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A mathematical model of interstitial transport and microvascular exchange Taylor, David G. 1990

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A M A T H E M A T I C A L M O D E L O F I N T E R S T I T I A L T R A N S P O R T A N D M I C R O V A S C U L A R E X C H A N G E By David G. Taylor B. A. Sc. (Engineering Science) University of Toronto A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S C H E M I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A March 1990 © David G. Taylor, 1990 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 refer-ence 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. Chemical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: z o / a f / S a Abstract A generalized mathematical model is developed to describe the transport of fluid and plasma proteins or other macromolecules within the interstitium. To account for the effects of plasma protein exclusion and interstitial swelling, the interstitium is treated as a multiphase deformable porous medium. Fluid flow is assumed proportional to the gradient in fluid chemical potential and therefore depends not only on the local hydrostatic pressure but also on the local plasma protein concentrations through appropriate colloid osmotic pressure relationships. Plasma pro-tein transport is assumed to occur by restricted convection, molecular diffusion, and convective dispersion. A simplified version of the model is used to investigate microvascular exchange of fluid and a single 'aggregate' plasma protein species in mesenteric tissue. The interstitium is approximated by a rigid, rectangular, porous slab displaying two fluid pathways, only one of which is available to plasma proteins. The model is first used to explore the effects the interstitial plasma protein diffusivity, the tissue hydraulic conductivity, the restricted convection of plasma proteins, and the mesothelial transport characteristics have on the steady-state distribution and transport of plasma proteins and flow of fluid in the tissue. The simulations predict significant convective plasma protein transport and complex fluid flow patterns within the interstitium. These flow patterns can produce local regions of high fluid and plasma protein exchange along the mesothelium which might be erroneously identified as 'leaky sites'. Further, the model predicts significant inter-stitial osmotic gradients in some instances, suggesting that the Darcy expression invoked in a number of previous models appearing in the literature, in which fluid flow is assumed to be driven by hydrostatic pressure gradients alone, may be inadequate. Subsequent transient simulations of hypoproteinemia within the model tissue indicate that ii the interstitial plasma protein content decreases following this upset. The simulations there-fore support (qualitatively, at least) clinical observations of hypoproteinemia. Simulations of venous congestion, however, demonstrate that changes in the interstitial plasma protein con-tent following this upset depends, in part, on the relative sieving properties of the filtering and draining vessels. For example, when the reflection coefficients of these two sets of boundaries are similar, the interstitial plasma protein content increases with time due to an increased plasma protein exchange rate across the filtering boundaries and sieving of interstitial plasma proteins at the draining boundaries. (This eifect is supported by the clinical observation that interstitial plasma protein content in liver increases during venous congestion.) As the reflection coefficient of the draining boundaries decreases relative to that of the filtering boundaries, there is a net loss of plasma proteins from the interstitium, resvdting in a decrease in the total interstitial plasma protein content over time (i.e., the familiar 'plasma protein washout'). Further, the model predicts increased fluid transfer from the interstitium to the peritoneum during venous congestion, supporting the clinical observation of ascites. Finally, the model is used to study the effects of interstitial plasma protein convection and diffusion, plasma protein exclusion, and the capillary transport properties on the transit times of two macromolecular tracers representative of albumin and 7-globulin within a hypothetical, one-dimensional tissue. As was expected, the transit times of each of the tracers through the model tissue varied inversely with the degree of convective transport. Increasing the interstitial diffusivity of the albumin tracer also led to a moderate decrease in the transit time for that tracer. The capillary wall transport properties, meanwhile, had only a marginal effect on the transit time for the range of capillary permeabilities and reflection coefficients considered. However, these properties (and, in particular, the reflection coefficient) had a more pronounced effect on the ultimate steady-state concentration of the tracer in the outlet stream. It was the interstitial distribution volume of a given tracer that had the greatest impact on the time required for the outlet tracer concentration to reach 50 % of its steady-state value. This was attributed to the increased filling times associated with larger interstitial distribution 111 volumes. These findings suggest that the :gel chromatographic effect' observed in some tissues could possibly be explained on the basis of varying distribution volumes. Table of Contents Abstract ii List of Tables x List of Figures xii Acknowledgements xvi Quotation xvii Dedication xviii 1 Introduction 1 2 Physiological Overview of the Microvascular Exchange System 6 2.1 The Capillary Wall and Basement Membrane 8 2.1.1 General Description 8 2.1.2 Transport Pathways Across the Capillary Barrier 10 2.1.3 Quantifying Transport Across the Capillary Wall 12 2.2 The Interstitium 14 2.2.1 Structure and Composition 14 2.2.2 Volume Exclusion within the Interstitium 19 2.2.3 Characterizing Interstitial Swelling 21 2.3 The Lymphatic System 23 2.3.1 General Description 23 2.3.2 The Terminal Lymphatic Vessels 25 v 3 Formulation of the General M odel of Interstitial Transport 28 3.1 A Continuum Representation of the Interstitium 28 3.1.1 Plasma Protein Partitioning within the Interstitium: Exclusion 30 3.1.2 Tissue Strain, Volumetric Dilation, and Tissue Compliance 37 3.1.3 A Constitutive Relationship between Fluid Flow and Fluid Chemical Po-tential 42 3.1.4 Protein Transport Mechanisms within the Interstitium 44 3.2 Mass Balance Equations for Sohd, Fluid, and Solute Species 48 3.2.1 Material Balance on the Solid and Immobile Fluid Phases 49 3.2.2 Material Balance on the Fluid Phase 50 3.2.3 Material Balance on Protein Species k 51 3.2.4 Summary of Governing Equations 51 3.3 Concluding Remarks 54 4 Steady-State Exchange i n Mesenteric Tissue 56 4.1 Defining the System 57 4.2 Case Studies 68 4.3 Numerical Procedure 70 4.4 Results and Discussion 74 4.4.1 A Specific Case of Interstitial Transport 74 4.4.2 Fluid Exchange across the Boundaries of the Interstitium 81 4.4.3 Plasma Protein Exchange across the Interstitial Boundaries 83 4.4.4 Interstitial Plasma Protein Convection, Diffusion, and Distribution . . . . 85 4.4.5 Comparison of Model Predictions to Experimental Data 90 4.5 Concluding Remarks 94 5 Transient Exchange in Mesentery Following a Systemic Upset 97 5.1 The Governing Equations 97 vi 5.2 Case Studies 100 5.3 Numerical Procedures 102 5.4 Results and Discussion 103 5.4.1 Transient Exchange hi Sustained Hypoproteinemia 103 5.4.2 Transient Exchange During Sustained Venous Congestion 127 5.4.3 Clinical Observations of Hypoproteinemia and Venous Congestion . . . . 145 5.5 Concluding Remarks 147 6 A Prel iminary Study of Tracer Transport through the Interstitium 149 6.1 Introduction 149 6.2 Denning the System 150 6.3 Case Studies 153 6.4 Numerical Procedures 155 6.5 Results and Discussion 155 6.5.1 The Effect of Capillary Boundary Conditions on Tracer Transit Time . . 155 6.5.2 The Effect of Tracer Distribution Volume on Globulin Transit Times . . . 159 6.5.3 The Effect of Interstitial Diffusivity on Albumin Transit Times 161 6.6 Concluding Remarks 165 7 Summary of Conclusions 166 8 Recommendations 171 Nomenclature 173 References 179 A One Dimensional Approximat ion to the Two Dimensional M o d e l Mesenteryl91 A . l Introduction 191 A.2 One-Dimensional Approximations to the Two-Dimensional Equations 192 vii A.2.1 Conservation of Fluid Mass 192 A.2.2 Conservation of Interstitial Plasma Proteins 193 A.2.3 Boundary Conditions 196 A.2.4 Non-Dimensional Form of the Equations 196 A.2.5 Influence of the 1-D Approximation on Characterizing the Mesothelial Transport Properties 198 A.3 Case Studies 200 A.4 Numerical Procedures 201 A.5 Results and Discussion 201 A.5.1 General Comparison of the 1-Dimensional and 2-Dimensional Simulations201 A. 5.2 Effect of H c f f on Exchange in the 1-D Simulations 208 A. 6 Concluding Remarks 214 B A n O v e r v i e w o f t h e C o m b i n e d F i n i t e E l e m e n t - F i n i t e D i f f e r e n c e T e c h n i q u e 2 1 7 B. l Introduction 217 B.2 Solving for the Spatial Variation in Concentration Using Finite Elements . . . . 218 B.3 Solving for the Temporal Variation in Concentration Using Finite Differences . . 221 B.4 Guidelines for Selecting Grid and Time Step Sizes 222 B. 5 Validation of Simulator 224 B. 5.1 Validation of the Fluid Mass Balance Equation and Starling Boundary Conditions in a One-Dimensional Mesentery 225 B.5.2 Validation of the Transient Solute Mass Balance Equation in a One-Dimensional Mesentery 227 C A P r e l i r n i n a r y S t u d y o f I n t e r s t i t i a l P l a s m a P r o t e i n D i s p e r s i o n 2 2 9 C. l Introduction 229 C2 Defining the System 229 C.3 Case Studies 231 V1U C.4 Numerical Procedure 231 C.5 Results and Discussion 232 C. 6 Concluding Remarks 236 D Program Listings 238 D. l Parameter List for Steady-State and Transient Simulators 238 D.2 Two Dimensional Simulator: MES8N0D.F0R 240 D.3 One-Dimensional Simulator: MESDISP.FOR 274 D*4 One-Dimensional Transient Simulator: TRANS.FOR 293 D.5 Two Protein Steady-State Simulator: MESDISP2.FOR 313 D.6 Two Protein Transient Simulator: TRANS2P.FOR 336 ix List of Tables 3.1 Summary of Model Equations 53 4.1 Parameter Values for Steady-State Analysis in Mesentery 71 4.2 Dimensionless Parameter Values for Steady-State Simulations 72 4.3 Boundary Fluid Fluxes for Steady-State Simulations 81 4.4 Boundary Plasma Protein Fluxes for Steady-State Simulations 84 4.5 Mean Interstitial Plasma Protein Concentrations for Steady-State Simulations . . 86 4.6 Ratio of Plasma Protein Convection to Diffusion for Steady-State Simulations . . 89 5.1 Fluid Chemical Potential Differences in Hypoproteinemia 105 5.2 Transient Fluid Fluxes Following H\"poproteinemia Assuming a Permeable Mesothe-lium 113 5.3 Transient Plasma Protein Fluxes Following Hypoproteinemia Assuming a Per-meable Mesothelium 118 5.4 Fluid Chemical Potential Differences in Venous Congestion 128 5.5 Transient Fluid Fluxes Following Venous Congestion Assuming a Permeable Mesothelium 136 5.6 Transient Plasma Protein Fluxes Following Venous Congestion Assuming a Per-meable Mesothelium 139 6.1 Effect of Capillary Wall Boundary Conditions on Steady-State Tracer Outlet Concentration 156 6.2 Effect of Capillary Wall Sieving on Tracer Transit Times 159 6.3 Effect of Interstitial Distribution Volume on Tracer Transit Times 160 6.4 Effect of Interstitial Tracer Diffusivity on Tracer Transit Times 161 x A . l Comparison of Predicted Boundary Fluid Fluxes for the One-Dimensional and Two-Dimensional Mesenteric Models 205 A.2 Comparison of Predicted Boundary Plasma Protein Fluxes for the One-Dimensional and Two-Dimensional Mesenteric Models 206 A.3 Comparison of Predicted Ratios of Plasma Protein Convection to Diffusion at the Permeable Boundaries for the One-Dimensional and Two-Dimensional Mesen-teric Models 207 A.4 Effect of Hcff on Average Fluxes at Permeable Boundaries for the One-Dimensional Tissue Model 213 C . l Effect of Mechanical Dispersion on Fluid Exchange within Mesentery Assuming Q Equal 9.117 235 C.2 Effect of Mechanical Dispersion on Plasma Protein Exchange within Mesentery Assuming a Equal 9.117 236 C.3 Effect of Mechanical Dispersion on the Ratio of Plasma Protein Convection to Dispersion within Mesentery Assuming a Equal 9.117 237 List of Figures 2.1 The Microvascular Network 7 2.2 Vesicular Channels within the Capillary 9 2.3 Transport Pathways across the Capillary Wall 11 2.4 A Conceptual Picture of the Interstitium • 15 2.5 Hierarchy of Collagen Structures 16 2.6 Compliance Relationships For a Typical Tissue 22 2.7 The Lymphatic Network Within the Bat Wing 24 2.8 The Structure of the Terminal Lymphatic 26 3.1 Elementary Volume of Interstitium before and after Spatial Averaging 32 3.2 A n Elementary Volume of Interstitium Before and After Increased Hydration . . 36 3.3 Linear Deformation of a Differential Segment 38 3.4 Differential Expansion of a Volume Element 39 4.1 Schematic Diagram of a Cross-Sectional Slice of Mesentery 58 4.2 Schematic Diagram of Tissue Segment Studied 59 4.3 Elementary Volume of Mesenteric Interstitium 62 4.4 Permeable Boundary Adjacent the Interstitium in Mesentery 65 4.5 Surface Plots of Interstitial Plasma Protein Distribution in Mesentery 76 4.6 Flux Patterns within Mesenteric Interstitium 78 4.7 Mesothelial Fluxes as a Function of Position 80 4.8 Comparison of Mesothelial Fluxes as a Function of Interstitial Plasma Protein Diffusion 82 4.9 Steady-State Interstitial Plasma Protein Concentration Profiles 88 xii 4.10 Experimental Measurement of Interstitial Plasma Protein Distribution in Mesen-tery 92 4.11 Comparison of Model Predictions to Experimentally Determined Concentration Profile in Mesentery 93 5.1 Transient Fluid Flux Following Hypoproteinemia and Assuming an Impermeable Mesothelium 106 5.2 Transient Plasma Protein Flux Following Hypoproteinemia and Assuming an Impermeable Mesothelium 110 5.3 Transient Plasma Protein Distributions Following Hypoproteinemia and Assum-ing an Impermeable Mesothelium I l l 5.4 Mesothelial Fluid Flux Distribution Following Hypoproteinemia Assuming Mesothe-lial Transport Properties Equal to Those of the Arteriolar Capillary 114 5.5 Transient Fluid Fluxes Following Hypoproteinemia Assuming Mesothelial Trans-port Properties Equal to Those of the Arteriolar Capillary and £ Equals 1 . . . . 115 5.6 Interstitial Fluid Chemical Potential Distribution Following Hypoproteinemia Assuming Mesothelial Transport Properties Equal to the Arteriolar Capillary and £ Equals 1 116 5.7 Transient Plasma Protein Fluxes Following Hypoproteinemia Assuming Mesothe-lial Transport Properties Equal to Those of the Arteriolar Capillary and £ Equals 1 120 5.8 Transient Plasma Protein Distributions Following Hypoproteinemia and Assum-ing Mesothelial Transport Properties Equal to Those of the Arteriolar Capillary • 121 5.9 Mesothelial Fluid Flux Distribution Following Hypoproteinemia Assuming a Highly Permeable Mesothelium and a £ of 1 124 5.10 Transient Plasma Protein Distributions Following Hypoproteinemia and Assum-ing a Highly Permeable Mesothelium 125 xiii 5.11 Transient Fluid Flux Following Venous Congestion and Assuming an Imperme-able Mesothelium 129 5.12 Transient Plasma Protein Flux Following Venous Congestion and Assuming an Impermeable Mesothelium 131 5.13 Transient Plasma Protein Distributions Following Venous Congestion and As-suming an Impermeable Mesothelium 132 5.14 Transient Fluid Fluxes Following Venous Congestion Assuming Mesothelial Trans-port Properties Equal to Those of the Arteriolar Capillary and £ Equals 1 . . . . 135 5.15 Transient Plasma Protein Fluxes Following Venous Congestion Assuming Mesothe-lial Transport Properties Equal to Those of the Arteriolar Capillary and £ Equals 1 137 5.16 Transient Plasma Protein Distributions Following Venous Congestion and As-suming Mesothelial Transport Properties Equal to Those of the Arteriolar Cap-illary 138 5.17 Transient Fluid Fluxes Following Venous Congestion Assuming a Highly Perme-able Mesothehum and £ Equals 1 141 5.18 Transient Plasma Protein Fluxes Following Venous Congestion Assuming a Highly Permeable Mesothehum and £ Equals 1 143 5.19 Transient Plasma Protein Distributions Following Venous Congestion and As-suming a Highly Permeable Mesothehum 144 6.1 Effect of Capillary Wall Sieving on Tracer Breakthrough Curves 158 6.2 Effect of Tracer Distribution Volume on Globulin Breakthrough Curves 163 6.3 Effect of Interstitial Diffusivity on Albumin Breakthrough Curves 164 A . l Schematic Diagram of a One-Dimensional Element of Mesentery 192 A.2 Schematic Diagram of a One-Dimensional Element of Mesentery 194 xiv A.3 Steady-State Interstitial Plasma Protein Concentration Profiles for the One-Dimensional Model of Mesentery 202 A.4 Comparison of Mesothelial Flux Distributions for the One-Dimensional and the Two-Dimensional Model of the Mesentery 204 A.5 Effect of H e f f on Fluid and Plasma Protein Flux Distributions Across the Mesothe-lium 210 A.6 Effect of H e f f on Fluid and Plasma Protein Flux Distributions Across the Mesothe-lium for the One-Dimensional Model and £ Equal 0.0 211 A.7 Comparison of Fluid and Plasma Protein Flux Distributions Across the Mesothe-lium for the Two-Dimensional Model and the One-Dimensional Model Assuming H c f f Equals 1.5 x 10~3 cm 212 A.8 Effect of H c f f on Interstitial Plasma Protein Distribution 215 C . l Effect of Mechanical Dispersion on Plasma Protein Distribution within Mesentery Assuming a Equal 0.9117 233 C.2 Effect of Mechanical Dispersion on Plasma Protein Distribution within Mesentery Assuming a Equal 9.117 234 xv Acknowledgements I would like to express my sincere gratitude to the following individuals: • my advisors, Drs. J .L. Bert and B.D. Bowen, for their direction, enthusiasm, and under-standing; • the other members of my committee, Drs. K . L . Pinder, R. Pearce, D. Brooks, J . Piret, D. Thompson, and L . Smith, for their helpful suggestions throughout this work; • Dr. R. Reed of the University of Bergen, Norway, whose clinical expertise is much appre-ciated; • Dr. M. Olsen. for his advice on some of the numerical procedures used in this dissertation; and • Dr. Clive Brereton, Dr. Frank Laytner, and Mr. Clive Chappie, for their friendship and support. This work was supported by a U .B .C. Research and Development Grant, the B.C. Health Care Research Foundation, and the Natural Sciences and Engineering Research Council of Canada. xvi I will praise thee; for I am fearfully and wonderfully made: marvellous are thy works, and that my soul knoweth right well. My substance was not hid from thee, when I was made in secret, and curiously wrought in the lowest parts of the earth. Thine eyes did see my substance, yet being unperfect; and in thy book all my members were written, which in continuance were fashioned, when as yet there were none of them. Ps. 139: 14-16. xvii This thesis is dedicated to my parents, Ron and Vi, my wife, Charmavne. and my children, Joseph and Kara, whom I love. xviii Chapter 1 Introduction The systemic blood circulation consists of a complex network of vessels that form a closed loop, passing through the various body tissues before completing the circuit. Blood, driven by the pumping action of the heart, travels through a set of small, permeable blood vessels where it exchanges fluid and solutes, including the plasma proteins, with the surrounding tissue. Fluid and solutes are drained from the tissue spaces by an additional circulatory system, called the lymphatics. This exchange is essential both for providing nutrients to the tissue cells, and for removing metabolic wastes from the cells' environment. Further, the exchange of fluid and plasma proteins between the blood, the tissue space and the lymph plays an essential part in balancing fluid within the body. The various physiological elements involved in the exchange of materials within tissues constitute the microvascular exchange system. A disturbance to the system, be it from an extrinsic source (such as a burn or hemorrhage) or an intrinsic one (such as venous congestion or hypoproteinemia), compromises the health and well-being of the individual. For example, following a burn, large quantities of fluid may shift from the blood stream to the tissues. The resultant loss of blood volume can be life threatening. A fundamental understanding of the forces and mechanisms governing exchange is therefore of interest to physiologists and clinicians alike. During exchange, fluid and plasma proteins encounter three principle resistances: the cap-illary wall and basement membrane, the tissue space (i.e., the interstitium), and the lymphatic wall. These are the major barriers encountered during the transfer of materials from the blood stream to the lymphatic circulation. 1 Chapter 1. Introduction 2 To describe the mechanisms governing mass exchange within the microvascular exchange system and, ultimately, to predict its response to physiological upsets, the transport character-istics of each of the resistances must be known. Due to the complexity of the system, and as a complement to experimental studies, mathematical models have been developed to describe microvascular exchange. Much of the effort has been directed to modelling the transport prop-erties of the capillary wall (see, for example, [30, 31, 58, 71, 75, 81] ). However, over the years researchers have identified the interstitium as another important component of the microvas-cular exchange system. General models of this system must therefore include mathematical descriptions of the interstitium and its physicochemical properties. Two basic modelling approaches have been adopted. In the first of these, the microvascular exchange system is reduced to a set of subsystems, or compartments. Material is exchanged be-tween compartments according to the driving forces present (such as differences in fluid chemical potential or solute concentration between compartments) and the transport properties of the interveiung boundary. Each compartment is assumed to be homogeneous; i.e., spatial hetero-geneities in the material properties of that part of the system represented by the compartment are not accounted for. Furthermore, the compartment is assumed to be well-mixed, so that mcoming material is instantaneously dispersed throughout its entire volume. Therefore, the solute concentrations, fluid pressures, and fluid volume associated with a given compartment represent average quantities. Because of the well-mixed assumption invoked in compartmental models, the driving forces for mass exchange between compartments will, in general, differ from the local driving forces found in the real system. This limits the model's ability to simulate the real system, particularly under transient conditions. In addition, compartmental models tell us nothing about mass transport within an individual compartment and its effect on the overall behavior of the system. However, the assumption of a well-mixed, homogeneous compartment simplifies the modelling problem immensely, because it reduces the number of parameters needed to characterize the system (since the transport properties of the compartment itself are neglected), and because Chapter 1. Introduction 3 it simplifies the mathematical description of the system. Hence, complex phenomena, such as tissue swelling, can be included fairly easily in these models. For these reasons compartmental models are frequently used to simulate whole organ and body fluid and plasma protein exchange under both normal and pathological states [108, 18, 14, 3. 70]. Recent advances in microfluorometry, electron microscopy and digital image analysis now permit much more detailed experimental studies of interstitial fluid and protein transport than were previously possible [61, 40. 115], including measurements of interstitial plasma protein gradients. Mathematical models of interstitial transport are therefore required to interpret this expanding body of experimental data. The requisite model must include mathematical descrip-tions of the physicochemical properties of the interstitium, such as plasma protein exclusion and interstitial swelling characteristics, which impact on fluid and protein transport. It must also be able to predict possible variations in the distribution of fluid pressure and protein within the interstitial space [40, 61, 110!. Compartmental models are incapable of this. Such detailed descriptions are only possible with a distributed (i.e., spatially varying) model of interstitial transport. Unlike compartmental models, the distributed models of the microvascular exchange system do not assume that the various body compartments are well-mixed so that, in principle at least, these models more closely describe the real system. Distributed models can therefore be used to investigate the influence of mass transport within a given compartment (such as the blood or the interstitial space) on microvascular exchange. In addition, since the distributed models eliminate the artificial dispersion caused by the well-mixed assumption, they better describe transient processes. The advantages associated with the distributed models are not without their costs. First, these models require far more detailed information about the structure, transport properties and spatial distribution of the various compartments. This leads naturally to a larger number of system parameters which need to be quantified, such as the interstitial hydraulic conduc-tivity, plasma protein effective diffusivity, and capillary vessel diameter. More often than not, Chapter 1. Introduction 4 many of these quantities must be estimated due to a lack of experimental data. In such in-stances it is necessary to conduct numerical experiments to determine the sensitivity of the model's predictions to the values assumed for the estimated parameters. Given the degree of uncertainty associated with these estimates, the results from distributed models are more often qualitative than quantitative. Despite these limitations, distributed models provide a powerful tool for investigating the mechanisms governing interstitial transport and their influence on microvascular exchange. A number of distributed models have already been proposed to describe fluid and/or protein transport within the interstitium. These models vary both in detail and in complexity. Blake and Gross [22] and Fleischman et al. [36] investigated fluid exchange within idealized tissues consisting of ordered arrays of capillaries. In both cases the interstitial space was treated as an isotropic, homogeneous, rigid porous medium. In addition; interstitial fluid flow was described by a form of Darcy's Law in which the authors assumed that the local fluid flux is proportional to the local gradient in hydrostatic pressure. Hence both models neglect the influence of osmotic pressure gradients on local fluid movement. Furthermore, neither model considers protein transport within the interstitium. Several investigators have addressed protein transport through the interstitial space. For example, Baxter et al. [7] assumed that protein transport occurs strictly by diffusion. Convec-tive contributions were not accounted for. Fry [43] considered both convection and diffusion in his model of interstitial transport of multiple protein species. However, Fry's model requires prior knowledge of the fluid velocities throughout the interstitial space. Furthermore, it makes no attempt to describe the effect of interstitial swelling on protein transport. Salathe and Venkataraman [87| presented equations to describe both fluid and protein trans-port within the interstitium. Again, fluid flow was assumed proportional to the gradient in hydrostatic pressure. The equation of protein transport included both convective and diffusive terms. However, their model does not distinguish between those regions of the interstitium which are accessible to protein and those from which protein is excluded. Hence, their model Chapter 1. Introduction 5 neglects the intrinsic heterogeneities within the interstitium resulting from plasma protein ex-clusion. Furthermore, the model is limited to steady-state conditions. It is therefore incapable of predicting the time-dependent response of the interstitial fluid and protein distributions to a variety of systemic perturbations. Each of the models cited above provides insights into various aspects of interstitial transport. However, in each case the mathematical model is limited in scope. The objective of the present work, therefore, is to develop a more general mathematical model which describes the combined effects of interstitial swelling and plasma protein exclusion on the transient re-distribution of fluid and any number of macromolecular species within the interstitium. Local fluid flow is related to the gradient in total fluid chemical potential rather than hydrostatic pressure alone. Thus fluid movement is linked to gradients in solute concentration through associated colloid osmotic pressure gradients. Protein transport occurs by convective, dispersive and diffusive mechanisms, thereby providing further linkage between fluid and solute behavior. As a consequence, the equations governing fluid and protein movement within the deforming interstitium must always be solved as a coupled set rather than as the independent equations often assumed in previous analyses. This dissertation is divided into seven remaining chapters. Chapter 2 provides an overview of the physiology of the microvascular exchange system. In Chapter 3, the general model of interstitial transport is developed. Chapter 4 applies a simplified version of the general model to investigate the mechanisms governing the steady-state exchange of fluid and macromolecules within mesenteric tissue. In Chapter 5, the analysis is extended to transient conditions and considers the response of the model system to two specific systemic perturbations. Chapter 6 adds a further dimension to the problem by investigating the simultaneous transport of multiple plasma protein species through the interstitium. Finally, Chapters 7 and 8 summarize the findhigs and ramifications of the dissertation and recommend several additional studies. C h a p t e r 2 P h y s i o l o g i c a l O v e r v i e w o f t h e M i c r o v a s c u l a r E x c h a n g e S y s t e m Fluid and various solute species contained within blood are transported to the body tissues and organs via a complex network of vessels fonning the systemic blood circulation. Upon entering a specific organ, blood passes through a system of small, permeable blood vessels that constitute the microcirculation (see Figure (2.1)). It is here that nutrients and metabolic wastes exchange between the blood and the tissues cells. In addition, fluid and various macromolecules (in particular, the plasma proteins) are transported across the walls of the exchange vessels to enter the surrounding tissue space called the interstitium. The blood capillaries are the principal vessels responsible for exchange between the blood and the interstitium. However, the blood vessels supplying the capillaries, namely the arterioles. and those which drain the capillary bed, i.e., the venules, are also known to participate in the exchange process [82]. The exchange vessels are of minute dimensions; capillary diameters, for example, average 6 /J.m in humans [46]. In addition to the blood vasculature, the body contains another circulatory network, called the lymphatic system, that drains fluid and solutes from the interstitial space. The lymphatic vessels return material to the systemic circulation, emptying into the venous portion of the latter network in the vicinity of the heart [46]. The exchange vessels of the blood vasculature (namely, the arterioles, the capillaries and the venules), the interstitium, and the tissue drainage system (such as the teiminal lymphatic vessels) constitute the microvascular exchange system. Based on this anatomical definition, the microvascular exchange system can be viewed as a series of resistances that fluid and solutes encounter in their journey from blood to lymph. These resistances may be loosely defined 6 Chapter 2. Physiological Overview of the Microvascular Exchange System Figure 2.1: The network of permeable vessels constituting the microcirculation. Blood enters via the arteriolar vessel (A) . A portion of this is drawn into the terminal arteriole (TA) , passes through the network of capillaries (C), is taken up by the terminal venule ( T V ) , and returned to the venule (V) . During this time fluid and solutes, including plasma proteins, leak from the blood to the surrounding tissue spaces [109]. Chapter 2. Physiological Overview of the Microvascular Exchange System 8 as the capillary wall and basement membrane, the interstitium, and the wall of the terminal lymphatic vessel. A discussion of each of these three components follows. 2.1 The Capillary Wall and Basement Membrane 2.1.1 General Description The capillary wall is composed of a single layer of flattened endothelial cells that rest on a specialized region of the interstitial matrix call the basement membrane or basal lamina [90, 13]. The latter structure consists largely of specialized forms of collagen that are not to be found elsewhere within the interstitial matrix [13] (see Section 2.2.1 for a further discussion of collagen). The basal lamina carries a net negative charge. It is believed to both provide mechanical support to the endothelial cells and to act as an additional transport barrier [112]. Together, the capillary wall and basal lamina act as a semi-permeable membrane that separates the blood and the interstitial compartments. Fluid and solutes selectively pass from the blood to the interstitium, driven by the local differences in the hydrostatic pressures, colloid osmotic pressures, and solute concentrations between the two compartments. The endothelial cell consists of the aqueous cytoplasm of the cell interior surrounded by a plasma membrane, the latter being comprised largely of lipids and protein. Within the cytoplasm are small spherical bodies 60 to 80 nm in diameter, called plasmalemmal vesicles [90J. These appear open on the lurninal (blood) and interstitial surfaces of the cell and as free bodies within the cytoplasm [112]. The vesicles axe thought to play a role in the transfer of macromolecules across the endothelial barrier. Several mechanisms have been suggested, including the shuttling of material from the luminal surface to the interstitial side by individual vesicles. It is also postulated that several vesicles may fuse to form temporary water channels across the width of the cell (see Figure (2.2)). Evidence suggests that vesicular uptake of macromolecules is selective [112]. For example, vesicles found in the microvessels of adipose tissue will take up native ferritin, but not native albumin, although the latter is smaller. The outer surface of the endothelial cells are covered with delicate, negatively charged fibers, Chapter 2. Physiological Overview of riie Microvascular Exchange System 9 Figure 2.2: This figure depicts vesicular transport pathways within a cross-section of an endothelial cell resting on the basement membrane below. Proposed vesicular transport mechanisms include the formation of temporary fluid channels due to fusion of several vesicles bridging the endothelial cell (modified after [103]). thought to be glycosaminoglycans, that form a coat 10 to 20 nm thick [90]. This felt-like cover also lines the inner surface of the vesicles. It is thought to serve as ait additional diffusion barrier, repelling like-charged particles such as the red blood cells. Adjacent endothelial cells meet at intercellular clefts that are typically 10 to 20 nm wide [44]. In all tissues except the brain, the intercellular clefts and plasmalemmal vesicles provide the major transport pathways for water and macromolecules [83]. However, certain portions of the clefts may be sealed due to contacting of apposing cells. In the case of the capillaries of the brain, the contacting cells fuse, eliminating the cleft altogether. Such seals prohibit the transport of larger molecules through intercellular junctions, confining exchange along this pathway to water, salts, and other small molecules [32]. Individual capillary vessels fall into one of three classifications, depending on the structural characteristics of their endothelia: namely continuous, fenestrated, or discontinuous capillaries. Continuous vessels are common to the microvascular beds of the lung, the nervous system. Chapter 2. Physiological Overview o f the Microvascular Exchange System 10 skeletal muscle, and skin, among others [90]. As the name implies, the endothelia of these vessels form a continuous layer 0.2 to 0.3 /xm thick, interrupted only by the intercellular clefts. The basement membrane is likewise continuous. Fenestrated vessels are characterized by the presence of disk shaped regions, typically 60 to 80 nm in diameter, located on the vessel wall. These regions, called fenestrae, are due to an attenuation of the endothelial cell to a thickness of 6 to 8 nm [90]. The attenuated cellular matter forms a diaphragm, the structure of which differs from the rest of the cell membrane in that it is thought to be composed largely of hydrophilic elements [112]. In some cases, such as the glomerular capillaries of the kidney, the fenestrae lack diaphragms altogether. The basement membrane of fenestrated vessels is continuous. The enhanced permeability of these vessels to plasma proteins suggests that the fenestrae provide a major pathway for the transport of macromolecules across the capillary wall [84]. Fenestrated vessels are found within the micro vasculature of the pancreas, the endocrine glands, and the gastrointestinal tract. Discontinuous vessels, also called sinusoids, are identified by large gaps in the endothelial layer and basement membrane. Fenestrae hundreds of nm in diameter may also be present [90]. While their structure would suggest that discontinuous vessels are highly permeable to various plasma proteins, lymph composition from tissues containing these vessels indicates that sieving of certain plasma protein species occurs even here [93]. 2.1.2 Transport Pathways Across the Capillary Barrier Several transport pathways have been identified for passage of fluid and various solute species across the capillary wall. These are summarized below [84]: 1. through the cell itself which includes two layers of cell membrane and the intervening cytoplasm; 2. within the endothelial cell membrane by lateral diffusion through intercellular junctions or lipid phase vesicular channels; Chapter 2. Physiological Overview of the Microvascular Exchange System 11 3. through interendothelial junctions in the aqueous extracellular phase (these pathways con-sist both of highly restrictive channels that are virtually impermeable to plasma proteins and less restrictive channels that permit exchange of these macromolecules); 4. via endothelial cell fenestrae; and 5. by vesicular transport, which includes shuttling of material within individual vesicles (i.e., transcytosis) and the fusion of several vesicles to form temporary fluid filled channels across the cell. These pathways are illustrated in Figure (2.3). (I) (2) (3aab)(5a) (5b) capillary •( (4a) (4b) Figure 2.3: This figure illustrates a cross-sectional slice of the capillary wall. Trans-port pathways across the capillary wall include direct routes across the cell (1), through the cell membrane (2), via intercellular pathways (3), across fenestrae (4), and via vesicular mechanisms (5a, 5b) (modified after [31]). Transport directly through the cell matter is limited to the diffusion of water and small lipid soluble molecules. Convective transport of fluid along this path is negligible [84]. In all likelihood virtually all respiratory gases are exchanged directly tlirough the cell. In addition, substantial amounts of fatty acids and other lipids cross the capillary wall. However, these Chapter 2. Physiological Overview of the Microvascular Exchange System 12 substances cannot penetrate the ceD cytoplasm and are therefore limited to transport through the cell membrane and lipid vesicles. Finally, the lipid insoluble materials, including small ions and the plasma proteins, are limited to the paracellular pathways (i.e., across open fenestrae, in fluid filled vesicles and across aqueous vesicular channels, and via interendothelial junctions). The permeability of these substances decreases with increasing molecular size, suggesting that these pathways display sieving characteristics [84]. Charge may also play a role in detemiining solute permeability. For example, the capillaries of the brain are more permeable to transferrin than to albumin, although transferrin is a larger molecule [83]. Transferrin, however, carries a smaller net negative charge. 2.1.3 Quantifying Transport Across the Capillary W a l l i We have seen that the capillary wall offers several different routes for the transport of material. While attempts have been made to delineate between these different pathways (see, for example, [83]), transcapillary exchange is typically quantified using expressions analogous to those for porous membranes. These describe mass exchange rates in terms of both the principal driving forces present and lumped parameters that characterize transcapillary resistance. Fluid is driven across the capillary wall by differences in the effective fluid chemical potential from one side of the barrier to the other. This driving force can be resolved into two principal components: a hydrostatic pressure difference and an osmotic pressure difference. The latter reflects the reduction in fluid chemical potential due to the presence of solute species within the fluid. The osmotic pressure of a particular solute species is typically a nonlinear function of the solute concentration. Each solute species present in the plasma and interstitial fluid can potentially influence fluid exchange across the capillary wall. In fact, the degree to which a particular solute species alters transcapillary fluid exchange depends on the ease with which the given solute crosses the capillary wall. Only those solutes to which the capillary wall is impermeable exert their entire osmotic pressure. The effective Chapter 2. Physiological Overview of the Microvascular Exchange System 13 osmotic pressure of those solutes that can penetrate the capillary barrier varies inversely with the solute's permeability. The fraction of the total osmotic pressure of a solute species i that acts on the capillary membrane is represented by the reflection coefficient for that solute species, cr\. A a of 0 indicates that the solute's permeability across the capillary wall is equal to that of water [80]. If the membrane is completely impermeable to a given solute, a equals 1. Most small lipid-insoluble solutes, such as NaCl, have reflection coefficients below 0.1, while a for most plasma proteins approaches 0.9 - 1.0 [80]. Further, since the capillary wall is very permeable to these small solutes and ions, any differences in their osmotic pressures across the membrane are quickly dissipated [71]. It is the plasma proteins, then, that contribute most to the overall osmotic driving force for fluid exchange across the capillary wall. If we treat the array of plasma proteins as an aggregate species exerting an overall osmotic pressure of LT and having an effective reflection coefficient of cr, then the fluid flux across the capillary wall, j v , is given by the Starling equation [81]: JV = L P [ P P - [ P ^ - c ^ I P - [ n^j j ] , • (2.1) where L p is the hydraulic conductance of the capillary membrane, and P p and [Prat]b are the hydrostatic pressure in the plasma and in the interstitial space adjacent the boundary, respectively. IP and [IImt]b denote the plasma protein osmotic pressures in the plasma and in the interstitial space adjacent the capillar}- wall, respectively. We will now turn to the exchange of plasma proteins across the capillar}- wall. Again, for convenience, we will limit the discussion to a single (possibly 'aggregate') species. More detailed discussions can be found in any one of many reviews on the subject [80, 82, 84, 31, 71, 93, 75. 58]. Assuming that plasma protein convection and diffusion occur along the same paracellular pathways, the exchange of these substances is described by the nonlinear flux equation (see, for example, [71]): [ C P _ rnint] -p e] Chapter 2. Physiological Overview of the Microvascular Exchange System 14 where j s is the local flux of plasma proteins from blood to tissue, C p and [ C m t ] b are the plasma protein concentrations in the plasma and the interstitial plasma protein distribution volume adjacent the boundary respectively, and where Pe is a modified Peclet number, denned as p e = ( l _ - 0 ) j v D v ; D refers here to the permeability of the capillary wall to the plasma proteins. The modified Peclet number indicates the relative contributions of convection and diffusion to the total ex-change of plasma proteins. As Pe approaches oo, the exchange is dominated by convection. A Pe of 0, on the other hand, indicates purely diffusive exchange. Equations (2.2) and (2.3) have been used to describe transcapillary macromolecular exchange in a range of tissues (see, for example, [80]). 2.2 The Interstitium 2.2.1 Structure and Composition The interstitium has been likened to a three-dimensional meshwork of fibrous elements em-bedded in a gel-like substance, referred to as ground substance, created by soluble polymers in an aqueous solution [26, 53] (see Figure (2.4)). The interstitium is therefore a composite of elements, each element contributing to the overall behavior of this medium. The principal components detennining the gross characteristics of the interstitium include the following: col-lagen, elastin, the glycosaininoglycan and proteoglycan elements, and the interstitial plasma proteins. Each of these will now be discussed briefly. Collagen Collagen is the primary structural protein of the body [57]. It is formed from a precursor molecule, procollagen, that consists of three extended polypeptide chains wound to form a triple helix [4]. The helical configuration is stabilized by interchain hydrogen and covalent bonds [57]. The procollagen molecules combine to form the collagen monomer, a rod-like molecule 300 /xm long and having a diameter of approximately 1.5 nm [26]. The Chapter 2. Physiological Overview of the Microvascular Exchange System 15 Figure 2.4: A n artist's concept of the interstitium shows the fibrous collagen mesh-work [ 1 3 ] . Chapter 2. Physiological Overview of the Microvascular Exchange System 16 monomers spontaneously form aggregates through covalent bonding and crosslinks [o7], yielding the collagen fibrils. The fibrils further combine to give collagen fibers (Figure (2.5)). Collagen 33S&3c^&2SX5o?2S^ monomer Collagen fiber Figure 2.5: The hierarchy of collagen elements is shown above. Procollagen combines to give the collagen monomers that aggregate to give collagen fibrils which, in turn, combine to yield the collagen fiber (modified after [26]). Numerous distinct collagen types have been identified within the interstitia of various tissues [5, 13]. The extent to which a particular collagen type is found within the interstitium varies from tissue to tissue. Each of the collagen types, however, forms molecules of similar structure and dimensions. Collagen is polyampholytic; that is, it is capable of bearing both positive and negative charges. The former are due to amino groups present in collagen, while the latter are attributed to carboxyl groups [57]. However, at physiological pH most of these are neutralized, so that collagen bears only a slight positive charge. Functionally, collagen fibers provide tensile strength to the tissue, resisting changes in tissue Chapter 2. Physiological Overview of the Microvascular Exchange System 17 volume when stressed along the longitudinal axis of the fiber [4j. This is due to the covalent cross-linkages that form between collagen molecules [26]. By forming a meshwork, the fibers tend to immobilize the polymers responsible for the interstitial gel (i.e., the glycosarninoglycans and proteoglycans) [26]. Finally, collagen is partly responsible for the exclucling properties of the interstitium [13], to be discussed later. Elastic Fibers While collagen fibers impart tensile strength to a tissue, elastic fibers provide it with elasticity [13]. Elastic fibers occur in small quantities (relative to collagen content) in most interstitia, with the possible exception of certain specialized tissues such as the greater arteries [53] that display a high degree of elasticity. Elastic fibers consist of two principal components: an amorphous mass of elastin surrounded by microfibrils of protein [13]. Elastin is one of the most apolar proteins known [57], providing it with a hydrophobic nature [13]. At physiological pH, elastic fibers contain approximately 0.56 ml of water per ml of elastin [13]. Most of this water is likely accessible to small molecules and ions, such as sucrose, urea, sodium, and chloride. Larger molecules, such as the plasma proteins, however, are thought to be excluded from this fluid space. Glycosarninoglycans and Proteoglycans Glycosarninoglycans are linear polymer chains of disaccharide units common to all tissues [4]. Essentially all of the charge groups associated with these polymers are ionized at physiological pH [57]. Glycosarninoglycans therefore at-tract counterions, thereby creating a Donnan distribution of mobile ions that exert an osmotic pressure [4]. One of the most prevalent of the glycosarninoglycans is hyaluronate. In its hydrated state, hyaluronate forms an unbranched, random coil that occupies a solvent domain some 1000 times greater than the polymer volume [4]. Further, the mutual repulsion of negative charges present along the hyaluronate chains tends to expand the coil [13]. Therefore, even at concentrations Chapter 2. Physiological Overview of the Microvascular Exchange System 18 as low as 0.1 percent by weight, hyaluronate molecules become entangled [4]. It is the entan-glement of hyaluronate and other glycosarninoglycans and proteoglycans that gives the ground substance its gel-like properties. Gersh and Catchpole [47] also identified 'water-rich, colloid poor' regions within .the interstitial matrix, leading some to postulate the existence of free fluid channels within the interstitium (see, for example [104]). However, the original study made no mention of such continuous structures; rather, the authors simply identified heterogeneities in glycosaminoglycan distribution within the matrix [4]. Other early ultrastructural studies have identified transient, submicroscopic fluid vacuoles within the interstitium, but later studies have not confirmed their existence [4]. In fact, the preparative procedures used in many of these studies are known to extract ground substance [53], suggesting that the 'free fluid phase' may well be an artifact of these early experiments. However, as Aukland and Nicolaysen [4] point out, this does not preclude the possibility of heterogeneities within the interstitial gel, due to local rarefactions in polysaccharide content, that might provide preferential channels for fluid and solute transport. Except for hyaluronate, glycosarninoglycans exist tn vivo not as free polymers, but covalently bound to a protein core [13]. Such structures are termed proteoglycans. These can further bind to hyaluronate molecules to form proteoglycan aggregates, having molecular weights in excess of 2 xlO8 Daltons. At physiological pH, proteoglycans display a high charge density [13]. These structures are also known to bind to collagen [26]. The glycosaminoglycan and proteoglycan elements contribute to the interstitium's resistance to bulk fluid movement [13]. Their water retaining properties also enhance the stability of coUagen-glycosarninoglycan solutions, resisting volume changes under compression [26]. This has been demonstrated experimentally using prepared solutions of collagen and hyaluronate, for example. In vitro mixtures of thermally precipitated collagen and hyaluronate produce structures that resist compression during centrifugation [13]. The Interstitial Plasma Proteins Plasma proteins represent a broad group of macro-molecules. Various types of these are transported across the endothelial membrane into the Chapter 2. Physiological Overview of the Microvascular Exchange System 19 interstitium. They range in size, displaying Stokes' radii anywhere from 1 to 11 run [13]. At physiological pH, most plasma proteins carry a net negative charge [13]. Interstitial plasma proteins exert an osmotic pressure that is a nonlinear function of plasma protein concentration. However, the major part of the osmotic pressure is due to a single species - albumin [13]. Albumin is the most plentiful of the plasma proteins, constituting approximately 60 % of the serum protein content in humans [46]. It has a molecular weight of 6.6 X l O 4 Daltons and a Stokes' radius of 3.5 nm [13]. With an isoelectric point at a pH of 4.7, albumin bears a net negative charge at physiological pH. Experimental studies of extravascular albvunin indicate that significant quantities of this protein he outside of the blood stream, largely in the interstitia of muscle and skin [13]. This suggests that the interstitium may act as a reservoir for osmotically active macromolecules [13]. 2.2.2 V o l u m e E x c l u s i o n w i t h i n t h e I n t e r s t i t i u m As mentioned earlier, the various components of the interstitium, particularly the glycosarnino-glycans, occupy a volume in solution that far exceeds the volume of the polymers themselves. Even at low concentrations, the solvent domains associated with these polymers overlap to cre-ate a meshwork of molecular dimensions [53]. A given interstitial solute species will distribute throughout only those spaces in the meshwork that have dimensions larger than the solute itself. The remaining regions of the meshwork are inaccessible to the solute. As a consequence, the space available to certain interstitial solutes (i.e., the solute's distribution volume) is consider-ably less than the total interstitial fluid volume. This phenomenon has been termed volume exclusion. The glycosarninoglycans have traditionally been identified as the principal components re-sponsible for the exclusion of plasma proteins from regions of the interstitium [4]. The fraction of total fluid volume inaccessible to plasma proteins in hyaluronate solutions, for example, can be significant, even at low concentrations. A 0.5 % by weight solution of hyaluronate excludes Chapter 2. Physiological Overview of tie Microvascular Exchange System 20 albumin from 25 % of the solution space. As the hyaluronate concentration is raised to 1.5 % by weight, the excluded volume increases to 75 % of the solution volume [13]. However, because of its abundance relative to the glycosarninoglycans in certain tissues such as dermis, collagen may well be the major source of plasma protein exclusion in some instances [13]. Exclusion bears upon the processes within the interstitial space. It is the effective concen-tration of a given solute species (i.e., the concentration based on the solute's distribution volume rather than the total fluid volume) that determines its chemical activity, which in turn affects chemical equilibria, osmotic properties, solubilities, and driving forces for diffusion within the system [27]. By treating the interstitial solute species as spheres contained in a random meshwork of rods, Ogston and co-workers [72] developed the following equation to calculate volume exclusion: fe = i_e-Kr-+r<)/r<]2v'c', (2.4) where fe is the excluded volume fraction, r s and rr are the solute radius and radius of the rods making up the meshwork, respectively, Vf is the partial specific volume of the rod material, and Cf is the mass concentration of rods in the system. This analysis would suggest that exclusion increases with increased concentration of excluding species (i.e., the rods) and increased solute radius, but decreases with increasing rod diameter. Similar expressions have been developed for single rod-sphere and sphere-sphere systems (see [13] for details). The above analysis of exclusion considers only geometric factors. However, since the gly-cosarninoglycans are negatively charged, electrostatic effects may also play a role in determining the exclusion properties of specific tissue-solute systems. This may be true, in particular, for tissues such as cartilage that display a high interstitial charge density [53]. In fact, the exclusion of low molecular weight anionic tracers has been demonstrated, but the effect is unpredictable Chapter 2. Physiological Overview of the Microvascular Exchange System 21 2.2.3 Characterizing Interstitial Swelling The glycosarninoglycans and the collagen fibers are the principal components within the in-terstitium that determine the mechanical properties of connective tissues [4, 53, 117, 89]. It is generally thought that the glycosaminoglycan element provides the tissue with its swelling tendency by virtue of its osmotic activity. As a polymer solution, the glycosarninoglycans (and their aggregates) exert an osmotic pressure that tends to imbibe fluid. The charged groups associated with the polymers create a mutual repulsive force that may further tend to expand the network [4]. The stiff collagen meshwork, on the other hand, imparts rigidity to the tissue, acting to limit volume changes within the interstitium. The relative influences of the glycosarninoglycans and of the collagen on tissue hydration are well demonstrated experimentally. Degradation of the collagen by chemical treatment causes umbilical cord to swell [53]. Destruction of hyaluronate in swollen tissue, on the other hand, leads to a reduction in tissue hydration [53]. Theoretical interpretations of the swelling process, however, are clouded in controversy (see, for example, [53, 89, 107]). Much of the confusion seems to he i n the delineation of the various forces acting on the system into the mechanical components responsible for deformation (i.e., the mechanical stresses within the system) and the forces responsible for fluid exchange within the system (namely differences in fluid chemical potential between vascular and tissue compartments). Typically, the swelling properties of a tissue are characterized by an experimentally deter-mined relationship between the equihbrium tissue hydration and the interstitial fluid pressure, i.e., the tissue compliance relationship. The interstitial fluid pressure within a tissue is mea-sured at various states of hydration using microneedles, wicks, or implanted capsules. The major problem i n such experiments lies with interpreting the reading provided by the pressure measuring device (see, for example, [107]). Again, there seems to be a great deal of confu-sion regarding whether such devices measure an equivalent interstitial fluid chemical potential, which would include both hydrostatic and colloid osmotic pressures, or whether they isolate the hydrostatic component. A fundamental understanding of the operation of these pressure Chapter 2. Physiological Overview of the Microvascular Exchange System 99 Compliance = AIFV/AP; 1 _ I IFV Figure 2.6: This figure shows the general trend in the change in interstitial fluid vol-ume (IFV), given along the x-axis, following a change in interstitial fluid pressure, shown on the y-axis (modified after [5] ) . measuring devices within tissues is therefore needed before tissue hydration data can be reliably interpreted. The general shape of the pressure-volume curves typical of tissues is given in Figure (2.6). Generally, the change in tissue hydration per unit change in interstitial fluid pressure is low at the lower tissue hydrations, increasing as the tissue becomes swollen. The high resistance to tissue hydration in the initial part of the curve suggests a mechanism to ward off edema formation. Specifically, a small change in interstitial hydration is accompanied by a substantial increase in the interstitial fluid pressure. According to the Starling equation (see Eq. (2.1)), this increase in interstitial fluid pressure reduces the driving force for fluid transport from the blood to the tissue space, thereby reducing the threat of severe tissue swelling [13j. Chapter 2. Physiological Overview of the Microvascular Exchange System 23 2.3 The Lymphatic System 2.3.1 General Description In contrast to the abundance of data on the exchange vessels of the blood vasculature, there appears to be a dearth of information regarding the operation of the lymphatic network. Several reviews [4, 117, 77, 49] are available in the literature, however, and the reader is referred to these for more detailed discussions of this system. Only a brief description will be provided here that focuses on the withdrawal of interstitial fluid and plasma proteins by the permeable vessels of the lymphatic network. Lymphatic vessels occur in most tissues; exceptions include the brain, the retina, and bone marrow [5]. Unlike the arterio-venous blood system, the lymphatic network typically begins with bulbous terminal lymphatic vessels located in close proximity to the blood capillaries. These bulbous structures are typically 20 to 80 /mi in diameter, although they can reach diameters of 720 ^m in some tissues (e.g., the bat wing) [49!. The terminal lymphatic vessels are unevenly distributed within the microcirculation, being more prominent at the venous side of the microvascular bed where the blood vessels are most permeable [77]. Further, they occur less frequently than the blood capillaries [5]. Figure (2.7) illustrates the structure and orientation of these vessels within the microcirculation. Fluid and solutes that have been withdrawn from the tissue space are carried along the lymphatic network via collecting vessels. These empty their contents into the left and right subclavial veins [46], thereby returning fluid and solutes to the blood circulation. In the average human, an estimated 25 % to 50 % of the total circulating plasma proteins are returned to the blood circulation along this route on a daily basis, while 2 to 4 liters of fluid enter the lymphatics from the interstitial space each day [46]. Chapter 2. Physiological Overview of the Microvascular Exchange System 24 Figure 2.7: The lymphatic vessels within the bat wing are illustrated above in solid black. The system begins with the bulbous terminal lymphatic vessels, located close to the blood capillaries. These drain into collecting vessels that eventually return fluid and solutes to the blood vasculature (modified after [49]). Chapter 2. Physiological Overview of the Microvascular Exchange System 25 2.3.2 The Terminal Lymphatic Vessels The terminal lymphatic vessels are responsible for chaining fluid and solutes from the interstitial space. The wall of the terminal lymphatic vessel is similar to that of the blood capillary in that it consists of a single layer of fiat endothelial cells [77]. However, it differs from the capillary wall in several respects [49]. First, the interendothelial junctions appear more loose, the cells overlapping each other at times. The basement membrane is poorly developed or absent altogether. Furthermore, while the endothelial cells of the tenninal lymphatic vessels contain vesicles, fenestrae have not been observed. The terminal lymphatics display irregular geometries with bulbous sacs and constricted regions along the vessel length. The vessels are easily collapsed, making pressure measurements within terminal lymphatics difficult [77]. The wall of the terminal lymphatic vessel is anchored to the surrounding interstitial matrix by fine strands of reticular fibers and collagen [49] (see Figure (2.8)). It is thought that the anchoring filaments aid in the withdrawal of fluid from the interstitial space. As fluid accumu-lates within the tissue spaces, the tissue expands, placing the anchoring filaments under tension. This tensile stress keeps the lymphatic vessel from collapsing under the increased tissue fluid pressure associated with the accumulation of fluid there. The terminal lymphatic is then able to withdraw fluid and solutes from the interstitium [49, 77]. Interstitial fluid is thought to cross the terminal lymphatic wall via diffusion through the endothelial cell, by vesicular pathways, and through the intercellular junctions [49]. The relative importance of these pathways in lymphatic filling is, as yet, unknown. A number of theories for the filling of the terminal lymphatic vessels have been proposed, mcluding vesicular, osmotic pressure driven, and hydraulic (i.e., hydrostatic pressure) driven mechanisms. To date, there is little to no experimental evidence to support the first two hypotheses [5]. However, it has been demonstrated experimentally that lymph flow increases with increased interstitial fluid pressure in a number of tissues, including dog hindpaw, the small intestine, the fiver, the myocardium, and rat kidney [5j. Hence, it is frequently assumed that the rate of lymph formation in the teirninal vessel is a direct function of the local tissue fluid pressure (see, for example, [108, 14]). Chapter 2. Physiological Overview of the Microvascular Exchange System 26 Figure 2.8: T h e structure of the terminal lymphatic vessel is shown, illustrating the anchoring filaments that serve to keep the vessel patent under increased tissue fluid pressure [49]. Chapter 2. Physiological Overview of the Microvascular Exchange System 27 In the absence of injury, tissue swelling, or muscular activity, the intercellular junctions are typically closed. However, given any of these circumstances, the junctions open to permit large particles to pass through [49]. Experiments with labelled particles suggest that both intercellular junctions and vesicular mechanisms serve as routes for macromolecules. although the relative importance of these two pathways is debated [49]. However, it is typically assumed that the composition of lymph in the terminal lymphatic vessel is the same as the interstitial fluid in the adjacent tissue space [117]. The walls of most collecting lymphatic vessels contain smooth muscle [5]. The collecting vessels propel fluid and solutes along the network in response to both extrinsic forces (such as limb movements, respiratory pressure variations, and massage) and spontaneous, coordinated contractions of the muscle within the vessel walls [49]. The intrinsic contractile behavior of the lymphatic vessels appears driven by the increased stress (hoop pressure) within the vessel walls that accompanies the uptake of fluid from the surrounding interstitial space. The amplitude of the contraction is proportional to the degree of wall stretch [77]. Lymph flow within the collecting vessels remains uni-directional by virtue of one-way valves found within the vessels [77]. These valves occur in abundance along the lymphatic network; the average spacing between valves ranges from 2.3 mm to 4.0 mm in the upper arm in humans, for example [49]. The valves of the larger vessels can withstand back-pressures as high as 60 rnmHg [77], far above typical pressure drops reported within the lymphatic network (see [49]). Chapter 3 Formulation of the General Model of Interstitial Transport 3.1 A Continuum Representation of the Interstitium At the microscopic level, the transport of fluid and plasma proteins through the interstitium represents an extremely complex process. Fluid and plasma proteins interact as they traverse the interstitial space along tortuous pathways. Furthermore, plasma proteins may encounter barriers resulting from electrostatic forces and/or the architectural configuration of various structural components, such as hyaluronate, proteoglycans, collagen, and elastin, all of which exclude proteins from regions of the interstitium. These structural components deform under a complex set of forces as the tissue hydration changes. A detailed description of interstitial transport is impractical. Instead, we adopt the concept of a continuum to represent the interstitium (see [106, 8, 28] for details). Here each principle phase, such as fluid or structural elements, is represented by a hypothetical continuum which is distributed throughout the interstitium. The properties of these continua, as well as the pro-cesses occurring within them, represent spatial averages of the properties and processes found at the microscopic level. The characteristic dimension of the elementary volume over which this averaging procedure takes place is large relative to the microscopic dimensions (represented, for example, by the diameter of a collagen fiber bundle), yet small relative to the characteristic dimension of the system as a whole (such as the total distance traversed by fluid and plasma proteins in their journey from the blood to the lymphatic circulations). The resulting averaged properties are assigned to the point about which the elementary volume is centered. The vol-ume is then centered about an adjacent point, and the averaging process is performed again. The procedure is repeated throughout the domain, transforming the complex, heterogeneous 28 Chapter 3. Formulation of the General Model of Interstitial Transport 29 system into a hypothetical continuum to which the laws of differential calculus apply. The averaging process introduces parameters associated with the continuum, such as the intersti-tial fluid conductivity, the effective protein diffusivities, and the excluded volume fractions. These represent the averaged effects of the complex structure and molecular interactions at the level of the microscale. The parameters are then used to describe interstitial transport when approximating the real system by the continuum. The principle of spatial averaging is applied here to analyze the transient flow and distri-bution of fluid and any number of macromolecular species through the interstitium, which is treated as an isotropic, deformable porous medium. Since crystalloid solutions are exchanged rapidly, compared to the plasma proteins [71], any disturbances to the system with respect to small ion distribution is likely dissipated quickly. Therefore, interstitial gradients in small ion concentrations and the influence of tissue cells on fluid exchange will be neglected here. Hence the analysis cannot describe hypertonic fluid resuscitation, for example. Since the to-tal plasma protein concentration in plasma is small (6 gm/dl in humans [46].), the interstitial fluid and plasma proteins form a dilute, incompressible solution. The sohd components of the interstitium are also considered incompressible. Hence interstitial deformation results from the spatial reorientation (e.g. bending) of the sohd elements relative to each other. Exclusion is accounted for by assigning different distribution volume fractions to the various plasma protein species. These distribution volume fractions are functions of the sohd phase volume fraction, and therefore vary with interstitial hydration. Fluid and protein transport parameters may vary between individual distribution volume fractions and with interstitial hydration. In the remainder of this section we present mathematical descriptions of each of the following aspects of the interstitial continuum: 1. plasma protein exclusion and its effect on local protein partitioning, colloid osmotic pres-sures, and fluid chemical potential; 2. the relationship between fluid transport and fluid chemical potential; and Chapter 3. Formulation of the General Model of Interstitial Transport 30 3. protein transport mechanisms wi thin the interstit ium. These relationships are then used in the following section to develop the mass balance equations which govern the time-dependent distributions of solids, fluid and protein species i n a deforming interst i t ium. 3.1.1 Plasma Protein Partitioning within the Interstitium: Exclusion Plasma protein exclusion i n tissues can be substantial; for example, a lbumin is excluded from 60 percent of the total interstitial volume i n canine smooth muscle [13]. Because of exclusion, the effective concentration of an interstitial plasma protein species (i.e., its mass per unit volume of available space) is higher than its concentration based on the total fluid volume. The effective concentration plays an important role i n interstitial fluid and protein transport because it detennines the protein osmotic pressures, convective protein fluxes, and the diffusional driving force wi th in the interst i t ium [27]. A complete description of interstitial transport must therefore include a treatment of exclusion and its effect on local plasma protein distribution and fluid chemical potential. We w i l l now discuss how the principle of volume averaging can be employed to describe exclusion of multiple solute species. Figure (3.1) is a schematic diagram of a typical elementary volume centered about some point wi th in the interst i t ium, over which the averaging process has been performed. The volume element contains m protein species and m+3 distinct volume fractions: 1. a total mobile fluid volume fraction (n° ) ; 2. m volume fractions corresponding to the distribution volume fractions for each of the m protein species ( n k , k = l , 2 , . . . m ) ; 3. a sohd phase volume fraction (n s ) comprised of structural elements such as collagen, gly-cosarninoglycans, proteoglycans, and elastin, which form the sohd skeleton of the porous structure; and Chapter 3. Formulation of the General Model of Interstitial Transport 31 4. an immobile water volume fraction consisting of water trapped and/or bound to the solid phase (n™). The immobile fluid phase will, in general, depend on the amount and composition of the solid phase. For a given tissue, then, n™ will be a function of ns: n™ = Fs(ns), (3.1) where F s is an empirical function relating the two volume fractions. If, for example, we assume that the immobile fluid is largely made up of the intrafibrillar water of collagen and that the collagen is uniformly distributed throughout the solid phase, then n m is directly proportional to the solid phase volume fraction: n™ = /T • n s, (3.2) where f3~ is an experimentally determined constant of proportionality. The remaining distribu-tion volume fractions are indexed such that n° > n1 > n 2 > ... > n1""1 > n m . (3.3) We also define a set of incremental volume fractions, £n k, where 6n k = n k - n k + 1 , k = 0,1,2,m - 1. (3.4) That is, the incremental volume fraction £nk represents the difference between the distribution volume fractions of species k and species k+1. Note that, by this definition, £n m equals n m . As we will see shortly, these incremental volume fractions are needed to describe the plasma protein and fluid pressure distributions within the volume element. The total excluded volume for a given protein species depends on the amount of interstitial solid components present in the elementary volume [53j. Therefore the fraction of total mobile and immobile fluid from which plasma protein species k is excluded, n^, is a function of ns: nck = k = 1,2,..., m; (3.5) Chapter 3. Formulation of the General Model of Interstitial Transport 32 Figure 3.1: Elementary volume of interstitium before (A) and after (B) volume aver-aging. The various incremental volume fractions, tfn1, distribution volume fractions, n 1 , solid phase volume fraction, n s , and immobile fluid phase volume fraction, n , m , are associated with the point, P, in the continuum about which the elementery volume is centered. The heterogeneous interstitium is thereby transformed into a multiphase continuum. Chapter 3. Formulation of the General Model of Interstitial Transport 33 where F is an empirical function for the kth protein species. By definition, the sum of the distribution volume fraction and the excluded volume fraction for species k must equal the total fluid volume fraction (i.e., 1 — n s). That is, n k = 1 - n s - n * . (3.6) It therefore follows from Eqs. (3.5) and (3.6) that n k = 1 - n s - F k ( n s ) , (3.7) while from Eqs. (3.4) and (3.7) we have Snk = F k + 1 ( n s ) - F k (n s ) , k = l , . . . , m - 1. (3.8) Hence all pertinent fluid volume fractions ( n k , 6 n k , n e k , and n m ) may be expressed in terms of n° for a given tissue using Eqs. (3.1), (3.5), (3.7), and (3.8). To describe fluid and protein transport through the interstitium, we must make some as-sumptions regarding the distribution of proteins and the variation of fluid chemical potential throughout the incremental volume fractions within the elementary volume. Consider first the distribution of plasma proteins within the volume element. The incremental volume fraction 6n° contains no protein (see Figure (3.1)). Each subsequent incremental volume fraction con-tains an additional protein species. Let C ^ 1 represent the concentration of protein species k in far1. Assuming that, for each volume element, the protein has a uniform concentration C k within its distribution volume n k , then Qk.1 = C k ) ! > k ) (3.9) and & ' 1 = 0, 1 < k. (3.10) Hence only one value, C k , is needed to describe the local concentration of protein species k throughout all of the incremental volume fractions in the elemental volume which are accessible to that species. Chapter 3. Formulation of the General Model of Interstitial Transport 34 Assuming an isothermal system free of external forces, the chemical potential of the fluid in far1, pi, is [62] ul = / C R + V ^ P 1 + R T • In (TWXL) , (3.11) where u%f is the reference chemical potential, equal to the chemical potential of the pure fluid at standard conditions, and P 1 are the fluid's partial molar volume and hydrostatic pressure in far1, respectively, R is the universal gas constant, T is the absolute temperature, 7 W is the activity coefficient for the fluid, and is the mole fraction of fluid in far1. For dilute solutions, variations in V^, are negligible. Using the Gibbs-Duhem relation, the chemical potential can be expressed alternatively in terms of hydrostatic and colloid osmotic pressures [117] as fil = /x^ + V w ( ? l - £ n k ) . 1 = 1 , 2 , m , (3.12) where V ] w has been replaced by V w , the molar volume of pure fluid, due to the assumption of a dilute solution. The colloid osmotic pressure exerted by plasma protein species k in far1, TI k' 1, is a function of the protein concentration in far1, i.e., H k > 1 = G k (C k ' 1 ) , (3.13) where G k is the coUoid osmotic pressure relationship for plasma protein species k. By virtue of Eqs. (3.9), (3.10) and (3.13), n k , l = I J k ? 1 > k, (3.14) and n k l l = 0, l < k . (3.15) In fai° we have /z°w = / C f + V w ( P ° - n ° ) (3.16) where P° is the hydrostatic pressure in fat0, and II 0 is the sum of any additional osmotic terms associated with fai°, such as Donnan and polysaccharide osmotic pressure contributions. Chapter 3. Formulation of the General Model of Interstitial Transport 35 If the relative amounts of these osmotically active components are known, then II0 can be determined through an appropriate osmotic relationship. Assuming that the fluid throughout the elementary volume is in thermodynamic equilibrium, it follows that m p° - n° = P 1 - n 1 = P 2 - n 1 - n 2 = ... = p m - ^ n k . (3.17) k=i Hence the fluid hydrostatic pressure in any of the incremental volume fractions may be expressed in terms of a single hydrostatic pressure ( P M ) and the various osmotic pressures (nk, k = 0. 1.2....m). The concepts and definitions presented here are best illustrated with a simple example. Consider an elementary volume of interstitium containing a single plasma protein species k, as shown in Figure (3.2A). By definition, the sum of the sohd phase volume fraction, nE, the excluded volume fraction, n^, and the plasma protein's distribution volume fraction, n k, equals 1. Since we are considering only a single plasma protein species, it follows from Eq. (3.4) that 6n k equals n k. Also, since we have assumed local thermodynamic equilibrium within the elementary volume, the sum of the hydrostatic and colloid osmotic pressures in the distribution volume, P K - lfk, equals the sum of pressures in the excluded volume, P ° — 11°. Figure (3.2B) shows the same elementary volume following an increase in hydration accom-panying, for example, an increase in local fluid hydrostatic pressure. If we assume that the local concentration in protein species k remains the same, then the local fluid chemical poten-tial in n k, P K — n k , will increase. Since the fluid within the volume is in local thermodynamic equihbrium, P ° — n° will also increase by the same amount. Since the hydration has increased, the local sohd phase volume fraction is reduced. This results in a reduction in the excluded volume fraction as well, since the latter depends only on the amount of sohd phase present in the volume. Consequently, the volume available to the plasma proteins increases, so that n k increases. We will now discuss how such a change in tissue hydration may be described in more rigorous terms. Chapter 3. Formulation of the General Model of Interstitial Transport 36 Figure 3.2: A n elementary volume of interstitium containing a single plasma protein species k before (A) and after (B) an increase i n local hydration. Following hydra-tion, the local solids phase volume fraction, n &, decreases, resulting i n a decrease in the excluded phase volume fraction, n e k , as well. B y definition, the fraction of tot a l fluid available to the plasma proteins, n^, increases. Chapter 3. Formulation of the General Model of Interstitial Transport 37 3.1.2 Tissue Strain, Volumetric Dilat ion, and Tissue Compliance Changes in interstitial hydration result from a net flow of fluid into or out of the interstitium. Swelling is therefore linked to the forces governing fluid exchange, including interstitial hy-drostatic and colloid osmotic pressures. As interstitial sohd components deform there is an accompanying change in the sohd stress. The system maintains mechanical equilibrium by a concomitant change in the hydrostatic pressure of the interstitial fluid [53, 89, 117]. This change in hydrostatic pressure, together with the change in interstitial hydration and possi-ble net exchange of plasma proteins, alters the driving forces for fluid exchange. Interstitial swelling therefore involves a complex set of coupled processes that depend on the mechanical characteristics and transport properties of the microvasacular exchange system. A rigorous examination of swelling in a porous medium requires a complete description of the stress distribution throughout the medium, together with constitutive relationships between sohd stress and deformation. This type of analysis has been used to describe fluid movement in deformable porous rocks [19, 28]. To apply the principles underlying this theory to biolog-ical systems in turn requires detailed information regarding the mechanical properties of the interstitium and its boundaries. Such information is not available for most tissues. Therefore a simpler - albeit less rigorous - approach is adopted which follows the method used by Terzaghi [98] to analyze land subsidence following the removal of large volumes of groundwater [28]. The method assumes that the local deformation at any point in the interstitium is a function of the local fluid hydrostatic pressure (see [89]). The problem of swelling then reduces to de-termining the distribution of fluid pressure within the interstitium. This is accomplished by solving the set of transport equations which are developed in Section 3.2. Fatt and Goldstick [35] and Friedman [41] have used similar approaches to study swelling in corneal stroma. In these cases, however, swelling is linked to a 'swelling pressure', rather than the hydrostatic fluid pressure. In addition, their analyses are limited to a single dimension. The present analysis ap-plies to the case of isotropic, tlu-ee-dimensional swelling in which the influence of shear stresses is neglected. As such it represents only a first approximation to the complete description of Chapter 3. Formulation of the General Model of Interstitial Transport 38 Figure 3.3: Linear deformation of a differential segment, dxj, by an amount d U j . The following discussion is limited to the case of small deformations described by classical elastic theory. Therefore the equations do not apply, for example, to the development of se-vere edema. We begin with a brief description of deformation theory. Let dU; represent the deformation in the x; direction of a small element of initial length dx; (see Figure (3.3)). By definition, the local solid strain in the xi direction, ei, is equal to the change in length of the element divided by its initial length, i.e., 5Uj dxi (3.18) Similar equations apply for the strains in the x 2 and x 3 directions. The local volumetric dilation, ev, of an infinitesimal volume element, dV° , undergoing deformation is defined as d V 1 - dV° dV° (3.19) where d V 1 is the volume of the deformed element (see Figure (3.4)). For small strains, ev is equal to the sum of the individual linear strains [102]: (3.20) i = l Chapter 3. Formulation of the General Model of Interstitial Transport 39 That is, <v = ^ , (3-21) where the right hand side of Eq. (3.21) is written in tensor notation (see [99]). Figure 3.4: Volumetric dilation of a differential volume element from an unstrained volume, d V ° , to a strained volume, d V 1 . To relate the volumetric dilation, ev, to the local fluid hydrostatic pressure we begin with Terzaghi's concept of effective stress, which forms the basis for describing deformation in porous media [19, 59, 101, 67]. The total stress is set equal to the sum of the local fluid pressure and an effective stress responsible for the deformation of the solid skeleton as follows [102]: c-^o-f-PSz, (3.22) where CT;J and <7-jff are the components of the total stress tensor and the effective stress ten-sor, respectively, P denotes the local hydrostatic fluid pressure, and 5;J is the Kronecker delta function (see [99]). The negative sign in front of the pressure term results from defining the c Chapter 3. Formulation of the General Model of Interstitial Transport 40 pressure as positive in compression, while the remaining stresses are defined as positive in ten-sion. Furthermore, Eq. (3.22) assumes that the pressure in the fluid creates a normal stress of equal magnitude in the sohd skeleton, and that both the fluid and the sohd skeleton are incompressible. The effective stress then represents the additional stress within the sohd phase that causes the sohd components to reorient themselves relative to each other, resulting in the deformation of the medium. If, for the small range of volume changes considered, we neglect any changes in the overall stress in the system which might occur, for example, due to changes in the applied stresses at the interstitial boundaries, then A ( r f = AP5ij. (3.23) Equation (3.23) implies that the local volumetric dilation is a function only of the local hydro-static pressure within the system. This function is provided by the interstitial compliance, fi, [4] defined here as where A V is the change in the interstitial fluid volume, relative to a reference volume, in re-sponse to a change in fluid hydrostatic pressure, relative to the corresponding reference pressure. The compliance can be expressed in terms of the volumetric dilation, ev, by dividing Eq. (3.24) by the reference volume. Then, in the limit of infinitesimal volumes, the specific compliance, fi, is "(P) = ^ - ( 3 2 5) In the multiphase system proposed here the question arises as to which of the m+1 hydro-static pressures (corresponcling to the m+1 incremental fluid volume fractions) are to be used in the compliance relationship. Following Lewis and Schrefler [67], the average local hydrostatic pressure, 1 m k=0 Chapter 3. Formulation of the General Model of Interstitial Transport 41 will be used. The exact form of Eq. (3.26) will depend on the experimental method used to measure interstitial fluid pressures when deterrnining the compliance of a given tissue. For example, the average pressure given above assumes that the measuring device (such as a mi-cropipette) cannot distinguish between the hydrostatic pressures in the various fluid phases, and hence yields a composite value. Together, Eqs. (3.21), (3.25) and (3.26) define the relationship between the local hydrostatic pressures in each of the incremental volume fractions, the local linear strains, and the local vol-umetric dilation. These equations will be used when developing expressions for the distribution of fluid pressure and various plasma protein species within the deformable interstitial space. To detennine the geometry of the deformed medium, it is necessary to calculate the linear displacements, U; , i = 1,2,3, throughout the interstitial space as functions of the local average hydrostatic pressure. The spatial components of the deformed medium are then evaluated from these displacements, i.e., x! = x; + TJi(xi), (3.27) where x- is the X j location after deformation of a point that was originally positioned at xi. Hence TJi(xi) represents the total displacement of a point from its original (unstressed) position, x;. The displacements are found by introducing the solid displacement potential, $, where [59j ^ = U, (3.28) $ is therefore related to ev by (see Eq. (3.18)) d 2 $ (3.29) Sx;2 v-Upon deterrnining the local average fluid pressure from the transport equations and Eq. (3.26), the volumetric dilation, eV ) the displacement potential, 4?, and the individual local solid phase displacements, U;, are calculated from Eqs. (3.25), (3.29), and (3.28) respectively. The deformed geometry of the interstitial space is then calculated using Eq. (3.27). Together, these equations describe the local interstitial deformation associated with variations in local fluid hydration. Chapter 3. Formulation of the General Model of Interstitial Transport 42 3.1.3 A Constitutive Relationship between Fluid Flow and Fluid Chemical Po-tential At low Reynolds numbers, the creeping flow of a homogeneous Newtonian fluid through an isotropic porous medium is described by Darcy's Law: *, = - * " • £ . <»•*» where j ° . is the total local volumetric fluid flux in the x; direction, K° is the hydraulic conduc-tivity of the porous medium-fluid system, and P is the fluid hydrostatic pressure. The hydraulic conductivity is a function of the structure of the porous medium and the absolute viscosity, p, of the fluid. However, the specific hydraulic conductivity, K' , equal to K/p, is a material property of the porous medium and therefore does not depend on the type of fluid flowing within the system [31, 66, 60]. In deforming porous media, j° represents the fluid flux relative to the moving solids [20]. It is related to the absolute fluid flux, q° . (where q°. is the fluid flux relative to stationary coordinates), and the local sohd phase velocity, v s ; (taken with respect to the same set of stationary coordinates), by qw,=Jw 1 +n 0 -v^ ) (3.31) where, as before, n° is the mobile fluid volume fraction. In the case of solutions, the presence of solutes influences the flux of solvent. This interac-tion is described by the phenomenological relationships of irreversible thermodynamics. These relationships, which have been used to quantify mass exchange across the vascular wall, relate the fluid flux to the colloid osmotic pressure and hydrostatic pressure driving forces present in the system. The exact way in which the colloid osmotic and hydrostatic pressures within the interstitial space affect the local interstitial fluid flux remains unresolved. However, as a start-ing point, it is assumed here that the local fluid flux through an incremental volume fraction is proportional to the local gradient in fluid chemical potential there, i.e., Chapter 3. Formulation of the General Model of Interstitial Transport 43 While Eq. (3.32) is a postulate only, it should provide a satisfactory first approximation for quantifying the effect of colloid osmotic pressure gradients on the solvent flux. This dependence of fluid flux through a porous medium on the gradient in fluid chemical potential was first proposed by Biot [21], and has been suggested by several researchers to describe interstitial fluid transport [57, 26, 69]. However, Eq. (3.32) does not consider the influence of solute mobility in determining the effective colloid osmotic pressure driving fluid within the interstitium. It will therefore most likely over-estimate the effect of colloid osmotic pressure gradients on fluid flow. Further research is needed to determine the influence of osmotic pressures on interstitial fluid flow and, hence, the appropriate form of Eq. (3.32). Since we have assumed local thermodynamic equflibrium with respect to the fluid chemical potential in each of the incremental volume fractions, the total local fluid flux for our system is A = -K° •^(pm-Enk)- (3-33) Equation (3.33), because it incorporates colloid osmotic effects in the fluid flow relationship, represents a more general version of Eq. (3.30). The assumption of local thermodynamic equi-hbrium in fluid chemical potential implies that the local driving force for fluid flow in each of the incremental volume fractions within an elementary volume of interstitium is the same. That is M = M = ... = M ( 3 3 4 ) dxi dx\ dxi The local fluid flux associated with the distribution volume fraction n k , expressed in terms of hydrostatic and colloid osmotic pressure gradients, is then J ^ = ~ K k ^ ( P m " £ n k ) ' ( 3 - 3 5 ) where is the local fluid flux and K k is the hydraulic conductivity associated with the dis-tribution volume fraction of protein species k. It follows from Eqs. (3.34) and (3.35) that the fraction of the total volumetric fluid flux that is associated with the distribution volume fraction n k is •k K k -o J w ^ ^ o J w r (3-36) Chapter 3. Formulation of the General Model of Interstitial Transport 44 In general, K°, and hence Kk, will vary with interstitial hydration (see, for example, [53]). For a discussion of interstitial hydraulic conductivity and its dependence on hydration and on sohd phase composition, the reader is referred to the recent review by Levick [66]. The local fluid flux through any of the protein distribution volume fractions can then be calculated using Eqs. (3.33) and (3.36). 3.1.4 P r o t e i n T r a n s p o r t M e c h a n i s m s w i t h i n t h e I n t e r s t i t i u m The transport of a solute through a porous medium occurs via convective and diffusive mech-anisms. The relative contributions of these two processes to the overall solute flux will depend on the fluid velocities within the medium and the system's transport properties with respect to that particular solute. When modelling the interstitial transport of plasma proteins it is often assumed that molecular diffusion dominates [38, 65,.7]. However, given the comparatively high hydraulic conductivity of certain tissues, such as tumours [61], cases may exist in which convec-tive transport plays an important role. In addition, convective transport in porous media can result in mechanical dispersion which, while bearing a resemblance to diffusion, is dependent on the solute convective velocities [8]. A general description of interstitial protein transport must consider the possible contributions of each of these mechanisms to the overall protein flux. Molecular diffusion is the result of random thermal motions of the solute. When coupled with convective transport, the diffusive flux represents the solute flux relative to the convective component. In a porous medium the apparent diffusive flux of the solute is somewhat hindered, due to both the increased pathlength of the tortuous channels that the solute must follow and the reduced cross-sectional area available to the solute due to the presence of the sohd matrix In dilute solutions, interactions between solute molecules are negligible [91]. The local diffusive flux of protein species k through the interstitium is then described by Fick's Law: [88]. <9Ck (3.37) dx; where j is the diffusive flux of protein species k in the X ; direction. The effective molecular Chapter 3. Formulation of the General Model of Interstitial Transport 45 diffusion coefficient, D ^ , is typically less than the protein's free difhision coefficient, reflecting the hindering effects of the matrix components. Therefore, as interstitial hydration increases, DrfT approaches the free chffiision coefficient [53, 4, 27]. Comparison of the diffusion of plasma proteins and various dextrans within tissues suggests that charge and molecular size also play a major role in determining the effective diffusivity of individual macromolecular species [61]. The local convective velocity of an interstitial plasma protein may be somewhat less than the local fluid velocity, due to the hydrodynamic interaction between the protein molecule and the sohd matrix [92, 87, 61]. This phenomenon has been analyzed, from a theoretical standpoint, for the case of neutrally buoyant spheres travelling through narrow cylindrical channels (e.g. [23, 31]). The extent to which a particle is hindered (given by the ratio of the local particle velocity to the local fluid velocity, v") depends both on the position of the particle relative to the wall and the ratio of the particle radius to the channel radius, A [23]. Brenner and Gaydos [23] estimate the mean velocity of particles in the channel for cases where A is less than or equal to 0.2. Their analysis reveals two opposing effects. On the one hand, the velocity ratio v" decreases with increasing particle radius, due to hydrodynamic interaction. However, the larger particles are also restricted, due to steric exclusion, to the more central portions of the flow field where the local fluid velocities are higher. For this range of A, Brenner and Gaydos predict mean particle velocities that are greater than the mean fluid velocity within the channel, even though the local particle velocities are always less than the local fluid velocities. Because of the complex geometry of the interstitium, the extent to which this type of inter-action influences interstitial protein transport is unknown. Based on the foregoing discussion however, we assume that the mean convective velocity of protein species k in the x; direction, vk., is related to the mean interstitial fluid velocity within the protein's distribution volume, v 'pby vck; = (3-38) where the fluid and protein velocities are defined relative to the sohd phase velocity. The Chapter 3. Formulation of the General Model of Interstitial Transport 46 convective hindrance of the protein, £ k , is less than or equal to 1. Since £ k is a function of A, it will vary with interstitial hydration. The mean interstitial fluid velocity in the X ; direction is related to the fluid flux through the distribution volume, j k . , by < = (3-39) The total convective flux of protein species k in the X j direction, j k , relative to the moving solid phase is 4 = n k - £ k - v V C k , ( 3 - 4 0 ) which states that the convective solute flux is equal to the net fluid flux through the protein's distribution volume fraction (n k • v k . ) times the protein's convective hindrance ( £ k ) and the local protein concentration (C k ) . This expression can be rewritten in terms of the total fluid flux, j ° . , relative to the solid phase by noting that ^ • < = § - J w ; - (3-41) Equation (3.40) then becomes j k ? K o Ck , (3.42) where the bracketed term may be identified as the retardation factor, R k , [92, 61] associated with protein species k. In light of the preceding discussion, it does not necessarily follow that the mean convective velocity of the protein m exceeds the mean velocity of some larger protein n. According to Eqs. (3.38) and (3.41), the ratio of these two velocities depends on the quantity ( £ m K m n n ) / ( f n K n n m ) . This suggests a new, alternative mechanism for the 'gel chromatographic effect' where the mean transit time through the interstitium for larger protein molecules is less than that for smaller proteins [104]. This mechanism is quite different from the one proposed by Watson and Grodins [104], who divided the interstitial space into a 'gel phase', in which proteins move by restricted diffusion, and a 'free fluid phase', in which proteins are transported by convection and free diffusion. In their model the smaller proteins, which access a greater percentage of Chapter 3. Formulation of the General Model of Interstitial Transport 47 the gel phase, are retarded compared to the bigger proteins, which are largely restricted to the free fluid channels. Equations (3.38) and (3.41) together suggest that protein exclusion in a continuum may be sufficient to account for the gel chromatographic effect without introducing open channels in the description of the interstitium. The contribution of mechanical dispersion to the total interstitial plasma protein flux has neither been addressed experimentally nor theoretically. The mechanical dispersive flux arises from variations in the true microscopic convective velocity of the protein from the mean con-vective velocity given by Eq. (3.38). This includes the phenomenon of Taylor dispersion [96, 97] which results from local velocity profiles within a given channel, and the fact that the protein, because of its finite size, cannot access the entire channel cross-section [23]. Mechanical disper-sion in porous media also results from deviations in the microscopic flow paths of the solute particles from the direction of bulk convective flow [8]. Like diffusion, mechanical dispersion tends to spread an advancing solute front. It is therefore generally assumed that the mechanical dispersive flux obeys Fick's Law (see Anderson for details [1]): j U = - n k ^ , (3.43) where j ^ . is the flux of protein species k in the X ; direction resulting from mechanical dispersion, and $ k is the protein's coefficient of mechanical dispersion, a second rank tensor. In general, the dispersive flux is some fraction of the total convective flux. It is therefore significant only when the magnitude of the convective protein flux is large compared to the diffusive flux. Mechanical dispersion is a function of both the local convective protein velocity and the structure of the porous medium. The latter effect is characterized by a set of parameters, the longitudinal and transverse dispersivities (QJ and a t). For an isotropic medium, t?k is related to the components of the mean protein convective velocity and the dispersivities by [8] -k -k tfk = a k |v k | tfs + ( a k - a k ) (3.44) where |v k | is the magnitude of the mean convective velocity of the protein, i.e., (3.45) Chapter 3. Formulation of the General Model of Interstitial Transport 48 While the dispersivities depend on the medium's pore geometry, no analytical expressions exist to link these parameters to other appropriate material properties, such as the solid phase volume fraction and the medium's hydraulic conductivity. Hence, in practical applications, the dispersivities are adjusted to give the best possible agreement between experimental observa-tions and model predictions [1]. Since the dispersivities reflect, for example, the tortuosity of the pathways available to the various plasma protein species, they will vary with tissue hy-dration. Furthermore, since the pathways for different protein species will vary as a result of exclusion, the dispersivities may also be expected to vary amongst protein species. The sum of the diffusive flux (Eq. (3.37)), the convective flux (Eq. (3.42)), and the mechan-ical dispersive flux (Eq. (3.43)) gives the total protein flux at any point within the interstitium, relative to the moving solid phase. Equations (3.37), (3.42), and (3.43) will be used in Section 3.2.3 to develop expressions for the transient distributions of the various macromolecular species within the interstitial space. 3.2 Mass Balance Equat ions for Sol id , F l u i d , and Solute Species In the previous section we developed mathematical expressions for the flow of solid, fluid and protein species within a deformable interstitium. We will now develop the equations that describe the transient distribution of these phases within the interstitium. The equations de-scribing fluid transport through a porous medium subject to small deformations have been applied in a number of fields, including groundwater hydrology and soil mechanics [19, 8. 101]. The equations are based on differential mass balances for the solid and fluid phases, combined with an appropriate description of deformation. A similar approach is adopted here. Likewise, the equations describing the distribution of various plasma protein species within the interstitial space are based on differential mass balances on each of the protein species contained within the hiterstitium. Because of the linkages existing between the fluid flux and the protein os-motic pressures, between the convective protein flux and the fluid flux, and between the various transport properties and the tissue hydration, the material balances result in a set of coupled Chapter 3. Formulation of the General Model of Interstitial Transport 49 partial differential equations which must be solved simultaneously. We will now consider each of the material balances individually. 3.2.1 Mater ia l Balance on the Solid and Immobile F lu id Phases The solid and immobile fluid phases have much the same impact on mass flow within the interstitium in that they both reduce the volume available to mobile fluid and plasma proteins. Furthermore, given that the immobile fluid phase volume fraction is a function of the solid phase volume fraction, it is convenient to consider the two as a single composite phase (n s + F s (n s )) when carrying out mass balances on the various components within the interstitium. Assuming that the density of the solid phase is constant, a material balance on the solid and immobile fluid phases within a differential volume of interstitium gives d(n s + F') g([ns + F 5 ] - v S i ) . dT~ = ~ dl, ' ( 3 ' 4 6 ) where [ns + F s ] • Vj. is the net flux of the composite phase, per unit volume, at a point within the interstitimn. Equation (3.46) states that the net rate of change in the composite phase volume per unit volume of interstitium at some point is equal to the net flux of the phase at that point. The solid phase velocity, v s., relative to a fixed coordinate system, is related to the solid displacement in the x; direction, TJ;, by [101J = (3-47) Since the volumetric dilation, ev, is equal to d\Ji/dxi, then dvs. dev (3.48) dxi dt ' The local solid phase velocity can therefore be related to the local average hydrostatic pressure through the compliance relationship (Eq. (3.25)). Together, Eqs. (3.46) and (3.48) describe the distribution of solid material and immobile fluid in response to variations in the local average hydrostatic pressure. Chapter 3. Formulation of the General Model of Interstitial Transport 5 0 3.2.2 M a t e r i a l Balance on the Flu i d Phase Consider the flow of fluid-protein solution within the interstitium. For dilute solutions, varia-tions in density can be neglected. Furthermore the volumetric flux of solution is approximately equal to the total solvent flux, q° . . A material balance on the total fluid-protein mixture within a differential volume of interstitium then gives di dx-, • v ; The total mobile fluid volume fraction, n° , can be rewritten in terms of the sohd phase volume fraction, i.e., n° = 1 - n s - F s (n s ) . (3.50) Furthermore, the total solvent flux relative to fixed coordinates, q^., may be expressed in terms of solvent flux relative to the sohd phase, j ° . , the sohd phase velocity, v s ; , and n s using Eqs. (3.31) and (3.50), i.e., < = J w i + ( l - n s - F s ( n s ) ) - v S i . (3.51) The second term of Eq. (3.51) represents the flux of mobile water at the sohd phase velocity. Equation (3.49) can now be expressed in terms of j ° . , n s , F s , and v S i using Eq. (3.51) to give 5(n s + F s ) d([ns + F s ] - v s ; ) + o dt dx; Since the sohd phase and immobile fluid phases are conserved, the sum in square brackets is zero (see Eq. (3.46)). Also, the second term in Eq. (3.52) is equal to dev/dt, so that Eq. (3.52) becomes 5T--5x7 ( 3 - o 3 ) which states that the rate of volumetric dilation at a given point in the interstitium is equal to the net rate of fluid inflow to that point. Finally, using the expressions for j° and ev developed earlier (see Eqs. (3.35), (3.24) and (3.25)), Eq. (3.53) can be rewritten in terms of the local average hydrostatic pressure and the Chapter 3. Formulation of the General Model of Interstitial Transport 51 local fluid chemical potential to give 1 T = 5 « ( ' - - S 4 (3'54) 3.2.3 Material Balance on Protein Species k The net rate of increase in the mass of protein species k contained in a differential element within the interstitium is equal to the net diffusive, dispersive and convective flows of protein into the element. In a fixed coordinate reference frame - ± (»M«S + DSA] f ) - £ + B X ] ( ? ) (3,5, where — n k(i9 k + D k f f6jj)c?C k/dXj is the total dispersive and diffusive flux, n k V s . C k is the convec-tive protein flux at the solid phase velocity, and R k j ° . C k is the additional convective protein flux due to the motion of the fluid relative to the solid phase. is the retardation factor for protein species k, denned in Section 3.2.2. Equation (3.55) is combined with the equation for volumetric dilation (Eq. (3.48)) to give g(n k C k ) k p > T , c?(nkCk) fifRi&C") dt  + dt '  Si d^  + d^ Equation (3.56) may be interpreted as follows. The first term represents the net rate of change in protein content, per unit volume of interstitium, within the element. The second term represents the change in protein content associated with deformation within the interstitium. The third term represents the net convective flow of protein, at the solid phase velocity, out of the element, while the fourth term is the remaining convective flow associated with the solute motion relative to the solid phase. The final term represents the net dispersive and diffusive flows leaving the element. 3.2.4 Summary of Governing Equations Table (3.1) summarizes the equations describing fluid, solids, and protein transport in a deform-ing interstitium. Alongside are fisted the primary dependent variables obtained as the solution Chapter 3. Formulation of the General Model of Interstitial Transport 52 of each equation. The equations and dependent variables are grouped into one of three cate-gories: those describing interstitial deformation, those describing interstitial fluid transport and distribution, and those pertaining to solute transport and distribution within the interstitium. The first category includes equations for the conservation of sohd phase and local sohd phase velocity (Eqs. (3.46) and (3.47)), relationships linking the geometry of the deformed interstitium to the local volumetric dilation (Eqs. (3.27) and (3.28)), and constitutive relationships express-ing volumetric dilation as a function of the local average hydrostatic pressure (Eqs. (3.25) and (3.26)). The second category consists of an equation for the conservation of fluid mass within the system (Eq. (3.54)), expressions relating local fluid fluxes to gradients in local fluid hydro-static and coUoid osmotic pressures (Eq. (3.35)), and coUoid osmotic relationships (Eq. (3.13). The final category is comprised of the conservation equations for the various protein species (Eq. (3.56)), expressions hoiking each of the distribution volume fractions to the sohd phase volume fraction (Eq. (3.7)), and relationships used to define the various protein and fluid fluxes and velocities (Eqs. (3.36), (3.37), (3.38), (3.39) and (3.43)). Together these equations form a coupled system that must be solved simultaneously. For example, the total local volumetric fluid flux, j ° . , is a function of the local colloid osmotic pressures, II k, k = l ,2 , . . .m, and hence the local concentrations of the various plasma protein species, C k , k = l ,2, . . .m. In the next three chapters, we demonstrate how the method of Finite Elements can be used to solve sets of coupled equations, based on those shown in Table (3.1), for a number of simplified circumstances. U N K N O W N GOVERNING EQUATION L O C A T I O N IN T E X T I. INTERSTITIAL DEFORMATION n • o J ( 1 i = 1,2,3 -°V> = 0 '1 W Y I ' 1 " , ' > . /Tij. (3.46), Sec. 3.2.1 U i , 1=1,2,3 « . ,= ^ . ' £g. (3.47), Sec. 3.2.1 * i t _ y. £<?. (3.28), Sec. 3.1.2 % _ " £ 9 . (3.29), Sec. 3.1.2 P „ A 3 1 7 1 " . ' ' / £ 9 . (3.25), Sec. 3.1.2 = 1,2,3 m ' - l _ T ^ f ' (3.26), Sec. 3.1.2 = ^ 1 2-*=o&n £ 0 . (3.17), Sec. 3.1.1 p ° - n ° = P 1 - n 1 = p 3 - n ' - n J =... = P M -x)™ = 1 n* II. FLUID TRANSPORT II** = 1,2 m , = £ ( * ° & (J"" - Er=, "')) • Eq. (3.54), Sec. 3.2.2 Jl,i= 1,2,3 nk = C*(C*). xTij. (3.13), Sec 3.1.1 j " — j£L (p m _ yj"1 II*) El4 (3 33), Sec. 3.1.3 III. PLASMA PROTEIN TRANSPORT C , A = l , 2 , . . , m i ^ . J . f ) l » U + J , l A-.-lasiW « n „ t „ + f i * , - « l c O (3-6), Sec. 3.1.1 * .* .» '= 1.2,3; j = 1,2,3 * l B r « + X , ^ H ^ K T U " " ' ' + ^ H C J £ g . (3.56), Sec. 3.2.3 « i „ » = 1,2, 3 ' ' = «f + U -at) Eq. (3.44), Sec. 3.1.4 . = ^ ' (3.39), Sec. 3.1.4 . k V - - ^ - (3.38), Sec. 3.1.4 J ' ° " l ~ 1 , 2 , 3 (3.36), Sec. 3.1.3 Jitl — K" Jul' Table 3.1: Summary of the model equations. Chapter 3. Formulation of the General Model of Interstitial Transport 54 3.3 Concluding Remarks In the preceding two sections, mathematical relationships were developed to describe the tran-sient flow and distribution of fluid and plasma proteins within the interstitium. resulting in a system of coupled, nonlinear partial differential equations which must be solved simultaneously (see Table (3.1)). Despite the complexity of the model, it is limited in several respects. First, the description of interstitial deformation applies to small strains only (on the order of ten percent). The model is therefore unsuitable for analysing extreme cases of edema formation. However, the model could be expanded to consider large deformations by introducing a more general and, as a result, a more complex finite deformation theory (e.g., [21]). Furthermore, the model uses a compliance relationship to characterize deformation, which assumes that any change in volume is a function of the hydrostatic pressure within the system. This neglects the influence of shear stresses on deformation. However, we are interested primarily in the effect of volume changes on the various transport properties and material characteristics of the inter-stitial space (such as the hydraulic conductivity, effective diffusivities, and various distribution volume fractions), rather than a description of the deformed geometry of the interstitium. We therefore consider this approach a reasonable first approximation to the complete theory devel-oped by Biot [19]. A more detailed analysis of deformation would require additional information about the material properties of the various interstitial components, such as their stress-strain characteristics. Fluid flow within the interstitium is assumed to be proportional to the gradient in fluid chemical potential alone, thus neglecting any coupling between fluid flow and solute chemical potential, for example. The theory presented here could easily be modified to include these additional effects, given better information about the nature of fluid transport in the intersti-tium. Previous models of interstitial fluid transport have considered the effect of hydrostatic pressure gradients only [22, 36, 87]. Therefore, because it includes the influence of colloid os-motic as well as hydrostatic pressure gradients, the interstitial fluid flux representation given here is considered to be more general than that offered by any previous interstitial transport Chapter 3. Formulation of the General Model of Interstitial Transport 55 models. However, further research in the area of fluid flow within osmotically active, partially restricting matrices (such as the interstitium) is needed. Despite its limitations, the model describes the combined effects of a number of interstitial properties (such as exclusion and swelling characteristics) and transport mechanisms (such as protein convection, (hffusion, and dispersion) on interstitial fluid and plasma protein transport. It therefore provides a far more comprehensive description of interstitial transport than has been offered by any of the previous models. The model can be used to study numerous aspects of interstitial transport over a wide range of physiological conditions. When combined with mathematical descriptions of fluid and protein exchange across the capillary wall and, where c appropriate, the lymphatic wall, the model provides a tool to investigate the sensitivity of the microvascular exchange system to any number of parameters characterizing its transport behaviour. The next three chapters give examples of the model's utility in this respect by investigating fluid and plasma protein exchange, both in mesentery and a hypothetical tissue, under steady-state and transient conditions and for a number of systemic upsets. Chapter 4 Steady-State Exchange i n Mesenteric Tissue In the previous chapter we presented a general mathematical model describing the transport and distribution of fluid and macromolecules within the interstitium, which is treated as a mul-tiphase, deforming porous medium. In this chapter, a simplified version of the model, in which the interstitium is approximated as a rigid porous medium containing a single plasma protein species, is combined with mathematical descriptions of transport across the interstitial bound-aries to study steady-state fluid and protein exchange within the mesentery. The mesentery was selected both for its simple geometry and because a number of its transport parameters have been measured. Furthermore, the mesentery remains a popular tissue for experimental studies of microvascular exchange [30] and interstitial transport [115, 40]. The mesentery consists of a thin sheet of loose connective tissue, the upper and lower surfaces of which are bounded by a serous membrane (the mesothehum) made of a single layer of epithelial cells [50]. In some respects, then, fluid and plasma protein exchange across the mesothehum may resemble that of the capillary wall. While it is generally accepted that fluid and plasma proteins are able to cross this boundary and enter the surrounding peritoneal fluid [50, 37], the mesothehum's exchange properties remain poorly defined. Therefore, one objective of this study is to explore the potential influence of the mesothehum on the steady-state exchange of fluid and a single plasma protein species within the mesentery. Three scenarios are considered: 1. the mesothehum is impermeable to fluid and proteins; 2. the mesothehum's transport properties are identical to those of the capillary wall; and 3. the resistance of the mesothehum to fluid and plasma protein exchange is substantially 56 Chapter 4. Steady-State Exchange in Mesenteric Tissue 57 lower than that of the capillary wall. These three cases were selected so as to encompass a wide range of mass transport character-istics. We also investigate the sensitivity of microvascular exchange in the mesentery to a select number of interstitial parameters, namely the tissue hydraulic conductivity, the protein effective diffusion coefficient and the protein convective hindrance. The possible influence of dispersion on interstitial transport is not considered in this study. (In fact, the effects of mechanical dispersion on mass exchange within this model tissue are tentatively in Appendix C.) The model equations are recast in dimensionless form. This reduces the number of interstitial parameters that must be varied from the three listed above to two equivalent dimensionless groups. A brief description of the model follows. 4.1 Defining the System Figure (4.1) is a schematic diagram of a cross-sectional portion of mesenteric tissue of uniform thickness. For simplicity we will assume that the conditions in the tissue are independent of the z-direction, thereby limiting the flow field to the two remaining dimensions. The interstitium is bounded left and right by arteriolar and venular capillaries respectively. It is assumed that the tissue thickness, H , is small relative to the distance, L , separating the vessels, so that the system can be approximated by the two-climensional rectangular domain shown in Figure (4.2). In addition, the hydrostatic pressure and plasma protein concentration in each of the vessels are assumed uniform along the vascular walls. The upper and lower boundaries of Figure (4.1) represent the mesothelial layers. For simplicity, and for lack of additional information, it is assumed that the peritoneal fluid is well-mixed, semi-infinite in extent, and subject to a uniform hydrostatic pressure along the length of the mesothehum. The following notation will apply in the remainder of the paper A superscript '0' denotes a quantity associated with the total mobile fluid phase (denned below), while a superscript '1' identifies a parameter associated with the accessible volume phase (also defined below). Figure 4.1: Schematic diagram of a cross-sectional slice of mesentery. The shaded area represents the region of interest. Chapter 4. Steady-State Exchange in Mesenteric Tissue 59 Y H/2-Arteriolar-end capillary 0 boundary -H/2 Peritoneal fluid Mesothelial boundary [ Finite element grid Mesothelial boundary Peritoneal fluid Venular-end capillary boundary Figure 4.2: Schematic diagram of the tissue segment studied. The system is assumed symmetric about the x-axis; hence only the upper half of the tissue is modelled. T h e finite element grid is superimposed on this portion of tissue. T h e aspect ratio ( H / L ) is 0.1. For simplicity, the curvature associated with the vessels' walls is neglected. Chapter 4. Steady-State Exchange in Mesenteric Tissue 60 Parameters associated with the arteriolar capillary, the venular capillary, and the mesothehum are identified by superscripts 'art', 'ven', and 'mes', respectively. Finally, a superscript 'b' will be used to identify parameters associated with a general permeable boundary or with the well-mixed fluid on the luminal side of the boundary. The mathematical description of interstitial transport developed in the previous chapter is applied here, along with these additional simplifying assumptions. 1. Since detailed information on the swelling properties of mesentery is unavailable, it is assumed that the tissue behaves as a rigid porous medium. Furthermore, for lack of additional information, the material properties of the tissue are considered to be spatially invariant. 2. In this study we are not concerned with the relative transport rates of individual plasma protein species within the interstitium. Therefore, the array of interstitial plasma pro-tein species is treated as a single aggregate displaying averaged properties. In fact, sev-eral steady-state simulations were performed to investigate the effects of treating the plasma and interstitial fluid as aqueous solutions containing two different osmotically ac-tive plasma proteins representative of albumin and globulin. The results of that study, not presented here, identified albumin as the dominant osmotically active plasma protein, when these species are present in physiological concentrations. These fmdings substanti-ate the notion that albumin is the major contributor to the coUoid osmotic driving forces within tissues. 3. In light of assumption 2, the interstitium contains two distinct mobile fluid phases, only one of which is accessible to proteins (see Figure (4.3)). The accessible volume fraction is denoted by n 1 , while the total mobile fluid volume fraction is represented by n°. In addition to the two mobile fluid phases, the interstitium contains a 'solid' phase, n s , composed of elements such as hyaluronate. elastin, collagen and proteoglycans, and an immobile fluid phase, n m . It is recognized that, under some conditions, hyaloronate Chapter 4. Steady-State Exchange in Mesenteric Tissue 61 may be mobile [5]. However, provided that the relative amount of mobile hyaloronate is small, and assuming that interstitial hyaloronate lost to the circulation is replaced, so that the physicochemical properties of the tissue does not change, it is reasonable to neglect movement of this component. Further, the intrafibrillar water of the collagen is included in the immobile phase, due to the comparatively low hydraulic conductivity of the intrafibrillar spaces [66]. 4. The total interstitial hydraulic .conductivity, K° , is divided between the accessible fluid phase and the excluded mobile fluid phase according to their proportionate share of the total mobile fluid volume. Hence, K> = ^ . K ° . (4.1) Equation (4.1) therefore neglects any variations in flow resistance between the accessible fluid phase pathways and the pathways of the excluded fluid phase. Given further in-formation about the relative resistances of these two pathways, an alternative expression relating K 1 to K° can be substituted for Eq. (4.1). 5. Interstitial protein transport occurs via molecular diffusion and restricted convection only; i.e., mechanical dispersion is not considered. This represents a significant limitation only in convectively dominant problems. 6. Body forces, such as gravity, are neglected. 7. The system is at steady-state. Since the tissue is assumed rigid, a material balance on the fluid within a differential volume of interstitial space gives (see Eq. (3.53) in Chapter 3) where is the local total fluid flux in the x; direction. This fluid flux is related to the gradient in hydrostatic pressure in the accessible fluid phase. P 1 , and the gradient in plasma protein Chapter 4. Steady-State Exchange in Mesenteric Tissue 62 Sn° n TL Figure 4.3: A schematic diagram of an elementary volume of interstitium illustrating the different volume fractions associated with any one point in the continuum representation of the interstitial space. Chapter 4. Steady-State Exchange in Mesenteric Tissue 63 osmotic pressure, II 1, according to Eq. (3.33) in Chapter 3. That is, (4.3) The local protein osmotic pressure depends on the local plasma protein concentration in the accessible volume, C 1 , according to a polynomial relationship: n 1 = A 1 • c1 + A 2 • ( c 1 ) 2 + -43 • ( c a ) : (4.4) Substituting Eq. (4.3) in Eq. (4.2) gives d2{P1 - n1) d2(P1 - n1) dx2 + dy2 -: 0. (4.5) The local flux of plasma proteins within the interstitial space consists of a convective com-ponent, j c ; , and a diffusive component, j d . . The first of these is given by Eq. (3.42) in Chapter 3: K \ - P • — • i° - C 1 Jc; — C. j£0 J w ; ^ , (4.6) where £ is the convective hindrance. The (diffusive flux is defined by Eq. (3.37) in that same chapter: Jd; - nx D efT d C 1 (4.7) A material balance on the plasma proteins within a differential volume of interstitium then gives (see Eq. (3.56) in Chapter 3) n J D ^ C 1 c W dx2 + dy 2 K 1 K° .0 d& ^ .0 d& J w i dx ' J w ' ay = 0. (4.8) The first term in Eq. (4.8) represents the net diffusive flux of plasma proteins at a point within the interstitium, per unit volume of interstitial space. The second term is the net convective protein flux, per unit volume of interstitium, at that point. Since the system is at steady-state, the net accumulation of plasma proteins at the point (the right-hand side of Eq. (4.8)) is zero. Boundary conditions are needed to complete the description of fluid and protein exchange. Their forms depend on the physical nature of the boundaries themselves. We will consider three Chapter 4. Steady-State Exchange in Mesenteric Tissue 64 boundary types in our system: a symmetry boundary (i.e., from (0.0) to (0,L) in Figure (4.2)). an impermeable boundary (i.e., from (0,H/2) to (L.H/2), corresponding to the case where the mesothelium is treated as an impermeable barrier), and the permeable boundaries (i.e., from (0,0) to (0,H/2) and from (L,0) to (L,H/2), corresponding to the arteriolar and venular capillary wails, respectively, and from (0,H/2) to (L,H/2), for those cases where the mesothelium is permeable). In the case of a symmetry boundary, the gradients in plasma protein concentration and fluid chemical potential normal to the boundary are zero. That is. and ^(P 1 - n 1 ) " dx •lx + b ' ^(P 1 dCl~ •lx + b dc1' dx dy dy • ly = 0, J b • L = 0, (4.9) (4.10) where l x and l y are the x and y components of the unit outward normal to the boundary (see Figure (4.4)), and where implies an interstitial quantity evaluated at a point along the boundary. At an impermeable boundary the fluid flux and plasma protein flux normal to the boundary are zero. That is, (4.11) and ([idxlb + Li c j b ) • ix + (Lidjb + l i c j b ) • i y = o. (4.12) Upon substituting Eqs. (4.3), (4.6). and (4.7) into these last two expressions, Eqs. (4.11) and (4.12) reduce to Eqs. (4.9) and (4.10), respectively, cited for the symmetry boundary. Chapter 4. Steady-State Exchange in Mesenteric Tissue 65 Figure 4.4: A n elementary volume representing a point in the interstitial continuum adjacent a permeable boundary. T h e fluid film of infinitesimal thickness is in local equil ibrium with the fluid in the accessible space at that point in the continuum. T h e vectors l x and l y represent the x and y components of the unit outward normal, n. Chapter 4. Steady-State Exchange in Mesenteric Tissue 66 Fluid flow across a permeable boundary is described by Starling's Law. This is equated to the total fluid flux to the boundary from within the interstitium to give -K1 d{?1 - n 1) dx lx + ' ^ P 1 - n 1) dy L p(I p l ] b - p b - b ( [ n l - n b ) ) (4.13) where L b and <xb are the hydraulic conductivity and reflection coefficient, respectively, of the boundary, while P b and f l b represent the hydrostatic and colloid osmotic pressures on the luminal side of the boundary. Plasma protein exchange across a permeable boundar)' obeys the nonlinear flux equation [76, 71]. Equating the plasma protein flux across the boundary with the sum of the diffusive and convective protein fluxes through the available space to the boundary then gives - j i 1 D 8Cl [ C r j b - C b e( - P g ) 1 - e(- P e) dC1 •lx + dy •1, K 1 (4.14) K° V r ^ J b " • L"™TJb - ' / L J b where C b is the plasma protein concentration on the luminal side of the boundary, and where Pe, the local Peclet number for the boundary, is denned by ( [ j ° J b - l x + [ j ° , ] b - l . v ) - ( l - - b ) Pe D b (4.15) D b represents the boundary's permeability to plasma proteins. The boundary conditions defined by Eqs. (4.13) and (4.14) assume that a thin fluid film exists between the boundary and the interstitial space (see Figure (4.4)) . This fluid is in local equilibrium with the interstitial fluid within the perivascular region of the accessible space, and is therefore at a hydrostatic pressure [Px]b and protein concentration [C 1 ]^ By this assumption we circumvent the need to distinguish between the transport properties of the boundary segment exposed to the accessible space from those of the boundary segment exposed to the excluded Chapter 4. Steady-State Exchange in Mesenteric Tissue 67 space. It therefore represents a mathematical convenience rather than a physiological condition. However, since it is impossible to distinguish between these two segments when measuring the transport parameters for a given permeable boundary, Eqs. (4.13) and (4.14) are considered reasonable approximations to the conditions prevailing in vivo . In the case of m different plasma protein species, it is assumed that the fluid film is in equilibrium with the material contained in the distribution volume m. Hence, the thin film concentration of each plasma protein species is equal to the concentration within that protein's distribution volume, C k . The mathematical formulation is therefore consistent with the fact that it is the distribution volume concentration of a plasma protein species, and not the concentration based on the total fluid volume, that determines, for example, the interstitial osmotic pressure influencing fluid exchange across the capillary wall [26]. To miiiirnize the number of independent parameters that must be evaluated in the numerical simulations, the equations governing fluid and protein transport are recast in dimensionless form. This is accomplished by introducing the following nondimensional parameters: P = P / P " 4 , C = C / C " 1 , fi = n/P"", i = x / L , y = y / L , H = H / L , a = (K° • P ^ / D ^ , f3 = K V K ° , j ° . - j W i L / D e f f , j d i = j d j L / P e f f C " 1 ) , j c ; = j C i L / ( D e f f C a r t ) , M = A a • C ^ / P ^ , A 2 = A 2 • ( C a n ) 2 f P a T t , A 3 = A 3 • (Can)3/Paxt, = (L£ • L ) / K ° , and D b = ( D b • L J / D ^ . The governing equations and auxiliary relationships then take the following form. 1. Fluid transport within the interstitium: (4.16) (4.17) (4.18) 2. Plasma protein transport within the interstitium: d2Cl d2c' Jw x Q - 1 JW dC 1 (4.19) n dx2 cK-2 w* dx dy Chapter 4. Steady-State Exchange in Mesenteric Tissue 68 U = £ - / 3 - J ° i - C 1 , dx-, ' The boundary conditions axe rewritten in dimensionless form as fohows. 1. Conditions at a symmetry boundary or impermeable boundary: dx •lx + <9(pa - ii 1) dv • lv - 0, J b dCl dx •4 + J6 dCl dy •ly = 0. 2. Fluid flow across a permeable boundary: ^ ( P 1 -n 1) dx • l x - r ^ ( P 1 - n 1) dy lv £ p ( [ p 1 b - p b - b - ( M b - r f b ) ) 3. Protein transport across a permeable boundary: !i°-.-/ '-f-e i] b-n 1-+ ( [ j ° y - / 3 - £ - C 1 ] b - n 1 1 - e(- P e b ) 9C 1 <9x a c 1 Ur -u, P e b = ( l - a b ) - i f e D b Equations (4.16) to (4.26) fully describe the system. (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) 4.2 Case Studies Values for the various system parameters are reported in Table (4.1). while Table (4.2) lists values for the corresponding dimensionless parameters used in the numerical simulations. The Chapter 4. Steady-State Exchange in Mesenteric Tissue 69 values presented in Table (4.1) were drawn from the literature, where available. However, several of the variables had to be estimated, including the permeabilities of the vascular walls, D™1 and D v e n , the plasma protein reflection coefficients, cr1""1 and cr v e n , the immobile fluid volume fraction, n1™, and the accessible volume fraction, n 1 . In addition, values for the tissue dimensions, H and L , the mesothelial transport parameters, and the peritoneal fluid properties were assumed. The permeabilities D*" and D v e n were assigned values of 2.4 X 1 0 - 8 cm/s and 3.6 X 1 0 - 6 cm/s, respectively, which he within the range of capillary permeabilities to albumin reported for a variety of tissues (see [83]). Furthermore, these values were selected so that the ratio D a r t / D v e n equaled the ratio Lp^/Lp 6 1 1 reported for mesentery [39]. The protein reflection coefficients for the arteriolar and venular capillaries were both allotted a value of 0.85, which falls within the normal range reported for continuous capillaries (see [83]). It was assumed that the principal component of the immobile fluid volume was the in-trafibrillar water of the collagen. The immobile volume fraction was then calculated assuming that the specific volume of intrafibrillar water equalled 1.14 cm 3 /gm of collagen [66], and that the volume fraction of collagen in mesentery equalled that found in subcutaneous tissue. This yielded a value of 0.128 for n m . The accessible fluid volume fraction was assigned a value of 0.68, which lies within the range reported for albumin in skin (see [18]). Since , by definition, the sum of n s , n1™ and n° equals 1.0 (see Figure (4.3)), a value of 0.089 for n s [66] implies that n° equals 0.783. H was assumed to equal 3 X l O - 3 cm, which is the same order of magnitude as the mi-crovessels. L was assigned a value of 3 x l O - 2 cm. The peritoneal fluid was assumed to be at atmospheric pressure, with a plasma protein content of 0.015 gm/cm 3 . The transport param-eters L™", D m c s . and cr m c s were varied to simulate three different boundary conditions along the mesothehum. In boundary condition 1, Lp"" and D m c s were set to zero, thus describing an impermeable boundary, fn boundary condition 2, L™", D m e s , and cr"1" were set equal to the corresponding parameters for the arteriolar capillary. Finally, in boundary condition 3, L™" Chapter 4. Steady-State Exchange in Mesenteric Tissue 70 and D m e s were assigned values 100 times greater than Lp^1 and D 8 " , respectively, while cr™" was assigned a value of zero, thereby reducing substantially the resistance of the mesothehum. compared to the resistances of the other two permeable boundaries. According to Eqs. (4.18), and (4.20), the dimensionless parameters £ and a are key to characterizing the interstitial transport of fluid and plasma proteins. The first of these, the convective hindrance, is a measure of the local convective velocity of the solute relative to the local fluid velocity (see Chapter 3). The parameter a, on the other hand, is a measure of the resistance of the interstitium to plasma protein diffusion, relative to its resistance to fluid flow. Together, these two parameters determine the relative role of convection and diffusion in transporting plasma proteins through the interstitium. A series of numerical simulations were performed to investigate the coupled effects of £ and a on microvascular exchange within the model tissue for each of the three mesothelial boundary conditions outlined above. For each boundary condition, £ was assigned values of 1.0, 0.5, and 0.0, while a was given values of 9.117, 0.9117, and 0.09117, resulting in a 3 x 3 x 3 factorial study. The results of the study are presented in Section 4.4. 4.3 Numerical Procedure A form of the Finite Element Method using isoparametric elements [59J was used to solve the system of coupled partial differential equations presented in Section 4.1. The interstitial space was first divided into a set of rectangular subdomains of variable dimensions (see Figure (4.2)). To enhance the accuracy of the solution, the element size was reduced in the vicinity of the interstitial boundaries, where fluid pressure and solute concentration gradients were typically greatest. Associated with each element were eight nodes representing discrete locations within the domain. The dependent variables, P 1 and C 1 , were approximated in each element by a set of biquadratic interpolating functions, which in turn depended on the nodal values of P 1 and C 1 . By following the Galerkin procedure [59], the partial differential equations were reduced to a set of coupled algebraic expressions for these nodal values (see Appendix B for details). The er 4. Steady-State Exchange in Mesenteric Tissue Parameter Value Tissue Source A i 2.8 x IO5 dyne - cm/gm plasma [64] A 2 2.1 x IO6 dyne - cm 2 /gm 2 1.2 x IO7 dyne - cm 3 / gm 3 plasma [64] A 3 plasma [641 0.06 gm/cm 3 human serum [46] c m e s 0.015 gm/cm 3 - Assumed 0.06 gm/cm 3 human serum " [46] 2.4 x 10~8 cm/s - See text 0 - 2.4 x I O - 6 cm/s - See text D v e n 3.6 x 10~8 cm/s - See text H 3.0 x IO" 3 cm - Assumed K° 3.1 x I O - 1 2 cm4/(dyiie - s) 3.0 x 10"2 cm mesentery [111] in [66] L - Assumed T art L p 1.35 x I O- 9 cm3/(dyne - s) 0 - 1.35 x 10~ s cm 3/(dyne - s) mesentery [39] T mcs p - See text T ven 2.02 x I O - 9 cm3/(dyne - s) 6.8 x I O - 8 cm 2 /s mesentery [39] n 1 - D ^ mesentery [38] n™ 0.128 subcutaneous See text n s 0.089 subcutaneous [66] n 1 0.68 rabbit skin [86] in [13] part 2.942 x IO4 dyne/cm 2 mesentery [63] pmcs 0.0 dyne/cm 2 - Assumed p vcn 1.667 x 104 dyne/cm 2 mesentery [63] Tjart 2.707 x IO4 dyne/cm 2 - Eq. (4.4) in text 0.472 x IO4 dyne/cm 2 - Eq. (4.4) in text Jjven 2.707 x IO4 dyne/cm 2 - Eq. (4.4) in text 0.85 - See text 0 - 0.85 - See text o- v e n 0.85 - See text 0.0 - 1.0 - See text Table 4.1: Values of model parameters assumed in the simulations. Chapter 4. Steady-State Exchange in Mesenteric Tissue 72 Parameter Value 0.571 A 2 0.261 A 3 0.881 Qart 1.0 Qmes 0.25 c v e n 1.0 part 0.0072 pjmes 0. - 0.72 D v e n 0.0180 H 0.1 T art ^ P • 13.06 T mcs 0. - 1306. f ven 19.55 n m 0.128 n s 0.089 n 1 0.68 part 1.00 pmcs 0.00 pvcn 0.57 a 0.09117 - 9.117 P 0.87 jjart 0.92 jjmes 0.16 jTven 0.92 ^art 0.85 0.00 - 0.85 C T v e n 0.85 0.0 - 1.0 Table 4.2: Values of dimensionless parameters assumed in the simulations. Chapter 4. Steady-State Exchange in Mesenteric Tissue 73 system of algebraic equations was then solved using a banded matrix technique [100]. Because of their coupled nature, the fluid and protein material balance equations were solved iteratively. A n initial guess of C 1 was used to calculate the local gradients in plasma protein osmotic pressure, by differentiating Eq. (4.17). The hydrostatic pressure distribution was then calculated using Eq. (4.16) and its corresponding boundary conditions. Using this solution of P 1 and Eq. (4.18), the local fluid fluxes throughout the interstitial space were determined. A n updated estimate of C 1 was then obtained by solving Eq. (4.19), subject to the assigned concentration boundary conditions. The iteration procedure was terminated when one of the following two criteria was met. 1. The change in the value of the dependent variables at each node during successive itera-tions satisfied the conditions p.1. - pi . ^ < I O - 5 , ( 4 . 2 7 ) and PJ i.max C-1- - & i 1J 1-1.. f l i.max < 1 0 - 5 , (4.28) where denotes value of some variable tb at node j , calculated in the ith iteration, and where V'i.max represents the maximum nodal value of ip from that iteration. 2. The total number of iterations exceeded 200. In the latter case convergence was not achieved to within the specified tolerance, and the solutions were rejected. Under-relaxation techniques [24] were used where needed to achieve convergence. As an additional check of the numerical solution's accuracy, overall material balances were performed around the boundaries of the system. In all cases, the total inflow of fluid and plasma proteins equalled the total outflow, to within .005 percent. Finally, numerical tests were performed to determine the sensitivity of the solution to grid size. Increasing the grid density from 501 to 971 elements produced less than one percent change in the calculated fluid and protein exchanges across each of the permeable boundaries. Thus, all of the results presented hi the next section were produced with a grid having 501 elements. Chapter 4. Steady-State Exchange in Mesenteric Tissue 74 4.4 Results and Discussion The large number of simulations performed makes it impractical to discuss each one in detail. Instead the following discussion focuses on selected examples of how the model can be used to investigate microvascular exchange in this system. The discussion is divided into four parts. In the first part we demonstrate with a specific example how the model can be used to predict fluid and plasma protein flow patterns and plasma protein transport mechanisms within the interstitium. The second part discusses the influence of f, a. and the mesothelial transport properties on the net fluid exchange across each of the permeable boundaries while the third part considers the influence of these parameters on net plasma protein exchange across the boundaries. In the final section we discuss the effects of £, a. and the mesothelial transport parameters on plasma protein distribution and transport within the interstitial space. 4.4.1 A Specific Case of Interstitial Transport The analysis of fluid and plasma protein transport within the interstitium is complex, due to the coupled nature of the transport equations and the nonlinear effects arising from the osmotic activity of the interstitial plasma proteins. In the following discussion we will seek a mechanistic interpretation of the model predictions for the specific case where £ equals 0.5, a equals 9.117, and the mesothelial transport properties are described by boundary condition 2. However, this interpretation is only possible with the detailed description of interstitial fluid and plasma protein flow patterns and plasma protein distribution afforded by the model itself; the results are not intuitively obvious. The solution of Eqs. (4.16) and (4.19), with the appropriate auxiliary equations and bound-ary conditions, yields the steady-state distributions of both the dimensionless hydrostatic pres-sure, P 1 , and the dimensionless plasma protein concentration. C 1 . throughout the interstitial space. Combining tins information with the expressions for the local interstitial fluid flux (Eq. (4.18)). the local convective protein flux (Eq. (4.20)) and the local diffusive protein flux (Eq. (4.21)) gives a complete description of fluid and plasma protein transport within the modelled Chapter 4. Steady-State Exchange in Mesenteric Tissue 75 interstitivirn. Figure (4.5) presents plots of the distribution in the interstitium of the dimensionless hy-drostatic pressure and the dimensionless total plasma protein concentration, C 1 , where n 1 ct = (T^ r c 1 ' <4-29) i.e., the plasma protein concentration based on the total fluid volume. C 1 represents the concen-tration that would be calculated, for example, from measurements of fluid and plasma protein content within the interstitium using ultraviolet absorbance techniques. Figure (4.5) also con-tains a plot of the distribution of P 1 — IP, which is a measure of the local fluid chemical potential (see Eq. (3.12) in Chapter 3). In each plot the x / L axis corresponds to the symmetry boundary in Figure (4.2), while the y / L axis hes on the arteriolar capillary wall. The concentration plot reveals a local ridge of high plasma protein content near the arteriolar capillary, and a trough near the venular capillary corresponding to a local region of low plasma protein concentration. As we will see, the profile is a direct consequence of the transport of fluid- and plasma proteins from the arteriolar end of the interstitium into the peritoneal fluid and the entry of fluid and proteins from the peritoneum to the interstitium at the venular end of the system, together with the sieving properties of the mesothehum. p m e s — <rmes • n m e s , is a measure of the driving force for fluid exchange at the mesothehum. Its value hes between part_0.art.nart ^ pven^ven.Tjven Therefore, fluid entering the interstitium from the arteriolar capillary is drawn to the mesothehum due to the lower chemical potential of the peritoneal fluid, carrying with it plasma proteins. The plasma proteins are sieved at the mesothehum where then-concentration builds up. At the venular end of the system, fluid is drawn into the interstitium from the peritoneum and then removed from the interstitium by the venular capillary. Again proteins are sieved at the mesothehum so that the fluid-plasma protein solution entering the interstitium is diluted somewhat, causing the local washout of proteins seen in the surface plot. Interstitial plasma proteins carried by the fluid to the venular capillar}' once again build up due to the sieving properties of this boundary. Hence, the peritoneum acts here as both an infinite source and an infinite sink for fluid and plasma proteins. A substantial portion of the fluid and Chapter 4. Steady-State Exchange in Mesenteric Tissue 76 (iii) Figure 4.5: Surface plots of the distri b u t i o n of (i) dimensionless to t a l concentration, (ii) dimensionless hydrostatic pressure, and (iii) dimensionless fluid chemical po-tential (expressed as an equivalent pressure, P 1 — I I 1 ) in the interstitium, for the case where £ = 0.5, a = 9.117, and the mesothelial transport properties are defined by boundary condition 2. The x/L axis represents the symmetry boundary, while the y/L axis lies on the arteriolar-end capillary boundary. Note that the tissue's aspect ratio, H, is exaggerated in these figures to provide greater detail. Chapter 4. Steady-State Exchange in Mesenteric Tissue 77 plasma proteins entering the interstitium from the arteriolar capillary are transported into the peritoneum at the arteriolar end of the system. Some fraction of this re-enters the interstitium in the vicinity of the venular capillary, bypassing the central region of the interstitium altogether. The local gradient P 1 - IT1 gives the driving force for fluid flow within the interstitium. Comparing the surface plots of P 1 and P 1 — IT1 (Figures (4.5) (ii) and (iii), respectively), it is clear that the colloid osmotic pressure contributes significantly to the overall driving force for interstitial fluid transport. The local ridge of high plasma protein concentration in the vicinity of the arteriolar capillary creates a local minimum in fluid chemical potential there, while the region of low plasma protein content produces a local maximum in fluid chemical potential in the vicinity of the venular capillary. Therefore, while the gradients in fluid hydrostatic pressure would suggest a flow of fluid from the arteriolar end of the system to the venular end, the gradients in fluid chemical potential produce a complex recirculating flow pattern. This is illustrated in Figure (4.6) (i). However, since the fluid chemical potential varies only marginally in the central regions of the interstitium, the fluid flow associated with the recirculation is comparatively small. It is also apparent from these surface plots that, for this case at least, the gradients in the y-direction are small compared to those in the x-direction, indicating that the mesentery acts here as a one-dimensional tissue. In fact, this is investigated for all of the cases considered here in detail in Appendix A and taken advantage of in subsequent chapters. The convective plasma protein flux pattern follows that of the fluid. However, the diffusive flux pattern must be calculated from the plasma protein concentration distribution using Eq. (4.21). The latter pattern, illustrated in Figure (4.6) (ii), reveals that the chffusive protein flux also recirculates. However, the diffusive flux often occurs in a direction opposite to the local convective plasma protein flux, particularly in the vicinity of the capillary boundaries. The sum of these two flow patterns gives the net protein flow pattern within the interstitium shown in Figure (4.6) (iii). The combined convective and diffusive patterns produce a net flow of plasma proteins from the arteriolar end of the system to the venular end. Note that the plasma protein transport characteristics could have been presented in terms of local Peclet numbers, given by Chapter 4. Steady-State Exchange in Mesenteric Tissue 78 "3 2 ' 5 f" -» —. — - " ^ *• . -• -» ^ *~ ^ ^ ^ ^ * — -. -« »- ». *. ^ *• 1 V v* V 1 vw V », r P ^ ^ i : £ £ £ ^ £ ^ *f—zz3 — - — — ^ -^(a.-) Figure 4.6: Flux patterns for the case where £ = 0.5, a = 9.117, and the mesothelial transport properties are denned by boundary condition 2. Plot (i) shows the fluid flux pattern, or equivalently, the convective plasma protein flux pattern within the interstitial space. Plot (ii) is the diffusive flux, and plot (iii) illustrates the total (convective plus diffusive) plasma protein flux. The arrows show the local directions of the fluxes at the positions corresponding to their origins and their lengths are proportional to the magnitudes of the local flux vectors. Note that the tissue aspect ratio, H, is exaggerated in these figures to provide greater detail. Chapter 4. Steady-State Exchange in Mesenteric Tissue 79 the ratio of the magnitude of the local convective flux to the magnitude of the local diffusive flux. However, the Peclet number fails to account for direction. Therefore, in Figure (4.6), we have chosen to present the predicted flux patterns. The opposing effects of the convective and diffusive plasma protein fluxes on the net protein transport are further illustrated in plots (iii) and (iv) of Figure (4.7). These panels of Fig-ure (4.7) show, respectively, the local interstitial convective and diffusive plasma protein fluxes normal to the mesothelial boundary as a function of position, x, along the mesothelium. At the arteriolar end of the boundary the convective flux transports plasma proteins to the mesothe-lium from within the interstitium, while the chffusive flux draws protein from the mesothelial boundary into the adjoining interstitial space. These trends are reversed near the venular end of the mesothehum. The lack of convective and diffusive protein transport to the mesothelium in the central portions of the boundary implies that these fluxes run parallel to the boundary in this region. The stun of the local convective and diffusive plasma protein fluxes normal to the mesothehum gives the net protein flux crossing the boundary, as a function of position x (see Figure (4.7) (ii)). For this particular case the magnitude of the normal interstitial convective plasma protein flux to the mesothehum is greater than that of the normal interstitial diffusive flux of plasma proteins from that boundary, resulting in a net transport of plasma proteins into the peritoneum. Associated with this steady-state condition, and as a result of the resistance of the mesothelial barrier to plasma protein transport, there is local high concentration of in-terstitial plasma proteins near the arteriolar end of the system, and a local dilution of plasma proteins near the venular end. From the above example it is clear that transport within the system can be complex. In some cases this yields surprising behavior that could be subject to misinterpretation. Consider, for example, the fluid and plasma protein exchange across the mesothelial boundary when its transport properties are defined by boundary condition 2, with £ = 1.0, and a = 9.117. Panels (i) and (ii) of Figure (4.8) show these fluxes as a function of x. Despite uniform mesothelial transport properties, there is a localized region of high fluid and plasma protein exchange, Chapter 4. Steady-State Exchange in Mesenteric Tissue 80 -5 | . . . . 1 -S I . • • ' • 1 OO IO O 1 DIMENSIONLESS DISTANCE " DIMENSIONLESS DISTANCE Figure 4.7: Panels (i) and (ii) represent local dimensionless fluid fluxes and plasma protein fluxes crossing the mesothelium, as a function of position along the bound-ary, when £ = 0.5, a = 9.117, and the mesothelial transport properties are given by boundary condition 2. Panels (iii) and (iv) show the local dimensionless con-vective protein flux and the local dimensionless diffusive protein flux reaching the mesothelium from within the adjacent regions of the interstitium. T h e s u m of (iii) a n d (iv) yields the net protein flux crossing the mesothelium (panel (ii)). A negative value represents a flux directed into the interstitial space, while a positive quantity denotes a flux directed into the peritoneal fluid. Chapter 4. Steady-State Exchange in Mesenteric Tissue 81 located at approximately x = 0.2, that could be erroneously interpreted as a 'leaky site' in the mesothelial layer. 4.4.2 Fluid Exchange across the Boundaries of the Interstitium Table (4.3) lists the average fluid fluxes crossing each of the permeable boundaries for the various cases studied. Note that, with £ = 1, a = 9.117, and the mesothelial transport properties denned by boundary condition 3, the simulation failed to converge to the required tolerances. Hence, no numerical results are reported for this case. (In fact, in this case the solution suffered from oscillations from one iteration to the next, suggesting that alternate choices for the relaxation parameters could possibly alleviate the problem.) £ a Boundary Condition 1 Boundary Condition 2 Boundary Condition 3 Art. Ven Mes Art Ven Mes Art Ven Mes 1.0 0.09117 -0.04 0.04 — -0.22 0.06 0.01 -0.31 0.09 0.01 1.0 0.9117 -0.40 0.40 — -2.30 0.63 0.08 -2.87 0.65 0.11 1.0 9.117 -4.49 4.49 — -25.16 6.84 0.92 No Convergence 0.5 0.09117 -0.04 0.04 — -0.22 0.06 0.01 -0.31 0.09 0.01 0.5 0.9117 -0.37 0.37 — -2.26 0.61 0.08 -2.87 0.66 0.11 0.5 9.117 -4.30 4.30 — -24.90 6.46 0.92 -26.14 2.47 1.18 0.0 0.09117 -0.03 0.03 — -0.22 0.06 0.01 -0.31 0.09 0.01 0.0 0.9117 -0.34 0.34 — -2.21 0.60 0.08 -2.86 0.66 0.11 0.0 9.117 -2.42 2.42 — -20.20 5.70 0.72 -25.58 2.92 1.13 Table 4.3: Average dimensionless fluid fluxes across permeable boundaries. A nega-tive value indicates a flux entering the interstitium, while a positive value denotes a flux leaving the interstitial space. As seen in Table (4.3), an increase in a led consistently to an increase in the net fluid exchange across each of the permeable boundaries. For example, increasing a from 0.9117 to 9.117, with £ equal to 0.5 and the mesothelial transport properties given by boundary condition 2, increases the fluid exchange across each of the permeable boundaries by an order of magnitude. It should be noted that a, which is defined as K°Part/Dee-, was increased by increasing the Chapter 4. Steady-State Exchange in Mesenteric Tissue 82 nUEMSJCNLBS DISIMCE HMENSJQNLESS DCSXUCS F i g u r e 4.8: C o m p a r i s o n o f t h e l o c a l d i m e n s i o n l e s s fluid fluxes a n d p l a s m a p r o t e i n fluxes a s a f u n c t i o n o f p o s i t i o n a l o n g t h e m e s o t h e l i u m f o r v a r i o u s v a l u e s o f £, a s s u m -i n g a = 9.117, a n d t h e m e s o t h e l i a l t r a n s p o r t p r o p e r t i e s a r e d e f i n e d b y b o u n d a r y c o n d i t i o n 2. I n p a n e l s ( i ) a n d ( i i ) £=1.0, i n p a n e l s ( i i i ) a n d ( i v ) £=0.5, a n d i n p a n e l s (v) a n d (vi) £=0.0. Chapter 4. Steady-State Exchange in Mesenteric Tissue 83 value of K ° . Furthermore, L b is defined as L b L / K ° . To maintain L b constant, L b was increased by a proportionate amount. The increase in fluid exchange accompanying an increase in a is therefore attributed to the enhanced fluid transport properties of both the interstitium and the permeable boundaries. As previously mentioned, the fluid exchange rate within the system depends on the values of P b and rib, which are the driving forces, as well as the transport properties of each of the permeable boundaries. While £ affects the plasma protein transport mechanisms within the interstitium it does not influence the transport properties of the permeable boundaries, nor does it alter the fluid chemical potential in the blood or the peritoneal fluid. In these cases, where the principal resistances to fluid flow are at the boundaries, a change in £ generally had only a marginal effect on the net fluid exchange to or from the interstitium. However, £ did influence substantially the distribution of fluid flux crossing the mesothehum, since it affected the distribution of interstitial plasma proteins and therefore the distribution of interstitial fluid chemical potential. This is illustrated in panels (i), (iii), and (v) of Figure (4.8), which show the distribution of fluid fluxes crossing the mesothehum when £ equals 1.0, 0.5, and 0.0, respectively, for the case where a equals 9.117 and the mesothelial transport properties are defined by boundary condition 2. Enhancing the transport characteristics of the mesothehum typically led to a moderate increase in fluid exchange across the arteriolar capillary, due to the increased capacity for the system to exchange fluid with the peritoneum. Consider, for example, the case where £ = 0.5, and a = 0.9117. Altering the mesothelial transport properties from those given by boundary condition 2 to those of boundary condition 3 increased the fluid flux across the arteriolar capillary from 2.26 to 2.87. 4.4.3 Plasma Protein Exchange across the Interstitial Boundaries Table (4.4) reports the average plasma protein fluxes across the permeable boundaries for each of the 26 successful simulations. The enhanced fluid exchange associated with an increase Chapter 4. Steady-State Exchange in Mesenteric Tissue 84 in a produced a concomitant increase in the convective plasma protein exchange across the permeable boundaries, thereby increasing the total exchange of plasma proteins within the system. £ a Boundary Condition 1 Boundary Condition 2 Boundary Condition 3 Art Ven Mes Art Ven Mes Art Ven Mes 1.0 0.09117 -0.005 0.005 — -0.034 0.000 0.002 -0.046 0.001 0.002 1.0 0.9117 -0.060 0.060 — -0.345 0.069 0.014 -0.430 0.029 0.020 1.0 9.117 -0.673 0.673 — -3.774 0.396 0.169 No Convergence 0.5 0.09117 -0.005 0.005 — -0.034 -0.000 0.002 -0.046 -0.001 0.002 0.5 0.9117 -0.056 0.056 — -0.339 0.064 0.014 -0.430 0.029 . 0.020 0.5 9.117 -0.645 0.645 — -3.736 0.512 0.161 -3.920 0.130 0.189 0.0 0.09117 -0.005 0.005 — -0.034 -0.000 0.002 -0.046 0.001 0.002 0.0 0.9117 -0.051 0.051 — -0.332 0.060 0.014 -0.430 0.028 0.020 0.0 9.117 -0.364 0.364 — -3.030 0.433 0.130 -3.840 0.144 0.184 Table 4.4: Average dimensionless plasma protein fluxes across permeable bound-aries. A negative value indicates a flux entering the interstitium, while a positive value denotes a flux leaving the interstitial space. Material balances dictate that the net amount of plasma proteins entering the venular capil- . lary and the peritoneum must equal the net amount of plasma proteins entering the interstitium from the arteriolar capillary. In general the exchange of plasma protein across the arteriolar capillary was predominantly convective. Since £ had negligible effect on the net fluid influx across the arteriolar boundary, it had little impact on the net amount of plasma proteins en-tering the system. However, since £ had a strong influence on the distribution of fluid flux crossing the mesothehum, it also influenced the distribution of plasma protein fluxes crossing that boundary (see panels (ii), (iv), and (vi) of Figure (4.8)). The influence of the mesothehum on net plasma protein exchange paralleled its influence on net fluid exchange across each of the permeable boundaries. For example, a change from boundary condition 2 to boundary condition 3, with £ = 0.5 and a = 0.9117, increased plasma protein exchange across the arteriolar capillary from 0.339 to 0.430. Again, this behavior is attributed to the increased capability of the interstitium to exchange material with the Chapter 4. Steady-State Exchange in Mesenteric Tissue 85 peritoneum. 4.4.4 Interstitial Plasma Protein Convection, Diffusion, and Distribution According to Eq. (4.20), interstitial plasma protein convection is directly proportional to the interstitial fluid flux available to transport proteins, as well as the local concentration of plasma proteins within the interstitium. Plasma protein diffusion, on the other hand, is proportional to the local gradient in plasma protein concentration (see Eq. (4.21)). Therefore, the influence of £, a, and the mesothelial transport properties on protein convection and diffusion within the interstitium will depend upon the effect of these parameters on each of the interstitial fluid flux, the local interstitial plasma protein concentration, and the interstitial plasma protein gradients. Consider first the influence of the parameters on convective plasma protein transport. Since an increase in a typically enhanced fluid flow through certain regions of the interstitium (par-ticularly in the vicinity of the arteriolar capillary), such a change promoted protein convection there. Likewise, reducing the resistance of the mesothehum to fluid and protein exchange in-creased fluid transport through these regions of the interstitial space. However, such a change also tended to decrease the average value of C l within the entire interstitium (see Table (4.5)); i.e., as the resistance and plasma protein sieving properties of the mesothehum were decreased, the interstitial fluid composition approached that of the peritoneal fluid. For example, a change from boundary condition 2 to boundary condition 3, holding £ and a constant at 0.5 and 0.9117 respectively, reduced the average value of & within the interstitium from 0.54 to 0.18. (It is worth noting that those mean dimensionless interstitial concentrations in the range of 0.31 to 0.37 predicted by a number of the simulations agree closely with the typical value of 0.33 reported by Drake and Gabel [45].) The overall influence of the mesothehum on plasma pro-tein convection therefore depended on the relative magnitudes of the two opposing effects of increased fluid flow and reduced interstitial plasma protein concentration. Finally, while £ had only a limited influence on the magnitude of net exchange of fluid between the interstitium and the vascular system, it determined the degree of convective hindrance for plasma protein Chapter 4. Steady-State Exchange in Mesenteric Tissue 86 transport (see Eq. (4.20)). It therefore played a key role in deteimining the degree of plasma protein convection within the interstitium. Q Boundary Boundary Boundary Condition 1 Condition 2 Condition 3 1.0 0.09117 0.74 0.31 0.19 1.0 0.9117 0.61 0.55 0.18 1.0 9.117 0.24 0.37 No Convergence 0.5 0.09117 0.74 0.31 0.19 0.5 0.9117 0.69 0.54 0.18 0.5 9.117 0.42 0.54 0.19 0.0 0.09117 0.75 0.31 0.19 0.0 0.9117 0.77 0.53 0.18 0.0 9.117 0.95 0.52 0.20 Table 4.5: Mean values for the total plasma protein concentration, C , for each of the simulations. The plot of C 4 (Figure (4.5) (i)) reveals comparatively small gradients in the y direction. This suggests that averaging the concentration in this dimension will still provide a reasonable picture of the plasma protein distribution within the interstitium. Furthermore, a y-averaged profile approximates more closely the plasma protein distributions obtained experimentally using, for example, ultraviolet light absorbance techniques [40, 115]. We will therefore refer to these averaged profiles in our discussion of diffusion within the interstitial space. The y-averaged plasma protein concentration profile of each of the 26 cases is given in Figure (4.9). A single plot is reported for each of the nine possible combinations of £ and mesothelial boundary conditions. Each plot contains up to three curves corresponding to the three values of a considered. Comparing plot (ii) to plot (iii), for example, it is evident that enhancing the transport properties of the mesothehum reduced the plasma protein concentra-tion gradients within the central regions of the interstitium, suggesting reduced diffusion there. In Figure (4.9) (ii), with a equal to 9.117, the high convective flux of plasma proteins encoun-tered a barrier at the mesothehum, creating a local buildup of proteins that promoted diffusion Chapter 4. Steady-State Exchange in Mesenteric Tissue 87 towards the central portions of the interstitial space. As the mesothelium became more per-meable to fluid and proteins, the plasma protein buildup was ehminated, and the diffusive flux was reduced, producing the corresponding profile in plot (iii). A decrease in a promoted plasma protein diffusion relative to convection within the inter-stitium, since a is a measure of the resistance of the interstitium to diffusion relative to its resistance to fluid flow. Furthermore, a reduction in a resulted in less protein exchange across the permeable boundaries, as discussed earlier. The enhanced protein diffusivity, relative to fluid conductivity, and the reduced quantity of plasma proteins entering the interstitium caused a flattening of the interstitial plasma protein concentration profiles in all of the plots. With £ equal to zero, interstitial plasma protein transport was limited to diffusion alone (see Eq. (4.21)), so that the protein concentration profiles were often altered substantially from those in which protein convection occurred. For example, the local buildup of plasma proteins due to the high convective plasma protein flux to the mesothehum discussed earlier (see plots (ii) and (iv)) is absent in plot (viii) where plasma protein transport is by diffusion only. It is not generally possible, however, to identify the dominant transport mechanism on the basis of the averaged concentration profile alone. Compare, for example, the curves in plot (vi), corresponding to a value of 0.5 for £ to the curves in plot (ix) in which £ equals 0.0. The curves closely resemble one another. However, in the former case, the ratio of the average interstitial plasma protein convective flux in the £ direction to the diffusive flux in that direction ranges from 1.52 to 2.34 in the vicinity of the arteriolar capillary, indicating significant plasma protein convection in this region for all values of a considered (see Table (4.6)). In plot (ix), however, interstitial plasma protein transport is by cliffusion alone. Based on the above discussion, it is clear that no single parameter can be identified that fully characterizes fluid or plasma protein transport within the interstitium. It is the com-bined influence of the various transport parameters that determine the relative importance of interstitial plasma protein convection to diffusion. This is illustrated in Table (4.6). No clear Chapter 4. Steady-State Exchange in Mesenteric Tissue B O U N D A R Y ( X N O T I C N 1 B C U N B A K Y (XMJTTTON 2 BCXJJ>TDAKY ( X M J T I C N 3 f=l£> o i l i i / \ 1 t - y £=05 £=0u0 ix . . . . i . . . . Figure 4.9: The thickness-averaged dimensionless total concentration, C t / C a r t , as a function of position, x / L . The nine plots correspond to the nine different combina-tions of boundary conditions (columns) and values of £ (rows) studied. Each plot contains up to three curves corresponding to the three values of a considered (i.e., the solid line corresponds to a equal to 0.09117, the dotted line corresponds to a equal 0.9117, and the chain-dot line corresponds to a equal to 9.117). Chapter 4. Steady-State Exchange in Mesenteric Tissue 89 a Boundary Condition 1 Boundary Condition 2 Boundary Condition 3 Art Ven Mes Art Ven Mes Art Ven Mes 1.0 0.09117 -1.22 -1.21 — -1.69 -0.99 -2.26 -3.54 -0.97 23.22 1.0 0.9117 -1.35 -1.21 — -1.34 -1.21 -1.34 -7.89 -1.21 5.73 1.0 9.117 -64.51 -1.21 — -2.71 -1.21 -1.23 No Convergence 0.5 0.09117 -1.55 -1.53 — -5.28 -0.98 9.16 2.34 -0.93 0.92 0.5 0.9117 -1.67 -1.53 — -1.89 -1.53 -1.98 1.52 -1.53 0.75 0.5 9.117 -9.22 -1.53 — -2.30 -1.53 -1.62 1.80 -1.53 0.71 Table 4.6: Ratio of average plasma protein convection to average plasma protein diffusion normal to each of the permeable boundaries, evaluated in the intersti-tial space adjacent the respective boundaries. A negative values indicates that convection and diffusion are in opposite directions. trend appears relating the ratio of convective to diffusive protein transport to a, the mesothe-lial transport properties, and non-zero values of £. However, the data reported in Table (4.6) emphasize the importance of convection in the model's prediction of interstitial plasma protein transport for all cases in which £ is non-zero. Finally it is noted that, under certain circumstances, the local concentration of interstitial plasma proteins in the accessible space, C1, exceeded that in the blood. Consider, for example, the case where a=9.117, £=1 .0 , and the mesothelial transport properties are given by boundary condition 2. The buildup of plasma proteins at the mesothehum discussed earlier caused C 1 to reach a value of 0.8 at x=0.2, which corresponds to a value of 1.07 for C 1 . A more dramatic concentrating effect was observed for the case where the mesothehum was assumed impermeable, a equalled 9.117, and £ equaled 0 (corresponding to diffusion only within the interstitium). In this case the large convective flux of plasma proteins crossing the arteriolar capillary caused a buildup of protein in the interstitial space, until the plasma protein gradient was sufficient to transport proteins by diffusion at the same rate as they entered the interstitium at the arteriolar capillary. Chapter 4. Steady-State Exchange in Mesenteric Tissue 90 4.4.5 Comparison of Model Predictions to Experimental Data To date, there is little experimental data in the literature describing the distribution of native interstitial plasma proteins within a specific tissue. Recently, however, Friedman and Witte [40] measured the interstitial plasma protein concentration profile in rat mesentery. We will therefore refer to this data set in the following discussion. Furthermore, B.J. Barber of the University of Wisconsin has improved on the technique used by Friedman and Witte and is currently using it to determine plasma protein content and distribution within mesentery. It is therefore expected that even better data will be available in the near future. Friedman and Witte employed ultraviolet light absorbance techniques and fluorescent tracers to determine local interstitial plasma protein content and interstitial fluid content, respectively, as a function of position in the ileal mesentery of the rat. A segment of the tissue bounded by arteriolar and venular microvessels was selected for study, where the distance separating the two vessels was approximately 295 pm. The experimental determination of the local interstitial plasma protein content was based on the fact that aromatic amino acids have maximum light absorption at a wavelength of 280 nm and negligible absorption at 320 nm. Therefore, by performing two measurements of hght absorbance using these wavelengths, the authors were able, in principle, to distinguish between the absorbance due to the interstitial plasma proteins the absorbance associated with other non-specific material. Local fluid volume was determined by measuring the hght intensity from the fluorescent tracers (in this case, sodium and FITC-dextran) that distribute rapidly throughout the entire interstitial fluid volume. From the measurements of local plasma protein content and fluid content, and employing some simplifying assumptions regarding the geometry of the tissue (e.g., that the thickness of the mesenteric tissue segment is constant), the authors estimated the variation in the local concentration of interstitial plasma proteins (i.e., Ct) with position. The results of the experimental study are presented in Figure (4.10). Due to the considerable scatter in the data, the authors calculated two profiles based on the upper and lower limits in the scatter, as well as an average profile lying between these two limits. The top graph of Chapter 4. Steady-State Exchange in Mesenteric Tissue 91 Figure (4.10) shows upper and lower bounds of the concentration profile associated with local fluctuations in the measurements, while the bottom graph plots the mean value of these two curves. The boundary parameters and interstitial transport parameters of the model were adjusted to obtain a reasonable fit between the model predictions of C 1 and the mean concentration profile shown in Figure (4.10). This was done for two different scenarios: one in which it was assumed that interstitial plasma protein transport occurs by diffusion alone (i.e., £ equal to 0), and one in which both convection and diffusion take place (i.e., a non-zero value for £) . In both cases, the parameter values were detemiined by trial-and-error using only a few iterations. A rigorous least-squares fit was not attempted. Hence, it is conceivabe that other choices of parameters might lead to even better agreement between model predictions and experimental data. With £ equal to zero, a reasonable match between the model predictions and experimental data was obtained by adjusting the following parameters as stated, keeping the other variables at their baseline values: Lp^ = 1.5 X 10- 8cm 3/(dyne-s), Lp""31 = 3.0 x 10- 8cm 3/(dyne-s), LJJ"" = 1.0 x 10- 9cm 3/(dyne-s), cr** = 0.85, <rven = 0.80, tr™5 = 0.70, P v e n = 2.207 X 10 4dyne/cm 2, C m e s = 3.0 gm/dl, K° = 3.0 x 10- ncm 4/(dyne-s), and D e f f = 2.0 x 10- 7 cm 2 /s . I £ * and Lp™ are therefore somewhat higher than reported for mesentery, but not outside the general range of values reported in the literature [71]. A similar profile could also be obtained assuming £ equalled 0.35 and the assigning these same parameters the following values: Lp^1 = 1.4 x 10- 8cm 3/(dyne-s), Lp™ = 1.6 X 10- 8cm 3/(dyne-s), L ^ = 5.0 X 10- 9cm 3/(dyne-s), o** = 0.75, <rven = 0.70, o^" = 0.51, P v e n = 2.207 x 10 4dyne/cm 2, C m e s = 3.6 gm/dl, K° = 3.0 x 10 _ 1 1crn 4/(dyne-s), and = 1.0 X 10~ 7cm 2/s. Again, while and L £ m are elevated, they remain within the range reported in the literature. The arteriolar and venular capillary reflection coefficients, meanwhile, are somewhat lower than reported in the literature for mesentery. But again, the values he within the range reported for single capillaries in frog mesentery, for example [71]. The resulting profiles for these two cases are compared to the experimental data Chapter 4. Steady-State Exchange in Mesenteric Tissue 92 Figure 4.10: The upper graph shows the maxima and minima associated with the ex-perimental determination of interstitial plasma protein concentration distribution in rat mesentery by Friedman and Witte [40]. The lower graph plots the average between these two. Chapter 4. Steady-State Exchange in Mesenteric Tissue 93 in Figure (4.11). 1.0 o *—t % E -iz; w o % o o CO CO w o t—t CO 2 0.0 0.0 1.0 DIMENSIONLESS DISTANCE Figure 4.11: The model predictions of C* assuming £ is zero (solid line) and assum-ing £ is 0.35 (dotted line) are compared here to the mean concentration profile determined by Friedman and Witte. In both cases, fluid and plasma proteins enter the interstitial space from the two vascular compartments and leave the interstitium via the mesothehum. Clearly, when £ is zero, all plasma protein transport is by diffusion alone. However, when £ is 0.35, there is substantial convective transport of plasma proteins within the interstitium. For example, the ratio of convection to diffusion at the arteriolar, venular and mesothelial boundaries is 3.36, 2.56, and Chapter 4. Steady-State Exchange in Mesenteric Tissue 94 1.63 respectively. It is also interesting to note that neither of these scenarios agrees with Friedman's and Witte's interpretation of the data. These authors assumed that fluid and plasma proteins entered the interstitium across the arteriolar wall, some of the proteins then crossing the mesothehum to cause the local gradient in concentration near that vessel. However, in contrast to the model predictions, they further assumed that proteins were transported by convection to the venular vessel where they were reabsorbed into the blood. It is clear from Figure (4.11) that reasonable agreement between experimental data of Fried-man and Witte and model predictions is possible assuming drastically different interstitial plasma protein transport mechanisms. In both cases, however, the hydraulic conductivities of the vascular boundaries had to be increased by an order of magnitude, while decreasing the reflection coefficients for these vessels somewhat, to match the experimental data. More importantly (and contrary to opinions expressed by some [115, 74]), it is evident from this ex-ample that, without reasonable estimates of the transport properties of arteriolar, venular and mesothelial boundaries, one cannot draw definitive conclusions regarding interstitial plasma protein transport mechanisms from concentration profiles in mesentery. 4.5 Concluding Remarks In the preceding sections we applied a simplified version of the general model of interstitial transport developed in Chapter 3 to study the influence of a number of transport parameters on microvascular exchange in mesentery. The analysis was limited in several respects. First, the simplified model failed to account for possible deformation resulting from pressure gradients within the interstitium. The extent to which this limits the analysis depends on the deformation characteristics of the mesentery, which remain poorly defined. Second, the study focussed on steady-state exchange only. Since the model considered only a single 'average' plasma protein species, it neglected the possible influence of several distinct plasma protein species on the overall exchange of fluid and proteins in the system. Thirdly, values for a number of the model parameters were unknown and had to be estimated from the best available data. Chapter 4. Steady-State Exchange in Mesenteric Tissue 95 Finally, the largest fluid and plasma protein fluxes occurred in the vicinity of the arteriolar and venular capillaries, which were approximated by rectangular boundaries. The vessels' curvature may have to be considered to provide a more accurate description of fluid and plasma protein exchange in these regions. The findings of the study are therefore hypothetical. However, several points are noted which warrant further attention. These are summarized below. 1. A recent experimental study of the movement of labelled albumin in rat mesentery sug-gests that convection plays a significant role in interstitial plasma protein transport [115, 74]. Our numerical investigation further suggests this even at reduced values of convective hindrance, £. Hence, diffusion models [38, 7] may represent an oversimplifica-tion of interstitial plasma protein transport in this tissue. However, the model also shows that steady-state interstitial plasma protein concentration profiles alone yield insufficient information to determine the principal mechanisms of plasma protein transport within the interstitium. In some cases where plasma protein transport was predominantly convective, the profiles are virtually indistinguishable from those in which plasma protein transport is purely diffusive. These profiles are strongly influenced by the transport properties of the mesothehum, for example. Further information about the mesothehum's exchange char-acteristics, as well as other system parameters, is needed to interpret interstitial plasma protein distribution data (see, for example, [40]). 2. Because it is influenced by osmotic as well as hydrostatic pressure gradients, the hydrody-namics within the interstitium can be quite complex, culminating, for some circumstances, in the recirculation of fluid within the mterstitium. The hydrodynamics, in combination with the sieving properties of the bounding walls, can also result in irregularities in the distribution of fluid and plasma protein fluxes crossing a permeable boundary, such as the mesothehum, even when the boundary's transport properties are uniform. This could lead to the erroneous identification of 'leaky sites' within the system. Chapter 4. Steady-State Exchange in Mesenteric Tissue 96 3. The coUoid osmotic pressure gradients exert a strong influence on the flux patterns within the interstitium, suggesting that the Darcy expression evoked in a number of previous models [22, 36, 87], which considers hydrostatic gradients only, is inadequate for describing interstitial fluid transport. The model presented here can be adapted readfly to simulate microvascular exchange in a variety of tissues. The changes might include, for example, the addition of a lymphatic vessel as an interstitial boundary, the inclusion of multiple plasma protein species in the analysis, and extension to transient conditions. In fact these changes are incorporated into the model formulation in subsequent chapters. In this way the model provides a powerful tool to investi-gate microvascular exchange under transient conditions and for other tissue systems, providing insights into the behavior of the system that may not be identified readily in laboratory studies. Chapter 5 Transient Exchange in Mesentery Following a Systemic Upset In the previous chapter we studied the steady-state exchange of fluid and plasma proteins within a segment of mesentery as a function of interstitial transport mechanisms (i.e., restricted convection and molecular diffusion) and the transport properties of the mesothelial layer. In this chapter we will extend the analysis to consider the transient behavior of the system following a systemic perturbation. Specifically, we will look at exchange within the mesenteric slab in response to two different upsets: a sustained reduction in plasma protein concentration in the blood (i.e., hypoproteinemia), and a sustained elevation in systemic blood pressure (i.e., venous congestion). As before, the system response to these perturbations will be investigated as a function of mesothelial transport properties and interstitial transport mechanisms. Since the tissue segment is assumed to be rigid, however, edema formation will not be addressed here. . The remaining portion of this chapter is divided as follows. In Section 5.1 we present the transient version of the system equations. Section 5.2 specifies the cases making up the study, while Section 5.3 outlines the numerical procedures employed in the simulations. A discussion of the results is found in Section 5.4. Finally, Section 5.5 summarizes the findings of the investigation. 5.1 T h e Governing Equations In Chapter 4 the mesentery was treated as a two-dimensional, rectangular slab. The results of that study suggested that, in many cases at least, the twc>-dimensional tissue could be adequately approximated by an equivalent one-dimensional system. This suspicion was fur-ther substantiated by a series of numerical experiments in which the simulations performed 97 Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 98 i n Chapter 4 were repeated assuming a one-dimensional mesentery'. The development of the one-dimensional equations and the results of that analysis are presented i n Appendix A . Based on those findings, al l subsequent simulations have assumed the one-dimensional geometry. Consider first the material balance equation for the interstitial f luid. Since the tissue is assumed to be rigid and the fluid is incompressible, the local interstitial fluid flux adjusts in-stantaneously to any changes that occur in the interstit ial colloid osmotic pressure distribution. Hence, the interstitial fluid mass balance equation is the same as for the steady-state case. That is, the sum of the net local efflux of interstitial fluid, per unit volume of interstitium, and the net loss of interstitial fluid to the peritoneum, per uni t volume of intersti t ium, must equal zero. Hence, for the one-dimensional mesentery we have f + | . j ? « = 0 , (5..) where j ° is the local interstitial fluid flux at some point x i n the system, H is the tissue thickness, and j™ 6 5 is the local fluid flux crossing either of the two mesothelial boundaries at that same point. (By virtue of the symmetry of the system, the fluid fluxes across the upper and lower mesothelial boundaries are identical.) The local interstitial fluid flux is given by the extended Darcy expression: where, as before, the colloid osmotic pressure, II1, is related to the local interstitial plasma protein concentration, C 1 , v ia a third-order polynomial . The fluid exchange rate between the intersti t ium and the peritoneum, meanwhile, is described by Starling's Law: jmes = jmes j p l _ pmes _ ^ j j l _ jj m e s j j ^ 3) Substituting Eqs. (5.2) and (5.3) into E q . (5.1) gives the final form of the fluid mass balance equation: d 2 ( P l ; n l ) - ^ [ p l - p m e s - *™ (nl - n m e s)] = o- 0-4) Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 99 A material balance on the plasma proteins within a differential volume of interstitium under transient conditions gives the following: IP ;o dC1 2 a s ^ Jw dx H where j™es is given by the nonlinear flux equation, i.e., •mes _ /-, m « \ -mes [C1 - C m C SeXp(-Pem c S)] J s - { 1 ~ a ) - J w ' [ l - « p ( - P e — ) ] ' ( 5 - 6 ) Pe being the modified Peclet number given by Eq. (4.15) of Chapter 4. Substituting Eq. (5.6) into Eq. (5.5) then gives 0 dC1 _ _ _ r l - m e s J w ' dx H ^ C 1 2 [C1 - C™exp(-Pe—)] , 5C1 + )J. [ L ^ . p e — ) ] ~ - n ^ T - ( 5 J) The first set of terms of Eq. (5.7) found within the square brackets represents the net convective efflux of plasma proteins from a point within the interstitium, per unit volume of interstitial space, while the second term is the net diffusive efflux of proteins, per unit volume of interstitium (mechanical dispersion effects are neglected here). The third term represents the net loss of plasma proteins to the peritoneal fluid (j™"), per unit volume of interstitium. The sum of these three terms equals the local net rate of decrease in interstitial plasma proteins per unit volume of interstitium, given by the right-hand-side of the equation. Equations (5.4) and (5.7) must be combined with the pertinent set of boundary and initial conditions. The boundary conditions at the arteriolar and venular capillaries remain unchanged from the steady-state analysis, and so are given by Eqs. (4.13), (4.14), and (4.15) of Chapter 4. The initial conditions, meanwhile, can be calculated by solving the steady-state versions of the transport equations using appropriate boundary conditions. The interstitial fluid and plasma protein mass balance equations are cast in dimensionless form using the same set of dimensionless groups as before, along with the dimensionless time, t, equal to t Deff/L2. The fluid mass balance equation then becomes d 2 ( P i _ f j i ) 2 dx2 H - ^ t ™ " [P1 - p m c s - <rmcs (n 1 - nm c s)j = o. (5.8) Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 100 The plasma protein mass balance equation, meanwhile, is given by •r0 dC1 _ - ra - 1  J w ' dx H J w , d'C1 1 -1F- + 2 fc1 - Cm e sexp(-Pem e s)| + - • (1 - <rmes) • j m e s • f : Y-1 = 0, (5.9) H 1 - exp(-Pemes)] where Pem e s is given by Eq. (4.26), and where j™" is = a • Lp""* [P1 - P m e s - <rmcs (ii 1 - n m e s)] . (5.10) This completes the mathematical formulation of the transient mass balance equations. 5.2 Case Studies As was mentioned at the beginning of this chapter, two systemic perturbations were simulated, namely the case of sustained hypoproteinemia and that of sustained venous congestion. These two upsets are discussed below. Hypoproteinemia Hypoproteinemia is characterized by a drop in the plasma protein con-centration within the blood. In the simulations presented here, it was assumed that C"*1 and C v e n fell instantaneously to 50 % of their original value (that is, from 6 gm/dl to 3 gm/dl). C m e s , on the other hand, was kept at its original value of 1.5 gm/dl. While an instantaneous drop in plasma protein content is not representative of a typical pathological state, it does provide a reasonable starting point for simulating the effects of a injection of saline into the vascular system, for example, provided that the time course for the injection is much shorter than the response time of the system. Venous Congestion The arteriolar and venular capillary pressures can be related to the venous and arterial blood pressures ( P V E N and P A R T , respectively) through the following resistance relationships [108]: part = pVEN + k i (pART _ pVENJ ^ ( 5 u ) Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 101 pvcn _ pVEN + k z ^pART _ pVEN j ; ( 5 1 2 ) where kj and k 2 are the fractions of the systemic resistances associated with the branch of the blood vasculature from the arteriolar end of the network to the heart and from the venular end of the network to the heart, respectively. Given that, in the simulations, P*"* equals 2.942 x l O 4 dyne/cm2 (22 mmHg) while P v e n equals 1.667 XlO 4 dyne/cm2 (12.5 mmHg), and assuming that P A R T and P V E N are 1.337 XlO 5 dyne/cm2 (100 mmHg) and 1.605 x lO 4 dyne/cm2 (12 mmHg), respectively, [108], then ki becomes 0.1136 while k 2 assumes a value of 0.0057. During venous congestion, P V E N is elevated. This results in an increase in both P 0 1 1 and P v c n according to Eqs. (5.11) and (5.12). In the venous congestion case studies it is assumed that P V E N increases to 3.342 x lO 4 dyne/cm2 (25 mmHg), raising and P v e n to 4.4825 x l O 4 dyne/cm2 (33.52 mmHg) and 3.400 x lO 4 dyne/cm2 (25.43 mmHg), respectively. Hence the arteriolar capillary pressure increases by 52 % of its original value, while the venular capillary pressure increases by 103 % of its baseline value. Again, it is assumed that these shifts in hydro-static pressures occur instantaneously, so that the simulations provide only a first approximation to the onset of venous congestion. Computer Simulations The three different mesothelial boundary conditions described in Chapter 4 were simulated to determine the influence of this boundary on the transient response of the system to each of the two systemic perturbations cited above. In addition, the plasma protein convective lundrance was varied to consider two extreme cases of interstitial plasma protein transport: pure diffusion (£ equal to 0), and full convection (£ equal to 1). Al l other system parameters were maintained at their baseline values during the simulations. Hence a, for example, remained 0.9117. These alterations in £ and the mesothelial transport properties resulted in a 2 X 3 factorial study for each of the perturbations considered. Finally, it is noted that the initial conditions of each of the simulations were calculated from the steady-state model, assuming pre-perturbation conditions, while the final steady-state conditions were determined using the same model and assuming the perturbed conditions Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 102 prevailed. 5.3 Numerical Procedures As before, the finite element method was used to reduce the fluid mass balance equation to a set of coupled, algebraic expressions that could be solved iteratively using matrix techniques. The interstitial plasma protein mass balance equation, however, contains both spatial and temporal terms, hi this case the finite element method was applied to the spatial terms of the equation, while a Crank-Nicolsen finite difference scheme was used to approximate the temporal term. A detailed discussion of this combined technique, as it applies to the plasma protein mass balance equation, can be found in Appendix B. The interstitial plasma protein concentration distribution and the interstitial hydrostatic pressure field were determined using the following iterative procedure. A fully explicit finite difference formulation was used to obtain a first estimate of the plasma protein concentration distribution at some time At after initiation of the system upset, using the specified initial con-ditions. This initial estimate of C 1 was used to update the hydrostatic pressure distribution, P1, using the finite element formulation. Having an estimate of both C 1 and P 1 at the new time, the Crank-Nicolsen finite difference scheme could then be used during subsequent iterations at this same time step to obtain new estimates of C 1 at At. Upon each iteration, the appropriate finite element matrices and vectors were revised to reflect the updated estimates of the plasma protein concentration distribution and hydrostatic pressure field. The iterative procedure was repeated until the convergence requirements outlined in Section 4.3 of Chapter 4 were met. This overall process was repeated at each new time step to determine the plasma protein con-centration distribution and hydrostatic pressure field as functions of space and time. Typically, the system required less than 10 iterations to achieve convergence at any one time-step. The simulation specifications were as follows. The domain was divided into 25 quadratic elements (i.e., each element contained 3 nodes) to give a total of 51 nodes within the one-dimensional tissue space. This choice of step size was based on the favourable results of the Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 103 one-dimensional simulations performed in Appendix A. Lagrange basis functions were used to approximate the spatial variations of C 1 and P 1 . The initial time-step size was chosen so that the Courant number did not exceed a specified value (see Section B.4 in Appendix B for details). For a number of cases, the value of the initial Courant number was varied over a range of values from 0.0001 to 0.01 to assure a consistent estimate of the dependent variables. The validity of the transient simulations was further coimrmed by allowing selected simulations to reach steady-state. These estimates of the new system steady-state conditions were then compared to the steady—state conditions calculated by the one-dimensional steady-state simulator. In all cases, the two predictions showed excellent agreement. 5.4 Results and Discussion We will now consider, individually, the results of the transient simulations of hypoproteinemia and venous congestion. In each case we will address the effects of the mesothelial transport properties and the interstitial transport mechanisms on the transient exchange of fluid and plasma proteins within the system, as well as their effect on the distribution of interstitial plasma proteins over time. 5.4.1 Transient Exchange in Sustained Hypoproteinemia The transient exchange rates of fluid and plasma proteins and the changes in interstitial plasma protein distribution within the mesenteric tissue segment following a drop in the vascular plasma protein content are all affected by the transport properties of the mesothelial layer and the mechanisms governing interstitial plasma protein transport. However, before discussing how these factors influence the behavior of the system during hypoproteinemia, it seems appropriate to consider briefly the effect that this perturbation has on the overall driving forces for fluid and plasma protein exchange within the system. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 104 T h e E f f e c t o f H y p o p r o t e i n e m i a o n L u m i n a l D r i v i n g F o r c e s T h e o v e r a l l d r i v i n g forces for the exchange o f fluid a n d p l a s m a p ro t e in s w i t h i n the m o d e l t issue consist o f t h e differences i n t h e effective fluid c h e m i c a l p o t e n t i a l a n d p l a s m a p r o t e i n c o n c e n t r a t i o n , r e spec t ive ly , b e t w e e n e a c h o f three l u r n i n a l fluids i n the s y s t e m (i .e . , the a r t e r i o l a r c a p i l l a r y fluid, t he v e n u l a r c a p i l l a r y fluid, a n d the p e r i t o n e a l fluid). C l e a r l y , a d r o p i n p l a s m a p r o t e i n content i n t h e b l o o d reduces t h e ove ra l l d r i v i n g force for diffusive exchange o f p l a s m a p r o t e i n s i n the s y s t e m . L i k e w i s e , t h i s r e d u c t i o n i n vascu la r p r o t e i n con ten t w o u l d t e n d to r e d u c e the ne t c o n v e c t i v e exchange o f p l a s m a p ro t e in s , s ince less p r o t e i n w o u l d a c c o m p a n y the fluid t r a n s p o r t e d across the vascu la r b o u n d a r i e s . H o w e v e r , the convec t ive t r a n s p o r t o f p l a s m a p r o t e i n s , a n d h e n c e t h e t o t a l exchange o f p ro t e in s , also depends o n the t o t a l v o l u m e o f f l u i d e x c h a n g e d b e t w e e n t h e var ious l u m i n a l c o m p a r t m e n t s . H e n c e , we m u s t also cons ide r h o w t h e lower vascu la r p l a s m a p r o t e i n c o n c e n t r a t i o n i m p a c t s o n the d r i v i n g forces for fluid exchange w i t h i n the t issue segment . T h e d imens ion le s s effective fluid c h e m i c a l p o t e n t i a l o f the l u m i n a l fluid assoc ia ted w i t h a p e r m e a b l e b o u n d a r y b , / Z ^ , is g i v e n b y P b — erh Ilb. F o l l o w i n g a d r o p i n p r o t e i n c o n c e n t r a t i o n i n the p l a s m a , the effective p l a s m a p r o t e i n o s m o t i c pressure , <rb l l b , decreases b y a n e q u a l a m o u n t i n b o t h the a r t e r i o l a r a n d the v e n u l a r c a p i l l a r y . T h e effective c o l l o i d o s m o t i c pressure o f the p e r i t o n e a l fluid, h o w e v e r , r ema ins u n c h a n g e d . H e n c e , f o l l o w i n g the d r o p i n p l a s m a p r o t e i n c o n c e n t r a t i o n i n t h e b l o o d , p,^ increases from 0.218 t o 0.692, p^ increases from -0.215 t o 0.259, a n d flQf* r e m a i n s -0.136 for b o u n d a r y c o n d i t i o n 2 a n d 0 for b o u n d a r y c o n d i t i o n 3. T h e ove ra l l d r i v i n g force for the e x c h a n g e o f fluid from one l u m i n a l c o m p a r t m e n t to a n o t h e r is g i v e n b y the difference i n the effective f l u i d c h e m i c a l p o t e n t i a l b e t w e e n the t w o c o m p a r t m e n t s . T a b l e (5.1) l is ts these for the var ious p a i r s o f c o m p a r t m e n t s . T h e f o l l o w i n g gene ra l observa t ions are m a d e . 1. W h i l e b o t h p^]: a n d /2^™ inc rease , the difference b e t w e e n the t w o , p,^ - p^f1, r e m a i n s u n c h a n g e d from p r e - p e r t u r b a t i o n t o p o s t - p e r t u r b a t i o n i n a l l cases. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 105 State Boundary Condit ion 1 Boundary Condit ion 2 Boundary Condition 3 ' s f T * e f f ' S i T ^eff > « f f * e f f ^eff * e f f ^c f f »*eff ^ t f f ^eff ^c f f _ Pre-Upset 0.433 0.433 0.354 -0.079 0.433 0.21S -0.215 Post-Upset 0.433 0.433 0.828 0.395 0.433 0.692 0.259 Table 5.1: Fluid chemical potential differences between the various luminal com-partments before and after the initiation of hypoproteinemia. 2. The magnitude of p,^ — pt'^s increases in all cases where the mesothehum is permeable. AefP ~~ /•'ST J U S O increases in all cases where the mesothehum is permeable. Furthermore, Aeff 1 ~~ changes from a negative value to a positive value in each of these cases. We will return to these general observations in later discussions of fluid and plasma protein exchange during hypoproteinemia. Mass Exchange Assuming an Impermeable Mesothelial Layer Fluid Exchange The transient fluid exchange rates across the arteriolar boundary, for those cases in which the mesothehum is impermeable, are illustrated in Figure (5.1). Since the tissue segment and fluid are both incompressible, the fluid exchange at the venular boundary is equal in magnitude to that at the arteriolar boundary, and hence is not shown. The net driving force for fluid exchange within the system is p,^ — p^. As mentioned earlier, this quantity remains unaffected by the drop in vascular plasma protein content. In addition, because the mesothehum is impermeable, fluid entering the interstitium from one vascular compartment must pass through the entire interstitial space before re-entering the blood at the other end of the tissue segment. Therefore, the overall effective hydraulic resistance of the system remains constant and equal to the sum of the two vascular wall resistances and the total resistance associated with the interstitial space. Given this, one would expect no change in fluid exchange within the system following the Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 106 Figure 5.1: The average transient fluid flux across the arteriolar capillary wall fol-lowing hypoproteinemia is shown assuming an impermeable mesothelium and (i) £ equal to unity, and (ii) £ equal to zero. In both cases the fluid flux is normalized with respect to its initial value prior to the upset. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 107 drop in systemic plasma protein concentration, provided that the transcapillary fluid exchange and interstitial fluid flux were both independent of interstitial coUoid osmotic pressures. The transient changes in fluid exchange within the tissue segment must therefore reflect the varia-tions in the interstitial osmotic pressure distribution, and hence the interstitial plasma protein distribution, with time. For the case where £ is unity, the system experiences a marginal increase in fluid exchange shortly after the drop in vascular plasma protein concentration. (At t equal to 0.1, or 15 minutes after initiation of hypoproteinemia, the dimensionless fluid flux crossing the arteriolar boundary has increased in magnitude by only a factor of 1.004, from -0.3982 to -0.3999.) This is illustrated in panel (i) of Figure (5.1), which shows the transient flux across this boundary, normalized with respect to its value prior to the onset of hypoproteinemia. The marginal rise in fluid exchange rate is attributed to the increase in f i ^ , which is only partially offset by a concomitant increase in P 1 . The effective interstitial osmotic pressure at the boundary, a0** •IT a r t , meanwhile, is still near its pre-upset value since insufficient time has elapsed to significantly reduce the concentration of interstitial plasma proteins there. Hence, the arteriolar fluid exchange rate, given by Starling's Law, is slightly greater following the upset. Subsequently, as the local plasma protein content near the boundary decreases with time, the dimensionless fluid flux across this boundary also declines so that, at t of 5.0 (i.e., 12.5 hours), it is -0.3836. This represents 48.4 % of the total drop that occurs before the system reaches its new steady-state value of 0.3662. When £ is zero, the transient fluid exchange within the system follows a different pattern (see panel (ii) of Figure (5.1)). Again, there is a marginal increase in the fluid exchange rate across the arteriolar boundary (i.e., at t equal to 0.1 units, the dimensionless fluid flux has increased in magnitude from 0.3378 to 0.3400). However, in this case the magnitude of the dimensionless fluid flux across this boundary continues to increase with time until it reaches a new steady-state value of 0.3462. which represents a 2.5 % increase over the initial value of 0.3378. At t equal 5.0, the arteriolar fluid flux has undergone 59.5 % of the total increase from initial to final steady-state conditions, indicating that the relaxation time for this case may be Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 108 somewhat less than, though of the same order of magnitude as, that when £ equals one. The fact that the the fluid exchange rate within the system increases with time when £ is zero, but decreases with time when £ is one, indicates that, with an impermeable mesothehum, the interstitial plasma protein transport mechanisms play a significant role in determining the transient fluid exchange within the model system following the onset of hypoproteinemia. As mentioned earlier, this can only be attributed to differences between the transient adjustments in the interstitial coUoid osmotic pressure distributions for the two cases. Plasma Protein Exchange and Interstitial Plasma Protein Distribution The tran-sient plasma protein exchange between the vascular and interstitial compartments is coupled to the fluid exchange between these via the convective transport of the macromolecules across the vascular boundary. In addition, plasma proteins enter the interstitium from the vascular compartment by diffusion. The relative importance of these two transport mechanisms depends on the magnitude of the fluid flux across a given permeable boundary, the degree of sieving at the boundary, and the differences in plasma protein concentration on either side of the bound-ary. Assuming that convection dominates, the total plasma protein flux across a permeable boundary b from vascular to interstitial compartments, expressed as a dimensionless quantity, is equal to (1 — erb) • j w • C b . Likewise, if the exchange is from the interstitium to the vascular compartment and assuming that convection dominates, the plasma protein flux is given by ( l - t r b ) . j « .[C 1 ]!, . Figures (5.2) (i) and (ii) Ulustrate the transient plasma protein exchange across the arteriolar and venular capiUaries assuming £ equal to 1 and £ equal to 0, respectively. In each case the flux is normalized with respect to its value prior to the upset. Consider first the case where £ equals 1. During the entire transient phase and subsequent steady-state, the plasma protein transport from the arteriolar capiUary to the interstitium is predominantly convective. Hence the transient flux of plasma proteins across this boundary foUows the general trend of the transient fluid flux profile there. However, the former profile is further characterized by a dramatic reduction in the plasma protein exchange rate immediately after the perturbation, Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 109 due to the reduced vascular concentration of plasma proteins. At the venular boundary, convective plasma protein exchange likewise dominates during the transient period and the subsequent new steady-state. However, since the interstitial plasma protein concentration near the boundary is greater than the vascular plasma protein concentra-tion during the transient period, the plasma protein flux across the venular boundary exceeds the plasma protein flux across the arteriolar boundary. This results in a net exchange of plasma proteins from the interstitium to the blood and subsequent reduction in interstitial plasma pro-tein content. Furthermore, while the dimensionless arteriolar plasma protein exchange rate is only 4.7 % greater than the steady-state value at t of 5.0, the venular exchange rate is still 66.3 % greater than its steady-state value. The length of the transient period is therefore determined by the time required to remove the excess plasma proteins from the interstitial space by way of the venular capillary. This is further illustrated in the transient interstitial plasma protein distributions shown in the left panel of Figure (5.3). Within the interstitium itself, both convection and diffusion play significant roles during the entire transient period. For example, the ratio of convection to diffusion within the interstitial space adjacent the arteriolar boundary varies from -1.15 to -1.33 from a t of 0.1 units to the new steady-state. The negative values for these ratios indicate that diffusion and convection occur in opposite directions. When £ is 0, plasma protein exchange across the arteriolar and venular capillaries is, like-wise, predominantly convective. The arteriolar plasma protein exchange rate drops slightly below, and then slowly rises to, the ultimate steady-state value so that, at t equal 5.0, the dimensionless arteriolar plasma protein exchange rate is approximately 1 % less than at steady-state. Meanwhile, the dimensionless plasma protein flux across the venular boundary first rises above its initial value of 0.0507, then steadily decreases until reaching a new steady-state value of 0.0260. By t equal 5.0, the venular plasma protein exchange rate is approximately 69 % greater than the final steady-state value. The length of the transient period when £ is zero is therefore close to that when £ is one. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 110 ART VEN 0.0 5.0 0.0 5.0 DIMENSIONLESS TIME DIMENSIONLESS TIME Figure 5.2: The average transient p lasma prote in flux across the arteriolar and venu-lar capil lary walls following hypoproteinemia is shown assuming a n impermeable mesothel ium and (i) £ equal to uni ty , and (ii) £ equal to zero. In b o t h cases the prote in flux is normal ized w i t h respect to its in i t i a l value p r io r to the upset. T h e dot ted line i n each case represents the new steady-state value. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 111 Figure 5.3: The transient total dimensionless plasma protein concentration distribu-tions ( C 1 ) following hypoproteinemia and assuming an impermeable mesothelium are shown for (i) the case where £ is 1 (left panel) and (ii) for the case where £ is 0 (right panel). In each case the solid line represents the i n i t i a l condition, the dotted hne is at t equal 0.5, the chain-dot line is at t equal 2.5, the dashed line corresponds to t equal 5.0, and the chain-dash line represents the fin a l steady-state condition. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 112 Because the plasma protein content in the blood decreases following the onset of hypopro-teinemia, the rate of plasma protein exchange across the arteriolar capillary drops. Meanwhile, the interstitial plasma protein concentration adjacent the venular boundary is near its initial condition. The net exchange rate across the venular boundary therefore exceeds the exchange across the arteriolar boundary during the course of the transient period so that, once again, there is a net loss of plasma proteins from the interstitium to the blood. This is reflected in the concentration profiles in the right panel of Figure (5.3). However, unlike the case where £ is one, the fluid exchange rate within the system remains elevated above its initial condition despite the washout of plasma proteins from the interstitium. Mass Exchange Assurning Mesothelial Transport Properties Similar to Those of the Vascular Walls In these simulations, it is assumed that the mesothelial transport properties are identical to those of the arteriolar capillary wall. Since the mesothehum is permeable, fluid and plasma proteins may be exchanged between the arteriolar capillary and the peritoneum, the venular capillary and the peritoneum, and the arteriolar and venular capillaries. In addition, the over-all effective resistance of the tissue segment to mass exchange depends on the flow patterns within the interstitium itself, since fluid and plasma proteins are able to bypass regions of the interstitium via the peritoneum. In fact, the steady-state analysis of Chapter 4 suggests that, when the mesothehum is permeable, the majority of fluid and plasma proteins exchanged between the various luminal compartments passes through only a small portion of the interstitial space. Further, intersti-tial fluid and plasma proteins located in the central portions of the tissue segment need only travel a short distance to reach the mesothelial surface. Hence, when all bounding surfaces are permeable, the system can achieve its new steady-state following the onset of hypoproteinemia much more quickly here than for those cases in which the mesothelial layer is impermeable. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 113 F lu id Exchange In the previous discussion of luminal driving forces it was noted that, fol-lowing the drop in vascular plasma protein concentration, p^ — / i j f 5 and p ^ — both increase in magnitude, suggesting that the fluid exchange between these respective compart-ments should increase following the systemic upset. In addition, it was noted that /x^p — p ^ changes from a negative quantity to a positive one, which would imply a reversal in the direction of fluid exchange between the venular capillary and the peritoneum. In fact these trends are observed both when £ is zero and when £ is one, as discussed below. Consider first the case in which the convective hindrance, £, is unity. Initially following the drop in vascular protein, there is a substantial increase in the rate of fluid exchange across each of the three permeable boundaries (see Table (5.2)). Further, the fluid exchange rate across the venular capillary changes direction, so that the vessel moves from a state of fluid re-absorption to one of fluid filtration. Likewise, the direction of the mesothelial fluid flux near the venular boundary changes direction, as shown in Figure (5.4). The system has reached steady-state with respect to fluid exchange by t equal 3.0 units (i.e., 7.5 hours), as illustrated in Figure (5.5). £ Period Boundary Condition 2 Boundary Condition 3 Art Ven Mes Art Ven Mes 1.0 Pre-Upset -2.387 0.656 0.087 -3.551 0.762 0.139 1.0 Post-Upset -5.567 -3.253 0.441 -8.600 -6.243 0.742 1.0 Steady-State -5.527 -3.247 0.439 -8.305 -5.238 0.677 0.0 Pre-Upset -2.313 0.632 0.084 -3.570 0.803 0.138 0.0 Post-Upset -5.439 -3.187 0.431 -8.630 -6.273 0.745 0.0 Steady-State -5.444 -3.180 0.431 -8.319 -5.263 0.679 Table 5.2: T h e average transient fluid fluxes across the permeable boundaries fol-lowing hypoproteinemia, for the mesothelial boundary conditions 2 and 3. In each case the table reports the flux prior to the upset ('pre-upset'), at t equal 0.001 post-upset ('post-upset'), and at the new system steady-state ('steady-state'). A negative flux indicates a flow into the interstitium. Upon examining the distribution of P 1 - ft1 at t equal 2.5 (see Figure (5.6)), it is clear that Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 114 x a I 0 I DIMENSIONLESS DISTANCE o 1 DIMENSIONLESS DISTANCE Figure 5.4: The dimensionless transient fluid flux distribution across the mesothe-li u m following hypoproteinemia, assuming mesothelial transport properties equal to those of the arteriolar capillary and a £ of 1, are shown at the pre-perturbation state (panel (i)), and at a t of 0.001 (panel (ii ) ) , 0.05 (panel ( i i i ) ) , 0.5 (panel (iv)), and at the final steady-state (panel (v)). Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 115 -55 - 43 DIMENSIONLESS TIME DIMENSIONLESS TIME DIMENSIONLESS TIME Figure 5.5: The average transient fluid fluxes across the permeable boundaries fol-lowing hypoproteinemia, assuming mesothelial transport properties equal to those of the arteriolar capillary and a £ of 1, are shown in the three panels above. Panel (i) shows the fluid flux across the arteriolar capillary, panel (ii) corresponds to the fluid flux across the venular capillary, and panel (iii) represents the net fluid flux across the mesothelium. In each case the fluxes are normalized with respect to their respective pre-perturbation values. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 116 0.25 •-3 -< CO E -O cu o • — I PO o Q t_ CO CO PO o CO 2. PO s 0.00 -0.25 DIMENSIONLESS DISTANCE Figure 5.6: T h e dimensionless interstitial fluid chemical potential distribution (P 1 — II 1) is shown at t equal 2.5 following hypoproteinemia for the case where the mesothelial transport properties equal those of the arteriolar capillary and £ is 1. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 117 the majority of interstitial fluid flow occurs in the regions near the blood vessels; the central regions of the tissue space are relatively quiescent. Movement of fluid within the interstitial space near the vessels is towards the central regions of the tissue. Conservation of fluid mass, meanwhile, is satisfied by a local, concomitant exchange of fluid across the mesothelial boundary. The transient fluid exchange within the system for the case where £ equals 0 is much the same as when £ is one. There is a rapid increase in fluid exchange across each of the permeable boundaries immediately following the onset of hypoproteinemia, as shown in Table 5.2. Again, the venular capillary shifts from a state of fluid re-absorption to one of fluid filtration. Further, the transient fluid flux distribution across the mesothelial boundary parallels that for £ equal to one, and so is not shown. However, the fluid fluxes reach their new steady-state values by t equal 1.0 (i.e., 2.5 hours), indicating a shorter transient period than that found when £ is one. Plasma Prote in Exchange and Interstitial Plasma Prote in Distribution When £ is equal to one, the plasma protein transport across each of the permeable boundaries is predomi-nantly convective. Therefore, since both the arteriolar and venular capillaries are filtering fluid, the plasma protein exchange rates across these boundaries follow the respective transient fluid fluxes. For example, the plasma protein flux across each of these boundaries reaches a new steady-state value at the same time as the fluid fluxes. However, since the vascular plasma protein content is lower subsequent to the perturbation, the increase in the plasma protein fluxes across these boundaries is not as pronounced as increase in the fluid exchange rates (see Table (5.3)). The exchange rate of plasma proteins across the mesothehum is also enhanced following the onset of hypoproteinemia. Again, since the exchange is largely convective, the plasma protein flux distribution across the mesothehum is qualitatively the same as the fluid flux distribution. The transient plasma protein fluxes across each of these boundaries is illustrated in Figure (5.7). In this case, the system reaches a new steady-state by a dimensionless time of 3.0. Since, during the transient period, the total efflux of plasma proteins across the mesothehum Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 118 £ Period Boundary Condition 2 Boundary Condition 3 Art Ven Mes Art Ven Mes 1.0 Pre-Up set -0.358 0.072 0.014 -0.533 0.762 0.025 1.0 Post-Upset -0.417 -0.244 0.049 -0.645 -0.468 0.148 1.0 Steady-State -0.415 -0.244 0.033 -0.623 -0.393 0.051 0.0 Pre-Upset -0.347 0.063 0.014 -0.535 0.036 0.025 0.0 Post-Upset -0.408 -0.239 0.049 -0.647 -0.471 0.151 0.0 Steady-State -0.408 -0.239 0.032 -0.624 -0.395 0.051 Table 5.3: The average transient plasma protein fluxes across the permeable bound-aries following hypoproteinemia, for the mesothelial boundary conditions 2 and 3. In each case the table reports the flux prior to the upset ('pre-upset'), at t equal 0.001 post-upset ('post-upset'), and at the new system steady-state ('steady-state'). A negative flux indicates a flow into the interstitium. exceeds the influx of protein across the other two boundaries, there is a net loss of plasma pro-teins from the interstitium to the peritoneum: This is to be expected for the following reasons. Prior to the onset of hypoproteinemia, the net plasma protein transport into the system is zero because steady-state conditions prevail. Immediately following the upset, there is increased fluid exchange from the blood to the interstitium. The concentration of plasma proteins within the fluid is more dilute, however, so that convective transport of plasma proteins across the vascular boundaries does not increase in proportion to the increased fluid fluxes. At the same time, the fluid exchange from the interstitium to the peritoneum increases by an amount equal to the increase in fluid exchange across the blood vessels. This increases the convective trans-port of plasma proteins from the interstitium to the peritoneum. However, the interstitial fluid crossing the mesothehum has virtually the same plasma protein concentration as that prior to the onset of hypoproteinemia, so that the increase in plasma protein exchange across this boundary is substantial. Since the vascular fluid replacing the interstitial fluid is somewhat diluted, compared to the conditions before the upset, there is a net loss of plasma proteins from the interstitial space. Furthermore, plasma protein exchange across the mesothehum does not Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 119 reach its new steady-state value until the wash-out of interstitial plasma proteins is complete. The transient dimensionless plasma protein distributions assuming £ is one are shown in the left panel of Figure (5.8). By a dimensionless time of 0.001, the profile has been altered substantially in the region of the venular blood vessel, so that the gradient in interstitial plasma protein concentration changes direction following the shift from plasma protein re-absorption to filtration at that boundary. Furthermore, by this time there is a slight increase in the local in-terstitial plasma protein concentration near the arteriolar and venular capillaries. This appears to be similar, qualitatively, to the buildup of plasma proteins seen in several of the steady-state cases of Chapter 4 and is presumably due to the same effects, namely the combination of signif-icant convective plasma protein transport within the interstitium and the sieving of proteins at the mesothehum. However, these local maxima in interstitial plasma protein concentration are soon dissipated as interstitial plasma proteins continue to be lost to the peritoneum. (Compare, for example, the profiles at a t of 0.001 and 0.05.) It is still conceivable that, in some circum-stances, irregularities in the mesothelial fluid and plasma protein flux distributions, similar to those discussed in Chapter 4, might occur as transient phenomena during hypoproteinemia. As Figure (5.6) showed, convective transport of plasma proteins is directed towards the central regions of the tissue, with the largest convective velocities occurring near the blood vessels. Based on the concentration gradients of Figure (5.8), then, plasma protein convection and djffusion continue to oppose one another in the vicinity of the vascular boundaries during the entire transient phase. Furthermore, there is very httle plasma protein transport in the central portions of the tissue throughout that period. The fact that diffusion and convection oppose one another may offer one explanation for the longer transient period of fluid exchange when £ is one, compared to that when £ is zero. The overall transient exchange of plasma proteins within the system when the interstitial convective hindrance of zero is much the same as for the case where £ is unity (see Table (5.3)). Once again, the plasma protein transport across each of the boundaries is largely convective. The transient distributions of dimensionless plasma protein fluxes across the mesothehum are Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 120 3 0 5 o 5 DIMENSIONLESS TIME DIMENSIONLESS TIME DIMENSIONLESS TIME Figure 5 .7: The average transient plasma protein fluxes across the permeable bound-aries following hypoproteinemia, assuming mesothelial transport properties equal to those of the arteriolar capillary and a £ of 1 , are shown i n the three panels above. Panel (i) shows the protein flux across the arteriolar capillary, panel (ii) corresponds to the protein flux across the venular capillary, and panel (iii) rep-resents the net protein flux across the mesothelium. In each case the fluxes are normalized w i t h respect to their respective pre-perturbation values. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 121 Figure 5.8: The transient dimensionless t o t a l plasma protein concentration distribu-tions (C*) following hypoproteinemia and assuming that the mesothelial transport properties are equal to those of the arteriolar capillary are shown for (i) the case where £ is 1 (left panel) and (ii) for the case where £ is 0 (right panel). In each case the solid line corresponds to the i n i t i a l condition, the dotted line is at t equal 0.001, the chain-dot line is at t equal 0.05, the dashed line corresponds to t equal 0.5, and the chain-dash line represents the final steady-state condition. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 122 similar here to the case where £ is one and so are not shown. The transient distribution of interstitial plasma proteins assuming £ is zero is illustrated in the right panel of Figure (5.8). Since the venular boundary shifts from a re-absorbing to a filtering one, and since the only means of interstitial plasma protein transport is by diffusion, the gradient is forced to reverse directions immediately following the onset of the perturbation. Further, because the plasma protein exchange across the permeable boundaries is largely con-vective, there is a net loss of plasma proteins from the interstitium to the peritoneum, for the same reasons given when £ is one. The mean interstitial plasma protein concentration therefore decreases with time, as illustrated in Figure (5.8). It is clear from these results that, although the transient variations in the interstitial plasma protein distributions depend strongly on the value of £ assumed, the overall transient exchange of fluid and plasma proteins within the system does not. This further emphasizes the fact that, when the mesothehum is permeable, the interstitium contributes less to the overall resistance within the system, so that the interstitial plasma protein transport mechanisms have less impact on the behavior of the system as a whole. Mass Exchange Assuming a Highly Permeable Mesothel ium In the final set of simulations of hypoproteinemia, it is assumed that the mesothelial transport properties are given by boundary condition 3 of Chapter 4; that is, the mesothehum is much more permeable than the capillary walls and offers no sieving of proteins (crTnes is zero). Fluid Exchange Once again, the simulations suggest that the transient fluid exchange within the model mesenteric tissue is affected little by the interstitial plasma protein transport mech-anisms (see Table (5.2)). Further, the general trends are similar to those found when the mesothelial transport properties niimic the arteriolar capillary, except that the fluxes are typi-cally an order of magnitude larger and the time to reach steady-state is an order of magnitude smaller. The reduced time for the transient is attributed to the lower mass transfer resistance of the mesothelial layer. Hence, following an initial increase immediately after the onset of Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 123 hjpoproteinemia, fluid exchange across each of the boundaries declines slightly over time to reach a new steady-state value by a t of 0.5 (i.e., 1.25 hours). The dimensionless fluid flux distribution across the mesothelial boundary for the case where £ is one is shown in Figure (5.9). Figure (5.9) shows that the majority of the fluid exchange across the mesothehum occurs in close proximity to the blood vessels. However, although it is not apparent in the figure, there is some exchange in the central portions of the tissue segment as well. Again, the trend is virtually identical when £ is zero. Following the drop in vascular plasma protein content, then, the interstitial fluid flow is directed towards the central regions of the tissue, independent of the value of £. Again, the fluid material balance constraints are met by the appropriate fluid exchange across the mesothehum. Plasma Protein Exchange and Interstitial Plasma Protein Distribution Since plasma protein transport is largely convective, the transients follow the fluid flux behavior, except that the increase in protein exchange across the arteriolar boundary is limited by the fact that the filtering fluid contains less plasma proteins following the upset. Again, the transient plasma protein exchange across the permeable boundaries is affected only marginally by the transport mechanisms within the interstitial space. Both when £ is one and when £ is zero, plasma proteins leave the interstitium by way of the mesothelial boundary. Since the exchange across this boundary is largely convective, the distribution of the plasma protein flux across the mesothehum follows closely the fluid flux profile, and so is not shown here. The transient distribution of plasma proteins for the two values of £ investigated are shown in Figure (5.10). In both cases the massive plasma protein fluxes across the mesothehum in the vicinity of the blood vessels reduce the plasma protein content in those regions. This causes the local depletion of interstitial plasma proteins in the vicinity of the arteriolar and venular boundaries over time, so that diffusive transport tends to move interstitial plasma proteins from the central regions of the tissue towards the vascular boundaries. For the case where £ is one, this is counteracted in part by a convective flux of plasma proteins towards the central portions of the tissue. However, when plasma protein transport is limited to Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 124 30 g -3a 30 -8 -30 30 0 1 DIMENSIONLESS DISTANCE o i DIMENSIONLESS DISTANCE Figure 5.9: The dimensionless transient fluid flux distribution across the mesothe-lium following hypoproteinemia, assuming the mesothelial transport properties are given by boundary condition 3 and £ equals 1, are shown at the pre-perturbation state (panel (i)), and at a t of 0.001 (panel (ii)), 0.05 (panel (iii)), 0.5 (panel (iv)), and at the final steady-state (panel (v)). Chapter 5. Transient Exchange in Mesentery Following a Systenvc Upset 125 Figure 5.10: The transient dimensionless total plasma protein concentration distri-butions (C*) following hypoproteinemia and assuming a highly permeable mesothe-l i u m (boundary condition 3) are shown for (i) the case where £ is 1 (left panel) and (ii) for the case where £ is 0 (right panel). The solid line corresponds to the i n i t i a l condition, the dotted line is at t equal 0.001, the chain-dot line corresponds to t equal 0.05, the dashed line is at t equal 0.5, and the chain-dash hne corresponds to the final steady-state conditions. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 126 difrusion, the interstitial plasma protein concentration distribution undergoes further alterations so that, by the time the system has reached its new steady-state, there is a net diffusion of interstitial plasma proteins from the vascular boundaries towards the center of the interstitial space in the regions adjacent the blood vessels. Because the magnitude of the shift is small and limited to the region adjacent the capillary walls, it is not apparent in Figure (5.10). Once again the interstitial plasma protein content decreases following hypoproteinemia, both when £ is one and when £ is zero. This occurs for the same reasons presented earlier when discussing plasma protein exchange assuming the mesothelial transport properties equal those of the arteriolar capillary wall. Summary of Hypoproteinemia Simulations In all cases considered here, the onset of hypoproteinemia led to a washout of interstitial plasma proteins. The transient behavior of the system during the washout, however, depended on the transport properties of the mesothehum and, to a lesser degree, on the interstitial plasma protein transport properties. When the mesothehum is assumed to be impermeable, fluid and plasma proteins must traverse the entire interstitial length in their journey from the arteriolar end of the system to the venular end. However, when it is assumed that the mesothehum is permeable, the distance travelled by material leaving the interstitium is reduced substantially. As a result, the response time of the model system to hypoproteinemia varied inversely with the permeability of the mesothelial layer. Further, when the mesothehum is impermeable, the interstitium represents a substantial portion of the total resistance to mass exchange within the system. Hence, the behavior of the system as a whole is influenced to a great degree by the conditions prevailing within the interstitial space. For example, in this case the transient changes to the interstitial colloid osmotic pressure distribution had a significant effect on the transient fluid exchange within the tissue segment. The transient fluid exchange across the boundaries of the system therefore varied according to changes in the interstitial plasma protein distribution, which depended Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 127 further on the interstitial plasma protein transport properties. However, when the mesothehum is assumed to be permeable, the behavior of the system with respect to mass exchange is dominated by the transport properties of this boundary. In addition, the interstitial plasma protein transport mechanisms have less impact on the transient distribution of interstitial plasma proteins as the transport properties of the mesothehum are enhanced. 5.4.2 T r a n s i e n t E x c h a n g e D u r i n g S u s t a i n e d V e n o u s C o n g e s t i o n In this set of simulations, the system response to sustained venous congestion is studied as a function of the mesothelial transport properties and interstitial transport mechanisms outlined in the investigation of hypoproteinemia. Once again, it is instructive to consider first the effect that venous congestion has on the driving forces for fluid and plasma protein exchange between the luminal compartments. T h e Effect o f V e n o u s C o n g e s t i o n o n the L u m i n a l D r i v i n g F o r c e s During venous congestion, the venous pressure rises, resulting in an increase in hydrostatic pressure throughout the microcirculation. However, according to Eqs. (5.11) and (5.12), the incremental increases in arteriolar and venular capillary pressures are not equal. In fact, the arteriolar pressure increases by 1.538 x l O 4 dyne/cm 2 (11.5 mmHg), while the venular pressure rises by 1.725 x l O 4 dyne/cm 2 (12.9 mmHg). The hydrostatic pressure of the peritoneal fluid, meanwhile, is assumed to remain at 0 dyne/cm 2. Since the vascular plasma protein content is unchanged following the perturbation; the effective fluid chemical potential in each of the two blood vessels of the tissue segment increases according to the change in the hydrostatic pressures. The effective chemical potential of the peritoneal fluid is unaffected by the systemic disturbance. Further, the driving force for diffusive plasma protein exchange between the various luminal compartments is unchanged following venous congestion. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 128 State Boundary Condition 1 Boundary Condition 2 Boundary Condition 3 "rfT ''eff * c f f * s f f " t f f * e f f ^ e f f " i f f * e f f ^zfl >*e« l*cS Pre-Upset 0.433 0.433 0.354 -0.079 0.433 0.218 -0.215 Post-Upset 0.370 0.370 0.602 0.232 0.370 0.738 0.368 Table 5.4: F l u i d chemical potential differences between the various luminal com-partments before and after the initiation of venous congestion. Table (5.4) lists the differences in effective fluid chemical potential for the various pairs of compartments both before and after the onset of venous congestion. The following general observations are made. 1. While in all cases both p^ and /i^p increase, the difference between the two, p^ — /i^p, decreases. 2. p^ff - p^ increases in all cases where the mesothehum is permeable. 3. The magnitude of /Z p^ — p^ also increases in all cases where the mesothehum is perme-able. Furthermore, p^1 — p^* changes from a negative value to a positive value in each of these cases. We will return to these general observations in later discussions of fluid and plasma protein exchange during venous congestion. Mass Exchange Assuming an Impermeable Mesothel ium Flu id Exchange Based on the information of Table (5.4), the fluid exchange rate within the system for this set of mesothelial boundary conditions is expected to decline following the onset of venous congestion, since p^ — p^ decreases with the increase in systemic blood pressure. In fact this trend is observed, both for the case of £ equal to one and for the case where £ is zero. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 129 DIMENSIONLESS TIME 0.9 0.8 0.0 5.0 DIMENSIONLESS TIME Figure 5.11: T h e average transient fluid flux across the arteriolar capillary wall following venous congestion is shown assuming an impermeable mesothelium and (i) £ equal to unity, and (ii) f equal to zero. In bo th cases the fluid flux is normalized with respect to its steady-state value prior to the upset. T h e dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 130 With £ equal to one, the dimensionless fluid flux across the arteriolar capillary wall drops from its pre-upset value of-0.3982 to 86 % of that (i.e., -0.3422) by t equal 0.1 (i.e., 15 minutes). This flux declines further with time to -0.3401, then gradually increases to its new steady-state value of -0.3423 (see panel (i) of Figure (5.11)). The transient fluid exchange within the system when £ is zero somewhat different. Again, by a t of 0.1, the dimensionless arteriolar fluid flux has dropped from its original value of -0.3378 to 85 % of that (i.e., -0.2879). However, by a t of 0.5 (i.e., 1.25 hours), this flux has risen shghtly to -0.2884 and, by steady-state, has reached -0.2885. Hence, in this case fluid exchange within the system is very close to its steady-state by 1.25 hours (see panel (ii) of Figure (5.11)). Note that, when £ is zero, the interstitial plasma protein distribution remains virtually unchanged following the onset of venous congestion (see Figure (5.13)). Since the interstitial plasma protein washout is less here than when £ is one, the system response time is shorter. Plasma Protein Exchange and Interstitial Plasma Prote in Distr ibut ion Consider first the case where £ is one. Plasma protein transport across the arteriolar and venular bound-aries is predominantly convective during the transient period and subsequent steady-state. The rate of plasma protein exchange across the arteriolar boundary therefore closely follows the fluid flux pattern there. Shortly after the perturbation, the dimensionless plasma protein flux across this boundary drops from its original value of -0.05973 to 86 % of that, or -0.0512. It declines further to -0.509, then slowly rises to eventually reach its new steady-state value of -0.0514. At the venular end, the dimensionless plasma protein flux drops from 0.05973 to approximately 83 % of that, or 0.0498 shortly after the onset of venous congestion. By t equal to 0.5, the protein flux has dropped to 0.0497. From this point it rises slowly to achieve the steady-state value of 0.0514. This is illustrated in panel (i) of Figure (5.12). These transients have the following effect on the interstitial plasma protein concentration distribution. Under the initial conditions, interstitial plasma protein diffusion is from the venu-lar boundary towards the arteriolar boundary, as illustrated in the left panel of Figure (5.13). Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 131 ART VEN 1.0 0.9 0.8 -i -' 1 ' • 0.0 5.0 Cd 3 o CO CO 2 o g D I M E N S I O N L E S S T I M E D I M E N S I O N L E S S T I M E Figure 5.12: The average transient plasma protein flux across the arteriolar and venular capillary walls following venous congestion is shown assuming an imper-meable mesothelium and (i) £ equal to unity, and (ii) £ equal to zero. In both cases the protein flux is normalized w i t h respect to its steady-state value prior to the upset. The dotted line i n each case represents the new steady-state value. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 132 Figure 5.13: The transient dimensionless total plasma protein concentration distribu-tions (C 4) following venous congestion and assuming an impermeable mesothelium are shown for (i) the case where £ is 1 (left panel) and (ii) for the case where f is 0 (right panel). In each case the solid line represents the initial condition, the dotted line is at t equal 0.1, the chain-dot line is at t equal 0.5, the dashed line corresponds to t equal 5.0, and the chain-dash line represents the final steady-state condition. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 133 Since fluid flows from the arteriolar capillary to the venular capillary, interstitial plasma pro-tein convection and diffusion are in opposite directions throughout the tissue space. As the fluid flow through the system decreases following the onset of venous congestion, so does the convective transport of plasma proteins through the interstitial space. With the decrease in convective plasma protein transport, diffusion within the interstitium tends to diminish the gradient in interstitial plasma protein concentration. Hence the interstitial plasma protein con-centration near the arteriolar wall increases while the plasma protein concentration near the venular boundary decreases. Now the transport of plasma proteins into the system across the arteriolar boundary is largely convective, so that it is approximately equal to (1 - era r t) • j° • C a r t . By the time the system achieves a new steady-state, this influx of plasma proteins must be balanced by the efflux of proteins at the venular boundary. This latter quantity is approximately equal to (1 — 0-art) • j w • [C 1]ven ) where [ C 1 ] v e n is the interstitial plasma protein concentration in the accessible space adjacent the venular boundary. Since o-^ equals o-ven, [ .C 1 ] v e n must eventually equal the plasma protein concentration in the arteriolar vessel to satisfy continuity. Hence the plasma protein concentration in that region eventually increases to its original value, as seen in the left panel of Figure (5.13). When £ is zero, convective plasma protein exchange across the permeable boundaries domi-nates once again. Hence, the trends in the transient dimensionless plasma protein fluxes follow7 the variations in dimensionless fluid exchange. Specifically, the dimensionless protein flux across the arteriolar boundary drops from -0.0507 to -0.0432 by a t of 0.1 units. This flux then in-creases slowly with time to eventually reach the steady-state value of -0.0433. At the venular boundary, the dimensionless plasma protein flux drops from 0.0507 to 0.0434 by t equal to 0.1 units, and continues to drop, albeit slowly, to eventually reach the steady-state value of 0.0433 (see panel (ii) of Figure (5.12)). The onset of venous congestion has a marginal effect on the interstitial plasma protein distribution when interstitial plasma protein transport is restricted to diffusion only (see the Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 134 right panel of Figure (5.13)). The drop in transcapillary transport of plasma proteins into the interstitium at the arteriolar end of the system results in the net loss of plasma proteins from the interstitium via the venular capillary. Since this reduction in plasma protein transport into the system is sustained, the net diffusive flux of plasma proteins through the interstitium must drop as the system approaches its new steady-state. Hence, the interstitial plasma protein concentration gradient must be slightly less under the new steady-state conditions, compared to the initial state. However, by the time the system has reached steady-state, conservation of plasma proteins within the system dictates that [ C 1 ] v e n equal C 8 1 * . The interstitial plasma pro-tein concentration adjacent the arteriolar boundary therefore increases to satisfy the constraints imposed by continuity and the fact that diffusion of interstitial plasma protein transport is less under the new steady-state conditions. Mass Exchange Assuming Mesothelial Transport Properties Similar to Those of the Vascular Walls F l u i d Exchange The shifts in the effective chemical potential of the various luminal fluids following venous congestion suggest that subsequent fluid exchange between the blood and the peritoneum should increase and that fluid flow across the venular capillary wall should change direction. In fact, this is observed both when £ is one and when it is zero (see Table (5.5)). Figure (5.14) shows the transient fluid exchange across the three permeable boundaries following venous congestion and assuming that £ is unity. The fluid fluxes increase dramatically following the onset of venous congestion, then continue to rise more slowly, so that the system achieves steady-state-by a t of 5.0. However, when £ is zero, the system reaches steady-state with respect to fluid exchange almost immediately after the onset of the perturbation. The transient distribution of fluid exchange across the •mesothelial boundary is qualitatively the same as shown in Figure (5.4), both when £ is one and when it is zero. Fluid exchange across the mesothehum in the vicinity of the venular capillary shifts direction so that, following the onset of venous congestion, fluid is transported from the interstitium to the peritoneum Chapter 5. Transient Exchange in Mesentery" Followuig a Systeimc Upset 135 o 0 5. DIMENSIONLESS TIME a i -DIMENSIONLESS TIME a 5 DIMENSIONLESS TIME Figure 5.14: The average transient fluid fluxes across the permeable boundaries following venous congestion, assuming mesothelial transport properties equal to those of the arteriolar capillary and a £ of 1, are shown in the three panels above. Panel (i) shows the fluid flux across the arteriolar capillary, panel (ii) corresponds to the fluid flux across the venular capillary, and panel (iii) represents the net fluid flux across the mesothelium. In each case the fluxes are normalized with respect to their respective pre-perturbation values. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 136 £ Period Boundary Condition 2 Boundary Condition 3 Art Ven Mes Art Ven Mes 1.0 Pre-Upset -2.387 -0.656 0.087 -3.551 -0.762 0.139 1.0 Post-Upset -5.857 -4.134 0.500 -9.232 -8.070 0.865 1.0 Steady-State -5.971 -4.278 0.513 -9.143 -7.347 0.825 0.0 Pre-Upset -2.313 0.632 0.084 -3.570 0.803 0.138 0.0 Post-Upset -5.720 -4.048 0.488 -9.259 -8.100 0.868 0.0 Steady-State -5.708 -4.024 0.487 -9.153 -7.364 0.826 Table 5.5: T h e a v e r a g e t r a n s i e n t fluid fluxes a c r o s s t h e p e r m e a b l e b o u n d a r i e s f o l l o w -i n g v e n o u s c o n g e s t i o n , f o r t h e m e s o t h e l i a l b o u n d a r y c o n d i t i o n s 2 a n d 3. I n e a c h c a s e t h e t a b l e r e p o r t s t h e flux p r i o r t o t h e u p s e t ('pre-upset'), a t t e q u a l 0.001 p o s t - u p s e t ( ' p o s t - u p s e t ' ) , a n d a t t h e n e w s y s t e m s t e a d y - s t a t e ( ' s t e a d y - s t a t e ' ) . A n e g a t i v e flux i n d i c a t e s a flow i n t o t h e i n t e r s t i t i u m . along the entire length of the mesothelial boundary. P l a s m a P r o t e i n E x c h a n g e a n d I n t e r s t i t i a l P l a s m a P r o t e i n D i s t r i b u t i o n Table (5.6) lists the transient plasma protein fluxes across each of the three boundaries following venous congestion. Figure (5.15), meanwhile, shows the dimensionless plasma protein flux across the three boundaries as a function of time and assuming that £ is one. Because the plasma pro-tein exchange across the boundaries is predominantly convective, the transient plasma protein exchange parallels the transient fluid exchange behavior. However, as this figure illustrates, the relative changes in protein fluxes across the permeable boundaries are more dramatic than the relative changes in fluid exchange rates. Further, the transient behavior assuming £ is zero is qualitatively the same as when £ is one, except that the length of the transient period is somewhat shorter. Both when £ equals zero and when it equals one, the time needed for the mesothelial boundary to achieve steady-state with respect to plasma protein exchange is longer than the period required for the other two boundaries. Since the arteriolar and venular capillaries filter fluid and plasma proteins following the upset, and since the transport of proteins across these Chapter 5. Transient Exchange in Mesentery Following a Systenvc Upset 137 Figure 5.15: The average transient plasma protein fluxes across the permeable boundaries following venous congestion, assuming mesothelial transport proper-ties equal to those of the arteriolar capillary and a £ of 1, are shown in the three panels above. Panel (i) shows the protein flux across the arteriolar capillary, panel (ii) corresponds to the protein flux across the venular capillary, and panel (iii) represents the net protein flux across the mesothelium. In each case the fluxes are normalized with respect to their respective pre-perturbation values. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 138 Figure 5.16: The transient dimensionless total plasma protein concentration distr i -butions (C l ) following venous congestion and assuming mesothelial transport prop-erties equal to those of the arteriolar capillary are shown for (i) the case where £ is 1 (left panel) and (ii) for the case where £ is 0 (right panel). I n each case the solid line represents the i n i t i a l condition, the dotted line is at t equal 0.001, the chain-dot line is at t equal 0.05, the dashed line corresponds to t equal 0.5, and the chain-dash line represents the final steady-state condition. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 139 £ Period Boundary Condition 2 Boundary Condition 3 Art Ven Mes Art Ven Mes 1.0 Pre-Upset -0.358 0.072 0.014 -0.533 0.762 0.025 1.0 Post-Upset -0.878 -0.620 0.059 -1.385 -1.210 0.188 1.0 Steady-State -0.896 -0.604 0.077 -1.371 -1.102 0.124 0.0 Pre-Upset -0.347 0.063 0.014 -0.535 0.036 0.025 0.0 Post-Upset -0.858 -0.607 0.055 -1.389 -1.215 0.192 0.0 Steady-State -0.856 -0.604 0.073 -1.373 -1.105 0.124 Table 5.6: The average transient p lasma pro te in fluxes across the permeable bound-aries following venous congestion, for the mesothel ial boundary condit ions 2 and 3. I n each case the table reports the flux pr ior to the upset ( 'pre-upset ') , at t equal 0.001 post-upset ( 'post-upset'), and at the new system steady-state ( 'steady-state'). A negative flux indicates a flow into the in te rs t i t ium. boundaries is largely convective, the length of the transient associated with plasma protein exchange is approximately equal to that for fluid exchange. However, as indicated in the plasma protein distributions of Figure (5.16), a considerable increase in the total interstitial plasma protein content occurs following the initiation of venous congestion. This filling period is much longer than the time required for the fluid exchange rates within the system to adjust. Further, plasma protein exchange across the mesothelial boundary cannot reach steady-state until the interstitial plasma protein concentration distribution stabilizes. The left-hand panel of Figure (5.16) shows the transient distributions of the aimensionless interstitial plasma protein concentration assuming £ is equal to one. In this case, there is a local buildup of interstitial plasma proteins in the vicinity of the arteriolar and venular vessels shortly after the onset of venous congestion, similar to the buildup observed during the early stages of hypoproteinemia. This is likely due to the same causes cited in that case, namely the combination of plasma protein convection within the interstitium and sieving at the mesothelial boundary. Furthermore, since interstitial fluid flow is directed towards the center of the tissue, interstitial plasma protein convection and diffusion generally act in opposite directions, as evidenced by the interstitial plasma protein concentration gradients of the left Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 140 panel of Figure (5.16). When £ is zero, the gradient near the venular boundary changes direction following the perturbation (see the right panel), reflecting the shift in plasma protein exchange across that boundary. In contrast to the situation during hypoproteinemia, we have already noted that the inter-stitial plasma protein content following venous congestion increases with time for both values of £ (see Figure (5.16)). This occurs for the following reason. The average plasma protein concentration in the interstitium prior to the upset is less than that in the serum. Following the perturbation, the convective flux of plasma proteins from the blood into the interstitium rises. The convective flux of plasma proteins from the interstitium to the peritoneum likewise increases. However, since the sieving properties of the three boundaries are the same (i.e., a*** = <rven = cr1 1 1 6 5), and since the interstitial plasma protein concentration is less than that of the serum, the concentration of plasma proteins in the mcorning fluid is higher than that of the fluid leaving the interstitium. In addition, since the rate of fluid flow into the interstitium equals the rate of fluid flow out of the interstitium at all times (because the interstitium is rigid), the convective flow of proteins into the tissue space must then exceed the convective flow of proteins out of the interstitium, thereby increasing the interstitial plasma protein content. Mass Exchange Assuming a Highly Permeable Mesothel ium F l u i d Exchange The data of Table (5.4) suggest that the fluid exchange within the system should increase following the onset of venous congestion. In fact, as illustrated in Table (5.5), this is true for both values of £ investigated. The normalized transient fluid fluxes across the permeable boundaries are illustrated graphically in Figure (5.17) for the case where £ is one. A similar pattern is seen when £ is zero, except that, once again, the transient period is somewhat shorter when plasma protein transport occurs by cfiffusion alone. The transient distribution of fluid fluxes across the mesothehum is qualitatively the same as that of Figure (5.9), both when £ is one and when it is zero. Therefore, the fluid flow across the mesothehum in the vicinity of the venular capillary changes direction so that, following Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 141 Figure 5.17: The average transient fluid fluxes across the permeable boundaries following venous congestion, assurning a highly permeable mesothelium (boundary condition 3) and a £ of 1, are shown in the three panels above. Panel (i) shows the fluid flux across the arteriolar capillary, panel (ii) corresponds to the fluid flux across the venular capillary, and panel (iii) represents the net fluid flux across the mesothehum. In each case the fluxes are normalized with respect to their respective pre-perturbation values. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 142 the onset of venous congestion, fluid movement is from the interstitium to the peritoneum. Furthermore, interstitial fluid flow is directed from the vascular boundaries towards the center of the tissue segment in both cases. Plasma Protein Exchange and Interstitial Plasma Protein Distribution The ex-change of plasma proteins across the various boundaries is largely convective for both values of £. Hence, the transient behavior of the system with respect to plasma protein exchange follows its pattern of fluid exchange. Further, the transient distribution of plasma protein fluxes across the mesothehum is qualitatively the same as that shown in Figure (5.9). Figure (5.18) shows the normalized transient plasma protein exchange across the three boundaries for a £ of one. In each case, there is an initial rise in the plasma protein flux, followed by a decay in the exchange rate. A similar behavior is observed when £ is zero, except that, once more, the length of the transient period is shorter. The transient plasma protein distribution for the two values of £ are shown in Figure (5.19). Whereas, when the mesothehum behaves as a sieving boundary the interstitial plasma protein content increases following venous congestion (see Figure (5.16)), in this case the mean intersti-tial plasma protein content decreases with time. Following the onset of venous congestion, the fluid exchange across the vascular boundaries increases, so that the convective flux of plasma proteins also increases. However, as fluid enters into the interstitium from the vascular com-partments, plasma proteins are sieved. In contrast, the fluid crossing the mesothelial boundary is not filtered so that, as the fluid flux across the mesothehum increases, there is a net loss of plasma proteins from the interstitium to the peritoneal fluid. This demonstrates that the sieving properties of the drainage system within a tissue (in this case, the mesothehum) play a major role in determining the ratio of interstitial plasma protein concentration to vascular plasma protein concentration. Once again the simulations suggest that, when both interstitial plasma protein convection and diffusion occur, they oppose one another. The comparatively high interstitial plasma protein content initially found in the vicinity of the venular boundary is soon depleted, as Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 143 Figure 5.18: The transient plasma protein fluxes across the permeable boundaries following venous congestion, assuming a highly permeable mesothelium (boundary condition 3) and a £ of 1, are shown in the three panels above. Panel (i) shows the protein flux across the arteriolar capillary, panel (ii) corresponds to the protein flux across the venular capillary, and panel (iii) represents the net protein flux across the mesothelium. In each case the fluxes are normalized with respect to their respective pre-perturbation values. The dotted line represents the new steady-state value in each case. Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 144 2 o 2 to o 2 O o CO CO 2 o CO 2 DIMENSIONLESS DISTANCE oo 0.0 10 DIMENSIONLESS DISTANCE F i g u r e 5.19: T h e t r a n s i e n t d i m e n s i o n l e s s t o t a l p l a s m a p r o t e i n c o n c e n t r a t i o n d i s t r i b u -t i o n s ( C 4 ) f o l l o w i n g v e n o u s c o n g e s t i o n a n d a s s u m i n g a h i g h l y p e r m e a b l e m e s o t h e -l i u m ( b o u n d a r y c o n d i t i o n 3) a r e s h o w n f o r ( i ) t h e c a s e w h e r e £ i s 1 a n d ( i i ) f o r t h e c a s e w h e r e £ i s 0. I n e a c h c a s e t h e s o l i d l i n e r e p r e s e n t s t h e i n i t i a l c o n d i t i o n , t h e d o t t e d l i n e is a t t e q u a l 0.001, t h e c h a i n - d o t l i n e is a t t e q u a l 0.005, t h e d a s h e d l i n e c o r r e s p o n d s t o t e q u a l 0.01, a n d t h e c h a i n - d a s h l i n e r e p r e s e n t s t h e final s t e a d y - s t a t e c o n d i t i o n . Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 145 the venular capillary shifts from a re-absorbing to a filtering vessel. W h e n interstitial plasma protein transport is l imited to diffusion, there is a change i n gradient i n interstitial plasma protein concentration some distance from the venular boundary shortly after the perturbation begins. This is due to the increased protein exchange from the interst i t ium to the peritoneum there. However, adjacent the venular capulary wall, plasma protein diffusion is directed into the interst i t ium, due to the transfer of proteins the from blood to the interst i t ium at that boundary. Since this shift i n the interstitial plasma protein concentration gradient occurs over a very short distance just outside of the venular boundary, it is not discernable i n Figure (5.19). S u m m a r y o f V e n o u s C o n g e s t i o n S i m u l a t i o n s A s in the case of hypoproteinemia, the length of the transient period varied inversely wi th the permeability of the mesothelial layer. Further, when it is assumed that the mesothehum is impermeable, the transient behavior of the fluid and plasma, protein fluxes entering and leaving the system are strongly influenced by the interstitial plasma protein transport mechanisms and by the transient distribution of plasma proteins wi th in the interstitial space. Once again, this is due to the greater role that the interst i t ium has here i n determining the overall resistance of the system. In contrast to the simulations of hypoproteinemia, however, some of the simulations of venous congestion predicted an increase i n the interstitial plasma protein content with t ime. A determining factor for this behavior appears to be the plasma protein sieving properties of the filtering boundary relative to the sieving properties of the draining boundary. This provides a possible explanation for the differences i n the interstitial plasma protein content within selected tissues observed i n clinical settings during venous congestion, and is discussed further below. 5.4.3 C l i n i c a l O b s e r v a t i o n s o f H y p o p r o t e i n e m i a a n d V e n o u s C o n g e s t i o n According to the simulations presented above, the rate of fluid exchange wi th in the model tissue segment typically increases following hypoproteinemia. However, because of the reduced plasma Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 146 protein content in the filtering fluid, the net rate at which plasma proteins enter the interstitial space is less than that prior to the upset. This causes a 'washout' of plasma proteins from the interstitium and leads to a somewhat reduced interstitial plasma protein content by the time the system has achieved a new steady-state. These findings are in keeping with the clinical observations reported by Witte and co-workers [114]. The authors state that hypoproteinemia is followed by a lowering of interstitial plasma protein content. The simulations of venous congestion revealed several interesting phenomena that might shed further light on clinical observations of this state. Clinical data show that, in most pe-ripheral tissues, the plasma protein concentration in lymph decreases with the onset of venous congestion, suggesting that the interstitial plasma protein concentration likewise drops [113]. Witte and co-authors attribute the washout of plasma proteins to the increased filtration rates that accompany an elevation in systemic pressure. However, the simulations presented here demonstrate that an increased nitration rate does not assure washout of plasma proteins from the interstitium of mesentery, for example. On the contrary, the majority of the simulations predicted an increase in interstitial plasma protein content within the model tissue. Only in those cases in which the sieving of proteins at the draining boundary (e.g., the mesothehum) was less than the sieving at the filtering boundaries did the interstitial plasma protein content decrease following venous congestion. The simulations suggest that it is the relative sieving properties of the draining and filtering boundaries, and not simply the filtration rate, that determines whether plasma protein washout occurs. Since it is generally thought that the terminal lymphatics do not sieve proteins to any great extent [77], it follows that, in most tissues, we would expect a decrease in plasma protein content to accompany enhanced nitration, as seen clinically. The notion that the relative sieving properties are important in determining changes in interstitial plasma protein content is further supported by the fact that the plasma protein content in hepatic lymph (and so, presumably, the plasma protein content in the interstitium of the fiver) increases under venous congestion [114]. Since the exchange vessels of the fiver are sinusoids, far less sieving occurs across the Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 147 vascular walls in this tissue, so that the sieving properties of the filtering and the draining boundaries are similar. One other characteristic of venous congestion is the accumulation of fluid in the pleural cavity. This pathological state is called ascites [51]. The increased fluid exchange across the mesothehum predicted by the model seems to support this clinical observation. Further, the simulations indicate that this increase accompanies a shift from fluid re-absorption to fluid filtration at the venular capillary, due to the increase in hydrostatic pressure in that vessel. This suggests that there may be some hmiting value for venular hydrostatic pressure, corresponding to the shift in the direction of transcapillary fluid flux there, that leads to ascites. 5.5 Concluding Remarks In this chapter we studied the response of the model tissue segment to two systemic pertur-bations, namely hypoproteinemia and venous congestion. Since the simplified version of the model employed in these simulations does not include all of the features of the microvascular exchange system (for example, interstitial swelling is neglected) the results are, at best, quali-tative. However, the simulations reveal several interesting features of the model microvascular exchange system. These are summarized below. 1. When the mesothehum is permeable, the trends in fluid and plasma protein exchange following a systemic perturbation can be anticipated by considering the effect that the given upset has on the effective chemical potential of the luminal fluids. In such instances the interstitium may not be the major resistance within the system. However, when the mesothehum is impermeable, fluid and plasma proteins must cross the entire interstitial space in their journey from the filtering vessel to the re-absorbing vessel, so that the interstitium comprises a large fraction of the system's total resistance to mass exchange. In these cases the distribution of interstitial plasma proteins plays a greater role in de-termining the overall behavior of the model tissue. Further, the length of the transient period following an upset is typically shorter for those cases in which the mesothehum is Chapter 5. Transient Exchange in Mesentery Following a Systemic Upset 148 permeable. 2. Following hypoproteinemia, the interstitial plasma protein content of the tissue segment typically decreases with time, due to a decrease in plasma protein exchange across the vascular boundaries and an increase in the total rate of fluid exchange within the system. This is in keeping with qualitative clinical observations. 3. Following venous congestion, the fluid exchange rate and plasma protein exchange rate within the system both typically increase. However, the change in interstitial plasma protein content depends, in part, on the relative sieving properties of the filtering and draining boundaries. When the reflection coefficients of these two sets of boundaries are similar, the interstitial plasma protein content increases due to the increased protein exchange rate across the filtering boundaries and sieving of interstitial plasma proteins at the draining boundaries. As the reflection coefficient of the chaining boundaries decreases relative to that of the filtering boundaries, there is a net loss ..of plasma proteins from the interstitial space, resulting in a decrease in total interstitial plasma protein content over time. These results are supported by the clinical observation that interstitial plasma protein content in the liver increases during venous congestion. Since this tissue is serviced by sinusoids, the sieving properties of the filtering blood vessels and the draining lymphatic vessels are similar. (It should be noted, however, that both the filtering and draining vessels of the liver offer little sieving of plasma proteins.) In addition, the model predicts an increase in fluid transfer from the mesentery to the peritoneal fluid, supporting the clinical observation of ascites formation. The trends in system behavior predicted by the simplified model are, for the most part, in keeping with the limited number of clinical observations associated with hypoproteinemia and venous congestion discussed here. Furthermore, the simulations provide a clearer picture of the relationship between the vascular wall transport properties, the interstitial transport properties, and the transient behavior of the system as a whole following these upsets. Chapter 6 A Prel iminary Study of Tracer Transport through the Interstitium 6.1 Introduction Thus far we have considered the transport of a single, aggregate plasma protein species through the interstitium of a model tissue. In fact, numerous types of macromolecules are exchanged between the blood and the interstitium under normal conditions. In addition, certain clinical procedures, such as chemotherapy, involve the exchange of small quantities of foreign substances between the blood stream and a particular organ or tissue. The exchange of multiple solute species within the microcirculation is therefore of interest to physiologists and clinicians alike. This chapter presents a study of the relative exchange rates of two different macromolecular tracers representing albumin and 7 — globulin within a hypothetical, one-dimensional tissue. Specifically, the study investigates the time required for the concentration of the tracer in the outlet stream to reach 50 % of its steady-state value, subsequent to its introduction in the blood at a dimensionless concentration of 0.01, as a function of the following: 1. the transport properties of the capillary wall to the tracer, 2. the transport properties of the tracer through the interstitial space, and 3. the interstitial distribution volume of tracer. The results of the study suggest that the distribution volume of a particular solute species can play a major role in determining its rate of transport through the interstitium. Further, the exclusion properties of the interstitium can create conditions for a 'gel-chromatographic effect', whereby larger macromolecules pass through the interstitial space more quickly than smaller 149 Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 150 macromolecules. (This mechanism is to be distinguished from the 'free-fluid phase - gel phase' mechanism proposed by Watson and Grodins [104] that lacks substantial structural evidence.) The chapter is divided into five remaining sections. Section 6.2 describes the model tissue segment and the equations applying to the transient movement of a solute species through that tissue segment. Section 6.3 outlines the particular numerical experiments constituting the study, while Section 6.4 outlines the numerical procedures employed to solve the relevant mathematical expressions. The results of the study are then discussed in Section 6.5. Finally, Section 6.6 summarizes the ramifications of these findings to the interpretation of transient tracer experiments and the delivery of substances to specific tissue sites. 6.2 Defining the System Consider a flat, thin, sheet-like tissue analogous to the mesenteric tissue of Chapter 4 and bounded left and right by a blood capillary and a terminal lymphatic vessel, respectively. It is assumed that the upper and lower surfaces of the tissue are impermeable and that the mtervening interstitium is both homogeneous and isotropic. Furthermore, the tissue properties and conditions are considered uniform in the direction parallel to the vessels' axes, so that the model tissue can be treated as a one-dimensional system. Contained within the tissue is an aggregate plasma protein species representative of the various osmotically active plasma protein species found in vivo. These display the same char-acteristics as the aggregate plasma proteins of Chapters 4 and 5. It is further assumed that, at some time te, a macromolecular tracer is introduced into the blood vessel. Since this macro-molecule is present in minute quantities only (i.e., one percent of the total plasma protein concentration), its contribution to the system osmotic pressure is negligible and therefore it does not alter the exchange of fluid or the exchange of aggregate plasma protein species within the system. Given this, the interstitial fluid flux depends only on the local gradient in fluid hydrostatic pressure, P 1, and aggregate protein osmotic pressure, II1, according to Eq. (5.2). The fluid material balance is then given by Eq. (5.4), with the added constraint that L™" is Chapter 6. A Prehminary Study of Tracer Transport through the Interstitium 151 zero. With these assumptions, the transient tracer distribution problem is uncoupled from the steady-state problem of fluid and aggregate plasma protein transport. The material balance expression for the aggregate plasma protein species is therefore given by the steady-state version of Eq. (5.7), assuming further that j™" (given by Eq. (5.6)) is zero. Meanwhile, the transient distribution of interstitial tracer concentration, C 2 , is described by an analogous form of Eq. (5.7), assruning once more that no exchange of tracer occurs across the upper and lower surfaces of the tissue segment. Furthermore, since the tracer's osmotic pressure is negligible compared to that of the aggregate plasma protein species, then the fluid chemical potential in the tracer's distribution volume is equal to P 1 — II1. The exchanges of fluid and solutes across the capillary wall are described by the same set of boundary conditions as presented in Chapter 4 (see Eqs. (4.13), (4.14), and (4.15)). However, the conditions prevailing at the lymphatic vessel warrant some discussion. It is typically assumed [3, 14, 70. 108] that, under normal conditions, the flow of fluid across the lymphatic wall is proportional to the interstitial hydrostatic fluid pressure, P 1. That is, [j«L= L ? m -([ p l L- p I , m ) ' ^ where L^-P 1 5 ™ is some reference lymph drainage rate. This type of relationship can be viewed as a specific form of the Starling relationship in which the reflection coefficient of the lymphatic wall, cr , y m, is zero. If we further assume, for simplicity, that the hydrostatic pressure within the lymphatic vessel,Plyin, is zero, we then have li°L=LiMp lL- <6-2' Solutes, meanwhile, are assumed to cross the lymphatic wall by unhindered convection [3,14, 75, 108]. The valve-like behavior of the overlapping endothelial cells of the lymphatic wall prevents back-flow of solutes from the lymph to the tissue space. Hence, assuming that a thin fluid film separates the lymphatic wall and the interstitium (analogous to that at the vascular boundary), the rate of exchange of aggregate proteins across the vessel wall ( [ j s ] ^ ) , for example, is • Ic 1 ' J , J l v m L J w lvm . (6.3) lym Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 152 A similar expression applies for the exchange of a particular tracer species 2, namely, lym lym lym (6.4) An alternative boundary condition for the exchange of a given solute species k at the lym-phatic vessel wall might be where ^ • |C k ] is equivalent to the average plasma protein concentration based on the total n L J lym mobile fluid volume fraction. In this fashion, Eq. (6.5) eliminates the 'thin film' assumption. However, since all previous simulations used the thin film approach, it is retained here as well. The dimensionless form of the mass balance equations can be found in Chapters 4 and 5. Note that the tracer mass balance equation is non-dimensionalized with respect to the parameters for the aggregate plasma protein species. For example, the dimensionless diffusivity of tracer species k is n ^ - D ^ / D ^ . The dimensionless forms of Eqs. (6.2) and (6.3), meanwhile, are, respectively, die lym and I lym L"w J lym L J lym ' ^ ^ where lb™ = L ^ L / K 0 . As before, all pressures and concentrations are normalized with respect to P ^ and C 1 , a r t (the arteriolar concentration of the aggregate plasma protein species), respectively. Equations (6.6) and (6.7) are combined with the appropriate dimensionless forms of the aggregate plasma protein mass balance equation, the tracer mass balance equation, the fluid mass balance equation, the capillary wall boundary conditions, and the initial conditions to describe the movement of both the aggregate plasma protein species and the tracer through the interstitium. Chapter 6. A Prehminary Study of Tracer Transport through the Interstitium 153 6.3 Case Studies In order to carry out the numerical simulations, the various model parameters must first be assigned values. These parameters can be divided into three groups: those parameters charac-terizing fluid and aggregate plasma protein exchange across the capillary wall and the transport of these materials through the interstitium; those parameters characterizing the transcapillary exchange and interstitial transport of the tracers; and the parameters characterizing mass ex-change across the lymphatic wall. The first group of parameters were assigned the same set of values as in the steady-state analysis of fluid and aggregate plasma protein exchange in mesentery (see Table (4.1) in Chap-ter 4), assuming further that the aggregate protein convective hindrance, £, equalled 0.5. The properties of the aggregate plasma protein species are therefore close to, but not identical with, the values described below for the albumin tracer. The following tracer parameters were assumed. The capillary wall permeabilities to the albumin and globulin tracers, D 8 1 1 , 0 1 1 5 and D a r t , g l o b , assumed values of 2.4 x 10~8 cm/s and 1.39 X 10 - 8 cm/s respectively. These values fall within the range reported in the literature for these two species [83]. The values of the reflection coefficients for the albumin tracer and the globulin tracer, <ralb and , were 0.89 and 0.91 respectively, based on data reported for dog hindpaw [82]. The distribution volume fractions assumed for the two tracers, meanwhile, were 0.68 and 0.5, respectively, based on rabbit skin data [13]. The effective interstitial diffusivity of the albumin tracer, D*^, assumed a value of 1.0 x 10 _ r cm2/s, based on the data of Fox and Wayland [38]. Since no value was available for the interstitial diffusivity of globulin, D ^ b , this parameter was assigned a value such that D^/D^* 5 equalled D a r t - o l b / D B r t , g l o b . This yielded a Bf°h of 0:58 X K T 7 cm2/s. The product £ k • / 3 k for the albumin and globulin tracers was varied over a range of values by establishing first the lower limits for £ k and / 3 k individually. The lower limit for £ k was arbitrarily set equal to 0.25. The lower limit for / 3 k , meanwhile, was set equal to the ratio n k /n°, thereby assuming that the flow conductivity of the interstitial space was uniformly distributed Chapter 6. A PreUminary Study of Tracer Transport through the Interstitium 154 throughout the various mobile fluid volume fractions. Under this assumption the regions of the interstitial space accessible to the macromolecular tracers were no more conductive to fluid than the excluded regions of the interstitium. On this basis, a lower limit for / 3 k - £ k of 0.21 for alburnin and 0.16 for globulin was determined. However, in all likelihood, macromolecules excluded from portions of the interstitial matrix will be limited to pathways of higher conductivity, leading to somewhat larger values of /3k • £ k . Hence, additional values of /3k • £ k equal to 0.5 and 0.9 were assumed for each of the two tracers during the sensitivity analyses discussed below. According to Eqs. (6.2) and (6.3), mass exchange across the lymphatic vessel wall is charac-terized by a single parameter - the lymphatic hydraulic conductance, Lp5™. Since no information could be found in the literature to quantify Lp7™, a set of simulations was first performed to investigate the influence of this parameter on the exchange of fluid within the system. Increas-ing Lpy111 from 1.35 x 1 0 - 9 cm3/(dyne-s) to 1.35 x 10~7 cm3/(dyne-s) caused the fluid exchange rate within the system to increase by less than 7 %, indicating that fluid flow was relatively insensitive to the value of in this range. The lymphatic hydraulic conductance was there-fore arbitrarily assigned a value of 1.35 x 1 0 - 9 cm3/(dyne-s), equal to the value applied to the arteriolar boundary. Having established the reference values for the model parameters, a series of numerical simulations was performed to investigate the effect of several system parameters on the transport rates of the alburnin and globulin tracers through the model tissue. These system parameters are summarized below: 1. the transport characteristics of the capillary wall to each of the two tracers; 2. the interstitial distribution volume fraction of the globulin tracer; and 3. the diffusivity of the albumin tracer. In each case, the product /3k - £ k of the tracer was varied over the range of values described earlier to provide a factorial design. In all, 21 transient simulations were performed, as well as 21 corresponding steady-state runs. Chapter 6. A Prehminary Study of Tracer Transport through the Interstitium 155 6.4 Numerical Procedures A numerical procedure similar to that reported in Chapter 5 that combined the finite element and finite difference methods was used here to solve for the transient distribution of tracer throughout the interstitium. Again, the interstitium was divided into 25 elements and 51 nodal points. The initial time step size was determined by specifying an initial Courant number of 0.05. As a further check of the validity of the numerical solution, one of the simulations was repeated assuming an initial Courant number of 0.025, thereby doubling the number of time steps performed during the run. As a result, the dimensionless time taken for the tracer concentration at the lymphatic vessel to reach 50 % of its steady-state value, t50%, changed by less than .05 %. In another test, the initial Courant number was reduced from 0.05 to 0.01, resulting in a 5-fold increase in the number of time steps taken during the simulation. The reduced time step produced no significant change in the model predictions. 6.5 Results and Discussion This section is divided into three parts. The first discusses the effect of the capillary wall transport properties on the transport rates of the globulin and albumin tracers, assuming various values of /3k • £ k for these macromolecules. The second part considers the effect of the globulin interstitial distribution volume on the transport of that tracer for different values of / ? g l o b • £ g l o b . Finally, the third part of this section discusses the effect of interstitial diffusivity on the transport rate of the albumin tracer as a function of /3 a l b • £ a l b . 6.5.1 The Effect of Capillary Boundary Conditions on Tracer Transit Time The first set of simulations explored the combined effect of the /? k • £ k and the capillary wall transport properties on the exchange of each of the two tracers. Two 2 x 3 factorial studies were performed in which each tracer was subjected to first the globulin boundary conditions and then the albumin boundary conditions for each value of /3k - £ k considered. (Recall that the Chapter 6. A PreUminary Study of Tracer Transport through the Interstitium 156 alburnin boundary condition corresponds to a reflection coefficient of 0.89 and a permeability, D 1 " 1 , of 2.4 X 1 0 - 8 cm/s at the capillary wall, while the globulin boundary condition implies a reflection coefficient of 0.91 and a permeability of 1.39 X 10~8 cm/s.) Tracer Boundary Condition £ • /3 Steady-State Outlet Concentration 0.16 0.00091 Globulin Globulin 0.50 0.00091 0.90 0.00091 0.16 0.00113 Globulin Albumin 0.50 0.00113 0.90 0.00113 0.21 0.00113 Albumin Albumin 0.50 0.00113 0.90 0.00113 0.21 0.00091 Albumin Globulin 0.50 0.00091 0.90 0.00091 Table 6.1: T h e effect of capi l lary w a l l boundary condit ions on the steady-state con-centrat ion o f the tracers i n the outlet ( lymphat ic ) stream. Table (6.1) presents the steady-state dimensionless concentrations for the albumin and the globulin tracers in the lymphatic vessel (i.e., their plasma/lymph ratios) as functions of and the capillary wall boundary conditions. Recall that the outlet concentration in the lymphatic vessel equals the interstitial concentration within the tracer's distribution volume in the vicinity of the lymphatic vessel. It is clear from Table (6.1) that the steady-state concentration of a tracer is determined by the boundary conditions at the capillary wall, and not the transport Chapter 6. A Prehminary Study of Tracer Transport through the Interstitium 157 mechanisms within the interstitial space. In fact, the outlet concentration is determined largely by the capillary wall reflection coefficient since, in the simulations presented here, the principle mechanism for the transcapillary exchange of tracer is convective transport. Hence, the influx of some tracer k into the system is proportional to (1 - o*). Under steady-state conditions and for this model tissue, then, the flux of tracer across the lymphatic vessel wall must also be proportional to (1 — cr^), so that the outlet stream composition is determined by the degree of sieving at the capillary wall. Table (6.2) presents the dimensionless time required for the tracer's outlet stream concen-tration to reach 50 % of its steady-state value, t50%, as a function of /3 • £ and the capillary wall reflection coefficient for both the globulin tracer and the albumin tracer. The breakthrough curves associated with these simulations are shown in Figure (6.1). In each case iso% decreases with increasing /3-£ , due to the enhanced convective transport of the tracer accompanying such an increase. In addition, when the capillary wall is assigned the more permeable (albumin) transport properties, the time required for each tracer to reach 50 % of its steady-state value decreases marginally (i.e., by less than 0.5 %) in each case. Hence, the capillary transport prop-erties exert a stronger influence on the ultimate steady-state outlet concentration than on the transit times through the interstitium, for the range of permeabilities and reflection coefficients considered here. Of greater significance is the difference in transit times between the albumin and globulin tracers. In all cases, the globulin tracer reaches 50 % of its steady-state value in a significantly shorter time than the albumin tracer, even when both tracers are subject to the same boundary conditions (and hence achieve the same ultimate outlet concentrations), and despite the fact that the globulin interstitial diffusivity is less than that of albumin. For example, when /3-£ is 0.90 and assuming globulin boundary conditions, the t50% for the globulin tracer is 1.636, compared to a t50% of 2.115 for the albumin tracer. Therefore, only two model parameters remain to account for this difference in transit times: the tracer distribution volume and interstitial diffusivity. These are investigated separately below. Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 158 1.2. HX 0 DIMENSIONLESS TIME DIMENSIONLESS TIME Figure 6.1: The breakthrough curves for various values of are shown (I) for glob-ulin, assuming globulin boundary conditions; (II) for globulin, assuming alburnin boundary conditions; (III) for albumin, assuming albumin boundary conditions; and (IV) for albumin, assuming globulin boundary conditions. In each case the top (chain-dot) curve corresponds to 13 • £ equal to 0.90, the middle (dotted) curve corresponds to /3 • f equal to 0.50, and the lower (solid) curve corresponds to 0 • f equal to 0.16 for globulin and 0.21 for albumin. Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 159 Tracer £ • 8 t5Q% t 5 0 % Assuming Albumin B . C . Assuming Globulin B . C . 0.21 2.355 2.349 Albumin 0.50 2.254 2.247 0.90 2.119 2.115 0.16 2.064 2.057 Globulin 0.50 1.853 1.849 0.90 1.639 1.636 Table 6.2: Transit times of Albumin and Globulin tracers as functions of /3 • £ and the capillary wall transport properties. 6.5.2 The Effect of Tracer Distribution Volume on Globulin Transit Times Table (6.3) shows the t50% for the globulin tracer assuming a distribution volume of first 0.50 and then 0.68, and compares these values to the t50% for albumin (which has a distribution volume of 0.68). In each case globulin boundary conditions prevail. The breakthrough curves for these cases are illustrated in Figure (6.2). The increase in globulin distribution volume results in a dramatic increase in the transit time for that tracer. In fact, when both the globulin tracer's distribution volume and /3-£ equal those of the albumin tracer, the £50% for the globulin tracer exceeds the t60% for the albumin tracer, due to the globulin tracer's lower interstitial diffusivity. Assuming a /3 • £ of 0.50, for example, the t50% for globulin increases by 29 %, from 1.853 to 2.394, as the tracer's distribution volume is raised from 0.50 to 0.68. This is to be compared to the t50% of 2.247 for the albumin tracer at the same /3 • £ and a distribution volume of 0.68. The rise in transit time accompanying the increase in distribution volume is attributed to the increased capacity of the interstitium to contain the given tracer. Altering the distribution volume from 0.50 to 0.68 represents a 36 % increase in the interstitial volume available to the globulin tracer. It is not surprising, then, that an increase in the distribution volume leads to a Chapter 6. A Prelwwiary Study of Tracer Transport through tie Interstitium 160 t50% ^50% ^50% ^ l o b = 0.50 n 8 l o b = 0.68 Alburnin Tracer n' 0.16 0.50 0.90 2.064 1.853 1.639 2.607 2.394 2.177 2.355* 2.254 2.119 * evaluated at 0 • £ = 0.21 Table 6.3: The effect of interstitial distribution volume on the transit time of a glob-ulin tracer through the interstitium. The last column of values presents the transit times for the albumin tracer assuming the same capillary boundary conditions as those for the globulin tracer, and assuming that the distribution volume of the albumin tracer equals 0.68. concomitant rise in the length of the transient for a given tracer, since the tracer must fill the available interstitial space before steady-state conditions prevail. This finding offers an alternative mechanism for the 'gel chromatographic effect' discussed in Chapter 3. Recall that some experimental data suggests that, in certain instances, larger probes pass through the interstitial space more rapidly than smaller ones [48]. To date, only one paper has addressed this phenomenon from a theoretical standpoint [104]. In that work, the authors assumed that the interstitium contained a 'free-fluid phase', in which macromolecules moved by convection and diffusion, and a 'gel phase', in which the transport of macromolecules was limited to restricted diffusion alone. Assuming, then, that the larger molecules were limited to the free-fluid phase while smaller molecules penetrated both phases, the transit time through the interstitium for the smaller tracer could conceivably exceed that of the larger probe. As was mentioned in Chapter 2, the concept of continuous, distinct free-fluid and gel phases lacks solid evidence. Macromolecular exclusion, on the other hand, is well documented (see, for example, [13]). Hence this latter mechanism for the 'gel chromatographic effect' requires no additional assumptions regarding the structure of the interstitium, and so is preferred over the Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 161 'gel phase - free-fluid phase' mechanism of Watson and Grodins [104]. It is also conceivable that variations in the convective hindrances for various macromolecular species may, under certain conditions, create conditions for the gel chromatographic effect, as described in Chapter 3. Again, this mechanism does not rest on a 'gel phase - free-fluid phase' model of the interstitium. 6.5.3 The Effect of Interstitial Diffusivity on Albumin Transit Times Finally, consider the effect of interstitial diffusivity on the transit time of the albumin tracer through the interstitium. Table (6.3) presents t 5 0 % for the various values of /? • £, assuming three different values of D^: 0.58 X 10 - 7 cm2/s (i.e., equal to Df^ b), 1.0 x 10~7 cm2/s, and 1.5 x 10 - 7 cm2/s. The breakthrough curves are shown in Figure (6.3). £ • P ho% t 5 0 % t 5 0 % = 0.58 X IO"7 cm2/s = 1.00 x IO"7 cm2/s = 1.50 x IO - 7 cm2/s 0.21 2.564 2.349 2.246 0.50 2.389 2.247 2.174 0.90 2.173 2.115 2.081 Table 6.4: The effect of interstitial diffusivity on the transit time of an albumin tracer through the interstitium. In general, varying the interstitial diffusivity according to these amounts had only a small to moderate effect on the value of t 5 0 % for the alburnin tracer. For example, increasing the diffusivity from 0.58 x l O - 7 cm2/s to 1.0 x l O - 7 cm2/s, assuming a/3-£ of 0.50, reduced f 5 0 % by less than 6 %. Further increases in the tracer diffusivity had an even less pronounced effect on the transit time of the macromolecule. For example, increasing the diffusivity of the albumin tracer from 1.0 x 10 - 7 cm2/s to 1.5 xl0~ 7 cm2/s, at a f3 • £ of 0.50, only dropped t 5 0 % by an additional 3.2 %. This trend was observed for all values of 0 • £ considered in the study. Furthermore, even at the higher diffusivity, the transit time for the aiburnin tracer exceeded Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 162 the transit time for the globulin tracer for all values of 0 • £ investigated. Within the limits of this study, then, it appears that interstitial diffusion has less impact on the transit time of an interstitial macromolecule than does the interstitial distribution volume of that species. Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 163 1-2 DDffiNaONLESS TIME Figure 6.2: The breakthrough curves for various values of /3-£ and globulin boundary conditions are shown (i) for albumin; (ii) for globulin, assuming a distribution vol-ume of 0.50; and (iii) for globulin, assuming a distribution volume of 0.68. In each case the top (chain-dot) curve corresponds to /3-£ equal to 0.90, the middle (dotted) curve corresponds to /3 • £ equal to 0.50, and the lower (solid) curve corresponds to /? • £ equal to 0.16 for globulin and 0.21 for albumin. Chapter 6. A Preliminary Study of Tracer Transport through the Interstitium 164 i . z o 8 1 § I f n sir f # o o o z o z w o z o u z o CO z CJ ;s DIMENSIONLESS TIME Figure 6.3: T h e b r e a k t h r o u g h c u r v e s for a l b u m i n , a s s u m i n g v a r i o u s va lues o f ft • £, a r e s h o w n for (i) T>f£ = 0.58 x I O " 7 c m 2 / s , (ii) T>f£ = 1.00 x 1 0 - 7 c m 2 / s , a n d (iii) = 1.50 x I O " 7 c m 2 / s . I n e a c h case t h e t o p ( c h a i n - d o t ) c u r v e c o r r e s p o n d s to (3 • £ e q u a l to 0.90, the m i d d l e ( d o t t e d ) c u r v e c o r r e s p o n d s to /3 • £ e q u a l to 0.50, a n d t h e l ower (sol id) c u r v e c o r r e s p o n d s to /3 • £ e q u a l to 0.16 for g l o b u l i n a n d 0.21 for a l b u m i n . Chapter 6. A PreUminary Study of Tracer Transport through the Interstitium 165 6.6 Concluding Remarks This chapter presented the results of a preliminary study that investigated the effects of intersti-tial convection and diffusion, interstitial distribution volume, and capillary transport properties on the transit times of two macromolecular tracers representative of albumin and 7 — globulin for a specific set of interstitial fluid flow conditions. The findings are summarized below. 1. As to be expected, the transit time of the tracers varied inversely with the degree of convective transport within the interstitium. 2. Increasing the interstitial diffusivity of the alburnin tracer also led to a moderate decrease in the transit time for that tracer. 3. The capillary transport properties had only a marginal effect on the transit times of the tracers, for the range of capillary permeabilities and reflection coefficients consid-ered. However, these properties (and, in particular, the reflection coefficient) had a more pronounced effect on the ultimate steady-state concentration in the outlet stream. 4. The interstitial distribution volume of a given tracer had the greatest influence on the time required to achieve steady-state. This is attributed to the increased filling times associated with the larger interstitial distribution volumes. These findings suggest that the 'gel chromatographic effect' [48] observed in some tissues could possibly be explained on the basis of varying distribution volumes, rather than the hypothetical 'gel phase -free-fluid phase' model proposed by Watson and Grodins [104]. Clearly, much more experimental and theoretical research is needed before the interstitial transport of multiple tracer species can be well characterized. However, this study suggests that the relative transport rates of different macromolecules is governed by a number of interstitial properties, including the interstitial distribution volume. This may ultimately bear clinical import, particularly in the use of macromolecular carriers for drug delivery. Chapter 7 Summary of Conclusions In this dissertation, mathematical relationships are developed to describe the transient flow and distribution of fluid and various macromolecular solute species within the interstitium, yielding a system of coupled, nonlinear partial differential equations. The resultant mathematical model describes the combined effects of a number of interstitial properties (such as exclusion and swelling characteristics) and transport mechanisms (such as solute convection, cliffusion, and dispersion) on mass transport within the interstitium. Despite the complexity of the model, it is limited in several respects. First, the description of interstitial deformation applies to small strains only (on the order of ten percent), and so is not suited to analyzing extreme cases of edema formation. Further, the model uses a compliance relationship to characterize swelling which assumes that any change in volume is a function of the interstitial hydrostatic pressure. It therefore neglects the influence of shear stresses on interstitial deformation. However, since the model concerns itself primarily with the effect that swelling has on the various transport properties and material characteristics of the interstitial space (such as the hydraulic conductivity, effective diffusivities, and various distribution volume fractions), rather than a description of the deformed geometry of the interstitium, this approach provides a reasonable first approximation to the complete theory of deformation for porous systems developed by Biot [19]. The use of Biot's theory to describe interstitial swelling must await further experimentation to quantify the material properties of the various interstitial components. Interstitial fluid flow is assumed to be proportional to the gradient in fluid chemical poten-tial alone, thus neglecting any coupling between fluid flow and solute chemical potential, for 166 Chapter 7. Summary of Conclusions 167 example. However, the theory developed here could be modified to include these additional effects, given better information about the nature of fluid transport within the interstitium. Further research in the area of fluid flow within osmotically active, partially restricting matri-ces is therefore needed. However, because the interstitial fluid flow expression presented here includes the influence of colloid osmotic as well as hydrostatic pressure gradients, it is considered more general than that offered in previous models [87, 36, 22]. Despite these limitations, the model of Chapter 3 provides a far more comprehensive de-scription of interstitial transport than that offered by any of the previous models to appear in the literature. Its strength lies in the general, self-consistent, and self-contained nature of the mathematical formulation. It therefore provides a framework in which to further understand the interstitium and its role in microvascular exchange. Subsequent chapters of the dissertation have used simplified versions of the general model to conduct theoretical investigations of microvascular exchange under normal and pathological states. In Chapter 4, for example, the model is used to describe the steady-state exchange of fluid and plasma proteins in mesenteric tissue, which is treated as a two-dimensional, rigid system. This tissue was selected both for its simple geometry and because it is a popular tissue for experimental studies of interstitial transport and microvascular exchange. The array of plasma protein species found in vivo was approximated by a single, 'aggregate' species that displayed average properties. The simulations of Chapter 4 indicate that convective transport of plasma proteins is sig-nificant, even at reduced values of convective hindrance, £. This supports a recent study of the movement of labelled albumin in rat mesentery suggesting that convection plays a significant role in interstitial plasma protein transport within that tissue [74]. However, the simulations also show that exact nature of interstitial plasma protein transport cannot be determined from protein distributions alone. The model predictions also reveal that the hydrodynamics within the interstitial space can Chapter 7. Summary of Conclusions 168 be complex, resulting, for example, in the development of fluid recirculation patterns. The hy-drodynamics can also lead to irregularities in the distribution of fluid and plasma protein fluxes across a permeable boundary, such as the mesothehum, even when the boundary's transport properties are uniform. This behavior, which is strongly influenced by the transport properties of the mesothelial layer, could lead to the erroneous identification of 'leaky sites' within the system. Finally, the model predicts significant interstitial osmotic pressure gradients in some in-stances, suggesting that the Darcy expression evoked in a number of previous models [22, 36, 87], that considers hydrostatic gradients only, is inadequate for describing interstitial fluid transport. The analysis of Chapter 4 is extended to transient perturbations in Chapter 5. Again, a rigid model of mesenteric tissue is used although, in this case, the two-dimensional tissue is replaced by a one-dimensional analogue. Two systemic perturbations are considered: namely hypoproteinemia and venous congestion. The simulations of Chapter. 5 demonstrate that, assuming a permeable mesothehum, the trends in fluid and plasma protein exchange can be anticipated by considering the effect that a particular upset has on the effective chemical potential of the luminal fluids. In these instances the interstitium is not the major resistance within the system, due to by-passing. However, when the mesothehum is impermeable, fluid and plasma proteins must cross the entire interstitial space in their journey from the filtering vessel to the re-absorbing vessel, so that the interstitium comprises a large fraction of the system's total resistance to mass exchange. In these cases the distribution of interstitial plasma proteins plays a greater role in determining the overall behavior of the system. The simulations indicate further that, following hypoproteinemia, interstitial plasma protein content decreases, while the rate of fluid exchange within the tissue increases. This is in keeping (qualitatively, at least) with clinical observations of hypoproteinemia. In the case of venous congestion, however, the change in interstitial plasma protein content depends, in part, on the relative sieving properties of the filtering and draining vessels. When the reflection coefficients Chapter 7. Summary of Conclusions 169 of these two sets of boundaries are similar, the interstitial plasma protein content increases due to the increased plasma protein exchange rate across the filtering boundaries and sieving of interstitial plasma proteins at the draining boundaries. This effect is further supported by the clinical observation that interstitial plasma protein content in fiver increases during venous congestion. Since this tissue is serviced by sinusoids, the sieving properties of the filtering blood vessels and the draining lymphatic vessels are similar. The simulations also predict that, as the reflection coefficient of the chaining boundaries decreases relative to that of the filtering boundaries, there is a net loss of plasma proteins from the interstitium, resulting in a decrease in the total interstitial plasma protein content over time (i.e., the familiar 'plasma protein washout'). In Chapter 6 a one-dimensional model of a hypothetical tissue was used in a preliminary study investigating the effects of interstitial plasma protein convection and diffusion, plasma protein exclusion, and the capillary transport properties on the transit times of two macro-molecular tracers representative of albumin and 7-globulin. As was expected, the transit times of each of the tracers through the model tissue varied inversely with the degree of convective transport. Increasing the interstitial chffusivity of the albumin tracer also led to a moderate decrease in the transit time for that tracer. The capillary wall transport properties, meanwhile, had only a marginal effect on the transit time for the range of capillary permeabilities and reflection coefficients considered. However, these properties (and, in particular, the reflection coefficient) had a more pronounced effect on the ultimate steady-state concentration of the tracer in the outlet stream. It was the interstitial distribution volume of a given tracer that had the greatest impact on the time required for the outlet tracer concentration to reach 50 % of its steady-state value. This was attributed to the increased filling times associated with larger interstitial distribution volumes. These findings suggest that the 'gel chromatographic effect' [48] observed in some tissues could possibly be explained on the basis of varying distribution volumes, rather than the hypothetical 'gel phase - free fluid phase' model proposed by Watson and Grodins [104]. Chapter 7. Summary of Conclusions 170 Finally, in Appendix C, we investigate the possible influence of mechanical dispersion on mass exchange within the model tissue. While it influenced the distribution of interstitial plasma proteins to some extent, mechanical dispersion had less impact on the overall exchange of fluid and plasma proteins within the system. Clearly, much more experimental and theoretical research is needed before the interstitial transport of fluid and multiple solute species can be well characterized. However, it is hoped that the work presented here offers some further insight into the mechanisms governing interstitial transport and microvascular exchange. Continued research in this area will not only contribute to a fundamental understanding of the operation of the microvascular exchange system, but will assist clinicians in developing more effective techniques for fluid resuscitation and drug delivery. Chapter 8 Recommendations In the preceding chapters we investigated the combined effects of a number of system parameters (such as the interstitial hydraulic conductivity, the interstitial plasma protein diffusivity, and the plasma protein convective hindrance) on the steady-state and transient exchange of fluid and various plasma protein species within a model tissue representative of mesentery. However, many questions regarding the nature of the interstitium and its influence on mass exchange within tissues remain unanswered. These include, among others, the effect of interstitial swelling on microvascular exchange and the nature of interstitial fluid flow. To address these and other questions, the analysis presented in this dissertation might be extended to include the following. 1. The equations describing interstitial deformation should be incorporated into the numer-ical simulations to include the influence of tissue swelling on microvascular exchange. Alternate expressions suitable for large changes in interstitial hydration should be sought out and applied, where possible. 2. The analysis of mass exchange in mesentery should be extended to other tissue mod-els. Such models would include, for example, a more rigorous description of lymphatic drainage. 3. The effect of local gradients in interstitial coUoid osmotic pressure on local interstitial fluid flow should be investigated, possibly by introducing an "effective interstitial reflection coefficient' analogous to the capillary wall reflection coefficient into the extended Darcy flux expression. However, a rigorous theoretical description of interstitial fluid flow is to be desired over the introduction of an arbitrary parameter such as this. 171 Chapter 8. Recommendations 172 4. F inal ly , the prehminary study of the transient movement of multiple tracer species through the interst i t ium should be expanded to consider other species having a broader range of solute transport characteristics. Nomenclature SYMBOL DESCRIPTION UNITS A;, i=l,2,3 first, second, and third virial coefficients of coUoid F • L ' / M 1 osmotic pressure relationship for aggregate plasma protein species C b plasma protein concentration of luminal fluid associated M / L 3 with boundary b C ^ 1 local concentration of plasma protein species k in incremental M / L 3 volume fraction 1 (6nl) C k local concentration of plasma protein species k in species' M / L 3 distribution volume fraction (nk) Cr Courant number D permeability of membrane boundary to aggregate plasma L/0 protein species Da local dispersion coefficient of interstitial plasma protein species k L2/6 D ^ local effective diffusion coefficient of interstitial plasma protein species k L 2 /# F k function relating excluded volume fraction for plasma protein species k to the sohd volume fraction (ns) F s function relating the inrmobfle fluid phase volume fraction to the sohd volume fraction (ns) G k function relating the osmotic pressure of plasma protein species k (LTk) to its concentration (C k) H mesentery thickness L H e f f effective resistance thickness for one-dimensional mesentery L 173 Nomenclature 1 7 4 SYMBOL DESCRIPTION UNITS j k local convective flux of protein species k in x; direction, relative M/(L 2 • 0) to moving solids local diffusive flux of protein species k in x; direction, relative M/(L 2 • 0) to convective flux j ^ . local mechanical dispersive flux of protein species k in Xj direction, M/(L 2 • 0) relative to convective flux j s transcapillary plasma protein flux M/(L 2 • 0) j k total local flux of plasma protein species k in Xj direction, M/(L 2 • 0) j v transcapillary fluid flux L/0 j k . local volumetric fluid flux in Xj direction through distribution L/0 volume of protein species k, relative to the moving solids j° local total volumetric fluid flux in Xj direction, relative to L/0 moving solids j W i local non-specific fluid flux in x; direction, relative to moving L/0 solids K k local interstitial hydraulic conductivity associated with distribution L 4 / (F- 0) volume of protein species k K° local total interstitial hydraulic conductivity L4/(F • 0) K' local interstitial specific hydraulic conductivity L 2 L distance separating arteriolar and venular capillaries L L p hydraulic conductance of membrane boundary L3/F-0 l x ; x;th component of outward normal, n, of boundary Al maximum dimension of finite element n unit outward normal of boundary n k local distribution volume fraction of protein species k n e k local excluded volume fraction of protein species k Nomenclature 175 SYMBOL n n° P1 Pe q° R R k tot R T t tk W o V 1 vk DESCRIPTION local total mobile fluid volume fraction local immobile fluid volume fraction local solid phase volume fraction local fluid hydrostatic pressure in incremental volume fraction 1 (tW) average local hydrostatic fluid pressure Peclet number local total volumetric fluid flux in Xi direction, relative to stationary coordinates universal gas constant retardation factor of plasma protein species k effective hydraulic resistance of in y direction for one-dimensional mesentery effective diffusive resistance of in y direction for one-dimensional mesentery absolute temperature time time for breakthrough curve of species k to reach 50 % of its steady-state value local sohd phase displacement in X; direction partial molar volume of fluid in incremental volume fraction 1 (for1) molar volume of pure fluid local mean convective velocity of protein species k in x; direction, relative to sohd phase velocity (vs;) local sohd phase velocity local superficial convective solute velocity UNITS F / L 2 F / L 2 L/6 F/(MOL • T) 6 • F / L 3 6/1 T e 6 L L 3 /M0L L 3 /MOL L/6? L/0 Nomenclature 176 SYMBOL DESCRIPTION UNITS Vgd local superficial dispersive solute velocity L/6 vk. local mean fluid velocity in x\ direction through distribution volume L/0 of protein species k, relative to solid phase velocity (v8.) v" ratio of particle velocity to local fluid velocity, for arbitrary spherical L/6 particle travelling in cylindrical channel local fluid mole fraction in incremental volume fraction 1 (far1) Xj local spatial coordinate L x- local spatial coordinate of deformed medium L Q ratio of interstitial resistance to plasma protein diffusion to interstitial resistance to fluid flow (K^P011 /D^) Qk longitudinal dispersivity of protein species k in interstitium L a k transverse dispersivity of protein species k in interstitium L 0k ratio of hydraulic conductivity in distribution volume k to total interstitial hydraulic conductivity (K k/K°) 0' ratio of immobile fluid phase volume fraction to solid phase volume fraction (n^/n5) 7 W fluid phase activity coefficient A difference sign 6% Kronecker delta function 6nl incremental volume fraction 1 e; local solid strain in the Xj direction £v local volumetric dilation of interstitium t?j- local coefficient of mechanical dispersion for protein species k L2/6 X ratio of particle diameter to channel diameter for arbitrary spherical particle travelling in cylindrical channel Nomenclature 177 SYMBOL DESCRIPTION UNITS u^ff effective fluid chemical potential of luminal fluid associated with F / L 2 boundary b Uy, general solvent chemical potential F • L/MOL fr], local chemical potential of fluid in incremental volume fraction 1 (cm1) F • L/MOL fi^f reference fluid chemical potential F • L/MOL £ k local convective hindrance of protein species k n1^1 local osmotic pressure of protein species k in incremental volume F / L 2 fraction 1 (/in1) ITk local osmotic pressure of protein species k averaged over its F / L 2 distribution volume fraction (nk) cr° reflection coefficient of membrane boundary b cr^ j component of total stress tensor in interstitium component of effective stress tensor in interstitium c; finite element weighting function $ sohd displacement potential function <j)\ finite element basis function fi interstitial compliance function fi specific interstitial compliance function [•]b interstitial quantity evaluated at boundary b Superscripts and Subscripts alb albumin anal analytical solution of dependent variable art arteriolar capillary b unspecified permeable boundary Nomenclature Superscripts and Subscripts glob globulin grid finite element grid quantity int interstitium lym lymphatic mes mesothehum P plasma simul numerical simulation solution of dependent variable tot total 'effective' quantity for one-dimensional mesentery ven venular capillary 7 dimensionless quantity (see text for specific definitions) References [1] Anderson, M.P. 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Appendix A One Dimensional Approximat ion to the Two Dimensional M o d e l Mesentery A . l Introduction The mathematical model developed earlier to describe interstitial transport and microvascular exchange in mesentery treats the tissue as a two-dimensional structure. However, the fact that the distance separating the arteriolar and venular vessels is an order of magnitude greater than the tissue thickness provokes the question: can the behavior of the tissue segment be adequately described by a one-dimensional model. As initial evidence that this in fact is the case, one need only examine the surface plots of interstitial fluid pressure and interstitial plasma protein concentration from the two-dimensional simulation presented earlier (see Figure (4.5) of Chapter 4). In this case the gradients in the transverse (y) direction are insignificant compared to those in the longitudinal (x) direction. However, this represents the results of only one of 26 simulations. Hence a detailed study was undertaken to determine under what conditions a one—dimensional description of the system would prove adequate. The presentation will take the following form. In Section A.2 the mathematical expressions describing interstitial fluid and plasma protein transport are developed. These equations, along with the various boundary conditions, are then cast in dimensionless form. Section A.3 describes the simulations performed in this study, while Section A.4 outlines the numerical procedures used. In Section A.5 the results of the study are presented and the ramifications discussed. Section A.6 summarizes the work. We will now develop the one-dimensional model of interstitial transport in mesentery. 191 Appendix A. One Dimensional Approximation to the Two Dimensional Model Mesentery 192 A.2 One—Dimensional Approximations to the Two