange of O2 and C O 2 . Some of the averaged physical characteristics of red blood cells are given in Table 2-3. RBC Average Dimensions Diameter 8.5 - 9 ( urn ) Surface Area 150-170 (u,m 2) Volume 80-90 ( urn 3) Table 2-3: Average geometric properties of normal red blood cells in humans [Burton, 1965; Altman and Dittmer, 1971 ]. The percentage of the total volume of blood occupied by cells is called hematocrit, Hct, and is expressed as, Hct(%) = 100 [2-1] V V RBC ~ V P L / where VRBC and VPL are the volumes of red blood cells and plasma, respectively. The normal range of hematocrit values is around 37 to 44% for women and 38 to 49% for men [Ganong, 1991; Guyton, 1991; Reference Man ICRP 23, 1975]. Corresponding to an average hematocrit of 40%, if the intracellular volume of the RBCs is assumed to be all water, it will account for about 2.1 1 or 5% of the total body water. 2.4.2.2 Plasma As presented in Section 2.2 (see also Fig. 2-1), plasma averages between 45 to 50 ml/(kg-BW) of total body weight corresponding to a volume of about 3000 to 3500 ml for humans [Guyton, 1991; Altman and Dittmer, 1971]. By weight, slightly more than 90% of plasma is Chapter 2: Physiological Overview 17 plasma is water while plasma proteins account for about 7%. The remaining percentage is made up of organic and inorganic solutes [Berne and Levy, 1988]. The principal solutes normally found in plasma were presented in Table 2-1. The small ions present in plasma are mainly sodium and chloride, moderate concentrations of bicarbonates (about 30 mEq/1) and relatively low concentrations of potassium, calcium, magnesium, phosphates, sulfates and organic acids. The organic and inorganic solutes of plasma listed in Table 2-1, contribute to a total osmotic pressure of about 6000 mmHg and generate a plasma osmolarity of ~ 300 mOsm/1. This osmolarity of plasma is equivalent to a physiological saline solution of 0.9% (by weight). The majority of small solutes are transported relatively rapidly across the highly permeable capillary membrane; therefore, the plasma proteins are the primary contributors to the osmotic pressure governing the exchange of water between the plasma and the interstitial compartments. This pressure, referred to as the colloid osmotic pressure, is about 23-27 mmHg. Albumin as well as various globulins synthesized by the parenchymal cells of the liver make up the majority of the proteins found in plasma. Since the molecular weight of albumin is considerably less than that of the globulins (i.e., 69,000 vs. 90,0000 - 160,000 g/mol), albumin is the single most important contributor to the plasma colloid osmotic pressure. 2.5 Characterization of the Interstitial Compartment 2.5.1 Structure of the interstitium The interstitium is made up by the connective and supporting tissues of the body and is spatially located outside the vasculature, parenchymal cells (tissue cells) and the lymphatic vessels. The composition and structure of the interstitium has been the subject of several reviews [e.g., Aukland and Reed, 1993; Bert and Pearce, 1984]. Although the composition and structure of the interstitium varies from tissue to tissue, there are basic characteristics and functions that are representative of interstitia of most tissues. Therefore, as a generalized description, the interstitium can be thought of as a three-dimensional meshwork composed of a complex aggregation of protein fibers, glycoconjugates and carbohydrate polymers. The main roles of the interstitium are to provide a three-dimensional mechanical support matrix for cells and different types of blood vessels and an adequate medium for transporting all the solutes exchanged between plasma, cells and the lymphatics. Based on these two functions, Chapter 2: Physiological Overview 18 most of the interstitia can be essentially divided into two phases: 1) the interstitial fluid and 2) the structural molecules of the interstitial matrix. 2.5.1.1 Interstitial fluid The fluid found in the interstitium results from a balance between the fluid that is filtered from the vasculature across the capillary membrane and the fluid that is not drained by the lymphatic system. The dissolved species in the interstitial fluid, such as small ions, proteins, hormones, O2, CO2, etc., are similar to those found in plasma. However, as shown in Table 2-1, the protein concentration in the interstitium is lower than that of plasma. Additionally, due to the Donnan effect, the interstitial fluid has a slightly higher concentration of small anions and a lower concentration of small cations. 2.5.1.2 Structural molecules present in interstitium The main interstitial structural molecules are collagen fibers, elastic fibers and glycosaminoglycans (GAGs). The collagen fibers have a high tensile strength, especially when stressed longitudinally, [Aukland and Reed, 1993] and they thus provide the mechanical strength of the tissues. They are soft flexible fibers that have a collagen molecule as their functional unit. Collagen, a monomer with a rod-like shape having a length of about 300 pm and a diameter of 1.5 nm [Comper, 1984], is the most important structural molecule in the body. At physiological pH values, the collagen bears a slightly positive charge although the presence of amino and carboxylic groups within its structure provides this molecule with polyamp ho lytic properties (i.e., it has both positive and negative charged regions). Several collagen monomers form collagen fibril aggregates through covalent bonds. These fibrils combine further to form collagen fibers. By means of covalent cross-linked bonds, the collagen fibers form a meshwork that tends to immobilize other structural molecules such as proteoglycans, and, consequently, confer further resistance to changes in the volume of tissues. The collagen fibers account for 70 to 75% of the dry weight of normal adult skin [Uitto etal., 1984]. The elastic fibers contain two types of structural components, elastin fibers and microfibrils. Elastin is an insoluble protein that accounts for about 70 to 90% of the mass of the elastic fiber. It can best be described as a rubber-like material that is responsible for the elasticity of the tissue. The hydrophobic chains of amino-acids that form the protein are cross-linked in Chapter 2: Physiological Overview 19 random configurations. The additional components of the elastic fibers - the microfibrils - are composed mainly of glycoproteins. They can be associated with the elastin fibers or, alternatively, can form independent aggregates. The elastic fibers account for only a small fraction, about 2-4%, of the weight of dry skin in normal adults [Uitto et al., 1984]. The glycosaminoglycans (GAGs) occur in the body as hyaluronan, the only G A G that is a free polymer in solution, or as proteoglycans, that are aggregates of GAGs covalently linked with a core protein. The GAGs are highly hydrophylic substances. The most common of the GAGs, hyaluronan (a high molecular weight, unbranched polymer), when placed in a solution, coils and absorbs water such that it will occupy a volume as high as 1000 to 10000 times its own unhydrated volume [Aukland and Reed, 1993]. 2.5.2 Physical characteristics of the interstitium 2.5.2.1 Volume exclusion The presence of numerous interstitial macromolecules, particularly hyaluronan and collagen-based species, creates a macromolecular crowding of the interstitial space. As mentioned above, these solutes generate expanded interstitial mesh-like structures that, under certain conditions, exceed the volume of polymers themselves. Consequently, the space available to other interstitial species is less than the total interstitial volume, i.e., a given interstitial solute will distribute itself only outside the meshwork or, alternatively, through those spaces of the meshwork that have dimensions larger than that of the solute. This phenomenon of geometrical interstitial exclusion was first described by Ogston and Phelps [1961] and refers to the fact that two solid structures cannot occupy the same confined volume at the same time. The magnitude of the exclusion effect depends on the geometrical configuration of not only the excluding species but also the excluded one. Thus, the geometrical exclusion phenomenon is relevant only for species with high molecular weight such as proteins and not for, e.g., small ions. In addition to this geometrical exclusion, due to the fact that GAGs are negatively charged at normal pH values, electrostatic factors are also involved in selectively excluding other negatively charged solutes (e.g., proteins) found in interstitium. For the reasons described above, the interstitial volume can be divided into two fractions, the excluded volume, VJT,EX , and available volume, VIT,AV , such that Vrr = VIT.AV + V I T E X [2-2] Chapter 2: Physiological Overview 20 where Vrr is the total interstitial volume. The magnitude of the excluded volume has an important role to play in the dynamics of transcapillary exchange. Due to exclusion, the effective protein concentration in the interstitium is much higher than the value that would be estimated i f it were assumed that all the fluid in the interstitium was available. 2.5.2.2 Interstitial compliance The interstitial compliance reflects, to some extent, the elastic properties of the interstitial space. When the volume of the interstitium increases, the interstitial matrix attempts to limit this expansion, causing an increase in the interstitial pressure. The equation relating the interstitial hydrostatic pressure and the interstitial volume is known as the interstitial compliance. Generally, this equation assumes that the change in interstitial fluid volume, AVrr , is proportional to the change in interstitial fluid pressure, APrr, i.e., A V I T = FCOMPI • A P ^ [2-3] where FCOMPI is a proportionality constant called the interstitial compliance. Most of the measurements of interstitial compliance come from animal studies [e.g., see Guyton, 1965; Reed and Wiig, 1981]. The compliance of human tissues has not been studied as extensively. A compliance curve is generated by the measurement of several pairs of interstitial pressure-volume data that correspond to specific conditions of tissue hydration. An example of such a curve obtained from experiments on dog skin is presented in Fig. 2-4. As shown in this figure, for the dehydration region (i.e., volume less than normal), the curve has a steep slope. As the interstitial volume increases, the slope decreases until the curve becomes almost flat. The reciprocal of different segments of this slope represents the interstitial compliance, FCOMPI. For the region of normal to low interstitial volume (i.e., the dehydration region), the compliance is low. A low compliance implies that small changes in interstitial fluid volume produce large changes in interstitial pressure. Correspondingly, the high compliance observed in the overhydration region means that large changes in volume result in only small changes in interstitial pressure. Moderate compliance values are obtained for a moderately hydrated interstitium. Similar conclusions were drawn from compliance studies involving several other animals [Wiig and Reed, 1981; 1985; 1987]. Chapter 2: Physiological Overview 21 Figure 2-4. The change from normal of interstitial fluid volume AVrr (ml/g dry weight) vs. the change from normal of interstitial pressure, A P I T (mmHg), for experiments with dogs. For a discussion related to the different regions depicted in this compliance curve, see main text From Reed [1995]. 2.6 Characterization of the Lymphatic System Although the lymphatics are not accounted for as a distinct compartment in the model, their anatomical position and physiological role justifies their description immediately following the characterization of the interstitial compartment. The lymphatic system consists of a network of vessels that parallels the arterio-venous system. The lymphatic vessels drain excess fluid, proteins, bacteria, particulate material (e.g., in lungs) and immune cells from the interstitium back into the blood. Therefore, the lymphatic system has the dual role of preventing fluid and solute accumulations in the interstitial space (i.e., preventing edema formation) and acting as an 'immune-surveillance system' [Holtz, 1996]. The lymphatic vessels are classified as initial (terminal) lymphatics, lymphatic collectors or central lymphatic conduits. Chapter 2: Physiological Overview 22 The initial lymphatics are small vessels with diameters between 15 and 150 um (depending on the anatomical location of the vessel) and are involved directly in the collection of interstitial fluid in the microvascular system. They are located near the capillary beds. A detailed review of the micro-anatomical characteristics of the initial lymphatics has been presented by Schmidt-Schonbeim [1990]. As shown schematically in Fig. 2-5, these vessels have a single layer of overlapping discontinuous endothelial cells. The majority of initial lymphatics (with the exception of those in the brain, spinal cord and ocular space) have open junctions between adjacent cells. It was observed that the endothelial lining is irregularly tethered to its surrounding tissue by collagen-anchoring filaments [Leak, 1987; 1968]. The unattached luminal side of the cells forms flaps that float freely above the junction [Schmidt-Schonbeim, 1990]. This structural arrangement allows the initial lymphatics to operate as microvalves between the interstitium and the lumen of the lymphatics. When the interstitial pressure exceeds that in the initial lymphatics, the valves open allowing the entry of fluid, small ions and macromolecules, but when the pressure in the lymphatic vessel becomes greater than that in the surrounding tissue, the flaps overlap preventing the outward movement of lymphatic fluid. Specific mechanisms that allow the movement of lymph into and/or through the initial lymphatics have never been observed microscopically. It has been proposed, however, that the compression of the initial lymphatics that are inherent when normal bodily activities are performed (e.g., during skeletal muscle contraction, respiratory movements, etc.) determine the uptake and transport of interstitial fluid in these vessels [Olszewski and Engeset, 1980; Aukland and Reed, 1993]. The collecting lymphatics drain the incoming fluid from several initial lymphatics. These vessels reach diameters up to 600 urn and are characterized by two types of segments. Segments similar to the initial lymphatics in that they have a discontinuous endothelial layer (and can uptake fluid directly from the interstitium) are often called lymphatic precollectors, while the other type of segments have walls that resemble those of small veins and are called lymphatic collectors. Both types of segments are endowed with bileaflet valves positioned in the direction of lymph flow such that the back flow of lymph is prevented (see the lymphatic segment on the far right in Fig. 2-5). It has been proposed that the collecting lymphatics are organized as discrete contractile units surrounded by circular layers of smooth muscle and nerve endings [Olszewski and Engeset, 1980]. Each contractile unit is separated from the next one by a bileaflet valve and exhibits a Chapter 2: Physiological Overview 23 peristaltic contraction with a synchronized opening and closing of the valve. So far, this is the most accepted mechanism of unidirectional lymph propulsion. Figure 2-5: Schematic diagram of the initial lymphatic and collecting vessels. FromHoltz, [1996]. The large central lymphatic conduits, which are the final vessels in the lymphatic architecture, carry the lymphatic fluid back into the blood circulation via the jugular veins. The composition of lymph undergoes some modification in its route from the initial lymphatics to the central conduits. The lymphatic nodes positioned throughout the lymphatic system, besides being involved in the formation and differentiation of lymphocytes, act as filters for bacteria and particles. From the perspective of this study, and based on other literature information [Aukland and Nicolaysen, 1981], the non-excluded interstitial fluid and the lymphatic fluid are considered identical in composition. Furthermore, the lymphatics are not approached as a distinct fluid compartment; rather they will be treated as passive conduits of interstitial fluid integrated into the microvascular exchange system. Chapter 3: Transcapillary and Cellular Exchange 24 Chapter 3: TRANSPORT ACROSS THE CAPILLARY AND CELLULAR MEMBRANES 3.1 Introduction In order to provide a better understanding of the system being studied in this thesis, Chapter 2 presented a general overview of the characteristics of the different fluid compartments that are assumed to represent the body. The partition of fluid and solutes amongst these compartments is determined ultimately by the various exchange processes that occur between them. The purpose of the present chapter therefore is to discuss in detail the two primary sites of exchange within the body, namely the capillary and cellular membranes. In particular, generalized equations, representative of all tissues, as well as the governing driving forces that cause material exchange across both membranes, will be introduced. To guide the discussion that follows, the presentation will be conceptually separated into two distinct parts, one for each of the two types of membranes involved. Part I: Transcapillary Exchange 3.2 Structure and Function of the Microcirculation When a generalized description of the circulatory system was presented earlier, it was shown that the microcirculation is an integral part of this system. A l l the extracellular exchanges of materials between the circulating blood and the various types of interstitia of the body occur in the capillary beds that comprise the microvascular exchange system (MVES). The morphological design of the microcirculation is such that it favors efficient exchanges of fluid and solutes between the blood and the surrounding tissues and organs. Accordingly, in these exchange vessels (e.g., capillaries and venules), the velocity of blood is slowest (< 1 mm/sec) while the surface area of exchange is highest [Renkin and Crone, 1996; Guyton, 1991]. Normally the blood flows intermittently through individual capillaries. This process is termed vasomotion and is due to an intermittent opening and closing of the precapillary sphincters. The on-off pattern of blood flow repeats itself every few seconds under normal circumstances, but can be severely disturbed during conditions such as hemorrhage (when the capillary circulation can shut off for as long as 15 minutes). In order to assure that the nutritive Chapter 3: Transcapillary and Cellular Exchange 25 demands of the tissues are met, each organ may auto-regulate to some extent its own blood supply by causing dilations or contractions of the smooth muscles of the metarterioles and the precapillary sphincters. This control is based on the specific local requirements (mainly oxygen demand, but also to some extent, the local concentration of nutrients, an excess of metabolic products, etc.) [Guyton, 1991]. A maximized transfer of nutrients is assured by the fact that, generally, the capillary-to-tissue cell distances, are extremely low, i.e., between 20-50 um [Renkin and Crone, 1996; Guyton, 1991]. Additionally, the microcirculation of each organ is anatomically tailored to serve the organ's particular needs. Thus, in skeletal muscle, the capillaries are positioned along the muscle fibers [Renkin and Crone, 1996] while, as will be shown later in Chapter 8, typical renal microvessels surround the renal tubules as in interwoven arrangement [e.g., Guyton, 1991; Krieger and Sherrard, 1991]. Despite variations in the structural configuration of different capillary beds, similar arrangements serve the same exchange purposes. Therefore, a general characterization, in terms of pathways for transport or governing forces that cause mass exchange across individual capillaries, is possible. 3.3 The Capillary Wall and its Transport Pathways The capillary wall is about 0.2 to 0.5 um thick [Guyton, 1991; Renkin and Crone, 1996] and is composed of a unicellular layer of endothelial cells supported by a specialized region of collagenous fibers termed the basement membrane (or basal lamina) [Simionescu and Simionescu, 1984; Bert and Pierce, 1984]. The capillary wall acts as a semi-permeable membrane that separates the blood from the interstitium. Transport of fluid and solutes between the blood and interstitium can take place as a consequence of local transmembrane differences in concentration and/or hydrostatic and colloid osmotic pressures or, by more specific mechanisms such as via plasmalemmal (endoplasmic) vesicles. The endoplasmic vesicles are believed to play a role in selectively shuttling substances (including macromolecules) through the capillary wall [Wissig and Charonis, 1984]. Generally, material can selectively permeate the capillary wall via three main routes: directly through the cell membranes, through channels within the cells, or through the spaces between adjacent cells [Renkin, 1977; Schneerberger, 1992]. The several transport pathways identified for the passage of fluid and small molecules [Renkin, 1985; Schneerberger, 1992] can be summarized as follows: intercellular clefts, fenestrae, plasmalemmal vesicles, discontinuous vessels and stable pores. These pathways are illustrated in Fig. 3-1. Chapter 3: Transcapillary and Cellular Exchange 26 (1) (2) (4a) (4b) _ (3) . Figure 3-1. Transport of different types of solutes (see below) through the multiple pathways identified for transcapillary exchange. The transport pathways in the capillary endothelium include: (1) transport through the cell membranes; (2) transport through intercellular clefts, across small or large pores; (3) transport across fenestrae that can be specialized pathways through the cell or areas where the two membranes of the endothelial cell are fused; and (4) continuous (4a) or discontinuous (4b) vesicular shuttling of substances. Modified from Curry [1984]. The intercellular clefts are openings present between two adjacent endothelial cells. Typically they are around 10-20 nm across [Fung, 1984] and, in most tissues, they provide the major pathway for the transport of water and macromolecules. Sometimes, these clefts are intermittently closed due to the small areas of contact between two adjacent cells. Moreover, in some microvascular beds, such as the microvasculature of the brain, these clefts are completely closed. The fenestrae are formed by a reduction of the capillary wall to a small thickness of 6-8 nm as a result of the fusion of the membranes of two endothelial cells. In some exchange sites, such as the glomerular capillaries of the kidney, the fenestrae are completely open. The spherical bodies of the plasmalemmal vesicles belong to the cytoplasm of the endothelial cells. These vesicles are believed to shuttle back and forth between opposing cell surfaces, thus delivering their fluid and solute contents to the opposite surface. A temporary fusion of a number of vesicles, such that an open water channel through the capillary wall is created, has also been described [Wissig and Charonis, 1984]. The discontinuous vessels are referred to as sinusoids and are most commonly found in liver or spleen. They are characterized by large discontinuities in the endothelium and its basement membrane and thus confer the entire organ with an increased permeability for solutes. Chapter 3: Transcapillary and Cellular Exchange 27 As a consequence, the reflection coefficient for proteins in the liver microvasculature is almost zero, compared to a characteristic value of about 0.9 in skeletal muscle. The degree to which several types of transport pathways occur is dependent on the location and function of the microvascular bed in question. Continuous vessels are commonly found in the microvascular beds of lung, nervous system, skeletal muscles and skin [Simionescu and Simionescu, 1984]. Fenestrated vessels are predominant within the microvasculature of pancreas, endocrine glands and the gastrointestinal tract. As previously mentioned, the sinusoid vessels are typical of liver. 3.4 Transcapillary Transport Equations Owing to the multitude of the capillaries present in the body, their overall function must be averaged. Therefore, in all subsequent quantitative analyses of transport in the M V E S , only average values for capillary pressure, blood flow rate, material transport rates, area available for transport, etc., will be used. Throughout most of the medical literature, discussions about the different pathways available for capillary exchange, as presented in the previous section, remain largely qualitative. The most accepted methods of quantifying the transcapillary mass exchange still use expressions derived for transport across porous membranes. Thus, the capillary membrane is treated either as a homoporous or as a heteroporous barrier [e.g., Rippe, 1989; McNamee and Wolf, 1991]. Mass exchange rates for fluid, small ions and macromolecules (i.e., proteins) are therefore assumed to be given by the ratio of a lumped driving force divided by the transport resistance of the transcapillary barrier. 3.4.1 Fluid transport across the capillary wall The principal driving forces governing fluid flow across the capillary membrane between the vascular space and the neighboring interstitium are the hydrostatic and colloid osmotic pressure differences. These driving forces are known as Starling forces and are depicted in Fig. 3-2. The hydrostatic pressure in the capillary along with the interstitial colloid osmotic pressure promote fluid movement from the capillary (i.e., fluid filtration), while the hydrostatic pressure in the interstitium together with the plasma colloid osmotic pressure drive fluid into the capillary (i.e., fluid absorption). Chapter 3: Transcapillary and Cellular Exchange 28 Osmotic and colloid osmotic pressures Any of the solutes found in the plasma and interstitium that cannot freely cross the endothelial barrier can, by means of exerting a transcapillary osmotic pressure, influence fluid transport across the capillary membrane. The degree to which a solute affects transcapillary fluid flow depends in part upon the permeability of the capillary wall for that solute. The fraction of the total osmotic pressure exerted by a species upon the capillary membrane is given by the reflection coefficient, a. Thus the osmotic pressure difference, Arcs, exerted by a solute S having a reflection coefficient as across a semi-permeable membrane is given by the following relationship derived from the van't Hoff law: A7ts = R T a s ( C s > i - C s , 2 ) [3-1] where (Cs.i - Cs,2) is the concentration difference of solute S across the membrane while R and T are the absolute temperature and the universal gas constant, respectively. CAPILLARY Capillary hydrostatic pressure Plasma (colloid) osmotic pressure Pc • i ft P,PL '. Interstitial hydrostatic pressure P I T Interstitial colloid osmotic pressure ft P,IT Figure 3-2. The Starling forces that contribute to fluid exchange across the capillary membrane. Chapter 3: Transcapillary and Cellular Exchange 29 A reflection coefficient of 1 denotes complete impermeability of the capillary membrane for a given solute. A value of 0 indicates a solute permeability across the capillary equal to that of water, i.e., the solute moves freely through the capillary pores [Parker et al., 1984]. Most lipid-soluble gases such as O2 and CO2 have reflection coefficients near zero [Renkin and Crone, 1996], while the values for proteins, except in some microvascular beds such as liver, are found to be nearly unity. The capillaries of skeletal muscle have reflection coefficients for proteins around 0.95 to 1 [e.g., Renkin, 1988; Renkin and Crone, 1996]. The values for the small lipid-insoluble solutes, such as Na + , K + and glucose, are somewhat more uncertain. Traditionally, it was assumed that most of the latter species have low reflection coefficients (i.e., a < 0.1) and therefore any differences in their concentrations between plasma and interstitium are dissipated rather quickly [Michel, 1984; Wolf, 1980]. More recent reports, however, suggest reflection coefficient values for small ions to be as high as 0.3 to 0.5 [Wolf and Watson, 1989]. Theoretically, all the osmolites typically found in plasma or interstitial fluid could exert a total osmotic pressure of about 6000 mmHg across a membrane permeable only to water [Kleinman and Lorentz, 1984]. Of this value, only about 25 to 30 mmHg are provided by the presence of proteins. Owing to the high permeability of the capillary for most solutes other than plasma proteins, under normal conditions these latter species are the predominant contributors to the transcapillary osmotic driving force. To differentiate it from the total osmotic pressure exerted by the extracellular solutes, the osmotic pressure due to the plasma proteins is termed the colloid osmotic pressure. Gibbs-Donnan effect The contribution of plasma proteins to the colloid osmotic pressure difference is actually greater than would be predicted i f only the number of dissolved protein molecules in plasma is considered. This added pressure difference is due to the Gibbs-Donnan effect and occurs as a result of the retention in the capillary of various protein species (mainly albumin), which, at physiological plasma pH values, are negatively charged. In order to neutralize these charges, a large number of plasma cations (mainly N a + by virtue of its abundance in plasma, but to some extent also K + , C a 2 + and Mg + 2 ) surround the proteins. These cations are not chemically bound to the macromolecular species, but are held in their proximity by electrostatic forces. As a result, the number of osmotically active species prevented from crossing the endothelial barrier is Chapter 3: Transcapillary and Cellular Exchange 30 higher and consequently the colloid osmotic pressure exerted in the presence of these proteins must also be higher. In light of the above discussion, the Donnan effect therefore has the following implications: 1) it contributes to an additional osmotic pressure difference between the plasma and interstitium, and 2) it causes an asymmetric distribution of ions between the two compartments. The consequence of this statement is the partition of small ions shown in Fig. 3-3, which depicts a higher concentration of non-diffusible positive small ions in the plasma compartment, and a correspondingly higher accumulation of anions in the interstitium. This partition is directly related to the concentration of proteins in plasma; e.g., it is expected that, during hemodilution when the concentration of plasma proteins decreases, the Donnan effect will decrease in importance. Traditionally, the asymmetric distribution of charges across the capillary membrane has been determined by accounting for the Na + , K + and C f ions, and the plasma proteins. Such a description is not useful if other positive and negative ions are considered to take part in transcapillary transport. A detailed description on how the Donnan effect was accounted for in the current model, where multiple ionic species are present, is given in Chapter 4. For now it will be assumed that the Donnan contribution to the colloid osmotic pressure generated by plasma proteins is incorporated in the term TCP.PL of Fig. 3-2. INTERSTITIUM CAPILLARY Cat+ ^ An" PLASMA C a t + An" Figure 3-3. Schematic representation of the Donnan partition of small ions across the capillary. The size of the symbols, reflect a higher concentration of non-diffusible positive charges, Cat+, on the plasma side and the corresponding higher accumulation of non-diffusible negative charges, An", on the interstitial side. For more discussion see text. Chapter 3: Transcapillary and Cellular Exchange 31 Based on the overview given above regarding the various driving forces that act upon the capillary membrane, the rate of fluid movement across this membrane is given by the classic equation based on Starling's hypothesis (Starling, 1890). Thus, i f a generic protein species (representative of the multitude of proteins found in plasma) has a reflection coefficient o and if the effect of small solutes can be neglected, then the fluid flow rate across the capillary membrane, JIT, can be written as: JIT = LcapS• [(P c - P J T ) - o ( 7 r P P L - 7 i p 1 T)] [3-2] Equation 3-2 delineates the balance between the hydrostatic and colloid osmotic forces in governing transcapillary fluid movement. The symbols that appear in Eq. [3-2] are defined as follows: Leap and S are the hydraulic conductivity and the exchange surface area of the capillary, respectively; Pc and PIT are the capillary and interstitial hydrostatic pressures, respectively; while 7tp.PL and TCp r^r represent the plasma and interstitial colloid osmotic pressures, respectively. The subscript IT used for the transcapillary fluid flow indicates that, under normal conditions, there is a positive filtration of fluid from plasma toward the interstitium. The hydraulic conductivity, L c a p , is an intensive parameter of the capillary wall that is functionally connected to the variable surface area available for transport, S. Usually, transport equations that have the form of Eq. [3-2] take into account only the product of the above two terms; that product is expressed as kp.the capillary filtration coefficient. 3.4.2 Protein transport across the capillary wall Because proteins are the primary contributors to compartmental osmotic pressures, the rate of protein transport across the capillary membrane has an important role in determining the partition of the extracellular volume between plasma and interstitium. There are several equations available to describe the transcapillary transport of macromolecules depending on the assumptions made with respect to the number of pathways available for transport. The Kedem and Katchalsky [1958] equation assumes that the diffusion and convection of proteins from plasma to interstitium occurs as two independent processes. This equation does not make any assumptions with respect to the geometry of the channels within the porous membrane. Thus, the transcapillary transport of protein, Qn, from plasma toward the interstitium is given by, Chapter 3: Transcapillary and Cellular Exchange 32 QIT = J I T ( l - a ) - ( C p - P L + 2 C p j T - A V ) + PS-(c p , P L - c P J T , A V ) [3-3] where the first r.h.s. term is the convective contribution and the second term, the diffusive contribution. The terms cp,pL and Cpjx.Av are the protein concentrations in plasma and in the available interstitial volume, respectively. Recall from Chapter 2 that only a certain fraction of the interstitial volume, i.e., about 25%, is available for protein distribution. In the convective term, it is assumed that the appropriate protein concentration being carried by the fluid flow, JIT, is the average value for the two compartments. PS in the diffusive term of Eq. [3-3] represents the permeability-surface area product for proteins; i.e., it is the product of the protein permeability through the capillary, P, and the total capillary area that is available for exchange, S. An alternative equation for the transport of proteins (and generally for the transport of any neutral species), was proposed by Patlak et al. [1963] as follows: c _ c e x p ( - J l T ( 1 " a ) ^ CP.PL ^P,IT,AVC XPV „ „ ) QIT = Jrr-0~°)- [ T n_JS 1 [3-4] l - e x p K 1 ^ 1 G ) ) PS where all the symbols in this equation were defined previously. Equation [3-4] is obtained by solving the one-dimensional convection-diffusion equation for a uniform cross-sectional channel. This equation can be rearranged into the following form proposed by Bresler and Groome [1981]: Qrr = Jrr - 0 " °")CPL + ^ ~ CP.rr,AV ) t3"5] J I T . ( l - o ) 1 - exp(— ) PS Equation [3-5] is not equivalent to the sum of uncoupled diffusion and convection proposed by Kedem and Katchalsky [1958] (i.e., Eq. 3-6), although it gives similar predictions at high flow rates [Parker et al., 1984]. 3.4.3 Transport of small ions across the capillary To this author's knowledge, the Kedem and Katchalsky equation is the only expression that has been proposed and used for the transcapillary movement of small ions. Considering a generic species, ION, that represents one of the multitude of cations and anions found in plasma and interstitium, the transcapillary movement of each ionic species can be expressed as, Chapter 3: Transcapillary and Cellular Exchange 33 N W = J ^ . ( l - o I 0 N ) . ( [ I ° N ] p L ^ [ I ° N ] l T ) + P S 1 0 N . ( [ I O N ] P L - [ I O N ] I T ) [3-6] where [ ION ]PL and [ ION ] IT are the ionic concentrations in the plasma and interstitium, respectively; while CJION and PS ION are the reflection coefficient and the permeability-surface area product, respectively, for this small solute. Several slightly modified forms of Eq. [3-6] can be found in the literature, where the average ionic concentration inside the channel (see the first term of this equation) is replaced by either the logarithmic average or simply by the ionic concentration at the plasma interface, i.e., [ ION ]PL . Based on experimental studies, it was concluded that the algebraic average provides a better approximation of the experimental ionic transport data in the M V E S [Parker et al., 1984]. The above description completes the quantitative analysis of the contributions of the capillary membrane to fluid and solute redistribution in the extracellular environment. The governing equations for transcapillary fluid and solute transport were provided in relation to the two extracellular compartments: plasma and interstitium. At steady state, there are also continuous exchanges of fluid and solutes across the cellular membrane between the red blood cells and the surrounding plasma as well as between the tissue cells and the interstitial fluid. The next part of this chapter deals with an analysis of this exchange. Part II: Cellular Exchange 3.5 Theoretical Considerations The discussion that follows will be primarily concerned with the characterization of the transport between a cellular compartment (ICV) and its extracellular environment (ECV). Due to the fact that the E C V and ICV are in direct contact, any disturbance in the extracellular volume or composition will have repercussions on the intracellular fluid and solute properties. The cells in the body possess unique mechanisms for transporting fluid and small solutes across the cellular membrane. These transport mechanisms are central to the maintenance of a constant cellular volume and, generally speaking, body homeostasis. Regulation of cellular volume implies a close control of intracellular water activity. Mammalian cells have a water content of somewhere between 75-85% of their total weight [MacKnight, 1984(b)]; therefore, the control of cellular water implies a control of cellular volume. Even though the cellular membrane Chapter 3: Transcapillary and Cellular Exchange 34 is highly permeable to water, under normal physiological conditions, the water content and, implicitly, the cellular volume are maintained within very narrow limits. 3.5.1 Cell membrane The average difference in fluid composition between the ICV and E C V compartments was presented previously in Table 2-1 (see Chapter 2). The partition of solutes between the two environments is brought about by the existence of an outer cellular membrane called the plasma membrane of the cell, or simply the cell membrane. Although the presence of the cellular membrane as a distinct physical barrier between the two environments is generally accepted, its role in cell volume regulation is still under debate. Most researchers agree, however, that the membrane not only assures a physical separation between the E C V and ICV fluids, but it also provides a controlled communication between the interior and exterior of the cells [e.g., MacKnight, 1994; Greger and Windhorst, 1996]. To attain the ionic partition illustrated in Table 2-1, the cell membrane must maintain very large ionic gradients and, implicitly, must develop a membrane potential. Additionally, the ions have to be transported across the membrane in a controlled fashion in agreement with the cellular requirements. As will be described in this section, some of the control mechanisms for differentiating the cytosol from the extracellular environment reside within the membrane itself. From a structural point of view, the cell membrane consists of a lipidic bilayer arrangement [see e.g., Matthews, 1998]. The phospolipids of the membrane arrange themselves in a bilayer sandwich-type structure, with the polar (hydrophylic) regions oriented toward the interior or exterior of the cell, while the non-polar hydrophobic tails are pointing toward each other. The proteins that form complex aqueous pores and/or channels through the membrane protrude through the lipidic layer. By weight, about 30% of the membrane is composed of lipids while the remainder is primarily proteins [Greger and Windhorst, 1996, Matthews, 1998]. However, the lipidic portion offers a higher available area for transport [Jensen, 1989]. 3.5.2 General characteristics of transport through the cellular membrane Transport through the cellular membrane has been the topic of several extensive reviews [e.g., Tosteson, 1981; MacKnight, 1994]. It is currently accepted that the solutes and water can permeate the cellular membrane by either passing through the lipid bilayer or, alternatively, directly through the protein pathways. Chapter 3: Transcapillary and Cellular Exchange 35 There are two basic types of transport across the cell membrane: 1) passive transport and, 2) active transport. Passive transport represents the simple physical diffusion of a solute down its concentration and/or electrochemical gradient and can, in turn, be sub-divided into the following three categories: a) Solubility-diffusion (or simple diffusion) - the net movement of a species down a concentration or chemical potential gradient (e.g., movement of water) b) Electrodiffusion - the movement of ionic species driven by both concentration and electrical potential differences (e.g., transport of Na + , K + and Cl") c) Facilitated diffusion - involves a specific interaction between the transported solute and another solute that crosses the cellular membrane. A particular subset of facilitated diffusion can occur when two solutes move simultaneously across the membrane in either the same direction (cotransporf), or the opposite direction (countertransport). Active transport requires the presence of a membrane component that transports ions through the membrane against their concentration (or electrochemical) differences. In order to achieve this type of transport, the cell must expend energy. The most common type of active transport is the one associated with the electrogenic Na +-K +-pump. Generally, the literature dealing with the structure and function of the cellular membrane describes the protein channels as highly selective for the passage of a given small ion. This implies that the ionic permeability is at least partially determined by the specific function of the membrane proteins that make up the channels. It would be expected therefore that some protein macromolecular aggregates embedded in the lipidic bilayer might constitute elaborate 'membrane transport devices'. So far, however, very few such devices have been observed and none have been analyzed to the point where their function can be quantitatively described. 3.5.3 Cell membrane permeability An important consideration in the transport of solutes across the cell membrane is its permeability to various solutes as well as to water. For porous membranes, the net flow of water across the membrane is due mostly to the osmotic pressure difference between the intra- and extracellular compartments [Weiss, 1998]. Consequently, the water permeability is characterized by the so-called osmotic permeability coefficient, pw, expressed in cm/s and defined as: Chapter 3: Transcapillary and Cellular Exchange 36 R T L Pw = — [3-7] m,w were Lv is the hydraulic conductivity (cm/(mmHg-s)) and V m , w is the partial molar volume of water (cm3/mole). For ions that are passively transported across the lipidic membrane, the ionic permeability refers to an electro-diffusive mechanism, characterized by the diffusive permeability coefficient, PION, which is given by PION = ^ION ~, [3-8] dm where DION (cm2/s) is the diffusion coefficient of the ionic species, ION; d m is the membrane thickness (cm) and k i o N is the partition coefficient for that ion between the lipidic phase of the membrane and the aqueous phase of the intra- or extracellular medium. This permeability is determined by the relative lipid-water solubility of the diffusing solute, the size of the membrane pores and the electric charge of the membrane, as well as the dimensions of and the charge carried by the diffusing solutes. The reported membrane permeability values for the solutes of primary interest to this work are presented in Table 3-1 for RBCs and skeletal muscle cells. As is apparent from the discrepancies seen in this table, the true values of these membrane permeabilities are somewhat uncertain. The permeability of water is usually estimated from transient osmotic data. There are basically two approaches used to determine the membrane permeability of ions: the flux method in which the net rate of movement of a radioactive tracer through the cell membrane is measured, and the electrical method in which the electric currents carried by ions moving through the membrane are measured. Both of these methods have some significant disadvantages. The flux method has a low time resolution and sensitivity, while in the electrical method, the ion flow affects the membrane potential, which, in turn, can alter the permeability value. Usually, both of these methods can be applied to either intact, relatively large cells such as RBC and skeletal muscle cells, or, as is often the case, to 'artificial membranes', i.e., cell-membrane lipid layers which are subjected to measurements after the proteins have been extracted. The solute permeability values presented in Table 3-1 are, in general, extremely low if one thinks in terms of the diffusion of the same solutes through aqueous layers having thicknesses comparable to those of the cell membranes. This includes the water permeability, which has the highest value reported in this table. However, the rate at which water movement, Chapter 3: Transcapillary and Cellular Exchange 37 for instance, occurs through the cell membrane is extremely rapid when the dimensions of real cells are considered. A sample calculation for a RBC, whose typical volume is about 90 um 3 and surface area is 160 um 2 [Burton, 1965], shows that the total volume of water that can diffuse in and out of the cell per second is about 80 times greater than the cell volume itself. (The calculation was performed based on the permeability value for water reported by Dick [1970] which is presented in Table 3-1.) Species RBC Permeability (cm-s"1) Skeletal Muscle Cell Permeability (cm-s1) Water 5 x 10"3 [House, 1974] 3x 10"2[Dick, 1970] 1 x 10"2 [Jensen, 1989] Sodium, Na + 4x 10"10 rHoffman, 1987] (1-4) x 10"10 [Altman and Dittmer, 1971] 4.5 x 10"7[Jakobsson, 1980] 2x 10"8 [Jensen, 1989] Potassium, K + 10"10 [Hoffman, 1987] 5x 10"5[Jakobsson, 1980] 5 x 10"5 [Fromkin, 1973] 2 x IO"6 [Jensen, 1989] Chloride, Cl" 10"8 [Altman and Dittmer, 1971] 5 x 10"5 [Jakobsson, 1980] 4x 10"6 [Jensen, 1989] Proteins and Sucrose 0.0 [Greger and Windhorst, 1996] 0.0 [Altman and Dittmer, 1971] Table 3-1: Reported permeabilities for red blood cells and skeletal muscle cells (see table entries for references) Chapter 3: Transcapillary and Cellular Exchange 38 3.6 Exchanges of Ions and Water across the Cellular Membrane A l l the previously mentioned transport mechanisms operate concurrently to assure the normal distribution of ions across the cellular membrane. It is not possible to present a complete account of all these mechanisms here. The present work therefore will rely on a simplified, but still reasonable, model of cell volume regulation (CVR). The full C V R model is given as a part of the model formulation in Chapter 4. The intention of this section is to present only the ion transport equations needed to provide a basic understanding of cellular exchange. The final goal of this section is to build a reasonable picture of a typical cell, which is able to control its volume while actively exchanging solutes across its enclosing membrane. First, the characteristics of a typical cell, the partition of ions across its membrane and the accepted transport mechanisms that bring about this partition will be discussed. Following this, the transmembrane ionic transport for such a cell will be detailed in a hierarchical fashion, i.e., from the simple electro-diffusional transport of two ions across the membrane, up to a more complex case where three ions participate in both active and passive processes. 3.6.1 Mechanistic description of a typical cell Figure 3-4 illustrates the typical distribution of ions and their transport directions across the cell membrane of muscle cells. Of all the solutes present in the intra- and extracellular fluids, Na + , K + and Cl" are the primary species involved in transport across the cellular membrane. Additionally, they govern the electrical properties of cells, i.e., the generation of a membrane potential. Similar to the approach of other authors [e.g., Wolf, 1980; Jakobsson, 1980], it is therefore assumed that only these three species are transported across the cellular barrier. The permeabilities, which control the passive transmembrane transport of these ions are pNa, PK and pci, respectively. The remaining solutes are considered to be fixed and mobile anions, (FA) and A, respectively, or fixed and mobile cations, (FC) and C, respectively, confined to either the interior (ICV) or exterior (ECV) of the cell. None of these other solutes can cross the cellular membrane; however, they contribute to both the ICV and E C V total osmolarities and thus are important contributors to the osmolarities of the intra- and extracellular environments. As a result of differences in the distribution of these charged species between the ICV and ECV, a resting cellular membrane potential, V m , is generated. By convention, the cellular membrane potential is measured with reference to the potential outside the cell; therefore, V m for Chapter 3: Transcapillary and Cellular Exchange 39 the type of cell described in Fig. 3-4 (i.e., skeletal muscle cells) is somewhere between -85 and -90 mV, with the interior of the cell being negatively charged. The simplest and most accepted cell membrane transport mechanisms for Na + , K + and Cl" ions are 1) electrodiffusion due to concentration and electrical potential differences and 2) active transport. Based on these two primary mechanisms, the three major ions are transported across membrane as follows. The sodium ion diffuses passively from the E C V to the ICV as a consequence of concentration and electric driving forces (the dashed line of Fig. 3-4 indicates the electrodiffusional transport). This ion is also transported back out of the cell through active transport, by means of a Na +-K +-pump (indicated by the solid line of Fig. 3-4). The permeability of the membrane to potassium is about two orders of magnitude higher than for sodium. K + is transported out of the cell by electrodiffusion down a concentration gradient and against an electrical potential difference. The potassium ion is pumped back into the cell by active transport. The chloride ion, with its relatively high permeability (similar to that of potassium), is the only negative species transported across the cell membrane. This ion is transported passively from the E C V to the ICV because of a concentration difference, and from the ICV to the E C V due to a potential difference. The net rate of exchange of Cl" should be just sufficient to maintain an electrically neutral transport of all ions across the membrane. Water moves freely across the cell membrane. The instantaneous shift of water is determined only by changes in the ICV or E C V osmolarity. The total osmolarity in the extracellular or intracellular medium is given by the sum of all the osmolites present in the E C V or the ICV, respectively, at any given time. The equations that govern the transport of ions across the cell membrane are presented next. 3.6.2 Transport of ions due to electrical and diffusional driving forces As part of a model of cellular exchange, quantitative expressions for passive and active ionic transports are required. In order to detail these transports, it will be assumed that the partition of ions between the E C V and ICV is equivalent to that given in Fig. 3-4, which reflects the normally measured concentrations of these ions in the body (see also Table 2-1). Furthermore, due to their concentration differences across the membrane, a transmembrane potential difference, V m , is generated. Chapter 3: Transcapillary and Cellular Exchange 40 Intracellular fluid (ICV) - 1 0 mEq/l Na MEMBRANE - 85 to -90 mV + Extracellular fluid (ECV) 2 K ~150mEq/ l K pu -10" cm/s K ~4mEq/ l CI" P c , ~ 1 0 c m / s FA FC p -10 cm/s + - N a Na -139 mEq/l 3 Na K ~4.5 mEq/l CI -107 mEq/l A C Figure 3-4: Distribution of ionic species across the cellular membrane as a consequence of the different types of transport mechanisms. Representative average concentrations and membrane permeabilities are presented for the transfering species. Na + , K + and CI" are the only ions that participate in solute exchange across the membrane. The 'fixed charges' FA, FC, A and C cannot pass through the membrane; they contribute, however, to the total osmolarity of the ICV and the ECV. The dashed and solid lines indicate the electrodiffusional and active transport mechanisms, respectively. The Na +-K +-pump, represented by the closed circle in the membrane, is an electrogenic pump, i.e., it extrudes 3 ions of N a + out of the cell for every two K + ions brought into the cell. The function of this pump requires consumption of energy by the cell. The dotted line used for the transport of CI" indicates that this ion moves back and forth across the membrane at a rate dependent on N a + and K + movement. Water is assumed to move freely across the cell membrane. More details are given in the main text. Chapter 3: Transcapillary and Cellular Exchange 41 The transport rate of potassium by electrodiffusion, Td,K, that takes place from the ICV to the ECV, is given by the following expression: T = m RT [K] F-V (—) a RT ICV [K] ECV F-V RT [3-9] e -1 Similarly, the electro-diffusional flux of chloride from the E C V to ICV, T^ci, is given by: _ F - V m Td.ci ~ J^ J,™ Pci [ C l ] E C V - [ C l ] I C V - e 1-e R T F-V ( - — ) RT [3-10] In Eqs. [3-9] and [3-10], the ionic concentrations for potassium, [K], and chloride ion, [CI], in the extracellular (ECV) and intracellular (ICV) media are expressed in (mEq/l), the permeabilities for potassium and chloride are P K and pa, respectively, expressed in (cm/s); while F, R and T are Faraday's constant, the gas constant and the absolute temperature, respectively. As written, Eqs. [3-9] and [3-10] account for a charge of -1 for the chloride ion and +1 for the potassium ion. These equations are based on the Goldman-Hodgkin-Katz [Goldman, 1943; Hodgkin and Katz, 1949] 'constant field' formulation of the ionic current that passes through the membrane, which in turn is derived from Nernst-Plank electrodiffusion theory subject to assumptions that the membrane is homogenous, the electric field within the membrane is constant, and the individual ions move through the solution independently. A detailed derivation of the general form of the above equations is presented in Appendix A. If K + and CI" ions were the only species transported across the cell membrane, a steady-state distribution of the two ionic species, K + and CI", could exist across the cellular membrane when their electrical gradient driven transports counterbalance exactly their concentration gradient driven transports, i.e., when Td,K - Ta,ci = 0. Furthermore, this dynamic equilibrium condition can be characterized by a specific transmembrane potential called the Nernst potential. Both ionic species, K + and CI", are at equilibrium if their Nernst potentials are the same. If E K is the Nernst equilibrium potential for K + while Eci is the corresponding potential for CI", equilibrium is attained when the transmembrane potential equals the Nernst potentials of both ions as follows: Chapter 3: Transcapillary and Cellular Exchange 42 V m = E K = Hi n (Ekv ) = E c i = ^ln(S^) [3-11] r L K Jicv * [C1JE C V The hypothetical result obtained above is not representative, however, of the cells in the body. As discussed in the previous section, the cellular membrane is also permeable to the sodium ion. Thus, sodium ions are also transported into the intracellular environment by both concentration and electrical difference driving forces according to _ F - V m Xl.Na ~ PNa F-V e R T -1 [3-12] If nothing prevents the movement of this ion from the E C V to the ICV, its intracellular accumulation will cause swelling and eventual bursting of the cell. The presence of this third transportable species, i.e. Na + , has the following consequences [Matthews, 1998]: 1) the species that participate in transport (i.e., Na + , K + or Cl") can no longer achieve an electro-diffusional equilibrium, i.e., a Nernst equilibrium does not exist for this case, 2) the resting membrane potential, V m , must now incorporate a contribution due to sodium transport across the cell membrane, and 3) an expenditure of cellular energy is required to actively extrude sodium from the intracellular environment. In order to prevent the intracellular accumulation of N a + as it diffuses down its concentration and electrical gradients, a mechanism, other than simple electrodiffusion has to actively extrude this ion from the cell at a rate high enough to counterbalance the inward electrodiffusion. Stated otherwise, the resulting osmotic disturbance and accompanying cellular swelling caused by the N a + ions must be prevented by achieving an apparent impermeability of the cellular membrane to this ion. This continuous extrusion of N a + from the cell is provided by what is known as the Na+-K*-pump which utilizes a high energy phosphate compound (ATP) as a source of metabolic energy. This mechanism is referred to as the pump-leak hypothesis and was first described by Wilson [1954] and further elaborated by Leaf [1956] and Tosteson and Hoffman [I960]. Cumulative evidence from researchers ascribing to the pump-leak hypothesis [Post, 1957; Skou, 1957; Tosteson, 1984] shows that the pump extrudes N a + in exchange for K + via N a + - K + -ATPase in the ratio 3Na +: 2K + . Chapter 3: Transcapillary and Cellular Exchange 43 The pump is characterized therefore by: a) a pumping rate (or the 'velocity' of the pump) symbolized by RP, which is a proportionality constant between the active transport flux of sodium and the intracellular N a + concentration, and b) a fixed pumping ratio (the 'transporting capacity' of the pump) symbolized by p which represents the number of sodium ions extruded from the cell per number of potassium ions introduced into the cell at each cycle performed by the pump, i.e., p= 3/2 [Post, 1957]. For the sodium ion, the flux due to active transport, Ta,Na (which alternatively can be called simply the pump flux) is given by T a , N a = RP[Na] I C v [3-13] Hence, the active transport flux of potassium ions, T ^ K , can be obtained from T a , K = RP[Na]icv/p [3-14] One of the requirements for a cell to achieve a steady-state ionic distribution, therefore, is that the total net flux, TION , of both N a + and K + across the cell membrane must be zero. Taking the cellular environment as the reference these conditions are respected i f T N a = T d N a - T a N a = 0 [3-15] and T K = T A , K - T d K = 0 [3-16] where the flux of a given ion due to electrodiffusion, Ta, and the flux due to active transport, T a , are expressed in mmol/(s-cm2). 3.6.3 The resting membrane potential As previously mentioned, the unequal distribution between the extra- and intracellular media of all three ions (i.e., Na + , K + and CI") assumed to cross the cellular membrane, determines the value of the cellular membrane potential which, at steady state, is called the resting membrane potential, V m . The value of V m reflects a balance between the competing influences of the equilibrium Nernst membrane potentials for Na + , K + , and CI". The actual membrane potential, V m , assumes an intermediate value between these extreme equilibrium potentials having the electrical driving forces for the transport of the potassium, sodium and chloride ions given by ( V m - E K ) , ( V m - E N a ) and ( V m - E a ) , respectively. Chapter 3: Transcapillary and Cellular Exchange 44 The following three factors determine the value of the resting potential, V m [Matthews, 1998]: 1) the ionic concentrations inside and outside the cells, which determine the equilibrium potential for each small ion, 2) the cellular membrane permeability for a given small ion, which will establish the relative importance of that ion in determining the value of V m , and 3) the presence (or absence) of an electrogenic pump that actively transports certain ions into or out of the cell. Since the potassium permeability through the cellular membrane is about two orders of magnitude higher than that of sodium, it is expected that the resting membrane potential will be close to the Nernst equilibrium potential for K + . This is the reality for skeletal muscle cells (and generally most animal cells) that bear characteristics similar to the cell schematically depicted in Fig. 3-4, where it is shown that V m i s in the range of -85 to -90 mV [Matthews, 1998]. The first quantitative relationship between the transmembrane potential, ion concentrations and ion permeabilities was developed by Goldman [1943] for the case where electrodiffusion is the only mechanism of transmembrane transport. His constant field equation is known simply as the Goldman equation. When only three species (i.e., Na + , K + and Cl") cross the cellular membrane, the equation can be written as v - R T i n ( P N a • [ N a ] g c v + p K ' [ K ] e c v + P a ' [ C 1 k v ) [3-17] F PNa • [ N a J i c v + PK • [KJicv + Pa " [CIJECV where V m is the cellular transmembrane potential expressed in mV. A l l the other symbols are as previously described. The derivation of this equation is presented in Appendix A. Conversely, i f other species in addition to the three ions previously considered are also transported across the cell membrane, Eq. [3-17] can be extended to incorporate the contributions of these ions as follows: RT, P N 3 • [Na]E C V + p K • [K] E C V + P c i • [C1]ICV + V p C a t • [Cat]E C V + Y PM, • [An] I C V V m = — ln( ^ ) [3-17a] * PNa • [Na] I C V + p K • [K] I C V + p c l • [CIJECV + 2_ Peat • [CatJicv 2 ^ • [An] E C V where the symbols Cat and An represent all other diffusible cations and anions, respectively. Equation [3-17a] is seldom used to establish the membrane potential of a cell due to the unknown permeability values of the many other possible ions that could participate in cellular transport. Chapter 3: Transcapillary and Cellular Exchange 45 The Goldman equation, Eq. [3-17], was extended, first by Moreton [1969] and then by Jakobsson [1980], to incorporate the contribution of the Na +-K +-pump. This extended equation, which is implicit in V m , has the following form: RT In P N a • [Na] E C V + p K • [ K ] E C V + p c l • [C1] I C V + RP • [Na] ICV (1 -1 /p) ( F V m / R T ) P N a • [Na] I C V + p K • [K] I C V 1 + p c l • [C1]E C V + RP • [Na]I( (1-1 /p) [3-18] I C V (F • V m / RT) where all the terms were previously defined. A detailed derivation of this equation in its final form, starting from the Goldman equation, is also provided in Appendix A. Equations [3-9], [3-10], [3-12] and [3-18] presented above, can be used together to quantitatively determine the exchange rates of various small ions across the cell membrane in the presence of a cell membrane potential. However, in order to define the steady state for the cell schematically depicted in Fig. 3-4, there are two basic constraints that must be imposed on the bulk intra- and extracellular environments: 1) the electroneutrality condition and 2) the isotonicity condition. 3.6.4 Electroneutrality condition The electroneutrality condition imposes an additional constraint on the movement and distribution of ions across the cell membrane. It states that, despite a continuous movement of ions across the cellular membrane, the intracellular and the extracellular environments must be electrically neutral, i.e., £ ( z I 0 N • [ION]) l c v = £ ( z I 0 N • [ION]) E C V = 0 [3-19] ION ION where [ION] and ZION are, respectively, the molar concentration and charge of the ionic species ION (including the non-diffusing F A FC, A and C species). Equation [3-19] should, more accurately, be based on the activities of the ions in the solution and not their concentrations such that the chemical interactions between the species are accounted for. The choice of using the concentrations instead of the activities of the ions is a mathematical simplification employed by several authors [MacKnight, 1994; Jakobsson, 1980; Wolf, 1980] to avoid the uncertainties related to the type of interactions found in the intracellular environment. Moreover, for relatively large cells such as red blood cells there is experimental Chapter 3: Transcapillary and Cellular Exchange 46 evidence that the activities of Na + , K + and Cl" ions are essentially the same as their molar concentrations (i.e., the intracellular medium of the R B C behaves as an infinitely dilute solution) [Hoffman, 1987;Edzes and Berendsen, 1975]. Based on the above discussion, it can be fairly reasonably assumed that, under normal physiological conditions, the bulk intracellular concentration of cations is equal the bulk intracellular concentration of anions. 3.6.5 Osmotic pressure, osmolarity and isotonicity condition Generally, the osmotic and hydrostatic pressures are important factors in determining the distribution of water among the body compartments. In the majority of cells within the body (with a few exceptions such as some epithelial cells of the renal tubules), the cellular membrane cannot withstand a difference in hydrostatic pressure and, therefore, the movement of water across the cell membrane occurs only by osmosis. By definition, osmosis is the passive movement of water across a semi-permeable membrane from a solution with a higher water concentration (i.e., a lower concentration of solutes) to a solution with a lower water concentration (i.e., a higher concentration of solutes). The theoretical osmotic pressure of a solution assumes that the membrane is impermeable to the dissolved solutes and, hence, is proportional to the number of osmoles (Osm) found in that solution. Very often, in the dilute solutions normally found in the human body, the osmolality of the solution (i.e., the number of solute particles per unit weight of solution) and the osmolarity (i.e., the number of solute particles per unit volume of solution) are assumed to be equal. The effective osmotic pressure of a solution is dependent on both the osmolarity of the solution and the permeability of the membrane for the solutes. In general, the higher the permeability of a membrane to a given solute, the lower will be the effective osmotic pressure of the solution containing that solute. The experimental methods available for the measurement of osmolarity (e.g., solution freezing point) determine the theoretical, not the effective, osmotic pressure. In order to differentiate between the effective and theoretical osmolarity in cell physiology, it is customary to use the term tonicity. By definition, a solution is isotonic with a cell i f it has no effect on the cell volume when the cell is placed in that solution. If the solution causes the cell to shrink (i.e., if water is lost from the cells), it is considered to be hypertonic; if Chapter 3: Transcapillary and Cellular Exchange 47 the solution causes the cells to swell (i.e., i f incoming water increases the volume of the cell), it is said to be hypotonic. As previously discussed, the cellular membrane of most cells is highly permeable to water, but has a much reduced permeability to all other non-lipid species such as ions. This differential permeability of the cellular membrane is the cause of the rapid movement of water between the intra- and extracellular solutions, when the two media have different osmolarities. It is reasonably assumed that, under the conditions which occur within the body (dilute solutions, constant temperature, etc.), the rate of osmosis is directly proportional to the concentration (or osmolarity) difference between the non-diffusible species on the intra- and extracellular sides of the cellular membrane. The isotonicity condition imposes a constraint for osmotic equilibrium between the cells and their environment and, implicitly, for the distribution of water across the cell membrane. This condition states that water will be at equilibrium if the total concentration of impermeant intracellular solute is the same as the total concentration of impermeant extracellular solute, i.e., ICV tonicity equals E C V tonicity, or 2 ( a I O N • [ION]) I C V = £ ( a I 0 N • [ION]) E C V [3-20] ION ION where OION is the reflection coefficient of species ION. The left hand side of Eq. [3-20] represents the intracellular osmolarity, while the right hand side is the extracellular osmolarity. If the volumes of the ICV and E C V are Vicv and V E C V , respectively, we can determined the amount of osmotically active species in the two media as O s m I C V = V ] C V - £ [ I O N ] I C V [3-21] ION and O s m E C V = V E C V - £ [ I O N ] E C V [3-22] ION where Osmicv and OsmEcv are the number of milliosmoles in the intra- and extracellular compartments, respectively. The volume of water shifted across the cellular membrane can be obtained from the isotonicity condition, i.e., Eq. [3-20], by assuming that any disturbance in the extracellular or Chapter 3: Transcapillary and Cellular Exchange 48 intracellular osmolarity will cause an instantaneous shift of water across the cellular membrane. This assumption is based on experimental evidence showing that erythrocytes and skeletal muscle cells behave as perfect osmometers for a wide range of hypertonic and hypotonic external media (for reviews, see House, 1974; Weiss, 1998). The algebraic equation describing the volume of water, Vw, shifted into or out of the tissue cells, at any instant, has the following form: v _ O s m E C V . V I C V - O s m I c v . V E C V [ 3 _ 2 3 ^ (Osm I C V +Osm E C V ) According to Eq. [3-23], the volume of water shifted across the cellular membrane, Vw, is positive when the extracellular osmolarity increases thereby causing the cells to shrink. Alternatively, when the extracellular osmolarity decreases, the term Vw is negative causing the cells to swell. The equations presented above summarize the simplest, most accepted mathematical descriptions for the transport of ions and water across the cell membrane. The equations are representative of typical mammalian, spherical, non-excitable cells. It is expected that the relative magnitude of the different types of transport vary amongst the multitude of cells found in the human body. This chapter summarized the transport of water, small ions and/or proteins across the two important transport barriers found in the body, i.e., the capillary and cellular membranes. As part of this work, the mathematical descriptions of cellular and transcapillary exchanges presented in this chapter will be incorporated into a compartmental model that is intended to predict fluid and solute movements in the human and/or animal body. A description of this model is the objective of the next chapter. Chapter 4: Model Formulation 49 Chapter 4: M O D E L F O R M U L A T I O N 4.1 Introduction As a complement to experimental studies of resuscitation protocols, there is a need to develop, as well as validate, mathematical models capable of predicting the fluid, protein and small ion shifts that take place between the vascular, interstitial and cellular compartments following the administration of a given type of solution. A number of mathematical models have emerged over the past few years that address the issue of body fluid distribution and transport with the particular goal of predicting blood volume changes following hemorrhage. Significant modeling effort has been directed toward describing the mechanisms of blood restitution and, most importantly, the requirements of different resuscitation schemes [Carlson et al., 1996; Mazzoni et al., 1988]. However, prior to modeling mass transport in a highly perturbed state such as burn or hemorrhage, it would be prudent to create and validate an initial model based on the normal physiological behavior of the system, including the estimation of parameters which are difficult to measure or unavailable from the literature. The present chapter presents the formulation of a compartmental model that describes fluid, protein and small ion exchanges in the human body. This model is concerned with the two main sites of exchange in the body namely, the microvascular exchange system (MVES) and the cellular exchange. The necessary physiological background required for understanding the model was provided in Chapters 2 and 3. A mathematical description of microvascular exchange is a subject of ongoing interest to our group. In an attempt to accurately describe the human M V E S , compartmental as well as distributed models have been considered [e.g., Bert et al., 1988; Taylor et al., 1990]. Initially these models were formulated for different animals; but, as more information became available, they have gradually been modified to accommodate transport in humans. Historically, these models have been developed in a hierarchical fashion; from simple to complex. New fluid compartments and/or new species and their corresponding transport equations were introduced when the availability of new physiological information allowed an increase in complexity without comprising the accuracy of these models. To simulate traumatic conditions such as burns required a further increase in model complexity [e.g., Ampratwum et al., 1995]. The addition of Chapter 4: Model Formulation 50 cellular exchange as a subset to the M V E S compartmental models is the primary new feature introduced in this work. The present chapter is concerned primarily with the following two topics: • a general review of previous models involving transport of fluid and solutes within the M V E S and/or between the intra- and extracellular fluids. • a description of the mass balances, transport equations and constitutive relationships of the human model proposed in this work. 4.2 Literature Review 4.2.1 Compartmental MVES models Guyton was among the first physiologists to emphasize an analytical approach to the understanding of physiological control. His most complex system analysis- of body fluid dynamics and its control [e.g., Guyton et al., 1973], involved not only the circulatory dynamics, but also many other aspects including the dynamics of free and gel fluids in tissues, the intracellular fluid and the endocrine control of body fluid volume. Useful simulations with this model and the predicted results for fluid distribution and transport in the human body are available at http://phvs-main.eduAVORKSHOP/MODELS/ [Coleman, 2000]. Subsequent to Guyton's initial formulation, Wiederhielm [1979] developed a M V E S model that had the following important feature: in addition to plasma proteins, it considered that the interstitium also contains other osmotically active substances, which exhibit unusual volume exclusion characteristics. This model was employed to simulate the steady-state and transient responses to perturbations in arterial and venous pressures, plasma oncotic pressure and interstitial mucopolysacharide content as well as to disruptions of the lymphatic system. As a follow-up to the Wiederhielm model, Bert and Pinder [1982] introduced a constant value for interstitial volume exclusion based on the observation that the volume from which albumin is excluded remains unchanged, even during tissue swelling. The Bert and Pinder model was used for studying both the steady-state and transient responses to different forms of edema. Parts of this model are found incorporated in all other models subsequently developed by this group. Using the same basic approach, Bert et al. [1988] developed a dynamic mathematical model to describe the distribution and transport of fluid and plasma proteins between the circulation, the interstitial space of skin and muscle, and the lymphatics in the rat. Two theories Chapter 4: Model Formulation 51 of transcapillary transport were investigated and tested, i.e., a homoporous 'Starling Model' and a heteroporous 'Plasma Leak Model'. It was concluded, based on statistical fitting of the model predictions to the available experimental data, that the 'Plasma Leak Model' gave a slightly better description of microvascular exchange. However, due to the fact that the 'Plasma Leak Model' requires a larger number of transport parameters which are usually not available experimentally, it was abandoned in future studies in favor of the Starling-type models. Chappie et al. [1993] employed a 'Coupled Starling' or 'Patlak' model of transcapillary exchange to describe the distribution and transport of fluid and albumin in the circulation, interstitium and lymphatics of humans. Xie et al. [1995] also used the 'Coupled Starling Model' and a statistical fitting procedure involving clinical data to determine the transport parameters during normal human microvascular exchange. This model was successful in simulating the transient responses of the normal M V E S to cases where saline and albumin solutions were infused. In all previous models of microvascular transport formulated by our group [Bert et a l , 1988; Chappie et al., 1993], it was assumed and reasonably justified that, under specific conditions, the cellular compartments did not play a significant role in the exchange process, i.e., the small ions and fluid in the cells were considered at all times to be at equilibrium with those in the plasma and interstitium. Nevertheless, important clinical situations, such as hemorrhage with its altered cellular activity and use of hypertonic resuscitants, require the consideration of a cellular component which plays an active role in exchange involving fluids and ions during short-term fluid regulation. The electrophysiological basis for this requirement (e.g., cell membrane depolarization, cellular edema) is well established now by studies pioneered by Shires et al. [1972] and Nakayama et al. [1985]. 4 . 2 . 2 Cellular volume regulation (CVR) models There are only a few mathematical models available that allow a quantitative analysis of cellular volume regulation. This is partly due to the experimental difficulty in obtaining sufficient information to both formulate and validate any mathematical description of cellular behavior. Thus, the predictions provided by these models are only speculative, at best. Some modeling progress has been made in describing the steady-state behavior of a typical mammalian cell bathed in an infinite extracellular volume [e.g., Jakobsson, 1980]. However, there have been only a few successful attempts to formulate models that predict the transient behavior of a Chapter 4: Model Formulation 52 cellular compartment embedded in a finite extracellular volume [e.g., Carlson et al., 1996], as is required in the present study. The steady-state models described below refer to a single generalized cell (i.e., cellular compartment) characterized by a constant area of transport and defined cellular membrane permeabilities. The cell is bathed by an infinite extracellular medium with a constant composition. Tosteson and Hoffman [1960] pioneered the first steady-state model of a red blood cell that controls its cation composition and volume by the action of a Na +-K +-pump. Their model was intended to establish the processes responsible for counterbalancing the osmotic forces that produce a natural, constant tendency of cells to swell. Simple diffusion equations were written and solved, one at a time, for Na + , K + and Cl". The equilibrium Nernst potential for Cl" was considered to provide the constant transcellular membrane potential. The contribution of sodium or potassium to the cell membrane potential was not specifically taken into account in any of their equations. The Na +-K +-pump was considered to exchange one sodium ion from inside the cell for one potassium from outside, i.e., a non-electrogenic pump with p =1. The mathematical models that followed extended the Tosteson and Hoffman model to include more precise descriptions of membrane permeabilities, ion fluxes and the Na+-K+-pump [Stein, 1967; Fromkin, 1973]. Wolf [1980] developed a mathematical model for a red blood cell in which he incorporated net steady-state fluxes for N a + and K + derived by integrating the Nernst-Planck equation under constant field conditions [Plonsey, 1969], together with a mathematical expression for the Na+-K+-pump [Tosteson, 1964]. He also assumed a variable osmotic coefficient for hemoglobin. A computer program associated with this model allowed the prediction of the steady-state water and electrolyte distributions between the interior and exterior of the cell. Although it represented a major step forward in the understanding of cell volume regulation, his model had several weaknesses. For example, it assumed that extracellular electroneutrality and osmolarity are due only to contributions from the Na + , K + and Cl" ions; it is well established, however, that other ions present inside or outside the cells are important determinants of water movement across the cellular membranes. Although the values chosen for these ions respect external electroneutrality, the isotonicity condition between the exterior and interior of the cell was not satisfied. Also, only discrete values were ascribed to the membrane potential (as imposed by the chloride equilibrium condition). As was standard practice at the time, the model equations were not solved simultaneously. Chapter 4: Model Formulation 53 One of the most detailed models of cellular volume regulation is the one proposed by Jakobsson [1980]. His model consists of transport equations for each of the permeant species assumed to cross the cellular membrane, as well as equations for bulk internal electroneutrality within the cell and for the condition of isotonicity between the intracellular and extracellular fluid. This was the first model where the transport equations together with the electroneutrality and isotonicity conditions were solved simultaneously. By using this approach, Jakobsson was able to determine the steady-state cellular transport parameters. Modifications of his initial steady-state model allowed Jakobsson to further follow the transient behavior of the cellular volume under a variety of postulated conditions. His analysis stressed the interdependence of passive and active ionic fluxes, membrane potential, cellular composition and cellular volume, as well as the need to quantitate all of these variables in order to interpret experimentally induced changes in cellular volume. One important feature of Jakobsson's model was his extension of the steady-state equations to account for the dynamics of cell volume regulation. Jakobsson's model will be extensively used in this work; therefore, a complete development of both his steady-state and time-dependent equations are presented in Appendix B. The models enumerated so far rely on known and/or estimated values of the cellular membrane permeability in order to describe the transport of small ions through a cellular membrane. Due to inherent experimental difficulty in determining the cellular membrane permeabilities (see Table 3-1), there is some concern about the accuracy of the literature values. As an alternative to the previous modeling approach, several mathematical models have appeared over the past decade that describe the volume regulation of the epithelial cells of renal tubules by using the so called 'equivalent circuit' approach [Civan and Bookman, 1982; Duszyk and French, 1991; Gordon and Macknight, 1991; Strieter et al., 1992]. These models have the ability to incorporate a very detailed description of both the primary and secondary transports (i.e., transmembrane transport by simple electrodiffusion and facilitated co-transport of two different ions, respectively). The contributions of K +-C1" and Na + -Cl" co-transport as well as Cl" counter-transport to the steady-state cellular membrane potential were discussed extensively in a paper by Gordon and Macknight [1991]. These models are based on the assumption that the net electric current carried by an ion across a polarized barrier is given by the sum of a constant field electric term and a contribution from a specialized transfer function within that barrier. The cellular membrane therefore is essentially represented as an electric capacitor. The only appreciable currents are considered to be those due to the transmembrane passage of Na + , K + and Chapter 4: Model Formulation 54 CI" for which KirchofFs first law can be applied (i.e., the sum of currents carried across the membrane at any time is zero). Such models are mainly applied to describe transport across the membrane of epithelial and endothelial cells, and consequently are of little practical value to the present study. 4.2.3 Mathematical models of whole-body fluid and solute exchange In some early models of fluid volume regulation, Wolf [1982] and Mazzoni et al. [1988] described the movement of water, protein and crystalloids between plasma and interstitial fluid compartments and took into consideration only the fluid exchange with cells. The exchange of solutes between the intracellular and extracellular compartments was not accounted for. The more complex model of Carlson et al. [1996] added several perturbations hypothetically descriptive of hemorrhage to a precursor model. Although this latter model describes fluid, protein and small ion exchanges between the extra- and intracellular compartments, no true validation was attempted. Additionally, the magnitudes of the imposed perturbations were adjusted to fit their specific set of experimental data. Although novel in concept, because the Carlson et al. model lacks validation, the reliability of its predictions are unknown. The primary goal of the present study is to extend our previous modeling efforts [Bert et. al, 1988; Bowen et al., 1989; Chappie et al., 1993], as well as those of other authors [Jakobsson, 1980; Mazzoni et al., 1988; Wolf, 1982] to develop a compartmental model which describes fluid and solute exchanges between the plasma, interstitium and cells. Furthermore, to have confidence in our description of the underlying physiological mechanisms which might contribute to effective plasma replacement, we will first test this model against experimental data which take into account solely the effects of different types of infusions, i.e., in the absence of any type of trauma such as hemorrhage or burn injury. 4.3 Model Description The purpose of this section is to develop a conceptually up-to-date mathematical model which describes the fluid and solute (i.e., protein and small ions) exchanges between circulating blood and interstitium, taking into account the exchanges occurring with the cells associated with these spaces. Chapter 4: Model Formulation 55 4.3.1 General considerations In the present study, the fluid and solute exchanges in the body are described in terms of a compartmental model. Hence, all variables are considered to be spatially invariant, and the dynamic behavior of the system can be represented by a set of ordinary differential equations. The transcapillary fluid exchange is based on Starling's hypothesis presented in Chapter 3, i.e., hydrostatic and osmotic pressures govern fluid flow across the capillary membrane. The transcapillary transport of proteins is described by the Patlak equation for the transport of a neutral solute through a homoporous membrane. The transport of small ions is assumed to take place through uncoupled diffusion and convection and is described by the Kedem-Katchalsly equation. Based on their molecular weight, the solutes introduced to the body by infusion will be considered to be macromolecules or small ions. Hence, their transcapillary transport is also described by either the Patlak or Kedem-Katchalsky equations. The Donnan effect exerted across the capillary membrane is accounted for in both the fluid transport and small ion transport equations. The cellular exchange portion of the model assumes an instantaneous shift of water between the cells and their corresponding extracellular compartments based on rapid osmotic equilibration. The movement of ions across the cell membrane described by the Nernst-Planck equation, is influenced by changes in the concentration difference driving forces and in the cell membrane potential. Active transport of N a + and K + ions is also accounted for by the inclusion of an electrogenic Na +-K +-pump. 4.3.2 Model compartments The model consists of the set of compartments presented schematically in Fig. 4-1. Two interconnected homogenous compartments, namely the plasma compartment and the generalized interstitial compartment, constitute the extracellular fluid. The two compartments exchange fluid, proteins and small ions across a semi-permeable membrane, the capillary wall, as well as, via the lymphatics. As was discussed in Chapter 2, although there may be an anatomical barrier between the interstitium and the lymphatic vessels, due to an incomplete elucidation of the mechanisms responsible for lymphatic transport, the lymphatics are treated as membraneless, passive conduits for fluid and solute transport from the interstitium to plasma. Chapter 4: Model Formulation 56 MION,PER JPER JISL L U m • m J Q R M RES RES RES ION.RES INTERSTITIAL CELLS TC M ION.TC INTERSTITIAL FLUID J,T IT M ION.IT, M ION.L ^ PLASMA 1 ! 1 RBC *RBC ION.RBC 'UR M ION.UR Figure 4-1: Schematic of a general model of fluid, protein, small ion and additional solute (introduced through infusion) exchange between plasma, interstitium and cells. Arrows indicate fluid (J), protein (Q) , ion ( M ) and solute ( R ) transport rates in relation to the compartments participating in mass exchange. The subscripts are defined in the main text as well as in the Nomenclature section found near the end of the thesis. Chapter 4: Model Formulation 57 Each of the two main fluid compartments, plasma or interstitium, contains an embedded cellular fluid compartment. As a consequence, the interstitial compartment is composed of interstitial fluid having a volume Vrr, and tissue cells, whose fluid volume is VTC- The generic term 'tissue cells' describes cells with lumped properties, which represent the diversity of cells included in the generalized interstitium. Additionally, for a first formulation of this model, the volume of endothelial cells is also lumped into the generalized 'tissue cell' compartment. Since about 60 to 70% [Koushanpour, 1976] of the interstitial intracellular water is found in the skeletal muscle cells of the body, it is assumed in this work that the tissue cells have the same properties as skeletal muscle cells. The vascular compartment is made up of plasma having a volume VPL , and red blood cells (RBC) with a fluid volume VRBC- The R B C volume is related to the plasma volume via the systemic hematocrit, Hct, given by H c t - ^ £ B £ 100 (%) [4-1] (VRBC + ^ P L ) 4.4 Model Assumptions All models that purport to describe biological systems involve varying degrees of simplification. Compartmental models, especially, are often criticized for excessively idealizing the system under study. Usually they are formulated such that a complex, real system is reduced to a set of sub-systems or compartments where the spatial heterogeneity of material properties is largely ignored while the physical boundaries between different compartments are described in terms of lumped parameters. Nevertheless, owing to their relative mathematical simplicity and minimal parameter requirements, such models are the only reasonable choice for simulating whole body fluid and solute exchanges during normal and pathophysiological states (e.g., hemorrhage or burns, followed or not by fluid resuscitation). In order to minimize such criticisms, it is important to clearly specify the assumptions upon which the model is formulated and, furthermore, to base such assumptions on reliable physiological information. The set of assumptions that form the basis of this model, as well as their justifications, are listed below. • A l l compartments, including the cellular sub-compartments, are considered to be well-mixed with spatially constant descriptive parameters, i.e., mean values for descriptive properties (e.g., transport parameters) and dependent variables (e.g., concentrations of specific solutes) are used at any given time. Chapter 4: Model Formulation 58 This assumption is consistent with the type of data normally measured experimentally for both intra- and extracellular fluid compartments, [e.g., Vick, 1970; Houser, 1980]. Additionally, the use of spatially-invariant compartments has proven to be successful in several previous modeling studies [e.g., Wiederhielm, 1979; Wolf, 1980; Bert et al., 1988; Carlson et a l , 1996]. • A l l proteins in the system are assumed to have the same properties as albumin with an average molecular weight, MWp = 67000 g/mol These species generate oncotic pressures and are exchanged only between plasma and interstitium (i.e., there is no protein exchange across the cell membrane). Additionally, they are responsible for generating a Donnan effect across the capillary membrane. In all of the M V E S models previously developed by this group [e.g., Chappie et a l , 1993], albumin was used as a surrogate protein to represent the collective influence of all plasma proteins. This assumption was justified by the fact that albumin accounts for more than 50% of the total plasma protein mass in humans, and generates approximately 65% of the total plasma colloid osmotic pressure. In the present work, however, similar to what was done in other more recent models [e.g. Carlson et a l , 1996], the model accounts for the concentration and amount of total proteins present in the extracellular fluid. • The same conductive pathways in the capillary membrane serve as sites for both fluid and protein transport between plasma and interstitium. This assumption implies the use of the Patlak [Patlak et a l , 1963] equation for coupled transcapillary transport of fluid and macromolecular species which is valid over a wide range of hydrostatic and colloid osmotic pressure differences across the membrane. The formulation applies to homogenous membranes in which convective and diffusive transport occur within identical channels that exhibit a constant reflection coefficient for the protein. Chappie et a l , [1993] found that Patlak's representation of transcapillary fluid and protein transport was the most suitable for compartmental models. • Five types of mobile ions are accounted for the in the two extracellular compartments: Na + , K + , C 2 + (a species representative of all other positive ions), C I and A" (representative of all other anions). Sodium, potassium and chloride represent the bulk of ions in the human body. These ions are the primary contributors to the total osmolarity of the body fluids; of the total 300 mOsm/1 typically reported for the fluids of the body, approximately 280 mOsm/1 are provided by the combined Chapter 4: Model Formulation 59 contribution of these three ions [e.g., Kleinman and Lorentz, 1984]. Additionally, they are the main species transported across the capillary and cellular membranes. The cationic species C 2 + and the anionic species A" were introduced in order that the electrical neutrality requirements of both the plasma and interstitium were respected. Since most of the additional positive charge in these two fluid compartments is due to M g 2 + and C a 2 + [Kleinman and Lorenz, 1984], a charge of +2 was attributed to the C 2 + species. The negative species A" represents other anions such as acetates, carbonates, lactates, etc. Hence, a negative charge of -1 was assumed for this species. A l l of these ions participate in transcapillary exchange. Additionally, they are distributed on either side of the capillary membrane according to a Donnan equilibrium that will be discussed later in this chapter. • The transport of small ions across the capillary membrane takes place through uncoupled convection and diffusion. This assumption implies the use of the equation for diffusive-convective transport of fluid and small solutes proposed by Kedem and Katchalsky [1958]. Although the capillary channels are not well defined structurally, an increasing number of researchers [e.g., Wolf, 1996; Meinild et al., 1998] report the presence of additional (either diffusive or convective) transcapillary pathways for solutes and/or water other than the typical inter-endothelial junctions. Such an assumption, therefore, is not unreasonable and seems to be increasingly employed in modeling studies [e.g., Carlson et al., 1996]. • Five types of ions are also accounted for in the two intracellular compartments: Na + , K + , F C 2 + , Cl" and FA". The only small ions transported across the cellular membrane between the intra- and extracellular compartments are sodium, potassium and chloride. A l l other intracellular cations besides N a + and K + are denoted as F C 2 + , and all anions other than Cl" are denoted as FA". These species are considered to be present in fixed amounts, i.e., they do not cross the cell membrane. Thus, any changes in the concentrations of these species are due solely to cellular swelling or shrinking. Such an assumption is in agreement with other modeling studies [e.g., Wolf, 1980; Jakobsson, 1980]. • The transport of N a + and K + across the cell membrane occurs by both electrodiffusion and active transport (i.e., by means of a Na+-K+-pump). Chapter 4: Model Formulation 60 This type of cellular transport, was described in some detail in the previous chapter, and reflects our current understanding of cellular transport. As in a previously reported model [Jakobsson, 1980], the Na +-K +-pump is characterized by a constant ratio of N a + to K + transport, p = ([Na+]out/[K+]in) = 3/2, and a constant rate, RP. The flux of N a + or K + due to active transport is dependent on the pump rate and the intracellular sodium and extracellular potassium concentrations. • Chloride is not transported across the cellular membrane by electrodiffusion in response to its electrochemical gradient and no active transport is ascribed to this ion. Since CI" is the only negative ion considered to cross the cell membrane, the transport of this ion is correlated to N a + and K + exchanges through electroneutrality considerations, i.e., the rate of CI" transfer is equal to the net rate of N a + and K + exchange. • Changes in the cell transmembrane potential, V m , take place instantaneously as a direct consequence of the redistribution of ions across the cell membrane. Several authors who modeled cellular transport [e.g., Wolf, 1980; Jakobsson, 1980], have used a similar approach. This assumption is based on the fact that, compared with the amount of ions normally present inside and outside the cells, only a minute separation of charges is required to produce a change in the membrane potential [Matthews, 1998]. Additionally, changes in membrane voltage due to the redistribution of charges on either side of the cell membrane occur within milliseconds, while changes in intracellular concentrations take place in the order of seconds to minutes. Thus, at every instant, the transmembrane potential is in a quasi-steady state dictated by the ionic concentration differences across the cell membrane, the membrane permeabilities for small ions, and the magnitude of the active transport term. • Changes in cell volume are directly related to the change in cellular water content and are assumed to occur instantaneously (i.e., the cellular membrane is assumed to be freely permeable to water). Such an assumption has as its basis the high water content of most mammalian cells, i.e., on the order of 80-90% [MacKnight and Leaf, 1977; MacKnight, 1994]. As in other models [Jakobsson, 1980], the water shifts across the cell membrane are imposed by the isotonicity condition between the extra- and intracellular environments. Since, in most cells, the cellular membrane permeability for water is about three to five orders of magnitude higher than for small ions [Jakobsson, 1980], these shifts are assumed to take place instantaneously [MacKnight and Leaf, 1977; MacKnight, 1994]. One consequence of this justifiable assumption is that the model does Chapter 4: Model Formulation 61 not predict the dynamics associated with the movement of cellular fluid based on differences in tonicity. • The model accounts for a basic requirement neglected in previous models [e.g., Tosteson and Hoffman, 1960; Wolf, 1980], namely that the electroneutrality of the bulk intracellular solution must be respected. • The cellular characteristics are taken to be different for red blood cells and tissue cells according to experimental information [Hoffman, 1987; Kleinman and Lorenz, 1984]. These include the intracellular concentrations of small ions as well as such cellular transport parameters as the ion permeabilities. In addition to the above statement, it was also considered that the tissue cells have the same characteristics as skeletal muscle cells. This assumption is based on the fact that 65% of the total tissue mass available for capillary exchange is attributed to skeletal muscle [Landis and Pappenheimer, 1963]. Additionally, the muscle cells represent about 30% of the total normal body weight or alternatively about 70% of the ICV [Guyton, 1991; Koushanpour, 1976]. It is expected, therefore, that the skeletal muscle cells are an important source of water mobilization when the interstitial fluid is osmotically disturbed. • The properties of the cellular and capillary membranes, e.g., the transport parameters, are unaffected by the type of perturbation (e.g., different types of non-traumatic infusions) simulated in this study. Unfortunately, there is not enough experimental information to either confirm or contradict the above statement. However, this assumption will be reconsidered appropriately, if validation of the model proves to be unsatisfactory. • A l l species introduced by infusion that are not already constituents of the system will have transport characteristics similar to either proteins or small ions depending on their molecular weight. According to the above assumption, the following arbitrary classification of solutes has been made: - solutes with molecular weights greater than or equal to 1000 g/mol (e.g., dextrans) are transported in the M V E S via the same mechanisms as the ones described for proteins, solutes with molecular weights less than 1000 g/mol (e.g., glucose) are transported in a similar fashion as the small ions. Chapter 4: Model Formulation 62 In both cases, however, the specific transport parameters of the solute (e.g., reflection coefficient, permeability-surface area product, etc.) will be obtained from the literature. • For simplicity and due to lack of sufficient data, the model does not account for the complex endocrinal and neural effects that might take place during the type of perturbation simulated in this work (i.e., fluid overload and/or external loss of blood through hemorrhage). 4.5 Model Equations To predict the interdependent fluid, protein and small ion distribution and transport in the vascular, interstitial and intracellular compartments, the model requires, as a next step, mass balance equations for each species and compartment considered, as well as descriptions of the transcellular and transcapillary membrane exchanges. The discussion that follows is given in reference to the schematic diagram of the four fluid compartments illustrated in Fig. 4-1. 4.5.1 Mass balance equations for extracellular compartments The dynamic behavior of the two extracellular compartments, plasma (PL) and interstitium (IT), is based on the following mass balance equations. A l l the symbols used are defined at the end of this section and also in the Nomenclature section near the end of the thesis. Fluid balances: d V P L = J RES J IT ^ L J UR [4-2] ^ = J I T — J L — J P E R — J I S L [4-3] Protein balances: "QPL dt dt Infused macromolecular species (e.g., dextran) balances: dR = QRES-QIT + Q L [4-4] = Q I T - Q L [4-5] P^L dt dR IT dt = R R E S - R I T + R L - R U R [4-6] R I T - R L [4-7] Chapter 4: Model Formulation 63 A total of ten additional mass balance equations, five for each extracellular compartment, describe the transport of the small ions, Na + , K + , C 2 + , CI" and A". Since the balances of all ions in a given compartment, either PL or IT, have a similar form, in order to avoid repetition, the generic subscript ION is used. Small ion balances: d M dt d M ION.PL _ fyfioN.RES - M l O N . I T + M l O N . L - M l O N . R B C — M l O N . U R [4-8] I Q N ' I T = M ION.IT - M ION.L - M ION.TC - M ION.PER [4-9] dt In Eqs. [4-2] - [4-9], V, Q, R and M represent the compartmental fluid volume, protein, infused macromolecular species (e.g., dextran) and ion contents, respectively; while J, Q, R and M represent the rates of transport of fluid, protein, infused species and small ions, respectively, into or out of the compartment. The subscripts PL, IT, RBC and TC denote plasma, interstitial, red blood cell and tissue cell compartments, respectively; L indicates lymph; while ION is a generic term used to describe any of the ionic species (i.e., ION = Na + , K + , C 2 + , CI" or A"). The subscript RES stands for resuscitation when a time-dependent resuscitation rate constitutes an input to the model. PER indicate the loss of fluid through perspiration, while ISL denotes the insensible losses (e.g., losses of solute-free water from lungs or through skin), for specific cases where these losses are considered, i.e., when time-dependent rates of these latter fluids constitute known or predictable outputs from the interstitial compartment (see also Fig. 4-1). UR represents the urine production rate. 4.5.2 Mass balance equations for intracellular compartments The behavior of the two intracellular compartments is described by a set of six time-dependent ordinary differential equations corresponding to each of the three ions participating in cellular transport (i.e., Na + , K + and CI") in both compartments. Two additional algebraic equations account for water shifts to and from the cells, in response to external changes in osmolarity. This approach of describing cellular exchange follows that previously formulated by Jakobsson [1980]. Given below are the mass balances describing the tissue cell compartment, TC. Chapter 4: Model Formulation 64 Intracellular sodium balance: dM Na.TC dt V Na.TC-A(|)T c.([Na] I T -[Na]T C.exp(A(J)T C)) (exp(A<t>TC)-l) RP T c . [Na ] T C [ 4 - 1 0 ] Intracellular potassium balance: dM K.TC dt - A — - ^ T C -p K T C .A(j) T C . ([K] [ T -[K] T c .exp(A(t) T C )) . [Na] T C (exp(A<|)TC)-l) • + R P T C •" [ 4 - 1 1 ] ' T C Intracellular chloride balance: d M c i . T c d M N a J C d M K T C -ir=-^r+-ir [4-12] In Eqs. [ 4 - 1 0 ] - [ 4 - 1 2 ] , M N a T C , M K T C and M C 1 T C represent the sodium, potassium and chloride ion contents of the tissue cell compartment; A T C is the membrane surface area of this compartment; pNa and PK denote the permeabilities for N a + and K + , respectively; RP is the rate of the Na +-K +-pump; while p is the pump ratio. [Na], [K] and [Cl] are, respectively, the sodium, potassium and chloride concentrations for either the intracellular medium (i.e., tissue cells), subscript TC, or the extracellular medium (i.e., interstitium), subscript IT. A<j> is the dimensionless cell membrane potential, defined as Act) = F V m / R T , where the ratio F/RT is 2 6 . 7 mV for mammalian cells at body temperature, while V m is the dimensional cell membrane potential. The first term on the right hand side of Eqs. [ 4 - 1 0 ] and [ 4 - 1 1 ] represents transport due to electrodiffusion and establishes the interdependence between the ion permeabilities, the intra-and extracellular concentrations, and the transmembrane potential. The second right-hand-side term of these two equations describes active transport and states that the transport rate associated with the Na +-K +-pump is a linear function of the intracellular sodium concentration. The area term in Eqs. [ 4 - 1 0 ] and [ 4 - 1 1 ] represents the membrane surface area of the entire cellular compartment which is available for transport. This area is assumed to be constant, i.e., an increase or decrease in cellular volume would stretch or compress the cellular membrane without affecting the area (i.e., the number of pores or exchange sites) available for exchange (R.K. Reed, personal communication). Chapter 4: Model Formulation 65 Equation [4-12] states that the chloride ion crosses the membrane at a rate sufficient to maintain intracellular electroneutrality (assuming that the compartment is initially electroneutral). Therefore, the CI" mass balance equation implicitly includes the electroneutrality condition assumed for the internal environment of either type of cell. According to the assumptions mentioned in the previous section, the other positive, F C 2 + , and negative, FA", species are not transported across the cellular membrane. Consequently, the transport rates across the cell membrane for these two fixed species are zero, their intracellular contents are constant and their concentrations are determined solely by the changes in cellular volume. The algebraic equation describing the volume of water shifted at any instant into or out of the tissue cells, Vw, TC, has the following form: O s m r r . V T C ' W.TC Osm^.Vpj. [4-13] (Osm I T +Osm T C ) where Osmrr and Osnvrc are the total osmolarities of the interstitial and tissue cell compartments, respectively. This equation is a direct consequence of the isotonicity condition, i.e., the external osmolarity equals the intracellular osmolarity, and shows that any disturbance in the extracellular osmolarity will cause an instantaneous water shift across the cellular membrane. With each water shift, the interstitial volume, V IT, is updated instantly to (V IT + Vw, TC), independent of the terms on the right hand side of Eq. [4-3]. The tissue cell transmembrane potential, A(j)xc, is described by a non-linear algebraic equation employed initially by Jakobsson [1980]. As presented in Chapter 3, this equation is a modified form of the Moreton equation for the transmembrane potential [Moreton, 1969] and can be written as: A())TC - In FNa.TC [Na] I T + p K T C [ K ] I T + p c l T C [C1]T C + R P T C [Na] TC 1 1 'TC A<|). TC [Na] T C + p K T C [ K ] T C + p c l T C [ C l ] I T +RP T C [Na] PTC TC- A(t)1 [4-14] A l l of the symbols used in Eq. [4-14] were defined earlier. Chapter 4: Model Formulation 66 A similar set of five equations describes the behavior of the R B C compartment. In summary, a total of 22 ordinary differential equations for fluid, protein, infused species, and five type of small ions, Eqs. [4-2] to [4-12], together with two algebraic water-shift equations of the type of Eq. [4-13], and two non-linear membrane potential equations, similar to Eq. [4-14], describe the interdependent fluid, protein, and small ion distribution and transport in the vascular, interstitial and intracellular compartments. These equations are solved simultaneously in order to obtain a complete description of the time-dependent behavior this system. The numerical and computational procedure for solving the above system of equations is given in the next chapter. 4.5.3 Transport equations The mass balance equations presented above require a number of transport equations that characterize fluid, protein and small ion exchanges across the capillary and cell membranes as well as through the lymphatics. A generalized discussion of these transport equations was presented in Chapter 3. However, the formulation of the model, which includes contributions of small ions to extracellular osmotic pressures as well as Donnan considerations, requires revision of some of these equations. 4.5.3.1 Transport across the capillary membrane Fluid, protein and small ions are transported across the capillary wall by filtration, diffusion and convection. The magnitude and direction of the transcapillary fluid movement is determined by differences in hydrostatic pressure, colloid osmotic pressure exerted by proteins and infused macromolecules, and osmotic pressure provided by the small ions. Thus, the rate of fluid filtration, JIT (ml/h), from the plasma to the interstitial compartment is given by the extended form of the Starling equation: Jrr = k F [ P c — ^IT — ^ C ^ P . P L ~ ^ P IT) — ^ C J i o n ( 7 I I O N PL — ftI0NIT — TCIOND) — a R (ftR P L ~TCR.IT) ION [4-15] where ICF (ml/mmHg^-h"1) is the fluid filtration coefficient representing the hydraulic conductivity of the capillary membrane; Pc and PIT (mmHg) are the hydrostatic pressures in the capillary and interstitium respectively; TCPPL, ~ R P L and TCPIX, TCrit (mmHg) are the colloid Chapter 4: Model Formulation 67 osmotic pressures exerted by the proteins and infused dextran in the plasma and interstitial compartments, respectively; TC 1 0 N P L and TC I O N n (mmHg) represent the osmotic pressures exerted by the small ions in plasma and interstitium, respectively; while T C i o n D (mmHg) indicates the osmotic contribution of the small ions restricted in their movement due to Donnan constraints. A more detailed description of the small ion contribution to the Donnan effect and the way in which this effect was accounted for in the model is given in Section 4.5.4.4. a, OR, and CTION are the average reflection coefficients for proteins (i.e., albumin), dextran and small ions, respectively. The rate of albumin transport across the capillary membrane, Qn (g/h), is governed by the Patlak equation [Patlak et al., 1963; Bresler and Groom, 1981], which non-linearly couples the protein transport from the circulation to the interstitium with the fluid exchange: c _ c exp(- J l T- ( 1~ g )) Q I T = J I T.(l-a).[ — p ] [4-16] l _ e x p ( - i l l ^ ) ) PS In Eq. [4-16], CP,PL and CP,IT,AV (g/ml) are the protein concentrations in plasma and available interstitial volume. As discussed in more detail in Section 4.5.4.3, CPJT.AV is the effective interstitial protein concentration calculated as the interstitial protein content divided by the interstitial volume available to macromolecular species (see also Section 2.5.2.1). PS (ml/h) is the protein permeability-surface area product of the capillary. The argument of the exponential operator represents a modified Peclet number given as the ratio of convective to diffusive protein transport, i.e., Pe = 1^ (1 - a) / PS. A similar expression is used to describe the rate of transport of the macromolecular species, R IT (g/h), e.g., dextran, introduced through infusion: c - c cxp( J I T ( 1 ~ C T R ) ) ^R,PL ^R.IT.AV _ „ J Rrr = J i t . ( 1 - O r ) . [ R ] [4-17] l - e x p ( - J " ( 1 " q R ) ) P S R where the subscript R stands for dextran while all the other symbols were defined for Eq. [4-16]. Chapter 4: Model Formulation 68 It was assumed in the present study that the transport rate of small ions across the capillary membrane, MION.IT (mmol/h), occurs through separate convective and diffusive pathways [Kedem and Katchalsky, 1958] and therefore is described as: MION.IT - J I T - ( 1 - C T I 0 N ) . ( [ I Q N ] p l + [ I Q N ] l T ) + PS I O N . ( [ ION] P L — [ION] I T — A I O N D ) [4-18] where [ION] P L and [ION]IT (mmol/ml), are the ion concentrations in the plasma and interstitium, respectively; and PSION (ml/h) is the capillary permeability-surface area product for small ions. For a given ion, A I O N D (mmol/ml), represents the concentration difference across the capillary membrane caused by the Donnan effect which is described in Section 4.5.4.4. Equation [4-18] is used to represent the transcapillary transport of all the small ions present in the system except for A". It was assumed that the transport of species A" occurs at a rate that is just sufficient to maintain an overall electroneutral transport across the capillary wall. Therefore, M a ^ = ( Z Na - M N a . I T + Z K • M K . I T + ZQ • M c . I T + Z Q • M c l . I T Z A where z is the charge of the ionic species. According to the measurements reported by Crone and Christensen [1981], the capillary endothelium has a very low electrical resistance. If no active transport is considered across the endothelial cells, the magnitude of the Donnan potential difference can be calculated as being 1-2 mV, which is not expected to influence the ionic transport significantly. Hence, for simplicity, the transcapillary membrane potential was not accounted for, and the contribution of an electrical term in Eq. [4-18], describing the electrodiffusion of small ions, was ignored. 4.5.3.2 Transport through the lymphatics Tissue fluid, proteins, ions, and infused macromolecules that accumulate as a result of transcapillary exchanges are drained back into the circulation by the lymphatic system, at a fluid flow rate JL (ml/h), protein rate Q L (g/h), ion rate M L (mmol/h), and dextran rate R.L(g/h) respectively. It is assumed that no accumulation of material occurs in the lymphatics; consequently, the transport of fluid and solutes toward the vascular compartment by this mechanism takes place instantaneously. At steady state, the filtration flow rate equals the Chapter 4: M o d e l Formulation 69 lymphatic flow rate while the net rate at which each individual solute is exchanged across the capillary membrane equals the rate at which it is returned by the lymphatic fluid. The equations for lymph flow used in the model are based on previous work done by this group on modeling the M V E S exchange in both animals [Bert et al., 1988] and humans [Chappie et al., 1993; Xie et al., 1995], Under normal conditions, the lymph flow rate, JL, is always positive. Thus, lymph always flows from the interstitium toward the terminal lymphatics. The equations describing lymphatic transport assume that the changes in lymph flow rate from its baseline value depend on the hydration status of the interstitium. Therefore, the lymph flow rate varies linearly with the interstitial hydrostatic pressure under both overhydrated and slightly dehydrated conditions but ceases when the interstitial pressure becomes equal to or falls below that corresponding to the excluded volume of the interstitium. For the case of an overhydrated interstitium, i.e., when Vn ^ V I T N L , JLIS described by the following relationship: JL = JL,NL + L S • [P I T - ] , P I T > P ^ [4-20] Equation [4-20] shows that JL increases from the normal steady-state value JL.NL, proportionally with the change in the interstitial hydrostatic pressure by a factor L S , where L S (ml/mmHg-h) is referred to as the lymph flow sensitivity. JL,NL, the lymph flow rate under normal steady-state conditions corresponds to the interstitial hydrostatic pressure, PIT.NL- PIT is the actual interstitial hydrostatic pressure. For the dehydrated case, where V I T > N L > V I T > V I T E X , JL is given by, J L = J L.NL P - P IT IT.EX P - P IT.NL 1 IT, EX P > P > P IT.NL - IT MT.EX [4-21] where PIT,EX is the hydrostatic pressure when the interstitial volume reaches the excluded volume VIT,EX (i.e., VIT.AV = 0). Once the interstitial volume reaches the dehydration limit, VIT,EX, the lymph flow ceases completely. Thus, for any V I T < V I T E X : J L =o, P < P MT - R ' IT.EX [4-22] Chapter 4: Model Formulation 70 The composition of the initial lymph is always the same as that of the corresponding interstitial fluid. Because of the non-sieving character of the lymphatic wall, the reflection coefficient a for protein (and ions) is zero and therefore there is no colloid osmotic pressure difference exerted between the interstitium and lymph. Thus, the lymphatic transport of proteins from the interstitium into the main circulation, Q L (g/h), occurs only by convection, i.e.: Q L - J L . c P I T [4-23] A similar approach is taken when describing the lymphatic transport of an infused species such as dextran as well as the small ions, i.e., R L and MION.L , respectively. Thus, R L - J L . C R I T [4-24] MIONX = J L . [ ION] I T [4-25] where all of the symbols were defined previously. Equation [4-25] is employed to represent the lymphatic transport of all small ions present in the system except for A". As previously assumed for transcapillary transport (see Eq. [4-19]), it was also assumed that the transport of A" via the lymphatics occurs at a rate that is just sufficient to maintain an overall electroneutral lymph return. Therefore, ' , Z N • MNS.L + z K • MK.L + z c • MC,L'+ z c ] • MQ,L . R . MAL=-(— - - ) [4-26] Z A where all the symbols were previously defined. In order to complete the model description, the next section presents the additional algebraic equations needed to define certain compartmental variables (e.g., concentrations, colloid osmotic pressures, etc.), or the constitutive relationships that exist between these variables (e.g., compartmental compliance, Donnan partition, etc.). 4.5.4 Constitutive relationships 4.5.4.1 Capillary hydrostatic pressure and circulatory compliance As discussed in Chapter 2, all the vessels of the circulatory system, including the capillaries, are, to some extent, distensible. An increase in hydrostatic pressure in any segment of the circulation dilates the vessel and decreases its resistance to blood flow. Hence, the blood flow rate as well as the quantity blood that is stored in the vessel is increased. In order to avoid Chapter 4: Model Formulation 71 the inherent complexity associated with the process for controlling capillary pressure (i.e., the vasomotor tone, capillary recruitment, etc.), the model makes the assumption that any changes in plasma volume will affect the capillary pressure proportionally. This assumption implies the existence of a vascular compliance. The term vascular (circulatory) compliance means, literally, the ratio of the change in vascular volume to a change in the vascular filling pressure. Hence, the circulatory compliance, FCOMPC, can be expressed as: A V FCOMPC = — S L [4_27] AP C where A V P L and APc are the changes, relative to their normal values, of the plasma volume and capillary pressure, respectively. The reciprocal of the compliance, Pc, COMP, can then be used to determine the capillary pressure from: = Pc.NL + Pc.COMP (VpL — VpL .NL ) [4-28] where the subscript N L indicates the normal, steady-state value. The normal values of all variables are discussed in Section 4.6. There is insufficient literature data to allow a direct determination of PC.COMP for humans. Thus Chappie et al. [1993] used data from rats [see also Bert et al., 1988] to estimate an approximate human PC.COMP value of 0.0096 mmHg/1. When the plasma value falls bellow 2367 ml, the V P L value at which Pc = 3 mmHg, it is assumed that the capillary pressure remains at 3 mmHg, i.e., Pc cannot take a value lower than this arbitrarily imposed minimum. 4.5.4.2 Interstitial pressure and interstitial compliance The notion of interstitial compliance was briefly explained in Section 2.6.2.2 in relation to the physical characteristics of the interstitium. Similar to the vascular compliance, the interstitial compliance FCOMPI relates the changes in interstitial volume, A V u with the changes in interstitial pressure, APrr. However, unlike the vascular compliance, data from several animal studies [Wiig and Reed, 1981; 1985; 1987] indicate that there is no unique constant FCOMPI for the entire range of interstitial volume changes. This conclusion is corroborated by one human study [Stranden and Myhre, 1981]. Stranden and Myhre's data, however, are too scattered to establish a compliance curve for the human interstitium. Therefore, as a compromise, Chappie et Chapter 4: Model Formulation 72 al. [1993] used animal data to develop a 'most-likely' relationship for the interstitial compliance of humans as follows. Based on pressure-volume experimental data from rats, Bert et al. [1988] established separate compliance relationships for the skin and muscle of these laboratory animals. These compliance relationships determined for the rat were used in conjunction with the available human data. The experimental data showed that an increase in the interstitial volume by 210% from normal, yielded an increase in the interstitial hydrostatic pressure of 2.9 mmHg in humans [Stranden and Myhre, 1981] and a corresponding increase of 2.4 mmHg in rats [Reed and Wiig, 1981], The human compliance relationship was, therefore, developed by using the rat-to-human scaling ratio of (2.9/2.4). Hence, each pair of (Pu, RAT-VIT, RAT) data can be scaled to the appropriate human values as, 2.9 PlT.HUMAN — PlT,HUMAN,NL = (^IT.RAT — PlT.RAT.NL ) ' (~"T^ [4-29] 2.4 and V = V r V IT.HUMAN,NL>. IT,HUMAN IT.RAT V ) \P JyJl IT.RAT.NL where all the symbols have been previously defined. The compliance relationship is separated arbitrarily into three regions: a 'dehydration segment', an 'intermediate segment', and an 'overhydration segment' as shown in Fig. 4-2. The range of interstitial volume values and the compliance relationships corresponding to each of the above-mentioned segments are as follows: Dehydration segment: PIT = -0.7 + 1.96 x 10"3' [Vrr - 8.4 x 103], Vrr < 8.4 x 103 ml [4-31] Overhydration segment: P I T - 1.88+ 1.05 x lO" 4 ' [V IT-12 .6 x 103], V I T > 12.6 x 103 ml [4-32] Intermediate segment 8.4 x 103 < V i T < 12.6 x 10 3ml For interstitial volumes in the range of 8.4 x 103 < V I T ^ 12.6 x 103 ml, the interstitial pressure is obtained by interpolating the discrete PIT and V I T data generated from Eqs. [4-29] and [4-30] above (see Section 5.3). In all of the above relationships, PIT is expressed in mmHg and Vrr is expressed in ml. Chapter 4: Model Formulation 73 Interstitial Volume, V I T (I) Figure 4-2. The 'most-likely' human compliance curve (solid line) with its three distinct segments for dehydration, moderate hydration and overhydration. The symbol (filled circle) indicates the reference pair of PIT-VIT data that correspond to the normal values of Vrrand PIT (i.e., 8.4 1 and -0.7 mmHg, respectively). At a minimal dehydration state (i.e., the value at which VIT becomes equal to the excluded volume, VIT,EX) the lymph flow equals 0.0. The starting value for overhydration is Vrr= 12.6 1. Data correspond to normal humans as reported by Chappie [1993]. Chapter 4: Model Formulation 74 4.5.4.3 Compartmental concentrations and colloid osmotic pressure relationships The concentrations of various solutes within a given compartment are expressed by simple algebraic equations that relate the amount of the solute and the compartmental volume. Thus, for example, the concentration of plasma proteins, CP,PL, expressed in g/1, is given by C P . P L = ^ E L [4-33] VPL Similar simple equations can be written for the plasma concentrations of dextran, CR,PL, and small ions, [ ION]PL, expressed in g/1 and mmol/1, respectively. Correspondingly, for the interstitial fluid compartment, the interstitial protein concentration, CPJT (g/1), is given as CP.IT = [4-34] VJJ and similar relationships describe the dextran, CRJT, and small ion, [ ION ] r r , concentrations in the interstitium, expressed in g/1 and mmol/1, respectively. As described in Chapter 2, it is the effective interstitial concentration of proteins (and any other high molecular weight solute) that governs exchange across the capillary. For the two types of macromolecular species, i.e., proteins and dextran, these effective concentrations are given by the distribution of their respective amounts in the available interstitial volume, VIT,AV, as CP,IT,AV ~ v [4-35] IT, AV and C R . r r = T ^ E L - [4-36] IT.AV respectively, where V I T A V = V I T - E x . The subscripts A V and E X indicate the available and excluded interstitial volumes, respectively. According to Bell et al. [1980] and Bert and Pinder [1982], the excluded interstitial volume accounts for about 25% of the total interstitial volume. The effective concentrations described by Eqs. [4-35] and [4-36] must be taken into account when determining the interstitial colloid osmotic pressure. Chapter 4: Model Formulation 75 Several empirical relationships between the plasma concentration of albumin (or total protein) and the colloid osmotic pressure exerted by these macromolecules have been proposed. Landis and Pappenheimer [1963] established separate relationships for the colloid osmotic pressures exerted by either albumin, or alternatively, by the total plasma proteins. By fitting available experimental data, Chappie [1993] established a simple relationship between the plasma albumin concentration and the colloid osmotic pressure exerted by all plasma proteins. Similar to Chappie's approach the following simple relationship was found in this work to correlate the total plasma protein concentration and the total colloid osmotic pressure measured in plasma: C P.PL = 2.7-7CP.PL [4-37] where the units of colloid osmotic pressure are mmHg and those of protein concentration are g/1. For a normal plasma protein concentration of about 70 g/1, Eq. [4-37] predicts a total plasma colloid osmotic pressure of 25.9 mmHg, which compares well with the value of 25.6 mmHg predicted by the corresponding Landis and Pappenheimer [1963] equation. Hence Eq. [4-37] is adopted here. Due to lack of sufficient information, the same correlation will be used to relate the total interstitial colloid osmotic pressure and the available interstitial protein concentration, i.e., CP,IT,AV = 2.7 ' - P ; i T [4-38] The colloid osmotic pressures exerted in both extracellular compartments by the infused dextran are given by the following empirical relationships proposed by Webb [1992]: where ~R.PL and ~R.IT are expressed in mmHg, while the dextran concentration in plasma, CR.PL, and the available interstitium, CR.IT.AV, are expressed in g/1. From Eq. [4-39] it can be calculated that, in agreement with other literature reports, a solution of 38 g/1 of Dextran 70 is iso-oncotic with plasma [Tollofsrud, 1997] (i.e., generates the same colloid osmotic pressure as the normal concentration of plasma proteins). TTR.PL- 0.2688 ' CR.PL + 1.101 x 10 " 2 CR.PL2 [4-39] and -R.IT = 0.2688- CR.IT.AV + 1.101 x 10 "2 CR,I T.AV 2 [4-40] Chapter 4: Model Formulation 76 When the dextran species is introduced into plasma by infusion, the total plasma colloid osmotic pressure, n P L , is given by the sum of the individual colloid osmotic pressures exerted by the two plasma macromolecules (proteins and dextran), i.e., ripL = 7rp,PL + 7CR,PL [4-41 ] The osmotic pressures exerted by the small ions in both plasma and interstitium are assumed to respect the van't HofF law of ideal solutions such that, 7CION,PL=RT- [ I O N J P L [4-42] and TT-ioN, IT = R T • [ION]rr [4-43] where the R is the universal gas constant and T the absolute temperature. Equations [4-43] and [4-42] therefore imply that interactions between the solutes in the two fluid compartments can be ignored and, furthermore, that each mole of a given small ion exerts an osmotic pressure of 19.3 mmHg. 4.5.4.4 Mathematical representation of the Donnan distribution As described in Chapter 3, because the capillary barrier is only semipermeable to proteins, the plasma proteins are preferentially retained in the capillary lumen. Therefore, at any time, there will exist a protein concentration difference, A(Prot), between the plasma and the interstitium. The negatively-charged proteins on each side of the membrane will associate with an electrically equivalent number of positively-charged species, namely Na + , K + and C 2 + . An excess of these cations thus becomes effectively immobilized on one side of the membrane, thereby establishing a concentration difference of all the small ions across the capillary wall. There is no mathematical description of the Donnan partition for cases that involve the presence of more than sodium, potassium and chloride ions. A procedure that accounts for the Donnan effect in the presence of all five extracellular ionic species considered in this model is outlined below. If A(Cat) represents the sum of all the differences in cation concentrations across the capillary membrane, then, at steady state, A(Cat) - A(Na)D + A(K) D + A(C) D [4-44] where Chapter 4: Model Formulation 77 where A(Na)o — (Na)pL - (Na)rr A ( K ) D = ( K ) p L - ( K ) I T A (C )D = ( C ) P L - (C)rr [4-45] [4-46] [4-47] In Eqs. [4-44] - [4-47], A(Na)D, A ( K ) D and A (C )D are the Donnan transcapillary differences in the concentrations of sodium, potassium and other cations, respectively, where all the concentrations are expressed in mEq/l. Since it is the presence of total plasma proteins that creates a A(Cat) distribution across the capillary, it is assumed here that a proportionality constant, kpr 0t, relates the total cations concentration difference with the total protein concentration difference, A(Prot), as follows: A(Cat)= kprot A(Prot) where A(Prot) also expressed in mEq/l, is [4-48] A(Prot) = CP,PL CP,IT,AV z p M W P (CP,PL) (CP,IT,AV) where zp and MWp are the charge and molecular weight, respectively, of the proteins, while (CP,PL) and (CPJT.AV) represent the total protein concentrations in plasma and interstitium, respectively, expressed in mEq/l. Note that, the charge zp for the extracellular proteins was assumed to be the same as that of albumin, i.e., z P = -17. Based on the relationships [4-44] and [4-48], Eqs. [4-45], [4-46] and [4-47] can be rewritten for the non-steady-state as follows: Donnan distribution for Na+: A(Na)D = kprot (CP.PL) (Na) PL (Na ) P L +(K) P L +(C) P L Donnan distribution for tC: ( K ) P L (CP,IT,Av) (Na) IT (Na) I T +(K) I T +(C) I T A ( K ) D - k p r o t (CP.PL)-( N a ) P L + ( K ) P L + ( C ) p L Donnan distribution for C2+: (C), (CP,IT,AV ) (K) IT ( N a ) I T + ( K ) I T + ( C ) I T A ( C ) D = kprot (CP.PL) /PL ( N a ) P L + ( K ) P L + ( C ) P L (CP,IT,Av) (QIT (Na) 1 T +(K) I T +(C) I T [4-49] [4-50] [4-51] Chapter 4: Model Formulation 78 Equations [4-49] - [4-51] assume that the ability of the proteins to associate with a given type of cation is proportional to the mass fraction of that cation in the plasma or interstitium. At steady state, the condition of electroneutrality requires that the differences in positive charges across the capillary membrane be equal to the differences in negative charges, i.e.: A(Cat) = A(Prot) + A(C1)D + A ( A ) D [4-52] where A(C1)D and A(A )D are, respectively, the transcapillary concentration differences for the chloride ion and the anionic species A", expressed as mEq/1. By substituting Eq. [4-48] into Eq. [4-52], the following relationship, which relates the transcapillary anionic concentration differences to the protein concentration difference, is obtained: AfProt) (k P r o t -1) = A(C1)D + A(A) D Consequently, the Donnan distributions for the negative charges can be written as: [4-53] Donnan distribution for Ct: A(Cl) D = (k P r o t - l ) (CP,PL ) (Cl) PL ( C 1 ) P L + ( A ) (CP,IT,AV ) (C1) I T PL ( C 1 ) I T + ( A ) I T [4-54] Donnan distribution for A A(A) D = (k P r o t - l ) (CP,PL)' (A) PL (CP,IT,AV )" ( A ) I T [4-55] ( C 1 ) P L + ( A ) P L * ™ V ' ( C 1 ) I T + ( A ) I T Equations [4-49] - [4-51], [4-54] and [4-55] express the contribution of the Donnan effect to the distribution of small ions across the capillary membrane; these were taken into account in describing the transcapillary fluxes of fluid and small ions in the model (see Eqs. [4-15] and [4-18]). 4.5.5 Fluid and solute inputs and outputs Figure 4-1, shows a number of fluid and solute rates that are either introduced into the body by infusion or lost from the body through renal output and sensible/insensible losses. The plasma compartment is the recipient of infusate and the source of renal elimination. The sensible and insensible losses, i.e., PER and ISL, respectively, occur from the interstitial compartment. Chapter 4: Model Formulation 79 Inputs The infusion rates are normally obtained from experimental or clinical data. Depending on the specific needs of the patient (in case of a trauma) or the experimental protocol employed (in the case of testing an infusate), the composition, volume and rate of infusion are varied during the resuscitation procedure. These inputs are accounted for in the model by simple time-dependent source terms in the appropriate mass balance equations. Thus the rate of fluid input by infusion can be written as jRES=f(t) [4-56] This fluid may contain species that already exist in the body such as albumin and small ions, that are infused at the rates of Q^s and MRES, respectively. In some instances, resuscitation with alternative colloidal compounds, which are foreign to but well tolerated by the body, is desired. The rate of infusion of such materials is symbolized as R RES . The infusate R can be any macromolecular species such as Dextran 40, Dextran 70 and HES (hydroxyethyl starch), etc., but because of its importance as a colloidal resuscitant, a brief background about the characteristics and use of Dextran 70 is given in Section 4.5.6. Outputs Under normal conditions, fluid is lost from the body as urine at a rate JUR, through perspiration (sensible loss) at a rate JPER, and through insensible losses at a rate JISL-The solutes eliminated in urine include all five species of small ions found in plasma, but not proteins. Although Dextran 70 is eliminated in urine after breakdown in the body, these losses occur only after 3 to 4 h post-infiision [Haljamae, 1993; Arrfors and Buckley, 1989]. Such times are beyond those normally simulated by the model. For initial model validation, the urinary rate, JUR, as well as the rate of small ion output, M I O N U R , are obtained from experimental data. A new kidney model that can predict both fluid and solute renal outputs will be presented later in Chapter 8. Data for fluid and ion rates lost through perspiration were obtained from the literature. The only ions involved are Na + , K + and Cl". For a normal, 70 kg human, the perspiration rate is usually very small, i.e., about 2 ml/h of fluid and contains sodium, potassium and chloride at concentrations of 45, 4 and 49 mEq/1, respectively [e.g., Krieger and Sherrard, 1991; Kleinman Chapter 4: Model Formulation 80 and Lorenz, 1984]. The 'insensible losses' represent the combined losses of solute free water mainly through respiration but also, to some extent, through skin. In humans, JISL occurs at a normal rate of about 40 ml/h. 4.5.6 Characterization of Dextrans, Dextran 70 Dextrans are plasma substitutes used for intravenous infusion. By virtue of their relatively high molecular weight, they have good plasma retention. When infused they exert a colloid osmotic pressure across the capillary membrane and thus they induce a transcapillary shift of water that results in plasma expansion, hemodilution and improved microcirculatory flow. They are used therefore for hypovolemic conditions (e.g., burn, hemorrhage, trauma). For reviews on the use of these colloidal resuscitants, see Arrfors and Buckley [1989], Webb [1992] and Haljamae [1993]. The two types of dextrans most commonly used in clinical practice are Dextran 70 (commercial name Macrodex) and Dextran 40 (commercial name Rheomacrodex). These natural colloids are composed of fractions with a wide variety of molecular weights. The number in their nomenclature represents the average molecular weight. Thus, Dextran 70 is a compound whose average molecular weight of about 70 kDa is very similar to that of albumin, 67 kDa. The typical concentration of Dextran 70 in solutions used for infusion is 60 g/1 [Tollofsrud, 1997]. The colloid osmotic pressure exerted by 60 mg/ml of Dextran 70 is about 65 mmHg, i.e., slightly more than two times the normal value exerted by the plasma proteins. With time, Dextran 70 is completely degraded to smaller fractions that are eliminated from the body. Between 4 and 6 h post-infusion, about 60% of the dextran infused is still above the normal renal threshold for elimination [Arrfors and Buckley, 1989]. 4.6 Steady state conditions The normal steady-state conditions are those corresponding to a 'Reference man' [Reference Man ICRP 1975] defined as a healthy male in supine position who that has a weight of 70 kg and a height of 170 cm. These values are summarized in Table 4-1 and briefly described below. The plasma and interstitial volumes as well as the hematocrit value are specified in the available literature [Wiederhielm, 1974; Noddeland et a l , 1984; Fauchald et a l , 1985(a); Fauchald et a l , 1985(b)]. Chapter 4: Model Formulation 81 Variable Value Source Body weight, BW 70 kg Plasma volume, V P L 3200 ml [Fauchauld et al., 1985(a,b)], [Wiederhielm, 1979] Red blood cell volume, VRBC 2133 ml calculated from Hct data Hematocrit, Hct 40.0 % [Kleinman and Lorenz, 1984] Interstitial volume, Vn 8400 ml [Fauchauld et al., 1985(a,b)], [Wiederhielm, 1979] Excluded interstitial volume, Vn\Ex 2100 ml assumed based on [Reed et al., 1989] Tissue cell volume, V T C 28000 ml calculated as 40% BW Plasma hydrostatic pressure, P c 11 mmHg [Chappie et al., 1993] Plasma colloid osmotic pressure, - P - P L 25.9 mmHg [Fadnes et al., 1986], [Xie etal., 1995] Plasma protein concentration, cP > P L 70 g/1 [Carlson et al., 1996], [Kleinman and Lorenz, 1984] Interstitial hydrostatic pressure, Pn -0.7 mmHg [Wiig and Reed, 1987], [Xie etal., 1995] Interstitial colloid osmotic pressure, 7 r P - I T 14.7 mmHg [Fadnes et al., 1986] [Fauchauld et al., 1985(a,b)] Interstitial protein concentration, CPJT 29.8 g/1 calculated as per Section 4.5.4.3 Protein concentration in the available volume, cP,rr,AV 39.7 g/1 calculated as per Section 4.5.4.3 Protein reflection coefficient, a 0.988 [Xie etal., 1995] Permeability-surface area product for protein, PS 73.0 ml/h [Xie etal., 1995] Dextran 70 reflection coefficient, a D E X 0.5 [Haraldsonetal., 1982], [Curry, 1984] Permeability-surface area product for Dextran 70, PSDEX 18.25 ml/h [Garlick et al., 1970], [Altman and Dittmer, 1971] Reflection coefficient for small ions, a I 0N 0.05-0.5 [Curry, 1979], [Wolf and Watson, 1989] Permeability-surface area product for small ions, PSION 73,000-365,000 ml/h assumed based on [Yudilevich et al., 1968] Fluid filtration coefficient, kF 121.1 ml/mmHg-h [Xie etal., 1995] Lymph flow sensitivity, L S 43.1 ml/mmHg-h [Xie etal., 1995] Lymph flow rate, J L 75.7 ml/h [Xie etal., 1995] Table 4-1. Initial steady-state compartmental values for the 70 kg 'Reference man' (see text for a more detailed discussion). Chapter 4: Model Formulation 82 As in previous modeling studies [Bert and Pinder, 1982; Bert et al., 1988; Reed et al., 1989], it was assumed that proteins are excluded from 25% of the total interstitial fluid volume. The volume of the red blood cell compartment was calculated based on the hematocrit. For the 'lumped' tissue cell compartment, it was considered that all these cells bear the properties and characteristics of skeletal muscle cells and constitute about 40% of the total body weight or approximately 28 1 [Landis and Pappenheimer, 1963; Arfors and Buckley, 1989]. The normal compartmental hydrostatic and colloid osmotic pressures for plasma and interstitium employed by Xie et al. [1995] and Chappie et al. [1993] in formulating the human microvascular exchange model, are also given in Table 4-1. A l l these compartmental pressure values are obtained from the available literature and correspond to a combined tissue compartment and the general circulation. The same table shows a typical value of the plasma protein concentration available from literature sources [Kleinman and Lorenz, 1984] as well as the calculated interstitial protein concentrations (see Eqs. [4-33] - [4-35]). The transport properties of the capillary wall, that separates the plasma and interstitial compartments, include the capillary filtration coefficient, k F , the capillary permeability-surface area products for proteins, Dextran 70 and small ions, PS, PS D E X and PS ION, respectively, and the reflection coefficients for proteins, Dextran 70 and small ions, a, O D E X and C>ION, respectively. The fluid and protein transport coefficients together with the normal lymph flow, J L , and the lymph flow sensitivity, LS, were estimated by Xie et al. [1995] by statistical comparison of model predictions with clinical data available for humans. The literature value for PS D E X is about 0.20 to 0.25 times the PS for albumin [Garlick et al., 1970; Altman and Dittmer, 1971]. A value of 0.25 - PS was assumed for the present study. According to Haraldson et al. [1982] and Curry [1984], CTDEX ranges from 0.30 to 0.55 depending on the Dextran 70 concentration. For the concentration range commonly used in clinical practice (i.e., 6% Dextran 70 solution), CTDEX is at the higher end of this range. Thus, we assumed a value of ODEX= 0.5. There is little reliable information about the two transport parameters associated with the transport of small ions across the capillary. Based on the studies reported by Yudilevich [1963], it can be estimated that the value for PS ION is between 1000 and 5000 times higher than the corresponding permeability-surface area product for proteins, PS. Most physiological textbooks state that the small ions have a very low transcapillary reflection coefficient, i.e., less than 0.1. However, Wolf and Watson [1989] measured values of CTION ranging from 0.1 to 0.5 for cat hindlimb, while Curry [1979] obtained an average reflection coefficient for small ions of 0.05 for Chapter 4: Model Formulation 83 frog mesentery. In their hemorrhage model, Carlson et al. [1996] assumed _ I O N values of 0.045 for N a + and CI" and a value of 0.1 for all other small solutes. Thus, it is anticipated that CION should lie in the range of 0.05 - 0.5 as listed in Table 4-1. Since both PSJON and GION are imprecisely known, one of the objectives of the Chapter 6 will be to describe their estimation using available experimental data. Table 4-2 shows how the small ions are normally partitioned between the intra- and extracellular compartments. The values selected for these ion concentrations as well as the properties ascribed to these cells are approximate, representative of a fairly large number of cells, but do not exactly characterize any of them. They are the result of combining experimental information with additional calculations. Table 4-2 was assembled based on the following considerations. Extracellular concentrations: The extracellular concentrations for Na + , K + and CI" ions in plasma are obtained from experimental measurements in rats [Onarheim, 1995]. This reference was chosen in order to maintain the same source for initial conditions and for comparison with model predictions as will be discussed in Chapter 6. Similar concentrations have been reported for rabbits [Kunze, 1977], dogs [Vick et al., 1970] and humans [Kleinman and Lorenz, 1984]. The extracellular values for C 2 + and A" in plasma were calculated by solving simultaneously the following two algebraic equations: £ ( z I O N P L • [ION] P L ) + = 0 [4-57] ION.PL M W P O- 2 ([ION] P L) + - ^ = S P L [4-58] ION.PL M W p where [ION]pL and ZION.PL are the molar concentration and charge, respectively, of the ionic species ION (i.e., ION = Na + , K + , C 2 + , CI", A") present in plasma; while CP,PL, Z p and MWp are the molar concentration, charge and molecular weight of the proteins (i.e., albumin) in plasma. O and SPL are the osmotic coefficient and plasma osmolarity, respectively. Eq. [4-57] represents the condition of electroneutrality for the plasma compartment while Eq. [4-58] expresses the relationship for total plasma osmolarity. The unknowns in the above equations are the molar concentrations for the A" and C 2 + species. Chapter 4: Model Formulation Tissue Cells (TC) Ref. 2 2 2 2 CJ Tissue Cells (TC) [mM] 10.0 150.4 13.6 174.0 o 122.3 ! 126.3 300.3 Tissue Cells (TC) [mEq/1] 10.0 150.4 27.1 187.5 o 183.5 i • 187.5 375.0 Tissue Cells (TC) N r H + r n + TH r H 1 Interstitium (IT) Ref. C,R2 2 2 CJ C,R2 C,R2 C,R2 Interstitium (IT) I 135.0 10.2 149.8 107.2 42.7 VO o 150.5 300.3 Interstitium (IT) [mEq/1] 135.0 ITi 20.5 160.0 107.2 42.7 10.1 160.0 320.0 Interstitium (IT) N r H + r H + ? ( l l r H Plasma (PL) Ref. 2 2 CJ 2 CJ CJ Plasma (PL) [mM] 139.7 10.4 154.8 105.7 41.7 o 148.4 303.2 Plasma (PL) [mEq/1] 139.7 20.7 165.1 105.7 41.7 17.7 165.1 330.2 Plasma (PL) N r H + r H + + l r H r H Red Blood Cells (RBC) Ref. 2 2 2 u Red Blood Cells (RBC) 11.0 140.0 13.5 164.5 73.2 65.5 j 138.7 303.2 Red Blood Cells (RBC) [mEq/1] 11.0 140.0 27.0 oo r H 73.2 104.8 1 1 1 00 r-r~( 356.0 Red Blood Cells (RBC) N r H + r H + + l VO r H f 1 1 Charged Species Positive Charges Na Cor FC TOTAL Negative Charges U A or FA Proteins TOTAL TOTAL CHARGES Chapter 4: Model Formulation 85 The value for the plasma osmolarity was obtained from experimental measurements for rats [Onarheim, 1995]. Following the modeling practice of previous authors [Wolf, 1982], both the intra- and extracellular media were assumed to be ideal solutions for which the van't Hoff law applies; therefore, an osmotic coefficient, O = 1 was assumed for all solutes, i.e., ions and proteins, in plasma. The molar and normal protein concentrations given in Table 4-2 were calculated based on the known mass concentration of protein in plasma and by considering for this species an average charge of zp = -17 and a molecular weight of Mw,p = 67000 g/mol [Kleinman and Lorenz, 1984]. As previously mentioned, a charge of +2 was assumed for all positive ions other than N a + and K + and a charge of -1 was attributed to the A" species. The concentrations of small ions in the interstitium were calculated from Donnan considerations across the capillary membrane based on the relationships presented in Section 4.5.4.4. A l l of the extracellular ion and protein concentrations presented in Table 4-2 are in good agreement with corresponding values reported in the literature, [Kleinman and Lorenz, 1984]. Intracellular concentrations: Values for the intracellular concentrations of Na + , K + and the intracellular non-diffusible positive species F C 2 + , were available in the literature [Hoffman, 1987; Kleinman and Lorenz, 1984] for tissue and red blood cells. The intracellular chloride concentration for red blood cells was calculated using the ratio [Cl"]RBc/[Cr]pL = 0.69 previously reported by Hoffman [1987]. Numerous negative species, mainly proteins, constitute the internal environment of the cells. The generic term FA" was used to describe these non-diffusible species. Its concentration and average charge were calculated by simultaneously solving two algebraic equations. The first equation is based on the bulk internal electroneutrality condition: 2>ION,RBC •[ION] R B C ) + z F A R B C - [ F A ] ^ = 0 [4-58] ION.RBC where ZION, RBC and [ ION]RBC have been defined previously, while ZFA,RBC and [FA]RBC are the unknown average charge and molar concentration of the non-diffusible species FA". The second equation imposes the isotonicity condition between the cells and their surroundings, i.e., SPL = SRBC [4-59] where SRBC represents the total osmolarity inside the red blood cells, i.e., Chapter 4: Model Formulation 86 SRBC= Z([I0N]RBC) + [ F A ] R B C [ 4 -60 ] ION.RBC A similar approach as for red blood cells was adopted to calculate the intracellular concentration and charge of FA" for the tissue cells. The only exception made was with respect to the intracellular chloride ion, whose internal concentration was obtained from literature reports on skeletal muscle cells [Kleinman and Lorenz, 1 9 8 4 ] . The above steps complete a simplified picture of the small ion partition between the intra- and extracellular media as well as between the plasma and interstitium. They provide an initial set of numerical values that are in a reasonable physiological range when compared to experimental data [Kleinman and Lorenz, 1 9 8 4 ] . For each type of cellular compartment, four other cellular membrane parameters are required in order to completely describe the cellular exchange process. These parameters are the three membrane permeabilities, one for each of the ions transported across the cell membrane, i.e., pNa, PK and pci, and the rate, RP, of the Na +-K +-pump. Based on known intra- and extracellular concentrations for a particular type of cell, i.e., RBC or tissue cell, the system of membrane transport equations given by Eqs. [ 4 - 1 0 ] and [ 4 - 1 1 ] coupled with the non-linear equation giving the membrane potential (Eq. [ 4 - 1 4 ] ) were solved for steady-state conditions (i.e., the left-hand-side accumulation terms of Eqs. [ 4 - 1 0 ] and [ 4 - 1 1 ] were set to zero). Initial literature value for p a and one of the other permeabilities, pNa or PK , together with the intracellular and extracellular concentrations presented in Tables 4 - 2 , were assumed known. The unknowns were then the other permeability pNa or pK, RP and A<j> (where Acj)= F V m /RT) . The membrane parameters obtained by this procedure are presented in Table 4 - 3 . A l l the compartmental values and parameters presented above represent the steady-state values of the system. This requirement was verified by running the computer program associated with the transient model for an extended period of time and in the absence of any perturbations to the system in order to evaluate whether there were significant changes in any of the dependent variables. Additionally, when perturbations to the system are considered, i f all the inputs exactly equal the outputs from the system, the model predicts a return of all the compartmental variables to the normal steady-state. As part of the previous work, most the transport parameters descriptive of mass exchange in the M V E S (with the exception of those related with the transport of small ions and of the infused macromolecular species) were subjected to a detailed sensitivity analysis. In this work, Chapter 4: Model Formulation 87 although results are not presented, similar sensitivity analysis studies were done for the transcapillary parameters related to the Dextran 70 transport (i.e., P S D E X and G D E X ) . Additionally, the variation of the parameters describing the transport across the cell membrane (i.e., the cell membrane permeabilities and RP) for a wide range outside the values presented in Table 4-3, were also investigated. From this modeling exercise it was concluded that, the model predictions are relatively insensitive to changes in the above-mentioned transport parameters. Thus, all the values presented in Tables 4-1 to 4.3, are reasonable choices for the parameters described. Membrane Parameter Red 1 Hood Cells Muscle Cells Value Source Value Source P N a (cm/sec) 4 x l f J - 1 0 [Hoffman, 19871 5.95 x 10"7 (C) PK (cm/sec) 1 0 -io 3.82 x lO - 1 0 [Hoffman, 1987] (C) 5x 10"5 [Jakobsson, 1980] pci (cm/sec) 10 - 8 [Hoffman, 1987] 5 x 10"5 [Jakobsson, 1980] RP PK 15.03 (C) 0.548 ( Q RP PNa 14.35 (C) 46.05 ( Q V m (mV) -9 -9.78 [Hoffman, 1987] (C) -84 -90 -87.8 [Nakayama et al., 1984] [Shires et al., 1972] ( Q Table 4-3: Calculated (C) and literature values for steady-state cellular membrane parameters. Where available comparative literature values are provided in addition to the calculated values. 4.7 Normal conditions for subjects other than the 'Reference man' a) Humans In studies that involve humans having a body weight different than that of the generic 'Reference man', all the extensive properties considered in the model are scaled by a weight ratio, WR [Ampratwum, 1995], given as W R - Weight of Human { Weight of 'Reference man' Chapter 4: Model Formulation 88 The variables that are subjected to scaling based on weight include the compartmental fluid volumes, amounts of proteins and ions, interstitial compliances as well as all the global transport parameters with the exception of the capillary reflection coefficient. b) Animals The scaling approach employed when the model is used for animal studies can be summarized as follows: 1. The animal plasma volume together with the intensive physiological properties of the vascular and interstitial compartments are considered to be the same as those of the normal 'Reference man' (e.g., V P L , HUMAN = VPL/SCALED ANIMALS TCP,PL,HUMAN = TCP.PU'SCALED ANIMALS [ ION]PL, HUMAN = [ION]PL,-SCALED ANIMALS CtC.). 2. A l l the extensive physiological properties of plasma, interstitium and tissue cells are scaled by the ratio (VPL,HUMAN / VPL,ANIMAL)- Thus, for example, the value of the interstitial volume of the scaled animal is calculated as, V = _VHUMAN_ VIT,'SCALED ANIMAL' IT.ANTMAL L H U z - J "PL,'ANIMAL' Note that the subscript ' A N I M A L ' refers to the actual compartmental value of a real animal obtained usually from experimental information. The subscript 'SCALED A N I M A L ' refers to the animal variable that is scaled to the appropriate human value. 3. The interstitial compliance and all the transport parameters, other than a values, are scaled based on the interstitial volume ratio of scaling, Rscaiing, given by, v „ v IT,'SCALED ANIMAL' r . , _ , R Scaling = [4-63] IT.HUMAN Thus, for instance, the new values for the transport parameters can be obtained from an equation having the following form: Transport Parameter,SCALED A N I M A L , = R S c a l i n g • Transport Parameterm^ [4-64] An explicit example of the scaling procedure is provided in the Appendix C, where it is applied to determine compartmental values representative of dogs. Chapter 5: Numerical Methods 89 Chapter 5: NUMERICAL METHODS AND COMPUTATIONAL PROCEDURES 5.1 General Aspects From a mathematical point of view, the overall model describing the four homogenous compartments (see Section 4.5), consists of twenty-two ordinary time-dependent differential equations (accounting for balances of fluid volumes, ions, proteins and macromolecules) coupled with two implicit non-linear algebraic equations (corresponding to the cellular transmembrane potentials) and two explicit algebraic equations (describing the changes in cellular volume). These equations are linked via several auxiliary algebraic equations such as compliance relationships and equations for osmotic pressures. To solve the overall mathematical model, a computer program was developed using the Fortran-77 language as applied by the Microsoft Fortran PowerStation®compiler and programming environment [Microsoft, 1993]. The output of the Fortran program was connected to the Microsoft Excel® spreadsheet software for convenient handling and processing of the large data files. The calculations were performed in double precision. 5.2 Conceptual Design of the Program The main objective in designing the program was to achieve a high degree of adaptability and robustness for the various simulation scenarios of interest, such as different fluid infusions, inclusion of kidney function and hemorrhage. Adaptability was assured by incorporating into the main program, easy to modify, 'specific task oriented' subroutines for input and output data files, calculation of the normal physiological values, managing the numerical solution of the different types of model equations, calculation of the normal physiological values, etc. The robustness of the program is provided by the numerical techniques employed. The complete computer code is listed in Appendix H. The subroutines in order of appearance in the program, and their functions are: CONS Determines the various constants used in the equations of the model, such as scaling ratios, parameters required for calculating the cellular area available for transport, etc. This subroutine also determines the Chapter 5: Numerical Methods 90 compartmental fluid volumes for cases involving humans with different body weights. FIT Establishes the compliance relationship (Chapter 4), by using cubic splines (via subroutine SPLINE) to fit the experimental pressure-volume data for the region of moderate overhydration (see Section 5.3 for a brief description of the method of splines). INIT Based on the compartmental volumes established above, assigns/calculates the initial values for all the model variables and parameters describing the extracellular compartments (i.e., calculates the normal unperturbed values for e.g., glucose, albumin and dextran concentrations, their osmotic pressures, etc., in plasma or interstitium). PUMP M O D E L M E M B R P Supplies the cellular membrane parameters, such as ion permeabilities, pump rate and ratio. Defines the complete set of algebraic model equations for all species and for both the intra and extra-cellular compartments. Solves the non-linear membrane potential equation for the red blood and interstitial cells, using Brent's method via the external functions PFUNC, ZBRENT and the root bracketing subroutine Z B R A K (see Section 5.3 for a description of Brent's method). H E M This subroutine is used only in the case of hemorrhage simulations, to calculate the plasma and RBC volumes lost during hemorrhage. ODEINT Implements the integration of the system of twenty two ordinary differential equations (ODEs). The subroutine is based on the Runge-Kutta-Fehlberg method with adaptive error control but uses an alternate set of coefficients given by Cash and Karp [Press et al., 1992] (see also Chapter 5: Numerical Methods 91 subroutines R K C K , RKQS and DERIVS called by ODEINT). Section 5.3 provides an overview of this ODE solution technique. B A L A N C E Calculates the water shift across the cellular membrane caused by disturbances in extracellular osmolarity. It also updates accordingly all the compartmental fluid volumes. ASSIGN; OUTPUT These two subroutines handle the output data files by storing the variables in a matrix (ASSIGN) and setting up the proper format for the output as an Excel spreadsheet (OUTPUT). To obtain an accurate solution, the non-linear membrane potential equations as well as the algebraic equations for cell volume adjustment were solved simultaneously with the system of differential equations. This was achieved by calling the non-linear equation solver (MEMBRP) as well as the subroutine handling the cellular volume updates (BALANCE) , for each local time step of the Runge-Kutta-Fehlberg method (ODEINT). Thus the membrane potential together with the ion concentrations, albumin contents, fluid fluxes and cellular volumes were continuously updated until the integration converged over the imposed local time step. 5.3 Numerical Methods Three numerical techniques are used in the present work, i.e., spline interpolation, non-linear equation solving via Brent's method and a variant of the Runge-Kutta-Fehlberg method to solve the system of ODEs. An overview of these methods is given next. a) Cubic Spline Interpolation Given a set of data values (x;, y;; i=l,...,N), cubic spline interpolation makes use of the second derivative, y , to fit a function y of the form [Press et al., 1992]: y = A Y i +By i + 1 +Cy] + D y ; i + l ' [5-1] to the data set, where the coefficients A, B, C and D are given by: Chapter 5: Numerical Methods 92 A = JL±1_JL; B = 1 _ A ; c - ^ A ' - A X x ^ - X j ^ a n d D - i c B ' - B X x ^ - X i ) 2 [5-2] x i + 1 - X i 6 6 Equations [5-1] and [5-2] show that y has a cubic dependence on x through the coefficients C andD. In order to use Eq. [5-1], the second derivatives must be evaluated. Using the continuity condition imposed on the first derivative yi ' , at a given point x; situated at the boundary of two adjacent intervals [XJ.I x;] and [x;, x;+i], yields the following system of equations for the interior points i=2,...,N-l, + ^ y « v v - [ « 1 6 3 6 x i + 1 - X i X i - x M which allows the calculation of the second derivatives. The set of N - 2 linear equations for N unknowns y i ; [i=l,...,N]; requires the specification of boundary conditions at xi and XN- Generally, three options are available, the so-called natural, clamped and fitted boundary conditions [Bowen, 1994]. In the present work, the 'natural' condition, which sets the second derivatives to zero in the end points, was used at both boundaries. The system of equations [5-3] and the two end conditions are tridiagonal; therefore it can be readily solved using well-known methods such as the Thomas algorithm [Bowen, 1994] (see subroutine T D M A called by SPLINE in Appendix H). The general strategy in using the cubic spline interpolation is comprised of two steps. First, based on a set of given data (x;, y,; i=l.. .N), the second derivatives are calculated via the procedure outlined above. Second, to find the corresponding y value for a desired argument x, an interval search method (e.g., bisection) is applied to locate x in the interval [xi, X j + i ] . Then Eqs. [5-1] and [5-2] are used to determine y. Note, the first step has to be executed only once as long as the set of data points remains the same. b) Non-linear Equations: Brent's Method There are several techniques available to solve non-linear equations [Press et al., 1992]. Among them, algorithms based on the first derivative of the equation, such as the Newton-Raphson method, are extremely popular, due especially to their very fast (i.e. quadratic) convergence in the vicinity of the root. However, the Newton-Raphson method has several disadvantages as well. First, it requires the evaluation of the first derivative, which, furthermore, Chapter 5: Numerical Methods 93 must be continuous and non-zero in the neighbourhood of the root. In other words, the function cannot have a local extremum near the root. Second, far from the root, where the higher-order derivatives are important, the Newton-Raphson method may fail to converge to the correct result. Thus, the 'global' convergence of the method is poor, unless an excellent initial guess is provided. In the initial development of the present program, the Newton-Raphson algorithm coupled with root bracketing was implemented to solve for the non-linear membrane potentials at each time step of the ODE solution. Unfortunately, due to the above-mentioned problems, the failure rate of the method was unacceptably high for the diverse conditions that were simulated. Therefore, the need for a more robust, but also fast, algorithm arose. One of the best algorithms for solving one-dimensional non-linear equations quickly and with robust convergence properties*, is the so-called Brent's method [Press et al., 1992]. This technique makes use of root bracketing, bisection and inverse quadratic interpolation to combine the safety of bisection with the speed of a higher-order procedure. Assuming a general non-linear equation y = f(x), inverse quadratic interpolation means that the method uses three prior root estimates to obtain x as a quadratic function of y. The new root estimate is then determined by setting y = 0 in this quadratic relationship. Without entering into a detailed mathematical treatment, the next root estimate is given by the formula [Press et al., 1992]: x = b + —, [5-4] Q where P = S[T(R - T)(c - b) - (1 - R)(b - a)] Q = (T - 1)(R - 1)(S -1), with s = f (b) T = f W a n d R = £(b) f(a)' f(c) f(c) In the above equations, x is the next root estimate, b is the current estimate, while a and c are two prior estimates. * In fact it is guaranteed to converge as long as the function can be evaluated in the initial interval known to contain a single root [Press et al., 1992]. Chapter 5: Numerical Methods 94 Brent's algorithm was implemented in the program as the ZBRENT external function in the form given by Press et al. [1992] (see subroutine MEMBRP) . This routine proved to be extremely reliable and fast for all the simulations tested in the present study. c) System of Ordinary Differential Equations: the Runge-Kutta-Fehlberg Method. The general theory behind the Runge-Kutta method of ODE integration is well known [Bowen, 1994]. In order to reduce the computational effort, various Runge-Kutta methods with adaptive step-size control have been developed [Press et al., 1992]. Among them, the stepsize adjustment algorithm originally developed by Fehlberg (known otherwise as the embedded Runge-Kutta formula) gained popularity due to its efficiency and robustness. In essence, this method relies on the observation by Fehlberg, that a certain combination of the six function evaluations required by a fifth-order method* reduces the order to four (i.e. the fourth-order formula is embedded in the fifth-order method). Furthermore, the difference between the general form fifth-order formula and the embedded fourth-order formula provides an error estimate and, hence, a basis for step-size control [Bowen, 1994; Press et al., 1992]. The general fifth-order Runge-Kutta formula is: y n + , = Yn +c,k, +c 2 k 2 +c 3 k 3 +c 4 k 4 +c 5 k 5 +c 6 k 6 +0(h6) [5-7] The embedded fourth-order (Fehlberg) formula is expressed as: y L =y n+ci~i + c ; k 2 + c ; k 3 + c ; k 4 + c ; k 5 + c ; k 6 + o ( h 5 ) [5-8] where k, = h f ( x n , y n ) k 2 - h f ( x n + a 2 h , y n + b 2 I k , ) k 6 - h f ( x n + a 6 h , y n + b 6 1 k 1 +... + b 6 5 k 5 ) . In Eqs. [5-7] to [5-9], h is the stepsize, and a2,...,C6* are constants. Table 5-1, shows the Cash-Karp values for the constants &2,.. . ,C6*. * A method is called w-th order if the leading error term is of 0(hn+1). Note also that the number of function evaluations required by Runge-Kutta formulas with an order M higher than four, are typically M+1 or sometimes M+2. Chapter 5: Numerical Methods 95 i Ci * 1 37 378 2825 27648 2 1 5 1 5 0 0 3 3 3 9 250 18575 To 40 40 621 48384 4 3 3 9 6 125 13525 5 10 "lO 5 594 55296 5 1 11 5 70 35 0 277 ~54 2 _ 27 27 14336 6 7 1631 175 575 44275 253 512 1 8 55296 512 13824 110592 4096 1771 4 j = 1 2 3 4 5 Table 5-1: Cash-Karp coefficients for the Runge-Kutta-Fehlberg method [Press et al., 1992]. The error of the Runga-Kutta-Fehlberg method is estimated as the difference between Eqs. [5-7] and [5-8]; i.e., A = y n + 1 - y ; + 1 = 2 > i - c * ) k i - t 5 - 1 0 ] i=l According to Eq. [5-8], the error A of the Runge-Kutta-Fehlberg method, scales with step-size as h 5. Therefore, i f a certain step-size hi produced an error A i , then the required step-size ho to obtain the desired accuracy (or error) Ao, can be estimated as: h 0 =h, rA V ' 5 [5-11] Equation [5-11] controls the step-size h of the Runge-Kutta-Fehlberg algorithm as a function of the relative error (so-called adaptive step-size control). The subroutine ODEINT uses the Runge-Kutta-Fehlberg algorithm with Cash-Karp coefficients to solve the set of twenty-two differential equations (see subroutines RKQS and R K C K ) . This technique was found to be reliable and fast for all the simulation scenarios considered in the present study. Chapter 6: Model validation based on animal studies 96 Chapter 6: MODEL VALIDATION BASED ON ANIMAL STUDD2S 6.1 Introduction Confidence in the human model presented in Chapter 4 can only be gained if the predictions of this model compare well with available experimental data. In the present chapter, an initial validation of the model will be carried out based on measured data obtained from several animal experiments. The chapter is organized according to the following topics, which all play an important role in the validation exercise: • Evaluation of infusion fluids The fluids that have been employed in the various animal infusion experiments of relevance here include: isotonic solutions, either normal saline (NS) or Ringer's type solutions (RS); hyperosmotic saline solutions (HS); and hypertonic colloidal solutions (HSD). • Estimation of the permeability-surface area product and reflection coefficientfor small ions As was suggested in the previous chapter, the model parameters which have the highest degree of uncertainty are those related to the transcapillary transport of small ions. Hence, prior to model validation, it becomes necessary to use part of the available literature data to estimate these two essential transport parameters. • Validation of the model using measured data obtainedfollowing the infusion of animals with normal saline (NS), Ringer's solution (RS), and/or hyperosmotic solutions (HS) A first validation of the model is based on information obtained from NS, RS and HS infusions in nephrectomized animals. The data were selected only for animals that were not subjected to any previous trauma (i.e., infusions were administered to otherwise healthy animals). • Discussion of the model prediction following infusions with NS, RS and HS 6.2 Solutions for Infusion One of the purposes of fluid therapy is to replace fluid and/or solute deficits in the body that have resulted from a traumatic condition (e.g., hemorrhage, burns, etc.). The goal in external fluid administration is to re-establish an adequate circulatory volume and, implicitly, an adequate cardiac output, blood pressure, as well as oxygen delivery to the tissues. Chapter 6: Model validation based on animal studies 97 Over the past couple of decades, there has been a transition in fluid therapy from blood products toward plasma substitutes. This departure from the use of blood products is due to economic factors and the availability of blood, as well as concerns about the possibility of these products carrying pathogenic organisms and producing unwanted immunizations. Based on their composition, the various alternative fluids that can be used for infusions can be categorized as crystalloid or colloid solutions. A further classification of both of these types of solutions is based on their osmolarity relative to plasma. Thus, the solutions used in clinical practice are hypo-osmolar (lower osmolarity than plasma), iso-osmolar (approximately the same osmolarity as plasma), or hyperosmolar (higher osmolarity than plasma). The last two categories of infusion fluids are extensively used in the validation studies presented in this chapter and, therefore, will be discussed in more detail in the following two sections. 6.2.1 Iso-osmolar saline (Ringer's) solutions The use of isotonic solutions is based on studies done by Ringer more than a century ago [see Weisberg, 1962]. Isotonic solutions have about the same osmolarity as the internal body fluids, i.e., intra- and extracellular fluids. It is expected, therefore, that their infusion does not alter the osmolarity of the extracellular fluid and hence causes no significant osmotic shifts of water across the capillary membrane or into or out of the cells. Infusion of large volumes of these solutions results in an initial increase in plasma volume followed by a secondary transport of fluid from plasma into interstitium because of differences in hydrostatic pressure. As a consequence, the retention of the isotonic infusate in plasma is poor; although the net result of infusion is an expansion of the total extracellular volume, this expansion occurs at the expense of an increased interstitial volume and potential edema formation. A great many of the solutions used for infusion are described in the medical literature as belonging to the class of iso-osmolar solutions. The typical compositions of two of these solutions are presented in Table 6-1 and these are discussed briefly below. Isotonic 0.9% sodium chloride solution, also known as normal saline (NS), isotonic saline or physiological sodium chloride, contains approximately 154 mEq/l of NaCl. Despite its name, however, this solution is not exactly in 'physiological' balance with the internal body fluids. As shown in a previous chapter (see Table 2-1), the sodium and chloride concentrations in the extracellular plasma and interstitial fluids (e.g., about 140 mEq/l and 100 mEq/l, respectively) are lower than those of NS. It is expected, therefore, that when NS is infused, it will Chapter 6: Model validation based on animal studies 98 induce some minor fluid and small-ion shifts between the different fluid compartments (mainly PvBCs). In addition to its limited intravascular retention that is typical of iso-osmolar solutions, the iso-osmolar NS solution has been reported to cause hemodilution, renal potassium loss, and aggravation of metabolic acidosis [Rocha E Silva et al., 1987]. To overcome the excess of sodium chloride in isotonic saline, several Ringer's type solutions (RS) have been developed over the years. As shown for the lactated Ringer's example given in Table 6-1, these contain a more balanced combination of different small ions. Other Ringer's type solutions include acetated-RS, Hartman's solution, Fox's solution, etc. Owing to their great variety, the exact composition of the infusion fluids that are used in this work for model validation and/or prediction will be indicated when discussing specific experimental protocols. 6.2.2 Hyperosmolar/hypertonic solutions The osmolarity of these solutions is higher than that of plasma. Infusion of hyperosmolar solutions increases plasma osmolarity and therefore causes an osmotic water shift into the plasma compartment from the interstitium and also from the cells (both RBCs and tissue cells). The overall effect is a much higher initial increase in the osmolarity and volume of the extracellular fluid than can be expected from iso-osmolar solutions. As shown in Table 6-1, the hypertonic solutions can consist of only concentrated NaCl (hypertonic saline) or a combination of a concentrated saline solution with a colloidal compound (hypertonic colloidal solution). a) Hypertonic NaCl solutions (HS) Numerous experiments carried out during the past several years have proven the ability of HS infusions to rapidly increase the plasma volume by two to three times the infused volume [Maningas et al., 1989; Onarheim et al., 1989; Wade et al., 1990; Onarheim, 1995; Carlson et al., 1996]. However, although HS solutions have proven to be effective in acute traumatic situations involving experimental animals, they can cause hypernatremia, hyperosmolarity, and hypokalemia, conditions that may impose limitations on their use as resuscitants [Shires et al., 1995]. Similar to RS infusions, the plasma retention of HS resuscitants is short-lived. Chapter 6: Model validation based on animal studies 99 Type of Solution Total Concentration (mEq/l) Cations (mEq/l) Anions (mEq/l) Colloid (g/1) ISOTONIC (300 ± 3 0 ) Normal Saline (NS) Lactated -Ringer's (L-RS) 308 274 Na + : 154 Na + : 130 K + : 4 Ca + 2 : 3 CI": 154 CI": 109 H C 0 3 ' : 28 HYPERTONIC SALINE ( > 500 ) Sodium Chloride 3% Sodium Chloride 7.5% 1026 -2400 Na + : 513 Na + : 1200 CI": 513 CI": 1200 HYPERTONIC COLLOIDAL SOLUTION ( > 500 ) 6% Dextran 70 in 7.5 NaCl -2400 Na + : 1200 CI": 1200 Dex 7 0: 0.06 Table 6-1: Examples of compositions for isotonic and hypertonic fluids used most commonly for experimental infusion studies. b) Hypertonic/hyperoncotic solutions The marked hemodynamic improvement obtained by the addition of colloids such as dextrans in a hypertonic saline solution has been demonstrated in numerous experimental studies. The results are very promising for a variety of traumatic conditions such as hemorrhage [e.g., Sondeen et al., 1990(a)], trauma [e.g., Vassar and Holcroft, 1992] as well as burns [e.g., Kramer, 1986]. An extended literature review of more than 800 studies on this topic can be found at http://www.hvpertonic.utmb.edu/. The hypertonic component (i.e., the high concentration of NaCl) of these solutions has the effect previously described for HS solutions, i.e., of rapid water shifts into the vascular compartment. The hyperoncotic component (i.e., the colloidal substance) that has a relatively good retention in the vascular space, maintains an elevated vascular volume for a longer period of time. As a result of these effects, much lower volumes of hypertonic/hyperoncotic solutions Chapter 6: Model validation based on animal studies 100 are required to attain hemodynamic stability, compared with their RS and HS counterparts [e.g., Smith, 1985; Kramer, 1986]. Despite the numerous studies cited above, the use of hypertonic solutions or their crystalloid counterparts is still controversial [Shires et al., 1995; Tjallefsrud, 1997]. As pointed out in a recent study [Tcllefsrud, 1997], the main difficulty in establishing the optimum fluid for resuscitation and/or the optimal resuscitation protocol lies in a lack of agreement as to what constitutes an ideal physiological outcome from fluid therapy. There are a multitude of interdependent factors that influence the exchange of fluid and solutes between the vascular and interstitial compartments, as well as between the extracellular and cellular spaces. Among these factors, the physicochemical properties of the participating fluid compartments, the properties of the membranes that separate them, and the chemical or electro-chemical gradients governing transport are just a few. The full interaction between these factors is often difficult to assess from experimental studies, which are usually able to investigate only a limited number of measurable variables at a time. However, due to their ability to generate large quantities of additional information, validated mathematical models can serve as useful tools for better understanding the roles played by the many factors implicated in physiological exchange processes. 6.3 Estimation of the Permeability-Surface Area Product and the Reflection Coefficient for the Transcapillary Transport of Small Ions As indicated in Table 4-1 (see Chapter 4), the permeability-surface area product, PS ION, and the reflection coefficient, OION, which control the transport of small ions across the capillary barrier are not well-defined. Although several authors have investigated and suggested acceptable ranges for these two parameters, there is little agreement about their precise values [Carlson et al., 1996; Curry, 1984; Wolf, 1982]. Because earlier M V E S models proposed by this group [Chappie et al., 1993; Xie et al., 1995] were primarily concerned with fluid and protein redistribution, optimized values for PS ION and OION were not determined. Thus, as part of the current study, it was necessary to make use of a portion of the available experimental data to obtain more reliable estimates of these two missing transport parameters. To estimate PS ION and GION and to help validate the present model, its predictions were compared with the experimental results from a previously published report by Wolf [1982]. Wolfs experimental data were selected for this purpose because 1) the data were collected from animals not subject to any type of previous trauma, 2) they include hyperosmotic infusions Chapter 6: Model validation based on animal studies 101 where significant intercompartmental ion transfers occur, and 3) the study reports numerous data that are time-dependent and hence provide a more rigorous test of the model. To minimize the number of parameters to be determined, it is assumed that the same values of PSION and OION apply to all of the ionic species (Na +, K + , C 2 + , CI" or A") that are exchanged between the plasma and interstitial compartments. Thus, the parameters obtained can be considered as average values for all five ions. 6.3.1 Experimental information In Wolfs [1982] study, normovolemic, nephrectomized dogs were infused for 6 minutes with an iso-osmotic saline solution (NS). Following a stabilization period of 45 minutes, the animals were then infused for 3 minutes with a hypertonic saline solution (HS). The resuscitation protocol reported by Wolf and subsequently used as an input to the model, is given in Table 6-2. Plasma volume and plasma osmolarity were measured continuously over a period of 3 hours. According to Wolf, all the infused volumes and times are comparable to commonly used clinical resuscitation protocols, i.e., short infusion times and relatively low infused volumes. Type of Infusion Osmolarity (mOsm/1) Volume Infused (ml/kg) Duration of Infusion (minutes) Normal Saline Solution (NS) 303.2 23.15 6 Hyperosmolar NaCl (HS) 2030 9.25 3 Table 6-2. Resuscitation protocols used by Wolf [1982 ]. 6.3.2 Initial conditions In Chapter 4, where the formulation of the model was first introduced, the initial conditions were given for a 70 kg 'Reference man'. For parameter estimation and for a first validation of the model, the current study relies on data from studies involving different animal species. Throughout this work, in order to make use of as much of the available experimental information as possible, there will be a continuous transition from human to animal Chapter 6: Model validation based on animal studies 102 compartmental values. Detailed calculations exemplifying the scaling procedure used for Wolfs dogs are given in Appendix C. The general scaling procedure will also be briefly mentioned just once in this section with reference to Wolfs study. Thereafter, only tabulated values for the initial compartmental values will be given where appropriate. The initial values for the fluid compartments at the start of each infusion experiment in dogs are given in Table 6 - 3 . The initial whole body plasma and interstitial volumes were specified in W o l f s [ 1 9 8 2 ] report. As was pointed out in the previous chapter, based on studies by Bert and Pinder [ 1 9 8 2 ] , proteins were assumed to be excluded from 2 5 % of the normal interstitial volume. The normal hematocrit values were also reported in these experimental studies, while the corresponding volumes for the R B C compartment were calculated based on hematocrit according to Eq. [4 -1 ] . The tissue cell volume was assumed to constitute about 4 0 % of the total body weight. Table 6 - 3 also summarizes the properties of the lymphatic system and of the capillary membrane which separates the plasma and interstitial compartments. The majority of the parameters characterizing these two exchange pathways were obtained by scaling the values obtained by Xie et al. [ 1 9 9 5 ] for humans to the appropriate animal values. The basis for this scaling was the assumption that the capillary density is about the same for all species and, hence, the area available for exchange is proportional to the volume of the interstitium, such that: Transport parameterANIMAL = VIT,ANIMAL Transport parameter^jj^ [6-1 ] V VIT.HUMAN^ To complete the description, the permeability-surface area product, PSION, used to describe the rate of diffusive ion (i.e., Na + , K + , C 2 + , C f or A") transfer across the capillary membrane, is also shown in Table 6 - 3 . This value was estimated according to the procedures outlined in Section 6 . 3 . 3 . The interstitial compliance relationships were obtained by scaling, on a weight basis, the compliance relationships developed for the human M V E S model by Chappie et al. [ 1 9 9 3 ] to the appropriate dog values. In accordance with our past modeling practice, the compliance relationship was separated into three regions: the 'dehydration segment', the 'intermediate segment' for moderate hydration and the 'overhydration segment' (see Chapter 4 ) . The range of interstitial volume values and the compliance relationships corresponding to each of the above-mentioned segments are as follows: Chapter 6: Model validation based on animal studies 103 For dehydration, V u < 4160.0 ml: Prr= - 0.7 + 3.9 x 10"3 (VIT - 4160.0 ml). For moderate hydration, 4160.0 ml < Vr r< 6240.0 ml: Mathematical interpolation for Prr versus VIT experimental data. For overhydration, Vrr ^ 6240.0 ml: PIT= 1.88 + 2.1 x 10"4 (VIT - 6240.0 ml). Variable Value for Dog Source Body weight, B W 26.0 kg rWolf, 1982] Plasma volume, VPL 49.2 ml/kg TWolf, 19821 Hematocrit, Hct 43.9 % [Wolf, 1982] Red blood cell volume, VRBC 30.8 ml/kg calculated based on Hct Interstitial volume, Vrr 159.2 ml/kg [Wolf, 19821 Excluded interstitial volume, VIT.EX 40.0 ml/kg calculated based on [Bert and Pinder, 1982] Tissue cell volume, VTC 400.0 ml/kg calculated based on 40% B W Fluid filtration coefficient, kp 2.3 ml/kg-mmHg-h calculated based on [Xie etal., 1995] Permeability-surface area product for protein, PS 1.4 ml/kg-h calculated based on [Xie et al., 1995] Permeability-surface area product for small ions, PSION 4200 ml/kg-h see Section 6.3.3 Normal lymph flow rate, JL, NL 1.4 ml/kg-h calculated based on [Xie et al., 1995] Lymph flow sensitivity, LS 0.8 ml/kg-mmHg-h calculated based on [Xie et al., 1995] Table 6-3: Initial steady-state compartmental values for Wolfs [1982] dogs. Chapter 6: Model validation based on animal studies 104 A l l the compartmental variables presented in Table 6-3, are extensive properties (i.e., they depend on the mass of the system). The initial values of various intensive properties are presented in Table 6-4. These are maintained constant for the simulation of all resuscitation protocols (animal or human). The initial hydrostatic and oncotic pressures as well as the protein concentrations shown are those previously presented in Table 4-1. These normal steady-state conditions correspond to a generic tissue compartment and the general circulation. The average reflection coefficients o (for proteins) and OION (for small ions) are, respectively, the value reported by Xie et al. [1993] and that estimated by curve-fitting in Section 6.3.3. A simplified initial partition of sodium, potassium and chloride ions between the interstitial and plasma compartments is also shown in Table 6-4. The considerations, including the Donnan equilibrium, taken into account in determining these concentration values, are the same as those described in Section 4.5.4.4. Similarly, the cellular membrane transport parameters, i.e., pNa, PK and pci, and the contribution of the Na +-K +-pump to the active transport term, R P , are unchanged from the values previously given. A l l the values presented in Tables 6-3 and 6-4 are the steady-state values obtained by running the computer program associated with the model in the absence of any perturbation (i.e., fluid inputs and/or outputs), for a sufficient length of time until none of these values changed beyond a small pre-established error criterion. Hence, the modeled system was at steady state prior to simulating any disturbance due to external fluid and solute infusion. The infusion protocols reported for Wolfs experiments constitute inputs to the model as already summarized in Table 6-2. The right-hand sides of Eqs. [4-6] and [4-7], i.e., the mass balance equations for dextran in plasma and interstitium, were set to zero since they are not applicable to simulating either the work of Wolf [1982] or the other animal studies used in this chapter. As well, the urinary outputs of fluid and small ions were set to zero, and, because of the short-infusion periods studied here, the output terms for the perspiration/respiration rates of fluid and small ions are assumed negligible. Chapter 6: Model validation based on animal studies 105 Variable Plasma Compartment Interstitial Compartment Value Source Value Source Protein concentration 70 g/1 [Kleinman and Lorenz, 1984] 29.8 g/1 Table 4-1 and Section 4.5.4.3 Protein concentration in the available volume 39.7 g/1 Table 4-1 and Section 4.5.4.4 Hydrostatic pressure 11 mmHg [Xie et a l , 1995] -0.7 mmHg [Xie etal., 1995] Colloid osmotic pressure 25.9 mmHg [Xie et al., 1995] 14.7 mmHg [Xie etal., 1995] Albumin reflection coefficient 0.988 [Xie et al., 1995] Small ions reflection coefficient 0.15 estimated in Section 6.3.3 Sodium concentration 139.7 mEq/1 [Onarheim, 1995] 135.0 mEq/1 [Kleinman and Lorenz, 1984] Section 4.5.4.4 Potassium concentration 4.7 mEq/1 [Onarheim, 1995] 4.5 mEq/1 [Kleinman and Lorenz, 1984] Section 4.5.4.4 Chloride concentration 105.7 mEq/1 [Onarheim, 1995] 107.2 mEq/1 • [Kleinman and Lorenz, 1984] Section 4.5.4.4 Table 6-4. Initial steady-state compartmental values common to humans and all experimental animals. 6.3.3 Estimation of parameters In the parameter estimation procedure, pairs of discrete values in the ranges of 1000 < P S I O N / P S < 5000 (where P S is the protein permeability-surface area product) and 0.05 < OION ^ 0.5 were selected as inputs to the simulation program. With all transport parameters now defined, the ordinary differential equations governing the model were integrated from t = 0 to t = 4 h, corresponding to the time course of Wolfs replicated normal saline ( N S ) and hyperosmotic solution ( H S ) infusion experiments. In the simulations, the solution volumes and compositions, given in Table 6-2, were injected into the plasma compartment at a constant rate over the reported infusion periods. Based on the generated results, separate sums-of-squares-of-differences between the predicted and experimental plasma volumes and osmolarities were calculated. To obtain an overall sum-of-squares value for each pair of parameters, the volume and osmolarity sums were first normalized Chapter 6: Model validation based on animal studies 106 with respect to their minimum values over the ranges investigated, weighted by their respective number of data points, and then added together. These combined sum-of-squares values are plotted as a function of P S I O N / P S and OION in Fig. 6-1. Figure 6-1 shows that the sum-of-squares surface is minimal and relatively insensitive to the values of the two parameters over the more constrained ranges of 2000 < P S I O N / P S < 4000 and 0.1 < OION ^ 0.2. Furthermore, there is a clear global minimum at the point P S I O N / P S = 3000 and OION = 0.15. The model predictions, obtained with P S I O N / P S = 3000 and OION = 0.15, and the experimental data for plasma volume and plasma osmolarity reported by Wolf are shown in Figs. 6-2 and 6-3, respectively. Figure 6-2 shows the experimental changes in blood volume measured by Wolf [1982] for an NS infusion followed, about 40 minutes later, by an HS infusion. As can be seen in this figure, plasma volume, V P L , increases in both experiments by about 54% above control immediately after the administration of NS at 303 mOsm/1, followed by a further experimental increase of about 65 or 105% after HS administration at 2030 mOsm/1. If the total infusate volumes had simply remained in the plasma compartment in each case, the expected increases would have been 58% and 23%, respectively. Thus, clearly, the HS resuscitant is causing the recruitment of significant amounts of fluid from other reservoirs such as the interstitium and the plasma and tissue cells. However, the volume expansion is short-lived in both cases; as soon as the infusion ends, fluid begins leaking out of the plasma compartment such that, at least in the case of the HS infusion, a relatively stable plasma volume is reached at about 1 h after fluid administration was terminated. Wolfs plasma osmolarity results are compared to the model predictions in Figure 6-3. Once again the agreement between the model results and the experimental measurements is very good. This is not surprising since both the plasma volume and osmolarity data were used in the estimation of PS ION and CJION. It should be noted that the plasma osmolarity remains near its baseline value during the normal saline infusion, but increases rapidly during the hyperosmotic infusion. The slight decrease in plasma osmolarity that occurs immediately post-NS infusion is due to the difference in composition between the NS infusion and the plasma composition, as discussed earlier in Section 6.2. At the end of the HS infusion, the osmolarity decreases sharply, as indicated by both the predicted and experimental data. Additionally, both experimental data and simulations reach an apparent steady state of about 10% above the control value within 0.5 to 1.0 hour after the infusion stops. Chapter 6: M o d e l validation based on animal studies 107 Figure 6-1. Combined sum-of-squares-of-differences between the model predictions and Wol f s [1982] plasma volume and plasma osmolarity measurements as a function of P S I O N / P S and CTION, where P S is the permeability-surface area for proteins. The optimal solution is obtained for P S I O N / P S = 3000 and O I O N = 0.15. Chapter 6: Model validation based on animal studies 108 Time (h) Figure 6-2. Comparison of the model predictions for plasma volume changes vs. time with experimental data from Wolf [1982]. The solid line represents the model predictions for NS followed by HS infusion while the symbols represent two sets of experimental data points for the same infusion protocol. Chapter 6: Model validation based on animal studies 109 40 bfj s g 30 -3 • £ 20 -] M a O io-3 c3 PH 0 -«:• T—r—-——r --rn—r—T -T ----T -I -T-—- I I I 1 I I I I I I I I I •10 . . . . NSA A O O : C4 HSi l U 3 t i i i i i i i i i i 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time (h) Figure 6-3. Comparison of the model predictions for changes in plasma osmolarity vs. time with experimental data from Wolf [1982]. The dashed line represents the model predictions for NS followed by HS infusion while the symbols represent two sets of experimental data points for the same infusion protocol. Chapter 6: Model validation based on ariimal studies 110 6.4 Model Validation Based on Additional Animal Studies One important aspect of model validation is that the model should provide a good representation of the experimental data used in the estimation of model parameters. This aspect was shown to be satisfied in the previous section. However, a more rigorous validation requires that, with the same set of estimated parameters, the model should be capable of predicting the results of other, independent, experimental studies. Based on the information available in the two additional studies, the data generated in the infusion experiments of Onarheim [1995] on normal rats and Manning and Guyton [1980] on normal dogs were selected for this purpose. These authors report the changes to such variables as plasma volumes, hematocrits, osmolarities, and ion and protein concentrations, as well as interstitial, extracellular and cellular volumes following infusions of either Ringer's (RS) or hyperosmolar (HS) solutions. As part of the validation procedure, the agreement, with respect to both transient and steady-state behavior, between the model predictions and these sets of available experimental data was analyzed. Whenever extensive properties (i.e., those that depend on the amount of a particular variable) are compared within a simulation or between simulations, they are done so on a relative (or percentage) basis. 6.4.1 Experimental information In Onarheim's experiments, normovolemic rats were infused continuously for a period of about 20 minutes with either acetated Ringer's solution (270 mOsm/1) or with hypertonic saline solution (2400 mOsm/1). The infused volumes were selected such that the same amount of sodium was administered in both types of infusion protocols. To allow better assessment of the alterations in interstitial and intracellular fluid volumes following infusion, Onarheim chose to administer larger amounts of fluid over longer infusion times than are commonly used in clinical studies. Onarheim reports initial and final values for plasma volume, total extracellular volume, plasma sodium, potassium and chloride concentrations, hematocrit and plasma osmolarity. These were measured prior to fluid administration and 1 h after the start of the infusion, at which time an apparent steady state had been reached. Initial and post-infusion intracellular volumes for different tissues were calculated as the difference between the measured total tissue fluid volumes (determined by drying) and tissue extracellular fluid volumes (determined using tracers). Chapter 6: Model validation based on animal studies 111 Manning and Guyton performed experiments in which dogs were infused with lactated Ringer's solution (270 mOsm/1) in amounts equivalent to 5%, 10% and 20% of the animals' body weight. The infusion times ranged from 30 to 60 minutes. In this study, the time-changes in blood volume, total extracellular volume, and plasma protein concentration were followed. As was the case in Wolfs experiments, the animals used in these two studies were nephrectomized. Additionally, the fluid and/or solute redistribution between plasma and interstitium after RS or HS infusions is followed for cases where there was no previous trauma. The initial compartmental values as well as the interstitial compliance relationships determined for Onarheim's rat and Manning and Guyton's dog are given in Tables 6-5 and 6-6, respectively. The resuscitation protocols reported in these two studies and used as inputs to the model are presented in Table 6-7. Variable Rat [Onarheim, 1995] Dog [Manning and Guyton, 19801 Body weight, B W (kg) 0.255 20.0 Plasma volume, V P L (ml/kg) 25.8 45.0 Hematocrit, Hct (%) 47.3 32.9 Red blood cell volume, VRBC (ml/kg) 23.1 22.1 Interstitial volume, Vrr (ml/kg) 174.2 296.3 Excluded interstitial volume, VIT,EX (ml/kg) 43.5 74.0 Tissue cell volume, V T C (ml/kg) 400.0 396.0 Fluid filtration coefficient, k F (ml/kg-mmHg-h) 2.5 4.3 Permeability-surface area product for protein, PS (ml/kg-h) 1.52 2.57 Permeability-surface area product for small ions, P S I 0 N (ml/kg-h) 4560.0 7710.0 Normal lymph flow rate, JL,NL (ml/kg-h) 1.57 2.66 Lymph flow sensitivity, LS (ml/kg-mmHg-h) 0.896 1.52 Table 6-5. Initial steady-state compartmental values used in animal validation studies. Chapter 6: Model validation based on animal studies 112 Region of Curve Range of V I T (ml) and corresponding relationship for P u (mmHg) Rat Dog Dehydration V I T < 44.42 ml PIT= - 0.7 + 0.37 (Vrr - 44.42) Vrr < 5926 ml p I T = . 0.7 + 2.78 x 10"3 (V IT - 5926) Moderate Hydration 44.42 ml < V i T < 66.63 ml Mathematical interpolation of Prr versus V I T experimental data 5926 ml < V i T < 8888 ml Mathematical interpolation of PIT versus Vrr experimental data Overhydration V I T > 66.63 ml P I T = 1.88 + 1.98 x 10"2 (V IT - 66.63) Vrr>8888 ml P i T = 1.88 + 1.48 x 10"4 (V IT - 8888) Table 6-6: Interstitial compliance relationships for 'scaled rat' [Onarheim, 1995] and 'scaled dog' [Guyton and Manning, 1980]. Species Type of Infusion Osmolarity (mOsm/1) Volume Infused (ml/kg) Duration of Infusion (minutes) Rat Hyperosmolar NaCl (HS) Acetated Ringer's Solution (RS) 2400 270 10 100 18 18 Dog Lactated Ringer's Solution (RS) 270 50 30 Lactated Ringer's Solution (RS) 270 100 60 Lactated Ringer's Solution (RS) 270 200 40 Table 6-7: Resuscitation protocols used as inputs to the model in the validation studies. Chapter 6: Model validation based on animal studies 113 6.4.2 Comparisons with Onarheim's data Onarheim [1995] measured several system variables before and approximately 1 h after the start of 18-minute infusions with either HS (2400 mOsm/1) or RS (270 mOsm/1). A comparison of the model predicted values of these variables with the experimentally measured data is presented in Table 6-8. In the simulations, the infused volume was added to the plasma compartment at a constant rate over the infusion period, in accordance with Onarheim's experimental protocol. Variable Measured Control Acetated Ringer's Solution 270 mOsm/1 Hypertonic Solution 2400 mOsm/1 Experiment Model Prediction Experiment Model Prediction Plasma Volume (% Change) 0.0 ±3.1 37.9 ±4.5 35.0 27.5 ±4.2 26.7 Hematocrit (%) 47.4 ± 0.3 38.4 ±0.5 40.2 40.1 ±0.7 38.8 Plasma Osmolarity (mOsm/1) 303.3 ±2.0 294.1 ±0.5 296.7 333.0 ±2.3 337.8 [NalpL (mmol/1) 139.7 ± 0.8 139.5 ±0.4 140.3 154.8 ±0.3 158.6 [ K ] P L (mmol/1) 4.7 ±0.1 4.5 ±0.1 4.6 6.4 ±0.1 4.5 [CTJPL (mmol/1) 105.7 +0.8 108.0 ±0.4 108.7 126.6 ±0.7 130.8 Extracellular Volume (% Change) 0.0 +0.3 61.5 ±3.0 46.9 28.0 ±3.5 25.4 Tissue Cell Volume (% Change) 0.0 +0.9 -0.8 ± 2.3 1.5 -8.4 ± 1.9 - 10.1 Table 6-8. Comparison between model predictions and Onarheim's [1995] experimental data at 60 minutes after fluid infusion. Chapter 6: Model validation based on animal studies 114 One hour after beginning the fluid administration, when an apparent steady state was reported, the model predicts that the HS infusion produces a 27% increase in plasma volume above the control value, while the almost order-of-magnitude larger RS infusion caused only a 35% increase. These results are in good agreement with the experimental measurements, which indicate 28 ± 4% and 38 ± 5% plasma volume expansions, respectively. The simulations also show that, for both types of solutions infused, the plasma volume reaches its maximum expansion at the end of the infusion period. At this intermediate time, the plasma volume almost doubled with the simulated infusion of 10 ml/kg HS and increased only to about three times the initial volume for the 100 ml/kg RS infusion. If all of the infused fluid had simply been retained by the plasma compartment, the expected expansions would have been about 40% and 390%, respectively. Thus, as was the case in Wolfs [1982] experiments, Onarheim's HS infusion engenders additional fluid recruitment far beyond the volume administered and, following the infusion period for both types of resuscitants, fluid rapidly leaks from the plasma compartment such that an apparent steady state is reached in less than 1 h. According to Table 6-8, blood hematocrit falls from 47% to a new steady-state value of about 40% for both the RS and HS infusions. For the RS case, Hct is predicted to drop continuously up to the end of the infusion period because of an elevated plasma volume and a relatively constant RBC volume. However, as the plasma volume returns toward its baseline value (see above), Hct is partially restored to a final predicted value of about 40%. This end-point, obtained by simulation, agrees closely with the measured value of 38% reported by Onarheim. Good agreement between the measured Hct value and the model prediction was also obtained for the HS infusion. In this case, the final plasma volume is lower, but the R B C volume is also reduced, leading to a similar steady-state Hct. Table 6-8 indicates that, at 1 h post-infusion, the plasma osmolarity increases for the HS infusion but decreases slightly for the RS infusion. For the HS case, there is a model predicted peak increase in plasma osmolarity corresponding to about 40 mOsm/1 occurring at the end of the infusion period. This is largely due to the osmotic contribution of the ionic species infused. After the HS infusion was terminated, the plasma osmolarity decreases rapidly toward a new steady state. At steady state, the model predicted a value of 338 mOsm/1 which compares well with the 333 mOsm/1 reported by Onarheim. For the RS infusion protocol, a maximum decrease of approximately 6 mOsm/1 in plasma osmolarity is predicted during the infusion period. According to the simulations, the minimum in plasma osmolarity corresponds to the peak in Chapter 6: Model validation based on animal studies 115 plasma volume and occurs at the end of the infusion. This decrease is due partly to dilution of plasma by the large volume of the slightly hypotonic RS infused (270 mOsm/1 in RS vs. 303 mOsm/1 in plasma). After the infusion stops, as the fluid and small ions are re-equilibrated between plasma and interstitium, a slight increase in overall plasma osmolarity toward the baseline value takes place. At about half an hour after the infusion has ceased, the model predicts a steady-state osmolarity of 297 mOsm/1 which is in good agreement with the experimental value of 294 mOsm/1. The plasma osmolarity trends predicted for Onarheim's essentially isosmotic and hyperosmotic infusions are very similar to those measured by Wolf [1982] (see Fig. 6-3). For the RS infusion, the predicted plasma electrolyte concentrations for sodium, potassium and chloride ions at 1 hour post-infusion are in good agreement with the values reported in Onarheim's experimental study. In fact, for this case, neither the experiments nor the model were expected to give concentration values that are significantly changed from normal for any of the ionic species. For the HS infusion on the other hand, the model predicts that the plasma sodium increases from a baseline concentration of 140 mmol/1 up to about 160 mmol/1 immediately post infusion. This increase is a direct result of the high sodium content of the infused HS. Once the concentration difference between the plasma and interstitium begins to dissipate, the concentration of plasma sodium decreases gradually. When steady state is achieved, the model indicates a plasma sodium concentration of 159 mmol/1 which compares well to the value of 155 mmol/1 reported experimentally. A similar predicted trend is observed for the chloride ion, whose concentration increases from an initial steady-state value of 106 mmol/1 up to 135 mmol/1 immediately post-infusion, followed by a return to 131 mmol/1 when the system reaches its new steady state. This value is in good agreement with the measured value of 127 mmol/1. For K + , the model predicts that, following HS infusion, the concentration falls from the baseline value of 4.7 mmol/1 to 4.4 mmol/1 immediately after the infusion stops, and returns to a value of 4.5 mmol/1 at the new steady state. As can be seen in Table 4, there is a disagreement between the computed and Onarheim's reported potassium ion concentration for reasons we cannot fully explain. Other series of experimental studies by the same the author, with HS/Dextran infusions following hemorrhage or burn, show that serum potassium often tends to decrease [Onarheim et al., 1989; Onarheim et al., 1990]. Onarheim [1995] also measured the changes in total extracellular fluid volume following RS and HS infusions. The simulation of the HS case, in close agreement with the experimental data, predicts a 25% increase in E C V 1 h post-infusion. However, for the RS infusion, the model Chapter 6: Model validation based on animal studies 116 predictions underestimate the experimental values by about 15% (i.e., the model-predicted change is 47% compared to the experimental value of 62%). If, as one would anticipate for an essentially iso-osmotic infusion, little fluid exchange with the cells occurs, then the increase in extracellular volume would have been 50%, which is about 3% greater than the change predicted by the model. Onarheim [1995] also found that, for both the RS and HS infusions, E C V values measured at the end of the infusion period were very close to the final steady-state values obtained at 1 h. The model predicted a similar extracellular volume behavior; a linear increase during the infusion period followed by an almost instantaneous attainment of steady state immediately after the infusion was completed. In contrast, the predicted plasma and interstitial volumes continued to change significantly following the infusion, only approaching their new steady states about 1 h after the infusion began. These results were expected for the isotonic RS infusion where little transport to or from the cells takes place, but they also offer experimental justification for the assumption of a very rapid shift of fluid between the intra- and extracellular spaces in the case of the HS infusion. Finally, Onarheim [1995] also provided information about the change in the cellular fluid volume of skeletal muscle following RS and HS infusions. For the RS case, the model predicts a small increase in the volume of the tissue cell compartment because the administered fluid is slightly hypotonic; the measurements corroborate this prediction, indicating virtually no change in the tissue cell volume. According to the simulations for the HS case, the tissue cells shrink gradually throughout the HS infusion, reaching an apparent steady state immediately after the infusion stops. At the final steady state, the model predicts about a 10% decrease in the volume of these cells, a value which is in good agreement with the measured 8.4% decrease reported by Onarheim. 6.4.3 Comparisons with Manning and Guyton's data Figure 6-4 shows a comparison between the simulated and experimental blood volume changes following RS infusions, according to the experimental protocol reported by Manning and Guyton [1980]. The three panels (A, B , and C) in the figure correspond to Lactated Ringer's solution infusions equivalent to 5% body weight (BW), 10% B W and 20% BW, respectively. The trends and the magnitudes of the predicted fluid volumes are in good agreement with the experimental data, except near the end of the infusion period for the 20% BW case. Chapter 6: Model validation based on animal studies 117 RS Infusion RS Infusion & 100 6 8 0 60 H 5 40 o % 20 o 3 o-|l T — i — i — | — i — i — i — i — i — | — i i i • i | i 1 1 r i i i I i i i—i i i i i > • • i • 0 1 2 B 1 RS Infusion 100 1 ! | A ' ' / i\. / i\ / i \ / ; \ / r 6 80 H 60 -\ 3 40H o > 20 H o S o - i—i—i—i—I—i—r v. -i—i—I—i—i—i—i—i—|—r 0 1 I I I I I I I I I ' I I 1 1 2 3 4 T i m e (h) Figure 6-4. Comparison of the model predictions for blood volume change vs. time with experimental data from Manning and Guyton [1980]. The lines represent the predictions of the model while the symbols indicate the experimental results for A: 5% of body weight (BW); B: 10% of B W and C: 20% of B W RS infusions. Chapter 6: Model validation based on animal studies 118 Note, however, that the maximum measured blood expansion is essentially the same as for the 10% B W case, even though the 20% B W infusion involved the addition of twice as much fluid over a 33% shorter period. The following values were reported by the authors at 5 hours after the infusion terminated: about a 14% increase for the 5% B W infusion, a 23% increase for the 10% B W infusion and a 25% for the 20% B W infusion. The corresponding simulated percentage increases at 5 hours post-infusion are 19%, 24% and 27% for the 5%, 10% and 20% B W infusions, respectively. The simulations and the measurements both indicate an increase in blood volume during RS infusion followed by stabilization at an elevated steady-state volume within approximately 1 hour post-infusion. Manning and Guyton [1980] provided information about plasma protein concentration changes after RS infusions. Figure 6-5 shows their experimental results along with our model predictions. For all three levels of infusion, there is always a decrease in plasma protein concentration as a result of plasma expansion. According to the simulations, the maximum decrease is achieved immediately at the end of the infusion period. After the infusion is completed, the plasma protein concentrations increase slowly toward the control value. However, for the 5 hours post-infusion period studied, the protein concentrations remain well below the initial value. At 5 hours post-infusion, the experimental results indicate 14%, 20% and 28% decreases from the control level for the 5%, 10% and 20% BW infusions, respectively. In good agreement with the experiments, the model-predicted values at this time are 16%, 21% and 27% below control, respectively, for the corresponding infusions. Manning and Guyton [1980] also measured total extracellular volume changes as part of their infusion experiments with dogs. According to Fig. 6-6, the experiments and simulations both show that, during RS infusions, the total extracellular volume increases proportionally with the fluid infused, for the duration of the infusion period. However, once the infusion is terminated, a new elevated steady-state E C V level is almost immediately reached. A similar behavior was observed by Onarheim in his RS and HS infusion experiments. The post-infusion predictions are in reasonable agreement with the measured E C V values for all three of Manning and Guyton's experiments. Chapter 6: Model validation based on animal studies 119 RS Infusion •70 -1—i—I—i—i—i—|—i—i—i—i—i—J—i—i—i—i—i—|—i—i—i—i—i—|—i—i—i—i—i—|—i—i—i—i—i—J 0 1 2 3 4 5 Time (h) Figure 6-5. Comparison of the model predictions for plasma protein concentration change vs. time with experimental data from Manning and Guyton [1980]. The lines represent the predictions of the model while the symbols indicate the experimental results for the A: 5% of BW; B: 10% of B W and C: 20% of B W RS infusions. Chapter 6: Model validation based on animal studies 120 RS Infusion 100 ^ 80 A 60 I 60 u £ 40 > U 20 w I I I UJ I ' l l T — i — I — i — i — i — i — r ~i—i—i—r A 0 "~|—i—i X. i—i—|—i—i—i—i—i—|—i—i—i—i—i—|—i—r 0 1 2 1 1 I 1 4 RS Infusion 100 80 -60 60 U -& 40 -_ > -u 20 -m w -i—i—i—I—i—i—i—i—i—I—i—i—i—i—i—I—i—i—i—i—r B A mm • 100 ^ 80 H 60 6 60 ^ 40 H > u w 2 0 H 0 - 1 — I — I — I — I — I — I — I — I — I — p 0 1 R S I i i fusion - i—i—i—|—i—i—i—i—r 2 3 T — i — i — | — i — i — r 4 i i IJJ—' i ' ' T—i—i—i—i—I—r T — i — i — r !* • * « • • * c J Figure 6-6. Comparison of the model predictions for extracellular volume change vs. time with experimental data from Manning and Guyton [1980]. The lines represent the predictions of the model while the symbols indicate the experimental results for the A: 5% of BW; B: 10% of BW and C: 20% of BW RS infusions. Chapter 6: Model validation based on animal studies 121 6.4.4 Summary of validation study The results presented above suggest that the model's predictions are well supported by the experimental results for hyperosmotic as well as iso-osmotic or essentially iso-osmotic solutions. The comparisons for fluid volumes and solute concentrations proved to be satisfactory for the first 1-5 hours post-infusion, and no additional volume compensatory mechanisms needed to be accounted for in the mathematical formulation. The experimental data provided by Wolf [1982] for NS and HS infusions, as well as by Manning and Guyton [1980] for RS infusions, provided information about the transient phase of plasma expansion in dogs. As shown in Figs. 6-2 and 6-4, the time-course of the predicted plasma volume changes was in good agreement with the experimental data. As well, the model provided excellent predictions of the steady-state plasma volume changes in rats measured by Onarheim for RS and HS infusions. These results provide confidence that this improved four-compartment model has the ability to provide reliable predictions of plasma expansion, even for cases where ionic resuscitants cause significant cellular volume changes. 6.5 Implications of the Model One of the main advantages of the validated model is its ability to predict simultaneously a large number of both experimentally accessible and difficult to measure (or experimentally inaccessible) variables. This wealth of information can help contribute to a better understanding of the phenomena occurring at both the microvascular and cellular levels following resuscitation. Based on the predictions of the model (some of which were used in the previous section for model validation), the compartmental fluid and solute changes which occur after infusion of either HS or RS/NS will be discussed in more detail in this section. 6.5.1 Iso-osmotic solution (NS or RS) infusions Infusion with iso-osmotic solutions represents the less complex of the two cases studied using the model. Since no osmotic disturbance is present at the boundary between the extra- and intracellular compartments, the cells play an essentially passive role and the vascular and interstitial compartments are the only ones which undergo significant changes in volumes and protein concentrations. Chapter 6: Model validation based on animal studies 122 Infusion period: Corresponding to Wolfs experiments, the computed plasma volumes increase during NS infusion to about 50% above the control value, a peak which is slightly less than that predicted i f all the infused fluid is retained in plasma. Based on his transient experimental data, Wolf reported a 91% vascular retention for NS when plasma expansion is at its peak. According to the simulations, which indicate a similar value of about 88%, the incomplete retention of infused fluid is due to a continuous fluid shift from the vascular to the interstitial compartment caused by an increased hydrostatic pressure and decreased colloid osmotic pressure in the vasculature. As illustrated in Fig. 6-7 for this experiment, the model predicts an increase of about 16 mmHg in the plasma hydrostatic pressure (Fig. 6-7, panel A) and a decrease of about 8 mmHg in the vascular colloid osmotic pressure (Fig. 6-7, panel B). The model also predicts (although it is not shown here) that, during NS infusion, the amount of protein in plasma remains near the control value; however, the protein concentration decreases due to plasma dilution. Manning and Guyton [1980] similarly reported that, while the plasma protein content remained essentially constant, a decrease in plasma protein concentration occurred for each of the RS infusions they studied (see Fig. 6-5). Simulations of their experiments also demonstrate that increased hydrostatic pressures and decreased colloid osmotic pressures in plasma favor fluid filtration across the capillary barrier toward the interstitium. As illustrated in Fig. 6-8, which shows model simulations for the relative transcapillary fluid flow corresponding to Manning and Guyton's experimental conditions, fluid is shifted from the vascular to the interstitial compartment at an increasing rate throughout the infusion period. The model predicts that the fluid continues to be filtered into the interstitium more rapidly than it can be removed by the lymphatics. As shown in Fig. 6-8, the relative transcapillary flow increases much more dramatically than does the lymphatic flow. The net result is the accumulation of fluid in the interstitium. The compliance-engendered increase in the interstitial hydrostatic pressure causes slightly higher fluid and protein return rates through the lymphatics. Post-infusion period: In agreement with the experimental results, the simulations for all the cases considered show a near equilibration of fluid, proteins and small solutes between the plasma and interstitium within about 0.5 to 1 hour post-infusion. When final steady states are achieved, the hydrostatic pressures are increased and the colloid osmotic pressures are decreased in both the vascular and the interstitial compartments. No experimental information was available regarding the interstitial hydrostatic and colloid osmotic pressure changes. Chapter 6: Model validation based on animal studies 123 ( g j T u n i i ) ajrissajd oijmsojpAH ( S H 1 1 1 1 1 1 ) QJnssajj OIJOUISQ piono3 Chapter 6: Model validation based on animal studies Pi i i i | i i i | i i i | i i " | " • ' | 1 1 1 I i I i i i i i i i i i i i CO I ( 1f/If) A V O I J iiduiXq 3Ap.-p-jj "\~ i i I i i i i i i i I i I 1 1 1 I i i i I i I 1 1 1 I i i i I i i i I i i 1 o o o o o o o o o o o o o C N < — i o O N O o r - ~ ^ o < O T i - m c N ' - H ( ^ ^ f / 1 ^ ) M O T j p i t i T j AjK^idBOSireix aAirep-^j <+-< <U CD o J : > <—' +-> CD 00 CD u . 5 c *o 8 "5b & h o o J2 m — i .fl O C l-H e g C D CD £ o t" e t - "53 fc <u „ H <u £ .2 O eg c CD e o o - .s a <*> & c 3 £ o -2 j f l O o ps! o vi CD c c rt ;rs rt rt r « 1/5 C sP O 0 s O CN ^ -rt O H * ^3 fl O - ~ rt T3 00 § .£ o« 1 t s § CD •s <» so O J ^ « ^3 — oo •a vO o CD 3 E pq 2 * a3 o a- £ CD Chapter 6: Model validation based on animal studies 125 The simulations for the NS infusion case, based on the experiments by Wolf [1982] and shown in Fig. 6-7 (C and D), suggest, however, an increase of about 1.5 mmHg in interstitial hydrostatic pressure and an approximately 2 mmHg decrease in the interstitial colloid osmotic pressure for the post-infusion steady-state period. The increased interstitial hydrostatic and decreased colloid osmotic pressures are, as mentioned before, a direct consequence of fluid being shifted from the vasculature. From Fig. 6-5, which showed comparisons between computed and experimental plasma protein concentrations for Manning and Guyton's [1980] experiments, it can be observed that, following a decrease during infusion, the protein concentration returns toward its baseline value. However, both the simulations and experiments demonstrate that the protein concentrations remain 15-30% below control for the entire post-infusion period, depending on the particular experimental protocol. In all cases, the net effect of RS infusion is to produce hemodilution. In accordance with experimental values reported by Manning and Guyton [1980], the model output shows that, at steady state, only about 10-20% of the Ringer's solution infused is retained in the vascular compartment. Similarly, based on data from Onarheim [1995], a 10% isosmotic fluid retention within the vasculature was calculated from his measurements and also predicted by the model. 6.5.2 Hyperosmotic solution (HS) infusions For the HS infusions, according to the model predictions for both infusion conditions described; namely, short-term [Wolf, 1982], and long-term with large infusion volume [Onarheim, 1995], the plasma volume increases up to a maximum, which coincides with the end of the infusion period and accounts for far more than the volume of fluid infused. This condition is followed by a decrease in plasma volume with a significantly reduced retention of infusate at steady state. In accordance with the transient experimental data reported by Wolf [1982] and presented in Fig. 6-2, the model predicts an increase in plasma volume of about 70% at its peak, the equivalent of about three times the infused volume. Furthermore, for Onarheim's [1995] experiments involving larger volumes and longer infusion times, the simulations show a plasma volume elevation of more than two times the infused volume at the end of the infusion period. At this time, plasma osmolarity is increased by approximately 40 mOsm/1 as compared to controls. Chapter 6: Model validation based on animal studies 126 Infusion period: During HS infusion, at least for the two experimental conditions simulated in this work, the model suggests the following events that account for fluid and solute shifts between compartments. The highly hyperosmotic state created in plasma by the HS infusion has an initial impact on the red blood cells as well as on the transcapillary transport of small ions and fluid. The red blood cells reduce their fluid volume in order to achieve an osmotic balance with plasma. As shown in Fig. 6-9 (solid line), the simulations corresponding to Onarheim's [1995] experimental protocol indicate that the volume of these cells is rapidly reduced by about 15% from control during the infusion. Additionally, the infused 7.5% NaCl solution creates large ionic concentration differences across the capillary wall. Measurements of plasma N a + and Cl" concentrations during the transient infusion period were not reported in any of the experimental studies. For these experiments, however, the simulation predicts increases of about 20 mmol/1 and about 30 mmol/1 in the plasma sodium and chloride concentrations, respectively. The hemodynamic effects of HS infusion are short-lived since the small ions leak rapidly through the highly permeable capillary wall. The model predicts that these large differences in ion concentration begin to dissipate quickly (within seconds after the beginning of infusion) by both an enhanced transcapillary transport of small ions toward the interstitium by diffusion and a fluid absorption from the interstitium due to the increased plasma osmolarity. 0.0 0.5 1.0 1.5 2.0 Time (h) Figure 6-9. Model predictions for changes in cell volume vs. time following HS infusion. The simulations shown are for RBC (solid line) and tissue cells (dashed line), and are performed according to the experimental protocol described by Onarheim [1995]. Chapter 6: Model validation based on animal studies 127 As a consequence of both of these effects, the interstitial osmotic pressure increases and in turn has an osmotic impact on the tissue cells. As shown by the model predictions in Fig. 6-9 (dashed line), fluid is also fairly rapidly mobilized from the tissue cells into the interstitial reservoir. The simulations demonstrate that the shift of fluid from the tissue cells takes place throughout the infusion period, independent of whether the infusion is short-term [Wolf, 1982] or long-term [Onarheim, 1995]. The fluid mobilized from these cells causes an increase in the interstitial volume throughout this period. However, the interstitial fluid participates also in elevating the plasma volume, mainly by absorption to plasma (driven by increased transcapillary osmotic pressure differences) and, to a lesser extent, through an enhanced lymphatic transport (due to elevation of the interstitial hydrostatic pressure). Simulations of the transcapillary fluid flow, for both Onarheim's [1995] HS experiment, presented in Fig. 6-10(A), and Wolfs [1980] infusions, Fig. 6-10(B), show a continuous absorption of fluid into plasma from the interstitium during HS infusion. According to the simulations, the fluid absorption to plasma lasts as long as the HS solution is infused, regardless of the duration of the infusion. The progressive increase in the interstitial volume (not shown) paralleled by a continuous fluid absorption into the vasculature suggest that the two- to three-fold plasma expansion relative to the infused volume, noted at the end of the infusion period, is due to fluid recruited from the cellular compartments and mainly from the tissue cells. Post-infusion period: Both the experimental results and the model predictions demonstrate that the redistribution of fluid and small ions between plasma, interstitium and cells is essentially complete within 30 minutes post-infusion. When the post-infusion steady state is achieved, the plasma volume remains elevated compared to its pre-infusion control value. The experimental results for the two HS studies of Wolf [1982] and Onarheim [1995], as corroborated by the simulations, show an approximate 30% elevation in plasma volume (see Fig. 6-2 and Table 6-8) at steady state. The data and the simulations corresponding to Onarheim's [1995] experiments indicate that, when steady state is reached, the final plasma expansion is about 70%o of the infused fluid volume. Clearly, for this case, the HS resuscitant results in a greater expansion of the vasculature than does an essentially isosmotic infusion which, as was mentioned earlier for Onarheim's RS experiment, results in only 10% of the infused fluid being retained in the plasma compartment. Chapter 6: Model validation based on animal studies 128 H H t—J - 1 V o • 1—( o H > 3 ID 100 50 H 0 - 1 -50 H -100 1 1 1 1 1 1 1 1 . 1 1 1 1 1 1 1 1 1 1 1 . 1 1 1 1 1 Filtration i i i | i i i i | i i i i ^ A ; Absorption l l l k HS I 1 k -i i 0.0 0.5 1.0 Time (h) 1.5 o I-H H > 3 100 0 -200 -300 0.0 1.0 2.0 Time (h) 3.0 2.0 1 1 1 1 1 1 r v • '. Filtration ^ ^ ^ ^ ^ ^ I i . i . I . . . . B - Absorption - . . . " f t . . HS4 h i : i — j — i — i — i — i — j — i — i — i — i — 4.0 Figure 6-10. Model predictions for the relative transcapillary fluid flow vs. time after HS infusion. The simulations are performed according to the experimental protocols described by Onarheim [1995] (panel A) and Wolf [1982] (panel B). Chapter 6: Model validation based on animal studies 129 As shown in Fig. 6-9 for Onarheim's experimental protocol, immediately after the HS infusion was terminated, the R B C volume increases slightly up to about 10% below control where it remains for the entire post-infusion period. These changes reflect directly the osmotic conditions in the plasma. At the final steady state, the model also predicts a 10% volume decrease for the tissue cells. Based on his experimental measurements (see Table 6-8), Onarheim reported an approximately 10% decrease in muscle cell volume. The experimental results as well as the interpretation of the model predictions presented above indicate that, following an HS infusion, the increase in the steady-state plasma volume is caused by the recruitment of cellular fluid as well as by the infused fluid. 6.6 Conclusions from Validation Studies In a first validation, for all the cases explored, the model predictions compared well with experimental results for compartmental fluid volumes as well as small ion and protein contents. The experimental data for validation were chosen so that comparisons with several variables were possible. An important aspect of the overall exchange process, which the model helps to clarify, is whether the increase in plasma volume that results from an HS infusion is due to fluid transfer from the cellular compartments alone, or i f changes in interstitial volume are involved as well. The model simulations suggest that, during infusion with hyperosmolar NaCl solutions, all of the fluid that contributes to an increase in plasma volume is recruited from the infusate and from the cellular compartments, mainly from the tissue cells. The simulations also predict a continuous absorption of fluid into plasma throughout the HS infusion period, independent of its duration and the volume of the infusate. In agreement with other studies [Cervera and Moss, 1974; Mazonni et al., 1988; Onarheim et al., 1990; Wolf, 1982], this model predicts that once the infusion period is over, the elevated plasma volume rapidly declines such that, at the new steady state, plasma retention of the infusate is poor. The model simulations indicate that the transitory nature of plasma elevation is strictly dependent on the duration of the infusion; i.e., a longer infusion period results in a longer duration of fluid absorption into the vasculature. Among other factors, this might be one of the reasons why Onarheim et al. [1989] reported hemodynamic improvements when lower resuscitation rates and increased durations of HS infusion were used. However, more experimental data and analyses are required in order to confirm these results. Chapter 7: Model validation based on human studies 130 Chapter 7: MODEL VALIDATION BASED ON HUMAN STUDEES 7.1 Introduction A further logical step in the validation of a model which yields the dynamics of mass exchange in the body would be to compare the model predictions with the experimental results obtained from studies of fluid infusion in humans. To this author's knowledge, an attempt to predict the changes associated with fluid balances in humans following different types of infusions by using a compartmental model such as the one described in Chapter 4 has never been carried out before. Particularly for humans, where the experimental and clinical studies involve a limited number of invasive measurements, a validated human model would be a welcome addition to help understand the complex interactions that exist between the various fluid compartments of the body. The proposed purpose of the present chapter is, therefore, to provide a validation of the human model. The experiments selected for this validation exercise involved administration of two types of infusions: 1) isotonic normal saline (NS) and 2) hypertonic saline with Dextran 70 (HSD). As discussed in the previous chapter, despite the large amount of information already published about HSD infusions, they are still under intensive clinical investigation in order to meet regulatory approval for use in humans. The HSD simulations represent the greater challenge of the two since the hypertonic solution causes a significant alteration of the cellular compartments, while the added dextrans contribute, along with proteins, to the plasma and interstitial oncotic pressures. Additionally, the implications of administering HSD infusions were examined for both normal and 10% hemorrhaged subjects. As was the case in the previous chapter, data from infusions on otherwise healthy subjects (including those with a 10% hemorrhage) were chosen for validation. As an added feature, compared with the validation described in Chapter 6, the model must also be updated to account for the effects of external losses of fluid and solutes through urine. The rest of the chapter is organized as follows. First, the minor changes required by the model in order to describe losses of blood are discussed. Then, a comparison is presented between the simulated results and measured data for two sets of human infusion experiments. The experimental data include the changes with time of plasma and interstitial fluid volumes, plasma and interstitial colloid osmotic pressures, hematocrit, plasma solute concentrations (both Chapter 7: Model validation based on human studies 131 sodium and total osmolarity) and transcapillary flow rates. This validation is then followed by a brief discussion concerning the implications and applicability of the model. Specifically, the model is used to investigate mechanisms associated with the redistribution and transport of fluid and solutes administered following a mild blood removal, and to speculate on the relationship between the timing and amount of fluid resuscitation and subsequent blood volume expansion in an attempt to optimize fluid resuscitation protocols. 7.2 Model Validation for Human Studies 7.2.1 Model description The only modifications required for the human model described in Chapter 4 are those related to the removal of whole blood from the vascular compartment. Whole blood loss implies the loss of red blood cells with their cellular contents of small ions, as well as the loss of plasma volume together with its proteins and small ion constituents. This scenario is represented schematically in Figure 7-1, where the symbol H E M signifies the rates of all the lost materials described above. The model relies on the time-dependent values taken by the hematocrit (calculated from Eq. [4-1]) to establish, based on the known rate of whole blood loss, both the rates of plasma volume loss and red blood cell loss. A l l the terms that describe these rates were accounted for in the mass balance equations for the plasma and red blood cell compartments (see Eqs. [4-2], [4-4],[4-5], [4-8] and [4-9] for the plasma compartment and Eqs. [4-10], [4-11] and [4-12] for the RBC compartment). 7.2.2 Available experimental data The validation of the model for human subjects is carried out by comparing the model predictions with measured data provided by two independent experimental studies. The first study, by Watenpaugh et al. [1992], reports measured and calculated isotonic plasma volume expansion data for normal humans, following short infusions of normal saline (NS). The data were collected as function of time up to three hours post-infusion. The large number of measured physiological variables such as plasma volume, plasma oncotic pressure, renal output, together with calculated values for the net whole-body transcapillary fluid transport and interstitial fluid volume, make this study an ideal choice for model validation. Chapter 7: Model validation based on human studies 132 RES M k J k J k ION.PER k PERM ISL INTERSTITIAL CELLS TC M IQNJC INTERSTITIAL FLUID J,T 'IT M ION.IT \M IQN.L M RES PLASMA ION.RES RBC RBC ~M ION.RBC t HEM 'UR M. ION.UR Figure 7-1. A schematic diagram depicting the compartments and flows which comprise the model. The four compartments include the plasma and red blood cells as well as the interstitial fluid and its associated cells. J represents the flow of fluid, R represents the flow of Dextran, M represents the transport of small solutes and H E M refers to whole blood loss via hemorrhage. The subscripts are as follows: IT represents the interstitial fluid, TC represents the tissue cells, R B C denotes the red blood cells, L corresponds to the lymph, U R represents urine, ION represents all the small solutes individually and RES corresponds to resuscitation fluids. PER and ISL corresponds to perspiration and solute free insensible losses, respectively. Chapter 7: Model validation based on human studies 133 The second study, by Tollofsrud et al. [1997], is more complex and involves fluid administration of hypertonic saline containing 6% Dextran 70 (HSD) to normal human subjects. This latter study reports the effects of small bolus HSD infusions on fluid and solute distribution among the different body compartments (i.e., plasma, interstitium and cells). A total of five women (averaging 62 kg) and four men (averaging 74 kg) in resting position, all healthy individuals with no history of renal dysfunction, were subjected to fluid administration. Furthermore, two distinct cases were investigated by Tollofsrud et al. [1997]: one in which the HSD infusion was administered to normovolemic humans, and another where the same volume of HSD solution was administered to 10% (by volume) hemorrhaged volunteers. The resuscitation protocol performed by Watenpaugh et al. [1992] is presented in Table 7-1. Stated briefly, this protocol consists of an intravenous infusion of about 30 ml/kg of NS solution; specifically a rate of 100 ml/min was administered for 24 minutes. The average inputs to and outputs from the subjects are shown in Table 7-1; i.e., inputs in the form of infused fluid and solutes, and outputs such as urine as well as other losses such as sensible (perspiration) and insensible (e.g., respiration) losses. A l l of these quantities were measured as part of the experimental data except for the sensible and insensible fluid losses that were estimated by Watenpaugh et al. to be around 46 ml/h*. Of this, for lack of better data, the value of about 0.03 ml/(kg-h) reported in the literature [e.g., Altman and Dittmer, 1971; Krieger and Sherrard, 1991] for a normal 70 kg human, was considered to be a reasonable approximation for the perspiration rate. The normal concentrations of small ions eliminated via perspiration are also given in Table 7-1. The distribution of solutes between the compartments under study was recalculated to account for compartmental small ion values that are more descriptive of humans [Tollofsrud et al., 1997] and is presented in Appendix D. In order to accommodate the different body weights of the volunteers involved in these two experimental studies, the coefficients and properties of the model fluid compartments were scaled to the average weight and height of the individuals in each test. The weight and height of the volunteers in Watenpaugh's study averaged 76.9 kg and 177 cm, respectively. * Note that the sensible/insensible losses estimated by these authors are the same on a per kg basis as those of a normal human, i.e., ~ 0.6 ml/(kg-h). 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The experimental protocols for the two separate but related studies carried out by Tollofsrud et al. [1997]