A COMPARTMENTAL MODEL OF HUMAN MICROVASCULAR EXCHANGE By Shuling Xie B. Sc. (Biomedical Engineering) Shanghai Jiao Tong University, China A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CHEMICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1992 © Shuling Xie, 1992 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. b1-’.9 e4M’1, Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Qc-evbc-’r /3 / 992 Abstract To study the distribution and transfer of fluid and albumin between the human circula tion, interstitium and lymphatics, a dynamic mathematical model is formulated. In this model, the human microvascular exchange system is subdivided into two distinct com partments: the circulation and the interstitium. Fluid is transported from the capillary to the interstitium by filtration according to the Starling’s hypothesis, while albumin is transported passively by coupled diffusion and convection through the same chan nels that carry the fluid. Data for parameter estimation are taken from a number of studies involving human microvascular exchange and include information from normals, nephrotics, heart failure patients, and also information from experiments on both nor mals and patients following saline or albumin solution infusions. Transport parameters are determined by fitting model predicted results to available measurements from the literature. The best-fit parameters obtained are LS = 0.9888 ± 0.002, Pc, 0 mL .mmHg’h’, and = JL,O = = 11.00 ± 0.03 mmHg, PS 75.74 mL.h ‘. 43.08 ± 4.62 rnL.mmHg’•h’, = 73.01 mLh , KF 1 = 121.05 Simulation of the available experimental data using these parameters gave a reasonable fit in terms of both trends and absolute valiles. All of the best-fit parameter values are in reasonable agreement with estimated values based on experimental measurements where comparisons with literature data are possible. The fully described model is used to simulate the transient behaviour of the system when subjected to an intravenous infusion of albumin and the predicted values compare reasonably well with the experimental infusion data of Koomans et al. (1985). The coupled Starling model is able to successfully simulate the transport mechanisms of human microvascular exchange system. 11 Table of Contents Abstract ii List of Tables viii List of Figures x Acknowledgement xiii 1 Introduction 1 2 Physiological Principles Of The Microvascular Exchange System 5 2.1 Capillaries 6 2.1.1 Arrangement of the Capillaries 6 2.1.2 The Composition and Properties of the Blood 9 2.1.3 Structure of the Transport Barrier and the Transport Mechanisms 10 2.1.3.1 Capillary Wall 10 2.1.3.2 Basement Membrane 14 2.2 Interstitial Compartment 14 2.2.1 14 2.2.2 Structure and Composition of the Interstitium 2.2.1.1 Collagenous Fibres 16 2.2.1.2 Elastic Fibres 18 2.2.1.3 Glycosaminoglycans and Proteoglycans 18 2.2.1.4 Interstitial Fluid 19 Physicochemical Properties of the Interstitium 111 20 2.3 3 4 2.2.2.1 Compliance 20 2.2.2.2 Exclusion 22 2.2.2.3 Colloid Osmotic Behaviour 25 Lymphatic System 26 2.3.1 Terminal Lymphatics 26 2.3.2 Mechanism of Lymph Formation 28 Model Formulation 30 3.1 Introduction 3.2 General Assumptions of Compartmental MVES Models 3.3 Coupled Starling Model 3.4 Constitutive Relationships 3.4.1 Circulatory Compliance 3.4.2 Interstitial Compliance 3.4.3 Colloid Osmotic Pressure Relationship 3.5 Normal Steady-State Conditions 3.6 Summary 33 Parameter Estimation And Data Analysis 4.1 Parameters to be Determined 4.2 Additional Relationships between the Unknowns 4.3 4.2.1 Steady-state Balances at Normal Conditions 4.2.2 Albumin Clearance Relationship 4.2.3 Parameters to be Optimized 4.2.4 Parameter Search Ranges 59 • Analysis of Data 4.3.1 55 . . 61 63 Experimental Data 65 iv 4.3.2 5 4.3.1.1 Set A: Saline and Albumin Infusion 4.3.1.2 Set B: Acute Saline Infusion 4.3.1.3 Set C: Saline Infusion 4.3.1.4 Set D: Heart Failure 71 4.3.1.5 Set E: Nephrotic Syndrome 73 4.3.1.6 Set F: Saline infusion before extracorporeal circulation 74 . . 67 . 69 . . Parameter Estimation Strategy 71 76 4.4 Parameter Estimation Procedure 78 4.5 Numerical Methods and Computer Programs 79 4.5.1 Transient Solutions 79 4.5.2 Steady-state Solutions 79 4.5.3 Computer Programs 80 Results And Discussion 82 5.1 Introduction 82 5.2 Results 83 5.2.1 Best-fit Parameters for the Coupled Starling Model 83 5.2.2 The Feasibility of Combining Different Data Sets 83 5.2.3 Simulations Using Best-fit Parameters 89 5.2.3.1 Transient Responses of 11 PL, GPL and VPL after Saline or Albumin Infusions (Sets A, B and C) 90 5.2.3.2 Simulations of Heart Failure (Set D) 98 5.2.3.3 Simulations of V 1 vs. [PL and fl 1 vs. PL 11 in Nephrotic Syndrome (Set E) 5.2.3.4 5.2.4 99 Summary of the Simulations using Best-fit Parameters Sensitivity Analysis 101 101 V 5.2.4.1 Lymph Flow Sensitivity 5.2.4.2 Albumin Reflection Coefficient 5.2.4.3 Capillary hydrostatic pressure at normal steady-state — LS 104 — u 5.4 6 - 105 Pc,o 5.3 105 5.2.5 Péclet Number 106 5.2.6 Verification of Fit 107 5.2.7 Residual Analysis 108 Validation of the Best-fit Parameters 110 5.3.1 Lymph Flow Sensitivity 110 5.3.2 Albumin Reflection Coefficient 5.3.3 Permeability-Surface Area Product 5.3.4 Fluid Filtration Coefficient 5.3.5 Normal Lymph Flow — — LS — — KF a . — . 113 PS 114 . . JL,O Simulations of a Single Intravenous Infusion of Human Albumin Conclusions 115 116 117 121 7 Future Work 123 Bibliography 125 Nomenclature 134 Appendices 137 A Raw Experimental Data 137 B Calculation Of Error Propagation 144 C Basic Concepts Related To Statistical Analysis 147 D Surface Plot And Contour Plot 149 E Simulations At Best-Fit Parameters 151 F List Of Computer Programs 166 F.1 Parameter List of Steady-State and Transient Simulators 166 F.2 Listing of FORTRAN function XDFUNC 167 F.3 Listing of program PATDYN 208 vii List of Tables 2.1 Chemical composition of plasma in the human 11 2.2 Classification of connective tissue 15 3.1 Mathematical descriptions of the interstitial compliance relationship 3.2 Normal steady-state conditions for the “reference man” 52 3.3 Coupled Starling model 54 4.1 Unknowns in the coupled Starling model 4.2 Bounds on u for Pc,o equal to 7 to 11 mmHg 63 4.3 Experinwntal data from Hubbard et at. (1984) 67 4.4 Experimental data from Hubbard et al. (continued) 4.5 Experimental data from Hubbard et al. (continued) 4.6 Experimental data from Doyle et al. (1951) 70 4.7 Experimental data from Mullins et al. (1989) 71 4.8 Experimental data at steady-state for patients with heart failure (1985) 73 4.9 Experimental data from Rein et al. (1988) 76 5.1 Best-fit parameters corresponding to three tissue compliance relationships 5.2 Best-fit LS and u, as well as their confidence intervals for individual data . . 56 . . 68 . 68 . sets 5.3 84 86 Experimental data of interstitial hydrostatic pressures vs. lymph flow in the leg superficial lymphatics 5.4 47 110 Lymph flow rates in the leg superficial lymphatics and in the thoracic duct 111 viii 5.5 Experimental estimates of PS values 114 5.6 Experimentally determined KF values 116 A.1 Raw experimental data from Hubbard et al. (1984) 138 A.2 Raw experimental data from Hubbard et al. (continued) . 138 A.3 Raw experimental data from Hubbard et al. (continued) 139 A.4 Raw experimental data from Doyle et al. (1951) 140 A.5 Raw experimental data from Mullins et al. (1989) 141 A.6 Raw experimental data from Fauchald (1985) 141 A.7 Raw experimental data from Noddeland et al. (1984) A.8 PL 11 vs. V 1 for patients with nephrotic syndrome A.9 H vs. PL 11 for patients with nephrotic syndrome ix . 141 142 143 List of Figures 2.1 Schematic of the circulatory system in human body 7 2.2 The general patten of the capillary network 8 2.3 Structure and types of capillaries found in the human body 2.4 Scanning electron micrographs of collagenous and elastic fibres in human skin 13 17 2.5 Relationship between interstitial fluid volume and interstitial fluid pressure 21 2.6 Schematic diagram depicting the exclusion phenomenon 23 2.7 Schematic of the lymphatic system 27 Microstructure of the terminal lymphatics 28 3.1 Schematic diagram of the compartmental model of the MVES 34 3.2 Schematic of the Patlak model 38 3.3 Experimental data of compliance of human lower limb subcutaneous tissue 46 3.4 The “most-likely” human interstitial compliance relationship 3.5 Relationship between albumin concentration and total colloid osmotic 48 pressure 50 4.1 Experimental data for patients with nephrotic syndrome 75 5.1 Best-fit LS and u, as well as their confidence intervals for individual data sets 5.2 85 Contour plots of OBJ during early and late responses for Hubbard’s data (1984) 88 x 5.3 Simulations of a 100 mL saline or albumin infusion with 1.4 L fluid intake during waking hours 5.4 92 Simulations of a 200 mL saline or albumin infusion with 1.4 L fluid intake during waking hours 94 5.5 Simulations of acute saline infusion in selected patients 96 5.6 Simulation of 2 L normal saline infusion within 2 hours 97 5.7 Simulations of steady-state Vj vs. PL 11 and V 1 vs. P in heart failure patients 100 5.8 Simulations of V 1 vs. 5.9 Steady-state effects of graded reduction of plasma oncotic pressure on fluid PL 11 and Hi vs. PL 11 in nephrotic syndrome patients 102 and protein exchange 103 5.10 Sensitivity analysis for LS 104 5.11 Sesitivityanlysis for a 106 5.12 Sensitivity analysis for P 0 107 5.13 Residual plot for the best-fit parameters of compliance relationship #3 109 5.14 Plot of 112 .JL vs. P 1 5.15 Effect of Koomans’ (1985) albumin infusion on select microvascular ex change variables 119 D.1 Surface and contour plots of the objective function E.1 150 Simulations of a 100 mL saline or albumin infusion using compliance rela tionship #1 152 E.2 Simulations of a 100 mL saline or albumin infusion using compliance rela tionship #2 153 E.3 Simulations of a 200 mL saline or albumin infusion using compliance rela tionship #1 154 xi E.4 Simulations of a 200 mL saline or albumin infusion using compliance rela tionship #2 155 E.5 Simulations of acute saline infusion in selected patients using compliance relationship #1 156 E.6 Simulations of acute saline infusion in selected patients using compliance relationship #2 157 E. 7 Simulation of 2 L normal saline infusion within 2 hours using compliance relationship #1 158 E.8 Simulation of 2 L normal saline infusion within 2 hours using compliance relationship #2 159 E.9 Simulations of steady-state V 1 vs. and V 1 vs. P in heart failure PL 11 patients using compliance relationship #1 E.10 Simulations of steady-state V 1 vs. PL 11 160 and V 1 vs. P in heart failure patients using compliance relationship #2 E.11 Simulations of Vj vs. PL 11 and Hi vs. PL 11 161 in nephrotic syndrome patients using compliance relationship #1 E.12 Simulations of Vj vs. FL 11 162 and H 1 vs. PL 11 in nephrotic syndrome patients using compliance relationship #2 163 E.13 Effect of Koomans’s (1985) albumin infusion on select microvascular ex change variables using compliance relationship #1 164 E.14 Effect of Koomans’s (1985) albumin infusion on select microvascular ex change variables using compliance relationship #2 xii 165 Acknowledgement First of all, I would like to express my sincere gratitude to Drs. J.L.Bert and B.D.Bowen, for their enthusiastic supervision throughout the course of this work, and for their gra ciousness when I became a new mother. Second, I am indebted to Drs. P. Englezos, G. Wong, N. Lee and T. Nicol, for their helpfully suggestions in statistics. Similarly, I also give my special thanks to Dr. R. Reed of the University of Bergen, Norway, for the clinical information. In addition, I gratefully acknowledge that this work was supported by the Natural Sciences and Engineering Research Council of Canada and the Norwegian Council for Science. Finally, I would like to dedicate this work to my husband Xiao Ping Yang, for his love and encouragement. xli’ Chapter 1 Introduction By virtue of the unique properties of water as a solvent and as medium in which a wide range of exchanges and metabolic reactions are possible, it plays a very important role in maintaining homeostasis. Water, as the major component of all living tissues, typically makes up about 60% of the total body weight. It distributes in two major compartments — the intracellular compartment within the cells, which retains about 60% of the total body water, and the extracellular compartment, which retains about 40% of the total body water. The extracellular water, in turn, exists as two compartments, namely, the plasma water in the circulating blood and the interstitial water bathing the tissue cells. The plasma water represents about 10% of the total body water, and the interstitial water about 30% [95]. The word fluid, which will be used throughout this dissertation, identifies the mixture of water and non-protein components dissolved in the water. The maintenance of the composition and volume of this aqueous phase is critical to the vital functions of the human body. The regulation of composition and volume, to a large extent, can be described by system and control theory. Feedback control is the typical control strategy used for regulating the microvascular exchange system, which consists of the capillaries, the interstitium and the lymphatic system. Using the feedback signals, the microvascular exchange system is capable of reacting to changes in its environment by passively altering its own properties so that the pre-perturbation state can be achieved again. Thus, under normal conditions, the composition and volume of the plasma and interstitial fluid remain 1 Chapter 1. Introduction 2 in a very narrow range and the body functions normally. The utilization of mathematical models in the description of the biological behaviour of systems is now common practice. The advantages of mathematical models are selfevident. The human body is well recognized to be a complex control system. When physicians attempt to diagnose and correct the malfunctioning of the system, they are faced with the problem that most physiological and biochemical state variables are in fluenced by too many factors to be grasped by the unaided human mind. Therefore, reliable mathematical models as a complementary approach to physiological experimen tation is of potential importance to aid the clinician in gaining insight into these intricate interactions. Several models of microvascular exchange have been developed [7, 32, 106]. These models require information available in the literature concerning the normal steady-state -values of volume and protein content and transport parameters, and use this- information, along with relevant auxiliary relationships, to predict the responses of the microvascular exchange system after perturbations. The current model uses the same general compart mental approach and mechanistic descriptions as many of these earlier models. However, it differs from the latter in one important respect: whereas the previous models assume the values of transport parameters a priori, in the current model these are determined on the basis of a statistical fit of the model predictions to selected experimental data. The present work is a continuation of an earlier study carried out by Chapple [13]. Chapple modelled the microvascular exchange system by using the uncoupled Starling model and the plasma leak model to represent the transcapillary exchange of fluid and proteins. The coupled Starling model is used in the current study. The coupled Starling model was found to yield the best statistical fit between the model predictions and the experimental data for nephrotic patients [14]. Furthermore, it is generally thought that the coupled Starling model provides the most accurate description of the transcapillary protein flux Chapter 1. Introduction 3 [73]. The objectives of the current study are: 1. to formulate a coupled Starling model for describing the microvasdillar exchanges in humans; 2. to assess and assemble all of the available literature data for the response of the microvascular exchange system to both diseaseinduced and artificially applied per turbations; 3. to design a reliable parameter estimation procedure; 4. to obtain a set of best-fit transport parameters so that the model is fully described and can be used to simulate various experimental or hypothetical situations; 5. to validate the mathematical model. If a reliable model of microvascular exchange system in the human can be developed, the potential benefits and uses are manifold. One of the foreseeable benefits is that these kinds of models could be used for pretesting therapeutic procedures on individual patients in certain disease states, e.g. for the on-line simulation of trial fluid therapy for burn injured patients and patients undergoing extra-corporal circulation, etc.. Bert et al. [4] have developed a dynamic model to describe the distribution and transport of fluid and plasma proteins between the circulation, interstitial space of skin and muscle, and the lymphatics in the rat under burn condition. The thesis is organized in the following manner. Chapter 2 reviews the basic phys iology relevant to the microvascular exchange system. This chapter provides a physical basis for better understanding the transport mechanisms of fluid and proteins within the system. Chapter 3 develops a compartmental model based on the coupled Starling rep resentation of the transcapillary fluid and protein transfers. The necessary constitutive Chapter 1. Introduction 4 relationships and the normal steady-state conditions of the system are also presented in this chapter. Chapter 4 describes the optimization procedure designed to obtain the best-fit transport parameters. The experimental data used in statistical fitting are as sessed and the numerical techniques used in the current work are reviewed. Chapter 5 presents the results of the parameter estimation procedure and the simulations obtained using these best-fit parameters. Statistical procedures are used to evaluate the reliability of the best-fit parameters. Finally, the parameters and model are validated from a phys iological point of view. Chapter 6 summarizes the conclusions drawn from the current study and Chapter 7 makes recommendations for future work. For the convenience of the reader, all mathematical symbols used throughout the thesis are listed and defined in a special Nomenclature section following the text. Chapter 2 Physiological Principles Of The Microvascular Exchange System Many physiological systems involve the maintenance of the milieu interieur, or home ostasis, so that optimal function can be fulifihled. Milieu interieur is defined by Claude Bernard [84] as the pervasive extravascular and extracellular space in which all cells bathe. The cardiovascular system, which consists of the heart and a series of blood vessels forming a closed network, plays an important role in enabling homeostasis to be achieved. It supplies oxygen and nutrients necessary for life to the tissues and carries away metabolic waste products from the tissues. Oxygenated and nutrient rich blood is pumped out of the left ventricle of the heart, flows through arteries and arterioles, then reaches the capillary beds, which are the primary sites where materials, e.g. oxy gen, plasma proteins, vitamins, hormones, heat, etc., are transported across the capillary walls to the surrounding tissue space called the interstitium. Meanwhile, metabolic-endproducts, e.g. carbon dioxide, water and heat, are collected from the tissue. As a result of these exchanges, blood becomes oxygen depleted and is collected by venules before being transported, via large veins, back to the right atrium. Rhythmic muscle contractions and one-way valves force the blood to flow in one direction. When the muscles in the wall of the right ventricle contract, blood is pumped into the pulmonary artery and hence passes to the lungs where the transfer of gasses with the inspired air occurs. Oxygenated blood then returns to the left atrium by the pulmonary veins, finally emptying into the left ventricle. On the other hand, interstitial fluid containing plasma proteins drains from the interstitium and is returned to the blood circulation via the lymphatic system. The 5 Chapter 2. Physiological Principles Of The Microvascular Exchange System 6 entire circulatory system is illustrated in Fig. 2.1. The current investigation is concerned with the microvascular exchange system. It consists of three components: capillaries, interstitium, and the lymphatic system. The structure and functions of these components will be discussed in this chapter. 2.1 2.1.1 Capillaries Arrangement of the Capillaries The microcirculation consists of arterioles, capillaries, and venules. Capillaries are tubes of 5 to 10 m in diameter and their walls consist of a single layer of flattened endothelial cells, 0.1 to 0.3 jim in thickness, with nuclei that sometimes bulge into the lumen [59]. The so-called “capillary count” (i.e. the number of capillaries visible in a cross-section of a tissue per square millimeter of area) has been measured in red skeletal muscle in dog, and has been found to be about 1000 capillaries/mm . Thus the distance between blood 2 capillaries averages only about 140 tm and individual cells are seldom more than 40 — 80 gum from a capillary surface [93]. These numerous microscopic vessels form a complex meshwork which has the general pattern shown in Fig. 2.2. Most physiologists agree that the local circulation rate should be in accordance with the needs of every tissue as well as the variant physiological states of the subject. Therefore the capillaries are able to open or close depending on the local requirements of the system. Whether the capillary is off or on is controlled by the precapillary sphincter. When the sphincter muscle contracts, the channel closes and vice versa. The dilation and constriction of precapillary sphincters produce a continuously changing pattern of flow through the capillary network. In a particular segment of the capillary bed, the blood may flow rapidly through one channel for a period of time, then cease to flow or even flow in the opposite direction, depending on which sphincters are open. Normally, only around 5 — 10% of the total available Chapter 2. Physiological Principles Of The Microvascular Exchange System Capillaries Arlercs to head and upper exItemilies Capiltties Figure 2.1: Schematic of the circulation systems in human body [17]. 7 Chapter 2. Physiological Principles Of The Microvascular Exchange System 8 Meriole Arteriovenous anastomosis True capillaries Thoroughfare channel Venule Figure 2.2: Schematic diagram showing the general pattern of the capillary network [53]. chapter 2. Physiological Principles Of The Microvascular Exchange System 9 capillaries in the human adult at rest are open [93]. But during exercise, the number of open capillaries can increase 10-fold or even 20-fold to fulfill the metabolic requirement. By virtue of its enormous membrane surface and relatively low velocity of blood flow, capillary beds undertake more than 90% of the fluid and solute transfers between the circulation and tissues, while the other part of the circulation, e.g., the small arterioles and venules, only account for a small amount of the duty. 2.1.2 The Composition and Properties of the Blood Blood is a transport medium for nutrients and wastes. A constancy of blood composition is vital for survival and the mechanisms that operate to keep it constant are vital processes of homeostasis. Blood is a suspension of erythrocytes, leukocytes, platelets, and other particulates iii a. complex solution of dissolved gases, salts, proteins, carbohydrates, and lipids. The specific gravity of whole blood ranges from 1.055 to 1.065 [85]. In a 70kg healthy adult, the average blood volume is considered to he about 5 litres, which is approximately 7% of the body weight. Rapid loss of large amounts of blood has very serious consequences. With a rapid loss of 800 mL (about 15% of the total blood volume), a drop in arterial blood pressure is prevented by constriction of arteries and veins. However, the heart rate will increase and the cardiac output will fall. With greater blood loss, for example 30 — 40% of the total blood volume, a state of shock will be induced. When a sample of blood is drawn from a person, it is initially set aside in a test tube. Heparin is added to prevent blood clotting. Then the tube is placed in a centrifuge to accelerate separation. After a few minutes, the blood is separated into two layers. Blood cells sink to the bottom and their volume is normally close to 45% of the total volume. The straw-colored supernatant liquid, called plasma, is a cell free fluid. The specific gravity of plasma varies between 1.028 and 1.032 [85]. The ratio of blood cells to Chapter 2. Physiological Principles Of The Microvascular Exchange System 10 total blood volume is expressed as the hematocrit. Extremely low hematocrit, say 0.15, is called anemia, while the opposite is called polycythemia. After removing the cellular elements from the blood, an analysis of the chemical composition of plasma typically yields the results depicted in Table 2.1. Despite the fact that chemical substances are constantly entering and leaving the blood stream, its general composition is relatively uniform in the higher-order animals. As can be seen from the table, about 90% (by weight) of the plasma is water. Proteins account for about 7 - 9% of the total weight. Some of these proteins are also found elsewhere in the body, but when they occur in blood, they are called plasma proteins. Plasma proteins exert an osmotic pressure (around 25 mmHg) which influences the passage of water and other solutes through capillaries. The globulins, classified as a , a 1 , 2 and ‘y globulins, have a wide range of molecular weights varying from 100,000 to 450,000 [53]. Albumins constitute the majority of the plasma proteins (55 - 64%) and possess &relatively low molecular weight of the order of 68,000. Also, albumins are not transported freely across the intact vascular endothelium. Therefore, they provide a significant colloid osmotic pressure gradient between the blood and the interstitium which greatly influences the transmembrane exchanges of water and other solutes. 2.1.3 Structure of the Transport Barrier and the Transport Mechanisms The transport rates of water and solutes across the capillary wall are determined by three factors: transmural driving forces, transport barrier properties and its surface area. Here, the transport barrier means the capillary wall and the basement membrane. 2.1.3.1 Capillary Wall Capillaries of different tissues vary considerably, both anatomically and functionally. For example, in the skin, there are arteriovenous anastomoses (AVAs) which are wide-bore, Chapter 2. Physiological Principles Of The Microvascular Exchange System Constituent Proteins Albumin Total globulin Transferrin Haptoglobin Hemopexin Ceruloplasmin Ferritin Concentration (g/100 mL) 6.0 8.0 3.4 5.0 2.2 4.0 0.25 0.03 0.2 0.05 0.1 0.03 0.05 0.015 0.3 Nonproteins Water Cholesterol Glucose Urea nitrogen Uric acid Creatinine Iron 0.14 0.25 0.07 0.11 < 0.02 < 0.008 < 0.002 < 0.0002 - - - - - - - - - Table 2.1: Chemical compositions of plasma in the human [74]. 11 Chapter 2. Physiological Principles Of The Microvascular Exchange System 12 direct channels between the arterioles and venules (see Fig. 2.2). The AVAs are under neurogenic control; their shunting capabilities enable them to reduce heat loss through the skin during exposure to cold. The capillary wall is made up of a number of microscopic structures, some of which affect the transcapillary exchange. Those that have been observed and are thought by most physiologists to play a role in transcapillary exchange are intercellular clefts, fenestrae, and pinocytotic vesicles (Fig. 2.3). Intercellular clefts are the junctions between endothelial cells. They can be loosely joined to form large pores or tightly joined to form small pores. Fenestrae are formed by the stretching of some parts of the capillary wall. They are usually thin, disc-shaped diaphragms. Some of them are open and provide a minimally restrictive pathway through the endothelium, while others are closed with a slight central thickening or knob. Inside the cytoplasm of the eridothelium, there are rnaiy pinocytotic vesiçles which shuttle back and forth between opposing cell surfaces. They intake fluid and solutes from one side and release them at the other side. These vesicles may be important in protein transfer. Sometimes the vesicles fuse together and form a transitory open channel. The above structures determine the transport properties of the capillary wall. Accord ing to the degree of continuity, capillaries can be separated into three types: continuous, discontinuous and fenestrated capillaries (see Fig. 2.3). Continuous capillaries are widely distributed; characteristically, they are found in lung, brain, kidney as well as skeletal and smooth muscles. There are no recognizable intercellular openings in this type of capillary. Fenestrated capillaries have small gaps which are either closed as in endocrine glands and intestinal villi or open as in renal glomeruli. These gaps usually range from 40 — 60 nm in radius [100]. Discontinuous capillaries have large nonselective gaps, rang ing from 100 — 1000 nm in radius [100]. These gaps are sufficiently wide to allow large proteins and even cells to pass through. This type of capillary is typically found in liver, Chapter 2. Physiological Principles Of The Microvascular Exchange System 13 50 10—40 1. 2. 3. PLASMA CONTINUOUS CAPILLARY BASEMENT MEMBRANE TISSUE 50 2—6 F EN E STR AT ED CAPILLARY TISSUE 40 1 PLASMA — 60 5—11 5. DISCONTINUOUS CAPILLARY TISSUE 100— 1000 Figure 2.3: Structure and types of capillaries found in the human body. 1, plasmalemmal vesicles; 2, intracellular clefts; 3, fused plasmalemmal vesicle s; 4, fenestrae; 5, nonselective gaps. Unit of the radius is nm [100]. Chapter 2. Physiological Principles Of The Microvascular Exchange System 14 bone marrow and spleen. 2.1.3.2 Basement Membrane The basement membrane is composed of several matrix-specific components, including the structural protein collagen, the glycoproteins laminin and entactin, and a large hep arm sulfate proteoglycan, which covers the external surface of capillary endothelial cells, penetrated here and there in discontinuous capillaries but not in the other types. It provides surfaces on which epithelial cells adhere and assumes a polarized orientation. The thickness of the membrane averages 30 — 80 nm. The transport properties of the basement membrane are attributed to the presence of pores. The membrane is negatively charged, and prevents negatively charged proteins from passing to the interstitium freely [55]. The transport resistances of the capill-ary wall and the basement membrane are in series. Both contribute to the selectivity of the harrier. Based on the above discussion, mass transfer across the barrier can occur by means of convective mechanisms (e.g. filtration) or by passive mechanisms (e.g. diffusion or vesicular transport). 2.2 2.2.1 Interstitial Compartment Structure and Composition of the Interstitium The word interstitium is borrowed from Latin; it generally refers to the connective tissue space situated outside the vascular and lymphatic systems and the parenchymal cells. Half a decade ago, interstitium was perceived as an inert, metabolically inactive sub stance, which only provided support for cells and held tissues together. But this concept has been extensively revised. It is now appreciated that the interstitium not only pro vides a framework for parenchymal cells and a space for distribution of blood vessels Chapter 2. Physiological Principles Of The Microvascular Exchange System Type Myxoid Fibrous Elastic Adipose tissue Muscle Vascular tissue Cartilage Bone 15 Examples Nucleus polposus, synovium, areolar connective tissue Tendon, ligament, fascia, dermis Ligamentum nuchae, artery White fat, brown fat Striated muscle, smooth muscle Artery, capillary, vein Hyaline cartilage, fibrocartilage, elastic cartilage Compact bone, trabecular bone Table 2.2: Classification of connective tissue. and nerve fibers, but it also provides a suitable transport medium for nutrient and end metabolic waste products between cells and capillary blood. In physiological situations, various exchanges occur between the interstitium and the circulatory compartment and lymph, but these exchanges are in a dynamic equilibrium. According to their anatomical structure and physiological function, connective tissues can be classified into many types (Table 2.2). These tissues are extremely heterogeneous both in their cellular population and composition. The cell component may occupy from 5% (e.g. tendon, cartilage, bone) to 95% (e.g. muscle, fat) of the tissue volume. These cells serve various functions. Some of the cells synthesize and degrade the in terstitial components (e.g. bone cells), while the others maintain a specialized internal function (e.g. muscle cells have active contractile function). The compositon of intersti tia varies not only from one tissue to another, but also from domain to domain within the tissue. Most interstitia are intimate composites of two phases: the structural macro molecules and interstitial fluid. The structural macromolecules include collagens, elastin, glycosaminoglycans (GAGs) and proteoglycans. Interstitial fluid consists of water, salts, plasma proteins, 02, C0 , hormones, vitamins, etc.. The basic structure of interstitia 2 is similar. It can be described as an elastic, three-dimensional gel-like structure which chapter 2. Physiological Principles Of The Microvascular Exchange System 16 is produced by the complex aggregation of collagenous and non-collagenous components by covalent, electrostatic, and hydrogen bonds. The following discussion will emphasise the characteristic components of the interstitium (i.e. collagenous fibres, elastic fibres, GAG, proteoglycans), as well as the composition of the interstitial fluid. 2.2.1.1 Collagenous Fibres Collagenous fibres are the soft, flexible, white fibres which are a characteristic constituent of all types of connective tissues. In normal adult skin, they account for 70 — 75% of the dry weight [104]. They are composed of bundles of fibrils which can be visualized with the scanning electron microscope (Pig. 2.4 a). These bundles range from one to several hundred microns in diameter, but their texture may be loose enough to admit molecules as large as plasma proteins between the individual fibrils. These fibrils inter sperse with GAGs and can glide smoothly past each other. The collagen molecule is the basic structllral unit of collagenous fibres and the most common protein in the body. It is a triple-helical molecule that is assembled from the coiling of three peptidic a-chains. The arrangement of amino acids in each chain shows a characteristic periodicity (i.e. every third amino acid is glycine, see Fig. 2.4 b). Each chain contains approximately 1000 amino acids and has a molecular weight of around 100 kDa [29]. The collagen molecule is biosynthesized from a precursor known as procollagen. Pro collagen is enzymatically trimmed of its nonhelical ends and spontaneously assembles into fibrils extracellularly. These fibrils interact with each other in the direction parallel to the axis of each molecule and form the well organized collagen fibers. Collagenous fibres determine the strength and elasticity of the tissue. They demon strate poor elasticity but excellent mechanical strength. Chapter 2. Physiological Principles Of The Microvascular Exchange System 17 (b) (a) C 87 . 1 (c) Figure 2.4: Electron micrographs of collagenous and elastic fibres. (a) collage-. nous fibres have a regular banding patern(x45,000) [103]; (b) triple-helical collagen molecule; (c) elastic fibres in human skin after removal of other extracellular matrix components. The fibres are randomly oriented and of varying size [90]. Chapter 2. Physiological Principles Of The Microvascular Exchange System 2.2.1.2 18 Elastic Fibres Compared to collagenous fibres, elastic fibres account for only a small fraction of connec tive tissue, 2 — 4% of the dry weight in normal adult skin [104]. Nonetheless, they play an important role in terms of providing long-range, reversible elasticity and resilience. Elastic fibres contain two structural components. Under the electron microscope, the predominant component appears amorphous and is called elastin (70 90%); the secondary one has a distinct appearance and each unit is referred to as a microfibril (not shown on Fig. 2.4 c). Elastin is a unique protein which is composed of hydrophobic amino acid sequences. The individal hydrophobic chains crosslink and form a random configuration. This results in a very insoluble protein that possesses properties analogous to rubber. The microfibrils are composed of several glycoproteins that vary in size from 25k to 34k daltons. They can exist in fibrillar arrangements or as aggregates independent of their association with elastin. 2.2.1.3 Glycosaminoglycans and Proteoglycans Glycosarninoglycans (GAGs), formerly called mucopolysaccharides, distribute widely in most tissues. However, except for hyaluronate, GAGs do not exist in vivo as free poly mers. Instead, these GAGs normally covalently link to a core protein at the terminal end and form compounds which are called proteoglycans. The general structure of GAGs consists of three levels. The first level is an unbranched chain of repeating disaccharide units in which one of the monosaccharides is an amino sugar (hexosamine) and the other is usually a hexuronic acid. The type and number of disaccharide units vary in different GAGs. Four basic types of GAGs are recognized and defined according to the type of hexosamine: hyaluronic acid, chondroitin/dermatan sulfate, heparin/heparan sulfate, and keratin sulfate. The second level is the charged Chapter 2. Physiological Principles Of The Microvascular Exchange System 19 side groups connecting to the chain. These side groups, e.g. the carboxylate group (—COOj, the sulfate ester group (—0 — SOT), and the sulfamino group (—N — SO), are all negatively charged. The third level is the spatial arrangement of the backbone and the side groups. The backbone and the side groups form fibrils up to several hundred microns in length. The fibrils are folded every few hundred angstroms so that instead of being long linear molecules they are actually jumbled, folded, springlike coils occupying a space having a diameter of several hundred millimicrons [49]. GAGs interact with proteins, collagens and proteoglycans and form a continuous gel-like substance. GAGs demonstrate a highly hydrophilic character. Knowledge of the structure of the proteoglycans in tissues other than cartilage is incomplete. In cartilage, the average proteoglycan molecule consists of a core protein to which about 100 chondroitin sulfate chains and 30-60 keratin sulfate chains are attached. The core protein is approximately 250 kDa and about 300 urn in length. The protein portion constitutes only 5 — 10% of the entire molecule, and the rest of the molecule consists of complex carbohydrates. In the presence of hyaluronate, protein cores bind to the hyaluronate chain and form aggregates [16]. The high charge density permits these molecules to attract a large volume of water and contribute to the ability of proteoglycans to absorb compressive loads. 2.2.1.4 Interstitial Fluid Interstitial fluid is a filtrate of blood. It consists of the same nonparticulate components as plasma except it has lower macromolecule concentrations. Interstitial fluid is the continuous bathing media through which nutrients diffuse to the parenchymal cells and metabolic-end-products arrive at the lymphatics. Chapter 2. Physiological Principles Of The Microvascular Exchange System 2.2.2 20 Physicochemical Properties of the Interstitium The physicochemical properties of the interstitium are determined by the structure, com position, and physicochemical conditions of the compartment. In this section, the com pliance of the interstitial space, exclusion caused by the macromolecules, and colloid osmotic behaviour will be discussed. 2.2.2.1 Compliance From the previous section, we know that collagen molecules can glide a short distance along the axis of the fibre, but the amount of movement is limited by the interaction between the molecules. Also, collagen is in a ribbon-like arrangement in some tissues. Therefore, when an external tensile force is applied to this structure, the fibre will be stretched out. When the external force is removed, the fibre will be drawn back and regain its original state. This specific property is called elasticity. Moreover, elastic fibres like GAGs and proteoglycans have spring-like configurations; therefore they all possess an elastic behaviour similar to springs. The compliance of human tissues is a poorly understood property, but one which is very important to microvascular exchange. It involves the interaction between the fluid and the solid components of the interstitium, the state of charge of the structural components, the fibrillar organization, composition, etc.. For reasons of convenience, compliance (FCOMPI) in current study is defined by FCOMPI = //P 1 AV (2.1) where 1.V 1 is the change in interstitial volume when the interstitium is subjected to a hydrostatic pressure change, AP . To some extent, it determines the degree of elasticity 1 of the interstitial space. The interstitium has high compliance if small changes of inter stitial pressure can induce large changes in the interstitial fluid volume. When pressure Chapter 2. Physiological Principles Of The Microvascular Exchange System 21 C w z Cl, 4 —e +110 +120 4430 CHANGE IN LEG WEIGHT (per cent) Figure 2.5: Relationship of interstitial fluid pressure to change in leg weight during progressive increase in interstitial fluid volume. Weight is correlated with fluid volume. Each curve represents results from a separate dog leg [34]. is plotted against volume, the reciprocal of the slope of the curve is the compliance of the interstitial space at a certain volume. Even though knowledge of human tissue com pliance is incomplete, studies on other animal tissues suggest that the pressure-volume curve should be sigmoidal [34, 2], as exemplified by Fig. 2.5. Low compliance occurs at low and normal interstitial volume. This is one of the tissue’s passive mechanisms for preventing dehydration and edema. When the volume decreases, 1 P decreases signifi cantly. Thus the driving force for filtration to the interstitium increases and preven ts Chapter 2. Physiological Principles Of The Microvascular Exthange System 22 further dehydration. When the volume increases, Pj increases significantly to decrease fluid filtration and hence prevent edema. High compliance occurs at high V . This makes 1 the interstitium serve as an overflow reservoir to help maintain a constant plasma volume. At very high water contents, the tissue volume is constrained by the boundaries of the tissue which therefore exhibits, once again, a low compliance. At this stage, the plasma volume will increase due to the elevation in the driving force for absorption. 2.2.2.2 Exclusion Exclusion refers to fact that two or more objects of any volume can not overlap and occupy the same space at the same time. This phenomenon exists in the interstitium and was first described by Ogston and Phelps in 1961 [65]. To depict the exclusion phenomenon more clearly, two models are introduced. One is the sphere-and-rod model (Fig. 2.6 b). The spheres represent plasma proteins and the rods represent structural components of the interstitia. In this model, the center of the sphere is not accessible to a volume of 7rl(r 3 + rr) 2 surrounding the rod, where rr is the radius of the rod and 1 is its length; r 3 is the radius of the spere. This volume is called the excluded-volume (VEX). The other model is closer to the real situation in the interstitium. It is a sphere located inside a random network of rods. The excludedvolume fraction for this model has been calculated to be 1 — e_ , 2 (rS+T) where L is the total length of the rods per unit volume (Fig. 2.6 c). The above models can be used in hypothetical descriptions of the exclusion phenomenon, but because the organization within the interstitium is not well defined, they can be used only to estimate the excluded volume of the interstitium. VEX is determined by many factors. There are different sizes of collagen, elastin, GAGs and proteoglycans which contribute to limiting the volume available to the plasma proteins. The spatial arrangement of the fibres greatly affects the value of VEX. For example, a larger number of smaller fibres arranged randomly Chapter 2. Physiological Principles Of The Microvascular Exchange System A 23 8 ——--F 1; I I ‘ I ‘I 4 Figure 2.6: Schematic diagram depicting the exclusion phenomenon. (a) two spheres can’t occupy the same space at the same time; (b) sphere-and-rod model; (c) a sphere located inside a random network of rods [16]. Chapter 2. Physiological Principles Of The Microvasculai Exchange System 24 causes much greater exclusion than an equal weight of a smaller number of larger fibres in an orderly lineup [6]. In addition, exclusion is also affected by the nature of the charge on the proteins and the various structural components they interact with. For example, both GAGs and albumin are negatively charged. Because similarly charged entities repel one another, albumin is excluded by GAGs to a greater extent than other fibrils which have a similar size but are uncharged. Since the structural components differ from tissue to tissue, the exclusion effect is tissue dependent. The best way to study exclusion inside interstitium is through in vivo experimenta tion. Unfortunately, the results of only a few in vivo studies are available. One way to estimate VEX is to use a multiple-indicator technique. One indicator has a very small molecular size (e.g. sucrose, Cr-EDTA) and its exclusion other than by solid structures can therefore be neglected. The other indicator is the excluded material under investi -gation, e.g. albumin, globulin. It is generally believed that collagenous fibres play an important role in excluding plasma proteins. Other materials might also play an impor tant role. This issue is not yet clarified. In vivo studies show that the excluded-volume fraction by collagenous fibres ranges from 25% to 53% of the volume of the interstitial matrix [6, 5, 86]. Due to exclusion, the interstitial space available to the plasma proteins is less than the interstitial fluid volume; consequently, the effective protein concentration and chem ical activity of the protein is greater than the mass of plasma proteins divided by the interstitial fluid volume. The colloid osmotic pressure, which depends on the protein concentration, is therefore sensitive to the interstitial exclusion phenomenon. Chapter 2. Physiological Principles Of The Microvascular Exchange System 2.2.2.3 25 Colloid Osmotic Behaviour Because capillary walls restrict protein movement, changes in the concentration of body fluids will affect the movement of water between fluid compartments due to the phe nomenon of osmosis. Osmosis is the movement of water across a semipermeable mem brane from an area of lower solute concentration to an area of higher solute concentration. Although the term osmosis specifically refers to the movement of water only, to a smaller degree, osmosis also affects the movement of solutes. The forces of friction cause some solutes to be carried along with the water. This is termed solvent drag. The driving force for osmosis is the difference in osmotic pressures on both sides of the membrane. Colloid osmotic pressure is caused by a relative deficit of permeating molecules on one side of the membrane versus the other side. It should be mentioned that colloid osmotic pressure (or oncotic pressure) is the osmotic pressure exerted by proteins. This applies to the circulation as well as the interstitium. Albumin, for example, exerts oncotic pressure within the blood vessels and helps maintain the water content of the blood in the intravascular space. The normal plasma colloid osmotic pressure is approximately 25 — 30 mmHg, which consists of approximately 19 mmllg due to plasma proteins and 9 mmHg due to cations held by the Donnan equilibrium effect ‘. The direction of the colloid osmotic pressure is opposite to that of the hydrostatic pressure within the interstitium. Together they counteract the oncotic and hydrostatic forces within the capillary and maintain a dynamic equilibrium between the two com partments. There is a good correlation between protein concentration and colloid osmotic pres sure. This issue will be discussed in more detail in the next chapter. Experiments show 1 N egatively charged proteins attract positive ions and create a high ion density environment, conse quently causing osmotic pressure. This phenomenon is called the Donnan equilibrium effect [35]. Chapter 2. Physiological Principles Of The Microvascular Exchange System 26 that about 68% of the total colloid osmotic pressure is due to albumin, 6% is due to globulin, and the rest is due to the other proteins [53]. 2.3 Lymphatic System The lymphatic system is considered to be somewhat in parallel to the blood circulation. It is a network of terminal and collecting lymphatic vessels in the body that drain the interstitial fluid back to the blood circulation (Fig. 2.7). The drainage is important in fluid and plasma protein distribution and transport in the body. The cells bathed by the lymph, namely, lymphocytes, are primarily responsible for the specificity of immunolog ical responses. 2.3.1 Terminal Lymphatics The terminal (or initial) lymphatic is the part of the lymphatic system that is involved with collection of interstitial fluid in the microvascular exchange system. It is an irregular, microscopic, blind end conduit which consists of a single layer of overlapping endothelial cells. The differences between a terminal lymphatic and a blood capillary are that the former has a much thinner wall but its diameter is much larger (15 — 20 pm) and its basement membrane is highly attenuated and frequently absent completely. It is well recognized that terminal lymphatics are discontinuous, with open junctions between endothelial cells (except those in brain, spinal cord and ocular space). Some workers have even identified some ultrastructures around these junctions. For example, Leak [51, 50] observed anchoring filaments that attach the basement membrane to the adjacent collagenous and elastic fibres. The unattached sites around the junctions form flaps which are called lymphatic endothelial microvalves, as shown on Fig. 2.8 [90, 12]. The microvalves become more obvious during distension of the lymphatics. The anchoring Chapter 2. Physiological Principles Of The Microv.ascular Exchange System 27 Figure 2.7: Schematic of the lymphatic vessels in an area of bat wing at a bifurcation of two large vascular channels: terminal lymphatics (irregular black bulbs), collecting lymphatics (black bold lines), blood vessels (shaded), and capillaries (thin lines) [15j. Chapter 2. Physiological Principles Of The Microvascular Exchange System 28 I I Figure 2.8: Microstruture of the terminal lymphatics. When the anchoring filaments are tightened, the microvalves are open and the lymphatics are filled; when the filaments are loosened, the microvalves are closed and the lymphatics are emptied [90, 12]. - filaments and the microvalves are of potential importance in supporting the postulated transport mechanism for lymph which will be disscussed later. Collecting lymphatics into, which initial lymphatic fluids derived from the interstitium continuously drain are distinguished from terminal lymphatics by the appearance of macroscopic bileaflet valves and smooth muscle intima, and by the fact that they contract spontaneously [37]. The bileaflet valves are one-way valves and hence prevent backflow. The collecting lyinhatics finally converge at the thoracic duct and, through the duct, lymph is returned to the blood circulation. 2.3.2 Mechanism of Lymph Formation The question concerning the mechanism by which the interstitial fluid is transported into the terminal lymphatics has long been disscussed but still remains controversial. The first postulated mechanism is that lymph is formed by the periodic contraction Chapter 2. Physiological Principles Of The Microvascular Exchange System 29 of the terminal lymphatics which is caused by its own smooth muscle. This mechanism is supported by research on bat wing lymphatic endings [38, 62]. These endings have their own smooth muscle and rhythmic pressure pulsations have been recorded [38]. However, the bat wing is a notable exception. No similar observation has been reported in any other tissue. A detailed review of the microanatomy of terminal lymphatics in different organs has been presented [90]. The second postulated mechanism is considered to be more common. In this postula tion, lymph is formed by the contraction of the tissues surrounding the terminal endings. Muscle contraction, intestine motilities, skin tension, vasomotion, etc., stretch the an choring filaments and expand the terminal lymphatics. The unattached microvalves are opened, and with a small pressure gradient, interstitial fluid is pushed into the lymphatics. During the relaxation of these organs, the filaments loosen, the bileafiets overlap and the microvalves are closed. With the spontaneous contraction of the collecting lymphatics, lymph is then drained out of the terminal lymphatics. Initial lymph is generally assumed to have the same composition as interstitial fluid due to the large intercellular junctions and the incomplete basement membrane, both of which result in a nonsieving in the terminals. Some investigators have observed hyaluronate present in the prenodal lymph [77]. After the lymph passes through the lymph nodes, some components are degraded and the composition of the lymph changes. But whether the assumption that compositions of initial lymph and interstitial fluid are the same is correct or not still needs to be proven [2]. Chapter 3 Model Formulation 3.1 Introduction With the development of computer sciences, a new interdisciplinary approach for studying biological systems has appeared. Computer modelling has provided researchers with a powerful tool for better understanding intricate systems, which are generally influenced by too many factors to be grasped by the unaided human mind. In studies of the microvascular exchange system (MVES), many different mathemat ical models have been developed on the basis of the Starling’s hypothesis. To elucidate his hypothesis, Starling wrote [96]: Although the osmotic pressure of the pro teids of the plasma is so insignif icant it is of an order of magnitude comparable to that of the capillary pres sures; and whereas capillary pressure determines transudations the osmotic pressure of the pro teids determines absorption. Moreover, if we leave the functional resistance of the capillary wall to the fluid through it out of ac count, the osmotic attraction of the serum for the extravascular fluid will be proportional to the force expended in the production of the latter, so that at any given time, there must be a balance between the hydrostatic pressure of the blood in the capillaries and the osmotic attraction of the blood for the surrounding fluids. With increased capillary pressure there must be increased transudation until equilibrium is established at a somewhat higher point, when 30 Chapter 3. Model Formulation 31 there is a more dilute fluid in the tissue spaces and therefore a higher absorbing force to balance the increased capillary pressure. With diminished capillary pressure there will be an osmotic absorption of salt solution from the extravas cular fluid until this becomes richer in pro teids; and the difference between it (proteid) osmotic pressure and that of the intravascular plasma is equal to the diminished capillary pressure. The center of the hypothesis is that the fluid and protein exchanges across the capillary wall are governed by the hydrostatic pressures and colloid osmotic pressures on both sides of the wall. This is a fundamental basis of all mathematical models of the MVES. Based on their particular interests, investigators have set up different models which were used to investigate specific aspects of the system. For example, Rippe and Haraldsson [83] set up a mathematical model to investigate the role played by different sized pores - in the mass transfer across the capillary wall. Others, such as Arturson et al. [1], were more interested in the overall regulation of body fluid. Hence, in their models, not only transport mechanisms across the capillary wall were considered, but also the influence of hormone release and renal function. According to the spatial distribution properties of the parameters which describe the system, MVES models are divided into two types: lumped parameter or compartmental models and distributed parameter models. In compartmental models, all the variables under consideration are spatially invariant, i.e. any changes in those variables are consid ered equal and simultaneous throughout the volume for which the laws of conservation (mass, energy, and momentum) have been established. These models can be represented in terms of algebraic equations (static systems) or ordinary differential equations (dy namic systems). In the distributed parameter model, however, some or all of the variables are not the same throughout the whole system volume at any given time, i.e. the values Chapter 3. Model Formulation 32 of the variables are spatially varying. Ordinary differential eqilations (one-dimensional static systems) or partial differential equations (multi-dimensional and/or dynamic sys tems) are required to describe these types of models. Currently, our interest is in compartmental models of the MVES. A distributed pa rameter model, developed by other members of the research group [101, 102], was used to investigate the fluid and protein transfers inside an idealized interstitium. For corn parmental models, based on different assumptions about the structure and transport mechanisms of the capillary-interstitium interface, different mathematical descriptions of the fluid and protein transport across the capillary wall have been developed. Chapple [13] studied two such models, referred to as the uncoupled Starling model and the plasma leak model, and used these in an overall MVES model to simulate experimental data from nephrotic patients. In the uncoupled Starling model, albumin is assumed to pass through the capillary wall by convection along with the filtrate from the ircu1ation to the intersti tium, and by diffusion which takes place via a separate pathway. These two mechanisms are non-interacting and hence the name uncoupled Starling model. In the plasma leak model, it is assumed that two types of pores exist in the capillary wall, namely, the so-called small and large pores. Albumin is hypothesized to pass through the capillary wall by convection through the large pores and by diffusion through other parts of the wall, but is completely rejected by the small pores. In the current study, the coupled Starling hypothesis (or the Patlak formulation) will be used. This macromolecule trans port mechanism is considered to better reflect the true nature of transcapillary exchange. Details of the model formulation will be discussed in the following sections. Chapter 3. Model Formulation 3.2 33 General Assumptions of Compartmental MVES Models One of the goals of this work is to fomulate a mathematical model which describes the normal behaviour of the MVES. As we know, all computer models involve varying degrees of simplification of the real system. By making appropriate assumptions, a complex system can be replaced by a more simple model which simulates, at least approximately, the behaviour of the real system. This provides an economical and productive approach for studying real systems as long as the results of the simulations are valid. From Chapter 2, we know that the MVES consists of capillaries, interstitium, and terminal lymphatics. It is a complex biological system. Some structures and properties of the system are still unknown or controversial. Therefore, simplifying assumptions must be made. The goal of this chapter is to develop a compartmental model of the MVES. Thus, the first question which arises is how many compartments should the MVES be divided into? In the present model, the MVES is divided into two compartments, the circulation and a general tissue compartment (Fig. 3.1). The capillary-interstitium interface, which consists of the capillary wall and its basement membrane, lies between these two compart ments. It is obvious that lymph and blood can each be treated as a single compartment due to their physical separation from the interstitium. However, comparatively speaking, it is more problematic to lump the various tissues which exist in the human body into a general interstitial compartment due to their heterogeneities in properties, form and function (see Table 2.2). Iii previous modelling studies of rats [41, the interstitium was separated into individual skin and muscle compartments, since, on a weight basis, they are the two largest tissues in the body. However, this separation is not reasonable in human studies because of the lack of information on human tissues. The most complete Chapter 3. Model Formulation Figure 3.1: Schematic diagram of the compartmental model of the MVES. 34 Chapter 3. Model Formulation 35 set of information on humans is available for subcutaneous tissue and dermis. Fortu nately, 50 — 70% of the total interstitial fluid is stored in loose connective tissue such as skin and as little as 10% in muscle [2]. Also. results from the rat model indicate that skin and muscle compartments behave similarly [76]. Therefore, the properties of the general interstitial compartment will be assumed to be approximately equal to those of subcutaneous tissue. The microscopic geometry of both the capillary and the interstitium, which make up an exchange unit, also supports the compartmental assumption. Here, an exchange unit is defined as the cross-sectional area through which fluid and proteins must travel from the capillary to the nearest terminal lymphatic. The exchange unit for subcutaneous tissue has been estimated to be less than 8.1x10 2 i 4 m in area [113]. Assuming the exchange unit is a square, the distance between the capillary and the nearest terminal lymphatic will be around 0.03 1 um. The time required for fluid and solute molecules to diffuse across the exchange unit is a few seconds (unpublished results from I. Gates). But typical perturbations to the system usually last for hours. Compared with hours, a few seconds are instantaneous. Therefore, the compartmental method is an appropriate approach in MVES modelling. The second question concerns the assumptions made about each compartment. First, since we are dealing with lumped parameter models, each compartment is assumed to be homogeneous, i.e. whatever changes happen in one place occur throughout the whole compartment volume simultaneously. Other important assumptions include: • All plasma proteins are referenced to a single protein species, albumin. This is justified by the fact that albumin accounts for more than 50% of the total plasma protein in humans and, by virtue of its smaller molecular weight and greater net Chapter 3. Model Formulation 36 charge, it accounts for approximately 65% of the total plasma colloid osmotic pres sure. • Ions can pass through the interface freely, hence, the effect of ions can be neglected. • Cellular components are considered to be stable in both compartments. During the system perturbations investigated in the dllrrent study, the volume and composi tions of intracellular fluid are unchanged. Parenchymal cells are always in dynamic equilibrium with extracellular fluid. • Physicochemical properties of macromolecules, such as GAGs, proteoglycans, col lagenous and elastic fibres, remain unchanged throughout the perturbations. Com pliance of the interstitial compartment is assumed to follow the “most likely human compliance relationship which will be discussed later. Excluded-volume is - constant and is not affected by tissue edema. • Action of the kidneys in overall fluid balance is very rapid compared to the dynamics associated with the MVES. The MVES is not an isolated system. Thus the third question concerns the assump tions about the surrounding environment. The effects of nervous system and hormone release on the MVES is considered only indirectly in the current study. In summary, the MVES is simplified to two homogeneous compartments, the circu lation and the interstitium. Fluid and albumin are the only significant species within the system. Mass exchanges take place at the capillary-interstitium interface and at the terminal lymphatics due to passive driving forces. Chapter 3. Model Formulation 3.3 37 Coupled Starling Model When the principle of mass conservation is applied, the following equation applicable in both the circulation and the interstitium can be derived: Accumulation of S = Inflow of S — Outflow of S + Generation of S — Consumption of S where S denotes fluid or albumin content. If the generation and consumption terms equal zero, the above equation simplifies to Accumulation of S Inflow of S — Outflow of S (3.1) This form of the conservation of mass equation will be used to simulate perturbations involving a step change of one of the driving forces at zero time. According to the different equations which are used to define the flowrates of S in or out of the system, MVES models can be classified depending on the mechanism of transcapillary exchange. Several mechanisms, such as those described by the Patlak [10] as well as uncoupled Starling and plasma leak models, have been investigated in our research [13]. Because the Patlak formulation requires fewer fitting parameters than the plasma leak model and provides a more reasonable coupling of protein diffusion and convection than the uncoupled Starling model, it will be used throughout this work. The coupled Starling model, which is sometimes called the Patlak model, is a ho moporous model. In this model, the pores in the capillary membrane are assumed to be a single size and this size is characterized by the value of the albumin reflection co efficient (o). Fluid is transported from the capillary to the interstitium by filtration according to the Starling hypothesis described earlier. Solutes (i.e., albumin) are trans ported passively by diffusion and convection through the same channels that carry the fluid and thereby coupled. Interstitial fluid is drained back to the circulation by lymph. Chapter 3. Model Formulation BLOOD CAPiLLARY 38 INTERSTITIUM LWALJ P Figure 3.2: Schematic of the coupled Starling model. The hydrostatic pressure within the capillary is represented by a single value termed the capillary hydrostatic pressure, Pc (see Fig. 3.2). The quantitative analysis of the fluid filtration flow across a membrane was first introduced by Staverman [97] and was further developed by Kedem and Katchalsky [43]. In their analyses, Starling’s hypothesis was expressed by the following equation: JF=KF[PCPI7(11PL—llI)] (3.2) where JF denotes the fluid filtration rate; P and 11 denote hydrostatic and colloid osmotic pressures, respectively; and subscripts C, PL, and I denote capillary, plasma , and inter stitium, respectively. KF is the filtration coefficient . 1 (mL h’ .mmHg ) and its value is determined by the hydraulic conductivity and total area available for fluid transport. a is the albumin reflection coefficient and its value is determined by the proper ties (e.g. size, charge) of the solute molecule and the channel (i.e. pore) through which solute molecules Chapter 3. Model Formulation pass. If ci = 39 1, the membrane is perfectly impermeable to solute and the full osmotic pressure of the solution opposes filtration; if ci 0, the membrane allows solute molecules = to pass through freely and the osmotic pressure of the solution offers no resistance to filtration. The filtration rate is proportional to the net Starling driving force (bracketed term in Eq. 3.2). Albumin is transported across the membrane by diffusion with the superimposition of convection. Bresler and Groome [10] solved the one- dimensional convective-diffusion equation for a uniform cross-section pore to derive the following equation for transmem brane albumin transport: / Qs=JF1—ci). CFL — 1 exp(—Pe) CI,Av — exp(—Fe) 33 where Fe is the modified Péclet number given by Fe Here Qs = (1 — ci). (3.4) JF/PS denotes the albumin transport rate from the capillary to the interstitium (g/h), C denotes albumin concentration (g/L). CJ,Av is called the effective interstitial albumin concentration and is calculated as the interstitial albumin content divided by the intersti tial volume available to albumin (Vr,Av), which will be discussed later. PS is the product of membrane permeability to albumin and the membrane surface area (mL/h) and used to describe diffusive exchange across the capillary wall. The modified Péclet number, Fe, is therefore the ratio of the imposed (plug flow) velocity to the diffusion velocity of the solute. By rearranging Eq. 3.3, the albumin transport rate can be partitioned into two com ponents, i.e., Qs = JF (1 — ci) . CPL + i_e(x1p(Pe . (CPL — CI,Av). (3.5) Chapter 3. Model Formulation 40 where the first term is referred to as the convective component and the second term as the diffusive component. Equation 3.5 is not equivalent to the sum of the uncoupled convective and diffusive solute transports through a single channel [13], i.e.. Qs = JF (1 — a). [(CpL + CI,Av)/ ] + FS 2 (CPL — CI,Av). (3.6) Thus it can be seen that the convective and diffusive transfers of albumin in the Patlak model mutually influence each other. Interstitial fluid is drained back to the circulation compartment by lymph flow. As discussed in Chapter 2, bileaflet valves on terminal and collecting lymphatics ensure unidirectional lymph flow. Under normal conditions, the lymph flowrate, JL, is always assumed to be positive, i.e. lymph always flows from the interstitium to the terminal lymphatics. It is also assumed that the composition of the initial lymph is the same as that of the interstitial fluid. Therefore, there is no difference in colloid osmotic pressure between the lymph and interstitial fluids. Hydrostatic pressure in the interstitium is assumed to be the only driving force for lymph flow deviations from the normal baseline level. A linear relationship between the lymph flowrate and the tissue hydrostatic pressure of the type developed by Bert et al. [4] for their rat model is also employed in the current human model. During tissue overhydration, the lymph flowrate, JL, is assumed to increase proportionally to the change in interstitial hydrostatic pressure by a factor LS, which is called the lymph flow sensitivity 1 (mL.mmHg . h’). Thus, = JL,O where JL,O + L5 (Pi — ) when P 0 , 1 F 1 Pi,o (3.7) is the lymph flowrate at normal steady-state conditions (mL/h). As discussed in Chapter 2, this normal lymph flow is formed by the contraction of the tissues surround ing the terminal lymphatics, as well as by the spontaneous contraction of the collecting lymphatics. Chapter 3. Model Formulation 41 During tissue dehydration, another linear relationship is used, namely JL FI,EX JL,O 1 (P — PJ,Ex)/(PI,o — PI,Ex) when P 1 <P 0 , 1 (3.8) is the interstitial hydrostatic pressure when the tissue dehydrates till the intersti tial fluid volume is equal to the excluded volume (VI,Ex). This is considered to be the limit to which the tissue can be dehydrated. At this condition, the lymph flow, according to Eq. 3.8, ceases completely, i.e. JL = 0. Further dehydration is beyond the scope of the current discussion, because it would be accompanied by drastic changes in the struc ture and properties of the interstitium, e.g. cell dehydration, fiber and macromolecule breakdown, etc.. Such changes conflict with the assumptions made at the beginning of this chapter. Because of the non-sieving character of the terminal lymphatic walls (o is assumed to be zero), plasma proteins within the intersititium are assumed to be transported across the wall solely by convection with lymph flow. Therefore, the albumin exchange rate across the lymphatic wall, Qr, is given as the product of the lymph flowrate and the albumin concentration in the interstitium: QL 1 = Jr C (3.9) Equation 3.9 assumes that, once the plasma protein solution from the available volume enters the lymphatics, it mixes with the interstitial fluid leaving the remaining portions of the mobile fluid volume according to their relative volumes; therefore, C 1 is used instead of CI,Av. Now the mass balances for the circulatory and interstitial compartment can be writ ten more specifically. The fluid and protein balances, respectively, for the circulatory compartment become dVp/dt = JL — JF 1 +D (3.10) Chapter 3. Model Formulation 42 and dQpL/dt = QL — 2 Qs + D (3.11) where D denotes external perturbations to the system, e.g. fluid infusion (D positive), protein infusion (D 2 positive), urine output (D 1 and D 2 negative), etc.. For the interstitial compartment, the two balances become dV / 1 dt = JF — (3.12) JL and clQj/dt = Q (3.13) — If both compartments are at steady-state, then the left hand sides of Eqs. 3.10 set to zero, i.e. the fluid volumes (VPL and V ) and albumin contents 1 (QPL — 3.13 are and Qi) in both compartments remain constant. 3.4 Constitutive Relationships When the dynamic characteristics of the microvascular exchange system are studied, some variables are subject to variations with time until a new balance is obtained. These variations involve the changes in volume, concentration, hydrostatic and colloid osmotic pressures, etc.. Thus, in order to complete the description of the model, it is necessary to establish additional, constitutive relationships between these variables. In this section, three such relationships will be discussed, i.e. the circulatory compliance, the interstitial compliance, and the relationship between albumin concentration and colloid osmotic pressure. 3.4.1 Circulatory Compliance Due to the microscopic nature of the capillaries, the techniques for measuring capillary hydrostatic pressures, P , are not yet reliable. Therefore, it becomes necessary to relate 0 Chapter 3. Model Formulation 43 Pc to variables which can he measured directly. It is believed that Pc is dependent on the arterial pressure, FAA, Pc = and venous pressure, PVV according to [72]: [(Rvv/RAA) . FAA + Pvv]/[1 + (Rvv/RAA)} (3.14) where RAA and Rv are the precapillary and postcapillary resistances, respectively. The physical meaning of the above equation is obvious. However, the use of Eq. 3.14 is problematic, because the resistances are also unknowns and cannot be measured directly. (Only FAA and PVV can be measured easily.) Because the blood vessels (particularly on the venous side) are distensible, changes in arterial and venous pressures result in changes to another measurable output, the plasma volume, VPL. Circulatory compliance, FCOMPC, is defined as the ratio of the change in plasma volume to the change in capillary hydrostatic pressure, i.e., FCOMPC = AVPL/PC (3.15) which is similar to the definition of interstitial compliance (Eq. 2.1). From Starling’s hy pothesis, we know that there are other factors (e.g. colloid osmotic pressures, transport properties of the membrane) besides the capillary hydrostatic pressure which determine the plasma volume. Other active mechanisms associated with hormonal, neural or my ological behaviour which will not be considered directly in this study may also affect the circulatory compliance. In fact, for humans, there are insufficient data even to establish a quantitative relationship between Pc and VPL. Thus for the sake of simplicity, a linear relationship is assumed to apply, i.e. FCOMPC = LVPL//PC = constant or Pc = Pc,o + PC,GRAD• (VPL — VpLO) (3.16) Chapter 3. Model Formulation where PC,GRAD 44 is the reciprocal of the circulatory compliance. Since, to our knowledge, PC,GRAD has never been measured in humans, therefore, an estimate of its value is needed. Bert et al. [4] estimated PC,GRAD = 5.05 mmHg/mL in the rat microcirculation by statistically fitting a set of fluid volume distribution data as a function of total fluid volume. 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. In the rat, mmHg/mL corresponds to an elevation in P of 30.91 mmHg when of its normal level, (from 6.12 mL to 12.24 mL), i.e. PC,GRAD = VPL PC,GRAD = 5.05 is raised to twice 30.91 mmHg/6.12 mL. In human, with a normal plasma volume of 3200 mL (the normal values for human will be discussed in next section), to satisfy the scaling criterion, 30.91 mmHg/3200 mL 3.4.2 = PC,GRAD should be equal to 0.009659 mmHg/mL. Interstitial Compliance The interstitial compartment is considered to be a storage reservoir for body fluid. Inter stitial compliance, FCOMPI, therefore, is an important property which can markedly affect the distribution of body fluid. It is defined by Eq. 2.1. Although there is a general awareness of the importance of the interstitial compliance, very little actual information about it is available in the literature. Reed and Wiig [109, 79, 110, 111] have conducted a series of experiments to study the interstitial compliance relationship in mammals other than humans. After studying the compliance characteristics of skin and skeletal mus cle in rat, cat and dog, they concluded that the compliance of both skin and skeletal muscle follows a similar trend: low compliance during severe tissue dehydration, mod erate compliance between moderate dehydration and moderate overhydration, and high compliance during severe tissue overhydration. Experimental data from the rat [109, 79] were fitted and used in compliance relationships for the skin and muscle compartments Chapter 3. Model Formulation 45 in the rat microvascular model of Bert et al. [4]. The only experimental data on human tissue compliance known to the author is from Stranden and Myhre [99]. They studied 46 patients with unilateral leg edema. The tissue hydrostatic pressure, F , was measured in subcutaneous tissue by the wick-in-needle 1 technique. The subcutaneous tissue volume increase was calculated as the difference in volume between the edematous and the contralateral leg. The results are shown in Fig. 3.3. The “most-likely” human interstitial compliance #1 (see later discussion) is also plotted on the figure. The data are too scattered to assign a particular fit to them. However, it was found that the compliance varied significantly at different levels of interstitial tissue hydration: P 1 increased markedly with increasing subcutaneous tissue volume in patients with moderate edema (0 — 100% subcutaneous tissue volume increase), but insignificant further increase in P 1 was observed with additional edema (100 - 600 Under these circumstances, a so-called “most-likely” human interstitial compliance relationship is constructed on the basis of the information from Stranden and Myhre as well as that from Reed and Wiig. With a 210 % increase in interstitial fluid volume, the interstitial hydrostatic pressure increases 2.9 mmHg in human subcutaneous tissue [99], while in the rat skin, interstitial hydrostatic pressure increases only 2.4 mmHg [79]. The “most-likely” human interstitial compliance is therefore constructed by scaling the interstitial hydrostatic pressure of rat according to: PI,HUMAN — PI,HUMAN,O = (PI,RAT — (3.17) PJ,RAT,o). and by scaling the interstitial volume according to: VI,HUMAN = VI,RAT ( VI,HUMAN.O VI,J?AT,IJ ) (3.18) where the subscripts “HUMAN” and “RAT” have the expected meanings. The subscript “0” refers to the normal value. VI,HUMAN,O equals to 8.4 L. PI,RAT and V1,RAT are a Chapter 3. Model Formulation E E - 46 5- •. . 4- . . . . a.. 3• .‘ 2- .. • • C 3 0• 0, I. G -3- C I I I I I I I I 0 200 400 600 800 Subcutaneous tissue volume increase (%) Figure 3.3: The interstitial fluid pressure-volume relations hip in patients fol lowing arterial reconstruction for femoropopliteal ather osclerosis. • repre sents patients with postoperative edema. represents patie nts without post operative edema (mean±SD). represents healthy cont rols. Number of sub jects investigated is bracketed below [99]. The solid line is the “most-likely” human interstitial compliance relationship. Chapter 3. Model Formulation Dehydration: Intermediate: Overhydration: Compliance #1 Compliance #2 Compliance #3 47 Pi = —0.7 + 1.96154 x 1 (V 8.4 x i0) 3 10 cubic spline interpolation (see subroutine SPLINS) — 1 = 1.88 + 1.8 x 10— P (Vi 5 Pi = 1.88 + 5.0 x 1 (V 5 10— Pi = 1.88 + 1.05 x 1 (V 4 10— — — — 1.26 x iO) 1.26 x 10) 1.26 x i0) Table 3.1: Mathematical descriptions of the interstitial compliance relation ship. series of discrete data points obtained from the skin compliance relationship in the rat microvascular model [4]. The “most-likely” human interstitial compliance so generated is plotted as the solid line (compliance #1) in Figs. 3.3 and 3.4. Figure 3.3 shows that the selected curve is a reasonable representation of the experimental data points. The relationship is artificially separated into three regions: the “dehydration segment” 1 (V 8.4 L), the “intermediate segment” (8.4 L < V 1 < 12.6 L), and the “overhydration segment” (V 1 12.6 L). The confidence of the overhydration region is considered to be the lowest, therefore, two other overhydration compliances (compliance #2 and #3) are generated by arbitrarily increasing the slope in this region, while the other two regions remain the same as that in compliance relationship #1. The mathematical descriptions of the three compliance relationships are summarized in Table 3.1. 3.4.3 Colloid Osmotic Pressure Relationship Since colloid osmotic pressure is caused by the fact that solute molecules can not readily diffuse through the semipermeable capillary membrane, it is natural to assume that there is some kind of correlation between protein concentration and colloid osmotic pressure. In 1963, Landis and Pappenheimer [47] derived three empirical equations to relate albumin, globulin, and total protein concentrations to colloid osmotic pressure. However, in the Chapter 3. Model Formulation 48 5.0 0.0tzO -5.0 - 0.0 tOM Compliance curve 1 PJ’.P Curvej2 cQmpU..cc. w’x. 20.0 30.0 - -. 40.0 v ( 1 L) Figure 3.4: The “most-likely” human interstitial compliance relatio nship [13]. Chapter 3. Model Formulation 49 current model, the single protein species albumin has been chosen to be respresentative of all protein species because: 1. it is about 50 % of the total protein mass; 2. it actually accounts for 65 % of the total osmotic pressure by virtue of its smaller molecular weight and greater net charge; and 3. the measurements of its content and concentration are the most widely reported. Therefore, we chose to determine the relationship between the albumin concentration and the total colloid osmotic pressure so that, in case one of them is given, the other can be calculated. Chapple [13] has developed such a relationship by a least squares fitting of the data collected from the circulatory compartment of patients with nephrotic syndrome [31, 63, 241. Nephrosis is a kidney disease which causes a lowering of the blood protein level. The fitted relationship was forced to pass through the point CFL = PL = 11 0 wheu 0. First, second, third and fourth order polynomial curve fits were attempted. Because a difference in the variance of fit of less than 1 percent was found between the first and second order relationships and because the higher order polynomials produced non-monotonic behaviour within the range of interest, the linear relationship ‘—‘FL — 1.522 x 3 in— ,u TT 3.19 “FL was selected as the best choice. In Eq. 3.19, the units of CPL and FL 11 are g/mL and mmHg, respectively. This relationship as well as the experimentally measured data points are plotted on Fig. 3.5. Landis and Pappenheimer [47] had earlier determined the following relationship be tween total protein concentration (C) and total colloid osmotic pressure (H): fl = 0.21 x C + 1.6 x i0 x C 2 + 9.0 x 10-6 x C . 3 (3.20) Chapter 3. Model Formulation 50 50. C) C C) 10.0 20.0 II (mmHg) Figure 3.5: Relationship between albumin concentration and total colloid os motic pressure [31, 63, 24]. Chapter 3. Model Formulation 51 The good agreement between Eq. 3.19 and Eq. 3.20 over the concentration range of interest shows that neglecting protein differences does not affect the LI values significantly [14]. Because of the difficulties in collecting interstitial fluid, very little information about the interstitial compartment is available. If it is assumed that the relative osmotic activity of albumin to that of the total proteins is similar in both the interstitium and plasma, then it is reasonable to apply Eq. 3.19 to the interstitial compartment as well, i.e. CI,AV = 1.522 x i0 LI . (3.21) Note that the effective albumin concentration is used since albumin is excluded from some of the tissue volume. 3.5 Normal Steady-State Conditions In order to study the transient responses of the MVES after the system is disturbed, an initial point from which these transient responses deviate must be specified. This initial point is taken to be the normal steady-state conditions in the current study. As we know, different individuals possess different body weights, fluid volumes, pres sures, chemical compositions, etc.. To help normalize these individual differences, a “reference man” is introduced. The “reference man” is: • a healthy male 170 cm in height and 70 kg in weight; • in a supine position; hence location-dependent values such as colloid osmotic and hydrostatic pressures are taken at heart level of the thorax, if possible. To make comparison convenient, the normal steady-state conditions of the “reference man” in the current model are the same as those used by Chapple [13], and they are Chapter 3. Model Formulation Variable , mL 0 V VEX, mL ll, mmHg , g/L 0 C CAV,O, g/L Qo, g PA,o, mmHg Pv,o, mmHg ,o, mmHg 1 F Circulation 3200 — 25.9 39.4 — 126.1 5.92 24.54 52 Tissue 8400 2100 14.7 16.8 22.4 141.1 — — — -0.7 References [21, 106, 94, 56, 26] [76](assumed value) [23, 46, 64, 27] calculated from Eq. 3.19 and 3.23 calculated from Eq. 3.21 calculated [106] [106] [108] Table 3.2: Normal steady-state conditions for the “reference man”. listed in Table 3.2. There are several points about Table 3.2 that should be emphasized here. First, the normal fluid volumes in both compartments are confidently known. The values have been further verified by more updated information. Fauchald [25] measured the plasma volume by using injections of I - labelled albumin in 16 normal subjects, and found the normal range to lie between 2.8 L — 3.5 L. Applying the same technique, Noddeland et al. [63] measured the normal range of plasma volume at 3.1 L — 4.1 L. These workers also measured the extracellular fluid volume (ECV) allowing a calculation of the interstitial fluid volume as Vi,o = ECV — VPL,O. In this manner, Fauchald and Noddeland et al. found that the normal range of interstitial fluid volume lay between 6.0 L — 11.3 L and 7.1 L — 12.2 L, respectively. These measurements made us feel more confident about the fluid volume values listed in the table. The second point concerns the excluded-volume, VEX. The degree of exclusion is expressed in various ways. Sometimes, it is based on the matrix space instead of the “free-fluid” space (i.e. V ) 1 , while sometimes, it is based on the excluding material’s volume or weight alone. These different bases make it difficult to use the information available in the literature. Moreover, there is little information concerning exclusion in Chapter 3. Model Formulation 53 human tissues. Therefore, an assumed excluded-volume fraction (i.e. VEX/VI,o) of 25% is used. This value is close to that estimated by Bell et al. [3]. In their studies on dog hindpaw skin, albumin was excluded from an average of 26% of the sucrose space. Here, it is assumed that, since sucrose is a relatively small molecule, the sucrose space equals the interstitial fluid volume; in other words, the exclusion of sucrose can be neglected. 25% exclusion corresponds to an excluded-volume of 0.25 x8400 mL = 2100 mL. The third point concerns the effective albumin concentration in the interstitial com partment, CI,Av. It is defined as the ratio of interstitial albumin content to the interstitial fluid volume available as distribution space for that protein, i.e. CI,Av = /(V 1 Q — (3.22) VEX) It is the effective interstitial albumin concentration rather than the actual interstitial al bumin concentration that-determines the interstitial colloid osmotic pressure. Thus, given 111,0 (experimentally measured), CJ,Av,o can be calculated from Eq. 3.21, and accordingly, the interstitial albumin content, Q’,o from Eq. 3.22 if Vi,o and VEX are known. The in terstitial albumin concentration required in the lymph drainage expression (Eq. 3.9) is obtained from 1 C = Q / 1 Vj (3.23) A similar equation applies to the plasma albumin concentration, i.e. CPL 3.6 = QPL/VPL (3.24) Summary A summary of the complete set of equations required by the coupled Starling model is listed in Table 3.3. q -__- I H j Cl) C Cl) C . (I DI lIc —I ‘-(i)IC) CD ‘- IC) c_— I— CD CD H -‘ H II Cl’_- )—Le I II II II Cl) II S— I. --- + I e-l H ‘— I _— I . C3 1’9 I II i2I lJ C) H : -. CD q CD CD 0 — -. I- Cl) 0 C) H I—. CD 0 Chapter 4 Parameter Estimation And Data Analysis In this chapter we will first describe the design of the parameter optimization procedure, then present the experimental data collected from the literature, and finally discuss how to use these data to obtain the unknown parameters for the coupled Starling model. 4.1 Parameters to be Determined A review of the mathematical descriptions of the microvascular exchange system in Chap -ter- 3- reveals that there are some constants which are determined only by the physico chemical properties of the system. These constants are called the internal parameters of the MVES model. Their values used in the model affect the fluid and protein distri bution within the MVES significantly. For example, the fluid filtration coefficient, KF, the diffusive permeability coefficient, PS, the solute reflection coefficient, a, the normal lymph flow rate, JL,O, and the lymph flow sensitivity, LS, are all internal parameters which characterize the transport properties of the capillary wall and the terminal lym phatics; KF reflects the hydraulic conductivity, PS mirrors the permeability to albumin, and a represents the sieving property of the capillary wall. JL,O and LS characterizes the efficiency of the lymphatic system in removing accumulated interstitial fluid. These parameters are assumed to remain unchanged at the constant value assigned to them throughout the duration of whatever perturbation is applied to the MVES. One of the objectives of the current study is to determine the optimal values for these parameters. 55 Chapter 4. Parameter Estimation And Data Analysis Unknown LS a Pc,o KF PS JL,O 56 Description lymph flow sensitivity (mL.mmHg’.h ) 1 albumin reflection coefficient capillary hydrostatic pressure at normal steady-state (mmHg) the fluid filtration coefficient (mL.mmHg’.h ) 1 permeability-surface area product (mL .h’) normal lymph flow (mL.h’) Table 4.1: Unknowns in the coupled Starling model. Generally, there are two approaches for dealing with transport parameters in com partmental models: 1. assume values based on literature information for human or related animal MVESs; 2. treat them as unknowns and determine their values by statistically fitting model predictions to available measurements from the literature. Because some of the aforementioned parameters have not been measured directly in humans or even in animals, and some have only been measured inaccurately (these details will be discussed in Chapter 5), the latter approach is selected in the current study. The normal capillary pressure, Pc,o has also never been measured directly in humans. In addition, the constitutive relationship between the interstitial pressure and the in terstitial fluid volume, P 1 F(V ) 1 , can only be estimated (see Table 3.1). Thus, the capillary pressure, Pc,o, is added to the list of unknown model parameters listed in Ta ble 4.1, while the three interstitial compliance relationships discussed in Section 3.4.2 are introduced into the model one at a time to test how they influence the results of the parameter estimation procedure. Chapter 4. Parameter Estimation And Data Analysis 4.2 57 Additional Relationships between the Unknowns Of the six unknowns parameters listed in Table 4.1, three can be eliminated via additional relationships which exist between these parameters. These relationships will be discussed in the sections that immediately follow. 4.2.1 Steady-state Balances at Normal Conditions Two equations can be derived from the fluid and protein mass balances which must exist in the intersititial or circulatory compartments under normal steady-state conditions. At normal steady-state, the fluid balance can be written as JF,O — JL,O = 0 (4.1) QL,O = 0 (4.2) and the protein balance as Qs,o — When we substitute Eqs. 3.2, 3.3 and 3.9 into Eqs. 4.1 and 4.2, we obtain, respectively, the following two equations relating KF [Pco 0 , 1 P KF, — PS, JL,O, a (HPL,o — a and Pc,o: L[’,o)] — JL,O = 0 (4.3) and JF,O . (1 — a) . [CPLO 1 CIAvoexp(PeO) — JL,O 0 , 1 C = 0 (4.4) The degrees of freedom for a model is equal to the number of unknowns minus the number of relationships between these unknowns. Thus, when Eqs. 4.3 and 4.4 are introduced, only four degrees of freedom remain, i.e. only four unknown parameters need to be specified to completely characterize the model. Chapter 4. Parameter Estimation And Data Analysis 4.2.2 58 Albumin Clearance Relationship In addition, several workers have investigated the clearance rate of 125 1-radiolabelled (or 1-radiolabelled) 3 ‘ 1 albumin from the subcutaneous tissue in humans. It is assumed that injecting the tracer does not alter the normal steady-state conditions of the system. The interstitial albumin disappearance half-time, , 112 is defined as the time required to T eliminate 50% of the total introduced labelled albumin from interstitium to the lymphatic system or to the circulatory compartment. A small dose of labelled albumin (10 30 — [LCi) is injected subcutaneously, and its disappearance rate is registered by a Geiger counter at the injection site at various time intervals [22]. The percentage of the initial radioactivity is plotted as a function of time on a semilogarithmic graph allowing T 172 to be determined as the slope of the disappearance curve [48]. The rate of disappearance of albumin from the interstitial compartment is defined as the albumin turnover rate, A1bTQ (fraction lost pr hdur). Thus, A1bTO and T 112 are related by (1 — A1bTQ)T1/2 = 0.5 (4.5) The average value of T 112 obtained by several investigators [48, 39, 28] is 33.4 hours. From Eq. 4.5, A1bTQ is therefore calculated to be 2.05% per hour. A mass balance for the labelled species yields the following equation: (4.6) where the * superscipt indicates a tracer quantity. Upon substituting Eqs. 3.3 and 3.9, Eq. 4.6 becomes: dQ — — ( 0 C, — F,0 CZAV,o exp(—Peo) 1 exp(—Peo) — — According to the definition of albumin turnover rate, at t dQ*I = AlL iWTQ 17 VI,O fY* = 0, — L,0 (4 7) . Chapter 4. Parameter Estimation And Data Analysis 59 Also, at t = 0, there is no labelled albumin in plasma, so that CLO since CZ 0 = / Vj,o 0 Q, (at t = and CZAv,0 = /(V 0 Q, , 1 o — VEX), = 0. Furthermore, substituting into Eqs. 4.7 and 4.8 0) and then setting them equal yields (1 a) exp(—Peo) JL,O + V 0 , 1 exp(—Peo)1(Vi,o VJ,Ex) JF,O [1 . — . — — — AibTo — — 0 (4.9) — where — — 0 Fe (1 — a). JF,O Eq. 4.9 provides a third relationship between the unknown parameters of the model. Thus, of the six unknowns listed in Table 4.1, only three are independent. As a conse quence, only three parameters need to be determined by fitting available response data, the other three can then be calculated from Eqs. 4.3, 4.4 and 4.9. 42.3 Parameters to be Optimized The albumin reflection coefficient (a), the lymph flow sensitivity (LS) and the capillary hydrostatic pressure at normal steady-state (Pc,o) were selected as the three parameters to be determined by statistically fitting the model predictions to the experimental results. The selection of these three parameters was based on the following criteria: 1. the parameters must be physiologically important to the system; 2. the parameters should be sensitive to the estimation procedure; 3. the parameters must be independent each other. LS does not appear in the normal steady-state mass balances nor in the albumin clearance relationship; therefore, it must be selected. We don’t know which among KF, PS, JL,O, a and Pc,o is more important physiologically, but we do know that a ranges from 0 to 1. Hence, because of its convenient bounds, a was chosen as another parameter to be Chapter 4. Parameter Estimation And Data Analysis 60 optimized continuously within these bounds. Finally, Pc,o was chosen to be the third parameter investigated because its bounds can also be estimated, as is discussed in the next section. Because the optimization program (see Section 4.5.3) is robust enough to solve a three-dimensional problem and since a single set of the parameters is required for validation purposes, we chose to optimize LS, a and Pc,o continuously, instead of treating as a discrete variable in the statistical fitting procedure as Chapple (1990) did in his study using nephrotic syndrome data. Once values for a, LS and Pc,o have been specified, the remaining three parameters, KF, PS, and JL,O can be calculated from the following equations obtained by manipu lating Eqs. 4.3, 4.4 and 4.9: A1bTQ JL,O = (1—cr).ecp(—Peo) [1—exp(—Peo)J(VJ,o—VJ,Ex) P5= (1 ln[ — a). + JL,O CJ,o—(1—o-)CJ,Av,Q] (4.10) (4.11) —)CpL,o 1 CI,o—( and KF JL,O = Pc,o — Pi,o — a. (IIPL,O (4.12) — At this point, the model is completely defined. Thus, knowing the initial conditions of the system and the perturbation, we can solve the four coupled ordinary differential equations, Eqs. 3.10, 3.11, 3.12 and 3.13, to find the transient or steady-state responses of volumes and protein contents in both the interstitial and the circulatory compartments. In the process, the protein concentrations, osmotic pressures and hydrostatic pressures of the system are also calculated. Chapter 4. Parameter Estimation And Data Analysis 61 Parameter Search Ranges 4.2.4 To save computational costs, bounds that encompass the region within which physiolog ically feasible parameter values must lie were calculated. The upper and lower bounds on Pc,o are the same as those used by Chapple [13]. The upper limit was obtained by assuming that a trariscapillary pressure gradient of 1 mmHg is maintained under normal steady-state conditions. Thus, since Pi,o fli,o = when o 14.7 mmllg and = —0.7 mmHg, 25.9 mmHg, and since the maximum Pc,o is obtained PL,O = 11 1, then Pc,o < 11.5 mmHg. The lower limit was obtained on the basis of experimental measurements of venous capillary hydrostatic pressure (P,o 6 mmHg) [2]; thus, Pc,o > 6 mmHg. The search range of Pc,o, therefore, is chosen to be: 7.0 - As was discussed in Section pc,o 11.0 mmHg. LB is independent of the steady-state mass balance. Thus there is no restriction imposed on LS from these mass balance equations and LS may range from zero to infinity. To reduce computational costs, LS is assigned the bounds: 0 LS 500 mL mmHg’ hr’ . . In most cases, the value of LB which produces the best fit is significantly less than the artificially imposed upper limit. Because KF, PS, and accordingly. Since JF,O JL,O must be positive, the possible values of a must be restricted is positive (which is equal to JL,O at normal steady-state), then to ensure that KF remains positive in Eq. 3.2, Pc,o — Pi,o — 0-. (H,o — > 0 (4.13) or, 0 P, L[PL,O — — Pi,o 111,0 (4.14) Chapter 4. Parameter Estimation And Data Analysis Since a JL,O > < 1, therefore, 1 — a 62 0. And normal lymph flow is assumed to be positive, i.e., 0. To ensure that PS remains positive, then from Eq. 4.11 >0. (4.15) Solution of the above inequality accounting for the mathematical restriction which exists in the evaluation of logarithms, gives CPL,0 — I,0 1. <a CPL,0 To ensure that JL,O (4.16) remains positive, then according to Eq. 4.10 a) exp(—Peo) + exp(—Peo)](Vj,o VI,Ex) 0 , 1 V (1 [1 — . — 0 > 4 17 — Rearranging Eq. 4.11 gives, (1— a) JL,O —(1—a). CI,Av,oj —1 n (418) — PS (1 — a) — CPL,0 Taking exponentials on both side of the above equation gives, Ci,o 0 , 1 C e(l_JL0 — — (1 (1 — — a) CI,Av,o a) CPL,0 (4.19) . or, according to the definition of the Péclet number (Eq. 3.4), C 0 , 1 ,o 1 C e_Pe0 — — (1 (1 — — a) CpL,o a) CI,Av,o (4.20) Upon substituting Eq. 4.20, Eq. 4.17 can be simplified as 1 a> CPL,0 Also, since CJ,Av,o = [CPLO Q1,0/( V 0 , 1 — — C1,o VI,Ex) = (VIO — V1Ex) . ,V 1 o and Cj,o = (VIO _EX) . (CpL,o — CJ,Av)]. (4.21) /V then 1 Q 0 , CI,Av,o (4.22) Chapter 4. Parameter Estimation And Data Analysis P,o (mmHg) 7 8 9 10 11 63 clmjn amax 0.574 0.574 0.574 0.574 0.574 0.688 0.777 0.866 0.955 1.000 Table 4.2: Bounds on a for Pc,o equal to 7 to 11 mmflg. Substituting Eq. 4.22 into Eq. 4.21 yields a> CI,Avo — (4.23) . CI,Av,o The range of a values which simultaneously satisfies inequalities 4.14, 4.16 and 4.23 is found to be CPL,0 — 0 Ci, — . CFL,0 The upper bound is chosen to be either 1 or (Pc,0 PL,O 11 — — 111,0 } PI,o)/(llpL,o — ui,o), whichever is smaller. The latter expression, as can be seen, depends on P, . The actual bounds on 0 a for Pc,o values ranging from 7 to 11 mmHg are listed in Table 4.2. 4.3 Analysis of Data Ever since the general principles of the Starling hypothesis for transcapillary fluid ex change were widely accepted, measurements of the Starling forces and other related factors in the MVES have interested many investigators. After years of exploration, reliable approaches for measuring many of these factors have been developed. General knowledge of these measuring techniques will assist us in evaluating the quality of the available experimental data. For example, interstitial colloid osmotic pressure data can be obtained by: Chapter 4. Parameter Estimation And Data Analysis 64 1. direct sampling by micropipettes and catheters. The drawback of the method is that the applied suction could increase the net capillary filtration and thereby influence the interstitial protein concentration [86, 87]; 2. implanted capsules or wicks based on the assumption that a fluid compartment in contact with the interstitium will finally attain the same protein concentration and hydrostatic pressure as the free-fluid phase of the interstitium [23]; 3. noninvasive blister suction method [80]. This method is subject to the same criti cism as per # 1 above. The wick technique is the most widely used method and is generally thought to be the most representative and accurate. Once the interstitial fluid is collected, the interstitial colloid osmotic pressure can be measured directly by an osmometer, and protein concen tration can be determined by using a number of methods. Blood samples are easy to collect; therefore, plasma osmotic pressures and protein concentrations can be determined immediately using these same techniques. Alternative methods for the measurement of hydrostatic pressures in interstitium include [2]: 1. the use of chronically implanted perforated plastic capsules (with diameters of 1 cm or more); 2. use of a wick or a wick-in-needle (with a needle outer diameter of 0.6 mm); 3. the micropuncture technique (with a micropipette tip diameter of 2 — 4 [tm). In all these methods, the interstitium is connected to a low-compliance manometer through a fluid-filled tube. However, it is impossible to place a micropipette or nee dle inside the capillary because the capillaries have internal diameters of 5 — 10 m only. Chapter 4. Parameter Estimation And Data Analysis 65 Thus, to the knowledge of the author, no one has so far directly measured the capillary hydrostatic pressure. The methods for the measurement of plasma volume and interstitial volume are based on the dilution principle. A measured amount administered by injection. (Q) of a suitable tracer substance is After equilibrium throughout the whole body is attained and corrections are made for metabolic losses of the tracer, the concentration (C) of the substance is measured in a suitable sample of the body water. The volume of the compartment (V) is then given by the relation, V = Q/C. The dilution agents used in the measurement of extracellular fluid volume (ECV) include insulin, sucrose, mannitol, sodium thiocyanate and the radioactive ions, 35 SO, 82 Br Na+. Tracers used and 24 in the measurement of plasma volume include Evans blue dye, ‘ 1-labelled 3 1 albumin or Cr-labelled erythrocytes. The interstitial volume is then calculated as Vi 51 = ECV—VPL. In this section, we begin by preseiiting all of the available literature dataon humans which we hope to include in the parameter estimation procedure. We then analyze each data set in detail to determine whether it really provides information about the problem under study and whether the measured results can reflect the relationships between the dependent variables and the parameters we investigated. Finally, we discuss how these different data sets can be combined and used in the optimization procedure. 4.3.1 Experimental Data Six sets of useful data were found by an extensive search of the literature. The quantities reported in these independent studies include colloid osmotic pressures, fluid volumes, protein contents and protein concentrations, which were measured by applying the various techniques discussed in the previous section. Each quantity has a different set of units, e.g. pressure has units of mmllg, volume has units of mL, etc.. In order to make these different measurants comparable, we have chosen to convert all of them into percentage Chapter 4. Parameter Estimation And Data Analysis 66 changes, so that they are unitless and can, therefore, be compared directly. For the transient data, this conversion is based on the following equation: X%=XtX0 xlOO xo where X denotes the quantity measured in the experiment and subscripts t and 0 indicate quantities measured at a particular time and just prior to the perturbation (t = 0), respectively. For the steady- state data, the conversion is based on the equation: = X - XNQRM x 100 XNORM where X and XNORM represent the same variables measured in a patient and in a normal subject, respectively. This normalization procedure has several advantages. First, as was mentioned, it makes all the measured quantities unitless so that they can be compared. Second, it converts the magnitude of all the quantities into a similar range, again so that they are more comparable. For example, before conversion, Vj has a value of around 8400 mL and , a value of about —0.7 mmHg. It is difficult to compare 8400 mL with —0.7 mmHg. 1 F But, after conversion, their percentage changes are usually less than 100% as can be seen in sections which follow. Third, using percentage changes simplifies the normalization process because it eliminates the differences in baseline values of variables in different experiments. For example, consider the variable VPL. In one experiment, the initial plasma volume may be 2547 mL, while in another it may be 3227 mL. If percentage changes are used, the differences in starting values are eliminated automatically. Only the modified data, based on the normalization procedure discussed above, are presented in the section which follow. The raw experimental data are listed in Appendix A. For each of these six sets of results, the objectives, perturbations, quantities measured and sample sizes of the experiments are described in detail below. Chapter 4. Parameter Estimation And Data Analysis Time (hr) Saline, 100 mL N=4 Albumin, 100 mL N=4 Saline, 200 mL N=4 Albumin, 200 mL N=4 Mean SD + Mean SD + Mean SD ± Mean SD ± -1.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 -3.37 5.85 2.04 11.11 -3.73 16.94 13.28 12.51 3 -0.41 8.43 -0.41 10.19 -0.83 10.94 12.86 20.36 67 6 -2.86 7.09 -2.45 7.64 -1.66 11.71 6.22 14.87 9 -4.09 6.63 -1.64 9.73 2.07 19.88 8.30 17.93 12 -7.36 10.16 -4.50 6.33 -0.41 14.71 6.64 13.24 Table 4.3: Percentage change in plasma colloid osmotic pressure at room tem perature. t = 0 designates the end of the infusion period [41]. 4.3.1.1 Set A: Saline and Albumin Infusion To study albumin-induced plasma volume expansion, Hubbard et al. [41] infused eight healthy male volunteers with solutions of saline or saline plus albumin over a 1.5 hour period in a thermoneutral enviroment. Four of them were raiadomly assigned to a low dosage treatment, i.e. intravenous infusion of 100 mL saline or 100 mL saline with 25 g albumin, while the other four were assigned to a high dosage treatment, i.e. 200 mL saline or 200 mL saline with 50 g albumin. Just before the infusion and 1, 3, 6, 9 and 12 hours post-infusion, plasma volume was measured by dye dilution using indocyanine green, plasma osmotic pressure was measured using an oncometer, and total plasma protein content was determined by commercial tests. The results are tabulated in Tables 4.3, 4.4 and 4.5 (for FL, 11 VPL and QFL, respectively) in terms of percentage changes from pre-infusion values. Both average changes and standard deviations are provided. During the observation period (0-13.5 hours), an average net weight gain of 1.4 kg due to fluid intake occurred. This will be considered as part of the perturbation. Therefore, the total fluid inputs were 1500-1600 mL. The same perturbations are input into the model to predict the transient responses. Chapter 4. Parameter Estimation And Data Analysis Time (hr) Saline, 100 mL N=4 Albumin, 100 mL N=4 Saline, 200 mL N=4 Albumin, 200 mL N=4 Mean SD + Mean SD ± Mean SD ± Mean SD ± -1.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 1.80 38.10 11.04 41.82 5.00 19.75 13.25 18.41 3 0.59 38.25 10.30 45.62 4.50 17.37 11.88 14.79 68 6 3.52 39.17 11.66 45.10 5.00 16.25 7.85 15.61 9 7.41 39.99 10.82 43.14 3.38 18.41 8.72 14.84 12 6.56 39.35 10.52 43.34 5.24 17.67 9.74 16.36 Table 4.4: Percentage change in plasma volume at room temperature. t designates the end of the infusion period [41]. Time (hr) Saline, 100 mL N=4 Albumin, 100 mL N=4 Saline, 200 mL N=4 Albumin, 200 mL N=4 Mean SD ± Mean SD ± Mean SD ± Mean SD ± -1.5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 0.04 6.60 9.41 5.54 0.77 6.19 19.06 10.44 3 1.14 7.27 8.31 5.42 3.08 7.48 15.60 9.32 6 0.81 7.55 7.25 7.21 4.10 7.60 12.56 7.99 9 3.56 6.16 7.67 5.52 4.32 9.15 10.38 8.29 = 0 12 3.64 6.29 6.57 6.12 5.51 9.88 11.58 8.25 Table 4.5: Percentage change in plasma albumin content at room temperature. t = 0 designates the end of the infusion period [41]. Chapter 4. Parameter Estimation And Data Analysis However, because QPL independent. Only VFL VPL and x CPL, CPL, 69 the results presented in the three tables are not the measured quantities, are selected to be compared with the model predictions. 4.3.1.2 Set B: Acute Saline Infusion To study the effect of rapid intravenous infusion of physiologic saline solution on the pulmonary arterial and capillary pressure of otherwise normal human subjects, Doyle et al. [18] studied twelve adult men convalescing from noncardiac ailments. In each case, 900 — 1000 mL of normal saline solution were injected intravenously in 6.5 — 13 mm. Blood volumes (BV) and hematocrits (HCT) were measured both pre-infusion and post-infusion. Plasma volumes were then calculated from VPL = BV. (1 — HCT). In some patients, e.g. Patients #4, #8 and #10, the plasma volume kept increasing after the infusion, (which seems physiotogically unreasonable); while, in some other patients, e.g. Patients #3, #5, #9 and #12, the plasma volume response to the infusion was uncertain (i.e., only two points were measured). These data sets were therefore eliminated from further consideration, leaving only five sets to be used in the parameter estimation. These data, in terms of their absolute values as well as percent changes from normal conditions, are presented in Table 4.6, along with estimates of their standard error. Unfortunately, no replicated measurements were carried out; hence, approximations of experimental errors are used. The estimate of the standard error (SE) is based on the assumptions that an error of ±316 mL/m 2 of body surface area was associated with the measurements of blood volume [18] and that the hematocrit readings were accurate. The calculation of the estimated standard error is illustrated in Appendix B. Chapter 4. Parameter Estimation And Data Analysis Patient 1 2 6 7 11 # Remarks (rnL) Control l000mL/9.5mins 38 mm. after infusion Control l000mL/llmins 60 mm. after infusion l000mL/l3mins 15 mm. after infusion Control l000mL/llmins 30 mm. after infusion Control 900mL/9rnins 40 mm. after infusion 70 VPL AVPL 2547.4+331.3 3182.5±361.8 2766.0±342.8 3227.9±342.3 4031.2±366.1 3726.9+358.0 2382.2±273.7 2927.2+300.3 2711.2±290.4 2292.8±303.1 2956.9±336.1 2668.9±323.1 2606.0+286.9 3122.3+311.2 2817.2+297.7 0.00 24.93 8.58 0.00 24.89 15.46 0.00 22.88 13.81 0.00 28.96 16.40 0.00 19.81 8.10 SE % 26.01 30.45 27.58 21.21 24.59 23.33 22.98 26.72 25.27 26.44 31.71 29.48 22.02 25.13 23.33 Table 4.6: Percentage change in plasma volume after rapid saline infusion [18]. Chapter 4. Parameter Estimation And Data Analysis VFL t=0 hr (mL) 3200 - t=2.5 hr Percentage change (%) 3541 ±1177 10.66 +36.78 (g/L) 34 ±5 30 +5 -11.76 ±27.68 CPL 71 (g) 108.8 ±16 106.2 ±53 -2.39 +63.07 QPL Table 4.7: Experimental data from Mullins et al. [60]. Infusion starts at t 0, and lasts for 2 hours (N = 111). 4.3.1.3 = Set C: Saline Infusion In this study, Mullins et al. [60] examined 126 patients with multiple (but non-cardiac) diseases. In their experiments, 2 L of normal saline were injected intravenously over a two-hour period. Only 111 patients tolerated the entire dosage. Hemoglobin, hematocrit, total protein and albumin were measured before and after the infusion. Hemoglobin con tent is assumed to remain constant throughout the study. Hence, the percentage change of albumin concentration (ACpL%) and plasma volume (AVpL%) can be calculated (see Appendix B) and compared with the results predicted by the model using the same per turbation. All data in this set (Table 4.7) are quantitatively significant because each data point represents 111 repeat measurements. Also, since and CPL 4.3.1.4 QFL VPL x CPL, only VPL were selected for use in parameter estimation. Set D: Heart Failure Patients suffering different extents of heart failure were examined in two different studies. In Fauchald’s [25] experiment, 13 patients with heart failure were studied, 7 of whom had diuretic-resistant fluid retention with anasarca. Noddeland et al. [63] examined 22 patients who had angina pectoris in another independent experiment. FL 11 and H Chapter 4. Parameter Estimation And Data Analysis 72 were measured in both experiments. Noddeland et al. also measured the interstitial hydrostatic pressure F , and Pc was then estimated as the isogravimet’ric capillary pres 1 sure, Pc,iso . From the filtration equation JF know that only when JF = 0, Pc = = KF• [Pc — Pi — u (HpL — fl)j, we Pc,iso. Because the filtration rate is relatively low under normal condition, thus, the approximation of Pc,iso as Pc seems reasonable. They found a linear relationship between P’ and the right atrial pressure (RAP), i.e. Pc = 0.62RAP + 6.8. Pc was not measured in Fauchald’s experiment, therefore, this relation is used to estimate the capillary hydrostatic pressure Pc in his patients. VPL and ECV were also measured; accordingly, Vj was calculated from the difference between ECV and VPL. The results are listed in Table 4.8. The mean value of each group is used, so these data points are also quantitatively significant. Note that the pressures are normalized arithmetically with respect to the values for reference man, i.e., Hi,j,NORMALIZED = Hi,j + (Ho — Hj,o), (4.24) where the subscripts i and j indicate, respectively, the specific data point and the data set undergoing normalization, and H represents the normal colloid osmotic pressure of reference man. Also the fluid volumes are proportionally normalized with respect to the value for reference man, i.e., Vi,j,NORMALIZED = x , (4.25) where Vo represents the normal fluid volume of reference man. Physiologically speaking, during the early stage of heart failure, the MVES still be haves normally [105]. The most pathological condition studied in heart failure as part of this work is for the condition of anasarca. It is believed that at this stage of heart failure, ‘Isogravimetric capillary pressure equals the total pressure opposing net filtration of fluid from the capillaries, i.e. Pc,iso = P 1 + o (IIPL Hi). It is not a measure of the actual capillary pressure, in principle. — Chapter 4. Parameter Estimation And Data Analysis mmHg Hi mmHg Pc mmllg 1 mL V SD /V?7o SD PL 11 Anasarca (N=7) 20.3 8.1 20.1 13120 17340 56.2 206.4 Heart Failure (N=13) 23.4 10.4 16.91 12040 15420 43.3 183.6 Angina Pectoris (N=22) 24.0 12.2 11.45 7450 6750 -11.4 80.4 73 Normal 25.9 14.7 10.0 8400 — — — Table 4.8: Experimental data at steady-state for patients with heart failure. patients begin losing dry tissue. Consequently, the protein content in the interstitium will be severely lowered. As well, the tissue structure will change [57]. Patients in this condition are not considered to have normal MVES parameters and are therefore not considered in this study. Thus, the corresponding data point (see Table 4.8) has been eliminated from further consideration. In the parameter estimation procedure, PL, 11 H, and Pc are fixed at the values tabulated in Table 4.8, because these values are thought to characterize the different extents of heart failure. Thus, the only Starling force component which can be altered is the interstitial hydrostatic pressure. Parameter estimation in this case is based on the differences between experimental and predicted /V 1 values. 4.3.1.5 Set E: Nephrotic Syndrome The data from chronic nephrotic patients which were used in Chapple’s [13] study are also included in the current investigation. The nephrotic syndrome is characterized by proteinuria (i.e. excessive urinary excretion of plasma proteins) sufficient to induce hy poalbuminemia (i.e., a plasma albumin content below that of normal) and edema. Ex periments show that patients with nephrotic syndrome exhibit a normal microvascular Chapter 4. Parameter Estimation And Data Analysis 74 exchange behaviour, which is altered by changes only in the Starling forces [26, 91, 58]. The kidney allows protein loss but the rest of the system otherwise behaves normally. Koomalls [46], Fadnes [24], Noddeland [64] and Fauchald [27] have studied the tran scapillary forces and fluid distributions in chronic nephrotic patients. All of them reported data relating H to FL 11 Vj at different (see Fig. 4.1, upper panel). All of the data are normalized following PL 11 (see Fig. 4.1, lower panel), but only Fadnes et al. [24] measured the same procedure as that used in Chapple’s study [13], i.e., pressures (LI 1 and IIFL) are normalized arithmetically with respect to the normal steady-state values of reference man (Eq. 4.24); while the fluid volumes are normalized proportionally (Eq. 4.25). In the parameter estimation procedure, the circulatory compartment is assumed to behave as an infinite source/sink of both fluid and proteins and the interstitium is allowed to change its fluid and albumin contents to attain the steady-state corresponding to each perturbed 4.3.1.6 PL 11 value. Set F: Saline infusion before extracorporeal circulation Before 13 patients with coronary artery disease were operated on using extracorporeal circulation, Rein et al. [80] injected intravenously 1500 - 2000 mL of Ringer’s acetate. was measured subcutaneously on the chest using the blister suction method and ll represents the average colloid osmotic pressure during the suction period (1.5 hrs). PL 11 was measured in a blood sample collected from a cubital vein. P 1 was measured as well, but was not used in the parameter estimation, because, after investigation, we found that the error in P 1 controlled the fit. The reason for this is that Pj has the low negative baseline value of —0.7 mmHg in the model. Using the best-fit parameters of tissue compliance relationship #3 (which will be discussed in Chapter 5), the injection of 1750 mL Ringer’s acetate will cause an elevation in P 1 of around 1.63 mmHg. This corresponds to a AP % 1 = 1.63/(—0.7) = —233%. According to Eq. 4.26, this point will Chapter 4. Parameter Estimation And Data Analysis .75 50.0- 37.5- 25.0- •. 12.5 •• .‘ t 0.0 0.0 5.0 10.0 15.0 20.0 25.0 3( 1T(mmHg) - 20.016.012.0- .. $ 8.0- . a a 4.0- . 3.•a • • 0.0 5.0 •. • •. I I I I 10.0 15.0 20.0 25.0 30.0 (mmHg) Figure 4.1: Normalized data for patients with nephrotic syndrome [13]. The upper panel has 18 points and the lower panel has 66 points, all of which are included in the parameter estimation procedure. Chapter 4. Parameter Estimation And Data Analysis “FL t=0 hr t=3 hr Percentage change (%) (mmHg) 24.1 ±1.1 17.5 ±1.2 -27.4 ±8.3 76 ll (mmHg) 14.7 ±0.9 13.7 ±1.2 -6.8 ±13.9 Table 4.9: Experimental data from Rein et al. [80]. Blister suction method was used to measure Hi, hence, All 1 represents the average osmotic pressure during the suction period (1.5 hrs). contribute to the objective function by 5843 units. Compared with the OBJmjn of 74.57 units (from Table 5.1), we can see that this point will dominate the shape of the surface plot if it is included in the parameter estimation procedure. Hence, it was eliminated in the current study. The useful results are tabulated in Table 4.9. Although the MVES apparently behaves normally for these patients convalescent from non-cardiac diseases, we suspect that the blister fluid does not represent the true interstitial fluid. Therefore, we eliminated data set F from the parameter estimation procedure. 4.3.2 Parameter Estimation Strategy First, we analyse the suitability of the experimental data for the parameter estimation task at hand. Except for Set D and Set E, all the other experimental data sets use at least a 900 mL fluid infusion within a short period. Now, let us discuss how the system will likely be affected by 1000 mL fluid infusion. Assume that the 1000 mL of fluid is quickly redistributed to the interstitium after it is injected intravenously, and that the plasma volume is therefore essentially unchanged. This is the typical physiological response of the MVES to fluid infusion. According to the model, an extra 1000 mL of fluid in the interstitium will increase Pj from -0.7 to 0.86 mmHg, i.e. AP 1.5 mmHg. With LS Chapter 4. Parameter Estimation And Data Analysis 77 ranging from 0 to 300 mL.mmHg’•h’, the lymph flow, therefore, could increase up to 450 mL•h 1 above the normal level. Thus, the lymph flow sensitivity will affect the MVES lymph flow return rate moderately when the system is subjected to a perturbation at this value of fluid infusion. For an albumin infusion, 25 g of albumin in plasma will increase the plasma colloid osmotic pressure by approximately 5 mmHg, while 50 g will cause approximately 10 mmHg. Since JF = KF• (P — Pj — g. (IIPL — PL 11 to increase by Hi)), we know that a will influence the MVES filtration rate and, consequently, the plasma volume, plasma colloid osmotic pressure, etc., which are the quantities being compared between the model predictions and experimental results. This is only a simplified analysis; in the real model, many factors will be affected by these perturbations. Nevertheless, we can still conclude that the experimental data do provide the information about the parameters under study andean be used to estimate the parameters that we-are interested in. Second, we will discuss how these different data sets may be combined in the param eter estimation procedure. We have to keep in mind that, when these different data sets are combined, it is done so on the assumption that the transport parameters for these different groups of people are similar and independent of age, sex and degree of health. To combine these different data sets, the first question that must be asked is how to make the different measured quantities comparable. The solution to this issue, that is, the use of percentage changes, was discussed at the beginning of this section. The second question is how to evaluate the importance of each data point within each set. To solve this problem, each point was assigned a weight factor, W. If the point represents an individual value, then W = 1/SD. This means that points which have a high degree of uncertainty associated with them (i.e. high SD) are weighted less (i.e. small 147). If the point represents the average value of a measurement repeated n times, then W = n/SD, i.e., the standard error of the mean (SE = SD//) is used Chapter 4. Parameter Estimation And Data Analysis 78 to calculate the weight factor. This implies that if an average value is used, that point is quantitatively more important, and n times more weight should be given to that point. In addition, because most of the data sets concern transient responses, the model predictions must be compared with the experimental data point at that specific time. As a convenience, the vertical distance instead of the shortest distance (which was used in Chapple’s studies) between the prediction and measurement was selected for parameter estimation purposes. 4.4 Parameter Estimation Procedure To obtain the best-fit values of u, LS, and Pc,o, the weighted least squares fitting criterion is selected. Thus, the parameter estimation procedure is based on finding the values of the unknown parameters which will minimize an objective function, OBJ, formed by summing, over all data points, the squares of the vertical distances between the measured and predicted percent changes, i.e. wi,j(Ax%epij OBJ = — j 2 X%simjj) © given LS, a, Pc,o (4.26) i=1 j=1 where N denotes the number of data sets and M denotes the number of data points in the ith data set. The weight for each data point reflects the importance or accuracy of that point. The objective function is calculated iteratively inside the limits on a, LS, and Pc,o, and those values which produce the minimum objective function (OBJmin) are the best-fit parameters. Chapter 4. Parameter Estimation And Data Analysis 4.5 4.5.1 79 Numerical Methods and Computer Programs Transient Solutions To find the transient response of the MVES after a perturbation, the set of four simul taneous first—order nonlinear ordinary differential equations listed in Table 3.3 must be solved. These equations represent mass balances of fluid and proteins in both the in terstitial and the circulatory compartments. They are integrated over time using the Runge-Kutta-Fehlberg method with error control [40]. The advantage of employing this method is that the local time step size is adjusted (i.e. in sharply curved regions the step size will be smaller, while in slowly-changing regions it will be larger) so that the cumu lative error over the entire time interval can be maintained below a prespecified value. This saves computational time and costs. In the present study, the maximum allowable error is set to 0.01 mL and 0.01 g for fluid volume and protein content, respectively. 4.5.2 Steady-state Solutions The steady-state solutions are obtained by solving the set of simultaneous nonlinear algebraic equations that result when the accumulation terms of the ordinary differen tial equations are dropped. The numerical technique employed to accomplish this task is Newton’s iterative method [11]. In this method, the non-linear equations are trans formed to a set of linearized equations. The required partial differentials (to calculate the coefficients of the Jacobian matrix) are approximated by finite differences. The resulting linear algebraic equations are solved by Gauss elimination incorporating full pivot selec tion. The maximum allowable error is also set to 0.01 mL and 0.01 g for fluid volume and protein content, respectively. Chapter 4. Parameter Estimation And Data Analysis 4.5.3 80 Computer Programs The optimization routine used in current study is the UBC NLP, which is a nonlinear function optimization program. The parameter values which yield a global minimum in the objective function are obtained by invoking the subroutine GRG (in which the generalized reduced gradient method is employed) or subroutine NLPQL (in which a slightly modified version of the quadratic approximation method of Wilson et al. [89] is employed) from the interactive monitor program, NLMON. The monitor program provides an interface to the nonlinear optimizing routines. To use the monitor program, a FORTRAN function subprogram named XDFUNC must be supplied to evaluate the objective function. This function subprogram is listed in Appendix F. Since the current study deals with a constrained optimization problem, a subprogram named XDCONS must also be provided to evaluate the constraint functions; it is also listed in Appendix F. In addition, because the Objective function éan not be expressed analytically, by default, the monitor provides numerical approximations of the first and second partial derivatives using central differencing. The monitor is invoked by issuing the MTS command $RUN FUNC.0 + NA:NLMON where FUNC.0 is a file containing the compiled versions of the aforementioned subpro grams (i.e. XDFUNC and XDCONS). The monitor takes care of the difference in the calling sequences among the different routines. The global optimum is found by rerun ning the program using several different starting points. If all runs return the same point as the optimum, it is assumed that the optimum solution is the global one. Upper and lower bounds are set as the search ranges of the parameters. The step size used in the calculation of numerical derivative (DELTA) is set to l.D—3. Besides the minimum of the objective function, the monitor program can also provide Chapter 4. Parameter Estimation And Data Analysis 81 important statistical information about the best-fit parameters, such as covariance and confidence interval. Some basic concepts related to the statistical analysis are explained in Appendix C. For detailed information on how to use the monitor program, refer to the manual of UBC NLP. Appendix F also includes the routine that simulates the transient response of the uncoupled Starling model to a specified perturbation according to Koomans’ experiment and also determines the final steady-state conditions of the system following the distur bance. Chapter 5 Results And Discussion 5.1 Introduction Now that the mathematical model has been formulated (Chapter 3) and the parameter estimation procedure has been established (Chapter 4), in this chapter, the best-fit pa rameters for each independent data set will be presented separately. Following that, the possibility of combining these different data sets will be investigated. Then the best-fit parameters for the combined data set (including Sets A — E) for the coupled Starling model determined by employing the least squares method will be presented. With these statistically determined parameters, the model is fully described and is used to simulate variolls experimental situations. The simulations are compared with the experimental results in terms of both the trends and the fit. Because statistical fitting between the experimental data and the model predictions during parameter estimation is one of the major characteristics of the current work, a considerable amount of attention will be devoted to the parameter estimation procedures in the following discussion. Statistical analyses are included in the discussion to evaluate the reliability of the estimated parameters. Sensitivity analyses are carried out to inves tigate how the transport parameters influence the objective function. The correctness of the computer program and its ability to converge to a set of known parameters are also investigated (see Section 5.2.6). Residual analysis is carried out to investigate the distribution of the errors between the predicted and the experimental data. 82 Chapter 5. Results And Discussion 83 Finally, the model is validated by comparing the best-fit transport parameters with available literature values, and by comparing simulation predictions with the measured dynamic behaviour of nephrotics following an albumin infusion [45], a set of data which has not been used for and hence is independent of the parameter estimate procedure. 5.2 Results 5.2.1 Best-fit Parameters for the Coupled Starling Model One of the major goals of the current study was to obtain a set of best-fit mass exchange parameters for humans so that the model becomes fully described and can then be applied to various clinical and experimental situations. Using the estimation procedure described in Chapter 4 and applying it to all 138 data points, three sets of best-fit parameters corresponding to the three different tissue compliance relationships were obtained and are listed in Table 5.1. From the table, we can see that all three compliance relationships give a similar fit to the experimental data, with a maximum difference in OBJmjn of less than 6 units, or about 7%. Clearly, compliance relationship #3 produces the lowest OBJmin and hence will receive the highest degree of interest in the following sections. The best-fit Pc,o values were relatively consistent at about 11.0 mmHg. The best-fit u values increase slightly as the tissue compliance decreases during overhydration, i.e., as the compliance changes from relationship #1 to #3. The best-fit LS values decrease when the tissue compliance decreases during overhydration. 5.2.2 The Feasibility of Combining Different Data Sets As was discussed in Section 4.3.2, the validity of combining data from individual patients having different diseases to obtain a single, comprehensive set of parameters is based on Chapter 5. Results And Discussion Compliance .h’) 1 LS (rnL.mmHg C.I. (L,U) U C.I. (L,U) pc,o (mmHg) C.I. (L,TJ) PS (mL.h’) KF (mL.mmHg’.h’) JL,O (mL.h’) Fe OBJmin 84 # 1 53.04±0.31 (52.43,53.64) 0.9560±0.005 (0.9453,0.9666) 10.95±0.05 (10.86,11.04) 71.68 83.99 77.26 0.0487 79.67 2 52.57+5.27 (42.24,62.90) 0.9761±0.018 (0.9399,1.0000) 10.99±0.17 (10.66,11.33) 72.52 101.08 77.06 0.0254 77.68 # 3 43.08±4.62 (34.03,52.13) 0.9888±0.002 (0.9840,0.9936) 11.00±0.03 (10.95,11.05) 73.01 121.05 75.74 0.0116 74.57 Table 5.1: Best-fit parameters and their confidence intervals, as well as the associated transport coefficients corresponding to the three tissue compli ance relationships. C.I. denotes confidence interval (see Appendix C for its calculation); L and U denote lower and upper limits respectively. the assumption that all the subjects involved in those independent studies have similar microvascular exchange transport coefficients. To test the possibility of combining the data collected for the current study with those from nephrotic patients used by Chapple [13], each data set was analysed separately and the statistical best-fit parameters and their confidence intervals were obtained (Table 5.2). This test must be done before a single, valid set of parameters can be obtained. Because P 0 0 was treated as a discrete parameter during the parameter estimation procedure for nephrotic patients and because Pc,o = 11 mmHg is believed to produce the most reasonable fit to the experimental data [13], therefore, for the sake of convenience, in all cases shown in Table 5.2, tissue compliance relationship #3 with Pc,o = 11 mmHg was employed. In order to visualize the fits corresponding to each set of data, the confidence limits as well as the optimum values for LS and cr listed in Table 5.2 are plotted on Fig. 5.1. In the Chapter 5. Results And Discussion 85 000.0 500.0 0 Set A Hubbard et al.) XSet B Doyle et al.) V Set C Mullin.s et al. )@Set D Heart failure • 400.0 = 300.0- Set E Nephrotic syndrome) Combined data - 200.0- 100.0’ 0.0- I 0.5 0.6 I 0.7 0.6 L:J • 0.9 V 1.0 U Figure 5.1: Best-fit LS and a’, as well as their confidence intervals for individual data sets. The results correspond to tissue compliance relationship #3 with Pc,o = 11 mmflg. Chapter 5. Results And Discussion LS 0.0 232.5 0.0 79.0 42.7 39.9 Set # A B C D E Combined Confidence Interval 0.0 LS 55.4 0.0< LS 600.0 0.0 LS 172.4 0.0< LS 600.0 31.1 LS 54.3 39.5< LS 40.3 86 ci 0.9630 1.0000 0.9855 0.9886 0.9892 0.9829 Confidence Interval 0.9614 ci 0.9646 0.8332< ci 1.0000 0.9517 ci <1.0000 0.5744< ci 1.0000 0.9579 ci 1.0000 0.9765< ci <0.9893 Table 5.2: Best-fit LS and ci, as well as their confidence intervals for individual data sets. The results correspond to tissue compliance relationship #3 with Pc,o = 11. mmHg. “Combined” represents the best- fit parameters obtained when data Sets A, B, C and D are combined. ci direction (i.e., x-direction in the figure), the confidence intervals for these independent data sets all overlap, with the narrowest interval being that of Set A, ranging from 0.9614 to 0.9646, and widest being Set D, ranging from 0.5744 to 1.0. The best-fit ranged from 0.9630 (the best-fit ci of Set A) to 1.0 (the best-fit ci ci values of Set B); the majority were around 0.98. It can be said with certainty that the different subjects investigated in experiments A to E have similar albumin reflection coefficients (ci). In the LS direction (i.e., y-direction in Fig. 5.1), the best-fit LS values ranged from 0 to 232.5 mL.mmHg’.h’. Once again, all the confidence intervals for LS overlap each other to a great extent. Best-fit values for LS for Sets B and D have particularly large confidence intervals, covering the entire range we investigated (0 — 600 1 mL.mmHg . h’). The reason why some data sets are not sensitive to LS may be explained as follows. In the model, the lymph flow sensitivity affects the MVES by altering the lymph flow rate: i.e., JL = JL,O 1 + LS (P — F ) 0 , 1 . That the objective function is not sensitive to LS may result from the fact that, in Sets B and D, AP 1 is always close to zero. That is, the experimental data depict very small changes in V . These result in very small changes in P 1 1 so that JL does not change significantly. However, this is not the situation in most experiments Chapter 5. Results And Discussion 87 as discussed in Section 4.3.2, i.e., by approximation, 1 L of infused saline may cause an increase ill interstitial hydrostatic pressure of about 1.5 mmHg. In real systems, the lack of sensitivity of the LS parameter estimation may also relate to the time dependence of lymph flow (personal communication with R.K. Reed.). Unfortunately, there is no quantitative information about such effects nor have we included delays in lymph flow in the present model. This is perhaps one of its defects. For Sets B and C, data are collected in less than 2.5 hours and we suspect that the lymphatic system had not been fully triggerred during the period investigated. To prove this hypothesis, we separated the data of Set A into two parts; one is the so-called early response corresponding to t 3 hrs post-infusion; the other is the late response corresponding to 9 t < 12 hrs post-infusion. From Fig. 5.2, we found that, in the early response, the objective function is not sensitive to LS, i.e., the contour lines are vertical; while in the late response, the objective function is sensitive to LS. Even though the predicted best-fit LS of the late response equals to zero, Fig. 5.2 B provides much more information about the behaviour of the lymphatic system than does Fig. 5.2 A, suggesting that the lymphatic response has not been fully triggerred at the earlier time. The procedure used for generating the contour plots is detailed in Appendix D. When data Sets A, B, C and D are combined, we found that the best-fit LS and a are 39.9 1 .h and 0.9829, respectively, with 95% confidence intervals of 39.5 mL.mmHg LS 40.3 mLmmHg’.h’ and 0.9765 a 0.9893. It can be seen that the best-fit a and LS values and the confidence intervals for Set E (see Table 5.2) are very similar to those of the combined data set, and the 95% confidence intervals overlap to a large extent. Thus, we conclude that the MVES in uephrotic patients behaves normally and that the nephrotic data can be combined with the other data collected for parameter estimation purposes in the current study. Chapter 5. Results And Discussion 88 0.80 0.85 0.90 SIG Figure 5.2: Contour plots of OBJ during early and late responses for Hubbard’s data [41]. A represents the early response, t3 hrs; B represents the late response, 9 hrst12 hrs. Chapter 5. Results And Discussion 89 It should also be mentioned that the best-fit parameters for the nephrotic data ob tained for the coupled Starling model by applying the shortest distance least squares criterion were [14]: a = 0.996 LS .h mL.mmHg . 39.8 1 These parameters are similar to the best-fit parameters obtained by the current parameter estimation procedure. This agreement adds further to our confidence in the reliability of the current parameter estimates presented in Table s.2. In summary, there is a total number of 138 experimental data points which can be used for parameter estimation purposes. The subjects include healthy males, adult men convalescing from non-cardiac ailments, patients with some degree of heart failure, - -patients-with non-cardiac multiple-system diseases, patients with nephrotic syndrome and normals. All of these subjects are considered to exhibit normal microvascular exchange behaviour. We can therefore say that the current study represents a general investigation of the human microvascular exchange system under normal conditions. 5.2.3 Simulations Using Best-fit Parameters By just looking at the OBJmjn value, it is difficult to tell how well the experimental data are fitted. Does the model predict the correct response trends after the microvascular exchange system is perturbed? Has each independent data set been uniformly well fitted? Or are some sets fitted very well, while others are poorly fitted? The best way to answer these questions is to compare each set of experimental data with the corresponding model simulation results obtained using the best-fit parameters. Chapter 5. Results And Discussion 5.2.3.1 Transient Responses of llpj, 90 CPL and VPL after Saline or Albumin Infusions (Sets A, B and C) To properly compare the model predictions with the experimental data, we must also understand how the MVES responds to the specific perturbations. The circulation system is regulated by three major factors: (1) neural mediators, (2) humoral mediators, and (3) the Starling forces. These three factors work together to make the circulation system self-regulating. There are many delicate pressure receptors and chemoreceptors on the walls of the blood vessels. If these receptors detect any changes in blood pressure and/or chemical composition, the nervous system responds to these changes directly (i.e. by stimulating nerve fibers to control cardiac functions and vessel contraction or relaxation); or indirectly (i.e. by stimulating the endocrine system to release hormones which also exert a profound influence on the blood vessels). However, neural and humoral mediatorsact primarily on the larger vessels of the circulation system. The microvascular exchange system, on the other hand, is locally controlled by conditions in the capillaries and in the tissues in the immediate vicinity of the capillaries. These conditions are generally referred to as the Starling forces. Suppose a certain amount of isotonic solution is injected intravenously. After the injection, the plasma volume expands, and its concentration of high molecular weight solute is diluted. Lundvafl et al. [54] suggested that, in man, there may be particularly efficient mechanisms for maintaining constant plasma volume via fluid transfer between the bloodstream and the large fluid reservoir of skeletal muscle and skin. The increased hydrostatic pressure caused by volume expansion and the decreased osmotic pressure due to solute dilution are considered to be the driving forces for eliminating extra fluid from the bloodstream and consequently maintaining constant plasma volume. After a hyperoncotic solution is injected intravenously, the Starling forces will alter in Chapter 5. Results And Discussion 91 the following manner. The capilliary hydrostatic pressure increases due to the increase in the plasma volume in addition to the capillary colloid osmotic pressure, therefore adding an additional driving force for fluid exchange. As we mentioned before, the circulation system has been shown to be self-regulating. If the capilllary wall is permeable to the solute, the solute molecules will penetrate to the interstitiurn until a balanced osmotic pressures on both sides of the wall is attained. If the capillary wall is not permeable to the solute, water is drawn into the bloodstream, thereby diluting the plasma to achieve the same result. Because the capillary wall is semi-permeable to albumin, both of the aforementioned movements occur when albumin solution is injected. At the same time, water as well as solute are removed from the bloodstream due to the increased capillary hydrostatic pressure. ‘Whether the plasma volume is increased or decreased depends on which effect dominates. Figures 5.3 — 5.6 demonstrate the model predictions of the transient responses of the circulatory system oncotic pressure and volume following saline infusions with or without albumin. The experimental protocols were described in Section 4.3.1. Since the lowest OBJmin value was obtained for compliance relationship #3, the best-fit parameters asso ciated with this compliance were used in generating all of the results shown in Figs. 5.3 — 5.6. Simulations using the parameters obtained for compliance relationships #1 and #2 are included in Appendix E. Figure 5.3 (A) demonstrates the percentage changes in plasma colloid osmotic pres sure and plasma volume with time during and after a saline infusion without albumin. The curves generated by the model correctly represent the response trends of the sys tern and are in good quantitative agreement with the experimental data. These curves show that the circulatory compartment remains relatively stable after the perturbation. Changes in PL 11 and VPL are relatively small compared with the large experimental er ror associated with the data. For a closed circulatory system, a 1500 mL fluid input, if Chapter 5. Results And Discussion 92 (A) ,J,J.iJ - 15.0—5.0 I I— -I- —25.C‘? I —1.5 0.0 1.5 60.0 35.010.0 —15.0— —40.0 I —1.5 0.0 1.5 I 11. I I I 3.0 4.5 3.0 4.5 I I 6.0 I 7.5 I 6.0 I 7.5 I 9.0 I 10.5 V. H I 9.0 1 10.5 Time(hrs) (B) 15.0- I tz —25.0—1.5 0.0 60.0 35.010.0.; —15.0—40.0- I I —1.5 0.0 I 1.5 I 3.0 I 4.5 I 6.0 I 7.5 I 9.0 I •1’ I 1.5 3.0 4.5 6.0 7.5 I F I 9.0 I 10.5 12.0 10.5 I 12.0 Time(hrs) Figure 5.3: (A) Simulation of a 100 mLsaline infusion. (B) Simulation of a 100 mL saline infusion with 25 g of albumin. Fluid intake during waking hours (0 13.5 hr) is 1.4 L Filled circle: experimental data point (data from Hubbard et al. [41]); solid line: model simulation. For the best-fit parameters of tissue compliance #3. — Chapter 5. Results And Discussion 93 all of it were retained in the circulatory compartment, would correspond to a 47% (= 1500/3200) increase in plasma volume. However, due to the self-regulating mechanisms of the circulatory system, the plasma volume increases by only 11.5% at 12 hours postinfusion, corresponding to an actual volume increase of 367 mL. Thus, most of the excess fluid is shifted to the interstitium. Plasma oncotic pressure decreases slightly (6.27%) due to the dilution effect of the infused saline. Figure 5.3 (B) represents the response of the subjects to a hyperoncotic albumin infusion. During the infusion period (-1.5 — 0 hr), the plasma colloid osmotic pressure increases due to the increase in albumin concentration. The negative slope of /H% between 0 — 0.64 hr postinfusion is higher than that after 0.64 hr. The steeper slope is due to the rapid loss of injected albumin from the vascular space [41]. The return of 11 PL to normal afterwards is mainly attributable to the dilution effect of the 1.4 L fluid intake during waking hours [41]. During the infusion period, the increase in plasma colloid osmotic pressure is greater than the increase in capillary hydrostatic pressure; therefore, the plasma volume increases. Figure 5.4 presents the percentage changes in PL 11 and VPL with time when the vol unteers (N=4) are subjected to higher dose infusions, i.e. a 200 mL saline infusion or a 200 mL saline infusion with 50 g albumin (data Set A). The transient responses are similar to those obtained with lower dose infusions. There is no reason for experimen tally measured PL 11 to increase to 2.1 % at 9 hours after the 200 mL saline infusion. This can only be explained in term of the large experimental error associated with the measurement. Overall, the simulation results fit the experimental data reasonably well within the error of measurement. Figure 5.5 (data Set B) illustrates the percentage change in plasma volume which occurs after 900 — 1000 mL of normal saline were injected intravenously within 6.5 — 13 minutes (specific information corresponding to individual patients is listed in Table 4.6). The model predictions are in good agreement with the experimental measurements. For Chapter 5. Results And Discussion 94 (A) 35.0 15.0 1; —5.0 —25.0 —1.5 0.0 .4 1.5 3.0 4.5 6.0 7.5 60.0 35.010.0 —15.0- 9.0 10.5 12.0 T Afl fl I I I —1.5 0.0 1.5 I I 3.0 4.5 6.0 I 7.5 I 9.0 I 10.5 I 12.0 Time(hrs) (B) 60.0 35.010.0-i —15.0—40.0 I I —1.5 00 I 1.5 I 3.0 I 4.5 I 6.0 I 7.5 I 9.0 I 10.5 I 12.0 Time(hrs) Figure 5.4: (A) Simulation of a 200 mL saline infusion. (B) Simulation of a 200 mL saline infusion with 50 g of albumin. Fluid intake during waking hours (0 13.5 hr) is 1.4 L. Filled circle: experimental data point (data from Hubba rd et al. [41]); solid line: model simulation. For the best- fit parameters of tissue compliance #3. — Chapter 5. Results And Discussion 95 patient #1, 1000 mL of normal saline was infused within 9.5 minutes. At the end of the infusion (the peak of the curve), VPL increased by 850 rnL (a 26.6% increase). Hence, at least 150 mL of the injected solution is transported to the interstitium during the infusion period (9.5 minutes). Compared with the experimental measurement, i.e., a 24.9% increase, the agreement is good. The model predicts the average filtration rate during this period is 947 mL/h, which is much higher than the normal filtration rate. At 38 minutes post-infusion, the model predicts a 10.9% increase in plasma volume while a 8.5% increase was measured experimentally. At t=2 hour, the VPL increase is only 224 mL (a 7% increase) and the system appears to be close to steady-state. The 7% increase in VFL is due to the assumption that within these 2 hours, water lost from skin and during respiration, as well urine output are negligible because no such information was presented by the investigators. - Figure 5.6- (data Set C) shows the percentage changes in plasma volume and plasma concentration following a 2 L normal saline infusion within 2 hours. During the infusion period (0 — 2 hour), increase), while VPL CPL is predicted to increase steadily to a maximum of 3929 mL (a 23% decreases steadily to a minimum of 32.4 g/L. Accordingly, expected to decrease because of the linear relationship between Afterwards, both VpL and CPL CFL and PL 11 FL 11 is (Eq. 3.19). tend to return to normal. All these trends match the physiological responses of the MVES after an isotonic solution infusion. The model predictions at half an hour after the infusion match the experimental measurements very well within the large experimental errors in terms of both absolute values and percentage changes. The model predicts that CPL decreases to 34.89 g/L ( VPL increases to 3659 mL (a 14.36% increase), and a 11.45% decrease); experimentally it was found that at 0.5 h post-infusion was 3541±1177 mL and CPL was 30±5 g/L. PL 11 Chapter 5. Results And Discussion 96 Patient #1 20.0 10.0 u.0 0.0 30.0 60.0 90.0 120.0 90.0 120.0 90.0 120.0 90.0 120.0 Patient #2 30.0 20.0 0.0 30.0 60.0 Patient #6 30.0 20.0 - 10.0 0o 0.0 30.0 60.0 Patient #7 30.0 0.0 0.0 30.0 60.0 Patient #11 30.0 20.0 10.0 nfl. 0.0 30.0 60.0 120.0 Time(mins) Figure 5.5: Simulations of acute saline infusion in selected patients [18]. Dot: experimental data point; solid line: model simulation. For the best-fit pa rameters of tissue compliance #. Chapter 5. Results And Discussion 97 70.040.010.0—20.0—50.00.0 1.0 2.0 3.0 4.0 5.0 3.0 4.0 5.0 30.010.0-10.0- F —30.0—50.()- 0.0 10 2.0 Time(hrs) Figure 5.6: Transient responses of plasma volume and plasma albumin con centration after a 2 L of normal saline infusion within 2 hours [60j. Filled circle: experimental data point; solid line: model simulation. For the best-fi t parameters of tissue compliance #3- Chapter 5. Results And Discussion 5.2.3.2 98 Simulations of Heart Failure (Set D) Chronic heart failure is the pathophysiological state in which an abnormality in cardiac function is responsible for the failure of the heart to pump at a rate commensurate with the requirements of the metabolizing tissues. It is frequently caused by a defect in myocardial contraction. As a result of the deficiency of contraction, the volume of blood delivered into the systemic vascular bed is chronically reduced, and a complex sequence of adjustments occurs that ultimately results in the abnormal accumulation of fluid in interstitia in the body. These adjustments include a rise in venous pressure. The increment in venous pressure is transmitted to the capillary level and therefore increases the transcapillary pressure. This will accelerate the transcapillary filtration rate (Eq. 3.2), and water will accumulate in the interstitium. As V 1 increases, Pi will increase and H will decrease. In turn, the rise in Pi and the reduction in llj both act to oppose further increase in JF• In addition, lymph flow increases with the elevation in P . As the falling 1 filtration and rising lymph flow approach each other, a new steady state is approached. The current model predicts that interstitial hydrostatic pressure decreases slightly to -0.85 mmHg for patients with angina pectoris, corresponding to V 1 = to 2.35 mmHg for patients with heart failure, corresponding to Vj 8.3 L, and increases = 17.0 L for tissue compliance #3. In the experiments, P 1 was observed to decrease to 0.6 mmHg lower than control level and Vj was observed to decrease to 7.5 L with a measurement error of +6.8 L for the former [64]. Also, it was reported that during the early stage of heart failure, some patients have reduced cardiac pump function without edema. This coincides with our predicted value which is around the normal range of V . For patients with heart 1 failure, P 1 was not measured and V 1 was observed to be 12.0±15.4 L [25], which contains the range of the predicted value. These basically match the adjustments which occur in heart failure as discussed above. Chapter 5. Results And Discussion 99 In addition, several studies on animal [19, 36] have shown that there exists a criti cal capillary hydrostatic pressure at which extravascular fluid accumulates very rapidly, and the critical pressure is approximately equal to the plasma colloid osmotic pressure. However, since there are only three points in this data set, it is difficult to predict such a critical pressure. But it is certain that when Pc exceeds 17 mmllg, massive edema occurs (see Fig. 5.7). 5.2.3.3 Simulations of V 1 vs. PL 11 and Hj vs. PL 11 in Nephrotic Syndrome (Set E) Interstitial edema in patients with chronic nephrotic syndrome is caused by the patho logical removal by the kidneys of plasma protein from the blood stream which eventually results in hypoproteinemia. Thern current model assumes that the lowered plasma colloid osmotic pressure- is the only cause of edema formation in nephrotic patients. Figure 5.8 was constructed by decreasing PL 11 step by step at a constant plasma volume, and predicting the steady- state interstitial volumes and interstitial colloid osmotic pressures which occur for each new PL 11 value. From this figure, we can see that both Vj and H are well fitted as PL 11 increases. Figure 5.9 shows the steady-state fluid fluxes (i.e. JF and JL) and protein fluxes (i.e. Qs and QL), as well as the albumin contents in both the interstitial and the circulatory compartments. Since the tissue begins dehydrating for HPL > 25.9 mmHg, the lymph flow relationship switches from Eq. 3.7 to Eq. 3.8. Thus, there is an obvious slope change around PL 11 equal to 25.9 mmHg. The predictions shown in Figs. 5.8 and 5.9 are helpful in investigating edema formation and its mechanisms. In the upper panel of Fig. 5.8, it can be seen that as PL 11 decreases from 28 to 12 mmHg, V 1 increases from 7.7 to 11.5 L. In other words, V 1 increases 3.8 L as HPL decreases 16 mmHg. As PL 11 falls, the transcapillary fluid flux and lymph flow both rise to about double their normal levels (see Chapter 5. Results And Discussion 100 22.5 25.0 rrP(mmHg) 25.0 20.0 15.0 10.0 . 5.0 I 7.5 10.0 I 12.5 15.0 17.5 20.0 P(mmHg) Figure 5.7: Simulations of steady-state Vi vs. 11 1 vs. Pc in heart failure PL and V patients [63, 25]. For the best-fit parameters of tissue compliance #3. Chapter 5. Results And Discussion 101 Fig. 5.9). Due to the loss of plasma proteins from the bloodstream (at HPL = 12 mmHg, 58.46 g), the interstitial protein content drops dramatically (at PL = 11 12 mmHg, QFL = Qi = 58.51 g). When PL 11 decreases to 12 mmHg, Pi 1 decreases to 4.1 mmHg. Thus it can be seen that protein washout (i.e., a net reduction in Qi) is the predominant edema preventing mechanism during this period. If HPL decreases further, protein washout can not prevent the further expansion of interstitial fluid volume. According to Fig. 5.8, as PL 11 decreases from 12 to 4 mmHg, the interstitial volume increases from 11.5 L to 43.5 L, which is a very severe state of tissue edema. At = PL = 11 4 mmllg, Qi = 19.5 g and P 1 5.1 mmllg. This shows that protein washout does not occur as extensively as when PL 11 is decreased from 28 to 12 mmHg because most of the protein has already been washed out, and the increase in tissue hydrostatic pressure plays a more important role in preventing further edema formation. However, due to the high tissue compliance, the 1 also fails to prevent the edema formation effectively. Thus, massive edema bcease in P formation takes place. A plasma colloid osmotic pressure of 12 mmHg can be regarded as the critical level below which severe edema occurs. To prevent edema formation, PL 11 must be maintained above this critical value. 5.2.3.4 Summary of the Simulations using Best-fit Parameters Based on the above discussion, it is concluded that the model predicts the correct trends and values after the MVES is perturbed. Each independent data set has been well fitted by using the best-fit parameters. 5.2.4 Sensitivity Analysis Though we have obtained three sets of best-fit parameters, we do not know how the variables (LS, a, and Pc,o) influence the objective function. If a slight deviation from Chapter 5. Results And Discussion 102 50.0- 37.5. 25.0- 12.5 fI 0.0 I 0.0 5.0 --I 10.0 15.0 20.0 25.0 30.0 PL (mmllg) 20.0- 18.0- . 12.0 8.0- .. . a . 4.0I •. 0.0 I 0.0 5.0 I I I 10.0 15.0 20.0 25.0 30.0 PL (mmHg) Figure 5.8: Simulations of Vi vs. IIPL and fl 1 vs. 1 IPL in nephrotic syndrome patients [24, 46, 64, 27]. For the best-fit parameters of tissue compliance #3. Chapter 5. Results And Discussion 103 350.0 280.0 210.0 140.0 70.0 0.0 .0 5.0 10.0 15.0 20.0 26.0 3 I.0 15.0 20.0 Z50 3 3.0 15.0 20.0 26.0 30.0 1.5 1.2 O .6 0.3 0.0 I — I 5.0 10.0 .0 1Ô.O 150.0 120.0 90.0 60.0 30.0 0.0 0.0 1r(mmHg) Figure 5.9: Steady-state effects of graded reduction of plasma oncotic pressure on fluid and protein exchange. For the best-fit parameters of tissue compli ance #3. The solid line in the last panel is the tissue protein content, the dashed line is the protein content of the plasma compartment. Chapter 5. Results And Discussion 104 250. C 50.0 70.0 LS (mi/rn mHg.h) Figure 5.10: Sensitivity analysis for LS. The analysis is conducted for tissue compliance #3, Pc,o 11.00 rnmHg and a = 0.9888. the best-fit value affects the objective function significantly, then the location of this value must be chosen very accurately. The study on whether the optimum fit is sensitive to changes in the fitting variables is called a sensitivity analysis. A sensitivity analysis is helpful in evaluating the degree of reliability of the best-fit parameters. 5.2.4.1 Lymph Flow Sensitivity — LS To analyse the sensitivity of the objective function to LS, Pc,o and a are fixed at their best-fit values, LS is varied on either side of its optimum point, and the corresponding OBJ value is calculated. The results obtained for compliance relationship #3 (i.e. Pc,o 11.00 mmHg, a = 0.9888) with LS varying from 10 110 mL.mrnllg’.lr’ are shown — in Fig. 5.10. The figure demonstrates that raising LS does not affect the fit very much. For example, OBJmtn is equal to 76.50 units when LS is at its best-fit value of 43.08 Chapter 5. Results And Discussion 105 1 mL•mmHg’• . h If LS increases to 60 , 1 mL•mmHg’• h the OBJ rises only to 80.83 units, 4.33 greater than OBJmin. If LS decreases to 20 rnL•mmHg’.h’, the OBJ increases to 115.01 units, 38.51 greater than OBJmim. Thus, it is clear that, if LS = 43.08 .h is not the “true” value of the lymph flow sensitivity, then the “true” 1 mL.mmHg value is “more likely” to be found at a greater value than at a lesser value. This analysis reveals an asymmetry in the objective function and suggests that regions on one side of the best-fit value are more likely than regions oi the other side of to yield the “true” value of the parameter. Such a conclusion can not be reached by simply examining the calculated confidence interval on LS (Table 5.1), which are symmetrically positioned about the bestfit value. ÔOBJ/8LS is fiat for LS ranging from 35 to 50 1 .h This interval mL.mmHg . coincides with the 95% confidence interval on LS (Table 5.1), which is interpretated as the probability of OBJmin falling inside it is 95%. 5.2.4.2 Albumin Reflection Coefficient — a The sensitivity analysis for a is carried out for tissue compliance relationship #3, Pc,o = 11.00 mmHg, LS = 43.08 mL•mmllg’.h , and a varying from 0.8 1 — 1.0. The results are plotted on Fig. 5.11. The objective function decreases from 172.46 to 80.14 as a is increased from 0.8 to 0.96. Then it displays a plateau around the best-fit value of a = 0.9888. This plateau exists over the range 0.96 < a < 1.00 where the OBJ values vary by less than 3.64 units. The optimum a does not appear to be biased towards either direction in the plateau region. 5.2.4.3 Capillary hydrostatic pressure at normal steady-state — Pc,o As it was mentioned in Section 3.3.1, the minimum value of Pc,0, below which the model breaks down, is restricted by the choice of a. With a set to 0.9888, the interval over which the capillary hydrostatic pressure could be investigated was limited to F,ü Chapter 5. Results And Discussion 106 0 0.80 0.85 0.90 0.95 1.00 0• Figure 5.11: Sensitivity analysis for o. The analysis is conducted for tissue. compliance #3, Pc,o = 10.8431 mmHg, and LS = 49.7848 mL•rnmH g’h’. 10.37 mmHg. 10.5 Pc,o Therefore, the interval for the sensitivity analysis was selected to be 15.0 mmHg. From Fig. 5.12, we note that the optimum Fc,o lies between 10.75 to 11.25 mmHg; the objective function increases dramatically when Pc,o is less than 10.75 mmHg or greater than 11.25 mmHg. Thus, even though it is still reason able physiologically for Pc,o to exceed the upper limit of 15 mmflg, a worse fit to the curren t experimental data is definitely obtained when Pc,o 11.5 mmHg. 5.2.5 Péclet Number The Péclet number is a dimensionless mass transfer parameter which describ es the ratio of convective to diffusive exchange. Here, the Péclet number is defined (according to Eq. 3.4) as Pe (1 —o}JF/PS, and it represents the importance of convection compared to diffusion as a mechanism for transcapillary albumin transfer. From Table 5.1, we note Chapter 5. Results And Discussion 107 275. 235.0 195.0 C 155.0 115.U 75.’ 12.0 12.5 13.0 13.5 0 (mmHg) P. Figure 5.12: Sensitivity analysis for F,o. The analysis is conducted for tissue compliance #3, LS = 43.08 mL-rnn11g’•h’ and o- = 0.9888. that the Péclet numbers at steady-state for all three compliance relationships are very low. Therefore, we conclude that, at steady-state, albumin is mainly transpo rted by diffusion. 5.2.6 Verification of Fit This test is used to check the correctness of the computer program and its ability to converge to a set of known parameters [112]. In this test, 138 “error-free” data points are generated by solving the model equations using the best-fit parameter values when the MVES is subjected to the same perturbations as those stated in Section 4.3.1. The data generated in this manner are assumed to have no inherent errors. Then the “error- free” data are inputted into the program as data to be fitted. The transpo rt parameters are assumed to be unknown and the same optimization procedure as that used with the real Chapter 5. Results And Discussion 108 experimental data is followed to find a set of new parameter estimates for these “errorfree” data. Theoretically, if the program is working properly, a set of parameters identical to the best-fit parameters used to generate the “error-free” data will be estimated. Using the best-fit parameters obtained for tissue compliance relationship #3, i.e., LS = 43.08 mLmmHgh’, o = 0.9888 and Pc,o = 11.00 mmHg, a set of results corresponding to all of the quantities which were measured experimentally were predicted by the mathematical model and substituted back into the optimization program. The best-fit parameters obtained in this way were: LS = 43.126 mL.mmHg’.h’ a = 0.98877 Pc,o = 11.000 mmHg with OBJmjn equal to 0.014885. As can be seen by comparing the above set of parameters with those listed in Table 5.1 for this case, the best-fit values are all reproducibleto at least three significant figures. Therefore, we are confident that the parameter estimation procedure is reliable. 5.2.7 Residual Analysis A check of the normal distribution of the errors can be made by constructing a residual plot. Here, the errors refer to the differences between the predicted and the experimental data, or more precisely, the residuals (see Appendix C for the definition of residual). If the residuals are normally distributed, then this plot should not reveal any obvious pattern. Although the fitting parameters obtained by the least squares procedure do not depend on a normal distribution of the errors, the calculation of the confidence interval will depend on this assumption. The true confidence intervals may differ greatly from the calculated values if the normality assumption is not satisfied. Therefore, it is worthwhile checking to determine whether the distribution of the errors is normal. Chapter 5. Results And Discussion 109 3 Xsim/SD Figure 5.13: Residual plot for the best-fit parameters of compliance relation ship X and X refer to the simulation value and the experimental measurement, respectively. A residual plot was constructed for the best-fit parameters of tissue compliance # 3 and is shown in Fig. 5.13. Here, the x coordinate is Xsjm/SD (unitless), i.e., the simulation value of the coresponding measurement divided by the standard deviation, and the y coordinate is (Xsim — Xexp)/SD (unitless), i.e., the standardized residual. From the figure, it is shown that 60 of the 138 standardized residuals are negative and 78 are positive. The residuals scatter at random around the zero line and no trends are observed. Thus, it appears that the residuals are normally distributed. The residual plot is also useful for highlighting major departures in the observed values of the data from anticipated patterns. Here, no pattern of systematic departure of the points around the zero line is evident. Thus, the residual plot in Fig. 5.13 suggests that each experimental data point is properly weighted. Chapter 5. Results And Discussion 110 Fi (mmHg) +2.0 +1.0 +1.0 +0.5 JL (mL/h ) 2.8 1.4 0.6 0.6 Table 5.3: Experimental data of interstitial hydrostatic pressures vs. lymph flow in the leg superficial lymphatics [98]. L denotes the lymph flowrate in the leg superficial lymphatics. 5.3 Validation of the Best-fit Parameters The best-fit parameters listed in Table 5.1 are calculated on the basis of a statistical fitting between the model predictions and the experimental measurements. In this section, we try to compare these best-fit parameters with estimated values available from the literature. Additionally, a comparison between model predictions and experimental data which have not been used in the optimization would also be very helpful to validate these best-fit parameters. Unfortunately, no other new data have been found by the author so far. Thus, Koomans’s data which were used for the same purpose in the previous study [13] are also used in the current study to make such a comparison. 5.3.1 Lymph Flow Sensitivity — LS To the knowledge of the author, no one has experimentally determined the lymph flow sensitivity for humans which is defined as aforementioned. Stranden et al. [98] found that there was a significant correlation between lymph flow and interstitial hydrostatic pressure. The data from their experiments on patients with local leg edema is proposed by the author for a comparison with LS as estimated in this work. The data are listed in Table 5.3. Note that the lymph flow rate measured in the experiment (denoted by jL) is Chapter 5. Results And Discussion LSL LSL TD TD Lymph flowrate (mL/h) 0.25 0.34 84.0 70.8 111 Source [67] [68] [20] [33] Table 5.4: Lymph flow rates in the leg superficial lymphatics and in the thoracic duct. LSL denotes the leg superficial lymphatics; TD denotes the thoracic duct. that in the leg superficial lymphatics, which have diameters in the range of 0.1 But lymph flow rate .JL 2 (JL) 0.4 mm. in Eq. 3.7 refers to the lumped whole body flow rate. Therefore must be converted to a total lymph flow in the thoracic duct which has a diameter of — 3 mm in the neck region (see later discussion). Table 5.4 shows some measurements of lymph flow in the leg superficial lymphatics (LSL) and in the thoracic duct (TD). If average values are used, the lymph flow rate in the thorac1cduct is around 260 77.4/0.295) times higher than that in the leg superficial lymphatics. Scaling LS is found to be 400 mL.mmHg’.h’ by linear fitting between JL and (P 1 L — by 260, F ) 0 , 1 . If the scale factor is arbitrarily altered to 100 (because of varying topography and caliber of the cannulated vessels), LS is found to be 154 1 mL.mmHg . h’ (see Fig. 5.14). In both cases, normal lymph flows are found to be negative, which is unreasonable and conflicts with the assumption in the current model and with physiological evidence. This might be due to a discrepancy in P 0 between the reference man and the patients involved in , 1 Stranden’s experiments. However, mathematically, this discrepancy does not affect the slope of the straight line, i.e. LS. Compared with the LS value estimated by the above procedure, the lymph flow sensitivity predicted by the model (which ranges from 43.08 to 53.04 mL.mmHg’.h ) 1 tends to be underestimated. The extent of underestimation is difficult to evaluate because Chapter 5. Results And Discussion 112 800.0600.0400.00 200.0- -o 0.0—200.0—400.0 I I —1.0—0.5 0.0 I 0.5 1.0 1.5 2.5 2.0 Pi(mmHg) Figure 5.14: Plot of 1 . Filled circles: experimental data obtained by 1 L vs. F scaling iL (Table 5.3) by 260; open circles: obtained using scale factor of 100; solid line: 1 —378 + 400 x (Fi P ,o); dashed line: JL = —145 + 154 x (F 1 L 1 —F ). 0 , 1 — the experimental data are too few and too scattered. Another quantitative criterion was also applied to validate the value of LS. This criterion is the ratio of maximal to basal lymph flowrate, JLRMB (i.e., The maximum lymph flowrate is assumed to occur at a cutoff of Vj = JL,MAX/JL,0). 20 L [23]; thus, according to the tissue compliance relationships, F 1 is calculated to be between 2.01 and 2.66 mmHg. If the estimated basal lymph flowrate and lymph flow sensitivity are assumed to be correct (see Table 5.1), according to JL = JL,O 1 + LS(F — ,F 1 o), JL,MAX is estimated to range between 220.5 and 232.1 mL/h for the different tissue compliance relationships. Thus, predicted values of JEMB range between 3.0 and 5.1. The experimentally observed values of JLRMB in animal are 5 to 10 [2]. Comparison between the values of predicted and experimental JLRMB suggest that the model predicted LS values are close to the lower bound of the experimentally determined LS. Chapter 5. Results And Discussion 5.3.2 113 Albumin Reflection Coefficient — a Experimental estimations of the reflection coefficient (a) have been based either on the pore estimation method [42] or on steady-state lymph flow analyses. Renkin [82] esti mated that small pores having an inner diameter of 40 A have a reflection coefficient for albumin of 0.95, and large pores have a a value of 0.45. Reed et al. [78] estimated that small pores of 45 A, 50 A, 60 A, and 80 A in radius have a values of 0.966, 0.919, 0.802 and 0.588, respectively. Pore dimensions estimated by many investigators indicate small pore populations with radii 40 — 50 A in subcutaneous tissue and 60 muscle; large pore populations have a a radius of around 200 A A in skeletal [73]. The number ratio of large to small pores is around 1:3500. Therefore, the whole body capillary membrane a for albumin of 0.936 estimated by Reed et al. [78] seems reasonable. The steady-state lymphatic analysis used to estimate a is based on the linear protein flux equations (Eq. 3.6) or the nonlinear equation (Eq. 3.5). Lymph ff6w as well as protein concentrations in lymph and plasma are measured at two or more steady lymph flow states produced by elevating venous pressure. The estimation of a is in reality a problem of solving two equations with two unknowns, a and PS. Rutili et al. [88] estimated a values ranging from 0.85 to 0.95 for total plasma protein in dog paw using different mathematical formulations. Renkin et al. [81] reported a values ranging from 0.98 to >0.99 based on their recent study on rat skin and muscle. Reflection coefficients estimated by both methods are in good agreement with the values obtained by statistical fitting in current study. It is concluded that the albumin reflection coefficient is close to unity. Chapter 5. Results And Discussion Species Dog Dog Cat Rabbit Human Human Method LA LA ID ID ID ID 114 PS (mL/h/lOOg wet tissue) 0.707 0.593 4.530 4.434 0.467 2.760 PS (mL/h) 269.36 225.93 1725.93 1689.35 177.93 1051.56 Source [82] [88] [92] [69] [33] [70] Table 5.5: Experimental estimates of PS values. The PS values were normal ized per 100 g wet tissue for comparison. LA denotes lymphatic analysis; ID denotes indicator dilution. Values in the fourth column are calculated by assuming that skin and skeletal muscle for the whole body weigh 10.1 kg and 28.0 kg, respectively. 5.3.3 Permeability-Surface Area Product — PS The permeability-surface area product (PS) is the most widely used index to describe the diffusive characteristics across capillary wall. The penneabilities vary for different solutes. The PS value used in the present study refers to that for albumin. Experimental estimates of the permeability-surface area product are typically ob tained by one of two methods. One is the steady-state protein flux analysis which was mentioned in the previous section. The other is the indicator dilution method, or sin gle injection, residue detection method. In the latter method, a single bolus of labelled albumin is injected intra-arterially and blood samples are collected at time intervals to analyse their radioactivity. An indicator dilution curve is then constructed to estimate the permeability-surface area product. The experimental estimates of PS available in the literature are quite controversial, especially those obtained by the indicator dilution method (Table 5.5). For the estimated skeletal muscle and skin for the whole body, weighing 28.0 kg and 10.1 kg respectively, the total whole body permeability-surface area product ranges from 178 to 1725.93 mL/h. Chapter 5. Results And Discussion 115 Paaske et al. [70, 92] pointed out in their reports that the indicator dilution PS values for both human and animal tissues were 3 to 10 times higher than those obtained by other methods [88, 82]. However, no explanation for this overestimation was given. Compared to the values listed in Table 5.1, which are close to 70 rnL/h, it seems that the PS estimated in the current study is about two times lower than expected, if human tissues have permeability coefficients similar to animal tissues. But, it should be kept in mind that the physiological measurements of PS are highly uncertain. 5.3.4 Fluid Filtration Coefficient — Experimental determinations of the capillary filtration coefficient (KF) in human have been limited to two methods: the venous occlusion method [57, 91, 54, 52] and the osmotic transient method [71, 44]. Both methods are developed based on the Starling equation, he., - JF=KF[Pc—PI—o(HPL—f11)]. The venous occlusion method is based on the assumption that a change in Pc is the only source for the change in fluid filtration rate (JF) when venous pressure is raised or lowered, at least at times near the perturbation. The osmotic transient method is based on the assumption that a change in PL 11 is the change in driving force which dominates JF variations after the injection of a hyperoncotic albumin solution. The transcapillary filtration rate is determined with the aid of a well-balanced volume piston recorder connected to the plethysmograph which can record the changes in tissue volume. Plasma volume is assumed to remain constant. The capillary filtration coefficient is calculated as the filtration rate divided by the concomitant pressure change. Table 5.6 presents the experimentally determined KF values for mixed tissue (skin and skeletal muscle) in human. The table contains a wide range of values. Most investigators Chapter 5. Results And Discussion KF (mL/(min.mmHg.100g soft tissue)) 0.0012 0.0036* 0.0046k 0.0058 0.0077 0.054 116 KF (mL/(mmllg.h)) 27.43 82.30 105.16 132.59 176.02 1234.44 Reference [91] [71] [44] [52] [57] [54] Table 5.6: Experimentally determined KF values. All these measurements were made in human by the venous occlusion method except * were measured by the osmotic transient method. Values in the second column are calculated by assuming that skin and skeletal muscle for the whole body weigh 10.1 kg and 28.0 kg, respectively. report a KF value of around 0.005 mL/(min.mmHg•lOOg tissue) by using the venous occlusion method with increases in venous pressure of 30 — 60 mmHg. However, Lundvall et al. [54] suggested that such high increases of venous pressure may lead to closure of the precapillary sphincters, thereby reducing the capillary surface area available for transcapillary exchanges. Also, it may cause large decreases in regional blood flow. If these postulates are tenable, they will cause an underestimation of the experimental KF values. Lundvall et al. reported a ten times higher KF value (0.054 mL/(min.mmHg.100 g tissue)) by increasing venous pressure by only 1.6 mmHg. A similar magnitude of KF was also estimated by Chapple [13] based only on fitting data from nephrotic patients. Whether such a high KF value is reasonable remains to be further confirmed. The KF values estimated in the current study, ranging from 83.99 to 121.05 mL.mmHg’.h’, are well within the range reported by most investigators. 5.3.5 Normal Lymph Flow — JL,O Direct measurements of lymph flow in human have been made mostly either in the leg lymphatics or the thoracic duct. Lymph flow rate in the leg superficial lymphatics is Chapter 5. Results And Discussion 117 simply measured by determining the volume of lymph collected in a calibrated syringe connected to the cannula during a measured period of time. Lymph flow is found to occur only simultaneously with the lymph pulse [67]. The normal lymph flow rate (JL,o) in the current model refers to the lumped whole body flow rate. Therefore, comparisons should only be made between the estimated JL,O and the lymph flowrate measured in the thoracic duct. Measured thoracic duct flows are of the order of 1 — 3 mL/min, or 60 — 180 mL/h [75]. The normal lymph flowrates predicted in the current study are within the lower side of the normal range at about 78 mL/h for all tissue compliance relationships. 5.4 Simulations of a Single Intravenous Infusion of Human Albumin In Section 5.2.3, model simulations of transient responses of the microvascular exchange - system subject to salineor albumin infusion, and simulat-ions of heart failure and nephrotic syndrome were presented. All these data have been used in the parameter estimation procedure; therefore, it is not surprising that these data are well represented by the model predictions. To test further that the parameters obtained based on the aforementioned data are generally applicable to simulations of a microvascular exchange system which exhibits normal behavior, a comparison was made between the simulation predictions and the dynamic results from an experimental study by Koomans et al. [45]. The data from Koomans’ study have not been used in the parameter estimation procedure. Hence, such a comparison is considered to be a partial validation of the model and its estimated transport parameters. Koomans’ experiment was designed to study the fate of a single intravenous infusion of human albumin in 10 patients with nephrotic syndrome. In their experiment, 60 g of human albumin in 300 mL of solution were infused continuously over a period of 1.5 Chapter 5. Results And Discussion 118 hours. Plasma volume, and colloid osmotic pressure in both interstitium and plasma were measured before and immediately after the albumin infusion, as well as at 1 and 24 hour post-infusion. Urinary albumin loss was observed to be 10.5 g/day before infusion and 26.4 g/day during the first day after infusion. Accordingly, the excess albumin excretion rate was 15.9 g/day. The 300 mL of infused fluid was observed to be excreted within the first 24 hours post-infusion. In addition, patients were reported to possess an average pre-infusion interstitial fluid volume of 18.25 L. Plasma volume, measured experimen tally, was very close to normal. Plasma and interstitial colloid osmotic pressures were determined from the steady-state simulations of nephrotic patients (see Fig. 5.8), which were 10.574 and 3.304 mmHg, respectively. Thus, the corresponding albumin contents in both circulatory and interstitial compartments could be calculated (albumin content = volume x albumin concentration, and albumin concentration can be determined from the colloid osmotic pressure relationship, see Eqs. 3.19 and 3.21). Starting with these initial conditions, the model predictions of the transient responses when the system was subjected to the same perturbations as the experimental counterpart, are presented in Fig. 5.15. Figure 5.15 shows that the model predictions are in good agreement with the experimental data in terms of both absolute values and trends. Both PL 11 and VpL follow a uniform pattern of initial increase and subsequent decrease which starts immedi ately after infusion. The pattern bf change in the interstitial fluid evidently differs from that in the plasma: after a slight initial increase, Il rises further during the 24 hours post-infusion. All these trends are consistent with those reported by the experimenters [45]. The bottom panel shows the distribution of albumin during and after infusion. The continuous accumulation of interstitial albumin mass during the 24 hours post-infusion proves that part of the infused albumin disappears into the interstitium where it causes an increase in tissue fluid oncotic pressure [45]. Finally, the good agreement between the model predictions and the experimental Chapter 5. Results And Discussion 119 25.0 S 4 20.015.0 10.05.0 - I 12.0 I 24.0 6.05.04.03.02.01.0D.C I 36.0 48.0 36.0 48.0 I 12.0 24.0 150.0 100.0a C’ QPL 50.0- QI 0.0- I 0.0 12.0 I 24.0 I 36.0 43.0 Time(hrs) Figure 5.15: Transient responses of UPL, [1, VPL as well as QPL and Qi after single infusion of 60 g of human albumin solution [45]. Solid lines represe nt the dynamic responses of the model. Dotted lines represent the steady -state values obtained by running the program for a long time. Experi mental data are shown as filled circles with error bars. Chapter 5. Results And Discussion 120 data in the form of absolilte values shows the suitability of the estimator we chose (Eq. 4.26). The best-fit transport parameters, which were estimated by minimizing the sum of squares of the differences between model predictions and experimental data in the form of percentage changes, are also applicable for predicting the absolute values. This, in turn, shows that the initial values of the reference man (see Table 3.2) are representative of those of the majority of human beings. Chapter 6 Conclusions Based on the discussion in the previous chapter, it is concluded that the coupled Starling model is capable of providing good descriptions of microvascular exchange and fluid and albumin distribution in the human. According to statistical fitting results, the best-fit to the available experimental information is obtained with tissue compliance relationship #3 and the transport parameters LS Pc,o = = 11.00±0.03 mmHg, PS = = 43.08 + 4.62 1 mL•mmHg h ’, a 73.01 mL•h’, KF = = 0.9888 ± 0.002, 121.05 mL• mmHg’.h , and 1 JL,O 75.74 mLh . Simulations of the available experimental data using these parameters 1 gave a reasonable fit in terms of both trends and absolute values. All of the best-fit parameter values so obtained were in reasonable agreement with estimated values based on experimental measurements where comparisons with literature data were possible. The fact that the albumin reflection coefficient is near unity indicates that transcapil lary osmotic forces are very similar in magnitude as the transcapillary hydrostatic forces. Therefore, osmotic effects play an important role in transcapillary exchange. In addition, the low values of the Péclet number for all three compliance relationships indicate that diffusion is the dominant mechanism of transcapillary protein transport under normal conditions. The good agreement between the best-fit parameters obtained by studying normal subjects and those obtained for chronic nephrotic patients suggests that the microvascu lar exchange system behaves close to normal in chronic nephrotic patients. For patients 121 Chapter 6. Conclusions suffering from 122 mild nephrotic syndrome, protein washout (i.e. a net reduction in in terstitial protein content) is the predominant edema preventing mechanism. For those suffering from severe nephrotic syndrome, all edema preventing mechanisms fail, leading to a rapid expansion of the interstitial fluid volume. Finally, the transient simulations of Koomans’ albumin infusion experimental data, which have been used as an independent test of the validity of the transport parameters obtained by statistically fitting between the model predictions and other experimental data for humans, further proved that the coupled Starling model can successfully simulate the transport mechanisms of the microvascular exchange system in the human. Chapter 7 Future Work The recommendations for future work include the following two aspects: 1. Further improve the current model for the normal microvascular exchange system. Improvements might include: (a) Replacing the “most-likely” human tissue compliance relationship by the ex perimentally determined tissue compliance relationship when clinical measure ments of this property become available. One can then investigate how close the -“most-likely” compliance relationship is to the real one. - (b) Dividing the generic tissue compartment into two compartments, i.e., a skeletal muscle compartment and a skin compartment, when the properties of these two separate compartments become available. (c) Considering the influence of cellular component. 2. Develop a new model to study the behavior of the microvascular exchange system after a burn injury. Because of the limited amount of information available for humans, the microvascular exchange system in the burn model should probably be divided into three compartments as the first step in developing a more comprehen sive model. The three compartments would be the circulatory, normal tissue and burned tissue compartments. The transport parameters obtained in current study could he used to describe the transport characteristics of the normal tissue. Some of the transport properties will be subject to a transient adjustment due to the 123 Chapter 7. Future Work 124 burn injury [8, 9]. Using available data from burn patients a similar optimization procedure is recommended for finding the optimal parameters which describe these transient adjustments. Such a model is of potential use as an aid in monitoring fluid resuscitation in burn injured patients. Bibliography [1] Arturson, G., Groth, T., Hedlund, A., and Zaar,B. Computer simulation of fluid resuscitation in trauma: First pragmatic validation in thermal injury. J. Burn Care Rehabil. 10:292-299,1989. [2] Aukiand, K., and Nicolaysen, G. Interstitial fluid volume: Local regulatory mech anisms. Physiol. Rev. 61(3) :556-638,1981. [3] Bell, D.R. 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Nomenclature Symbol Meaning Aib Albumin BV Blood volume (ml) C Concentration of albumin (g.1’) D Perturbation (g.h’ or ml.h ) 1 ECV Extracellular volume (ml) F( ) Units Function of FCOMFI Tissue compliance relationship HCT Hematocrit J Fluid transport rate (ml.h) K Fluid transport coefficient (ml.mmHg . 1 h’) Length (m) (ml.mmHg’) L Length or lower bound LS Lymph flow sensitivity OBJ Objective function value F Hydrostatic pressure (mmllg) FCOMPC Capillary compliance (ml.mmHg’) Fe Péclet number PS Permeability-surface area product of ) 1 (ml.mmHg.h ) 1 (ml.h capillary with respect to albumin Q Albumin content (g) 134 Nomenclature 135 Q Albumin transport rate ) 1 (g.h r Radius (m) RAP Right atrial pressure (mmHg) SD Standard deviation SE Standard error T Time U Upper bound V Volume W Weight factor H Colloid osmotic pressure a Albumin reflection coefficient Change Superscripts and Subscripts AV Interstitial volume available to albumin C Capillary D Diffusion EX Excluded exp Experimental value F Filtration GRAD Gradient HUMAN Human I Interstitial compartment (h) (ml) (mmHg) Nomenclature 136 iso Isogravimetric L Lymph max Maximum value mm Minimum value NORM Normal steady-state FL Plasma RAT Rat RMB Ratio fo maximal to basal level S Solute sim Simulation value TO Turnover rate o Normal steady-state 1/2 Half-time * Tracer properties Appendix A Raw Experimental Data 137 Appendix A. Raw Experimental Data Time (hr) Saline, 100 mL N=4 Albumin, 100 mL N=4 Saline, 200 mL N=4 Albumin, 200 mL N=4 Mean SD Mean SD Mean SD Mean SD -1.5 24.1 24.8 24.0 24.2 138 1 -0.8 ±0.4 0.5 ±1.6 -0.9 +2.3 3.2 +0.9 3 -0.1 ±1.0 -0.1 +1.4 -0.2 ±0.8 3.1 ±2.8 6 -0.7 ±0.7 -0.6 ±0.8 -0.4 +1.0 1.5 +1.6 9 -1.0 ±0.6 -0.4 +1.3 0.5 ±2.9 2.0 +2.3 12 -1.8 ±1.5 -1.1 +0.5 -0.1 ±1.7 1.6 ±1.2 Table A.1: Change in plasma colloid osmotic pressure (mmHg) at room tem perature. t = 0 designates the end of infusion. Time (hr) Saline, 100 mL N=4 Albumin, 100 mL N=4 Saline, 200 mL N=4 Albumin, 200 mL N=4 Mean SD Mean SD Mean SD Mean SD -1.5 2,727 ±883 3223 ±394 1 3 49 ±140 301 +160 161 ±223 427 +147 16 ±155 281 ±270 145 ±148 383 +36 6 96 ±154 318 +244 161 ±110 353 ±66 9 202 ±142 295 ±198 109 ±186 281 ±50 12 179 ±132 287 ±206 169 +155 314 ±95 Table A.2: Change in plasma volume (mL) at room temperature. t nates the end of infusion. = 0 desig Appendix A. Raw Experimental Data Time (hr) Saline, 100 mL N=4 Albumin, 100 mL N=4 Saline, 200 mL N=4 Albumin, 200 mL N=4 Mean SD Mean SD Meái SD Mean SD 4.5 236 234 139 1 0.1 +8.2 22.2 ±5.0 1.8 ±4.8 44.6 +13.0 3 2.7 +9.7 19.6 ±4.8 7.2 ±7.6 36.5 +10.7 6 9 12 1.9 8.4 8.6 ±10.4 ±6.9 ±7.2 17.1 18.1 15.5 ±9.1 +5.1 +6.6 9.6 f0T 12.9 ±7.8 ±11.4 ±13.0 29.4 24.3 27.1 +7.9 +8.8 +8.6 Table A.3: Change in total plasma albumin content (g) at room temperature. t = 0 designates the end of infusion. Appendix A. Raw Experimental Data Patient # Surface Area (m ) 2 1 1.82 2 1.60 3 1.68 4 1.65 5 1.63 6 1.65 7 1.71 8 1.95 9 1.93 10 1.76 11 1.71 Remarks 140 HCT (%) Control l000mL/9.5mins 38 mm. after infusion Control l000mL/llmins 60 mm. after infusion Control l000mL/l0mins Control l000mL/llmins 35 mm. after infusion Control l000mL/llmins Control l000mL/l3mins 15 mm. after infusion Control l000mL/llmins 30 mi after infusion Control 950mL/6.5mins 30 mm. after infusion Control 950mL/7.5mins Control 950mL/8.5mins 30 mm. after infusion Control 900mL/9mins 40 mm. after infusion 42.4 37.1 40.4 32.3 27.6 29.2 37.7 31.6 40.0 34.3 33.2 34.7 28.4 47.5 42.4 44.3 43.9 37.8 40.2 52.3 45.2 44.4 44.1 38.1 45.7 44.8 43.1 46.9 42.4 44.9 General Blood Volume (mL) 4422.6 50.59.6 4641.0 4768.0 5568.0 5264.0 4116.0 4905.6 4273.5 4983.0 5148.0 4026.1 4922.6 4537.5 5082.O 4867.5 4086.9 4753.8 4463.1 5635.5 6396.0 6493.5 5056.6 5867.2 5632.0 5737.6 5966.4 4907.7 5420.7 5112.9 Plasma Volume (mL) 2547.4 3182.5 2766.0 3227.9 4031.2 3726.9 2564.3 3355.4 2564.1 3273.8 3438.9 2629.0 3524.6 2382.2 2927.22711.2 2292.8 2956.9 2668.9 2688.1 3505.0 3610.4 2826.6 3631.8 3058.2 3167.2 3394.9 2606.0 3122.3 2817.2 Table A.4: Hemodynamic effect of rapid intravenous infusion of physiologic saline infusion. Appendix A. Raw Experimental Data HGB (g.100mL’) HCT () PL (g.lOOmL’) 0 141 Baseline 12.5+1.7 38.4±5.2 3.4±0.5 1/2 hour post-infusion 11.7+1.7 36.2+5.1 3.0+0.5 Table A..5: Mean concentration of hemoglobin and albumin, and mean hemat ocrits before and after saline infusion, (mmHg) Hi (mmHg) RAPT (mmHg) Pct (mmHg) VPL (mL/cm) Body Height (cm) VpL/VI V/ (mL) PL 11 Patients with Heart Failure 26.1+4.2 11.5±3.4 11.15 13.71 19.2±4.5 171.9+10.7 0.27±0.07 12224±6795 Patients with Anasarca 23.0+2.6 9.2+2.6 16.29 16.90 21.7±4.4 171.9±10.7 0.28±0.09 13322±7945 Normal Subjects 28.6±3.4 15.8±2.7 0 6.8 17.3±1.5 172.0±9.2 0.36±0.06 8266± 2536 Table A.6: 11 PL, H and VPL in patients with heart failure (N=13), pa tients with anasarca (N=7) and n normal subjects. calculated from H —0.4 x RAP + 15.8; t calculated from Pc = 0.62 x RAP + 6.8; * the cal culation of SD is the same as that described in APPENDIX B. (mmHg) 1 (mmllg) U RAP (rnmHg) Pc (rnmHg) VPL (rnL/cm) Body Height (cm) FL 11 VPL/VI Vj (mL) Angina pectoris 24.9±2.1 10.65±2.35 5.1±1.7 12.6±2.9 18.8±1.8 175±6 0.37±0.08 8892±3079 Controls 26.8±3.7 13.15±2.5 1.5±2.25 11.15±2.7 19.9±1.35 181±5 0.36±0.08 10005+3179 Table A.7: 11 1 and VFL in patients with angina pectoris and in controls. PL, H the calculation of SD is the same as that described in APPENDIX B. * Appendix A. Raw Experimental Data 1 I PL Table A.8: FL 11 (mrnHg) 9.2 10.7 11.7 12.7 13.7 13.7 14.2 14.2 18.7 18.7 20.7 20.7 21.7 24.7 25.7 25.7 26.7 26.7 142 Vi (L) 19.01 16.02 14.40 12.98 7.98 6.71 10.70 8.27 9.94 9.11 7.79 6.73 6.61 8.44 8.34 7.69 8.25 9.38 vs. V 1 for patients with nephrotic syndrome. Appendix A. Raw Experimental Data PL 11 (mmHg) 2.3 5.3 5.8 6.6 7.0 7.0 7.8 8.3 8.9 9.0 9.0 9.2 9.5 9.5 9.6 9.9 10.3 10.7 11.3 11.5 11.5 11.7 11.9 12.3 12.4 12.9 13.0 13.2 13.7 13.9 14.2 14.2 14.5 Table A.9: ll vs. Hi (mmHg) 2.3 5.9 1.9 2.6 3.4 2.9 0.9 2.7 1.7 4.9 4.9 3.2 4.4 3.9 0.7 1.6 29 5.7 5.4 3.9 3.9 4.7 4.0 1.9 4.5 4.3 3.4 7.0 5.7 7.2 5.7 4.7 4.9 PL 11 143 PL 11 (mmHg) 15.5 15.5 15.7 17.3 17.4 17.5 17.5 17.9 18.0 18.2 18.4 18.7 18.7 19.0 19.0 19.4 19.9 20.7 21.0 21.0 21.7 22.9 23.6 23.6 23.6 24.0 24.4 24.8 24.2 24.4 24.9 26.4 26.9 1 (mmHg) fT 6.4 6.4 3.7 5.9 7.7 5.9 3.9 8.0 8.6 8.2 8.4 5.2 6.7 6.9 9.9 7.7 I&.7 7.7 10.4 10.9 8.7 14.7 11.7 13.2 14.0 12.2 11.7 12.7 15.2 14.4 13.2 13.7 14.7 for patients with nephrotic syndrome. Appendix B Calculation Of Error Propagation Calculation of the propagation of experimental error is necessary in the following situa tions: 1. When the compared quantity is not directly measured in experiment. For example, blood volume and hemotocrits are measured experimentally, then plasma volume is calculated from their prodllct. The uncertainty associated with the calculated plasma volume is estimated on the basis of the standard deviations of blood volume - and hemotocrits; - 2. When measured quantities are further manipulated, e.g. being converted from absolute value to percentage change. The propagation of experimental error is calculated in the following manner. Assume that the desired quantity U is related to several directly measured quantities , x 1 x , 2 •.•, x, by the general equation , 2 U=f(xi,x (B.1) Each directly measured quantity has an associated experimental error of Ax, ••, Ax, respectively. The differential change in U for a differential change in each of the measured x’s is then given by dU = —dx + 1 1 ax 2 ax 144 +... + -‘—dx (B.2) Appendix B. Calculation Of Error Propagation 145 where -dx is the partial derivative off with respect to x taken with all the remaining x’s held constant. When Ax , Ax 1 , 2 ..., Ax are sufficiently small that higher order terms of the Taylor expansion can be neglected, the differentials dU, dx , dx 1 , 2 the finite increments AU, Ax , 2 , Ax 1 AU = ..., dx can he replaced by Ax 1 + 2 LA LA +... + 1 —Ax 2 ax (B.3) The maximum uncertainty of quantity U is calculated according to Eq. B.3. Examples In Mullins’ saline infusion experiment (see Table A.11), baseline hemoglobin concentration: HGBB = 0.125±0.017 g/mL, postinfusion hemoglobin concentration: HGBp = 0.117±0.017 g/mL, Assume hemoglobin remains constant throughout the experiment, baseliiie blood volume (BVB) is 5195 mL, then post-infusion blood volume (BVp) is BV = HGBB HGB x BVB Substituting the numbers into the above equation gives BV = (0.125/0.117) x 5195 = 5550.2 mL Apply Eq. B.3 ABVP = AHGBB P = Since plasma volume P 1561.3 mL (VPL) baseline hemotocrit: HCTB x BVB + HGBB x BVB x AHGBP = equals to BVPL x (1 — HCT), given 0.384±0.052; post-infusion hernotocrit: HCT = 0.362±0.051, we get Appendix B. Calculation Of Error Propagation baseline plasma volume: VPL,B post-infusion plasma volume: 146 3200 mL; = VPL,F 3541±1177 (SD ) mL. Therefore, 1 = [VPLF = VPL% — 1] 100 x 10.65% = VPL,B and standard deviation for AVPL% is = SDVPL% Since QPL = CPL x VPL, QPL,B 2 SD 16 g; /-GPL,B x VPL,B = post-infusion albumin content: of SD 3 = x 100 36.78 = VPL,B therefore, baseline albumin content: = 1 SD /CPL,P x VPL,P + 34 x 3.2 = QPL,P = 108.8 g. with a standard deviation of 30 x 3.541 1 SD CPL,P x = = = 106.2 g, with a standard deviation 5 x 3.541 + 30 x 1.117 Thus, the percentage change of albumin content is QPL% = [QPLP Apply Eq. B.3, the standard deviation for SDQPL% QFL% = QPL,B — — 53.0 108.8 1] x 100 — Q PL,B + + 106.2 108.82 = —2.39 is x SD 2 x 16 All these values correspond to those listed in Table 4.7. — — 63 07 = 53.0 g. Appendix C Basic Concepts Related To Statistical Analysis To permit a better understanding of the statistical analysis presented in chapter 5, several relevant basic concepts will be clarified here. Covariance Matrix The covariance matrix is defined as the inverse of the matrix of second partial deriva tives (i.e. the Hessian matrix) of the objective function, expressed as: where a 2{ } u { 2 X} a{X,Y} u{X,Y} a { 2 Y} is called the variance operator (read “variance of”); a{ , } is called the covariance operator (read “covariance of”); X and Y are two random variables. The variance measures the spread or dispersion of a probability distribution. The standard deviation of X is the positive square root of the variance of X, i.e. SD = a{X} = a2{X} The covariance provides a measure of the association between X and Y. If X and Y are independent, a{X,Y} = 0, however, the converse is not necessarily so. Confidence Interval The probability that a correct interval estimate of an unknown parameter X is obtained is called the confidence coefficient and is denoted by 1 147 — c. The interval, Appendix C. Basic Concepts Related To Statistical Analysis 148 L <X < U, within which the value of the parameter in question would be expected to lie is called a 100(1 — c) percent confidence interval for the parameter X. The interpre tation of this interval is that, if in repeated random samplings, a large number of such intervals are constructed, 100(1 — c) percent of them will contain the true value of X. In the current study, only the 95% confidence interval will be used. When the sample size is reasonably large, a variable with a sample mean X and a standard deviation SD has the 95% confidence interval [61] X—1.96xSD<XX+1.96xSD Residual Plot A residual (e) is the difference between an observed value (X) and the corresponding anticipated value (X), i.e. e = X — by the standard deviation, i.e. é X. A staudardized residual (é) is the residual divided = e/SD. A residual plot is a plot of standardized residuals (or residuals) plotted against the fitted value X. It often provides useful clues for the evaluation of the aptness of the model (see Chapter 5). Appendix D Surface Plot And Contour Plot The objective function (Eq. 4.19) is an unknown function of LS, a, and the intersti tial compliance relationship. It is determined by the differences between the simulation results and the experimental data. It is non-linear and can not be expressed as an ana lytical equation. Therefore, analytical methods can not be applied to find the location of OBJmin. To allow a visual estimate of the minimum region, surface and contour plots of the objective function are generated first. Then, starting with an initial point located in side the minimum region, a numerical optimization program is applied to find the best-fit parameters. The surface plot has three dimensions. In this case, the X—axis is a, the Y—axis is LS, and the Z—axis is OBJ. It is generated by increasing a and LS in incremental steps (a 20 x 20 grid is chosen) between their lower and upper bounds, and calculating OBJ at the given values of a, LS, Pc,o and interstitial compliance relationship by using Eq. 4.14. Figure D.1 shows three surface plots viewed from different angles. The figure was constructed by using 138 experimental data points. A contour plot is a projection drawing of the surface plot (Fig. D.1). Contour lines are drawn by connecting points having the same OBJ value on the projection. From the contour plot, it is easy to locate the minimum region of the objective function (e.g. the shaded area on Fig. D.1). 149 Appendix D. Surface Plot And Contour Plot 150 9OD 0’ 0 0.5 0.6 0.7 0.8 0.9 1.0 a Figure D.1: Surface plots of the objective function from different angles (a, b, and c); d shows the contour plot of the objective function. The shaded area is the minimum region of OBJ. For compliance relationship #3. Appendix E Simulations At Best-Fit Parameters 151 Appendix E. Simulations At Best-Fit Parameters 152 (A) 30.015.0: 0.0-1 15.0— —30.0—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 1 ‘.0 —1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 I I I I I I I Time(hrs) (B) G. liz 30.015.00.0-I —15.0—30.0—1.5 -rT p I I I I 0.0 1.5 3.0 4.5 6.0 0.0 1.5 3.0 4.5 6.0 I 7.5 I I 9.0 10.5 1 60.0 40.0 20.0 0.0 —20.0 —40.0 —1.5 Time(hrs) Figure E.1: (A) Simulation of 100 mL saline infusion; (B) Simulation of 100 mE albumin infusion; Fluid intake during waking hours (0 13.5 hr) is 1.4 L (N=4). For tissue compliance relationship #1. — Appendix E. Simulations At Best-Fit Parameters 153 (A) 30.015.0— 0.0 I— —15.0—30.C._T I —1.5 0.0 - 60.C 40.020.0- I 1.5 I I 3.0 4.5 6.0 I 7.5 I I 9.0 10.5 1 . 0 —-tffEH - —20.0—40.C —1.5 I 0.0 1.5 3.0 I 4.5 6.0 I 7.5 9.0 10.5 12.0 Time(hrs) (B) 30.015.0- —15.030.0I —1.5 0.0 I I I 1.5 3.0 4.5 1.5 3.0 4.5 I 6.0 I 7.5 I I 9.0 10.5 12.0 60.0 40.0 20.0 0.0 —20.0 —40.0 —1.5 0.0 Time(hrs) Figure E.2: (A) Simulation of 100 mL saline infusion; (B) Simulation of 100 mL albumin infusion; Fluid intake during waking hours (0 13.5 hr) is 1.4 L (N=4). For tissue compliance relationship #2. — Appendix E. Simulations At Best-Fit Parameters 154 (A) 30.015.0 — —1.5 III 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 1 0.0 3.0 4.5 6.0 7.5 9.0 10.5 12.0 9.0 10.5 I I I I 60.040.0 20:0 —20.0 —40.0—1.5 1.5 Time(hrs) (B) 30.0 I —1.5 0.0 1.5 3.0 4.5 I 6.0 7.5 I 12.0 60.0- IL— —20.0—40.0— —1.5 0.0 - I 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 Time(hrs) Figure E.3: (A) Simulation of 200 mL saline infusion; (B) Simulation of 200 mL albumin infusion; Fluid intake during waking hours (0 13.5 hr) is 1.4 L (N=4). For tissue compliance relationship #1. — Appendix E. Simulations At Best-Fit Parameters 155 (A) 3 0.0- 15.0 0.0 —15.0 I —30.0- —1.5 0.0 _40.0: —1.5 1.5 3.0 I 0.0 1.5 4.5 I 6.0 I 3.0 7.5 9.0 7.5 9.0 10.5 12.0 7.5 9.0 10.5 12.0 I 4.5 6.0 I 10.5 I 1 I Time(hrs) (B) 30.0 15.0 C. 0.0 —15.0 —30. —1.5 0.0 1.5 6 0.0- 40020.00.0-i —20.0- —40.0—1.5 I 0.0 I 1.5 I 3.0 I 4.5 I 6.0 Time(hrs) Figure E.4: (A) Simulation of 200 mL saline infusion; (B) Simula tion of 200 mL albumin infusion; Fluid intake during waking hours (0 13.5 hr) is 1.4 L (N=4). For tissue compliance relationship #2. — Appendix F. Simulations At Best-Fit Parameters Patient 156 #1 30.0 20.0 10.0 0.0 0.0 30.0 120.0 60.0 Patient #2 30.0 0.0 I 0.0 30.0 60.0 Patient #6 30.0 0.0 0.0 30.0 60.0 90.0 Patient #7 30.0 20.0 - 0.0 0.0 30.0 60.0 Patient 90.0 120.0 90.0 120.0 #11 20.0 10.0 0.0 0.0 30.0 60.0 Time(nilns) Figure E.5: Simulations compliance relationship #1. of acute saline infusion in selected patients. For tissue Appendix E. Simulations At Best-Fit Parameters 157 Patient #1 30.0 K 200 I 00 00 30.0 0.0 60.0 Patient #2 30.0 200 10•000 0.0 30.0 0.0 60.0 Patient #6 30 0 20.0 0.0 0.0 30.0 60.0 90.0 0.0 90.0 120.0 90.0 120.0 Patient #7 30.0 20.0 10.0 0.0 0.0 30.0 60.0 Patient #11 30.0 20.0 10.0 0.0 0.0 30.0 60.0 Time(mlns) Figure E.6: Simulations of acute saline infusion in selected patients. For tissue compliance relationship #2. Appendix E. Simulations At Best-Fit Parameters 158 75.0- .0 .0 L 5.0 50.025.0- Time(hrs) Figure E.7: Transient responses in plasma volume and plasma albumin con centration after 2 L of normal saline infusion within 2 hours. For tissue compliance relationship #1. Appendix F. Simulations At Best-Fit Parameters 159 75.0- oLo.o.oLo5.o 50.O- 25.0- 0 0 0 0 Time(hrs) Figure E.8: Transient responses in plasma volume and plasma albumin con centration after 2 L of normal saline infusion within 2 hours. For tissue compliance relationship #2. Appendix E. Simulations At Best-Fit Parameters 160 25.0 20.0 15.0 i:.: 20.0 22.5 25.0 PL 27.5 30.0 mmHg 25.0- 20.0- 15.0- 10.0 - 5.07.5 10.0 12.5 15.0 17.5 20.0 P (mmHg) Figure E.9: Simulations of steady-state V 1 vs. HPL and V 1 vs. Pc in heart failure patients. For tissue compliance relationship #1. Appendix E. Simulations At Best-Fit Parameters 161 25.0 20.0 15.0- 10.0I 5.0- I 20.0 I 22.5 25.0 PL 27.5 30.0 (mmHg) 25.0- 20.0 15.0 10.0- 5.0 I 7.5 10.0 I 12.5 I 15.0 17.5 20.0 P(mmHg) Figure E.1O: Simulations of steady-state Vj vs. IIPL and V 1 vs. failure patients. For tissue compliance relationship #2. Pc in heart Appendix E. Simulations At Best-Fit Parameters 162 100.0 80.0 60.0 40.0 20.0 0.0 I— 0.0 5.0 I I I 10.0 15.0 20.0 25.0 31 PL (mmHg) 20.016.0- . S 12.08.0- I. a I. 1; I 4.0- . . . •. •. 0.0 I 0.0 5.0 I I I I 10.0 15.0 200 25.0 30.0 ‘TrP(mmHg) Figure E.11: Simulations of V 1 vs. IIPL and H vs. ‘IPL in nephrotic syndrome patients. For the best-fit parameters of tissue compliance relationship #1. Appendix E. Simulations At Best-Fit Parameters 163 100.080.060.040.020.00.0 I 0.0 5.0 I 10.0 150 20.0 25.0 3 ).0 7rL(mmHg) 20.0 16.0- . 12.08.0- .. . a 4.0• 0.0 I 0.0 5.0 I I 10.0 15.0 20.0 25.0 30.0 P1. (mmHg) Figure E.12: Simulations of V 1 vs. 1 IPL and H vs. HPL in nephrotic syndrome patients. For the best-fit parameters of tissue compliance relatio nship #2. Appendix E. Simulations At Best-Fit Parameters 164 25.020.0- S 15.0- 4 10.05.0- I 12.0 6.05,0 4.0 3.0 2.0 I I 24.0 36.0 4 I.0 - $ - ‘-a .L - - 1.0- 0.0 - 0.0 12.0 24.0 36.0 48.0 12.0 24.0 36.0 48.0 150.0100.0- QPL 0 -..--. -. 50.0- QI 04 C, 0.0- 0.0 12.0 24.0 Time(hrs) 36.0 48.0 Figure E.13: Transient responses of colloid osmotic pressu res (IIPL and Hi), plasma volume (Vpjj, and protein contents (QPL and Qz) after single infusion of 60 g of human albumin solution. For the best-fit par ameters of tissue compliance relationship #1. Appendix E. Simulations At Best-Fit Parameters 165 25.0 S 4 20.015.010.0- 5.0 0.0 S I — 6.0 5.04.0 3.0 2.0’ 1.00.0- 12.0 I 24.0 I 36.0 4 [.0 - J I I - I 0.0 12.0 24.0 36.0 48.0 ).0 12.0 24.0 36.0 4 I.0 4.004 3.5. 3.0- 2.5150.0100.0- 0 QPL :- 50.00.0 QI - 0.0 12.0 24.0 36.0 48.0 Time(hrs) Figure E.14: Transient responses of colloid osmotic pressures (HPL and Hi), plasma volume (Vpjj, and protein contents and after Qj) single infusion (QPL of 60 g of human albumin solution. For the best-fit para meters of tissue compliance relationship #2. Appendix F List Of Computer Programs F.1 Parameter List of Steady-State and Transient Simulators C C A list of program variables of all programs listed C **************************************************** C C Variable key: C C ACC C ALPHA = Scaling factor for ROOT NV = Number of nonlinearequation C PIPL = Plasma colloid osmotic pressure C PISKIN C P15 = Accuracy limit - - Tissue colloid osmotic pressure = Tissue colloid osmotic pressure C PSNRN = Normal tissue hydrostatic pressure C QPLS = Plasma: albumin content_calculated C QSS = Tissue: albumin content_calculated C QPLNRM Plasma: normal albumin content C QSNRN = Tissue: normal albumin content C QTNRM = Normal total albumin content C QTOT = Whole body: albumin content C RHSF = Nonlinear equations to be solved C VPLNRN Plasma: normal volume C VIFNRM Tissue: normal volume C VTNRN = Normal total extra-cellular volume C VTOT = Whole body: volume C VEXS = Tissue excluded volume C YNEW = Initial solution estimates C YOLO = Final solution values C CAVS = Protein concentration in available volume 166 Appendix F. List Of Computer Programs C CPL = Protein concentration in plasma C CS = Protein concentration in tissue C iFS = Fluid filtration flowrate C JLS = Fluid lymph flowrate C PC = Capillary hydrostatic pressure C PCNRN = Normal PC C QPS = Protein flowrate across the capillary wall C QLS = Lymphatic protein flowrate C AS = Initial gradient of tissue compliance curve C BS = Final gradient of tissue compliance curve C KFS = Tissue filtration transport coefficient C LSNRM = Tissue basal lymph flowrate C LSS = Lymph flow sensitivity C P55 = Permeability-surface area product C SIGS = Albumin reflection coefficient C NPS = Number of points on tissue compliance curve C NPSN1NPS- 1 F.2 Listing of FORTRAN function XDFUNC1 C FUNCTION XOFUNC(X,N) C C C This function is used to calculate the value of the objective C function at specific values of SIGS, LSS end PCNRM. C IMPLICIT REAL*8CA-H,K,L,3,O-Z) C CONMON/BLKA2/CPLNRM , CSNRM , CASNRM CONMON/BLKB/JFS , AS COMMON/ELKO/QPS , QLS CONMON/BLKG/CPL ,CS, CAVS COMMON/BLKI/PIPL ,PIS COMMON/BLKF/PC ,PCNRM,PCGRAD CONNON/BLKH/LSNRM CONMON/BLKJ/VTNRN, VIFNRN, VPLNRN CONNON/BLKK/VEXS CONNON/ELKL/LSS ,KFS , SIGS , P55 167 Appendix F. List Of Computer Programs COMMON/BLKT/QTNRM , QSNNM , QPLNRM COMMON/BLKU/PSNRM ,PS COMMON/BLKBB/PIPNRN ,PISNRM COMMON/BLKZB/QTOT, VTOT COMMON/BLKO/VSP(14) ,PSP(14) ,AS,BS,NPS,NPSM1 COMMON/A/VPINIT , QPINIT , VSINIT , QSINIT C DIMENSION X(N) ,G(6) C LBS x(i)*40.D0 = SIGS X(2) = PCNRN X(3)*10.D0 = C PIPNRM=25 .900 PISNRM=14 .700 CALL SPLINS CALL COEFF C PLINIT = 25.900 PIINIT = 14.7D0 VIFINT = 8400.000 VPINIT = VPLNRM QPINIT = VPLNRM VSINIT = VIFINT QSINIT = FALB(PIINIT) VTINIT = VPINIT + VSINIT QTINIT = QPINIT + QSINII C * FALB(PLINIT) C * (VIFINT - VEXS) C C C Each data set is progrsned separately so that it can be called C separately if necessary. C HUB: Data set A (Data from Hubbard et al.) C DOYLE: Data set B (Data from DOYLE et al.) C MULLINS: Data set C (Data from Mullins et al.) C HF: Data set 0 (Data from heart failure patients) C NEPHRO: Data set E (Data from nephrotic patients) C 168 Appendix F. List Of Computer Programs SUM 169 0.00 = CALL HUB(i.D-2,4,SUM1,SUM) 0(1) SUM = SUM 0.00 = CALL DOYLE(i.D-2,4,SUM) 0(2) SUM = SUM 0.00 = CALL MULLIMS(i .D-2,4,SUM) 0(3) SUM = SUM 0.00 = CALL HF(1.D-2,.SD0,i.D-2,4,SUM) 0(4) SUM = SUM 0.00 = CALL MEPHRO(SUM) 0(5) = SUM 0(6) = 0(1)+G(2)+0(3)+G(4)+0(S) XDFUMC = 0(6) C RETURN END C SUBROUTINE XDCONS(X,N,M,ME,MMAX,G) C C C This subroutine is used to evaluate the constraints imposed on C the parameters to be estimated. C IMPLICIT REAL*6(A-H,K,L,0-Z) INTEGER N,M,ME,MMAX DIMENSION X(N) ,G(MMAX) C COMMON/BLKA2/CPLMRM , CSNRM , CASNRM COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKU/PSNRM ,PS COMMOM/BLKBB/PIPNRM ,PISNRM C IF(M.LE.0) GO TO 100 0(1) = (x(3)*io.Do - PSNRM)/(PIPNRN — PISMRM) — x(2)/lo.Do Appendix F. List Of Computer Programs 170 RETURN 100 CONTINUE JUMP = -M GO TO (1),JUMP RETURN 1 0(1) = (X(3)*10.OO — PSNRM)/(PIPNRN - PISNEM) — X(2)/10.OO RETURN END C SUBROUTINE HUB (EPS ,NV, SUM1 ,SUM) C C C This subroutine is used to evalute the sum of square of the error C contributed by Hubbard’s data only. C IMPLICIT REAL*8(A-H,K,L,J,O-Z) INTEGER FLAG C COMMON/BLKI/PIPL,PIS -- COMMON/A/VPINIT , QPINIT ,VSINIT ,QSINIT COMMON/BLKB/JFS , JLS COMMON/BLKO/QPS ,QLS COMMON/SLKG/CPL ,CS, CAVS COMMON/BLKF/PC ,PCNRM,PCGRAD COMMON/BLKH/LSNRM COMMON/BLKL/LSS ,KFS ,SIGS ,PSS COMMON/BLKU/PSNRM ,PS COMMON/BLKZB/QTOT ,VTOT C DIMENSION YOLD(4) ,YNEW(4) DIMENSION SOLN(6,2),EXP1(6,2,4),T1(6),SD1(6,2,4) EXTERNAL RHSFB C C Experimental data. C DATA T1/ -1.500,1.D0,3.D0,6.D0,9.O0,12.D0/ DATA EXP1/0.D0,-.033TD0,-.004100,-.O2BSDO,-.040900,-.OT3SDO, 1 0.00, .O1B000, .005900, .035200, .074100, .065600, Appendix F. List Of Computer Programs 2 0.00, .0204D0,—.0041D0,-.024500,-.0164D0,- .045000, 3 0.D0, .1104D0, .103000, .1166D0, .108200, .105200, 4 0.00,— .037300,—.0083D0,-.0166D0, .020700,-.004100, 5 0.00, .050000, .045000, .050000, .033800, .052400, 6 0.D0, .132800, .128600, .062200, .083000, .066400, 7 0.00, .132500, .118800, .078500, .087200, .097400/ DATA SD1/O.ODO, 5.85D—2, 8.430—2, 7.090—2, 6.63D-2,10.16D-2, 1 0.000,38. 1OD-2,38.25D-2,39. 17D—2,39.99D—2,39.35D-2, 3 0.ODO,11.11D-2,10.19D-2, 7.64D—2, 9.73D—2, 6.33D—2, 4 0.ODO,41.82D—2,45.62D-2,45, 100—2,43. 14D—2,43.34D-2, 6 0.ODO,16.94D-2,10.94D-2,11.710-2,19,88D—2,14.71D—2, 7 0.000,19.750—2,17.370-2,16.250—2,18.410-2,17.670—2, 9 0.000, 12.510—2,20.360—2,14.870—2,17.930-2,13,240—2, 1 0.000, 18.41D-2,14.79D—2,15.61D—2,14.84D-2,16,36D—2/ C DO 60 IPET1,4 C C Set initial conditions. C YNEW(1) = VPINIT YNEW(2) = QPINIT YNEW(3) = VSINIT YNEW(4) = QSIMIT C C Solve differential equations. C DO 30 I = 0,5 IP=I+1 TI = T1(IP) IF(I,EQ.0) GOTO 10 TIM T1(I) DT=TI-TIM TSTART=1 .D-1*DT TMIN1 . D-4*DT TMAX=DT CALL RK4C(RHSFB,NV,TIM,TI,YOLD,EPS,YNEW,NFTJNC,FLAG,IPET) IF(FLAG.EQ.0) GOTO 70 10 CALL AUXSAM(YNEW) SOLN(IP,1) PIPL 171 Appendix F. List Of Computer Programs SOLN(IP,2) DO 20 J = l,NV YOLO(J) 20 30 YNEW(1) = = YNEW(J) CONTINUE CONTINUE C C Calculate the sum of square of error. C 00 50 .1=1,2 SOLNOSOLN(1,J) 00 40 1=2,6 SOLN(I ,J)=(SOLN(I,J)-SOLN0)/SOLNO ERROR’SOLNCI , .1) -EXP1 (I, .1, IPET) SUN 40 50 60 SUM +4.00*(ERR0R**2)/(5D1(I,J,IpET)**2) CONTINUE CONTINUE CONTINUE RETURN C 70 WRITE(6,80) 80 FORMAT(1X,’ODE SOLVER FAILS 1 ‘) STOP END C SUBROUTINE DOYLE (EPS ,NV,SUM) C C C This subroutine is used to evaluate the sum of square of error C contributed by Doyle’s data only. C IMPLICIT REAL*8(A-H,K,L,J,O-Z) INTEGER FLAG COMMON/B/T2(3,S) CONNON/BLKI/PIPL ,PIS COMMON/A/VPINIT,QPINIT,VSINIT,QSINIT COMMON/BLKB/JFS ,J1S COMMON/BLKD/QPS ,QLS 172 Appendix F. List Of Computer Programs COMMON/BLKG/CPL ,CS, CAVS COMMON/SLKF/PC ,PCNRM,PCGRAD COMMON/BLKH/LSNRM COMNON/BLKL/LSS ,KFS , 5105 ,PSS COMMON/BLKI3/PSNRN ,PS COMMON/8LKZ8/QTOT ,VTOT C DIMENSION YOLO(4) ,YNEW(4) DIMENSION SOLN(3,S) ,EXP2(3,S) ,502(3,S) EXTERNAL RHSFC C C Experimental data. C DATA EXP2/O.000, .249300, .085800, + 0.000, .248900, .154600, + 0.000, .228800, .138100, + 0.000, .289600, .164000, + 0.000, .198100, .081000/ DATA S02/ .07i500,.078000,.073900, + .066300, .070900, .069300, + .069600, .076400, .073900, + .077300, .085700, .082400, + .064400, .069800, .066800/ C 00 60 IPET1,S C C Set initial conditions. C YNEW(1) = VPINIT YNEW(2) = QPINIT YNEW(3) = VSINIT YNEW(4) = QSINIT C C Solve the differential equations. C DO 30 I 0,2 1P1+1 TI = T2(IP,IPET)/60.D0 IF(I.EQ.0) GOTO 10 173 Appendix F. List Of Computer Programs TIM = T2(I,IPET)/60.D0 DTTI-TIM TSTART1 .O-1*DT TMIN1 .D-4*DT ThAXDT CALL RK4C(RHSFC,NV,TIM,TI,YOLD,EPS,YNEW,NFUNC,FLAG,IPET) IF(FLAG,EQ.O) GOTO 70 10 CALL AUXSAM(YNEW) SOLN(IP,IPET) DO 20 .3 = YOLO(J) 20 30 = YNEW(1) 1,NV = YNEWCJ) CONTINUE CONTINUE C C Calculate the sum of square of error. C SOLNOSOLN(1 ,IPET) DO 50 1=2,3 SOLN(I , IPET)(SOLN(I ,IPET)-SOLNO)/SOLNO ERROR=SOLN(I ,IPET)-EXP2(I,IPET) sUMsUM+(ERROR*ERR0R)/(5D2C.I ,IPET)*SD2(I,IPET)) 50 60 CONTINUE CONTINUE RETURN C 70 80 WRITE(6,80) FORMAT(1X,’ODE SOLVER FAILS 2’) STOP END C SUBROUTINE MULLINSCEPS ,NV,SUM) C C C This subroutine is used to evaluate the sum of square of error C contributed by Mullins’ data only. C IMPLICIT REAL*8(A-H,K,L,J,O-Z) INTEGER FLAG 174 Appendix F. List Of Computer Programs C COMMON/BLKI/PIPL , PIS COMMON/A/VPINIT , QPINIT ,VSINIT , QSINIT COMMON/BLKB/JFS , JLS COMMON/BLKD/QPS , QLS COMMON/BLKG/CPL ,Cs, CAVS COMMON/BLKF/PC ,PCNRM,PCGRAD COMMON/BLKH/LSNRM COMMON/BLKL/LSS , KFS , SIGS , PSS COMMON/BLKU/PSNRN ,PS COMMON/BLKZB/QTOT ,VTOT C DIMENSION YOLD(4) ,YNEW(4) DIMENSION SOLN(2,2) ,EXP4(2,2) ,T4(2) ,SD4(2,2) EXTERNAL RHSFE C C Experimental data. C DATA T4/ O.000,2.SDO/ DATA EXP4/O.ODO, .106500,O.ODO,-.117600/ DATA SO4/O.ODO,49.31D-2, O.000,27.68D-2/ C C Set initial conditions. C YNEW(1) = VPINIT YNEW(2) QPINIT YNEW(3) VSINIT YNEW(4) QSINIT C C Solve the differential equations. C IPET2 DO 30 I = 0,1 IP=I+1 TI T4(IP) = IF(I.EQ.0) GOTO 10 TIM = T4(I) DT=TI-TIM TSTART=l .D-1*DT 175 Appendix F. List Of Computer Pro,gTams TMIN=1 .D-4*DT TMAX=DT CALL RK4C(RHSFE,NV,TIM,TI,YOLD,EPS,YNEW,NFUNC,FLAG,IPET) IF(FLAG.EQ.0) GOTO 70 10 CALL AUXSAM(YNEW) SOLN(IP,i) SOLN(IP,2) DO 20 3 20 30 YNEW(1) = CPL 1,NV = YOLD(J) = = YNEW(3) CONTINUE CONTINUE C C Calculate the sum of square of error. C DO 60 3=1,2 SOLNOSOLN(1 ,J) DO 50 1=2,2 SOLN(I , J)’(SOLN(I ,J)—SDLND)/SOLNO ERROR”SOLN(I !4U ,3) SUM=SUM+i ii. D0* (ERROR*ERRDR) / (SD4(I, 3) **2) 50 60 CONTINUE CONTINUE RETURN C 70 80 WRITE(6,80) FORMAT(iX,’DDE SOLVER FAILS 3’) STOP END C ************************************* SUBROUTINE HF(ACC,ALPHA,EPS,NV,SUM) C ************************************* C C This subroutine is used to evaluate the sum of square of eorror C contributed by the data from heart failure patients. C IMPLICIT REAL*8(A-N,K,L,J,D-Z) INTEGER FLAG,FLAGG C 176 Appendix F. List Of Computer Programs COMMON/BLKA2/CPLNRM , CSNRM , CASNRM COMMON/BLKB/JFS , JLS COMMON/BLKD/QPS , QLS COMMON/BLKG/CPL ,Cs, CAVS COMMON/BLKI/PIPL ,PIS COMMON/BLKF/PC ,PCNRM,PCGRAD COMMON/BLKH/LSNRJ4 COMM0N/BLKJ/VTNR4 ,VIFNRM,VPLNRN COMMON/ELKK/VEXS COMMON/BLKL/LSS , KFS ,SIGS ,PSS COMMONJBLKT/QTNRM , QSNRN , PLNRM COMMON/BLKU/PSNRM ,P5 COMMON/BLKBB/PIPNRN ,PISNRM COMMON/BLKDD/ALBSTO COMNON/BLKZB/QTOT ,VTOT COMMON/VQPL/VPL ,QPL C DIMENSION YOLD(4),YNEW(4),YOLDA(2),YNEWA(2),YFINAL(2) DIMENSION A(4) ,B(4) ,C(4) ,XPVI(3),XPVPL(3),SD3(3) ,POINT(3) DIMENSION Y1(4),Y2(4),Y3(4),Y4(4) EXTERNAL RHSFA C C Experimental data. C DATA A/20.3,23.4,24.0,25.9/ DATA B/8.1,1O.4,12.2,14.7/ DATA C/20. 1,16.91,11.45,10.0/ DATA XPVI/O.562000,0.433200,—O.113500/ DATA SD3/2 .064400,1.836100,0.803900/ DATA POINT/7,00,13.D0,22.D0/ C TSTART1 . 0-1*0 .00500 TMIN1 .0-4*0.00500 TMAX=0 .0100 C C Set initial conditions. VSG = VIFNRN+3000.D0 DO 40 IPET2,3 177 Appendix F. List Of Computer Programs PIPL = A(IPET) P15 = B(IPET) PC C(IPET) = CPL = FALB(PIPL) PCC = PC VPL = FVPL(PCC) QPL = VPL*FALB(PIPL) QSG = VSG*FALB(PIS)*3.DO/4.OO YOLD(1) VSG = C C Solve the differential equation. C CALL ROOT(RHSFA,l,O.DO,YOLD,ACC,ALPHA,YNEW,FLAG) IF(FLAG.EQ,O) THEN PRINT*,’----ROOT FAILED----’ STOP ENDIF IF(VSG.LE.O.OO.OR.QSG.LE.O.OO) THEN PRINT*, ‘----NEGATIVE VOL/Q GENERATED----’ STOP ENDIF C C Calculate the sum of square of error. C VSG QS FALB(PIS) = YNEW(l)*FALB(PIS) = QTOT = PS FCOMPS(YNEW(1)) = QS + QPL Y1CIPET) = CHANGE (YNEW(l)-vIFNRM)/VIFNRN ERROR = 40 YNEW(1) = CIS = = YNEW(1)/1000.DO CHANGE-XPVI(IPET) SUN + p0INT(IPET)*ERR0R**2/SD3(IPET)**2 CONTINUE RETURN END C ************************ SUBROUTINE NEPHRO (SUM) 178 Appendix F. List Of Computer Programs C C C This subroutine is used to evaluate the sum of square of error C contributed by the data from nephrotic patients. C IMPLICIT REAL*8(A-H,K,L,J,O-2) INTEGER FLAG C COMMON/BLKA2/CPLNRM , CSNRM , CASNRM COMMON/BLKB/JFS JLS COMMON/BLKD/QPS , QLS COMNON/BLKG/CPL ,CS,CAVS COMMONIBLKI/PIPL ,PIS COMMON/BLKF/PC , PCNRM , PCGRAD COMMON/BLKH/LSNRM COMMON/BLKJ/VTNRM ,VIFNRM,VPLNRM COMMON/BLKK/VEXS COMMON/BLKL/LSS ,KFS,SIGS,PSS çoMMoN/T0T/vToT , QTOT COMNON/BLKU/PSNRM ,PS COMMON/BLKY/PIPPS(100) ,PISKIN (100) • NA ,NAM1 COMMON/BLKBB/PIPNRM , PISNRM COMNON/BLKDD/ALBSTO COMMON/BLKZ/DIFV(19) ,DPIPL(19) ,NN COMMON/BLKHH/PIPLPI (66) ,PII (66) C DIMENSION YOLD(2),YNEW(2),EXP6(66),SD6(66),EXP7(19) ,SD7(19) EXTERNAL RHSF C DATA ACC,ALPHA,NV/1.D-2,0.5D0,2/ C C Experimental data. C DATA EXP6/-0 .84354D+0,-0.59864D+0,—0.87075D+0,-0.82313D1-0, + -0.76871D+0,-0.80272D+0,-0.93878D+O,-0.81633D+0, + -0.884350+0,-0.66667D+0,—0.66667D+0,—0.78231D+0, + -0.70068D+O,—0.73469D+0,—0.962380—0,—0.89116D+0, + -0,80272D+0,-0.6l224D+0—0.63265D+0,—0.73469D+0, + -0.73469D+0,—0.68027D+0,—0.72789D+0,—0.87075D+0, 179 Appendix F. List Of Computer Programs + -0.69388D+0,—0.70748D+0,—0.76871D+0,-0.52381D+0, + —0.61224D+0,—0.51020D+0,—0.61224D+0,-0.68027D+0, + -0.66667D+0,—0.56463D+0,—0,56463D+0,-0.74830D+0, + —0.598640+0,-0.47619D+0,—0.59864D+0,--0.73469D+0, + —0.45578D+0,—0.41497D+0,-0.44218D+0,-0.42857D+0, + -0.64626D+0,--0.54422D+0,—0.53061D+0,-0.32653D+0, + —0,47619D+0,-0.272110+0,-0.47619D+0,—0.29252D+0, + -o .258500+0,-0.40816D+0, 0 .000000+0,-0.20408D+0, + —0. 10204D+O,-0.47619D—1,--0. 170070+0,—0.20408D+0, + —0.136060+0, 0.34014D—l,-0.20408D-1,—0,10204D+0, + -0.680270—1, 0.000000+0/ DATA SD6/0.1103D0,0.130300,0.108100,0.111900,0.1164D0, + 0.113600,0.109700,0.119700,0.114200,0.1319D0,0.131900,0.1225D0, + O.129200,0.126400,0.1086D0,0.1136D0,0.120800,0.1364D0, + O.134700,0.1264D0,0.126400,O.130800,O.126900,O.115300, + 0.108000,0.106900,0.101900,0.121900,0.114700,0.123000, + O.114700,O.109100,0.110200,0.118500,O.118500,0.103500, + 0.115800,0.172600,O.162600,0.151500,0,174300,0.177600, + 0.175400,0.176500,0.158800,0.167100,0.168200,0.184900, + 0.172600,0.189300,0.172600,0.187600,0.190400,0.178200, + 0.163200,0.146600,0.154900,0.159300,0.149300,0.146600, + 0.152100,0.166000,0. 161500,0. 154900,0.157700,0.163200/ DATA EXP7/ 0.126310+1, 0.907140+0, 0.714290+0, 0.545240+0, + -0.500000-1,—0.20119D+0, 0.27381D+0,-0,15476D-1, + 0.000000+0, 0.183810+0, 0.84524D—1,-0.72619D—1, + -0. 19881D+0,—0.213100+0, 0.47619D-2,—0.71429D-2, + —0,84524D—1,—0. 178570—1, 0.116670+0/ DATA S07/0.362900,0.337100,0.323000,0.310800,0.267500, + 0.256500,0,291000,0.270000,1. 111100,0.156200,0.149000, + 0.144700,0.135500,0.134500,0.145700,0.144800,0.139200, + 0.144000,0.153800/ C NA = NAM1 97 = PIPLUL NA = - 1 28.00 C C GUESS THE INITIAL VALUES OF QPL, QS, VPL, VS C QPLG = VPLNRM * FALB(PIPNRM) 180 Appendix F. List Of Computer Programs QSGO= (VIFNRN VPLG = VPLNRM VSG0 = VIFNRM DIV — - 400.00) * FALB(PISNRM) * 3.00/4.00 400.00 0.2500 = C SUMPIO .00 VSG=VSG0 QSGQSGO C C Solve the differential equations. C 00 20 IN PIPP CPL 1,66 PIPLPI(IN) = = = FALB(PIPP) YOLD(1) = VSG YOLD(2) = QSG CALL ROOT(RHSF,NV,0.O0,YOLO,ACC,ALPHA,YNEW,FLAG) IF(FLAG.EQ.0)THEN PRINT*, ‘*****RQQT FAILED! ****‘ STOP ENOIF VSG = YNEW(1) QSG = YNEW(2) IF(VSG.LE.O.O0.OR.QSG.LE.D.O0)ThEN PRINT*,’ IN = ‘,IN,’VSG = ‘,VSG,’QSG PRINT*,’** NEGATIVE VOL/Q GENERATED = ‘,QSG *****‘ STOP ENOIF C C Calculate the sum of square of error contributed by PISKIN. C VPLG = VTOT QPLG = QTOT CAVS = QSG PISKIN(IN) - - VSG QSG / (VSG = - VEXS) FPI(CAVS) (PISKIN(IN) -PISNRM) /PISNRH CHANGE = ERROR CHANGE-EXP6 (IN) SUMPISUNPI+ERROR**2/506 (IN) **2 20 CONTINUE 181 Appendix F. List Of Computer Programs C SUMIFV = 0.00 VSGVSG0 QSG=QSGO DO 40 1=1,19 PIPPD?IPL(I) CPL=FALB (PIPP) YOLO(i)VSG YOLO (2) =QSG CALL ROOT(RHSF,NV,0.O0,YOLD,ACC,ALPHA,YNEW,FLAG) IF(FLAG.EQ.0) THEN PRINT*, ‘*****ROOT FAILEIH*****’ STOP ENOIF VSG = YNEW(1) QSG = YNEW(2) IFQVSG.LE.O.O0.OR.QSG.LE.0.O0) THEN PRINT*,’ I = ‘,I,’VSG= ‘,VSG,’QSG= ‘,QSG PRINT*, ‘*****NEGATIVE VOL/Q GENERATED STOP ENOIF C Calculate the sum of square of error contributed by VIF. C C IF(PIPP.EQ. 17.400) THEN ERROR = 0.00 GO TO 35 END IF CHANGE= (YNEW (1) -VIFNRN) /VIFNRN ERROR= CHANGE-EXP7 (I) SUNIFVSUNIFV+ERROR**2/5O7 (I) **2 35 50 FORNAT(IX,SF1O.4) 40 CONTINUE SUN = SUN+SIJNPI+SUNIFV RETURN END C SUBROUTINE COEFF 182 Appendix F. List Of Computer Programs C C C This subroutine is used to calculate the values of the transport C parameters: LSNRN, P55, KFS. C IMPLICIT REAL*8(A-H,J,K,L,0-Z) COMMON/BLKA2/CPLNRN , CSNRM , CASNRM COMMON/BLKT/QTNRM QSNRM , QPLNRM COMMON/ELKDO/ALBSTO COMMON/BLKH/LSNRM COMMON/ELKF/PC , PCNRM , PCGRAD COMMON/BLKU/PSNRM ,PS COMMON/BLKK/VEXS COMMON/BLKBS/PIPNRM ,PISNRM COMMON/BLKL/LSS , KFS ,SIGS ,PSS COMMON/BLKJ/VTNRN,VIFNRM , VPLNRN C C Set the normal steady-state conditions. C PIPNRM = 25.900 PISNRM = 14.700 PSNRM FCOMPS(VIFNRM) = CPLNRM = FALB(PIPNRM) CASNEM = FALB(PISNRM) CSNRM = CASNEM * (1.00 QPLNRN = VPLNRM * CPLNRM QSNRM = VIFNRM * CSNRM QTNRM = QPLNRM + QSNRM VPLNRM + VSNRM = VPLNRM VSNRM = VIFNRM VTNRM = VPLNRM - VEXS / VIFNRM) C C Calculate LSNRM. EPNRM Ml = (CSNRM-(1 .00-SIGS)*CPLNRM)/(CSNRM-(1 .D0-SIGS)*CASNRM) = VIFNRM - VEXS IF(ABS(1.00-SIGS) .LT.1.0-8) THEN LSNRM ELSE = ALBSTO/ (CSNRM/ (CPLNRM-CASNRN) /W1+i . 00/VIFNRM) 183 Appendix F. List Of Computer Programs LSNRM = 184 ALBSTO/((i .OO-SIGS)*EPNRM/ (1 .OO-EPNRM)/W1+1 .OO/VIFNRM) ENDIF C C Calculate P55. C IF(ABS(i.DO-SIGS).LT.l.O-8) THEN PSS=CSNRM*LSNRM/ (CPLNRN-CASNRM) ELSE P55 Ci .DO-SIGS)*LSNRM/OLOGC1 .DO/EPNRM) = ENDIF C C Calculate KFS. C KFS PE LSNRM = = / (PCNRM - PSNRM - 5105 * CPIPNRM - PISNRM)) -DLOGCEPNRM) RETURN END C SUBROUTINE AUXSAMCY) C ********************** C C This subroutine is used to calculate all system variable values at C given compartmental protein and fluid contents. C INPLICIT REAL*8(A-H,J,K,L,O—Z) C COMMON/BLKB/JFS , AS COMNON/BLKO/QPS , QLS CONNON/BLKF/PC,PCNRM,PCGRAD COMNON/BLKG/CPL ,CS, CAVS CONNON/BLKH/JLSNRM COMMON/BLKI/PIPL ,PIS COMMON/BLKK/VEXS COMNON/BLKL/LSS ,KFS, SIGS, P55 COMMON/BLKT/QTNRM , QSNRN , QPLNRM COMMON/BLKU/PSNRN ,PS CONNON/BLKZA/PRESLO COMNON/BLKZB/QTOT ,VTOT Appendix F. List Of Computer Programs C DIMENSION Y(4) C CPL PC Y(2) I Y(1) = FCOMPC(Y(1)) = Cs Y(4)/ Y(3) = CAVS = Y(4)/ (Y(3) PIPL = FPI(CPL) PIS = FPI(CAVS) PS - VEXS) FCOMPS(Y(3)) = JFS = KFS JLS = .JLSNRM * (PC — PS LSS + * — SIGS* (PIPL (PS - - PIS)) PSNRM) IF(JLS .LT. JLSNRM) JLS=JLSNRM*(PS-PRESLO)/(PSNRM-PRESLO) ULJLS1O .OO*JLSNRM IF(JLS . GT.ULJLS) JLSUL.JLS EP= (1.00 — SIGS) * JFS / P55 IF(EP.GT.5O.00) THEN QPS=JFS* (1 DO-SIGS) *CPL . ELSEIF(EP.LT.-5O.OO) THEN QPSJFS*(1.OO—SIGS)*CAVS ELSEIF(ABS(O.D0-EP) .LT.1.D-8) THEN QPS=PSS* (CPL-CAVS) ELSE QPS = JFS *(1.oo-sIG5)*(CpL-CAv5*Exp(-Ep))/(1.oo-Exp(-Ep)) ENDIF QLS = JLS*CS RETURN END C SUBROUTINE AUXNEP (Y) C C IMPLICIT REAL*B(A-H,J,K,L,O—Z) C COMMON/BLKB/JFS , AS ,QLS ,PCNRM , PCGRAD COMMON/BLKG/CPL ,CS, CAVS 185 Appendix F. List Of Computer Programs COMMON/BLKH/JLSNRM COMMON/BLKI/PIPL,PIS COMMON/BLKK/VEXS COMMON/BLKL/LSS , KFS , SIGS ,PSS COMMON/TOT/VTOT , QTOT COMMON/ELKU/PSNRM ,PS COMMON/BLKZA/PRESLO C DIMENSION Y(2) C 3200.00 VDLPLA = VTOT VOLPLA = QPLAS QTDT PC = VOLPLA QPLAS = Y(1) + Y(2) + PCNRM = CS / Y(2) = Y(1) / CAVS = Y(2) PIPL = FPI(CPL) PIS = FPI(CAVS) PS CPL * (Y(i) VEXS) — FCDMPS(Y(1)) = JFS = KFS JLS = JLSNRN (PC * + — PS LSS * — SIGS (PS * (PIPL - PIS)) PSNRM) - IF(JLS .LT. JLSNRM) JLSJLSNRM*(PS-PRESLO)/(PSNRM-PRESLD) ULJLS=10 D0*JLSNRM . IF(JLS.GT.ULJLS) .JLS=ULJLS EP= (1.00 - SIGS) * iFS / p55 IF(EP.GT.50.DC) THEN QPS=JFS* (1. DO—SIGS) *CPL ELSEIF(EP.LT.-50.DD) THEN QPS=—JFS* (1.00-5105) *CAV5 ELSEIF(ABS(0.D0-EP).LT.1.D-8) THEN Qp5=PSS* (CPL-CAVS) ELSE QPS = iFS *(1.DQ-5I05)*(CPL—CAV5*EXP(—Ep))/(1.DQ-EXp(—EP)) ENDIF QLS = RETURN END .JLS * CS 186 Appendix F. List Of Computer Programs C 187 ****************************** SUBROUTINE TOMACA,B,C,O,X,N) ****************************** C C C This subroutine uses the Thomas algorithm to find the solution C of a tridiagonal matrix. C IMPLICIT REAL*8CA-H,O-Z) C C Thomas algorithm C DIMENSION ACN),B(N),C(N),O(N),X(N),PC1OI),QCiD1) C NM = N - PU) = —CU) / BU) QC1) = OC1) / BC.) 1 00 10 I IM 10 I—i = DEN AU) = PCIM) * PCI) = -CCI) / OEN QCI) = COCI) — B(I) + ACI) QCIM)) / OEN * CONTINUE XCN) = QCN) DO 20 II I = = N XCI) 20 2,N = - = 1,NM II PCI) * XCI+1) + QCI) CONTINUE RETURN END C ************************************ DOUBLE PRECISION PUNCTION FALBCPI) C ************************************ C C This function calculates Albumin concentration C CAlb) vs. C.0.Ppl relationship. C IMPLICIT REAL*8CA-H,O-Z) C - using a fitted Appendix F. List Of Computer Programs 188 CONMON/BLKCC/CALB1 ,CALB2 C FALB (CALB1 = + CALB2 * P1) / 1.0+3 RETURN END ********************************** C DOUBLE PRECISION FUNCTION FPI(C) C C C This function calculates colloid osmotic pressure C fitted C.O.P vs. (Alb) relationship. - using a C IMPLICIT REAL*8(A-H,O-Z) C COMMON/BLKN/CPI1 ,CPI2 C R 1.0+3 = C * FPI = CPI1 + CPI2 * R RETURN END C **************************************** DOUBLE PRECISION FUNCTION RHSFA(IJX,Y) **************************************** C C C This function evaluates the non-linear equations for a given C set of Y values. Same as RHSFB,RHSFC,RBSFD,RHSFE,RHSFF. C IMPLICIT REAL*8(A—H,J,K,L,O—2) C COMMON/BLKB/JFS , JLS CDMMON/BLKD/QPS , QLS DIMENSION Y(i) C CALL AUXALT(Y) RHSF3 = JFS RHSF4 = QPS — - GOTO(1O,20),I 10 RI-ISFA = RHSF3 JLS QLS Appendix F. List Of Computer Programs RETURN 20 RHSFA = RHSF4 RETURN END C DOUBLE PRECISION FUNCTION RHSFB(I,T,Y,IPET) C C IMPLICIT REAL*8(A-H,.J,K,L,O-Z) C CONMON/BLKB/JFS , JLS CONMON/BLKD/QPS , QLS DIMENSION Y(4) ,SAL1(4) ,PRO(4) C DATA SAL1/100.DD,100.OD,200.DD,200.DD/ DATA PRO? 0.00, 25.00, 0.00, 50.00? C CALL AUXSAM(Y) IF(T.LT.0.000)THEN RHSFI = .JLS RHSF2 = QLS RHSF3 = iFS RHSF4 QPS — iFS + SAL1(IPETV1.500 - QPS + PRO(IPET)*.6SDO/1.SDO — — AS QLS ELSEIF(T.GT.D.DDO.AND.T.LT. 13.SDD)TNEN RNSFI. = AS — iFS RHSF2 = QLS — QPS RHSF3 = iFS RHSF4 = QPS RHSF1 = JLS RHSF2 = QLS RHSF3 = iFS RHSF4 = QPS — — AS QLS ELSE - — - - iFS QPS JLS QLS ENDIF GDTO(10,2D,3D,40) ,I 10 RHSFB = RUSF1 = RHSF2 RETURN 20 RHSFB + 1400.DD?13.SDO 189 Appendix F. List Of Computer Programs RETURN 30 RHSFB = RHSF3 = RHSF4 RETURN 40 RHSFB RETURN END C DOUBLE PRECISION FUNCTION RHSFCCI,T,Y,IPET) C C IMPLICIT REAL*8(A-H,J,K,L,0—Z) C COMMON/BLKB/JFS,JLS COMMON/BLKD/QPS QLS COMMON/B/T2(3,S) DIMENSION Y(4) ,SAL2(5) C DATA SAL2/4*1000.D0,900.D0/ C CALL AUXSAM(Y) IF(T.LE.T2(2,IPET)/60.D0)THEN RHSF1 RHSF2 itS = QLS — — RHSF3 = IFS RHSF4 = QPS - RHSF1 = itS - RHSF2 = QLS RHSF3 = .TFS — RHSF4 = QPS - — JFS QPS ES QLS ELSE — JFS QPS JLS QLS ENOIF GDTO(10,20,30,40),I 10 RHSFC = RHSF1 = RHSF2 = RHSF3 RETURN 20 RHSFC RETURN 30 RHSFC RETURN + SAL2(IPET)/T2(2,IPET)*60.O0 190 Appendix F. List Of Computer Programs 40 RHSFC = RHSF4 RETURN END C ********************************************* DOUBLE PRECISION FUNCTION RHSFO(I,T,Y,IPET) ********************************************* C C IMPLICIT REAL*8(A-H,J,K,L,O-Z) C COMMON/BLKB/JFS ,JLS COMMON/BLKO/QPS, QLS COMMON/B/T2(3,5) DIMENSION Y(4) ,SAL3(2) ,PRO3(2) C DATA SAL3/4420 .0500,5996. 83D0/ DATA PRO3/29.3500,9.6500/ C CALL AUXSAM(Y) IF(T.LT.1:soo) THEN RHSF1 = .JLS - JFS + SAL3(IPET)/1.SDO RHSF2 RHSF3 = QLS - QPS + PRO3(IPET)*.6500/1.SDO = JFS RHSF4 — JLS = QPS = JLS - QLS ELSE RHSF1 — JFS RHSF2 = QLS - QPS RHSF3 = JFS - JLS RHSF4 = QPS - QLS ENDIF GOTO(10,20,30,40),I 10 RHSFD = RHSF1 = RHSF2 = RHSF3 = RHSF4 RETURN 20 RHSFD RETURN 30 RHSFD RETURN 40 RHSFD RETURN 191 Appendix F. List Of Computer Programs END ********************************************* C DOUBLE PRECISION FUNCTION RHSFE(I,T,Y,IPET) C C INPLICIT REAL*8(A—H,J,K,L,O-Z) C COMNON/BLKB/JFS , JLS COMMON/BLKO/QPS, QLS DIMENSION Y(4) ,SAL4(2) ,URI(2) C DATA SAL4/-2800 .00,2000.00/ OATA URI/O .000,1500.00/ C CALL AUXSAN(Y) IF(T.LT.2,000)THEN RHSF1 = ES RBSF2 = QLS — - .JFS + SAL4(IPET)/2.OO QPS RRSF3=JFS— JLS RHSF4 = QPS - QLS ELSEIF(T.GE.2,000.ANO.T.LT.24.000)THEN RHSF1 = JLS RHSF2 = QLS RHSF3 = JFS RHSF4 = QPS RHSF1 = JLS RNSF2 = QLS RHSF3 = iTS RHSF4 = QPS — — - - JFS QPS JLS QLS ELSE - - - - JFS QPS AS QLS ENOIF GOTO(1O,20,30,40) ,I 10 RUSFE = RHSF1 = RHSF2 = RHSF3 RETURN 20 RESFE RETURN 30 RHSFE RETURN - URI(IPET)/22.DO 192 Appendix F. List Of Computer Programs 40 RHSFE = RHSF4 RETURN END **************************************** C DOUBLE PRECISION FUNCTION RHSFFCI,T,Y) C C IMPLICIT REAL*8(A-H,J,K,L,O-Z) C CDMMON/BLKB/JFS , AS COMMON/BLKD/QPS , QLS DIMENSION Y(4) C CALL AUXSAM(Y) RHSF1 = JLS RHSF2 = QLS RHSF3 = JFS RHSF4 = QPS - - - - JFS + 1750.D0/2,SDO QPS AS QLS GO-TD(10,2&,30,40),I 10 RHSFF = RHSF1 = RHSF2 = RHSF3 = RHSF4 RETURN 20 RHSFF RETURN 30 RHSFF RETURN 40 RHSFF RETURN END C **************************************** DOUBLE PRECISION FUNCTION RHSF(I,X,Y) C **************************************** C IMPLICIT REAL*8(A-H,J,K,L,D—Z) C COMMON/BLKB/JFS , AS COMMON/BLKD/QPS , QLS DIMENSION Y(2) C 193 Appendix F. List Of Computer Programs CALL AUXNEP(Y) RHSF3 = JFS - JLS RHSF4 = QPS - QLS GOTO(10,20),I 10 RHSF = RHSF3 RETURN 20 RHSF = RHSF4 RETURN END C BLOCK DATA C C IMPLICIT REAL*8(A-H,K,L,O-Z) C COMMON/B/T2(3,S) COMNON/BLKF/PC ,PCNRM,PCGRAD CONNON/BLKJ/VTNRN,VIFNRN,VPLNR14 COMMON/BLKK/VEXS COMMON/BLKN/CPI 1, CPI2 CONNON/BLKO/VSP(14) ,PSP(14) ,AS,BS,NPS,NPSN1 CONNON/BLKZ/DIFV(19) ,DPIPL(19) ,NN COMMON/BLKHH/PIPLPI (66) ,PII (66) CONNDN/BLKCC/CALB1 , CALB2 COMMON/BLKDD/ALBSTO COMMDN/BLKZA/PRESLO C DATA VEXS/2.1D+3/ DATA PCGRAD/0 . 00965BD0/ DATA PRESLO/ 13.057700/ DATA VTNRN,VIFNRM,VPLNRN/11 .60+3,8.40+3,3.20+3/ DATA CPu ,CPI2/1 .754900—4,0.65684000/ DATA CALB1 ,CALB2/-0.2671730-3,0 .1522440+1/ DATA NPS,NPSN1/14,13/ DATA VSP/B.4D+3,8.92D+3,9.45D+3,9.970+3,10.500+3,1j.02D+3, + 11.550+3,12.07D+3,12.600+3,13.65D+3,14.700+3,16.80D÷3, + 21.000+3,25.230+3/ DATA PSP/-0.7000,0.3200,0.8600,1,15D0,1.37D0,1.5600, 194 Appendix F. List Of Computer Programs + + 1.69D0,1.80D0,1.8800,1.99D0,2.O1DO,2.04D0,2.1200, 2.20D0/ DATA DPIPL/9.2D0,1O.7D0,11.7D0,12.7D0,13.7D0,13.7D0, + + 14.2DO,14.2DO,17.4DO18.7DO,18.7DO,2Q.7DQ,2O.7DQ, 21.7D0,24.7D0,25.7D0,25.7D0,26.7D0,26.7D0/ DATA DIFV/19.O1D+3,16.02D+3,14.40D+3,12.98D+3,7.98D+3,6.71D+3, + 1O.7OD+3,8.27D+3,O.ODO,9.944D+3,9.11D+3,7.79D+3,6.73D+3 + 6.61D+3,8.44D+3,8.34D+3,7.69D+3,8.25D+3,9.35D+3/ DATA P11/ 2.3130, 5.9D0, 1.9D0, 2.6D0, 3.4D0, 2.9D0, + 0.9D0, 2.7D0, i.7D0, 4.9D0, 4.9130, 3.2D0, + 4.4D0, 3.9D0, 0.7D0, 1.6D0, 2.9D0, 5.7D0, + 5.4D0, 3.9D0, 3.9D0, 4.7D0, 4.0130, 1.9D0, + 4.5D0, 4.3D0, 3.4D0, 7.ODO, 5.7D0, 7.2D0, + S.7D0, 4.7D0, 4.9D0, 6.4D0, 6.4D0, 3.7D0, + 5,9D0, 7.7D0, 5.9D0, 3.9D0, 8.ODO, 8,6D0, + 8.2D0, 8.4D0, 5.2D0, 6.7D0, 6.9D0, 9,9D0, + 7.7D0,10.7D0, 7.7D0,10.4D0,1O.9D0, 8.7D0, + 14.7D0, i1.7D0,13.2D0,14,ODO,12.2130,j1.7D0, + 12.7D0,15.2DD,14.4D0,13.2D0,13.7D0,14.7130/ DATA PIPLPI/ 2.3D0, 5.3D0, 6.8D0, 6.6D0, 7.ODO, 7.ODO, + 7.8130, 8.31)0, 8.9D0, 9.0130, 9.ODO, 9.2D0, + 9.5D0, 9.5D0, 9.6D0, 9.9D0,10.3D0,10.7D0, + 11.3D0,i1.5D0,11.5D0,11.7130,11.9130,12.3130, + 12.4D0,12.9D0,13.ODO, 13.2D0, 13.7D0,13.9D0, + 14.2D0,14.2D0,14.5D0,15.5D0,15.5D0,15.7D0, + 17.3D0,17.4D0,17.5D0,17.5D0,17.9130,18.oDo, + 18.2D0,18.4D0,18.7D0,18.7D0,19.ODQ,19.oDa, + 19.4D0,19,9D0,20.7D0,21 .ODO,21 .0D0,21.7D0, + 22.9D0,23.6D0,23.6130,23.6D0,24.QDQ,24.4D0, + 24.8D0,24,2D0,24.4D0,24.9D0,26.4D0,26.9D0/ C DATA AS,BS/1.96154D-3,1 .81347D—5/ C DATA AS,BS/1 . 96154D—3,5.0000D-5/ DATA AS,BS/1.96154D-3,1 .05D-4/ DATA ALBSTO/0 . 020539D0/ DATA T2/ 0.D0, 9.5D0,47.5D0, + 0.D0,11.ODO,71.ODO, + 0.D0,13.0D0,28.0D0, + 0.DO,11.DDO,41.ODO, + 0.D0, 9.0D0,49.0D0/ 195 Appendix F. List Of Computer Programs END C ************************************* C ************************************* DOUBLE PRECISION FUNCTION FCOMPS(V) C C This function calculates the skin C compartments hydrostatic pressure. C IMPLICIT REAL*8(A-H,O-Z) C COMMON/BLKO/VSP(i4),PSP(14),AS,BS,NPS,NPSM1 C IF(V.LE.VSP(1)) GO TO 10 IF(V.GE.VSP(9)) GO TO 20 FCOMPS = FS(V) = PSP(i) + AS * PSPCP) + BS * RETURN iO FCONPS C V — VSPC1)) RETURN 20 FGOMPS KV — VSPC9)) RETURN END ******************* C SUBROUTINE SPLINS C C C This subroutine SPLINES the experimently C obtained PS vs. VS data set. C IMPLICIT REAL*B(A-H,O-Z) C COMMON/BLKO/XCi4) ,YC14) ,A1,BN,N,NM COMMON/BLKQ/QC1Oi) ,RC1O1) ,S(100) C DIMENSION HC100) ,AC1O1) ,BC10I) ,C(101) ,OC1O1) C 10 DO 10 I = 1,NM H) = XCI+i) CONTINUE — XCI) 196 Appendix F. List Of Computer Programs AC1) = 0,00 BU) = 2.00 CU) = MCi) 0(1) = 3.00 DO 20 I 20 * HO) * CCYC2) Y(1)) / MU) — — 197 Al) 2,NM = IP = 1+1 IM = I-i AU) = MCDI) BCI) = 2.00 CCI) = MCI) DCI) = 3.00 * (MCDI) * CCYCIP) MCI)) + YCI)) / MCI) — YCNM)) / MCNM) - — CYCI) — Y(IM))/MCIM)) CONTINUE ACN) = BCN) = 2.00 CCN) = 0.00 OCN) = -3.00 MCNM) MCNM) * * CCYCN) - BN) CALL TOMACA,B,C,O,R,N) 00 30 I = I QCI) = CYCIP) SCI) = CRCIP) IP 30 l,NM = + 1 — — YCI)) / MCI) MCI) — RCI)) / C3.00 * * (2.00 *RCI)+R(IP))/3.00 MCI)) CONTINUE RETURN END C **************************+****** C ********************************* DOUBLE PRECISION FUNCTION FSCZ) C C C This function evaluates skin compartment hydrostatic pressure - using a splined set of data. C IMPLICIT REAL*BCA-M,O-Z) C COMMON/BLKO/X(14) ,Y(l4) ,A1,BN,N,NM COMMON/BLI{Q/QClOl) ,R(101) ,S000) C 11 IF(Z.LT.X(1)) GO TO 30 Appendix F. List Of Computer Programs IF(Z.GE.X(NM)) GO TO 20 3 10 NM = K(I+J)/2 IF(Z.LT.X(K)) 3 = K IF(Z.GE.X(K)) I = K IF(J.EQ.I+1) GO TO 30 GO TO 10 20 INM 30 OX=Z—x(I) FS = Y(I) + OX * (Q(I) + OX * (RU) + ox * 5(I))) RETURN ENO C ************************************* DOUBLE PRECISION FUNCTION FCOMPC(V) C ************************************* C C This function is used to calculate capillary hydrostatic C pressure at given volume. CIMPLICIT REAL*B(A-H,O-Z) C COMMON/BLKF/PC,PCNRM , PCGRAO COMMON/BLKJ/VTNRM ,VIFNRM,VPLNRM C FCOMPC = PCNRN + PCGRAO * IF(FCOMPC.LT.2.O0) FCOMPC (V = - VPLNRM) 2.00 RETURN END C *************************************************** SUBROUTINE RK4C(F,M,A,B ,YA,EPS,YB ,NFUN,FLAG, IPET) C *************************************************** C IMPLICIT REAL*B(A-H,K,O-Z) INTEGER FLAG DIMENSION YA(M) ,YI(lO),YB(N),YBOLO(iO),YARG1(10),yARG2(10) DIMENSION K1(1G) ,K2(10) ,K3(iO) ,K4(10) C FLAG1 198 Appendix F. List Of Computer Programs NINT=1 NFUN=0 10 NFUN=NFUN+4*NINT*M Dfl (B-A) /NINT XIA DO 20 11,M 20 YI(I)=YA(I) 00 70 INT1,NINT 00 30 11,M K1(I)=Dx*Fa ,XI ,YI,IPET) 30 YARG1(I)=YI(I)+K1(I)/2.O0 XARG=XI+0X/2 .00 00 40 11,M K2(I)=DX*F(I ,XARG,YARG1 ,IPET) 40 YARG2(I)=YI(I)+K2(I)/2.D0 00 80 11,M K3 (I) 0X*F (I ,XARG,YARG2 ,IPET) 50 YARG1(I)=YI(I)+K3(I) xI=xI+0x 0060 11,M K4w=ox*Fa ,XI ,YARG1 ,IPET) 60 YI(I)YI(I)+(K1(I)+2.O0*(K2(I)+K3(I+K4(I))/6.oo 70 CONTINUE 80 YB(I)=YI(I) 00 80 11,M IF(NINT.EQ.1) GO TO 100 YBOIFM=0 .00 00 90 1=1,11 90 YBOIFM=OMAX1(DABS((YB(I)—YBOJ.D(I))/YB(I)),yBDIfl4) IF (YNOIFM . LT EPS) RETURN . IF(NINT.GT.10000) GO TO 120 100 110 DO 110 I=1,M YBOLO(I)=YB(I) NINT2*NINT GO TO 10 120 FLAG=0 RETURN ENO 199 Appendix F. List Of Computer Programs C 200 ********************************************** SUBROUTINE GAUSS(A,N,NOR,NOC,X,RNORM, IERROR) C C C Purpose: C Uses Gauss elimination with partial pivot selection to C solve simultaneous linear equations of form JA(*;X:;C:. C C Arguments: C A Augmented coefficient matrix containing all coefficients C and R.H.S. constants of equations to be solved. C N Number of equations to be solved. C NOR First (row) dimension of A in calling program. C NOC Second (column) dimension of A in calling program. C X C RNORM C IERROR Solution vector. Measure of size of residual vector ;C:-]A(*;X:. Error flag. C 1 Successful Gauss elimination. C =2 Zero diagonal entry after pivot selection. C IMPLICIT REAL*B(A-H ,O-Z) OIMENSION A(NDR,NOC) ,x(N) ,B(SO,51) C NM = N NP = N 1 - i + C C Set up working matrix B C 00 20 I 00 10 .3 20 1,NP = B(I.3) 10 1,N = = A(I,J) CONTINUE CONTINUE C C Carry out elimination process N-i times C 00 80 IC = i,NN K + 1 KP C = - - Appendix F. List Of Computer Programs C Search for largest coefficient in column K, rows K through N C IPIVOT is the row index of the largest coefficient C BIG 0.00 = DO 30 I SF K,N = OABS(B(I,K)) = DO 26 3 SF 26 KP,NP = OMAX1(SF,OABS(BCJ,K))) = CONTINUE AB OABS(B(I,K) = / SF) IF(AB.LE.BIG) GOTO 30 BIG AB = IPIVOT 30 I = CONTINUE C C INTERCHANGE ROWS K AND IPIVOT IF IPIVOT.NE.K C IF(IPIVOT.EQ.K) GOTO 50 00 40 3 TEMP K,NP = B(IPIVOT,J) = B(IPIVOT,J) B(K,J) = B(K,J) TEMP = 40 CONTINUE SO IF(B(K,K).EQ.O.DO) GOTO 130 C C Eliminate BCI,K) from rows K+i through N C 00 70 I QUOT = B(I,K) 60 KP,N = B(I,K) / B(K,K) 0.00 = DO 60 3 = KP,NP 8(1,3) = 6(1,3) — QUOT * B(K,J) CONTINUE 70 CONTINUE 80 CONTINUE IF(B(N,N).EQ.0.O0) GOTO 130 C C C Back substitute to find solution vector 201 Appendix F. List Of Computer Programs XCN) DO 100 II = 0.00 I N - IP II I = 90 IP,N = SUM = B(I,J) + * XCJ) CONTINUE XCI) 100 1 + DO 90 3 SUM B(N,N) 1,NM = SUM = / BCN,NP) = 202 (B(I,NP) = — SUM) / BCI,I) CONTINUE C C Calculate norm of residual vector, C-A*X C Normal return with IERROR 1 = C RSQ 0.00 = 00 120 I SUN DO 110 .1 SUM 110 1,N = SUM = + ACI,J) * XCJ) dONTIUE RSQ 120 l,N = 0.00 = RSQ = + CDABS(ACI,NP) - SUM)) ** 2 CONTINUE RNORM IERROR OSQRTCRSQ) = = 1 RETURN C C Abnormal return because of zero entry on diagonal, IERROR2 C 130 IERROR = 2 C C C find the solution of a set of simultaneous END C ********************************************** SUBROUTINE ROOTCF,M,X,YOLD,EPS,ALPHA,Y,FLAG) C ********************************************** C C This subroutine uses Newton’s method to RETURN RETURN Appendix F. List Of Computer Programs C find the solution of a set of simultaneous C non-linear equations. C IMPLICIT REAL*8(A-H,O-Z) INTEGER FLAG DIMENSION YOLD(M) ,Y(M),DY(1O),DELY(10),A(lo,11) C NP = N 1 + DO 10 I Y(I) 1,M YOLD(I) = DELY(I) 10 1.0—6 = * YOLO(I) CONTINUE DO 60 ITER FLAG 1,100 = 0 = DO 40 I 1,M DO 30 J = iMP IF(J.EQ.MP) GOTO 20 y(J) FUP y(j) = DELY(3) F(I,X,Y) = Y(J) Y(J) = FDOWN y(j) + A(I,J) 2.00 DELY(J) * F(I,X,Y) = = - y(3) + DELY(J) = (FUP = —F(I,X,Y) — FDOWN) / (2.00 * DELY(J)) GOTO 30 20 A(I,J) 30 CONTINUE 40 CONTINUE CALL GAUSS(A,M,10,11,DY,RNORM,IERROR) C FLAG 1 DO 50 I Y(I) = = 1,M YCI) + ALPHA * DY(I) IF(DABS(DY(I)).GT.EPS) FLAG 50 CONTINUE IF(FLAG.EQ,1) RETURN 60 CONTINUE RETURN END = 0 203 Appendix F. List Of Computer Programs C ********************** SUBROUTINE AUXALT(Y) C C C This subroutine is used to calculate the system variables from C compartmental fluid and protein contents. C IMPLICIT REAL*8(A-H,J ,K,L,O-Z) COMMON/BLKB/JFS , .315 COMMON/BLKO/QPS , QLS COMMON/BLKF/PC , PCNRM,PCGRAO COMMON/B LKG/CPL ,CS, CAVS COMNON/BLKH/JLSNRM COMMON/BLKI/PIPL ,PIS COMMON/BLKK/VEXS COMMON/BLKL/LSS ,KFS ,SIGS ,PSS COMMON/BLKZB/QTOT ,VTOT COMMON/BLKU/PSNRM ,PS COMMON/BLKZA/PRESLO COMMON/VQPL/VPL , QPL DIMENSION Y(i) C qis CS / Y(i) QIS = CAVS PS Y(l)*FALB(PIS) = / (Y(i) QIS = - vExS) FCOMPS(Y(Ij) = JFS = KFS JLS = JLSNRM (PC * + PS — LSS * - SIGS (PS — * (PIPL - PIS)) PSNRM) IF (.315. LT. JLSNRM) THEN JLSJLSNBN* (P5-PRESLO) / (PSNRM—PRESLO) ENOIF ULJLS1O .OO*JLSNRM IF(JLS GT.ULJLS) JLSULJLS . EP= (1.00 — SIGS) * iFS / P55 IF(EP.OT.SO.DO) THEN QPS.TFS* Ci .00—5105) *CPL ELSEIF (EP LT. -50.00) THEN . QPS=JFS*(1 .D0-SIGS)*CAVS ELSEIF(ABS(0.D0-EP).LT.l.O-B) THEN 204 Appendix F. List Of Computer Programs QPSPSS* (CPL-CAVS) ELSE QPS JFS *(1.O0—SIGS)*(CPL—CAVS*EXP(—Ep))/(j.D0-EXP(-Ep)) = ENDIF QLS = .JLS * CS RETURN END ***************************************************************** C SUBROUTINE OESOLV(F,M,A,B,YO,EPS,HSTART,HMIN,HMAX,YB,FLAG,IPET) C C IMPLICIT REAL*B (A-H, O-Z) DIMENSION YO(M) ,YA(10) ,YB(M) EXTERNAL F INTEGER FLAG BNAB-A HOLD=HSTART DO 10 I=1,M 10 YA(I)=Y0(I) 20 CALL RKF(F,M,X,YA,HOLD,YB,YOIF,IPET) GAZIMM BD0* (EPS*HOLD/ (BMA*YDIF) ) **O .2500 . HNEW=GAMMA*MOLD IF(GAMMA.GE.1.D0) GO TO 30 IF(HNEW.LT.HOLD/10.OO) HNEW=HOLD/10. IF(BNEW.LT.HMIN) GO TO 50 HOLDHNEW GO TO 20 30 IF (HNEW GT .5. D0*HOLO) HNEW5 O0*HOLD . IF(NNEW GT .HMAX) HNEWHMAX . IF(X+HOLD.GE.B) GO TO 70 XX+H OLD HOLDHNEW DO 40 11,M 40 YA(I)=YB(I) GO TO 20 50 FLAG=0 BX . 205 Appendix F. List Of Computer Programs DO 60 11,M 60 YB(I)YA(I) RETURN 70 FLAG=1 HSTART=HNEW HDLDB-X CALL RKF(F,M,X,YA,HOLD,YB,YDIF,IPET) RETURN END C SUBROUTINE RKF(F,M,X,YOLD,H,YNEW,YDIFM,IPET) C C IMPLICIT REAL*8(A-H,D-Z) DIMENSION YOLD(N) ,YNEW(M) ,YARG1(10) ,YARG2(10) REAL*8 K1(1D),K2(10),K3(10),K4(10),KS(10),K6(10) C DATA C21,C31,C32,C33/.2500, .375D0, .0937500, .2612500/ DATA C41,C42/.92307692307692307300, .87938097405553025700/ DATA C43,C44/-3 .2771961766044606100,3.32089212S62S8532300/ DATA C51 ,C52/2 . 03240740740740722D0, -B. 00/ DATA C53,CS4/7. 1734892787524364700,-.2058966B61598440SD0/ DATA C61 C62/ SDO , . ,— . 29629629629629629400/ DATA C63,C64/2.D0,-1 .3816764132S5360500/ DATA C6S,C66/ .4529727095S1656916D0,—.275D0/ DATA C71,C72/ . 118518518518518509D0, .51898635477582845400/ DATA 3,C74/ .506131490342016654D0, —. 1800/ DATA C75,CB1/.0363636363636363636D0, .00277777777777777778D0/ DATA C82 C83/— .0299415204678362568D0, , — .029199893673577883800/ DATA CB4,C85/.O200, .036363636363636363600/ DO 10 11,M K1(I)3*F(I,X,YDLD,IPET) 10 YARG1(I)YDLD(I)+C21*K1(I) XARG=X+C21*H DO 20 I=1,N K2(I)=H*F(I,XARG,YARG1 IPET) , 20 YARG2 (I) YDLD(I)+C32*K1 CI)+C33*K2 (I) XARG=X+C31*H 206 Appendix F. List Of Computer Programs 00 30 11,M K3(I)H*F(I,XARG,YARG2,IPET) 30 YARG1(I)Y0L0(I)+C42*Ki(I)+C43*K2(I)+C44*K3(I) XARGX+C41*H 00 40 I=1,M K4(I)H*F(I ,XARG,YARG1, IPET) 40 YARG2(I)=YOLO(I)+C51*K1(I)+C62*K2(I)+C53*I{3(I)+C54*J{4(I) XARG=X+H 00 50 11,M K5(I)=H*F(I ,XARG,YARG2,IPET) 50 YARG1(I)=YOLD(I)+C62*K1(I)tC63*K2(I)+C64icK3(I)+C65*K4(I) 1 +C66*I{5(I) XARGX+C61*H 00 60 11,M 60 K6(I)=H*F(I,XARG,YARG1,IPET) YDIFMO .00 00 70 11,M YNEW(I)Y0L0(I)+C7i*Ki(I)÷C72*K3(I)+C73*J(4(I)÷C74*K5(I)+ 1 C75*K6(I) YDIFOABS(C81*Ki(I)+C82*K3(I)+C83*K4(I)÷C84*K5(I)+C55*K6(ifl) 70 IF(YDIF.GT.YOIFM) YDIFM=YDIF RETURN END C ************************************* DOUBLE PRECISION FUNCTION FVPL(PCC) C ************************************* C C This function is used to calculate the plasma volume at given C capillary hydrostatic pressure. C IMPLICIT REAL*8(A-H,O-Z) COMMON/BLKF/PC,PCNRM,PCGRAD COMMON/BLKJ/VTNRN , VIFNRM, VPLNRM C FVPL = VPLNRM + (PCC - PCNRM)/PCGRAO IF(FVPL.LT.2372.O0) FVPL RETURN END = 2372.00 207 Appendix F. List Of Computer Programs F.3 Listing of program PATDYN C Filename: PATDYN C This progam is used to simulate the transient responses of the C microvascular exchsnge system after a single intravenous infusion C of human albumin. Unless specify, subroutines and functions called C in this program are the same as those listed in Section F.2 with C the same names. C IMPLICIT REAL*8(A-H,K,L,J,O—Z) INTEGER FLAG,FLAG6,FLAG7,FLAGG COMMON/BLKA2/CPLNRM ,CSNRM , CASNRM COMMON/BLKB/JFS , JLS CCMMON/BLKD/QPS ,QLS COMMON/3LKG/CPL,CS,CAVS COMMON/5LKI/PIPL ,PIS COMMON/BLKF/PC ,PCNRM,PCGRAD COMMON/BLKH/LSNRN COMMON/BLKJ/VTNRN ,VIFNRM , VPLNRM COMMON/BLKK/VEXS COMMON/BLKL/LSS , KFS , SIGS ,PSS COMMON/BLKT/QTNRM, QSNRM , QPLNRM COMMON/BLKU/PSNRM ,PS COMMON/BLKBB/PIPNRM ,PISNRM COMMON/BLKDD/ALBSTO COMMON/BLKZB/QTOT , VTOT COMMON/BLKZJ/EXPLVL (4) ,EXCPL(4) ,EXDLPI (3) ,EXDLCA(3) COMMON/BLKZK/EXPIPL(4) ,EXPII(3) ,EXCAV(3) DIMENSION SOLN(16,769),T(lOOl),SOL(iOOi) ,SOLF(16) DIMENSION YOLO(4),YNEW(4),YOLOA(2),YNEWA(2),YFINAL(2) DIMENSION YEX1(4),YEX9(4),YEX3(4) ,YEX2(3),YEX11(3), + YEX1T(3) ,YEX18(3) EXTERNAL RHSF,RHSFA C DATA ACC,ALPHA,EPS,DT,NP,NV/i,O-6,.500,i.000,O.OSOO,769,4/ FLAG7 = 0 FLAGS = 0 TSTART1 .0-1*0 .00500 TMIN=l .0-4*0.00500 208 Appendix F. List Of Computer Programs TMAXDT C C Set LSS, SIGS, and PCNRM paremeter values. C LSS 43.08086800 = SIGS 0.9887763700 = PCNRM 11.000 = CALL SPLINS CALL COEFF C C Set initial fluid volumes end protein contents. C PLINIT = 10.57400 PIINIT = 3.30400 VIFINT = 18250.00 VPINIT = VPLNRM QPINIT = VPLNRM VSINIT = VIFINT QSINIT = FALB(PIINIT) VTINIT = VPINIT + VSINIT QTINIT = QPINIT + QSINIT YNEW(1) = VPINIT YNEW(2) = QPINIT YNEW(3) = VSINIT YNEW(4) = QSINIT * FALB(PLINIT) * (VIFINT — VEXS) C C Solve differential balances. C 00 30 I TIM TI 1,1940 (I-1)*OT = = = I * DT IF(I.EQ,1) GOTO 14 CALL DESOLV(RHSF,NV,TIM,TI,YOL0,Ep5,TSTP.RT,TMIN,TMAX, + YNEW,FLAG) IF(FLAG.EQ,0) GOTO 33 CALL AUXSAM(YNEW) C C C Store results at select time intervals. 209 Appendix F. List Of Computer Programs 14 IF(TI.LT.3.DO.OR.TI.EQ.3.000)THEN IK=(I—1)/4 +1 REALI RI I = (REALI = RIK — 1.00)14.00 + 1.00 1K CHK RI = RIK - ELSE IK=(I-i)120+ 13 REALI RI I = (REALI—l.O0)/20.O0 = RIK = 1K CHK = RI + 13,00 RIK - ENOIF IF(CHK.NE.0.O0) GOTO 56 15 CALL AUXSAN(YNEW) T(IK)0T*(I—1) 56 SOLN(1,IK) = YNEW(1) SOLN(2,IK) = YNEW(2) SDLN(3,IK) = PC SOLN(4,IK) = PIPL SOLN(5,IK) = CPL SOLN(6,IK) = PIS SOLN(T,IK) = CS SOLN(8,IK) = CAVS SOLN(9,IK) = PS SOLN(1O,IK) = .TFS SOLN(11,IK) = JLS SOLN(12,IK) = QPS SOLN(14,IK) = QLS SOLN(15,IK) = YNEW(3) SOLN(16,IK) = YNEW(4) DO 20 J = YOLO(J) 20 30 - 1,NV = YNEW(J) CONTINUE CONTINUE C C Calculate final steady-state variable values and store them. 210 Appendix F. List Of Computer Programs VTOT = YNEW(1) + YNEW(3) QTOT = YNEW(2) + YNEW(4) YOLOA(1) = YNEW(3) YOLDA(2) = YNEW(4) CALL ROOT(RHSFA,2,1000.0O,YOLOA,ACC,ALPHA,YFINAL,FLAGG) CALL AUXALT(YFINAL) SOLF(1) = VTOT - SOLF(2) = QTOT - SOLF(3) = PC SCLF(4) = PIPL SOLF(S) = CPL SOLF(6) = PIS SCLF(7) = CS SOLF(8) = CAVS YFINAL(1) YFINAL(2) PS SOLF(9) SOLF(10) = JFS SOLF(11) = .TLS SOLF(12) = QPS SOLF(14) = QLS SOLF(1S) = YFINAL(1) SCLF(16) = YFINAL(2) C C Output variables stored and associated times. C 00 112 1=1,109 WRITE(6,91)T(I) ,SOLN(4,I),SOLN(6,I) ,SOLNC3,I) ,SOLNC9,I) 91 112 FORMAT(6X,F6.3,SX,F7.4,5X,F7.4,SX,F7.4,5X,F7.4) CONTINUE WRITE(6,4i)SOLF(4),SOLF(6),SOLF(3),SOLF(9) 41 FORMAT(17X,F7.4,5X,F7.4,5X,F7.4,5X,F7.4) 00 lii 1=1,109 WRITE(6,95)T(I),SOLN(15,I),SOLN(1,I),SOLN(16,I),SOLN(2,I) 95 FORMAT(6X,F6.3,SX,F10.3,SX,F7.2,SX,F7.3,5X,F7.3) 111 CONTINUE 45 FORNAT(17X,F10.3,5X,F7.2,SX,F7.3,5X,F7,3) WRITE(6,45)SOLF(15) ,SOLF(1) ,SOLF(16) ,SOLF(2) 00 113 1=1,109 WRITE(6,99)T(I),SOLN(10,I),SOLN(11,I) ,SOLN(12,I),SOLNC14,I) 99 FORMAT(6X,F6.3,5X,F8.2,5X,F8.2,5X,F8.3,5X,F8.3,SX,F8.3) 211 Appendix F. List Of Computer Programs 113 CONTINUE WRITE(6,49)SOLF(1O) ,SOLF(11) ,SOLFC12) ,SOLF(14) 49 FORNAT(17X,F8.2,5X,F8.2,5x,F8.3,5x,F8.3,5x,FB.3) 00 117 1=1,109 WRITE(6,58)T(I),SOLN(5,I),SOLN(5,I),5OLN(7,I) 68 117 FORMAT(6X,F6.3,5X,F7.S,5X,F7.5,5X,F7.5) CONTINUE WRITE(6,24)SOLF(6) ,SOLF(8) ,SOLF(7) 24 FORMAT(17X,F7.5,5X,F7.5,6X,F7.5) STOP C 33 WRITE(6,40) 40 FORNAT(1X,’OOE SOLVER FAILS! ‘) STOP END C *************************************** DOUBLE PRECISION FUNCTION RHSF(I,T,Y) C C *************************************** - - - C This function evaluates the non-linear C equations for a given set of Y values. C IMPLICIT REAL*8CA-H,J,K,L,O-Z) CONMON/BLKB/JFS ,AS COMNON/BLKO/QPS , QLS DIMENSION YC4) C CALL AUXSAN(Y) IF(T.LT. 1.600)THEN RHSF1 = JLS - .JFS + 200.00 RHSF2 = QLS - QPS + 40.00*0.6500 RHSF3 = JFS — JLS RHSF4 = QPS - QLS ELSEIFCT.GE. 1.500.AND.T.LE.26.500)THEN RHSF1 = JLS RHSF2 = QLS - QPS RHSF3 = JFS - JLS RHSF4 = QPS — - JFS QLS - - (300.00/24.00) (15.aoo/24.Do)*o.65D0 212 Appendix F. List Of Computer Programs ELSEIF(T.GT.25.500.AND.T.LE.92.639D0)THEN RHSF1 = JLS RUSF2 = QLS RHSF3 RHSF4 JFS — — — iTS QPS JLS = QPS RHSF1 = AS RHSF2 = QLS - QPS RHSF3 = IFS - AS RHSF4 = QPS — QLS ELSE - - JFS QLS ENDIF GOTO(10,20,30,40),I 10 RHSF = RHSF1 RETURN 20 RBSF = RHSF2 RETURN 30 RHSF = RHSF3 RETURN 40 RHSF = RETURN END RHSF4 - — (15.8D0/24.D0)*o.e500 213
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A compartmental model of human microvascular exchange Xie, Shuling 1992
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Title | A compartmental model of human microvascular exchange |
Creator |
Xie, Shuling |
Date Issued | 1992 |
Description | To study the distribution and transfer of fluid and albumin between the human circulation, interstitium and lymphatics, a dynamic mathematical model is formulated. In this model, the human microvascular exchange system is subdivided into two distinct compartments: the circulation and the interstitium. Fluid is transported from the capillary to the interstitium by filtration according to the Starling’s hypothesis, while albumin is transported passively by coupled diffusion and convection through the same channels that carry the fluid. Data for parameter estimation are taken from a number of studies involving human microvascular exchange and include information from normals, nephrotics, heart failure patients, and also information from experiments on both normals and patients following saline or albumin solution infusions. Transport parameters are determined by fitting model predicted results to available measurements from the literature. The best-fit parameters obtained are LS = 43.08 ± 4.62 mL• mmHg⁻¹•h⁻¹, õ = 0.9888 ± 0.002, Pc,0 = 11.00 ± 0.03 mmHg, PS = 73.01 mL•h⁻¹, KF = 121.05 mL•mmHg⁻¹•h⁻¹, and JL,O = 75.74 mL•h⁻¹. Simulation of the available experimental data using these parameters gave a reasonable fit in terms of both trends and absolute values. All of the best-fit parameter values are in reasonable agreement with estimated values based on experimental measurements where comparisons with literature data are possible. The fully described model is used to simulate the transient behaviour of the system when subjected to an intravenous infusion of albumin and the predicted values compare reasonably well with the experimental infusion data of Koomans et al. (1985). The coupled Starling model is able to successfully simulate the transport mechanisms of human microvascular exchange system. |
Extent | 3318149 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-12-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0058573 |
URI | http://hdl.handle.net/2429/2973 |
Degree |
Master of Applied Science - MASc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1992-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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