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A compartmental model of microvascular exchange in humans Chapple, Clive 1990

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A COMPARTMENTAL MODEL OF MICROVASCULAR E X C H A N G E IN HUMANS By Clive Chappie B. Sc. (Chemical Engineering) University of Surrey, England. A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S C H E M I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A April 1990 © Clive Chappie, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CHEMICAL ENGINEERING The University of British Columbia Vancouver, Canada D a t e 16 th A p r i l 1990 DE-6 (2/88) ABSTRACT A mathematical model describing the transport and distribution of fluid and plasma proteins between the circulation, the interstitium, and the lymphatics, is formulated for the human. The formulation parallels that adopted by Bert et al.{5] in their model of microvascular exchange in the rat. The human microvascular exchange system is subdivided into two distinct compart-ments: the circulation and the interstitium. Both compartments are treated as homo-geneous and well-mixed. Two alternative descriptions of transcapillary exchange are investigated: a homoporous "Starling Model" and a heteroporous "Plasma Leak Model". Parameters which characterize fluid and protein transport within the two models are de-termined by a comparison (quantified statistically) of the model predictions with selected experimental data. These data consist of interstitial fluid volumes and colloid osmotic pressures measured as a function of plasma colloid osmotic pressure for subjects suffer-ing from hypoproteinemia. The relationship between this fitting data and the model transport parameters is investigated using a visual "graphical optimization technique" and additionally, in the case of the Starling Model, by use of a non-linear optimization technique. Both the Starling Model and the Plasma Leak Model provide good representations of the fitting data for several alternative sets of parameter values. The ranges of pa-rameter values obtained generally agree well with those available in hterature. The fully determined model is used to simulate the transient behaviour of the system when sub-jected to an intravenous infusion of albumin. All alternative "besi-fii," parameter sets determined for both models produce simulations which compare reasonably well with the n experimental infusion data of Koomans et al. [42]. The predictions of both models compare favourably not only with the available ex-perimental data but also with the known behavioural characteristics of the human mi-crovascular exchange system. However, no conclusions may be drawn regarding which of the alternative transcapillary transport mechanisms investigated provides the better description of human microvascular exchange, although it appears likely that diffusion of proteins plays a significant role in both. Final model selection and choice of fitting parameters await the availability of more and better microvascular exchange data for humans. Analysis of both the Starling Model and Plasma Leak Model indicates that the mi-crovascular system is capable of regulating the interstitial fluid volume over a fairly wide range of transport parameter values. The important model-predicted passive regulatory mechanisms are tissue "protein washout", which reduces its colloid osmotic pressure,and a low tissue compliance which increases the hydrostatic pressure of the interstitium as it becomes hydrated. It would therefore seem that the human microvascular system can be regarded as a fairly "robust" system when considering its ability to regulate inter-stitial fluid volume (i.e., small changes in the values of transport parameters, such as the capillary wall permeability, have little effect on the conditions and operation of the system). in Table of Contents A B S T R A C T ii List of Tables x List of Figures •: Acknowledgement XV 1 I N T R O D U C T I O N 1 2 P H Y S I O L O G I C A L O V E R V I E W 5 2.1 T H E C A R D I O V A S C U L A R S Y S T E M • 5 2.1.1 A General Description 5 2.1.2 The Composition and Properties of Blood 6 2.1.3 Classification of the Blood Vessels 7 2.1.4 Architecture of Capillary Circulation 8 2.2 T R A N S C A P I L L A R Y T R A N S P O R T 10 2.2.1 Transport Mechanisms 11 2.2.2 Forces Influencing Trans capillary Exchange of Fluid and Proteins 14 2.3 T H E INTERSTITIUM 15 2.3.1 Properties of the Interstitium 19 2.3.1.1 Exclusion . . . . ; 19 2.3.1.2 Tissue Compliance 21 2.4 T H E L Y M P H A T I C S Y S T E M 23 i v 2.4.1 The Terminal Lymphatics 25 2.4.2 Lymph Propulsion 27 2.5 THE NEPHROTIC SYNDROME 28 3 MODEL FORMULATION 31 3.1 INTRODUCTION 31 3.2 RELATIONSHIPS COMMON TO BOTH THE SM AND THE PLM . . 36 3.2.1 Colloid Osmotic Pressure Relationship 36 3.2.2 Concentration Relationships 38 3.2.3 Lymphatics 39 3.2.4 Interstitial Fluid Hydrostatic Pressure & Interstitial Comphance . 40 3.3 THE STARLING MODEL 45 3.3.1 Compartmental Mass Balances 47 3.3.2 Circulatory Comphance 48 3.4 THE PLASMA LEAK MODEL 48 3.4.1 Compartmental Mass Balances 51 3.4.2 Arterial and Venous Capillary Compliances 51 3.5 NORMAL STEADY-STATE CONDITIONS 54 3.5.1 Body Fluid Distribution 54 3.5.2 Interstitial Excluded Volume 57 3.5.3 Plasma Colloid Osmotic Pressure 57 3.5.4 Interstitial Colloid Osmotic Pressure 58 3.5.5 Interstitial Fluid Hydrostatic Pressure 60 3.5.6 Capillary Hydrostatic Pressure 60 3.5.7 Arterial and Venous Capillary Hydrostatic Pressures 62 3.5.8 Summary of Normal Steady-state Conditions 63 v 4 PARAMETER ESTIMATION PROCEDURE 64 4.1 INTRODUCTION 64 4.2 PARAMETER ESTIMATION PROCEDURE 67 4.2.1 Parameters to be Determined 67 4.2.2 Data for the Statistical Fitting Procedure 70 4.2.3 Steady-state Balances at Normal Conditions 73 4.2.4 Albumin Clearance Relationship 74 4.2.5 The Starling Model 75 4.2.5.1 Parameters to be Empirically Determined 75 4.2.5.2 Parameter "Surface" Generation 79 4.2.5.3 Parameter Search Regions 81 4.2.5.4 Objective "Surface" Grid Size 83 4.2.5.5 Parameter Surface Normalization 83 4.2.6 The Plasma Leak Model . 88 4.2.6.1 Parameter Search Regions 91 4.2.6.2 OBJ "Surface" Generation Procedure 92 4.3 NUMERICAL TECHNIQUES AND COMPUTER PROGRAMS . . . . 93 4.3.1 Transient Analysis 93 4.3.2 Steady-State Solutions 94 4.3.3 "Shortest Distance" Least-squares Technique 94 4.3.4 Interpolating Data with Cubic Splines 95 4.3.5 Computer Programs 95 5 RESULTS AND DISCUSSION 97 5.1 INTRODUCTION 97 5.2 THE PARAMETER ESTIMATION PROCEDURE 98 vi 5.2.1 Definition of "Best-fit" Parameters 98 5.2.2 Irregularities in OBJijv "Surfaces" 99 5.2.3 The Effect of LS Search Region Size on "Surface" Normalization . 100 THE STARLING MODEL 102 5.3.1 General Shape of OBJUi and OBJifv "Surfaces" 106 5.3.2 "Best-fit" Steady-state Vj vs. Hp£ and Hj vs. Upr, Simulations . 106 5.3.3 OBJiii Insensitivity to Tissue Compliance Overhydration Gradient 110 5.3.4 Relationship between Tissue Compliance Characteristics and their Steady-state Vj vs. Upr, Simulations 113 5.3.5 Validation of the Starling Model 114 5.3.5.1 Lymph Flow Sensitivity - LS 116 5.3.5.2 Lymph Flowrate under Normal Steady-state Conditions ~ JL,NORM 116 5.3.5.3 Permeability-Surface Area Product - PS 116 5.3.5.4 Capillary Wall Solute Reflection Coefficient - er 118 5.3.5.5 Capillary Filtration Coefficient - KF 119 5.3.5.6 Maximal to Basal Lymph Flowrate - J ^ 3 119 5.3.5.7 Steady-state Transcapillary Pressure Gradient - PTC • • 121 5.3.5.8 Combined "Surface" Objective Function Value - OBJ^^ 121 5.3.5.9 Transient Response Time - TRES 122 5.3.6 Summary of the Starling Model Results 135 THE PLASMA LEAK MODEL 137 5.4.1 General Shape of OBJ^ and OBJ{fv "Surfaces" 137 5.4.2 "Best-fit" Steady-state Vj vs. UPL and Hj vs TS.PL Simulations . . 139 5.4.3 Selection of "Best-fit" Results 144 5.4.4 OBJ "Surface" Dependence on RF/R 144 vii 5.4.5 Validation of the Plasma Leak Model 146 5.4.5.1 Ratio of Filtration Coefficient to Reabsorption Coefficient -RF/R 146 5.4.5.2 Lymph Flow Sensitivity - LS 147 5.4.5.3 Lymph Flowrate under Normal Steady-state Conditions - JL,NORM 147 5.4.5.4 Permeability - Surface Area Product - PS 147 5.4.5.5 Filtration Coefficient - KF 147 5.4.5.6 Reabsorption Coefficient - KR 148 5.4.5.7 Plasma Leak Coefficient - KPLL 148 5.4.5.8 Maximal to Basal Lymph Flowrate - jf^B 148 5.4.5.9 Transient Response Time - TRES 149 5.4.6 Summary of the Plasma Leak Model Results 155 6 CONCLUSIONS 156 7 R E C O M M E N D A T I O N S 160 Nomenclature 162 Bibliography 166 Appendices 172 A S M OBJ "Surfaces" 172 B P L M OBJ "Surfaces" 218 C S M Albumin Infusion Simulations. 264 v u i D P L M Albumin Infusion Simulations. 295 E Computer Programs 310 E.l Parameter List of Steady-State and Transient Simulators 310 E.2 Listing of Program SM-A 313 E.3 Listing of Program SM-B 317 E.4 Listing of Program SM-C 334 E.5 Listing of Program PLM-A 360 E.6 Listing of Program PLM-B 364 E.7 Listing of Program PLM-C 382 ix List of Tables 2.1 Some protein components of human plasma 6 3.1 Experimentally determined interstitial hydrostatic fluid pressures 42 3.2 Summary of the Starling Model and the Plamsa Leak Model equations. . 53 3.3 Composition of "skin" in "reference man" 55 3.4 Interstitial fluid distribution 56 3.5 Experimentally determined HPL,NORM values 58 3.6 Experimentally determined T1I,NORM values 59 3.7 Normal steady-state conditions for a 70 kg human male 63 4.1 Model parameters to be determined by statistical fitting 68 4.2 Maximum value of Rd/i as a function of PPL,C,NORM 83 4.3 PPL,A,NORM and PPL,V,NORM values investigated and associated minimum values of RF/R 92 5.1 Starling Model OBJIJ. "best-fit" parameters and associated transport co-efficients 103 5.2 Starling Model OBJijv "best-fit" parameters and associated transport co-efficients 104 5.3 Starling Model OBJcomb "best-fit" parameters, transport coefficients, and other associated behaviour characterizing criteria 105 5.4 Various experimentally determined values of permeability-surface area prod-uct 117 x 5.5 Sensitivity of T R E S to changes PCOMPPLTC 125 5.6 Plasma leak model O B J c o m b "best-fit" parameters, associated transport parameters, and other behaviour characterizing criteria 138 5.7 Sensitivity of the transient response time, TRES , to changes in the value of venous capillary compliance, PCOMPPL,V 150 xi List of Figures 2.1 The general arrangement of the capillary network 9 2.2 Transport pathways across the capillary wall 11 2.3 The endothelium of a mouse diaphragm containing abundant vesicles. . . 13 2.4 The Starling forces 15 2.5 The features of the various levels of collagen organization 17 2.6 The general organization of the interstitial matrix 20 2.7 Examples of steric exclusion 22 2.8 Experimentally determined rat tissue comphance relationships 24 2.9 Lymphatic and blood vessels in an area of bat's wing 26 2.10 The structure of the terminal lymphatic wall. 28 2.11 The nephrotic syndrome 30 3.1 A schematic representation of the microvascular exchange system 35 3.2 Albumin concentration vs. oncotic pressure relationship 37 3.3 The "most likely" human comphance relationships. . . 43 3.4 The Starling Model 46 3.5 The Plasma Leak Model 49 4.1 Normalized Vi vs. HPL data 71 4.2 Normahzed 11/ vs. HPL data 72 4.3 A hypothetical relationship illustrating the disadvantages of using either a vertical distance or horizontal distance least-squares fitting criterion. . 80 4.4 Objective function "surface", OBJifv, at a PPL,C,NORM of 10 mmHg. . . 85 xii 4.5 Objective function "surface", OBJui, at a PPL,C,NORM of 10 mmHg. . . . 86 4.6 Objective function "surface", OBJcomb, at a PPL,C,NORM of 10 mmHg. . . 87 4.7 A flow diagram of the "minimization procedure" used to determine the "best-fit" parameters for the Starling Model 89 5.1 Three contour plots of OBJijv surfaces at a PPL,C,NORM °f 9 mmHg. . . 101 5.2 SM best "best-fit" fitting data simulations 107 5.3 SM worst "best-fit" fitting data simulations 108 5.4 Alternative tissue compliance OBJT\{ "surfaces" at a PPL,C,NORM of 10 mmHg for the Starling Model Il l 5.5 Starling Model OBJrii "best-fit" predictions of the steady-state Vj vs. TLpi, and H r vs. UPL relationships 11 2 5.6 Alternative compliance relationships investigated and their corresponding OBJ{fv "best-fit" predicted Vj vs. Upi relationships 115 5.7 Effect of Koomans' albumin infusion on select microvascular exchange vari-ables. For the OBJ comb "best-fit" Starling Model at a PPL,C,NORM of 11 mmHg 127 5.8 Effect of Koomans' albumin infusion on select microvascular exchange vari-ables. For the OBJcomb "best-fit" Starling Model at a PPL,C,NORM of 7 mmHg 128 5.9 Effect of Koomans' albumin infusion on all microvascular exchange vari-ables. For the OJ3Jcomb "best-fit" Starling Model at a PPL,C,NORM of 11 mmHg 131 5.10 Effect of Koomans' albumin infusion on all microvascular exchange vari-ables. For the OBJCOTnb "best-fit" Starling Model at a PPL,C,NORM of 7 mmHg 132 xm 5.11 Effect of a Koomans' albumin infusion on select microvascular exchange variables. For the OB Jamb "best-fit" Starling Model at a PPL,CJ*ORM of 10 mmHg 133 5.12 Effect of a SCALED Koomans' albumin infusion on select microvascu-lar exchange variables. For the OBJcomb "best-fit" Starling Model at a PPL,C,NORM of 10 mmHg 134 5.13 Alternative PLM OBJ{fv surfaces generated at a RF/R 5% above R^JR. • 140 5.14 Alternative PLM OBJm surfaces generated at a RF/R 5% above RFJR. . 141 5.15 PLM "best-fit" simulations of fitting data (associated with Rd/i = 1.00). 142 5.16 PLM "best-fit" simulations of fitting data (associated with Rd/i = 0.45). 143 5.17 An example of a PLM OBJcomb "surface" containing a distributed minimum. 145 5.18 Effect of a Koomans' albumin infusion on select microvascular exchange variables. For the OBJcomb "best-fit" Plasma Leak Model for a PPL^AJ^ORM of 24.54 mmHg, a PPL,V,NORM of 5 .92 mmHg 151 5.19 Effect of a Koomans' albumin infusion on all microvascular exchange vari-ables. For the OBJcomb "best-fit" PLM with a PPL,A,NORM of 26.54 mm Hg, a PPLXNORM of 7.92 mmHg, an RF/R of 0.2091, Rd/i of 1.00 and a LS of 38.6 m /^(mmHg.hr) 153 5.20 Effect of a Koomans' albumin infusion on all microvascular exchange vari-ables. For the O B J c o m b "best-fit" PLM with a PPL,A,NORM of 26.54 mm Hg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2413, Rd/i of 0.45 and a LS of 31.8 mi/(mmHg.hr) 154 xiv Acknowledgement I would like to express my sincere gratitude to the following individuals: • my advisors, Drs. J.L.Bert and B.D.Bowen, for their advice, understanding and much appreciated editing; • Dr. B.D.Bowen, once again, for the few games of squash he did allow me to win; • Dr. R.Reed of the University of Bergen, Norway, for the clinical information I otherwise would have never known existed; • Dr. Clive Brereton, Dr. David Taylor, and Mr. El-Marghani Besher for their friendship and concern; • Shiva for the caring friendship that kept me sane; • Joanne Moorhen for her faith in me and her love (and her skill at photocopying); • my parents for always believing in me. xv Chapter 1 INTRODUCTION Approximately one-sixth of the human body's tissue volume consists of space be-tween parenchymal cells. This extravascular-extracellular space is referred to as the interstitium and contains, among other things, fluid which acts as the tissue cells' environment. The maintenance of this fluid, both in terms of composition and volume, is of the utmost importance in ensuring correct body function. The regulation of the interstitial fluid volume is a complex phenomenon governed by forces and processes operating within the microvasculature, the interstitium, and the lymphatic system. Under normal conditions, any excess fluid within the circulation leaks into the in-terstitial spaces, thereby avoiding excessive loading of the cardiovascular system. The excessive accumulation of fluid within the interstitium is termed edema. The body's homeostatic mechanisms (i.e., those which maintain the internal environ-ment of the body constant at conditions optimum for cell function) constantly guard against edema formation or its opposite dehydration, both of which can be extremely damaging to the individual, if not fatal. The microvascular exchange system is capable of reacting to a change in its environment by passively altering its own properties; this ability is termed autoregulation; To understand the dy-namics of interstitial fluid volume regulation it is essential to have knowledge of the physical properties of this autoregulatory system. To this end the physiology 1 Chapter 1. INTRODUCTION 2 relevant to the system is reviewed in Chapter 2. Over the past decade or so, with the advent of increasingly inexpensive and more widely available computational resources, several mathematical models of microvas-cular exchange have been developed. These include both general models describing the overall behaviour of the normal intact system and a number that specifically address the exchange within a certain tissue or the effect of a certain trauma on microvascular exchange[46, 68, lj. A wide variety of approaches has been adopted in the formulation of mathematical models of complex systems; these include de-terministic and stochastic, linear and non-linear, lumped and distributed forms. Modelling of the microvascular exchange system has generally focused on the use of compartmental models, which are a particular subclass of the lumped determin-istic model form. A systems approach is taken in the development of the current model, the main advantage being that all factors are considered simultaneously. In contrast, the clinical experimentalist is generally restricted, by both resources and technological limitations, to monitoring only a few system variables at any one time. Hence the systems approach may emphasize the potential importance of certain system ele-ments, or their interconnections, that might have previously seemed insignificant to the clinician or medical research worker. Untested assumptions may be highlighted, possibly indicating the need for either improved experimental design or specific new experiments. The systems approach involves first the development of a reasonable functional de-scription of the system based on experimentally determined facts and logical postu-lates. The completed description is then translated into mathematical expressions. Subsequently the model can be simulated on a computer by use of the appropriate Chapter 1. INTRODUCTION 3 numerical techniques and algorithms. The simulation results can then be com-pared with experimental data, testing the feasibility of the hypothesized system mechanisms. It should be noted, however, that an approach of this type can never completely validate a particular postulate because the possibility of a non-unique model cannot be eliminated. Nevertheless, postulated models may be tested in this fashion and obviously incorrect forms either modified or abandoned. In fact, much information may be gained by investigating the nature of the mismatch between the predictions of the postulated model and the available experimental data, which may, in turn, suggest other more appropriate hypotheses. Hence, mathematical modelling can be used as a complement to physiological experimentation. However, if sufficient reliable experimental data are available and the postulated model is validated by comparision with them, the model may be assumed to be an acceptable representation of the real system. If this assumption is made, the model can be used not only in a descriptive fashion but also to predict the quantitative behaviour of the system. If a realistic model of microvascular exchange in the human can be developed the potential benefits and uses are numerous. These include the personal tailoring of resuscitation protocols to the specific needs of individual burn patients and also the fluid management of those undergoing extra-corporal circulation, to name but a few. Increasing use of micro-computers in both the hospital ward and in the laboratory enables mathematical models to be developed for direct clinical application. The model presented here serves only as a starting point for the development of more comprehensive models describing microvascular exchange during trauma or disease. Nevertheless, it is a necessary step toward achieving this goal. Previously published models of human microvascular exchange have incorporated Chapter 1. INTRODUCTION 4 transport parameters evaluated at normal steady-state conditions[l, 6 8 ] . The model developed here is different in that the transport parameters are determined by a statistical comparison of predicted results with select experimental data covering a wide range of conditions. Both the model formulation and the parameter estimation closely parallel previous work by Bert et al.[5] in which they developed a model of the microcirculation for the rat. In Chapter 3 a general model of microvascular exchange is formulated with two alternative capillary transport mechanisms. A statistically based parameter esti-mation procedure is developed in Chapter 4 and the numerical techniques utilized throughout the current work are reviewed. Chapter 5 presents and discusses the results of the parameter estimation procedure and also compares these results to existing experimental data. Chapter 6 presents the conclusions of the study pre-sented here. Finally, Chapter 7 contains recommendations regarding further work in this area. Chapter 2 PHYSIOLOGICAL OVERVIEW 2.1 THE CARDIOVASCULAR SYSTEM 2.1.1 A General Description The cardiovascular system delivers blood to all parts of the body. The blood carries oxygen, metabolic fuels, salts, hormones, and heat to every living cell and simultaneously collects wastes such as carbon dioxide, urea, water, and excess heat from the cells. The needs of the tissues and cells of the body vary according to their changing levels of activity. The cardiovascular system is designed to accommodate the varying needs of all body tissues by delivering volumes of blood that are always adequate for their survival. Viewed simplistically, the cardiovascular system can be regarded as a closed sys-tem. The system is self-regulating, constantly striving to maintain homeostasis (an optimal internal environment for body cell function). This regulation is achieved by a system of control mechanisms which are activated by negative feedback. The cardiovascular system has two component parts. First, there is the heart which functions as a variable pump. The second part of the system is the blood vessels, a complex system of hollow, branching, elastic tubes. The contraction of the right and left ventricles of the heart provide the energy for the circulation of blood in the pulmonary and systemic circuits, respectively. 5 Chapter 2. PHYSIOLOGICAL OVERVIEW 6 Upon expulsion from the left ventricle, blood enters the aorta, which subsequently branches into the arteries, arterioles and finally the capillaries. At this level nutri-ents and wastes are exchanged between blood and tissue cells. This site of exchange is referred to as the microcirculation. Following the exchange process the blood is returned to the right side of the heart via the venous blood vessels. Blood is then pumped into the pulmonary circulation where carbon dioxide is exchanged for oxy-gen in the lungs and is then returned to the heart. 2.1.2 The Composition and Properties of Blood Blood consists of a plasma medium and a number of suspended components. The plasma is chiefly water containing small dissolved ions and various protein species. The proteins are generally categorized, on the basis of size and function, into one of two major groups either as albumin or as one of the various globulins. Albumin is the predominant species both in terms of mass content and osmotic activity. Table 2.1 shows the relative concentration, molecular weight and the contribution to the total colloid osmotic pressure of the two most important protein groups in blood plasma. Component Cone. % of Total Mol.Wt Approx.II Protein in Plasma Total protein 7.0 100 25.0 Albumin 3.6 51 69000 16.4 Globulins 0.9 14 ^150000 ^1.3 Fibrinogen 0.3 4 340000 0.2 Table 2.1: Some protein components of human plasma. Colloid osmotic pres-sure, II, values are reported in mmHg and concentrations in grams per 100 ml of plasma[44]. Chapter 2. PHYSIOLOGICAL OVERVIEW" 7 The most abundant of the larger suspended components in the blood are the red blood cells. Their main function is to transport of oxygen within the cardiovascular circuit. There are also a smaller number of white blood cells present which act to combat and remove any foreign matter or organisms that might enter the body. 2.1.3 Classification of the Blood Vessels Prior to discussion of the circulation's function, it is helpful to understand the roles of the various vessels which make up the cardiovascular system. The blood vessels are generally classified into four types on the basis of different anatomical features and physiological functions. First are the aorta and arteries, whose function is to transport blood under high pressure to the tissues. As would be expected, these vessels have thick musculoe-lastic walls and blood flows rapidly through them. Second are the arterioles. These vessels are the last small branches of the arterial system, and act as control valves through which blood is released into the capillaries. The arterioles are capable of either complete vessel closure or dilation to a cross-sectional area several times larger than normal. This ability enables the arterioles to regulate the blood perfusion of the capillary beds. Next are the capillaries. As mentioned earlier, the function of the capillaries is to exchange fluid, nutrients, and other substances between the blood and the tissue space surrounding the cells. These vessels form a complex network of microscopic, branching and interconnecting, thin-walled, permeable tubes within the tissue. The membranous capillary wall selectively influences the exchange of materials between blood and tissue cells. Chapter 2. PHYSIOLOGICAL OVERVIEW 8 The fourth type of blood vessels is the veins. The venules (small veins) collect the blood from the capillaries and then gradually combine to form progressively larger veins. The veins act as conduits, allowing blood to return to the heart from the tissues. Due to the very low pressures existing in the veins, the venous vessel walls are relatively thin. Nevertheless, the walls are muscular, and their ability to expand or contract allows them to act as a variable capacity blood reservoir. 2.1.4 Architecture of Capillary Circulation In most tissues, the blood normally flows through only a small portion of the total available capillaries, about 5 % of capillaries being perfused at any given time. The capillaries that comprise this 5 % are constantly alternating. Within a particular tissue, the degree of capillary perfusion depends on the tissues current metabolic activity. The general arrangement of the capillary network within each tissue is shown in Figure 2.1. Referring to Figure 2.1, it is easy to see how the perfusion of the bed can vary so greatly. The true capillaries are not on direct flow paths from the arterioles to the venules. They emerge from vessels termed metarterioles, which branch from their parent arterioles. The capillaries of the network nearest to the venules are called venous capillaries, and those nearest the arterioles are called arterial capillaries. The direct route from arteriole to venule is referred to as the preferential channel. Within the skin there also exists a flow path that avoids the capillary bed entirely -the arteriovenous anastomosis. This is a short muscular vessel that joins arterioles to venules directly and whose main function is the regulation of heat loss from the body. The metarterioles have smooth muscle cells along their walls. The last one or two Chapter 2. PHYSIOLOGICAL OVERVIEW 9 Figure 2 .1 : The general arrangement of the capillary network. Chapter 2. PHYSIOLOGICAL OVERVIEW 10 cells encircle the capillary forming a structure known as the precapillary sphincter. Contraction of the muscle cells is controlled by sympathetic nervous and local chemical stimuli. The capillary walls lack smooth muscle or any other supporting tissue (barring the basement membrane, see Section 2.2). They act as passive tubes whose filling is dependent on the amount of blood entering, which in turn is controlled by the sphincters. Thus, when not perfused, the capillaries collapse as the blood drains out into the venules. 2.2 TRANSCAPELLARY TRANSPORT The capillary wall is composed of a thin layer of endothelial cells, which act as a membranous barrier separating the luminal (blood) compartment from the inter-stitial compartment. Under the influence of local driving forces, such as differences in hydrostatic pres-sures and solute concentrations, fluid and solutes selectively pass through the capil-lary wall. Variations in wall structure, and therefore its permeabihty characteristics, may occur from tissue to tissue and at times even along a single microvessel. Figure 2.2 illustrates the ultramicroscopic structure of the capillary wall. The wall is composed of a layer of cells typically 0.1 to 0.5 um thick, surrounded by a basement membrane. The luminal surface of the endothelial cells is covered in a fine coat of fibres, thought to be glycosaminoglycans (see Section 2.3). This covering may act as a diffusion barrier and also carries a net negative charge. This charge acts to repel hke-charged particles and therefore hinders their passage across the capillary wall. Chapter 2. PHYSIOLOGICAL OVERVIEW 11 The basement membrane is mainly composed of collagen fibers (see Section 2.3) which form a honeycomb structure, and is thought to provide mechanical support for the endothelial cells. It also carries a net negative charge and is believed to behave as diffusion barrier, as well as a resistance to fluid flow. 2.2.1 Transport Mechanisms There are many proposed mechanisms and pathways for the transport of fluid and solutes across the capillary wall. Figure 2.2 illustrates the most important of these. They are briefly reviewed below. Figure 2.2: Transport pathways include direct routes across the cell(l), through the membrane(2), via intercellular pathways(3), across fenestrae(4), and via vesicular mechanisms(5a,5b) [15]. - Intercellular Clefts Adjacent endothelial cells meet at intercellular clefts. These clefts are typically 10 to 20 nm wide. Regions where two apposing cells contact are referred to as Chapter 2. PHYSIOLOGICAL OVERMEW 12 tight junctions or maculae occludens. In certain microvascular beds complete closure of the cleft may occur, e.g., in the microvessels of the brain. This type of fused cleft is known as a zonula occludens. The degree to which these types of junctions occur is dependent on the particular location and function of the microvessel in question. - Fenestrae Fenestrae are formed by the attenuation of the capillary wall endothelial cells to a thickness between 6 to 8 nm and are generally disc-shaped, about 60 to 80 nm in diameter. In some cases the fenestrae are completely open, such as in the glomerular capillaries of the kidney. Fenestrae are thought to be the major pathway for the transport of protein across the endothelium. - Plasmalemmal Vesicles These are spherical bodies, typically 60 to 80 nm in diameter, which exist in the cytoplasm of the endothelial cells; open vesicles have also been observed on both the luminal and interstitial surfaces of the cells. Vesicles are thought to play a role in the transport of macromolecules across the endothelial barrier. Proposed mechanisms include the vesicles shuttling molecules from one side of barrier to the other. An alternative possibihty is the formation of temporary open fluid channels through the cytoplasm of the cell, this being due to the fu-sion of a number of vesicles stretching between the luminal and the interstitial surfaces of the cell. Figure 2.3 illustrates this mechanism of transport. - Discontinuous Vessels This type of vessel, also referred to as a sinusoid, is characterized by the pres-ence of large discontinuities in both its endothelium and basement membrane. This vessel type is found in the liver, the spleen, and the bone marrow. Chapter 2. PHYSIOLOGICAL OVERVIEW 13 — Stable Pores Although not as yet verified by electron-microscopic techniques, it has been proposed that a population of rigid, stable pores exists within the endothelium. This postulate attempts to account for the observation of a high capillary wall permeability to water and small solutes. - Free Diffusion Free diffusion through the endothelial wall may constitute a major pathway for the transport of both water and non-polar solutes. Lipid and lipid-soluble molecules are also thought to use this mechanism to traverse the cyctoplasm of the cell. Figure 2.3: The endothelium of a mouse diaphragm venule extends across the field, containing abundant vesicles. In the center a transendothelial channel can be seen[72]. Chapter 2. PHYSIOLOGICAL OVERVIEW 14 Therefore, in summary, it would seem that the transport of water and water-soluble molecules across the endothelial barrier is both a diverse and complex process. 2.2.2 Forces Influencing Transcapillary Exchange of Fluid and Proteins The transcapillary exchange of fluid and proteins occurs via either convective or diffusive mechanisms. The exchange is driven by differences in hydrostatic pres-sure and protein concentration between the luminal and interstitial compartments. Fluid exchange is therefore determined by the balance of the following forces: — the hydrostatic pressure within the capillary — the hydrostatic pressure within the interstitium — the oncotic pressure (i.e., the total colloid osmotic pressure exerted by all proteins present) of the blood plasma — and the oncotic pressure exerted within the interstitial compartment. Fluid exchange is also influenced by the physical characteristics of the intervening capillary wall. The forces listed above are referred to as the Starling forces. Both the first and final forces listed above drive fluid from the luminal compartment into the interstitium (i.e., filtration). Those listed second and third promote fluid flow in the reverse direction (i.e., reabsorption). These forces are pictorially illustrated in Figure 2.4. Under normal conditions the arterial end of the capillary exhibits a higher hydro-static pressure than the venular end. Therefore, when (under normal conditions) the relative magnitudes of all of the forces listed above are compared, it would seem that fluid filtration is favoured in the arterial segment of the capillary and fluid reabsorption in the venous segment. Fluid filtered from the capillar)' into the Chapter 2. PHYSIOLOGICAL OVERVIEW 15 CAPILLARY PRESSURE OSMOTIC PRESSURE OF PROTEINS Figure 2.4: The Starling forces. interstitium may be returned to the circulation hj either capillary reabsorption as Protein exchange between the luminal and interstitial compartments is usually as-sumed to occur via one of two alternative mechanisms: either by diffusion driven vective transport associated with the fluid flows detailed above. 2.3 T H E INTERSTITIUM The interstitium is often defined as the space existing between the capillary walls, the lymphatic walls, and the tissue cells. The principle components of the intersti-tium include the following: — Collagenous fibers. outlined above or via the lymphatic system (see Section 2.4). by a difference in protein concentration between the two compartments or by con-Chapter 2. PHYSIOLOGICAL OVERVIEW 16 — Glycosaminoglycans and proteoglycans. — Elastin. — Plasma proteins. — Interstitial fluid. These components, and the way in which they interact, dictate the physicochemical properties of the interstitium as a whole. Therefore, the properties of each of the components hsted above will be reviewed individually. Only then will the behavioural characteristics of the entire interstitium, and the role it plays in the microvascular exchange system, be discussed. Collagenous Fibers The building block of the collagen fiber is the collagen molecule, which was formerly called tropocollagen. Each collagen molecule consists of three coiled polypeptide chains. Microfibrils of collagen are formed by the aggregation of many collagen molecules. Collagen fibers are in turn formed by the aggregation of many of these microfibrils. The arrangement of the collagen molecules within the fiber is illus-trated in Figure 2.5. The collagen fibers exhibit a high tensile strength, particularly when stressed along their longitudinal axes. This great strength is attributed to the existence of stable covalent cross-linkages between the component collagen molecules[3] . Glycosaminoglycans and Proteoglycans The glycosaminoglycans are a family of negatively-charged polysaccharide chains of variable length. The chains are composed of repeating disaccharide units. The charge exhibited by the glycosaminoglycans is due to anionic groups existing within the disaccharide units and is fully present under normal physiological conditions. Chapter 2. PHYSIOLOGICAL OVERVIEW 17 Procollagen Collagen fiber Figure 2.5: The features of the various levels of collagen organization[l3]. Chapter 2. PHYSIOLOGICAL OVERVIEW 18 The most common of the glycosaminoglycans is hyaluronate (more recently referred to as hyaluronan), a high molecular weight, unbranched polymer that forms an extended coil in solution [3]. This configuration allows hyaluronate to occupy a hydrodynamic volume of between 1,000 to 10,000 times its own unhydrated volume. Hyaluronate is the only glycosaminoglycan to occur as a free polymer. The other glycosaminoglycan species found within the interstitium form proteoglycans by bonding with polypeptides [3]. Proteoglycans often form aggregates in the presence of hyaluronate. Elastin Elastin is present in most connective tissue and takes the form of rubber-like elastic fibers. As would be expected, the fibers impart an elastic behaviour to the tissues in which they are found. The elastic characteristics of the particular tissue increase with increasing elastin content. An example are the large arteries, where the elastin content is found to be in the order of 30%. Plasma Proteins Plasma proteins represent a broad group of macromolecules of various sizes. Those found in the interstitium originate in the luminal compartment, but both the total quantities and the species distribution vary between the luminal and interstitial compartments. This variation is due to the semi-permeable character of the capil-lary wall. At normal physiological pH most proteins carry a net negative charge. For the purposes of the current work the most important effect of the plasma proteins is their contribution towards the interstitial colloid osmotic pressure. As in plasma, the major contribution to the total colloid osmotic pressure is provided by albumin. Chapter 2. PHYSIOLOGICAL OVERVIEW 19 Interstitial Fluid The capillary wall is almost completely permeable to water and the dissolved ions it contains (i.e., the plasma minus the plasma proteins). Hence, the interstitial fluid possesses an almost identical composition to the plasma of the luminal compartment barring the aforementioned proteins. The structure of the interstitium is illustrated in Figure 2.6. It resembles a three-dimensional network of fibrous components imbedded in a gel-matrix. The fibers are mainly collagenous, but certain tissues also contain fibers of elastin. The collagen fibers endow the tissue with tensile strength and structural stabihty, giving the tissue resistance to changes in configuration and volume (see Section 2.3.1.2). The fibers also act to immobilize the tissue-gel (also called the ground substance) which is formed by the presence of hyaluronate and proteoglycans within the interstitial fluid. As already mentioned, both the proteoglycans and hyaluronate are extremely coiled molecules - they form a sponge of very fine reticular filaments which exhibits gel-like characteristics, and hence provides great hindrance to both the bulk flow of fluids and the diffusion of macromolecules through the interstitial space. Hyaluronate is the only glycosaminoglycan that has been proven not to be bound within the interstitial matrix[6]. 2.3.1 Properties of the Interstitium 2.3.1.1 Exclusion Mutual exclusion between spherical macromolecules takes place because two molecules cannot occupy the same space simultaneously and their centers cannot come closer Chapter 2. PHYSIOLOGICAL OVERVIEW 20 Figure 2.6: The general organization of the interstitial matrix. Components include collagen fibrils (large rods), hyalronate molecules (long unbranched heavy line) and proteoglycans[l3]. Chapter 2. PHYSIOLOGICAL OVERVIEW 21 than the sum of their radii[6]. Of course, exclusion takes place between non-spherical structures also. Other types of steric exclusion also occur within the interstitium. The fixed components of the interstitium, such as collagen fibers, elastin fibers, and the varoius glycosaminoglycans, reduce the space available to other components, such as the plasma proteins. The exclusion phenomenon is simply demonstrated by the model of a sphere in a random network of rods shown in Figure 2.7(C). The sphere represents a plasma protein molecule and the rods the other interstitial components. The volume that is not accessible to the protein molecule is termed the excluded volume. Processes within the interstitium are affected by the phenomenon of steric exclusion. As the volume available to a given component is reduced by exclusion, its effective concentration within the available space increases. This results in concomitant increases in both the osmotic pressure and the driving forces for diffusion within the system[14]. The magnitude of the exclusion effect depends on the geometry and concentration of both the excluded and excluding components. Exclusion within the interstitium is mainly attributed to the presence of collagen and hyaluronan. The volume excluded by hyaluronan is in the order of 10 times that of collagen when measured on a weight basis[3]. However, collagen is by far the more abundant of the two within the interstitial space. 2.3.1.2 Tissue Compliance The mechanical properties of the interstitium, are, for the most part, determined by the presence of collagen, polysaccharides, and interstitial fluid. The collagen fibers form a semi-rigid matrix within the tissue space which acts to hmit volume changes. Chapter 2. PHYSIOLOGICAL OVERVIEW 22 Figure 2.7: A, B, and C are all examples of steric exclusion. In C the available space for a sphere in a random network of rods is equal to the volume(dotted) within which the center of the sphere can move freely[6]. Chapter 2. PHYSIOLOGICAL O V E R V I E W 23 The polysaccharides, however, which may be interacting and/or linking with the collagenous fibers, exert an osmotic pressure within the interstitial fluid which confers a tendency towards hydration and swelling. This tendency is enhanced by the mutually repulsive charges present on the polymer molecules. An important property of the interstitium is the relationship between its fluid hy-drostatic pressure and its volume. This relationship is termed the tissue comphance. Often the comphance of a tissue is more specifically defined as the rate of change of fluid volume with respect to fluid pressure. Typical comphance curves for animal skin and muscle are are presented in Figure 2.8. These exhibit similar trends: a low comphance at low and normal tissue volumes and a high comphance at high volumes. The point on the comphance curve at which the interstitium "normally" operates is located within the low comphance region. Therefore, under normal con-ditions, a small increase in tissue volume produces a relatively large rise in tissue pressure. This pressure increase then acts to resist further movements of fluid into the interstitial space, thereby reducing the hkelihood of edema formation (i.e., an abnormal excess accumulation of fluid in the tissue spaces). 2.4 T H E L Y M P H A T I C S Y S T E M The principal function of the lymphatic system - from the perspective of the current study - is the transport of proteins from the interstitial space into the blood. The lymphatics also act as an additional route by which fluid may return to the circulation from the interstitium. The lymphatic system consists of a network of vessels which parallel those of the arteriovenous system. The lymphatic system begins with many closed-end tubes, the walls of which consist of a thin monolayer of endothelial cells. These vessels are Chapter 2. PHYSIOLOGICAL OVERVIEW 24 Figure 2.8: Experimentally determined rat tissue compliance relationships for skin and muscle. IFP and AIFV represent, respectively, the interstitial fluid hydrostatic pressure and the change in interstitial fluid volume from normal[69]. Chapter 2. PHYSIOLOGICAL OVERVIEW 25 called the terminal lymphatics, and are also referred to as the initial lymphatics. _ They are similar in calibre (within an order of magnitude) to the blood capillaries and are generally located in close proximity of them. Figure 2.9 shows the general arrangement of the terminal lymphatics relative to the blood capillary bed. The fluid collected within the terminal lymphatics is transported (see Section 2.4.1) into a series of collecting vessels which move the lymph through the system. The lymph is deposited into either the right or left thoracic duct (which are approxi-mately located at the base of the neck), then subsequently passed into the subcla-vian veins. The nature of the lymph changes as it travels through the lymphatic vessels. Lo-cated at certain points within the system are the lymph nodes. These filter foreign matter from the lymph and neutralize toxins, thereby altering the lymph composi-tion. However, for practical purposes interstitial fluid (in a well mixed non-excluded volume) and lymph may be considered identical in composition, the major differ-ence between the two being their locations. When the fluid bathes the cells it is called interstitial fluid, when it flows through the lymph vessels it is called lymph. Another feature of the lymphatics is the presence of endothelial valves. These valves are generally monocuspid or bicuspid, and function to allow only one-way flow of the lymph (i.e., drainage towards the central veins). 2.4.1 The Terminal Lymphatics The terminal lymphatics are most numerous in the regions surrounding the venular segments of the arteriovenous capillaries, these sections being the most permeable parts of the capillaries. A thin layer of endothelial cells form the wall of the terminal lymphatic. The Chapter 2. PHYSIOLOGICAL OVERVIEW" 26 Figure 2.9: Lymphatic and blood vessels in an area of bat's wing at a bifurcation of two large vascular channels: lymphatics(black), blood vessels(shaded), and blood capillaries (lines) [13]. Chapter 2. PHYSIOLOGICAL OVERVIEW 27 cells are loosely joined at intercellular junctions and may at times overlap (see Figure 2.10). The overlapping cells are thought to behave as a series of one-way valves, allowing fluid and protein movement from the interstitium into the terminal lymphatics, but not vice-versa. Both osmotically driven and vesicular transport mechanisms have been postulated as governing the filling of the terminal lymphatics. However, as of yet, neither theory has been widely accepted or substantiated^]. Although similar average fluid pressures have been determined for both the interstitium and the terminal lymphatics, it seems generally believed that interstitial fluid pressure is the main determinant of lymph flow[3]. The walls of the terminal lymphatics are fixed to the surrounding tissue by anchoring filaments. As the surrounding tissue pressure increases, the filaments pull taut to prevent the collapse of the filling vessels[13]. Under normal resting conditions the permeability of the terminal lymphatics to water and dissolved ions is close to that of blood capillaries. However, if the in-terendothelial junctions are opened - by swelling or activity - the permeability is increased[29]. 2.4.2 Lymph Propulsion Movement of fluid through the lymphatic system is governed by both the contractile activity of the smooth muscle, which surrounds all vessels, excluding the terminal lymphatics, and a number of extrinsic factors. These factors include muscular contraction of the surrounding tissues, massage and increased temperature. The smooth muscles surrounding the lymph vessels create a pumping action by contracting in a systematic fashion; the intra-lymphatic valves ensure the correct Chapter 2. PHYSIOLOGICAL OVERVIEW 28 Figure 2.10: The structure of the terminal lymphatic wall. direction of fluid movement. The strength of contractions is influenced by stretch-ing of the lymphatic vessel walls. Hence, as the vessels become distended due to excessive lymph loading the contractions exporting the fluid increase in vigour. 2.5 T H E NEPHROTIC SYNDROME A considerable portion of the data used within the current work is related to the study of transcapillary forces present in humans suffering with the nephrotic syn-drome. Hence the characteristics of this disease state are briefly reviewed here. The term nephrotic syndrome does not imply a specific etiologic diagnosis; rather it identifies a major renal syndrome whose recognition can precede a more definite diagnosis of the underlying cause. The nephrotic syndrome is characterized by proteinuria (i.e., excessive unrinary excretion of plasma proteins) sufficient to in-duce hypoalbuminemia (i.e., a plasma albumin content below that of normal) and edema. It is accompanied frequently by hyperlipidemia (i.e., a general elevation of Chapter 2. PHYSIOLOGICAL OVERVIEW 29 the concentrations of all hpids in the plasma) and lipiduria (i.e., the presence of hpids in the urine). Plasma volumes are reduced in some patients but, in patients with chronic edema they are generally within normal limits [25]. A more detailed description of the disease is presented in Figure 2.11. Chapter 2. PHYSIOLOGICAL O V E R V I E W 30 ? Decreased albumin -synthesis PROTEINURIA .1 • Hypoaiouminemia« -? Increased albumin catabolism Decreased plasma oncotic pressure Shift of capillary fluid into interstitial space \ Reduced plasma volume-I Reduced renal blood flow i Increased renin secretion Increased Increased interstitial vasopressin volume release Increased aldosterone secretion 1 Reduced sodium and water excretion -Continued ingestion of salt and water Salt and water retention Restoration of plasma volume EDEMA Figure 2.11: The nephrotic syndrome. Chapter 3 MODEL FORMULATION 3.1 INTRODUCTION In this chapter we will first establish a reasonable basis on which to formulate our model of the human microvascular exchange system. We will then review two alternative concepts of transcapillary exchange which are to be incorporated into the model. Next, we will write down phenomenological relationships which attempt to describe mathematically the transient behaviour of the system. Finally, since the governing differential equations require the specification of initial conditions, we will determine reasonable estimates of the normal steady-state (i.e., unperturbed) conditions representative of a 70 kg male human. The water contained within the human body can be conveniently subdivided into two major compartments: the extracellular water and the intracellular water. In-tracellular water is contained within the cells of the body. Extracellular water includes the water within the plasma, the interstitium, and the lymphatics. Tran-scellular fluid, which includes cerebrospinal fluid (that which surrounds the brain and spinal cord), intraocular fluid within the eyes, plural fluid around the lungs, peritoneal fluid which surrounds the organs, synovial fluid within the joints, and digestive fluids, is often classified as extracellular fluid. However, regulation of the transcellular fluid is a complex process and the fluid represents only a small fraction of the total body water. Hence, for the purpose of the model developed here it will 31 Chapter 3. MODEL FORMULATION 32 be neglected. Intracellular water content is governed by the balance of osmotic forces generated across the cell membranes. This osmotic force is determined, for the most part, by the concentration of small ions such as Na + . The capillary wall separating in-travascular fluid from extravascular fluid is almost completely permeable to these ions. Hence, if the body is treated as a balanced system (i.e., dietary intake of water and salt is equal to losses such as those associated with urinary output), then it can be assumed that the intracellular water content will remain unchanged. This water compartment is therefore also omitted from the model of microvascu-lar exchange developed here. The assumption of an ionically stable extracellular water compartment appears reasonable for the conditions considered herein, due to strong renal regulation of ion concentrations in chronic disorders such as the nephrotic syndrome. This assumption has been invoked in several previous models of microvascular exchange[68, 5]. However, the inherent limitations it introduces into the model should not be overlooked when investigating other physiological disorders or system perturbations. In the present model, the microvascular exchange system is subdivided into a num-ber of constituent compartments. Each compartment is assumed to be homoge-neous and well-mixed. Hence, any physical characteristic of a particular compart-ment, such as internal hydrostatic pressure, is represented by a single average value. The magnitude and direction of the material fluxes between compartments is then determined by the average driving forces (e.g., differences in protein concentrations, etc.) and the average properties of the compartmental boundaries. Not enough is known about the properties and the behaviour of the interstitium and the cap-illary wall to allow the development of an accurate non-compartmental model of Chapter 3. MODEL FORMULATION 33 microvascular exchange. The compartmental approach has both advantages and disadvantages. The well-mixed homogeneous assumption significantly simplifies the model, greatly reducing the number of constitutive mathematical relationships and characterizing transport parameters required for a full model description. This type of model also allows easy inclusion of fairly complex phenomena, such as tissue swelling. However, this approach may hmit the models' abihty to simulate the behaviour of the real sys-tem, because the average driving forces between compartments used in the model will generally differ from the local forces existing within the real system. This re-striction may be of particular concern when the model is used to simulate transient behaviour. As already stated in Section 2.2.2, fluid movement across the capillary wall is determined by the physical characteristics of the wall itself and the balance of hydrostatic and colloid osmotic pressures (which are dependent on plasma protein concentrations) acting across it. The primary objective of the model is to describe how fluid and plasma proteins are redistributed within the body following a system disturbance. For simplicity, it is assumed that the behaviour of all plasma proteins can be referenced to a single protein species, albumin. A schematic representation of the microvascular exchange system is presented in Figure 3.1. The microvascular exchange system is subdivided into two compart-ments: one representing the circulation and the other a general tissue compartment. It is well known, however, that the properties of skin and muscle, which represent the bulk of the interstitial compartment, vary considerably. The approximation introduced by treating both as one lumped unit is a weakness of the current model. Chapter 3. MODEL FORMULATION 34 The separation of this generic tissue compartment into individual skin and mus-cle units is a trivial matter with respect to the mathematical modelling required, but currently such an increase in model complexity would be inappropriate when considering the human. The reasons for this are: — an almost complete lack of quantitative physiological data regarding human muscle tissue, and — a concomitant increase in the number of characterizing transport parameters that must be estimated statistically using limited experimental data. Two alternative models of transcapillary exchange are investigated: the Starling Model (SM) and the Plasma Leak Model(PLM). One of our objectives is to try to determine which of the two can best describe the available data. These models differ in their proposed mechanisms of interstitial fluid and plasma protein transport across the capillary membrane. The Starling Model hypothesizes two basic mechanisms of capillary mass exchange. Fluid is transferred by filtration from the plasma to the tissue and plasma proteins are also transferred by convection with this flow. Proteins ma3' also be exchanged between the capillary and the tissue space by diffusion. The Plasma Leak Model is a more complex representation of capillar exchange and is based mathematically on the model developed by Wiederhielm[68]. It attempts to account for the variations in both plasma hydrostatic pressure and capillary wall fluid conductivity that have been observed to occur along the length of the capillary. It is postulated that fluid is filtered from the plasma to the tissue at the arterial end of the capillary and that it is transported by plasma leak in the same direction at the venous end of the capillary. It is also hypothesized that fluid is reabsorbed Chapter 3. MODEL FORMULATION Figure 3.1: A schematic representation of the microvascular exchange syst Chapter 3. MODEL FORMULATION 36 from tissue to circulation at the venous end. Filtration and reabsorption occur through small completely-sieving pores which are assumed to exist at both ends of the capillary, while plasma leak takes place through larger non-sieving channels existing in the venous capillary only. Plasma proteins are exchanged between the capillary and the tissue by either convection with the plasma leak or by diffusion across the capillary membrane. Both the SM and the PLM assume the lymphatics return both fluid and protein to the circulation from the interstitium. Hence, we shall first develop the math-ematical relationships common to both models and then detail the SM and PLM mathematical relationships that describe capillary transport mechanisms as well as the overall balance equations which are obtained in each case. 3.2 R E L A T I O N S H I P S C O M M O N T O B O T H T H E S M A N D T H E P L M 3.2.1 C o l l o i d O s m o t i c Pressure Re la t ionsh ip It is assumed that the osmotic behaviour of all the plasma proteins can be ref-erenced to a single species: albumin. In normal mammalian plasma, albumin is the predominant protein species, both in terms of content and osmotic activity. It represents about 50 percent of the of the total protein mass and contributes approximately 65 percent of the oncotic pressure. These facts point clearly to al-bumin as the appropriate choice for the "representative protein species". Another important consideration is that, in terms of measured concentration, albumin is the most widely reported protein in microvascular exchange studies. The assumption of a representative protein species also significantly reduces the number of transport Chapter 3. MODEL FORMULATION 37 coefficients required for a complete model description. Relationships between albumin concentration and albumin colloid osmotic pressure and also between total protein concentration (i.e., including all protein species normally present in human plasma) and total colloid osmotic pressure have been empirically determined by Landis and Pappenheimer[44]. However, if albumin is to be used as the representative protein species, a correlation between albumin concentration and total colloid osmotic pressure is required. 50.0 0.0 100 20.0 30 0 PLASMA COP. (mm Hg) Figure 3.2: Albumin concentration vs. oncotic pressure relationship. Hence, the following relationship for the albumin concentration, CPL, as a function of the total osmotic pressure, HPL, in plasma was determined by least squares fitting of the data of Geers et al.[26], Noddeland et al.\bl], and Fadnes et ai.!22] (see Figure 3.2) CPL = 1.522 x 1 0 ~ 3 n P L (3.1) Chapter 3. MODEL FORMULATION 38 where the units of colloid osmotic pressure are mmHg and those of albumin con-centration are g/m£. The above relationship was forced to pass through the point HpL = 0 when Cpr, = 0. A hnear relationship was selected after first fitting 1st, 2nd, 3rd, and 4th order polynomials through the data. A difference in the vari-ance of fit of less than 1 percent was found between the 1st and 2nd order fits and the higher order fits were found to be non-monotonic within the range of interest. Encouragingly, for a plasma colloid osmotic pressure of 25 mmHg, equation(3.1) predicts a CPL of 38.1 g/£, which compares well with the value of 36.0 g/£ provided by Landis and Pappenheimer[44]. Due to the lack of alternative information, this relationship for the plasma compartment was also assumed applicable to the tissue compartment, i.e. Cijcv = 1.522 x 10 _ 3n/ (3.2) where Cj^y represents the concentration of albumin in the interstitial space avail-able to it (see Section 2.3.1.1), and Hj represents its associated colloid osmotic pressure. 3.2.2 Concentration Relationships Relationships between effective protein (albumin) concentration, C, protein (albu-min) mass, Q, and compartment volume, V, may be written as follows: For the plasma compartment, CPL — QPL/VPL (3.3) and for the interstitial compartment, Ci = QI/VJ (3.4) Chapter 3. MODEL FORMULATION 39 and Ci,AV = Qi/VJlAV (3.5) where VhAV = V} - VIiEX (3.6) where the subscript "PL': indicates the plasma compartment and subscript "I" the interstitial compartment. The subscripts "EX" and "AV" indicate respectively which volumes are excluded and which are available to albumin. Exclusion sig-nificantly influences the effective concentration of the albumin within the inter-stitial compartment (see Section 2.3.1.1). Therefore failure to account for this phenomenon may lead to errors in the estimates of fluid and protein exchange. This is of particular concern if the model is used to simulate conditions involving significant reductions in tissue fluid volume, such as dehydration. 3.2.3 Lymphatics Within the microvasculature the two predominant sites of mass exchange are the capillary wall and the lymphatic wall. The equations describing trans capillary exchange will be developed later in the chapter. Little is known about the transport of fluid and proteins across the lymphatic wall. As mentioned in Section 2.4 it is thought that both fluid and plasma proteins pass into the lymphatics by way of interendothehal junctions. These junctions behave as one-way valves, allowing only unidirectional flow into the lymphatics. It is usually assumed that as the interstitial hydrostatic pressure, Pi, increases the junctions open wider, allowing increased flow across the lymphatic wall[29]. When attempting to quantify lymph drainage, the most commonly used assumption is that of a linear relationship between lymph flowrate and tissue hydrostatic pressure[5j. During tissue overhydration the lymph Chapter 3. MODEL FORMULATION 40 flowrate, J L , is therefore described by the following equation: JL = JL,NORM + LS.{Pi - PI ,NORM) (3.7) where JL,NORM is the lymph flowrate and PI,NORM the tissue hydrostatic pressure, both at normal steady-state conditions. The proportionality constant LS is referred to as the lymph flow sensitivity. A hnear relationship is also used to describe lymph flow under conditions of dehydration, as follows: JL = JL,NORM {PI — PI,EX )/(PI,NORM — PI,EX ) (3-8) Equation(3.8) assumes that lymph flow ceases entirely when the tissue hydrostatic pressure is reduced to PIJSX ( the tissue hydrostatic pressure corresponding to a tissue volume equal to that of the excluded volume). Since lymph flow is unidirectional, if a negative value is generated (i.e., at tissue volumes less than the excluded volume), it is then set to zero. Plasma proteins are thought to cross the lymphatic wall by convection. Hence, the flow of proteins passing from the tissue into the lymphatics, QL, is given by the product of the lymph flowrate, and the interstitial plasma protein concentration based on the total mobile fluid volume, C j : QL = JL-CI (3.9) Equation(3.9) assumes that, upon entering the lymphatics, the plasma protein solution from the available volume mixes with fluid leaving the remaining portions of the mobile fluid volume according to their relative volumes. 3.2.4 Interstitial Fluid Hydrostatic Pressure &: Interstitial Compliance Over the last three decades several methods have been developed for the mea-surement of interstitial fluid hydrostatic pressure, Pi. These include implanted Chapter 3. MODEL FORMULATION 41 capsule, wick, "wick-in-needle", and micropuncture techniques. In principle all of these methods can be classified as "fluid-equilibration" techniques (i.e., measuring the hydrostatic pressure within the free fluid at the probe tip). The wick-in-needle and micropuncture techniques appear to cause the least im-plantation trauma and also to produce the most consistent results, both within and between subject species. These two experimental methods also yield similar pressure values for a given tissue type. However, the measurement of Pj in man using the micropuncture method has been restricted to the nailfold of the fmgers[3]. Hence measurements of Pj determined by wick-in-needle techniques are chosen as representative in the current work. Several workers' measurements of normal Pj in the human are summarized in Table 3.1. Interstitial compliance describes the variation of interstitial fluid volume, V}, with changing hydrostatic fluid pressure, Pi (see Section 2.3.1.2). It is usually presented in the following form: Pi = FCOMP(Vj) (3.10) where FCOMP represents the functional relationship between Pi and Vj. Despite the obvious importance of tissue space compliance in the regulation of interstitial volume, the subject has received little attention to date. In fact, considerable controversy still surrounds the existing measurements of tissue compliance that have been made for experimental animal tissues. Almost no information is available regarding the compliance of human tissues. Several workers have experimentally determined the compliance characteristics of a number of different mammalian tissues [35, 55, 61]. In nearly all cases the basic features of the observed relationships are similar, including: a hnear hydration, around the normal control point and up to Chapter 3. MODEL FORMULATION 42 TISSUE LOCATION TECHNIQUE PI,NORM (mmHg) REF. Sub cutis Dorsum of Hand Wick-in-Needle 0.0±1.7 [70] Muscle Lower Leg Wick 1.0±3.0 [37] Subcutis Lower Leg Wick-in- Needle -0.7±1.3 [61] Subcutis Thorax Wick-in-Needle -1.3±1.6 [52] Table 3.1: Experimentally determined interstitial hydrostatic fluid pressures. 50-100% overhydration; a rapid increase in comphance toward infinity at about 50-100% overhydration; a reduction in comphance at very high levels of overhydration due to the restriction on further expansion imposed by fascias surrounding the tissues. Stranden and Myhre[61] have presented experimental measurements of the comph-ance of human lower limb subcutaneous tissue. Their results exhibit the sigmoid-hke shape generally observed in other mammalian tissues. However, their data are too scattered to draw conclusions regarding the exact nature of the relationship. No other data regarding human interstitial comphance are known to the author. The work of Reed and Wiig[55] on rat tissues has yielded a well defined comph-ance relationship for both muscle and skin; confidence in the general form of the relationship has been enhanced by similar findings for dog tissues[71]. In order to determine a "most likely" comphance relationship for human tissue, the following logic is followed. In Section 3.5.1 we will establish that "reference man" has an interstitial compart-ment comprised of 2.4 £ of fluid within the skin and subcutaneous tissue, 4.5 £ of muscle tissue fluid and a further 1.5 £ in other tissues. Therefore, since the bulk of the interstitial fluid resides in the muscle compartment, the shape of the rat muscle relationship determined by Reed and Wiig is used as a basis for the "most Chapter 3. MODEL FORMULATION 43 5.0-re B oo oo £X a, o 0.0 H 00 o Q -5.0 H Compliance Curve ffl J^pmpjiance CurveJf2_ Compliance.Curve #3 Co} oo oo -10.0-0.0 10.0 20.0 30 0 INTERSTITIAL FLUID VOLUME ( L ) 4 0 . 0 Figure 3.3: T h e " m o s t l i k e l y " h u m a n c o m p l i a n c e relat ionships. Chapter 3. MODEL FORMULATION 44 likely" human comphance curve. Stranden and Myhre [61] have provided data regarding the expected rise in Pi as human subcutaneous tissue moves from a nor-mal condition to an edematous state. Under conditions of normal tissue hydration the subcutaneous tissue hydrostatic pressure, PI^ORM , was measured to be -0.7 ± 1.3 mmHg and, at a 210% elevation in interstitial fluid volume, P/ was found to be 2.2 ± 2.0 mmHg. Thus, for human subcutaneous tissue, a 210% increase in Vj is associated with a 2.9 mmHg rise in Pj. From the rat muscle tissue comphance relationship determined by Reed and Wiig an elevation of 210% in tissue hydration above normal yields a concomitant rise in Pj of 2.4 mmHg. As a starting point in the construction of our "most-likely" human comphance relationship we have a series of discrete Pj, Vi data points for rat muscle tissue. The "most-likely" re-lationship is then generated by using the following two equations for scaling the rat Pi, VI values. The human pressure values are obtained from the rat pressures according to 2.9 Pl,BUMAN — Pl,HUMAN,NORM = (PlJlAT ~ Pl,RATJfORM )-7T7 (3-H) 2.4 where the subscripts "HUMAN" and "RAT" have the expected meanings. Corre-sponding rat tissue volumes are scaled by the human:rat ratio of normal interstitial fluid volumes, i.e. T / TT ,VitHUMAN,NORM v VI,HUMAN = vi,RAT\—T? ) K61*) *I,RAT,NORM The human comphance relationship so generated is presented graphically as the solid line in Figure 3.3. The flat segment of the curve representing overhydration is the region most prone to the influence of experimental error (due to the small gradient), hence this section can be regarded as the most "uncertain". In order to determine how this gradient might influence the parameter estimation proce-dure, the three alternative relationships presented in Figure 3.3 are investigated. Chapter 3. MODEL FORMULATION 45 Compliance curve #1 is generated as detailed above, and curves #2 and #3 are identical to curve #1 except for an arbitrary increase in the slope of the "overhy-dration segment" (i.e., at volumes above 10.4 £ the slopes are 1.8xl0-5, 5.0xl0~5, and 1.05xl0-4 mmHg/m£ for curves $1, #2 and #3, respectively). The section of the relationship representing dehydration is generated by continuing the curve tangentially downwards from the point representing normal conditions. For computational purposes the relationship was divided into three segments: two hnear relationships representing both dehydration and massive overhydration (i.e., greater than a 50 percent increase in interstitial volume) and cubic spline interpo-lation of intermediate data points (see Section 4.3.4). 3.3 T H E STARLING M O D E L The Starling Model (SM) is considered first, it being the simplest in concept and mathematical formulation. This model assumes the capillary wall to be homoporous and no distinction is made between its arterial and venous ends. The capillary is represented by a single "average" value of hydrostatic pressure referred to as the capillary pressure, PPL,C i and an associated pair of transport coefficients, these being the solute reflection coefficient, cr, and the filtration coefficient, KF. The Starling Model is presented schematically in-Figure 3.4. Starling's hypothesis is used to represent the fluxes of fluid and plasma proteins between the circulation and the interstitium. The flowrate of fluid across the capillary wall, J F , is described using the following Starling type equation: JF = KF.\PPLic - Pi- <T.(TLpl - Iii)} (3.13) In the above equation the driving force for fluid exchange is provided b)' differences Chapter 3. MODEL FORMULATION 46 BLOOD I 1 Figure 3.4: The Starling Model. between both the hydrostatic pressure, P, and colloid osmotic pressure, II, on either side of the capillary wall. The transport parameters Kp and cr characterize the physical properties of the capillary bed. The fluid filtration coefficient, Kp, is determined by the surface area available for fluid exchange and the hydraulic conductivity of the capillary wall. The protein reflection coefficient, a, may have a value ranging between zero and one; its value indicates the relative permeability of the capillary to plasma proteins. A value of zero indicates that the capillary wall is completely permeable to the plasma proteins, whereas a value of one indicates the opposite. Therefore, when both the pressures and transport coefficients are known, the direction and rate of fluid flow, Jp, can be determined. A positive value of JF indicates a net filtration of fluid from the circulation into the tissues, and a negative value corresponds to a net reabsorption from the tissues into the plasma compartment. Chapter 3. MODEL FORMULATION 47 Plasma protein exchange across the capillary wall is described using the two ex-pressions developed by Kedem and Katchalsky[40]: QF = J A l - * ) { C l ' A V + C p L ) (3-14) QD = PS.(CPL-CI,AV) (3.15) Equation(3.14) describes the convective transport of proteins across the capillary wall, Q F , assuming that the plasma protein solution may be treated as an ideal, dilute solution [40]. Equation(3.15) represents the diffusive flux of plasma proteins across the capillary wall, Qp- The diffusion coefficient PS is determined as the product of the exchange area available and the permeability of the capillary wall to plasma proteins. 3.3.1 Compartmental Mass Balances Mass balances, on both the fluid and protein, are conducted on the microvascular exchange system to obtain the differential equations which describe its transient behaviour. The system is assumed to be closed, hence, balances about it yield the following relationships: ^(VPL + VI) = 0 (3.16) and TAQPL + QI) = 0 (3-17) Balances may also be conducted about the interstitial compartment alone to obtain ~ ^ J F ~ J L (3.18) Chapter 3. MODEL FORMULATION 4 8 and ^ =QF + QD-QL (3.19) 3.3.2 Circulatory Compliance Plasma hydrostatic pressure within the capillary is mainly determined by the ratio of pre- to postcapillary resistances imposed by the surrounding smooth muscle. It is also influenced by blood flowrate, capillary size and a number of other factors. The tone of the smooth muscle, and hence the resistance it imposes, is influenced by many different stimuli, such as arterial pressure, certain hormones, etc.. Needless to say, the control of capillary pressure is an extremely complex phenomenon, and the modelling of its associated control systems is not within the scope of the current work. However, it is anticipated that changes in plasma volume will affect the capillary pressure in a passive manner. No quantitative information is available concerning a relationship of this type. The Starling Model requires the capillary to have a single "average" hydrostatic pressure, P P L . C • For the sake of simplicity, a linear relationship of the following form is assumed to apply between PPL,C and the plasma volume: PPL,C — PpL,C,NORM + PCOMPpLtc .(VPL — VPL,NORM ) (3.20) where PPL,C,NORM is the normal steady-state capillary hydrostatic pressure. Equa-tion^.20) introduces another parameter, the circulatory comphance, PCOMPPL^C • 3.4 T H E PLASMA L E A K MODEL The most general form of the Plasma Leak Model (PLM) is presented schemat-ically in Figure 3.5. The capillary is assumed to exhibit minimal heteroporosity Chapter 3. MODEL FORMULATION 49 (i.e., two pore sizes exist: small pores which do not allow proteins to penetrate and large pores which allow unhindered passage of plasma proteins through the capillary wall). In contrast to the SM, the PLM accounts for the hydrostatic pres-sure differential existing between the arterial and venous ends of the capillary. The upstream pressure is referred to as the arterial capillary pressure, PPL.A , aud the downstream pressure as the venous capillary pressure, PPL,V-BLOOD Figure 3.5: The Plasma Leak Model. Normally, fluid nitration, Jp, through the small pores occurs in the arterial segment of the capillary and reabsorption, J R , also through the small pores occurs in the venous segment. These fluid movements are described by the following equations: JF = KF.(PPL<A - P T - (UPL - II,)) o (3.21) and JR = KR.(P! - PPL,V - (H, - HPL)l (3.22) Chapter 3. MODEL FORMULATION 50 Equations(3.21) and (3.22) describe fluid movement through the protein imper-meable small pores. Hence, the reflection coefficient, a, is equal to unity. The transport parameter Kp is determined by the surface area available for fluid trans-port in the arterial segment of the capillary and the permeability of this wall to fluid flow through the smaller channels. Kp is determined by the same factors as Kp but relates to the properties of the venous capillary segment. Fluid transport also occurs via the large pores that exist only at the venous end of the capillary. This plasma leakage, JPLL, is related to the hydrostatic pressure driving force according to JPLL = KPLL.{PPLy - Pj) (3.23) where the transport parameter K~PLL is again obtained as the product of a surface area and a hydraulic conductivity. Note that the reflection coefficient of these large pores "is zero and hence JPLL is independent of the transcapillary colloid osmotic pressure difference. Uncoupled protein transport can occur via either diffusion across the capillary wall or by "plasma leakage" through the the large pores in the venous segment, i.e., QD = P S . ( C P L - C I J L V ) (3-24) and QPLL — JPLLCPL (3.25) Equation(3.24) describes the diffusive protein flux across the capillary wall. As with the Starling Model the diffusion coefficient is the product of the available exchange area and the permeability of the capillary wall to plasma proteins. QPLL represents Chapter 3. MODEL FORMULATION 51 the convective movement of protein along with the plasma leakage through the . capillary's large pores. 3.4.1 Compartmental Mass Balances As with the SM, protein and fluid balances may be conducted about both the entire system and a single compartment to obtain the equations which govern the system behaviour. Assuming a closed system, these balances for the PLM yield the following differential equations: jt(VPL + Vr) = 0 (3.26) and d for the whole system and dt(QpL + Qi) = 0 (3.27) = JF -f JPLL — JR — JL ^ (3.28) and dQ - = QPLL + Q D - Q L (3.29) dt for the interstitial compartment. 3.4.2 Arterial and Venous Capillary Compliances The PLM requires a value of arterial capillary hydrostatic pressure, PPL,A , a n d of venous capillary hydrostatic pressure, PPL,V • As with the SM (see Section 3.3.2), it is anticipated that changes in plasma volume will affect both of these capillary pressures in a passive manner. However, no quantitative information is available concerning the required compliance relationships. Chapter 3. MODEL FORMULATION 52 For the sake of simplicity, hnear relationships of the following form are assumed to apply between both PPL,A and PPL,V , and the plasma volume: PPL,A = PpLAJfORM + PCOMPPLtA.(VPL - VpL<N0RM ) (3.30) and PPL.V = PpLWORM + PCOMPPLy-{VPL - VpL,NORM ) (3.31) where PPL,A,NORM and PPL,V,NORM a r e the normal steady-state values of arte-rial and venous capillary hydrostatic pressures, respectively. Equations(3.30) and (3.31) also introduce two additional parameters, the arterial capillary compliance, PC0MPPL,A, and the venous capillary compliance, PCOMPpLy. Table 3.2 summarizes all of the equations which constitute the Starling Model and the Plasma Leak Model of microvascular fluid and protein exchange. Once a set of initial conditions have been specified (see Section 3.5), the set of equations which make up each model describe how the fluid and protein contents, pressures and concentrations of each compartment as well as the intercompartmental fluid and protein fluxes change as a function of time subsequent to an applied perturbation. Chapter 3. MODEL FORMULATION 5 3 . Relationships common to both models Tissue Circulation VITEX) Ci = QijVi CI>AV = QI/(VJ n, = Fn(chAV) Pi = Fn(Vj) JL = JL,NORM + LS.(Pi — PiTNORM ) Qp — PS.(CPL — Ci<Av) Qh — JL-CI Starling model ( 3 . 4 ) ( 3 . 5 ) ( 3 . 2 ) ( 3 . 1 0 ) ( 3 . 7 ) ( 3 . 1 5 ) ( 3 . 9 ) C P L = QPLIVPL ( 3 . 3 ) HPL = FN(CPL) ( 3 . 1 ) dVPL/dt = -dVrfdt ( 3 . 1 6 ) dQpLldt = -dQijdt ( 3 . 1 7 ) Tissue Circulation dVildt = J F - JL dQj/dt = QF + QD - QL JF = KF.{PPL,C - P i - <T{HPL ~ H , ) ) QF = JF-(1 - V).{[CPL + CI>AV)M Plasma Leak model ( 3 . 1 8 ) ( 3 . 1 9 ) ( 3 . 1 3 ) ( 3 . 1 4 ) PPL,C =Fn(VPL) ( 3 . 2 0 ) Tissue Circulation d-Vildi = J F + JPLL - J R - J L ( 3 . 2 8 ) dQildt = QPLL + QD - QL ( 3 . 2 9 ) JF = KF.(PPL,A - P i - ( H P L - 11,)) ( 3 . 2 1 ) JR = KR.(Pj - PPLy - (11/ - ILPL)) ( 3 . 2 2 ) JPLL — KPLL(PPL,V — Pi) ( 3 . 2 3 ) QPLL = JPLL-CPL ( 3 . 2 5 ) P P L , A =Fn(VPL) ( 3 . 3 0 ) i'pL.v = Fn(VPL) ( 3 . 3 1 ) Table 3 . 2 : Summary of the Starling Model and the Plamsa Leak Model equa-tions. Chapter 3. MODEL FORMULATION 54 3.5 NORMAL STEADY-STATE CONDITIONS In this section we will use existing data to establish reasonable values for the nor-mal steady-state conditions of an average human referred to as "reference man". These normal steady-state conditions are selected on the basis of the following assumptions: — "reference man" is male, 170 cm in height and weighs 70 kg. — "reference man" is supine; hence location-dependent values such as colloid osmotic and hydrostatic pressures, are taken at heart level on the thorax, if possible. — orthostatic (i.e., standing in upright position) effects are neglected. 3.5.1 Body Fluid Distribution Interstitial fluid volume can only be measured indirectly by the use of tracers, the ideal tracer being water molecules confined to the extracellular space. This ideal tracer is not available so several other small molecules such as sucrose, chloride ion, and Cr51-EDTA have been used in its place. Edelman and Leibman[20] have reported values of 45 m£/kg and 120 m£/kg as average fluid contents of the plasma and interstitial compartments, respectively. Considering "reference man" these values yield a plasma volume of 3.2 I and an interstitial volume of 8.4 I. Dickerson and Widdowson[17] have measured the extracellular water content of skeletal muscle, ECWSM, to be 183 grams per kg of wet tissue. The blood content of skeletal muscle, BCSMI is 30 m /^kg[47] and a typical value of cell volume fraction in blood, CF, is 0.46. Plasma density, pp^, is equal to 1.03 g/ml. Therefore the Chapter 3. MODEL FORMULATION 55 interstitial water/fluid content per kg of wet skeletal muscle, FCI^SM > is given by the following relationship: FCr ,SM — ECWSM — (1 — CF).BCSM-PPL (3.32) =166 g/kg An average human male posesses 2 8 kg of skeletal muscle tissue, WSM[73]. Hence, the total interstitial fluid volume associated with skeletal muscle, VI,SM,NORM can be calculated as follows. Assuming plasma and interstitial fluids have equal densi-ties, VI,SM,NORM = WSM-FCI,SM/PPL (3.33) = 4.51* The epidermis, dermis, and hypodermis which make up the skin are also important depositories of interstitial fluid. The average weights of the above tissues for "refer-ence man" are shown in Table 3.3. The average water content of skin,' WCs, (t.e., TISSUE WEIGHT epidermis 100g dermis 2500g hypodermis 7500g Table 3.3: Composition of "Skin" in "reference man" [73]. dermis and epidermis) is 61% on a weight basis[73]. Assuming that hypodermis may be treated as adipose tissue, it will have a water content,WCJTYO, °f approx-imately 15%[73]. Because of the high fat content of adipose cells, the assumption is made that all the water contained in the hypodermis is extracellular. Chapter 3. MODEL FORMULATION 56 The distribution of water in human skin has been measured as 86.8% extracellular, ECWFs, and 13.2% intracellular[67]. Due to a lack of alternative information, and the highly vascular nature of skin, the blood content of dermis and epidermis will be assumed equal to that of skeletal muscle. The blood content of hypodermis will be neglected. Epidermis, dermis, and hypodermis will be collectively referred to as "total skin". The interstitial water content of "total skin" is then calculated as: V^SKIN.NORM. = [{WEP + WDE).(WCS.ECWFs - (1 - CF).BCSM.PPL) +WHYO-WCHYO]/PPL = 2.39* (3.34) where WEp, WDE, and WHYO represent the weights of the epidermis, dermis, and hypodermis, respectively. Hence, the total body interstitial fluid distribution for "reference man" is as shown in Table 3.4. The fluid volume of the other tissues (primarily the internal organs) is found by difference. TISSUE INTERSTITIAL FLUID VOLUME Total Body 8.40 £ Skeletal Muscle 4.51 £ "Total Skin" 2.39 £ Other Tissues 1.50 £ Table 3.4: Interstitial fluid distribution. The above estimates compare well with others found in the literature. Wiederhielm[68] estimates the volume of interstitial fluid residing in the combined skin and mus-cle compartments to be 7.1 £; this is in close agreement with the value of 6.9 £ calculated here. However, the relative volumes of fluid located in skin and mus-cle estimated above are almost opposite to those presented by Wiederhielm. The Chapter 3. MODEL FORMULATION 57 discrepancy in the Vj estimates for skin appears due to differing treatments of the hypodermis in Wiederhiehn's work and here; that is, Wiederhielm treats all "skin" -epidermis, dermis, and hypodermis - as extracellular fluid rich, whereas we assume hypodermis to be similar in composition to adipose tissue - which is extracellular fluid poor. Also the difference in our estimates of skeletal muscle interstitial fluid volume is mainly due to the fact that Wiederhielms' source reports the extracellu-lar component of this tissue to be 12% of the total tissue weight[68] which is lower than the value used here. Our estimate of total interstitial fluid volume also compares well with that measured by Fauchald[25] in normal controls which is reported as 13.0 ± 2.0% of bodyweight. Considering "reference man" this yields a Vj of 8.8 ± 1.3 t. 3.5.2 Interstitial Excluded Volume Due to a lack of alternative information, 25% of the interstitial space is assumed to be excluded to albumin. This is same value used by Bert et al.[5] in their model of rat microvascular exchange. This value is expected to be a reasonable estimate for human tissue as well. 3.5.3 Plasma Colloid Osmotic Pressure The colloid osmotic pressure of plasma is the only one of the Starling forces which can be measured directly by blood sample, hence confidence in the accuracy of its value is high. Studies on edema mechanisms in the nephrotic syndrome by Fadnes[23], Koomans 143], Chapter 3. MODEL FORMULATION 58 Noddeland[52], and Fauchald[25], have all included measurements of the plasma col-loid osmotic pressure in normal controls, HPL,NORM • An average value of UpLjfORM for use in this study is calculated by an unweighted averaging of the mean values reported by each worker (see Table 3.5). The weighting of unity, regardless of the sample size of each worker, attempts to compensate for any experimental bias that might exist in a given set of results. Hence, for normal steady-state conditions RPL,NORM (mmHg) Reference 26.9±4.1 [23] 24.2±0.8 [42] 28.6±3.4 [52] 23.9±2.5 [25] Table 3.5: Experimentally determined HPL,NORM values. H-PL,NORM= 25.9 ±1.5 mmHg Using the albumin concentration vs. colloid osmotic pressure relationship deter-mined in Section 3.2.1, the concentration of albumin in plasma at normal steady-state conditions, CPL,NORM , is calculated to be 39.4 g/£. For a "reference man" having a 3.2 £ plasma compartment this leads to a normal plasma albumin content of 126.1 g. 3.5.4 Interstitial Colloid Osmotic Pressure A number of different experimental techniques exists for the determination of either the interstitial protein concentration or the interstitial colloid osmotic pressure, H/. These include the use of lymph samples, wick and blister suction techniques, and Chapter 3. MODEL FORMULATION 59 tissue osmometry, all of which are invasive in nature. The use of lymph fluid samples in determining 11/ values requires the assumption that the protein concentration of the prenodal lymph is representative of the interstitial fluid; this assumption is reasonable when considering subjects under steady-state conditions[3]. The other techniques avoid this assumption by direct sampling of the interstitial fluid. All the above techniques yield similar values of 11/,NORM for controls. For a comprehensive discussion of the aforementioned techniques the reader is referred to the review by Aukland and Reed[3]. The measurements of interstitial colloid pressures' used here are from the same studies of nephrotic syndrome mentioned in Section 3.5.3. and were obtained using implantable nylon wicks in subcutaneous tissue. These values of Hi,NORM are reported in Table 3.6. A representative average value is calculated in an identical fashion to that for Hp/, mentioned above . RI,NORM (mmHg) Reference 15.8±2.3 [23] 12.0±1.2 [42] 15.8±2.7 [52] 15.2±2.0 [25] Table 3.6: Experimentally determined H / J V O K M values. Hence, for normal steady-state conditions, RI,NORM = 14.7 ±1.1 mmHg Once again, using equation(3.2), the normal interstitial protein concentration in the available space is calculated to be 22Ag/£. For "reference man", who has a ' total interstitial fluid volume of 8.4 £ of which 25% is excluded to protein, this Chapter 3. MODEL FORMULATION 60 concentration in turn yields a value of 141.1 g for the normal interstitial albumin mass. 3.5.5 Interstitial Fluid Hydrostatic Pressure In Section 3.2.4 we reviewed several reported values of interstitial fluid hydrostatic pressure for both muscle and subcutaneous tissue (see Table 3.1). It seems to be generally accepted by physiologists that the value of PJ,NORM in skin and subcu-taneous tissue is slightly subatmospheric while that of muscle tissue is about at-mospheric. Considering the error associated with the experimental values reported (once again see Table 3.1) the arbitrary selection of the value of -0.7 mmHg (for subcutaneous tissue) reported by Stranden and Myhre[61] as the normal steady-state Pj of our generic tissue compartment seems reasonable. It also seemed to be an appropriate choice because it falls in the middle of the range generally reported for skin and muscle (i.e., -2 to 1 mmHg). 3.5.6 Capillary Hydrostatic Pressure Aukland and Nikotaysen^ ] suggest" that "The appropriate capillary pressure for the Starling equation should represent a functional average for the portion of the microvascular bed taking part in fluid exchange, weighted for the the surface area and conductivity of the various vessel segments". Both the venules and the venous end of the capillaries exhibit high permeabilities. Therefore, in keeping with the above definition, it may well be that the anatomical location of Ppj,,c must be weighted towards the venules. This requirement of weighting is satisfied by the "isogravimetric" capillar}7 pressure. 61 The isogravimetric capillary pressure, PPL,C,ISO , is not a measure of the actual capillary pressure, but is equal to the total pressure opposing net filtration of fluid from the capillaries under normal steady-state conditions, i.e. PpL,C,ISO = UpL,NORM ~ Tl-I.NORM + Pl,NORM (3.35) Equation(3.35) is derived from the Starling equation (equation 3.13) assuming that the solute reflection coefficient, cr, is equal to unity. Under normal steady-state conditions, the pressure obtained using equation(3.35) will equal the actual PpL,c,NORM > minus the net transcapillary filtration pressure, PTC, (i.e., the pres-sure that gives rise to a net filtration equal to the lymph flowrate). Since under normal conditions P^c appears to be less than 1 mmHg [2], PPL,C,ISO will closely approximate the required weighted value. Considering the assumed normal steady-state conditions, PPL,CJSO = 25.9 - 14.7 + (-0.7) = 10.5mmHg Therefore, the actual value PPL,C,NORM should be approximately equal to or less than 11.5 mmHg. However, for the purpose of this study the isogravimetric value for PPL,C,NORM is not particularly useful because of its underlying assumption that a is one. Since cr is to be determined in the present investigation, then PPL,C,NORM must also be regarded as an unknown parameter. It is clear from the Starling expression (equa-tion 3.13) that, if a is less than unity, then both PPL,C,ISO a n a PPL,C,NORM W M " fall accordingly. PPL,C,NORM values as low as 5.5 mmHg have been recorded by direct measurement of hydrostatic pressures in the venules of dog muscle using the Chapter 3. MODEL FORMULATION 6 2 micropuncture technique [ 2 ] . 3.5.7 Arterial and Venous Capillary Hydrostatic Pressures The normal hydrostatic pressures in the large arteries, PAA, and large veins, Pvv, are assumed to be 1 0 0 mmHg and 2 mmHg, respectively[68]. Lack of alternative information concerning the pressure distribution between venous capillaries, PPL,V , and arterial capillaries, PPL,A > forces the use of the following relationships[68]: which have been previously applied in modelling microvascular exchange in humans [ 6 8 ] . The constants 0 . 2 3 and 0 . 0 4 represent the fractions of the total large vessel pressure drop which take place between the arterial and the venous capillaries, respectively, and the large veins. Combination of the above equations and the assumed large vein and large artery pressures yields normal arterial and venous capillary pressures of 2 4 . 5 4 mmHg and 5 . 9 2 mmHg, respectively. PPLJL — Pvv + 0 . 2 3 ( P A ^ — Pvv) ( 3 . 3 6 ) PPL.V = Pvv + 0M{PAA - Pvv) ( 3 . 3 7 ) Chapter 3. MODEL FORMULATION 3.5.8 Summary of Normal Steady-state Conditions Table 3.7 summarizes the normal steady-state values used for "reference man". Circulation Tissue V, ml 3200 8400 VEX, ml — 2100 C, g/l 39.4 16.8 CAV, g/l — 22.4 Q,s 126.1 141.1 PPL,A , mmHg 5.92 — PPL,V, mmHg 24.54 — P, mmHg — -0.7 H, mmHg 25.9 14.7 Table 3.7: Normal steady-state conditions for a 70 kg human male. Chapter 4 PARAMETER ESTIMATION PROCEDURE 4.1 INTRODUCTION In this chapter we will first review the historical development of the model of human microvascular exchange detailed in the current work. We will then proceed to briefly outline the various investigations which led to the use of the parameter estimation procedure adopted here. The current model represents a direct extension of the model of rat microvascular exchange developed by Bert et al.[5]. In their study, estimates of unknown trans-port parameter values were obtained on the basis of a statistical fit of the model's predicted results to selected experimental data. Similar, but less extensive and varied data to that used in the rat model fitting procedure was available for the human. Bert et al. subdivided the microvascular exchange system into three compartments; the circulation, the skin tissue, and the muscle tissue. Therefore, a model of similar complexity was formulated for the human and subsequently programmed for the computer. The three compartment model of the microvascular exchange system has associated with it almost twice the number of transport parameters as its two-compartment equivalent (if formulated in the the manner indicated in Chapter 3). It was pro-posed to estimate all the unknown parameters by minimizing the error between the 64 Chapter 4. PARAMETER ESTIMATION PROCEDURE 65 predicted results and experimental data using a standard constrained optimiza-tion technique that iteratively searches for the parameter values yielding the lowest residual error by means of the quadratic approximation method of Schittkowski[60] (the same method adopted by Bert et al.). However, in our case, this approach proved unsuccessful. It became apparent that the data available for the human contained insufficient information for the determination of a unique set of best-fit parameter values for a three-compartment model (i.e., numerous parameter com-binations yielded equally good fits of the data available). This result led to a closer scrutiny of the rat modelling work by Bert et al., which, in turn, resulted in two interesting observations: - In all, three sets of data were used in the rat model fitting procedure, includ-ing: venous pressure, PPL,V , vs. interstitial colloid osmotic pressure, H/,5 and n^Af , data for both skin and muscle tissues and venous pressure vs. intersti-tial fluid volume, V ^ M , data for muscle alone. The Ppr,,v vs. UIJH and PPL,V vs. V/Af data sets were used simultaneously to determine muscle compart-ment transport parameters while the PpT,,v vs. H/ts data alone were used to determine the skin compartment transport parameters. It was interesting to note the marked similarity between the colloid osmotic pressure data sets for both skin and muscle (and that the final PPL,V vs. IIjjw and PPL.V vs. Hj^ best-fit parameter simulations were almost identical). - The tissue comphance relationships used for the skin and muscle compart-ments were very similar when normalized on a volume basis. It therefore became apparent that differences in the estimated transport parameter values for each compartment were due mainly to small differences in the protein Chapter 4. PARAMETER ESTIMATION PROCEDURE 66 concentration vs. colloid osmotic pressure relationships which had been experimen-tally determined for each of the three compartments and slight differences in the tissue comphance relationships. This information was not available for use with the human model. Consequently, the next step taken was to simplify the human model to a two-compartment representation of the microvascular exchange system (i.e., the circulation and a generic tissue compartment). The formulation of two alternative representations of a two-compartment model was detailed in Chapter 3. Following this simplification, and the concomitant reduction in the number of trans-port parameters for determination, the non-linear optimization technique employed still failed in its task to find best-fit parameter values. Each new set of guessed parameter values (which are required to initialize the optimization search) yielded a different set of best-fit parameters, indicating that, the multi-dimensional "sur-face" of the optimization function contained many shallow depressions. Various non-linear optimization techniques were employed in an attempt to locate a global residual error minimum. However, none was successful and, furthermore all of the alternative best-fit parameter values determined resulted in almost identical values of residual error. In an attempt to locate the global minimum best-fit parameters and to aid under-standing of the interaction between parameter estimates and the data used in their determination, the graphical minimization technique described below was proposed and used by the author. Chapter 4. PARAMETER ESTIMATION PROCEDURE 67 4.2 PARAMETER ESTIMATION PROCEDURE 4.2.1 Parameters to be Determined The development of the constitutive mathematical relationships defining both the Starling Model and the Plasma Leak Model was completed in Chapter 3. A sum-mary of these relationships is presented in Table 3.2. However, in both models, various parameters remain unknown. Estimates of each unknown must be estab-lished before model simulations can be performed. The unknown parameters in each model are listed in Table 4.1. It should be noted that, when considering the SM, the entire parameter estimation procedure is repeated for each of the alternative tissue comphance relationships generated in Section 3.2.4. This is, in effect, a partial sensitivity analysis of the influence of this factor on the results of the parameter estimation procedure. How-ever, the tissue comphance relationship is not considered to be a "fitting parame-ter". In the PLM parameter estimation procedure, only a single tissue comphance is considered, but the values of the normal venous and arterial capillary hydrostatic pressures are also varied in a similar partial sensitivity analysis. Further details regarding the treatment of these parameters will be given later this chapter. Estimates of the parameters are obtained by deterrnining which values achieve the best statistical fit between model predictions and observed experimental results. Unfortunately, little suitable data is available for use in the parameter estimation procedure. The most complete set of suitable experimental data concerns the vari-ation of interstitial colloid osmotic pressure, H/, and interstitial fluid volume, Vj, in patients with chronic nephrotic syndrome (see Section 2.5). For this disease both interstitial colloid osmotic pressure and interstitial fluid volume have been mapped Chapter 4. PARAMETER ESTIMATION PROCEDURE 68 The Starling Model 1 R~F - the fluid filtration coefficient 2 PS - the diffusion coefficient 3 JL,NORM ~ the basal lymph flowrate 4 LS - the lymph flow sensitivity 5 cr - the solute reflection coefficient 6 PpL,c,NORM - the normal hydrostatic capillary pressure 7 PCOMPPL,C ~ the circulatory compliance The Plasma Leak Model 1 Kp - the fluid filtration coefficient 2 PS - the diffusion coefficient 3 JL,NORM ~ the basal lymph flowrate 4 IrS - the lymph flow sensitivity 5 KR - the fluid reabsorption coefficient 6 KPLL ~ the plasma leak coefficient 7 PCOMPPL^A - the arterial compliance 8 PCOMPPL,V ~ the venous compliance Table 4.1: Model parameters to be determined by statistical fitting. against variations in plasma colloid osmotic pressure. The available data, in a nor-malized form, are summarized in Figures 4.1 and 4.2 (see Section 4.2.2). Each data point in the figures represents a patient with a certain degree of hypoalbuminemia (i.e., a plasma protein content less than that of normal). Since each patient is a chronic disease sufferer, it is assumed each individual is in a steady-state condition at the time of measurement. It is generally agreed that plasma volume does not vary significantly from its nor-mal value, VPL,NORM > m patients with chronic nephrotic syndrome[22, 59, 25]. Chapter 4. PARAMETER ESTIMATION PROCEDURE 69 Therefore, since both the SM and the PLM assume variations in capillary hydro-static pressure to be solely a function of plasma volume (see Sections 3.3.2 and 3.4.2, respectively), it follows that, when considering steady-state nephrotics, for the Starling Model PPL,C = PPL,C,NORM and for the Plasma Leak Model PpL,A = PpL,A,NORM PpL,V = PpL,V,NORM Hence, it is not necessary or even possible to determine either PCOMPPL,C or PCOMPPL>A and PCOMPPLRA using the data presented in Figures 4.1 and 4.2. Therefore values of circulatory, venous, and arterial comphance must be determined by alternative methods such as those discussed in Section 5.3.5.9. Thus, the use of nephrotic syndrome data in the fitting procedure reduces the number of unknown parameters to be determined for each model to six apiece. In addition, the transport parameters characterizing the behaviour of the microvas-cular exchange system are assumed to remain unchanged as an individual moves from a normal into a diseased condition (i.e., the physical properties of the sys-tem, such as the capillary wall permeability, are not altered by the progress of the nephrotic syndrome. It should be noted that this assumption is somewhat justified by the fact that, in numerous articles on nephrotics, little or no mention is made of potential changes in transport parameters). Therefore, relationships known to hold at normal steady-state conditions (see Section 4.2.3) as well as the albumin clearance relationship (see Section 4.2.4) can be used to further reduce the num-ber of unknown parameters. It should be remembered that the PLM is a more Chapter 4. PARAMETER ESTIMATION PROCEDURE 70 complex description of transcapillary exchange than the SM. However, PPL,CJ?ORM has been treated as a unknown parameter to be determined in the SM whereas in the PLM values of PPL,A,NORM and PPL,V,NORM a r e known (see Section 3.5.9). Nevertheless, as was mentioned earlier, the values of PPL^A^ORM and PPL,V,NORM are varied in a partial sensitivity analysis much in the same way as is the tissue compliance relationship in the SM procedure. 4.2.2 Data for the Statistical Fitting Procedure Koomans [43], Fadnes[22], Noddeland[52], and Fauchald[25] have made independent studies of the balance of transcapillary forces existing in the chronic nephrotic human. Each worker has presented data relating interstitial colloid osmotic pressure to plasma colloid osmotic pressure in patients suffering to varying extents from the disease. Each study has reported average values of both Hr and TLPL for normal control groups. In an attempt to minimize any experimental bias that might exist in a specific data set, all four are normalized arithmetically with respect to the average values of the normal controls of both Hr and Hpr, (see Sections 3.5.3 and 3.5.4). The following relationship clarifies the normalization procedure: Rx,l,NORMALIZED = Hx.i + {TLAVE,NORM ~ Ul,NORM ) (4-1) where the subscript "1" indicates the data set being normalized, the subscript "X" indicates the specific data point undergoing normalization, and HAvEjroRM represents the average colloid osmotic pressure of all the data sets combined. The values of UAVE,NORM (f° r either the plasma or the interstitial compartment) are calculated according to the following relationship: 1 4 U-AVE,NORM = ~ Ri.NORM 4 i=l Chapter 4. PARAMETER ESTIMATION PROCEDURE 71 30.0 25.0 20.0 15.0 10.0 0.0 5.0 i i r 100 15.0 20.0 25.0 30.0 lJpL (mmHg) Figure 4.1: Normalized Vj vs. TLpi data. Chapter 4. PARAMETER ESTIMATION PROCEDURE 72 20 0 16.0 W 12.0 -j £ ' 8.0 4.0 H 0.0 • o . <r • • 1 1 1 1 1 0.0 5.0 10.0 15.0 20.0 25.0 30.0 LTpx, (mmHg) Figure 4.2: Normalized ITj vs. UPL data. Chapter 4. PARAMETER ESTIMATION PROCEDURE 73 and correspond to the TLPL,NORM and HI,NORM values listed in Table 3.7. Fadnes et a/. [22] also present measurements of the variation of interstitial fluid volume with that of the plasma colloid osmotic pressure in chronic nephrotics. This data is also normalized before use within the fitting procedure, the fluid volume being proportionally normalized with respect to the interstitial fluid volume of "reference man" and the HPL values arithmetically normalized as above. It is these normalized data sets which are presented in Figures 4.1 and 4.2. Figures 4.1 and 4.2 illustrate the changes generally observed to occur as a healthy individual (i.e., H.PL,NORM = 25.9 mmHg, 1T.I,NORM = 14.7 mmHg, and ViTNORM = 8.4 £) moves into a worsening chronic nephrotic condition. Increased urinary excretion of plasma proteins reduces the colloid osmotic pressure of the plasma. Proteins are subsequently "washed-out" of the interstitial spaces by the transcap-illary fluid flux and the value of Hj is reduced, thereby, once again, balancing the Starling forces (see equation 3.13). The interstitial fluid volume is regulated in this fashion down to HPL of approximately 14 mmHg; the tissue becomes edematous below this value. 4.2.3 Steady-state Balances at Normal Conditions Under normal conditions, fluid and protein exchange is assumed to exist in a state of dynamic equilibrium. Therefore, considering the Starling Model, steady-state mass balances of fluid and protein about the interstitial compartment yield, respectively, the following relationships: JF,NORM — JL,NORM - 0 (4.2) Chapter 4. PARAMETER ESTIMATION PROCEDURE 74 and QF,NORM + QD,NORM — QL,NORM = 0 (4.3) Similarly, for the Plasma Leak Model: JF,NORM + JpLL,NORM ~ JR^ORM ' JL,NORM = 0 (4.4) and QPLL,NORM + QD,NORM ~ QL,NORM = 0 (4.5) 4.2.4 Albumin Clearance Relationship Albumin turnover rates in human subcutaneous tissue have been determined by a number of workers, both under normal and certain pathological conditions [38, 45]. Turnover rates reported by Hollender et al. [38], were determined by local disap-pearance rates of interstitially injected 7131-radiolabelled albumin. The clearance rates reported for normal controls, are assumed representative of steady-state val-ues. The average disappearance half-time, T1*"1*, is reported to be 33.4 hours, which is equal to an albumin turnover rate, Albxo, of 2.05% per hour. The disappearance half-time is assumed to be related to the albumin turnover rate as follows: (1 -AlbTof"' =0.5 Laggard [45] has also reported human subcutaneous tissue clearance ranging be-tween 29 and 40 hours, values which are in good agreement with that obtained by Hollender et al.. When considering either the Starling Model or the Plasma Leak Model, a tracer balance around the interstitial compartment yields PSC*I,AV + JL,NORM-CJ = Albxo-VI,NORM-C'i (4.6) Chapter 4. PARAMETER ESTIMATION PROCEDURE 75 where the asterisk indicates tracer properties. The first term of equation(4.6) repre-sentsthe diffusive flux of the tracer protein from the interstitial space to the plasma. The second term quantifies the lymphatic transport of the tracer also leaving the tissue, and the final term is equal to the turnover of the tracer (i.e., the amount leaving the interstitial compartment per unit time). Derivation of equation(4.6) requires the assumption that CPL remains negligible for the duration of the experi-ment. Note that, since Cj — Q'/VITNORM and C}AV = Q}/(VIINORM -Vi$x) then Cj^y = Cj.ViflORM/(VI,NORM — VI^EX )• Thus, the clearance relationship can be written in the following alternate form: Vj PS.(— ) +JL,NORM = A l b T 0 . V h N 0 R M (4.7) * I,NORM — VI,EX Currently no parallel measurements on human muscle tissue are known to the author. It is therefore assumed that the albumin turnover rate for subcutis is also representative of muscle tissue. The validity of this assumption is supported by the work of Reed and Wiig[55, 69], who found identical turnover rates in both the skin and muscle tissue of rats. 4.2.5 The Starling Model In total, seven parameters must be determined to complete the definition of the Starling Model (see Table 4.1). It is proposed that four be determined by statisti-cal fitting and the three remaining then calculated using the normal steady-state mass balances, equations(4.2) and (4.3), and the albumin clearance relationship, equation(4.7). Chapter 4. PARAMETER ESTIMATION PROCEDURE 76 4.2.5.1 Parameters to be Empirically Determined It has previously been established that circulatory compliance, PCOMPPL,C , can-not be determined from the available steady-state nephrotic data. Hence, at this stage the problem is reduced to the determination of six unknowns, three by sta-tistical fitting. For convenience, the following parameters were chosen for empirical determination: — Rd/i = QD,NORM IQL,NORM the ratio of the diffusive protein transfer rate across the capillary wall to that by convection into the lymphatics, both at normal conditions), — LS (the lymph flow sensitivity), and — PPL,C,NORM (the capillary hydrostatic pressure at normal steady-state condi-tions). PPL,C,NORM and LS are chosen for statistical determination because both are in-dependent of the normal steady-state mass balances and the albumin clearance relationship. The dimensionless group Rd/i is selected because limits existing on its value are known (i.e., Rd/i equals zero when all protein transport across the capillary wall is by convection and it is equal to unity when protein transport is solely by diffusion). Once Rj/i, LS, and PPL,C,NORM have been specified, the four remaining unknown parameters, Kp, PS, JL,NORM , and a, can be calculated using equations(4.2), (4.3), and (4.7) as follows. By definition Chapter 4. PARAMETER ESTIMATION PROCEDURE 77 which on rearrangement yields P C _ JL,NORM CJ^ORM Rd/i Q^ — n r> v 4- y) (sPL,NORM — ^IrAV.NORM Thus, once JLJIORM has been determined, equation(4.9) can be used to calculate PS. An expression for JL.NORM c a u be obtained by substituting equation(4.9) into the clearance relationship (equation 4.7). This gives r Albro.VijiQRM  JLjiORM - v-—— : c , „ „ „ „ jr,„ ( 4 - l u i ^ VI,lfORM ~V1^X '' CpL.NORM ~CI,AV.NOKM Explicit expressions for R~p and <r may then be obtained from the steady-state fluid and protein balances. The steady-state protein balance can be written in full as ,.. \ ,CPL,NORM + CitAV,NORM \ T {i—(T)\ )-JFJiORM +PS.{CPL ,NORM — Cj^NORM ) ~ JL,NORM -CijifORM = 0 (4.11) Noting that JFJIORM equals JL,NORM from the steady-state fluid balance and sub-stituting equations(4.9) and (4.10) into equation(4.11) yields after simplification and rearrangement cr = 1 - 1CI<NORM.{\ - Rd/i) ^ CpL,NORM + Ci^NORM Finally, Kp can be obtained from the filtration rate expression, equation(3.13), as Chapter 4. PARAMETER ESTIMATION PROCEDURE 78 JL,NORM ( A 10N K f = . (4.13) rpL,C,NORM ~ ^I,NORM ~ ^\^PL,NORM ~ "-I,NORM ) It is proposed to use both data sets, Vj vs. IIpx (Figure 4.1) and 11/ vs. TLPL (Fig-ure 4.2), first individually and then in combination, in the parameter estimation procedure. First, arbitrary values (which he within physiological limits) are as-signed to each of the parameters, Rj/i, LS, and PPL,C,NORM , selected for empirical determination. Then JL,NORM , PS, cr, and KF are calculated from equations(4.10), (4.9), (4.12), and (4.13), respectively, thereby completing the model description. As previously stated, the data available for comparative fitting represents a series of nephrotic steady-state conditions; it is therefore assumed that the plasma volume is fixed and equal to the normal steady-state value of 3.2 £[25]. It is also assumed that all the properties of the plasma are fixed at their normal levels, or in the case of the protein concentration and colloid osmotic pressure, at their perturbed values (see below). Note, this treatment therefore assumes the plasma behaves as an infinite source/sink of both fluid and proteins and the interstitium is allowed to change its fluid and albumin contents to attain each "steady-state" as shown in Figures 4.1 and 4.2. The interstitial compartment mass balances, equations(3.16) and (3.17), are set equal to zero (i.e., for steady-state conditions) and then solved as the plasma colloid osmotic pressure is progressively reduced by small increments from an initial value of 28 mmHg down to 6 mmHg (thus encompassing the full range of the available experimental data). The simulated Vj vs. Hp/, relationship is now available in the form of discrete points. The criterion used to gauge the "goodness of fit" of the simulation is the sum of squares of the shortest distances between model predictions Chapter 4. PARAMETER ESTIMATION PROCEDURE 79 and experimental data points[10]. This criterion is chosen for the following reasons: — Because of the shape of the interstitial fluid volume curve (see Figure 4.1). If the more standard criterion involving vertical distances is employed those data points having low values of plasma colloid osmotic pressues receive inordinate weight due to the large value of dVj/dTLpi, within that region. The other standard least-squares measure involving horizontal distances introduces sim-ilar difficulties when evaluating data at large values of Llpr, (due to the small value of dVi/dUpi). These three alternative fitting criteria as well as their potential drawbacks are illustrated in Figure 4.3. — Because it is recognized that both the independent and dependent variables have associated errors. In such cases, neither the vertical nor horizontal cri-teria have any statistical import and Brace[10] recommends that the shortest distance criterion be used. 4.2.5.2 Parameter "Surface" Generation Using the above technique, "goodness of fit" or sum of squares values may be calcu-lated for any specified set of parameter values. From this point onwards "goodness of fit" values will be referred to as objective function values, OBJ. Values gener-ated by comparison with Vj vs. UPL and 11/ vs. UPL data sets will be identified by appropriate subscripts (i.e., OBJifv and OBJJI^ respectively). The combination of parameter values yielding the lowest objective function value must now be determined. This problem can be considered as a four-dimensional minimization, the unknown parameters Rj/i, LS, and PPL,C,NORM representing three of the dimensions and the fourth being the value of the objective function. Chapter 4. PARAMETER ESTIMATION PROCEDURE 80 Figure 4.3: A hypothetical relationship illustrating the disadvantages of using either a vertical distance or horizontal distance least-squares fitting criterion. Chapter 4, PARAMETER ESTIMATION PROCEDURE 81 Interpretation of a four dimensional system is difficult; the problem is therefore simplified by reducing it to a series of three-dimensional minimizations. This is accomplished by mapping the value of the objective function across the full range of possible values of two of the three unknown parameters whilst holding the value of the third parameter constant. Because it has fairly well established physiological limits, PpL,c,NORM is selected as the parameter to be held constant. This mapping procedure is repeated for several different values of PPL,C,NORM , thereby represent-ing a section of four-dimensional "space" by a series of three-dimensional "surfaces". This type of approach allows the minimization results to be presented graphically and evaluated visually, providing insight into the nature of the relationship between the "best-fit" parameter estimates and the data used in their determination. 4.2.5.3 Parameter Search Regions For reasons of expediency and efficiency, bounds are calculated that encompass the region within which physiologically feasible parameter values must he, thus reducing the parameter search region. Both the upper and lower bounds on the value of PPL,C,NORM a r e established by physiological considerations. An estimate of the upper hmit is obtained by assuming that a transcapillary pressure gradient of 1 mmHg is maintained under normal steady-state conditions (i.e., the driving force for normal fluid flow across the capillary wall)[2]. Therefore, considering equation(3.13) we may write: PPL,C,NORM — Pi,NORM — A-{J^-PLJNORM — TLI,NORM) ' 1 mmHg Substituting the normal steady-state values established in Chapter 3 into the above yields: PPL,C,N0RM = 0.3 +(7.(11.2) Chapter 4. PARAMETER ESTIMATION PROCEDURE 82 By definition the maximum value of the solute reflection coefficient is unity, hence the following inequality holds: PPL,C,NORM < 11-5 mmHg The lower limit of normal capillary pressure is based on experimental measurements of venous capillary hydrostatic pressure (^ 6mmHg[2]). By definition PPL,C,NORM must exceed the venous capillary pressure (see Section 3.5.8). The value of LS may range between zero and infinity. However, since each region of "surface" mapped within the parameter estimation procedure has an associated computational cost, the range of LS values to be investigated requires an upper bound. This bound was estabhshed by an initial systematic search for the range within which the minimum OBJ{fv and O B J ^ values are generally located (i.e., 0< LS <200 m£/(mmHg.hr)). Rd/i is restricted by its very definition to a value between zero and one, but the value of PPL,C,NORM with which it is associated may restrict it further. This is true because the denominator in equation(4.13) must remain positive, due to the fact that neither the normal lymph flowrate, JL,NORM , nor the filtration coefficient, Kp, may be negative. Therefore the following inequality holds: PpL,C,NORM — Pl,NORM ~ c(IIpL,NORM ~ Ul,NORAf) > 0 Substituting equation(4.12) into the above and rearranging yields D , CpL,NORM +CiTAV,NORM N ,PpL,C,NORM ~ Pl,NORM u Kd/1 < 1 jr* -(1 - (-^  ^ j) /,NORM ^PL,NORM ~ ^I,NORM Hence the upper limit of Rj/i is a function of the capillary hydrostatic pressure. aptei 4. PARAMETER ESTIMATION PROCEDURE 83 This relationship is presented in Table 4.2 for selected values of PPL,C,NORM in the range investigated. P Ttmax PL,CJfORM K D / L (mmHg) ( - ) 11 1.0000 10 0.9178 9 0.7534 8 0.5890 7 0.4246 Table 4.2: Maximum value of Rd/i as a function of PPL,C,NORM • We have just established that as the value of Rd/i approaches R™/}* the denominator in equation(4.13) tends to zero and at the same time the value of Kp approaches infinity. Hence, each "surface" is generated over an R^/i range with an upper hmit a fraction below the maximum possible value. An extremely high Kp value implies either an excessive fluid filtration rate or a very small transcapillary pressure gradient; both possibilties seem unlikely. 4.2.5.4 Objective "Surface" Grid Size All objective function "surfaces" generated within the current work consist of a 20 x 20 grid of Rd/i, LS, and OBJ values. The number of grid points evaluated attempts to balance computational expense and an acceptable level of surface definition. 4.2.5.5 Parameter Surface Normalization Examples of the results obtained by this graphical minimization technique are pre-sented in Figures 4.4 and 4.5, which have been generated using the Vi vs. HPL Chapter 4. PARAMETER ESTIMATION PROCEDURE 84 and LT; vs. 1T.PL data sets, respectively. Each figure represents the same region of "parameter space", the only difference between them being the data set used to generate the objective function values. The regions containing the minimum objective functions are well defined on both "surfaces". However, the locations of these regions do not coincide. The "surfaces" must therefore be combined in order to determine which parameter values achieve the best statistical fit when considering both available data sets simultaneously. The "surface" combination procedure used here involves first normalizing both the OBJifv and OBJ^ values for the specified section of "parameter space" (i.e., corre-sponding to the same value of PPL,C,NORM a n a over the same LS and Rd/i ranges). This normalization is accomplished by application of the following equation: OBJNORMALIZED = ^oiL'OBJ^ ( 4 1 4 ) where the subscripts "HI" and "LOW" refer to the highest and lowest objective function values determined on all surfaces generated, respectively. This is done to ensure that comparisons can be made between all surfaces generated using a specific data set. The best fit is therefore denoted by an OB J NORMALIZED value of zero and the worst fit by a value of a half. The normalized objective function values, OBJ{fv and OBJ\j{, associated with each pair of Rj/i and LS values (at a given value of PPL,C,NORM ) a re then summed. What results is a hybrid "surface" (referred to as the OBJcomf, surface) on which the value nearest unity represents the worst fit of both data sets and that closest to zero the best fit. The above procedure ensures numerically equal contributions by both data sets in the selection of parameters. The hybrid "surface" resulting from the combination of the OBJifv and 0i?Jn, "surfaces" presented in Figures 4.4 and Chapter 4. PARAMETER ESTIMATION PROCEDURE 85 -1 1 1 — 1 1 0.000 0.183 0366 0-549 0.732 0.915 R d/1 Figure 4.4: Objective function "surface", 0BJifv, at a PPL,C,NORM °f 10 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Chapter 4. PARAMETER ESTIMATION PROCEDURE 86 Figure 4.5: Objective function "surface", OBJn{, at a PPL,C,NORM of 10 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Chapter 4. PARAMETER ESTIMATION PROCEDURE 87 Figure 4.6: Objective function "surface", OBJCOTnb, at a PPL,C,NORM of 10 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Chapter 4. PARAMETER ESTIMATION PROCEDURE 88 4.5 is shown in Figure 4.6. This iniiumization procedure yields a series of "best-fit" parameter sets, one set of Rd/i and LS values associated with each normal capil-lary pressure investigated. The graphical mirurnization technique outlined above is summarized by the flow diagram depicted in Figure 4.7. It can be seen in Figure 4.6 that a region exists within which all parameters yield equally low values of OBJeomt. To obtain the location of the absolute minimum, Schittkowski's [60] non-linear optimization algorithm was used with initial starting parameter values taken from within the "best-fit" region indicated on the contour plot. Several different sets of initial values were used in an attempt to locate the global minimum; this was deemed necessary due to the undulating nature of the OBJcomj, surfaces. In general, these initial values yielded the same set of final "best-fit" parameter values. 4.2.6 The Plasma Leak Model To determine the "best-fit" parameters for the Plasma Leak Model, an approach is adopted analogous to that used with the Starling Model. Eight parameters must be estimated to complete the full description of the Plasma Leak Model. Once again the steady-state nephrotic data are used in the fitting procedure, therefore neither PCOMPPL^A nor PCOMPPL,A can be determined at this stage. Combining this fact with equations(4.3), (4.4), and (4.7) once again reduces the number of parameters for statistical determination to three. Thus, as with the Starling Model, the problem becomes a four-dimensional mini-mization, the unknown parameters representing three dimensions and the objective function value the fourth. Chapter 4. PARAMETER ESTIMATION PROCEDURE 89 Specify PPL,C,NORM and tissue compliance^ relationship I Generate OBJijv and OBJnt "surfaces" Normalize "surfaces" and combine I Locate "surface" minimum Figure 4.7: A flow diagram of the "minimization procedure" used to determine the "best-fit" parameters for the Starling Model. Chapter 4. PARAMETER ESTIMATION PROCEDURE 90 Since it is generally more convenient to specify ratios of parameters rather than their absolute values the following parameters were selected for empirical determination: — Rd^ = QrjjfORM IQL,NORM the ratio of the diffusive protein transfer rate across the capillary wall to that by convection into the lymphatics, both at normal conditions), — LS (the lymph flow sensitivity), and ~ RF/R — KF/KR (*e-, the ratio of the filtration coefficient to the reabsorption coefficient). Once Rd/i, LS, and RF/R have been specified, the remaining transport coefficients, Kp, PS, JL,NORM, KR, and KPLL can be calculated as follows. Since PS.(CpL,NORM — CitAV,NORM ., .-\ Kd/i = j ^ (4.15) JL,NORM I,NORM then, upon rearrangement, N R , JL,NORM-CljfORM -Rd/l /A i c \ P S = n n (4"16) ^PL,NORM — ^IJLVJiORM Thus, once JLJIORM has been determined, equation(4.16) can be used to calculate PS. An expression for JL,NORM can be obtained by substituting equation(4.4) into the albumin clearance relationship (equation 4.7). This gives T AlbroVufoRM . JL.NORM = V l Cj.NORM .Rd/l ( 4 " 1 7 J 1 ' V Vl.NORM - ^I.EX )• IsPL.NOKM —<••1.AV.NORM Explicit expressions for Kpn,, Kp, and Kp may then be obtained from the steady-state fluid and protein balances. The steady-state protein balance (equation 4.5) Chapter 4. PARAMETER ESTIMATION PROCEDURE 91 may be rearranged into the following form: JpLL,NORM = (JL,NORM ClJfORM ~ P'S.[C'PL,NORM ~ CI^V^/ORM])/CPL,NORM (4.18) Once JPLL,NORM is available, KPLL can be calculated using U - VPLL,NORM 1 Q x KPLL - p p (4.19) rpLXJIORM — "I,NORM The steady-state fluid balance (equation 4.4) yields upon rearrangement jr JL,NORM -JPLL.NORM  R RF/R-(PPL,A,NORM ~pI,NORM ~UPL.NORM + n 7 J t f O R M )~(PI,NORM -pPL.V,NOBM-ul.NORM +nPL.NOKM ) (4.20) The filtration coefficient can then be calculated directly using KF = K R . R F , R (4.21) 4.2.6.1 Parameter Search Regions In an attempt to establish what effect a change in PPLAJ^ORM and/or PPL,V,NORM might have on the results of the parameter estimation procedure, several alternative values were investigated in a partial sensitivity analysis (as was the overhydration gradient of the tissue compliance relationship in the SM analysis). Note that, since negative values of KR are not physiologically possible, the denominator of equation Chapter 4. PARAMETER ESTIMATION PROCEDURE 92 (4.20) must remain positive. Hence, if PPL,A,NORM = 24.54 mmHg and PPL,V,NORM = 5.92 mmHg (see Section 3.5.9), we may write D ^ PlJfORM — PpL,V,NORM ~ Hj./yofiAf + UpL,NORM , . n ~ \ RF/R > ~B E T n ^ > rpL,AJiORM ~ •* IJfORM " LlpL,NORM + ^I,NORM The values of PPL,A,NORM and PPL,V,NORM investigated and the corresponding min-imum value of RF/R with which they are associated are listed in Table 4.3. PpL,A,NORM (mmHg) PpL,V,NORM (mmHg) X-F/R (-) 22.54 3.92 0.5465 24.54 5.92 0.3262 26.54 7.92 0.1609 Table 4.3: PPL,A,NORM a r*d PPL,V,NORM values investigated and associated mini-mum values of RF/R A lower limit on the value of RF/R has been established by considering the phys-iological limitations of the system. However, no upper limit exists. Therefore, several arbitrary values of RF/R are investigated (each corresponding to a spec-ified percentage increase above the minimum). The values investigated for each PpL,A,NORM > PPL,V,NORM pair considered are 1, 10, 30, 50, and 100 percent above the minimum values calculated using equation(4.22). The reason for considering these RF/R values is discussed in Section 5.4.1. In contrast to the Starling Model the value of R^/i in the Plasma Leak Model is unconstrained, except by its physical definition (i.e., Rd/i must he between zero and one). Hence, the full range of possible Rj/i values is investigated. The choice of the LS range investigated and the implications of that choice are identical to those described for the Starling Model (see Section 4.2.5.3). Chapter 4. PARAMETER ESTIMATION PROCEDURE 93 4.2.6.2 OBJ "Surface" Generation Procedure The graphical minimization procedure presented in Figure 4.7 is now repeated incorporating the following modifications. — RF/R replaces PPL,C,NORM 3 5 the parameter held constant during each three-dimensional mapping. — Only a single tissue comphance relationship (#1) is considered throughout the procedure. — Since confidence in the accuracy of the normal arterial and venous capillary pressures is low, several alternative values of these two constants are investi-gated. It should he noted that the same OBJ HIGH a n d OBJ LOW values determined during the SM investigation were also used in the normalization of the PLM surfaces. This ensures that the normalized OBJ values calculated are comparable between models. 4.3 NUMERICAL TECHNIQUES AND COMPUTER PROGRAMS 4.3.1 Transient Analysis Each model consists of a set of first-order differential equations representing the mass balances of fluid and protein in each compartment, along with their associated auxiliary and constitutive relationships. The equations must be integrated with respect to time, subject to imposed initial conditions. Analytical solution of these equations is net possible due to the non-linearity of the tissue comphance relationship. Hence the equations are integrated numerically Chapter 4. PARAMETER ESTIMATION PROCEDURE 94 using a Runge-Kutta-Fehlberg algorithm incorporating error control[28]. As the integration proceeds, the local time step size is adjusted to ensure that errors in solution values do not exceed specified limits. Since the inherent accuracy of the Runge-Kutta-Fehlberg method is high, equal to that of a fourth-order Taylor series expansion, it allows the use of fairly generous time steps. For the transient solutions presented in this study the maximum allowable error was set to 0.1 ml and 0.1 g for fluid and protein contents respectively. 4.3.2 Steady-State Solutions Steady-state solutions of the differential equations may be obtained using a tran-sient analysis by extending the solution time span until no further change in the dependent variable is detected. A more efficient alternative is to set the time deriva-tive portions of the differential equations to zero, and then to solve the resulting set of simultaneous non-linear algebraic equations. Newton's iterative method[28] is employed to accomplish this latter task, with finite differences approximating the required partial differentials. The resulting sets of linearized algebraic equations are solved using Gaussian Elim-ination [28] incorporating full pivot selection. Iterations continue until the changes in all dependent variables are within a specified tolerance (0.1 units). The solutions so generated are used to check the accuracy of the transient solutions. Note that, when simulating steady-state nephrotics, as in the parameter estimation procedure, it is only necessary to solve the interstitial compartment balance equations, since both the volume and protein concentration of the plasma are specified. Chapter 4. PARAMETER ESTIMATION PROCEDURE 95 4.3.3 "Shortest Distance" Least-squares Technique The simulated Vj vs. HPL or Hj vs. HPL relationship is in the form of discrete points. First the pair of simulated points closest to experimental point in question are located and subsequently joined by a straight line. The length of the straight line passing through the experimental point and normal to the straight hne joining the simulated points is then calculated, squared and added to the sum of squares. The error associated with this type of slope approximation technique is minimized by selection of a small step size incremement (0.25 mmHg) between the independent HPL values. 4.3.4 Interpolating Data with Cubic Splines A cubic polynomial (a spline)[28] is generated for each pair of adjacent points in the data set requiring interpolation. The coefficients of each spline are determined by matching the values, first derivatives, and second derivatives with those of the two adjoining splines. 4.3.5 Computer Programs All programs are coded in FORTRAN 77 and the listings are provided in Appendix E. A brief description of each program listed is given here, however, if more infor-mation is required the listings themselves should be consulted. The following descriptions occur in the same order as the listings appear in Ap-pendix E. - S M - A This driver program generates the grids of R^/i and LS values and calculates Chapter 4. PARAMETER ESTIMATION PROCEDURE 96 corresponding values oiOBJifv and OBJ^ for the Starling Model. It requires the subroutines of program SM-B to function. - SM-B This group of routines generates OBJifv and OBJ\-yi values for the SM when values of Rj/i, LS, and PPL,C,NORM have been specified. Note that the value of PPL,C,NORM is stored in the Block Data routine. - S M - C This routine simulates the transient response of the SM to a specified per-turbation and also determines the final steady-state conditions of the system following the disturbance. - P L M - A This driver program generates the grids of R^/i and LS values and calculates corresponding values of OBJ{jv and OBJui for the Plasma Leak Model. It requires the subroutines of program PLM-B to function. - P L M - B This group of routines generates OBJ{fv and OBJr^ values for the PLM when values of Rd/i, LS, and RF/R have been specified. - P L M - C This routine simulates the transient response of the PLM to a specified per-turbation and also determines the final steady-state conditions of the system following the disturbance. Chapter 5 RESULTS AND DISCUSSION 5.1 INTRODUCTION In this chapter we will first discuss several points of interest relating to the general behaviour of the parameter estimation procedure results. We will then present, discuss, and attempt to validate the "best-fit" parameters determined using the Starling Model and then repeat the procedure for the Plasma Leak Model. Distribution of the extracellular body fluids between the circulation and the in-terstitial space is normally maintained within narrow limits. The complex system regulating this partition is not completely understood at present, although the transcapillary balance of hydrostatic and colloid osmotic pressures has long been recognised as playing an important role. Recently biomedical researchers have begun to develop mathematical models of the interstitial fluid volume regulatory system to aid both in the interpretation of experimental data and the identification of system mechanisms/parameters which are quantitatively influential in the determination of system behaviour. The current work differs from previous efforts both in model formulation and the method used to estimate model transport parameters. Certain system parameters, such as the capillary wall solute reflection coefficient, are difficult to determine by clinical experimentation. Typically previous models 97 Chapter 5. RESULTS AND DISCUSSION 98 have used transport parameters evaluated under normal steady-state conditions. In contrast, the approach adopted here is to deterrnine which parameter values yield the best agreement (quantified statistically) between model predictions and experimental data for a selected disease state (i.e., chronic hypoproteinemia). The data used in the fitting procedure represent the regulatory system's steady-state response to a series of disturbances of increasing magnitude (i.e., a progressively worsening disease state). The primary objective of the current work is the deterrriination of model parameters which not only yield the best agreement between simulation results and experimen-tal data, but also appear reasonable when considered in; a physiological context. 5.2 THE PARAMETER ESTIMATION PROCEDURE Before discussing the numerical results of the parameter estimation procedure pre-sented in Chapter 4, several points of interest relating to the characteristics of the procedure itself will be reviewed. 5.2.1 Definition of "Best-fit" Parameters Each OBJ "surface" generated possesses a region within which all Rd/i, LS combi-nations for both the SM and the PLM can be regarded as yielding a good fit of the data set in question. The threshold OBJ value separating a "good" fit from a "bad" fit could be estimated if detailed information regarding the errors associated with the fitting data was available. However, no such information has been provided. Therefore, only the parameter values corresponding to minimum OBJ values are compared, but the fact should not be overlooked that the difference between the Chapter 5. RESULTS AND DISCUSSION 99 "best-fit" O B J value and those located nearby by on the "surface" may not be statistically significant. (For the SM, minimum O B J values represent the lowest obtained from three trials using a non-linear optimization algorithm - note that each trial generally yielded the same location - and, for the PLM, they represent selected points within the "best-fit" region.) This matter is nicely illustrated by the sum-of-squares results shown in Figure 4.6 for the SM. It can be seen that a fairly wide range of Rd/i, LS combinations yield low OBJcomb values (e.g., O B J c o m b <0.06 within the rectangular region defined by 0.80< Rd/i <0.90 and 35< LS <60). 5.2.2 Irregularities in OBJifv "Surfaces" The three-dimensional "surfaces" used to represent the objective function are gen-erated from a series of discrete parameter value "grid points". Therefore, the exact nature of the surface existing between these grid points is unknown (the lines shown joining the points in both the three-dimensional plots and the contour plots axe generated by a polynomial fitting procedure). This fact is of particular inter-est when considering the'OBJijv (and consequently the hybrid OBJcomb) surfaces. Visual inspection of these plots (see Figure 4.4 for example) gives an impression of a somewhat irregular surface (i.e., undulating, containing many depressions). This formation appears to be an artifact produced by a combination of the shortest distance least-squares algorithm used to determine objective function values and the shape of the tissue comphance relationship. This conclusion is reached after consideration of the following two observations. — As the gradient of the overhydration segment of the tissue comphance relation-ship is increased, which consequently reduces the gradient of the overhydration Chapter 5. RESULTS AND DISCUSSION 100 segment of the simulated Vj vs. UPL curves (see Section 5.3.4), the magni-tudes of the irregularities are increased. This behaviour is demonstrated in Figure 5.1. — The surface can be "smoothed-out" by progressively reducing the Upr, incre-ment size used in the generation of the simulated Vj vs. 1T.PL results (see Section 4.2.5.2). The increment size ultimately used in the present study represents a compromise between computational expense and OBJifv surface distortion. The irregularities in the OBJ{fv and OBJ^ surfaces can compromise the accuracy of the fitting procedure. Furthermore, since they create a large number of local "minima" their presence means that an optimizing algorithm cannot generally be trusted to seek out the location of a true global minimum. 5.2.3 The Effect of LS.Search Region Size on "Surface" Normalization The value of LS may range between zero and infinity. However, since each region of "surface" mapped during the parameter estimation procedure has an associated computational cost, the range of LS values to be investigated requires an upper bound. This bound was established by an initial systematic search for the range within which the minimum OB •/,-/„ and OBJj^ values are generally located and was conservatively found to be LS < 200 m£/(mmHg.hr) (see Section 4.2.5.3). This semi-arbitrary choice of search region has an important implication. Figures 4.4 and 4.5 show respectively the OBJijv and OBJn, "surfaces" obtained in the case of the SM for a PPL,C,NORM of 10 mmHg and tissue compliance relationship #1. It can be seen that, for both OBJifv and OBJua the objective function value Chapter 5. RESULTS AND DISCUSSION 101 o T : 1 1 s 1— 0.00 0.15 030 0.45 0.60 R d/1 Figure 5.1: T h r e e c o n t o u r plots of 0BJifv surfaces at a PPL,C,NORM of 9 m m H g . U p p e r : T i s s u e c o m p l i a n c e #1, C e n t e r : T i s s u e c o m p l i a n c e #2, B o t t o m : T i s -sue c o m p l i a n c e # 3 Chapter 5. RESULTS AND DISCUSSION 102 is still changing at the boundary where SL=200 m£/(mmHg.hr). Hence, if we consider the "surface" normalization procedure introduced in Section 4.2.5.5, it ap-pears likely that extension of the LS range will cause the value of OBJgi to rise. This increase, will in turn, affect the shape of the normalized surfaces, including the OBJcomb surface, and therefore, likely, the location of the "best-fit" parame-ters. However, the rate of change of objective function value with increasing L S at L5=200 m /^(mmHg.hr) is small in all cases (i.e., the "surfaces" have almost levelled-off) and so the location of a strong global minimum is unlikely to be af-fected. It is therefore assumed that only a small error is introduced by limiting the L S range in this way. 5.3 T H E STARLING M O D E L All of the OBJifv, OBJjii and OBJcomb surfaces generated for the Starling Model by discretely varying the normal capillary pressure and the tissue comphance re-lationship are given in Appendix A. The final numerical results of the parameter estimation procedure are presented in Tables 5.1, 5.2, and 5.3. Note that values of maximal to basal lymph flowrate ratio, j £ l B f B , the transient response time, TRES , and the transcapillary pressure difference, PTC •> are included in Table 5.3. Each of these system behaviour characterizing criteria will be introduced and discussed later in this section. Values of these additional parameters have only been calculated for the "best-fit" parameter sets associated with the minimum OBJcomb values, these having been determined using the greatest amount of information and so being of the most interest and significance. A number of general observations about the results will be made before discussing the parameter values presented in Table 5.3. Chapter 5. RESULTS AND DISCUSSION 103 Starling M o d e l - Compliance Relationship #1 PpL,C,NORM Rd/l LS JL,NORM PS CT KF OBJNI 11 0.843 163.2 81.9 67.97 0.915 56.3 68.7 10 0.729 >200. 88.2 63.27 0.853 76.5 68.3 9 0.598 >200. 96.7 56.90 0.782 102.3 67.8 8 0.497 >200. 104.4 51.08 0.727 186.6 67.5 7 0.378 >200. 115.3 42.91 0.662 406.6 67.7 Starling M o d e l - Compliance Relationship #2 PpL,C,NORM Rd/i LS JL,NORM PS CT KF OBJn, 11 0.843 163.2 81.9 67.97 0.915 56.3 68.7 10 0.729 >200. 88.2 63.27 0.853 76.5 68.3 9 0.598 >200. 96.7 56.90 0.782 102.3 67.8 8 0.497 >200. 104.4 51.08 0.727 186.6 67.5 7 0.378 >200. 115.3 42.91 0.662 406.6 67.7 Starling M o d e l - Compliance Relationship #3 PpL,C,NORM Rd/i LS JL,NORM PS CT KF OBJNI 11 0.843 163.2 81.9 67.97 0.915 56.3 68.7 10 0.729 >200. 88.2 63.27 0.853 76.5 68.3 9 0.598 >200. 96.7 56.90 0.782 102.3 67.8 8 0.497 >200. 104.4 51.08 0.727 186.6 67.5 7 0.378 >200. 115.3 42.91 0.662 406.6 67.7 Table 5.1: Starling Model OBJNI "best-fit" parameters and associated trans-port coefficients. Units: LS and KF are given in ml/(mmHg.hr); Rd/h OBJIFV, and cr are dimensionless; PS and JL,NORM are given in ml/hr; PPL,C,NORM is reported in mmHg. Chapter 5. RESULTS AND DISCUSSION 104 Starling M o d e l - Compliance Relationship # 1 PpL,C,NORM Rd/l LS JL,NORM PS CT Kp OBJjfy 11 1.000 50.0 74.6 73.44 1.000 149.2 37.3 10 0.908 73.1 78.7 70.36 0.950 1320.5 36.5 9 0.746 88.7 87.2 64.02 0.862 1936.6 36.4 8 0.583 87.7 97.7 56.09 0.774 2677.9 36.9 7 0.420 91.8 111.2 45.98 0.685 3974.6 37.2 Starling M o d e l - Compliance Relationship #2 PPL,C,NQRM Rd/l LS JLJJORM P S CT Kp OBJjfv 11 1.000 50.0 74.6 73.44 1.000 149.2 37.3 10 0.908 68.7 78.7 70.36 0.950 1320.5 36.2 9 0.746 84.9 87.2 64.02 0.862 1936.6 36.0 8 0.583 93.4 97.7 56.09 0.774 2677.9 36.0 7 0.420 94.2 111.2 45.98 0.685 3974.6 36.3 Starling M o d e l - Compliance Relationship #3 ,C,NORM R-d/i LS JL,NORM P S CT KF OBJif, 11 1.000 40.0 74.6 73.44 1.000 149.2 36.3 10 0.905 64.4 78.9 70.25 0.948 1012.8 35.0 9 0.732 59.6 88.0 63.40 0.854 676.0 35.2 8 0.578 64.4 98.1 55.82 0.771 1466.2 35.2 7 0.417 67.9 111.5 45.77 0.683 2411.8 35.5 Table 5.2: Starling Model OBJIFV "best-fit" parameters and associated trans-port coefficients. Units: LS and KF are given in ml/(mmHg.hr); Rd/i, OBJJU and <r are dimensionless; PS and JL,NORM are given in ml/hr; PPL,C,NORM is reported in mmHg. Chapter 5. RESULTS AND DISCUSSION 105 Starling Model - Compliance Relationship #1 PpL,CNORM Rd/l LS JL.NORM PS Kp jTtME TRES PTC OBJcomb 11 0.950 40.0 76.8 71.81 0.973 95.5 2.51 43.3 0.80 0.050 10 0.844 50.1 81.9 68.01 0.915 182.3 2.77 42.4 0.45 0.050 9 0.709 57.5 89.4 62.37 0.842 330.9 2.87 41.3 0.26 0.043 8 0.574 76.5 98.4 55.60 0.769 1078.4 3.26 39.8 0.09 0.043 7 0.420 124.7 111.2 45.98 0.685 3974.6 4.25 36.3 0.03 0.034 Starling Model - Compliance Relationship # 2 PpL,C,NORM P-d/l LS JL.NORM PS cr KP TRES PTC 11 0.950 40.0 76.8 71.81 0.973 95.5 2.56 42.5 0.80 0.044 10 0.851 50.0 81.5 68.27 0.919 200.6 2.84 42.4 0.40 0.044 9 0.720 62.5 88.7 62.87 0.848 436.6 3.11 41.5 0.19 0.034 8 0.576 85.0 98.7 55.71 0.770 1242.5 3.59 39.6 0.06 0.034 7 0.420 124.3 111.2 45.98 0.685 3974.6 4.36 36.2 0.03 0.027 Starling Model - Compliance Relationship #3 PpL,C,NOKM Rd/l LS JL.NORM PS cr Kp jRMB TRES PTC OBJ^t 11 1.000 40.0 74.6 73.44 1.000 149.2 2.77 43.3 0.50 0.030 10 0.868 40.0 80.6 68.91 0.928 266.2 2.63 42.2 0.30 0.031 9 0.719 53.1 88.8 62.82 0.847 424.2 2.97 41.3 0.21 0.024 8 0.582 70.0 97.8 56.04 0.773 2297.1 3.36 39.7 0.04 0.017 7 0.420 125.2 111.2 45.98 0.685 3974.6 4.70 36.2 0.03 0.027 Table 5.3: Starling Model 0BJcomb "best-fit" parameters, transport coefficients, and other associated behaviour characterizing criteria. Units: LS and Kp are given in ml/(mmHg.hr); Rd/i, 0BJcomb, JjfMB, and cr are dimensionless: PS and JL,NORM are given in ml/hr; PPL,C,NORM and PTc are given in mmHg; TRES is reported in hours. Chapter 5. RESULTS AND DISCUSSION 106 5.3.1 General Shape of OSJn,. and OBJifv "Surfaces" The general form of the OBJ^ "surfaces" is similar for all values of PPL,C,NORM investigated and is well illustrated by Figure 4.5. It can be clearly seen that the Uj vs. UPL data set contains considerable information regarding the most appropriate value of Rd/i for use with the Starling Model, but little regarding the value of LS (i.e., a valley of equally good fits lies across almost the entire range of LS values investigated, but exists within a fairly narrow band of Rd/i values). By comparing the OBJjif "surfaces" included in Appendix A it can be seen that, for each value of PPL,C,NORM considered, the "best-fit" R^/i exists close to its upper hrnit (see Section 4.2.5.3). As with the OBJn{ surfaces, the OBJiJV surfaces possess a fairly consistent shape. The general shape of all OBJifv surfaces generated (which can also be seen in Appendix A) is demonstrated in Figure 4.4. Once again the "best-fit" value of Rd/t lies close to the upper limit value. However, in contrast to the OB Jn, surfaces, their OBJifv counterparts have a smaller region of LS values yielding good data fits. It would therefore seem that the Vj vs UPL data contain more "fitting information" than does the Ux vs Upr, set. 5.3.2 "Best-fit" Steady-state V} vs. UPL and Uj vs. UPL Simulations The best and worst of the SM "best-fit" simulations (i.e., those associated with the highest and lowest OBJcomb values in Table 5.3) of the steady-state V/ vs. UPL and Uj vs UPL relationships are presented in Figures 5.2 and 5.3, respectively. The behaviour of the SM subject to each of these alternative parameter sets is quite similar. In fact, the predicted Ur vs. UPL relationship is almost identical in each Chapter 5. RESULTS AND DISCUSSION 107 30.0 Figure 5.2: SM best "best-fit" fitting data simulations (associated with a PPL,C,NORM of 8 mmHg and tissue compliance relationship #3) . Chapter 5. RESULTS AND DISCUSSION Figure 5.3: SM worst "best-fit" fitting data simulations (associated with PPL,C,NORM of 11 mmHg and tissue compliance relationship #1). Chapter 5. RESULTS AND DISCUSSION 109 case. The predicted Vr vs. UPL curves differ mainly in the section representing overhydration (note that, Figure 5.3 is generated using tissue comphance relation-ship #1 and Figure 5.2 using relationship #3, see Section 5.3.4). In both cases pronounced edema formation occurs below a Upr, of between 12 to 15 mmHg. This behaviour agrees well with clinical observation [3]. Considering both Vj vs UPL and Hr vs. UPL simulations in either Figure 5.2 or 5.3, it can be seen that over the UPL range spanning 27 to 12 mmHg the reduc-tion in Hj is equal to about 80% of that in UPL- Since each point on the simulated relationship represents a nephrotic patient in a steady-state condition, then by con-sidering equation(3.13), it becomes apparent that Pj must rise by approximately 0.2 mmHg for each mmHg drop in the value of UPL over this range (remembering that VPL is constant at the normal steady-state value of 3.2 I and therefore PPL,C is equal to PPL,C,NORM )• These calculations are only approximate because they as-sume that the normal fluid filtration flowrate remains unaltered as UPL falls within this range. This obviously is not the case since the value of Pj is increasing and therefore the lymph flowrate and the filtration rate must also be increasing. It does however illustrate the edema preventing mechanisms at work. It appears that, as the plasma colloid osmotic pressure is reduced, protein washout (i.e., a net reduc-tion of interstitial protein mass) and not the increase in tissue hydrostatic pressure is the predominant edema preventing mechanism when considering long term fluid volume regulation. However, as Hj drops below about 4 mmHg, further washout of interstitial proteins fails to prevent the pronounced edema whose formation is evident in the upper panels of Figures 5.2 and 5.3. Experimental studies have indicated that Uj and UPL fall by nearly identical Chapter 5. RESULTS AND DISCUSSION 110 amounts over the 12 < HPL < 27 range [3]. This near one to one drop in tis-sue and plasma colloid osmotic pressures seems unlikely because it offers no passive mechanism to account for the edema formation observed to occur at or below a HPL of 12 mmHg. In contrast, our simulations indicate that the interstitial com-partment begins to increase in volume as soon as HPL falls below its normal value. Due to the low tissue comphance around the normal steady-state value of Vj (see Section 3.2.4) the increase in tissue volume is small. This is because the inter-stitial fluid hydrostatic pressure, which opposes further fluid filtration, increases rapidly as fluid enters the interstitial spaces. However, when the interstitial fluid volume reaches about 10 £ (at a TLpi, of approximately 12 mmHg), the tissue be-comes much more compliant and, consquently, Pj increases far less responsively with further drops in HPL- Hence, the model's predictions suggest the existence of two passive regulatory mechanisms whose failure at Hp/, values less than 12 mmHg account for the clinically observed onset of massive edema. 5.3.3 OBJj^ Insensitivity to Tissue Compliance Overhydration Gradi-ent Table 5.1 shows that the OBJj^ "best-fit" parameters obtained for the Hj vs. HPL data set are completely independent of the tissue comphance overhydration gradient. This phenomenon is illustrated graphically in Figure 5.4 which contains contour plots of the three OBJu{ "surfaces" generated for a PPL,C,NORM of 10 mmHg (i.e., one "surface" for each tissue comphance relationship investigated). The contours of each surface are nearly identical across the entire region mapped and the location of the minimum OB JR. is the same in each case. The simulated fits of the Hj vs. HPL data obtained using the OBJu, "best-fit" Chapter 5. RESULTS AND DISCUSSION 111 o T 1 1 1 " n r — 1 1 , 1 1 T T1 0000 0.183 0366 0.549 0.732 0.915 Rd/l Figure 5.4: Alternative tissue compliance OBJIJ. "surfaces" at a PPL.CJIORM of 10 mmHg for the Starling Model- Upper (compliance relationship #1), Center (compliance relationship #2), Bottom (compliance relationship #3). Chapter 5. RESULTS AND DISCUSSION 112 Figure 5.5: Starling Model OBJ^ "best-fit" predictions of the steady-state Vj vs. UPL and 11/ vs. LTP L relationships at a PPL,C,NORM of 10 mmHg. Chapter 5. RESULTS AND DISCUSSION 113 parameters at a PPL,C,NORM of 10 mmHg are presented in Figure 5.5. The three simulated relationships (each generated using a different tissue compliance rela-tionship) coincide exactly. The reason for this behaviour becomes apparent when the simulated Vr vs. UPL relationships associated with the 0f?Jn, "best-fits" are plotted and compared also in Figure 5.5. The overlapping Vj vs. UPL simulations presented in this figure clearly show that across the entire range of UPL values inves-tigated the predicted interstitial fluid volume never increases enough to reach the "overhydration" section of the tissue compliance relationship (see Section 3.2.4). Hence, under these circumstances, the common gradient of this section of all three tissue compliance curves will not influence the behaviour of the system. It can also be seen that the use of OBJut "best-fit" parameter sets gives rise to unrealistic steady-state Vj vs. UPL relationships. 5.3.4 Relationship between Tissue Compliance Characteristics and their Steady-state Vj vs. UPL Simulations. Another interesting observation is the fact that, as the tissue compliance overhydra-tion gradient is increased, the minimum OBJ{fv value obtained decreases slightly (i.e., a somewhat better fit of the data set is obtained). This relationship be-tween the predicted curves for the Vj vs. UPL data set and the shapes assumed for the tissue compliance relationship is depicted in Figure 5.6. The figure shows the "best-fit" simulations associated with the minimum OBJifv values for each tissue compliance considered (at a PPL,C,NORM of 10 mmHg) in addition to the tissue compliance relationships themselves. It should be noted that the relative OBJifv values associated with each of the alternative tissue compliance "best-fit" steady-state Vj V S . UPL simulations is strongly influenced by a single data point (i.e., the Chapter 5. RESULTS AND DISCUSSION 114 point having the lowest value of Hp/,). The shape of the simulated Vj vs. UPL relationship appears highly dependent on the form of the tissue compliance curve used. This dependence suggests that the behaviour of the system is strongly influenced by the characteristics of the tissue compliance relationship. Two conclusions may be drawn from the above: 1. The forms of the tissue compliance relationship used throughout this work rep-resent "best estimates" (see Section 3.2.4). This fact combined with the fact that the behaviour of the model appears to be highly dependent on the form of the tissue compliance relationship used indicates that differences between these tissue compliance "best-estimates" and the true compliance relationship may produce unrealistic model behaviour. 2. Further clinical study of human tissue compliance characteristics is required to enhance the reliability of models of the microvascular exchange system. The concern voiced in the first point is somewhat allayed by noting that although the Vj VS. Upr, simulation appears strongly influenced by the shape of the tissue compliance relationship, all three compliances investigated yield good fits of the available data. It is therefore probable that all of the tissue compliance relationships used here provide reasonable representations of the true compliance behaviour. 5.3.5 Validation of the Starling Model The OBJcomb results presented in Table 5.3 yield fifteen alternative sets of best-fit parameters, all of which produce reasonably good fits of both data sets used in the parameter estimation procedure. In this section a partial validation of the Starling Model will be attempted by first comparing the transport coefficients with those Chapter 5. RESULTS AND DISCUSSION 115 30.0 0.0 5.0 -i i 1 r 10.0 15.0 20.0 25.0 30.0 LTpr (mmHg) 5.0 tc B B, w cc 73 oo oo Ui cc o E -oo o cc > w oo oo 0.0 - 5 . 0 --10.0 Compliance Curve =1 Cpmpjiance Curve j=2_ Compliance Curve #3 0.0 10.0 2 0 . 0 3 0 . 0 I N T E R S T I T I A L FLUID V O L U M E ( I ) 4 0 . 0 Figure 5.6: Alternative compliance relationships investigated and their cor-responding 0BJifv "best-fit"(PPL,C,NORM = 10 mmHg) predicted V> vs. I\.PL relationships. Chapter 5. RESULTS AND DISCUSSION 116 available in the literature and then by comparing certain aspects of the model's behaviour (i.e., its transient response to a specified perturbation and its predicted maximal to basal lymph flowrate ratio) when subject to these parameters. 5.3.5.1 Lymph Flow Sensitivity - LS. The lymph flow sensitivity is a phenomenological parameter within an assumed relationship incorporated into the model and may only be calculated indirectly. A more appropriate related criterion for gauging lymphatic function is the ratio of maximal to basal lymph flowrate, , which is discussed in Section 5.3.5.6. 5.3.5.2 Lymph Flowrate under Normal Steady-state Conditions — JL,NORM • Lymph flowrates in humans have been determined by a number of different ex-perimental techniques. These include direct measurements by the cannulation of lymphatic vessels in limbs and indirect methods by measuring the turnover rates of radio-labelled tracers in tissues. It appears to be generally accepted that the human whole body lymph flowrate under normal conditions is in the 2 to 4 £/da.y range[53]. The values predicted for JL,NORM range between 76.8 and 111.2 ml/hr, which is equal to 1.84 to 2.67 £/day. These values, therefore, compare well with the expected range of values. 5.3.5.3 Permeability—Surface Area Product — PS Experimentally determined PS values have been reported by various workers for a number of different animal species and tissues[12]. The values listed in Table 5.4 have been determined assuming all protein transfer occurs by diffusion and hence Chapter 5. RESULTS AND DISCUSSION 117 are automatically likely to be high. A conservative estimate of the range of whole Species Tissue PS* Dog Subcutaneous(paw) 0.102 Dog Subcutaneous(paw) 0.062 Dog Subcutaneous(paw) 0.056 Dog Skeletal muscle(paw/leg) 0.067 Rabbit Skeletal muscle 0.043 Table 5.4: Various experimentally determined values of permeability-surface area product. * Units of PS are ml/min per 100 grams of wet tissue[12]. body PS values for humans based on the specific PS values listed in Table 5.4 is obtained by assuming all protein transport occurs only within the skin (including subcutis) and skeletal muscle tissues (10.1 kg and 28.0 kg in weight respectively, see Section 3.5.1). Hence the range of PS expected is 983 - 2332 ml/hr. This is in poor agreement with the values obtained by fitting (45.98 - 73.44 ml/hr). The significance of and reason for this disagreement may be better understood if we recall equation(4.9) which states pg _ JL,NORM CJTIMORM -Rd/i CpL,NORM — CiTAV,NORM By definition the value of Rd/i must he between zero and one. The albumin con-centrations CiTAV,NORM and CPL,NORM have been well established experimentally. The value of CJ^ORM depends on CitAv,NORM and the interstitial volume excluded to albumin (see equation 3.4). Therefore, as the excluded volume tends to zero, CitffoRM approaches its maximum value of CitAV,NORM • Hence the maximum value of CINORM/(CpL,NORM ~ CI,AV,NORM ) is also well established. Experimental mea-surements of lymph flowrates in the human indicate whole body values ranging between 2 - 4 £/day[53]. The maximum value of JL,NORM is therefore expected to Chapter 5. RESULTS AND DISCUSSION 118 be 170 ml/hr. If equation(4.9) is assumed valid the likely upper limit on the value of PS based on the above arguments and on the normal albumin concentrations presented in Table 4.5 is = 170 * 22.38 * 1.0/(39.43 - 22.38) = 223.14 ml/hr This value is still far less than the smallest value calculated using the experimental information listed in Table 5.4. The PS values presented in Table 5.4 were calcu-lated using specific lymph flowrates from various animal studies[12]. The specific lymph flowrates used range between 0.015 and 0.063 m£/min per 100 g of wet tis-sue. However, the 2-4 £/day total body lymph flowrate reported for humans is equal to a specific lymph flowrate of only 0.0037 to 0.0074 m£/min per 100 g of wet tissue (assuming exchange occurs in only the skin and muscle tissues, i.e. 38.1 kg). The human specific lymph flowrates are over an order of magnitude lower than the animal values used in the calculation of the PS values presented in Table 5.4. It would therefore seem that the discrepancy between our statistically predicted PS values and those available from Table 5.4 can be attributed to differences between the specific lymph flowrates measured for animal tissues and those of man. It is also worth mentioning that PS values in humans or in animals have never been determined by direct experimental measurements. 5.3.5.4 Capillary Wall Solute Reflection Coefficient — a The value of a for proteins is typically measured by increasing the transcapillary solute flux until transport becomes purely convective[3|. The values obtained by use of this technique on skin and muscle tissue are about 0.85 and 0.9 for albumin and total proteins, respectively[3]. Values of a as low as 0.7 have been reported for Chapter 5. RESULTS AND DISCUSSION 119 albumin in human skeletal muscle[3]. In addition, values obtained for the lung are generally lower ranging between 0.5 and 0.6 for albumin[3]. Therefore, considering that the cr used in the present study represents an average value for all body tissues, the 0.685 - 0.928 range of "best-fit" values obtained for PPL,C,NORM values between 7 and 10 mmHg seems very reasonable. Only the tr values obtained for a PPL,C,NORM of 11 mmHg appear unrealistic. However, recent clinical investigations on the rat have indicated that the value of cr may in fact be close to unity (personal communication with R.K.Reed.) 5.3 .5 .5 Capillary Filtration Coefficient — Kp Typical measured values of Kp range between 0.06 and 0.48 ml/(hr.mmHg) per 100 grams of wet tissue for the human forearm and leg, the lower value being associated with measurements in orthostasis (i.e., when standing erect)[3]. As-suming that fluid transport occurs only in the skin (including subcutis) and skele-tal muscle tissues (10.1 kg and 28.0 kg in weight respectively, see Section 3.5.1), a conservative estimate of whole body Kp is calculated to range between 22.9 and 182.9 ml/(hr.mmHg). Values of KF estimated by statistical fitting agree well down to a PPL,C,NORM of 9 mmHg (predicted values range between 95 and 436 mf/(mmHg.hr)). Below this, estimated Kp values appear excessively high (1078 to 3975 m//(mmHg.hr)). 5.3 .5 .6 Maximal to Basal Lymph Flowrate - J ^ 8 To aid in the evaluation of these alternative sets of fitting parameters, two addi-tional quantitative criteria are calculated and assessed. The first of these is the ratio of maximal to basal lymph flowrate, J™13 (i.e., JL,MAXIJL,NORM)- The maximal Chapter 5. RESULTS AND DISCUSSION 120 flowrate is observed experimentally in animals to be in the order of 5 to 10 times that of the basal flowrate[2]. Since it is assumed that lymph flowrate is directly pro-portional to interstitial fluid hydrostatic pressure (equation 3.7), and that the latter is dependent on the interstitial fluid volume via the tissue comphance relationship (equation 3.10), then some form of arbitrary decision must be taken regarding the value of Vj at which maximal lymph flowrate is achieved (due to the fact that each tissue comphance relationship investigated possesses a small but finite gradi-ent in the region representing massive overhydration). The calculation of J L U I B is therefore based on the assumption that the maximum lymph flowrate occurs at an interstitial fluid volume of 20 £ (i.e., approximately the maximum normalized level of fluid expansion observed by Fadnes et al. [22] in nephrotic subjects as shown in Figure 4.1). This assumption, combined with the tissue comphance characteristics and the "best-fit" values of LS and JL,NORM > yields a value of J ^ L B . The ratio jRMB - g O D s e r v e ( i tQ increase as the value of PPL,C,NORM is reduced and/or as the tissue comphance overhydration gradient is increased. Predicted values of range between 2.5 and 5.0. Considering the assumptions inherent in the calcula-tion of J B M B , the predicted ratio is considered to be in reasonable agreement with the range of values experimentally observed. The relationship existing between the tissue comphance overhydration gradient and J B M B is a result of the assumptions made in the calculation of this ratio. Since the interstitial fluid volume at which maximal lymph flowrate occurs is fixed at 20 £, then the tissue pressure associated with this volume increases as the "overhydration" gradient is increased. This in turn produces a higher lymph flowrate. Chapter 5. RESULTS AND DISCUSSION 121 5.3.5.7 Steady-state Transcapillary Pressure Gradient — PTC The steady-state transcapillary pressure gradient, PTC, w a s introduced and defined in Section 3.5.6. However, to reiterate: PTC = PPL,C,NORM — Pi,NORM — <T{}TPL,NORM — UI,NORM) PTC is the force which drives fluid and its various dissolved components from the circulation into the interstitial spaces under normal steady-state conditions. The flux is required to ensure normal function of the tissue cells. The value of PTC is thought to be about 1 mmHg or less[21]. However, this figure represents only a rough estimate. The PTC values associated with the OBJcomb "best-fit" parameters range between 0.80 and 0.03 mmHg. Considering the importance of maintaining the transcapillary fluid flux, very small values of PTC would appear to indicate poor system design. Contrary to this, clinical experience (and the high human population) indicates that the system is well regulated about normal steady-state conditions. Therefore "best-fit" parameter sets determined at values of PPL,C,NORM below 9 mmHg (i.e., PTC <0.19 mmHg) are considered to be too "finely-tuned" and are disregarded as being unreasonable (i.e., PPL,C,NORM values of 9, 10 and 11 mmHg are considered to be reasonable according to this criterion). 5.3.5.8 Combined "Surface" Objective Function Value — 0 i ? J C O T n b Good fits of both data sets are obtained for all values of PPL,C,NORM investigated (i.e., OBJcomb <0.05 in all cases). As the tissue compliance overhydration gradient is increased, moving from compliance relationship #1 through to #3, better overall fits are obtained. This behaviour is inherited from the OBJijv "best-fit" parameters and has been previously discussed. Chapter 5. RESULTS AND DISCUSSION 122 It is generally observed that as the value of PPL,C,NORM is reduced, a better match between simulation and fitting data is obtained. However, for a given comphance relationship all OBJcomb values obtained are similar. This point is well illustrated in Figures 5.2 and 5.3. Figure 5.3 contains the "best-fit" simulations associated with the highest OBJcomb value presented in Table 5.3 (i.e., OBJcomb = 0.05, PPL,C,NORM = 11 mmHg, and comphance relationship #1) and Figure 5.2 contains the "best-fit" simulations associated with the lowest OBJcomb value (i.e., OBJcomb — 0.017, PPL,C,NORM = 8 mmHg, and comphance relationship #3). Both Hj vs. HPL simulations are almost identical and the difference in Vj vs. HPL simulations occurs solely in the overhydration segment of the relationship (see Section 5.3.4). It is therefore difficult to attribute any real significance to the small variations observed in the OBJCOmb "best-fit" values, particularly when the scatter associated with the data used for fitting is considered. 5.3.5.9 Transient Response Time — TRES The second additional criterion investigated, TRES, is used to gauge the transient behaviour exhibited by the model subject to "best-fit" parameter values. Koomans et al. [42] studied the fate of a single intravenous infusion of human albu-min in 10 nephrotic subjects. The infusion consisted of 300 ml of fluid associated with 60 g of albumin and took place continuously over a period of 1.5 hours. It was generally observed that nearly all initial increases in plasma volume and in-travascular albumin had disappeared after the initial 24 hour post-infusion period, this being due to a combination of urinary excretion and protein migration to the tissue spaces. The transcapillary oncotic gradient was observed to have returned to its pre-infusion value within 24 hours. Chapter 5. RESULTS AND DISCUSSION 123 TRES is equal to the post-perturbation time predicted by the model for the tran-scapillary oncotic gradient to return to within experimental error (i.e., the accuracy with which Koomans reports the experimental value of the transcapillary colloid osmotic pressure gradient, Al l , at 24 hours post-perturbation) of its pre-infusion value. This time is compared to the experimentally observed value of 24 hours and therefore acts as a further comparison of the suitability of each "best-fit" parameter combination. To enable the simulation of the microcirculatory response to the aforementioned albumin infusion, it was necessary to make the following assumptions: — A value of the circulatory compliance is estimated by scaling the value deter-mined for the rat by Bert et al.[5]. Scaling ensures that increases in capillary hydrostatic pressure are equal in both man and rat for equal fractional in-creases in the plasma volume. The validity of this assumption is investigated and discussed later in this section. — Urinary excretion of albumin was observed to increase by an average of 15.8 g/day above that of normal over the initial 24 hour post-infusion period. It is assumed that this increased excretion rate remains undiminished until all infused albumin is eliminated from the system. — The fluid infused with the albumin (300 m£) is eliminated by urinary excretion over the initial 24 hour post-infusion period. — Koomans' average subject is approximated by "reference man" in all respects except compartmental fluid and protein contents. Before simulation of the aforementioned perturbation is possible, initial values of compartmental fluid and protein contents representative of Koomans' average Chapter 5. RESULTS AND DISCUSSION 124 nephrotic patient must be estimated. This is accomplished for a given set of pa-rameter values by first obtaining the steady-state Vj vs. ILpr, and H/ vs. Upr, relationships as outlined in Section 4.2.5.3. The steady-state value of TLPL cor-responding to the average patient Vi is then determined (Koomans' subjects are reported to possess an average pre-infusion interstitial fluid volume excess of 9.85 £, yielding a Vj of 18.25 I), and finally the corresponding steady-state value of 11/ is found, thereby establishing the required values of fluid and protein contents. Hence, it can be seen that initial conditions required for the simulation are fully determined by the specification of Vj. To simulate the transient behaviour of the system the assumption of a constant : plasma volume invoked for steady-state chronic nephrotics must be abandoned. If the model is used to simulate transient behaviour, a further unknown parameter, the circulatory compliance, PCOMPpr,tc > must be specified. Due to a lack of data relating to the circulatory compliance as defined herein, an estimate of its value is obtained from the rat microcirculation model developed by Bert et a/.[5]. They provide an estimate of the rat circulatory compliance of 5.05 mmHg/ml. This value is scaled for use in the current human model by ensuring that equal fractional increases in plasma volume for both the rat and human give rise to equal elevations in capillary hydrostatic pressure. The human circulatory compliance is therefore estimated to be 0.009658 mmHg/ml, which is equivalent to an elevation in capillary hydrostatic pressure of 30.91 mmHg when the plasma volume is raised to twice its ; normal level. Since confidence in the circulatory compliance value is low, a sensitivity analysis is conducted to determine the quantitative influence of PCOMPpitc °" the transient Chapter 5. RESULTS AND DISCUSSION 125 behavibur of the system as characterized by TRES • The OBJcomb "best-fit" param-eters associated with a PPL,C,NORM of 10 mmHg and tissue comphance relationship #1 are used as the basis for this analysis. The results of the analysis are presented in Table 5.5. PCOMPPL,C Deviation from TRES (mmHg/ml) scaled value (hours) 0.018350 +90% 40.8 0.015453 +60% 40.9 0.011590 +20% 41.3 0.010624 +10% 41.4 0.009658 +00% 41.5 0.008692 -10% 41.5 0.007726 -20% 41.4 0.003863 -60% 41.0 0.000966 -90% 24.6 Table 5.5: Sensitivity of the transient response time, TRES, to changes in the value of circulatory compliance, PCOMPpic• Table 5.5 indicates that the transient response of the model as characterized by TRES is fairly insensitive to the value of PCOMPPL,C over a wide range of values about the estimated value, except at very low values. Hence, for the purpose of comparing the relative transient behaviour simulating abihties of each set of "best-fit" parameters, the circulatory comphance of 0.009658 mmHg/ml appears to be a reasonable value to use. Thus, all SM TRES times were calculated using this value of the circulatory comphance. It is interesting to note however that the lowest value of PCOMPPLC investigated yields the most reasonable value of TRES («- e., a value of 24.6 hours compared to the experimentally determined value of approximately 24 hours). As can be seen in Table 5.3, the transient response time TRES is observed to decrease Chapter 5. RESULTS AND DISCUSSION 126 with decreasing PPL,C,NORM - Values predicted for TRES range between 43.3 and 36.2 hours, all of which are in fair agreement with the experimentally determined value of approximately 24 hours [42]. The response time appears to be almost completely independent of the tissue comphance overhydration gradient. Koomans et al. [42] monitored the variation of five system parameters for a 24 hour period following an hour and a half long intravenous albumin infusion in nephrotic patients. The variables monitored were: 1. HPL - Plasma oncotic pressure 2. Hi - Interstitial oncotic pressure 3- VPL - Plasma volume 4. All - Transcapillary oncotic pressure gradient 5. AC - Transcapillary protein concentration gradient For the purpose of qualitative comparision of alternative OBJcomb "best-fit" model simulations with the experimental data provided by Koomans et al. [42], the tran-sient behaviour resulting from two of the "best-fit" parameter sets determined for comphance relationship #1 is depicted in Figures 5.7 through 5.10. These present the behaviour of the "best-fit" simulations obtained for the two extreme values of PPL,C,NORM (i.e., 11 mmHg and 7 mmHg). The remainder (i.e., PPL,C,NORM = 10, 9, and 8 mmHg) demonstrated intermediate behaviour and are included in Appendix C. Figures 5.7 and 5.8 plot the progress of the system variables moni-tored by Koomans. Figures 5.9 to 5.10 represent the same simulations and time periods as Figures 5.7 and 5.8, respectively, but show the variation of a number of additional microvascular system variables including the fluid and protein fluxes. Figures 5.7 and 5.8 clearly demonstrate good qualitative agreement between the Chapter 5. RESULTS AND DISCUSSION 127 48.0 TimeOirs) 96J0 X g 5.0-4.0-3-0-ao-OX) 24.0 48.0 Time(hrs) 7ZX) 96-0 5.0-S 4.0 > 3.0-ao-ao — i — 24.0 48.0 Time(hrs) — i — 72-0 96J0 Time(hrs) 96-0 0.0 24.0 48.0 Time(hrs) 7Z0 96X) Figure 5.7: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Star-ling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables mon-itored). Chapter 5. RESULTS AND DISCUSSION 128 30.0 20.0 H 10.0 0.0 0.0 — 1 — 24.0 4S.0 Time(hrs) 72.0 96J0 GO SC 5.0 4.0 3-0 2.0 0-0 —I 24.0 +ao Time(hrs) 72-0 96-0 5.0 © 4.0 > 3.0 2.0 O-O 24.0 4S-0 —1 72.0 96.0 Tirne(hrs) Figure 5.8: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the O-BJ^b "best-fit" Star-ling Model at a PPL,C,NORM °f ? mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables monitored). Chapter 5. RESULTS AND DISCUSSION 129 "best-fit" model predictions and the experimental data provided by Koomans et al.. It is interesting to note the large rise in the plasma colloid osmotic pressure; the simulated elevation in its value being in excess of that observed experimentally in all cases. This can be accounted for by considering the nature of the protein concentration/colloid osmotic pressure relationship used by the model (see Section 3.2.1). The albumin concentration of the plasma/interstitial fluid is correlated directly against the total colloid osmotic pressure exerted by all protein species normally present. An infusion of pure albumin therefore produces an exaggerated elevation in the colloid osmotic pressure of the compartment it enters (because the albumin concentration/colloid osmotic pressure relationship automatically adds the pressure contribution of the missing protein species to that of the albumin). Better agreement between the simulated response and the experimental data would be expected if the size of the simulated albumin infusion were scaled by the ratio of the osmotic pressure contribution of albumin to the total colloid osmotic pressure of normal plasma, i.e. by a factor of approximately 0.65[44]. Simulations of both the scaled and unsealed albumin infusions have been conducted for the PpL,cj*ORM — 10 mmHg "best-fit" parameters and are depicted in Figures 5.11 and 5.12, respectively. As anticipated, the fit of the data improves substantially when the amount of albumin infused is scaled downwards according to the recipe given above. The transient response time, TRES , associated with the simulation is "now reduced from 42.4 to 32.5 hours, which agrees better with the observed 24 hour period. However, it should be noted that the scaled infusion reduces the protein concentration below the true value and hence causes the diffusional driving forces to be in error. It is suspected that the true system behaviour lies somewhere between Chapter 5. RESULTS AND DISCUSSION 130 these two extreme simulations. Hence for the purposes of comparing simulations the unsealed versions have been retained. Within the range of values investigated, the influence of the tissue comphance overhydration gradient on the time-dependent responses of the system appears negligible (see Appendix C). This is to be expected considering the small interstitial volume change occurring. Figures 5.9 and 5.10 clearly show the transition from completely diffusive pro-tein transport to transport containing a substantial convective element. However, regardless of the transcapillary protein transfer mechanism, all albumin infusion simulations produce a similar series of responses within the microvascular exchange system as exemplified by the two figures. These are as follows. At the onset of the albumin infusion, the fluid filtration rate is reduced rapidly by the increasing colloid osmotic pressure of the plasma compartment, although this is somewhat offset by the increase in capillary hydrostatic pressure associated with the infused fluid (300 m£). Due to the large reservoir of fluid within the interstitial spaces the lymph flowrate is maintained at an almost pre-perturbation level. This in turn leads to build-up of fluid in the circulation as the lymphatics continue to transfer fluid from the interstitium. Capillary hydrostatic pressure then begins to increase which counteracts the decrease in filtration pressure. After several hours, the capillary pressure attains a level at which the rate of fluid filtered from the circulation again equals that entering from the lymphatics. Protein continues to migrate into the interstitial compartment via diffusive and/or convective pathways. This pattern of events and the associated time scales appear httle affected by which alternative "best-fit" parameter set is used within the model, and are therefore fairly indepen-dent of the ratio of convective to diffusive transcapillary protein transport over the Chapter 5. RESULTS AND DISCUSSION 131 z o o 23.0 20.0 15.0 10.0-5.0 0.0- — I 24.0 48.0 TIME(hrs) PL 600.0 m b3 400.0-200.0-OJO-£ -200.0 • 0.0 24.0 48.0 TIME(hrs) CO w 3 0. 24.0 48.0 TIMEChrs) O0 c 23-0 20 X) 15.0 10.0 5-0 0-0' 77 48.0 TIME(hrs) PL Figure 5.9: Effect of Koomans' albumin infusion on all microvascular exchange variables. For the 05J c o m b "best-fit" Starling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #1. Chapter 5. RESULTS AND DISCUSSION 132 s—v 230 Z 01 200 t-15.0 z o 10.0 o Q 50 FLU1 0.0 ' P f 600.0 40O0 W X 200.0 OX) Q -200JO 3 . J u. 48.0 TIMEXhrs) 48.0 TIMEXhrs) V "v. P L z w E -z o o ta E -O cc c 48.0 TIMEXhrs) 0, Q i P L ta X g &3 O cc Cu 4.0 20 rxo--2-0 46.0 TIMEXhrs) Q D Q L TIMEXhrs) Figure 5.10: Effect of Koomans' albumin infusion on all microvascular exchange variables. For the OBJcomb "best-fit" Starling Model at a PPL,C,NORM of 7 mmHg using compliance relationship # 1 . Chapter 5. RESULTS AND DISCUSSION 133 0.0 34.0 48.0 72J0 96.0 Time(hrs) 30.0 ou-1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 Time(hrs) Figure 5.11: Effect of a Koomans1 albumin infusion on select microvascular exchange variables (includes Koomans1 data). For the 0BJcomb "best-fit" Starling Model at a PPL,C,NORM of 10 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables monitored). Chapter 5. RESULTS AND DISCUSSION 134 s~t 30.0 -* P 20-0-A 10.0- ' o.o-t-0.0 24.0 48.0 Time(hrs) 72.0 96.0 OS 5.0-4.0-1? 2-0-0.0 24.0 48.0 Time(hr-s) 72.0 96.0 > 5.0-4.0-3.0- i r 2-0-0.0 24.0 48.0 Timefhrs) 72.0 96J0 -Ti 2O0-g <] 15.0-10X1-5-0-0.0-0.0 24.0 4ao Time(hrs) 72.0 96.0 30.0-r 20D-<i 10.0-0.0-0.0 24.0 48.0 Time(hrs) 72.0 96 X) Figure 5.12. Effect of a SCALED Koomans' albumin infusion on select mi-crovascular exchange variables (includes Koomans' data). For the 0i?Jcom6 "best-fit" Starling Model at a PPL,C,NORM of 10 mmHg using compliance re-lationship #1 (the dotted lines represent the final steady-state values of the variables monitored). Chapter 5. RESULTS AND DISCUSSION 135 range observed. Figures 5.9 through 5.10 also indicate that transcapillary protein transport is the rate-limiting step determining the systems re-equilibration time period following the albumin infusion. 5.3.6 Summary of the Starling Model Results The three criteria, OBJcomb, J™3, and TRES, all indicate that better agreement between model simulations and observed system behaviour is obtained as the value of PpL,c,NORM is reduced (i.e., from 11 mmHg down to 7 mmHg) and/or the tissue comphance overhydration gradient increased (see Table 5.3). A strong relationship is observed to exist between the shape of the tissue comph-ance curve and the "best-fit" steady-state Vj vs. HPL simulation results. This, in turn, indicates that, of those investigated, comphance relationship #3 is most representative of the true tissue comphance. It also introduces the possibility of determining a "best-fit" tissue comphance by statistical fitting of the Vj vs. HPL data; however, this avenue of investigation is not pursued further here (only a par-tial sensitivity analysis on the effect of the gradient of the overhydration section was conducted, see Section 5.3.4). The steady-state transcapillary pressure gradient, Pre, indicates that "best-fit" parameter sets determined at PPL,C,NORM values of below 9 mmHg produce an exchange system that is too "finely-tuned" (i.e., a small change in any of the Star-ling forces could either halt or reverse the direction of the normal transcapillary fluid flux which maintains the tissue cells). Therefore, the "best-fit" parameters associated with PPL,C,NORM values of less than 9 mmHg are considered to be unrea-sonable. Similarly, a PPL,C,NORM value of 11 mmHg appears to be too high because Chapter 5. RESULTS AND DISCUSSION 136 it yields a reflection coefficient of unity or near unity which is higher than the ex-perimental cr values usually reported. Thus, "best-fit" parameter sets determined for PPL,C,NORM values of 9 and 10 mmHg are considered to be the most appropri-ate for use with the Starling Model. This indicates diffusion as the major protein transport mechanism, accounting for upwards of 70% of transcapillary movement under normal steady-state conditions. The transient behaviour of the model is in good quahtative agreement with the available experimental data for all OBJcomb "best-fit" parameters sets determined. Predicted ranges of capillary wall solute reflection coefficient, tr, and normal steady-state lymph flowrates compare well with values presented in hterature. OBJcomb "best-fit" filtration coefficients, Kp, give the best agreement with hterature values at normal steady-state capillary pressures above 8 mmHg which is consistant with the "best" range of 9 < PPL,C,NORM < 10 mmHg selected above. Permeabihty-surface area products, PS, values agree poorly with those available from animal studies. However, this discrepancy appears attributable to differences between human specific lymph flowrates and the much higher values reported for animal tissues which have been used in the calculation of the available PS values. It should be remembered that no values of PS have been determined directly in experimental investigations. In general, "best-fit" parameter sets associated with normal capillary hydrostatic pressures of above 8 mmHg provide good representations, both quantitatively and qualitatively, of the experimental data provided by Koomans et al. regarding in-travenous albumin infusion of nephrotic patients. Both sets also yield reasonable maximal to basal lymph flowrate ratios (i.e., predicted range of 2.77 to 3.11 com-pared with an expected range of appoximately 5 to 10). Chapter 5. RESULTS AND DISCUSSION 137 Therefore, the estimation procedure has provided the location of a small section of "parameter space" within which all parameter value combinations yield good agreement between Starling Model predictions and the available experimental in-formation. Rather than determining one specific set of parameter values suitable for describing the available data, a "nest" or "family" of parameter combinations has been found. 5.4 THE PLASMA LEAK MODEL Table 5.6 contains the "best-fit" parameter sets which correspond to representative points within the "best-fit" regions (i.e., ORJcomfc < 0.06) existing on the OBJ comb surfaces generated using the Plasma Leak Model (see Section 5.4.3). All OBJifv, OBJ^, and OBJcomb PLM surface plots generated are given in Appendix B. A number of general observations regarding the results require comment before spe-cific comparisons are made between "best-fit" parameters and information available in literature. 5.4.1 General Shape of OBJRi and O B J i f v "Surfaces" For the Plasma Leak Model, the general shape of the O B J ifv surfaces appears to be highly dependent on the value of RF/R with which they are associated. However, it is not the absolute value of RF/R which is important but rather the percentage increase in its value above the minimum possible value it may hold (see Section 4.2.6.1). This point is nicely illustrated in Figure 5.13 where OBJifv surface contour plots of the RF/R = RF/R * l- u ^ plane are presented for the three alternative pairs of arterial and venous capillary pressures investigated (since R p ^ is dependent on the values of PPL,A,NORM and PPL,V,NORM , see Section 4.2.6.1). These surfaces Chapter 5. RESULTS AND DISCUSSION 1 3 8 8 8 8 8 o o V V co co o o V V t- o f-< d o o "« a «o g CD T cd co to e-s I s g g CS CN 6 GO CO CO CD i - i rH rH CO CO CO CO CO CO CO U O CO Q in cjq CD © d o O r i o o V V CO T CO •a co CO p O CO CO* r ^ Hf CO s § 2 0 0 t-i rH rH rH CO CO C » CH d d CO CO T-H rH CO CO d d »a 10 to iO to W9 to cs cs cs CO r t CD CS cd cs « <S Table 5.6: Plasma leak model OBJcomb "best-fit" parameters, associated trans- * port parameters, and other behaviour characterizing criteria. Units: LS, K F , K R , and KPLL are given in ml/(mmHg.hr); R^/i, OBJcomb, RF/R, and J ^ 5 are dimensionless; PS and JL,NORM are given in ml/hr; PAJVORM and PVJ*ORM are reported in mmHg; TRES is reported in hours. Chapter 5. RESULTS AND DISCUSSION 139 indicate that the fit obtained of the Vj vs. Hp/, data set is highly dependent on the value of the lymph flow sensitivity, but almost completely independent of the protein transport mechanism (i.e., Rd/i)-The shape of the OBJ^ surfaces exhibit a similar dependence on the percentage elevation of RF/R above its mimimum possible value. Figure 5.14 shows the three OBJnt surface contour plots for the same region of parameter space mapped in Figure 5.13 (i.e., RF/R = 1.05 * RF/R in each case). It can be seen that all three plots exhibit similar trends, that is, the best fit of the data set is associated with low values of LS and values of Rd/i above about 0.5. 5.4.2 "Best-fit" Steady-state V} vs. Upr, and H/ vs UPL Simulations The PLM's ability to fit the available steady-state V} vs. TLPL and 11/ vs Hpx, data is demonstrated in Figures 5.15 and 5.16. The first of these two figures shows the model simulations associated with the "best-fit" parameter set which includes no plasma leak mechanism (i.e., Table 5.6: PPL,AJ<ORM = 26.54 mmHg, PPL,VJ<ORM '= 7.92 mmHg, LS = 38.6 m£/(mmHg.hr), Rd/l = 1.0, RF/R = 0.2091) and the sec-ond figure shows the predictions made with the parameters indicating substantial plasma leakage (i.e., Table 5.6: PPL,A,NORM = 26.54 mmHg, PpL.vjfORM = 7.92 mmHg, LS = 31.8 m£/(mmHg.hr), R^/i = 0.45, RF/R = 0.2413). Both alterna-tive data sets predict almost identical behaviour. Hence, it appears that the two data sets do not contain sufficient information to determine which of the protein mechanisms is dominant. Chapter 5. RESULTS AND DISCUSSION 140 o o o of 0.0 0.2 0.4 0-6 QJB VO R d/1 Figure 5.13: Alternative P L M 0BJifv surfaces generated at a RF/R10% above RF/R- Upper: PPLA.NORM = 26.54 mmHg, PPL,V,NORM — 7.92 mmHg. Center: PPL .AjfORM — 24.54 mmHg, PPL.VJJORM — 5.92 mmHg. Lower: PPL,A,NORM — 22.54 mmHg, PPL,V,NORM = 3.92 mmHg. Chapter 5. RESULTS AND DISCUSSION 141 R d/1 Figure 5.14: Alternative P L M OBJm surfaces generated at a RF/R10% above RF/R- Upper: PPL,A,NORM = 26.54 mmHg, PPL,V,NORM = 7.92 mmHg. Center: PPL,A,NORM = 24.54 mmHg, PPL,V,NORM = 5.92 mmHg. Lower: PPL,A,NORM = 22.54 mmHg, PPL,V,NORM = 3.92 mmHg. Figure 5.16: P L M "best-fit" simulations of fitting data (associated with PPL,A,NORM — 26.54 mmHg, PpLyjioRhi = 7.92 mmHg, LS = 31.8 m£/(mmHg.hr), ity, = 0.45, RF/R = 0.2413 in Table 5.6). Chapter 5. RESULTS AND DISCUSSION 143 30.0 -r 25.0-20.0-Figure 5.15: P L M "best-fit" simulations of fitting data (associated with PpLAFORM = 26.54 mmHg, PPL,V,NORM = 7.92 mmHg, LS = 38.6 m£/(mmHg.hr), RD/L = 1.0, RF/R = 0.2091 in Table 5.6). Chapter 5. RESULTS AND DISCUSSION 144 5.4.3 Selection of "Best-fit" Results The "best-fit" parameters presented in Table 5.6 have been determined by selecting representative points in all OBJCOmb "surface" areas having an objective function value of less than 0.06. The use of this criterion assures that all parameter sets selected yield data fits (i.e., fits of the data used in the statistical fitting process) of a quality comparable to those of the Starling Model "best-fits" obtained using the same tissue comphance relationship (==£ 1). In contrast to the SM parameter estimation procedure a non-linear optimization routine was not employed to locate the global minimum in the P L M procedure. The reason for this change was the distributed form (i.e., several possible minima can be located visually) of some of the OBJcomb surfaces generated (see Figure 5.17, for example). Hence, it seemed more appropriate to select only a couple of representative points from the largest regions circumscribed by the OBJcomb — 0.06 contours. 5.4.4 OBJ "Surface" Depend ence on RF/R By comparing "surfaces" generated using alternative PPL,A,NORM and PPL,V,NORM values and hence different minimum RF/R values (see Appendix B), it can be seen that the shape of the "surfaces" and the range of objective function values oc-curring within them depend more on the percentage elevation of RF/R above its minimum value, than on the absolute value itself. OBJcomb "best-fit" parameters^ combinations are generally located within 30 to 50% above R^m-Chapter 5. RESULTS AND DISCUSSION 145 c Rd/l Figure 5.17: An example of a P L M OBJcomb "surface" containing a distributed minimum. PPL^NORM = 24.54 mmHg, PPL,V,NORM = 5.92 mmHg, and RF/R — 0.4241. Chapter 5. RESULTS AND DISCUSSION 146 5.4.5 Validation of the Plasma Leak Model Note that, only the "best-fit" parameters associated with minimum OBJcomb values are listed. For reference the related OBJ{jv and OBJu, "surfaces" are included in Appendix B . It should also be noted that experimental measurements are generally interpreted in terms of the Starhng Model. Hence, whereas clinically determined transport parameters are available for comparison with the SM, this not the case with the P L M . 5.4.5.1 Ratio of Filtration Coefficient to Reabsorption Coefficient -RF/R Intaghetta[39] has studied the filtration rates in the capillaries of the frog mesen-tery using the Landis micro-occlusion technique. His data indicate that venous capillaries in this tissue possess permeabilities 60% greater than those of arterial capillaries. Various studies[66, 16] have been conducted to determine the extent of regional differences in capillary surface area. These investigations have indicated that the relative surface areas of the venous and arterial capillaries can vary con-siderably between species, between tissues, and even regionally within a specific tissue. However, the venous surface area generally appears to be in the order of 2 to 6 times larger than the arterial capillary surface area. Combining the information presented above, the value of RF/R is expected to range between approximately 0.1 and 0.3. This range encompasses part of the "best-fit"' parameter value range (i.e., 0.21 - 0.71) determined for the P L M (see Table 5.6). Chapter 5. RESULTS AND DISCUSSION 147 5.4.5.2 Lymph Flow Sensitivity - LS See Section 5.3.5.1 for comment. 5.4.5.3 Lymph Flowrate under Normal Steady-state Conditions - JL,NORM The lymph flowrates predicted range between 74.6 and 108.5 ml/hr (1.8 - 2.6 l/d&y) which compare well with rates observed experimentally (see Section 5.3.5.2). 5.4.5.4 Permeability - Surface Area Product — PS The range of predicted values (48.1 - 73.4 ml/hr) is similar to that predicted for the Starhng Model (46 - 73.4 ml/day). See Section 5.3.5.3 for further discussion. 5.4.5.5 Filtration Coefficient — Kp Due to the fact the almost all clinical experiments relating to transcapillary trans-port have been interpreted in terms of the Starhng equation little experimental information is available for comparison with the P L M transport parameters pre-dicted here. However, values of Kp , Kp and KPLL have been estimated b}' Bert et al.[5] in their modelling work on the rat. These values may be scaled for comparison with the human values determined in the present study by assuming that the total capillary surface area of a given tissue is proportional to the tissue's interstitial "fluid volume. Bert et al.[5] determined separate values of these transport coefficients for> both skin and muscle tissue and therefore both will be scaled for comparison with our generic tissue value. Upon scaling , the rat values yield human Kp values of 46.1 and 51.2 m£/(mmHg.hr) for skin and muscle tissues, respectively. These Chapter 5. RESULTS AND DISCUSSION 148 are in reasonable agreement with the range of "best-fit" values (i.e., 14.7 to 27.7 ml/(mmHg.hr)) determined by fitting the sparse set of nephrotic syndrome data. 5.4.5.6 Reabsorption Coefficient - KR Estimated human values of KR are calculated from the work of Bert et cd.[5] as described in the previous section. These scaled values for rat skin and muscle are 146.1 and 127.5 m£/(mmHg.hr), respectively. Once again these values compare reasonably well with the range of "best-fit" human KR values determined here (i.e., approximately 56 - 103 m£/(mmHg.hr)). 5.4.5.7 Plasma Leak Coefficient - KPLL The scaled rat tissue values (see Section 5.4.5.5) of KPLL range between 13.5 and 11.7 m£/(mmHg.hr) for skin and muscle, respectively. These values are clearly higher than those determined here for the human (i.e., 0.0 to 3.0 m£/(mmHg.hr)). This difference is probably due mainly to the fact that, in the rat model protein transport was predicted to be solely due to plasma leakage whereas, in the present study, human transcapillary protein exchange is predicted to be substantially dif-fusive in all cases. 5.4.5.8 Maximal to Basal Lymph Flowrate -The predicted values of range between 1.85 and 2.50. These are clearly below but close to the expected range of 5 to 10. Based on trends apparent in the Starling Model analysis, it is anticipated that the predicted values of would increase Chapter 5. RESULTS AND DISCUSSION 149 if the "overhydration" gradient of the tissue comphance relationship were to be increased. 5.4.5.9 Transient Response Time - TRES Before the Plasma Leak Model can be used to simulate the transient behaviour of the system, two further unknown parameters, the arterial and venous comphances, must be specified. Due to a lack of alternative information about these parameters, estimates are obtained from values determined for the rat by Bert et al.{5]. The values reported are PCOMPpi^ = 0 m£/(mmHg.hr) (i.e., the arterial pressure is insensitive to circulatory volume changes) and PCOMPpi,y = 5.12 m m H g / m l The venous comphance is scaled for use in our human model as discussed in Section 5.3.5.9, yielding a human PCOMPPLy of 0.009850 m m H g / m l As with the Starhng Model, the sensitivity of the transient response time, TRES, to changes in the value of venous comphance was investigated and the results are reported in Table 5.7. These results indicate that the transient response of the model as characterized by TRES is fairly insensitive to PV,COMP over a wide range of values. Transient responses of the system to the simulated. albumin infusion are qualita-tively similar for all "best-fit" parameter sets, varying only in the relative contribu-tions of diffusive and plasma leak transport to the movement of proteins across the capillary wall. The "best-fit" parameter P L M transient simulations of Koomans' albumin infusion are included in Appendix D. A representative example of the albumin infusion simulations obtained using a set of the "best-fit" parameters is presented in Figure 5.18 along with Koomans' experimentally determined data. As was observed with the SM, excessive elevation of the plasma colloid osmotic Chapter 5. RESULTS AND DISCUSSION 150 PCOMPPLy (mmHg/ml) Deviation from scaled value TRES (hours) 0.018715 +90% 38.4 0.015760 +60% 38.4 0.011820 +20% 38.3 0.010835 +10% 37.7 0.009850 +00% 37.4 0.008865 -10% 37.3 0.007880 -20% 37.3 0.003400 -60% 37.1 0.000985 -90% 31.2 Table 5.7: Sensitivity of the transient response time, TRES-> *° changes in the value of venous capillary compliance, PCOMPpr,y • "Best-fit" values associ-ated with a PPLAJIORM of 24.54 mmHg, a PPLXNORM of 5.92 mmHg, and an Rd/i of 0.80 is used as the basis of the analysis. pressure occurs during the simulated infusion. The reasons for this behaviour have been discussed in Section 5.3.5.9. The "best-fit" TRES listed in Table 5.6 all agree well with the experimentally de-termined value of approximately 24 hours (the predicted range being 33.3 - 42.8 hours). All "best-fit" P L M parameter models exhibit a similar set of time-dependent re-sponses to the albumin infusion perturbation. For example, Figures 5.19 and 5.20 show the predicted responses obtained using for the sets of "best-fit" parameters from Table 5.6 with Rd/i values of 1.0 and 0.45 , respectively (i.e., an example in which the plasma leak protein transport mechanism is nonexistent and one in which it predominantes). The response depicted in Figure 5.19 can be qualitatively described as follows. At the start of the albumin infusion the fluid filtration rate into the tissues quickly falls Chapter 5. RESULTS AND DISCUSSION ^ 300-1 : " x i i : 1 1 0.0 24.0 4&0 7ZX3 96.0 Tlme(hrs) Figure 5.18: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^b "best-fit" Plasma Leak Model for a PPL^ORM of 24.54 mmHg, a PPLyNORM of 5.92 mmHg, an RF/R of 0.4241, Rd/i of 0.80 and a LS of 31.8 m£/ (mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Chapter 5. RESULTS AND DISCUSSION 152 due to the rapid increase of the plasma colloid osmotic pressure. This elevation of Upr, also increases the fluid reabsorption rate. However, the increase in the plasma volume due to the fluid infused along with the albumin (300 m£) raises the venous capillary hydrostatic pressure and, hence, tends to offset a small portion of the effects caused by this increase in plasma colloid osmotic pressure. At the same time diffusive transcapillary transport of protein begins to increase along with the increasing plasma albumin concentration. The fluid volume shifts which occur are relative^ small when compared to the quantity of fluid present in the interstitium. Therefore, neither the interstitial fluid volume, and consequently, nor the flowrate of lymph is much affected. Thus, fluid continues to enter the circulation via the lymphatics. All these changes tend to increase the plasma volume over the first few hours and thereby decrease its colloid osmotic pressure. As a consequence of this, diffusive protein transport declines. Also, as the plasma colloid osmotic pressure, decreases, the fluid reabsorption rate falls rapidly to almost zero and the filtration rate begins to increase acting to bring the system back to its normal operating point. This can be compared with the model's response to the same perturbation when the plasma leak protein transport mechanism dominates (see Figure 5.20). The basic response is identical in most respects. The fluid shifts follow a similar pattern to that described above because the plasma leak flowrate is generally small when compared to the other fluid flowrates. However, plasma leakage does allow a higher protein transport rate across the capillary wall than did the completely diffusive-case and, as would be expected, the value of TRES w a s significantly reduced (i.e., 42.8 hours for the diffusion only case compared to 33.3 hours for the 55% plasma leakage case just described). Chapter 5. RESULTS AND DISCUSSION 153 2 M 20.0 15.0 z 8 10.0 R 3.0 0.0 4ao TIME(hrs) 96.0 P L 1 6 0 0 . 0 -P 200.0 ri ao -20O0-24.0 48.0 TIMEXhrs) 72.0 'PLL 96JO 4 8 . 0 TIMEXhrs) Q PL co to X 3 w O CC O.0 4 8 . 0 TIMEXhrs) Q PLL QD Q L W CC co co w a a. 25.0 2 0 0 15.0 10.0-5.0 O X TIMEXhrs) 7 Z X 7T PL TT, PL,V Figure 5.19: Effect of a Koomans' albumin infusion on all microvascular ex-change variables. For the OBJ^ "best-fit" Plasma Leak Model with a PPL^NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2091, RD/L of 1.00 and a LS of 38.6 nrii/(mmHg.hr). Chapter 5. RESULTS AND DISCUSSION 154 z o 23.0 20.0 15.0 10.0 48.0 TIMEXhrs) PL 48.0 TIMEXhrs) ' P l i CO | H 150JO o o u f-o cc a. — i — 48.0 TIMEXhrs) —i— 72.0 PL 0, CO til X g o cc c 0.0-48.0 TIMEXhrs) Q F I X Q D Q L TIMEXhrs) Figure 5.20: Effect of a Koomans' albumin infusion on all microvascular ex-change variables. For the OBJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2413, Rd/i of 0.45 and a L S of 31.8 m^/(mmHg.hr). Chapter 5. RESULTS AND DISCUSSION 155 In all cases, as with the Starling Model, protein transport appears to be the step limiting the rate at which the system returns to pre-perturbation conditions. 5.4.6 Summary of the Plasma Leak Model Results All determined OBJcomb "best-fit" values of RF/R are located within 30 to 50% of the minimum possible value. This assures that both KF and KR are of similar magnitudes regardless of the arterial and venous capillary pressures. As PA,NORM is increased the "best-fit" value of Kp decreases, and as PV,NORM is increased the "best-fit" value of KR increases. Both of these changes tend to maintain the predicted steady-state fluid fluxes at an almost constant level. In all cases, plasma leakage plays only a minor role in the transfer of fluid across the capillary barrier. The plasma leak transport mechanism is not required to describe the fitting data. It is also interesting to note that diffusion generally appears to be the dominant transcapillary protein transport mechanism, which concurs with the results of the Starling Model analysis. Al l predicted transient response times, TRES , compare well with the experimentally determined value and are found to be fairly consistent between alternative "best-fit" models. The response times are anticipated to drop if the infusion is scaled down as discussed in Section 5.3.5.9. The lack of experimentally determined transport coefficients prevents further com-parison of the alternative "best-fit" parameter sets with clinical values as was done in the Starling Model analysis. Hence, without further applicable data relating to the system, we are left with a number of alternative parameter combinations which describe the available data well, but vary considerably in the relative importance of the plasma leak protein transport mechanism. Chapter 6 CONCLUSIONS Both the Starhng Model and the Plasma Leak Model are capable of providing good descriptions of interstitial fluid regulation in the human as characterized by the available experimental information. Although it is generally accepted that the properties of skeletal muscle and skin tissue (which together contain the bulk of the interstitial fluid and proteins) are different, the inclusion of both within a generic tissue compartment does not inhibit either of the models presented here from providing good descriptions of the available experimental data. The Starhng Model analysis indicates that the characteristics of the tissue comph-ance relationship play an important role in the regulation of the interstitial fluid volume. It also appears likely that the elevation of interstitial fluid hydrostatic pressure during hydration and the concomitant increase in lymph flowrate both play a significant role in prevention of edema in chronic nephrotics. The relation-ships incorporated into the current model appear to be fairly good representations of the true tissue comphance. However, more information regarding the exact form of the true human tissue comphance is required before a judgement may be made as to the relative importance of this edema preventing mechanism. For the Starhng Model, a volume of "parameter space" is determined within which all parameter value combinations yield results which are in good agreement with 156 Chapter 6. CONCLUSIONS 157 the experimental data used in the fitting procedure (i.e., yielding low OBJcomb values). However, several sections of this "space" are rejected on the grounds that they predict unreasonable system behaviour. Thus, a more limited volume of "pa-rameter space" is located within which all parameter combinations produce model behaviour which is consistent with all the available information. The existence of this restricted "volume" may indicate one of the following: — The available nephrotic data contains insufficient information with which to accurately locate the specific values of the "best-fit" transport parameters. — The system exhibits a certain degree of passive "self-regulation" (i.e., small variations in the values of the transport parameters about their normal oper-ating points produce negligible changes in the behaviour of the system). Intuitively, the latter of the two possiblities seems reasonable when considering the design of such an important regulatory system. However, the former possibility cannot be disregarded. The additional flexibility afforded the microvascular exchange model by the in-clusion of the more complex plasma leak description of transcapillary exchange makes the mapping of a "best-fit" parameter volume, as was done with the Star-ling Model, a difficult task. Therefore, only a limited number of discrete "points" within this volume have been located, each representing a combination of parameter values which, when incorporated into the model, describes the available experimen-tal data reasonably well. The results of the Plasma Leak Model analysis indicate that, although not generally dominant, the plasma leak mechanism likely plays an important role in the regulation of interstitial fluid volume. The results of both the Starling Model and Plasma Leak Model investigations indicate that diffusion is the Chapter 6. CONCLUSIONS 158 dominant mechanism of transcapillary protein transport, however, the role played by convection cannot be neglected. The transient simulations of Koomans' albumin infusion experiment have not been used as part of the parameter estimation procedure. All "best-fit" parameter sets determined for the SM and PLM models using the steady-state nephrotic data pro-duce transient simulations that compare well with Koomans' experimental data. The simulations are further improved when the infusion is scaled down to compen-sate for the overestimation of the colloid osmotic pressure caused by the albumin concentration vs. colloid osmotic pressure relationship used within the model (see Section 5.3.5.9). The results of these transient simulations are therefore regarded as encouraging, and confidence in the validity of the models is increased. Using the best, current experimental data, it is not possible to distinguish which of the two models investigated provides the better description of human microvascu-lar exchange. However, it is anticipated that as further experimental information becomes available better differentiation of the applicability of each model will be possible. The main discrepancy between the findings of the current work and those of pre-vious clinical investigations is the magnitude of the predicted permeability-surface area product, PS. The values determined by statistical fitting range between ap-proximately 50 and 100 ml/hr for both of the models investigated, whereas compa-rable coefficients obtained by scaling clinically determined values associated with other mammalian species are over an order of magnitude larger. This discrepancy can be attributed to the apparently large differences between human specific lymph flowrates and those associated with other animal tissues. These lymph flowrates Chapter 6. CONCLUSIONS 159 impact directly on the values of PS calculated. The predicted human and experi-mental animal PS values are therefore considered to be non-comparable. Both the Starling Model and the Plasma Leak Model are capable of accurately de-scribing the experimental data presented in the current work. Although the Plasma Leak Model represents a more realistic description of transcapillary exchange than the Starling Model (based on the observed behaviour of microvascular capillaries), it appears that this increase in model complexity is not required to describe the rather sparse set of experimental data presently available for nephrotic patients. However, it is anticipated that as more information about the behaviour of the human microvascular exchange system becomes available, the Plasma Leak Model may prove to be the more versatile of the two alternative descriptions of transcap-illary exchange. Chapter 7 RECOMMENDATIONS In this dissertation we have formulated and subsequently attempted to validate two alternative models of the human microvascular exchange system. Regarding future work, the following recommendations are made as suitable extensions to the current study. 1. Further validation of both the SM and the PLM by comparison of their pre-dictions with data available for non-nephrotic is required (thereby, also testing the assumption that the nephrotic syndrome does not alter the transport pa-rameters characterizing the system). Data suggested for this purpose include those derived from studies of heart failure, saline infusion studies, and ultra-filtration studies. 2. Closer collaboration with clinical investigators should be sought so that in-terests of theoreticians and experimentalists are more closely matched. In particular, more experimental information regarding tissue comphance char-acteristics, skeletal muscle properties, and lymphatic drainage could prove useful. 3. Mechanistic modifications could be made to the models presented here to better reflect the true nature of the human microvascular exchange system. Potentially advantageous modifications include the coupling of diffusive and 160 Chapter 7. RECOMMENDATIONS 161 convective transcapillary protein transport, a more rigorous description of lym-phatic drainage, and the addition of an intracellular water compartment. 4. A sensitivity analysis should be conducted to determine the influence on model behaviour of changes in the values of normal steady-state system properties such as the interstitial exclusion volume and normal interstitial fluid hydro-static pressure. Finally, it is the opinion of the author that further validation of the current models should be of a higher priority than the development of more elaborate descriptions of microvascular exchange in humans. Nomenclature Symbol Description Units Alb A lbumin BC Blood content ( m l - k g - 1 ) BWt. Bodyweight (kg) C Concentration of albumin (g .m£ _ 1 ) CF Cell fraction ECW Extracellular water content ( g . k g - 1 ) ECWF Extracellular water fraction Fn ( ) Function of FC F lu id content ( g - k g - 1 ) FCOMP Tissue compliance relationship J F l u i d transport rate ( m l . h - 1 ) K Transport coefficient ( m l . m m H g - 1 . h _ 1 ) OBJ Objective function value P Hydrostatic pressure (mmHg) PCOMP Capillary compliance ( m m H g . h - 1 ) PS Permeability-surface area product of ( m l . h - 1 ) capillary with respect to albumin diffusion Rj/i Ratio of diffusive to convective protein transport rate 162 Nomenclature 163 RF/R Ratio of filtration coefficient to reabsorption coefficient Q Albumin content (g) Q Albumin transport rate (g-h-1) LS Lymph flow sensitivity (ml .mmHg- 1 ^- 1 ) rpholf Tissue clearance half time 00 TRES Transient response time 00 V Volume (ml) W Weight (kg) WC Water content (fractional) n Colloid osmotic pressure (mmHg) a Protein reflection coefficient P Density (g.cm-1) Superscripts and Subscripts A Arterial capillary AA Artery ' AV Interstitial volume available to albumin AVE Average C Capillary comb Combined data D Diffusion DE Dermis EC Extracellular EP Epidermis Nomenclature 1 6 4 EX F HI HUMAN HYO I ifv ISO L LOW max mm NORM NORMALIZED PL PLL R RAT RES RMB S SM TO V Excluded Filtration Highest Human Hypodermis Interstitial compartment Interstitial fluid volume data Isogravimetric Lymph Lowest Maximum value Minimum value Normal steady-state Normalized Plasma Plasma leak Reabsorption Rat Transient response Ratio of maximal to basal rate Skin Skeletal muscle Turnover rate Venous capillary Nomenclature VV Large vein * Tracer properties Bibliography [1] ARTURSON,G., T .GROTH, A.HEDLUND, AND B .ZAAR. Potential use of com-puter simulation in treatment of burns with special regard to oedema forma-tion. Scand. J. Plast. Reconstr. Surg. 18 (1984) 39-48. [2] AUKLANDjK., AND G.NlCOLAYSEN. Interstitial fluid volume: local regulatory mechanisms, Physiological Reviews, 61 (1981) 556-643. [3] AUKLAND ,K., AND R.R.EED. Interstitial-lymphatic control of extracellular fluid volume, Unpublished, (1990). [4] B E N O I T , J . N . , C . A . N A V I A , A . E . T A Y L O R , A N D D . A . G R A N G E R Mathe-matical model of intestinal transcapillary fluid and protein exchange. In: A.P.Shepherd and D.N.Granger(eds.), Physiology of the Intestinal Circulation, N.Y. : Raven Press, 1984, p.275-287. [5] BERT, J . L . , B.D.BOWEN, AND R . K . R E E D . Microvascular exchange and in-terstitial volume control in the rat: model validation. Am. J.Physiol. 254(1988) H384-H399. [6] BERT , J .L . , AND R.H .PEARCE. The interstitium and microvascular exchange. 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[38] H O L L A N D E R , W . , P .REILLY AND B . A . B u R R O W S . Lymphatic flow in human subjects as indicated by the disappearence of J 1 3 1-labelled albumin from the subcutaneous tissue. J. Clin. Invest. 222 (1960) 732. [39] l N T A G L I E T T A , M . , AND B . W . Z w E I F A C H . Indirect method for measurement of pressure in blood capillaries. Circ. Res. 19 (1966) 199-205. [40] K A T C H A L S K Y , A . , AND P . D . C U R R A N . Nonequalibrium Thermodynamics in Biophysics. Cambridge: Harvard Press,1965. [41] K A T Z . M . A . , AND E . H . B R E S L E R . Osmosis. In:N.C.Staub and A.E.Taylor (eds.), Edema, Chapter 2, New York, N.Y. : Raven Press, 1984. Bibliography 1 6 9 [ 4 2 ] K O O M A N S , H . A . , A . B . G E E R S , W . K O R L A N D T A N D E . J . D O R H O U T M E E S . Albumin infused in nephrotics: Where does it go ? Neth. J. Med. 28 ( 1 9 8 5 ) 1 4 8 . [ 4 3 ] K O O M A N S , H . A . , W . K O R T L A N D T , A . B . G E E R S , A N D E . J . D O R H O U T M E E S . Lowered protein content of tissue fluid in patients with nephrotic syndrome: observations during disease and recovery. Nephron. 4 0 ( 1 9 8 5 ) 3 9 1 - 3 9 5 . [ 4 4 ] L A N D I S , E . M . , A N D J . R . P A P P E N H E I M E R Exchange of substances through the capillary walls. In: Handbook of Physiology - Circulation, Volume II - Section 2 , Md: American Physiological Soc, 1 9 6 3 . [ 4 5 ] L A N G A R D , N . The subcutaneous absorption of albumin in edematous states. Acta Medica Scand. 1 7 4 ( 1 9 6 3 ) . [ 4 6 ] M A Z Z O N I , M . C . , P . B O R G S T R O M , K - E . A R F O R S , A N D M . I N T A G L I E T T A . Dy-namic fluid redistribution in hyperosmotic resuscitation of hypovolemic shock. Am. J. Physiol. 2 5 5 ( 1 9 8 8 ) H 6 2 9 - H 6 3 7 . [ 4 7 ] M E L L A N D E R , S . , A N D B . O B E R G . Transcapillary fluid absorption and other vascular reactions in the human forearm during the reduction of ciculating blood volume. Acta.Physiol.Scand. 7 1 ( 1 9 6 7 ) 3 7 - 4 6 . [ 4 8 ] M I C H E L , C . C . Fluid movements through capillary walls. In: E.M.Renkin and C.C.Michel (eds.), Handbook of Physiology - The Cardiovascular System, Vol-ume IV - Microcirculation, Chapter 9 , Bethseda, Md: American Physiological Soc, 1 9 8 4 . [ 4 9 ] N l C O L L , P . A . , A N D A . E . T A Y L O R . Lymph formation and flow. Annu. Rev. Physiol. 38 ( 1 9 7 7 ) 7 3 - 9 5 . [ 5 0 ] N O D D E L A N D , H . Influence of body posture on transcapillary pressures in hu-man subcutaneous tissue. Scand. J. Clin. Lab. Invest. 4 2 ( 1 9 8 3 ) 1 3 1 - 1 3 8 . [ 5 1 ] N O D D E L A N D , H . , P . L U N D - J O H A N S E N , J . O F S T A D , A N D K . A U K L A N D . Inter-stitial colloid osmotic and hydrostatic pressures in human subcutaneous tissue during the early stages of heart failure. Clin. Physiol. 4 ( 1 9 8 4 ) 2 8 3 - 2 9 7 . [ 5 2 ] N O D D E L A N D , H . , S.M.RlSNES, AND H . O . F A D N E S . Interstitial fluid colloid osmotic and hydrostatic pressures in subcutaneous tissue of patients with nephrotic syndrome. Scand. J. Clin. Lab. Invest. 4 2 ( 1 9 8 2 ) 1 3 9 - 1 4 6 . [ 5 3 ] R E D D Y , N . P . . Lymph circulation: physiology, pharmacology, and biomechan-ics. Crit. Rev . Biomed. Eng. 1 4 ( 1 9 8 6 ) 4 5 - 9 1 . [ 5 4 ] R E E D , R . K . , S . J O H A N S E N A N D H . W l I G . Turnover rate of interstitial albumin in rat skin and skeletal muscle. Effects of limb movements and motor activity. Acta Physiol. Scand. 1 2 5 ( 1 9 8 5 ) 7 1 1 - 7 2 8 . Bibliography 1 7 0 [ 5 5 ] R E E D , R . K . , A N D H . W l I G . Compliance of the interstitial space in rats. I. Studies on hindhmb muscle. Acta Physiol. Scand. 113 ( 1 9 8 1 ) 2 9 7 - 3 0 5 . [ 5 6 ] R E E D , R . K . , A N D H . W l I G . Comphance of the interstitial space in rats. III. Contribution of skin and skeletal muscle interstitial volume to changes in total extracellular fluid volume. Acta Physiol. Scand. 121 ( 1 9 8 4 ) 5 7 - 6 3 . [ 5 7 ] R l P P E , B . , A . K A M I Y A , A N D B . F A L K O W . Simultaneous measurements of cap-illary diffusion and filtration exchange during shifts in filtration - absorption and at graded alterations in the capillary surface area product (PS). Acta Physiol. Scand. 104 ( 1 9 7 8 ) 3 1 8 - 3 3 6 . [ 5 8 ] R o S E L L I , R . J . , R . E . P A R K E R , A N D T . H A R R I S . Comparison between pore model predictions and sheep lung fluid and protein transport. Microvasc. Res. 11 ( 1 9 7 6 ) 1 - 2 3 . [ 5 9 ] R 0 S S I N G , N . Intra- and extravascular distribution of albumin and im-munoglobin in man. Lymphology. 11 ( 1 9 7 8 ) 1 3 8 - 1 4 2 . [ 6 0 ] S C H I T T K O W S K I , K . The nonlinear programming method of Wilson, Han and Powell with an ugmented Lagrangian-type hne search function. Numerische Math. 38 ( 1 9 8 1 ) 8 3 - 1 1 4 . [ 6 1 ] S T R A N D E N , E . A N D H . O . M Y H R E . Pressure - volume recordings of human sub-cutaneous tissue: A study in patients with edema following arterial reconstruc-tion for lower limb atherosclerosis. Microvasc. Res. 24 ( 1 9 8 2 ) 2 4 1 - 2 4 8 . [ 6 2 ] S l M I O N E S C U , M . , A N D N .SlMIONESCU .Ultrastucture of the microvascular wall: functional correlations. In: E.M.Renkin and C.C.Michel (eds.), Hand-book of Physiology - The Cardiovascular System, Volume IV - Microcirculation, Chapter 3 , Bethseda, Md: American Physiological Soc, 1 9 8 4 . [ 6 3 ] T l P T O N , I . H . , A N D M . J . C O O K . Trace elements in human tissues.II. Adult subjects from the United States. Health Phys.. 9 ( 1 9 6 3 ) 1 0 3 - 1 4 5 . [ 6 4 ] W E G R I A , R . D , H . Z E K E R T , R . W . E N T R U P , K . E . W A L T E R , D . P A I E W O N S K Y , D . M l E R Z W I A K , E . C . B E E C H W O O D , A N D G . H . M U E L H E I M S . Effect of an in-crease in systemic venous pressure on the formation and evacuation of lymph. Acta Cardiol. 19 ( 1 9 6 4 ) 1 9 3 - 2 1 6 . [ 6 5 ] W E I D E M A N N , M . P . Architecture. In: E.M.Renkin and C.C.Michel (eds.), Handbook of Physiology - The Cardiovascular System, Volume IV - Micro-circulation, Chapter 2 , Bethseda, Md: American Physiological Soc, 1 9 8 4 . [66 ] W E T Z E L , N . C . , A N D Y . Z O T T E R M A N . On differences in vascular colouration of various regions of normal human skin. Heart. 13 ( 1 9 2 6 ) 3 7 5 . [ 6 7 ] W I D D O W S O N , E . M . , A N D J . W . T . D I C K E R S O N . The effect of growth and func-tion on the chemical composition of soft tissues. Biochem. J. 77 ( 1 9 6 0 ) 3 0 - 4 3 . Bibliography 171 [68] WlEDERHIELM, C A . Dynamics of trancapillary fluid exchange: a nonlinear computer simulation. Microvasc. Res. 18 (1979) 48-82. [69] W l I G , H., AND R.K. R E E D . Compliance of the interstitial space in rats. II. Studies on skin. Acta Physiol. Scand. 113 (1981) 307-315. [70] WlIG,H. , AND H.NODDELAND. Interstitial fluid pressure in human skin mea-sured by micropuncture and wick-in-needle. Scand. J. Clin. Lab. Invest. 43 (1983) 255-260. [71] WlIG,H. , AND R.K.REED. Volume-Pressure relationship (comphance) rela-tionship of the interstitium in dog skin and muscle. Am. J. Physiol. 253 (1987) H291-293. [72] WlSSIG,S.L., AND A.S.CHARONIS. Capillary Ultrastructure. In: N.C.Staub and A.E.Taylor (eds.), Edema, Chapter 6, New York, N.Y.: Raven Press, 1984. [73] SNYDER,W.S.(Chairman) International Commission on Radiological Protec-tion. Task Group on Reference Man.: Pergamon Press, 1975. Appendix A SM OBJ "Surfaces" 172 Appendix A. SM OBJ "Surfaces" 173 o Rd / l Figure A.21: Objective function "surface", OBJIFV, at a PPL,C,NORM of 11 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SMOBJ "Surfaces" 174 R d / l Figure A.22: Objective function "surface", OBJUI, at a PPL,C,NORM of 11 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 175 Figure A.23: Objective function "surface", OBJCOMB, at a PPL,C,NORM of 11 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 176 R d / l Figure A.24: Objective function "surface", OBJIFV, at a PPL,C,NORM of 11 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 177 o d R d / l Figure A.25: Objective function "surface", OBJNI, at a PPL,C,NORM of 11 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 178 Figure A.26: Objective function "surface", OBJCOMB, at a PPL,C,NORM of 11 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 179 o d Rd / l Figure A.27: Objective function "surface", OBJIIV, at a PPL,C^ORM of 11 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SMOBJ "Surfaces" 180 0.0 0.2 0.4 0£ OS 1.0 Rd/l Figure A.28: Objective function "surface", 0BJni, at a PPL,C,NORM of 11 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 181 Figure A.29: Objective function "surface", OBJCOMB, at a PPL,C,NORM of 11 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 182 o R d/1 Figure A.30: Objective function "surface", OBJIFV. at a PPL,C,NORM of 10 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 183 Figure A.31: Objective function "surface", OBJn{, at a PPL,C,NORM of 10 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 184 Figure A.32: Objective function "surface", OBJCORNB, at a PPL,C,NORM of 10 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 185 Figure A.33: Objective function "surface". OBJi<v, at a PPL,C,NORM of 10 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 186 Figure A.34: Objective function "surface", OBJnr. at a PPL,C,NORM of 10 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 187 Figure A.35: Objective function "surface", OBJ^b-, at a PPL,C,NORM of 10 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 188 Figure A.36: Objective function "surface", OBJIFV, at a PPL,C,NORM of 10 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 189 i—" 1 i 1 1 0 .000 0 .183 0 .366 0 5 4 9 0.732 0 .915 R d/1 Figure A.37: Objective function "surface", OBJ^, at a PFL,C,NORM of 10 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 190 o 0.000 0.183 0.366 0.548 0.732 0.915 R d/1 Figure A.38: Objective function "surface", 0BJcomb,/ at a PPL,C,NORM of 10 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 191 Figure A.39: Objective function "surface", OBJIIV, at a PPL,C,NORM of 9 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 192 o R d / l Figure A.40: Objective function "surface", OBJn., at a PPL,C,NORM of 9 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 193 Figure A.41: Objective function "surface", OBJcomht at a PPL,C,NORM of 9 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 194 Figure A.42: Objective function "surface", OBJIIV, at a PPL,C,NORM of 9 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 195 Figure A.43: O b j e c t i v e function "surface", OBJUI, at a PPL,C,NORM of 9 m m H g w i t h c o m p l i a n c e relat ionship #2 (consisting of three a l t e r n a t i v e v i e w i n g angle t h r e e - d i m e n s i o n a l plots a n d a c o n t o u r m a p of the surface). Appendix A. SM OBJ "Surfaces" 196 Figure A.44: Objective function "surface", 0£?J c o m(,, a * a PPL,C,NORM of 9 mmHg with compliance relationship # 2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 197 Figure A.45: Objective function "surface", OBJ{fv, at a PPL,C,NORM of 9 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 198 o o. Rd/l Figure A.46: Objective function "surface", OBJ^, at a PPL,C,NORM of 9 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 1 9 9 Figure A.47: Objective function "surface", OBJCORNB, at a PPL,C,NORM °f 9 mmHg with compliance relationship # 3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 200 r — — — e — r | - * — > i - r 1 0 .000 0.117 0 .234 0 3 5 1 0 .468 0 5 8 5 R d/1 Figure A.48: Objective function "surface", 0BJifv, at a PPL,C,NORM of 8 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces 201 P R d/1 Figure A.49: Objective function "surface", OBJni, at a PPL,C,NORM of 8 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 202 Figure A.50: Objective function "surface", 0 5 7 ^ 6 , at a PPZ,C,NORM of 8 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 203 Figure A.51: Objective function "surface", OBJIFV, at a PPL,C,NORM of 8 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 204 Figure A.52: Objective function "surface", OBJ^, at a PPL,C,NORM of 8 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 205 i — ' — ' — r ~ — • • i — n • 1 0.000 0.117 0.234 0.351 0 .468 0 5 8 5 R d/1 Figure A.53: Objective function "surface", 0BJcornb, at a PPL,C,NORM of 8 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 206 o o 1 1 1 — I 1 1 0.000 0117 0234 0.351 0.468 0585 R d/1 Figure A.54: Objective function "surface", 0 B J i f v , at a PPL,C,NORM °f 8 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 207 Figure A.55: Objective function "surface", OBJm, at. a PPL,C,NORM °f 8 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 208 o 0.000 0-117 0234 0.351 0.468 0585 R d/1 Figure A.56: Objective function "surface", 0BJcomb, at a PPL,C,NORM of 8 mmHg with compliance relationship # 3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 209 Figure A.57: Objective function "surface", OBJLFV, at a PPL,C,NORM of 7 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 210 Figure A.58: Objective function "surface", 0-BJn,, at a PPL,C,NORM °f 7 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 211 T i 1 —r* -=—f— —1 OOOO 0.084 0.168 0.252 0.336 0.420 R d/1 Figure A.59: Objective function "surface", OBJ^t,, at a PPL,C,NORM of 7 mmHg with compliance relationship #1 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 212 i 1 1 r - — " i — 1 ' — i 0.000 0.084 0.168 0252 0.336 0.420 R d/1 Figure A.60: Objective function "surface", 0BJifv, at a PPL,C,NORM of 7 m m H g with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 213 o R d/1 Figure A.61: Objective function "surface", OBJu{, at a PPL,C,NORM of 7 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 214 Figure A.62: Objective function "surface", OBJcomb, at a PPL,C,NORM of 7 mmHg with compliance relationship #2 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 215 —""—>—"—r—=—=r-*— —=—•—i 0.000 0.084 0.168 0.252 0.336 0.420 R dA Figure A.63: Objective function "surface", 0BJijv, at a PPL,C,NORM °f ? mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 216 o Rd/l Figure A.64: Objective function "surface", OBJ^, at a PPL,C,NORM of 7 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix A. SM OBJ "Surfaces" 217 t — =-i—•I— r=—• T— • — i 0.000 0.084 0.168 0.252 0.336 0.420 R d/1 Figure A.65: Objective function "surface", 0BJcomb, at a PPL,C,NORM of 7 mmHg with compliance relationship #3 (consisting of three alternative viewing angle three-dimensional plots and a contour map of the surface). Appendix B P L M OBJ "Surfaces 218 Appendix B. PLM OBJ "Surfaces" 219 R d/1 Figure B.66: Objective function "surface", OBJifv, with PPL,A,NORM — 22.54 mmHg, PPL,v,NORM = 3.92 mmHg, and RF/R = 0.5520 (i.e., 1.01 * RFjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 220 Rd/ l Figure B.67: Objective function "surface", 0BJUi, with PPL,A,NORM — 22.54 mmHg, PPLy,NORM = 3.92 mmHg, and RF/R = 0.5520 (i.e., 1.01 * Rj7}R), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces 221 R d/1 Figure B.68: Objective function "surface", OBJcomb, with PPLA,NORM = .22.54 mmHg, PpL,v;NoRM = 3.92 mmHg, and RF/R = 0.5520 (i.e., 1.01 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 222 0.4 0. R d/1 Figure B.69: Objective function "surface", OBJifv, with PPL,A,NORM = 22.54 mmHg, PPL,V,NORM = 3.92 mmHg, and RF/R = 0.6012 (i.e., 1.10 * R^jR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. P L M OBJ "Surfaces" 223 R d/1 Figure B.70: Objective function "surface", OBJn{, with PPL,A,NORM = 22.54 mmHg, PpLynoBM = 3.92 mmHg, and R F / R = 0.6012 (i.e., 1.10 * R^jR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 224 d/1 Figure B.71: Objective function "surface", OBJcomh, with PPL,A,NORM = 22.54 mmHg, PPL,V,NORM = 3.92 mmHg, and R F / R = 0.6012 {i.e., 1.10 * i 2 $ £ ) , (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM O B J "Surfaces 225 Rd/l Figure B.72: Objective function "surface", OBJifv, with PPL,A,NORM — 22.54 mmHg, PPL,V,NORM = 3.92 mmHg, and RF/R = 0.7105 (i.e., 1.30 * RFjR), (con-sisting of three alternative "viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. P L M OBJ "Surfaces" 226 q Rd / l Figure B.73: Objective function "surface", OBJT^, with PPL,A,NORM — 22.54 mmHg, PPLXNORM = 3.92 mmHg, and RF/R = 0.7105 (i.e., 1.30 * RF}R), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 227 o d Rd/l Figure B.74: Objective function "surface", OBJrxmh, with PPL,A,NORM = 22.54 mmHg, PPLXNORM = 3.92 mmHg, and RF/R = 0.7105 (i.e., 1.30 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 228 Figure B.75: Objective function "surface", OBJifv, with PPL,A,NORM = 22.54 mmHg, PPL,V,NORM = 3.92 mmHg, and RF/R = 0.8198 (i.e., 1.50 * RF}R), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 229 Figure B.76: Objective function "surface", OBJn,, with PPL,A,NORM = 22.54 mmHg, PpL.vjfORM = 3.92 mmHg, and RF/R = 0.8198 (i.e., 1.50 * RFjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces 230 o d R d / l Figure B.77: Objective function "surface", OBJCOTnb, with PPL^NORM — 22.54 mmHg, PPL,V,NORM = 3.92 mmHg, and RF/R = 0.8198 (i.e., 1.50 * RpjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 231 o d o Rd/l Figure B.78: Objective function "surface", OBJi}v, with PPL^NORM — 22.54 mmHg, PPLyjioRM = 3.92 mmHg, and RF/R = 1.0930 (i.e., 2.66 * R^jR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 232 d/1 Figure B.79: Objective function "surface", OBJ^, with PPL,A,NORM = 22.54 mmHg, PpL,V,NORM = 3.92 mmHg, and RF/R = 1.0930 (i.e., 2.00 * RpjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 233 c c OO 02 0.4 0.6 0.8 1.0 R d / l Figure B.80: Objective function "surface", OBJcomb, with PPL,A,NORM = 22.54 mmHg, PPLXNORM = 3.92 mmHg, and RF/R = 1.0930 (i.e., 2.00 * R^p), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 234 d/1 Figure B.81: Objective function "surface", OBJiiv, with PPL,A,NORM '= 24.54 mmHg, PpL,vjfORM = 5.92 mmHg, and RF/R = 0.3295 {i.e., 1.01 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces 235 R d/1 Figure B.82: Objective function "surface", OBJni, with PPL,A,NORM = 24.54 mmHg, PPL,VINORM = 5.92 mmHg, and R F / R = 0.3295 (i.e., 1.01 * Rp%), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 236 R d/1 Figure B.83: Objective function "surface", OBJcomb, with PPL,A,NORM = 24.54 mmHg, PPL,vTNORM = 5.92 mmHg, and RF/R = 0.3295 (i.e., 1.01 * RpjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 237 R dA Figure B.84: Objective function "surface", OBJifv, with PPL,A,NORM — 24.54 mmHg, PPL,V,NORM = 5.92 mmHg, and RF/R = 0.3588 (i.e., 1.10 * RFjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 238 o R d/1 Figure B.85: Objective function "surface", OBJn,, with PPL,A^ORM — 24.54 mmHg, PPL,V,NORM — 5.92 mmHg, and R F / R = 0.3588 (i.e., 1.10 * R^jR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 239 R d / l Figure B.&6: Objective function "surface", OBJCOTnb, with PPL,A,NORM — 24.54 mmHg, PPLXNORM = 5.92 mmHg, and RF/R = 0.3588 (i.e., 1.10 * Rp}}\), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 240 Figure B.87: Objective function "surface", OBJi^.. with PPL,A,NORM — 24.54 mmHg, PPL,V,NORM = 5.92 mmHg, and RF/R = 0.4241 (i.e., 1.30 * RpjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 241 o R d/1 Figure B.88: Objective function "surface", OBJn,, with PPL,A,NORM = 24.54 mmHg, PPL,V,NORM — 5.92 mmHg, and RF/R = 0.4241 (i.e., 1.30 * RpjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 242 o d R d/1 Figure B.89: Objective function "surface", OBJ^f,, with PPL,A,NORM = 24.54 mmHg, PPL,v,NORM = 5.92 mmHg, and RF/R = 0.4241 (i.e., 1.30 * RF}R), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 243 Figure B.90: Objective function "surface", OBJ^, with PPL,A,NORM — 24.54 mmHg, PPL,V,NORM = 5.92 mmHg, and RF/R = 0.4893 (i.e., 1.50 * Rp}R), (con-sisting of three alternative viewing- angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces 244 q o R d/1 Figure B.91: Objective function "surface", OBJxi{, with PPL,A,NORM = 24.54 mmHg, PPL,V,NORM — 5.92 mmHg, and # F / / i = 0.4893 (i.e., 1.50 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface).-Appendix B. PLM OBJ "Surfaces" 245 0.0 0.2 0.4 0.6 0.8 1.0 R d/1 Figure B.92: Objective function "surface", 0BJcornb, with PPL,A,NORM — 24.54 mmHg, PPL,V,NORM = 5.92 mmHg, and RF/R = 0.4893 (i.e., 1.50 * Rp/p), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 246 Figure B . 9 3 : Objective function "surface", OBJIFV, with PPL,A.NOTUI = 24.54 mmHg, PPLXNORM = 5.92 mmHg, and RF/R = 0.6524 (i.e., 2.00 * R™R), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 247 3 b R d/1 Figure B.94: Objective function "surface", OBJn,, with PPL,A,NORM — 24.54 mmHg, PPL,V,NORM — 5.92 mmHg, and it>/H = 0.6524 (i.e., 2.00 * Rp}R), (con-sisting of three alternative -viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 248 R dA Figure B.95: Objective function-''surface'', OBJ^^, with PPL,A,NORM = 24.54 mmHg, PPL,V,NORM = 5.92 mmHg,, and RF/R = 0.6524 (i.e., 2.00 * RFjR), (con-sisting of three alternative viewing: angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 249 o d 0.0 0.2 0.4 0.6 0.8 1.0 R d/1 Figure B.96: Objective function "surface", 0BJifv, with PPL,A,NORM = 26.54 mmHg, PPLXNORM = 7.92 mmHg, and RF/R = 0.1625 (i.e., 1.01 * RpjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 250 o d R d/1 Figure B.97: Objective function "surface", 0 5 J n ; , with PPL,A,NORM = 26.54 mmHg, PPL,V,NORM = 7.92 mmHg, and RF/R = 0.1625 (i.e., 1.01 * RFj^), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 251 R d/1 Figure B.98: Objective function "surface", OBJ^j,, with PPL,A,NORM = 26.54 mmHg, PPLWORM = 7.92 mmHg, and R F / R = 0.1625 (i.e., 1.01 * R^R), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 252 Rd/l Figure B.99: Objective function "surface", OBJiiv, with PPL,A,NORM = 26.54 mmHg, PPL,V,NORM = 7.92 mmHg, and R F / R = 0.1769 (i.e., 1.10 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. P L M OBJ "Surfaces 253 R d/1 Figure B.100: Objective function "surface", 0i?J n . , with PPL,A,NORM — 26.54 mmHg, PpLyjtoRM = 7.92 mmHg, and RF/R = 0.1769 (i.e., 1.10 * RFjR), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 254 R d/1 Figure B.101: Objective function "surface", OBJCOTnb, with PPL,A,NORM = 26.54 mmHg, PPLyjfORM- = 7.92 mmHg, and RF/R = 0.1769 (i.e., 1.10 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 255 Figure B.102: Objective function "surface", OBJIFV, with PPL,A,NORM = 26.54 mmHg, PpLyjroRM = 7.92 mmHg, and R m = 0.2091 (i.e., 1.30 * (con-sisting of three alternative viewing-angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" - 256 O o R dA Figure B.103: Objective function "surface", OBJUI, with PPL,A,NORM = 26.54 mmHg, PpL,v,NORM = 7.92 mmHg, and R F / R = 0.2091 (i.e., 1.30 * R^jR), (con-sisting of three alternative viewing-angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 257 Rd/l Figure B.104: Objective function "surface", OBJcombi with PPL,A,NORM = 26.54 mmHg, PPLtvjfORM = 7.92 mmHg, and R F / R = 0.2091 (i.e., 1.30 * R%}%), (con-sisting of three alternative viewing-angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ- "Surfaces" 258 R d / l Figure B.105: Objective function "surface", OBJifv, with PPL,AJJORM = 26.54 mmHg, PPL,v,NORM = 7.92 mmHg, and RF/R = 0.2413 (i.e., 1.50 * Rp%), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces 259 0 Figure B.106: Objective function "surface", OBJ^, with PPL,A,NORM — 26.54 mmHg, PpLyTNORM = 7-92 mmHg, and RF/R = 0.2413 (i.e., 1.50 * RFjR), (con-sisting of three alternative viewing—angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 260 Figure B.107: Objective function "surface", OBJ^t, with PPL,A,NORM = 26.54 mmHg, PPLXNORM = 7.92 mmHg, and RF/R = 0.2413 (i.e., 1.50 * Rpjp), (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces" 261 R d/1 Figure B.108: Objective function ^surface", OBJ^, with PPL,A,NORM — 26.54 mmHg, PpL,v,NORM = 7.92 mmHg, and RF/R = 0.3217 (i.e., 2.00 * RFjR), (con-sisting of three alternative viewing-angle three-dimensional plots and a con-tour map of the surface). 262 q R d/l Figure B.109: Objective function.i'surface", OBJ*, with PPL,A,NORM = 26.54 mmHg, PpLyjioRM = 7.92 mmHg, and J2F / f l = 0.3217 (i.e., 2.66 * (con-sisting of three alternative viewing angle three-dimensional plots and a con-tour map of the surface). Appendix B. PLM OBJ "Surfaces 263 R d/1 Figure B.110: Objective function "surface", OBJcomb, with PPL,A,NORM — 26.54 mmHg, PPL,V,NORM = 7.92 mmHg,..and R F / R = 0.3217 (i.e., 2.00 * RF}R), (con-sisting of three alternative viewing-angle three-dimensional plots and a con-tour map of the surface). Appendix C SM Albumin Infusion Simulations. 264 Appendix C. SM Albumin Infusion Simulations. 265 Qi) C U 30.0 20.0 10.0 0.0 1 1 — 0.0 24.0 48.0 Time(hrs) 72.0 96.0 Q0 5.0 4.0-£ 3.0-1 2.0 0.0 24.0 F 48.0 Time(hrs) — i — 72.0 96-0 ao 24.0 48.0 Time(hrs) 72X) 96 •5? 0.0 24.0 48.0 Time(hrs) 72J0 96-0 48.0 Time(hrs) 96.0 Figure C.lll: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the 0-9JCOTnb "best-fit" Star-ling Model at a PPLIC,NORM of 11 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables mon-itored). Appendix C. SM Albumin Infusion Simulations. 266 w fr-iz: o a 250 20.0 15.0 10.0-5.0 0.0-24.0 TIME<hrs) P L -> 600 0 CO 400.0-200.0 OJO S -200.0 0.0 — I — 24.0 48.0 TIME(hrs) 72.0 96.0 •z. w o w f-o cc a, 150.0 50.0 oo H— 0.0 24.0 TIMEQirs) _ Q Q i PL w O a. 0.0 -2J) 24.0 48.0 TIME(hrs) TIME(hrs) Figure C.112: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the 0BJcomb "best-fit" Starling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #1. Appendix C. SM Albumin Infusion Simulations. 267 o.o 24.0 48.C Time(hrs) 72.0 96.0 5.0-00 4 .0 -3 .0 - . 2 .0- — I 24.0 — I 7 2 0 0.0 4ao Time(hrs) 96.0 5.0-O 4.0-III . > 3.0-2-0 H , r 0.0 24.0 48.0 7 2 0 96.0 TirneOirs) i i 1 i 1 0.0 24.0 4 a 0 72.0 96.0 Time(hrs) 30.0 ^ 20.0 CU) O 1 0 . 0 0.0 1 1 1 1 1 0.0 24.0 48.C 7 2 0 96.0 Time(hrs) Figure C.113: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Star-ling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #2 (the dotted lines represent the final steady-state values of the variables mon-itored). Appendix C. SM Albumin Infusion Simulations. 268 cu E -z o u V PL TIME(hrs) -> 600.0 CO w 2 0 0 . 0 -ao B - 2 0 0 . 0 1 48.0 TIME(hrs) 96-0 O to E -O CC CL 50.0-TIME(hrs) Q PL u CO CO X CO E -O CC a. 48.0 TIME(hrs) Q F Q D Q L 48.0 TIME(hrs) 77 77, PL Figure C.114: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the 0BJcomb "best-fit" Starling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #2. Appendix C. SM Albumin Infusion Simulations. 269 30-0 48.0 Time(hrs) 96.0 5.0-X £ 4.0-£ 3-0-2.0- —I 24.0 0.0 48.0 Time(hrs) 72.0 96-0 I I i 0.0 24.0 48.0 72.0 96.0 Tlme(hrs) 0.0 24.0 48.0 Time(hrs) 72.0 96.0 30.0-20.0-U ,00-0.0-0.0 24.0 48.0 Time(hrs) 72.0 96.0 Figure C.115: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the 0 B J C O T n b "best-fit" Star-ling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #3 (the dotted lines represent the final steady-state values of the variables mon-itored). Appendix C. SM Albumin Infusion Simulations. 270 TlME(hrs) PL — j 6 0 0 . 0 -2 0 0 . 0 ri oo 1 - 2 0 0 . 0 -0 .0 48.0 TIMEXhrs) e 03 CC P w cc IX 480 TIMEXhrs) 7T PL Figure C.116: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJ^b "best-fit" Starling Model at a PPL,C,NORM of 11 mmHg using compliance relationship #3. Appendix C. SM Albumin Infusion Simulations. 271 i 1 1 1 1 0.0 34.0 48.0 72.0 96.0 Time(hrs) 5.0 4X1 3.0 Z.0 0-0 —I 34.0 48.0 TirneOirs) 72-0 96-0 48.0 Time(hrs) 96.0 s s < 20.0 Time(hrs) 96.0 48.0 Time(hrs) 96.0 Figure C.117: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Star-ling Model at a PpL.cjiORM of 10 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables mon-itored). Appendix C. SM Albumin Infusion Simulations. 272 w f— z o o 3 6-23.0 20.0 15.0-10.0 3.0 0.0 48.0 TIME(hrs) -> 600.0 £ U) 4 0 0 0 ^ 200.0 £3 -200.0-S> 0.0 — I 24.0 TIME(hrs) 72.0 DO z o o w o ce a, 150.0 480 TIME(hrs) 0, in w w E-O a. 4.0-2.0-0.0 48.0 TIME(hrs) 96.0 QD Ox £ g u w CC 23.0 20.0 15.0 10 0 5.0-0.0 TIME(hrs) 7T 7T, PL PL.C Pi Figure C.118: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJ^f, "best-fit" Starling Model at a PPL,C,NORM of 10 mmHg using compliance relationship #1. Appendix C. SM Albumin Infusion Simulations. 273 3 0 . 0 4 8 . 0 Time(hrs) 96.0 5 . 0 -6 4 . 0 -B. 3 . 0 -2 . 0 -0 . 0 24 .0 4 8 . 0 Time(hrs) 7 2 . 0 9 6 . 0 5.0 -n 4 . 0 -> 3 . 0 -2.0 -1 1 i > 0 . 0 24 .0 4 8 . 0 7 2 . 0 9 6 . 0 TimeQirs) 30.0-1 o.o H 1 1 1 • 0 . 0 24 .0 4 8 . 0 7 2 , 0 9 6 . 0 Time(hrs) Figure C.119: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^b "best-fit" Star-ling Model at a PPL,C,NORM of 10 mmHg using compliance relationship #2 (the dotted lines represent the final steady-state values of the variables mon-itored). Appendix C. SM Albumin Infusion Simulations. 274 to z o 3 20.0-15.0-10.0-5.0 0.0 TIMEXhrs) P L oo <oo.° S 200.0-ri 0.0 Q -200.0 3 'J 48.0 TIMEXhrs) 96.0 SO 2 w o ca E-o cc 150.0-100.0 50.0 0.0 4 8 . 0 TIMEXhrs) Q PL u in bi X 1 CO S cc a 2.0-0.0 4 8 . 0 TIMEXhrs) QD QL TIME(hrs) Figure C.120: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the 0BJcomb "best-fit" Starling Model at a PPL,C,NORM of 10 mmHg using compliance relationship # 2 . Appendix C. SM Albumin Infusion Simulations. 275 Figure C.121: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^b "best-fit" Star-ling Model at a PPL,C,NORM of 10 mmHg using compliance relationship #3 (the dotted lines represent the final steady-state values of the variables mon-itored). Appendix C. SM Albumin Infusion Simulations. 276 PL TlME(hrs) — 600.0 CO X 200.0-= - 2 0 0 J T 48-0 TIME(hrs) CO I , , 0 0 24.0 48.0 72.0 96.0 TIME(hrs) Figure C.122: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJ^b "best-fit" Starling Model at a PPL,C,NORM of 10 mmHg using compliance relationship #3. Appendix C. SM Albumin Infusion Simulations. 277 o.o 24.0 48.0 Time(hrs) 72.0 96.0 a 5.0 4.0 3.0 H 2.0 0.0 24.0 48.0 Time(hrs) 72.0 96.0 5.0 © 4.0-> 3.0 2.0-f * 0.0 24.0 48.0 Time(hrs) 72.0 96.0 24.0 48.0 Time(hrs) 96.0 30.0-,—„ 20.0-ClS 10.0-< 0.0-0.0 24.0 48.0 Tirne(hrs) — i — 72.0 960 Figure C.123: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Star-ling Model at a PPL,C^ORM of 9 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 278 w f-z o o Q i 25.0 20 0 15 0-10.0 50 0.0 48.0 TIMEChrs) 96.0 PL 1 -3 600.0 CO 3 fa, O 400.0-200.0 0.0 -200.0 0.0 24.0 48-0 TIMEChrs) 96-0 CO f-Z w E-z o o w o cc a. o.o 24.0 — I 48.0 TIME(hrs) Q, PL Q i u CO X 3 a. o cc a. -- J 1——• 1 1 0.0 48.0 TIMEChrs) 96.0 QD 6l £ "£3 cc w cc a. 25.0 20.0-15-0-10.0 5.0-0.0 0.0 48J> TIMEChrs) 7T 7T, PL Pi>L.C p ; _ " Figure C.124: Effect of Koomans ' a lbumin infusion on a l l microvascular ex-change variables. For the O B J ^ b "best-fit" Star l ing M o d e l at a PPL,C,NORM of 9 m m H g using compliance relationship # 1 . Appendix C. SM Albumin Infusion Simulations. 279 Time(hrs) x-s 5.0 3 S 3.0 2.0 0.0 24.0 48-0 Time(hrs) —i— 72.0 96-0 5.0 20.0-15.0-<— 10X1-5.0-0.0-Time(hrs) TimeOirs) 96.0 96.0 3 Time(hrs) 96.0 Figure C.125. Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^b "best-fit" Star-ling Model at a PPL,C,NORM of 9 mmHg using compliance relationship #2 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Mbumin Infusion Simulations. 280 U3 t-2 O Q I 23.0 20.0 15.0-10.0-5.0 0.0 48.0 TIMEQirs) P L ~> 600.0 >< 200. Ei o.o 1.0- / -200.0-0.0 48.0 TIME(hrs) 72-0 9&J> J L o o w £-O cs Cu 150.0 48.0 TIME(hrs) Qi P L Q i CO =2 w t-o Cu Cu 4-0 2.0 -2.04 0.0 _0F_ Q D Q L 48.0 TIMEChrs) a >^ CO CO w cc cu 48-0 TIME(hrs) 7T 7T P L I Figure C.126: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJcomb "best-fit" Starling Model at a PPL,C,NORM of 9 mmHg using compliance relationship #2. Appendix C. SM Albumin Infusion Simulations. 281 5.0-f 4.0-g 3.0-2.0-0.0 24.0 4ao Time(hrs) 72-0 96.0 ao-0-0 24.0 4ao Time(hrs) 72.0 96.0 6 <3 0.0 24.0 48.0 Time{hrs) 72-0 96.0 30.0 20.0 O 10.0 4 0.0 0.0 24.0 480 Time(hrs) 72.0 96.0 Figure C.127: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the 0BJcomh "best-fit" Star-ling Model at "a PPL<C,NORM of 9 mmHg using compliance relationship #3 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 282 E-Z o u 23.0 20.0 15.0 10.0 Q 5.0 0.0 2 4 . 0 48.0 TIME(hrs) ' PL 1 eoao 400.0-200J ao S -200.0-5^ o.o " I .0- / 24.0 — I — 48.0 TIME(hrs) 720) CO 150.0 z o o w o a Cu 100.0 50.0 — I 24.0 48.0 TIME(hrs) 'i 72-0 96J> PL Qi CO EC X 3 W E— O CC Cu 48.0 TIME(hrs) 72J0 • QF QD QL TIME(hrs) Figure C.128: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJ comb "best-fit" Starling Model at a PPL,C,NORM of 9 mmHg using compliance relationship #3. Appendix C. SM Albumin Infusion Simulations. 283 96.0 Time(hrs) OS 5.0-4 .0 -2.0-0J0 24.0 4 a o Time(hrs) 72.0 96.0 5-0-4 .0-3 .0-2.0-0-0 24.0 4ao Time(hrs) 72.0 96-0 4 a o Tirne(rirs) 96.0 0.0 24.0 48.0 Time(hrs) 7 2 0 96.0 Figure C.129: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^ "best-fit" Star-ling Model at a PPL,C,NORM of 8 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 284 § 20.0-z o o 15.0-10.0-5.0-0 .0 -V PL TIME(hrs) •p 600.0 § 400.0-X fc. 200.0 0.0-9 - 200 .0 •J 0.0 48.0 TIME(hrs) J L CO E-O o o d c 48.0 TIME(hrs) _ Q Q i PL 'A ~ 0.0 - ^ o 4 8 0 TIME(hr5) QP_ QD QL CO cr CO CO CO cc Q. 20 .0 -15.0- ) r - . 10.0-1' 5.0-0.0-0.0 48.0 TlME(hrs) 9 6 . 0 77 PL 7T, p t . c Figure C.130: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the 0BJcomb "best-fit" Starling Model at a PPL,C,NORM of 8 mmHg using compliance relationship #1. Appendix C. SM Albumin Infusion Simulations. 285 Time(hrs) ^ 5.0 -j 1 1 1 1 OO 24.0 48.0 72.0 96-0 Time(hrs) 5.0 3 4 . 0 -> 3.0 Ir— 2.0-Q-0 24 O 4 a o Time(hrs) — i — 72.0 9 6 . 0 O L O 24.0 4 a o Time(hrs) 72.0 96 . 0 30.0-^ 20 .0-u 1 0 . 0 -o.o-o.o 2 4 . 0 48.0 Time(hrs) 72.0 96.0 Figure C.131: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Star-ling Model at a PPL,C,NORM of 8 mmHg using compliance relationship #2 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 286 w E -z o CJ p 5 20.0-15.0-10.0 5.0-0.0 TlME(hrs) 0J0 48.0 TIME(hrs) 72.0 CO E -•z. 8 w E -O CC Cu 150.0 100.0-0.0 48.0 TIME(hrs) Qi P L CO W Ed E -O CC Cu 4.0-2.0-48.0 TIMEChrs) QD Q L w a: :=> CO co w cc a. 25.0 20.0-15.0 10.0 H 5.0 0.0 48.0 TIMEChrs) 7T P L 7T, P L . C Figure C.132: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJ^b "best-fit" Starling Model at a PPLtc,NORM of 8 mmHg using compliance relationship #2. Appendix C. SM Albumin Infusion Simulations. 287 X 48.0 Time(hrs) 96.0 ^ 5.0-g 4.0-3-0-2-0- — I 72.0 0.0 24.0 48.0 Time(hrs) 96-0 5-0-4.0-3.0-2-0-1 1 , 1 0.0 24.0 4S.0 72.0 96.0 Time(rirs) i 1 1 1 1 0.0 24.0 48.0 72.0 96.0 Tirne(hrs) Figure C.133: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the 0BJcomb "best-fit" Star-ling Model at a PPL,C,NORM of 8 mmHg using compliance relationship #3 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 288 O : 4.U-CO CO X 2.0-S 5 Cu 48.0 TIMEChrs) QE QL -u 25.0 £ CO IX z> cn co co cc Cu 48-0 TIMEChrs) r r , PL Figure C.134: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the 0BJcomb Hbest-fit" Starling Model at a PPL,C,NORM of 8 mmHg using compliance relationship #3. Appendix C. SM Albumin Infusion Simulations. 289 i 1 1 1 1 0.0 24.0 48.0 72.0 96.0 Time(hrs) ^ 5.0-g 4.0-3.0-2.0 -0.0 24.0 48.0 Time(hrs) 72.0 96.0 Time(hrs) 96.0 Figure C.135: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJCOMB "best-fit" Star-ling Model at a PPL<C,NORM of 7 mmHg using compliance relationship #1 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 290 i-Z u f-z o o 0.0- 48.0 TlME(hrs) V P L 48.0 TIME(hrs) 96.0 v. z H O cj CO E-O ce 150.0 100.0-50.0 0J0 48.0 TlME(hrs) Q i PL Q. u CO CO X 3 CO o ce CL 0.0 48.0 TIME<hrs) 0F_ QD QL TIME(hrs) Figure C.136: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the 0BJcomb "best-fit" Starling Model at a PPL,C,NORM of 7 mmHg using compliance relationship #1. Appendix C. SM Albumin Infusion Simulations. 291 Time{hrs) 5.0 4.0 3.0-f 2.0 0.0 24.0 46.0 Time(hrs) 72.0 96-0 5.0-r 4.0-> 3.0-20-0.0 24.0 4 S . 0 Time(hrs) 72.0 960 Figure C.137: Effect of Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^b "best-fit" Star-ling Model at a PPL,C,NORM of 7 mmHg using compliance relationship #2 (the dotted lines represent the final steady-state values of the variables monitored). Appendix C. SM Albumin Infusion Simulations. 292 H 2 UJ 2 O O 48.0 TlME(hrs) V PL CO w 48.0 TIMEChrs) CO to X => -J co H O CC CL. 48.0 TIMEChrs) _Qr_ QD QL TIMEChrs) Figure C.138: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the O B J ^ b "best-fit" Starling Model at a PPL,CJ^ORM of 7 mmHg using compliance relationship #2. Appendix C. SM Albumin Infusion Simulations. 293 oo 2 4 . 0 +8 .0 Time(hrs) 7 2 . 0 9 6 . 0 GO 5 . 0 -4 . 0 -3 . 0 -2 . 0 -0X1 2 4 . 0 48.0 Time(hrs) 7 2 0 9 6 . 0 5 . 0 -> 3 . 0 -2 . 0 -ao 2 4 . 0 4ao TimeQhrs) 72X1 9 6 . 0 3 2 0 X 1 -1 5 . 0 -c 1 0 0 " g ' 5 . 0 -< 0 . 0 -0 .0 24X1 —1 4 8 . 0 Time(hrs) 72X1 9 6 . 0 3 0 . 0 \ 2 0 . 0 GO 0 . 0 T : 1 1 1 1 0 .0 2 4 . 0 4 8 . 0 7 2 . 0 9 6 . 0 Time(hrs) Figure C.139: Effect of K o o m a n s ' a l b u m i n infusion o n select m i c r o v a s c u l a r ex-c h a n g e variables (includes K o o m a n s ' d a t a ) . F o r the OBJ^mf, "best- f i t" S t a r -l i n g M o d e l at a PPL,C,NORM of 7 m m H g using c o m p l i a n c e r e l a t i o n s h i p #3 (the d o t t e d lines represent the final steady-state values of t h e variables m o n i t o r e d ) . Appendix C. SM Albumin Infusion Simulations. 2 9 4 to z o o TlME(hrs) t. I to eg X 3 600-0 200.0 til -200X1 3 48.0 TIMEChrs) CO t o X. o c c CL. — I — 24X1 48.0 TIMEChrs) QF_ QD "QL £ t o c c CO CO t o a . 71 PL 77, PL.C Figure C.140: Effect of Koomans' albumin infusion on all microvascular ex-change variables. For the OBJcomb. "best-fit" Starling Model at a PPL,C,NOR\I of 7 mmHg using compliance relationship #3. Appendix D P L M Albumin Infusion Simulations. 295 Appendix D. PLM Albumin Infusion Simulations. 296 96.0 Time(hrs) .BP 5.0 24.0 48.0 Time(hrs) 72.0 96-0 i 1 1 1 1 0.0 24.0 48.0 72.0 96.0 Time(hrs) GO X e 20.0 15.0 H 10-0 5.0 0-0 48.0 Time(hrs) 72X1 96.0 48.0 Time(hrs) 72.0 96.0 Figure D.141: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ comb "best-fit" Plasma Leak Model with a PPL,A,NORM of 22.54 mmHg, a PPL,V^ORM of 3.92 mmHg. an RF/R of 0.7104, Rd/i of 0.81 and a LS of 31.8 ml/(mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 297 25.0 ^ TIME(hrs) Figure D.142: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the OBJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 22.54 mmHg, a PPL,VJ{ORM of 3.92 mmHg, an RF/R of 0.7104, Rd/i of 0.81 and a LS of 31.8 m £ / ( m m H g . h r ) . Appendix D. PLM Albumin Infusion Simulations. 298 Time(hrs) 8 5.0 4.0 3.0-2.0 0.0 24.0 48.0 Time(hrs) — i — 72.0 96-0 5.0 4.0-O.0 24.0 48.0 72.0 96-0 Time(hrs) 48.0 Time(hrs) 96.0 30.0 ^ 20.0 g iQ.o-1 0.0 0.0 24.0 48.0 Time(hrs) — i — 72.0 96.0 Figure D.143: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ^b "best-fit" Plasma Leak Model with a PPL,A,NORM of 24.54 mmHg, a PPL,V,NORM of 5.92 mmHg, an RF/R of 0.4241, Rd/i of 0.55 and a LS of 31.8 m£/(mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 299 23.0 TIMEChrs) TIMEChrs) Figure D.144: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the 0BJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 24.54 mmHg, a PPL,V,NORM of 5.92 mmHg, an RF/R of 0.4241, Rd/i of 0.55 and a LS of 31.8 ml/(mmHg.hr). Appendix D. PLM Albumin Infusion Simulations. T I 1 1 I 0-0 24.0 48.0 7ZJ0 96.0 Time(hrs) • 1 ! 1 1 0-0 24.0 48.0 72J0 96.0 Time(hrs) Figure D.145: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Plasma Leak Model with a PPL,AJ*ORM of 24.54 mmHg, a PPL,V,NORM of 5.92 mmHg, an RF/R of 0.4241, RD/T of 0.80 and a LS of 31.8 ml/(mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 301 CO t-z o o o 5 TlME(hrs) P L ~ 600.0-co w X g p 3 200. tH -2O0-O-0.0 •'PU. 24.0 — I 48.0 96.0 TlME(hrs) CO E— Z O o CO E-o cr a SOLO 48.0 TlME(hrs) Q Qi PL CO CO X 3 CO E-O CL. 48.0 TIME(hrs) Q PLL QD QL CC CO cr CO CO CO cc c 7T 7T, PL P, 48 0 TlME(hrs) Figure D.146: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the OBJ^b "best-fit" Plasma Leak Model with a PPL^ORM of 24.54 mmHg, a PPL,V,NORM of 5.92 mmHg, an RF/R of 0.4241, RD/T of 0.80 and a LS of 31.8 m^/(mmHg.hr). Appendix D. PLM Albumin Infusion Simulations. 302 <3 0.0 24.0 48.0 Time(hrs) 720 96.0 Figure D.147: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 24.64 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2091, RD/, of 0.63 and a LS of 38.6 m£/(mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 303 w f-z o c j Q 25.0 20.0 15.0 10.0 48.0 TIMEChrs) ' P L •> 6 o a o m ••oo.o S 200.0 ri 0X1 S -200.0 0.0 48.0 TIME(hrs) 'PLL TIMEChrs) Figure D.148: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the 0BJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2091, Rd/i of 0.63 and a LS of 38.6 m£/(mmHg.hr). Appendix D. PLM Albumin Infusion Simulations. 304 2-0 H , 1 1 1 0.0 24.0 48.0 72X1 96-0 TimeOirs) 96.0 0.0 24.0 48.0 7 2 0 96.0 T1me(hrs) Figure D.149: Effect of a Koomans1 albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the 0BJcomb "best-fit" Plasma Leak Model with a PPLJI.NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2091, Rd/i of 1.00 and a LS of 38.6 m£ / (mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 305 p. oo § 200 Z 15.0 8 100 S i 0 " 3 0.0 4ao TIMEChrs) V PL ! to ri a 6oao-400 0 200.0-0-0 -200-0 0X> 48Xt TIME(hrs) 72.0 'PLL CO 2 O o w O tx Cu 150.0 24.0 48.0 TIMEChrs) 86.0 Qi PL Q i <-< 4X1-to ta X 2JD-•J tu 0-0-g s t- -2-0-o ( cc Cu 48.0 TIMEChrs) Q PLL QD QL IU cc Z> CO CO b] ec Cu 25.0 200-15.0 iao 5.0 0X1 48.0 TIMEChrs) 77, PL 77, PL,V P. Figure D.150: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the OBJcomb "best-fit" Plasma Leak Model with a PpL,AJ*ORM of 26.54 mmHg, a PpLyjtoRM of 7.92 mmHg, an RF/R of 0.2091, Rd/l of 1.00 and a LS of 38.6 m£/(mmHg.hr). Appendix D. PLM Albumin Infusion Simulations. 306 0.0 24.0 48.0 72.0 96.0 Time(hrs) 0.0 24.0 48.0 72.0 96.0 Time(hrs) 3ao-20.0-u 10.0-0.0-< 0.0 24.0 48.0 Time(hrs) 72.0 96.0 Figure D.151: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the 0BJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2413, Rd/l of 0.45 and a LS of 31.8 mf/(mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 307 & 25.0-§ 20.0-Z 15.0 H 8 J0.0-e 5.0-PL 48.0 TIME(hrs) -> 600.0 n 200.0-'PLL & , OJ3-9 -200.0 -24.0 48.0 TIME(hrs) 72.0 TIME(hrs) Figure D.152: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the OBJcomb "best-fit" Plasma Leak Model with a PPL,A,NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2413, Rdll of 0.45 and a LS of 31.8 m£/(mmHg.hr). Appendix D. PLM Albumin Infusion Simulations. 308 o.o 24 .0 4 8 . 0 TimeOirs) 72 .0 9 6 . 0 ofl X 60 X £ £ fe 5 . 0 4X> 3 . 0 -zn 0 - 0 2 0 . 0 15 .0 10 .0 5.0 - t 0 .0 24 .0 —I 48.0 7 2 0 9 6 . 0 Time(hrs) 9 6 . 0 Time(hrs) 9 6 . 0 Time(hrs) 0 . 0 24 .0 48.G TimeOirs) 9 6 . 0 Figure D.153: Effect of a Koomans' albumin infusion on select microvascular ex-change variables (includes Koomans' data). For the OBJ comb "best-fit" Plasma Leak Model with a PPL,A,NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2413, R D / L of 0.70 and a LS of 31.8 ml/ (mmHg.hr) (the dotted lines represent the final steady-state values of the variables monitored). Appendix D. PLM Albumin Infusion Simulations. 309 to. E~ z o o Q i 25.0 20.0 ISO 10.0-5-0 0.0 48.0 TIME(hrs) •7ZO V, PL. -> eoao co w 400.0-2000 H -2000 24.0 48.0 TIME(hrs) 72.0 'PLL o o w o 150.0 50.0-—I 24.0 —I 48.0 TIMEChrs) PL Q i CO CO s CO J— o (X. Cu 0.0 48.0 TIMEOirs) Q PU. Q D Q L CO cc CO (X Cu 25.0 20.0-15.0 10.0-5.0 0 0 0.0 TIME(hrs) 96-0 7T PL 7T , Figure D.154: Effect of a Koomans' albumin infusion on all microvascular exchange variables. For the 0BJcomb "best-fit" Plasma Leak Model with a PpL,A,NORM of 26.54 mmHg, a PPL,V,NORM of 7.92 mmHg, an RF/R of 0.2413, Rd/t of 0.70 and a LS of 31.8 ml/(mmHg.hr). Appendix E Computer Programs E. l Parameter List of Steady-State and Transient Simulators c ***************************************************** C A l i s t of program variables of a l l programs l i s t e d C ***************************************************** c C VARIABLE KEY: C C ACC = Accuracy l i m i t C ALPHA = Scaling factor f o r ROOT C UV = Number of N.L.E's C PIPL = Plasma c o l l o i d osmotic pressure_set C PIPLAS= " " " " _calc C PISKIN= Tissue " " " _calc C PSHRM = Normal tissue hydrostatic press C QDIF = Change from normal W.body protein cont. C QPLS = Plasma:Albumin content_calc. C pSS = tissue:Albumin content_calc. C QPLNRM= Plasma:normal Albumin content C QSNRM = tissue:normal Albumin content C QTOT = Whole body:Albumin content C RHSF = N.L.E's to be solved C VDIF = Change from normal body f l u i d cont. C VPLS = Plasma:volume_calc. 310 Appendix E. Computer Programs 311 C VSS = tissue:volume_calc. C VPLKRM= Plasma:normal volume C VIFNRM= tissue:normal volume C VTNRM = Normal t o t a l extra-cellular volume C VTOT = Whole body :volume C YHEW = I n i t i a l solution estimates C YOLD = Final solution values C CAVS = Protein concentration i n a v a i l , v o l . C CPL = " " " plasma C CS = " " " tissue C JFS = Fluid f i l t r a t i o n flowrate C JRS = F l u i d reabsorption " " C JPLLS = Flu i d plasma leak " " C JLS = Flu i d lymph C PC = Capillary hydrostatic press. C PIPL = Plasma c o l l o i d osmotic pressure C PIS = Tissue " " C QDS = Diffusive protein flowrate C QLS = Lymphatic C qFS = F i l t r a t i o n C AM = I n i t i a l gradient of muscle compliance curve C AS " " skin C BH Final " " muscle C BS = " " " skin " C CECF1.= Coeff's of ECFV vs. C.O.Ppl relationship C CPI1..= " " COP.vs.(Alb.) relationship C cqTl..= " " t o t a l Alb.cont. vs. C.O.Ppl relationship C relationship C VTNRM = Normal t o t a l extra-cellular volume C VTOT = Whole body :volume C YNEW = I n i t i a l solution estimates Appendix E. Computer Programs 312 c YQLD = Fina l solution values c CAVS = Protein concentration i n a v a i l , vol. c CPL " plasma c CS = " tissue c JFS F l u i d f i l t r a t i o n flowrate c JRS Flu i d reabsorption " " c JPLLS = Fl u i d plasma leak " " c JLS F l u i d lymph c PC Capillary hydrostatic press. c PIPL = Plasma c o l l o i d osmotic pressure c PIS Tissue c QDS Diffusive protein flowrate c QLS Lymphatic " " c QFS F i l t r a t i o n " " c AM I n i t i a l gradient of muscle compliance curve c AS " " " skin " " c BM Fina l " " muscle c BS " " " skin " " c CECF1.= Coeff's of ECFV vs. C.O.Ppl relationship c CPU. .= " COP.vs.(Alb.) relationship c CQT1..= " " t o t a l Alb.cont. vs. C.O.Ppl relationship c relationship c KFS Tissue f i l t r a t i o n transport coeff. c LSNRM = Tissue basal lymph flowrate c LSS Tissue lymph flow s e n s i t i v i t y c HPS Number of pts. on skin compliance curve c RPSM1 = HPS - 1 c PCNRM = Normal PC c PS Tissue hydrostatic press. c PSS Skin permeability to Alb.* s.area c of capillary Appendix E. Computer Programs 313 c QTNRM = Normal t o t a l Albumin content c SIGM = Muscle capillary n a i l Alb. re f l e c t i o n coeff c SIGS = Skin ti I I M it it c VEXM Muscle excluded volume c VEXS = Skin ti n c VIFNRM= NORMAL TOTAL INTERSTITIAL FLUID VOLUME c VM Muscle volume c VMNRM = Normal muscle volume c VPLNRM= Normal plasma volume c VS Skin ii c VIFNRM = Normal . skin volume c VTNRM = Normal to t a l extra-cellular volume E.2 Listing of Program SM-A IMPLICIT REAL*8(A-H,L,J,K,0-Z) COMMON/BLKBS/JFSS(40),JLSS(40) C0MM0N/BLKDS/QFSS(40),QDSS(40),QLSS(40) COMMON/BLKGS/PCS(40),PSST(40) COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKH/LSNRM COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKU/PSNRM COMM0N/BLKZ/DIFV(18),DPIPL(18),NN COMMON/BLKBB/PIPNRM,PISNRM COMMON/BLKK/VEXS C0MM0N/BLKA2/CPLNRM,CSNRM,CASNRM C0MM0N/BLKGG/CALIFV(40),CALPIP(30) C0MM0N/BLKHH/PIPLPIC66),PII(66) Appendix E. Computer Programs C0MM0N/BLKY/PIPPS(30) ,PISKIN(30) .NA.NAM1 C0MM0N/BLKKK/VPLS(40),VSS(40) COMMON/BLKDD/ALBSTO DIMENSION A0PT(21,21),G(1),X(2) C ******************************************* C Set i n i t i a l condtions C ***************************************************** PIPNRM = 25.9D0 PISNRH = 14.7D0 CALL SPLINS PSNRM = FCOMPS(VIFNRM) CPLNRM = FALB(PIPNRM) CASNRM = FALB(PISNRM) CSNRM = CASNRM * (1.D0 - VEXS / VIFNRM) C ********************************************** C Set LS (XA) and RDL (XB) values f or grid C points to be evaluated. C ********************************************** DO 10 1=1,21 XA = (1-1) * 10.DO + 50.DO X(l) = XA / 10.DO DO 20 J=l,21 XB = ( J - l ) * 0.1280D-1 X(2) = XB * 10.DO CALL GCOMP(G.X) AOPT(I.J) = G(l) WRITE(13,101)XA,XB,A0PT(I,J) 101 F0RMAT(3F10.5) 20 CONTINUE 10 CONTINUE STOP Appendix E. Computer Programs 315 END BLOCK DATA C **************************************** IMPLICIT REAL*8(A-H,K,L,0-Z) C **************************************** COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKJ/VTNRM,VIFNBM,VPLNRM COMMON/BLKK/VEXS COMMON/BLKN/CPIl,CPI2 COMMON/BLKO/VS(14),PS(14),AS,BS,NPS,NPSM1 C0MM0N/BLKZ/DIFV(18),DPIPL(18),NN COMMON/BLKCC/CALB1.CALB2 COMMON/BLKDD/ALBSTO C0MM0N/BLKHH/PIPLPK66) ,PII(66) COMMON/BLKZA/PRESLO COMMON/BLKZK/BIFVL.BIFVU,BPIIL.BPIIU DATA BIFVL,BIFVU,BPIIL.BPIIU/34.058273D0,87.568802D0, + 68.643616D0.126.749146D0/ DATA PCNRM,VEXS/10.D0,2.1D+3/ DATA PCGRAD/32.DO/ DATA PRESL0/13.0577D0/ DATA VTNEM,VIFNRM,VPLNRM/11.6D+3,8.4D+3,3.2D+3/ DATA CPI1,CPI2/1.75490D-4,0.656840D0/ DATA CALB1,CALB2/-0.267173D-3,0.152244D+1/ DATA NN/18/ DATA DPIPL/9.2DO,10.7DO,11.7DO,12.7DO,13.7DO,13.7DO, + 14.2D0,14.2D0,18.7D0,18.7D0,20.7D0,20.7D0, + 21.7D0,24.7D0,25.7D0,25.7D0,26.7D0,25.7D0/ DATA DIFV/19.01D+3,16.02D+3,14.40D+3,12.98D+3,7.98D+3,6.71D+3, + 10.70D+3,8.27Dt3,9.944D+3J9.11D+3,7.79D+3f6.73D+3, Appendix E. Computer Programs 3 1 6 + 6.61D+3,8.44D+3,8.34D+3,7.69D+3,8.25D+3,9.38D+3/ DATA PII/3.4D0,2.9D0,4.9D0,4.9D0,4.4D0,3.9D0, + 3.9D0,3.9D0,3.4O0,4.9D0,6.4D0,6.4D0, + 5.9DO,3.9DO,6.9DO,9.90DO,10.4DO,10.9DO, + 3.200,5.700,4.700,4.700,5.700,3.700,5.200,6.700, + 7.700,8.700, + 2.3DO,5.90,1.9D0,2.6D0,0.9D0,2.7D0,0.7OO,2.9D0, + 5.4D0,1.9D0,5.9D0, + 1.7D0,1.6D0,4.0D0,4.5D0,4.3D0,7.000,7.200,5.700, + 7.7D0,8.D0,8.6DO,8.2D0,8.4DO,7.7D0,10.7D0,11.7DO, + 12.2D0,11.7D0,12.7D0,13.2DO,14.D0,14.7DO,13.2DO, + 14.400,15.200,13.700,14.700/ DATA PIPLPI/7.D0,7.DO,9.D0,9.D0,9.5D0,9.5D0, + 11.5D0,11.5DO,13.D0,14.SD0,15.5D0, + 15.5D0,17.5D0,17.5D0,19.0D0,19.0D0,21.D0,21.D0, + 9.2D0,10.700,11.7D0.14.2D0,13.7D0,15.7D0, + 18.7DO,18.7DO,2O.7D0,21.7D0, + 2.3D0,5.3D0,5.8D0,6.6D0,7.8D0,8.3D0,9.6D0, + 10.3D0,11.3D0,12.3D0,17.3D0, + 8.9D0,9.9DO,11.9D0,12.4D0,12.9DO,13.2O0,13.9D0, + 14.2D0,17.4D0,17.9D0,18.0D0,18.2D0,18.4D0,19.4O0, + 19.9D0,23.6D0,24.D0,24.4D0,24.8D0,23.6D0,23.6D0, + 22.9D0,24.9D0,24.400,24.200,26.400,26.900/ DATA HPS,NPSM1/14,13/ DATA VS/8.4D+3,8.92D+3,9.45D+3,9.97D+3,10.50D+3,11.02D+3, + 11.55D+3,12.07D+3,12.60D+3,13.65D+3,14.70D+3,16.80D+3, + 21.00D+3.25.23D+3/ DATA PS/-0.70D0,0.32D0,0.86D0,1.15D0,1.37D0,1.56D0, + 1.69D0,1.80D0,1.88D0,1.99D0,2.0100,2.0400,2.1200, + 2.20D0/ DATA AS,BS/1.96154D-3,1.813470-5/ Appendix E. Computer Programs 317 DATA ALBSTO/0.020539DO/ END E.3 Listing of Program SM-B SUBROUTINE GCOMP(G.X) C **************************************** IMPLICIT REAL*8(A-H,K,L,J,0-2) INTEGER FLAG C0MM0N/BLKA2/CPLNRM,CSNRM,CASNRM COMMON/BLKB/JFS,JLS COMMON/BLKD/QFS,qDS,QLS COMMON/BLKG/CPL,CS,CAVS COMMON/BLKBS/JFSS(40),JLSS(40) C0MM0N/BLKDS/QFSS(40),QDSS(40),qLSS(40) C0MM0N/BLKGS/PCS(40),PSST(40) COMMON/BLKI/PIPL,PIS COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKH/LSNRM COMMON/BLKJ/VTNRM, VIFNRM, VPLNRM COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKT/VTOT,qTOT COMMON/BLKU/PSNRM C0MM0N/BLKY/PIPPS(30),PISKIN(30).NA.NAM1 COMMON/BLKBB/PIPNRM.PISNRM CDMMON/BLKDD/ALBSTO COMM0N/BLKGG/CALIFV(40),CALPIP(30) C0MM0N/BLKG5/CIFV1(200),CPIPL1(200) C0MM0N/BLKKK/VPLS(40),VSS(40) DIMENSION qPLS(40),qSS(40) Appendix E. Computer Programs DIMENSION G(1),X(2) DIMENSION Y0LD(2),YNEW(2) EXTERNAL RHSF DATA ACC,ALPHA,NV/1.D-7,0.5D0,2/ NA = 25 NAM1 = NA - 1 PIPLUL = 28.DO LSS = X(l) * 10.D0 RDL = X(2) / 10.DO CALL SPLINS C **************************************** C Set i n i t i a l f l u i d and protein contents Q **************************************** QPLG = VPLNRM * FALB(PIPNRM) QSG = (VIFNRM - 400.DO) * FALB(PISNRM) * 3.DO/4.DO VPLG = VPLNRM VSG = VIFNRM - 400.DO C **************************************** C Calcualte transport parameter values C **************************************** Wl = (CSNRM * RDL) / (CPLNRM - CASNRM) W2 = VIFNRM / (VIFNRM - VEXS) W3 = ALBSTO * VIFNRM LSNRM = W3 / (l.D0+Wl*W2) PSS = LSNRM * CSNRM * RDL / (CPLNRM - CASNRM) Z l = 2.DO * CSNRM * (l.DO - RDL) Z2 = CPLNRM + CASNRM SIGS = l.DO - (Zl / Z2) KFS = LSNRM / (PCNRM - PSNRM - SIGS * (PIPNRM - PISNRM)) DIV = 0.25D0 C *************************************** Appendix E. Computer Programs 319 C Set i n i t i a l plasma COP value C *************************************** PIPP = PIPLUL + l.DO DO 20 IN = 1,5000 PIPP = -l.DO * DIV + PIPP CPL = FALB(PIPP) Y0LD(1) = VSG Y0LD(2) = qSG C **************************************** C Solve d i f f e r e n t i a l equations C **************************************** CALL ROOT(RHSF,NV,0.DO,YOLD,ACC,ALPHA,YNEW.FLAG) VSG = YNEW(l) QSG = YNEW(2) IF(VSG.LE.0.D0.0R.QSG.LE.O.DO)THES PRINT*,'*********************************» PRINT*,'*********************************' PRINT*,'** NEGATIVE VOL/Q GENERATED ***•*' PRINT*,'*********************************» PRINT*,'*********************************' STOP ENDIF VPLG = VTOT - VSG QPLG = QTOT - QSG CPIPL1(IN)=PIPP CIFV1(IN)=VSG IF(PIPP.LT.4.D0) GOTO 64 CHEK = PIPP - IDINT(PIPP) IF(CHEK.GT.0.1D-4) . GOTO 20 Appendix E. Computer Programs 320 SNUM = -l.DO * PIPP + PIPLUL + l.DO II = IDINT(SNUM) C **************************************** C Store results corresponding to plasma COP set C **************************************** VSS(II) = VSG VPLS(II)= VPLG qss(n) = QSG QPLS(II)= QPLG PIPPS(II) = PIPP CALIFV(II) = VSS(II) CAVS = QSS(II) / (VSS(II) - VEXS) PISKIN(II) = FPI(CAVS) CALPIP(II) = FPI(CPL) JFSS(II) = JFS JLSS(II) = JLS QFSS(II) = QFS QDSS(II) = QDS QLSS(II) = QLS PCS(II) = PC PSST(II) = FCOMPS(VSS(II)) GOTO 20 50 IF(DIV.LE.1.5625D-2) GOTO 33 DIV = DIV / 2.DO SNUM = PIPP + l.DO PIPP = IDINT(SNUM) II = IDINTC -l.DO * PIPP + PIPLUL + l.DO) VSG = VSS(II) VPLG = VPLS(II) QSG = qss(n) QPLG = QPLS(II) Appendix E. Computer Programs 321 GOTO 20 33 WRITE(6,40)PIPP 40 FORMAT(IX,1 ROOT SOLVER FAILS! - WHEN PIPL = \G23.15) RETURN 20 CONTINUE 64 CALL CERROR(ERSUM) G(l) = ERSUM RETURN END SUBROUTINE CERROR(ERSUM) C *********************************** C This subroutine calculates the sum of the C squares of the errors - between calculated C and experimental values. C *********************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKZ/DIFV(18),DPIPL(18),NN C0MM0N/BLKHH/PIPLPK66) ,PII(66) C0MM0N/BLKY/PIPPS(30),PISKIN(30).NA.NAMl C0MM0N/BLKG5/CIFV1(200),CPIPL1(200) COMMON/BLKZK/BIFVL,BIFVU,BPIIL,BPIIU NPI = 66 NCP = 25 SUMPI=0.D0 DO 130 J=1,NPI IMIN=1 D1=DSQRT((PII(J)-PISKIN(1))**2+(PIPLPI(J)-PIPPS(1))**2) DMIN=1.D10 DO 120 1=2,NCP Appendix E. Computer Programs 322 D2=DSQRT((PII(J)-PISKIN(I))**2+(PIPLPI(J)-PIPPS(I))**2) DT0T=D1+D2 IF(DTOT.GE.DHIN) GO TO 110 IMIH=I DMIH=DT0T D1M=D1 D2M=D2 110 D1=D2 120 CONTIHUE IF(D1M.EQ.O.DO.OR.D2M.EQ.O.DO) GO TO 130 I=IMIN-1 IP=IMII D3=DSQRT((PISKIH(IP)-PISKIN(I))**2+(PIPPS(IP)-PIPPS(I))**2) AHG=DACOS((D1M*D1M+D3*D3-D2M*D2M)/(2.D0*D1M*D3)) SUMPI=SUMPI+D1M*DSIN(AHG) 130 COHTITOE HIFV = HB SUMIFV=O.DO DO 131 J=1,HIFV IMIH=1 Dl=DSqRT(((DTFV(J)-CIFVl(l))/14.D+3)**2+((DPIPL(J) + -CPIPL1(1))/20.0D0)**2) DMII=1.D10 DO 121 1=2,101 D2=DSQRT(((DIFV(J)-CIFVl(I))/14.D+3)**2+((DPIPL(J) + -CPIPL1(I))/20.0D0)**2) DT0T=D1+D2 IF(DTOT.GE.DMIN) GO TO 111 IMII=I DMII=DTOT D1M=D1 Appendix E. Computer Programs D2M=D2 111 D1=D2 121 CONTINUE IF(DlM.Eq.0.D0.0R.D2M.EQ.0.D0) GO TO 131 I=IMIN-1 IP=IMIN D3=DSQRT(((CIFVl(IP)-CIFVl(I))/14.D+3)**2+((CPIPLl(IP) + -CPIPL1(I))/20.0DO)**2) ANG=DAC0S((DlM*DlM+D3*D3-D2M*D2M)/(2.D0*D1M*D3)) SUMIFV=SUMIFV+D1H*DSIN(ANG) 131 CONTINUE EIFV = (SUMIFV * 30.DO - BIFVL) / (BIFVU - BIFVL) EPII = (SUMPI - BPIIL) / (BPIIU - BPIIL) ERSUM = SUMIFV*30.D0 RETURN END SUBROUTINE AUXSAM(Y) C *************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) C This subroutine calculates a l l system C variable values given compartmental C protein and f l u i d contents C **************************************** COMMON/BLKB/JFS,JLS COMMON/BLKD/QFS,QDS,QLS COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKG/CPL,CS,CAVS COMMON/BLKH/JLSNRM COMMON/BLKI/PIPL,PIS Appendix E. Computer Programs 324 COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKT/VTOT,QTOT COMMON/BLKU/PSNRM COMMON/BLKZA/PRESLO DIMENSION Y(2) VOLPLA = 3200. DO. VTOT = VOLPLA + Y(l) qPLAS = VOLPLA * CPL QTOT = QPLAS + Y(2) PC = PCNRM CS = Y(2) / Y(l) CAVS = Y(2) / (Y(l) - VEXS) PIPL = FPI(CPL) PIS = FPI(CAVS) PS = FC0MPS(Y(1)) JFS = KFS * (PC - PS - SIGS * (PIPL - PIS)) JLS = JLSNRM + LSS * (PS - PSNRM) IF(JLS.LT.JLSNRM) JLS=JLSNRM*(PS+PRESLO)/(PSNRM+PRESLO) ULJLS=10.DO*JLSNRM IF(JLS.GT.ULJLS) JLS=ULJLS QFS = JFS * (l.DO - SIGS) * (CPL + CAVS) / 2.DO qDS = PSS * (CPL - CAVS) QLS = JLS * CS RETURN END SUBROUTINE GAUSS(A,N,NDR,NDC,X,RNORM,IERROR) C **************************************** c C Purpose: C Uses Gauss elimination with p a r t i a l pivot selection to Appendix E. Computer Programs 325 c solve simultaneous li n e a r equations of form A*X=C c Argument s: c A Augmented coefficient matrix containing a l l coefficients c and R.E.S. constants of equations to be solved. c N Number of equations to be solved. c NDR F i r s t (row) dimension of A i n c a l l i n g program. c roc Second (column) dimension of A i n c a l l i n g program. c X Solution vector. c RNORM Measure of size of residual vector C-A*X c IERROR Error f l a g . c =1 Successful Gauss elimination. c =2 Zero diagonal entry after pivot selection. c *********************************** IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDR.NDC),X(N),B(50,51) NM = N - 1 NP = N + 1 C **************************************** C Set up working matrix B C **************************************** DO 20 I = 1,N DO 10 J = l.NP B(I,J) = A(I,J) 10 CONTINUE 20 CONTINUE C **************************************** C Carry out elimination process N-l times C **************************************** DO~80 K = l.NM Appendix E. Computer Programs 326 KP = K + 1 C **************************************** C Search for largest coefficient i n column K, rows K through N C IPIVOT i s the row index of the largest coefficient C **************************************** BIG = 0.D0 DO 30 I = K,N SF = DABS(B(I,K)) DO 25 J = KP.NP SF = DMAX1(SF,DABS(B(J,K))) 25 CONTINUE AB = DABS(B(I,K) / SF) IF(AB.LE.BIG) GOTO 30 BIG = AB IPIVOT = I 30 CONTINUE C **************************************** C INTERCHANGE ROWS K AND IPIVOT IF IPIVOT.NE.K C **************************************** IF(IPIVOT.EQ.K) GOTO 50 DO 40 J = K.NP TEMP = B(IPIVOT.J) B(IPIVOT.J) = B(K,J) B(K,J) = TEMP 40 CONTINUE 50 IF(B(K,K).Eq.0.D0) GOTO 130 C **************************************** C Eliminate B(I,K) from rows K+l through N C **************************************** DO 70 I = KP.N QUOT = B(I,K) / B(K,K) z ** ((wns - (dH'Dv)saya) + bsn = bsn aoniuioo o n (r)x * (r'i)v + wns = wns N'T = r OTT oa ocro = ms N'T = i ozi oa oa'o = bsn **************************************** 3 T = H0UH3I xtraaaa TBHIIOH 3 X+V-3 ' i o q . 3 9 A T B t i p x s a j j o m o a 3q . -BTn .0TB3 3 **************************************** 3 3MI1N0D OOT (i*i)g / (was - (dfl'na) = (i)x anNiifloo 06 (r)x * (r'i)a + wns = wns N'di = r 06 oa T + I = dl II - H = I oa-o = wns RH'T = 11 OOT oa (H'N)a / (dN'Ji)a = (N)X **************************************** 3 j:oq.oaA noxq.nxos p r r c i 0 3 . aqnaf asqtis 3[3Bg 3 **************************************** 3 OST oioo (oa-o-QH -(N'N)a)ii anaiiNoa os 3MIXN00 01 aONIXNOO 09 (r'x)a * xonb - (r'i)a = (r'i)a ds'dx = r 09 oa oa -o = (x'i)a sumiSoif] ja%ndiaoQ fl xrpuaddy Appendix E. Computer Programs 328 120 CONTINUE RNORM = DSORT(RSQ) IERROR = 1 RETURN C **************************************** C Abnormal return because of zero entry on diagonal, IERR0R=2 C **************************************** 130 IERROR = 2 RETURN END SUBROUTINE ROOT(F,M,X,YOLD,EPS,ALPHA,Y,FLAG) C **************************************** C This subroutine uses Newton's method to C f i n d the solution of a set of simultaneous C non-linear equations. C **************************************** IMPLICIT REAL*8(A-H,0-Z) INTEGER FLAG DIMENSION YOLD(M),Y(M),DY(10),DELY(10),A(10,11) MP = M + 1 DO 10 I = 1,M Y(I) = YOLD(I) DELY(I) = l.D-6 * YOLD(I) 10 CONTINUE DO 60 ITER = 1,100 FLAG = 0 DO 40 I = 1,M DO 30 J = l.MP 'lF(J.Eq.MP) GOTO 20 Appendix E. Computer Programs Y(J) = Y(J) + DELY(J) FUP = F(I,X,Y) Y(J) = Y(J) - 2.DO * DELY(J) FDOWN = F(I,X,Y) Y(J) = Y(J) + DELY(J) A(I,J) = (FUP - FDOWN) / (2.D0 * DELY(J)) GOTO 30 20 A(I,J) = -F(I,X,Y) 30 CONTINUE 40 CONTINUE CALL GAUSS(A,M,10,U,DY,RN0RM,IERR0R) FLAG = 1 DO 50 I = 1,M Y(I) = Y(I) + ALPHA * DY(I) IF(DABS(DY(I)).GT.EPS) FLAG = 0 50 CONTINUE IF(FLAG.EQ.l) RETURN 60 CONTINUE RETURN END SUBROUTINE TDMA(A,B,CtD,X,N) C **************************************** C This subroutine uses the Thomas algorithm C to f i n d the solution of a tridiagonal C matrix. C **************************************** IMPLICIT REAL*8(A-H,0-Z) Thomas algorithm DIMENSION A(N),B(N),C(N),D(N),X(N),P(101),Q(101) NM = N - 1 P(l) = -C(l) / B(l) Appendix E. Computer Programs q(l) = D(i) / B(l) DO 10 I = 2,K IM = 1-1 DEN = A(I) * P(IM) + B(I) P(I) = -C(I) / DEN Q(I) = (D(I) - A(I) * Q(IM)) / DEN 10 CONTINUE X(N) = Q(N) DO 20 II = l.NM I = N - II X(I) = P(I) * X(I+1) + q(i) 20 CONTINUE RETURN END DOUBLE PRECISION FUNCTION FALB(PI) C **************************************** C This function calculates Albumin C concentration - using a f i t t e d C (Alb) vs. C.O.Ppl relationship. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKCC/CALB1,CALB2 FALB = (CALB1 + CALB2 * PI) / l.D+3 RETURN END DOUBLE PRECISION FUNCTION FPI(C) C **************************************** C This function calculates c o l l o i d C osmotic pressure - using a f i t t e d Appendix E. Computer Programs 331 C C.O.P vs. (Alb) relationship. C **************************************** IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKN/CPI1,CPI2 R = C * l.D+3 FPI = CPU + CPI2 * R RETURN END DOUBLE PRECISION FUNCTION RHSF(I,X,Y) C **************************************** This function evaluates the non-linear equations f o r a given set of Y values. C **************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) COMMON/BLKB/JFS,JLS COMMON/BLKD/QFS,QDS,QLS DIMENSION Y(2) CALL AUXSAM(Y) RHSF3 = JFS - JLS RHSF4 = QFS + QDS - qLS G0T0(10,20),I 10 RHSF = RHSF3 RETURN 20 RHSF = RHSF4 RETURN END DOUBLE PRECISION FUNCTION FCOMPS(V) C **************************************** C This function calculates the skin Appendix E. Computer Programs 332 C compartments hydrostatic pressure. IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKO/VS(14),PS(14),AS,BS,HPS,HPSM1 IF(V.LE.VS(1)) GO TO 10 IF(V.GE.VS(10)) GO TO 20 FCOMPS = FS(V) RETURN 10 FCOMPS = PS(1) + AS * ( V - VS(1)) RETURN 20 FCOMPS = PS(10) + BS * (V - VS(10)) RETURN, END SUBROUTINE SPLINS C *************************************** C This subroutine SPLINES the experimently C obtained PS vs. VS data set. C **************************************** IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKO/X(14),Y(14),A1,BN,N,NM COMMON/BLKQ/Q(101),R(101),S(100) DIMENSION H(100),A(101),B(101),C(101),D(101) DO 10 1 = l.NM H(I) = Xd+1) - X(I) 10 CONTINUE Ad) ? = O.DO B(l) = 2.D0 * H(l) C(l) = H(l) D(l) = 3.DO * ((Y(2) - Y d ) ) / H(l) - Al) DO 20 I = 2.NM IP = 1+1 Appendix E. Computer Programs IM = 1-1 A(I) = H(IM) B(I) = 2.DO * (H(IM) + H(I)) C(I) = H(I) D(I) = 3.DO * ((Y(IF) - Y(I)) / H(I) - (Y(I) - Y(IM))/H(IM)) 20 CONTINUE A(N) = H(HM) B(N) = 2.DO * H(HM) C(N) = 0.D0 D(N) = -3.D0 * ((Y(N) - Y(NM)) / H(NM) - BN) CALL TDMA(A,B,C,D,R,N) DO 30 I = 1,NM IP = I + 1 Q(I) = (Y(IP) - Y(D) / H(I) - E(I) * (2.DO *R(I)+R(IP))/3.DO S(I) = (R(IP) - R(D) / (3.DO * H(I)) 30 CONTINUE RETURN END DOUBLE PRECISION FUNCTION FS(Z) C **************************************** C This function evaluates tissue compartment C hydrostatic pressure - using a splined C set of data. C **************************************** IMPLICIT REAL*8(A-H,0-Z) COMMON/BLK0/X(14),Y(14),A1,BN,N,NM C0MM0N/BLKQ/q(101),R(101),S(100) I = 1 IF(Z.LT.X(1)) GO TO 30 Appendix E. Computer Programs 334 IF(Z.GE.X(NM)) GO TO 20 J = NM 10 K = (I + J) / 2 IF(Z.LT.XOO) J = K IF(Z.GE.X(K)) I = K IF(J.Eq.I+l) GO TO 30 GO TO 10 20 I = NM 30 DX = Z - X(I) FS = Y(I) + DX * (Q(I) + DX * (R(I) + DX * S(I))) RETURN END E.4 Listing of Program SM-C Q **************************************** IMPLICIT REAL*8(A-H,K,L,J,0-Z) INTEGER FLAG,FLAG6,FLAG7,FLAGG C0MM0N/BLKA2/CPLNRM,CSNRM,CASNRM COMMON/BLKB/JFS,JLS COMMON/BLKD/qFS,qDS,qLS COMMON/BLKG/CPL,CS,CAVS COMMON/BLKI/PIPL,PIS COMMON/BLKF/PC.PCNRM.PCGRAD COMMON/BLKH/LSNRM COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKT/qTNRM,qSNRM,qPLNRM COMMON/BLKU/PSNRM,PS Appendix E. Computer Programs 335 COMMOH/BLKBB/PIPNRM,PISNRM COMMOH/BLKDD/ALBSTO COMMON/BLKZB/QTOT,VTOT C0MM0H/BLKZJ/EXPLVL(4),EXCPL(4),EXDLPI(3),EXDLCA(3) C0MM0N/BLKZK/EXPIPL(4),EXPII(3),EXCAV(3) DIMEHSION S0LN(16,769),T(1001).SOL(lOOl),S0LF(16) DIMEHSION Y0LD(4),YNEH(4),Y0LDA(2),YNEWA(2),YFINAL(2) DIMEHSION YEX1(4),YEX9(4),YEX3(4),YEX2(3),YEX11(3), + YEX17(3),YEX18(3) EXTERNAL RHSF.RHSFA DATA ACC,ALPHA,EPS,DT,NP,NV/1.D-7,.5D0,.01DO,0.005D0,769,4/ FLAG7 = 0 FLAG6 = 0 TSTART=1.D-1*DT TMIN=1.D-4*DT TMAX=DT C **************************************** C Set LSS and RDL parameter values C **************************************** LSS = 40.000DO RDL = 0.86800D0 CALL SPLIHS CALL COEFF(RDL) C **************************************** C Set i n i t i a l f l u i d volumes and protein contents C **************************************** PLINIT = 10.574D0 PIIHIT = 3.3O4D0 VIFTHT = 18250.DO Appendix E. Computer Programs PIDINT = PLINIT - PIINIT VPINIT = VPLNRM QPINIT = VPLNRM * FALB(PLINIT) VSINIT = VIFINT qSINIT = FALSCPIIHIT) * (VIFINT - VEXS) VTINIT = VPINIT + VSINIT QTINIT = qPINIT + QSINIT VTOT = VTINIT QTOT = qTINIT YNEW(l) = VPINIT YNEW(2) = QPINIT YNEW(3) = VSINIT YNEW(4) = QSINIT C **************************************** C Solve d i f f e r e n t i a l balances C **************************************** DO 30 I = 1,19200 TIM = (I-1)*DT TI = I * DT IF(I.EQ.l) GOTO 14 CALL RK4C(RHSF,NV,T(IM),T(I),Y0LD,EPS,YNEW,NFUNC,FLAG) CALL DES0LV(RHSF,NV,TIM,TI,YOLD,EPS.TSTART,TMIN,TMAX, + YNEW.FLAG) IF(FLAG.EQ.O) GOTO 33 CALL AUXSAM(YNEW) C **************************************** C Store results at select time intervals C **************************************** 14 IF(TI.LE.3.D0)THEN IK=(I-l)/40 +1 REALI = 1 Appendix E. Computer Programs 337 RI = (REALI - l.D0)/40.D0 + l.DO RIK = IK CHK = RI - RIK ELSE IK=(I-l)/200 + 12 REALI = I RI = (REALI - l.D0)/200.D0 + 12.D0 RIK = IK CHK = RI - RIK EHDIF IF(CHK.HE.O.DO) GOTO 56 CALL AUXSAM(YNEW) T(IK)=DT*(I-•1) SOLN(l.IK) = YNEW(l) S0LH(2,IK) = YNEW(2) SOLN(3,IK) PC S0L»(4,IK) PIPL SOLN(5,IK) = CPL S0LN(6,IK) PIS S0LH(7,IK) = CS S0LH(8,IK) = CAVS SOLHO.IK) PS SOLN(IO.IX) JFS SOLN(il.IK) = JLS SOUK 12, IK) = QFS S0LH(13,IK) = QDS S0LN(14,IK) = QLS S0LH(15,IK) YHEH(3) SOLN(16,IK) = YHEW(4) 56 DO 20 J = l.HV YOLD(J) = OTEW(J) Appendix E. Computer Programs 338 20 CONTINUE 30 CONTINUE VTOT = YNEW(l) + YNEW(3) QTOT = YNEW(2) + YNEW(4) YOLDA(l) = YNEW(3) Y0LDA(2) = YUEW(4) C **************************************** C Calculate f i n a l steady-state variable C values and store C **************************************** CALL ROOT(RHSFA,2,1000.DO,YOLDA,ACC,ALPHA,YFINAL,FLAGG) CALL AUXALT (YFINAL) SOLF(l) = VTOT - YFINAL(l) S0LF(2) = QTOT - YFINAL(2) S0LF(3) = PC S0LF(4) = PIPL S0LF(5) = CPL S0LF(6) = PIS S0LF(7) = CS S0LF(8) = CAVS S0LF(9) = PS SOLF(IO) = JFS SOLF(ll) = JLS S0LF(12) = QFS S0LF(13) = QDS S0LF(14) = QLS S0LF(15) = YFINAL(l) S0LF(16) = YFINAL(2) C **************************************** Appendix E. Computer Programs 339 c Output store variable values and associated C times C **************************************** DO 112 1=1,107 WRITE(6,91)T(I),S0LN(4,I),S0LH(6,I),S0LH(3,I),S0LH(9,I) 91 F0RMAT(6X,F6.3,5X,F7.4,5X,F7.4,5X,F7.4,5X,F7.4) 112 CONTINUE WRITE(6,41)S0LF(4),S0LF(6),S0LF(3),S0LF(9) 41 FORMAT(17X,F7.4,5X,F7.4,5X,F7.4,5X,F7.4) DO 111 1=1,107 WRITE(6,95)T(I),S0LN(15,I),S0LN(1,I),S0LN(16,I),S0LN(2,I) 95 F0RMAT(6X,F6.3,5X,F10.3,5X,F7.2,5X,F7.3,5X,F7.3) 111 CONTINUE WRITE(6,45)S0LF(15),S0LF(1),S0LF(16),S0LF(2) 45 FORMAT(17X,F10.3,5X,F7.2,5X,F7.3,5X,F7.3) DO 113 1=1,107 VRITE(6,99)T(I),S0LN(10,I),SOLN(ll,I),S0LN(12,I), + SOLN(13,I),S0LN(14,I) 99 F0RMAT(6X,F6.3,5X,F8.2,5X,F8.2,5X,F8.3,5X,F8.3,5X,F8.3) 113 CONTINUE VRITE(6,49)S0LF(10),S0LF(ll),S0LF(12), + S0LF(13),S0LF(14) 49 F0RMAT(17X,F8.2,5X,F8.2,5X,F8.3,5X,F8.3,5X,F8.3) DO 117 1=1,107 VRITE(6,58)T(I),S0LN(5,I),S0LN(8,I),S0LN(7,I) 58 FORMAT(6X,F6.3,5X,F7.5,5X,F7.5,5X,F7.5) 117 CONTINUE VRITE(6,24)S0LF(5),S0LF(8),S0LF(7) 24 F0RMAT(17X,F7.5,5X,F7.5,5X,F7.5) 33 WRITE(6,40) 40 FORMAT(IX,'ODE SOLVER FAILS! ') Appendix E. Computer Programs 340 STOP END SUBROUTINE COEFF(RDL) C ************************ IMPLICIT REAL*8(A-H,J,K,L,0-Z) C0MM0N/BLKA2/CPLNRM,CSNRM,CASNRM COMMON/BLKT/QTNRM,QSNRM,QPLNRM COMMON/BLKDD/ALBSTO COMMON/BLKH/LSNRM COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKU/PSNRM,PS COMMON/BLKK/VEXS COMMON/BLKBB/PIPNRM,PISNRM COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM C **************************************** C Set normal steady-state conditions and C calculate transport parameter values C **************************************** PIPNRM = 25.9D0 PISNRM = 14.7D0 PSNRM = FCOMPS(VIFNRM) CPLNRM = FALB(PIPNRM) CASNRM = FALB(PISNRM) CSNRM = CASNRM * (l.DO - VEXS / VIFNRM) QPLNRM = VPLNRM * CPLNRM QSNRM = VIFNRM * CSNRM QTNRM = QPLNRM + QSNRM VPLNRM = VPLNRM Appendix E. Computer Programs VSURM = VIFNRM VTNRM = VPLNRM + VSNRM Wl = (CSNRM * RDL) / (CPLNRM - CASNRM) W2 = VIFNRM / (VIFNRM - VEXS) W3 = ALBSTO * VIFNRM LSNRM = W3 / (l.D0+Vl*W2) PSS = LSNRM * CSNRM * RDL / (CPLNRM - CASNRM) Z l = 2.DO * CSNRM * (l.DO - RDL) Z2 = CPLNRM + CASNRM SIGS = l.DO - (Zl / Z2) KFS = LSNRM / (PCNRM - PSNRM - SIGS * (PIPNRM RETURN END SUBROUTINE AUXSAM(Y) C *************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) C **************************************** COMMON/BLKB/JFS,JLS eOMMON/BLKD/QFS,QDS,QLS COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKG/CPL,CS,CAVS COMMON/BLKH/JLSNRM COMMON/BLKI/PIPL.PIS COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKT/QTNRM,QSNRM,QPLNRM COMMON/BLKU/PSNRM,PS COMMON/BLKZA/PRESLO COMMON/BLKZB/QTOT,VTOT DIMENSION Y(4) C **************************************** 341 PISNRM)) (TOI)b'(TOI)d*(N)X'(H)a'(11)0*(N)3'(H)V H0ISK3Wia **************************************** 3 nniTC-ioSra sBmom 0 **************************************** 3 (Z-0 ,H-V)8*1V3H IIOIldHI **************************************** 3 • xx jq.Bin 3 TBUoSBxpTjq. B j o uotq.Tixos atpj. p u r y oq. 3 mqaTJoSrB SBmoTix au;q. sasv. aurq.iiojqTis s-rqx 3 **************************************** 3 (N'x*cTo'g'v)VHai aHimonans eras so * sir = sib (SAVO - ido) * ssd = sab ocrz / (SAY3 + ido) * (sols - ocrt) * s ir = sdb (01S3Hd+HHHSd)/(OlSHHd+Sd)*HHHSir=Sir (HHKSir *11'Sir )dl (RHHSd - sd) * ssi + wnnsir = s ir ((Sid - Idld) *SDIS - Sd - Od) * SdX = SdC ((£)A)SdWOO.i = Sd (SAVO)Idd = Sid (IdO)Idd = Idld (SXHA - (E)A) /(fr)A = SAVO (£)A / W i = SO ((r)A)0dHO0d = Od (T)A / (Z)A = IdO **************************************** 3 sq.traq.uo3 uraq.ojd pue p T U j j 3 TBauamqjBdnioa xoj. sajqeTJXA. maq.sAs aq.BTii3"[TB3 3 SUTBISOIJ is^ndmoQ 57 xipuaddy paq.q.Tj B 3UTSII - g j rassajd a x q o t n s o 3 p t o x i o o saa.BTn.OTBa uoxq.aTin.1 srqr , 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 (O)Idd fiOIIDHILi NOISIDHHd T1W.Q0. asa.. unman e-KTI / (Id * Z9TV3 + tSlVD) = STVi saTfo'tarvo/ooxia/KORHoo (Z-0'H-V)8*'rVHH IIOndHI * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 • d r c r s u o T q . B t a a T.dd-0"D - SA (qTV) 3 paq.q.xi B Soxst! - ttoxq.B.iq.tiaaiioa 3 luuinqrv saq.BTn.3TBa u o x q . a u u i STLTT 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 (id)aTra Noixoanj HoisiDHHd aianoa OHH Iranian anHixNoo oz (1)0 + (T+I)X * (I)d = ( I)X I I - I = I rn'i - 11 oz oa ( H ) 0 = ( N ) X ansiuioo ox saa / ((Hi)d * (i)v - (i)a) = ( i ) o una / ( 1 ) 0 - = (i)d (i)a + (wi)d * (i)v = naa i-i = HI &'z = 1 ox oa (i)a / (t)a = (T)& cm / (x)o- = (T )d T - H = RK smviSoij isindmoQ g- xrpuaddy Appendix E. Computer Programs 344 C C.O.P vs. (Alb) relationship. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKN/CPI1.CPI2 R = C * l.D+3 FPI = CPU •+ CPI2 * R RETURN END DOUBLE PRECISION FUNCTION RHSF(I,T,Y) C **************************************** C This function evaluates the non-linear C equations for a given set of Y values. C **************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) COMMON/BLKB/JFS,JLS COMMON/BLKD/QFS,QDS,qLS DIMENSION Y(4) CALL AUXSAM(Y) IF(T.LE.1.5D0)THEN RHSF1 = JLS - JFS + 200.DO RHSF2 = qLS - qDS - qFS + 40.D0*0.65D0 RHSF3 = JFS - JLS RHSF4 = qFS + qDS - qLS ELSEIF(T.GT.1.5D0.AND.T.LE.25.5D0)THEN RHSF1 = JLS - JFS - (300.DO/24.DO) RHSF2 = qLS - qDS - qFS - (15.8DO/24.DO)*0.65DO RHSF3 = JFS - JLS RHSF4 = qFS + qDS - QLS ELSEIF(T.GT.25.5D0.AND.T.LE.92.639D0)THEN Appendix E. Computer Programs 345 RHSFl = JLS - JFS RHSF2 = QLS - QDS - qFS - (15.8D0/24.D0)*0.65D0 RHSF3 = JFS - JLS RHSF4 = QFS + QDS - qLS ELSE RHSFl = JLS - JFS RHSF2 = QLS - QDS - QFS RHSF3 = JFS - JLS RHSF4 = QFS + QDS - QLS EHDIF G0T0(10,20,30,40),I 10 RHSF = RHSFl RETURN 20 RHSF = RHSF2 RETURN 30 RHSF = RHSF3 RETURN 40 RHSF = RHSF4 RETURN END BLOCK DATA **************************************** IMPLICIT REAL*8(A-H,K,L,0-Z) **************************************** COMMON/BLKF/PC,PCNRM.PCGRAD COMMON/BLK3/VTNRM,VIFNRM,VPLNRM COMMON/BLKK/VEXS COMMON/BLKN/CPI1,CPI2 COMMON/BLKO/VS(14),PS(14),AS,BS,NPS,NPSM1 Appendix E. Computer Programs 346 C0MM0H/BLKCC/CALB1,CALB2 COMMON/BLKDD/ALBSTO COMMON/BLKZA/PRESLO C0MM0N/BLKZJ/EXPLVL(4),EXCPL(4),EXDLPI(3),EXDLCA(3) C0MM0N/BLKZK/EXPIPL(4),EXPII(3),EXCAV(3) DATA EXPLVL/2.99D0,3.85D0,3.80D0,3.30D0/ DATA EXPIPL/9.3D0,13.5D0,12.6D0,9.8D0/ DATA EXCPL/10.4D0,24.IDO,23.4D0,15.4D0/ DATA EXPII/2.3D0,2.8D0,4.0D0/ DATA EXCAV/3.5D0.4.6D0.6.8D0/ DATA EXDLPI/7.0D0,10.7D0,5.8D0/ DATA EXDLCA/6.9D0,19.5D0,8.6D0/ DATA PCNRM,VEXS/10.0D0,2.1D+3/ DATA PCGRAD/0.00996D0/ DATA PRESL0/13.0577D0/ DATA VTNRM,VIFNRM,VPLNRM/11.6D+3,8.4D+3,3.2D+3/ DATA CPIl,CPI2/1.75490D-4,0.656840D0/ DATA CALB1,CALB2/-0.267173D-3,0.152244D+1/ DATA NPS.NPSMl/14,13/ DATA VS/8.4D+3,8.92D+3,9.45D+3,9.97D+3,10.50D+3,11.02D+3, + 11.55D+3,12.07D+3,12.60D+3,13.65D+3,14.70D+3,16.80D+3, + 21.00D+3.25.23D+3/ DATA PS/-0.70D0,0.32D0,0.86D0,1.15D0,1.37D0,1.56D0, + 1.69D0,1.80D0,1.88D0,1.99D0,2.01DO,2.04D0,2.12D0, + 2.20D0/ DATA AS.BS/1.96154D-3.1.81347D-5/ DATA ALBST0/0.020539D0/ END DOUBLE PRECISION FUNCTION FCOMPS(V) C **************************************** C This function calculates the tissue Appendix E. Computer Programs 347 C compartment hydrostatic pressure. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0H/BLK0/VS(14),PS(14),AS,BS,NPS,NPSM1 IF(V.LE.VS(D) GO TO 10 IF(V.GE.VS(9)) GO TO 20 FCOMPS = FS(V) RETURN 10 FCOMPS = PS(1) + AS * ( V - VS(1)) RETURN 20 FCOMPS = PS(9) + l.D-4 * (V - VS(9)) RETURN END SUBROUTINE SPLINS *************************************** This subroutine SPLINES the experimently obtained PS vs. VS data set. **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLK0/X(14),Y(14),A1,BN,N,NM C0MM0N/BLKQ/Q(101),R(101),S(100) DIMENSION H(100),A(101),B(101),C(101),D(101) DO 10 I = l.NM H(I) = X(I+1) - X(I) CONTINUE A(l) = 0.D0 B(l) = 2.DO * H(l) C(l) = H(l) DU) = 3.DO * ((Y(2) - Y(l)) / H(l) - Al) Appendix E. Computer Programs 348 DO 20 I = 2.NM IP = 1+1 IM = 1-1 A(I) = H(IM) B(I) = 2.D0 * (H(IM) + H(I)) C(I) = H(I) D(I) = 3.DO * ((Y(IP) - Y(I)) / H(I) - (Y(I) - Y(IM))/H(IM)) 20 CONTINUE A(K) = H(NK) B(N) = 2.DO * H(NM) C(N) = 0.D0 D(N) = -3.D0 * ((Y(N) - Y(NM)) / H(NH) - BN) CALL TDMA(A,B,C,D,R,N) DO 30 I = l.NM IP = I + 1 Q(I) = (Y(IP) - Y(I)) / H(I) - H(I) * (2.DO *R(I)+R(IP))/3.D0 S(I) = (R(IP) - R(I)) / (3.DO * H(I)) 30 CONTINUE RETURN END DOUBLE PRECISION FUNCTION FS(Z) C *************************************** C This function evaluates tissue compartment C hydrostatic pressure - using a splined C set of data. C **************************************** IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKO/X(14),Y(14)fAl,BN,N,NM C0MM0N/BLKQ/Q(101),R(101),S(100) 1 = 1 IF(Z.LT.X(1)) GO TO 30 Appendix E. Computer Programs IF(Z.GE.X(NM)) GO TO 20 J = NM 10 K = (I + J) / 2 IF(Z.LT.X(K)) J = K IF(Z.GE.X(K)) I = K IF(J.EQ.I+1) GO TO 30 GO TO 10 20 I = NM 30 DX = Z - X(I) FS = Y(I) + DX * (Q(I) + DX * (R(I) + DX * S(I))) RETURN END C **************************************** DOUBLE PRECISION FUNCTION FCOMPC(V) IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM C **************************************** C Calculate capillary hydrostatic pressure C **************************************** FCOMPC = PCNRM + PCGRAD * (V - VPLNRM) IF(FC0MPC.LT.2.D0) FCOMPC = 2.DO RETURN END C **************************************** C D i f f e r e n t i a l equation solver C **************************************** SUBROUTINE RK4C(F,M,A,B,YA,EPS,YB,NFUN,FLAG) IMPLICIT REAL*8(A-H,K,0-Z) Appendix E. Computer Programs INTEGER FLAG DIMENSION YA(M),YI(10),YB(M),YB0LD(10),YARG1(10),YARG2(10) DIMENSION Kl(10),K2(10),K3(10),K4(10) FLAG=1 NINT=1 NFUN=0 10 NFUN=NFUN+4*NINT*M DX=(B-A)/NINT XI=A DO 20 1=1, M 20 YI(I)=YA(I) DO 70 INT=1,NINT DO 30 1=1,M K1(I)=DX*F(I,B,YI) 30 YARG1(I)=YI(I)+K1(I)/2.D0 XARG=XI+DX/2.D0 DO 40 1=1,M K2(I)=DX*F(I,B,YARG1) 40 YARG2(I)=YI(I)+K2(I)/2.D0 DO 50 1=1,M K3(I)=DX*F(I,B,YARG2) 50 YARG1(I)=YI(I)+K3(I) XI=XI+DX DO 60 1=1,M K4(I)=DX*F(I,B,YARG1) 60 YI(I)=YI(I)+(K1(I)+2.DO*(K2(I)+K3(I))+K4(I))/6.DO 70 CONTINUE DO 80 1=1,M 80 YB(I)=YI(I) IF(NINT.EQ.l) GO TO 100 YBDIFM=O.DO Appendix E. Computer Programs DO 90 1=1,M 90 YBDIFM=DMAX1(DABS(YB(I)-YB0LD(I)),YBDIFM) IF(YBDIFH.LT.EPS) RETURN IF(NINT.GT.10000) GO TO 120 100 DO 110 1=1,M 110 YBOLD(I)=YB(I) NINT=2*NINT GO TO 10 120 FLAG=0 RETURN END SUBROUTINE GAUSS(A.N.NDR.NDC.X,RNORM,IERROR) C **************************************** C Purpose: C Uses Gauss elimination with p a r t i a l pivot selection to C solve simultaneous linear equations of form A*X=C C Arguments: C A Augmented coefficient matrix containing a l l coefficients C and R.E.S. constants of equations to be solved. C N Number of equations to be solved. C NDR F i r s t (row) dimension of A i n c a l l i n g program. C NDC Second (column) dimension of A i n c a l l i n g program. C X Solution vector. C RNORM Measure of size of residual vector C-A*X C IERROR Error f l a g . C =1 Successful Gauss elimination. C =2 Zero diagonal entry after pivot selection. C **************************************** IMPLICIT REAL*8(A-H,0-Z) Appendix E. Computer Programs 352 DIMENSION A(NDR,NDC),X(N),B(50,51) NM = N - 1 NP = N + 1 Q **************************************** C Set up working matrix B Q **************************************** DO 20 I = 1,N DO 10 J = l.NP B(I,J) = A(I,J) 10 CONTINUE 20 CONTINUE C **************************************** C Carry out elimination process N-l times C **************************************** DO 80 K = l.NM KP = K + 1 Q **************************************** C Search for largest coefficient i n column K, rows K through N C IPIVOT i s the row index of the largest coefficient C **************************************** BIG = 0.D0 DO 30 I = K,N SF = DABS(B(I,K)) DO 25 J = KP.NP SF = DMAX1(SF,DABS(B(J,K))) 25 CONTINUE AB = DABS(B(I,K) / SF) IF(AB.LE.BIG) GOTO 30 BIG = AB IPIVOT = I 30 CONTINUE Appendix E. Computer Programs 353 c **************************************** C INTERCHANGE ROWS K AND IPIVOT IF IPIVOT.NE.K C **************************************** IF(IPIVOT.EQ.K) GOTO 50 DO 40 J = K.NP TEMP = B(IPIVOT.J) B(IPIVOT.J) = B(K,J) B(K,J) = TEMP 40 CONTINUE 50 IF(B(K,K).EQ.O.DO) GOTO 130 C **************************************** C Eliminate B(I,K) from rows K+l through N C **************************************** DO 70 I = KP.N QUOT = B(I,K) / B(K,K) B(I,K) = O.DO DO 60 J = KP.NP B(I,J) = B(I,J) - QUOT * B(K,J) 60 CONTINUE 70 CONTINUE 80 CONTINUE IF(B(N,N).Eq.0.D0) GOTO 130 Q **************************************** C Back substitute to f i n d solution vector C **************************************** X(N) = B(N,NP) / B(H,S) DO 100 II = 1,NM SUM = O.DO I = N - II IP = I + 1 DO 90 J = IP.N STioaTTBiTTuaxs j o qas B j o uoxquTos aqq. p u r j 3 o% p o m a i a s,tioqjsaii s a s n aTrcq.TiojqTis spjr 3 **************************************** 3 (SVTi'A'VHcTW'SdH'cnOA'X'M'iJIQOH HfllluOHanS OHH jranxan Z = HOHHHI 0 £ T **************************************** 3 3=H0UH3I 'TBHOSBTP n o jLcvta o j a z 10 a s w e o a q u x a q . a j r e m i o t t q v 3 **************************************** 3 swim t = H 0 U H H I (bsn)iHosa = HHora aruiiuioa OZT z * * ((Has - (dH'Dv)sava) + dsn = O S H 3 M I X N 0 0 0TT (r)x * (r ' i)v + wns = was H ' T = r OTT oa oa*o = wns H ' T = i OZT oa oa'o = Gsra **************************************** 3 T = H0HH3I w q t f l uxc iq .a i T e u u o f l 3 X*V-3 '-zoqoaA TBtipTsaJC j o 1111011 aq-BTtiOTBS 3 **************************************** 3 anjiiuioo OOT (i*i)a / (wns - (dfl'i)a) = (i)x anHiuioo 06 (r)x * ( f i ) a + wns = wns SWVISOJJ is'indmoQ gr xmaaddy Appendix E. Computer Programs 355 C non-linear equations. IMPLICIT REAL*8(A-H,0-Z) INTEGER FLAG DIMENSION YOLD(M),Y(M),DY(10),DELY(10),A(10,11) MP = M + 1 DO 10 I = 1,M Y(I) = YOLD(I) DELY(I) = l.D-6 * YOLD(I) 10 CONTINUE DO 60 ITER = 1,100 FLAG = 0 DO 40 I = 1,M DO 30 J = l.MP IF(J.EQ.MP) GOTO 20 Y(J) = Y(J) + DELY(J) FUP = F(I,X,Y) Y(J) = Y(J) - 2.D0 * DELY(J) FDOWN = F(I,X,Y) Y(J) = Y(J) + DELY(J) A(I,J) = (FUP - FDOWN) / (2.D0 * DELY(J)) GOTO 30 20 A(I,J) = -F(I.X.Y) 30 CONTINUE 40 CONTINUE CALL GAUSS(A,M,10,11,DY,RNORM,IERROR) FLAG = 1 DO 50 I = I.M Y(I) = Y(I) + ALPHA * DY(I) IF(DABS(DY(I)).GT.EPS) FLAG = 0 50 CONTINUE Appendix E. Computer Programs IF(FLAG.EQ.l) RETURN 60 CONTINUE RETURN END DOUBLE PRECISION FUNCTION RHSFA(I,X,Y) C *********************************** C This function evaluates the non-linear C equations f or a given set of Y values. C **************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) COMMON/BLKB/JFS,JLS COMMON/BLKD/QFS,qDS,QLS DIMENSION Y(2) CALL AUXALT(Y) RHSF3 = JFS - JLS RHSF4 = QFS + QDS - QLS G0T0(10,20),I 10 RHSFA = RHSF3 RETURN 20 RHSFA = RHSF4 RETURN END ' SUBROUTINE AUXALT(Y) C *************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) C *************************************** COMMON/BLKB/JFS,JLS COMMON/BLKD/QFS,QDS,QLS COMMON/BLKF/PC,PCNRM,PCGRAD Appendix E. Computer Programs 357 COMMON/BLKG/CPL,CS,CAVS COMHOH/BLKH/JLSNRM COMHON/BLKI/PIPL,PIS COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS,SIGS,PSS COMMON/BLKZB/QTOT,VTOT COMMON/BLKU/PSNRM,PS COMMON/BLKZA/PRESLO DIMENSION Y(2) C **************************************** C Calculate system variables from compartmental C f l u i d and protein contents C **************************************** VOLPLA = VTOT - Y(l) QPLAS = QTOT - Y(2) PC = FCOMPC(VOLPLA). CPL = QPLAS / VOLPLA CS = Y(2) / Y(l) CAVS = Y(2) / (Y(l) - VEXS) PIPL = FPI(CPL) PIS = FPI(CAVS) PS = FCOMPS(Y(l)) JFS = KFS * (PC - PS - SIGS * (PIPL - PIS)) JLS = JLSNRM + LSS * (PS - PSNRM) IF(JLS.LT.JLSNRM) JI^=JLSNRM*(PS+PEESLO)/(PSNRM+PRESLO) QFS = JFS * (l.DO - SIGS) * (CPL + CAVS) / 2.D0 QDS = PSS * (CPL - CAVS) QLS = JLS * CS RETURN END Appendix E. Computer Programs 3 5 8 SUBROUTINE DESOLV(F,M,A,B,YO,EPS,HSTART,HMIN,HMAX,YB,FLAG) IMPLICIT REAL*8(A-H,0-Z) DIMENSION Y0(M),YA(10),YB(M) EXTERNAL F INTEGER FLAG BMA=B-A HOLD=HSTART X=A DO 10 1=1,M 10 YA(I)=YO(I) 20 CALL RKF(F,M,X,YA,HOLD,YB,YDIF) GAMMA=.8D0*(EPS*HOLD/(BMA*YDIF))**0.25D0 HBEW=GAMMA*HOLD IF(GAMMA.GE.l.DO) GO TO 30 IF(HNEW.LT.HOLD/10.DO) HNEW=H0LD/10. IF(HNEH.LT.HMIN) GO TO 50 HOLD=HNEW PRINT*,HOLD GO TO 20 30 IF(HHEW.GT.5.D0*H0LD) HNEW=5.D0*H0LD IF(HNEW.GT.HMAX) HNEW=HMAX IF(X+HOLD.GE.B) GO TO 70 X=X+HOLD HOLD=HNEW DO 40 1=1,M 40 YA(I)=YB(I) GO TO 20 50 FLAG=0 B=X DO 60 1=1,M 60 YB(I)=YA(I) Appendix E. Computer Programs 359 RETURN 70 FLAG=1 HSTART=HNEW H0LD=B-X CALL RKF(F,M,X,YA,HOLD,YB,YDIF) RETURN END SUBROUTINE RKF(F,M,X,YOLD,H,YNEW,YDIFM) IMPLICIT REAL*8(A-H,0-Z) DIMENSION YOLD(M),YHEW(M),YARG1(10),YARG2(10) REAL*8 Kl(10),K2(10),K3(10),K4(10),K5(10),K6(10) DATA C21.C31.C32,C33/.25D0,.375D0,.09375D0,.28125D0/ DATA C41,C42/.923076923076923073DO,.8793809740SS530257D0/ DATA C43,C44/-3.27719617660446061D0,3.32089212562585323D0/ DATA C51,C52/2.03240740740740722D0,-8.DO/ DATA C53,C54/7.17348927875243647D0,-.20589668615984405D0/ DATA C61,C62/.SD0,-.296296296296296294D0/ DATA C63,C64/2.DO,-1.3816764132553605D0/ DATA C65.C66/.452972709551656916D0.-.275D0/ DATA C71.C72/.118518518518518509D0,.518986354775828454D0/ DATA C73,C74/.506131490342016654D0,-.18D0/ DATA C75.C81/.0363636363636363636D0,.00277777777777777778D0/ DATA C82,C83/-.0299415204678362568D0,-.0291998936735778838D0/ DATA C84,C85/.02D0..0363636363636363636D0/ DO 10 1=1,M K1(I)=H*F(I.X,Y0LD) 10 YARG1(I)=Y0LD(I)+C21*K1(I) XARG=X+C21*H DO 20 1=1,M K2 (I) =H*F (I, XARG, YARG1) 20 YARG2(I)=Y0LD(I)+C32*K1(I)+C33*K2(I) Appendix E. Computer Programs 360 XARG=X+C31*H DO 30 1=1,M K3(I)=H*F(I,XAfiG,YARG2) 30 YARG1(I)=Y0LD(I)+C42*K1(I)+C43*K2(I)+C44*K3(I) XARG=X+C41*H DO 40 1=1,M K4(I)=H*F(I,XARG,YARG1) 40 YARG2(I)=Y0LD (I)+C51*K1(I)+CS2*K2(I)+C53*K3(I)+C54*K4(1) XARG=X+H DO 50 1=1,M K5(I)=H*F(I,XARG,YARG2) 50 YARG1(I)=Y0LD (I)+C62*K1(I)+C63+K2(I)+C64*K3(I)+C65*K4(I) 1 +C66*K5(I) XARG=X+C61*B DO 60 1=1,M 60 K6(I)=H*F(I,XARG,YARG1) YDIFM=O.D0 DO 70 1=1,M YNEW(I)=YOLD(I)+C71*K1(I)+C72*K3(I)+C73*K4(I)+C74+K5(I)+ 1 C75*K6(I) YDIF=DABS(C81*K1(I)+C82*K3(I)+C83*K4(I)+C84*K5(I)+C85*K6(I)) 70 IF(YDIF.GT.YDIFM) YDIFM=YDIF RETURN EHD E .5 Listing of Program P L M - A IMPLICIT KEAL*8(A-H,L,J,K,0-Z) C0MM0N/BLKBS/JFSS(40),JPLLSS(40),JRSS(40),JLSS(40) C0MM0N/BLKDS/QPLLSS(40),QDSS(40),QLSS(40) C0MMDN/BLKGS/PVS(40),PSST(40) oa-OT * oaow.ofrZfr -o = (z)i ******************************************** 3 paq.BTi.TeAa a q oq. squxod ptj3 3 J O I (TX) ica pws «(ax) s i *((z)x) uau 3 ***************************************** 3 (KHHdlA / SX3A - OO'T) * HHHSV3 = HHKS3 (HHHSId)tnVi = HHNSV3 (HHHdldOaTVd = HHHldO (KHHilA) SdWQM = WiSSd SHIldS nvo oarfrT. = mssid 0(16'SC = WHHdld ******************************************** 3 SUOT^ TTHICO TBfq.Ttrc qas 3 ******************************************** 3 (e)X'(T)9'(T2'TZ)XdOV HOISNHHia oxsaTV/aaxia/noHwoo (Ov)SSA*(Ofr)SldA/HSXia/HOMHOD THVK'VN*(OE)HIXSId*(08)SddId/JUna/H0HH03 (99)IId'(99)IcndId/HHyia/H0WW03 (OE)dldTVD*(0*)AdI1V0/99)na/H0HH03 WHHSV3' m&SQ' HHndD/ZVyia/HOMWOD SXHA/Xna/H0WH03 WHHSId'HHHdId/aaXia/H0HH03 HH*(8T)ldIda*(8T)Adia/Zyia/H0MW0D HHHSd/nxia/Howwoa SSd'SHX' STIdX' SdX* SSl/Tyia/H0HM03 MHHldA'WUHJIA'WHNXA/fN0HH03 WMSl/HXia/HOMWOS OTOOAd' WHKAd' WMVd' Ad * Vd/dX13/H0HH03 I 9 S sxrreiSoij la^ndxaoQ xrpuaddy Appendix E. Computer Programs 362 XA = (I-1)*0.05D0 X(l) = XA * 10.DO DO 20 J=l,21 XB = (J-l) * 10.D0 X(3) = XB / 10.D0 CALL GCOMP(G.X) AOPT(I.J) = G(l) WRITE(13,101)XA,XB,A0PT(I,J) 101 F0RMAT(3F10.5) 20 CONTINUE 10 CONTINUE STOP END BLOCK DATA C ************************************ IMPLICIT REAL*8(A-H,K,L,0-Z) C **************************************** COMMON/BLKF/PA,PV,PANBM,PVNRM,PVGRAD COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM COMMON/BLKK/VEXS C0MM0N/BLKN/CPI1,CPI2 C0MM0N/BLK0/VS(14),PS(14),AS,BS,NPS,NPSM1 COMM0N/BLKZ/DIFV(18),DPIPL(18),NN COMMON/BLKCC/CALB1,CALB2 COMMON/BLKDD/ALBSTO COMMON/BLKHH/PIPLPI(66),PII(66) COMMON/BLKZA/PRESLO COMMON/BLKZK/BIFVL,BIFVU,BPIIL.BPIIU DATA BIFVL,BIFVU,BPIIL.BPIIU/34.058273D0,87.568802D0, + 68.643616D0.126.749146D0/ Appendix E. Computer Programs 363 DATA PANRM,PVNRM,VEXS/24.54D0t5.92D0,2.1D+3/ DATA PVGRAD/32.D0/ DATA PEESL0/13.0577D0/ DATA VTNRM,VIFNRM,VPLNRM/11.6D+3,8.4D+3,3.20+3/ DATA CPI1,CPI2/1.75490D-4,0.656840D0/ DATA CALB1,CALB2/-0.267173D-3,0.152244D+1/ DATA NN/18/ DATA DPIPL/9.2DO,lO.TDO,11.7D0,12.7D0,13.7D0,13.7DO, + 14.200,14.200,18.700,18.71)0,20.71)0,20.700, + 21.7D0,24.7D0,25.7D0,25.7D0,26.7D0,26.7DO/ DATA DIFV/19.010+3,16.02D+3,14.4QD+3,12.98D+3,7.98D+3,6.71D+3, + 10.70D+3,8.270+3,9.9440+3,9.11D+3,7.79D+3,6.73D+3, + 6.61D+3,8.44D+3,8.34D+3,7.69D+3,8.25D+3,9.38D+3/ DATA PII/3.4D0,2.9D0,4.9D0,4.9D0,4.4D0,3.9D0, + 3.9D0,3.9D0,3.4D0,4.9D0,6.4D0,6.4D0, + 5.9D0,3.9D0,6.9D0,9.90D0,10.4D0,10.9D0, + 3.2D0,5.7D0,4.7D0,4.7D0,5.7D0,3.7D0,5.2D0,6.7D0, + 7.7D0,8.7D0, + 2.3D0,5.90,1.9D0,2.6D0,0.9D0,2.7D0,0.7D0,2.9D0, + 5.4D0,1.9D0,5.9D0, + 1.7DO,1.6DO,4.0DO,4.5D0,4.3D0,7.OD0,7.2D0,5.7D0, + 7.700,8.00,8.600,8.200,8.400,7.700,10.700,11.700, + 12.2D0,11.7D0,12.7D0,13.2D0,14.DO,14.7D0,13.2D0, + 14.4D0,15.2D0,13.7D0,14.7D0/ DATA PIPLPI/7.D0,7.D0,9.DO,9.D0,9.5D0,9.5DO, + 11.5DO,ll.BDO,13.DO,14.5DO,15.5D0, + 15.5DO,17.5D0,17.5DO,19.0D0,19.OD0,21.DO,21.D0, + 9.200,10.700,11.700,14.2D0,13.700,15.700, + 18.7D0,18.7D0,20.7D0,21.7D0, + 2.3D0,5.3D0,5.8D0,6.6DO,7.8DO,8.3DO,9.6D0, + 10.3DO,11.3DO,12.3DO,17.3DO, Appendix E. Computer Programs 364 + 8.9D0,9.9D0,11.9D0,12.4D0,12.9D0,13.2D0,13.9D0, + 14.2D0,17.4D0,17.9DO,18.OD0,18.2D0,18.4D0,19.4D0, + 19.9D0,23.6DOf24.D0,24.4D0,24.8D0,23.6D0,23.6D0, + 22.9D0,24.9D0,24.4D0,24.2D0,26.4D0,26.9D0/ DATA NPS.NPSMl/14,13/ DATA VS/8.4D+3,8.92D+3,9.45D+3,9.97D+3,10.50D+3,11.02D+3, + 11.550+3,12.070+3,12.600+3,13.650+3,14.700+3,16.800+3, + 21.00D+3.25.23D+3/ DATA PS/-0.70D0,0.32D0,O.86D0,1.15DO,1.37D0,1.56D0, + 1.69DO,1.80D0,1.88D0,1.99DO,2.OlDO,2.O4D0,2.12D0, + 2.20D0/ DATA AS,BS/1.961540-3,1.813470-5/ DATA ALBSTO/0.020539DO/ END E.6 Listing of Program P L M - B SUBROUTINE GCOMP(G.X) C **************************************** IMPLICIT REAL*8(A-H,K,L,J,0-Z) INTEGER FLAG C0MM0N/BLKA2/CPLNRM, CSNRM, CASNRM COMMON/BLKB/JFS,JPLLS,JRS,JLS COMMON/BLKD/qPLLS,qDS,QLS COMMON/BLKG/CPL,CS,CAVS COMMON/BLKBS/JFSS(40),JPLLSS(40),JRSS(40),JLSS(40) C0MM0N/BLKDS/qPLLSS(40),qDSS(40),QLSS(40) C0MM0N/BLKGS/PVS(40),PSST(40) COMMON/BLKI/PIPL,PIS COMMON/BLKF/PA,PV,PANRM,PVNRM,PVGRAD COMMON/BLKH/LSNRM Appendix E. Computer Programs 365 COMMDI/BLKJ/VTNRM,VIFNRM,VPLNRM COMMON/BLKK/VEXS COMMOH/BLKL/LSS,KFS.KPLLS,KRS,PSS COMMON/BLKT/VTOT,QTOT COMMOl/BLKU/PSNRM COMMDN/BLKY/PIPPSOO) ,PISKIN(30) ,HA,NAM1 COMMON/BLKBB/PIPNRM,PISNRM COMMON/BLKDD/ALBSTO C0MM0H/BLKGG/CALIFV(40),CALPIP(30) C0MM0N/BLKG5/CIFV1(200),CPIPL1(200) COMHOS/BLKKK/VPLS(40),VSS(40) DIMEHSION QPLS(40),QSS(40) DIMEHSION G(1),X(3) DIMEHSION Y0LD(2),YNEW(2) EXTERNAL RHSF DATA ACC,ALPHA,NV/1.D-7,0.5D0,2/ NA = 25 NAM1 = HA - 1 PIPLXJL = 28. DO LSS = X(3) * 10.DO RDL = X(l) / 10.DO RFR = X(2) / 10.DO CALL SPLINS C C Set i n i t i a l fluid and protein contents QPLG = VPLNRM * FALB(PIPNRM) QSG = (VIFNRM - 400.DO) * FALB(PISNRM) * 3.DO/4.DO VPLG = VPLHRM VSG = VIFNRM - 400.DO C **************************************** Appendix E. Computer Programs 366 C Calculate transport parameter values C **************************************** Wl = (CSNRM * RDL) / (CPLNRM - CASNRM) W2 = VIFNRM / (VIFNRM - VEXS) W3 = ALBSTO * VIFNRM LSNRM = W3 / (l.D0+Wl*W2) PSS - LSNRM * CSNRM * RDL / (CPLNRM - CASNRM) JPLLSN = (LSNRM * CSNRM - PSS * (CPLNRM - CASNRM))/CPLNRM KPLLS = JPLLSN / (PVNRM - PSNRK) Zl = PANRM - PSNRM - PIPNRM + PISNRM Z2 = PSNRM - PVNRM - PISNRM + PIPNRM KRS = (LSNRM - JPLLSN) / (RFR « Z l - Z2) KFS = KRS * RFR DIV = 0.25D0 PIPP = PIPLUL + l.DO DO 20 IN = 1,5000 PIPP = -l.DO * DIV + PIPP CPL = FALB(PIPP) YOLD(l) = VSG Y0LD(2) = QSG C ****************************************** Solve d i f f e r e n t i a l equations C ****************************************** CALL ROOT(RHSF,NV.O.DO,YOLD.ACC,ALPHA,YNEW,FLAG) VSG = YNEW(l) QSG = YBEW(2) IF(PIPP.LT.13.DO.AND.QSG.LT.0.01DO)THEN G(l) = 130.DO RETURN ENDIF Appendix E. Computer Programs 367 IF(VSG.LE.O.DO)THEH PRIST*,'*********************************» PRINT*,'*********************************> PRINT*,'** NEGATIVE VOL/Q GENERATED *****' PRINT*,'*********************************» PRINT*,'*********************************' STOP END IF VPLG = VTOT - VSG QPLG = QTOT - QSG CPIPL1(IN)=PIPP CIFV1(IN)=VSG IF(PIPP.LT.4.DO) GOTO 64 CHEK = PIPP - IDINT(PIPP) IF(CHEK.GT.0.1D-4) GOTO 20 SNUM = -l.DO * PIPP + PIPLUL + l.DO II = IDINT(SNUM) C ************************************** C Store results corresponding to plasma COP set C ************************************** VSS(II) = VSG VPLS(II)= VPLG QSS(II) = QSG QPLS(II)= QPLG PIPPS(II) = PIPP CALIFV(II) = VSS(II) CAVS = QSS(II) / (VSS(II) - VEXS) PISKIN(II) = FPI(CAVS) CALPIP(II) = FPI(CPL) JFSS(II)= JFS JPLLSS(II) = JPLLS Appendix E. Computer Programs 368 JRSS(II) = JRS JLSS(II) = JLS qPLLSS(II) = QPLLS qDSS(ii) = qDS qLSS(II) = QLS PVS(II) = PV PSST(II) = FCOMPS(VSS(II)) GOTO 20 50 IF(DIV.LE.1.5625D-2) GOTO 33 DIV = DIV / 2.DO SNUM = PIPP + l.DO PIPP = IDINT(SNUM) II = IDINT( -l.DO * PIPP + PIPLUL + l.DO) VSG = VSS(II) VPLG = VPLS(II) qsG = qss(n) qPLG = qPLS(n) GOTO 20 33 WRITE(6,40)PIPP 40 FORMAT(IX,'ROOT SOLVER FAILS! - WHEN PIPL = >,G23.15) RETURN 20 CONTINUE 64 CALL CERROR(ERSUM) G(l) = ERSUM RETURN END SUBROUTINE CERROR(ERSUM) C *********************************** C This subroutine calculates the sum of the C squares of the errors - between calculated Appendix E. Computer Programs 369 C and experimental values. C *********************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKZ/DIFV(18),DPIPL(18),HH COMMOH/BLKHB/PIPLPI(66),PII(66) C0MM0N/BLKY/PIPPS(30),PISKTJI(30).HA.NAMl C0MM0N/BLKG5/CIFV1(200),CPTPL1(200) COMMON/BLKZK/BIFVL,BIFVU,BPIIL,BPIIU NPI = 66 HCP = 25 SUMPI=0.D0 DO 130 J=1,NPI IMIN=1 D1=DSQRT((PII(J)-PISKIN(1))**2+(PIPLPI(J)-PIPPS(1))**2) DMIN=1.D10 DO 120 1=2,NCP D2=DSQRT((PII(J)-PISKIN(I))**2+(PIPLPI(J)-PIPPS(I))**2) DT0T=D1+D2 IF(DTOT.GE.DMIN) GO TO 110 IMIN=I DMIN=DTOT D1M=D1 D2M=D2 110 D1=D2 120 CONTINUE IF(DlM.EQ.0.D0.0R.D2M.Eq.0.D0) GO TO 130 I=IMIN-1 IP=IMIN D3=DSQRT((PISKIN(IP)-PISKIM(I))**2+(PIPPS(IP)-PIPPS(I))**2) ANG=DACOS((D1M*D1M+D3*D3-D2M*D2M)/(2.D0*DlM*D3)) SUMPI=SUMPI+D1M*DSIN(ANG) Appendix E. Computer Programs 370 130 CONTINUE NIFV = NN SUMIFV=0.D0 DO 131 J=1,NIFV IMIN=1 Dl=DSQRT(((DIFV(J)-CIFVl(l))/14.D+3)**2+((DPIPL(J) + -CPIPL1(1))/20.0D0)**2) DMIN=1.D10 DO 121 1=2,101 D2=DSQRT(((DIFV(J)-CIFVl(I))/14.D+3)**2+((DPIPL(J) + -CPIPL1(I))/20.0D0)**2) DT0T=D1+D2 IF(DTOT.GE.DMIN) GO TO 111 IMIN=I DMIN=DTOT D1M=D1 D2M=D2 111 D1=D2 121 CONTINUE IF(DlM.EQ.0.D0.0R.D2M.Eq.0.D0) GO TO 131 I=IMIN-1 IP=IMIN D3=DSqRT(((CIFVl(IP)-CIFVl(I))/14.D+3)**2+((CPIPLl(IP) + -CPIPL1(I))/20.0D0)**2) ANG=DAC0S((D1M*D1M+D3*D3-D2M*D2M)/(2.D0*D1M*D3)) SUMIFV=SUMIFV+D1M*DSIN(ANG) 131 CONTINUE EIFV = (SUMIFV * 30.DO - BIFVL) / (BIFVU - BIFVL) EPII = (SUMPI - BPIIL) / (BPIIU - BPIIL) ERSUH = SUMPI RETURN (SAV3)Idi = Sid (7d3)Idi = Idld (SX3A - (T ) A ) / ( Z ) A = SAVO ( I ) A / ( Z ) A = SO HHHVd = Vd WUHAd = Ad ( Z ) A + sndt) = loxb IdD * VldlOA = STIdb (T)A + VldlOA = IOXA oa•ooze = n<noA (Z)A HOISHHHia 01S3Hd/VZXia/H0HH03 MHHSd/fljna/HOHWOD XOXb'XOXA/IXia/HOHHOO SSd'SHX* STHX'SaS'SSl/TXia/HOHWOD SX3A/Xjna/H0WW03 Sid *Idld/IXia/HOHWOO WUHSir/Hyia/N0HM03 SAVO'S3'IdO/Oyia/HOKWOO OVHBAd * HHMAd'HUHVd* Ad'Vd/dJna/HOHHOO sib' sab * sndb/ana/HOHHOo sir * sHf' sndf' sar/axia/soHwoo **************************************** 3 33.TOq.uoo p x u x j ptre uraq .o .xd 3 TBquamq-rednioo atrq USATS SBUTBA aT.qBT.reA 3 maqsAs -rre saq.BTnoTBO a u r q u o i q u s 3tux 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 (z-o*Tx'r*H-v)8*iv3a xioiidHi * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 (A)wvsxnv 3Hixncraans aaa U S sureiSoij la^ndmoQ gr xrpuaddy Appendix E. Computer Programs 372 PS = FCOMPS(Y(l)) PS = FCOMPS(Yd)) JFS = KFS * (PA - PS - (PIPL - PIS)) JRS = KRS * (PS - PV - (PIS - PIPL)) JPLLS = KPLLS * (PV - PS) JLS = JLSNRM + LSS * (PS - PSNRM) IF(JLS.LT.JLSNRM) JLS=JLSNRM*(PS+PRESLO)/(PSNRM+PRESLO) IF(JPLLS.LT.0.DO)THEN QPLLS = JPLLS * CS ELSE QPLLS = JPLLS * CPL ENDIF QDS = PSS * (CPL - CAVS) QLS = JLS * CS RETURN END SUBROUTINE GAUSS(A,N,NDR,NDC,X,RNORM,IERROR) C **************************************** C C Purpose: C Uses Gauss elimination with partial pivot selection to C solve simultaneous linear equations of form A * X = C . C C Arguments: C A Augmented coefficient matrix containing a l l coefficients C and R.H.S. constants of equations to be solved. C N Number of equations to be solved. C NDR First (row) dimension of A in calling program. C NDC Second (column) dimension of A in calling program. C X Solution vector. (((x'r)a)sava*£S)TX¥Ha = is dfl'dx = r ss oa (OTi)a)SBva = is H * X = i oe oa oa'o = oia ******************************************* 3 q.ttaT3Tiiaco qsaSjeT, at(q. jo xaprrc aoi atfq. 3t xoAIdl 3 g t{3noax[q x sao.x *a ttnnrroo trt ^.tcaTOXiiaoa q.sa8.ret xo i tpieas 3 ****************************************** 3 T + X = dX H H ' I = X 08 oa ******************************************* o sanrtq. x-H ssaoo-id uoTquuiuicia 5.110 KTTZQ O ******************************************* o anfiiuioD oz 3QNIJJI03 Ot ( f i )? = (r'i)a dfi't = r ox oa S ' T = i oz oa ******************************************* 3 g xx.xq.-eiB Strcxxoa dn qas 3 ******************************************* 3 I + H = dH T - H = WK (T.9,09)a*(H.)x'(3aK,«ai)v HOisnawia (Z-0*H-V)8*TV3H IISIldHI *************************************************** 3 •uoTq.3aT.as q.OATd jaqxe jLzq.ua TBUOSBTP ojaz z~ 3 •uoTqerrcurtTa ssueg Tjixssaoong \= 3 •SBIJ aoxi3 HOWrai 3 X+V-3 JoqoaA TBtipTsaj 10 azxs 10 ajnseau, WHOHH SUTBJSOIJ i3indmoQ -g[ xjpuaddy Appendix E. Computer Programs 374 25 CONTINUE AB = DABS(B(I,K) / SF) IF(AB.LE.BIG) GOTO 30 BIG = AB IPIVOT = I 30 CONTINUE C **************************************** C INTERCHANGE ROWS K AND IPIVOT IF IPIVOT.NE.K Q ********************************************** IF(IPIVOT.EQ.K) GOTO 50 DO 40 J = K.NP TEMP = B(IPIVOT,J) B(IPIVOT.J) = B(K,J) B(K,J) = TEMP 40 CONTINUE 50 IF(B(K,K).EQ.O.DO) GOTO 130 C ********************************************* C Eliminate B(I,K) from rows K+l through N C ********************************************* DO 70 I = KP.N QUOT = B(I,K) / B(K,K) B(I,K) = O.DO DO 60 J = KP.NP B(I,J) = B(I,J) - QUOT * B(K,J) 60 CONTINUE 70 CONTINUE 80 CONTINUE IF(B(N,N).Eq.0.D0) GOTO 130 C ****************************************** C Back substitute to find'solution vector Appendix E. Computer Programs 3 7 5 c ******************************************* X(N) = B(N,NP) / B(N,N) DO 100 II = l.HM SUM = 0.D0 I = H - II IP = I + 1 DO 90 J = IP.N SUM = SUM + B(I,J) * X(J) 90 CONTINUE X(I) = (B(I.NP) - SUM) / B(I,I) 100 CONTINUE fj ****************************************** C Calculate norm of residual vector, C-A*X C Normal return with IERROR = 1 C ****************************************** Rsq = 0.D0 DO 120 I = 1,N SUM = 0.D0 DO 110 J = 1,N SUM = SUM + A(I,J) * X(J) 110 CONTINUE RSQ = RSQ + (DABS(A(I,NP) - SUM)) ** 2 120 CONTINUE RNORM = DSQRT(RSQ) IERROR = 1 RETURN C ******************************************* C Abnormal return because of zero entry on diagonal, IERR0R=2 C ******************************************* 130 IERROR = 2 RETURN Appendix E. Computer Programs END SUBROUTINE ROOT(F,M,X,YOLD,EPS,ALPHA.Y,FLAG) C **************************************** C This subroutine uses Newton's method to C find the solution of a set of simultaneous C non-linear equations. C **************************************** IMPLICIT REAL*8(A-H,0-Z) INTEGER FLAG DIMENSION YOLD(M),Y(M),DY(iO),DELY(10),A(10,11) MP = M + 1 DO 10 I = 1,M Y(I) = YOLD(I) DELY(I) = l.D-6 * YOLD(I) 10 CONTINUE DO 60 ITER = 1,100 FLAG = 0 DO 40 I = 1,M DO 30 J = l.MP IF(J.EQ.MP) GOTO 20 Y(J) = Y(J) + DELY(J) FUP = F(I.X.Y) Y(J) = Y(J) - 2.D0 * DELY(J) FDOWN = F(I.X.Y) Y(J) = Y(J) + DELY(J) A(I,J) = (FUP - FDOWN) / (2.DO * DELY(J)) GOTO 30 20 A(I,J) = -F(I,X,Y) 30 CONTINUE 40 CONTINUE CALL GAUSS(A,M,10,-1'1,DY,RN0RM,IERR0R) Appendix E. Computer Programs FLAG = 1 DO 50 I = 1,M Y(I) = Y(I) + ALPHA * DY(I) IF(DABS(DY(I)).GT.EPS) FLAG = 0 50 CONTINUE IF(FLAG.EQ.l) RETURN 60 CONTINUE RETURN END SUBROUTINE TDMA(A,B,C,D,X,N) C ************************************ C This subroutine uses the Thomas algorithm C to find the solution of a tridiagonal C matrix. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C *********************************** C Thomas algorithm C *********************************** DIMENSION A(N),B(N),C(N),D(N),X(N),P(10l),Q(101) NM = N - 1 P(l) = -C(l) / B(l) Q(l) = D(l) / B(l) DO 10 I = 2,N IM = 1-1 DEN = A(I) * P(IM) + B(I) P(I) = -C(I) 7 DEN Q(I) = (D(I) - A(I) * q(IM)) / DEN 10 CONTINUE X(N) = Q(N) DO 20 II = l.NM Appendix E. Computer Programs I = N - II X(I) = P(I) * X(I+1) + Q(I) 20 CONTINUE RETURN END DOUBLE PRECISION FUNCTION FALB(PI) C **************************************** C This function calculates Albumin C concentration - using a fitted C (Alb) vs. C.O.Ppl relationship. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKCC/CALB1.CALB2 FALB = (CALB1 + CALB2 * PI) / l.D+3 RETURN END DOUBLE PRECISION FUNCTION FPI(C) C **************************************** C This function calculates colloid C osmotic pressure - using a fitted C C.O.P vs. (Alb) relationship. C **************************************** IMPLICIT REAL*8(A-H,0-Z) COMM0N/BLKN/CPI1,CPI2 R = C * l.D+3 FPI = CPU + CPI2 * R RETURN END DOUBLE PRECISION FUNCTION RHSF(I,X,Y) Q **************************************** Appendix E. Computer Programs C This function evaluates the non-linear C equations for a given set of Y values. C **************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) COMMON/BLKB/JFS,JPLLS,JRS,JLS COMMOH/BLKD/qPLLS,qDS,qLS DIMEHSION Y(2) C CALL AUXSAM(Y) C RBSF3 = JFS + JPLLS - JRS - JLS RHSF4 = QPLLS + QDS - QLS G0T0(10,20),I 10 RHSF = RHSF3 RETURN 20 RHSF = RHSF4 RETURN END DOUBLE PRECISION FUNCTION FCOMPS(V) C **************************************** C This function calculates the skin C compartments hydrostatic pressure. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0H/BLK0/VS(14),PS(14),AS,BS,NPS,HPSM1 IF(V.LE.VS(1)) GO TO 10 IF(V.GE.VS(10)) GO TO 20 FCOMPS = FS(V) RETURN 10 FCOMPS = PS(1) + AS * ( V - VS(1)) RETURN Appendix E. Computer Programs 380 20 FCOMPS = PS(IO) + BS * (V - VS(10)) RETURN END SUBROUTINE SPLINS C *************************************** C This subroutine SPLINES the erperimently C obtained PS vs. VS data set. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLK0/X(14),Y(14).Al.BN.N.NM COMMON/BLKQAKlOl),R(101),S(100) DIMENSION H(100),A(101),B(101),C(101),D(101) DO 10 I = l.NM H(I) = X(I+1) - X(I) 10 CONTINUE A(l) = O.DO B(l) = 2.D0 * H(l) C(l) = H(l) D(l) = 3.DO * ((Y(2) - Y(l)) / H(l) - Al) DO 20 I = 2.NM IP = 1+1 IM = 1-1 A(I) = H(IM) B(I) = 2.D0 * (H(IM) + H(D) C(I) = H(I) D(I) = 3.D0 * ((Y(IP) - Y(I)) / H(I) - (Y(I) - Y(IM))/H(IM)) 20 CONTINUE A(N) = H(NM) B(N) = 2.DO * H(NM) C(N) = O.DO endix E. Computer Programs D(N) = -3.D0 * ((Y(N) - Y(NM)) / H(NM) - BH) CALL TDMAU.B.C.D.R.N) DO 30 I = l.HM IP = I + 1 Q(I) = (Y(IP) - Y(I)) / H(I) - H(I) * (2.D0 *R(I)+R(IP))/3.D0 S(I) = (R(IP) - R(I)) / (3.DO * H(I)) 30 CONTINUE RETURN END DOUBLE PRECISIOI FUNCTION FS(Z) C **************************************** C This function evaluates skin compartment C hydrostatic pressure - using a splined C set of data. C **************************************** IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKO/X(14),Y(14),A1,BH,H,HM C0MM0N/BLKQ/Q(101),R(101),S(100) 1 = 1 IF(Z.LT.X(1)) GO TO 30 IF(Z.GE.X(HM)) GO TO 20 J = NM 10 K = (I + J) / 2 IF(Z.LT.X(K)) J = K IF(Z.GE.X(K)) I = K IF(J.EQ.I+1) GO TO 30 GO TO 10 20 I = NM 30 DX = Z - X(I) FS = Y(I) + DX * (Q(I) + DX * (R(I) + DX * S(I))) RETURN Appendix E. Computer Programs END E.7 Listing of Program P L M - C c **************************************** IMPLICIT REAL*8(A-H,K,L,J,0-Z) INTEGER FLAG,FLAG6,FLAG7,FLAGG C0MM0N/BLKA2/CPLNRM,CSNRM,CASNRM COMMON/BLKB/JFS,JPLLS,JRS,JLS COMMON/BLKD/QPLLS,QDS,QLS COMMON/BLKG/CPL,CS,CAVS COMMON/BLKI/PIPL,PIS COMMON/BLKF/PA,PV,PANRM,PVNRM,PVGRAD COMMON/BLKH/LSNRM COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS.KPLLS,KRS,PSS COMMON/BLKT/QTNRM,QSNRM,QPLNRM COMMON/BLKU/PSNRM,PS COMMON/BLKBB/PIPNRM.PISNRM COMMON/BLKDD/ALBSTO COMMON/BLKZB/QTOT,VTOT C0MM0N/BLKZJ/EXPLVL(4),EXCPL(4),EXDLPI(3),EXDLCA(3) C0MM0N/BLKZK/EXPIPL(4),EXPII(3),EXCAV(3) DIMENSION S0LN(18,769),T(1001).SOL(lOOl),S0LF(18) DIMENSION YOLD(4),YNEW(4),Y0LDA(2),YNEWA(2),YFINAL(2) DIMENSION YEX1(4),YEX9(4),YEX3(4),YEX2(3),YEX11(3), + YEX17(3),YEX18(3) EXTERNAL RHSF,RHSFA DATA ACC,ALPHA,EPS,DT,NP,NV/1.D-7,.5D0,.0IDO,0.005D0,769,4/ FLAG6 = 0 Appendix E. Computer Programs FLAG7 = 0 TSTART=1.D-1*DT TMIN=1.D-4*DT TMAX=DT C *************************************** C Set RFR, LS, and RDL parameter values C *************************************** LSS = 31.82000D0 RDL = O.OOOOOODO RFR = 0.241470D0 CALL SPLIHS CALL COEFF(RDL,RFR) C *************************************** C Set i n i t i a l fluid volumes and protein contents C *************************************** PLINIT = 10.56689D0 PIINIT = 3.35855D0 VIFINT = 18250.DO PIDINT = PLINIT - PIINIT VPINIT = VPLNRM QPINIT = VPLNRM * FALB(PLINIT) VSINIT = VIFINT QSINIT = FALB(PIINIT) * (VIFINT - VEXS) VTINIT = VPINIT + VSINIT QTINIT = QPINIT + QSINIT YNEW(l) = VPINIT YNEW(2) = QPINIT YNEW(3) = VSINIT YNEW(4) = QSINIT C ************************************ C Solve differential, balances Appendix E. Computer Programs c ************************************ DO 30 I = 1,19200 TIM = (I-1)*DT TI = I * DT IF(I.EQ.l) GOTO 14 C CALL RK4C(RHSF,Sv,T(IM),T(I).YOLD,EPS,YHEW.HFUHC,FLAG) CALL DESOLV(RHSF,KV,TIM,TI,YOLD,EPS,TSTART,TMIN,TMAX, + YHEW.FLAG) IF(FLAG.EQ.O) GOTO 33 CALL AUXSAM(YKEW) Q ************************************* C Store results at select time intervals C *************************************** 14 IF(TI.LE.3.D0)THEN IK=(I-l)/40 +1 REALI = I RI = (REALI - l.D0)/40.D0 + l.DO RIK = IK CHK = RI - RIK ELSE IK=(I-l)/200 + 12 REALI = I RI = (REALI - l.D0)/200.D0 + 12.DO RIK = IK CHK = RI - RIK EHDIF IF(CHK.HE.O.DO) GOTO 56 CALL AUXSAM(YNEW) T(IK)=DT*(I-1) SOLH(l.IK) = YNEW(l) Appendix E. Computer Programs S 0 L N ( 2 , I K ) = Y H E W ( 2 ) S 0 L N ( 3 , I K ) = P V S 0 L N ( 4 , I K ) = P I P L S 0 L N ( 5 , I K ) = C P L S 0 L N ( 6 , I K ) = P I S S 0 L H ( 7 , I K ) = C S S 0 L N ( 8 , I K ) = C A V S S 0 L N ( 9 , I K ) = P S S O L N ( I O . I K ) = J F S S O U K 1 1 , I K ) = J L S S O U K 1 2 , I K ) = Q P L L S S 0 U K 1 3 . I K ) = QDS S O U K 1 4 , nc) = Q L S S O U K I E , I K ) = Y H E W ( 3 ) S O U K 1 6 , I K ) = Y N E W ( 4 ) S 0 U K 1 7 . I K ) = J P L L S S 0 U K 1 8 . I K ) = J R S 5 6 DO 20 J = l.NV YOLD(J) = YHEW(J) 20 CONTINUE 30 CONTINUE VTOT = YNEW(l) + YNEW(3) QTOT = YNEW(2) + TNEW(4) YOLDA(l) = YNEW(3) Y0LDA(2) = YNEW(4) C *************************************** C Calculate final steady-state variable C values and store C *************************************** CALL ROOT(RHSFA,2,1000.DO,YOLDA,ACC,ALPHA,YFINAL,FLAGG) CALL AUXALT(YFINAL) Appendix E. Computer Programs 386 SOLF(l) = VTOT - YFINAL(l) S0LF(2) = QTOT - YFINAL(2) S0LF(3) = PV S0LF(4) = PIPL S0LF(5) = CPL S0LF(6) = PIS S0LF(7) = CS S0LF(8) = CAVS S0LF(9) = PS SOLF(IO) = JFS SOLF(li) = JLS S0LF(12) = QPLLS S0LF(13) = QDS SOLF(14) = QLS S0LF(15) = YFINAL(l) S0LF(16) = YFINAL(2) S0LF(17) = JPLLS S0LF(18) = JES c ***************************************** C Output variables stored and associated times C *************************************** DO 112 1=1,107 WRITE(6,91)T(I),S0LH(4,I),S0LN(6,I),S0LN(3,I),S0LH(9,I) 91 F0RMAT(6X,F6.3,5X,F7.4,5X,F7.4,5X,F7.4,5X,F7.4) 112 CONTINUE WRITE(6,41)S0LF(4),S0LF(6),S0LF(3),S0LF(9) 41 F0RMAT(17X,F7.4,5X,F7.4,5X,F7.4,5X,F7.4) DO 111 1=1,107 WRITE(6,95)T(I),S0LN(15,I),S0LN(1,I),S0LN(16,I),S0LN(2,I) 95 F0RMAT(6X,F6.3,5X,F10.3,5X,F7.2,5X,F7.3,5X,F7.3) 111 CONTINUE Appendix E. Computer Programs WRITE(6,45)S0LF(1S),SOLF(i),S0LF(16),S0LF(2) 45 F0RMAT(17X,F10.3,5X,F7.2,5X,F7.3,5X,F7.3) DO 113 1=1,107 WRITE(6,73)T(I),S0LN(10,I),S0LH(17,I),S0LH(18,I), + SOLH(ll.I) 73 F0RMAT(6X,F6.3,5X,F8.2,5X,F8.2,5X,F8.3,5X,F8.3) 113 COHTIHUE WRITE(6,43)S0LF(10),S0LF(17),S0LF(18), + SOLF(ll) 43 F0RMAT(17X,F8.2,5X,F8.2,5X,F8.3,5X,F8.3) DO 114 1=1,107 WRITE(6,99)T(I),S0LH(12,I),S0LB(13,I),S0LN(14,I) 99 F0RMAT(6X,F6.3,5X,F8.3,5X,F8.3,5X,F8.3) 114 COHTIHUE WRITE(6,49)S0LF(12),S0LF(13),S0LF(14) 49 F0RMAT(17X,5X,F8.3,5X,F8.3,5X,F8.3) DO 117 1=1,107 WRITE(6,58)T(I),S0LH(5,I),S0LH(8,I),S0LH(7,I) 58 F0RMAT(6X,F6.3,5X,F7.5,5X,F7.5,5X,F7.5) 117 COHTIHUE WRITE(6,24)S0LF(5),S0LF(8),S0LF(7) 24 F0RMAT(17X,F7.5,5X,F7.5,5X,F7.5) DO 72 II = 1,4 IF(II.LT.4)THEH YEXl(II) = EXPIPL(II) YEX9(II) = EXCPL(II) YEX3(II) = EXPLVL(II) YEX2(II) = EXPII(II) YEXll(II) = EXCAV(II) Appendix E. Computer Programs YEX17(II) = EXDLPI(II) YEX18(II) = EXDLCA(II) ELSE YEXl(II) = EXPIPL(II) YEX9(II) = EXCPL(II) YEX3(II) = EXPLVL(II) ENDIF 72 COHTIHUE 33 WRITE(6,40) 40 FORMAT(IX,'ODE SOLVER FAILS! ') STOP EHD SUBROUTIHE COEFF(RDL,RFR) C ************************ IMPLICIT REAL*8(A-H,J,K,L,0-Z) C0MM0H/BLKA2/CPLHRM,CSHRM,CASHRM COMMON/BLKT/QTHRM,QSHRM,QPLHRM COMMOH/BLKDD/ALBSTO COMMOH/BLKH/LSHRM COMMOH/BLKF/PA,PV,PAHRM,PVHRM,PVGRAD COMMOH/BLKU/PSHRM,PS COMMOH/BLKK/VEXS COMMOH/BLKBB/PIPHRM.PISNRM COMMOH/BLKL/LSS,KFS.KPLLS,KRS,PSS COMMOH/BLKJ/VTHRM,VIFNRM,VPLNRM C *************************************** C Set normal steady-state conditions and C calculate transport parameter values C *************************************** PIPNRM = 25.9D0 PISHRM = 14.7D0 Appendix E. Computer Programs 389 PSNRM = FCOMPS(VIFNRM) CPLNRM = FALB(PIPNRM) CASNRM = FALB(PISNRM) CSNRM = CASNRM * (l.DO - VEXS / VIFNRM) QPLNRM = VPLNRM * CPLNRM QSNRM = VIFNRM * CSNRM QTNRM = QPLNRM + QSNRM VPLNRM = VPLNRM VSNRM = VIFNRM VTNRM = VPLNRM + VSNRM Wl = (CSNRM * RDL) / (CPLNRM - CASNRM) W2 = VIFNRM / (VIFNRM - VEXS) W3 = ALBSTO * VIFNRM LSNRM = W3 / (l.D0+Wl*W2) PSS = LSNRM * CSNRM * RDL / (CPLNRM - CASNRM) JPLLSN = (LSNRM * CSNRM - PSS * (CPLNRM - CASNRM))/CPLNRM KPLLS = JPLLSN /'(PVNRM - PSNRM) ZI = PANRM - PSNRM - PIPNRM + PISNRM Z2 = PSNRM - PVNRM - PISNRM + PIPNRM KRS = (LSNRM - JPLLSN) / (RFR * ZI - Z2) KFS = KRS * RFR RETURN END SUBROUTINE AUXSAM(Y) C *************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) C **************************************** COMMON/BLKB/JFS,JPLLS,JRS,JLS COMMON/BLKD/QPLLS,QDS,QLS COMMON/BLKF/PA,PV,PANRM,PVNRM,PVGRAD Appendix E. Computer Programs 390 COMMON/BLKG/CPL,CS,CAVS C0MM0N/BLKH7 JLSNRM COMMON/BLKI/PIPL,PIS COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS.KPLLS,KRS,PSS COMMON/BLKT/QTNRM, QSNRM, QPLNRM COMMON/BLKU/PSNRM,PS COMMON/BLKZA/PRESLO COMMON/BLKZB/QTOT,VTOT C ********************************** C Calculate system variable values from C compartmental fluid and protein contents C *************************************** DIMENSION Y(4) CPL = Y(2) / Y(l) PA = PANRM PV = FCOMPV(Y(l)). CS = Y(4)/ Y(3) CAVS = Y(4)/ (Y(3) - VEXS) PIPL = FPI(CPL) PIS = FPI(CAVS) PS = FC0MPS(Y(3)) JFS = KFS * (PA - PS - (PIPL - PIS)) JPLLS = KPLLS * (PV - PS) JRS = KRS * (PS - PV - (PIS - PIPL)) JLS = JLSNRM + LSS * (PS - PSNRM) IF(JLS.LT.JLSNRM) JLS=JLSNRM*(PS+PRESLO)/(PSNRM+PRESLO) QPLLS = JPLLS * CPL QDS = PSS * (CPL - CAVS) QLS = JLS * CS RETURN Appendix E. Computer Programs 391 END SUBROUTINE TDMA(A,B,C,D,X,N) C **************************************** C This subroutine uses the Thomas algorithm C to find the solution of a tridiagonal C matrix. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C ***************************************** C Thomas algorithm C **************************************** DIMENSION A(N)IB(N),C(N)>D(N),X(N),P(101)IQ(101) : NM = N - 1 P(l) = -CCD / B(l) 0(1) = D(l) / B(l) DO 10 I = 2,N IM = 1-1 DEN = A(I) * P(IM) + B(I) P(I) = -C(I) / DEN Q(I) = (D(I) - A(I) * Q(IM)) / DEN 10 CONTINUE X(N) = Q(N) DO 20 II - l.NM I = N - II X(I) = P(I) * X(I+1) + Q(I) 20 CONTINUE RETURN END DOUBLE PRECISION FUNCTION FALB(PI) C **************************************** C This function calculates Albumin H3Hl(0aS-f3TI)iI (A)HYSxnv nvo (fr)A HOISJOMia sib * sab' sndb/axia/iiowwoo sir'SHC ' s n d f ' sir/axia/HOMWOD (Z-0'T3*r ,H-V)8*TraH IIDIldMI **************************************** 3 •S8T1TBA I JO 3.3S U 9 A X 3 B XOJ. S U O T l B T l b a 3 jeantx-TioTi anq. saq.«tvrBAa xtoT+a'tnix S T t r i 3 **************************************** 3 (A'l'l)aSHH HOIXOKM H0ISI33Hd HiaQOa ana Human H * SId3 + TIdO = IdJ S+O'T. * 3 = H 3Id3'IIdO/HHia/HOHWOD (z-o*H-v)8*TraH nondwi **************************************** 3 • dTT^STioTa-BTaa (qi?) *SA d'O'D 3 paaaxy B 3UTSTI - a x o . s s a . x d DT^ OUISO 3 pxoxioo s a q . B T t U 3 T B 3 u o x i o n t i i STTJJ, 3 **************************************** 3 O) Ida HoiioKoa soisioaad xianoa ana Iranian e+a-T / (Id * Z3Tf3 + t3Tf3) = arva 23TV3'T aTV3/33Xia/H0WH03 (Z-0'H-V)8*TV3H IISIldHI **************************************** 3 •dTTfsuoxq.BT.a.1 l d d - 0 " 3 -SA (qiv) 3 paq .q . t i B SUTSTI - uoTqBJaTiaanoo 3 Z62 sureiSoij ja'indmoQ g- xrpuaddy Appendix E. Computer Programs RHSFl = JLS + JRS - JPLLS - JFS + 200.DO RHSF2 = QLS - QDS - QPLLS + 40.DO RHSF3 = JFS + JPLLS - JRS - JLS RHSF4 = QPLLS + QDS - QLS ELSEIF(T.GT.1.EDO.AND.T.LE.25.5D0)THEN RHSFl = JLS + JRS - JPLLS - JFS - (300.DO/24.DO) RHSF2 = QLS - QDS - QPLLS - (15.8D0/24.D0) RHSF3 = JFS + JPLLS - JRS - JLS RHSF4 = QPLLS + QDS - QLS ELSEIF(T.GT.25.5D0.AHD.T.LE.92.639D0)THEN RHSFl = JLS + JRS - JPLLS - JFS RHSF2 = QLS - QDS - QPLLS - (15.8D0/24.D0) RHSF3 = JFS + JPLLS - JRS - JLS RHSF4 = QPLLS + QDS - QLS ELSE RHSFl = JLS + JRS - JPLLS - JFS RHSF2 = QLS - qDS - QPLLS RHSF3 = JFS + JPLLS - JRS - JLS RHSF4 = QPLLS + qDS - QLS END IF GOTO(10,20,30,40),I 10 RHSF = RHSFl RETURN 20 RHSF = RHSF2 RETURN 30 RHSF = RHSF3 RETURN 40 RHSF = RHSF4 RETURN END Appendix E. Computer Programs 394 BLOCK DATA C **************************************** IMPLICIT REAL*8(A-H,K,L,0-Z) C **************************************** COMMON/BLKF/PA,PV.PANRM,PVNRM.PVGRAD COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM COMMON/BLKK/VEXS COMMON/BLKN/CPI1,CPI2 C0MM0N/BLK0/VS(14),PS(14),AS,BS,NPS,NPSM1 COMMON/BLKCC/CALB1.CALB2 COMMON/BLKDD/ALBSTO COMMON/BLKZA/PRESLO COMMON/BLKZJ/EXPLVL(4),EXCPL(4),EXDLPI(3),EXDLCA(3) C0MM0N/BLKZK/EXPIPL(4),EXPII(3),EXCAV(3) DATA EXPLVL/2.99D0,3.85D0,3.80D0,3.30D0/ DATA EXPIPL/9.3D0,13.5D0,12.6D0,9.8D0/ DATA EXCPL/10.4D0,24.IDO,23.4DO,15.4DO/ DATA EXPII/2.3D0,2.8D0,4.0D0/ DATA EXCAV/3.5D0,4.6D0,6.8D0/ DATA"EXDLPI/7.ODO,10.7D0,5.8D0/ DATA EXDLCA/6.9D0.19.5D0.8.6D0/ DATA PANRM,PVNRM,VEXS/26.54D0,7.92D0,2.1D+3/ DATA PVGRAD/0.009658D0/ DATA PRESL0/13.O577DO/ DATA VTNRM,VIFNRM,VPLNRM/11.6D+3,8.4D+3,3.2D+3/ DATA CPI1,CPI2/1.75490D-4,0.656840D0/ DATA CALB1,CALB2/-0.267173D-3,0.152244D+1/ DATA NPS.NPSMl/14,13/ DATA VS/8.4D+3,8.92D+3,9.4SD+3,9.97D+3,10.50D+3,H.02D+3, + 11.55D+3,12.07D+3,12.60D+3,13.65D+3,14.70D+3,16.80D+3, + 21.00D+3.25.23D+3/ Appendix E. Computer Programs 395 DATA PS/-0.70D0,0.32D0,0.86D0,1.15D0,1.37D0,1.56D0, + 1.69D0,1.8OD0,1.88D0,1.99DO,2.OlD0,2.O4D0,2.12DO, + 2.20D0/ DATA AS,BS/1.96154D-3,1.81347D-5/ DATA ALBST0/0.020539D0/ END DOUBLE PRECISION FUNCTIOH FCOMPS(V) C **************************************** C This function calculates the skin C compartments hydrostatic pressure. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLK0/VS(14),PS(14),AS,BS,NPS,HPSM1 IF(V.GE.VS(10)) GO TO 20 FCOMPS = FS(V) RETURN 10 FCOMPS = PS(1) + AS * ( V - VS(1)) RETURN 20 FCOMPS = PS(10) + BS * (V - VS(10)) RETURN END SUBROUTINE SPLINS C *************************************** C This subroutine SPLINES the experiaently C obtained PS vs. VS data set. C **************************************** IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLK0/X(14),Y(14),A1,BN,N,NM C0MM0N/BLKQ/Q(101),R(101),S(100) DIMENSION H(100),A(101),B(101),C(101),D(101) DO 10 I = l.NM Appendix E. Computer Programs H(I) = X(I+1) - X(I) 10 CONTINUE A(l) = O.DO B(l) = 2.D0 * H(l) C(l) = H(l) D(l) = 3.DO * ((Y(2) - Y(D) / H(l) - Al) DO 20 I = 2.NM IP = 1+1 IM = 1-1 A(I) = H(IM) B(I) = 2.D0 * (H(IM) + H(I)) C(I) = H(I) D(I) = 3.D0 * ((Y(IP) - Y(I)) / H(I) - (Y(I) - Y(IM))/H(IM)) 20 CONTINUE A(N) = H(NH) B(N) = 2.DO * H(NM) C(N) = O.DO D(N) = -3.D0 *' ((Y(N) - Y(NM)) / H(NM) - BN) CALX TDMA(A,B,C,D,R,N) DO 30 I = l.NM IP = I + 1 g(I) = (Y(IP) - Y(I)) / H(I) - H(I) * (2.DO *R(I)+R(IP))/3.D0 S(I) = (R(IP) - R(D) / (3.D0 * H(I)) 30 CONTINUE RETURN END DOUBLE PRECISION FUNCTION FS(Z) C **************************************** C This function evaluates skin compartment C hydrostatic pressure - using a splined Appendix E. Computer Programs C set of data. C **************************** IMPLICIT REAL*8(A-H,0-Z) COMM0N/BLKO/X(14),Y(14),A1,BN,N,NM C0MM0N/BLKQ/Q(101),R(101),S(100) 1 = 1 IF(Z.LT.X(D) GO TO 30 IF(Z.GE.X(NM)) GO TO 20 J = NM 10 K = (I + J) / 2 IF(Z.LT.X(K)) J = K IF(Z.GE.XOO) I = K IF(J.EQ.I+1) GO TO 30 GO TO 10 20 I = NM 30 DX = Z - X(I) FS = Y(I) + DX * (Q(I) + DX * (R(I) + DX * S(I))) RETURN • END DOUBLE PRECISION FUNCTION FCOMPV(V) IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKF/PA.PV,PANRM,PVNRM,PVGRAD COMMON/BLKJ/VTNRM,VIFNRM,VPLNRM FCOMPV = PVNRM + PVGRAD * (V - VPLNRM) IF(FC0MPV.LT.2.D0) FCOMPV = 2.D0 RETURN END SUBROUTINE RK4C(F,M,A,B,YA,EPS,YB.NFUN,FLAG) IMPLICIT REAL*8(A-B,K,0-Z) INTEGER FLAG DIMENSION YA(M),YI(10),YB(M),YB0LD(10),YARG1(10),YARG2(10) Appendix E. Computer Programs DIMENSION Kl(10),K2(10),K3(10) ,K4(10) FLAG=1 NINT=1 NFUN=0 10 NFUN=NFUN+4*NINT*M DX=(B-A)/NINT XI=A DO 20 1=1,M 20 YI(I)=YA(I) DO 70 INT=1,NINT DO 30 1=1,M K1(I)=DX*F(I,B,YI) 30 YARG1(I)=YI(I)+K1(I)/2.D0 XARG=XI+DX/2.D0 DO 40 1=1,M K2(I)=DX*F(I,B,YARG1) 40 YARG2(I)=YI(I)+K2(I)/2.D0 DO 50 1=1,M K3(I)=DX*F(I,B,YARG2) 50 YARG1(I)=YI(I)+K3(I) XI=XI+DX DO 60 1=1,M K4(I)=DX*F(I,B,YARG1) 60 YI(I)=YI(I)+(K1(I)+2.D0*(K2(I)+K3(I))+K4(I))/6.D0 70 CONTINUE DO 80 1=1,H 80 YB(I)=YI(I) IF(NINT.Eq.l) GO TO 100 YBDIFM=0.D0 DO 90 1=1,M 90 YBDIFM=DHAX1(DABS(YB(I)-YBOLD(I)),YBDIFM) Appendix E. Computer Programs 399 IF(YBDIFM.LT.EPS) RETURN IFCNINT.GT.10000) GO TO 120 100 DO 110 1=1,M 110 YBOLDCD=YB(I) NINT=2*NINT GO TO 10 120 FLAG=0 RETURN END SUBROUTINE GAUSSCA,N.NDR.NDC.X.RNORM,IERROR) C **************************************** C Purpose: C Uses Gauss elimination with partial pivot selection to C solve simultaneous linear equations fo form A*X=C C Arguments: C A Augmented coefficient matrix containing a l l coefficients C and R.E.S. constants of equations to be solved. C N Number of equations to be solved. C NDR First Crow) dimension of A in calling program. C NDC Second (column) dimension of A in calling program. C X Solution vector. C RN0RM Measure of size of residual vector C-A*X C IERROR Error flag. C =1 Successful Gauss elimination. C =2 Zero diagonal entry after pivot selection. IMPLICIT REAL*8CA-H,0-Z) DIMENSION A(NDR,NDC),X(N),B(50,51) NM = N - 1 NP = N + 1 C ********************************* Appendix E. Computer Programs 400 C Set up working matrix B C **************************************** DO 20 I = 1,N DO 10 J = l.NP B(I,J) = A(I,J) 10 CONTINUE 20 CONTINUE C *************************************** C Carry out elimination process N-l times C *************************************** DO 80 K = l.NM KP = K + 1 Q *************************************** C Search for largest coefficient in column K, rows K through N C IPIVOT is the row index of the largest coefficient C *************************************** BIG = O.DO DO 30 I = K,N SF = DABS(B(I,K)) DO 25 J = KP.NP SF = DMAX1(SF,DABS(B(J,K))) 25 CONTINUE AB = DABS(B(I,K) / SF) IF(AB.LE.BIG) GOTO 30 BIG = AB IPIVOT = I 30 CONTINUE C *************************************** C INTERCHANGE ROWS K AND IPIVOT IF IPIVOT.NE.K C *************************************** IF(IPIVOT.EQ.K) GOTO 50 Appendix E. Computer Programs 401 DO 40 J = K.NP TEMP = B(IPIVOT.J) B(IPIVOT.J) = B(K,J) B(K,J) = TEMP 40 CONTINUE 50 IF(B(K,K).EQ.0.D0) GOTO 130 C *************************************** C Eliminate B(I,K) from rows K+l through N C *************************************** DO 70 I = KP.N QUOT = B(I,K) / B(K,K) B(I,K) = O.DO DO 60 J = KP.NP B(I,J) = B(I,J) - QUOT * B(K,J) 60 CONTINUE 70 CONTINUE 80 CONTINUE IF(B(N,N).EQ.0.D0) GOTO 130 C *************************************** C Back substitute to find solution vector C *************************************** X(N) = B(N,NP) / B(N,N) DO 100 II = l.NM SUM = O.DO I = N.- II IP = I + 1 DO 90 J = IP.N SUM = SUM + B(I,J) * X(J) 90 CONTINUE X(I) = (B(I.NP) - SUM) / B(I,I) 100 CONTINUE Appendix E. Computer Programs 402 Q *************************************** C Calculate norm of residual vector, C-A*X C Hormal return with IERROR = 1 C *************************************** Rsq = O.DO DO 120 I = 1,H SUM = O.DO DO 110 J = 1,H SUM = SUM + A(I,J) * X(J) 110 CONTINUE RSq = RSQ + (DABS(A(I,NP) - SUM)) ** 2 120 CONTINUE RNORM = DSQRT(RSQ) IERROR = 1 RETURN C *************************************** C Abnormal return because of zero entry on diagonal, IERR0R=2 C *************************************** 130 IERROR = 2 RETURN END SUBROUTINE ROOT(F.M.X,TOLD,EPS,ALPHA,Y,FLAG) C **************************************** C This subroutine uses Newton's method to C find the solution of a set of simultaneous C non-linear equations. Q **************************************** IMPLICIT REAL*8(A-H,0-Z) INTEGER FLAG DIMENSION YOLD(M),Y(M),DY(10),DELY(10),A(10,11) Appendix E. Computer Programs MP = M + 1 DO 10 I = 1,M Y(I) = YOLD(I) DELY(I) = l.D-6 * YOLD(I) 10 CONTINUE DO 60 ITER = 1,100 FLAG = 0 DO 40 I = 1,M DO 30 J = l.MP IF(J.EQ.MP) GOTO 20 Y(J) = Y(J) + DELY(J) FUP = F(I,X,Y) Y(J) = Y(J) - 2.D0 * DELY(J) FDOWN = F(I,X,Y) Y(J) = Y(J) + DELY(J) A(I,J) = (FUP - FDOWN) / (2.DO * DELY(J)) GOTO 30 20 A(I,J) = -F(I,X,Y) 30 CONTINUE 40 CONTINUE CALL GAUSS(A,M,10,11,DY,RN0RM,IERR0R) FLAG = 1 DO 50 I = 1,M Y(I) = Y(I) + ALPHA * DY(I) IF(DABS(DY(I)).GT.EPS) FLAG = 0 50 CONTINUE IF(FLAG.EQ.l) RETURN 60 CONTINUE RETURN END DOUBLE PRECISION FUNCTION RHSFA(I.X.Y) Appendix E. Computer Programs Q **************************************** C This function evaluates the non-linear C equations for a given set of Y values. C **************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) COMMON/BLKB/JFS,JPLLS,JRS,JLS COMMON/BLKD/QPLLS,QDS,QLS DIMENSION Y(2) CALL AUXALT(Y) RHSF3 = JFS + JPLLS - JRS - JLS RHSF4 = QPLLS + QDS - QLS G0T0(10,20),I 10 RHSFA = RHSF3 RETURN 20 RHSFA = RHSF4 RETURN END SUBROUTINE AUXALT(Y) C *************************************** IMPLICIT REAL*8(A-H,J,K,L,0-Z) C **************************************** COMMON/BLKB/JFS,JPLLS,JRS,JLS COMMON/BLKD/QPLLS,QDS,QLS COMMON/BLKF/PA,PV,PANRM,PVNRM.PVGRAD COMMON/BLKG/CPL,CS,CAVS COMMON/BLKH/JLSNRM COMMON/BLKI/PIPL,PIS COMMON/BLKK/VEXS COMMON/BLKL/LSS,KFS,KPLLS,KRS,PSS COMMON/BLKZB/QTOT,VTOT Appendix: E. Computer Programs 405 COMMON/BLKU/PSHRH.PS COMMON/BLKZA/PRESLO DIMENSION Y(2) VOLPLA = VTOT - Y(l) QPLAS = QTOT - Y(2) PA = PANRM PV = FCOMPV(VOLPLA) CPL = QPLAS / VOLPLA CS = Y(2) / Y(l) CAVS = Y(2) / (Y(l) - VE1S) PIPL = FPI(CPL) PIS = FPI(CAVS) PS = FCOHPS(Y(D) JFS = KFS * (PA - PS - (PIPL - PIS)) JPLLS = KPLLS * (PV - PS) JRS = KBS * (PS - PV - (PIS - PIPL)) JLS = JLSSRM + LSS * (PS - PSNRM) IF(JLS.LT.JLSNRM) JLS=JLSNRM*(PS+PRESLO)/(PSNRM+PRESLO) QPLLS = JPLLS * CPL QDS = PSS * (CPL - CAVS) QLS = JLS * CS RETURN EHD SUBROUTINE DESOLV(F,M,A,B,YO,EPS,HSTART,HMIN,HMAI,YB,FLAG) IMPLICIT REAL*8(A-H,0-Z) DIMENSION YO(M),YA(10),YB(M) EXTERNAL F INTEGER FLAG BMA=B-A HOLD=HSTART X=A Appendix E. Computer Programs 406 DO 10 1=1,M 10 YA(I)=Y0(I) 20 CALL RKF(F,M,X,YA,HOLD,YB,YDIF) GAMMA=.8D0*(EPS*H0LD/(BMA*YDIF))**0.25D0 HNEW=GAMMA*HOLD IF(GAMMA.GE.l.DO) GO TO 30 IF(HNEW.LT.E0LD/10.D0) HNEW=HOLD/10. IF(HNEW.LT.HMIN) GO TO 50 H0LD=HNEW C PRINT*,HOLD GO TO 20 30 IF(HNEW.GT.5.D0*H0LD) HNEW=5.D0*H0LD IF(HNEW.GT.HMAX) HNEW=HMAX IF(X+HOLD.GE.B) GO TO 70 X=X+HOLD HOLD=HNEW DO 40 1=1,M 40 YA(I)=YB(I) GO TO 20 50 FLAG=0 B=X DO 60 1=1,M 60 YB(I)=YA(I) RETURH 70 FLAG=1 HSTART=HNEW HOLD=B-X CALL RKF(F,M,X,YA,HOLD,YB,YDIF) RETURN END Appendix E. Computer Programs 407 SUBROUTINE RKF(F,M,X,YOLD.H.YNEW.YDIFM) IMPLICIT REAL*8(A-H,0-Z) DIMENSION Y0LD(M),YNEW(M),YARGi(10),YARG2(10) REAL*8 Kl(lO),K2(10),K3(10),K4(10),K5(10),K6(10) DATA C2iIC31,C32,C33/.25D0,.37SD0,.09375D0,.2812SD0/ DATA C41,C42/.923076923076923073D0,.879380974055530257D0/ DATA C43,C44/-3.2771961766044606IDO,3.32089212562585323D0/ DATA C51.C52/2.03240740740740722DO.-8.DO/ DATA C53,C54/7.17348927875243647D0,-.2058966861598440SD0/ DATA C61,C62/.5D0,-.296296296296296294D0/ DATA C63,C64/2.DO,-1.3816764132553605D0/ DATA C65,C66/.452972709551656916D0,-.275D0/ DATA C71.C72/.118518518518518509D0,.518986354775828454D0/ DATA C73,C74/.5O6131490342016654D0,-.18D0/ DATA C75,C81/.0363636363636363636D0,.00277777777777777778D0/ DATA C82,C83/-.0299415204678362S68D0,-.0291998936735778838D0/ DATA C84,C85/.02D0,.0363636363636363636D0/ DO 10 1=1,M K1(I)=H*F(I,X,Y0LD) 10 YARG1(I)=Y0LD(I)+C21*K1(I) XARG=X+C21*H DO 20 1=1,M K2(I)=H*F(I,XARG,YARG1) 20 YARG2(I)=YOLD(I)+C32*K1(I)+C33+K2(I) XARG=X+C31*H DO 30 1=1,M K3(I)=H*F(I,XARG,YARG2) 30 YARG1(I)=YOLD(I)+C42+K1(I)+C43*K2(I)+C44*K3(I) XARG=X+C41*B DO 40 1=1,M K4(I)=H*F(I,XARG,YARG1) Appendix E. Computer Programs 408 40 YARG2(I)=Y0LD(I)+C51*K1(I)+C52*K2(I)+C53+K3(I)+C54*K4(I) XARG=X+H DO 50 1=1,M K5(I)=H*F(I,XARG,YARG2) 50 YARG1(I)=Y0LD(I)+C62*X1(I)+C63*K2(I)+C64*K3(I)+C65*K4(I) 1 +C66*K5(I) XARG=X+C61*H . DO 60 1=1,M 60 K6(I)=H*F(I,XARG,YARG1) YDIFM=0.D0 DO 70 1=1,M YNEW(I)=YOLD(I)+C71*K1(I)+C72*K3(I)+C73*K4(I)+C74*K5(I)+ 1 C75*K6(I) YDIF=DABS(C81*K1(I)+C82*K3(I)+C83*K4(I)+C84+K5(I)+C85*K6(I)) 70 IF(YDIF.GT.YDIFM) YDIFM=YDIF RETURN END 

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