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Microvascular exchange in human tissue Gates, Ian 1992

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Microvascular Exchange in Human TissueByIan GatesB.Sc. (Chemical Engineering) University of Calgary, Canada.A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF APPLIED SCIENCECHEMICAL ENGINEERINGWe accept this as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1992© Ian Gates, 1992In presenting this thesis in partial fulfillment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely available forreference and study. I further agree that permission for extensive copying of this thesis forscholarly purposes may be granted by the head of my department or by his or herrepresentatives. It is understood that copying or publication of this thesis for financial gainshall not be allowed without my written permission.Department of Chemical EngineeringThe University of British ColumbiaVancouver, British ColumbiaSeptember 1992AbstractA transient, spatially distributed mathematical model is developed describing theexchange of materials (fluid and solute) across the capillary membrane into the interstitialspace. The formulation includes a lymphatic sink which drains both fluid and solute fromthe tissue. This can be located anywhere within the tissue. The model is constructed incylindrical coordinates and consist of the capillary lying along the z axis and the tissueenvelope surrounding the capillary.The driving force for fluid motion is the fluid chemical potential. This is equal to thedifference between the local fluid hydrostatic pressure and the local colloid osmotic pressure.Starling's hypothesis governs fluid flow across the capillary wall. This states that the amountof fluid that crosses the capillary membrane is due to the transmembrane potential difference.The fact that solute may leak across the membrane promotes the use of a capillary membranereflection coefficient. In the tissue, the fluid motion is found from a modified Darcy's lawwhich makes use of the gradient in the fluid potential rather than the hydrostatic pressure. Inaddition, a tissue reflection coefficient is used.The study consists of an evaluation of the effect the physiological parameters have onthe system. This is presented in the form of a sensitivity analysis for steady state resultsonly. It is shown that the strength of the lymphatic sink is important in promoting fluidreabsorption back into the capillary and negative hydrostatic pressures (subatmospheric)throughout the tissue.Transient test are performed to evaluate the regulating mechanisms for capillary-tissue fluid balance. The capillary membrane, the colloid osmotic pressure, and thelymphatic sink are examined for their roles in maintaining fluid balance. It is found that thecolloid osmotic pressure acts as a negative feedback signal regulating the cycle of soluteconcentrations and fluid hydrostatic pressures throughout the tissue. The lymphatic sink isimportant as it provides a mechanism for lowering tissue pressures and removing solute fromthe interstitial space, thus lowering the tissue colloid osmotic pressure. The trends indicatedin the results compare well with results from Manning et al. (1983) and Taylor et al. (1973).ii lTable of ContentsAbstractList of Figures^ ViiiList of Tables xvAcknowledgement^ xvi1^Introduction ^12^Microvascular Exchange Physiology ^42.1 Body Fluid Compartments and the Circulation System ^52.1.1 The Blood and Plasma Proteins ^72.2 The Blood and Lymphatic Flow and Microvascular Exchange ^82.2.1 Blood Vessel Classification ^92.2.2 The Lymphatics^  102.2.3 Microvascular Exchange^  12The Capillary Wall and Basement Membrane^ 12Transcapillary Transport^  152.3 The Interstitial Space ^182.3.1 Volume Exclusion^  222.3.2 Geometry of Tissue  233^Model Development and Formulation^  243.1 Literature Review^  24iv3.2 The Interstitium as a Contiuum^  283.3 Model Formulation^  293.3.1 Model Geometry  293.3.2 Fluid Transport in the Fiber Matrix Porous Media^ 313.3.3 Solute Transport in the Fiber Matrix Porous Medium^ 353.3.4 The Lymphatic Sink^  403.3.5 Volume Exclusion and Surface Depletion^ 43Volume Exclusion (Inaccessible Volume)  43Surface Depletion^  473.4 Boundary Conditions  493.5 Fluid and Solute Exchange in the Capillary ^  513.6 Numerical Procedure and Solution Algorithm  543.7 Summary of Equations in Dimensionless Form^  564^Effects of System Parameters on Steady State Microvascular Exchange : ASensitivity Analysis^  594.1 Introduction  594.2 Problem Statement^  604.3 Parameter Values  604.4 Case Studies^  624.5 Discussion of Results and Sensitivity Analysis^  664.5.1 The Base Case^  664.5.2 Sensitivity Analysis  73Retardation Factor, ^  73Capillary Reflection Coefficient, a^  76Tissue Reflection Coefficient, at  80Solute Diffusion Coefficient, Ddiff^ 82Tissue Hydraulin Conductivity, K 84Capillary Membrane Diffusive Permeability, PS^ 88Lymphatic Sink Strength, LS^ 90Lymphatic Sink Pressure, PI, 92Lymphatic Sink Radial Position^ 94Capillary Membrane Filtration Coefficient, Lt, ^ 97Mechanical Dispersion^ 1014.5.3 Variable Capillary Membrane Filtration Coefficient, Lp (z) ^ 1034.5.4 High Flow Channels^ 1084.6 Conclusions^ 1135^Capillary - Tissue Fluid Balance and the Effects of Perturbations on MaterialExchange^  1165.1 Introduction  1165.2 Transcapillary Fluid Flow^  1175.3 Capillary - Tissue Fluid Balance : Mechanisms of Regulation^ 1225.3.1 Perturbed Tissue Solute Concentration^  1225.3.2 Perturbed Venous Hydrostatic Pressure  1275.3.3 Perturbed Capillary Solute Concentration^ 1345.3.4 Transient Analysis of the Effects of Osmotic Pressure^ 1465.4 Discussion and Conclusions^  1536^Conclusions and Recommendations^  157Nomenclature^  160viReferences^  165Appendices  171A The Petrov-Galerkin Finite Element Method^  171A.1^The Galerkin Finite Element Method  171Discretization^  173Interpolation  175Weak_ formulation^  176Formation of elemental matrices^  178Solution of the resulting algebraic equations^ 182Cylindrical geometry and variable properties  183A.2 The Petrov-Galerkin Upwinding Method^  183B^The Solution Algorithm and Under-relaxation Scheme^ 186B.1^The Solution Algorithm^  186B.2^The Under-relaxation Scheme  188C^Program Listing^  190vi iList of Figures2.1 Distribution of body fluid compartments. ^ 52.2 Structure of lymphatics in cat mesentery (taken from Schmid-Schonbein,1990). ^ 112.3 Structure of and transport mechanisms through capillary walls. ^ 142.4 Starling forces. 152.5 The interstitial space (taken from Bert and Pearce, 1984). ^ 192.6 Microvascular network of the frog sartorius muscle (taken from Dietrich andTyml, 1992). ^ 233.1 Model geometry. 303.2 Lymph flow versus average tissue hydrostatic pressure (taken from Taylor etal., 1973).  ^413.3 Conceptual picture of the lymphatic sink. ^ 423.4 Volume exclusion of solute. ^ 443.5 Partitioning of a representative elementary volume into the various volumefractions. ^ 453.6 Schematic^diagram^of (a)^surface^depletion^of spherical^and rod-likemolecules from fiber and (b) steric exclusion of sphere in random network offibers. ^ 483.7 Fluid and solute flow in capillary. ^ 524.1 Dimensionless^tissue^solute^concentration,^hydrostatic^pressure,^Pecletnumber, potential and velocity vector distributions for the base case at steady-state (huse). ^ 67viii4.2 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case withoutosmotic pressure effects but with the lymphatic sink at steady-state (nows).  724.3 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with theretardation factor, E, equal to 0.1 at steady-state.   754.4 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thecapillary reflection coefficient, a, equal to 0.1 at steady-state.   774.5 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thecapillary reflection coefficient, a, equal to 0.99 at steady-state.   794.6 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with the tissuereflection coefficient, at, equal to 0.0 at steady-state.   814.7 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with the solutemolecular diffusion coefficient, Ddiff , increased by an order of magnitudefrom the base value at steady-state.   834.8 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thehydraulic conductivity, K, decreased by an order of magnitude from the basevalue at steady-state.   854.9 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with theixhydraulic conductivity, K, increased by an order of magnitude from the basevalue at steady-state. ^  864.10 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thecapillary membrane diffusive permeability, PS, increased by a large amount(x10000) from the base value at steady-state.   894.11 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thelymphatic sink strength, LS, decreased by a factor of five from the base valueat steady-state.   914.12 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thelymphatic sink pressure, PL *, lowered to -0.4 (-12 mmHg) at steady-state. ^ 934.13 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thelymphatic sink repositioned at r* = 0.025 at steady-state.   954.14 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thecapillary membrane filtration coefficient, L I,, increased by a factor of twofrom the base value at steady-state. ^  994.15 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thecapillary membrane filtration coefficient, L1,, decreased by a factor of twofrom the base value at steady-state. ^  1004.16 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with theinclusion of mechanical dispersion at steady-state. ^  1024.17 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thecapillary membrane filtration coefficient, Lp , varied linearly from the basevalue at the arteriolar end to twice the base value at the venular end of thecapillary at steady-state. ^  1044.18 Step function for variable capillary membrane filtration coefficient. ^ 1064.19 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with a stepchange in the capillary membrane filtration coefficient, Lp , at z* = 0.50 fromthe base value at the arteriolar end to twice the base value at the venular endof the capillary at steady-state. ^  1074.20 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with a singlehigh flow channel centred at z* = 0.50 at steady-state.   1104.21 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with thedouble high flow channels centred at z* = 0.30 and 0.70 at steady-state.   1125.1 Starling forces across capillary wall. Taken from Taylor and Townsley(1987). ^  1205.2 Transcapillary fluid motion : Driving forces for base case. ^ 1205.3 Transcapillary fluid motion : Driving forces for LS reduced by a factor offive. ^  121xi5.4 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredtissue solute concentration at t = 1800 s.   1245.5 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredtissue solute concentration at t = 3600 s.   1255.6 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredtissue solute concentration at new steady-state.   1265.7 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with elevatedvenous hydrostatic pressure (P,- yen* 0.6667) at t = 120 s.   1295.8 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with elevatedvenous hydrostatic pressure ( Pvert* 0.6667) at t = 240 s.   1305.9 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with elevatedvenous hydrostatic pressure ( Pven* = 0.6667) at t = 600 s.   1315.10 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with elevatedvenous hydrostatic pressure (P„en * = 0.6667) at new steady-state.   1325.11 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredcapillary solute concentration (,cart* 0.1000) at t = 300 s.   1355.12 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredcapillary solute concentration (can* = 0.1000) at t = 600 s.   136xii5.13 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredcapillary solute concentration (cart * = 0.1000) at t = 1800 s.   1375.14 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredcapillary solute concentration (can* = 0.1000) at new steady-state.   1385.15 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with enlargedcapillary solute concentration (cart* = 1.2000) at t = 300 s.   1415.16 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with enlargedcapillary solute concentration (Cart = 1.2000) at t = 600 s.   1425.17 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with enlargedcapillary solute concentration (can * = 1.2000) at t = 1800 s.   1435.18 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with enlargedcapillary solute concentration (curt * = 1.2000) at new steady-state.   1445.19 Average tissue and lymphatic fluid solute concentrations through time.   1475.20 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with colloidosmotic pressure effects throughout the system and zero arteriolar capillarysolute concentration (cart * = 0.0000) at t = 600 s.   1485.21 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with colloidosmotic pressure effects throughout the system and zero arteriolar capillarysolute concentration (cart * = 0.0000) at t = 1200 s.   1495.22 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with colloidosmotic pressure effects throughout the system and zero arteriolar capillarysolute concentration (cart * = 0.0000) at t = 1800 s.   1505.23 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with colloidosmotic pressure effects throughout the system and zero arteriolar capillarysolute concentration (cart* = 0.0000) at t = 3600 s.   1515.24 Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with colloidosmotic pressure effects throughout the system and zero arteriolar capillarysolute concentration (curt * = 0.0000) at new steady-state.   152A.1 General description of elliptic boundary value problem.   173A.2 Discretized geometry using linearly interpolated rectangular elements.   174A.3 Rectangular element with linear interpolation.   175A.4 Node communication. ^  181xivList of Tables2.1 Electrolyte compositions of human body fluids (in meq/1 water) (taken fromGanong, 1989).  ^62.2 Plasma proteins and their approximate molecular weights (taken fromGanong, 1989).  ^82.3 Types of blood vessels in humans (taken from Ganong, 1989).  ^93.1 General model assumptions^  324.1 Parameter values. ^  634.2 List of cases.  654.3 Results from various cases. ^  1155.1 Results summary for transient cases performed at new steady-state. ^ 153B.1 Overall solution algorithm. ^  186B.2 Capillary hydrostatic pressure solution algorithm. ^  187xvAcknowledgementI would like to express my thanks to N.S.E.R.C. for funding me during my stay at theUniversity of British Columbia.I would also like to thank the following individuals:• Drs. J.L. Bert and B.D. Bowen for their help and guidance during the past two years,• Dr. Clive Brereton for free beer and some lively conversations,• Linda Hou and Katrina Gaarder (yes negative time) for the great chats,• Mike 'MTSman' Choi, Ian 'Spud' Wilson, and Rory 'Sheep' Todd for their friendship,• my parents for all their support and love over the years,• and Raksha, I thank you the most for the love and caring that makes you the mostimportant person in my life - LILLYANDIVY forever my love.xviChapter 1 : INTRODUCTIONChapter 1 : IntroductionThe circulating blood contains various solutes which are distributed via thecardiovascular system to the tissues within the human body. These solutes are transportedinto the space between the tissue cells (the interstitium). Nearly all of this material exchangebetween the circulatory system and interstitial space occurs in capillaries. This is termedmicrovascular exchange and involves specifically the transport of fluid and solutes (proteinsand electrolytes) across the capillary membrane, within the interstitial space, and out thelymphatics. The lymphatics drain fluid from the interstitium back into the circulation. Theregulation of the microvascular exchange system is very complex and depends largely on theproperties of the capillary membrane, the interstitium, and the lymphatics. The regulatorymechanisms of microvascular exchange protect against edema formation (excessaccumulation of fluid) or dehydration. The objective is to identify controlling features ofmicrovascular exchange system and their influence on the system dynamics for theprevention of edema or dehydration.The dominant forces driving fluid motion across the capillary wall and within theinterstitium are the hydrostatic and colloid osmotic pressures. The colloid osmotic pressurelowers the driving potential for fluid motion and is a non-linear function of the proteinconcentration. In this way, it acts to dilute protein. The driving potential is the differencebetween the hydrostatic pressure and the colloid osmotic pressure. This exists across thecapillary wall and within the tissue. Proteins are driven across the capillary membrane byChapter 1 : INTRODUCTION^ 2two main mechanisms. The concentration difference across the capillary membrane and thefluid motion induce both diffusion and convection respectively.The potential drop along the capillary causes fluid and solutes to be filtered into theinterstitial space at the arteriolar end of the capillary. Depending on the strength of thelymphatic sink, material may be reabsorbed back into the capillary at the venular end of thecapillary. An increase in the lymphatic sink strength enlarges the lymph drainage from thesystem. The filtration, reabsorption, and lymphatic drainage create recirculation patternswithin the tissue space.The amount of fluid which leaves via the lymphatic sink is a function of the localtissue hydrostatic pressure. Similarly, the rate at which solute that drains out through thelymphatic sink is both a function of the local tissue pressure and solute concentration. Themicrovascular exchange system obviously permits complex behaviours which are the resultof the fluid-solute-tissue matrix interactions in the presence of the pressure and concentrationfields.Experimental investigations of the microvascular exchange system are difficult due tothe scale of the system. Typical dimensions are in the range of tens to hundreds ofmicrometers. An alternative and complementary approach is the development of complexmathematical models describing microvascular exchange. These models can be used toexamine system dynamics and sensitivity and suggest directions for further experimentalresearch. Several mathematical models of microvascular exchange have appeared in the pastfew years (Baxter and Jain, 1989, 1990, 1991a, 1991b; Taylor et al., 1990b; Chapple, 1990).The two types of models in common use are compartmental and spatially distributed models.Compartmental models assume well-mixed, homogeneous compartments between whichmaterial exchange occurs. The equations governing transport between the compartments arefunctions of time only. In spatially distributed models, the material spatially distributes itselfthroughout the system space nonhomogeneously. The equations are functions of position inaddition to time.Chapter I : INTRODUCTIONIn this work, a transient, spatially distributed model is developed in cylindricalcoordinates to describe microvascular exchange of fluid and a single solute. Thedistributions of fluid velocity and solute concentration are determined. In this manner, theeffects of capillary membrane properties, osmotic pressure, the lymphatic sink, and highfluid conductivity channels can be examined. It also permits the use of the model to suggestkey features of the system that modulate and control hydrostatic pressure, fluid motion, andsolute field behaviour, possibly by some feedback mechanism. One of the primary uses ofsuch models is use as a tool to suggest possible further experimental work. Also,assumptions about the system may be tested and validated.In Chapter 2 the pertinent microvascular physiology is briefly outlined. Chapter 3presents the governing equations of microvascular exchange and model assumptions. Theappropriate boundary conditions are also presented. In Chapter 4, a sensitivity analysis ispresented detailing the influence of various physiological parameters on the microvascularexchange system. The effects of variable capillary membrane permeability and tissuehydraulic conductivity are also examined in this chapter. The latter allows us to investigatethe impact of high flow channels on the system. The controlling features of the capillarymembrane, the osmotic pressure, and the lymphatic sink are investigated in Chapter 5. Thetransient simulations performed here permit the identification of the roles of the capillarymembrane, the osmotic pressure, and the lymphatic sink within the system from the fluidbalance point of view. The results suggest that the osmotic pressure acts as a negativefeedback signal regulating capillary-tissue fluid balance. The final chapter, Chapter 6, listssome general conclusions and recommendations for further research work.Chapter 2 : PHYSIOLOGY OUTLINE^ 4Chapter 2 : Microvascular Exchange PhysiologyThis chapter discusses the underlying physiology of the microvascular exchangesystem. The circulation system is a closed loop through which blood flows. The heartpumps blood into the arteriolar circulation first via the aorta, then arteries, arterioles, andeventually capillaries. Microvascular exchange occurs between the capillaries and thesurrounding tissue. The blood returns to the heart by venules and then veins.The discussion of physiology is divided into five sections. The first section willoutline broadly the physiology of the circulatory system and body fluids. The second sectiondetails the flow of blood and lymph through the circulatory system and lymphatics and themajor microvascular mass exchangers, the capillaries. The third section presents theinterstitium, its constituent materials, and properties. In the fourth section, the flowproperties of membranes and the nature of capillary walls will be summarized. Theremaining section, section five, will briefly describe the structure and function of sometissues. The focus in that section will be on the geometrical configuration of capillarieswithin tissues.Chapter 2 : PHYSIOLOGY OUTLINE^ 52.1 Body Fluid Compartments and the Circulation SystemThe basic unit of machinery in all living creatures is the cell. Cells exist inmulticellular organisms in a sea of extracellular fluid. This fluid supplies cells with nutrientsand accumulates metabolic wastes. The extracellular fluid is divided into two compartments,the interstitial fluid and the blood circulation. In an average adult male human, theintracellular water (fluid within cells) constitutes about 40% of the total body weight(Ganong, 1989). The extracellular component, meanwhile, makes up about 20%. Aboutone-quarter of the extracellular fluid is the circulating blood plasma and the remaining three-quarters is the interstitial fluid (Ganong, 1989). This means the blood plasma and interstitialfluid comprise about 5% and 15% of the total body weight respectively. The distribution ofthe body fluid compartments are shown in Figure (2.1).Extracellular fluid20% body weightBlood plasma, 5% body weightInterstitial fluid, 15% body weight Intracellular fluid, 40% body weightFigure 2.1: Distribution of body fluid compartments.Chapter 2 : PHYSIOLOGY OUTLINE^ 6The composition of solutes (proteins and electrolytes) is considerably different ineach of the body fluid compartments. These differences in composition are largely due tothe character of the barriers between the fluid compartments. As seen from Table (2.1), theconcentration of protein anions in the interstitial fluid is significantly lower than that in theintracellular fluid and the blood plasma. Also the electrolytes Na + and Cl - arepredominantly found in the extracellular fluid whereas K+ is largely intracellular (Ganong,1989).Table 2.1: Electrolyte compositions of human body fluids (in meq/L water) (Ganong, 1989).Solute Blood Plasma Interstitial Fluid Intracellular FluidNa+ 152 143 14Cl - 113 117 smallK+ 5 4 157mg2+ 3 3 26Protein anions 16 2 74HCO3 - 27 27 10Besides supplying nutrients to cells and accumulating wastes, another function of thebody fluids is to provide buffering capacity. Buffering allows the intracellular andextracellular fluids to maintain a constant pH. For instance, the pH of the extracellular fluidis maintained at 7.40+0.05 (Ganong, 1989).The particular fluid compartments of interest in this work are the blood circulation(in the capillaries) and the interstitial space. The blood circulation system removes wastes(for example carbon dioxide) from and supplies nutrients (for example oxygen) to bodytissues. This is achieved via the cardiovascular system. The latter consists of the heart (apump) and a complex system of branching elastic tubes that distribute the blood throughoutthe body. The left ventricle of the heart pumps blood first through arteries, then arterioles,Chapter 2 : PHYSIOLOGY OUTLINE^ 7and ultimately capillaries. After mass exchange within the capillaries, the blood drainsthrough venules and then veins back to the right atrium of the heart. This closed circuit isthe systemic (major) circulation. In the pulmonary (lesser) circulation, the right ventriclepumps blood into the vessels of the lungs. This is the site of gas exchange.An additional circulatory system is the lymphatic circulation. Fluid and solutesexchange across the capillary wall into the tissue. Some of this exchanged material that isderived from the interstitium flows into the lymphatic vessels that drain via the thoracic ductinto the venous system. This fluid is known as lymph. More details about the lymphaticsand capillary exchange will be discussed later.2.1.1 The Blood and Plasma ProteinsThe blood is essentially a suspension of several cellular components - red and whiteblood cells and platelets - in plasma. Red blood cells carry oxygen bound to hemoglobin.The white blood cells are instrumental in the immune system for body defenses to viral andbacterial infections. Dissolved within the blood plasma are many ions, proteins, organic andinorganic molecules. These may serve several functions, for example, as nutrients,hormones, or aid in the transport of other compounds.Table (2.2) lists some plasma proteins found in blood. The most abundant proteinsare albumin, globulin, and fibrinogen. At the normal plasma pH of 7.40, these proteins arein their anionic forms. The proteins cannot easily traverse the capillary walls and thus exerta colloid osmotic pressure difference across the capillary wall. A reduction in the tissue-sidecolloid osmotic pressure tends to draw water back into the blood circulation from theinterstitial space (this effect actually manifests itself as a reduction in the localtransmembrane potential). The most abundant and osmotically active protein is albumin.Normally, over 50% of the plasma protein is albumin. Previous work (Taylor, 1990) onChapter 2 : PHYSIOLOGY OUTLINE^ 8microvascular exchange indicates that colloid osmotic pressure and its gradients havesignificant effects on fluid flows and solute distributions within the interstitium. Whenplasma protein levels are low (prolonged starvation or liver disease), this is known ashypoproteinemia. The decrease in protein osmotic activity leads to edema formation, i.e.,excess fluid in tissues.Table 2.2: Plasma proteins and their approximate molecular weights (Ganong, 1989).Plasma Protein^Molecular Weight (Daltons)Albumin 69000Hemoglobulin 64450Fibrinogen^ 340000131 - Globulin 90000y - Globulin 1560002.2 The Blood and Lymphatic Flow and Microvascular ExchangeBlood flows throughout the systemic circulation via different blood vessels. Thedriving force for flow is mainly the hydrostatic pressure gradient set up by the pumpingaction of the heart. To a lesser degree, diastolic recoil of the arterial walls, vein compressionduring exercise, and negative pressure generated in the thorax during inspiration alsocontribute to blood flow. The blood vessels are organized into various types based on theirsize.Chapter 2 : PHYSIOLOGY OUTLINE^ 92.2.1 Blood Vessel ClassificationTable (2.3) categorizes the various types of blood vessels. Upon leaving the heart,the blood enters immediately the aorta and then the arteries. These vessels are thick-walledand of large diameter. The walls of the aorta and arteries contain copious amounts of elasticmaterial and offer some pumping action as they recoil during diastole (Ganong, 1989).Table 2.3: Types of Blood Vessels in Humans (Ganong, 1989).Vessel LumenDiameterWall Thickness ApproximateTotal CrossSectional Area(cm2 )Blood VolumeFraction(systemic)Aorta 2.5 cm 2 mm 5 0.02Artery 0.4 cm 1 mm 20 0.08Arteriole 30 gm 20 gm 400 0.10Capillary 5 p.m 1 pm 4500 0.05Venule 20 pm 2 pm 40000.54Vein 0.5 cm 0 5 mm 40Vena cava 3.0 cm 1.5 mm 18The blood then flows into smaller diameter vessels called arterioles. These containless elastic tissue than arteries but contain more smooth muscle. This muscular actionprovides the main source for resistance to blood flow in the circulation. The arteriolessubdivide further into smaller vessels called metarterioles which then empty into thecapillary beds. The capillary bed is arranged as a complex random network of relativelyhighly permeable tubes within the tissue. This is the main site of material exchange betweenChapter 2 : PHYSIOLOGY OUTLINE^ 10the blood circulation and the interstitium. Capillaries have a wall thickness and diameter ofabout 1 1.1M and 5 pm respectively. The total available area for material exchange exceeds6300 m2 in a normal human adult (Ganong, 1989). Section (2.2.3) presents the structure ofthe capillary wall and describes its functions in detail.The blood drains from the capillary beds into venules and eventually into veins. Thewalls of venules and veins are thin and distend easily, but, however, do contain some smoothmuscle. This permits them to function as a variable volume blood reservoir (Ganong, 1989).2.2.2 The LymphaticsSome of the fluid from the interstitial space flows into the lymphatic circulation.Figure (2.2) illustrates the form and structure of a typical lymphatic terminal. Interstitialfluid enters the lymphatic system through a readily deformable lymphatic bulb into the initiallymphatics that have no smooth muscle and are not contractile. These vessels join to formcollecting lymphatics that may or may not contain smooth muscle and valves. A treestructure then follows as the lymphatic vessels converge to form bigger vessels. Lymphaticvessels are not necessarily paired with any blood vessel but are randomly distributedthroughout the interstitium (Schmid-Schonbein, 1990).It is not clearly understood how fluid and solutes are transported into the lymphaticsand how lymph is propelled within the lymphatic system. On average, about 50% of thetotal circulating protein recirculates via the lymphatics and 2-20 liters of fluid pass throughthe lymphatics daily (Ganong, 1989; Guyton et al., 1987). Material transport requires someform of potential difference. Two different forms of lymph pump are thought to existcorresponding to the two different observed lymphatic anatomies (Schmid-Schonbein, 1990).They are the intrinsic and extrinsic lymph pumps.Chapter 2 : PHYSIOLOGY OUTLINE^ 11The periodic compression of the microlymphatics by its own smooth muscle is themechanism proposed for the intrinsic lymph pump. Bat wing is the only known mammalianexample where this type of mechanism exists. The smooth muscle activity acts to fill thelymphatics by expanding the lymphatic ending. Lymph then empties into the collectinglymphatics. A requirement for this to work is that the membrane of the initial lymphaticbulb be permeable to material in one direction only. This is achieved through the use ofendothelial microvalves which prevent fluid from flowing out of the lymphatics (Schmid-Schonbein, 1990).Figure 2.2: Structure of lymphatics in cat mesentery(taken from Schmid-Schonbein, 1990).The extrinsic pump mechanism does not make use of smooth muscle to drive materialinto the initial lymphatics. The only way to achieve lymph flow in this case is via a pressuredrop from the interstitial space to the initial lymphatics or an active lymphatic membraneChapter 2 : PHYSIOLOGY OUTLINE^ 12pump operated by some cellular transport mechanism. Such a transport mechanism has notbeen conclusively identified as yet. Further, a pressure drop from the interstitium to theinitial lymphatics has not been firmly established. Pressure drops found experimentally tendto be small and periodic (Schmid-Schonbein, 1990).The second approach is the one adopted for this work. The lymph drainage rate isassumed to be a function of the local tissue hydrostatic pressure and lymphatic sink pressure.Tissues subjected to increased pressure drops between the interstitium and the initiallymphatics have enhanced lymph flows. For this reason, passive limb movement and skinmassage increase lymph flow. At present, the formation of lymph and dynamics related toits flow are not well understood (Schmid-Schonbein, 1990).2.2.3 Microvascular ExchangeThe basic functional unit of the microvascular exchange system is the capillary andits associated envelope of tissue. Also associated with this system unit are any lymphaticsthat may drain interstitial fluid. The lymph fluid eventually returns back to the circulationvia the right and left subclavian veins.Material (fluid and solute) exchange between the capillary and interstitium encountertwo main resistances, namely the capillary wall/basement membrane, and the interstitiumitself. Each of these resistances will be discussed in some detail in the following subsections.The Capillary Wall and Basement MembraneThe capillary wall consists of a single layer of endothelial cells. Figure (2.3)illustrates the structure of the capillary wall. The basement membrane (or basal lamina) is athin structure consisting primarily of a different specialised form of collagen than that foundChapter 2 : PHYSIOLOGY OUTLINE^ 13in the interstitial space. It surrounds and supports the blood vessels of the microvasculaturein an extracellular matrix envelope. In addition to its mechanical functions, the basementmembrane may also serve as a resistance to or conductance for material exchange (Bert andPearce, 1984).The capillary wall consists of a monolayer of epithelial cells. In general, capillarywalls are similar from tissue to tissue. Clear differences arise, however, when consideringthe relative molecular sizes of substances that may cross the capillary wall in differenttissues. In most tissues, water and relatively small solutes are the only materials that maycross the capillary walls with ease. The tissue fiber matrix restricts high molecular weightspecies such as plasma proteins due to their size. However, these molecules may transferinto the interstitial space by a variety of means.There is an assortment of different transport pathways across capillary walls that havebeen proposed. These are outlined in the following :• vesicular transportPlasma and substances in solution are taken up by endocytosis on the capillary lumina]side, transported across the endothelial cell interior, and then released into theinterstitium by exocytosis (Ganong, 1989). These vesicles may fuse to form aqueouschannels permitting passage of plasma (Bassingthwaite et al., 1989).• interendothelial cell cleftsMaterial transport may go through the endothelial cells or through the clefts betweenadjacent cells. These junctions usually vary in size from tissue to tissue but averagebetween 10-20 ptm except in the brain microvasculature where the junctions are nearlycompletely closed (the relatively impermeable blood-brain barrier). The structure ofthese pores is not well understood (Silberberg, 1988).• fenestrationsThe walls of some capillaries contain fenestrations. These are areas of the endothelialcell membrane that are stretched to form gaps typically between 20-100 nm in diameter.Interendothelial Vesicular TransportCell Clefts^or Channels FenestraeChapter 2 : PHYS'IOLOGY OUTLINE^ 14These fenestrations allow passage of relatively large molecules. In the liver, these gapsmay be of the order of 3000 nm diameter making the capillary wall very porous(Ganong, 1989).• passive diffusionThe capillary wall is very permeable to water and other small non-polar solutes. Lipidsoluble molecules also pass through the capillary wall but are hindered by the aqueousintracellular environment and thus can only pass through the capillary wall by travellingwithin the endothelial cell membrane of vesicles. These substances simply diffuse downtheir chemical potential gradient.DirectCapillaryWall CellsCDiu  ^Basement MembraneFigure 2.3: Structure of and transport mechanisms through capillary wallsThe mechanisms for transport across the capillary wall are summarized in Figure(2.3). There are other factors that affect transcapillary transport of solutes such as electriccharge, solute concentration, and pressure differences. The effect of albumin concentrationand hydrostatic pressure on capillary wall hydraulic conductivity has been studiedextensively (Dull et al., 1991; lida, 1990; and Parker et al., 1984).Chapter 2 : PHYSIOLOGY OUTLINE^ 15Transcapillaty TransportThe capillary wall exhibits sieving characteristics allowing it to be treated simply as amembrane. The various forms of transport across the capillary wall are not all fullyunderstood and so it is often treated mathematically as a porous membrane (Ogston andMichel, 1978; Curry, 1984; Taylor, 1990a). Here material exchange rates are expressed asthe product of a driving force, usually the potential gradient, and a conductivity constant(inverse resistance). The mathematical complexity of treating the membrane is simplifiedvia the use of the lumped resistance term.There are two main driving forces for fluid exchange across the capillary wall. Theseare the hydrostatic pressure and osmotic pressure gradients. The latter arises because ofsolute (protein) concentration difference across the capillary membrane. These two forcesfor fluid flow have been termed the Starling forces. The colloid osmotic pressure is usuallya non-linear function of the solute concentration and reduces the local fluid chemicalpotential. Figure (2.4) illustrates the roles of the Starling forces.HydrostaticPressurePc PtOsmoticPressureit 7CtMembraneCapillary^InterstitiumFigure 2.4: Starling ForcesChapter 2 : PHYSIOLOGY OUTLINE^ 16The Starling forces give rise to the Starling hypothesis (Taylor and Townsley, 1987) :VIM = [ P, - - (3{7r, - (2.1)where vfn is the transmembrane fluid velocity, Lp is the capillary wall filtration coefficient(also known as the capillary wall hydraulic conductivity coefficient), and a is the particularsolute's osmotic reflection coefficient. P(  and Pt are the capillary and interstitial hydrostaticpressures while ne and Itt are the capillary and interstitial colloid osmotic pressures,respectively. Note that the interstitial hydrostatic and colloid osmotic pressures used inequation (3.1) are evaluated immediately adjacent to the capillary membrane.The solute osmotic reflection coefficient, a, accounts for the difference between theeffective osmotic pressure difference which actually operates across the membrane and thecalculated osmotic pressure difference. If the reflection coefficient is equal to 1.0, the soluteis totally reflected from the membrane, i.e., the membrane is impermeable to the solute. Areflection coefficient of 0.0 means the solute has the same permeability as that of waterthrough the membrane, i.e., the only driving force for material exchange is the hydrostaticpressure gradient. Capillaries in the brain display a reflection coefficient of nearly 1.0(impermeable), while the liver sinusoids have a reflection coefficient of nearly 0.0 (totallypermeable) (Taylor and Townsley, 1987).In most capillary membranes, the protein reflection coefficient is somewhere between0.5 and 0.95 (Renkin, 1986). This is interpreted as meaning that the solutes are transferredinto the interstitium but that the effective colloid osmotic pressure driving force is a fraction(a) of the actual (measured) colloid osmotic pressure driving force.Chapter 2 : PHYSIOLOGY OUTLINE^ 17If there are two different means of transcapillary exchange, for example two differentpore sizes 1 and 2 (small and large pore sizes), this may be treated by an areal weightingbetween the two flow terms, i.e.,vfs = coL„,[P, — —^— rc )1+ (1— w)Lp2 [P, —^cs-2 (7t,— Icc )]^(2.2)where co (0<co<l) is the fractional area of pores having size 1.The Patlak equation (Patlak et al., 1963; Curry, 1987; Renkin, 1986) links solute fluxacross the capillary wall to the Starling hypothesis. For a membrane separating a capillaryhaving concentration c1 and an interstitial solution having concentration c2, the solute flux,js, is given by :PS(c, — c2 )Pe„,js = vf,(1— o-)c,+^ePe- —1where Pe,n is the modified capillary membrane Peclet number given by := vfm(1—Pe PSand PS is the diffusive component of the capillary permeability. When the convectivetransport contribution is zero or the capillary membrane completely reflects the solute,equation (2.3) reduces to the simple form :(2.3)(2.4)js = PS(c, —c2 )^ (2.5)Chapter 2 : PHYSIOLOGY OUTLINE^ 18The Patlak equation, (2.3), is a non-linear flux equation. Two assumptions to bear in mindabout its use are that, first, it is only applicable to a single solute species and second, thetransport pathways are all the same for both transport mechanisms (convection anddiffusion).The capillary wall Peclet number is important for indicating the relative roles ofconvection and diffusion in solute transport across the capillary membrane. For convection-dominated flows, Pe„, is greater than unity.In summary, Starling's hypothesis and Patlak's equation give expressions for the fluidand solute flux across the capillary wall respectively.2.3 The Interstitial SpaceThe interstitium is a three-dimensional network of fibrous connective tissuemolecules embedded in a gel-like matrix consisting of various polymers dissolved in theintercellular fluid. It includes all of the tissue space outside the capillaries, the lymphatics,and the cells themselves. The fluid and solutes flow around and through the cells and themolecular meshwork occupying the extracellular space. In this respect, the interstitial spaceis really a porous medium. The fibers impart mechanical strength and elasticity to theinterstitium. This allows for deformation and fluid accumulation within the interstitium.The resulting complicated nature and behaviour of tissue is evident. Figure (2.5) provides avisual depiction of the interstitial space.In the following, each of the various components in the interstitium will be brieflydiscussed.•^interstitial fluidThe largest component of the interstitial space is water. Some of the fluid is bound to thefibrous elements in the interstitium. In addition, the effective viscosity of the fluid isChapter 2 : PHYSIOLOGY OUTLINE 19Figure 2.5: The interstitial space (taken from Bert and Pearce, 1984).Chapter 2 : PHYSIOLOGY OUTLINE^ 20increased by the presence of mobile hyaluronan (formerly known as hyaluronate). Thismeans that even under very high pressures, a significant portion of the fluid is retained inthe tissue (Lank, 1983). The interstitial fluid originates from the capillary and either isreabsorbed back into the capillary or drains into the lymphatics. Like many porousmedia, the interstitial matrix is virtually impossible to describe rigorously from amicroscopic point of view. It is expected as fluid and solute traverse the interstitium thatsome channeling occurs. This phenomena is not well understood and has not beenconclusively proven (Bert and Pearce, 1984).• collagenous fibersCollagenous fibers impart structure and mechanical strength (in the form of tensilestrength) to the interstitium. These are long fiber bundles consisting of collagenmolecules (tropocollagen). A collagen fiber consists of an organized array of collagenmolecules. All of the molecules are arranged in parallel with many stable covalent cross-linkages between the molecules providing the high tensile strength (Bert and Pearce,1984; Laurent, 1987).• elastic fibersElastic fibers provide tissues with elasticity, conferring a rubber-like texture to tissues.The aorta and larger arteries are particularly rich in elastic fibers (Laurent, 1987). Themain elastic fiber found in tissues is elastin, which is constructed from the tropoelastinmolecule. Elastin is a three dimensional network of highly hydrophobic coiledmolecules jointed at many cross-links. When unstressed, elastin assumes a randomconfiguration, and contains about 0.56 ml water per ml elastin. This water is most likelyaccessible to smaller solutes (for example, glucose, urea, and sodium). However, largermolecules like colloid proteins are excluded from this space (Bert and Pearce, 1984).• glycosaminotdycans and proteoglycansThe properties these substances contribute to the interstitium are physicochemical innature. Glycosaminoglycans are hydrophilic charged polysaccharide chains containingChapter 2 : PHYSIOLOGY OUTLINE^ 21an amino sugar. An example of a glycosaminoglycan is hyaluronan (molecular weightusually between 106 and 107 Daltons) (Laurent, 1987). Glycosaminoglycans are usuallycovalently bound in tissue to polypeptides known as proteoglycans. Hyaluronan is anunbranched polysaccharide which forms an extended random coil when solubilized.Water is bound by hydrogen bonds to the hyaluronan molecule. The coil is furtherexpanded by the mutual repulsion of the negative charges throughout the hyaluronanmolecule. The volume occupied by the random coil is often greater than 1000 times thepolymer molecular volume. The effective viscosity of interstitial fluid is increasedconsiderably in the presence of hyaluronan. This is ascribed to entanglement that occursbetween different hyaluronan coils. At low hyaluronan concentrations, solutions may gel(Bert and Pearce, 1984). Proteoglycans form aggregates in the presence of and may bindto hyaluronan. These polymer chains are hydrophilic and have a high charge density.They also can bind to collagen fibers providing a stabilizing effect. These fibers tend tobind water due to their hydrophilic nature and restrict movement of the interstitial fluidand matrix (Bert and Pearce, 1984).• interstitial plasma proteinsThere are over 100 different plasma proteins (Bert and Pearce, 1984). The mostabundant plasma protein is albumin. About 60% of the total body albumin (approximatemolecular diameter is 7.5 nm) is contained in the extravascular space. This means thatthe interstitium provides a considerable reservoir for plasma proteins. At thephysiological pH of roughly 7.40, most of the plasma proteins are negatively charged.All of the plasma proteins exert a colloid osmotic pressure, but most of this is contributedby albumin. This work examines the concentration distributions of protein (albumin)within the interstitium.The response of the interstitium to any changes is the sum total of the effects of theperturbations on all of the components interacting with each other within the interstitialChapter 2 : PHYSIOLOGY OUTLINE^ 22space. Thus, the system behaviour is in general, very complex. In this work, severalassumptions are made, simplifying the system enormously. These are described in detail inChapter 3.The mechanical properties of the interstitial space are predominantly determined bycollagen and elastin fibers. The degree of hydration of tissue is largely determined byglycoaminoglycons and proteoglycans. As mentioned earlier, the resistance of tissue todeformation is due to the mechanical properties of the fiber matrix, the binding of the fluidto matrix elements, and the increased fluid viscosity due to hyaluronan. As a tissue deforms,the fibers rotate, stretch, and compress in the tissue volume. This movement exerts stress onthe fluid forcing it to be expelled from the matrix.2.3.1 Volume ExclusionVolume exclusion is the term applied to the phenomenon occuring when themeshwork flow domain limits solutes of larger dimensions than the matrix voids. Theinterstitial space consists of collagen, elastin, hyaluronan, and proteoglycans producing adense fibrous network of molecular dimensions. This means that larger solutes will berestricted from entering some regions of the interstitium because of steric exclusion. Theresult is a larger effective solute concentration due to the smaller possible occupationvolume. It is this effective solute concentration which will determine the osmotic drivingforce for fluid flow.An example illustrating the effect of volume exclusion is that of albumin exclusionby hyaluronan. Hyaluronan solutions of 0.5% and 1.5% by weight exclude albumin from25% and 75% of the solution volume respectively (Bert and Pearce, 1984).It is necessary to include solute exclusion in any mathematical description of theinterstitium due to its effects on solute concentration.Chapter 2 : PHYSIOLOGY OUTLINE^ 232.3.2 Geometry of TissueThis section briefly describes the arrangement of capillaries in tissues. This willserve as the basis for defining the functional unit used in the next chapter to formulate themicrovascular exchange model.Figure (2.6) presents a capillary network geometry from the frog sartorius muscle(Dietrich and Tyml, 1992). As can be seen, the capillaries are largely arranged in parallel toeach other. Single unbranched capillary lengths appear to average between 200-800 gm.Klitzman and Johnson (1982) determined that the average capillary length in the hamstercremaster muscle was 262 gm. The functional unit described in the next Chapter is based onthe single unbranched capillary length. This is assumed in Chapter 4 to be 300 gm.Intercapillary spacing has been experimentally found to be of the order of 40-60 gm(Intaglietta and Zweifach, 1971; Ganong, 1989). This is assumed to be 60 pm in Chapter 4.Figure 2.6: Microvascular network of the frog sartorius muscle(taken from Dietrich and Tyml, 1992).Chapter 3 : MODEL FORMULATION^ 24Chapter 3 : Model Development and FormulationThis chapter presents the model development and formulation. The first sectionreviews microvascular models found in the literature. The second section discusses thecontinuum approach for fluid and solute transport. The third section presents the governingequations of microvascular exchange. The associated boundary conditions are given in thefourth section. The fifth section briefly outlines the mathematical treatment of fluid andsolute exchange in the capillary. The numerical procedure is outlined in the sixth section.The final section summarizes the governing equations and boundary conditions.3.1 Literature ReviewA wide variety of mathematical models have been developed for differentphysiological systems. These models aid in our understanding of the fundamentals oftransport and material exchange in the human body. The expectation is that mathematicalmodels provide a framework for experimentation, especially with respect to determiningphysiological parameters for predictive behaviour and governing mechanisms of transportand regulation.Microvascular exchange has been receiving much attention during the last couple ofdecades. Mathematical models have been developed to investigate the underlyingmechanisms of pressure and flow regulation and transcapillary material exchange. Themodels are becoming increasingly complicated. They include such factors as propertyheterogeneity, colloid osmotic pressure, and lymphatic sinks, thus serving as more effectiveresearch tools.Chapter 3 : MODEL FORMULATION^ 25Early attempts at modeling microvascular exchange (Apelblat et al., 1974; Intagliettaand de Plomb, 1973; Salathe and An, 1976; An and Salathe, 1976) generally investigatedfluid exchange. These models did not include solute transport and thus assumed colloidosmotic effects were constant in the region of interest. Salathe and Venkataraman (1978)presented a capillary-tissue fluid exchange model taking variations in plasma and interstitialosmotic pressure into account due to protein convection and diffusion. Their model,however, neglects transcapillary protein exchange. They obtained analytical solutions usingperturbation methods. Weiderhielm (1979) performed a non-linear simulation of capillaryfluid exchange including the effects of volume exclusion due to glycoaminoglycans andproteoglycans. Plasma protein transfers into the interstitium via convection and diffusionthrough large pores at the venous end of the capillary. An analog computer model was usedto generate the simulations. This analysis is similar conceptually to a compartmental modelsince tissue and capillary variables were functions of time only (spatially invariant).Blake and Gross (1980, 1981) presented a series of papers outlining a model for fluidexchange in and between capillaries. The model assumes the capillaries are parallelcylinders of finite length. They assumed that there is Poiseuille flow in the capillaries andtranscapillary fluid exchange obeys Starling's hypothesis. Constant protein concentrationsthroughout the solution domain were assumed and analytical solutions were found for singleand multiple capillaries in parallel.Benoit et al. (1984) proposed a compartmental model for fluid and protein exchangein the rat intestine. They consider three compartments: the capillary, interstitium, andlumen. They used the Patlak equation to evaluate the protein flux across the capillarymembrane. Flessner et al. (1984, 1985a, 1985b) presented a series of papers discussing adistributed model of peritoneal-plasma transport. They use a combined compartmentalapproach in conjunction with a one-dimensional spatially distributed model for solutetransport within the peritoneal tissue. They assumed the colloid osmotic pressure is constantwithin each compartment. They also included lymphatic drainage of material via the use ofChapter 3 : MODEL FORMULATION^ 26an uptake rate expression. Their analysis treated volume exclusion and tortuosity effects bylumping them into an effective diffusivity. A retardation factor was also included to accountfor convective transport hindrance due to the intercellular matrix-solute interactions.Fry (1985) derived a one-dimensional finite element model (and an analytical one) todescribe transport of chemically reactive macromolecules across arterial tissues. He foundestimates for the tissue diffusivity and convective velocities by fitting the model toexperimental data.Bert et a/. (1988) presented a dynamic mathematical model describing thedistribution and transport of fluid and proteins among the three compartments: circulation,skin, and muscle interstitial spaces. Fluid also drained from the tissues forming lymph thatreturned to the circulation. They examined two mechanisms of transcapillary exchange: ahomoporous 'Starling' model and a heteroporous plasma leak model. The Starling modelproposes that fluid filters across a membrane from the circulation to the interstitium. Solutesmay cross the membrane by diffusion or by convection due to fluid filtration. The plasmaleak model proposes that fluid filters across the membrane through small completely-sievingpores. Solutes cross the membrane via large non-sieving pores at the venular end of thecapillary only or by diffusion along the length of the capillary. They concluded that theplasma leak model provides a better description of transcapillary exchange. Bert et al.(1989), Bowen et al., (1989), and Lund et al., (1989) presented a series of papers describinga dynamic compartmental model of microvascular exchange after burn injuries. This modelcontains four compartments: circulation, muscle, injured and non-injured skin.Recently, microvascular exchange models have shifted from compartmentalapproaches to spatially distributed formulations. In compartmental modeling, the materialexchange occurs between well-mixed homogeneous compartments. Rate constants governmaterial transfer between compartments. This means that properties in each compartmentare constant throughout the compartment, i.e., they represent an average value for thecompartment properties. The spatially distributed models do not assume that theChapter 3 : MODEL FORMULATION^ 27compartments are well mixed and homogeneous. This means that material transport within acompartment affects the systemwide behavior. These models thus approximate the realsystem more closely. This added benefit is offset by the increasingly more detailedinformation required about the structure and transport characteristics of each compartment.The mathematical complexity of spatially distributed models also increases. The lack of datain the literature forces the use of estimates for required parameters in mathematicalmicrovascular models. This is especially so for transport characteristics of the interstitiumand lymphatics. These models are used primarily for identifying general trends; they are notexpected to predict verifiable quantitative results.Baxter and Jain (1987) presented a transient, two-dimensional model formacromolecular transport in tissues. Their model only accounts for diffusive transport ofsolute and neglects convection. In a later paper, Jain and Baxter (1988) investigatedmechanisms of macromolecular transport in tumors. Their development is transient andincluded both convective and diffusive solute transport mechanisms within spherical tumors.This model was one-dimensional (radial) for both fluid and solute transport and assumed thatthe transport parameters and osmotic pressure are constant throughout the tumor. Theyextended their work (Baxter and Jain, 1989, 1990) by including a lymphatic drainage termand non-uniform heterogeneous perfusion of the tumor. They accomplished this byincorporating a necrotic core in the center of the tumor. They again assume that interstitialtransport parameters and osmotic pressure are constant within the solution domain. Baxterand Jain (1991a) further extended their model by introducing extravascular binding ofmacromolecules and metabolism within the tumor. All of their tumor models so far considerfluid and solute transport in spherical tumors in the radial direction only. They thereforeassumed that the lymphatic sinks and capillary sources exist continuously throughout thetumor. These models do not really address microvascular exchange at the capillary level astheir material balances are based on differential volumes which include many capillaries.Chapter 3 : MODEL FORMULATION^ 28In their most recent paper, Baxter and Jain (1991b) address microscopicmacromolecular transport from capillaries by examining a single capillary in the planeperpendicular to the axis of the capillary. They assumed constant membrane transportparameters, vascular pressure, and osmotic pressures in the solution domain. The modelsdescribed by Baxter and Jain above are increasingly complex but fail to account for theeffects of variations in the osmotic pressure throughout the geometry on fluid movement(and thus the solute concentration distribution).Taylor et al. (1990a, 1990b, 1990c) described a complex transient model ofmicrovascular exchange that potentially includes the combined effects of interstitial swellingand protein exclusion. Local osmotic pressure gradients also determine fluid velocitieswithin the interstitial space, i.e., fluid movement is a function of the protein concentrationdistribution. Transport of solute occurs by diffusion, dispersion, and convection. Thismodel, however, does not include the effects of lymphatic drainage. The model was appliedto both steady-state and transient transport in mesentery (Taylor et al., 1990a, 1990c).However, it neglected the effects of capillary pressure variations as well as swelling,dispersion, and property heterogeneity in the tissue and capillary membranes.This work extends the model proposed by Taylor et al. to include axial pressure andsolute concentration variations in the capillary, lymphatics, and property heterogeneitywithin the interstitial space and the capillary membrane. The tissue is assumed as rigid withconstant volume and, hence, swelling effects are assumed to be negligible.3.2 The Interstitium as a ContinuumAs described in Chapter 2, the interstitium is a complex structure consisting of a fibermatrix swollen with fluid. The hydrophilic proteoglycans and collagen form a meshworkthat permits fluid to percolate throughout the organic porous medium. Dissolved within theChapter 3 : MODEL FORMULATION^ 29interstitial fluid are various electrolytes and proteins. At the molecular scale, fluid andsolutes traverse the interstitial space through tortuous passages. The solutes may interactwith the passage walls throughout their journey and may be hindered by electrostatic forcesand the fiber matrix structure (volume exclusion). Any attempt to model the interstitium atthe molecular level is impractical since it is both impossible to adequately define thegeometry and measure variables (such as the pressure and concentration) at microscopicscales.The alternative approach moves to the macroscopic scale and considers theinterstitium as a continuum. The fluid and solid phases are not dealt with on an individualbasis, but considered, rather, as one continuous composite phase exhibiting averageproperties. This means that the properties of the fluid and solid phases are spatially averagedat some local scale and continuously distributed throughout the interstitial region. Examplesof averaged properties are the fluid hydraulic conductivity, the effective protein diffusivities,and the excluded volume fractions.3.3 Model FormulationThis section presents the geometry of the problem and the governing equations forfluid and solute transport in the interstitium, their associated boundary conditions, and modelassumptions. It also includes the mathematical treatment of volume exclusion and its effectson the colloid osmotic pressure and convective velocities.3.3.1 Model GeometryFor this work, a single capillary is assumed to be the fundamental unit ofmicrovascular exchange. The use of a single capillary model was first introduced by Krogh(1919). Here, the microvascular exchange unit is approximated as a rigid capillary of fixedChapter 3 : MODEL FORMULATION^ 30length and radius surrounded by an annular tissue space (the Krogh cylinder). Krogh (1919)used this geometry to investigate diffusive exchange of solutes. The Krogh cylinder wasused by Apelblat et al. (1974) to describe fluid and solute exchange across the capillary. Theuse of the Krogh cylinder approach implies that there is no intercapillary fluid or solutecommunication. This means that material can enter the system but may only exit viareabsorption or lymphatic drainage.For this work, the Krogh cylinder approach is adopted. Tissue swelling is ignored,that is, the tissue is assumed to be rigid. This implies that there is no accumulation of fluidin the system. Blood flows through the capillary due to a drop in the capillary pressure fromFart at z 0 to Pven at z = L. Fluid flows through the membrane based on Starling'shypothesis. Solute may be transported across the capillary membrane (based on Patlak'sequation) and then distribute itself freely throughout the tissue as a consequence of thediffusive, dispersive, and convective mechanisms. The fluid and solute can be withdrawnfrom the tissue via a lymphatic sink. This may be confined to a specific region or may bedistributed throughout the tissue space. The model geometry is presented schematically inFigure (3.1).Membrane77+^ Capillary - _ .....^......... _ .... .............. .___.....r = 0r = Rt,Tissuerr = Rtz = Lz = 0Lymphatic SinkFigure 3.1: Model Geometry.Chapter 3 : MODEL FORMULATION^ 31The cylindrical coordinate system is the natural system to use due to the modelgeometry. All governing equations are derived in this system of coordinates withaxisymmetry. This means that the lymphatic sink as shown in Figure (3.1) is a concentricshell surrounding the tissue space. The outer tissue boundaries (at the z = 0 and z = L planesand the concentric shell at r = Rt ) are assumed to be impermeable, i.e., there is no fluid orsolute transport across these boundaries. Material may only enter through the capillary walland then exit either through reabsorption back into the capillary or through the lymphaticsink. Capillary flow enters at z = 0 and exits at z = L. The lymphatic sink can be placed atany arbitrary location within the sink and is shown at the edge of the tissue envelope inFigure (3.1). The volume of the lymphatic sink may also be varied.The general model assumptions are listed in Table (3.1). These assumptions are usedin formulating the model and are thus an intrinsic part of the model.3.3.2 Fluid Transport in the Fiber Matrix Porous MediumWithin a differential element, the fluid mass balance equation has to be satisfied. Forcylindrical coordinates in the r and z directions only, this is given by (Bird et al., 1960) :dp1 1 d(P fv,)^fvz) p fQ = 0dt r  (3.1)where Q is a source term and pr. is the fluid density. Equation (3.1) simply states that theamount of fluid entering into the differential element by flow and from the source is equal tothat leaving the element by flow. The source term is used to implement the lymphatic sink.In this model, the tissue is assumed to be rigid and non-deformable. The fluid density isassumed constant and as a consequence, the mass balance equation simplifies to :Chapter 3 : MODEL FORMULATION^ 32Table 3.1 : General Model Assumptions.Model Assumption SummaryContinuum model formulation.Rigid capillary (not distensible).Fluid density is constant.Rigid interstitial space (not deformable).Fluid motion in tissue can be described by Darcy's law.Chemical potential driving fluid flow is given by (P-o-n).Effective solute drag velocity is some fraction of the fluid velocity (the retardationfactor,Interstitial space reflection coefficient assumed to be equal to 1.0. The effect of thisparameter on the system is examined in Chapter 4.Lymphatic drainage is treated simply as a sink. Solute is convected with the fluid outthe lymphatic sink.Starling's hypothesis governs fluid flow through capillary membrane.Patlak's equation governs solute flow through the capillary membrane.Thin film of fluid between capillary membrane and tissue porous medium (forcontinuity of pressure and solute concentration).Capillary pressure only a function of distance down capillary. Capillary pressure maybe assumed a linear function from the arterial to the venular pressures (Poiseuille'slaw).Capillary solute concentration only a function of distance down capillary. May beassumed as constant along the length of the capillary.Single aggregate protein represents all distinct protein species.Dispersion of solute is described by two parameters, the longitudinal and transversedispersivities.Lymphatic sink material removal rate is simply given by a linear equation in the localtissue hydrostatic pressure.Chapter 3 : MODEL FORMULATION^ 33dvr^+—avz -Q= 0dr^r^az (3.2)A general macroscopic equation that describes the flow of a fluid through ananisotropic porous medium is Darcy's law (Smith, 1990) given by :vf - vs = k V( + pfgz)^f^ (3.3)where k, 6, and !if are the anisotropic permeability and porosity of the porous medium, andfluid viscosity respectively. The constant g is the acceleration due to gravity. This empiricallaw expresses the fluid velocity, v1; relative to the solid matrix velocity, vs, due to animposed fluid pressure drop, Pt, and height difference, z, across the porous matrix.Neglecting the effects of height differences and with zero solid phase velocity, this may berewritten :vf =—KVP,^ (3.4)where K is the fluid hydraulic conductivity. Substituting Darcy's law, equation (3.4), intothe mass balance equation (3.2) yields :a ( aP) K, P, _ ( dPty Q(pi ) = 0K --L. --dr rr dr^r dr az - az (3.5)where Pt is the local tissue pressure. The lymphatic term, Q(P), is a function of the localtissue hydrostatic pressure. Equation (3.5) does not include the effects of the colloid osmoticpressure and therefore has to be modified.Chapter 3 : MODEL FORMULATION^ 34The driving force for fluid movement is the chemical potential gradient. In theabsence of osmotically active solute, the only driving force for fluid motion is then thehydrostatic pressure. Colloid osmotic pressure serves to reduce the local driving pressure forfluid flow across a membrane. The tissue may be considered as a stack of membranes thusallowing local osmotic pressure gradients throughout the tissue. Equation (3.5) is modifiedto include the effect of the colloid osmotic pressure as follows :K d K - CT 7C )ii+ a d(P - a It)K^st 0afr, — 0;71ar^'r^ar r dz) 1_ Q(pi ).^(3.6)dzwhere at is the local osmotic pressure and a t is the tissue reflection coefficient. If thereflection coefficient equals 0.0 then the driving force is simply the local hydrostatic pressuregradient. Solving equation (3.6) yields the tissue pressure which equals the fluid pressuresince equilibrium exists locally within the porous medium. The local fluid velocities in the rand z directions may then be determined from :d(P crtgt) vff = — K, aranda(P —^)vf.Z = — K2^"azEquations (3.6), (3.7), and (3.8) describe fluid transport in the interstitial spaceporous medium. They are not complete, however, since they do not include the effects ofvolume exclusion. The mathematical treatment of volume exclusion and its effects on theabove equations will be presented in a later section.(3.7)(3.8)Chapter 3 : MODEL FORMULATION^ 353.3.3 Solute Transport in the Fiber Matrix Porous MediumSolute transport occurs by three mechanisms in the tissue fiber matrix. These arediffusion, dispersion, and convection. All three modes of solute transport have to beincluded since at high convective velocities, convection and dispersion may becomesignificant.Fick's law gives the local diffusive flux of solute through the interstitial space :idiffs —DdiffVCs^ (3.9)where jdiffs and cs are the diffusive solute flux and solute concentrations respectively andDdiff is the anisotropic diffusivity tensor for the solute through the porous medium. Thesediffusion coefficients are usually less than the solute's free diffusion coefficients because ofthe impeding effects of the solid matrix. These hindering effects are due to both sterichindrance and electrostatic interactions between the solute and the solid matrix components.For albumin, the diffusion coefficients tend to be about 1.0x10 -12 m2/s in human tissue(Gerlowski and Jain, 1988). The effects of volume exclusion on solute diffusion is presentedin a later section.Convection is an important mechanism of solute transport in tissue. The mass flux ofsolute transported by convection,,leOtiv,s5 is given by :convs = V s,eff C^ (3.10)where vseff is the effective solute transport velocity. The local effective solute convectivevelocity (solvent drag velocity) is expected to be less than the local effective fluid velocity.This is because of several factors. First, the hydrodynamic interactions between the soluteand the solid matrix will retard solute flow. Second, the shape and size of the soluteChapter 3 : MODEL FORMULATION^ 36molecule will result in volume exclusion effects. This effect actually promotes increasedmean solute flow rates and will be explained later in more depth. The third and final effectis that of electrostatic charge. Repulsion or attraction of the solute to the solid matrixcomponents may increase or decrease the overall solute convective flow rate and relatesclosely to surface depletion effects. The first factor will be briefly discussed here while theothers will be discussed further in the section on volume exclusion.The hydrodynamic effects on solute convective transport result from the viscous fluidinteraction between the convected solute particles and stationary randomly arranged fibers ofthe solid matrix. This will hinder the movement of the solute front since the particles will berequired to travel a tortuous path within the fiber matrix from one location to another andwill travel at a slower speed than the trasnporting fluid due to viscous interaction with thesolid components. Brenner and Gaydos (1976) theoretically analyzed transport of neutrallybuoyant spherical particles in a Poiseuille flow in narrow capillaries. Their analysisconsiders specifically solute particles of radius of similar order of magnitude as the channelradius. They predict that two opposing effects will exist. First, the ratio of the particlevelocity to the fluid velocity at the particle center decreases as the particle size increases.This is expected since as the particle radius increases, more solute-solid matrix interactionswill occur. Second, as the size of the particles increases, they tend to remain in the centralregion of the fluid flow. This is the surface depletion effect which will be discussed in detailin the volume exclusion section. This effect may cause the mean solute velocities to begreater than the mean fluid velocities.It is required to define the local effective solute convective velocity. As mentionedabove, it is known to be somewhat less than the local fluid velocity. It is useful to define aretardation factor, which relates the effective solute convective velocity and the effectivefluid velocity by :vs = of^(3.11)Chapter 3 : MODEL FORMULATION^ 37The transport of solute by convection is now given as :Jconvs =^=^ (3.12)The retardation factor is a function of the apparent accessible pore space fraction. Itshould be also noted that the retardation factor is a function of the tissue hydration since theapparent accessible pore space volume varies with the tissue hydration.The third and final solute transport mechanism is mechanical dispersion. Thedispersive flux originates from microscale fluid velocity variations from the mean fluidvelocity and solute flowing into microscale pathways different from the direction of the bulkconvective flow. Mechanical dispersion spreads the solute front in a manner similar tomolecular diffusion. For this reason, it is assumed that dispersion can be expressed as aFickian process :Jdisps = —Ddsp VCs^ (3.13)where Ddisp is the mechanical dispersion tensor. This means that the dispersion term ofsolute transport can be coupled with the diffusion term as :j d„ = —D VCS^(3.14)where D is the sum of the molecular diffusion and mechanical dispersion coefficients.The dispersion coefficients are functions of the effective convective velocities andaverage pore size in the porous medium (Smith, 1990). The relationship between thedispersion coefficients and the fluid velocities and porous medium structure is given by(Bear, 1972) :Chapter 3 : MODEL FORMULATION^ 38Da^= a ijkl^ (3.15)where aijki is the anisotropic dispersivity (a fourth rank tensor) and is a function of theporous medium structure and Vics and v4 s. are the fluid velocities in the k and 1 directions.The product of the dispersivity tensor and the fluid velocities gives the dispersioncoefficients in each direction. The absolute magnitude of the velocity, 10, is given by :Iv,' = + vz2, (3.16)For an isotropic porous medium, equation (3.15) may be simplified to a function of twoparameters, a/wig and atran, the longitudinal and transverse dispersivities respectively (Bear,1972), i.e.,DdiryJj = a tranl l )51 8 ij + (a long — atran) 11 s l(3.17)where Sly is the Dirac delta function. Typically the value of ai„ g is usually taken to be ofthe order of magnitude of the grain size of the porous medium (Smith, 1990). Bothparameters are usually statistically estimated from experimental data to gain the best fit withmodel predictions. Such data is not available for human tissue so rules of thumb are used.For human tissue, the typical grain size of the solid matrix can be assumed to be the tissuecell diameter. This results in a longitudinal dispersivity of the scale 10 12m. The transversedispersivity is usually estimated roughly as being ten percent of the longitudinal dispersivity(Smith, 1990).Chapter 3 : MODEL FORMULATION^ 39In some porous media, for instance in petroleum reservoirs, dispersion exceedsdiffusion by about two orders of magnitude due to the highly convective flows. Since fluidvelocities tend to be less than 1.0x10 -7 m/s in humans (Baxter and Jain, 1989), using alongitudinal dispersivity of 10 1-1.M for human tissue suggests that dispersion is a secondarytransport effect which may be safely neglected for low convective velocities.The three modes of solute transport have been described mathematically above. Nowthe solute convective-dispersion equation will be presented describing the transientbehaviour of solute within the interstitial space. The derivation of the solute transportequation begins with a mass balance on a differential element. This is stated as follows : therate of accumulation of solute within a differential volume is equal to the net diffusive,dispersive, and convective transport of solute into the differential volume and the soluteinflux due to the source. This can be expressed as follows :Rate of^= diffusive I dispersive flux +convective flwc+ source termaccumulation (lids)^(i conv,^) (Q)(3.18)This is expressed mathematically as :dcs^d2c, D,7 dc^d2c^d(v,cs ) d(v, scs )— D +--+ Du_ s^ ^'^+QV. ,P)dt^dr2^r dr^- dz2^dr^dz s s'eif(3.19)where Dn. and Dr__ are the diffusive-dispersive coefficients in the r and z directionsrespectively. The source term, 0,(cseil , /1), is a function of the effective solute concentrationand local tissue hydrostatic pressure. This term is used to model a lymphatic sink in thetissue space. The left hand side of equation (3.19) is the rate of accumulation of solute term.The first three terms on the right hand side are the diffusive-dispersive terms in the r and zChapter 3 : MODEL FORMULATION^ 40directions. The fourth and fifth terms represent the convective transport components in the rand z direction respectively. The final term represents the lymphatic drainage of fluid andsolute which is treated as a sink in this formulation.The convective terms contain the effective solute convective velocities. These termsare calculated from the equations (3.7) and (3.8). The potential, (Pratirt), is obtained fromthe fluid conservation equation (3.6). This equation, however, is a function of the localcolloid osmotic pressure which is also a function of the effective solute concentration. Thismeans that the fluid pressure equation and the solute transport equation are coupled and mustbe solved simultaneously. The colloid osmotic pressure is a non-linear function of the soluteconcentration, i.e.,7r, (cs.t.ff (3.20)and this complicates the solution of the governing equations. The effective soluteconcentration, csxg; is evaluated from the actual fluid volume the solute may occupy.The equations governing fluid and solute movement in the interstitial have beenpresented. In the following sections, the implementation of the lymphatic sink will bediscussed and the mathematical treatment of volume exclusion and its inclusion in the aboveequations will be presented.3.3.4 The Lymphatic SinkThe fluid drainage out through the lymphatics is treated as a volumetric sink. Thefunction governing the removal of material by the lymphatic sink is unknown. Taylor et al.(1973) report data for the lymph flow rate as a function of the average tissue hydrostaticpressure in a dog's thigh. The function is shown in Figure (3.2). As can be seen, it is nearlyChapter 3 : MODEL FORMULATION^ 41linear below -2 mmHg and levels off when tissue hydrostatic pressures become sufficientlypositive.252015LL10ra^50 -6^-4^-2^0^2^4Tissue Hydrostatic Pressure (mmHg)Figure 3.2: Lymph flow versus average tissue hydrostatic pressure(taken from Taylor et al., (1973).Baxter and Jain (1990) treated the lymphatic sink drainage as a linear function in the tissuehydrostatic pressure. Since there is no data for humans, for this work, as a firstapproximation, the lymphatic fluid depletion rate is assumed to be a simple linear function ofthe tissue hydrostatic pressure given by :Q(P,) = — LS(P,— PL )^ (3.21)This means that the lymphatic drainage is proportional to the difference between the localtissue pressure and the lymphatic sink pressure, PL. The lymphatic drainage function isnegative since the governing equations (3.6) and (3.19) are presented as having a source.The local amount of solute being drained by the lymphatic sink is then given by :Chapter 3 : MODEL FORMULATION^ 42Qs(cs.eff ,P, )= Q(P,^ (3.22)This gives the mass of solute leaving the system per unit time.The sink is implemented as a term where fluid and solute are simply removed fromthe system at that point. This is analogous to a binding term. This allows the fluid andsolute to communicate with the rest of the tissue above and beyond the sink. The tissue fluidflow velocities at the sink are not then necessarily equal to the lymphatic convective velocitysince the fluid does not all necessarily exit the system via the sink but rather flows throughthe local volume associated with the sink.The conceptual picture of the lymphatic sink is displayed in Figure (3.3). In thisrepresentation, the sink is located roughly in the middle of the tissue parallel to the capillary.It is important to bear in mind that the model developed here is in cylindrical coordinatesalthough Figure (3.3) does not reflect this.Capillary FlowLymphatic SinkDrainageFigure 3.3: Conceptual picture of the lymphatic sink.Chapter 3 : MODEL FORMULATION^ 433.3.5 Volume Exclusion and Surface DepletionThis section presents : 1) definitions of volume exclusion and surface depletion, 2)their effect on material transport, and 3) their implementation into the governing equationspresented above.Volume Exclusion (Inaccessible Volume)Volume exclusion refers to the phenomenon that occurs when the macromoleculesare restricted from certain regions of the porous medium because the opening sizesconnecting these regions are smaller than the size of the macromolecule. This excludesmacromolecules from a fraction of the pore space - both in these small pores and any largerpores they give access to. There are two effects of volume exclusion. The first is a relativeadvancement of the solute front with respect to the fluid front. This is because the fluid fronthas to pass through all porous regions whereas the solute may bypass certain regions due tovolume exclusion. This results in the fluid taking effectively longer flow paths than thesolute. The second is that the effective solute concentration in the fiber matrix is greater thanthe concentration expected if the solute were distributed throughout the entire volume. Intissues, this effect cannot be ignored; for example, albumin is excluded from 60% of the totalinterstitial space (dermis) due to volume exclusion (Bert and Pearce, 1984).Figure (3.4) illustrates volume exclusion through a two-dimensional porous medium.As can be seen, the solute particles are excluded from some regions of the space due to thesmall pore sizes. The implementation of volume exclusion mathematically is facilitatedthrough the use of an representative elementary volume. This is displayed for a single solutein Figure (3.5) The available fraction of space 'seen' by the fluid is f,. The accessiblevolume fraction available to the solute (space the solute can 'see') is 1;t. The excludedvolume fraction is thus given by :^ • • •••  • •• •• •^••• ••Flow andSoluteFlow•• •• ,-\••• •Excluded Volume^ Accessible Volume •Restricted EntryChapter 3 : MODEL FORMULATION^ 44Figure 3.4: Volume Exclusion of Solute.fex =^ (3.23)The volume fractions must sum to unity, that is :fs, + fex + ; + fso =1^ (3.24)where fso and^are the fractions of the volume occupied by the solid matrix and theimmobile fluid bound to the solid matrix respectively.The solid phase consists mainly of the large polymers mentioned in Chapter 2, that iscollagen, glycosaminoglycans, proteoglycans, and elastin. The dominant glycosamino-glycan, hyaluronan, is hydrophilic and readily binds the interstitial fluid. A more extensiveChapter 3 : MODEL FORMULATION^ 45description of partitioning the representative elementary volume for multiple solutes isprovided in Taylor (1990a). sois;-av• Solute MoleculeFigure 3.5: Partitioning of a representative elementary volume into thevarious volume fractions.The various volume fractions will in general be both functions of the tissue hydrationand the fiber matrix properties. Now it remains to be seen how volume exclusion isincorporated into the governing equations of fluid and solute transport mathematically.The fiber matrix, through volume exclusion, restricts the accessible fluid volumeavailable to solutes. This means that the effective solute concentration within the elementaryvolume is somewhat greater because it occupies a smaller fluid space. The solute can accessonly thels' t fraction of the elementary volume. This means the effective solute concentrationin the elementary volume is given by :cs eff^Cs (3.23)Chapter 3 : MODEL FORMULATION^ 46The osmotic pressure is determined by the effective solute concentration. This is then givenby := F(c.s.eff^(3.24)where F is typically a low order polynomial in culi:The mechanisms of solute transport will be affected as follows. The diffusive-dispersive flux of solute can only transport material through the volume fraction available tothe solute. This gives a diffusive-dispersive flux as follows :d = —LDVes,eif^ (3.25)The convective terms are also affected because volume exclusion has an influence onthe fluid velocities. The fluid can only flow through the fraction of pore space available toit, that is L. ,. The effective fluid velocities through the fiber matrix porous medium will thenbe given by :V f = VI^ (3.26).1 avThis, however, is not incorporated into the fluid pressure equation. This is because theexperimentally determined hydraulic conductivities are assumed to have the volumeexclusion effects lumped into them. Volume exclusion, however, does affect the solute dragvelocity since it cannot 'see' all of the fluid flow due to the excluded fluid volume fex .Therefore, the effective solute convective velocity has to be modified as follows :Chapter 3 : MODEL FORMULATION^ 47= .fs^ vfad (3 .27)The final form of the solute transport equation including volume exclusion effects is thengiven by :Cks.eff^4 1 /,. ,,eff Cs ., If )^1/,,,,ff Cs ,ejf )^ + + 4 =dt^dr^dzs ,J, DD„dr2(92c,,eff  + Dn. dc eff( d2 r dr 4- Da ai l2'eff i+ Qss 'e(Cff 'Pt )(3.28)This equation is solved to obtain the effective solute concentrations. The soluteconvective velocities are obtained indirectly from the fluid conservation equation which is afunction of the solute concentrations via the colloid osmotic pressure. The colloid osmotic,in turn, is a function of the effective solute concentration. The coupled equations (3.6) and(3.28) must be solved simultaneously.Surface DepletionTo some extent, surface depletion is very similar to volume exclusion in that someportion of the elementary volume is not available to the solute. While volume exclusionrefers to an inaccessible pore space due to the sizes of the macromolecules and the pores inthe fiber matrix, surface depletion refers to the steric exclusion of macromolecules from thepore wall (Sorbie et al., 1991). Volume exclusion is more likely to occur in lowpermeability porous media whereas surface depletion is more dependent on the shape andorientation of the molecule. It is expected that surface depletion will be less for sphericalglobular molecules than long rod-like molecules of equivalent molecular mass.(w)Random Network of RodsChapter 3 : MODEL FORMULATION^ 48Figure (3.6a) illustrates surface depletion using two molecules: a rod and sphere anda sphere in a random network of rods. Rod-like molecules will not be allowed to freelyrotate within a layer having the same thickness as the length of the rod-like molecule. Thislayer is referred to as the depleted layer (Sorbie, 1990). Thus the excluded volume is thelayer of fluid surrounding the rod having an annular radius equal to the length of the rod.This is dependent on the angle that the rod-like molecule makes with the fiber. Figure (3.6b)displays the available space for a spherical molecule in a random network of fibers. Thespace the molecule can move freely within is enclosed by the dotted line.(a) -^Free rotationDepleted layer thickness = 3Rod-like moleculeSpherical moleculeS^>111< Hindered rotation FiberFigure 3.6: Schematic diagram of (a) surface depletion of spherical androd-like molecules from fiber and (b) steric exclusion of sphere in randomnetwork of fibers.Similar to volume exclusion, surface depletion will lead to an advancement of themacomolecular solute front relative to a low molecular mass tracer. Also, the lowerconcentrations of the solute at the pore wall will lead to a lower solution apparent viscosityadjacent to the wall.Chapter 3 : MODEL FORMULATION^ 49In this model, the volume exclusion (inaccessible pore space) and surface depletioneffects are assumed to be lumped together into the volume fractions presented in Figure 3.5.First, this simplifies the analysis, and second, there is no data available to differentiatebetween the two phenomena in the human interstitium.3.4 Boundary ConditionsTo solve the above governing equations, the behaviour of the fluid and solute at theboundary has to be specified. As outlined above, the problem is posed in the cylindricalcoordinate system.At an impermeable boundary, the fluid boundary condition is given by :d(P, — afri) do(3.29)where n signifies the coordinate normal to the boundary. For the solute, an impermeableboundary is represented by :dc—D "ff^Cn,eff s,effnn an + V (3.30)This implies that there is no movement of solute across the boundary by any of the variousforms of transport: diffusion, dispersion, and convection.The mathematical treatment of the capillary membrane boundary conditions requiresthe assumption of an infinitesimally thin layer of fluid between the capillary membrane andthe tissue porous medium. In this thin layer, the pressure and solute concentration are inequilibrium with the pressure and solute concentration in the accessible pore space. This isChapter 3 : MODEL FORMULATION^ 50merely a mathematical convenience but it is required for continuity from the capillarymembrane to the tissue space.At the capillary membrane, fluid continuity is required across the membrane. Thismeans the amount of fluid flowing across the membrane according to Starling's hypothesis(Renkin, 1986) :vs;„ =^— P, — o-(7r, - 7r, )]^(3.31)must be the same as that indicated from Darcy's law for flow into the tissue, i.e.,v Da^K '903t a 1 7r t) f„^nn^domembrane(3.32) Equations (3.31) and (3.32) are coupled to provide the following boundary condition :d(PeK ^domembrane= L „ — P, —^— 70] (3.33)This boundary condition is complicated by the fact that the capillary pressure, Pc, thetissue pressure, P1, the tissue osmotic pressure, 710 the capillary membrane hydraulicconductivity Lp , and the capillary membrane reflection coefficient, a t, may all be functionsof the location down the membrane. This means an iterative procedure has to be used tosatisfy the fluid capillary boundary condition. The solution algorithm is presented in theSection 3.6.Chapter 3 : MODEL FORMULATION^ 51The capillary membrane solute boundary condition states that the amount of solutepassing through the membrane is given by Patlak's equation (2.3) (see Section 2.2.3). This isgiven by (Curry, 1984) :dc.s ' eff v^PS(Ce— Cs.eff )Pe.J"D" do^'Leff('^= V 11eff(1— S)Ce+^ePe"' —1(3.34)where cc is the local capillary solute concentration and the membrane Peclet number, Pem,isgiven by :V1.'"Pe.= 1.'"PS(3.35)This states that the diffusive, dispersive, and convective flux of solute into the tissue is equalto the amount passing through the capillary membrane given by Patlak's equation.3.5 Fluid and Solute Exchange in the CapillaryAt the arteriolar end of the capillary, as fluid and solute pass through the membraneinto the tissue, the flow rate and concentration of solute decrease with downstream distance.If fluid and solute are reabsorbed back into the capillary at the venular end, the capillaryfluid flow rate and solute concentration will increase with z. This section describes how thefluid and solute change in the capillary are treated mathematically.Figure (3.7) illustrates a differential length of the capillary for the change in a generalvariable W.Chapter 3 : MODEL FORMULATION^ 52V2nRcdzz TW(z).A dz^W(z+dz).ArV2nRedzFigure 3.7: Fluid and solute flow in capillary.Realistically, the pressure and solute concentration are functions of both the radial, r,and longitudinal, z, distances within the capillary. Also, red blood cells moving down thecapillary will cause pressure fluctuations along the capillary length. In this model, it isassumed that the pressure and solute concentration are functions of z only and the pressurefluctuations caused by the passage of red blood cells produce, in an average sense, a linearpressure profile down the capillary (Apelblat et al., 1974). The latter assumption is quitedrastic but simplifies the analysis substantially. The small amount of fluid leaking into thetissue determined from simulations presented in Chapter 4 (less than 0.1%) indicates that thepressure drop down the capillary is very nearly linear and suggests that the flow isPoiseuille-like. This implies that radial pressure gradients are negligible.Assuming that on average, flow obeys Poiseuille's law in the capillary yields thenormal capillary fluid flow rate at the capillary membrane, v„, e 1mentbnine, as= R(3, d2 P.V " 'el membrane^I 6 p dz2(3.36)Chapter 3 : MODEL FORMULATION^ 53For fluid continuity, this can be set equal to the fluid flow rate across the capillary membraneaccording to Starling's hypothesis yielding a second order ordinary differential equationgiven by :d2Pc^r^ = [13 — P — o-, (re —701dz2(3.37)where a2 = 1 6Lp pIR 3 For these simulations, the pressures at the arteriolar and venularc•ends (Part and Pven) of the capillary are specified. This means that solution of the pressurein the capillary requires the solution of a second order boundary value problem. It is alsonoted that the solution of equation (3.37) requires the tissue hydrostatic and osmoticpressures, the membrane filtration coefficient, and the fluid reflection coefficient asfunctions of the length down the capillary. This forces an iterative solution for the capillaryand tissue pressure profiles at the membrane-tissue interface.It is noted that a2 is very small (typically about 10 -4). This implies that the capillarypressure profile is very nearly linear from the value at the arteriolar end, Part, to the value atthe venular end, Pven. Assuming that the capillary pressure is linear from the inlet to theoutlet end removes the requirement for iterative solution, reducing computational costsconsiderably. This has been shown to be a good assumption since the amount of fluidleaking into the tissue is small relative to the flow of fluid (typically < 0.1%) through thecapillary wall.A solute mass balance on the differential length in Figure (3.7) yields :s (z)2n-Redz^(3.38)Chapter 3 : MODEL FORMULATION^ 54where Qe is the volumetric fluid flow down the capillary and the amount of solute passingthrough the membrane, Js (z), is given by Patlak's equation (3.34). This is a first orderdifferential equation describing the solute concentration down the capillary as a function onthe amount that transfers across the membrane, Js (z), and the capillary radius. The initialcondition for this boundary condition is the arteriolar inlet solute concentration, cart. Itwas found from simulations performed that the solute concentration changes by less than0.01 % along the length of the capillary. The capillary solute concentration was thereafterassumed as constant along the capillary.The mathematical model is now complete. It shall be now applied for several casespresented in the following chapters.3.6 Numerical Procedure and Solution AlgorithmThe coupled equations (3.6) and (3.28) are solved using the Petrov-Galerkin finiteelement method (Hughes, 1978; Brooks and Hughes, 1982; Yu and Heinrich, 1986, 1987,1988). In the finite element method, the system geometry is divided into elements ofvarying dimensions. The regions expected to have the highest gradients are subdivided intofine elements while those expected to have relatively lower gradients are subdivided withrelatively coarse elements. The method spatially discretizes the problem by specifying nodeswithin each element at which the dependent variables (Pt and Cseff) are evaluated. Thedependent variables are approximated within each element by interpolating or shapefunctions. In this work, rectangular elements with simple linear basis functions were used.The weighted residual analysis of the weak form of the partial differential equations (seeAppendix A for further details about the weak formulation) yields a set of algebraicequations for the dependent variables at each node. These equations can be solved using anystandard system of equations solver (Press et al., 1986).Chapter 3 : MODEL FORMULATION^ 55The numerical solution is complicated because there is convective domination withinthe interstitial space for the solute transport equation (3.28). This changes the nature of thepartial differential equation (3.28) from elliptic to hyperbolic-like. The Petrov-Galerkinstreamline upwinding method was adopted to deal with the dominance of convection. In thismethod, artificial diffusion is added to the basis functions. This can be viewed as amodification of the weighting functions for the convective terms. The Petrov-Galerkinmethod seeks to add the optimal amount of artificial diffusion such that the accuracy ismaximized. A more detailed description of the Petrov-Galerkin finite element method ispresented in Appendix A.The fluid conservation and solute transport equations are coupled. This means thatthe tissue hydrostatic pressure and solute concentrations have to be solved iteratively. At aparticular time step, the solute concentrations are assumed within the interstitial space.Usually this is estimated from the values at the past two time steps. The colloid osmoticpressure is then calculated and the tissue hydrostatic pressure is subsequently obtained from .equation (3.6) subject to the appropriate boundary conditions. If the capillary hydrostaticpressure is not assumed as linear down the capillary length, then a separate iterativeprocedure occurs for the solution of the hydrostatic pressures in the capillary, equation(3.37), and within the tissue, equation (3.6). The fluid velocities are then evaluated withinthe tissue and the solute transport equation is solved using the Petrov-Galerkin methodsubject to the imposed boundary conditions. The procedure is repeated until the relativechange in the solute concentrations and hydrostatic pressures are less than a specifiedtolerance, typically 10 -6 , or when the iterations exceeded a specified limit (100).A modified version of the dominant eigenvalue under-relaxation technique suggestedby Orbach and Crowe (1971) is employed. This is discussed in further detail in Appendix B.As a check on the numerical solution accuracy, material balances were performed about thesystem boundaries. The relative material balance errors for fluid and solute were always lessthan 0.5%. The number of elements was also tested to ensure accuracy of the numericalChapter 3 : MODEL FORMULATION^ 56solution. Increasing the number of nodes from 1020 (50x21) to 2000 (80x25) produced lessthan 0.01% difference in the results. All cases performed in Chapters 4 and 5 were producedusing 1020 nodes.The program listing may be found in Appendix C.3.7 Summary of Governing Equations in Dimensionless FormReference ValuesPressureConcentrationLengthDiffusion coefficientTissue hydraulic conductivityPartCantLDm*K„fArteriolar hydrostatic pressureArteriolar solute concentrationCapillary lengthMolecular diffusion coefficientBase tissue hydraulic conductivity value.Dimensionless Variables. rr = —L. 7r7r =Panzz =-L•^Dddii Dii =udiffC• = ^P. = P—Can pad= ^Pe,=1/11T(1—c7)Kref PSv.Lv. —^K Pref arttDdis.r —^Chapter 3 : MODEL FORMULATION^ 57Fluid Flow in Tissue'dr^dr*( K. a(P,* —a/ 7r/+r^dr^az^az^Krd(Pis 0-17r*J + a  (K. ^cri7r1 +^^* * '^* efPa,L2rr Q(Ps) = °co(P,*)=—Po„Ls(P,* —P;)K*„.1C P a(19, * —= r^ef a ^arsVI,z =az*Solute Transport in TissueKriCrefPan d(Pf* 6, l,•^•dc.s.eff  + PartKrej d( lir,s,effCs.eff) PartKref^,s,effCs* .eff) =az ^Ddiff^ar*^Ddiff^az*d2c: 1 d c:^D*D*^"ff +a2 c`"ff L2+dr* 2 r ^ar az*2 D do anFluid Flow Boundary Condition at Membrane membraneLL^*^{p^_Kref —Chapter 3 : MODEL FORMULATION^ 58Fluid Flow Boundary Condition at Impermeable BoundaryP, - =0an `Solute Transport Boundary Condition at MembraneCacs^Pa,Krefv„,eff •^ Pa,,Kn,f.eff^cr)c, + D L PS(e*, ere cs eff)Pe„,D""^=an^Ddiff^ddiff^('- —1s'eff^D iffSolute Transport Boundary Condition at Impermeable Boundaryc*^P K vn,effs,tff +  art rtf ^n a an*^Ddiff^cs = 0Osmotic Pressure relationship*^ rt^ff^a rt^s,e ,effffk1 a (cs,e*^+ k2 c2 (c * ) 2 + k3c^)3=Partwhere kb k2, and k3 are fitted parameters dependent on the particular solute species.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EX(WANGE^59Chapter 4 : Effects of System Parameters on Steady StateMicrovascular Exchange : A Sensitivity Analysis4.1 IntroductionChapter 3 presents the formulation of the model equations governing themicrovascular exchange system. Using this model we will first evaluate the effects of thevarious system parameters on the system. In the tissue, these include parameters that affectthe lymphatic sink (LS and EL), tissue fluid motion via the osmotic pressure (a s), hydraulicconductivity (K), and solute transport via diffusion (D(J&, dispersion (Ddjsp), andconvection (0. In addition, fluid motion and solute transport across the capillary wall arealtered with adjustments in the diffusive permeability (PS), capillary membrane filtrationcoefficient (Lp), and the capillary reflection coefficient (a). The position of the lymphaticsink will also be investigated.The effects of a variable capillary membrane filtration coeffcient, Lp, and reflectioncoefficient, a, along the capillary membrane and spatially variant tissue hydraulicconductivity, K(r,z), will also be examined briefly. The capillary membrane filtrationcoefficient and reflection coefficient are assumed to be a known functions which arespecified along the length of the capillary wall. The high flow channels are implemented asa locally increased tissue hydraulic conductivity.In this chapter, the problem will be first summarized. The parameter values arethen discussed and then the case studies performed listed. The results are then presentedChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^60and discussed in detail for the base case values (the base case). The results from thesensitivity analysis are then presented to evaluate the influence of the physiologicalparameters. The results obtained by using a variable capillary membrane filtrationcoefficient and tissue hydraulic conductivity are reported.4.2 Problem StatementThe system under investigation is displayed schematically in Figure (3.1). Theblood plasma flows through the capillary leaking fluid and solute into the tissue through thecapillary membrane. As a first approximation, the capillary radius is assumed constantalong its length. The tissue envelope surrounding the capillary is also assumed to be ofconstant outer radius. Fluid and solute may be reabsorbed at the venular end of thecapillary but may also leave the system via the lymphatic sink which can be placedanywhere within the tissue space. The tissue is assumed to be perfectly rigid. This meansthere are no deformations due to swelling.The problem is stated as follows: given the arteriolar and venular capillaryhydrostatic pressures and the arteriolar solute concentration, find the solute concentrationand hydrostatic pressure distributions throughout the tissue subject to Starling's hypothesisand Patlak's equation governing fluid and solute transport across the capillary membrane.The other boundaries are impermeable to both fluid and solute. The behaviour will dependon the physiological parameters chosen for the simulation.4.3 Parameter ValuesTable (4.1) lists the parameter values used in the numerical simulations. Thesevalues are typical for tissues found in the literature. Many values (e.g., /7), PS, a, at ,^,Chapter 4 : EFFE(7TS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^61andm were taken from estimates presented by Taylor (1990a). The value of the capillarypermeability, PS, was assumed to be 2.4x10 -10 m2/s. The capillary wall protein reflectioncoefficient, a, was assigned the value 0.85. This value is typical for albumin in capillaries(Ballard and Perl, 1978). The tissue protein reflection coefficient, at , is assumed for mostcases to be equal to 1.0. To test the effect this parameter has on the system, it was variedbetween 0.0 and 1.0 in three cases. There are no data available indicating a better estimatefor this parameter. The immobile and accessible fluid volumes, Am andfm, were obtainedby assuming that the main component of the immobile fluid volume is the intrafibrillarwater associated with the collagen (Taylor, 1990a).The tissue hydraulic conductivity, K, is estimated to be 3.1x10 -16 m2 /Pa s. Thisvalue is typical for subcutaneous tissue (Levick, 1987). The value of the solute diffusivitywas taken to be that of albumin in normal tissue. This was assumed to be 1.0x10 -12 m2/s(Gerlowski and Jain, 1988).The geometry of the system is shown in Figure (3.1). The length of the capillary, L,is taken to be 300 rim. This is based on the average capillary length determined byKlitzman and Johnson, (1982). The radius of the capillary, Rc, is set as 3 p.m. This istypical for capillaries (Ganong, 1989). The radius of the tissue envelope, RI, was assumedto be 30 Jim. This is comparable to the capillary spacing data reported by Intaglietta andZweifach (1971). The pressure drop from the arterial inlet to the venular outlet of thecapillary is taken to be 25 mmHg. The inlet arteriolar hydrostatic pressure is assumed to be30 mmHg. These values are typical for arterial regions in capillaries (Brace and Guyton,1977; Ganong, 1989). The concentration of the solute at the arteriolar end of the capillaryis taken to be 35.9 mg/ml (Bert and Pearce, 1984).The lymphatic sink drains fluid and solute according to the simple linear form givenby equation (3.20). Using data in Chapple (1990), the value of LS is estimated to be1.24x10-8 m3 fluid/m3 tissue Pa•s. The value of the lymphatic pressure is still underinvestigation. It is assumed to be near the limit of the most negative hydrostatic pressuresChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^62found in tissue where the lymph flow approaches zero. Using various techniques the tissuehydrostatic pressures have been reported to be as low as -9 mmHg (Guyton et al., 1987).The negative value implies that the lymphatics are below atmospheric pressure. Resultsfrom Taylor et al. (1973) (see Figure (3.2)) suggest that the no flow lymphatic sinkpressure is near -6 mmHg. Therefore, the base case value for the lymphatic sink pressurefor this work is chosen as -6 mmHg. The value is assumed to be temporally invariant andconstant throughout the lymphatic sink. The effect of the lymphatic sink pressure isinvestigated by lowering its value to -9 mmHg and -12 mmHg. The location of thelymphatic sink was assumed to be at the peripheral edge of the tissue envelope.The osmotic pressure is assumed to be a simple polynomial function of the proteinconcentration :nt (c.s.eff = 57.182c,,es. —1.2388cs2.,ff + 0.050849c,34-^ (4.1)This gives the colloid osmotic pressure (in Pascals) as a function of the effective soluteconcentration (kg/m 3 ). This was taken from Bert et al. (1988) for albumin in skin.4.4 Case StudiesA sensitivity analysis was performed to investigate the effects of various systemparameters on the microvascular exchange system. These comparisons were for steady-state cases only. These results also illustrate the effect the parameters have on thecapillary-tissue fluid balance. The fluid balance and its regulation will be examined furtherin Chapter 5.The influence of the lymphatic sink can be investigated from two perspectives. Thefirst is to simply vary the strength of the sink, that is vary LS and/or PL. The second is toactually move the sink within the tissue domain. In general, the sink may be placedChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICRO VASCULAR EXCHANGE^63anywhere in any orientation within the tissue space. For simplicity, we shall assume thelymphatics always run parallel to the capillary. This means only the radial position of thesink is varied. As the sink is moved radially, the lymphatic sink volume reduces becausethe system is in cylindrical coordinates. There are two methods to maintain the strength ofthe sink: either by enlarging the sink volume or by simply increasing the value of LS. Inthis work, the value of LS is increased to reflect the reduction in the lymphatic sink volumeas it is moved radially. To examine the effects of the sink position on the system, the sinkis placed at various dimensionless radial distances. The baseline position for the sink isagainst the outer edge of the tissue envelope.Table 4.1 Parameter Values.Parameter^Value^ ReferenceCart^35.9 mg/ml Bert and Pierce (1984)Part 30.0 mmHg^Brace and Guyton (1977)Pven^5.0 mmHg Ganong (1989)fim 0.128 Taylor (1990a)fst^0.680^ Taylor (1990a)^0.0-1.0 See texta^0.85 Taylor (1990a)at 0.0-1.0^ See textPS^2.4x10-1° m/s Taylor (1990a)L 1.35x10-1° m/Pa s^Taylor (1990a)PDdi•^1.0x10-12 m2/s Gerlowski and Jain (1988)K 3.1x10-16 m2/Pa s^Levick (1987)LS^1.24x10-8 m 3/m 3 Pa s Chapple (1990)PL^-6.0 mmHg^Taylor et al. (1973)Rc^3µm Ganong (1989)R t^30 tm^Intaglietta and Zweifach (1971)L 300 tm Klitzman and Johnson (1982)aton.e^1.0x10-5 m See textatran^1.0x10-6 m^ See textChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^64The effects of mechanical dispersion are also examined. The longitudinal andtransverse dispersivities are estimated from the size of the average grain size in the tissueporous medium. At the scale of the present simulations, the longitudinal dispersivity isestimated to be roughly 1x10 -5 m. This is based loosely on the size of the typical humantissue cell in the interstitium. The hydrodynamics and transport of solute occur around thecell bodies. The framework of cells forms the porous medium through which flow occurs.The scale of the continuum model (of the order of micrometers) suggests that an averagecell dimension be used rather than the diameter of the collagen molecule (of the order ofnanometers) for instance. As a first approximation, the transverse dispersivity is estimatedas ten percent of the longitudinal dispersivity, that is 1x10 -6 m.The cases performed are listed in Table (4.2). These are all steady-state cases.Steady-state was determined when the tissue solute concentrations and hydrostaticpressures changed by less than a specified tolerance between consecutive time steps. Theresults from these cases will be used to investigate the effects of the various parameters onthe system. The first case is the base case. This case uses the parameters listed in Table(4.1). The other 28 cases use the same parameters except one is varied to evaluate theinfluence of that parameter on microvascular exchange. For example, the case LS x 0.2indicates that this case used a value of the lymphatic sink strength, LS, equal to 2.48x10 -9m3/m3 Pa s instead of the base value 1.24x10 -8 m3/m3 Pa s. The case in which dispersion isincluded is case 17, called disp. The two cases Lp to 2Lp (linear) and Lp to 2Lp (step)denote linear and step variations in the capillary filtration coefficient along the length of thecapillary respectively. The functions vary from the base value of Lp in Table (4.1) at thearteriolar end of the capillary to twice this value at the venular end of the capillary. Thecases ms3 and ms6 each denote the repositioned sink at r* = 0.025 and r* = 0.055. For thebase case, the sink is placed at the outer edge of the tissue envelope, i.e., at r* = 0.10. Eachcase and its results are discussed in detail in the following section.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^65Table 4.2 List of CasesNo. Case Explanation1 base Base Case.2 nows No osmotic pressure effects, with lymphatic sink.3 LS x 0.2 Lymphatic sink permeability reduced by a factor of five.4 LS x 5 Lymphatic sink permeability increased by a factor of five.5 a = 0.1 Capillary reflection coefficient equal to 0.10.6 a = 0.5 Capillary reflection coefficient equal to 0.50.7 a = 0.99 Capillary reflection coefficient equal to 0.99.8 = 0.1 Retardation factor equal to 0.10.9 t = 0.5 Retardation factor equal to 0.50.10 PS = 0.0 Capillary diffusive permeability equal to zero.11 PS x 100 Capillary diffusive permeability increased by a factor of one hundred.12 PS x 10000 Capillary diffusive permeability increased by a factor of ten thousand.13 at = 0.0 Tissue reflection coefficient equal to zero.14 at = 0.5 Tissue reflection coefficient equal to 0.50.15 DdifiA 0.1 Solute diffusion coefficient reduced by a factor of ten.16 Ai; frx 10 Solute diffusion coefficient enlarged by a factor of ten.17 disp Dispersion included as transport mechanism.18 K x 0.1 Tissue hydraulic conductivity reduced by a factor of ten.19 K x 10 Tissue hydraulic conductivity enlarged by a factor of ten.20 P f .= -0.40 Lymphatic sink pressure set equal to -12 mmHg.21 P1,= -0.30 Lymphatic sink pressure set equal to -9 mmHg.22 L,, x 0.5 Capillary filtration coefficient reduced by a factor of two.23 L, x 2 Capillary filtration coefficient enlarged by a factor of two.24 L, to 2L„(linear)Linear capillary filtration coefficient variation from arteriolar to venular end.From base value to twice the base value.25 L„ to 2L„(step)Step capillary filtration coefficient variation from arteriolar to venular end.From base value to twice the base value.26 K1 Single high flow channel at z * = 0.5.27 K2 Two high flow channels at z * = 0.3 and 0.7.28 ms3 Lymphatic sink radial position equal to r * = 0.025.29 ms6 Lymphatic sink radial position equal to r * = 0.055.Chapter 4 : EFFE(7TS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^664.5 Discussion of Results and Sensitivity AnalysisIn this section, the results for the base case will first be discussed in detail. Figure(4.1) present the results from the base case. This forms the basis for comparison with allother cases and provides insights into the influences of osmotic pressure and the lymphaticsink.In the first section, the results for the base case will be presented. This will becompared to a case where the osmotic pressure contributions are switched off through thesystem. The effects of other physiological parameters on the system will next be presented,in the form of a sensitivity analysis, in the second section. Here the effect of aphysiological parameter is studied by varying the parameter and observing the change inthe results from the base case. The third section presents effects of a variable capillarymembrane filtration coefficient (L,p) and the fourth section examines the influence of highflow channels on the hydrodynamics and solute transport in the interstitium.4.5.1 The Base CaseFigure (4.1) displays the results from the base case. This case uses all the basevalues specified in Table (4.1) and includes the colloid osmotic pressure effects and thelymphatic sink. The capillary is located at the upper edge of each plot. The inlet arteriolarend of the capillary is at the left end and the outlet venular end is at the rightend of the plot. The distance into the tissue increases in the downwards direction. Thearrangement is displayed in Figure (3.1). The lymphatic sink is located at the lower edgeof the plots. It is important to note that the distance into the tissue scale has beenexaggerated (the tissue outer radius is actually one-tenth of the length of the capillary).The first contour plot contains the dimensionless solute concentration given by :0.0^0.10.010.02:0.03:0.04:0.050.06-0.07:0.08:0.09:0.10(13-cnr)/Parti :i. 7 i  1 I 1^I^I^1...co^co^co^a^co co ocoC3 coto r- co .-1 C3 40 ..:CA CO^0^0^0C3^C.5^CI^CO * V.^I Ic;t 61^c; 6^c;^6 c;. 1"-r- • .....0.010.0203 0.03to0.052 0.060 0.070.08■00 0.09:• 0.10Peclet Number......... . .... ....... ..................................................... .  ...................... ............................................. .^47 `..^40.•. 4.°,‘X^.•..^.4?>,^.•. .496s.% ..^01 •^•I..^42^• VI 6'%^1 1ClcoClcoNcotoCO0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0--.....^qd,^-•0.99 199^6:949^,^0.1^0.2^0.3 0.4 0.6^0.6^0.70.010.02:0.030-04:0.05:0.06--07-0.08-:Hydrostatic Pressure01.1.1^0.2^0.3^OA^0.6^0.6^0.7^ i!)3^10.9i ^1.0**-0.^ 0.00008000.010.02:0.03:0.04:0.050.07:0.080.090.100.0chapter 4 . EFFE(TS OF SYSTEM PARAMETERS ON MI(WOVAS(1ILAR EX(7IANGE^67Solute ConcentrationVelocity Vector FieldII W W ^kiiiii%••••••• ^A A AA A AAANN••■•■• 0.01 0.020.030.040.050- Osi0.07:0.08:0.09:0.101 1 1 A A AAAA‘1 1 1 AAAA ^I A A ^1 A A I' A • 4.^A ■0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.00.8^0.9^1.0Figure 4.1: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the base case at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON Mk7ROVAR'ULAI? EXCHANGE^68CsCs = —Cart(4.2)where Cart is the inlet arteriolar capillary solute concentration. The dimensionless tissuehydrostatic pressure is displayed in the second plot. The dimensionless hydrostatic pressureis given by :P * = 1,19(^(4.3)The third plot displays the Peclet number distribution which is calculated as follows :Pe = Iv, eff1L Dwhere2^+1 Vs'eff I = 17r"'•eff^vz-s'effand D is the total diffusion coefficient including molecular and dispersive contributions.The Peclet number distribution is directionless but reflects the importance of convectionrelative to the total diffusive and dispersive transport. Near the impermeable boundaries,the Peclet numbers attain their lowest values indicating that there is less convectivetransport near these boundaries. The Peclet numbers are not equal to zero at theimpermeable boundaries because there is fluid motion tangential to the boundary.The fourth plot displays the dimensionless potential :(4.4)(4.5)Chapter 4 : EFFECTS OF SYSTEM PARAMETERS' ON MICROVASCULAR EXCHANGE^69(P-0-7r)Pan(4.6)within the tissue. The potential drives fluid motion. Finally, the velocity vector field isdisplayed in the fifth plot. The length of the velocity vectors indicates the fluid velocitymagnitude (approximately 1 mm = 3x1 0-7 m/s).From the solute concentration contour plot, it is seen that the c: increases from theend of the tissue adjacent to the capillary inlet to the end nearest the capillary outlet (rangeis from about 0.45 to just greater than 0.72). The solute is washed out to a higher degree atthe arteriolar end of the tissue than the venular end due to the higher transmembrane fluidvelocities at the arteriolar end of the tissue. This leads to higher solute concentrationgradients towards the venular end of the tissue than in the central and arteriolar portions ofthe tissue. The solute is removed from the system by the lymphatic sink. The averagedimensionless solute concentration within the tissue is 0.5258. This corresponds to adimensional solute concentration of 18.88 mg/ml. This is similar to the average interstitialspace protein concentrations (21 mg/ml) estimated by Landis and Pappenheimer (1963).The dimensionless solute concentration in the lymph drainage is lower than the tissue-wideaverage at 0.5134 (18.43 mg/ml).The tissue hydrostatic pressure distribution is very similar to the soluteconcentration distribution. The hydrostatic pressure is lowest at the arteriolar end of thetissue and increases towards the venular end of the tissue. The gradients in the hydrostaticpressure are greatest near the capillary and drop off as the sink is approached. This isespecially so at the venular end of the tissue near the capillary. This is explained by therelatively large solute concentration gradients at the venular end of the tissue near thecapillary. The hydrostatic pressure responds to the solute concentration via the colloidosmotic pressure. Near the sink, the hydrostatic pressures fall to negative values. ThisChapter 4 : EFFE(IS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^70means that in this region, the tissue is subatmospheric. The dimensionless average tissuehydrostatic pressure is equal to -0.0255 (-0.7650 mmHg). This falls in the range of tissuehydrostatic pressures for tissues (Guyton et al., 1987). The dimensionless lymphatic sinkpressure is -0.20. The relatively more positive hydrostatic pressures in the region of thelymphatic sink cause the removal of fluid. The lymph drainage rate is equal to 5.47 1/day.This is in the range of lymph drainage rates observed experimentally. Mortillaro andTaylor (1976) report lymph flows between 5.53 and 8.64 1/day for similar hydrostaticpressure drops down the capillary as that used in this work.The Peclet number distribution indicates that solute transport is largely convectivelydominated. This is especially so near the capillary and within the central portions of thetissue. The Peclet number distribution passes through a maximum in the central portion ofthe tissue at approximately z* = 0.60. The range in the Peclet numbers for the base case areroughly between just above 0 and 120. The Peclet number distribution also indirectlyreflects the relative magnitudes of the fluid velocities (see equation (4.4)).The potential is entirely negative within the tissue. The range in the dimensionlesspotential is very small (approximately 0.015). Most of the loss of potential occurs acrossthe capillary membrane whereas potential gradients in the tissue are relatively small. Thissuggests that the capillary membrane resistance to fluid flow is much greater than the tissueflow resistance. The potential drops from the arteriolar end of the tissue near the capillaryto the venular end of the tissue near the lymphatic sink. This results in the fluid flowpattern displayed in the velocity vector plot. Fluid filters across the capillary wall into thetissue along the entire length of the capillary and is removed from the system by thelymphatic sink. From the velocity field plot, the fluid velocities are clearly seen to begreater at the arteriolar end of the tissue and drop along the length of the capillary.The results from the base case illustrate that lower solute concentrations andhydrostatic pressures are at the arteriolar end of the capillary. It is important to bear inmind that the potential and not the hydrostatic pressure drives the fluid motion. TheChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^71hydrostatic pressure shifts in response to the solute concentration distribution. The greateramounts of solute at the venular end of the tissue lead to increased colloid osmotic pressureactivity in this region. From the results, the importance of the osmotic pressure indetermining fluid motion is apparent. In the absence of the colloid osmotic pressure, thehydrostatic pressure distribution is identical to the potential distribution.For comparison purposes, a case without the effects of the colloid osmotic pressurethroughout the system (including the capillary) was performed. This case was termedvows. Here, the solute concentration via the colloid osmotic has no effect on fluid motion.The results from this case are presented in Figure (4.2).From the results, it is clear that the solute concentration distribution becomes moreone-dimensional in the radial direction. The solute concentrations change little within thetissue with the highest values at the arteriolar end of the tissue near the capillary. Thelowest values occur at the venular end of the tissue near the lymphatic sink. Thedimensionless average tissue solute concentration is 0.7462. The tissue hydrostaticpressure and potential distributions are identical. This is because there are no colloidosmotic pressure gradients within the tissue. The tissue hydrostatic pressures are loweredwithin the tissue (average dimensionless hydrostatic pressure equals -0.1014) whencompared with the base case average pressure (-0.0255). This results in lowered lymphdrainage rates (3.37 l/day) from the system than the base case (5.47 1/day).It is interesting to note that the fluid velocities across the capillary membrane areslightly greater for this case than the base case up to z 0.40 and beyond z -a: 0.90. In thecentral portion of the capillary length (0.40<z<0.90), the transcapillary fluid velocities arelower than the base case. This is reflected in the Peclet number distribution. The valuesare higher at the extreme arteriolar and venular ends and lower in the central portion of thecapillary than the base case values. The fluid velocities within the tissue are lower than thebase case fluid velocities. As a consequence, the convective transport of solute is loweredand therefore the more uniform solute concentration distribution results. Also the velocity0.01 ^0.02: ..............0.03:0.04: t., ..•.,.••'Hydrostatic Pressurec4^P20..,.... 0•••■ 000.9^1.0Chapter 4 : EFFE( TV OF SYSTEM PARAlifElERS ON MICROVAS(VLAR EX(TIANGE^720.010-02:0.03-0 .06-• ...............................................................^0.7463^0.74630.08-f .........................................  :^............. : .................. . ................... : '''''''''''''''''''''''''''''''0.07 :-. 0.7462 ------ 0.7462 --------0 7462 ----- 0 7462 ----O '1462 ---------O 7462—'—7.4.! ''' ................................................................................................................^00.04:9•99:` ................................................... 1------- 0:7461----- 0 ...... -----0.7461 .... ;;....0.05 30 7469^0.7463^0.7463................................. ...... .......... . '1........-...._ . ...... . ..............Solute Concentration0 . 0 0.1^0.2 0.3^0.4 0.5 0.6 0.7 0.6 0.9 1.00.0^0.1^0.2^0.9^0.4^0.5^0.6(^/ i^1^!/^/^/ tEPeclet Number..-■E--,0, 0.06-^ 0.037N. .............................................to0.05-• ......................0.04 .^ 39.9908---39.9908-39.9908 .............^...........O•^0.02 , ^.^. ...^ --•-r.,....^—.... ..... ...„.....• 0.07-, ^.....^ ..••----.-4^ .•"0.08-O 0.09: .........0.10 ^to^ 0.6^0.7^0.6^0.9tzt.,.•I I ^O.::;^:^•Nr^ •••■..., o ooA? 0,-,.0^:^o o7 / 7CY CY11-710 . 1^0.2^0.3^0.4^0.5^0.6^0.7^0.8Velocity Vector Field0.01•.........................t0.0^0.1     i•45.....1 0.2^0.3^0.4^0.50.01 ..........0.02- ....0.03:0.04:0.05 70.08-'0.09-0.10 - 0.01.01.00••.0.09:0.100.7^0.8 0.9........................... .... . . . ..... . . .......19 '0 6...-*.•••P—arr)/P.rt11111111111iiitiltH111111;141;;;;;;;;;;;;;; NI III 1 1 1 I 1 1 I I 1 I 1 1 A 1 A 1 1 1 1 A 1 1,11A1,AAA1A1,1,111 AAA AA 1 A 1 1 1I 1 1 1 1 1 1 1 1 1 1 1 A 1 1 1 1 t AA A A 4,4,4,4,4,4,4,4, 4,4,4,4,4,4,4,4,4,4,A AA A A 1 1 I 1II 11111111111 AA AAA AA AA A AAAAAAAAAAAAAANAAAAAA A A 1 1 1I 111111 A A A AA 4,4,4,4,4,4,4,4, AAA A AAAAAAA.AA.AAAAAAAAA AAAAAIII 1 1 1 1 1 t , , 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,.4,4,4,11 AAAAAAA AAA A A A 11111I 1 1111 A A A AAAA AA %NANA AANAAAAAAAA AA•NNA■AAA AA AAA A I I1  1 1 A   ..  1 1 1 1IA A0.010.02-0.04-0.05:0.06-0.07-0.08-0.10 AA 10.0^0 .1^0.2^0.3^0.4^0.5^0.6^0.7^0.8Distance down Capillary (dim)Figure 4.2: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case without colloid osmotic pressureeffects but with the lymphatic sink at steady-state.0.9^1 .0Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EX( RANGE^73magnitudes change less along the capillary length than for the base case. This results inless dramatic solute washout from the capillary.The influence of the colloid osmotic pressure is clearly illustrated by comparing thetwo above cases. The presence of the colloid osmotic pressure causes the tissue hydrostaticpressure to respond to the solute concentration distribution. This means that the tissuehydrostatic pressures becomes more positive and as a result, increases the lymphaticdrainage. The amount of solute removed from the tissue through the lymphatic sink risesleading to lowered solute concentrations within the tissue.4.5.2 Sensitivity AnalysisThe following summarizes the effects physiological parameters had on the system.These simulations indicate the parameters most influential on the system.Retardation Factor,The impact of the retardation factor on the system behaviour was evaluated viathree cases. First the base case where is set equal to 1.0. The second and third cases wereperformed with equal to 0.1 and 0.5 respectively.As the retardation factor is reduced, the amount of convective transport of solutedecreases relative to the diffusive transport. If the diffusion coefficient is sufficiently large,diffusive transport begins to dominate the solute behaviour and solute concentrationgradients throughout the region are reduced. This can be seen in Figure (4.3) for aretardation factor equal to 0.10. For the base case, the solute dimensionless concentrationrange is from roughly 0.47 to 0.70 (the range is thus 0.23). This range drops to about 0.13and 0.03 for retardation factor values of 0.5 and 0.1 respectively. Also at reduced values ofChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^74the retardation factor, the solute concentration gradient profile is more uniform within thecentral portion of the tissue. The highest gradients and solute concentration values arealong the capillary wall, especially towards the venular end of the tissue. Thedimensionless average tissue concentration of solute increases as the retardation factordecreases to 0.7132 for = 0.1.The pressure distribution displays similar characteristics as the solute concentrationdistribution. As the retardation factor is reduced, the hydrostatic pressure range is reducedand the solute distribution is more uniform. The average pressure in the tissue increases asthe solute hindrance increases. The lower pressure range and more uniform distribution arethe results of the increased solute concentration throughout the tissue space. The colloidosmotic pressure values and gradients throughout the tissue are functions of the local soluteconcentration. In response to the increased colloid osmotic pressure created by elevatedsolute concentrations, the hydrostatic pressure distribution increases. The dimensionlesshydrostatic pressure rises to 0.0406 (1.2180 mmHg) for i, = 0.10. The increase in pressureand solute concentrations leads to a slight increase in the range of the potential distribution.The potential distribution is shifted, at lower retardation factor values, in the negativedirection.The amount of lymph drainage increases as the retardation factor is reduced. For= 0.10, the lymph drainage rate is equal to 8.19 1/day. This is due to an increasedhydrostatic pressure throughout the tissue. The rise in the solute concentrations result inincreased osmotic pressure activity and thus higher hydrostatic pressures. Using the basevalues for the sink strength (LS and PL), there is no reabsorption of fluid back into thecapillary for all of these cases = 0.1, 0.5, and 1.0). The enlarged lymph flow causes anincrease in fluid filtration across the capillary membrane. The transcapillary flowvelocities are roughly one-third greater than the base case values. Since the soluteconvective velocities are directly affected by the retardation factor, the Peclet number. I•^• •-n--•0.3 OA^0.6^0.6^0.7^0.8^0.9^1.0Solute ConcentrationO o^. ...A^-^3ooar 0cP-.^ 10.2 \ 0.3 0.40.01 ^0.02:0.03-0.04:0.05:0.06:0.07:0.08 .0.10^. ^0.0^0.1Hydrostatic Pressure0.01 . 0.02:0.03:0.04:0.05:0.06-0.07-0.08-0.09-0.100.0^0.1^0.20.5 0.6 0.7 0.8 0.9 1.00 0Chapter 4 : EFFE('TV OF SYSTEM PARAMETERS ON MICROVASCULAR EX('HANGE^75:Vs0.010• ..0 32_ .........................................................................................................................................................:...........ra .......-• 0.07 Z-9996 9.91^:... ..... . g. co03^.............,E-■0.04,---39991--...., 7..,...0.05- ............^.-.'....................... -----.--". ..-- ........ . .^'.....^4C--2, 0.06-^ 4 -90.7^0.8Peclet Number0 0.08:-O 0.09-• 0.10 .... . ..-.---,-.-..-  .. -j^cl^ , ^ i -^--i• •-•-----,--•4.$ 0.0^0.1^0.2 0.3 0.4 0.5^0.6co- ...IA(P-°70/Part0.01,^0.03:0.05:--.............' 7/ 71/1/1  /Y i 7 :II/facbA* ..^4o^.:^coco^•^cs. a)^:^CO0.0270.067^ O.^:^(Cob^i^03^•^C13^ CO^0^•^0CO^ 0 : •■•4/ : 40 40 40^40 CO : CO0.077 0-0.09-^7^ ( .•al.^ci• ^1di^ci.^T.^•^a;0.08:0.10 I^ .. , .. .. .4. I^I^1 0.0 0.1 .2 0.3 0.4^0.5^0.6 0.7^0.8^0.9^1.0Velocity Vector Field.....0.9^1.00.080.09:0.0 6-:0.03:0.04-0.05-0.07-0.02:0.01 .1 1 1155^15    1 1 AAAAA‘‘A■ ............................ ■■ 4.AAAA 1 11 1 1 1 1 A AAAAAAAA 4.4.■ ••■■■■■■■ 4. 4. 4.%■ 4. 4.■ 4.4.•■4.4.•■•AAAAA 1 11 1 1 1 1 AAAAAA■• ........ ■•■■ ..... 4. 4.■■■■■■■■■■■■AAAA 1 11 1 1 1 1 A A AAA AAAAAAAANAA.•••■■■•■•■••■■■•■•AAAAAAA 11 1^1 1 Ai^ASAAAAAAAAAAAAAAAANAAAAA A A A A A A A A A 1 1 11 1 111 1 I 11111^III ;^1^1^  5551AAA 1 1A A^10.00.8^0.9^1.00.10 -0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)Figure 4.3: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the retardation factor,^equalto 0.1 at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EX(1-IANGE^76distribution is decreased and has a much smaller range. This means that for lower values ofthe retardation factor, as expected, greater portions of the tissue are undergoing greaterdiffusive transport. It is important to bear in mind that the Peclet numbers indicate theamount of convective solute tranport relative to diffusive-dispersive transport. Theconvective term makes use of the solute convective velocity and not the absolute fluidvelocity.Capillary Reflection Coefficient, aThe effect of the capillary reflection coefficient is evaluated using the base case andthree other cases (a = 0.10, 0.50, and 0.99).Figure (4.4) presents the results for the capillary membrane reflection coefficientequal to 0.1. As can be seen from the results, the tissue-side capillary membrane soluteconcentration and pressure profiles are considerably different from the base case (base, a =0.85). The solute concentration range and gradients are reduced within the tissue space.The dimensionless average tissue solute concentration, however, increases to 0.7150. Thegradients tend to become increasingly one-dimensional (in the longitudinal z direction) asthe capillary reflection coefficient is reduced.Similarly, the pressure range and associated gradients also decrease as the reflectioncoefficient is decreased and the distribution becomes more one-dimensional in thelongitudinal direction. The hydrostatic pressure distribution for the lowered reflectioncoefficients looks very similar to the solute concentration distribution. The hydrostaticpressure distribution is shifted in the positive direction. This is due to the increasedosmotic activity in the tissue because of the greater solute concentrations. Thedimensionless average tissue hydrostatic pressure rises to 0.04282 (1.2846 mmHg). Thisresults in an increase in the lymph drainage rate to 8.25 1/day. The potential distribution isshifted in the negative direction but has a very similar range as a is reduced. RadialI •^I •0.2 1.00.90.1 0.7 0.80.3^0.4^0.6^0.60.01 . 0.02:0.03-0.047:0.05-0.08-0.07-0.08-0.09-0.10 0.0Hydrostatic PressureChapter 4 : Eh-FE(:TS' OF SYSTEM PARAMETERS ON MI( RU ^EXCHANGE^77Solute Concentration0.01 .^0.02-0.03-' •0.09:0.08-^ to0.08:0.05-0.10  t   a ..\\:\ 0.04-0.07-. •-^*".^.0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^4 . g■•\' : , 1 i 1^0 -›^--. co•^.^6.^.^•^•1.01.00.1^0.2^0.9^0.4^0.5^0.6- --1377 (./ . 1/1, I, / : . /co^co^-^ce^•^0r. : co^co^coco co^o .. cvco co co^co o o oco' co so v^co^co/^O^ci^e; d O/ cS/ / (P-070/Part0.7 0.8 0.9 1.0Peclet Number1^0:4 7 39. .. ... 9..0..8. '''''''''''' :9-9. ''' '' ............................................................................................ ............... ....... .co000:... 01-....^0.03 ................4.'^-19 9^.. .........^u-,......a0.07^ ..*--^/---• 964• 0.08-.^co:.: coCI6'o 0.09co4) e0V^0.10 .....0.0^a'-6-i..fA0.4^0.5^0.6^0.7^0.8^0.0......-•0.3cn0 .1^0.20.01 . 0.02:0.03:0.04 .0.05:0.06:0.07:0.08:0.09:0.100.0Velocity Vector Field0.1011111: 111111;;;; ;:;;;;;;;;;;^11 11 11 11111 AA A AIAAAAAAAAAAAAAAAA•AAAAAAAAAAAAAA 1 1 11 1 1 1 1 I 1 1 A A AA AANAAA•••••••■••••••■•••••••■••AA A A A 11 1 1 1 A AA AA AAAAAAAA••••••••••••••■•■•••••••ANAAA A 11 1 1 1 A AllA AAAA•••••••••••••••••••••••••■•■■•ANA A 1 11 1 1 A A AAAAA••••••••••••••••••■••••••••••■•■■■■ Al 11 111 AAA%A•••• ........ •• ......... ...... ••••NAAA 1 1111 AAAA•■•••• .................. ....... •••••AA 111 I A ^  • A A 1 10.01 . 0.02:0.03:0.04-0.05-omai0.07:0.08:0.09:0.0I•^I - ^0.1 0.2^0.3 0.4^0.5^0.6^0.7Distance down Capillary (dim)0.8^0.9 1.0Figure 4.4: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the capillary reflectioncoefficient, a, equal to 0.1 at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVA.S'CULAR EX(1-IANGE^78gradients are greater especially at the arteriolar and venular ends of the tissue near thecapillary wall.These results are expected since a reduction in the capillary reflection coefficientlowers the osmotic pressure driving force acting against the hydrostatic pressure differenceacross the membrane. This means the transcapillary fluid velocities are increased. Alsofrom equation (3.34) it is seen that less solute is reflected at the capillary membrane as a isreduced, thus more solute is convected through the capillary wall. For a = 0.10, of thesolute that would otherwise have freely convected across the capillary membrane, only 10percent is reflected back into the capillary flow. The increased transcapillary fluid flowvelocities and reduced solute reflection combine to convect more solute into the tissuespace resulting in the higher average tissue solute concentration. This, in turn leads toelevated tissue hydrostatic pressures and the increased lymph flow rate.The increase in the capillary reflection coefficient to 0.99 leads to increased soluteconcentration gradients within the tissue. The results for this case are displayed in Figure(4.5). The solute concentrations especially at the arteriolar end of the tissue are lower thanthose found in the base case (the dimensionless average tissue solute concentration is0.5056). The range in the solute concentrations is increased. The hydrostatic pressuredistribution is similarly increased in range and has a slightly lower average tissuehydrostatic pressure equal to -0.0328 (-0.984 mmHg). The potential distribution isrelatively unchanged but is shifted in the positive direction and consequently the velocityfield and magnitudes are very similar to the base case. This is expected since the base casereflection coefficient is equal to 0.85.With a reflection coefficient equal to 0.99, this means that 99% of the solute thatwould be transported convectively across the capillary membrane is reflected. Thereduction in the solute crossing the membrane can be seen in the results since the soluteconcentrations at the arteriolar end of the tissue are reduced. This is the region of highest0.7 0.6Solute Concentration0.01 .0.0270.0370.04:0.05-0.06-0.07-0.08:0.0970.10 . - . -^- --- - . -102 0.3^0.4 0.5^0.6• 499-499 49990.10.0, •0.9^1.0Chapter 4 : EFFE(TV OF SYSTEM PARAMETER.S" ON MAROVASCULAR EXCHANGE^790.01 .0.02-0.03-0.04-0.05-00.07- 4400 0000.00:^ •0.10▪ 1-•^...•0.0^0.1^0.2^0.3^0.4^0.6^0.6^0.7^0.8 0.0^1.0Peclet Numberr_.9 0.04-^.... -- "PO^.....-- 19 \-,. ....A 0'0^co7--- I: rv^0.02 , ........ ............ . ........................................................93 0.03 '`-'^0.05 .-.^.9^•^61^ cP :4)^c:o^coO -....^1 e oi0.07-7•2 0.06-;0". r a)•to to\ l?O 0.09-o 0.10 ^ •0 0.08- 1^r.---.------T -----,--- ....... - ^ II ^.0 0 0.1^^ -..2 0.3^0.4^0.5^0.6 0.7in....,Hydrostatic Pressure.... .....^...............05000.01 e^rres-rt ri% - .......................0.8^0.9^1.00.8(P-crrr)/Par,0.03: I^1^1^1^I^I^10.047^0.05-^:^o) a al a CD 0 aa0^0^r- co .-.^0.06: 1,-^g^co a) co^co^co0 0 co^co^02 el el0.077 c; c;^c::; 6 c;^6^6^0.08 :^ Ii^1^I I I0.0970.10^.... ... , -4.---.^11-1-4-1-rvt...-,,,-Lrrve-r•r4L-m-,-,jr-r,^I-.^I.^i ^0 . ^0.1^01.2^0.3^0.4^0.5^0.6 0.7 0.8^• . ^0.9 1.00.01 .0.02:0.03:0.0470.0570.08:0.0770.08-0.09-0.100.0Velocity Vector Field0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)I III iiiiiWAAAAAAAAAAAA111AAAAA11AAAA  1 A A A1 A ^A4, 4.^ 4' ^‘.^ ■^ .. • A^ . A I^ .. 1^ . A A^ ..0.9^1.0Figure 4.5: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the capillary reflectioncoefficient, a, equal to 0.99 at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^80solute convection across the capillary wall. Solute, however, may still be transportedacross the wall by diffusion (the diffusive permeability PS).Tissue Reflection Coefficient, atThe effect of the tissue reflection coefficient is evaluated using the base case (at =1.0) and two other cases (at = 0.0 and 0.50). Using at equal to zero implies that the soluteconcentration distribution and gradients do not influence fluid motion and hence thehydrostatic pressure (and potential) fields within the tissue.The range in the solute concentration distribution is slightly reduced as atapproaches zero. The results for a t = 0.0 are shown in Figure (4.6). The soluteconcentration gradients in the radial direction are also reduced. The concentrationgradients grow more uniform within the arteriolar end and central portions of the tissue asat is reduced. The dimensionless average tissue solute concentration increases very slightlyto 0.5492 for at = 0.0.As a consequence of the increasingly more uniform solute concentration distributionas at is reduced, the colloid osmotic pressure distribution plays a reduced role towards fluidmotion. At at equal to zero, the solute concentration distribution plays no role in the fluidmotion. This is true within the tissue but is not, however, at the capillary wall whereStarling's hypothesis still governs fluid flow across the capillary membrane. In the limit ofat = 0.0, the hydrostatic pressure and potential distributions are identical within the tissue.At lowered at, the range of the hydrostatic pressure is greatly reduced. When al = 1.0 (thebase case), the dimensionless pressures range from about -0.1 to 1.0. At a t =0.5, thedimensionless pressure range changes to lie between 0.1 to 0.55. The hydrostatic pressureresponds to the reduced solute concentration gradients via the colloid osmotic pressure.The distribution changes from one that resembles the concentration distribution to one thatresembles the potential distribution. This means that the maximum hydrostatic pressure. • • - •k...-V ..••1 /I . • •7 .1 ..•• 7 .•..-••••0^ ---'%^.:^42 I 4.,47 :^0 :^40^:^0^:^,f,'°^0P . ..1'^-^:::^:^...^:^....^.^••••^.^•••• :^'1,'''', : -^: CO • 00 : 41'.. : . .1)..•: ... •,.•••I • -.^1^.^CI'. :^6^:^Ci^•^ci o^.• . o-/ /^/  ^z. ,--.,^.(1. , ^ r••• .1.0.1 0.90.30.2 OA 1.00.80.5^0.8 0.70.010.020.030.040.050.060.070.080.090.100.0Hydrostatic Pressure.^:11^..... •• •^... ... :••.--.-:^l 1.00.8 0.9Chapter 4 : EFEE(71S SYS1EM PARAMETERS ON M1CROVASCULAR EXCHANGE 81Solute Concentration^0.01 . ^0.02-0.03:0.04:0.05:0.08:0.07:a.138 70.09:0.10^0.0^0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0Peclet NumberO0.01, ................................................................................................................ . .......................N:93 00.0 32-0.05-00.07:^19.9954..a)^0.06.:O0.08:. ........O 0.09-0 0.10^4.*4^0 . 0 1 1^1.2^0.3^.4•cl....0 0ir020 .10.010.02^0.03.:0.040.05:0.06:0.07-0.0Velocity Vector Field1 1 I 1 1 1 1 I 1 1 1 1 11^1 1 1 1 1 iiiii l^iA^^11111 1 1 11 1 1I1 1 1 1 1 1 1 1 1 1 1 1 1 1 .1 t I l'illAltAAAAA'1"I‘AtAAAAAA'il t t 1 1 1 1111111 1 1 1 I 1A1 1 AillAiAA■■■■■■■■■■■A■■■■AAANAAS 1 1 1 1 1'I 1 1 1 1 1 1 1 t AWAll•■■AAAAAANA■••■•■■■■■WAWAAA1 1 1II 1 1 1 1it■■■■■■■Il■4.4.■■■■4•■■■■■4■■4.4.4.■4.4.4■■■■■■ 1 11'1 1 1 "( AltAtAAAAA■•■■•■■■•■■■■■■•■■■■■■■■■AAAAAA 1 I 1 1Ii1l■■■•■% 1  ■■•■■■••■•■ ■•■■■••■■■■ •••■•••••■■•■ ■AANA 1 1 111 1 .WAANNANN•■■ A. ■ ^ .1/4 ■•■•■■NtAII1 t t ^  AA* 1F••-• • •- •-•^•^• 1 1-•-•-• •0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)Figure 4.6: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the tissue reflection coefficient,at, equal to zero at steady-state.i l ss tiss  s l t  tr ti s, r st ti  r ss r , l t ,096°`1-r-•^ ..^•0.5^0.6 0.7^0.8^0.9 1.00.01 ^0.02.0.037,"0.04:0.05-0.08:0.07-0.08:0.09:0.10 . •10.0^0.1Chapter 4 : EFFECTS' OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^82shifts along the capillary wall towards the arteriolar end as at is reduced. The averagetissue hydrostatic pressure becomes more negative with reduced a t since it approaches thepotential distribution leading to lower lymph drainage rates. For at = 0.0, thedimensionless average tissue hydrostatic pressure drops to -0.1162 (-3.486 mmHg) and thelymph flow is 2.87 1/day. Correspondingly, as a t is reduced, the potential distribution isshifted in the positive direction and the range in its values is lowered since the soluteconcentration distribution range reduces.There is no fluid reabsorption back into the capillary as a t is reduced and velocitymagnitudes decrease substantially within the tissue space and are reduced at the arteriolarend of the capillary membrane. At the venular end of the capillary, the fluid velocities areslightly greater for reduced a t than the base case values. This means that there is morefluid filtering into the tissue in this region with lowered a t. As can be seen from the valuesin the Peclet number distribution, the fluid velocities in the tissue are reduced as a t isreduced.Solute Diffusion Coefficient, Dd11.The effect the diffusion coefficient had on the system was investigated using thebase case and two other cases where the diffusion coefficient was increased and decreasedby an order of magnitude.Figure (4.7) displays the results obtained with diffusion coefficient increased tenfold. An increase in the diffusion coefficient affects the results in a similar manner to thereduction of the retardation factor. The range in the solute concentration is reduced and thedimensionless average tissue solute concentration rises to 0.7151. The increase in thediffusion coefficient means that the solute can counter the convective transport to a higherdegree. This means that there is very little solute washout from the arteriolar region of theHydrostatic Pressureo^ o^oa- .•.^0 cr• •o o o0 o o0.3 0.4 0.60.01 }0.02:0.03-0.04:0.05-0.06:0.07:0.08:0.09:0.10  000.0^0.1^0.2 1.00.8^0.7^• 0.8 0.9. ...... . ........ . .....0.4^4.5^0.6^0.7(P-01)/Port440.8^0.9^1.0Chapter 4 : EFFE('IS OF SYSTEM PARAMETERS ON MICROIAS(111,A1? EX(IIANGE^83Solute Concentration0^ 0^ 0^00 0.ts cOC3^CO^CO• t  ^ .^0.0^ 0.3^0.4^0.5^0.6^0.7^0.8^0.90.01 .^0.02-0.03:0.04:0.05:0.06:0.07:0.08-0.09 -0.100.1 0.2 1.007s.1" Peclet Number0.01 ^ ......... ....... .... .. .. 40/................................................RI^0.03: .............................. 5.998603(.....•^0.04^3-9991 ......_^................0.05. ..............^----s••• ---.....3 0.06-^•-...^aO 0.07: 1-99gs^-.- • .^-.9.99•IO 0.08:O 0.10^ . 4 \-",^0 0.09-._. -1 ^...1^0.0^0.1 0.2 0.3....•^- 2 .............................................................^0.01 . ...... ......... Vii : ..711 ^7 .. i i0-02:.....-"./coz0 -^•^0^0^"1^au^ea^:^Ii.^ st,• co0.06-^ 40 • 0 0 0 0 0 0co :6 : 6 oi 1 i^1^:^/I----1^. 1 • • 10.0^0.1^0.20.070.08-0.09-0.10   •-•-7 •-•-■-• 4--/-,.....././ 0/• V6 0i 01 :. 61 1-1 -0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Velocity Vector Field0.01 0.02.0.03:0.04:0.05:0.06:0.07:0.08:0.09:0.10 •0.20.0 0.1 0.8 0.946f§b^:^• C5^•^ce, . 40CO 03 CO CO^: C0 00.05-1.0• •^• •^ • ^ • ^0.3^0.4 0.5 0.6^0.7Distance down Capillary (dim)1 1 1 1 1 1 1 AIWAAAAllt•l■■■•■■■■■■■■•■■■NAAAAANNAAA 1 1 11 1 1 1 1 1 1 l t AAAAA%•■■••■•■■•••■■■■■■■■■■••■•■■•AAAAA t I 11 I 1 1 1 t l AlAAA•■•■•■■■■■•■■■■■■■■■■■■■■■■•••■••■■■ t 1 11 1 1 1 1 A‘AANN■41N■■■■•■■•■•■ .............. ■■■■■•■•■■1A1 11 1 1At ANNN•■■•••■■■•■■ ..............................1 1 tAA••■ ■••■■ ............................ •1/4■■ ■ 4■Al 1 11 1 I l ^  AA 1 1 1     ■NA 1 1II III Iti^11111111 ;1;^; ; ; ; 1^; ;Figure 4.7: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the solute molecular diffusioncoefficient, Ddill; increased by an order of magnitude from the base value at steady-state.Chapter 4 : EFFE(7TS OF SYSTEM PARAMETERS ON MICRO VASCULAR EXCHANGE^84capillary since solute can easily diffuse back to the capillary membrane. This leads to themore uniform solute distribution throughout the tissue clearly displayed in Figure (4.7).The solute concentration distribution yields a similarly distributed osmotic pressurefield. The hydrostatic pressure responds to the colloid osmotic pressure resulting in adistribution similar to the solute concentration distribution. Due to the higher soluteconcentrations, the osmotic pressure is greater throughout the tissue. This leads to morepositive hydrostatic pressures throughout the tissue and therefore greater lymph flow. Theaverage tissue hydrostatic pressure is 0.0408 (1.224 mmHg).The potential distribution is shifted in the negative direction due to the increase inthe solute concentrations via the osmotic pressure. The potential gradients are stronger inthe radial direction than the base case. This is due to the greater lymph flow (8.23 1/day)caused by the more positive hydrostatic pressures in the tissue. The fluid velocities aregreater throughout the tissue and at the capillary membrane.It is interesting to note that an increase in the solute diffusion coefficient by a factorof ten has very similar transcapillary flow velocities to that when the retardation factor isreduced by a factor of ten. This is because the effect on solute transport is similar for bothcases, i.e., the diffusive contribution is effectively increased by ten-fold. As expected, thedrop in the range of the Peclet number indicates that the diffusive transport is increased.Tissue Hydraulic conductivity, KThe effect of the tissue hydraulic conductivity is tested in three cases. The basecase uses the value indicated in Table (4.1). Two other cases are performed with K reducedand then enlarged by an order of magnitude respectively. The results discussed below aredisplayed in Figures (4.8) and (4.9) (K x 0.1 and K x 10 respectively).The main effect of an adjustment in the tissue hydraulic conductivity is a change inthe potential gradients within the tissue. From the results, a reduction in K by a factor ofSolute Concentration0.010.02-0.03:0.04-0.05:0.08:0.07-0.08-0.09-0.100.010.02:0.030.04-0.05-0.08-0.07:0.08:0.09-0.100.70.0 0.8 0.9^1.0Hydrostatic Pressure0.1^0.2^4.3^4.4^0.60.7 0.9 1.00.0^0.1 0.2^0.3 0.60.50.44-;^0.00304a)• 0.07:0.08:O 0.09-• 0.10 ^(I^0.05-^•-..^a.o6:719 9^ cb.Ds4 00.1^0.2^0.30.010002.030^........................-9008...-........................................^4................................... ....... ......... . ... (,..0.04- ................... ...39 ... Co0.4^C1.5^0.6^0.7Peclet Number0.80.9^1.0Chapter 4 : EFFE( 'TV OF SYS7EA1 PARAMETERS ON MR.1?0VASCULAR EX( RANGE^(50.010.02: .-"..0.03:0.04: 40.05- CD/3.. CO00.06-0.07702o-0c,30.0870.09: --;0.10 ..... ..^, ^0.0^0.1i^ii^7CO^03^CO^CO 0^0CO CO CO 0^03 COCO^0 02 0 Ni•e, v,^;",^v^v^.it.• c; 4:5t(I(^I( ^0.8•_e ,0.2 0.3^0.4^0.5^0.6^0.7Velocity Vector Field•(P-air)/P0.010.02:0.03:0.0470.05:0.08:0.07:0.08:0.0970.100.01 1^ AAAAA'1 11A AAA■■•■■■■••■■■■■■•■■■■■■■%1ji^1 I 1 1 1 1^1 I '1^ All^tAAAAA1 I I I 11AI A AAA A AAAAAAA■■■■■■■■■■■■■■■■■ ........ ■■A1 11 1 I AAA AAAAANN■■■■■■■■■■■■■ .....................1 1 1 I 1 AAA A■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ A1 1 1 AAA^ A I^ .........■ A I^ ....•0.9 1.0Figure 4.8: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the hydraulic conductivity, K,decreased by an order of magnitude from the base value at steady-state.11 A A  I 1 A A ■ I A A  A A A0.1^I•^I ^ 0.2^• I0.3^0.4 0.5 0.6 0.7 0.6Distance down Capillary (dim)Solute Concentration•51990.010.02:0.03:0.04:0.05:0.08:0.07-0.08-0.09-0.10041990.0^0.1 0.2^0.3^0.4^- ^0.8 0.90.7 1.00.5^0.60.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.00.010.02-0.03-0.04-0.05:0.08:0.07:0.08:0.09:0.100.0Hydrostatic Pressure......._....^. ............o0Co^oci^a coc- co^oi^ts, ca•-• • .. .. e. ..: /.:0.6^0.7^0.80'.9CO031.011/4a• ^0.1^0.2 0.3^0.4^0.5^• 0.6Velocity Vector Field0.7^0.80J 0.9 1.0lliapter 4 : EFFE(7S OF SYSTEM PARAMETER.S' ON MI(I{OVAS(ULAR EX(1-IANGE^86ID^0.01• 0 .02_ .................. ....._ ........^-------. ... .z.:7-CO0.037_____,..................... ■-.1,........---^--'..s.-----........CO...^0.04-t ......."^'4P^ .•• -0E-.0'0.05 ^*-. 4.).. '49\^%..^.0\0.077---.^-,. 06, •00CP •.. icaoa, 0.06-O-*V0°cS^.....14 0.0^0.1^0.2to....A0.010.02:0.03:0.04:0.05:0.06:0.07:0.08:0.09:0.100.0kkk AAAAAAANANIIIAAAAAA‘■1 1 1^AAA ^1 1 AAA\ 1 A 1 A ^A A ■ ■0.0^0.1^0.21.0Figure 4.9: Dimensionless tissue solute concentrations, hydrostatic pressure, Peclet number,potential, and velocity vector distributions for the case with the hydraulic conductivity, K,increased by an order of magnitude from the base value at steady-state.Peclet Number0.50.01 0.02-0.030.04-0.05-0.06-0.07:0.08:0.09:0.100.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)0.8Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EX(TIANGE^87ten increased the range in the potential by just under a factor of ten. This is expected sincethis corresponds to an increase in the tissue fluid flow resistance. The combined decreasein the hydraulic conductivity and the increase in the potential gradients yield slightly lowerfluid velocities throughout the tissue. This means that less solute is convected into theinterstitial space. The range in the solute concentration distribution is reduced as thehydraulic conductivity is reduced. At the arteriolar end of the tissue, the soluteconcentrations are slightly higher than the base case values. This however, is not the caseat the venular end of the tissue where the solute concentrations are slightly lower than thebase case values. This produces a solute distribution with lower longitudinal and radialgradients throughout the tissue than the base case. The dimensionless average tissue soluteconcentration, slightly higher than the base case value, is equal to 0.5403.The increase in K results in a lower range in the potential within the tissue. Thefluid velocities increase and result in a greater solute range throughout the tissue. Thisproduces a less uniform solute concentration distribution throughout the tissue.The hydrostatic pressure distribution is very similar to the solute concentrationdistribution. For the case with decreased K, the more uniform solute concentrationsthroughout the tissue result in a more uniform hydrostatic pressure distribution. The rangein the pressures is decreased with increasing portions of the tissue becomingsubatmospheric. Hydrostatic pressure gradients are low in the central and arteriolarportions of the tissue. The dimensionless average tissue hydrostatic pressure for thereduced hydraulic conductivity case is -0.0268 (-0.8040 mmHg). The lymphatic drainagedecreased to 5.27 1/day. When the hydraulic conductivity was increased by an order ofmagnitude, the dimensionless tissue hydrostatic pressure increased very slightly to -0.0253(-0.7590 mmHg). The lymph drainage increased slightly to 5.51 1/day. The smaller changein the average tissue hydrostatic pressure for an order of magnitude increase in K than theorder of magnitude decrease in K implies that the tissue fluid flow resistance is notdominating the fluid flow structure at the current base value of K listed in Table (4.1). AsChapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^88K decreases further, the tissue potential drop increases and the hydraulic conductivitybegins to control the fluid flow patterns to a greater extent. Therefore, only reductions inthe tissue hydraulic conductivity (or increasing the capillary membrane filtrationcoefficient) would alter the flow patterns significantly.As K is decreased, the fluid velocities within the central portions and arteriolar endof the tissue are reduced despite the increase in the potential gradients. This is reflected inthe Pecelt number distribution. The reason the fluid velocities are largely reducedthroughout the tissue is due to the increased resistance of the tissue porous medium. Thefluid velocities are similarly decreased at the capillary wall except at the extreme venularend of the capillary where fluid velocities are slightly elevated from the base case values.Capillary Membrane Diffitsive Permeability, PSFour cases were used in evaluating the impact of the diffusive permeability throughthe capillary wall, PS. As well as the base case, the following cases were also investigated:PS = 0.0, PS x 100, and PS x 10000. Figure (4.10) displays the results for the last case.Moderate changes in PS only affect the solute concentration distribution (and thusthe pressure and potential distributions) slightly. Increases in PS increase the diffusivetransport of solute through the capillary membrane yielding higher solute concentrations atthe tissue-side of the capillary membrane and thus throughout the tissue. This is mostpronounced at the venular end of the capillary. This is expected since the convectivetransport of solute is minimal at the venular end of the capillary. The small changes in theresults for moderate changes (x100) in PS indicate that transport across the capillarymembrane is convectively dominated. If there were reabsorption of fluid back into thecapillary then increasing PS would most affect the central portions of the capillary near thezero point where the fluid velocities are the lowest.Chapter 4 : EFFE(71). OF SYS1EM PARAMETERS ON MICROVASCULAR EXCHANGESolute Concentration0.01 ^0.02,0.03:0.04:0.05-890.010.02„0.03-0.04-.0.05-0.0ei0.0770.08:0.09:0.10^0.08-^ ..-0 ^0^ ...--^42-?.47^ .....^IP>^ .-^0.07- >a,?^. es^--.^ee^- .^0.08- e 1,9  ^e^-0.0970.10    i ^0.0^0.1^0.2 0.3^ ......... ^......... .^.....;^ci-.a....*^ ......*^0.4^0.6 0.6 0.7^HydrostaticP^ressureo..osto■0.01..............4004-so^0.0520^0.0520^- 0. 0520(Lie0.060.090.080.03= 11-0,5•0043 .054 7.-0.070.00300................. .....................^.......0.0^0.1^0.2^0.3^0.4^0.6^0.6^0.7^0.6^0.9^1.000:00:1. ............................................................................................................................................af4 0.0340 0_02    ........^39'9908^39.990839.9906^39.9908cfebk........0.21.00.9 1.0Peclet Number......................2 0.06-0.070.08--o 0.09-0 10' .........cl 0.0^0.102....0.3^0.4^0.5^0.6^0.7 0.8^0.9(P-0-70/Part..-*.-'..'7/ is I / / ...." .. -..-'.^.-"^.."^.-".ca^:^ca^.:^al^:"^ca.^ .- : co of . ca.c:^ar^: 07^.. 02 .^42^..^V^:o.^:/ 0^.. O^.. O.r0.2 0.9^0.4^-, ^t^I i .--.;,--.-..--.--. 0.5 0.6 0.7^0.8^0.9iz)0.01^0.02,0.03:0.04 ,0.05-0.07-0.08:0.09:........ 0.10 0.10.01.00.0Velocity Vector Fieldilliiiiiiiiiiiiiirn1111111111;;;;;;;;;;;;;;;;;; IIII I I I 1 1 I 1 I I 1 I 1 I I I 1 I 1 AtIlItIII1 1 1 AtAliAlAWIIIIIIII 1 I 1 1 I I 1 1 VIAAAAAA AAAAAAAAAAAAAAAAA%A.k.lAtt 1 'I 1 1 1 11 I 1 1 I 1 1 I 1 1 tAttAlANA ■■•■■■■■■■■■•■■•■■■•■■ 1 AAA'. I I I1 I I I 1 IAAVAANAA.1%ANAA■••■■■■■■■■■••■•■■■NNAWAIIIII I 1 I 1 A tAAAANt■■■■•■ ■■■■■■■■■•■■■■■■•■■■••■■■ 1 1 1 1 1I VIAAAAANNN■■■ •••••••■•••••■■•••••■•• ■•■•■AAAAIIIWA A A A l••••••••••••••■■■■•■••••■•••• ■•■■■■■•1 1 1 1 I1 1 IlAA■■••■■■■■■ ••••■■■■■■•■■■•■•■■•■•• ■NNAAA 1 1 1 I^ I ^ I ^ 1^•••--• I ^0 .1 0.2,.----. -- • • 19--rm,--.--,--,--,-,--,--,--,--^.-10.3 0.4^0.5 0.6^0.7^0.8 0.9Distance down Capillary (dim) 1.0Figure 4.10: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the capillarymembrane diffusive permeability, PS, increased by a large amount (x10000) from the basevalue at steady-state.Chapter 4 : EFFECTS' OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^90Extreme changes in the diffusive permeability (for example, x 10000) alter thesolute concentration range and average tissue concentration values significantly. The soluteconcentration range is reduced dramatically to roughly 0.010 and gradients are reducedthroughout the tissue. The increased diffusive permeability raises the dimensionlessaverage tissue solute concentration to 0.7355. This rise in the solute concentrationdistribution leads in turn to a rise in the hydrostatic pressure distribution as it responds tothe increased colloid osmotic pressure throughout the tissue. The dimensionless averagehydrostatic pressure is 0.05209 (1.5627 mmHg). This results in an increase in the lymphdrainage to 8.60 I/day. The range in the potential is largely unaffected. The main change isthat the radial gradients are slightly greater throughout the tissue especially at the arteriolarend near the capillary. The potential distribution is shifted in the negative direction due tothe increased solute concentration values (via the colloid osmotic pressure). Fluidvelocities are greater along the capillary membrane and within the tissue due to theincreased lymph flow.Lymphatic Sink Strength, LSThe strength of the sink was varied in two other cases. In the first case, the sinkstrength, LS, is reduced by a factor of five while in the second case it is increased by afactor of five. This allows us to make some generalizations on the behaviour of the systemwith regard to the strength of the lymphatic sink.The solute concentration distribution is increased as the sink strength is reduced.The dimensionless average solute concentration is equal to 0.5653 for the reduced value ofLS. This is expected since less solute is being dragged out through the sink and less soluteis being convected across the capillary membrane. The range in the solute concentrationchanges little with a drop in LS. These results are presented in Figure (4.11). Increasingthe value of LS lowers the average solute concentration to 0.5065. The radial gradients are0.3 0.4^0.50.10 1  ^ ...0.0^0.1^0.2 0.7 0.9 1.00.60.01 .^0.0270.03:0.04:0.05-0.08:0.0770.08:0.097Hydrostatic Pressure0.010.02-0.03-0.04=0.05-0.06:0.07:0.08:0.09:0.100.0 1.0I •^• I • 4-4-• -• 4-4444 44-44 •i•^•i^• ..0.4 0.5^0.6^0.7 0.8 0.9^1.00.01 . 0.02:0.0370.04:0.0570.06:0.07:0.08:0.09-0.100.0•0.1^0.2^• I .^•1•0.3 0.4^0.5^0.6^0.7^0.8^0.9Velocity Vector Field4oco co^:co coAl^10 co^coco 02 IV^c‘i a:co^c3O. l^CI/^a 6^a 6I 1^i^i((1)-070/P.„i^1^I0^0 0 C30;43^0 e3CO 07^0303^e3 e) e3a al^a 6i^1 1^11 i^I1.0Chapter 4 : EFFEt 'TS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^91Solute ConcentrationPeclet Number0.01V• 0 .027____-- ....................................CO 0.0342 0.04- ................. ..^OaP.: 0- 05- D^.... ' .90y 0.06- -.99s10• 0.0770.08:061\C.)^0.09:0 ..\^• 0.10  1 ^ i.4.1^0.0^0.1 0.2 0.346co........ •0.01 . 0.02-0.03=0.04-0.05.:0.08-0.07-0.08-0.09-0.100.0I ^%1 I^I^jiiiIIIAIAII^AA  ^AAANNN1 1 A A AA A AAAAAAA%■••■•4 4.44%% 44. 4.% 4 . .....1 II 1111 AAAAAAAA% • 1 1 1 1 AAAAAAA ^1111 lANA 44%%% . I I AI ^A 4' ^1 I ^. ^ •I^ I^ I   I ^  , •0.1^0.2 0.3 0.4 0.5^0.6 0.7^0.8^0.9Distance down Capillary (dim)I1.0Figure 4.11: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the lymphatic sinkstrength, LS, decreased by a factor of five from the base value at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^92increased relative to the longitudinal gradients within the tissue as the sink strength isincreased.The hydrostatic pressure distributions are very similar to the solute concentrationdistributions. As the sink strength is increased, the pressure distribution shifts lower but islimited by the lymphatic pressure, PL. The average tissue hydrostatic pressures are 0.0924(2.772 mmHg) and -0.0575 (-1.725 mmHg) for the cases with LS reduced and increased bya factor of five respectively. As the sink strength is reduced, the potential distribution isshifted in the positive direction. The sink affects the fluid velocities within the tissue and atthe capillary membrane. As expected, as the sink strength is reduced the zero pointsapproaches the mid-point z* = 0.5. With a sufficiently strong sink, there is no reabsorptionof fluid back into the capillary and all the fluid is drained out through the lymphatics. Thisoccurs with the base case. The case with LS reduced by a factor of five displays fluidreabsorption back into the capillary. The zero point occurs at roughly z * = 0.94.As expected, as LS is increased, the lymph flow increases. The lymph drainageflows are equal to 1.91 and 21.42 1/day for the cases where LS is decreased and increasedrespectively. Numerical convergence was difficult to obtain in cases where LS wasincreased. This is because the fluid leaving the system via the lymphatic sink is limited bythe amount of fluid that can enter the system. This is dictated by the value of the capillaryfiltration coefficient, L/), which is prescribed at the capillary membrane. This resultindicates that, given a fixed value of the capillary filtration coefficient, there is a lymphflow limit that cannot be exceeded.Lymphatic Sink Pressure, PLThe base case value of the lymphatic sink pressure is -0.20 (-6 mmHg). Two othercases were performed with the lymphatic sink pressure reduced to -0.30 (-9 mmHg) and -0.010.02Da 0.030.04E-■ 0.050 0.060.070.08• 0.090.100.0^0.1^0.2^0.3^0.4^0.5^0.6 0.7 0.8 0.9 1.0• • .................................................................................(71apier 4 : EFFE(71V OF SYSTEM PARAMETERS ON MI(ROVAS( 7ULAR EX(1-IANGE^93Solute Concentration0.01 .^0.0270.0370.0470.05-0.06-0.07-0.08-0.09-0.100.0 0.1 0.2^r:0.1%^ 94^ I ....^i.:.% ^.......1  \ ..:**:4  ^I 40.3 ...4 0.5^0.6^...7^0.8^0.9 1.0Hydrostatic Pressure0.010.02:0.0370.04:0.0570.06:0.07:0.08:0.09:0.10 0.0 0.1 0.2^0.3^0.4 0.5^0.6^0.7^0.8 0.9^1.0-74^Peclet Number0.010.0270.03:0.04:0.05-(P-cnr)/P„tco^at^a^a^co^a^a^a -. cn an 1,- a ;.".;^o ..., •-,1 ...1n U:3^40^;27^471"^an goci^6 ci c; ci ci^ci^..--t i i^i^i^i 1^ I (^• •-1^i 411 4--4--.^4-, . 4 4-, i^I^t^%^t ^ I ^I ^0.2 0.3 0A 0.5^0.6^0.7^0.8^0.90.06-0.07-0.08-0.09-0.100.0 1.0Velocity Vector Field0.8^0.9^1.00.01 . 0.0270.0370.04:0.05:0.0670.0770.087,0.09-0.101*I A1 AWiii-kkAkk‘Ax%AAAAN••••■■••■•■■••■••■•■•••■■■••■■A A 1N -A■1■1.‘‘^1 I1 1 I I 1^AAAAAAAANNA1 I A 1 AAA 1 1 A I A A 1 I A 0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)Figure 4.12: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the lymphatic sinkpressure, PL *, lowered to -0.4 (-12 mmHg) at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^940.40 (-12 mmHg). The latter case represents an extremely low lymphatic sink pressure andis presented in Figure (4.12).Decreasing the lymphatic sink pressure increases lymph drainage (see equation(3.20)). Consequently, with more negative lymphatic sink pressures, the average soluteconcentrations in the tissue are slightly lower than the base case. This is especiallynoticeable in the tissue away from the capillary (r* > —0.03). In addition, the range in thesolute concentrations are reduced slightly. The change in the solute concentration fieldcauses a drop in both the hydrostatic pressure distribution and its range. The averagepressure drops from -0.0255 (-0.7650 mmHg) for the base case to -0.1800 (-5.4000 mmHg)for a lymphatic sink pressure of -0.40 (-12 mmHg). As expected, the potential distributionis shifted in the negative direction and its range decreased as PL is reduced. The velocityfield is virtually unchanged in most of the tissue although fluid velocities are greater thanthose of the base case at the venular end of the capillary. This is a result of the increasedlymph drainage. The lymph flow rises to 7.14 1/day for PL * = -0.40 (-12 mmHg). Theincreased lymph flow leads to lower solute concentrations within the tissue space. Thedimensionless average solute concentration is equal to 0.4901 for Pi, * = -0.40 (-12 mmHg).Lymphatic Sink Radial PositionThe position of the sink was varied in two other cases. These two cases place thesink at the radial positions r* = 0.025 and r* = 0.055 (half of tissue envelope thickness).The results for the former case are shown in Figure (4.13). The sink is always maintainedas parallel to the capillary. When the sink is repositioned to a reduced radius, the volumeof the sink is lowered because the lymphatic sink is represented by a concentric shell of thesmae radial thickness about the capillary. To provide a valid comparison, either the volumeof the sink or the sink strength has to be enlarged. Here, the sink strength will be increasedto reflect the reduction in sink volume with lower radii. The volume of theHydrostatic Pressure. ...........................................................^0.0000..............................................-0.0500...............................0.05000.0^0.1^0.2^0.3^0.4^0.6^0.6^0.7^0.8^0.9^1.00.01.0.02:0.03-0.04-0.05-0.08-0.07-0.08.0.09-0.10(P-o•n)/Part11110'.^1 ^0.4 0.5^0.60.01 .0.02:0.03:0.04:0.05-0.060.070.06-009.:0.100.0^0.1^0.2^0.91cc0cs), 0.8^0.90.7 1.0(lzapter 4 : EFFE(IS OE SYSTEM PARAMETERS ON MI(ROVASCULAR EX( RANGE^950.020.03:.0.01^Solute Concentration. ............... ...................................................549 ................. .............0.04 7ira9o.499s- o- ... .....o.o6-0.135 ;^..... ..... ^0.1999^....0.09-o.os-0.07-.10    i     u  0.0^0.1 0.2 0.3^0.4^0.5^0.6^0.7^0.8^ ........................................................................0. 01.......... ..0 . .4_,,-7:-.......:••••■^0.04 ........ ...^.9 \ -.^.• 0^..........^V^..-"'co V.E..' 0.05 .^-.. -.90N^-.^0 : a •^49° -* .........3 0.06: 44^:-. 0 0 ct; 94. /0.07:^•^6' c* *.^ :••• 0.08: .•.O 0.09-O 0.10 ^.....+4^0.0 8.1^ I 1.1•..--r-w• ,1-11-rn -1-1-1-1-T -, II-. 'VT- -,-11 .-r,--.--'7-11-T-T-11 ^I,^.--11 • -1 w in 1 I w ■ wy 1 , ^0.2^0.3^0.4^0.5^0.6^0.7 0.8^0.90•••■0.9^1.0Peclet Number1.00.010.02-0.04:0.05:0.06:0.07:0.08-0.09-0.100.0Velocity Vector Fielditilliniiiit I 1 1 AIAIAIANAAA A NN A A AN A ■ % .i. A A A A1 '1 A AAAANNANAANNNNA.A.NN •■•■••■•■■••■•••■■■•■•■■■■ '1/4 A A^ ' Ii• / 1 1 I 1 1^ 1114■ i .1 1 11^ ../ .., de 1 I^ ^iI^ ...^ • • . ..... .,,--,, .---.--.-.-r-.-,,r,.--,,--.-,-,-,--,--,-^. .•. 0.1^0.2 0.3^0.4^0.5^0.6^0.7^0.8^0.9 1.0Distance down Capillary (dim)1 1 1 1 1 AAANANNN1 IAAAAAN ^1 1 AA ^I AI'' Figure 4.13: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the lymphatic sinkrepositioned at r* = 0.025 at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICI?OVASCULAI? EXCHANGE^96sink when positioned at r* = 0.025 is 3.55 times less than the volume of the sink at theouter tissue envelope. Therefore the sink located at r* = 0.025, the value of LS is increasedby the factor 3.55. Similarly, the sink strength for the sink located at r* = 0.055 wasincreased by the factor 1.70.From the results, it is seen that the solute concentration and hydrostatic pressuredistributions are not affected significantly by the movement of the sink. The soluteconcentrations lower as the sink moves closer to the capillary. The radial gradients aregreater near the venular end of the tissue and increase near the capillary as the sink movescloser to the capillary. Despite the presence of the sink, the solute and fluid can 'see' theremainder of the tissue beyond and can pass into and out of these regions. This explainswhy there is solute on the other side of the sink into the tissue for the two repositioned sinkcases. The solute gradients in these regions are far less than the gradients that existbetween the capillary and sink. The average solute concentrations are 0.5006 and 0.5167for the r* = 0.025 and r* = 0.055 positioned sinks respectively.The hydrostatic pressure distributions are very similar to the solute concentrationdistributions for both cases. The gradients beyond the sink diminish as the sink is movedtowards the capillary. The dimensionless average tissue hydrostatic pressures for the r* =0.025 and r* = 0.055 positioned sinks are -0.0761 (-2.2833 mmHg) and -0.0322 (-0.9660mmHg) respectively. The lymph flows increase as the sink is moved towards the capillaryto 18.38 and 9.69 1/day for the r* = 0.025 and r* = 0.055 positioned sinks respectively. It isimportant to bear in mind that the sink strengths have been increased for each case due tothe sink geometry. This the reason the average tissue pressures for these cases are less thanthe base case value and yet provide more lymph flow. With increased sink strengths, thetissue beyond the sink would contain less and less solute (and lower gradients) and thehydrostatic pressure would fall to that slightly above the the lymphatic sink pressure (alsowith lower gradients). With a reduction of the sink strength, there is greater solute storedin the outer tissue space and the hydrostatic pressure increases.Chapter 4 : EFFE(TS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^97From Figure (4.13), as the sink becomes closer to the capillary, the potentialdistribution shifts in the negative direction. The range of the potential distribution issimilar to all the previous runs. There is a dip in the potential along the sink whichbecomes more pronounced at the venular end of the tissue. As can be seen from the fluidvelocity plots, the sink at r* = 0.025 affects the flow patterns. Fluid enters the sink fromthe capillary-side and from tissue-side of the lymphatic sink. The radial component of thefluid velocities increases in the neighborhood of the sink. This is indicated indirectly in theplot of the Peclet numbers. There is a local ridge of higher Peclet numbers in the vicinityof the sink reflecting the greater convective transport at the sink.The tissue space beyond the lymphatic sink stores solute. This means that thereexists the possibility to access this solute if there is a sudden shortage or that solute will stillbe drained via the sink even if there were none in the plasma. This would occur until thestored solute were depleted. In this manner, the tissue beyond the sink acts as acapacitance.Capillary Membrane Filtration Coefficient, L,pThe effect of the capillary membrane filtration coefficient, Lp, on the system wasexamined from the base case and two other cases where it was varied as half the base valueand twice the base value (Li,  0.5 and Li, x 2 respectively). A reduction in the capillarymembrane filtration coefficient increases the fluid flow resistance of the capillarymembrane. This means that the flow rate decreases for a given potential drop across themembrane. The results for the increased and decreased filtration coefficients (L x 2 andLP x 0.5) are shown in Figures (4.14) and (4.15) respectively.The range in the solute concentration and gradients increase as the capillarymembrane filtration coefficient is increased (or capillary fluid flow resistance is reduced).The average solute concentration decreases as Li) increases (0.4311 for Li) x 2 and 0.6053Chapter 4 : EFFE(:TS OF SYSTEM PARAMETERS ON MICROVASCULAR EX(11ANGE^98for Li, ^0.5). The reduction in Li) reduces the fluid velocities across the capillarymembrane. The reduction in Li,  a factor of two reduces the transcapillary flowvelocities by roughly a factor of two. This means that the fluid velocities in the tissue willalso be reduced and thus diffusion will play a relatively greater role for solute transport inthe tissue. This explains the increased average tissue solute concentration for the case Lp x0.5.The hydrostatic pressure distributions are very similar to the solute concentrationdistributions. As a consequence of the ranges in the solute concentration distributions, atincreased Lp, the hydrostatic pressure range increases and high gradients occur, particularlyat the venular end of the tissue near the capillary. A reduction in Lp also lowers theaverage tissue hydrostatic pressure. The average tissue hydrostatic pressures are -0.0205 (-0.6150 mmHg) and -0.0262 (-0.7860 mmHg) for the cases Lp x 2 and Lp x 0.5 respectively.This results in the following lymph flow rates : 5.76 l/day for Lp x 2 and 5.37 I/day for Lp x0.5. The reduced solute removal by the lymphatic sink also explains the higher soluteconcentrations in the tissue for lowered values of Lp .With enlarged LP' the potential gradients increase and the range in the tissue risesand is shifted slightly in the positive direction. This is expected since the decrease in thecapillary membrane filtration resistance will increase the potential drop throughout thetissue space relative to that in the capillary wall. In other words, the potential distributionwill be less uniform. For reduced Li) , at the arteriolar end of the tissue, the velocities acrossthe membrane are significantly lower than those for the base case. The opposite is true atthe extreme venular end of the capillary where flows are slightly greater than the base case.An increase in Lp changes the flow structure significantly. As shown for the case withincreased Lp, reabsorption back into the capillary starts to occur. The zero point occurs atabout z* = 0.92. This is expected since with increased Lp more fluid can enter the system.The transcapillary flow velocities are much greater in the arteriolar and central portions ofthe capillary. At the venular end of the tissue, the reabsorption is apparent. The strength ofHydrostatic Pressure• • ......4.6^0.6 0.7r,0.8^0.9^1.00.010.02.0.03:0.04:0.05:0.08:0.07-0.08-0.09-0.10 0.0 0.1^0.2^0.3, 0p00.00000.4(71apier 4 : EFFE('IS OF SYSTEM PARAMETERS ON MI(R(JVAS(111,A1? EX(71ANGE^99Solute Concentration0.01 .0.020.03-0.04-0.05-0.08-0.07-0.08-0.09-0.100.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^ 1.0Eaco0.02 ^.(0 0.03-0-01. - .................. . ................................... ..................................^'---.......^.. .-^ //9.9? 20^...•'.^1 :.--(E' 0.04- .2.4,O0.05:"43,9 6^Be 0,0.06-^.°4908 40-.7tO..-4 0.077 - -.a^--• coio^,:jco....O0.08:^ ••. Co-^0O 0.09:CO• 0.10 ^ i .... l'. . .. i .  ^ ••• . . , .-.. .^. . . . 1 . . . .1 . .^ ...-/-. . ,--. . .--.-. . 1-,-. . . .-. . . . i . . . .. .-. . . , ^CS^ i ^4., 0.0 0.1 0.2^0.3^0.4^0.5^0.6^0.7 0.8 0.9^1.0Peclet Number0.01 ^0.02.0.03:0.04:0.05-0.06-0.07:0.08:0.09:0.10 ^ ,0.0 0.1^0.▪ 20.01^0.02.0.03:0.04:0.05:0.08:0.07:0.08-0.09-0.10 •^i0.0^0.1 0.2(P-aiT)/P,„/^I^ia r1 D.CO 0^0C■7^tl 0O c; c;t t^i^L "-- r,^J  , ^I 0.3^0.4 0.5 0.6Velocity Vector Field\-‘, ^0.3^0.4^0.5 0.6^0.7^0.8^0.9Distance down Capillary (dim)co^ co40••-•0.7^0.8^0.9M'o;1 1 1 1 1^AAAAAAA ^1 1 1^AAAAAA ^1 1 1 A 1 1 A 1 1 A ^1 A 1.01.0Figure 4.14: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the capillarymembrane filtration coefficient, Up, increased by a factor of two from the base value atsteady-state.V 0.01 0 0.02--•CO^0 .03 :-...^---- ._^--- .^**---._•ceE--1^004- - -.... <9 ....‘ -.. 0 \•-•.05-  0_0 0.06:..9‘O . AS^-^0^--^fpO 0.07:^r I?"^0O 0.08: ..--0 0.097'O 0.10 -010...Peclet Number0.9^0.4^0.5^0.6^0.7.........................................................0.0^0.1^0.2 0.8^0.9^1.00.010.02:0.03:0.04-.0.05 -0.06:0.07-0.06-0.09-0.100.0Solute Concentration.............0-0?0.9,^0.2^0.9^0.4^0.5 0.6 0.70.1Hydrostatic Pressure0.0^0.1^0.2^0.9^0.4^0.5^0.50.01 .0.02-0.03-0.04-0.05-0.06:0.07:0.08:0.09:0.100.7 0.8 0.0 1.0COi^i^I^I^I ^I(1 •^% ^. ^  0.2 0.9 0.4^0.5^0.6 0.7(7hupter 4 : EFFE( . 1S OF SYSTEM PARAMETERS ON MICROVAMTULAR EXCHANGE^1 000.9Velocity Vector Field^  A A -A A • lk^♦^AAA ANANAAAAAA 41/4 AAANA■ 411 1 1 1 1 A AA AAAAAAAAAAAAAAA .........................1 t 1 1 AAA ^1 1 1 AA ♦ 11 1 AN ^A ♦ A1 A A♦ ♦ANA^ A A A A AA A^ A 1 ♦ I0.01 ^0.02.0.03-0.04:0.05:0.06:0.07-.0.08-0.09:0.10^•i0.0^0.1(P-air)/P07^:: i 1^1^I^t■ 03^ 0) 0) 03CO 0) CO inN.,^co COI o a)..... v.^v v.ci O 6 160.80.010.02-0.03-0.04:0_05:0.06-0.07-0.08-0.09-0.101.00.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)0.8^0.9^1.0Figure 4.15: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the capillarymembrane filtration coefficient, Up, decreased by a factor of two from the base value atsteady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON M1CROVAS(7ULAR EX(711ANGE^101the sink is constant and so fluid starts to flow back into the capillary. This shows that withthe increased L11,  fluid is entering the system to allow a large amount of lymphdrainage (at fixed sink strength) and reabsorption. The fluid velocity magnitudes areindirectly indicated in the Peclet number distribution. As expected, for increased Lp, thefluid velocities are greater throughout the tissue and at the capillary wall.In summary, if Lp is increased, the fluid velocities rise due to the reduction in theresistance to flow at the membrane. This is propagated throughout the tissue. An enlargedfiltration coefficient promotes convection throughout the tissue lowering the soluteconcentration distribution and leading to increased lymph flow. With reabsorption of fluidback into the capillary, there are higher than usual solute concentration gradients at thevenular end of the tissue. This is because convective transport is operating in the oppositedirection to diffusive transport in this region.Mechanical DispersionThe inclusion of dispersion in the problem simply increases the effective diffusioncoefficient. The results are displayed in Figure (4.16). This case involves an added degreeof complexity because the dispersion coefficient is a function of the local fluid velocitieswhich are in turn a function of the solute concentrations via the osmotic pressure term. Theincrease in the effective diffusion coefficient lead to slightly higher solute concentrationvalues and lower concentration gradients within the tissue. The inclusion of dispersionincreases the dimensionless average tissue solute concentration to 0.6842 from 0.5258 forthe base case (this is without dispersion). The solute concentration gradients are reducedwithin the tissue especially in the arteriolar end of the tissue. As a consequence of theincreased solute concentration distribution, the hydrostatic pressure distribution respondsand falls slightly. The average dimensionless tissue pressure is -0.0258 (-0.7740 mmHg)compared to the base case value of -0.0255 (-0.7650 mmHg). The lymph drainage rate('hapier 4 : EFFE('TS )1. - SYSTEM PARAMETERS ON M1(71?()VAS(11LAR EXCHANGE^102Solute Concentration0.01 .0.02:0.03:0.04:0.05:0.08:0.07:0.08:0.09:0.10 0.0^ 1^10.1^0.2 0.3 0.4^0.5^0.6^0.7Hydrostatic Pressure0.010.02:0.03:0.04:0.05:0.08:00.07- ..... ..... ..............................................................0.09:^0.10-rw^0.0 0.1^0.2^0.3^0.4^0.5^0.6-o,. 1 ..^..4^0.8 .^. 0.9 1.00.7E;.E.,. ........................................................................................_.E-.O0.04-19.99540.05- ................^19.9954.............................................................. • ' ^. ........ - • ................. :t4: ;. ^.19 -904.... ... ... .... ...93920.07: ...... .^.lir/..........O 0.09^.... -..^-0... ^.......^19.9954^.19-995 ---199-"-^S.^I‘ ..0.067- ^.... ._4a1^-.......o 0.08:...;^0.0^0.1^0.2^0.3^...• r.....----e .........O 0.10in0.4^0.5^0.6^0.7la-cz P-cr,WPart (: Iao^0^co^a)40^0O fa-0^./..to^...0 0.?to 1,0.06:^.-*.^0' 40 40•^.^4t30.077.. CY 0"^ci^6^.••^o'O. 0 9 701000 ^7, ^,..L : LI j____,.^I ^I •/ .0.0^0.1 0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9Velocity Vector FieldWW1 HIM11411111111MN ■fln ^ 1 1 I 1 11 1^1 1 1 1 1^1 1^1 Al 1^AAAAAAAAWAA.1•44‘4■AAAAA^1 1 11 1 t 1 1 1^1 '1^41 At^■■■■•■•■•■■■ •■••••■•■■ '4^11^1 11 1 1 1^1 1 41^AA^4•■•■■•• 4. 4. 4..4^4.4.NNN 4. 44^• 11 1 1^1 1^'1^‘A'1^••••■■•■•■• 4.■4.44■ 4.4.4.■■■^4. 4. 4 ■ 4.‘‘ 4^"1^I 11 1 1 1^ANAAAN■■■•■■•■■•■ ...............^■ ■■■^Ai 1 11 1 1 '1^■■•••■•■•■■••■■ ............... 4....■•■■• 4 4 '1^11 44 •■■ •■•■•■■■ ..................... AA 11 1 1 AA  ''II^ ,•^• •--1^,.0.1^0.2 0.3 0.4^0.5^0.6^0.7Distance down Capillary (dim)Figure 4.16: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the inclusion ofmechanical dispersion at steady-state.Peclet Number1.01.00.01.0.02-0.03-0.04-0.05L.0.06i0.070.080.090.10 0.0 0.8^0.9^1.0Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON M1CROVASCULAR EXCHANGE^103increases as a result of the increased solute concentration to 7.70 1/day. The potentialdistribution is shifted in the negative direction due to the greater osmotic pressuresthroughout the tissue. The velocity field, although similar to the base case, has greatertranscapillary velocities than the base case in response to the enlarged lymph drainage. ThePeclet number distributions are difficult to compare since the effective diffusion-dispersioncoefficient is a function of the local fluid velocities and is therefore a function of position.It is apparent, however, that convection plays a reduced role in solute transport from thePeclet number distribution. This is especially so at the arteriolar end of the tissue near thecapillary where the highest fluid velocities are found.4.5.3 Variable Capillary Membrane Filtration Coefficient, Liz)In this section, the capillary membrane filtration coefficient is considered to be aknown function of the length down the capillary. It is known that the capillary fluidconductance increases towards the venular end of the capillary network. This allows formore fluid and solute interchange between the circulation and the interstitial space atlocations where the filtration coefficient is high.As a first approximation, a linear function is assumed between values specified atthe arteriolar and venular ends of the capillary. The value at the arteriolar end is chosen tobe the base value given in Table (4.1). The value at the venular end is taken to be twice thebase value. This value is arbitrary but indicates general trends for linearly increasingcapillary filtration coefficient. Since Lp increases towards the venular end of the capillary,the fluid flow resistance decreases along the capillary length.The linear function for LI) leads primarily to greater fluid velocities across thecapillary membrane along its length. The results are presented in Figure (4.17). Thispromotes more lymph drainage in the system (5.64 1/day). The increased fluid velocities0.01 .0.02:0.03:0.04:0.05-0.06-0.07-0.08--•-•-v-vv-l-ITITTY0.1^0.2^0.3^0.4^0.5^0.6^0.7 0.8^0.9^1.00.09-0.10 0.0■0.2 0.3^0.40.01 .0.02:0.03:0.04-,0.05:0.08:0.07:0.08:0.09:0.10 0.0^0.11-• 1-7-1 ^0.5^0.60.01 .^0.02:0.03:0.04:0.05-0.06-0.07-0.08-0.09-0.i0^(P-a7r)/P„,.......... "7 lCb^71^t■ Cbc0 05Ch 03CS Ch^I ^ 610.01 . 0.02:0.03:0.04:0.05:0.08:0.07:0.08:0.09-0.100.01 1 A1 1 ^1 A ^111 1 IAA%1 1 AAAA ^I^1^1 1 1 1 1 1 1 11 1 H^,3/4^■ ‘^A‘IT 1 1 1 A 1 1 AAAAAAAAAANANNNNNNNNI 11 1 1 IANAA1 ^0.1^0.2•0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)IkAAAAAANA-A-xl--V-A-A-4r-N-1,-*-1-1ANNAN 4,NN ..........N N ................0.8 0.9A AN1.0'1=1('hapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MAROVAS(VLAR EX(.71ANGE^104Solute ConcentrationHydrostatic Pressure.................................................COa;Peclet Number0.01^ 0.02  ''^...................................................0....CO0.04- ............•--..^ -..E-■.....0.05-^--..^ 9e.9-.9 ..^(5,^...-.'904, • e•3 0 .06 _ Z D 9I nO - 5^,1 ,0.07: •..... • .^ 0.08:--..^...^ :C)^0.09;^'•. .0, 0.10   %•..^.^..... .•^.....^...,•..0.0^0.1 . 0.2^0.3 -......0r4^r.5. .•--.---r^-,--r-r4"7"--,--,--1.6 ^0.7^0.84:1GO.-■0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Velocity Vector FieldFigure 4.17: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the capillarymembrane filtration coefficient, Lp , varied linearly from the base value at the arteriolar endto twice the base value at the venular end of the capillary at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^105are localized to near the capillary and the central portions of the tissue. At the extremearteriolar and central portions of the tissue near the lymphatic sink, the flow velocities arelargely unchanged. Although the increased flows through the capillary wall lead to highertransport of seived solutes through the membrane, the increased lymph drainage removessolute from the system. The net effect is a lowering of solute concentrations in the tissue.This is particularly noticeable in the arteriolar end of the tissue (z* < 0.5) where increasedsolute washout drags more solute from the tissue space to the lymphatics where it isremoved from the system. Although the average hydrostatic pressure increases slightly,more of the tissue is subatmospheric due to the lower solute concentrations in the tissue.The average tissue concentration and tissue hydrostatic pressure are 0.5029 and -0.0250 (-0.7500 mmHg) respectively. The dimensionless average tissue hydrostatic pressure overthe region of the lymphatic sink is equal to -0.0356 (-1.068 mmHg). The averagehydrostatic pressure at the sink for the base case is equal to -0.0405 (-1.2100 mmHg). Themore positive value for this case explains why the lymph drainage (5.64 1/day) is greaterthan the base case value (5.47 1/day). As expected, the lymph flow rate for this case liesbetween the base case and the case Li) x 2 values.The range in the potential is largely unchanged but is shifted in the positivedirection. This is due to the drop in the solute concentration distribution which leads tolower colloid osmotic pressures in the tissue.In addition, the filtration coefficient along the capillary was also varied as a stepfunction (at z * = 0.5). It was found that the discontinuity occurring at the step causedconsiderable numerical difficulties. The following functional form solved this problem :L^L p( Z . ) = p ,an ( P •V  ^p^I ±^M(Z -11.$1 (4.7)Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON M1CROVASCULAR EXCHANGE^106Lp a ^00p,ven m = 10m = 50m = 100••1I.0.2^0.4^0.6^0.8Distance down Capillary (dim)10Figure 4.18: Step function for variable membrane filtration coefficient.where m governs the quality of the step and varies typically between 10 and 100. Thisproduces an effective step function at z* = 0.5 without the discontinuity and is displayed inFigure (4.18). The arteriolar capillary filtration coefficient, Lp,art, was set to the base casevalue while the venular end value, Lp yen, was set to twice the base case value. The valueof the parameter m was set to 50.The results, shown in Figure (4.19), were found to be very similar to those foundfor the linear case described earlier. Here, the sudden increase in the filtration coefficientresulted in a region of lower solute concentrations in the region z* = 0.53 to e = 0.64. Thisis because of the washout of solute occurring here due to the higher fluid velocities acrossthe membrane. The cushioned step function that exists between z* = 0.47 and z* = 0.53produces a local maximum in the solute concentration at roughly z* = 0.50. As can be seenin the velocity vector plot, this corresponds to a minimum in the transcapillary fluidvelocity across the membrane. Thus the minimal amount of washout of the solute from themembrane occurs here. The reduction of the membrane flow resistance leads to increasedfluid flow across the capillary wall after e = 0.53. This promotes more washout of solute0 0.09-0 0.10 .. ^ci 0.0 0.1^0.21.00.90.80.1^0.2^0.3^0.4^0.5^0.6^0.01 ^0.0270.03:0.04:0.05-0.06 -0.07-0.08-0.09-0.10 0.0‘. ,---- • , • • ,^•0.7.21:1^03-070/P,„0.01 ^0.02.0.03: i7^70.047^ : :0.05: ca^co N.^a ^aco co0.06 a7 a) co co co0.077^ co co^ co a0.10 : /^.. (^I^( ^ci^cie^O0.08- / / / I1 ^ 10.09-...„---, .------..-•-• , •-• .-----,^,0.0^0.1 0.2^0.9 0.4^0.5^0.6^0.7^0.8 0.9cb/1.0( luipter 4 EFFE(IS OF SYSTEM l'ARAME1ERS ON MICROVASCULAR EX(7-1ANGE^107Solute Concentration0.1 0.2^0.3^0.4Peclet Number0.06:-0 0.07••-,0.08- •Hydrostatic Pressure0.010.02.70.03:0.047;0.0570.06:0.07:0.08:0.09:0.10 .^ ^ ,•0.0zE01'^0°- $3°21 - .).  ......................................y^0.03- ^..-1, ..^...v.-.. .fr.-,-- --,•••• ... .......712. 0 oe\9^- . .. ------------***. i ''' '''.‘::.:"-----":7..................... ..^.....................................2^0.04---- ..........0.3^0.4^0.5^0.6^0.7^0.8Velocity Vector Field0.010.02-0.03:0.04:0.05:0.08:0.07:0.08-0.09:0.10itIIII 1111;;;;;;;;;;;; ; ;;;13/41;11;;;;;^A A A1 1 II A A 1 AAAAAAAAAAAAAAAAAAA AAAAAAAAAA •••••••••••1 1 1 A A AAAAANA ••••••••••• AAAAAAAANAA ....... AAAAAA1 1 1 A A AAAANAAAAA ......... AAAAAAA ..................A IS.S.A I♦ A '1• •-•-•-•-•••••-•-•T0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Distance down Capillary (dim)Figure 4.19: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions with a step change in the capillarymembrane filtration coefficient, Lp, at z* = 0.50 from the base value at the arteriolar end totwice the base value at the venular end of the capillary at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICRO VASCULAR EXCHANGE^108from the capillary membrane and a minimum in the solute concentration at this point. Thiscan be seen as a local solute concentration minimum at z* 0.56. In the region z* > 0.53,the transcapillary fluid velocities decrease further down the capillary length resulting in lesssolute washout from the capillary. This produces a maximum in the solute concentration atthe venular end of the capillary.The lymph drainage rises to 5.71 1/day. The hydrostatic pressure distribution is, asexpected, similar to the solute concentration distribution. At z* = 0.50, there is a localmaximum in the hydrostatic pressure. At z* = 0.53, just after the step change in thefiltration coefficient, there is a minimum in the hydrostatic pressure. These correspond tothe maximum and minimum found in the solute concentration distribution. The averagetissue solute concentration and hydrostatic pressure are 0.5161 and -0.0256 (-0.7680mmHg) respectively. As with the case above, the increased lymph drainage rate is due tothe relatively more positive dimensionless hydrostatic pressures in the lymphatic sinkregion (-0.0334) when compared to that of the base case (-0.0405). The increased flowvelocities after the step change in Lp are reflected in the Peclet number distribution. Thepotential distribution rises due to the drop in the solute concentration distribution.4.5.4 High Flow ChannelsThe high flow channels are implemented within the tissue as an increase of thetissue hydraulic conductivities in specified regions of the tissue space. This means that thefluid flow resistance is reduced in these regions. The spatial heterogeneity of the hydraulicconductivity requires that the fluid-pressure equation (3.6) be expanded as :Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^109K^P,d2( — (Yi n( ) K„. ^(7,70 + K a2 (Pi afzi +n.dr2^r^dr^g^dz2aK,r a(P, - cr,r, ) dIC,,  01 - Crt 71-1) Q(pi) = 0dr^dr^dz^az(4.8)The two new terms thus created contain the gradients in the hydraulic conductivities andcan be treated in the same way as the convective terms in the solute transport equation.When the hydraulic conductivity is constant throughout the tissue, these two extra termsreduce to zero.Two different physical situations were simulated. The first had the high flowchannel perpendicular to the capillary membrane centred at zs = 0.5. The dimensionlessthickness of the single high flow channel is approximately 0.10 or 10% of the length of thecapillary. This may be visualized as a cylindrical fin (of higher flow conductivity) aboutthe capillary. The second case is similar with two channels centred at z" = 0.3 and z* = 0.7.Each of the high flow channels is taken to be approximately 0.05 dimensionless lengthunits thick or 5% of the capillary length. For both cases, the hydraulic conductivity in thehigh flow channels are set to twice the base case value.As seen from earlier results, the primary location of the fluid flow resistance is thecapillary membrane. The tissue hydraulic resistance is small compared to that across thecapillary wall. This leads to the high potential drop across the membrane relative to thatacross the tissue. To accomodate higher velocities across the capillary membrane, thefiltration coefficient was doubled where the high flow channel met the capillary membrane.The tissue hydraulic conductivity was increased by a factor of two in the high flowchannels. This is sufficient to indicate the general trends. It is important to note that thehigh flow channels were not considered to be free flow channels but simply rather asregions with increased tissue conductivity. This means that the potential still acts as thedriving force for fluid flow in the high flow channels. If the high flow channels are0.010.020.03-0.04-0.050.06-0.07-0.08-0.09-0.100.0 0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8 0.9 1.0Hydrostatic Pressure0.010.02=0.03-^\0.04-0.05-0.06- 00.07-0.08-0.09-0.10  0.0^0.1v 0O08000.7^O.B^0.9^X1.0:.^•••0.000r0.2^0.3^0.4^0.5^0.60.01 ^0.02-0.03-0.04 -0.05-0.06-0.07-0.08-0.09-0.100.0 0.1 0.2[f l i t^ O^CI ^ I, ^0,.3....t....,.1^J0.5^I ..L ^ ,...^.1.^ 4 0.6^0.7(P-a7r)/P1zt0.8I.^I0.9 1.0('haptr'r 4 : EFh'E( '7:1' OF SYS'I EM I'AK4afETEKS ON MN R(1 VASCULAR EXCHANGE^110Solute ConcentrationPeclet Numberm 0.01^-.-   ^• ^ _^,v,. .........0.02- _..-- ••_.-•^ •:_____::a^ 0.03  h 0.04-' ._..........^.^7EH 0.05.1^o^ m.Ua 0.08-^gg^•^'• • :^t^_ -C 0.07-^4,^ce - =--- ch •• ".^........ ^^ .z^ 59 9882^•--. ^^.-^^0.10^.....I^__Ty,^0.0 0.1 0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9hAVelocity Vector Fieldilll^l^ti'''.'' .'^^^^^^,^^^^t 1 1 t t \ \ \ \ \ \ t ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ,'1 \ \ ^I \ I' '4. ^I^ I^ , ..--.r-...ice I ^ j... ^ ,^ I .^ .--. --r..T^^ -T^-r ------------- j - •--•--r.....0.0^0.1 0.2 0.3^0.4 0.5 0.6 0.7 0.8 0.9^1.0Distance down Capillary (dim)Figure 4.20: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with a single high flowchannel centred at z` = 0.50 at steady-state.1.00.01 0.020.03-0.04-0.05-0.06-0.07-0.08-0.09-0.10Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^111considered to be true free flow channels then the driving force for fluid motion would bethe hydrostatic pressure. The lymphatic sink region is also considered to be a high flowchannel. This aids in the distribution of fluid to the lymphatic sink.As expected, the increased conductivities in the high flow regions lead to higherfluid flow velocities in the high conductivity regions. This promotes greater soluteconvection in these regions. With the increased flow velocities, there is greater solutewashout from the capillary membrane into the tissue. This is seen in the results for the casewith the single high flow region (Figure (4.20)) where the concentrations drop near thecapillary membrane in the high conductivity region. The dimensionless average tissuesolute concentration is equal to 0.5307. The solute leaking through the membrane issufficient to produce a local maximum in the solute concentration in the area of z* 0.40 atthe capillary wall. Here, fluid velocities are still limited by the base values of the capillaryfiltration coefficient and hydraulic conductivity. The hydrostatic pressure distributionclearly responds to the solute concentration which has a similar distribution. Thedimensionless average tissue hydrostatic pressure rises to 0.0068 (0.2040 mmHg) and0.0128 (0.3840 mmHg) for the single and double high flow channel cases respectively.The lymph flow, as expected, increases with the presence of high flow channels. Thelymph flows are 6.57 and 6.78 1/day for the single and double high flow channelsrespectively. The potential distribution is largely unaffected by the presence of the highflow channels but does have larger radial gradients in the high flow channel as indicated bythe velocity field. The magnitudes of the fluid velocities are greater in the high flowchannel. This is also indicated in the Peclet number distribution where the higher Pecletnumbers are in the high flow channel areas and the lymphatic sink. The results from thesingle high flow channel are shown in Figure (4.20).For the case with two high flow channels, both have fluid propelled towards thelymphatic sink. The results are displayed in Figure (4.21). The trends indicated by the twochannel case are very similar to that of the single channel case.0.010.02.0.03:0.040.05-0.060.07-0.08-0.09-0.10Chapter 4 : EFFE(IS OF SYS2EM PARAMETERS ON M1(ROVASCULA1? EX(7IANGE^112Solute Concentration0.01 0.02.0.03:0.04:0.05:0.06-0.07-0.08:0.09-0.10(P-a7T)/P.rt0.0 1.00.010.02-0.03-0.04-0.05-0.06-0.07:-0.08-0.09-_0.10N.'s)•, .....^,  ^I0.2^0-3^0.4^0.5^0.6^0.7^0.8^0.90.0^0.1 1.0Hydrostatic PressureEPeclet Number..^ 0.2........ 0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0-E-■^0.047. ......r,,0.06-^. ----.^r•°0 0.097CS, 0.0^0.1t   1 i  8O 0.077 ..O 0.087CO ............ -....-- N'003 0.03::^0.05- 8♦ co0.10 -^0.02-r- 1.'0.01 ^ -a^:^„ ............... . ... ..to^;13co (0 .r?. _.:L9 ..  ....• te_^... ....... ......................... ..:,,,r,i.....^---..... ..... .. .................^1.CD_aClecp ..........^ 1^/-*.1^I^1^a a a^ato t^ a -1co co co r.^co CO clO .6^6^Oi i i 1I 1 i 0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0:.90.01 0.02:0.03:0.04:0.05:0.06:0.070.080.090.10Velocity Vector Field4444I444M4;M,111111 11^ 1\‘‘AAAAAA ■■■■■■■■•■■ ...I I 1 I I A AAAAAAA‘. .4,‘■■■■■■•■■•■ ............IIIIA AAAAAAAA ^1111111A••■•■ 111 A ^I I A A■A■ ..... ^AA A• -A0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)0.8^0.9^1.0Figure 4.21: Dimensionless tissue solute concentrations, hydrostatic pressure, Pecletnumber, potential, and velocity vector distributions for the case with the double high flowchannels centred at z * = 0.30 and 0.70 at steady-state.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON 1111(ROVASCULAR EXCHANGE^1134.6 ConclusionsThe model developed in Chapter 3 was used for steady-state analysis of fluid andsolute transport in human tissue. The influence of the physiological parameters on themicrovascular exchange system were examined in detail. The results from the manysimulations are summarized in Table (4.3). Some conclusions based on the results aboveare summarized in the following.1. The numerical simulations indicate that both convection and diffusion contribute tosolute transport. Convection appears to be the dominant mode of solute transport.Solute distributions were considerably different for different values of the retardationfactor and diffusion coefficient.2. The osmotic pressure plays an important role in microvascular exchange. Thedifferences in solute concentrations throughout the tissue yield osmotic pressuregradients which serve to reduce fluid motion. The hydrostatic pressure responds to thesolute concentration distribution via the osmotic pressure. The heterogeneity of theosmotic pressure throughout the interstitium cannot be ignored and should be taken intoaccount in any models of microvascular exchange.3. The inclusion of the lymphatic sink provides the mechanism for negative pressures(below atmospheric) in the tissue. As the strength of the sink increases, the amount ofmaterial reabsorbed back into the capillary is reduced and the zero point shifts towardsthe venular end of the capillary. With a reduced sink strength, the amount ofreabsorption increases. At the limit when the sink strength is equal to zero, the amountof fluid filtered into the tissue equals that being reabsorbed.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICRO VASCULAR EXCHANGE^1144. Dispersion plays a secondary role as a solute transport mechanism. It acts to increasethe effective diffusion coefficient, thus lowering solute concentration gradients. Thenature of dispersion in the interstitial space is not well understood but has beendescribed by two parameters, the longitudinal and transverse dispersivities which aredependent on the porous medium grain size and fluid properties. It is unknown at whatscale the representative grain size should be chosen.5. Fluid pressure, solute concentration, and potential distributions are determined largelyby transport properties of the capillary membrane. The range in the potential has beenshown to be due to the relative fluid flow resistance of the capillary membrane to thatof the tissue. The primary fluid flow resistance is located in the capillary membrane.6. An increasing capillary membrane filtration coefficient along the length of the capillaryinduces greater fluid flow across the capillary membrane due to lower membrane flowresistance. This leads to increased lymph drainage and thus lower solute concentrationswithin the tissue. The hydrostatic pressure drops in response to the solute concentrationdistribution.7. The many results indicate that the microvascular exchange system produces complexfluid flow and solute distributions which are highly reliant on the values of thephysiological parameters.Chapter 4 : EFFECTS OF SYSTEM PARAMETERS ON MICROVASCULAR EXCHANGE^115Table 4.3 : Results from Various Cases.No. Case Average c, Average Pi Average c, Lymph Flow(1/day)1 base 0.5258 -0.0255 0.5134 5.472 vows 0.7462 -0.1014 0.7460 3.373 LS x 0.2 0.5653 0.0924 0.5555 1.916 LS'x 5 0.5065 -0.0575 0.4911 21.425 a = 0.1 0.7150 0.0428 0.7139 8.256 a = 0.5 0.5990 -0.0002 0.5922 6.5171_ a = 0.99 0.5056 -0.0328 0.4914 5.178 = 0.1 0.7132 0.0406 0.7123 8.199 = 0.5 0.6137 -0.0002 0.6088 6.6010 PS = 0.0 0.5258 -0.0255 0.5134 5.4711 IS x 100 0.5265 -0.0252 0.5141 5.4814 l'Sx 10000 0.7355 0.0521 0.7349 8.6015 at = 0.0 0.5492 -0.1162 0.5460 2.8716 at = 0.5 0.5346 -0.0710 0.5265 4.2515 Ddif•x 0.1 0.4120 -0.1185 0.3854 2.7516 Ddifrx 10 0.7151 0.0408 0.7148 8.2317 disp 0.6842 0.0258 0.6836 7.7018 K x 0.1 0.5403 -0.0268 0.5327 5.2719 K x 10 0.5232 -0.0253 0.5100 5.5120 P1 = -0.40 0.4901 -0.1800 0.4794 7.1421 P f = -0.30 0.5078 -0.1029 0.4961 6.3022 L„ x 0.5 0.6053 -0.0262 0.5992 5.3723 L„ x 2 0.4311 -0.0205 0.4081 5.7624 L„ to 2L„(linear)0.5029 -0.0250 0.4940 5.6425 L„ to 2L„(step)0.5161 -0.0256 0.5097 5.7126 K1 0.5307 0.0068 0.5186 6.5727 K2 0.5116 -0.0128 0.4998 6.7828 ms3 0.5006 -0.0761 0.5251 18.3829 ms6 0.5167 -0.0322 0.5163 9.69Chapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 116Chapter 5 : Capillary - Tissue Fluid Balance and the Effects ofPerturbations on Material Exchange5.1 IntroductionThe roles of the interstitial space, the lymphatic sink, and osmotic pressure have notbeen clearly identified in terms of the capillary-tissue fluid balance. By the capillary-tissuefluid balance, it is meant the regulation of the transcapillary fluid flows and lymph drainage.These flows affect maintenance of the interstitial space and plasma volumes.Starling (1896) postulated that transcapillary fluid exchange is the result ofdifferences in both the hydrostatic and osmotic pressure across the capillary membrane. Thecolloid osmotic pressure acts to oppose the hydrostatic pressure driving force for fluidmotion. The steady-state equilibrium of the Starling forces is thus dependent on the soluteconcentrations within the capillary and tissue via their colloid osmotic pressurecontributions. Landis (1927) experimentally confirmed Starling's hypothesis.The thrust of the work in this area is to isolate the mechanisms of regulation of thecapillary-tissue fluid balance. The microvascular exchange system is maintained in a stablestate through various feedback mechanisms. This chapter makes use of the model developedin Chapter 3 to examine capillary-tissue fluid balance and its regulation. The nature of theeffect the osmotic pressure, the capillary membrane, and the lymphatic sink has oncontrolling and maintaining fluid balance will be discussed.Chapter 5 :CAPILLARY - TLY.SlIE FLUID BALANCE^ 117The first section (5.2) will investigate fluid flow through the capillary membrane.The driving forces for fluid movement through the capillary wall will be identified anddisplayed for the base case and the case where the lymphatic sink conductivity, LS, has beenreduced by a factor of five (LS x 0.2). These two cases were also examined in Chapter 4.The second section, (5.3), will describe some transient cases that were performed in order toexamine the effect on fluid balance. The final section will briefly list some conclusionsbased on the findings.5.2 Transcapillary Fluid FlowStarling's hypothesis relates the amount of fluid flowing across the capillarymembrane to the driving forces and the membrane fluid conductivity. This is given by(Taylor and Townsley, 1987) :vp, = {/De — — a(re )] (5.1)where Lp is the capillary membrane filtration coefficient, Pc and P1 are the capillary andtissue hydrostatic pressure respectively, and n(. and 7rt are the capillary and tissue osmoticpressure respectively. The reflection coefficient, a, is given by :epctivecr=^Ar theoretical(5.2)The reflection coefficient relates the value of the theoretical (measured) osmotic pressuregradient to the effective gradient operating across the membrane. If the reflection coefficientChapter 5 :CAPILLARY- TISSUE FLUID BALANCE^ 118is equal to 1, the the driving force for fluid motion across the membrane is the transcapillarypotential given by :AT = (13; — re )—(P, —70^ (5.3)This means that there is no leakage of solute across the membrane, in other words, it is 100percent reflected from the membrane. The actual measured osmotic pressure then acts tooppose fluid motion. If there is leakage of solute across the membrane, then the osmoticgradient across the membrane is lowered. Therefore, only some fraction, a, of the measuredosmotic pressure gradient will operate across the membrane opposing fluid motion. Thisleads to the effective operating osmotic pressure gradient. If the reflection coefficient isequal to zero, then solute may pass unsieved through the membrane. This means therewould be no osmotic pressure gradient across the membrane and the hydrostatic pressurewould be the only driving force for fluid motion. Typically, in human capillaries, a rangesbetween 0.75 and 0.95 (Renkin, 1977). There are some cases where the reflectioncoefficient is nearly zero (the liver sinusoids) and those where it is nearly 1 (the blood-brainbarrier) (Ganong, 1989).The driving force for fluid motion in the tissue is the potential given by := P, -^ (5.4)where Pt and nt are the tissue hydrostatic and colloid osmotic pressure respectively. Thetissue reflection coefficient, a t , relates the fact that the tissue porous medium may beconsidered as a stack of membranes. In this sense, solute transport will be hindered incertain regions by the fibrous meshwork of biopolymers causing osmotic pressure gradientsthroughout the tissue. The solute can still leak throughout the tissue by the various transportmechanism, diffusion, dispersion, and convection. This means that a fraction, a t, of theChapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 119measured osmotic pressure gradient (the effective osmotic pressure gradient) will operate tooppose fluid motion.In past analyses, the driving force for fluid motion in the tissue has been thehydrostatic pressure only (Taylor and Townsley, 1987). The results from Taylor (1990) andthis work (see Chapter 4) have demonstrated that the colloid osmotic pressure is sufficientlylarge and spatially variable that it must be included in any analysis of microvascularexchange. It plays a significant role in material exchange throughout the tissue. Taylor andTownsley (1987) have used a simplistic Starling force analysis of flow across the capillarywall. This is displayed in Figure (5.1). Here the tissue hydrostatic pressure drops along thelength of the membrane and the colloid osmotic pressure is considered nearly constant alongthe capillary. The transcapillary potential difference is seen to be constant and positive (fluidfiltration) down the length of the capillary. From their analysis, they conclude that fluidreabsorption is a transient phenomenon which will not occur at steady-state. In this work, ithas been found that the hydrostatic pressure increases along the length of the capillary inresponse to the solute profile (via the osmotic pressure) along the membrane. The soluteconcentrations increase along the capillary membrane due to the lower washout that occurs atthe venular end of the tissue. The transcapillary potential difference does not remainconstant but is maximal at the arteriolar end of the capillary and decreases along thecapillary. The tissue-side potential decreases along the capillary. The hydrostatic pressures,osmotic pressures, and transcapillary potential difference are displayed in Figures (5.2) and(5.3) for the base and LS x 0.2 cases.10 -0.8--a 0.6-V2CU  0.4-vOLT.., 0.2-• — -0.0L-C) -0.2-Chapter 5 :CAPILLARY - TISSUE FLt III) BALANCE^ 12030 -25 -ea 20 -15-1 0 ---A— Capillary Hydrostatic Pressure—C—Tissue Hydrostatic Pressure—A— Capillary Osmotic Pressure—A— Tissue Osmotic Pressure- Transcapillary Potential Difference5-c/)V-5--10-0.0^0.2^0.4^0.6^0.8^1.0Length down Capillary (dim)Figure 5.1: Starling forces across capillary wall. Taken from Taylor and Townsley (1987).—v-- Capillary Hydrostatic Pressure—0— Tissue Hydrostatic Pressure- Capillary Osmotic Pressure- Tissue Osmotic Pressure—v— Tissue Potential—4— Transmembrane Potential Difference-0.4- V---V-V---1/^V^I^• I0.0^0.2^0.4^0.6 0.8^1.0Distance down Capillary (dim)Figure 5.2: Transcapillary fluid motion : Driving forces for base case.—A— Capillary Hydrostatic Pressure—0— Tissue Hydrostatic Pressure—A— Capillary Osmotic Pressure—a—Tissue Osmotic Pressure-V- Tissue PotentialTransmernbrane Potential Di (TerenceAA -A A •0^Chapter 5 :CAPILLARY- TISSUE FLUID BALANCE^ 1211 .00.8-,:$ 0.6r.)0.4`mac„ 0.20.0-0.20.0^0.2^0.4^(16^0.8^1.0Distance down Capillary (dim)Figure 5.3: Transcapillary fluid motion : Driving forces for LS x 0.2.As can be seen from the results, the transcapillary potential difference remains positive forthe base case confirming the results shown in Chapter 4 for no fluid reabsorption. Theresults in Figure (5.3) display clearly the fluid reabsorption occurring at the venular end ofthe capillary (after about z* = 0.92) where the transcapillary potential difference becomesnegative. Both of these results are steady-state results. The reabsorption of fluid back intothe capillary for the latter case conflicts with Taylor and Townsley's suggestion thatreabsorption occurs only transiently until the tissue forces equilibrate. The amount of fluidthat flows across the membrane, however, is very close to the lymph drainage rates. Thissuggests that the amount of reabsorption is expected to be far less than fluid filtration intothe tissue. This agrees well with results suggested by Intaglietta and Endrich (1979).Chapter 5 :(APILLARY - TISSUE FLUID BALANCE^ 1225.3 Capillary - Tissue Fluid Balance : Mechanisms of RegulationThe roles of material exchange across the capillary membrane, lymphatic drainage,and the osmotic pressure will be examined in this section. The goal here is to identifymechanisms of control observed from transient simulations of microvascular exchange usingthe model formulated in Chapter 3. These controlling features of the microvascularexchange system will be investigated by implementing several perturbations to the system.The first section will deal with a sudden drop in the tissue solute concentrations and itsimpact on the flow structure. The second will examine the effects of an elevated venouspressure on the system. The third section will examine the effects of a sudden drop orincrease in the capillary solute concentration on the system. The final section will brieflysummarize the results from two cases which examine the role of the osmotic pressure on thesystem. This is done by perturbing the system after it has reached steady-state (the basecase) with zero plasma solute concentration. Two simulations are performed. The firstincludes the effects of osmotic pressure throughout the system and the second does not.5.3.1 Perturbed Tissue Solute ConcentrationThe tissue solute concentration generates an osmotic pressure which opposes fluidreabsorption back into the capillary. If the solute concentration is high at a particularlocation in the tissue adjacent to the capillary membrane, then the corresponding osmoticpressure will reduce the local tissue potential. This will promote fluid filtration from thecapillary into the tissue. The effects a lowered tissue solute concentration will have on thesystem will be examined in this section.This condition was implemented in the model by dropping the capillary soluteconcentration but maintaining the capillary osmotic pressure at its normal values found fromChapter 5 :(APILLARY - TLVUE FLUID BALAME^ 123normal capillary solute concentrations. This will maintain the driving forces (colloidosmotic pressure) from the capillary side for fluid exchange but will lower the concentrationof solute in the tissue since there is less solute effectively passing through the membrane.This is purely a contrivance to achieve the desired conditions for lowered soluteconcentrations in the tissue.Figures (5.4)-(5.6) display the transient results for the case where the effectivedimensionless capillary solute concentration is equal to 0.10. The colloid osmotic pressurein the capillary is maintained at the value for a dimensionless solute concentration equal to1.0 (the base value). This is a reduction in the solute concentration by a factor of ten whichwill lead to lower the solute concentrations within the tissue. This condition is implementedafter the sytem has reached the base case steady-state conditions presented in Chapter 4.As can be seen from the results, the drop in the effective solute concentration actingacross the capillary membrane results in lower tissue solute concentrations. The solute isstill washed out of the regions near the capillary to the sink and builds at the sink initially(see t = 1800 s). After one hour the solute concentrations are beginning to equilibrate withinthe tissue and the solute build-up at the sink is far reduced. At the new steady-state, thetissue solute distribution is lower (average concentration is equal to 0.0620) and there is abuild-up of solute near the venular capillary wall. It is important to bear in mind that theosmotic pressures in the capillary have not been changed to correspond with the effectivesolute concentrations transporting across the membrane.The transient behaviour of the hydrostatic pressure is similar to the soluteconcentration distributions. This is because the hydrostatic pressure responds to the soluteconcentration via the colloid osmotic pressure. The hydrostatic pressures become negative(subatmospheric) throughout the tissue with the maximum values at the venular end of thecapillary. These shift from the lymphatic sink region early in the transient response to nearthe capillary at steady-state. The average tissue hydrostatic pressure at the new steady-stateis -0.1288 (-3.864 mmHg).('hapter 5 :(APILLARY - TLS:SVE FLUID BALANCE0.010.02-0.03-0.040.05-0.08:0.07-0.08:0.09-0.100.0124Hydrostatic Pressure.............................................................................................................................-0.1350^-01.350-------0.1350..................... ............ - .......................... .......... ; ....... ....-:::: ........... .. .. ....................................0NI' ..^( .  ,^.-'^■ A,0.7^0.8^0.9^1.00.010.02-0.03=0.040.05.=0.06-0.070.08-0.09-0.10o o 0.1•0.2 0.3 0.4 0.5 0.6Peclet Number0.01061^0•^..0  32 ...............................................................................................COf•-■.....^0.04 ................^--....0.05 ..^I,g^• . R*9o 8g.."E 0.060.07^•^5'^a^..^1' .^I^-..a 0.08el .. : ... ..,.....: .... ...2 _0 0.09• 0.10 ......^0.0^0.1^0.2a.....Q'Po.D ed.0.3 0.4 0.5^0.6aoctayr- "^• • • • • ..0.7 0.8 0.9^1.0,(P-07)/parti i0.010.03^ ..:0.05N.0.020.04^49-O- a0.06 0^0^0^0^0^00.07^ -.co^Z aa a et. a0.08 a 6^--'^-40 .„ 00.096 0/ a^c; 0.10 ^I 6jI0.0^0.1^ji,_L^60.2^ - y -•-•. 1  ^i0.3^0.4 0.5^0.6^0.7^0.8^0.9Velocity Vector Fie ld........ • i i 1 jI0.010.02-0.03-0.04-0.05-0.06-0.07:0.08-0.09-0.10A t ANAA ■‘AAASA%^•■■■■••..1.00.0 0.1 0.2 0.3^OA^0.5^0.6^0.7Distance down Capillary (dim)0.8 0.9^1.0Figure 5.4: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with lowered tissue soluteconcentration at t = 1800 s.t.]cr)^so^cocoa•. •^•-• • . . -•^•^.^•^ .0.6^0.7^0.8 l^0.9 1.0N0.01 .0.02:0.03:0.04:0.05-0.06:0.07:0.08:0.09:0.10 l'hapwr^- TISSUE FLUID BALAME^125Solute ConcentrationPeclet Number0.4 0.50.010.02:0.03:0.04:0.05:0.06:0.07:0.08:0.09:0.10 ^0.0 0.10(13-070/P.„0CT,es0^. ^I ^^0.8 0.9 1.00.0 0.70.1^0.2 0.3^0.4^0.5^0.6Hydrostatic Pressure0.8^0.9^1.00.010.02.0.03:0.04:0.05:0.06:0.07-0.08:0.09:0.100.0^0.1 0.2........ •g •nr,•••yr-sg...15-.•-r-r-Ig.\\\.••0.3^0.4^0.5^0.6'•^• 0•oV‘,4s,^•^o^cs^' .....^ v .• ....^.^.• •^. ..r.•0.7^0.8a 0ti0.9^1.0E:0zo0.01.. "-0.02 .11: ................................... a................................-•-;la^00..00:;_ .................... .............^.e..9.9,7/21- :.::: ....... ...7.9.99 0O 0.07:..-■ O0 ..89;...,i ^• 0.10 .. .'"' • 0.10.2^0.30.04.20....Qi^i ^.. ,^.c   ...,   ,...•••1^ .1O 0 c;^c;1^1 1 ii •^(^• 1^1^1 ^I0.21 1 I 1 1 1 1 1 1 1....... l WIAAAN1 1 1 1 1 1 1 Al tAll ILA tt■AA■■■■■ Velocity Vector Field..... -0.20.010.02:0.03:0.04:0.05:0.06:0.07:0.08:0.09-0.100.0^0.1I^I^g^. --10.3 0.4 0.5 0.6^0.7^0.8^0.9Distance down Capillary (dim)1 1 1 1 1 1 l AAAA'1 ,1••■•■■■•1 1 1 1 l'INI.■■■■■■■■. ^1 1 1 llAt■■■ ^1 1 AAS,■■■ 11 I ^1    1.00.3 0.4 0:5 0.6 0.7Figure 5.5: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with lowered tissue soluteconcentration at t = 3600 s.Solute Concentration?,7o6-o^ \-•o^ o.,eoo0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.00.01O.02-0.03 20.04-0.05:0.06:0.07:0.08:0.0970.10 0.0-0( ltapter 5 :(7APILLARI - 11.11VIE FLUID BALANCE^ 1260.010.02-0.03-0.04-0.05:0.06:0.07:0.08:0.09:0.10 .....0.0^0.1^0.2^0.3Hydrostatic Pressure...........^......... ...^....... . ............. . .....43^.... ..... .. ...^ -,......... ----... .......^--zeaOA N^....),0.5 •^.6 6^1%.^\^\.._.7 -.8 _.9 ....1: 1.0V 0.0 ^O 0.02 ...,so^0.03 ..................... 29:59510 04-s. ^•-......,0    , .................................................1-, .^ 20..C.),^0.05-■. .......... . '74).990.06- -, •0.9^ ca4?O a.137:C.)^0.097....O 0.10 ^C 0.08:ell...•46.........o.ot.___-----0.02 .0.03:0.0470.057 0.067Velocity Vector Field0.03-0.02:0.01 . I IIIIIIIAAAAAAAAANN^I I I I I I^I six^..MIA AAAAAAASS ^0.04 •IIAAAAAA ^0.05 1 11 AAA 0.06- I A 1 ^0.07- I A AA0.08- I A A  0.09- I0.10Peclet Number......tz0.1^0.2(1)-°10/P.rt7:'al■ 0(0CD 0 C33-,•1i ,----„..^.--io^ce^•ce ct$coce..............Ca.0.8^0.9^1.010.9^1.0eee.1.1.1.1^ e1.1.11 '.' I I...IAt\oi\,^0.0^ I 0.3^0.4 0.5 0.6^0.7c;^ci^ci^60.08:^ i i I I I0.09- 1 ^•i^i )^I0.10     1 ( .^•1^• I^ 1.^-I0.0^8.1... m z■ 1-.:0.07: ci^... ..., --.0.2 0.3 0.4 0.5 0.6 0.7 0.80^000.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)0.8 0.9^1.0Figure 5.6: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with lowered tissue soluteconcentration at new steady-state.Chapter 5 :CAPILLARY - TLSISVE FLUID BALANCE^ 127The drop in the tissue solute concentrations lead to lower osmotic pressuresthroughout the tissue. This is particularly important adjacent to the capillary membrane. Ascan be seen in the potential distributions, the lowest potential regions shift from beingentirely at the the lymphatic sink to being at the venular capillary membrane. This causesfluid reabsorption back into the capillary as can be seen in the fluid field structure. This is adirect consequence of the drop in the osmotic pressure in the tissue.The drop in the tissue osmotic pressure causes the tissue-side potential to increase inmagnitude. This then at some point exceeds the capillary potential and then fluidreabsorption occurs. As the tissue solute concentrations are further lowered transiently, theamount of reabsorption increases. This can be seen in the results as the zero point (the pointon the capillary membrane where the normal fluid flow is zero) shifts up the capillary fromz* 0.82 at 1800 s to z* 0.79 at the new steady-state. The increasing amount ofreabsorption due to the reduction in the tissue osmotic pressures leads to lower fluidvelocities across the capillary membrane. This is clearly displayed in the Peclet numberdistribution. The convective transport drops off into the tissue from maximum values at thearteriolar capillary membrane. There is slight increase in the region of reabsorption at thevenular end of the capillary. The lymph drainage drops to 2.42 1/day at the new steady-state.5.3.2 Perturbed Venous Hydrostatic PressureThe venous hydrostatic pressure provides an additional controlling factor for fluidfiltration and reabsorption. This can be understood if a rise in the venous pressure isconsidered. This will lead to a rise in the hydrostatic pressures along the capillary. Elevatedvenous pressures may occur because of venous obstructions or in heart failure. If thecapillary solute concentration, and thus the capillary colloid osmotic pressure, is largelyunchanged, then an elevated venous pressure will increase the potential to be overcome forfluid reabsorption. In fact, this would lead to increased fluid filtration and lymph drainage.Chapter 5 :(APILLARY - TISSUE FLUID BALANCE^ 128The results shown in Figure (5.7)-(5.10) are the transient results for elevated venouspressure. The venous hydrostatic pressure is increased suddenly to four times its originalvalue. This corresponds to a venular pressure of 0.6667 (20 mmHg). Steady-state isachieved rapidly (< 2000 s) unlike the case performed in section 5.3.2 above. The initialstate corresponds to the base case performed in Chapter 4.Initially, with increased venous pressure, the fluid filtration rates across the capillarymembrane increase rapidly and the tissue pressures increase. Subsequent to the perturbation,the lymph flow increases to 6.43 1/day. This leads to greater removal of solute from thetissue space and thus the solute concentration drops. This can be seen at t = 120 s in Figure(5.5) where the solute concentrations are dropping throughout the tissue and radial gradientshave formed near the capillary wall. The maximum dimensionless hydrostatic pressures areas high as 0.1000 (3.000 mmHg) at the extreme venular end of the tissue. The dimensionlessaverage tissue hydrostatic pressure is increased to -0.0209 (-0.6270 mmHg).As a result of the lowered solute concentrations within the interstitium, thehydrostatic pressure begins to drop in response to the lower colloid osmotic pressures. As aconsequence, the lymph drainage rates begin to fall. At t = 240 s, the lymph flow drops to6.00 1/day. The fluid velocities across the capillary and within the tissue also drop. Thesolute concentrations and hydrostatic pressures continue to drop through time resulting indecreasing lymph flows. The new steady-state lymph flow (5.89 1/day) is higher then theinitial state value (5.47 I/day).At the new steady-state, the solute concentrations are lowered within the tissue (thedimensionless average tissue solute concentration equals 0.4997) and the range is lower thanthe initial state. The solute gradients appear to be more uniform throughout the tissue. Themaximal gradients occur at the venular end of the tissue but they are lower than gradients inthis region for the initial condition (the base case). This is because there is greater and moreuniform washout of solute into the tissue along the length of the capillary. This results in themore uniform solute concentration distribution within the tissue.Chapter 5 :CAPILLA1?Y -TISSUE FLUID VALANCE 129Solute Concentration0•4.3%^(0W..• CO%CIA CO........^.•..1 \^•• •-,r^I^ I^•^• • TT^0.4^0.6 0.6 0.7 0.80.010.02:0.03:-0.04-0.050.06-0.07-0.08-0.09-0.100.0 0.1^0.2 0.9^1.001:2BO ^, ^0.30tO0.010.02.70.0370.0470.05:0.0670.07:0.06:0.09-0.100.008.1^0.2^0.3..........1.00.6^0.7^0.8Hydrostatic PressurePeclet Number0.01 ...0.04- ............• 0.02 .• 0.0 3 -'^•.^4).90,9.........................................................................................^ ....................^..•^..............49,9^ *fa 945 4730 0.07-• .0.10 ^0^•O 0.09-° 0.06: '9.89sit0 .08- -• ••• \(Ls^0.13 0.9.•1.00.0 0.1^0.2^0.9^0.4^0.5V0.01 0.02.0.0370.0470.05:0.06-0.0770.0870.0970.10^ ANAII%All^AAAA 1• AAA I,^ ♦ A0.0^0.1^0.2 0.8^0.9^1.00.01 ^0.02.70.03:0.0470.0570.06:0.07:0.08:0.0970.100.0..... • • -(P - an)/Parti 1^C731co^co^co raeat.. coz,. t... e; 07€7 CO CO^dci^O^ci ii ^ 1^++>^( 1^1 1^/ 0 .1^0.2 0.3 0.4^0.5 0.6^0.7 0.8^0.97 ..1coA. uor30.^ C•3/ 0::)1.0Velocity Vector Field1 AAA% AAAAAAAAAAAAAAAAAAAAAAAAA 1 A 1 1 11 1 1 1 1 A A A A AAAAAAAAAAAAAA■A■■■•■‘‘‘A.A.ANAAAAAAAAA A 1 I1 1 1 1 1 AAAAAAAAAAAA.A.A‘A.■A. ♦ ♦ ♦ ♦ ♦ ♦ 44. 44. 41. ♦ ♦ ♦ ♦ A.AAAAAAAA A 11 1 1 A AAAANAAAANANA.A.A.A.A. ............... A.NA.A.NANAAAAA 1 '1 1 'AAA1 1 A ^1 A A ^A• r.. •-•-•0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)Figure 5.7: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with elevated venous hydrostaticpressure (Peen ` = 0.6667) at t = 120 s.( '/whirr 5 :(APILLARY -11.11111E 1-11111) BALAMI .Solute Concentration0.01 .0.02:^\ \^1t^::0.03: A0^ A 0 a0.04 -^ o 0^'.^0 -  . •".---'.0o "01:1 o .^o : 00a _.0.05 - cr. N : 4 ^u .-"o ^(3  -ro a a(Los o^0^00.10^, \^1^ , . \^• i^*I0.oca^.-0^0.02: ..................... . .......... ....................... ..... 2..a. ... 2...........5..9.; .6. .2......... . . - - - - - ''........ . . .. . ... Aisti ......... .......09.el 0.037--73.9.990-1............----^ .SO▪ .......^...... ......^---- ‘7.9^'- ...... 1.....  - - . . ..^• 09E-.^0.05:,...., 02 13.06- 49990 0.07^if.•,b.•o0.09-..:....^°II....^..^.....^..^....0 .08 :•••..^.....2.ca0 0.10 -  ^i- - 0.0^0.1 0.6^.8^ .90.7 ,-:.:• ^0.2^0.9^0.4^0.5^ 1.0....,0:i60cb/a caeb -,0.07- a0.08-^ /0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7iVelocity Vector Field0.00.8^0.9Figure 5.8: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with elevated venous hydrostaticpressure (I've; = 0.6667) at t = 240 s.0.02-0.04:0.05-0.06:0.07:0.08:0.09-0.100.0^0.10.01\^ \^/ -.-a " ..^-------O 0 AA O 47;^47 i e..-- -ct■^-3ra -sro o^to\  ^ \ \^ ^ I^ 10.2^0.3^4.4^0.5^0.6 0.7 0.8 0.11 f^1.0Hydrostatic Pressure0.01. . .. .0.010.02.0.03:0.04:0.05:0.06:0.07:0.08:0.09:0.10:^0.10 /   ,.f    I0.09/0,^0,ci 0.0.010.02.0.03:0.04:0.05:0.08-(P-o-rr)/P.r,111111111111111 ;M:;111;;;;;;;;;; : ; ; ;; ; ;■; ; ;II■1 1 1 1 1^1^SASSAAAAAAAAAAAANAAAANAAAAAA AAASAA^1 11 1 1 1 1 l '1 l A A SAAAAAAAA NAAAAAAAAAAAAAAANAAAANAA A ASS 1 11 1 1 SAAAAAANAAAAAA .................. AAAAAAA A AS 1 11 1 1 1 SAAAAANAAAAAAA ..................... AAAAAAAS 1 11 111 1  ^11^ A1^1 'AI 1♦ ♦ ♦ A 10*^ &a0.1^0.2 0.3^0.4 0.5^0.6^0.7Distance down Capillary (dim)01 ‘ot, z.I 0.8 0.9 1.01.00.09:0.08-i •^• i^\ i ^(^(^.::0.07-0.0^0.1^0.2 0.3^0.4^0.5 0.6 0.7 0.8 0.9^1.0i •Peclet Number1300.0^0.1^0.2^0.9^0.4^0.5^0.6 0.9 1.00.7 0.80.010.03:^390.04- ..................0.05-0.06 t9.99:19 98540.07-0.09-0.10.....................................ea-.990e.....................................................................................................................................................................................0.02. ...... ^....^ .........................................• •0.8^0.9^1.0litipwr 5^- TISSUE FL(II) EALAME^ 131Solute Concentration0.3^0.01 ^0.02.:0.047:0.05:0.06-0.07-0.08-0.09-0.10 0.0 0.1^0.2 0.7^0.8^0.0^1.00.010.03:0.04:0.05:0.06:0.07:0.08:0.09:0.100.0^0.1Hydrostatic PressureA°-0,^ -0^*-- 0 ^0^0*-00 .20^vl •^o ••..^0 0-°°\ 0-.^0^•0• ***^-1  ^•0.2^0.3 0.4^0.5^0.6^0.7^0.8o :aO6 :Peclet Number03-010/P.„.. is if i i i  i  i  i • etr-A.7.^a^•^Cb ^a^ca^•^aco co co ....,cr, co co^co^to^ts.^a' ^02^r) c) c) cr)^:^C'3/ CS/^6^6^6 ^6o^ I^11,---.41^i LL:^,• i^•^.  ^___...r,^,___. ._..___.^(•0.5^0.6 0.7 0.8 0.9^0.01 .^0.02-0-03:0.04:0.05:0.06-0.07-0.08:0.09-0.100.0^0.1O0.2 0.3^0.4 1.0Velocity Vector Fieldti^I I I 1 I 1 1 1 1 11 1^1^III^;^1 1 1^; II I 1 I 1 I I WA AAAAAAAAAAAAAAAANAAAAAAAAAAAAAAAAA I I 1 I1/ 1 I I A 1 AAAAAAAA■AANA•A■A ■•■■■ ■••••■■■•WAAAA All 1I 1 I AAAA AAAAAAAA••■■■■•■■■ ■•■■■••■■■■■•■■■■■ AAAA I I 1IIIAAAAAA■N■N•A%■■■ .......... ....... ••■■■NNAAAA A 1 II I 1 AAAANN■•■ •■■■■■•■■■■■■■■■■■■■■■■■•■■■■•■ WAI II I A   AAA I I 1     AI1-^-,-, --- • t^1^ 10.3 0.4 0.5 0.6^0.7Distance down Capillary (dim)Figure 5.9: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with elevated venous hydrostaticpressure (sPven* = 0.6667) at t = 600 s.0.01.^0.02,:0.03:0.04-0.06:0.07:0.08:0.09:0.100.0 8.1 0.24.3 OA0.20 .1 4.5^4.6^4.7^0.8^0.9^1.00.01 0.02.0.03:0.04:0.05:0.06:0.07:0.08:0.09-0.101.00.0 0.1\0^• • ^070^*rt0tip35.40 ' ,9. .s, 0^4".o^• 0 • 4)-^• .9^0 .0.2^0.3^OA 0.5^0.6^0.7^0.8^0.01.^0.02-0.03:0.04:0.05:0.06-0.07.0.08-0.10 ^0.0Hydrostatic Pressure1.0( liapter 5 :(APILL4RY -^FLUID BALAME^ 132Solute ConcentrationPeclet NumberTIT,i .. • • Iel 0.03-^va0.05-0.04 --------------------................................................................................................................... 39 - 9908 --..„--...0.9.9 S9*0.08..-90.../0.02 ..........................^.^ ......3 0.06-^/9.99,54 -0.07; 04,4oo/^\---%^. „-- . • .^ I s. 0^0.10 ^.... --; • •i ......3 - ,(P-air)/Part0.03:0.02: /0.04- A.9C2S4a^413wm0.06-^ 07^(.7(c• (17(.7^(I47^iCI^COCO^CO0.09:0.10     4 I^ 4,^ 1^2 i^I 2  ^1 •^7 i0.07 0./ 0^0; 0.^0. a'(^0.08; / 1^1 1^10.0^0.1^0.2 0.3 0.4^0.5^0.6 0.7 0.8 0.9^1.0Velocity Vector Fieldt 11 I^11 111111 111 11111111111 11 1:11 11 111 1 1111111111'0 .1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Distance down Capillary (dim)Figure 5.10: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with elevated venous hydrostaticpressure (Peen * = 0.6667) at steady-state.2-T:0.010 04-3 0.0^0.1^0.2^0.3^0.4^ .5^0.6 .7V20.8^0.90.01 ^0.02.0.03-0.04:0.05-0.06:0.07:0.08:0.09:•0.100.01 1 1 1 1 1 1 1 1 1 t 1N1 A■Al%•■■■■••••■•••■■■■■■■At AAA 1 1 1 1 11 11 1 1■■•1At■■■••••■44.••••••••••••••4'■4.•■■■■•■ 1 1 1 11 1 1 I 1 14.4.4.■•••••••••••••••••••••••••••••■•■•■■ 1 1 11 f 1 1 .1 1■••■••••••••••••••• .......... ••••••••••• 1 A 1 114 4.4.4.   4.4.44.111 1■■■4 ^  4 t t t 1 11 11 I  .. • •Chapter 5 :CAPILLARY - TLSISVE FLUID BALANCE^ 133As expected, the hydrostatic pressure distribution is very similar to the soluteconcentration distribution. The gradients are lower within the tissue and the range isreduced. The dimensionless average tissue hydrostatic pressure rises slightly from the initialstate to -0.0251 (-0.7530 mmHg) , however, greater portions of the tissue becomesubatmospheric as the venous hydrostatic pressure is increased. This was not expected butcan be explained by tendency of the tissue hydrostatic pressures to respond to the tissuesolute concentrations. The lower, more uniform solute concentration distribution leads to amore uniform hydrostatic pressure distribution throughout the tissue. Although the averagehydrostatic pressure for elevated venous pressure is higher than the initial state value, agreater portion of the tissue is subatmospheric.The radial gradients in the potential are increased and the distribution become slightlymore positive with the increase in the venous pressure . The increased radial gradients areexpected since the elevated venous pressure promotes more fluid filtration at higher fluidvelocities across the capillary membrane.In summary, the results from this case illustrate that the system initially responded tothe increased venous pressure by increased tissue hydrostatic pressure and associated lymphflow. This, subsequently, reduced the solute concentrations within the tissue leadingtransiently to lower hydrostatic pressures in the interstitium and thus lower lymph drainage.The lowered solute concentrations at the tissue-side of the capillary membrane decrease thetransmembrane potential thus providing the tendency to lower fluid filtration rates across thecapillary wall. These results follow trends found in experimental studies (Johnson andRichardson, 1974; Mortillaro and Taylor, 1976).When the system had reached the new steady state, the lymph flow had reduced untilit was just greater than the initial state. The system has shown that despite the large increasein the venous pressure, the solute concentration, hydrostatic pressure, and potentialdistributions readjust themselves to provide a modest increase in the average tissuehydrostatic pressure and lymph drainage. This safety factor mechanism against edema isChapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 134supported by the work of Johnson and Richardson (1974). It is important to bear in mindthat the potential drives fluid motion throughout the system. The tissue potential distributionincreases (by about 0.04) with the rise in the venous pressure. In terms of the range of thetissue potential, this is a significant amount. This further confirms that the capillary wallbears the dominant fluid flow resistance in the system. In other words, the potential dropacross the capillary increases to a larger degree than the tissue potential.5.3.3 Perturbed Capillary Solute ConcentrationIn this section, the results from two cases will be presented. In the first, thedimensionless capillary arteriolar solute concentration will be decreased to 0.10 (the basecase value is 1.0). The corresponding capillary osmotic pressures will accordingly bereduced in response to the lower solute concentrations. In the second case, the dimensionlesscapillary arteriolar solute concentration will be increased to 1.2. The first case correspondsto hypoproteinemia, that is when plasma protein levels are low. This may occur because ofliver disease (low hepatic protein synthesis) or nephrosis (elevated loss of protein in urine).The perturbation in the capillary solute concentration is implemented when the steady-stateconditions achieved in the base case in Chapter 4 have been reached.The transient results for the case where the blood solute concentration is dropped to0.10 are displayed in Figures (5.11) to (5.14). As the solute concentration in the blood isreduced, the plasma colloid osmotic pressure exerted is lowered. This means that thetranscapillary potential difference becomes more positive leading to greater amounts of fluidfiltration. Initially this results in increased lymphatic drainage rates. In this manner, theblood volume decreases and more fluid passes through the interstitial space and out throughthe lymphatics. Later, the solute concentrations in the tissue begins to drop as seen at t = 300s (Figure (5.11)) due to the increased lymphatic withdrawal. The maximum tissue soluteconcentrations occur near the lymphatic sink. The tissue hydrostatic pressure falls inet?0(.7upter 5 :(APILLARY - TISSUE FLUID BALANCE 1350°187: .............................0.08:0 -05-s-0.15000.043L -- 0.10000.020.010.20000.09:Solute Concentration0.1000 ^0.10000.1500 0.1600^0.15000.2000 ..^.............0.20000.1500 .. 0.0.1499.4.1.1!51.......................................................................................................................0.0 0.1^0.2 0.3^0.4 0.6^0.6 0.7^0.8 0.0^1.00.010.02 40.03 Hydrostatic Pressure ^ 0.1600-- 0.1600 ^0.1600^0.05:0.06 0.1200 ^0.1200--0.1200^-0.1200^-0.1200^-0.1200 ^...................................13 -07; ......................................... ...............................................0.00-^-0.0800-----o.0800-----0.0800 ^............0.09:.....................0.10.......... ^...... .............0.0^0.1EPeclet Number0^--.v 0.02.....^0.04ID(0^0.03 _ .......................................................................... I.:- .^ ....^.....::::::::::-..---^................................................................. ..--*"...^0.4)^. .....9480Z-_0.05-,. ....^3,9^ -...,.,^ .-2 o.oe-^...... .--..^-99^--...^sa 067^ ^.-0 0 07 -^." Oe..4^'O 0.08-O .09-.. .^•••••••f:1^0.10IS T▪ ITTIrr-v-TT IllIIr ,1--•-• V ^.:.......GI^0.0^OA^0.2^0.3^0.4^0.5^0.6^0.7...0.2^0.3 0.4 0.5^0.8 0.7 0.8 0.9 1.00.8 0.9^1.00.010.02;0.03:0.04:0.05:0.06:0.07:0.08-0.09;0.100.00.01.0.02:0.03:0.04:0.05-0.06:0.07:0.05-0.09;0.100.0II1 11111111 A A A A AAAAA AAAA■A‘tAA■■•••■•••AAAA AAAA111 1 1 1 1 I At Ali ■■■■•■•NNNNNNN■■■■■■■■■•■■■ ■■•■AA A A 1 11 1 1 1 A A A A AA AAAANA■■■•••• ••••••••••••■■•■■■•■AAA A A 11 1 1 1 A A A AAANN•■■■■%s% .................. ■•■%N■AA A A A 11 1 A A AA AN•■•■■■■■■■■■■■■■■■■■■■■ •■•■■■■■•■■•■■•A A A 11 1 A A AtAA•■•■■■ .......................... 44.4.■‘‘AA AA 1IA■NA AA A■■A........0.1^0.20.1^0.2^0.3^0.4^0.5^0.6^0.7Velocity Vector Field0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)ii(P-a-70/Par,acv0.80.80.90.91.01.0Figure 5.11: Dimensionless tissue solute concentration, hydrostatic pressure, Pecletnumber, potential and velocity vector distributions for the case with loweredcapillary solute concentration t(2arl* = 0.1000) at t = 300 s.Chapter 5 :(APILLARY - TISSUE FLUID BALANCE^ 1360.1^Solute Concentration i 70 . 0 :7  ..........^ ...^0.0800^....................................0 .07-^............................•. ............. 0.08 0.1200^0.12000.09-0:1200 ----0.1200^0.1200^.. ..... .-0.10^..........................................................................................0...........0.0^0.1 0.2^0.3 0.4^0.6 0.6^0.7^0.8 0.9^1.00.01- Hydrostatic Pressure0.02Z....1^0.04^39.99nti^ ...........0.05: ...... ......... ...O-......-...-....._.„..4, 0.06- al,•99_0^0.07 -...I ' g5'0 0.08-O 0.09--••.O 0.10^el^' -T '-4-,^0.0^0.1^0.2^0.3VI..-■E:E1O 0.01 ,..................................................................Si•^0.03 ..-•................ .............................................................. .. ...................... ..^ .. ..........-•../. C^ . ,0.03-0.06: ........^00. . .. ;102. .0. .0^.. ......... .... .... . . ......^.... ..... ......... ... ................ .... . ^-0.1000-0  120. .0. ........... . . .. .0. . . 10. 0. .. .....................0.04::::::::"-. .^................. ............0.0^........................................................................... ....0....0.8-0-0..„...........................060.08L-.. 0... .... .... ........-----0.10000.1^0.2^.. .^0 °43°....------.-0.09:0.10"^;-------" 0.8^0.90.3^0.4^0.5^0.60.4Peclet Number0.5^0.6 0.7 0.840ti0.9 1.01.0Q(P-an)/P0.01^ art 0.03-0.047:^./...-...07../...- 7 .177 ..:1 . 7 if0.02-1*^ .0.05- .-'^& I^<a : °^o- 00^0... 1\^•^to t,..^• .1.■:'0.06.:^ .• 0-^:^.... : 4.`. 0^0 0t.^••./ : ...I^•-•^. .i..i^0^00.07-^.- 0. •-■ --/d d 6^ci.-,0.10^.. ...^7^.0.06:..-*" i d^di^i 0.09- •0.010.02:0.03-0.04:0.05:0.08-0.08:0.100.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)0.0^0.1 0.2^0.3111 1111111111111111111111M ^1111I 1111111111111111111111A11111111111111111111111111111 11111 11111111111111•••••■■•■1•■•■•■■1111111111111 11111 AAA 1 AAAN■■■■■■■ ■•■•••■•••■■■■■•■■■ 111 1111 11 111 AAAAA••■•■••■■ ■■■■■■■■■■■■■■■•■•■■■■■■■•AAA 111111 AAAA•••■■•■■■■■■•■■■ ................ ■■•■■•■•AA 111111  1IIAA  1111 A 11' 1 .....0.4^0.5^0.6^0.7^0.8^0.9^1.0Velocity Vector Field0.8^0.9^1.0Figure 5.12: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with lowered capillary soluteconcentration (can* = 0.1000) at t = 600 s.Solute Concentration--0.2^0.3^OA^0.5^0.6^0.7^0.8^0.9^0.01^0.02.0.0370.0470.05-0.06-0.07-0.08-0.09-0.10  0.0 0.1 1.0Hydrostatic Pressure0.01 0.0370.04-0.05-0.7^0.8^0.9 1.00.067^ .--....0.077 .--'0.08 ..."-0 .09- • -•0.10 - ^•I^•-1-•-•-■••-•-•-•-•--u•-•0.0^0.1 0.2^0.3^0.4^0.5^0.6P-oir)/P0.01 .0.02-0.03-0.040.05-0.0770 . 0 9 70 .1 0 ^0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7Velocity Vector Field1.00.90.2 0.3 0.4 0.5 1.00.6 0 . 8 0.90.70.07:0.08:0.09:0-100.010.02-0.03-0.04-0.05-0Am=fi^11 1 111111111 11 4 11111 4^4 11^lAtt'lltill■;;1^I 1 I I 1 I 1 1 1 t^ I 1 1III 1 1 1 1 I AllAitAIIIAAAAAAA%•••■■■■11NAAAIAIAAAIAAIII1 t 1 1 1 1 IAAAAAAAAAAAA■•■•■■■■•■•■•■••■■■•■•AAAAtillI 1 I 1 1 AtAtA%lA‘A .W. 4.■■•■■■■■■•••■■■■■■••■•■■•■■■■1 I1 Ill l`iAAAA••■•••■•••••••••••••••••••••■•■•At‘ t 1 IIt 1.1A 'ttA■ ■■•■••••••••••••••••••••••••••••••■All 1 1IlitAAA••■•••■••■■■••• -------------------------------IS ^■NA(liapter 5 :(APILLAI?Y - TLSIYUE FLUID 11ALAN(7E^ 137Peclet Number.4-..-------4----■•••..... ....2^0.06-^........ --...^.....q ......" 0.04 :°C-. .9906. 1vt^0.03.Z..................3-.g....9-9-0..8....................................................9...9.9..0..e............U.•••••••./------0.05- -----------------^39 39.9908^3•^ 44,0g 0.07  ••s,.. I0.06-^9 89540 0.09-- ^ .. .° 0.10 03^ g ^0.10.01O• 0.02^0.00 . 0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Distance down Capillary (dim)Figure 5.13: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with lowered capillary soluteconcentration (can* = 0.1000) at t = 1800 s.0.0^0.1^0.01 . ^0.02-0.03:0.04:0.05:0.08:0.0770.08:0.09:0 .10^ •i•^0.0^8.1^0.2^0.3^0.4^0.6^0.6 0.7^0.8^0.9^1.00.01 .0.02-0.03-0.0470.05:0.06:0.07:0.08:0.09:0.10Hydrostatic. ..................^...... ..................--ao9soPressure. ........0•ocis.o^-42.9^4)410.80.70.2 0.3^0.4^0.5^0.6 0.9^1.0Q0.010.020.03-0.04-0.050.08-0.07-0.06-0.0970.100.0(liamer 5 :CAPILLARY - TISSUE FLUID BALAN(T^ 138Solute ConcentrationPeclet Number^ 0.01--. ..N1.1.b:d---.39.0908I^.0.......................... .. .......... ... 913 ...................................................................................................93^,,,,.,0.057 ....................^..'..... ...^CP••■••"'"-------^0 02-^...2 0.06-^39.9908-----39-9908 .--.^94/• 0.07: ...."•.-■0.4^0.5^0.6Velocity Vector FieldI 1 1 I I^1 1 i it I 1 1^I 1 1 1 1 1 1 11 1^1 44^; iiiiiiiiiiiiiiii0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Distance down Capillary (dim)Figure 5.14: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with lowered capillary soluteconcentration (can* = 0.1000) at new steady-state.0 OS:^ 19'9964  O 0.09- ....... _.O 0' 10 .i.a^0.0^0.1 ^0.3go.....0.7 0.8^0.9^1.00.010.02:0.03:0.04:0.05:0.08:0.07:0.08:0.09:0.101 1 1 1 1 1 1 1 1 1 1 1 t t lItAill'ilANAAAAAAAIAAAAANtitAll1 1 1 11 1 1 1 1 1 1 1'1 1 IttAAl•■•••••■••■•••••••••••••••• 1 A1 1 1 11 1 1 It 1 1 t t IAA■••••••••••••••••••••••••••••••••I'll11 1 1 1 1 1 1A1AAWN••••••••■•••■•••■••■•••••■•••■•••1 11 1 1 1 WAAAAA•••••••••••••• ....... •••■•■••••••••1 t 1 11 1 1 litAAAAA••••••••••■•••■■■■■■■■■••••••••NNA1 t 1 11 1 1••••••••••••• .......................... •••••1It    ••■Chapter 5 :CAPILLARY - TLSSUE FLUID BALANCE^ 139response to the drop in the tissue solute concentrations. At t = 300 s, the tissue hydrostaticpressures have become subatmospheric throughout most of the tissue. As a result thelymphatic sink removes less and less fluid and solute from the tissue thus reducing the fluidflow rates at the capillary membrane.The solute concentrations reduce further with time. The hydrostatic pressureresponds to the lowered solute concentrations by becoming increasingly more negativethroughout the tissue. As a consequence, the lymph flow rates lower and therefore thetranscapillary fluid velocities decrease with time. This means that solute removal rate outthe lymphatics is dropping with time.At steady-state (t 4600 s), Figure (5.14), the maximum solute concentration of thesolute distribution has shifted to the venular end of the tissue near the capillary. The excessamount of solute that had built up in the venular end of the tissue near the lymphatic sink dueto the initial condition has now been depleted and solute removal is now largely dependenton the amount of solute being transported from the capillary membrane. As expected, thetissue hydrostatic pressure distribution resembles the solute concentration distribution. Thelow solute concentrations within the tissue cause the tissue to be entirely subatmospheric.The new steady-state solute concentration distribution has lower gradients throughout thetissue. The range is also decreased to roughly one-tenth of the initial condition values. Thisis reasonable since the blood solute concentration was dropped to 10% of its original value.The dimensionless average tissue solute concentration at the new steady-state is equal to0.0495. The hydrostatic pressures are also lowered within the tissue. The dimensionlessaverage tissue hydrostatic pressure drops to -0.0949 (-2.847 mmHg). As a result of thelower hydrostatic pressures, the lymphatic drainage rate is reduced to 3.59 I/day (from theoriginal 5.47 1/day).These results are confirmed by the experimental findings of Manning et al. (1983) indogs. They found that the lymph flow decreased as the capillary protein content was reducedto very low levels.Chapter 5 : C:AI'ILLAK Y - 7LS SUE FLUID BALAN(:E^ 140The results for the increased capillary solute concentration are displayed in Figures(5.15)-(5.18). Here, the trends are opposite to those of the previous case. As expected, moresolute is transported into the tissue leading to higher solute concentrations and thus elevatedcolloid osmotic pressures. The transmembrane potential, as a result, decreases until, near thevenular end of the capillary, it eventually becomes negative. At this point, fluid reabsorptionback into the capillary begins to occur. At t = 0, corresponding to the base case, the solute iscompletely filtered into the tissue spaceFigure (5.18) display the results at t = 300 s. As can be seen, the soluteconcentrations are increased within the tissue. Also, the solute concentration gradients at thevenular end of the tissue near the capillary increase due to the fluid reabsorption andconsequent tissue-side protein filtration occurring in this region. Washout of solute from thearteriolar region of the tissue produces lower solute concentrations here with relatively lowergradients. The tissue hydrostatic pressure responds to the solute concentration distributionand has increased gradients at the venular end of the tissue near the capillary. Thehydrostatic pressures become more positive. This leads to an associated increase in thelymphatic drainage. Fluid velocities increase within the tissue as a consequence of the rise inthe lymph drainage. The rise in the solute concentrations within the tissue produce a drop inthe potential distribution via the colloid osmotic pressures. As can be seen from the fluidflow field, fluid reabsorption is occurring. The zero point is located at z * = 0.70.At t = 600 s, Figure (5.16), the solute concentration has increased further within thetissue. The solute concentration and hydrostatic pressure gradients have increased further atthe venular end of the tissue near the capillary. Also the maximum values of theconcentrations and pressures occur in this region. The rise in the hydrostatic pressuresresults in an enlarged lymph drainage rate. This leads to a slight reduction in the fluidreabsorption back into the capillary. This is reflected by the shift in the zero point down thecapillary to z* = 0.75............0.00 00Chapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 141Solute Concentration0.5999............................................................................................ .............................. ......................................^.. .................................................0.5999^0.5999^0.5999. .^0.5999^•^ 0'S9An........ ........0.010.02-0.03-0.04-0.05:0.06:0 07:0.08:0.0970.100.0 0.1^0.2 0.3^0.4^0.5^0.8 0.7 0.8 0.9 1.00.01 .0.03 .0.04:0.05:0.06-0.07-0.08-0.09-0.100.0Hydrostatic Pressure........0.0000......................................................................................0.0000^0.0000^... .. .....^..............................................................................................................0.20000.00000.1^0.20.00001.00.3^0.4^0.0^0.6^0.7^0.8^0.90.010-020.03...1^0.04• 0-052 0.06O 0.07O 0.080 0.09O 0.10CiPeclet Number...................... ...-a^ it. f a...... - ::ts; '-'ill; * . *TO \.....% trO^..• 1)4;^00^io ai of0to14) 0 ^03CO0.0.............................. ....................co'V \ V \ 'D^•11:%^0 0cz, ca-.2 040.1^0.2^0.3^OA^0.5^0.6^0.7^0.8ics^co0,^ i0a: r 4.co coi I. t  . I  I ; •^.1 C4CD^••••0.9^1.00.01.0.02.:0.03:0.04-0.05- cb00.067-^to0.07 NI.O0.08: I0.09-0.01 .0.02:0.03-0.04:0.05:0.06-0.07-V0.06-0.09:0.10---,Jr.--0.0^0.1^0.2^0.30.10 •^ ...... - ., --1-.111 ^  A A^k^k1 1 111 AAAANC■ .........^11AlA♦♦ ^IA ■■•■ A A ♦ ^N- . - a,,0^0^to•cl u3 tou) r.- cel.^1 Nt.o^Ooi II441t^t ^OA^0.5 0.6 0.7^0.8Velocity Vector FieldP-o-70/13. t1co0 00.9^1.00.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8Distance down Capillary (dim)0.9Figure 5.15: Tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with enlarged capillary soluteconcentration (ca ,/ * = 1.2000) at t = 300 s.1.04.7^4.8 1.00.9Peclet NumbervO 0.01• 0.020.030.041-4 0.052 0.06• 0.070.080 0.09O 0.100.0^0.1^02^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.00.01,-0.01 ^0.02,:0.03:0.04:0.05-0.08-0.07-0.08-0.09-0.100.02:0.03:0.04-0.05-^co0.06i0.07:0.05:0.09:0.10 ^ 1-• •^u•s •^• I-'r.-..0.0^0.1^0.2^0.3^ 0.5•-•^•-• 0.6^0.7 ^• ^• Distance down Capillary (dim)0.01 1 1 1 1 I ^1 1 1 I^AAAAA• . .0.1 0.2 0.3^0.4^0.5^0.6^0.7Velocity Vector Field000.8^ . e e I0.9^1.0coaAn0toa00.8 1.00.9Chapter 5 :('APILLARY - TISSUE FLUID BALANCE^ 142...........^Solute Concentration ....................................................0.02:0.030.010.6899^............. .....^......... ................... ......o.s999^... ......... ..............................^....499^_0 64990.5999................................... 69 0-649a0.04:0.6909eg0.08:0 :0 65: .............0.07:0.09 70_10^0.0^0.1^0.2^0.3^4.4Hydrostatic Pressure0.0000.00001.0Figure 5.16: Tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with enlarged capillary soluteconcentration (can* = 1.2000) at t = 600 s.0.6^0.60.010.02-0.03-0.04-0.05:0.06:0.07-.0.08-0.09:0.100.0^0.10. 000 00.2^0.3^0.4^0.5^0.6^0.7.... .2000. .. . . .. . .... . . . . . .. . .. .................. .. . . .. . .. . .. ... . . .. .. ... ..... ... .... . .. .... .. ... . .... . ... . . . .. . .. . .0.0000. ..... .. . .. ...... .. .. . ..... . .. . . . . . . . .. .. .. . ... .. ... . .. . .. . . . . ... .... . . ... . ..0.00000.8^0.9Chapter 5 :(APILLARY - TLVUE FLUID BALAN(1.Solute Concentration0.01 .0.02-0.03-0.04-0.05:0.08:0.07:0.08:0.09:0.100.0^0.1 0.2^0.3^0.4^0.5^0.6 0.7^0.8 0.90.08:1.01430.2^0.3^0.4^0.50.10.01 0.02;0.03:0.04:0.05:0.08:0.07-0.08-0.09-0.100.010.02.0.03:0.04:0_05:0.06-0.07-0.09:0.100.01.0Peclet NumberErig0.01 '............................0.02-2 .......................... .. ..-A..-..... ..3 ^ -00O 0.03-, .... ^ .. I ^*-. ';‘) ^%.^0 a,fa 0.04^.*0....■^ ...s.';^-. .0\^- 0^*-^*0cP E-g 0.05:-o01. 4:)0,^SONtO 0401^0.06: '-.^`," • • .... ............... . ........ .............................^.... ..... ......................................................................................................................................•0 0 0 000 00 ■■•••••0.5 O.7^0.5^0.9Hydrostatic Pressure0.6 1.0a10.1^0.2 0.3^0.4 0.8 0.9 1. 0). ^ t^)0.7 0.8 ^-0.100.09:• •1' • '^..... I*^ i......^ •^l• -4.3 0.1 0.2.10.0 0.3 OA 0.50...CA0.01 .^0.02-0.04:0.05:0.06-(1.07-0.08-0.09-0.10 ^0.0(P-onWP,,,1 ^ia a^a^co^o^o.., co to e- CO0^0^0^0 7•^‘1,40 40 40 10 10^40o c;^IS 01^I^1^1 1^It ^) ^, t ^ )1 ^1. ^I 1^. I0.5 0.6^0.7Velocity Vector FieldAA A ^A N. N4.I^I •00.2 0.3^ .4^0.5^0.6^0.7Distance down Capillary (dim)Figure 5.17: Tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with enlarged capillary soluteconcentration (cart* = 1.2000) at t = 1800 s.1.00.01 1 AAAAAA'AAA' ^t I^1^1 A I AAA 'IAA ^0.10.8 0.9...^................^.......... .. .. ........................................... ....... .................. ............................ .moo.4.6^4.6^4.7^4.6 1.00.90.01^0.02.0.03:0.04:0.05:0.06:0.07:0.08:0.09:0.100.0^0.1•0.2^0.30^ 1^1. •0.5^0.6^0.7^0.8^0.90.40atto-(P-a10/Part030to0330coto1.0l'hapter 5 :(APILLARY - TISSUE FLUID BALAN(T 1940.010.020.030.040.050.060.070.080.090.10Solute Concentration.....^...........^........ ....^...............- • •^• •^............. ......................................... ..... ....... ... .. . .0.. ....... . .......0. ...iiii: :s.- -----^ .0>.96 *-.7,??.6.--.,------.-__0.6^.. °---..^ ---.. 1 ........^age^....^-..•••••^•0.0^0.1 0.2 0.30.010.020.030.040.050.060.070.080.090.10 ^0.0 4.1^4.2^4.30.4 0.5 0.6Hydrostatic Pressure0.7 0.8 0.9 1.00.010.03:0.040.02 ^Peclet Number....................................................^ ...... ................................... .. '•^4&:;\0.05.7 •SS0.06: co co^00.07-0.08-0.09-0.100.0^0.1^0.2^0.3^0.4^0.5^0.6to0atoco^'co^to .co co•to ...^co^coco ta^co :0.7 0.8 0.9 1.0Velocity Vector Field^ ...■••1 r ".■ -1r •-•-0.1^0.2^0.3^0.4^0.5^ 0.7^0.8Distance down Capillary (dim)Figure 5.18: Tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with enlarged capillary soluteconcentration (c,,/ * = 1.2000) at steady-state.0.01 0.02,0.0370.0470.05:0.06:0.07:0.08-0.09-0.10littt0.04' 4u•0.9 1.0Chapter 5 :CAPILLARY- TISSUE FLUID BALAN(7E^ 145The solute continues to build-up, although more slowly, with time at the venular endof the tissue near the capillary membrane. The associated increase in the colloid osmoticpressure at the tissue-side of the capillary membrane results in a increasingly more positivetransmembrane potential. As a consequence, the tendency for fluid reabsorption begins tolessen and the zero point continues to move down the capillary. At t = 1800 s, the zero pointis at z* 0.80. At steady-state, Figure (5.18), the solute concentration distribution has highergradients than the initial state, especially at the venular end of the tissue near the capillary.The hydrostatic pressure distribution resembles the solute concentration distribution. Thegradients are particularly steep at the extreme venular end of the tissue adjacent to themembrane. The Peclet numbers are higher throughout the tissue than the initial stateindicating increased fluid velocites and thus a higher degree of convective transport. At thenew steady-state, the dimensionless average tissue solute concentration is equal to 0.6881.The average hydrostatic pressure is equal to 0.0189 (0.5670 mmHg) and the resulting lymphdrainage is 6.34 liday.From the above simulations it is clear that the lymphatic sink also plays a role in fluidbalance regulation. When reabsorption occurs due to the elevated capillary soluteconcentrations (via its colloid osmotic pressure), then less fluid is withdrawn from the tissuevia the lymphatic sink. As the solute concentration builds-up at the capillary wall, thetransmembrane potential becomes more positive and the amount of reabsorption decreasesand the hydrostatic pressures rise. The lymph drainage increases as a result. An equilibriumis established based on the fact that the increased solute removal by the lymphatics offsetsthe high solute flux across the capillary membrane. The amount of reabsorption is dependenton the transmembrane potential which is in turn is a function of the solute concentrationswithin the tissue. With a suffficiently strong sink, solute concentrations would be loweredleading to fluid reabsorption.Chapter 5 :CAPILLARY - TISSUE FLUID RALAIWE^ 1465.3.4 Transient Analysis of the Effects of Osmotic PressureThe influence of the osmotic pressure on the system was examined in this section.Two cases were performed; the first including colloid osmotic pressure effects and thesecond without any colloid osmotic pressure driving terms throughout the system. Thesystem was perturbed at steady-state (the base case solution) by resetting the capillary soluteconcentration to zero. This means that there is no longer any solute entering the system andthus solute concentrations will fall in the tissue due to the lymphatic drainage.The average tissue and lymphatic fluid solute concentrations are displayed in Figure(5.19). As can be seen, the solute concentrations drop rapidly to zero. For the case wherethe effects of the osmotic pressure are not included, the osmotic pressure gradients may betaken as equal to zero. For the case under the influence of the colloid osmotic pressureeffects, fluid is dragged out through the lymphatics causing relatively high gradients in theosmotic pressure, especially at the venular end of the tissue. The hydrostatic pressuresthroughout the tissue begins to fall in response to the depletion of solute. Consequently, thelymphatic drainage drops therefore lowering fluid velocities throughout the interstitium andacross the membrane. Initially, the hydrostatic pressures are greater throughout the tissue forthe case with osmotic pressure than the case without it. This means during these times,lymphatic drainage is greater in the case with osmotic pressure than the case without. This iswhy the solute concentrations for the case with osmotic pressure are lower, i.e., initially, thesolute is dragged out of the tissue at a faster rate than the case without the osmotic pressure.As the solute concentrations drop within the tissue, the colloid osmotic pressure playsa smaller role in determining the hydrostatic pressure and the profiles becomeindistinguishable.Osmotic Pressure Effects On— (3— Average Tissue Solute Concentration—0— Lymphatic Fluid Solute ConcentrationOsmotic Pressure Effects Off—6— Average Tissue Solute Concentration—9— Lymphatic Fluid Solute ConcentrationChapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 147•20.50.4•a0.30.2OC.)a0.10.00^1000^2000^3000Time (s)Figure 5.19 : Average tissue and lymphatic fluid solute concentrations through time.From Figure (5.19), it appears that the colloid osmotic pressure enhances soluteremoval from the system at early times for higher solute concentrations.The transient contour plots for the case with colloid osmotic pressure effects includedare displayed in Figure (5.20)-(5.24). The inital state is the steady-state base casedistributions presented in Chapter 4. The solute removal by the lymphatic sink creates astrongly radial solute concentration distribution. Also the relatively greater washout ofsolute at the arteriolar end of the tissue promotes lower solute concentrations at the arteriolarend of the tissue near the capillary membrane.Initially in the transient response, the hydrostatic pressure distributions are similar tothe solute concentration distributions. At low solute concentrations, the hydrostatic pressuredistribution approaches the potential distribution. This is because the colloid osmoticpressure approaches zero. For the case without osmotic pressure effects, the hydrostaticpressure distribution is constant through time and is identical to the potential distribution.The potential distribution for the case with osmotic pressure shifts in the positive direction ascaChapter 5 r(Al'ILLARY - TISSUE FLUID KALAME^ 1480.010.020.03-0.04- 0.0200^0.0200^0.05.3 ^0.0400_ 0.040O^.0.0600^0•0400.........^..........^.......... -^........................................----0 0400..........g::::-^0.0800 ----0.0800-----01-0800^.o.otsoo=-=cs.os000.osoo .. ..............................0 :078 .......... (.3.:(,).9...........9...c.T......................................................... ..... ... ........... ...... ................... .....o .t000-----o.l000.........---^0 1.^.^. ...-...:-.O.:i.i...O..6.....4.0..0 --°.-1.2.° ... .....^.........----^0 .1.-6.0..!!. ...........-----. ..............................0.0^0.1^0.20.01-0.03:^ 0.1400 ^0.04-0:0: ............^200^:0.11 200 .........^0:12051 ......... 70:12.00 ......... 0.1.208  00-0.1000 ------0.1000 ---- -- -0.1000 --- -0.1000 ---- -010-0 .......................0.05:..................................................................^...........0.09:-0.10^.--0.0800.......... .................0.-0^0.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.80.01 Peclet Number0.02" .................. - .............................2.30.03-, ..............................2s .............0.05-■. ....^39-9008^.......E-•0^0.06-^..........^..9.94  0.07 - 9.... e0.2^\I • • • ..... i ^0.3^0.4 0.5^0.6^0.8(1)-670/Part ......... ...‘.1'.• ..711/0-7.7 (if i 7 if / is40Nk :^. o^0 : o^o ^o--■.--------^ in^43coCO 03^0... 40^40 400.^ . 4-4ci^d^-,d^•-■c;^-,d^ •^( ^i ^li ^i ^I ^1^i^ ^.0.8^0.9I 1 1 1 1 A A 1 A A A A A A 1AAAAAA AA •■••••■■■••■■■■•■•ANAAAAAI III I I 1 A AA AlAA AAAAA••• •■•■■■■■•■■■■■■■■■•••■AA A A 1 I1 1 I 1 AAAAAAAA‘•••■■■•■■•■■ • • • • ..... • • • • -0 ■■•NA AA A I 11 1 1 A A A AAA••••■ ....... ■■•■■■■•■■■■■■■■ ....... ■■■■■1 II 1 I 1 A AAA•■•■■■■ ......................... ■■■•A% 1 1 11 A ^AlIA0.2^.0.3^0.4 0.5^4.6^0.7Distance down Capillary (dim)Figure 5.20: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with colloid osmotic pressure effectsthroughout the system and zero arteriolar capillary solute concentration (cart* = 0.0000) at t= 600 s.Solute Concentration0.3^0.4^0.6^0.6^0.7^0.8^0.0^1.0Hydrostatic Pressure..^....•. •........ ... .........^•41)cb .A (^cb.0.7^0.90.9^1.01.00.010.02:0.03:0.04-0.05:0.08:0.07:0.08:0.09:0.100.010.02:0.03:0.04:0.05-0.080.07-0.08:0.09:0.100.0 0.1 0.2^0.4 0.5•0.6 0.7Velocity Vector Field••••A,1.00.0 0.1 0I^0i 4.9^1 .0Chapter 5 :(APILLARY - TISSUE FLUID BALANCE,^149Solute Concentration. . .. . .... . . . . . ........ . . . ...... . .. . ... .. . .... . ...... .. .... . ..... .... . ..... . .... . ........... . ... ..... . .. . .... . .. . .. . ...... . ... . .. ... .......... . .. .. ......... .............. . ...... ............ . . . .. ......... .... .. . ..... ... . .... . .... .... . .... . . . . ........... . ... .... ......... ........... .................^.........- -^- 0.020 0 70.300 7. .2. .7 0 .0 sto. 0.^.......0.01000.01000.7^0.8^0.9^1.00.01 .^0.02-0.03- ..........................................................................................................................-0.1100- 0.1100 --0.1100- 0.11000.07-^01050- 01 050- 0.1050 ----0.1050---- - ... 1-10 ?.?.. --_-0-.1.0-0-0 .....................................................................................0.04-0.05:. ...................................................0.06:0.06:.,..................................................................... .....0....1.0..0..0. ........... 0..1.0....0.-.0^:0..1 i.) ... ..^........^.._---^..--- -^......^.9..6.0 ....- - - - ...t.)0: ......0.........---"--0.09:0.0 t0.02:0.03-0.05-0.06 0.07:0.08-0.09-0.10  0.0 0.1^0.2 0.3^0.4 0.5^0.6Hydrostatic Pressure0.01..^0.02-0.03::0.04-0.05-0.06-0.07-0.05:0.100.0Peclet Numbers9-99°60.2^0.3^0.4^0.5^0.6^0.7..........-Ii----^!„----• 7,-- -7 -isi is i ..-, ....^• • f0.^:^.....„, ,„,.^,.,,^.^0/^C ;^CS 0.C). . . I • • 1 • • • I., ,,...0 co ..:^ 4^I . I^I^:(p_owpart,E0.01c^ 0.02193......~E-...../;(It:. ............................. ........ ................. -3.9O 0.06^... ... 9..0..8. ................ 9..9..0. ...................9.05-• ................•PI4  9.07^190.08:^.9954LI^0.09. .......10^. ....4 -1Cr^0.0^0.1.tiCI0.1^0.2 0.3^0.4 1.00.8 0.9 1.07 /I/0 10)co^•.,... :^-..-, 0.0.^//0.5^0.6^0.7^0.8^0.9.....^ ........ ,................. ..... ....•0.010.02:0.03:0.04-0.057.0.0670.0770.0870.09:0 100.0^0.1^0.2 0.3^0.4^0.5^0.6Velocity Vector Field10.7^0.6^0.9111 1 11 11 4 111111 4 4 4 4 4 4 4 4 ; 4^4 - ; ; 4^I 1fT TI TIIT 1 1 1 IIIIIttAtt 11111A AttAttAttAttAtit I I 1 1 11 T I I 1 1 1I I 1 AA At AttAIAIAAAANA•■■••■••■■•■AAIIAti 1 I 11 1 1 1 1 1 1 t^AAAAAAANt ••••■•■•••••••••••••■■•■■ At 1 11 1 1 1 1 1^AllAtAtt•••••••••■••••••■•■■■■•••••••■■ 1 11 1 1 1^At ■•■•••••••••••■•■■■■■■■•■■■■•••••■■■111lItiltAtt%•■••■•■•■• ........... ■■■■■■•••••AAN 1 11 1^ANA••••■■•■■ .................................II I. • 10.2^0.3^0.4^0.5^0.6^0.7^0.8--^I ^0.9Distance down Capillary (dim)0.0^0.11.01.0Figure 5.21: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with colloid osmotic pressure effectsthroughout the system and zero arteriolar capillary solute concentration (curt = 0.0000) at t= 1200 s.(lutpter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 1500010109- ................ ............. ... .. ..^.0.08^..^..........0 . 03-.................................................................................................................................................................0.05:0.07:0.02-0.01.....•-'--- 0.0020^.................... .............................................^Solute Concentration0.01,..... -0.1040^-0.1..............^...............................................0.05 -0.03 ".0.04--0.02-.^..... .......  ,.,..i.<40-....f  .....^..0.10  ^„...-"----04-0^..................•.-* .-".0.09 -0.08-0.07.: ................-....._:... - 0.1020-0.1020^_........................................... .............. ..........13.1020^------ - " :---....° ,° 9-:^...--...........^„5,0 . ..- - ,., age_...•■•2^ -^--,2:..:0.06- 7.02.:.---...^0.0^0.1^0.2^0.3^0.4^4.5^4.6^0.7^0.8^0.9^1.0Peclet Number0.01V0 0.02...................:.1^0.04 ^E-'3 0.03..7 :: 3-..^ ..39.9908^39.9908...... 3..9....9.9-0-9. ............................. 3-9....9 90-. ...60.05- ......................•39.9908 .-'•.„.•-.....-<Pi'''.----2 0.06- ---^4-o^0.07 -^ •••- 1...,co 0.08: .:0^0.09- ..... ._ „,..-----0^0.10 -  ..... ....--'+2N^0.0^0.1^0.2^0.3^0.4^0.5^0.6...,Q0.01 . ^0.02-,0.03:-0.04:0.05:^4?^7 7 ; 7•..^o ^a ^no/^0.06- 0" -..^0°^• a^: 0-^..- 0.07- •-,:--- 0' ti •^•••-0.10 - ^ I ^ ,^ I /i^ii•.^ol .:^6cf0.090.08 -• .10.0 0.1 0.2 0.3^- -------r---- : ---- I - j•  0.4^0.5^0.6^0.7Velocity Vector Field0.1^0.20.9Figure 5.22: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with colloid osmotic pressure effectsthroughout the system and zero arteriolar capillary solute concentration (cart` = 0.0000) at t= 1800 s.Hydrostatic Pressure..........................................0.7^0.8 0.9^1.0P-criT)/Part0.8^0.9 1.00.010.03:0.04-0.05:0.060.07 40.08-0_b .00.0111111111 iiiiiitt1111111111;;;;;;;;;;;;;;;;;;;;;;1 I 1 1 1 1 1 1 1 1 1 I I 1 littliltiANAiAlAAAAAAAAAAW l t I I 1 11 1 1 1 1 1 1 1 1 I t t l t 11AWAtIltAAA• •■■■■■AAAANAAAAAA Al 1 1 1 1I 1 1 1 I I I t I t AAAA'1AAl'1%%A ‘ANA■‘AN■■■■■■•■■•■■■At 1 1 1 11 I 1 I WI AAAAAAAAN‘NNNNNNN,■•■••■■■■■■••■••■•■■■ 1 t I 11 1 1 I I t t AA AAAA AAA•■••••■■■■•■•■•■•■■ •••••••• ■ANA At IA 1 1V 1 1 ItIA‘tAAANA■ •••• ■•■•■■■■■•■■■■• •••••••• %A‘i'll 11 1 I AAA■N■••■•••■ ..... ..................... ■NAA  1 IIAAN■ANAI0.3^0.4, ^0.5^0.6^0.7 0.8Distance down Capillary (dim) 1.0o'0.80.7('hapter 5 :CAPILLARY -TLS1JE FLUID /?ALAN( T^ 151Solute Concentration0.010.020.030.040.050.080.070.080.090.100.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Hydrostatic Pressure0.01 0.02:0.03;0.04-0.05;0.06-0.07-0.08-0.09;0.100.0^0.1^0.2 ( ^  ...---.. l^...-' ia ^a•A a^a cv o:^•A a o-. ...,^I ^:^ 1:5^ 4 ^0.5^0.6^0 .7^(3.8' • . . . .. . .. ..... ....Peclet Number....■VSI 1• 0.02-^E-;^0.057. ..........................4:111 .. 40:s. ........................... ................. ....... .. ............ .. .9..9; ..c....................... ..... j111111111 . ..........19.„., 0.06;V 0.07-^co 0.09; ........^ 0.1045f/▪ 30.01 .0.02Lilli>,,/1 ......1(T..../' 7 ....../ i/// ../. i//'^..../0.030^a^a0.^ .-05-, /^0•^A ^:^0..... at •.•^O^.^aa' • A^: a o - IA0.06- .• A.^.• •A • aa' • '^• : A/^I 0. .^o/^i0.09-0.10 ^ I ^ ■•^r- ^. (^. ^----.-1•i^• ^i  ^••..^. •-•-,^I- ■0.0 0.1 0.2 0.3 0.4 0.5^0.6Velocity Vector Field^1111111111111111111111111111;11;;;;;;;;;;;;;;;;III ii till 1 1 1 1 1 1 1 A 1 I^AIAAAAAAAAAAAAAAAAAA^1 A 1 1 1 1 11 1 1 1 1 1 1 1 1 1 Iii A AA AA AAAAAAAAAANANAAAA AAAA AA A A 1 1111111111111 A A A %A AlAAAAAAAAAAAAAAAAAAAAAAAAAA A A 1 1 1 11 1 1 1 1 II 1 A A AAAAAAAWAA■AAAAAAANANAANAAAAAAAA AA A 1 11 1 1 1 1 1 A IA A A A AAA AAAAAAAAAANAAAAAAANAAANA AA AA A AAA 1 11 1111A AA AA AAAAAAAAAAAAAAAAANANAAAAANANAA AA A A A A11f 1 1 AA AAAAANAAAAAA.AA.A.A. ................ ANNAN AI 114. ^ AAA 1 r•-•00.1 0.2^0.3^ .4 0.5^0.6^0.7^0.8 0.9Distance down Capillary (dim)Figure 5.23: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with colloid osmotic pressure effectsthroughout the system and zero arteriolar capillary solute concentration c( art* = 0.0000) at t= 3600 s.0.0^0.1^0.2^0.3^0.4^0.5^0.6(P-a7r)/P„t•.................................^"0.7 0.8 0.9 1.00.010.02-0.03-i0.04-0.05;0.08:0.07:0.08:0.09:0.100.0 1.0 1.00.9 1.00.7 0.8 0.9 1.0Chapter 5 :(7APILL4RY - TISSUE FL(IID BALANCE^ 152Solute Concentration9s5 g*.-•.................^..0.4^0.5^0.60.01 .0.02: ............ ••^t)^........-^-...• ..... 7 ..... r^I ..^700.03^00.04 09^..-^0 :^a^:^a 00.05,___„../ ^..0 : 0^•••• :041? ^a020.08 .^ .-- O. ^....^.:^•••• •-•//^: 0- : 0-^0:. 7 0.09:0.10 ^ 1^il^/ .  -.----- •  ^- I I^ 1 K.--I0.08:i•0.0 0.1 0.2^0.3^0.4(P-crir)/Pare^0.01^0.02.0.03-0.04i0.05-0.08-0.07:0.08:0.09:0.10 0.0" • 'TT"^0.1^0.2^0.3^0.4 0.5^0.6^0.7^0.8 1.00.08: 0.09:0.10 ^ , -- .^. I --^• I-,^.-,-.-1,--. I    0.06-^ 0. ^•-•^--^..•.^`./ •^0- : 0- O.0.03:^943°^ : / 7 ..--^70.04: AP 0^.^0^• 0 a0.05: 0 a •0.07- '^ • / /0.02 ------o.3•^.^.........0.0 0.1^0.2^0.3 0.4^1 I •^1^I^ ^1 ,^: • a 0Hydrostatic Pressure..., .^41/ CI0.5 0.6 0.7^O.8^OA^1.0E700.08^Peclet Number0.02^ ..^..... ^......001•r-:^•0.05.:.......................................39.9"8-39.9908......... ................................ .............-140 0.06:0.07,€0^...^........*3^0.03-.. .......................................02 0.04 39.9908------. ....................... ..--'...:O 0.09; ...... -._O 0.10.tea^0.0^0.1^0.2^0.3coA0.5^0.6 0.7^0.6 0.9^1.0Velocity Vector Field0.01 . 0.02:0.03-10.04-0.05:0.08:0.07:0.08:0.09:0. 101I111110.01^1111111111111W t111111111 4^; 4 ;^; ; ; ; ;^; ;^; ;I^1 1^1 1 1 1 1^1 1^1^1 1 1 1 1 1 1 1 1 Al^ANA^1 tAAAA11 1 1^1 1^11^1 1^1 1^1 1 1^1 1 1 1 1 1 1^1^1 1^11^I 1^1 1 1 I II 1 t AIIINAIIIIIIA■I■A■■••AAAAAI■IAIIIIIII1^1 1^1 1 I I I I 1 1 1 AAAA 1 • 1 •■•■■•••■•■•■•■•1■lI1A11IIII 11^1 1 1 It AlAA IIAANAANN■•■•■■■■•■■■■•■•■■■•■■•■ t t 1 1 1^11^1 1^1 t AASAAA••••■••■••••■•■■■••■•■•■■•■■ANA 1^11^1 t 1 AANNN••■ ♦ • ♦^♦^ ♦ ♦ ♦ •••■At t I^1I 1 ■1 1^1.1.0.1^0.2^0.3^0.4^0.5^0.6^0.7Distance down Capillary (dim)• .10.8^0.9 1.0Figure 5.24: Dimensionless tissue solute concentration, hydrostatic pressure, Peclet number,potential and velocity vector distributions for the case with colloid osmotic pressure effectsthroughout the system and zero arteriolar capillary solute concentration (can* = 0.0000) atnew steady-state.Chapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 153the solute concentrations decrease. This is a direct result of the reduction of the osmoticpressure and its gradients within the tissue through time. The velocity vector pattern issimilar through time. There is no fluid reabsorption during the simulations.5.4 Discussion and ConclusionsThe average tissue solute concentration and hydrostatic pressures, lymphatic soluteconcentrations, and lymph drainage rates for the cases performed in this Chapter are listed inTable (5.1). These values are those at the new steady-state established after the perturbation.Table 5.1: Results summary for transient cases performed at new steady-state.Case Average c, Average Pt Average c, Lymph Flow(Uday)Ps ,„„ * = 0.6667 0.4997 -0.0251 0.4971 5.90c; lowered 0.0620 -0.1289 0.0612 2.42Cart* = 0.10 0.0495 -0.0949 0.0490 3.59car,* = 1.20 0.6881 0.0189 0.6492 6.34cart* = 0.0Osmotic Pr. On0.0000 -0.1014 0.0000 3.37c11 ,/* = 0.0Osmotic Pr. Off0.0000 -0.1014 0.0000 3.37Elevated venous hydrostatic pressure leads to higher tissue hydrostatic pressures.This means that the lymphatic sink will drain more material from the system. The fact thatthe lymphatic sink can accomodate large flow rates points to it being a regulator of the fluidbalance. As the tissue hydrostatic pressure increases, there is increased lymph drainage.This allows for the removal of excess fluid in the interstitial space which would increasetissue pressures. As pointed out by Guyton et al. (1987), the lymph drainage may increaseChapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 154up to 20 times before edema occurs. It then levels off as indicated by the work of Taylor etal. (1973). This supports the idea that the lymphatic sink initially handles the increase in thefluid filtration through the capillary membrane maintaining the fluid balance and tissuevolume. Once the tissue hydrostatic pressures are sufficiently positive, the lymph flow doesnot increase further indicating that it is saturated. The reason for this is not understood. It isimportant to remember that the model does not take into account tissue swelling. Withvenous pressure elevation it is expected that tissue swelling would occur. This may explainthe modest decrease in the solute concentration in the tissue. Mortillaro and Taylor (1976)observed that the lymphatic protein concentration fell about 20% for a similar elevation ofvenous pressure as performed in this work. This indicates a reduction in the interstitialprotein concentration. Elevated venous pressure would expectedly lead to tissue swellingresulting in enlarged tissue hydration. This would lower observed tissue proteinconcentrations.The lymphatic sink is also instrumental in providing the negative pressuresthroughout the tissue and lowering the solute concentrations within the tissue space. Withsufficient sink drainage, the solute concentrations would be reduced to low values in thetissue. This is because the transcapillary exchange of solute is limited by the capillaryfiltration and reflection coefficients. Eventually, the tissue solute concentrations will be lowenough that fluid reabsorption occurs. This is because of the phenomenon due to loweredtissue-side capillary membrane osmotic pressures discussed above. Once fluid reabsorptionoccurs, eventually, the hydrostatic pressure will drop in the tissue. Thus the lymphaticdrainage will be reduced and then consequently, the solute concentrations in the tissue willstart to increase once again. This will lead to a reduction in fluid reabsorption back into thecapillary and tissue fluid hydrostatic pressures will start to increase. The cycle of control isthen repeated as the lymph sink starts to drain more fluid and solute again. The soluteconcentration acting through the osmotic pressure provides the feedback signal to the systemto adjust the flow pattern. The signal that controls the lymphatic sink drainage is the tissueChapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 155hydrostatic pressure. Any increase in the tissue hydrostatic pressure leads to an increasedlymph drainage and thus a lowered solute concentration. Lower tissue solute concentrationslead to lower tissue osmotic pressures which in turn then result in higher tissue hydrostaticpressures via the transport through the capillary membrane phenomenon.The control of the interstitial fluid balance is closely tied with that of the bloodvolume. The lymphatic sink and capillary membrane through their transport propertiesregulate the flow of fluid and solute between the plasma and tissue compartments. Theosmotic pressure, as mentioned earlier, provides a negative feedback signal controlling thefluid balance across the capillary wall via the transcapillary potential difference. This leadsto the control of the blood volume. When the venous pressure rises, more fluid filters intothe interstitial space. This in itself reduces the blood volume. As the tissue soluteconcentrations are lowered due to removal by the lymphatic sink, this promotes increasedfluid reabsorption. This will increase the blood volume.The following conclusions can be made :1. The osmotic pressure plays an important role as a feedback signal for fluid balanceregulation. It serves to regulate the amount of fluid reabsorption into the capillary.2. The controlling mechanisms for capillary-tissue fluid balance arise from the cyclingeffects of the hydrostatic pressure and solute concentrations within the tissue and thecapillary. This can be established as follows. Lowered tissue solute concentrations resultin a reduction of the tissue hydrostatic pressure due to fluid reabsorption. This meansless flow exits via the lymphatics and thus the solute concentrations increase leading toless fluid reabsorption. The hydrostatic pressure rises because of both the lesserreabsorption and as a consequence of its response to the increasing osmotic pressure.This leads to greater lymph drainage and eventually lower solute concentrations.3. The lymphatic sink serves as a controlling feature of the capillary-tissue fluid balance.Its role is to remove excess fluid and solute from the interstitial space. The regulation ofthe solute concentrations in the tissue are key for the occurrance of fluid reabsorptionChapter 5 :CAPILLARY - TISSUE FLUID BALANCE^ 156back into the capillary. This is directly dependent on the local tissue hydrostaticpressure.Chapter 6 : CONCLUSIONS AND RE(OMMENDATIONS^ 157Chapter 6 : Conclusions and RecommendationsA transient, spatially distributed two-dimensional model of the microvascularexchange system has been developed and successfully implemented in cylindricalcoordinates. For the base case parameters selected, the calculated values for the averagetissue solute concentrations, hydrostatic pressures, and lymph drainage are in the range ofvalues expected for human tissue. The effect of these physiological parameters on thesystem have been investigated. It appears that some of the parameters have a larger impacton the system than others.In particular, the capillary membrane filtration coefficient and lymphatic sinkstrength dominate the fluid flow structure of the system. The capillary membrane filtrationcoefficient limits the amount of fluid entering the system. The high fluid flow resistance ofthe membrane relative to that in the tissue suggests that fluid flow is controlled entirely bythe capillary filtration coefficient. This is clearly shown by the significantly greater potentialdrop across the membrane relative to that across the tissue. The strength of the lymphaticsink in combination with the high flow resistance of the capillary membrane provides theopportunity for negative hydrostatic pressures (subatmospheric) within the interstitial space.The lymphatic sink also provides the primary mechanism for removal of solute from thetissue. Fluid reabsorption may be promoted based on the values of LS, PL, and Lp. Theaverage tissue solute concentration, hydrostatic pressure, and potential distributions arelargely functions of the transport properties of the capillary membrane and the lymphaticsink.Chapter 6 : CONCLUSIONS AND RECOMMENDATIONS^ 158Dispersion was not found to play a major role as a transport mechanism based on theassumed dispersivity values. This was found despite the highly convective nature of thesolute transport across the capillary membrane and within the tissue. High convectivetransport promotes greater dispersion.From the results, the colloid osmotic pressure cannot be ignored in any formulationof microvascular exchange. Gradients in the colloid osmotic pressure are significant in thetissue and contribute to the fluid motion. The hydrostatic pressure distributions are usuallyquite similar to the solute concentration distributions. This is because the hydrostaticpressure responds directly to the solute concentrations via the osmotic pressure. This wouldoccur for at greater than zero. Of course, as the tissue reflection coefficient is reduced, theeffect of the colloid osmotic pressure on hydrostatic pressure distribution would diminish.The fluid potential is the driving force for fluid movement throughout the system.The capillary-tissue fluid balance is regulated by a combination of the osmoticpressure, the lymphatic sink, and transport through the capillary membrane. The colloidosmotic pressure serves as a negative feedback signal controlling the influx of fluid into theinterstitial space by maintaining the transcapillary potential difference. The lymphatic sinkalso acts as a regulatory mechanism for fluid balance. It serves to reduce soluteconcentrations within the tissue thus reducing the transcapillary potential and promotingfluid reabsorption. This lowers the hydrostatic pressures within the interstitial space thusreducing the lymph drainage, which would eventually lead to higher solute concentrationsand therefore less reabsorption. The hydrostatic pressure would then begin to increase onceagain. This is intimately tied to maintenence of the blood plasma volume.The following recommendations are made for possible future efforts as extensions ofthe present work :Chapter 6 : CONCLUSIONS AND RECOMMENDATIONS^ 1591. The model is limited in that the tissue is assumed rigid and nondeformable. This meansthe effects of hydration and fluid accumulation on the fluid balance cannot be studied.This does not permit the observation of edema within the tissue. The inclusion ofswelling in the model formulation would enhance the model's utility and aid in observingthe regulation of interstitial fluid volume. This may be implemented as a firstapproximation as being one dimensional swelling in the radial direction. This could beaveraged over the entire length of the tissue thus producing a radial increase or decreasein volume depending on the compliance relationships and fluid balance. In this manner,the need for calculating stresses at each element in the tissue is avoided simplifying theproblem greatly.2. The description of the lymphatic sink may be extended to be more realistic. In thepresent model, it is assumed as a simple linear function of the local tissue hydrostaticpressure. This may be too simple in that there is no limit on the lymphatic flow which isobserved experimentally (Taylor et al., 1973).3. The tissue reflection coefficient and the implications of such a parameter should beexamined from a theoretical and experimental viewpoint. The emphasis would be onderiving or approximating the relationship between the tissue reflection coefficient andthe retardation factor and volume exclusion fractions and its impact on osmotically activesolute particles moving through a fibrous porous medium.4. It is also felt that existing models describing microvascular exchange should first bevalidated by experimental work. The construction of more complex and elaboratemodels may not be adding new understanding to the mechanisms of solute and fluidexchange in the microvascular system unless experiments are done validating them. Atpresent, there is little experimental data to draw from for model validation.NOMENCLATURE^ 160NomenclatureSymbol Description^ Unitsa^Dispersivity^ mA^Area m2c^Solute concentration^ kg m-3dim^DimensionlessD^Solute diffusion coefficient^ m2 s-i6^Dirac delta function or small distancef^Volume fractiong^Acceleration due to gravity^ m s-2j^Solute flux^ kg m-2 s-1k^Anisotropic porous medium permeability^mK^Fluid hydraulic conductivity^ m2 pa-I s-IL^Length^ mr,„^Capillary fluid filtration coefficient^m Pa-1 s- 1LS^Lymphatic sink strength^ m3 fl ni-3 tis Pa-1 s-1A^Fluid viscosity^ Pa s-1NOMENCLATURE^ 161n^Outward normal directionColloid osmotic pressure^ PaP^Hydrostatic pressure PaPe^Peclet numberPS^Solute capillary wall diffusive permeability^m s-1Q^Lymphatic sink drainage or^ s-Ivolumetric flow rate m3 s-1p^Fluid density^ kg m-3r^Radial coordinate directionR^Capillary or tissue radiusa^Solute reflection coefficientif no subscript, then for capillary membraneif subscipt t then for tissuet^TimeDimensionless timeT^Fluid driving potential, (P-an)^ Pa0^Tissue porosityv^Velocity^ m s-iW^General variableRetardation factorz^Longitudinal coordinate directionNOMENCLATURE^ 162AppendicesSymbol Description^ UnitsA^Vector or matrix of elemental function, a(s), valuesa^Upwinding parameterC^Convective matrixD Diffusion coefficient^ m2f^Function defined in solution domainF^Vector of elemental function,f valuesElemental Peclet numberh^Elemental characterisitic length^ mh(s)^Function defined along element boundaryH Vector of elemental function, h(s), valuesK Stiffness matrixA^Eigenvaluesn^Basis or trial functionsN Vector of elemental trial functionsp^Perturbation (upwinding) functions^Coordinate along boundary segmentt^Interpolation functionT^Transient matrixu^General unknown variableU Vector of elemental unknown variablesNOMENCLATURE^ 163v^Fluid velocity^ m s- Iw^Weighting or test function0J^Relaxation factorsx^x-directiony^y-directionSubscripts and Superscripts (including Appendices)Symbol Descriptionart^Arteriolar capillary quantityav^Available volume fraction seen by fluidc^Capillary quantityd^Combined diffusive and dispersive componentsdiff^Diffusive componentdisp^Dispersive componentDa^Darcy's lawe^Elemental quantityqif^Effective quantityex^Excluded volume fraction1^Fluid quantityG^Global matrix quantityith directionim^Immobile fluid volume fractionNOMENCLATURE^ 164nth directiondirection or dummy iteration variableith directionL^Lymphatic quantitylong^Longitudinal directionm^Membrane quantityn^Outward normal directionr^Radial coordinate directionref^Reference quantitys^Solute quantityso^Solid volume fractionst^Volume fraction solute can seeSt^Starling's hypothesist^Tissue quantitytran^Transverse directionyen^Venular capillary quantityz^Longitudinal coordinate directionDimensionless quantityREFERENCES^ I65ReferencesApelblat, A., A. 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Eng. 24 : 2201-2215(1987).APPENDICES^ 171AppendicesAppendix A : The Finite Element Method and the Petrov-Galerkin Method.Appendix B : Solution algorithm and under-relaxation technique.Appendix C : Program Listing.APPENDICES^ 172Appendix A : The Finite Element Method and the Petrov-GalerkinUpwinding Method.A.1 The Galerkin Finite Element MethodIn this section, a brief introduction of the basic theory of the Galerkin finite elementmethod will be presented. The Galerkin finite element method applies the method ofweighted residuals (MWR). The general procedure for implementing the finite elementmethod is presented by applying the method to two-dimensional transient solute transportequation in rectangular coordinates . The treatment the equivalent equation in cylindricalcoordinates will then be discussed. A detailed introduction of the finite element method maybe found in Bickford (1990).The two dimensional transient solute transport equation, an elliptic partial differentialequation, is stated as follows :in C2^ (A.1)u = g(s)^ on L 1^(A.2)du - F u(s)u = h(s)^ on L2^ (A.3)dndu^du^du D(1-1.21+ v^ + v —dt^x ar^' dy= + a-T2121+ f(u,x,y)where u is the unknown variable which is sought and .f(x,y) is an arbitrary functionprescribed within the solution domain O. The first and second terms are the temporal andAPPENDICES^ 1 73convective terms respectively. The third term is the diffusive-dispersive component and thelast term may be considered as a source term. The essential boundary conditions (Dirichletboundary condition) are specified at the boundary segment LI. The natural boundaryconditions (Robins boundary condition) are specified along the boundary segments L2. Ifa(s) is set to zero, then this is referred to as a Neumann boundary condition. The generalgeometry of the problem is described in Figure (A.1).LIL2Figure A.1: General description of elliptic boundary value problem.The general procedure for implementing the finite element method can beapproached in five steps (Bickford, 1990) : 1) discretization, 2) interpolation, 3) weakformulation, 4) formation of elemental matrices, 5) solution of resulting algebraic equation.DiscretizationThe finite element method, like the finite difference method, approximates the soluteof the differential equation at a finite set of point (or nodes) in the solute domain. Theobjective of these methods is to set up an approximate algebraic form of the differentialequation at each node. It is required that the position of nodes be chosen (discretization)APPENDICES^ 174within the solution domain. In the finite element method, the differential equation issatisfied locally within subregions called elements. There is then a requirement that at theboundary of two elements, for instance, the differential equation is satisfied for bothelements and at the mutual boundary. At the absolute boundary of the solution domain, theboundary conditions (A.2) and (A.3) have to be satisfied. A typical discretization for thegeometry above is displayed in Figure (A.2).NodesElementsFigure A.2: Discretized geometry using linearly interpolated rectangular elements.The region is discretized using linearly interpolated rectangular elements. This means thatcurved boundaries have to be approximated with straight line segments. This introduces aninherent npproximation error into the formulation. More nodes are placed within regionsexpecting relatively high gradients.The elements are constructed from any number of nodes (> 3 for two-dimensionalproblems). For linearly interpolated elements, three node elements are known as triangularelements whereas four node elements are known as rectangular elements. With additionalnodes within an element, say eight in a rectangular element, the degree of accuracy increasesand the element boundary may follow a quadratic function shape. For this work, linearlyinterpolated rectangular elements were used.APPENDICES'^ 175InterpolationThe solution at any particular location within each element is approximated by aninterpolation scheme using the values of the solution at the nodes. The linear interpolationfunctions are the simplest to implement. This can be visualized as a plane (the solutionsurface) above the element. This is shown in Figure (A.3).Linear representationof u(x,y) in elementFigure A.3: Rectangular element with linear interpolation.The approximation to the solution within the element, ue(x,.y), is expressed mathematically(Yu and Heinrich, 1987) as :4ue (x,A= to; (A.4)APPENDKES^ 1 76where ui is the solution value at node i and n i is the interpolation function (also known as theshape or trial function) of node i. For the space trial functions defined over the elementregion 0<y..!Xy, the interpolation functions are given by (Yu and Heinrich, 1987) :(x,y) = t 1 (9t2 (1 . )n2 (x, = t2 (x)t, (y)n3 (x,y)= t2 (x)t,(y)n4 (x,y)= t,(9t2 (y)(A.5)where the numbers refer to the nodes in Figure (A.3) and the functions ti(•) is theinterpolation function along the boundary for each node and are given by :t, (z) (1_ Lz^(A.6)t2 (z).AzIt is here that linear interpolation is being used to define the variation of the unknownvariable, u, within the element. If more nodes are placed within the element, then higherorder interpolation functions may be used.Weak formulationThe weak formulation is found by first multiplying the differential equation by anappropriate test function, w(x,y), and then integrating it over the solution domain :APPENDICES^ 177au^du^u^d'u +^ -w(x,y)[ —+ vx —+ v --D1^) (u,x,y)1112 = 0dt^dx Yady^dx2 ay(A.7)This equation satisfies the differential in an average sense within the solution region. It doesnot require that the differential equation be satisfied exactly at every point. In this manner,equation (A.7) is a weaker statement of the problem than equation (A.1). The test function isany arbitrary continuous function since at the solution, the term in the square bracketsvanishes. However, the test function must hold to the property that it vanishes on LI , i.e.,the portion of the boundary at which essential boundary conditions apply. From Green's firstidentity (the two-dimensional form of the divergence theorem), we have (Bickford, 1990) :d'u^uf w(—ax2 + —ay2 )(IQ = an ds - wrw du + aldf2dx dx ay ay (A.8)where n is the outward pointing normal and L is the line segment bounding the region.Substituting equation (A.8) into equation (A.7) gives :flw^ + Jw x — + v , u\du^d (K2+at^L.,^dx^ayD ff (—d" + —dw )(IQ = f w —dnds + ivfdx dx dy dy^du12(A.9)The bounding segment, L, is composed of the two segments L 1 and L2. Since the testfunction vanishes on L1 then the first term on the right hand side in equation (A.9) onlyapplies about the line segment L2. The imposed boundary condition, equation (A.3) maythen be substituted into equation (A.9) resulting in :APPENDICES^ 1 7 8duw du^au \2+ w v — +v — dt.2+at55 ( X ax^Y ayf2 (A.10)dwD w(h(s)- a(s)u)ds +1-1wt. (K2—du + —aw —du )(152=ff(—ax ax^dy ayOr(ffw.du ,r,--ale+ fi w duv x — + v au— (K2+at^ax Y ay, ■D55( aw du + dw du) df2+ i wauds = i whds + if wfdS2ax ax ay dycl ^o(A.11)This is the final weak form of the differential equation. It includes the natural boundaryconditions that may exist if there are any natural boundary conditions on the boundary.Using the appropriate interpolation functions for the variable u and test functions, w, it ispossible to generate a set of algebraic equations describing the system. This is performed inthe next section.Formation of elemental matricesThe unknown variable a is approximated by equation (A.4). This may be rewritten inmatrix notation as :4ujx,y)= n = UT N = N T U(.i=1(A.12)APPENDKE.V^ 179In the Galerkin method, the test function is taken to be the same as the unknown variable,thus equation (A.11) becomes :ff u du , ± jf u vx du +v df2+dt dx y dy 1DirraU)2 ±dU 2) df2+ uu2ds = uhds + 55 ufd.C2.)61 dx )^dy(A.13)Substitution of the interpolated approximations, equation (A.12), into equation (A.13) yieldsthe following discrete form :Tff^aN  U,,d.C1+ If U,7:1V(vx dATT +v XV' jUed.C2+dt^dx^dyDffr.aN^(je+Lf dN aNT  U jdn +dx dx0^ dY dY e(A.14)fU,T,'NuNTUedv= 5 UU Nhds + .11U,,TNT df2The unknown variable, U,, may be taken outside of the integrals resulting in a system offour equations for each node in an element. This is written as :de^is^dNT^dNT\ dn+N^df2+ N v^v ^dt x dx( dAT aNT dN dNT Dfs ^) (11.2+ ir NaNT ds U, = (A.15)dx dx dy dyNhds + ff NidnAPPENDICES'^ 180The functionfcan be approximated by its interpolant within the element, i.e.,= N T F^ (A.16)Also, a(s) and h(s) may be approximated by their linear interpolants along the boundarysegments, i.e.,a= co; +^= N T Ah= hini + h = N T H^(A.17)where nodes i and j are along he boundary segment. These approximations may besubstituted into equation (A.15) to give :[„ .2+ ff N(v x dNT v aN T JJ,^dt^dxDff aN aN T dN aNT  )(K2 + NNT AN T ds1U =dxf NN T Hds + ff 1VN T F (K2L^St(A.18)Or[T,]U e +[C e + K , + AJU =[Fe + H^ (A.19)where Te ,^Ke, and A E are the temporal matrix, convective, stiffness, and natural boundarycondition matrices respectively. Fe and He are the load and natural boundary conditionvectors respectively. The dot signifies the time derivative of the unknown variable. TheAPPENDICES^ 181integration of the interpolation functions in equation (A.18) is handled easily by a Gaussquadrature method (Press et al., 1986).The temporal derivative may be handled by a finite difference backward or Crank-Nicolson method. If a backward difference method is used, then (A.19) is rewritten as :, T+Ce i K9 +^4-1-.-Ut +Fe + He ]At^ At e(A.20)The unknown variable may be evaluated from this equation for the nodes within an element.However, since each node may have communication with other nodes outside the particularelement, then the equations have to be solved simultaneously for all other elements. Forfour-node, rectangular elements, each node will be in communication with four nodes. Thisis displayed schematically in Figure (A.4).0Element 1^Element 2Element 3^Element 4Figure A.4: Node communication.This means that the center node c will form four separate equations with the four elements 1,2, 3, and 4. Each equation will form elemental entries which apply globally to the entirewhereKG = L Telements AtI A„elements onboundarysegments(A.22)and= E Ft/et+ Fe j+ I Heelements 'At^elements onboundarysegments(A.23)APPENDICES^ 182system and thus they are incorporated into a global stiffness matrix. In this manner, theglobal stuffness equations are formed from the sum of the elemental matrices :[KG }t4;+°: = V;^ (A.21)Solution of the resulting algebraic equationsThe system of equations described in equation (A.20) may be solved using any of thedirect of iterative system of equations solvers (Press et al., 1986; Bickford, 1990). For thiswork, a banded direct method solver (Gauss-Jordan elimination) was used.Natural boundary conditions are automatically dealt with in the Galerkin finiteelement method. Essential boundary conditions have to be implemented separately. Sincethe value of u is known at the boundary segment for an essential boundary condition, thenthe equation associated with this node may be eliminated and the global stiffness matricesadjusted to reflect this change. On the other hand, the equation may be retained butmanipulated such that it solves for the specified boundary value. This is discussed further inSmith, 1990.APPENDICES^ 183Cylindrical Geometry and Variable PropertiesThe problem for cylindrical geometry can be handled in a similar manner to theabove formulation for rectangular coordinates. Deriving the problem in cylindricalcoordinates introduces an additional term :1 duarwhich can be simply treated as a convective-like term in the formulation. This is similar tothe situation with spatially variant properties. As can be seen from equation (4.8), thisintroduces two convective-like terms into the formulation.A.2 The Petrov-Galerkin Upwinding MethodIt was found that the straight-forward Galerkin finite element method was insufficientto solve the solute transport problem due to the high convection within the system. Thedominance of the convective terms alter the nature of the differential equation from that ofelliptic to that of hyperbolic-like. Its effect was observed in the form of 'wiggles' and largescale oscillations in the calculated solutions. Convective dominance produces large gradientsin the solution which can only be dealt with by making the finite element mesh very fine orby implementing an upwinding method. Upwinding techniques have been the most popularmethod employed for convection dominated finite difference problems (Richtmyer andMorton, 1967).The upwinding formalism developed by Yu and Heinrich (1986) is used in this work.This is largely based on the streamline upwinding technique first implemented by BrooksAPPENDICES^ 184and Hughes (1982). Artificial diffusion is added in the direction of flow. The method willsimply be presented here. Detailed theoretical justification of the Petrov-Galerkin upwindingformulation may be found elsewhere (Brooks and Hughes, 1982; Yu and Heinrich, 1986;Donea, 1984).The upwinding technique is implemented as a modification of the weighting functionof the convective term. This modified weighting function vanishes for the other terms andso can be applied to all terms thus defining a consistent finite element formulation. Theweighting functions are considered to be of the form :w,=n,+ p, (A.24)where n are the interpolation functions given by equation (A.5) and p is a perturbationfunction that adds the appropriate amount of artificial diffusion. Yu and Heinrich (1987)give the perturbation function as :h dn. v d (A.25)n.).= oc—211v11 Y dywhere h is a characteristic element length, a is a parameter which optimizes the size of theperturbation function, and 'MI is the Euclidean norm of the average velocity in the element.Equation (A.25) represents additional anisotropic diffusion added to the convective term ofthe solute transport equation of the form :avh d2u2 aVAPPENDICES^ 185where is the coordinate in the flow direction. This can be arrived at by substituting theweighting function in equation (A.24) into the finite element equation (A.18). The optimalvalue of the 'parameter a has been found to be given by (Yu and Heinrich, 1987) as :a = coth(1— —22 y(A.26)where y is defined for each element as :y _ D^ (A.27)where D is the local diffusion coefficient. The charactersitic element length for rectangularfinite elements is given by (Yu and Heinrich, 1987) :h = 11-11 11 (iv xi& +11) YIAY)^(A.28)where Ax and Ay are shown in Figure (A.3).The perturbed weighting functions are easily incorporated into the program andintegrated along with the other terms in the finite element equation (A.18). The upwindingroutines incorporated into the program constructed for this work were tested successfullyagainst the examples provided in Yu and Heinrich (1987).APPENDICES^ 186Appendix B : The solution algorithm and under-relaxation scheme.B.1 The Solution AlgorithmThe solution algorithm is presented in Table (B.1). The convergence tolerance wasset as 10 -6 for both the solute concentrations and the hydrostatic pressures. If the problemdid not converge within 100 iterations, the procedure was terminated.Table B.1: Overall solution algorithm.Step0. Set initial estimates for the solute concentrations and hydrostatic pressureswithin the tissue space.1. Calculate the colloid osmotic pressure throughout the interstitium.2. Obtain the tissue potential by solving the fluid conservation equation using thehydrostatic pressures for the sink drainage term.3• Evaluate the fluid velocities within the tissue and across the capillary wall.4. Calculate the hydrostatic pressure distribution throughout the interstitium.5• Obtain the solute concentrations by solving the solute transport equationthroughout the tissue space. Use the last estimates of the solute concentrationsfor the sink drainage term.6. Compare the solute concentrations and hydrostatic pressures from the previousestimates and if they differ by less than the tolerance, then the procedure iscomplete and this time step is complete. If not, go to step 1. and repeat theprocess.This procedure has to be repeated for each time step. The initial guess of the valuesat the following time step may be estimated from the first two terms of the Taylor's seriesexpansion :APPENDICES^ 187du= u` + At Tt(B.1)where u may be the solute concentrations or the hydrostatic pressures. This provides a roughestimate of the variable at the next time step. This is not very efficient at the first few timesteps since the estimate of the temperal derivative is not very good. At later times when agood estimate of the time derivative has been evaluated, equation (B.1) accelerates theprocess to steady-state.If the hydrostatic pressure in the capillary is not assumed linear, then the solution ofthe fluid conservation equation, equation (3.6), coupled with Starling's hypothesis, equation(3.33), has to be performed iteratively. This is because the the capillary hydrostatic pressureis required as a known value in the Starling hypothesis for the membrane boundary conditionfor the solution of the fluid conservation equation. On the other hand, the capillary pressureis a function of the tissue-side membrane hydrostatic pressure and requires its value for thesolution of the capillary hydrostatic pressure. The solution algorithm used for this work islisted in Table (B.2).Table B.2: Capillary hydrostatic pressure-tissue hydrostatic pressure solution algorithm.Step1. Using prior estimates of the tissue-side (from the last time step or overalliteration), solve for the capillary hydrostatic pressure using equation (3.36).2. Using the newly evaluated capillary hydrostatic pressure, solve for the tissuehydrostatic pressure distribution obtained from the solution of the fluidconservation equation.3. Repeat steps 1. and 2. until the tissue-side membrane hydrostatic pressures areless than the specified tolerance.APPENDI('ES'^ 188If the solute concentration is allowed to vary down the capillary, the identicalsolution is adopted as above for the capillary and tissue hydrostatic pressures. Here thesolute transport equation and Patlak equation have to be satisfied as well as the solutedepletion or addition to the capillary.B.2 The Under-relaxation techniqueThe fluid conservation and solute transport equation are coupled due to the colloidosmotic pressure. The osmotic pressure is a non-linear function of the solute concentration.This can cause problems for successful convergence and so an under-relaxation method wasadopted to attempt to ensure convergence if possible. The technique used was the dominanteigenvalue method suggested by Orback and Crowe (1971). This method often acceleratesthe convergence of the problem. The method is presented as follows. At two consecutiveiterations k and k+1, the change in the solution may be given by the vectors duk and duk+/.The general variable u may be the solute concentrations or hydrostatic pressures. The vectorduk+i, for example, is evaluated as the difference in the solution between the k and (k+1)stiterations. The general dominant eigenvalue method of Orbach and Crowe (1971) suggeststhat the dominant eigenvalues of the solution matrix can be roughly estimated from theseconsecutive differences :A. = du. k-1du g*.(B.2)where A, are the estimated eigenvalues. The relaxation factors are then calculated from :1co = ^ (B.3)1— A;APPENDICES^ 189and the accelerated solution evaluated from := ( —^k+1 W i lli^ (B.4)This solution is then used in the next iteration. As the residual difference between theiterations diminishes as convergence is achieved, the eigenvalues approach zero and thus therelaxation parameters approach unity. This acceleration procedure was implemented in theprogram constructed as part of this work.APPENDICES^ 190Appendix C : Program Listing.The program constructed was written in C and run on both an IBM 3090 using theMetaware High C compiler and then on a IBM RS/6000 using the standard IBM C compilersupplied with the unit./*MVE .0AXISYMMETRIC PROBLEM - SOLVES 2D ADVECTIVE DISPERSIONEQUATION TRANSIENTLY IN POROUS MEDIA USING THE PETROV-GALERKIN FINITE ELEMENT METHOD OSMOTIC PRESSUREEFFECTS TAKEN INTO ACCOUNT.THE ARGUMENTS TO THIS PROGRAM ARE AS FOLLOWS :mve cvpd b 1WHERE :c IS THE CONCENTRATION OUTPUT RESULTSv IS THE VELOCITY FIELD OUTPUT RESULTSp IS THE PRESSURE OUTPUT RESULTSd IS THE PECLET OUTPUT RESULTSb IS THE CAPILLARY PRESSURE RESULTS FILE NAME1 IS THE LOG FILE NAMEcvpd IS A SINGLE FILEWRITTEN BY IAN GATESCHEMICAL ENGINEERINGUNIVERSITY OF BRITISH COLUMBIAMAY 1992*/#include <stdio.h>#include <stdlib.h>#include <math.h>#include <float.h>#define I int#define F float#define D double#define C char#define V void#define PI 3.14159265352#define PI2 6.28318530704/*PROGRAM CONSTANTS :NM : NUMBER OF NODESEM : NUMBER ELEMENTSMM : NUMBER OF NODES ABOUT DOMAIN PERIMETER (ON BOUNDARIES)IM : NUMBER OF NODES ON CAPILLARY WALL (>nz)NJ : NUMBER OF COMMUNICATION INTERVALS ALONG CAPILLARY WALL (>IM)HM : NUMBER OF ODE'S TO SOLVE IN BVP SOLVER (DO NOT ADJUST)RW : NUMBER OF NODES IN R DIRECTION FOR RESULTSZW : NUMBER OF NODES IN Z DIRECTION FOR RESULTSr/#define NM 615#define EM 545#define MM 280#define IM 65#define NJ 103APPENDICES^ 191#define KM 4#define RW 25#define ZW 105/* DEFINED DATA STRUCTURES*1typedef struct (I error, nf, k;D err, t, x[HM], tt[NJ], xx[NJ], ) M;typedef struct (I iter;D p[N14], r[NM], z[N14], ptINMI, pcap[IM], qcap[I14]; } N;typedef struct (I iter;D c[NM], ccap[I14], ctis[IMI, fmem[IM], mpe[IM]; ) 0;typedef struct (D m[NM][NM], v[N14]; } U;typedef struct {I typ_lp, mnod[IM],D asq, c_art, sigma, pi_cap, pa, ps_d, diff,1p0, 1pl, lref, cref, pref, wall_th, wall_th_d,r[I14], z[IM1, hp[I14], op[IM], vm[IM], cP[IK]. ct[IK]; ) membr;typedef struct (I cam;D mu, rho, por, hind, ls, vol_tis, alfl, alft, diff, sigmag,plymph, Jo, fex, fat, fav, fs, fim, lref, cref, pref, kref,hi_k, 10_k;^media;typedef struct (D q[NJ], r[NJ], s[NJ];^vec3;typedef struct (D rr[N14], rz[NM], zz[NM); I vdiff;typedef struct (D v[6]; ) W;/*GENERAL DATA CHECKING ROUTINES*1I safechk(I, I, I, I, I, I);I datachk(I*);/*FINITE ELEMENT STIFFNESS MATRICES ROUTINES0* atif(I, I, I, I, D*, 0*, I*, D*, D*, D*, 0*, D*, D*, 0*, D*, D, I, I, I, I);/*MODIFY STIFFNESS ROUTINE FOR BC'S*/U modstif(I, I*, D*, D*, I*, D*, 0*, I*, I*, D*, D*, D*, I, I);/*DIFFUSION AND DISPERSION MATRICESvdiff dspc(I, D, D, D, D*, 0*, I);/*FLUID PRESSURE AND VELOCITY ROUTINESN itpruruz2d(I, I, I, I, I, I, I, I, media, membr, I*, 1*, 0*, 0*, D*, D*,0*, D*, I*, D*, I*, I*, D*, D*, I*, I*, I*, D, D, D, D, I, I, I, I, I);D *cmpd(I, membr, D, D);D *mempr(membr, D, D*, D, D);/*SOLUTE CONCENTRATION ROUTINES*/O itcsccap2d(I, I, I, I, I, I, I, I, media, membr, I*, D*, D*, 0*, D*,D*, 0*, D*, D*, D*, D*, D*, I*, D*, I*, I*, ID*, 0*, 1*, I*, I*,D, D, I, I, I, I, I);D *cmcd(I, membr);D *memcs(membr, D, D*, D, D. D);/*USER DEFINED 1p() FUNCTION.APPENDICES^ 192D 1p(I, D, D, D);/*MASS BALANCE^ massb(I, I, I, I, media, membr, I*, I*, D*, D*, D*, D*, D*, D, D, I, I);/*AVERAGE QUANTITIES FOR SOLUTION DOMAIN*1^ domavg(I, media, I*, D*, D*, D*, D*, I, I);/*COLLOID OSMOTIC PRESSURE ROUTINED *osmopr(I, media, D*);/* ODE AND BVP SOLVER ROUTINES (PRED-CORR ODE SOLVER)*M pdcr(I, membr, D*, D, D, I, vec3, vec3, vec3, I, I);D *rk4(I, membr, D, D, D*, D, D, D, I);/*CUBIC SPLINE ROUTINESvec3 fspl(I, D*, D*, I, I, D, D);D *tridiag(I, D*, D*, D*, D*);D evalcs(I, vec3, D*, D*, D);D devalcs(I, vec3, D*, D);D d2evalcs(I, vec3, D*, D);D ddpoly(D*, D * );/*MATH FUNCTION ROUTINES* /D sq(D x){ x *= x;return x; )D norm(D x, D y){ D z = sqrt(sq(x)+sq(y));return z; }D coth(D x)D ql, q2;if (x^0.0) {puts(" E: coth()-> Domain error.\n");exit(0); }if (fabs(x) > 10.0){ x = 1.0; }else {ql = exp(x);q2 = exp(-x);if (ql q2)( x = (ql+q2)/(ql-q2);else( x^1.0/x; } }return x;DATA RECORDING ROUTINES^ outdata(I, I, I, I, I, I*, I*, F*, F*, D*, D*, D*, D*, I, D, C*, D*, FILE*);^ outuv2d(I, I, I, I, I, I*, I*, F*, F*, D*, D*, D*, D*, I, D, C*, D*, D*, FILE*);^ outcoll(I, I, I, D, D*, D*, D*, C*, FILE*);^ outcapp(I, I, D, D*, D*, D*, D*, D*, D*, D*, C*, FILE*);^ contour(I, I, I, I, I, D*, D, C*, FILE*);^ header(D, D, D, membr, media, FILE*);/*MATRIX ROUTINES AND ITERATIVE SOLVERSAPPENDICES^ 193D *solver(I, I, D*, D*);D *matvecbw(I, I, I, D*, D*);/* ELEMENTAL STIFFNESS MATRIX INTEGRATION AND CONSTRUCTION ROUTINESD 11(D, D);D 12(D, D);D 13(D, D);D d11(D, D);D d12(D, D);D d13(D, D);D d211(D, D);D d212(D, D);D d213(D, D);W fnrzt(D, D, D, D);W fmrzt(D, D, D, D);W fwrzt(D, W, W, W, W, W);W fdndr(D, D, D, D);W fdndz(D, D, D, D);W fdmdr(D, D, D, D);W fdmdz(D, D, D, D);W fd2mdr2(D, D, D, D);W fd2mdrdz(D, /4 D. D);W fd2mdz2(D, D, D, D);W fd3mdr2dz(D, D, D, D);W fd3mdrdz2(D, D, D, D);D gw(I, I);D gp(I, I);D gw2(I);D gw3(I);D gw4(I);D gw6(I);D gw8(I);D gw16(I);D gp2(I);D gp3(I);D gp4(I);D gp6(I);D gp8(I);D gp16(I);/*M A I N ( )READS IN DATA AND IMPLEMENTS SOLUTIONS OF EQUATIONS AT EACHTIME STEP. RECORDS CONVERGED SOLUTION INTO A TEMPORARY FILEIN CASE OF SYSTEM SHUTDOWN.main(I argc, C *argv[])I 1, j, k, 1, m, n, 11, jj, bw, cdt, ndt, ict, crz, diap, doitpr,ic=1, irec, geom, ihof, ndnm, ndnmout, elnm,nncbc, necbc, nnpbc, nepbc, nesink, nrec, nr, nz,nrout, nzout, ngp, cmaxit, vmaxit, icontour, lincap, wtf,relaxc, relaxp, se, pert, pert_osmp;I drec[20], inchk[50];I cbcnod[MM], cbcele[mm], cbcgrp[MM], cbc1[MM], cbc2(MM],pbcnod[MM], pbcele[MM], pbcgrp[MM], pbcl(MM], pbc2[MM],isink[EM], iout[RW*ZW], nsink[NM], eldef[EM][6];I flow(NM];F wl, w2, w3, w4, w5;F rot[IM], zot[IM];D avgas, ctol, co, dd, ddt, dgamma, dr, dz, dt, dto, dti,dtmax, dtmin, effvol, en, eo, hc, hh, p_art, p_ven,pert_p_art, pert_p_ven, pert_c_art, pert_c_osmp, pn, ndvel,maxcc, maxco, maxpe, maxop, maxdc, maxdp, lref2d,t, ti, tf, rg, rg2, zg, ur, uu, uz, vtol, uc, ww, xm;D trec[20];D cbcndv[MM], cbceva[MM][3], cbcevh[MM][3],pbcndv[MM], pbceva[MM][3], pbcevh[MM][3];D edr[EM], edz[EM];D r[NM], z[NM), pe[NM], hp[NM], op[NM], cm[N14], cn[NM], dc[NM],drreff[NM], drzeff[NH], dzzeff[NM];D *p_q;C line[80], title[60];FILE *in, *outr, *outb, *outl, *outq, *outu, *tmpf;vdiff d;media pp;membr mem;APPENDICES^ 194N pruruz;O co;/*^•/puts(" mve()if ((argc ==( icif (ic == 1)puts(puts(puts(puts(puts(puts(puts(puts(puts(:\n");2) II (argc == 5))= 0; )* F: mve MODEL_fMODEL_f =▪ PRES• POTL▪ PECL▪ CONC▪ VEL▪ CAP_f▪ LOG_fPRES_POTL_PECL_CONC_VEL_f CAP_f LOG_f.\n");input model file");Pressure results");Potential results");Peclet number results");Concentration results");Velocity results");Capillary data file");Log file");exit(0); )/*OPEN INPUT*/if ((in = fopen(argv[1], "r")) == NULL) (printf("\n E: mve()-> %a not found. \n", argv[1]);exit(0); }/•INITIALIZE VARIABLES AND VECTORS*/ict^ic^1;ndnm ndnmout = elnm = nr nz 0;nncbc = nncbc = nnpbc = nepbc = nesink = pert = pert_osmp 0;bw = cdt = ndt crz = disp = doitpr = jj^as = 0;geom = ihof = cmaxit = vmaxit = ngp = 0;irec = nrec = icontour = wtf = relaxc = relaxp = 0;pert_p_ven = pert_p_art = pert_c_art = pert_c_osmp = 0.0;dgamma avgas = co = p_art p_ven pn = 0.0;dr = dz = ddt = dt = dto dti = dtmax dtmin 0.0;t = ti = tf = lref2d = effvol^ndvel = 0.0;dd = he = hh = uc = ur = uu uz = ww = xm = 0.0;ctol = vtol^0.0;maxcc = maxco maxdc = maxdp maxpe = maxop 0.0;for (i=0;i<50;i++)( inchk(i] = 0; )for (i=0;i<NM;i++) {nsink[i] = flow[i] = 0;r[i] = z[i] = op[i] = hp[i] = 0.0;d.rr[i] = d.rz[i] = d.zz[i] = 0.0;drreff(i] = drzeff[i] = dzzeff[i] = 0.0;cn[i] = cs.c[i]^cm[i] = dc[i] = 0.0;pruruz.r[i) = pruruz.z[i] = pruruz.p[i] = 0.0; }for (i=0;i<MM;i++) (cbc1[1] = cbc2[i] = pbc1[1] = pbc2[i] = 0;cbcele(i] = pbcele[i] = 0;cbcnod(i] = pbcnod[i] = 0;cbcgrp[i] = pbcgrp[i] = 0;cbcndv[i] = pbcndv[i] = 0.0;for (j=0;j<3;j++) {cbceva[i][j] = cbcevh[i][j] = 0.0;pbceva[i][j] = pbcevh[i][j] = 0.0; ))for (i=0;i<EM;i++) (isink[i] = 0;edr[i] = edz[i] = 0.0;for (j= 0 ;i< 6 ;i++){ eldef[i][j] = 0; } }for (i=0;i<RW*ZW;i++){ iout[i] = 0; )puts(" Work vectors and matrices initialized.");/*READ MODEL DATA*/ic = 0;while ((feof(in) == 0) && (ic < 120)) {/*^*/fgets(line, 80, in);/*^*/if (strncmp(line, "$end input", 10) == 0) (inchk[0] = 1; )elseif (strncmp(line, "$beg input", 10) == 0) (inchk[1] = 1;fgets(title, 80, in);puts(title); )else= 0) (APPENDICES^ 195if (strncmp(line, "$prob size", 10) == 0) {inchk(2] = 1;fscanf(in, "%d %d %d %d %f %f", &ndnm, &elnm, &nr, finz); )elseif (strncmp(line, "$transient", 10) == 0) {inchk[4] = 1;fscanf(in, "%d %f %f %f", &ss, &wl, &w2, &w3);t = ti = wl;tf = w2;dt = dto = dti = w3;fscanf(in, "%e %e %d', &wl, &w2, &ndt);dtmax = wl;dtmin = w2; )elseif (strncmp(line, "$max iterations", 15) == 0) (inchk[5] = 1;fscanf(in, "%d %d", ficmaxit, &vmaxit); )elseif (strncmp(line, "$dispersivity", 13) == 0)inchk[6) = 1;fscanf(in, "ese %e %e %d", &wl, &w2, &w3, &crz);pp.alfl = wl;pp.alft = w2;pp.diff mem.diff = w3;if ((pp.alfl 1= 0.0) I) (pp.alft != 0.0))disp = 1; )else{ disp^0; ) }elseif (strncmp(line, "$fluid sink LS", 14) == 0) (inchk[7] = 1;fscanf(in, "%e", &wl);pp.ls = wl; }elseif (strncmp(line, "$porosity", 9) .= 0) {inchk[8] = 1;fscanf(in, "%f", &wi);pp.por = wl; )elseif (strncmp(line, "$fluid viscosity", 16)inchk[9] = 1;fscanf(in, "%f", &wl);pp.mu = wl; )elseif (strncmp(line, "$relaxation", 11) == 0) (inchk[10] = 1;fscanf(in, "%d %d", &relaxc, &relaxp); )elseif (strncmp(line, "$hydraulic cond", 15) == 0) (inchk[11) = 1;fscanf(in, "%e %e", &wl, &w2);pp.lo_k = wl;pp.hi_k w2; }elseif (strncmp(line, "$conc sink", 10) == 0) {inchk[12] = 1;nesink = 0;for (i=0;i<elnm;i++) (fscanf(in, "%d %d", &j, &k);isink[i] = k;if (k == 1)nesink++;^}^}printf(" %d elemental sinks specified.\n", nesink);^)elseif^(strncmp(line,^"$mem refl coef",^14) == 0)inchk[13]^= 1;fscanf(in,^"%f",^&wl);mem.sigma = wl; }elseif^(strncmp(line,^"$fluid density",^14) == 0) {inchk[14)^= 1;fscanf(in,^"%f",^&wl);pp.rho = wl; 3elseif^(strncmp(line,^"$tis refl coef",^14) == 0)inchk[3)^= 1;fscanf(in,^"%f",^&wl);pp.sigmag = wl;elseif^(strncmp(line,^"$diff mem PS",^12)^== 0) (inchk(15]^= 1;APPENDICES^ 1 96fscanf(in, "%e", awl);mem.ps = wl; )elseif (strncmp(line, "$geometry", 9) == 0) (inchk[16] = 1;fscanf(in, "%d", &geom);switch (geom) {case 0: puts(" Rectangular Coord w/o Upwinding.");break;case 1: puts(" Cylindrical Coord w/o Upwinding.");break;case 2: puts(" Rectangular Coord w/ Upwinding.");break;case 3: puts(" Cylindrical Coord w/ Upwinding.");break; } }elseif (strncmp(line, "$cap osm pr", 11) == 0) (inchk[17] = 1;fscanf(in, "%f", awl);mem.pi_cap = wl; )elseif (strncmp(line, "$node definitions", 17) == 0) (inchk[18] = 1;for (i=0;i<ndnm;i++) (fscanf(in, "%d %f %f %d", &j, &wl, &w2, &k);r[j] = wl;z[j]^w2;flow[j] = k; )printf(" %d node definitions set.\n", ndnm); )elseif (strncmp(line, "$elem definitions", 17) == 0) (inchk[19] = 1;for (i=0;i<elnm;i++) (fscanf(in, "%d %d %d %d %d", &j, &k, &l, fira, an);eldef[J][0] = k;eldef[J][1] = 1;eldef[J][2] = m;eldef(J][3] = n; )printf(" %d element definitions set.\n", elnm); )elseif (strncmp(line, "$conc node bc", 13)^0) (inchk[20]^1;= 0;while(strncmp(fgets(line, 80,in), "$e", 2) != 0) {sscanf(line, "%d %e", &J, awl);cbcnod[i] = j;cbcndv[i] = wl;i++; )nncbc = i;printf(" %d concentration nodal BC's specified.\n", nncbc); )elseif (strncmp(line, "$conc elem bc", 13) == 0) (inchk[21]^1;= 0;while(strncmp(fgets(line, 80, in), "$e", 2) != 0) {sscanf(line, "%d %d %d %d %e %e %e %e", &j, &k, &l, &m, &wl,&w2, &w3, &w4);cbcele[i] = j;cbcgrp[i] = k;clocl[i] = 1;cbc2[1] = m;cbceva[i][0] = wl;cbcevh(1][0] = w2;cbceva[i][1] w3;cbcevh[i][1] = w4;i++; )necbc = i;printf(" %d concentration elemental BC's specified.\n", necbc); }elseif (strncmp(line, "$conc node is", 13) == 0) {inchk[22] = 1 ;fgets(line, 80, in);if (strncmp(line, "$$file", 6) == 0) (for (1=0;1<ndnm;i++) (fscanf(in, "%d %e", &j, &wl);cn[j]^cm[j] = wl; ) }elseif (strncmp(line, "$$init", 6) == 0) {fscanf(in, "%e", &wl);for (1.0;1<ndnm;i++)( cn[i] = cm[i] = wl; } )APPENDICES^ 197printf(" %d concentration IC's specified.\n", ndnm); )elseif (strncmp(line, "$cap conditions", 15) == 0) (inchk[33] = 1;fscanf(in, "%f %f %f %d", &wl, &w2, &w3, &Uncap);mem.c_art wl;p_art w2;p_ven w3; )elseif (strncmp(line, "$pres node bc", 13) == 0) (inchk[23] = 1;= 0;while(strncmp(fgets(line, 80,in), "$e", 2) 1= 0) (sscanf(line, "%d %e", &j, &wl);pbcnod[i] = j;pbcndv[i] = wl;pruruz.p[J] = wl;i++; }nnpbc = i ;printf(" %d pressure nodal BC's specified.\n", nnpbc); }elseif (strncmp(line, "$pres elem bc", 13) == 0) {inchk[24] = 1;i = 0;while(strncmp(fgets(line, 80, in), "$e", 2) != 0) (sscanf(line, "%d %d %d %d %e %e %e %e", &j, &k, &l, &m, &wl,&w2, &w3, &w4);pbcele[i] = j;pbcgrp[i]^k;pbcl[i]^1;pbc2[i] = m;pbceva[i][0] = wl;pbcevh[i][0] = w2;pbceva[i][1] = w3;pbcevh[i][1] = w4;i++; }nepbc = i;printf(" %d pressure elemental BC's specified.\n", nepbc); }elseif (strncmp(line, "$write results", 14) == 0) {inchk[25] = 1;switch (ss) {case 0 :nrec = 1;for (i=0;i<20;i++) {drec[i]^1;trec[i] = 0.0; }break;default :fscanf(in, "%d", &nrec);for (i=0;i<nrec;i++) {fscanf(in, "%f", &wl);trec[i] = wl;if (trec[i] == 0.0){ drec[i] = 1; }elsedrec[i] = 0; }break; ))elseif (strncmp(line, "$contour", B) == 0) {inchk[26] = 1;fscanf(in, "%d", &icontour); }elseif (strncmp(line, "$ref values", 11) == 0) (inchk[27] = 1;fscanf(in, "%e %e %e %e", &wl, &w2, &w3, &w4);pp.lref^mem.lref = wl;pp.cref = mem.cref^w2;pp.pref = mem.pref = w3;pp.kref = w4;puts(" Reference values set."); }elseif (strncmp(line, "Stolerance", 10) == 0) {inchk[28) = 1;fscanf(in, "%e %e", &wi, &w2);ctol = wl;vtol = w2; )}elseif (strncmp(line, "$frac volumes", 13) == 0) (inchk[29] = 1;fscanf(in, "%f %f %f %f %f",^&w2, &w3, &w4, &w5);APPENDICES^ 198pp.fex = wl;pp•fst = w2;pp.fav = w3;pp•fs = w4;pp.fim = w5; }elseif (strncmp(line, "$conv hindrance", 15) == 0)inchk[30] = 1;fscanf(in, "%f", &wl);pp.hind = wl; )elseif (strncmp(line, "$gauss points", 13) == 0) (inchk[31] = 1;fscanf(in, "%d", &ngp);= 0;switch(ngp) {case 2 :case 3 :case 4 :case 6 :case 8 :case 16: i 1;break; }printf(" Using Gaussian integration of order %d.\n", ngp);if (i == 0) (puts("\n E: mve() - > Incorrect Gauss Points (2,3,4,6,8,16allowed).\n"):exit(0); ) }elseif (strncmp(line, "$asq", 4) == 0) (inchk(32] = 1;fscanf(in, "%e", &wl);mem.asq = wl; }elseif (strncmp(line, "$hof", 4) == 0) (inchk[34] = 1;fscanf(in, "%d", &ihof);switch (ihof) (case 0: puts(" Using lower order trial functions.");break;case 1: puts(" Using higher order trial functions.");break; } )elseif (strncmp(line, "$node results", 13) == 0) {inchk(35] = 1;fgets(line, 80, in);if (strncmp(line, "$$coll", 6) == 0) {wtf = 1;nrout = nr;nzout = nz;ndnmout = ndnm;puts(" Record nodal results at collocation points."); }elseif (strncmp(line, "$$file", 6) == 0) {wtf = 0;fscanf(in, "%d", &nrout);for (i=0;i<nrout;i++)( fscanf(in, "%f", &rot[i]); )fscanf(in, "%d", &nzout);for (i=0;i<nzout;i++){ fscanf(in, "%f", &zot(i]);ndnmout = nrout*nzout;puts(" Record nodal results at specified points."); })elseif (strncmp(line, "$osm pres", 9) == 0) (inchk[36] = 1;fscanf(in, "%d", &pp.osm); )elseif (strncmp(line, "$dgamma", 7) == 0) {inchk[37] = 1;fscanf(in, "%f", &wl) ;dgamma = wl; )elseif (strncmp(line, "$lymph cond", 11) == 0) (inchk[38] = 1;fscanf(in, "%f %e", &wl, &w2);pp.plymph = wl;= w2 )elseif (strncmp(line, "$wall th", 7) == 0) (inchk[40] = 1;fscanf(in, "%e", &wl);APPENDICES^ 199mem.wall_th = wl; }elseif (strncmp(line, "$perturbation", 13) == 0) {inchk[41] = 1;fscanf(in, "%d %f %f %f %f %d", &pert, &wl, &w2, &w3, &w4, &pert_osmp);pert_p_art = wl;pert_p_ven = w2;pert_c_art = w3;pert_c_osmp = w4;if (pert != 0)( puts(" System perturbation will occur after SS reached."); )else( puts(" No specified system perturbation."); ))elseif (strncmp(line, "$cap wall Lp", 12) == 0) {inchk[39] = 1;fscanf(in, "%d %e %e", &mem.typ_lp, &wl, &w2);mem.1p0 = wl;mem.1p1 = w2;if ((mem.1p0 == 0.0) && (mem.1p1 == 0.0)) {puts("\n B: mve()-> Lp = Zero.\n");exit(0);^}ic++; )fclose(in);/* */if (safechk(ndnm, elnm, nr, nz, nzout, nzout) 1= 0) (puts("\n Run Aborted.");exit(0); }if ((ic = datachk(&inchk[0])) != 0) (printf("\n B: mve()-> %d input items missing. Run aborted.\n", ic);exit(0); }/* */puts("\n Model input data check passed.\n");if (argc == 2) (puts("\n T: mve()-> Data check. No analysis performed.\n");exit(0); }/*MEMBRANE NODE NUMBER, LOCATION, AND ESTIMATED PRESSURE*/xm = (p_art-p_ven);for (i=0;i<nz;i++) {k = i*nr;mem.mnod[i] = k;mem.r[i] = r[k];mem.z[i] = z[k];mem.hp[i] = p_art-xm*mem.z[1]; }/*TISSUE OUTER RADIUS*/rg = r[nr-1];zg = mem.z[nz-1]-mem.z[0];rg2 = sq(rg);/*GET dr(] AND dz[] VECTORS*/for (i=0;i<elnm;i++)edr[i] = fabs(r[eldef[i][1]]-r[eldef[i][0]]);edz[i] = fabs(z[eldef[i][3]]-z[eldeffi][0]]); )/*ASSIGN OUTPUT ELEMENTS FOR OUTPUT NODE LOCATIONS*/for (i=0;i<elnm;i++)for (j=0;j<nzout;j++)for (k=0;k<nrout;k++) (1 = j*nrout+k;if (iout[1] == 0) (if ((rot[k] >= r[eldef[i][0]])&& (rot[k] <= r[eldef[i][2]])&& (zot[j] >= z[eldef[i][0]])&& (zot[j] <= z[eldef[i][2]]))( iout[1] = i; } ) )))puts(" Assigned output element-node numbering.");/*DATA CONVERSIONS TO MAINTAIN CORRECT UNIT DIMENSIONS*/bw = 2*nr+1;/*DIMENSIONING TERMS*/lref2d = sq(pp.lref)/pp.diff;ndvel = pp.hind*pp.fst/pp.fav;APPENDICES^ 200mem.pa_d = mem.ps*mem.lref/mem.diff;mem.wall_th_d = mem.wall_th/mem.lref;/*ASSIGN NODAL SINX VALUES AND GET AVERAGE SINX AREA*/avgas = 0.0;for (i=0;i<elnm;i++) {if (isink[i] == 1) {avgas += (edr[1]*edz[i]);for (j=0;j<4;j++)^{ nsink[eldef[i][j]] = 1;^} )if (nesink 1= 0){ avgas *= sq(pp.lref)/((D)(nesink)); }else ( avgas = 0.0; )printf(" Lymphatic sink area = %10.6e\n", avgas);/*GET EFFECTIVE VOLUME AND TOTAL TISSUE VOLUME*/effvol = (1.0-pp.fs)/pp.fst;pp.vol_tis = 0.0;for (i=Ori<nz-lri++) (dr = 0.5*(mem.r(i]+mem.r[i+1]);dz = (mem.z[i+1]-mem.z(i));pp.vol_tis += dz*(rg2-sq(dr)); }pp.vol_tis *= (pp.lref*pp.lref*pp.lref*PI);printf(" Total tissue volume = %10.6e\n", pp.vol_tis);/*OPEN OUTPUT FILES*/outr = fopen(argv(2), "w+");outb = fopen(argv[3], "w+");/*WRITE (DISSPLA) HEADER DATA TO OUTPUT FILES*/fprintf(outr, " %d\n", nrec);header(rg, zg, pp.lo_k, mem, pp, outr);header(rg, zg, pp.lo_k, mem, pp, outb);/*CLOSE FILES*/fclose(outr);fclose(outb);switch (ss) {case 0 :printf(" Steady-state run. \n");break;default :if (ss == 1)( puts(" Temporal stability requirement (Co < 1) enforced."); }else( puts(" Temporal stability requirement (Co < 1) notenforced."); }printf(" Initial time = %10.4f\n Final time^= %10.4f\n", ti, tf);printf(" Time step will be increased every %d cycles.\n", ndt);break; )printf("\n %s Run start.\n", title);if (ss 1= 0) {outl = fopen(argv(4], "w+");fputs(title, outl);fputs("\n Model Run Log :\n\n", outl);fputs("^TOT t^INC t^MAXCO^MAXPE^MAXDC:NMAXDP:N\n", outl);fputs("----\n", outl);fclose(outl); }/*START MODEL RUN*/= cdt = 0;/* */do (^raku:ii++;cdt++;irec = 0;ti = t;puts("\n ^ ) rif (ss != 0){ printf("Time Step %4d :\n", ii); )/*SWITCH OFF ALL OSMP EFFECTS IN TIS AND CAP IF DESIREDAPPENDICES^ 201./if (pp.osm^0){ mem.sigma pp.sigmag^0.0; }CHECK IF TO RECORD THIS TIME STEP.1switch (Bs) {case 0 :irec^1;dt = ddt = 0.0;break;default :if (pert == 0) {if ((t+dt) > tf) {irec = 1;dto = dt;dt = tf-t; }for (i=nrec-1;1>-1,1--) (if (((t+dt) > trec[i]) && (drec[i] "irec = 1;dto = dt;dt = trec[i]-t;for (i*id> -1;i -- ){ drec[j] =^)) }if (dt > dtmax) {irec = 0;dt = dtmax; )if (dt <= 0.0) {irec = 0;printf("\n W: mve()-> dt <= Zero. Reset to dtmax.\n");dt = dtmax; )ddt = dt/lref2d;break; )/*SET PREDICTOR VALUES AND PRESSURE AT TIME t+dt*1for (i=0;i<ndnm;i++){ cs.c[i] = cm[i]^cn[i]+ddt*dc[i]; }/*START ITERATIVE SCHEME*1jj^0;eo = 1.0e-02;do {JJ++;doitpr^0;maxco = maxdc = maxop maxpe = 0.0;printf(" Conc Iter %3d :\n", jj);/*OSMOTIC PRESSURE EFFECT./j = 0;p_q = osmopr(ndnm, pp, &cs.c[0]);for (i=0;i<ndnm;i++) {op[i] = *p_q++;if (op[i] > maxop) {maxop = op[i];j = i; ))printf(" Max Osmotic Pressure = %10.6f at node %d.\n", maxop, j);/*SET MEMBRANE OSMOTIC PRESSURE./for (i=0;i<nz;i++){ mem.op[i] = op[mem.mnod[i]]; }/*PRESSURE DISTRIBUTION AND FLUID VELOCITY FIELD*1if (pp.osm != 0){ doitpr = 1; )else{ doitpr = 0; )if (ii < 2){ doitpr = 1; }switch (doitpr) {case 0 :puts(" Membrane-Tissue Pressure-Velocity variables0)) {converged.");break;case 1 :pruruz = itpruruz2d(ndnm, elnm, nnpbc, nepbc,nr, nz, bw, vmaxit, pp, mem,APPENDICES^ 202&eldef[0][0], &flow[0], &edr[0], &edz[0],fir[0], &z[0], &oP[0], &hP[i], &pbcnod[0],&pbcndv[0], &pbcele[0], &pbcgrp[0], fipbceva[0][0],&pbcevh[0](0], &pbc1[0], &pbc2[0], &nsink[0],p_art, p ven, ctol, vtol,lincap, relaxp, ihof, geom, WP);/*SET IMPROVED NODAL PRESSURE BOUNDARY CONDITIONS*/for (i=0;i<nnpbc;i++){ pbcndv[i] = pruruz.p[pbcnod[i]];/*SET MEMBRANE FLUID VELOCITIES*/for (i=0;1<nzii++) {mem.vm[i) = pruruz.r[mem.mnod[i]];mem.hp[i] = pruruz.p[mem.mnod[i]]; }break;/*CALCULATE VIRTUAL DIFFUSION COEFFICIENTS*/if ((disp == 1) II (ii < 2)) {d = dspc(ndnm, pp.alfl, pp.alft, pp.diff,&pruruz.r[0], &pruruz.z[0], crz);/*GET DIMENSIONLESS DIFF-DISP COEFFICIENTS*/for (i=0;1<ndnm;i++) (drreff[i] = pp.fst*d.rr[i]/pp.diff;if (crz == 1){ drzeff[i] = pp.fst*d.rz[1]/pp.diff; Idzzeff[i] = pp.fst*d.zz[i]/pp.diff; } I/*GET MAX COURANT AND PECLET ELEMENTAL NUMBERSAND CHECK MAX PERMISSABLE dt FOR STABILITY*/k = 1 = 0;for (i=0;i<elnm;i++) {dr = dz = ur = uz = 0.0;for (J=0;J<4;i") {dr += pp.fst*d.rr[eldef[i][J]];dz += pp.fst*d.zz[eldef(i][i]];ur += ndvel*fabs(pruruz.r[eldef[i][il]);uz += ndvel*fabs(pruruz.zfeldef[i][i]l); Idr *= 0.25;dz *= 0.25;ur 0.25;uz *= 0.25;uu = norm(ur, uz);dd = norm(dr, dz);if (uu != 0.0) (hh = pp.lref*(ur*edr[i)+uz*edz[i])/uu;co = uu*dt/hh;pn = uu*hh/dd;else (co = pn = 0.0; )if (co > maxco) {maxco = CO;UC = uu;he hh;k= i;if (pn > maxpe) (maxpe = pn;1 = i;^Iprintf("\n Max Peclet number = %10.6f at element %3d.\n", maxpe, 1);if (ss != 0) {printf(" Max Courant number = %10.6f at element %3d.\n", maxco,if ((maxco > 0.95) && (ss == 1)) (puts(" Stability violation (Max Co < 1.0).");ii--;dt = 0.9*hc/uc;t = ti;if (dt < dtmin) (outl = fopen(argv[8], "a+");puts("\n E: mve() - > Time increment too small.k);Run aborted.\n");fputs("\n E: mve()-> Time increment too small.Run eloorted.\n", outl);APPENDICES^ 203fclose(outl);goto aljo; )printf("\n W: mve()-> Reducing time increment to dt =%10.6f.\n", dt);goto raku; 3)/*USE ITERATIVE SOLVER WITH SINK TERMS ON THERIGHT HAND SIDE FOR SOLUTE DISTRIBUTION*Ca^itcsccap2d(ndnm, elnm, nncbc, necbc, nr, nz, bw,cmaxit, pp, mem, fieldef[0][0],fiedr[0], fiedz[0], &r[0], &z[0],&drreff[0], fidrzeff[0], &dzzeff[0],&pruruz.r[0], &pruruz.z[0],&pruruz.p[0], &cn[0],&cbcnod[0], &cbcndv[0],&cbcele[0], ficbcgrp[0], &cbceva[0][0], &cbcevh[0][0],ficbc1[0], &cbc2(0],&naink[0], ctol, ddt, crz, ss, ihof, geom. ngp);/*SET IMPROVED NODAL CONCENTRATION BOUNDARY CONDITIONS*/for (i=0;i<nncbc;i++){ cbcndv[i]^cs.c[cbcnod[i]]; )/*SET CAPILLARY AND TISSUE-SIDE MEMBRANE CONCENTRATIONS*1for (i=0;i<nz;i++) {mem.ct[i] = cs.c[mem.mmod[i]];mem.cp[i]^cs.ccap(iJ; )/*GET DEVIATION AND THEN APPLY RELAXATION*1for (i=0;i<ndnm;i++) {dc[i] = fabs(cs.c[il-cm[1]);if (dc[i] > maxdc)( maxdc = dc[i]; ))if (relaxc != 0) (en = 0.0;for (i=0;i<ndnm;i++){ en += fabs(dc[i]); )ww = dgamma/(1.0+en/e0);if ((eo = en) == 0.0)( eo = 1.0e-02; )for (i=0;i<ndnm;i++) (cs.c[i] = ww*cs.c[i]+(1.0-ww)*cm[i];cm[i] = cs.c[i]; )}else (for (i=0;i<ndnm;i++)( cm[i] = cs.c[i]; ))/* */printf(" Conc Iter %3d : MDeltaC = %7.3e\n", jj, maxdc);} while ((maxdc > ctol) && (jj < cmaxit));if (jj >= cmaxit) {puts(" Did not converge. Possibly try smaller time step");puts(" or implement relaxation parameter.\n Program terminated.\n");exit(0);/*SOLUTION CONVERGED*/for (i=0;i<ndnm;i++) (if (cs.c[i] < 0.0)( cs.c[i] = 0.0; ))switch (ss) (case 0 :printf("Steady-state solution completed in %3d iterations.\n",in; break;default :t += dt;printf("Solution completed in %3d iterations.\n", jj);printf("Time Step %4d : t = %10.6f : dt^%10.6f completed.\n",t, dt);break; )/*CALCULATE MAX CHANGE IN CONCENTRATION SOLUTION THROUGH TIME*/if (as != 0) (maxcc maxdc = maxdp 0.0;for (i=0;i<ndnm;i++) (uc^cs.c[i]-cn[i];APPENDICES^ 204&eldef[0][0],&edz(0),uu = pruruz.p[i]-bp[i];if (fabs(cs.c[i]) > maxcc)( maxcc = cs.c[i]; )if (fabs(uc) > fabs(maxdc)) {k = i;maxdc = uc; }if (fabs(uu) > fabs(maxdp)) {1 . i;maxdp = uu; }if (ddt != 0.0){ dc[i] = uc/ddt; }else( dc[i] = 0.0; )cn[i] = cs.c[i];bp[i]^pruruz.p[i];cs.c[i] /= effvol; })if (maxcc != 0.0)( uc = fabs(maxdc/maxcc); )else( uc = fabs(maxdc); )if ((uc < ctol) && (ss != 0)) (printf(" Max temporal relative change in solute conc. = 9610.6e\n", uc);puts(" Solute steady-state distribution has been achieved.\n");puts(' Will output steady-state data to amod.inp input Ms.");puts(" Will implement perturbation if specified.");printf(" Time to steady-state = 9610.6f\n", t);irec = 1;jj = ss;ss = 3;ti = 0.0;dt = dti; }/*CALCULATE PECLET NUMBER - THIS IS NOT THE ELEMENTALPECLET NUMBER (USES REFERENCE LENGTH r[0]*pp.lref)xm = pp.hind*pp.fst/pp.fav;for (i=0;i<ndnm;i++) (uu = xm*norm(pruruz.r[i], pruruz.z[i]);dd = pp.fst*norm(d.rr[i], d.zz[i]);if (uu != 0.0)( pe[i] = uu*pp.lref/dd; )else( pe[i] = 0.0; ) )/*MASS BALANCE FOR FLUID AND SOLUTE*/massb(elnm, nr, nz, nesink, pp, mem, &eldef[0][0], &isink[0],&edr[0), &edz[0], &cs.c[0], &pruruz.p[0], &cs.fmem[0],pruruz.qcap[0], avgas, ngp, geom);/*GET TISSUE AVERAGE PRESSURE AND SOLUTE CONCENTRATION5/domavg(elnm, pp, &eldef[0][0], &edr[0], &edz[0), &cs.c[0],&pruruz.p[0], ngp, geom);/*CONTOUR OUTPUT DATA1,/if (icontour != 0) (outq = fopen("zcon.dat", "w+");contour(0, ss, ndnm, nr, nz, &cs.c[0], t, title, outq);contour(1, ss, ndnm, nr, nz, &pruruz.p[0], t, title, outq);if (pp.osm != 0) (contour(2, ss, ndnm, nr, nz, &op[0], t, title, outq);contour(3, as, ndnm, nr, nz, &pruruz.pt[0], t, title, outq); }printf(" contour()");fclose(outq); )/*OUTPUT CONVERGED RESULTS5/if ((nrec == 0) II (irec == 1)) (printf(" Recording data : ");outr = fopen(argv[2], "a+");outb = fopen(argv[3], "a+");switch (wtf) (case 0:outdata(0, ss, ndnmout, nrout, nzout, &iout[0],&rot(0], &zot[0], &r[0], &z[0], &edr[0],ihof, t, title, &cs.c[0], outr);APPENDICES^205outdata(1, as, ndnmout, nrout, nzout, &iout(0],&eldef[0][0], &rot(0], &zot(0], &r[0] , fiz(0], &edr(0],&edz(0], ihof, t, title, &pruruz .p(0], outr);outdata(2, ss, ndnmout, nrout, nzout, &iout(0],&eldef(07(01, &rot(0], &zot[0], &r(0] , &z[0], &edr[0],&edz[0], ihof, t, title, &pe[0], outr);outdata(3, as, ndnmout, nrout, nzout, &iout(0],&eldef[0](0], &rot[0], &zot[0], &r[0], &z[0], &edr(0],&edz(0],case 1:ihof,^t,^title,break;&pruruz.pt(0], outr);title,title,outr);outr);outcoll(0,outcoll(1,outcoll(2,as,as,ss,ndnmout, t,t,t,&r[0],&r(0],&r(0],&z(0],&z[0],&z(0],&cs.c(0],&pruruz.p(0],&pe(0], title,ndnmout,ndnmout,outr);title, outr);outcoll(3,break; )as, ndnmout, t, &r[0], &z(0], &pruruz.pt(0],outuv2d(wtf, as, ndnmout, nrout, nzout, &iout(0], &eldef[0](0],&rot(0], &zot[0], &r(01, &z[0], &edr(0], &edz[0],ihof, t, title, &pruruz.r[0], apruruz.z(0], outr);outcapp(nz, ss, t, &mem.z[0], &pruruz.pcap(0], &pruruz.qcap[0],fimem.vm[0],&cs.ccap(0], fics.mpe(0), &cs.fmem[0], title, outb);fclose(outr);fclose(outb);printf(".\n"); }/*OUTPUT TO RUN LOG FILE*1if (ss != 0) {outl^fopen(argv[4], "a+");fprintf(outl, "%3d %10.4f %10.4f %6.3e %6.3e %+6.3e:%3d %+6.3e:%3d",ict++, t, dt, maxco, maxpe, maxdc, k, maxdp, 1);if (pert == 0) {switch(irec) {case 0: break;case 1: fprintf(outl, " R");break; ) }fprintf(outl, "\n");fclose(outl); )if (ss == 3)ss = 0; )/*WRITE RESULTS TO INTERMEDIATE FILE IN CASE OF SHUTDOWNOR FOR NEW START-UP FILE FOR OTHER RUNS:1tmpf = fopen("zmod.tmp", "w+");/*^./fputs("$beg input\n", tmpf);fputs(title, tmpf);fputs("$prob size\n", tmpf);fprintf(tmpf, "%d %d %d %d\n", ndnm, elnm, nr, nz);fputs("$transient\n", tmpf);fprintf(tmpf, "%d %f %f %f\n%e %e %d\n",ss, ti, tf, dt, dtmax, dtmin, ndt);fputs("$max iterations\n", tmpf);fprintf(tmpf, "%d %d\n", cmaxit, vmaxit);fputs("$dispersivity\n", tmpf);fprintf(tmpf, "%e %e %e %d\n", pp.alfl, pp.alft, pp.diff, crz);fputs("$fluid sink LS\n", tmpf);fprintf(tmpf, "%e\n", pp.ls);fputs("$porosity\n", tmpf);fprintf(tmpf, "%f\n", pp.por);fputs("$fluid viscosity\n", tmpf);fprintf(tmpf, "%f\n", pp.mu);fputs("$fluid density\n", tmpf);fprintf(tmpf, "%f\n", pp.rho);fputs("$relaxation\n", tmpf);fprintf(tmpf, "%d %d\n", relaxc, relaxp);fputs("$geometry\n", tmpf);fprintf(tmpf, "%d\n", geom);fputs("$dgamma\n", tmpf);fprintf(tmpf, "%f\n", dgamma);APPENDICES^ 206fputs("$lymph cond\n", tmpf);fprintf(tmpf, "%f %e\n", pp.plymph, pp.jo);fputs("$wall th\n", tmpf);fprintf(tmpf, "%e\n", mem.wall_th);fputs("$cap wall Lp\n", tmpf);fprintf(tmpf, "%d %e %e\n", mem.typ_lp, mem.1p0, mem.1p1);fputs("$osm prea\n", tmpf);fprintf(tmpf, "%d\n", pp.osm);fputs("$mem refl coef\n", tmpf);fprintf(tmpf, "%f\n", mem.sigma);fputs("$tis refl coef\n", tmpf);fprintf(tmpf, "%f\n", pp.sigmag);fputs(”diff mem PS\n", tmpf);fprintf(tmpf, "%e\n", mem.ps);fputs("$cap osm pr\n", tmpf);fprintf(tmpf, "%f\n", mem.pi_cap);fputs("$ref values\n", tmpf);fprintf(tmpf, "%e %e %e %e\n",pp.lref, pp.cref, pp.pref, pp.kref);fputs("$tolerance\n", tmpf);fprintf(tmpf, "%e %e\n", ctol, vtol);fputs("$frac volumes\n", tmpf);fprintf(tmpf, "%f %f %f %f %f\n",pp.fex, pp.fst, pp.fav, pp.fa, pp.fim);fputs("$conv hindrance\n", tmpf);fprintf(tmpf, "%f\n", pp.hind);fputs("$gauss points\n", tmpf);fprintf(tmpf, "%d\n', ngp);fputs("$contour\n", tmpf);fprintf(tmpf, "%d\n", icontour);fpute("$asq\n", tmpf);fprintf(tmpf, "%e\n", mem.asq);fputs("$hof\n", tmpf);fprintf(tmpf, "%d\n", ihof);fputs("$perturbation\n", tmpf);fprintf(tmpf, "%d %f %f %f %f %d\n",1, pert_p_art, pert_p_ven, pert_c_art, pert_c_osmp, pert_osmp);fputs("$hydraulic cond\n", tmpf);fprintf(tmpf, "%10.6e %10.6e\n", pp.lo_k, pp.hi_k);fputs("$conc sink\n", tmpf);for (i=0;1<elnm;i++)fprintf(tmpf, "%d %d\n", 1, isink[i]); }fputs("$node definitions\n", tmpf);for (i=0;i<ndnm;i++)fprintf(tmpf, "%4d %10.6f %10.6f %4d\n", 1, r[i], z[i], flow[i]); }fputs("$elem definitions\n", tmpf);for (i=0;1<elnm;i++)( fprintf(tmpf, "%4d %4d %4d %4d %4d\n",eldef[i][0], eldef[i][1], eldef[i][2], eldef[i][3]); }fputs("$conc node bc\n", tmpf);for (i=0;1<nncbc;1++)fprintf(tmpf, "%4d %10.6e\n", cbcnod[i], cbcndv[i]); }fputs("$e\n", tmpf);fputs("$conc elem bc\n", tmpf);for (i=0;i<necbc;i++)fprintf(tmpf, "%4d %4d %4d %4d",1, cbcgrp[i], cbc1[1], cbc2[1]);for (j=0;1<2;J++)fprintf(tmpf, " %10.6e %10.6e",cbceva[i][J], cbcevh[i][j]); }fprintf(tmpf, "\n"); )fputs("$e\n", tmpf);fputs("$conc node ic\n", tmpf);fputs("$$file\n", tmpf);for (1=0;1<ndnm;i++)fprintf(tmpf, "%4d %10.6e\n", i, cs.c[i]); )fputs("$cap conditions\n", tmpf);fprintf(tmpf, "%10.6f %10.6f %10.6f %d\n",mem.c_art, p_art, p_ven, lincap);fputs("$pres node bc\n", tmpf);for (i=0;1<nnpbc;i++)fprintf(tmpf, "%4d %10.6e\n", pbcnod[i], pbcndv[i]); }fputs("$e\n", tmpf);fputs("$pres elem bc\n", tmpf);for (1=0;1<nepbc,i++)fprintf(tmpf, "%4d %4d %4d %4d",pbcgrp[1), pbc1[1], pbc2(i));for (j=0;j<2;J++)fprintf(tmpf, " %10.6e %10.6e",pbceva[l][j], pbcevh[i][i]); }fprintf(tmpf, "\n"); )APPENDICES^ 207fputs("$e\n", tmpf);fputs("$write results\n", tmpf);switch (ss) (case 0 :break;default :fprintf(tmpf, "%d\n", nrec);for (i=0;i<nrec;i++)( fprintf(tmpf, "%10.6f\n", trec[i]); }break; }fputs("$node results\n", tmpf);switch (wtf) {case 0: fputs("Wile\n", tmpf);fprintf(tmpf, "%d\n", nrout);for (1=0;i<nrout,i++)fprintf(tmpf, "%f\n", rot[i]); }fprintf(tmpf, "%d\n", nzout);for (1.0;i<nzoutti++)( fprintf(tmpf, "%f\n", zot[i]); }break;case 1: fputs("$$coll\n", tmpf);break; }fputs("$end input\n", tmpf);/*^*/fclose(tmpf);/*ENLARGE TIME INCREMENT EVERY ndt TIME STEP CYCLES*1if (ss := 0) (if (irec == 1){ dt = dto; )if (cdt^ndt) (cdt^0;dt *= 2.0; }ddt dt/lref2d; }if ((pert !. 0) && (ss == 0)) {puts(" Initiating perturbation in Part, Pven, Cart, and Osmp.");ss = jj;pert = ii = 0;t^0.0;p_art pert_p_art;p_ven = pert_p_ven;mem.c_art = pert_c_art;mem.pi_cap = pert_c_osmP;pp.osm = pert_osmp;printf(" P_art^= %10.6f\n", p_art);printf(" P_ven = %10.6f\n", P_ven);printf(" C_art = %10.6f\n", mem.c_art);printf(" Osmp_cap^= %10.6f\n", mem.pi_cap);switch (pp.osm) {case 0 : printf(" Osmotic effects off.\n");break;case 1 : printf(" Osmotic effects on.\n");break; ))/*^*/while ((t < tf) && (ss := 0));/*^*/outl^fopen(argv[4], "a+");fputs("\n Model run complete.\n", outl);fclose(outl);rename("zmod.tmp", "amod.inp");aljo:^/*^*/printf(" %s Run complete.\n", title);O itcaccap2d(I ndnm, I elnm, I nncbc, I necbc, I nr, I nz, I bw, I cmaxit,media pp, membr mem,I^D *rd, D *zd, D *pr, D *pz,D *rr, D *rz, D *zz, D *ur, D *uz, D *ph, D *co,I "bn, D *bv, I *be, I *bg, D *ba, D *bh, I "bl, I *b2,I "sk, D ctol, D ddt, I crz, I ss, I ihof, I geom, I ngp)I 1, j, k, 1, n, m;I bcn[MM], bce[MM], bcg[MMI, bc1[MM], bc2(MM];I nsk[NM], eldef[EM][6];D aa, bb, cc, dd, 'sigmas, lref2d, lsvol, jovol, ndvel, mcerr, xm, *p_q;D edr[EM], edz(EM], bcv(MM], bca[MM][3], bch[MM][3];D r[NM], z[NM], drr[NM], drz[NM], dzz[NM), qt[NM],per[NM], pez[NM], vr(NM), vz[NM], hp[N14), glymph[NM];O cs;U cln;/*^*/for (i=0;i<NM;i++) (r[i] = *pr++;APPENDICES^ 208z[i] = *pz++;nsk[i] = *sk++;drr[i] *rr++;drz[i] = *rz++;dzz[i] = *zz++:vr[i] = *ur++;vz[i] = *uz++;hp[i] *ph++;qt[i] = *co++;for (j=0;j<NM;j++){ c1b.m[i][j] = 0. 0; ))for (i=0;i<EM;i++) (edr[i]^*rd++;edz[i] = *zd++;for (j=0;j<6;j++)( eldef[i][j] = *pe++; ))for (i=0;i<MM;i++) {bcl[i] = *b1++;bc2[1] = *b2++;bce[i] = *be++;bcn[i] *bn++;bcg[i] = *bg++;bcv[i] = *by";for (j=0;j<3;j++) {bca[i][j] = *ba++;bch[i][j] = *bh++; ))/* */puts(• Itcsccap()");isigmas = 1.0-mem.sigma;lref2d = sq(pp.lref)/pp.diff;ndvel = pp.lref*pp.hind*pp.fst/(pp.fav*pp.diff);lsvol = pp.pref*lref2d*pp.ls;jovol = lref2d*pp.jo;/*GET DIMENSIONLESS EFFECTIVE CONVECTIVE VELOCITIES*/for (i=0;i<ndnm;i++) {per[i] = ndvel*vr[i];pez[i] = ndvel*vz[i]: }/*GET NODAL LYMPH FLOW*/if (pp.ls != 0.0) (for (i=0;i<ndnm;i++) (if (nsk[i] 1. 0) {if ((xm^lsvol*(hp[i]-pp.plymph)) > 0.0)( qlymph[i] = jovol+xm; )else( qlymph[i]^jovol; ))else( qlymph[i] = 0.0; )) )else {for (i=0;i<ndnm;i++)( qlymph[i] = 0.0; ))/*TEMPORAL AND/OR SINK MATRIX*/P-4 = stif(2, ndnm, elnm, bw, &r[0], &x[0], &eldef[0][0], &edr[0], &edz[0],&drr[0], &drz[0], &dzz[0], &per(0], &pez[0], &per[0],pp.lref, crz, ihof, geom, ngp);for (i=0;1<NM;i++) {for (j=0;j<NM;j++){ cln.m[i][j] = *p_q++; ))*PREVIOUS TIME STEP CONTRIBUTION*/if (ss != 0) (p_q = matvecbw(0, ndnm-I, bw, &cln.m[0][0], &qt(0]);for (i=0;i<ndnm;i++)( qt[i] = (*p_q++)/ddt; ))else {for (i=0;i<ndnm;i++){ qt[i] = 0.0; }}/*SUM TEMPORAL AND SINK MATRICES*1if (pp.ls != 0.0) (for (i=0;i<NM;i++)for (j=0;j<NM;j++) {if (ss != 0)( c1n.m[i][J] *= (1.0/ddt+qlymph[i]): )APPENDICES^ 209e lse( cln.m[i][j] *= qlymph[i]; )))/*SUM DIFF-DISP STIFFNESS AND TEMPORAL-SINK MATRICES*/p_q = stif(0, ndnm, elnm, bw, &r[0], &z[0], &eldef[0][0], &edr[0J, &edz[0],&drr[0], &drz[0], &dzz[0], &per[0], &pez(0], &per[0],pp.lref, crz, ihof, geom, ngp);for (i=0;i<NM;i++) {for (j=0;j<NM;j++){ cln.m[i][i] += `1).4++; }}/*SUM TEMPORAL-SINK-DIFF-DISP AND CONVECTIVE MATRICES*/p_q = stif(1, ndmm, elnm, bw, &r(0], &z(0], &eldef[0][0], fiedm[0], Gedz(°],&drr[0], &drz[0], &dzz[0], &per[0], &pem(0], &per(0),pp.lref, crz, ihof, geom, ngp);for (i=0;i<NM;i++) {for (j=0;j<NM;j++)( cln.m[i][J] += *P_4++; ))/*GET MEMBRANE PECLET NUMBER*/for (i=0;i<nz;i++) (cs.mpe[i] = fabs(vr[mem.mmod[i]] *isigmas/mem.ps);cs.ccap[i] = mem.cp[i]; )/* */puts("\n Membrane-Tissue Concentration Iterations :");k = 1 = m = n = 0;do (mcerr = 0.0;printf(" %3d:", 1+1);/*GET TISSUE-SIDE SOLUTE CONCENTRATION USING INTEGRATEDPATLAK EQUATION*/for (i=0;i<nz;i++) (if (cs.mpe[i] > 1.0e-06) {if (cs.mpe[i] > 100.0)( aa = 0.0; yelse( aa = mem.ps_d*cs.mpe[i]/(exp(cs.mpe(i])-1.0);else{ aa = mem.ps_d;if (mem.vm[i] >= 0.0) {bb = aa+per[mem.mmod[i]]*isigmas;cc = aa+per[mem.mmod[i]];dd = cc*mem.wall_th_d/drr[mem.mmod[i]];xm = bb/cc;mem.ct[i] = (xm+(1.0-xm)*exp(-dd))*mem.cp[i]; }else {bb = per[mem.mmod[i]]-aa;cc = per[mem.mmod[i]]*isigmas-aa;dd = cc*mem.wall_th_d/drr[mem.mnod[i]];xm = bb/cc;mem.ct[iJ = (xm+(1.0-xm)*exp(dd))*mem.cp[i];^)/*CAPILLARY MEMBRANE CONCENTRATION DISTRIBUTION*p_q cmcd(nz, mem);for (i=°;i<nz;i++) (if ((mem.cp[i] = *p_q++) < 0.0)( mem.cp[i] = 0.0; ) )/* */for (i=0;i<nz;i++) {if (fabs(mem.cp[i]-cs.ccap[i]) > mcerr)( mcerr = fabs(mem.cp[i]-cs.ccap[i]);cs.ccap[i]^mem.cp[i];printf(" mcemr=%6.3e\n", mcerr);1++;) while ((mcerr > ctol) && (1 < cmaxit));cs.iter = 1;/* */for (i=0;i<nz;i++)( cs.c[mem.mmod[i]] = mem.ct[i]; )/*SET TISSUE NODAL BC*/for (1=0;i<nz;i++) (for (j=0;j<nncbc;j++) (if (bcn(j) == mem.mnod(1))APPENDICES'^ 210{ bcv(jJ^mem.ct(i]; )))/*SET TYPE 3 BOUNDARY CONDITION VALUES FORMEMBRANE BC AND WALLS.,t/for (i=0;i<necbc;i++) {switch(bcg[i]) (case 0: /* RO<r<RG, z=0 */for (J= 0 ;J<2;J++) (switch (j) (case 0 : k = bcl[i];break;case 1 : k = bc2[1.];break; }bca[i][j] = -pez(k]/dzz[k];bch[i][1] = 0.0; )break;case 1: /* RO<r<RG, z=L */for (j= 0 ;j<2;j++) (switch (j) {case 0 : k = bcl[i];break;case 1 : k^bc2[iJ;break, }bca[i][j] = -pez[k]/dzz[k];bch[i][J] = 0.0; )break;case 2: /* r=RG, 0<z<L */for (J=0;j<2;J++) (switch (j) {case 0 : k = bc1[1];break;case 1^k = bc2[i];break; }bca[i][j] = -par[k]/drr[k];bch[i] (j] = 0.0; }break;case 3: /* r=RO, 0<z<L, MEMBRANE BC */for (j=0;j<2;j++) (switch (j) {case 0 : k = bcl[i];break;case 1 : k = bc2[1];break; }for (m=0;m<nz;m++) (if (mem.mnod[m]^k)n = m; }}if (cs.mpe[n] > 1.0e-06) {if (cs.mpe[n] > 100.0)( xm = 0.0; }else( xm = cs.mpe[n]/(exp(cs_mpe(n]) - 1.0);}else( xm mem.ps_d; )if (per(k] > 0.0) (bca[i][j] = -pp.lref*(per[k]+mem.ps_d*xm);bch[i][1] = -pp.lref*(per[Wisigmas+mem.ps_d*xm)*cs.ccap[n]; )else (bca[i][j]^-pp.lref*(per[k]*isigmas-mem.ps_d*xm);bch[i][1] = -pp.lref*(per[10-mem.ps_d*xm)*cs.ccap[n]; ) )break;case 4 : /* PRESCRIBED AT INPUT */break; 1)/*MODIFY CONC STIFFNESS MATRIX FOR ELEMENTAL BOUNDARY CONDITIONS*1if (necbc != 0) (cln = modstif(necbc, &eldef[0)[0), &r[0), &z[0],&bce(0), &bca[0][0], &bch(0)[0], &bc1[0], &bc2[0],&edr[0], &edz[0], &cln.m[0][0], ihof, ngp); }/*SUM TYPE 3 BC VECTOR AND PREVIOUS TIME STEP VECTOR*/for (i=0;i<ndnm;i++)( cln.v[i] += gt[i]; }/*NODAL BOUNDARY CONDITIONS (TYPE 1)*/for (i=0;i<nncbc;i++) {for (j=0;j<ndnm;j++) (if (1 == bcn[i]) (cln.m[bcn[i]][j]cln.v[bcn[i]]else= 1.0e+15;= 1.0e+15*bcv[i]; )( cln.m[bcn[i]][J] = 0. 0; ) } )APPENDICES^ 211/*GET CONCENTRATION DISTRIBUTION*/p_ca^solver(ndnm, bw, ficln.m[0][0], &cln.v[0]);for (i..0;i<ndnm;i++)( cs.c[i] = *p_q++ ; )/*^*1for (i=0;i<nz;i++) (cs.ctis[i]^cs.c[mem.mnod[i]];cs.ccap[i] = mem.cp(i]; )/*SOLUTE FLOW ACROSS MEMBRANE*/for (i=0;i<nz;i++) (if (cs.mpe[i] > 1.0e-06) {if (cs.mpe[i) > 100.0)( xm = 0.0; }else( xm = cs.mpe[i]/(exp(cs.mpe[i])-1.0); )}else{ xm mem.ps; }cs.fmem[i]^vr[mem.mnod[i]]*isigmas*mem.cp[i]+mem.ps*xm*(cs.ccapli)-cs.ctis[i]); }/* */return co;}N itpruruz2d(I ndnm, I elnm, I nnpbc, I nepbc, I nr, I nz, I bw, I vmaxit,media pp, membr mem, I *pe, I *fl, D *rd, D *zd, D *pr, D *pz,D *po, D *ph, I *bn, D *bv, I *be, I *bg, D *ba, D *bh, I *bl, I *b2, I *sk,D p_art, D p_ven, D ptol, D vtol, I lincap,I relaxp, I ihof, I geom, I ngp)I i, j, k, 1, m, n, hiflow;I bcn(MM), bce[MM), bcg[MM], bc1[MM], bc2[MM];I nsk[NM], flow[NM), eldef[EM][6];D en, eo, ls_d, jo_d, lref2k, piatemu, prefl, mperr, mverr, xm, ww, StpcjiD xx[IM], yy[IM], verrEIMI, vmem[im), vtis[IM], edr[EM], edz[EM];D bcv[MM], bca[MM][3], bch[MM][3];D dp[NM), r[NM], z[NM], dkrr[NM], dkzz[NM], krr[NM], kzz[NM],kkr[NM), kkz[NM], 0P(NM], lsk[NM], ld[NMI, hp[Nm], sig[NM];D ln[NMUNMI;N pruruz;U pin;vec3 v_p;/*^*/for (i=0;i<NM;i++) (nsk(i] = *sk++;flow[i] = *fl++;r[i]^*pr++;z[i]^*pz++;op[i] = *po++;hp[i] = *ph++; }for (i=0;i<EM;i++) {edr[i] = *rd++;edz[i] = *zd++;for (j=0;j<6;j++)( e1def[i][J] = *pe++; ))for (i=0;i<MM;i++)bc1[1] = *b1++;bc2(i] = *b2++;bce[i] = *be++;bcn[i] = *bn++;bcg[i] = *bg++;bcv[i] = *bv++;for (j=0;j<3;j++) (bca[i][j] = *ba++;bch[i][j] = *bh++; )}/* */switch (geom) (case 0: /* NO UPWINDING - RECT */geom = 0;break;case 1: /* NO UPWINDING - CYL */geom = 1;break;case 2: /* UPWINDING - RECT */geom = 0;break;case 3: /* UPWINDING - CYL */geom = 1;break; )APPENDICES^ 212/* *1puts(" itpruruz()");hiflow^0;lref2k = sq(pp.lref)/pp.kref;prefl = pp.pref/(pp.lref);ls_d = lref2k*pp.ls;jo_d lref2k*pp.jo;/*HYDRAULIC CONDUCTIVITY*/for (i=0;i<ndnm;i++) {switch (flow[i]) {case 0 : /* LO FLOW CHANNEL */default :krr[i] = kzz[i] = pp.lo_k;kkr[i] = kkz[i] = pp.lo_k/pp.kref;sig[i] = pp.sigmag;break;case 1 : /* HI FLOW CHANNEL */hiflow = 1;krr[i] = kzz[i] = pp.hi_k;kkr[i] = kkz[i] = pp.hi_k/pp.kref;sig[i] = pp.sigmag;break;^case 2 : /* SINK REGION^*/krr[i] = kzz[i] = pp.hi_k;kkr[i] = kkz[i] = pp.hi_k/pp.kref;sig[i]^pp.sigmag;break; } )for (i=0;i<ndnm;i++){ sig[i] *= op[i]; }/*GRADIENTS IN HYDRAULIC CONDUCTIVITY :Krr IN r DIRECTIONKzz IN z DIRECTION*1if (hiflow != 0) {for (i=0;i<nz;i++)for (j=0;j<nr;j++)k = i*nr+j;xx[j] = r[k];yy[j]^krr[k]; )v_p = fspl(nr, &xx[0], &yy(0], 0, 0, 0.0, 0.0);for (j=0;j<nr;j++)k = i*nr+j;dkrr(k] = devalcs(nr,-^&xx[0], r[k]); ))for (i=0;i<nr;i++)for (j=0;j<nz;j++)k = j*nr+i;xx[J] = z[k];yy(j] = kzz[k]; )vp = fspl(nz, &xx(0], &yy(0), 0, 0, 0.0, 0.0);for (j=0;j<nz;j++)k = j*nr+1;dkzz[k] = devalcs(nz, vp , &xx(0], z[k]); ))printf(" dkrr()/dkzz()"); )/*GET CONDUCTIVITY STIFFNESS MATRIX*/p_q = stif(0, ndnm, elnm, bw, &r[0], &z[0], &eldef[0][0], &edr[0], &edz[0],&kkr[0], &kkz(0], &kkz[0], &dkrr[0], &dkzz[0], &kkz[0],pp.lref, 0, ihof, geom, ngp);for (i=0;i<NM;1++)for (j=0;j<NM;j++)ln[i][j] = *p_q++; ))/*SUM CONDUCTIVITY AND ANISOTROPIC CONDUCTIVITY (CONVECTIVE-LIKE) MATRICESft 1if (hiflow != 0) {p_q = stif(1, ndnm, elnm, bw, &r[0], &z[0], &eldef[0][0], &edr[0],&kkr[0], &kkz[0], &kkz[0], &dkrr(0], &dkzz[0], &kkz[0],pp.lref, 0, ihof, geom, ngp);for (i=0;i<NM;i++)for (j=0;j<NM;j++)I 1n[i][J1 -= lref2k*(*p_q++); )) )/* */puts("\n Membrane-Tissue Pressure-Velocity Iterations : ) ;eo = 1.0e-02;k = 1 = m = n = pruruz.iter^0;domperr mverr = 0.0;&edz[0],APPENDICES^ 213printf(" %3d:", 1+1);/*CAPILLARY MEMBRANE PRESSURE DISTRIBUTION*/if (lincap != 0) {xm = (p_art-p_yen)/(mem.z[0]-mem.z[nz-1]);for (i=0;i<nz;i++){ pruruz.pcap[i] = p_art+xm*mem.z[i]; }}else (p_q = cmpd(nz, mem, p_art, p_ven);for (i=0;i<nz;i++){ pruruz.pcap[i] = *p_q++; })/*SET POTENTIAL TYPE 3 BOUNDARY CONDITIONS d(hP-sigmaeoP)/dn = 0.0*/for (i=0;i<nepbc;i++) {switch(bcg[i]) {case 0: /* RO<r<RG, z=0 */for (J=0;J<2;J++) {switch (j) {case 0 : k = bc1[1.);break;case 1^k = bc2[1.];break; }bca[i][j] = 0.0;bch[i][J] = 0.0; }break;case 1: /* RO<r<RG, z=L */for (j=0;j<2;j++) (switch (j) {case 0 : k = bc1[1];breakicase 1 : k = bc2[1];break; }bca[i][j] = 0.0;bch[l][J] = 0.0; }break;case 2: /* r=RG, 0<z<L */for (j=0;j<2;j++) (switch (j) {case 0 : k = bol[i];breaktcase 1 : k = bc2(i];break; )bca[i][j] = 0.0;bch[i][j] = 0.0; }break;case 3: /* MEMBRANE BC */for (j=0;j<2;j++) {switch (j) {case 0 : k bc1[1];break;case 1 : k = bc2[1],break; }for (m=0;m<nz;m++) {if (mem.mnod[m] == k){ n = m; ))switch (flow[k]) {case 0 :default :xm = lref2k*lp(mem.typ_lp,mem.1p0, mem.lpl, z(10);break;case 1 :xm = lref2k*lp(0, mem.lpl,mem.lpl, z[k));break; )bca[i][j] = xm;bch[i][J] = xm*((pruruz.pcap[1:]-mem.sigma*mem.pi_cap)+(mem.sigma-pp sigmag)*op[k]); }break;case 4 : /* PRESCRIBED AT INPUT */break; )//*MODIFY PRESSURE STIFFNESS MATRIX*1if (nepbc != 0) {pin = modstif(nepbc, &eldef(0)(0), &r[0], &z[0],&bce[0], &bca[0][0], &bch[0][0], &bc1[0], &bc2[0],&edr(0], &edz[0), &ln(0)[0], ihof, ngp); }/*SUM CONDUCTIVITY AND SINK MATRICES*/if (pp.ls != 0.0) {/*SINK MATRIX CONTRIBUTION*/for (i=0;i<ndnm;i++) (if (nsk[i] != 0)APPENDICES^ 214( lsk[i]^la_d; )else( lsk[i] = 0.0; ))p_q^stif(3, ndnm, elnm, bw, fir[0], fiz[0], &eldef[0][0], &edr[0],&edz[0],Qkkr[0], &kkz[0], &kkz[0], &kkz[0], &kkz[0], &kkz(0],pp.lref, 0, ihof, geom, ngp);for (i=0;i<NM;i++) {for (j=0;j<NM;j++){ pin.m[i][j] += lsk[i]*(*P_g++); }}/*SINK VECTOR CONTRIBUTION (TAKEN FROM LHS OF FEM EQN)for (i=0;i<ndnm;i++) {if (nsk[i] != 0){ lsk[i]^ls_d*(pp.plymph-sig[i])-jo_d; }else( lsk[1] = 0.0; ))p_q = stif(4, ndnm, elnm, bw, &r[0], &z[0], &eldef[0][0], fiedr[01,&edz[0],&kkr[0], &kkz[0], &kkz[0], &kkz(0], &kkz[0], &lsk(0],pp.lref, 0, ihof, geom, ngp);for (1=0;i<NM;i++){ lsk[i] = (*p_q++); }for (i=0;i<ndnm;i++) (if (nsk(i] !=. 0)( pin.v[i] += lsk[i];^}/*NODAL BOUNDARY CONDITIONS (TYPE 1)for (i=0;i<nnpbc;i++) (for (j=0;j<ndnm;j++)if (j == /Derail) (pin.m[bcn[i]][J]^1.0e+15;pin.v[bcn[i]]^1.0e+15*(bcv[1.]-sig[j]); }else( pin.m[bcn[i]][J] = 0.0; ) })/*GET PRESSURE (hP) AND POTENTIAL (hP-sigma*oP) DISTRIBUTIONSp_q = solver(ndnm, bw, &pin.m[0][0], &pin.v[0]);for (i=0;i<ndnm;i++) (pruruz.pt[i] = *P_g++;pruruz.p[i] = pruruz.pt[i]+sig[i]; }/*FLUID FLOW VELOCITY FIELDfor (i=0;i<nz;i++)for (j=0;j<nr;j++) {k = i*nr+j;zz[j] = r[k];yy(j] = pruruz.pt [k]; )v_p^fspl(nr, &xx[0], &yy(0], 2, 1, 0.0, 0.0);for (j=0;j<nr;j++)k = i*nr+j;pruruz.r[k]^-krr[k]*prefl*devalcs(nr, v_p, &xx[0], r[kl); }}for (i=0;i<nr;i++) (for (j=0;j<nz;j++) (k = j*nr+i;xx[J] = z[k];yy[j] = pruruz.pt[k]; )v_p = fspl(nz, &xx[0], &YY[0], 1, 1, 0.0, 0.0);for (j=0;j<nz;j++) (k^j*nr+i;pruruz.z[k] = -kzz[k]*prefl*devalcs(nz, v_p, &xx[0], z[k]);printf(" uv2d()");/*CHECK FLUID CONTINUITY ACROSS MEMBRANEfor (i=0;i<nz;i++) (k = mem.mnod[i];vmem[i] = 1p(mem.typ_lp, mem.1p0, mem.lpl, z[k]) *PP.Pref * (pruruz.p[k]-pruruz.pcap[i]-mem.sigma*(op[k]-mem.pi_cap));vtis[i] = pruruz.r[k];verr[i] = (vmem[i]-vtis[i]);if (fabs(verr[i]) > fabs(mverr)){ mverr = verr[i]; }}/*GET PRESSURE DEVIATION AND THEN APPLY RELAXATIONfor (1=0;1<ndnm;i++) (APPENDICES^ 215dp(i]^pruruz.p[i]-hp[i];if (fabs(dp(i]) > mperr){ mperr = fabs(dp(i]); ))if (relaxp != 0) {en = 0.0;for (i=0;i<ndnm;i++){ en += fabs(dp(i]); )ww = 1.0/(1.0+en/eo);if ((eo = en) == 0.0){ eo = 1.0e-02; }for (1=0;i<ndnm;i++) {pruruz.pfil^ww*pruruz.p(i]+(1.0-ww)*hp[i];hp[i] = pruruz.p[i]; ) }else {for (i=0;i<ndnm;i++){ hp[i] = pruruz.p[i]; ) }/*SET MEMBRANE HYDROSTATIC PRESSURE*/for (1.0;i<nz;i++){ mem.hp(i] = pruruz.p(mem.mnod[i]]; }/*CHECK IF PRESSURES ARE BELOW SINK PRESSURE LIMIT AND SWITCHOFF SINK IF THIS IS CASE AND SET PRESSURES FOR THOSE NODESEQUAL TO LYMPHATIC SINK LIMIT VALUE*/k = 0;for (i=0;i<ndnm;i++) (if (nsk(i) I. 0) (if ((pruruz.p(i]-pp.plymph) < 0.0) {k = 1;bcn[nnpbc] = i;bcv[nnpbc] = pp.plymph;nnpbc++;nsk[i] = 0; }}}/* */if ((mperr < ptol) II (fabs(mverr) < vtol))( pruruz.iter = 1; }else{ pruruz.iter = 0; }if (lincap != 0){ pruruz.iter = 1; }if (k != 0)( pruruz.iter = 0; )/* */if (k != 0)( puts(" Recycle - sink pressure violation."); )else( printf(" mperr=%6.3e mverr=%+6.3e\n", mperr, mperr); }1++;) while ((pruruz.iter == 0) && (1 < vmaxit));pruruz.iter = 1;if (1 >= vmaxit) (puts(" W: itpruruz()-> Convergence not achieved.");puts("^ Try different Lp or K values.");puts(" Node List : ");for (i=0;i<ndnm;i++) (if (fabs(dp[i]) > ptol)( printf("^Node : %4d : Deviation = %+10.6e\n*, mem.mnod[i],dP(1.]); })printf("^Max pressure deviation^= %10.6e\n", mperr);printf("^Max velocity deviation = %+10.6e\n", mverr); }/*GET FLUID FLOW RATE ALONG CAPILLARY*1if (lincap != 0) (piatemu = (PI*pp.pref/(8.0*pp.mu*pp.lref))*(p_art-p_ven)/(mem.z(0]-mem.z[nz-17);for (i=0;i<nz;i++) (xm = pow(pp.lref*mem.r(i], 4.0);pruruz.qcap[i] = -piatemu*xm; ))else {piatemu = PI*pp.pref/(8.0*pp.mu*pp.lref);for (1=0;1<nz;i++) (xx[i]^z[mem.mnod[i]];yy[i] = pruruz.pcap(i]; }vp^fspl(nz, &xx[0], &YY[0], 2, 2, 0.0, 0.0);for (i=0;i<nz;i++) (xm^pow(pp.lref*mem.r[i], 4.0);pruruz.gcap(i) = -piatemu*xm*devalcs(nz, vP, &xx[0], mem.z[i]); }}/*^*1APPENDICES^ 216return pruruz;D *Istif(I lct, I ndnm, I elnm, I bw, D *pr, D *pz, I *pc, D *rd, D *zd,D *rrd, D *rzd, D *zzd, D *ur, D *uz, D *pf,D lref, I crz, I ihof, I goon, I ngp){• i, j, k, ii, jj, wind;I eldef[EM][6];D aa, alpha, gam, vravg, vzavg, dravg, dzavg, unorm, dnorm,hh, rr, zz, rp, zp, rw, zw, dr2, dz2, drdz, ww, ro, *P_4;D estif[6][6];D edr[EM], edz(EM];D f(NM], drr(NM], drz[NM], dzz(NM], r[NM], z[NM], vr[NM], vz[NM];D stifn(NMUNM];D nrzt, wrzt, dndr, dndz, edrr, edrz, edzz, evr, evz;/*^*/for (i=0;i<NM;i++) {f[i]^= *pf++;r[i]^= *pr++;z[i]^= *pz++;drr[i]^*rrd++;if (crz == 1){ drz[i] = *rzd++; }dzz(i1 = *zzd++;vr[iJ = *ur++;vz[i] = *uz++; }for (i=0;i<EM;i++) {edr[i] = *rd++;edz[i] = *zd++;for (j=0;j<6;j++){ eldef[i][j] = *pe++; } }/*INITIALIZE VARIABLES, MATRICES, AND VECTORS*/wind = 0;aa = ro = 0.0;for (i=0;i<NM;i++) (for (j=i;j<NM;j++)( stifnlil(J1 = stifn[J][i] = 0.0; }}/*STIFFNESS MATRIX AT t=t*/switch (geom) {case 0: /* NO UPWINDING RECT */geom = 0;wind = 0;break;case 1: /* NO UPWINDING CYL */geom = 1;wind = 0;break;case 2: /* UPWINDING - RECT */geom = 0;wind = 1;break;case 3: /* UPWINDING CYL */goon = 1;wind = 1;break; )/*FORM ELEMENTAL STIFFNESS MATRIX*/switch (lct) {case 0 :^/* LAPLACE */for (1=0;1<elnm;1++) {/* */dr2^0.5*edr(i];dz2 = 0.5*edz(i];drdz = dr2*dz2;for (i= 0 ;J< 6 ;J++) (for (k=j;k<6;k++)( estif[j][k] = estif[k][i] = 0.0; }}for (j= 0 ;J< 4 ;J++) (edrr.v[j] = drr[eldef[i][j]];edrz.v01 = drzfeldeffil[J]];edzz.v[j] = dzzIeldef[i][1]); )^/*^*/for (k=0;k<ngp;k++)zp = gp(ngp, k);zw = gw(ngp, k);zz = dz2*(zp+1.0);APPENDICES^ 217for (j=0;j<ngp;j++) {rp = gp(ngp, j);rw = gw(ngp, j);rr^dr2*(rp+1.0);/* */ww = rw*zw;if (geom == 1)( ro = r(eldef[i](011+rrt }/*^5/if (ihof == 1) {nrzt = fmrzt(rr, zz, edr[1], edz[1]);dndr = fdmdr(rr, zz, edr[i], edz[i]);dndz = fdmdz(rr, zz, edr[i], edz[i]); )else {nrzt = fnrzt(rr, zz, edr(i], edz(i]);dndr = fdndr(rr, zz,^edz(i]);dndz = fdndz(rr, zz, odr(il, edz(i]); }/*^5/for (ii=0;ii<4;ii++) (for (jj= 0 ;jj<4;jj++) {estif(iiMJJ] =(wW*edrr.v(iil*dndr.v[ii]*dndr.v(jj));(ww*edrz.v[ii]*dndr.v(ii)*dndz.v[jj));(ww*edrz.v[11]*dndz.v[ii]*dndr.v(iJ]); )(ww*edzz.v(iil*dndz.v[ii]*dndz.v(jj1);(ww*edrr.v(ii)*nrzt.v(ii]*dndr.v(jj1/ro); )^}/*SCALE TO INTEGRATION DOMAIN5/forif (crz == 1) {estif[ii][jj] +=estif[11]1JJ] +=estif[ii][jj] +=if (geom == 1)estif[ii][JJ] -=(ii=0;ii<4;ii++) (for (jj=0;jj<4;jJ++){ estif(iilljj] *= drdz; }}/*INCORPORATE INTO GLOBAL STIFFNESS MATRIX AT t*/for (j=0;j<4;j++) (for (k=0;k<4;k++){ stifn[eldef[1][J]][eldef[i][k]] =estiffJ100;printf(• lapl()*);break;case 1^/* CONVECTIVE */for (i=0;i<elnm;i++) {/*^5/dr2 = 0.5*edr(i);dz2 = 0.5*edz(i];drdz dr2*dz2;for (j=0;j<6;j++) (for (k=j;k<6;k++){ estif(j][k] = estif(k](j] = 0.0; }}for (j=0;j<4;j++) (evr.v(j] = vr[eldel[i][j]];evz.v[j] = vz[eldef[1][J]];edrz.v[j] = drr[eldef[i][j]];edrz.v[j] = drz[eldet[i][J]];edzz.v(j] = dzz(eldef(i)(j]); )/* */if (wind == 1) {vravg = vzavg = dravg = dzavg = 0.0;for (j=0;j<4;j++) (vravg += fabs(evr.v(j]);vzavg += fabs(evz.v(j]);dravg += edrr.v(j];dzavg += edzz.v(j); )vravg *= 0.25;vzavg *= 0.25;dravg *. 0.25;dzavg *= 0.25;unorm norm(vravg, vzavg);dnorm = norm(dravg, dzavg);if (Imo= != 0.0) {hh =lref*(fabs(vravg)*edr(i)+fabs(vzavg)*edz[11)/unorm;gam 0.5*unorm*hh/dnorm;alpha = coth(gam)-1.0/gam;APPENDICES^ 218dndz); }aa = alpha*hh/(2.0*unorm); )else{ aa = 0.0; } }else{ aa . 0.0; }/*^./for (k=0;k<ngp;k++) {zp^gp(ngp, k);zw gw(ngp, k);zz = dz2*(zp+1.0);for (j=0;j<ngp;j++) {rp^gp(ngp, j);rw = gw(ngp, j);rr = dr2*(rp+1.0);/*ww rw*zw;/*^./if (ihof == 1) {nrzt = fmrzt(rr, zz, edr[i], edz(i]);dndr fdmdr(rr, zz, edr[i], edz[i]);dndz = fdmdz(rr, zz, edr[i], edz[i]); )else (nrzt^fnrzt(rr, zz, edr[i], edz[i]);dndr = fdndr(rr, zz, edr[i], edz[i]),dndz = fdndz(rr, zz, edr[i], edz[i]); }if (wind == 1)( wrzt = fwrzt(aa, evr, evz, nrzt, dndr,else(ww*evr.v[ii]*wrzt.v[ii]*dndr.v[JJ]);( wrzt = nrzt; )/*^./for (ii=0;ii<4;ii++)for (jj=0;jj<4;jj++) (estif[ii][Jj] +=estif[ii][JJ] +=(ww*evz.v[iil*wrzt.v[ii]*dndz.v[jj]); )) ) }/*SCALE TO INTEGRATION DOMAINfor (ii=0;ii<4;ii++) (for (jj=0;jj<4;JJ++)estif[ii][JJ) drdz; }}/*INCORPORATE INTO GLOBAL STIFFNESS MATRIXfor (j=0;j<4;j++) {. for (k=0;k<4;k++){ stifn[eldef[i][i]][eldef[i][k]l +_estif[j][k]; )) )printf(• cony()•);break;case 2 :^/* TEMPORAL STIFFNESS MATRIX */case 3 :^/* SINK STIFFNESS MATRIX */case 4 :^/* LOAD VECTOR STIFFNESS MATRIX */for (i=0;i<elnm;i++)/*^./dr2^0.5*edr[i];dz2 = 0.5*edz[i];drdz = dr2*dz2;for (J=0;J<6;j++) (for (k=j;k<6;k++)( estif[J][k] = estif[k][J]/*^*/for (k=0;k<ngp;k++) (zP = gp(ngp, k);zw = gw(ngp, k);zz^dz2*(zp+1.0);for (j=0;j<ngp;j++) {rp = gp(ngp, j);rw = gw(ngp, j);rr = dr2*(rp+1.0);^/*^.1ww = rw*zw;/*^*/if (ihof == 1)( nrzt = fmrzt(rr, zz, edr[i], edz[i]); )else( nrzt = fnrzt(rr, zz, edr[i], edz[i]); }/*^*/for (ii=0;ii<4;ii++) (= 0. 0; ))APPENDICES'^ 219for (jj=0;jj<4;jj++)estiflii][JJ] +=(ww*nrzt.v[ii)*nrat.v[JJ]); }) ) }I.SCALE TO INTEGRATION DOMAIN*/for (ii=0;ii<4;ii++)for (jj=0;jj<4;JJ++){ estif[ii][JJ] *= drdz; ))/*INCORPORATE INTO GLOBAL STIFFNESS MATRIX AT t*/for (j=0;j<4;j++)for (k=0;k<4;k++)Stifn[eldef[i][J]][eldef[i][k]] +_estif[J][k]; }} }switch (lct)case 2case 3 : printf(" temp()/sink()");break;case 4 : printf(" func()");break; )break; }*/(lct == 4) {p_q = matvecbw(0, ndnm-1, bw, &stifn[0][0], &f[0]);for (1=0;i<1414;1++)( f[i] = *p_q++; )return &f[0]; )/* */return &stifn[0][0];U modstif(I nebc, I *pe, D *pr, D *pz, I *be, D *ba, D *bh, I *bl, I *b2,D *rd, D *zd, D *st, I ihof, I ngp)I i, j, k, ii, jj;I lnod[3], gnod[3], bce[MN], bc1[MM], bc2[M14];I eldef[EM][6];D gamma, cgamma, sganma, ss, rr, zz, sp, sw, sqn1n2, ds, ds2, ww;D edr[EM], edz[EM], r[NM], Z[1414];D he[3], se[3][3], ae[4][4], bca[MM][3], bch[MM][3];W nrzt;U my;/*^*/for (i=0;i<NM;i++)r[i] = *nr++;z[i] *pz++;for (j=0;j<NM;j++)( mv.m[i][j] = *st++; }}for (i=0;i<MN;1++)bol[i] = *b1++;bc2[i] = *b2++;bce[i] . *be++;for (j=0;j<3;j++) {bca[i][J] = *ba++;bch[i][j] = *bh++; }}for (i=0;i<EM;i++)edr[i] = *rd++;edz[i] = *zd++;for (j=0;j<6;j++)eldef[i][j] = *pe++; ) }/*INITIALIZE VARIABLES*/for (i=0;i<3;i++)lnod[i] = gnod[i]for (i=0;i<NM;i++)mv.v[i] = 0.0; }= 0 ; )/*INCORPORATE TYPE 3 BOUNDARY CONDITIONS*/for (i=0;i<nebc;i++) (for (j=0;j<4;j++) (if (bcl[1] == eldef[bce[i]][j]) {lnod[0] = j;gnod[0] = born]; )if (bc2[i] == eldef[bce[i]][j]) {lnod[1] = j;gnod[1] = bc2[i]; } )for (j=0;j<3;j++) (he[j] = 0.0;for (k=j;k<3;k++)( ae[1][k] = ae[k][J] = se[J][k] = se[k][J] = 0.0; )}/*ifAPPENDICES^ 220/*^*/ds = sqrt(sq(r[bc1[1]]-r[hc2[1]])+sq(z[bc1[1]]-z[bc2[1]]));ds2^0.5*ds;if (r[bc1[1]] == r(bc2[1])) (gamma = 0.5*PI;cgamma = 0.0;sgamma = 1.0; )elseif (z[bcl[i]] == z[bc2[1]])gamma = 0.0;cgamma = 1.0;sgamma = 0.0; }else {gamma = atan(fabs(r[bcl[i]]-r[bc2[1]])/fabs(z[bcl[i]]-z[bc2[1]]));cgamma = cos(gamma);sgamma = sin(gamma); }for (j=0;j<ngp;j++) {sp = gp(ngp, j);sw = gw(ngp, j);as = ds2*(sp+1.0);rr = ss*cgamma;zz ss*sgamma;ww = sw;/*^*/if (ihof == 1){ nrzt = fmrzt(rr, zz, edr[bce[i]], edz[bce[i]]); }else( nrzt = fnrzt(rr, zz, edr[bce[i]], edz[bce[i]]); }sqn1n2 = sg(nrzt.v[lnod[0]])+eg(nrzt.v[lnod[1]]);/* *1for (1i=0;13.<2;11++) (for (jj=0;jj<2;JJ++) (se[il][ii] += (ww*nrzt.v[lnod[li]]*nrzt.v[lnod[ji]]);ae[ii][jj] += (ww*bca[l][11]*nrzt.v[lnod[11]]*seinln2);/*SCALE TO INTEGRATION DOMAIN*/for (ii=0;ii<2;11++) {for (jj=0;jj<2;jj++) {se(ii][ii] *= ds2 ;ae[ii][jj] *= ds2; ))/*GET h CONTRIBUTION*/for (ii=0;ii<2;ii++) {for (jj=0;jj<2;JJ++){ he[ii] += se[11.1[Jj)*bch[i][JJ); }}/*INCORPORATE INTO STIFFNESS MATRIX AND MIS VECTOR*/for (ii=0;ii<2;ii++) {mv.v[gnod[ii]] += he[ii];for (jj=0;jj<2;jj++){ mv.m(gnod[ii]][gnod[jj]] += ae[11][JJ]; )) )) ) }vdiff/* */printf(" modstif()");/* */return mv;dspc(I ndnm, D al, D a2, D diff, D *vr, D *vz, I crz)11;D vel, vrz, vr2, vz2;vdiff d;/*^*/if ((al == 0.0) && (a2 == 0.0)) {for (i=0;i<ndnm;i++) {d.rr[iJ = diff;d.rz[1] = 0.0;d.zz[i] =^)1else (for (i=0;i<ndnm;i++) (vrz = fabs(vr[1]*vz[1]);vr2^sq(vr[i));vz2 = sq(vz[i]);vel = sqrt(vr2+vz2);if (vel == 0.0) {d.rr[i] = diff;d.rz[i] = 0.0;d.zz[i] = diff; )APPENDICES^ 221elsed.rr[i]^(a2*vz2+al*vr2)/vel+diff;if (crz == 1)( d.rz[i] = (al-a2)*vrz/vel; }else{ d.rz[i] = 0.0;d.zz[i]^(a2*vr2+al*vz2)/vel+diff; 1)/*^*/printf(" dspc()");/*^*/return d;/* USER DEFINED 1p() FUNCTION.*/D 1p(I typ, D 1p0, D 1pl, D x){ D 1p=0.0;switch (typ)case 0 :1p = 1p0;break;case 1 :1p = (1.0-x)*1p0+x*lpl;break;case 2 :if (x <= 0.50)( 1p = 1p0; )elsebreak;case 3if ((xelse( 1p = 1pl;< 0.90) II (x > 0.95))1p = 1p0;{ 1p = 1pl;break;case 4 :1p = 1p0+(lp1-1p0)/(1.0+exp(-50.0*(x-0.5)));break;return 1p;MASS BALANCE*/^ massb(I elnm, I nr, I nz, I nesink, media pp, membr mem, I *pc, I *sn, D *rd, D *zd,D *pc, D *ph, D *pf, D qcap, D avgas, I ngp, I geom)I i, j, k, 1, er, ez;I isink[EM], eldef[EM][6];D dr2, dz2, lref3, uc, uu, rr, rp, rw, zz, zp, zw,xm, fqm, sqm, qfi, qfo, qmf, qms, qsi, qso, qsf, gas, p, c,pa, ca, va, pavg, cavg, vavg, vol, ww;D fmem[IM), edr[EM], edz[EM], co[NM], hp(NM];D epr, ecs, nrzt;/* */for (i=0;i<NM;i++) (co[i] = *pc++;hp[i] = *ph++; )for (i=0;i<EM;i++)(isink[i] = *sn++;edr[i] = *rd++;edz[i] = *zd++;for (j=0;j<6;j++)( eldef[i][J] = *pe++;for (i=0;i<IM;i++){ fmem[i] = *pf++; }er = nr-1;ez = nz-1;switch (geom) (case 0: /* NO UPWINDING - RECT */geom = 0;break;case 1: /* NO UPWINDING - CYL */geom = 1;break;case 2: /* UPWINDING - RECT */geom 0;break;case 3: /* UPWINDING - CYL */geom = 1;APPENDICES^ 222break; }/*FLUID AND SOLUTE MASS BALANCE*1puts(" Mass Balance : ");lref3 = pp.lref*pp.lref*pp.lref;qfi = qfo = qmf = qms = qsi = qso = qsf = qss = 0.0;/*MEMBRANE FLUID FLOW AND SOLUTE TRANSPORTqfi, qfo : m3/sqsi, qso : Kg/sfor (i=0;i<ez;i++) (k = i*er;xm = edz[k]*pp.lref;rr = 0.5*(mem.r(i)+mem.r[i+1])*pp.lref;uu = 0.5*(mem.vm[i]+mem.vm[i+1]);uc = 0.5*(fmem[i]+fmem[1+1]);fqm = PI2*uu*rr*xm;eqm = PI2*uc*rr*xm*pp.cref;qmf += fqm;gins += sqm;if (fqm >= 0.0) {qfi += fqm;qsi += sgm; )else {qfo += fqm;(ISO += dgM; 3)/rprintf(" Fluid into membraneprintf(" Fluid out of membraneprintf(" Net fluid into membraneprintf(" Solute into membraneprintf(" Solute out of membraneprintf(" Net solute into membraneprintf(" Fractional fluid flow into membrane/*SINK FLUID FLOW AND SOLUTE TRANSPORTqsf : m3/sqss : Kg/s• %+10.6e\n", qfi);= 1:6+10.6e\n", qfo);• %+10.6e\n", qmf);• %+10.6e\n", qsi);= 16+10.6e\n", qso);= 96+10.6e\n", gins);= 96+10.6e\n", qfi/qcaP);*/pavg = cavg = vol = 0.0;for (i=0;i<elnm;i++) (if (isink[i] ==. 1) (dr2 = 0.5*edr[i];dz2 = 0.5*edz[i];xm = dr2*dz2;for (J=0;i< 4 ;J++) {epr.v[j] = hp[eldef[i][J]1;ecs.v[j] = co[eldef[i][j]];pa^ca = va = 0.0;for (k=0;k<ngp;k++) (zp = gp(ngp, k);zw^gw(ngp, k);zz = dz2*(zp+1.0);for (j=0;j<ngp,j++) (rp = gp(ngp, j);rw = gw(ngp, j);rr = dr2*(rp+1.0);./WW = rw*zw;nrzt = fnrzt(rr, zz, edr[i], edz[1]);p = c = 0.0;for (1.0;1<4;1++) (p += epr.v[11*nrzt.v[1];c += ecs.v[1]*nrzt.v[1];switch (geom) (case 0 :pa += ww*p;ca += WW* C;va += ww;break;case 1 :pa += ww*PI2*rr*p*pp.lref;ca += ww*PI2*rr*c*pp.lref;va += ww*PI2*rr*pp.lref;break; ) 3)pavg += (lref3*xm*pa);cavg += (lref3*xm*ca);vol += (lref3*xm*va); 1)APPENDICES^ 223if ((avgas := 0.0) && (vol != 0.0)) {pavg 1= vol;cavg 1= you;if ((pa pp.pref*pp.ls*(pavg-pp.plymph)) > 0.0){ qsf = -pp.vol_tis*(pa+pp.jo); )else( qsf = -pp.vol_tis*pp.jo; }qss qsf*cavepp.cref;vavg = qmf/(((D)nesink)*avgas);printf(" Net fluid out sink^ 96+10.6e\n", qsf);printf(" Net solute out sink = 96+10.6e\n", qss),printf(" Avg solute cone out sink^= 96+10.6e\n", cavg);printf(" Avg cony velocity out sink = %+10.6e\n", vavg);printf(" Avg tissue pressure at sink 16+10.6e\n", pavg);printf(" Volume of sink^ 96+10.6e\n", vol); }else{ puts(" No sinks in geometry."); )printf(" Fluid Mass Balance Residual^9610.6e\n", 00a-qsf);printf(" Solute Mass Balance Residual^9610.60\n", qms-qss);}V domavg(I alum, media pp, I *pe, D *rd, D *zd, D *pc, D *ph, I ngp, I goon)I i, j, k, 1;I eldef[EM][6];D dr2, dz2, lref2, rr, rp, rw, zz, zp, zw, xm, p, c,pa, ca, va, pavg, cavg, vol, ww;D edr[EM], edz[EM], co[NM], hP[NM];W epr, ecs, nrzt;/* */for (i=0;i<NM;i++) (co il] . *pc++;hp[i) = *ph++; }for (i=0;i<EM;i++)(edr[i]^*rd++;edz[i]^*zd++;for (j=0;j<6;j++)eldef[i][j] = *pe++; ))switch (geom) {case 0: /* NO UPWINDING - RECT */geom 0;break;case 1: /* NO UPWINDING - CYL */geom 1;break;case 2: /* UPWINDING - RECT */geom 0;break;case 3: /* UPWINDING - CYL */geom 1;break; )/*GET AVERAGE QUANTITIES IN SOLUTION DOMAIN*/lref2^pp.lref*pp.lref;1* *1pavg = cavg = vol = 0.0;for (1.0;i<elnm;i++) (dr2^0.5*edr[i];dz2 = 0.5*edz[i];xm = dr2*dz2;for (j=0;j<4;j++) (epr.v[j] = hp[eldeffinfl];ecs.v[j] = co[eldef[i][j]]; }pa = ca = va = 0.0;for (k=0;k<ngp;k++)zp = gp(ngp, k);zw = gw(ngp, k);zz = dz2*(zp+1.0);for (j=0;j<ngp;j++) (rp = gp(ngp, j);^rw^gw(ngp, j);rr = dr2*(rp+1.0);/*^*/ww = rw*zw;nrzt = fnrzt(rr, zz, edr[i], edz[1]);p = c = 0.0;for (1=0;1<4;1++) {p += epr.v[1]*nrzt.v[1];c^ecs.v[1]*nrzt.v[1]; }switch (geom) (case 0 :APPENDICES^ 224pa += ww*P;ca += ww*c;va += ww;break;case 1 :pa += ww*PI2*rr*p*pp.lref;ca += ww*PI2*rr*c*pp.lref;va += ww*PI2*rr*pp.lref;break; } ))pavg += lref2*xm*pa;cavg += lref2*xm*ca;vol^lref2*xm*va; }pavg /= vol;cavg /= vol;printf(" Average Tissue Solute Concentration = %10.5e\n", cavg);printf(" Average Tissue Pressure^= %10.6e\n", pavg),OSMOTIC PRESSURE ROUTINE.USE EFFECTIVE SOLUTE CONCENTRATION (DUE TO VOLUME EXCLUSION).THIS IS THE CONCENTRATION BEING CALCULATED (SO NO ADJUSTMENT).*/D *osmopr(I n, media pp, D *pc)I i, jtD cr, a[5], c[NM], op[NM];/* */for (i=0;i<NM;i++) (c[i]^*pc++;op[i] = 0.0; )/*CONVERT TO EFFECTIVE CONCENTRATION IN AVAILABLE FLUID VOLUME*/if (pp.osm := 0) {a[0] = 0.0;cr pp.cref/pp.pref;a[1] = 57.18198*cr;cr *= pp.cref;a[2] = -1.238832*cr;cr *= pp.cref;a(3]^0.050849*cr;/*DIMENSIONLESS FORMS of c[) and op[]*/for (i=0;i<n;i++) (op[i] = a[3];for (j=2;j>-1;j--)( op(i) = op[i]*c[i]+a[j]; }} }printf(" osmp()");/*^*/return &op[0];D *cmpd(I n, membr m, D p_art, D p_ven)I i, j, maxit;D bc, err, tol, xm;D xbc[3], fbc[3], p[H14), pm[IM];M s_f;vec3 v_f, v_p;/*^*/maxit = 200;tol = 1.0e-6;/* */v_p = fspl(n, &m.z[0], &m.hp[0], 2, 2, 0.0, 0.0);v_f^fspl(n, &m.z(0], &m.op[0], 2, 2, 0.0, 0.0);/* */p[0] = p_art;p[1] = (p_art-p_ven)/(m.z[0)-m.z[n-1));bc p_ven;/*GET FIRST TWO VALUES*/for (j=0;j<2;j++) (s_f^pdcr(2, m, &p[0], m.z[0], m.z[n-1), NJ-2, v_f, v_p, v_p, n, 1);if (s_f.error == 1) (printf("\n E: pdcr()-> Failed beyond t = 1610.6f.\n", s_f.t);exit(0); }err = bc-s_f.x[0];xbc[J) = p[l];fbc[j]^s_f.x[0];p[1] *= 0.9; )APPENDICES^ 225if (fabs(err) < tol) goto done;/*SECANT METHOD UNTIL CONVERGENCE*/j^2;do (j++;if ((xm^fbc[1]-fbc[0]) )= 0.0){ p[1] = xbc[1]-(fbc[1]-bc)*(xbc[1]-xbc[0])/xm; }else( P[1] = xbc[1]; }s_f = pdcr(2, m, &p[0], m.z[0], m.z[n-1], NJ-2, v_f, v_p, v_p, n, 1);if (s_f.error^1) {printf("\n E: pdcr()-> Failed beyond t^9610.6f.\n", s_f.t);exit(0); )err = bc-s_f.x[0];xbc[0] = xbc[1];fbc[0] = fbc[1];xbc[1] = p[1];fbc[1] = s_f.x[0];) while((fabs(err) > tol) && (j < maxit));if (j > maxit) (printf("\n Not converged after Ifid iterations.\n", maxit);exit(0); )*/(s_f.error == 1) (printf("\n E: pdcr()-> Failed beyond t^%10.6f.\n", s_f.t);exit(0); )err bc-s_f.x[0];/*vf/*fordone:^/*if/*ifels/ **/= fspl(s_f.k, &s_f.tt[0], &s_f.xx[0], 2, 2, 0.0, 0.0);*/(i=0;i<n;i++)Pm[i] = evalcs(s_f.k, v_f, &s_f.tt[0), &s_f.xx[0],*/(fabs(s_f.err) > tol)printf(" cmpd(merr=%6.3e)", fabs(s_f.err)); )e( printf(" cmpd()"); }*/m.z[i]); )return &pm[0];}D *cmcd(I n, membr m){I i;D c[HM], ccap[IM];M s_f;vec3 v_u, v_v, v_w;/* */v_u^fspl(n, &m.z[0], &m.cp[0], 2, 2, 0.0, 0.0);v_v^fspl(n, &m.z[0], &m.ct[0], 2, 2, 0.0, 0.0);v_w = fspl(n, &m.z[0], &m.vm[0], 2, 2, 0.0, 0.0);/*^*/c[0] = m.c_art;/* */s_f^pdcr(1, m, &c[0], m.z[0], m.z[n-1], NJ-2, v_u, v_v, v_w, n, 0);/* */v_u = fspl(s_f.k, &s_f.tt[0], &s_f.xx[0], 2, 2, 0.0, 0.0);/*^*/for (i=0;i<n;i++){ ccap[1] = evalcs(s_f.k, v_u, &s_f.tt[0], &s_f.xx[0], m.z[i]); }/*^*/printf(" cmcd()");/*^*/return &ccap[0];PREDICTOR-CORRECTOR ODE INTEGRATION ROUTINESs_f^CONTAINS ANY ERROR CONDITIONS, THE NUMBER OF FUNCTIONEVALUATIONS, k, THE SOLUTION AT tf, AND THE tt AND xoVECTORS.*/M pdcr(I n, membr m, D *p_x, D tt, D tf, I nci, vec3 v_f, vec3 v_p, vec3 v_q,I nn, I pc)11, j;D err, nerr, t[6], ci, ci24, xm;D pi[6], bp[6], cp[6], ct[6], vm[6], f[6][HM], x[6][HM], x4p[HM];M s_f;/*^*/a_f.err = 0.0;APPENDICES^ 226s_f.error = s_f.nf^s_f.k^0;xm = 19.0/270.0;ci^fabs(tf-tt)/((D)(nci)-1.0);ci24 = ci/24.0;for (i=0;i<n;i++){ m[0][1] = *p_x++; }s_f.xx[s_f.k) = x[0][0];s_f.k++;for (i=0;i<nci;i++){ s_f.tt[i] = (p)( 1 )*ci; }/*FOURTH ORDER RUNGE KUTTA ROUTINE FOR FIRST FOUR POINTSfor (j=0;j<3;j++) (tt += ci;switch (pc) {case 0 :^cp[j]^evalcs(nn, v_f, &m.z[0], &m.cp[0], tt-ci);ct[j] = evalcs(nn, v_p, &m.z[0], &m.ct[0], tt-ci);vm[j]^evalcs(nn,^&m.z[0], &m.vm[0], tt-ci);p_x^rk4(n, m, tt-ci, ci, fix[J][0], cp[j], ct[j}, vm[J] pc);break;case 1 :hp[j] = evalcs(nn, v_p, &m.z[0], &m.hp[0], tt-ci);Pi[J]^evalcs(nn, v_f, &m.z[0], &m.op[0], tt-ci);p_x = rk4(n, m, tt-ci, ci, &x[J][0], P1[J], hp[J], 0.0, pc);break;s_f.nf += 4;for (i=0;i<n;i++){ x[j+1][1]^*p_x++;s_f.xx[s_f.k]^x[j+1][0];s_f.k++; }/*PREDICTOR-CORRECTOR INTEGRATION ROUTINE*1for (1=4;i> - 14--) {t[i] = tt+(D)(i-3)*ci;switch (pc) {case 0cp[j]^evalcs(nn, v_f, &m.z[0], &m.cp[0], t[i]);ct[j]^evalcs(nn, v_p, &m.z[0], &m.ct[0], till);vm[j] = evalcs(nn, v_q, &m.z[0], &m.vm[0], t[i]);break;case 1 :hp[j] = evalcs(nn, v_p, &m.z[0], &m.hp[0], t[i]);Pi[J] = evalcs(nn, v_f, &m.z[0], &m.op(0], t[i]);break; ])for (j=0;j<4;j++)switch (pc) {case 0 :p_x^memcs(m, t[j], &m[J][0], cp[j], ct[j], vm[j]);break;case 1 :p_x = mempr(m, t[J], &x[J][0], pi[j], hp[J]);break; }s_f.nf++;for (i=0;i<n;i++){ f[j][1] = *p_m++; } }/*^*1while (tt < (tf-ci)) {err = nerr = 0.0;t[4)^tt+ci;switch (pc) {case 0 :cp[4]^evalcs(nn, v_f, &m.z[0), &m.cp[0], t[4]);ct[4] = evalcs(nn, v_p, &m.z[0], &m.ct[0], t[4]);vm[4] = evalcs(nn, v_q, &m.z[0], &m.vm[0], t[4]);p_x = memcs(m, t[4], &x4p[0], cp[4), ct[4], vm[4]);break;case 1 :hp[4] = evalcs(nn, v_p, &m.z[0], &m.hp[0), t[4]);pi[4] = evalcs(nn, v_f, &m.z[0], &m.op[0], t[4]);p_x = mempr(m, t[4], &x4p[0], pi[4), hp[4]);break; }s_f.nf++;for (i=0;i<n;i++){ f(4)(i) = *p_x++; }for (i=0;i<n;i++){ x4p[i] = m[3][1]+c124*(55.*f[3][1]-59.*f[2][1]+37.*f[1][1]-9. * f [0][i)); )/* x[4][] = x4c[] */APPENDICES^ 227for (i=0;i<n;1++) {x[4] [1] = x[3][i]+c124*(9.*f[4][i]+19.*f[3][1]-5.*f[2][1]+f[1][11);/*ABSOLUTE ERROR ESTIMATE*/nerr^xm*fabs(x4p[il-x[4][1]);if (nerr > err)( err = nerr; ))if (err > s_f.err)( s_f.err = err; )tt += ci;s_f.xx[s_f.k]^x[4][0];s_f.k++;for (J=0;i< 4 ;J++) (t[j] = t(j+1];for (i=0;i<n;i++) (x(j] (1] = x[J+1][1];f[j][i] = f[j+1][i]; )) )for (i=0;i<nri++){ s_f.x[i] = x[4][i]; )/*^*/return s_f;rk4() IS THE STANDARD FOURTH ORDER RUNGE-EUTTA SYSTEM OFORDINARY DIFFERENTIAL EQUATION INTEGRATION ROUTINE. THISIS REQUIRED FOR THE PREDICTOR-CORRECTOR ROUTINE SINCE IT ISA MULTI-STEP ALGORITHM.*/D *rk4(I n, membr m, D t, D ci, D *p_x, D xl, D x2, D x3, I pc)I i;D ci2, ci6;D z[HM], g[HM], 8111[13141, cl [HM] ;/*^*/ci2^ci/2.0;ci6^ci/6.0;/* */for (i=0;i<n;i++) (z[i] = *p_x++;sm[i] = 0.0; }switch (pc) (case 0 :p_x = memcs(m, t, &z[0], xl, x2, x3);break;case 1 :p_x = mempr(m, t, &z[0], xl, x2);break; }for (i=0;i<n;i++)( q[i] = *p_x++; )for (i=0;i<n;i++) (sm[i] += q[i];g[l]^z[i]+c12*q[i]; )switch (pc) (case 0 :p_x = memcs(m, t+ci2, &q[0], xl, x2, x3);break;case 1 :p_x = mempr(m, t+ci2, &g[0], xl, x2);break; }for (i=0;i<n;i++)( q[i] = *p_x++; )for (i=0;i<n;i++) {sm[i] += 2.0*q[1);g[i] = z[i]+ci2*q[i]; }switch (pc) (case 0 :p_x^memcs(m, t+ci2, &q[0], xl, x2, x3);break;case 1 :p_x = mempr(m, t+ci2, &g[0], xl, x2);break; }for (i=0;i<n;i++)( q(i] = *p_x++; )for (1=0;i<n;i++) {sm[i] += 2.0*q[i];g[i]^z[i]+ci*q[i]; )switch (pc) (case 0 :p_x = memcs(m, t+ci, &q[0], xl, x2, x3);break;APPENDICES^ 228case 1 :p_x = mempr(m, t+ci, &g[0], xl, x2);break;for (i=0;i<n;i++){ q[i] = *p_x++1 )for (i=0;i<n;i++)sm[i] += g[i];z[i] += ci6*sm[i]; )return &z[0];D *mempr(membr m, D t, D *px, D pi_mem, D p_tiss)I i;D p, x[HM], dxdt[HM];/*^*/for (i=0;i<HM;i++){ x[i] = *px++; }/*^*/p = p_tiss-m.sigma*(pi_mem-m.pi_cap);/*asg*lp*(p_tiss-p_cap-sigma*(pi_mem-pi_cap)*/dxdt[0]^x[1];dxdt[1]^m.asq*lp(m.typ_lp, m.1p0, m.lpl, t) * (P-X[0]);printf("", x[0],x[1],dxdt[0],dxdt[1]);/*^*/return &dxdt[0];D *memcs(membr m, D t, D *px, D c_cap, D c_tis, D v_mem)I 1;D pe, isigmas, xm, x[HM], dxdt[HM];/*^*1for (i=0;i<HM;i++){ x(i)^*px++;/*Js = (PS.Pe/(e(Pe)-1))(c_cap-c_tis)+Jv(1-sigmas)c_cap*/isigmas^1.0-m.sigma;xm^PI2*m.r[0]*m.lref;pe fabs(v_mem*isigmas/m.ps_d);if (pe > 1.0e-06) {if (pe > 100.0){ dxdt[0] = -(v_mem*isigmas*c_cap)*xm; Ielsedxdt[0] = -(m.ps_d*(c_cap-c_tis)*pe/(exp(pe)-1.0)+v_mem*isigmas*c_cap)*xm; I/else{ dxdt[0]^-(m.ps_d*(c_cap-c_tis)+v_mem*isigmas*c_cap)*xm; Iprintf("", x[0],dxdt(0]);/*^*/return &dxdt[0];NATURAL, CLAMPED, OR FITTED END POINTS CUBIC SPLINE. IF n < 6THEN ONLY NATURAL OR CLAMPED BOUNDARY CONDITIONS CAN BE APPLIED.O : NATURAL1 : CLAMPED (1ST DERIVATIVE SPECIFIED : ldv, udv)2 : FITTED END POINTS*/vec3 fepl(I n, D *px, D Spy, I lbc, I ubc, D ldv, D udv){I i, k, kp, km, nm=n-1, norm=n-2;D a[NJ], b[NJ], c[NJ], d[NJ), h[NJ], x[NJ], YINJ1, *P_f;vec3 v_f;/*^*/for (i=0;i<NJ;i++) {= *px++;y[i] = *py++; }/*^*/if ((n < 6) && (lbc == 2)) lbc = 0;if ((n < 6) && (ubc^2)) ubc = 0;/*^*/for (i=0;i<nm;i++)h[i]^x[i+1]-x[i]; )art)) = 0.0;switch (lbc) {case 0:b[0]^1.0;c[0]^0.0;d[0] = 0.0;APPENDICES^ 229break;case 1:b[0] = 2.0*h[0];c[0] = h[0];d[0] = 3.0*((Y[1]-y[0])/h[0]-1dv);break;case 2:b[0] = -h[0];c[0] = h[0];d(0] = 3.0*sq(h[0])*ddpoly(&x[0], &y[0]);break; }for (k=1;k<nm;k++)kp = k+1;km = k-1;a[k] = h[km];b[k] = 2.0*(h[km]+h[k]);c[k] = h[k];d[k] = 3.0*([Y[kP]-Y[h])/h[k]-(y(k]-Y[km])/h[km]); )c[nm] = 0.0;switch (ubc) {case 0: a[nm] = 0.0;b[nm] 1.0;d[nm] = 0.0;break;case 1:a[nm] = h[nmm];b[nm] = 2.0*h[nmm];d[nm] = -3.0*((y[nrq-y[nmm])/h[nmm]-udv);break;case 2:a[nm] = h[nmm];b[nm] = -h[nmm];d[nm] = -3.0*sq(h[nmm])*ddpoly(&x[n-4], &y[n-4]);break; )/*TRIDIAGONAL LINEAR EQUATION SOLVER/p_f = tridiag(n, &a[0], &b[0], &c[0], &d[0]);/*^*/for (i=0;i<n;i++)v_f.r[i] = *P_f++; }for (k=0;k<nm;k++) {kp = k+1;v_f.q[k] = (y[kp]-y[k])/h[k]-h[k]*(2.0*v_f.r[k]+v_f.r(kp])/3.0;v_f.s[k] = (v_f.r[kp]-v_f.r[k])/(3.0*h[k]); )/*^*/return v_f;}D *tridiag(I n, D *a, D *b, D *c, D *d){ I i, im, nm=n-1;D xmt, x[NJ], p[NJ], g[NJ];/*^*/a++ ;p[0] = -(*c++)/(*b);q[o] = (*d++)/(*b++);for (i=1;i<n;i++) {im = i-1;xmt = (*a)*p[im]+(*b++);p[i] = -(*c++)/xmt;q[i] = ((*d++)-(*a++)*q[im])/xmt; )x[nm] = q[nm];for (i=nm-1;i>-1;i--)x[i] = pfil*x(i+1]-4-q[i]; }/*^*/return fix[0];}D evalcs(I n, vec3 v, D *px, D *PY, D zz){I i, j, k, nm=n-1;D xt, zy, x[NJ], Y[N,1];C *mssg;mssg = "\n W: evalcs()-> %f out of range.\n";/*^*/for (i=0;i<NJ;i++) {x[i] = *px++;Y[i] = *PY++; )/*PLUS-MINUS TWO PERCENT OF RANGE*1APPENDICES^ 230xt = 0 02*(x[nm]-x[0]);if (zz < (x[0]-xt)) {= 1;printf(mssg, zz); }elseif (zz > (x[nm]+xt)) (= n-2;printf(mssg, zz); )else { i = 0;j = nm;while (j > 1+1) (k = (i+j)/2.0;if (zz < x[k]) j = k;else 1 = k; ) }zy = zz-x[i];zy = y[i]+zy*(v.g[i]+zy*(v.r[i)+zy*(v.e[i])));/*^*/return zy;}D devalca(I n, vec3 v, D *px, D zz)I i, j, k, nm=n-1;D xt, zy, x[NJ];C *mssg;mssg = "\n W: devalcs()-> %f out of range.\n";/*^*/for (i=0;i<NJ;i++)( x[i] = *px++; }/*PLUS-MINUS TWO PERCENT OF RANGE*/xt = 0.02*(x[nm)-x[0]);if (zz < (x[0]-xt)) {= 1;printf(masg, zz); )elseif (zz > (x[nm]+xt)) (= n-2;printf(mssg, zz); )else ( i = 0;j = nm;while (j > 1+1) (k = (1+J)/ 2 . 0 ;if (zz < x[k]) j = k;else i = k;zy^zz-x[i];zy^v.g[i]+zy*(2.0*v.r[i]+zy*(3.0*v.s[i]));/* */return zy;D d2evalca(I n, vec3 v, D *px, D zz)I i, j, k, nm=n-1;D xt, zy, x[NJ];C *mssg;mssg = •\n W: d2evalcs()-> 96f out of range.\n";/*^*/for (i=0;i<NJ;i++)( x[i] = *px++; )/*PLUS-MINUS TWO PERCENT OF RANGE*/xt = 0.02*(x[nm]-x[0]);if (zz < (x[0]-xt)) (= 1;printf(mssg, zz); )elseif (zz > (x[nm]+xt)) (i = n-2;printf(mssg, zz); )else (^0;j = nm;while (j > i+1) (k = (i+j)/2.0;if (zz < x[k]) j^k;else 1 = k; ) )zy = zz-x[i];zy = 2.0*v.r[i]+zy*3.0*v.s[i];/*^*/return zy;) }ddpoly() SUPPLIES THE THIRD DERIVATIVE OF THE FITTED NEWTON'SDIVIDED DIFFERENCE POLYNOMIAL. INSTEAD OF SETTING UP A DIVIDEDAPPENDICES^ 231DIFFERENCE TABLE, ONLY THE REQUIRED COEFFICIENTS DESCRIBING THEINTERPOLATING POLYNOMIAL ARE CALCULATED. THIS REDUCES MEMORY USEAND CALCULATIONS.*1D ddpoly(D *px, D *py){I i, j;D x[6], y[6];for (i=1;i<5;i++)x [i] = *px++;y[i]^*py++; }for (j= 1 ;i< 4 ;J++) (for (i=4;i>j;i--)( y(1) = (Y[1] -Y[1 - 1))/(x[i] -x(i -j)); )1RETURN THIRD DERIVATIVE/6.0*1return y[4];}D *solver(I n, I bw, D *pa, D *pb)I i, j, k=0, nm=n-1, ink, index(NM], ierror=0;D max, xm;D x(NM], b[NM], maxr[NM], a[NM]IEM];char *mssg;swag^"\n ERROR: solver()-> Diagonal element = 0.0•;for (i=0;i<NM;i++) (b[1] = *pb++;for (j=0;J<NM;J++)( a[i](1) = *Pa++; ))/*^*/for (1=0;1<n;1++) (index[1] = 1;max = 0.0;for (j=i;j<i+bw;j++) (if (fabs(a[i][j]) > max)( max = fabs(a[1](J)); ))maxr[1] = max; )for (i=0;i<nm;1++) (max = 0.0;for (j=i;j<n;J++)if (fabs(a[index[J])(1))/maxr[index[J]] > max) (k = j;max = fabs(a[index[j]][1])/maxr[index[j]]; }}ink = index[k];index[k] = index[i];index[i] = ink;for (j=i+l;j<n;j++) (xm = a[index[J]][1]/a[ink][i];for (k=i+1;k<i+bw;k++)( a[index[J]][k] -= xm*a[ink][k]; )b[index[J)) -= xm*b[index[i]];a(index(jMi] = 0.0; ))for (j=nm;j>0;j--) (for (i=j-l;i>-l;i--) (xm = a[index[i]][JI/a[index[j)][j];a[index[i]][j] -= xm*a[index[j]][j];b[index[i]] -= xm*b(index[j]); ))for (i=0;i<n;i++) {if (a[index[1]][1] == 0.0){ ierror = l;continue; )x[i] = b[index[1]]/a(index[1]][1]; )if (ierror == 1) (puts(mssg);exit(0); )/*^*/return &x[0];D *matvecbw(I nb, I ne, I bw, D *pa, D *pv)I i, j, bw2p;D b[NM], v[NM), a[NM][NM];bw2p = bw/2+2;for (i=0;i<NM;i++)v(i) = *pv++;b[i] = 0.0;for (j=0;j<NM;J++) (a[i][i] = *Pa++; ))/*^*/for (1=nb;i<=ne;i++)for (j=i-bw2p;j<=i+bw2p;j++)APPENDICES^ 232if (j < nb) continue;if (j > ne) continue;b[1] += 4(1][i]*v[1]; )}return &b[0];)D 11(D x, D dx){ D y = 1.0-x/dx,return y;}D 12(D x, D dx)D y = x/dx;return yr)D 13(D x, D dx)D y = 4.0*(x/dx)*(1.0-x/dx);return y;)D dll(D x, D dx)D y = -1.0/dx;return y;}D d12(D x, D dx)D y = 1.0/dx;return y;}D d13(D x, D dx)D y = (4.0/dx)*(1.0-2.0*(x/dx));return y;)D d211(D x, D dx)D y = 0.0;return y;)D d212(D x, D dx)D y = 0.0;return yr)D d213(D x, D dx)D y = -8.0/s4(dx);return y;}W fnrzt(D r, D z, D dr, D dz)D 11r, 12r, 11z, 12z;W v;llr = 11(r, dr);12r = 12(r, dr);11z = 11(z, dz);12z = 12(z, dz);/* *1v.v[0) = 11r*11z;v.v[1] = 12r*11z;v.v[2] = 12r*12z;v.v[3] = 11r*12z;return v;}W fmrzt(D r, D z, D dr, D dz)D 11r, 12r, 11z, 12z;W v;llr = 11(r, dr)+13(r, dr);12r = 12(r, dr)+13(r, dr);llz = 11(z, dz)+13(z, dz);12z = 12(z, dz)+13(z, dz);/* *1v.v[0] = 11r*11z;v.v[1] = 12r*11z;v.v[2] = 12r*12z;v.v[3] = 11r*12z;return v;)W fwrzt(D aa, W evr, W evz, W przt, W dpdr, W dpdz)APPENDICES^ 233I i;W a, v;/*^*/for (1=0;i<4;1++) {aa*(evr.v(i)*dpdr.v[1]+evz.v[i]*dpdz.v[i]);v.v[i] = przt .v [ 1] +a .v [1] ; }/*^*/return v;)W fdndr(D r, D z, D dr, D dz){D dllr, dl2r, 11z, 12z;W v;dllr = d11(r, dr);dl2r = d12(r, dr);llz = 11(z, dz);12z = 12(z, dz);/*^*/v.v[0] = dllr*11z;v.v[1] = d12r*11z;v.v[2] = d12r*12z;v.v[3] = dllr*12z;return v;)W fdndz(D r, D z, D dr, D dz){D 11r, 12r, dllz, d12z;W v;llr = 11(r, dr);12r = 12(r, dr);dllz^dll(z, dz);d12z = d12(z, dz);/*^*/v.v[0] = 11r*dllz;v.v[1] = 12r*dllz;v.v[2] = 12r*d12z;v.v[3] = 11r*d12z;return v;)W fdmdr(D r, D z, D dr, D dz){D dllr, d12r, 11z, 12z;W v;dllr = dll(r, dr)+d13(r, dr);dl2r = d12(r, dr)+d13(r, dr);llz = 11(z, dz)+13(z, dz);12z = 12(z, dz)+13(z, dz);/*^*/v.v[0] = dllr*11z;v.v[1] = d12r*11z;v.v[2] = dl2r*12z;v.v[3] = dllr*12z;return v;)W fdndz(D r, D z, D dr, D dz)D 11r, 12r, dllz, d12z;W v;llr = 11(r, dr)+13(r, dr);12r = 12(r, dr)+13(r, dr);dllz = dll(z, dz)+d13(z, dz);d12z = d12(z, dz)+d13(z, dz);/*^*/v.v[0] = 11r*dllz;v.v[1] = 12r*dllz;v.v[2] = 12r*d12z;v.v[3] 11r*d12z;return v;)W fd2mdr2(D r, D z, D dr, D dz)D d211r, d212r, 11z, 12z;W v;d211r = d211(r, dr)+d213(r, dr);d212r = d212(r, dr)+d213(r, dr);llz = 11(z, dz)+13(z, dz);12z = 12(z, dz)+13(z, dz);/*^*/v.v[0] = d211r*11z;v.v[1] = d212r*11z;^APPENDICES^ 234^v.v[2]^d212r*12z;v.v[3] = d211r*12z;return v;}W fd2mdrdz(D r, D z, D dr, D dz)D dllr, dl2r, dllz, d12z;W v;dllr = dll(r, dr)+d13(r, dr);dl2r = d12(r, dr)+d13(r, dr);dllzd12z= dll(z,d12(z,dz)+d13(z,dz)+d13(z,dz);dz);/*^*/v.v[0] = dllr*dllz;v.v[1] = dl2r*dllz;v.v[2] = dl2r*d12z;v.v[3] dllr*d12z;return v;W fd2mdz2(D r, D z, D dr, D dz)D 11r, 12r, d211z, d212z;W v;llr = 11(r, dr)+13(r, dr);12r = 12(r, dr)+13(r, dr);d211z = d211(z, dz)+d213(z, dz);d212z = d212(z, dz)+d213(z, dz);/*^*/v.v[0] = 11r*d211z;v.v[1] = 12r*d211z;v.v[2] 12r*d212z;v.v[3] = 11r*d212z;return v;W fd3mdr2dz(D r, D z, D dr, D dz)D d211r, d212r, dllz, d12z;W v;d211r = d211(r, dr)+d213(r, dr);d212r = d212(r, dr)+d213(r, dr);dllz = dll(z, dz)+d13(z, dz);d12z = d12(z, dz)+d13(z, dz);/*^*/v.v[0]^d211r*dllz;v.v[1] = d212r*dllz;v.v[2] = d212r*d12z;v.v[3] = d211r*d12z;return v;W fd3mdrdz2(D r, D z, D dr, D dz){D dllr, dl2r, d211z, d212z;W v;dllr = dll(r, dr)+d13(r, dr);dl2r = d12(r, dr)+d13(r, dr);d211z = d211(z, dz)+d213(z, dz);d212z = d212(z, dz)+d213(z, dz);/*^*/v.v[0] = dllr*d211z;v.v[1]^dl2r*d211z;v.v[2]^dl2r*d212z;v.v[3] = dllr*d212z;return v;D gp(I n,^I i)D z=0.0;switch^(n)^{case 2^: z = gp2(i);break;case 3^: z gp3(i);break;case 4^: z = gp4(i);break;case 6^: z = gp6(i);break;case 8^: z = gp8(i);break;casereturn z;16^: z = gp16(1);break;^}D gw(I n,^I i){D w=0.0;switch^(n)case 2^: w = gw2(i);break;APPENDICES^ 235case 3^w gw3(i);break;case 4 : w = gw4(i);break;case 6 : w = gw6(i);break;case 8 : w = gw8(i);breakicase 16 : w = gw16(1);break; )return w;TWO POINT GAUSS-LEGENDRE FORMULA*/D gp2(I i){Dswitch (i) {case 0 : z = 0.577350269189626/break;case 1 : z = -0.577350269189626;break; )return z;)/*TWO POINT GAUSS-LEGENDRE FORMULA*/D gw2(I 1){D w=0.0;switch (i) (case 0 : w^1.000000000000000;break;case 1 : w = 1.000000000000000;break; )return w;THREE POINT GAUSS-LEGENDRE FORMULA*/D gp3(I i){D z=0.0;switch (i) (case 0 : z^0.000000000000000;break;case 1 : z^0.774596669241483;break;case 2 : z^-0.774596669241483;break; )return z;}/*THREE POINT GAUSS-LEGENDRE FORMULA:1D gw3(I i)D w=0.0;switch (i) {case 0 : w = 0.888888888888889;break;case 1 : w = 0.555555555555556;break;case 2 : w 0.555555555555556;break; )return w;FOUR POINT GAUSS-LEGENDRE FORMULA*1D gp4(I i){D z=0.0;switch (i)case 0 : z = 0.339981043584856;break;case 1 : z = -0.339981043584856;break;case 2 : z• 0.861136311594053;break;case 3 : z = -0.861136311594053;break; )return z;)/*FOUR POINT GAUSS-LEGENDRE FORMULA*/D gw4(I 1)D w=0.0;switch (i) {case 0 : w = 0.652145154862546;break;case 1 : w = 0.652145154862546;break;case 2 : w 0.347854845137454;break;case 3 : w = 0.347854845137454;break; )return w;SIX POINT GAUSS-LEGENDRE FORMULAAPPENDICES^ 236*/D gp6(I 1){D z=0.0;switch (i) {case 0 : z = 0.932469514203152;break;case 1 : z = -0.932469514203152;break;case 2 : z = 0.661209386466265;break;case 3 : z^-0.661209386466265;break;case 4 : z^0.238619186083197;break;case 5 : z = -0.238619186083197;break; }return z;SIX POINT GAUSS-LEGENDRE FORMULA*/D gw6(I i){D w=0.0;switch (i) (case 0 : w^0.171324492379170;break;case 1 : w = 0.171324492379170;break;case 2 : w = 0.360761573048139;break;case 3 : w = 0.360761573048139;break;case 4 : w = 0.467913934572691;break;case 5 w = 0.467913934572691;break; }return w;}/ *EIGHT POINT GAUSS-LEGENDRE FORMULA*/D gp8(I i){D z=0.0;switch^(i)^(case 0^: z = C.960289856497536;break;case 1^: z = -0.960289856497536;break;case 2^: z = 0.796666477413627;break;case 3^: z = -0.796666477413627;break;case 4^: z 0.525532409916329;break;case 5 z = -0.525532409916329;break;case 6^: z = 0.183434642495650;break;returncasez;7^: z = -0.183434642495650;break;^}/*EIGHT POINT GAUSS-LEGENDRE FORMULA*/D gw8(I i)D w=0.0;switch (i)^{case 0^: w = 0.101228536290376;break;case 1^: w = 0.101228536290376;break;case 2^: w 0.222381034453374;break;case 3^: w = 0.222381034453374;break;case 4^: w = 0.313706645877887;break;case 5^: w = 0.313706645877887;break;case 6^: w = 0.362683783378362;break;returncasew;7^: w 0.362683783378362;break;^}SIXTEEN POINT GAUSS-LEGENDRE FORMULA*/D gp16(I i){D z=0.0;switch^(i)^{case 0 : z = 0.095012509837637440185;break;case 1 : z -0.095012509837637440185;break;case 2 : z 0.281603550779258913230;break;case 3 : z = -0.281603550779258913230;break;case 4 : z = 0.458016777657227386342;break;case 5 : z = -0.458016777657227386342;break;case 6 : z = 0.617876244402643748447;break;case 7 : z = -0.617876244402643748447;break;case 8 : z = 0.755404408355003033895;break;case 9 : z = -0.755404408355003033895;break;case 10 : z = 0.865631202387831743880;break;case 11 : z -0.865631202387831743880;brsak;APPENDICES^ 237case 12 : z = 0.944575023073232576078;break;case 13 : z -0.944575023073232576078;break;case 14 : z = 0.989400934991649932596:break;case 15 : z = -0.989400934991649932596;breaki }return z;SIXTEEN POINT GAUSS-LEGENDRE FORMULA*1D gw16(I 1)D w=0.0;switch (i)^{case 0^: w = 0.189450610455068496285;break;case 1^: w 0.189450610455068496285;break;case 2^: w = 0.182603415044923588867;break;case 3^: w = 0.182603415044923588867;break;case 4^: w = 0.169156519395002538189;break;case 5^: w = 0.169156519395002538189;break;case 6^: w = 0.149595988816576732081;break;case 7^: w = 0.149595988816576732081;break;case 8^: w = 0.124628971255533872052;break;case 9^: w = 0.124628971255533872052;break;case 10^: w = 0.095158511682492784810;break;case 11 w = 0.095158511682492784810;break;case 12^: w = 0.062253523938647892863;break;case 13^: w = 0.062253523938647892863;break;case 14^: w = 0.027152459411754094852;break;returncasew;15 w = 0.027152459411754094852;break;^}V outdata(I pc, I ss, I n, I nrout, I nzout, I *pi, I *pe, F *rp, F *zp,D *pr, D *pz, D *dr, D *dz, I ihof, D t, C *title, D *data, FILE *out)I i, j, k, 1, el;I iout[RW*ZW), eldef[EM][6];F rot[IM], zot[IM]:D rr, zz, sum, edr[EM], edz[EM], r[NM], ZINN], f[NM];W nrzt;/*for (i=0;i<NM;i++)r(i) = *pr++;z[i] = *pz++;f[i] = *data++; }for (i=0;i<IM;i++) {rot[i] = * r13++:zot[i] = * zP++: }for (i=0;i<EM;i++)edr[i] = *dr++;edz(i] = *dz++;for (j=0;j<6;j++)( eldef[1][J] = *pe++; ) }for (1=0;i<RW*ZW;i++){ lout[i] = *pi++; )if (ss == 3)ss = 0; }fprintf(out, "%d %d %d %d\n", pc, nrout, nzout, n);/*TITLE 1 AND 2fputs(title, out);switch (pc)case 0: /* CONC */fprintf(out, "Concentration of Solute");break;case 1: /* PRES */fprintf(out, "Pressure Distribution");break;case 2: /* PECL */fprintf(out, "Peclet Distribution");break;case 3: /* POTL */fprintf(out, "(P-#sp&)/Pref");break; Iswitch (ss) {case 0 :fprintf(out, " at Steady-State.$\n");break;default :fprintf(out, " at t^%10.6f$\n", t);APPENDICES^ 238break; )/*TITLE X, Y, AND Z AXES*Ifputs("Distance into Tissue, dimensionless$\n", out);fputs("Distance down Capillary, dimensionless$\n", out);switch (pc) {case 0: /* CONC */fputs("Concentration, dimensionless$\n", out);break;case 1: /* PRES */fputs("Pressure, dimensionless$\n", out);break;case 2: /* PECL */fputs("Peclet Number$\n", out);break;case 3: /* POTL */fputs("(P-#sp&)/PrefS\n", out);break; )for (i=0;i<nzout;i++) {for (j=0;j<nrout;j++) {k = i*nrout+j;el = iout[k];rr^rot[J]-r[eldef[el][0]];zz = zot[il-z[eldef[el][0]];if (ihof == 1){ nrzt = fmrzt(rr, zz, edr[el], edz[el]); }else{ nrzt^fnrzt(rr, zz, edr[el], edz[el]);sum = 0.0;for (1=0;1<4;1++){ sum += nrzt.v[1]*f[eldef[el][1]);fprintf(out, " %10.6f %10.6f %15.12f\n",rot[j], zot[i], sum); })switch (pc) {case 0: printf(" CONC");break;case 1: printf(" PRES");break;case 2: printf(" PECL");break;case 3: printf(' POTL");break;V outuv2d(I wtf, I ss, I n, I nrout, I nzout, I *pi, I *pe, F *rp, F *zp,D *pr, D *pz, D *dr, D *dz, I ihof, D t, C *title, D *ur, D* uz, FILE *out){I i, j, k, 1, el;I iout[RW*ZW], eldef[EM][6];F rot(IM], zot[IM];D rr, zz, sr, sz, edr[EM], edz[EM], rINM), z[NM], vr[NM], vz[NMJ;W nrzt;/*^*/for (i=0;i<NM;i++) {r[i] *pr++;z[i] = *pz++;vr[i] = *ur++;vz[i] *uz++; }for (i=0;i<IM;i++) {rot(i] = *rp++;zot[i] = *zp++; )for (i=0;i<EM;i++) (edr[i] = *dr++;edz[i] = *dz++;for (j=0;j<6;j++){ eldef(i)[J]^*Pe++; ) }for (i=0;i<RW*ZW;i++){ iout[i] = *pi++ ; )if (as == 3)ss = 0 ; }/*^*/fprintf(out, "%d %d %d %d\n", 4, nrout, nzout, n);/*TITLE 1 AND 2*/fputs(title, out);fprintf(out, "Velocity Field");switch (ss) {case 0 :fprintf(out, " at Steady-State.$\n");break;default :fprintf(out, " at t = %10.6f$\n", t);break; )/*APPENDICES'^ 239TITLE X, Y, AND Z AXES*/fputs("Distance into Tissue, dimensionless$\n", out);fputs("Distance down Capillary, dimensionless$\n', out);fputs(" $\n", out);switch (wtf)case 0 :for (i=0;i<nzout;i++) {for (j=0;j<nrout;j++) (k i*nrout+j;el = iout[k];rr = rotIJI-rfeldeffel][0]);zz^zot[i]-z[eldef[el][0]];if (ihof^1){ nrzt = fmrzt(rr, zz, edr[el], edz[e1]); )else{ nrzt^fnrzt(rr, zz, edr[el], edz[el]);sr = sz = 0.0;for (1...0;1<4;1++) (sr += nrzt.v[1]*vr[eldef[el][1]];sz += nrzt.v[1]*vz[eldef[el][1]]; )fprintf(out, ' %10.6f %10.6f %10.6e %10.6e\n",rot[J], zot[i], sr, sz); ))break;case 1 :for (i..0;i<n;i++)( fprintf(out, "%10.6f %10.6f %10.6e %10.6e\n",r[i], z[i], vr[i), vz[i]); )break; )printf(" VEL");V outcoll(I pc, I ss, I n, D t, D *px, D *py, D *pf, C "title, FILE *out)I i;if (ss^3)( ss = 0; )fputs(title, out);switch (pc) (case 0: /* CONC */fprintf(out, "Concentration of Solute");break;case 1: /* PRES */fprintf(out, "Pressure Distribution");break;case 2: /* PECL */fprintf(out, "Peclet Distribution");break;case 3: /* POTL */fprintf(out, "Potential Distribution");break; }switch (ss) (case 0 :fprintf(out, " at Steady-State.S\n");break;default :fprintf(out, " at t^%10.6f$\n", t);break; }fputs("Distance into Tissue, dimensionless$\n", out);fputs("Distance down Capillary, dimensionless$\n", out);switch (pc) {case 0: /* CONC */fputs("Concentration, dimensionless$", out);break;case 1: /* PRES */fputs("Pressure, dimensionless$", out);break;case 2: /* PECL */fputs("Peclet Number$", out);break;case 3: /* POTL */fputs("(P-#8P&)/Pref$", out);break; }fprintf(out, "\n %d\n", n);for (i.=0;i<n;i++)fprintf(out, " %10.6f %10.6f %15.12f\n", *px++, *py++, *pf++); )switch (pc)case 0: printf(" CONC");break;case 1: printf(" PRES");break;case 2: printf(" PECL");break;case 3: printf(" POTL");break;}APPENDK7ES^ 240V outcapp(I n, I as, D t, D *pz, D *pc, D *qc, D *vm, D *cc, D *pe, D *vs, C *title, FILE*out){I 1;if (as == 3){ ss = 0; }fputs(title, out);fprintf(out, "Capillary Variables");switch (as) {case 0 :fprintf(out, " at Steady-State.$\D");break;default :fprintf(out, ' at t = %10.6f$\n", t);break; )fprintf(out, " %d\n", n);fputs('^z^Pcap^Qcap^Vmem^Ccap^mPe^Vsol\n', out);for (1=0;i<n;i++)( fprintf(out, " 966.4f %8.6f %8.6e %8.6e %8.6e %8.6e %8.6e\n",*pz++, *pc++, *qc++, *vm++, *cc++, *pe++, *vs++); )printf(' CAPP");}V contour(I pc, I as, I n, I nr, I nz, D *pz, D t, C *title, FILE *out)I i, j, k;D z[NM];/*^*1for (i=0;i<NM;i++){ z(i) = *pz++; }if (as == 3){ ss = 0; }/*^*1fputs(title, out);switch (pc) (case 0: /* CONC */fprintf(out, "Concentration of Solute");break;case 1: /* PRES */fprintf(out, "Pressure Distribution");break;case 2: /* PECL */fprintf(out, "Peclet Distribution");break;case 3: /* POTL */fprintf(out, "Potential Distribution");break; )switch (ss) (case 0 :fprintf(out, " at Steady-State.\n");break;default :fprintf(out, " at t = %10.6f\n", t);break; )/^*/for (i=flz;i>071--) (fprintf(out, "I ");for (j=0;j<nr;j++) (k^(i-1)*nr+j;fprintf(out, "%5.4f ", z(k)); }fprintf(out, "I\n"); }V header(D rg, D zg, D knn, membr mem, media pp, FILE *out)fprintf(out, "AR = %3.1f:%d$\n", (zg/rg), 1);fprintf(out, "Rf = %4.2f$\n", pp.hind);fprintf(out, "#s& = %4.2f$\n", mem.aigma);fprintf(out, "#s&g = %4.2f$\n", pp.sigmag);fprintf(out, "PS = %4.2E m/s$\n", mem.ps);fprintf(out, "D = %4.213 m2/8$\n", pp.diff);fprintf(out, "Lp = %4.2E m/Pa.s$\n", mem.1p0);fprintf(out, "K = %4.213 m2/Pa.s$\n", knn);fprintf(out, "LS = %4.22 m3/m3.Pa.s$\n", pp.ls);I safechk(I ndnm, I elnm, I nr, I nz, I nzout, I nzout)I err=0;C *ml, *m2, *m3, *m4, *m5, *m6, *m7, *m8;ml = " E: safechk()-> NM too small. Set > %d.\n";m2 = " E: safechk()-> EM too small. Set > %d.\n";m3 = " E: safechk()-> IM too small. Set > %d.\n";APPENDICES'^ 241m4 = " B: safechk()-> MM too small. Set > %d.\n";m5 =^B: safechk()-> RW too small. Set > %d.\n";m6 =^B: safechk()-> ZW too small. Set > %d.\n";m7 =^B: safechk()-> nr > nz.^Set nr < nz.\n";m8 = " B: safechk()-> NJ < IM. Set NJ > IM.\n";/*^*/if (ndnm+2 > NM) {printf(ml, ndnm);err = 1; }if (elnm+2 > BM) {printf(m2, elnm);err = 1; )if (nz+2 > IM) (printf(m3, nz);err = 1; )if ((2*nr+2*nz) > MM) (printf(m4, 2*nr+2*nz);err = 1; )if (nrout+2 > RW) (printf(m5, nrout);err = 1; )if (nzout+2 > ZW) {printf(m6, nzout);err = 1; }if (nr > nz) (printf(m7);err = 1; )if (NJ < IM) (printf(m8);err = 1; )/*^*/return err;I datachk(I *pc)I 1, err=0, nerr=0, chk[50];C *mssgl, "item."";mssgl^" E: datachk()-> %s card missing.\n";for (1=0;1<50;1++)( chk[i] = *pe++; )for (1=0;1<50;1++) (switch(i) (case 0 : if (chk[1] == 0) {err = 1;item = "$end input"; }break;case 1 : if (chk[i] == 0) (err = 1;item = "$beg input"; }break;case 2 : if (chk[i] == 0) (err = 1;item = "$prob size"; )break;case 3 : if (chk[i] == 0) {err = 1;item = "$tis refl coef"; }break;case 4 : if (chk(i] == 0) (err = 1;item = "$transient"; }break;case 5 : if (chk[i] == 0) (err = 1;item = "$max iterations"; )break;case 6 : if (chk[i] == 0) {err = 1;item = "$dispersivity"; }break;case 7 : if (chk[i] == 0)err = 1;item = "$fluid sink LS"; }break;case 8 : if (chk[i] == 0) ferr = 1;item = "$porosity"; }break;case 9 : if (chk(i] == 0) (err = 1;item = "$fluid viscosity"; }APPENDICES^ 242break;case 10: if (chk[i] == 0) {err^1;item = "$relaxation"; }break;case 11: if (chk[i] == 0) {err = 1;item = "$hydraulic cond"; )break;case 12: if (chk[i] == 0) {err = 1;item = "$conc node sink"; }break;case 13: if (chk[i] == 0) {err = 1;item = "$mem refl coef"; }break;case 14: if (chk[i] == 0) (err^1;item^"$fluid density"; )break;case 15: if (chk[i] == 0) {err^1;item = "$diff mem PS"; }break;case 16: if (chk[i]^0) {err = 1;item = "$geometry"; )break;case 17: if (chk[i] == 0) (err = 1;item^"$cap osm pr"; )break;case 18: if (chk[i] == 0) {err = 1;item = "$node definitions"; )break;case 19: if (chk[i] == 0) (err = 1;item = "$elem definitions"; }break;case 20: if (chk[i] == 0) (err = 1;item = "$conc node bc"; )break;case 21: if (chk[i] == 0) (err = 1;item = "$conc elem bc"; )break;case 22: if (chk[i) == 0)err = 1;item = "$conc node is"; )break;case 23: if (chk[i] == 0) (err = 1;item = "$pres node bc"; }break;case 24: if (chk[i] == 0) (err = 1;item = "$pres elem bc"; )break;case 25: if (chk[i] == 0) {err = 1;item = "$write results"; )break;case 26: if (chk[i) == 0) {err = 1;item = "$contour"; }break;case 27: if (chk[i] == 0)err = 1;item = "$ref values"; )break;case 28: if (chk[i] == 0) (err = 1;item = "$tolerance"; )break;case 29: if (chk[i] == 0) {err = 1;item = "$frac volumes";break;243APPENDICEScase 30: if^(chk[i]erritembreak;==^0)^{=^1;= '$conv hindrance"; )case 31: if^(chk[i]erritembreak;==^0)^{=^1;= "$gauss points";^}case 32: if^(chk[i]erritembreak;==^0)^{=^1;.^"$asq";^)case 33: if^(chk[i]erritembreak;==^0)^(= 1;= "$cap conditions"; }case 34: if^(chk[i]erritembreak;== 0)^(= 1;= "$hof";^)case 35: if^(chk[i]erritembreak;.= 0)^(= 1;= "$node results"; )case 36: if^(chk[i]erritembreak;== 0)=^1;= "$osm pres";^}case 37: if^(chk[i]erritembreak;== 0)^{=^1;= "$dgamma"; }case 38: if^(chk[i]erritembreak;== 0)^{= 1;= "$lymph cond"; )case 39: if^(chk[i]erritembreak;== 0)^{= 1;= "$cap wall Lp"; )case 40: if^(chk[i]erritembreak;== 0)^{=^1;= "$wall th";^}case 41: if^(chk[i]erritem==^0)^(=^1;= "$perturbation"; }break; )if (err != 0)nerr++;printf(mssgl, item); }err = 0; )return nerr;}

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