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A compartmental model of human microvascular exchange Xie, Shuling 1992

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A COMPARTMENTAL MODEL OF HUMAN MICROVASCULAREXCHANGEByShuling XieB. Sc. (Biomedical Engineering) Shanghai Jiao Tong University, ChinaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESCHEMICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust, 1992© Shuling Xie, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of b1-’.9 e4M’1,The University of British ColumbiaVancouver, CanadaDate Qc-evbc-’r /3 / 992DE-6 (2/88)AbstractTo study the distribution and transfer of fluid and albumin between the human circulation, interstitium and lymphatics, a dynamic mathematical model is formulated. In thismodel, the human microvascular exchange system is subdivided into two distinct compartments: the circulation and the interstitium. Fluid is transported from the capillaryto the interstitium by filtration according to the Starling’s hypothesis, while albuminis transported passively by coupled diffusion and convection through the same channels that carry the fluid. Data for parameter estimation are taken from a number ofstudies 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 parametersare determined by fitting model predicted results to available measurements from theliterature. The best-fit parameters obtained are LS = 43.08 ± 4.62 rnL.mmHg’•h’,= 0.9888 ± 0.002, Pc,0 = 11.00 ± 0.03 mmHg, PS = 73.01 mLh, KF = 121.05mL .mmHg’h’, and JL,O = 75.74 mL.h ‘. Simulation of the available experimentaldata using these parameters gave a reasonable fit in terms of both trends and absolutevaliles. All of the best-fit parameter values are in reasonable agreement with estimatedvalues based on experimental measurements where comparisons with literature data arepossible. The fully described model is used to simulate the transient behaviour of thesystem when subjected to an intravenous infusion of albumin and the predicted valuescompare reasonably well with the experimental infusion data of Koomans et al. (1985).The coupled Starling model is able to successfully simulate the transport mechanisms ofhuman microvascular exchange system.11Table of ContentsAbstract iiList of Tables viiiList of Figures xAcknowledgement xiii1 Introduction 12 Physiological Principles Of The Microvascular Exchange System 52.1 Capillaries 62.1.1 Arrangement of the Capillaries 62.1.2 The Composition and Properties of the Blood 92.1.3 Structure of the Transport Barrier and the Transport Mechanisms 102.1.3.1 Capillary Wall 102.1.3.2 Basement Membrane 142.2 Interstitial Compartment 142.2.1 Structure and Composition of the Interstitium 142.2.1.1 Collagenous Fibres 162.2.1.2 Elastic Fibres 182.2.1.3 Glycosaminoglycans and Proteoglycans 182.2.1.4 Interstitial Fluid 192.2.2 Physicochemical Properties of the Interstitium 201112.2.2.12.2.2.22.2.2.32.3 Lymphatic System2.3.1 Terminal Lymphatics2.3.2 Mechanism of Lymph Formation4 Parameter Estimation And Data Analysis4.1 Parameters to be Determined4.2 Additional Relationships between the Unknowns4.2.1 Steady-state Balances at Normal Conditions4.2.2 Albumin Clearance Relationship4.2.3 Parameters to be Optimized4.2.4 Parameter Search Ranges4.3 Analysis of Data4.3.1 Experimental DataComplianceExclusionColloid Osmotic Behaviour2022252626283 Model Formulation3.1 Introduction3.2 General Assumptions of Compartmental MVES Models3.3 Coupled Starling Model3.4 Constitutive Relationships3.4.1 Circulatory Compliance3.4.2 Interstitial Compliance3.4.3 Colloid Osmotic Pressure Relationship3.5 Normal Steady-State Conditions3.6 Summary30303337424244475153555557575859• . . 616365iv4.3.1.1 Set A: Saline and Albumin Infusion . . . 674.3.1.2 Set B: Acute Saline Infusion 694.3.1.3 Set C: Saline Infusion . . 714.3.1.4 Set D: Heart Failure 714.3.1.5 Set E: Nephrotic Syndrome 734.3.1.6 Set F: Saline infusion before extracorporeal circulation 744.3.2 Parameter Estimation Strategy 764.4 Parameter Estimation Procedure 784.5 Numerical Methods and Computer Programs 794.5.1 Transient Solutions 794.5.2 Steady-state Solutions 794.5.3 Computer Programs 805 Results And Discussion 825.1 Introduction 825.2 Results 835.2.1 Best-fit Parameters for the Coupled Starling Model 835.2.2 The Feasibility of Combining Different Data Sets 835.2.3 Simulations Using Best-fit Parameters 895.2.3.1 Transient Responses of 11PL, GPL and VPL after Saline orAlbumin Infusions (Sets A, B and C) 905.2.3.2 Simulations of Heart Failure (Set D) 985.2.3.3 Simulations of V1 vs. [PL and fl1 vs. 11PL in NephroticSyndrome (Set E) 995.2.3.4 Summary of the Simulations using Best-fit Parameters 1015.2.4 Sensitivity Analysis 101V5.2.4.1 Lymph Flow Sensitivity— LS 1045.2.4.2 Albumin Reflection Coefficient— u 1055.2.4.3 Capillary hydrostatic pressure at normal steady-state -Pc,o 1055.2.5 Péclet Number 1065.2.6 Verification of Fit 1075.2.7 Residual Analysis 1085.3 Validation of the Best-fit Parameters 1105.3.1 Lymph Flow Sensitivity— LS 1105.3.2 Albumin Reflection Coefficient— a . . . 1135.3.3 Permeability-Surface Area Product— PS 1145.3.4 Fluid Filtration Coefficient— KF . 1155.3.5 Normal Lymph Flow— JL,O 1165.4 Simulations of a Single Intravenous Infusion of Human Albumin 1176 Conclusions 1217 Future Work 123Bibliography 125Nomenclature 134Appendices 137A Raw Experimental Data 137B Calculation Of Error Propagation 144C Basic Concepts Related To Statistical Analysis 147D Surface Plot And Contour Plot 149E Simulations At Best-Fit Parameters 151F List Of Computer Programs 166F.1 Parameter List of Steady-State and Transient Simulators 166F.2 Listing of FORTRAN function XDFUNC 167F.3 Listing of program PATDYN 208viiList of Tables2.1 Chemical composition of plasma in the human 112.2 Classification of connective tissue 153.1 Mathematical descriptions of the interstitial compliance relationship . 473.2 Normal steady-state conditions for the “reference man” 523.3 Coupled Starling model 544.1 Unknowns in the coupled Starling model . . 564.2 Bounds on u for Pc,o equal to 7 to 11 mmHg 634.3 Experinwntal data from Hubbard et at. (1984) 674.4 Experimental data from Hubbard et al. (continued) . 684.5 Experimental data from Hubbard et al. (continued) . 684.6 Experimental data from Doyle et al. (1951) 704.7 Experimental data from Mullins et al. (1989) 714.8 Experimental data at steady-state for patients with heart failure (1985) 734.9 Experimental data from Rein et al. (1988) . 765.1 Best-fit parameters corresponding to three tissue compliance relationships 845.2 Best-fit LS and u, as well as their confidence intervals for individual datasets 865.3 Experimental data of interstitial hydrostatic pressures vs. lymph flow inthe leg superficial lymphatics 1105.4 Lymph flow rates in the leg superficial lymphatics and in the thoracic duct 111viii5.5 Experimental estimates of PS values 1145.6 Experimentally determined KF values 116A.1 Raw experimental data from Hubbard et al. (1984) 138A.2 Raw experimental data from Hubbard et al. (continued) . 138A.3 Raw experimental data from Hubbard et al. (continued) 139A.4 Raw experimental data from Doyle et al. (1951) 140A.5 Raw experimental data from Mullins et al. (1989) 141A.6 Raw experimental data from Fauchald (1985) 141A.7 Raw experimental data from Noddeland et al. (1984) . 141A.8 11PL vs. V1 for patients with nephrotic syndrome 142A.9 H vs. 11PL for patients with nephrotic syndrome 143ixList of Figures2.1 72.2 82.3 132.4172.5 212.6 232.7 2728Schematic diagram of the compartmental model of the MVESSchematic of the Patlak modelExperimental data of compliance of human lower limb subcutaneous tissueThe “most-likely” human interstitial compliance relationshipRelationship between albumin concentration and total colloid osmoticpressure4.1 Experimental data for patients with nephrotic syndrome 755.1 Best-fit LS and u, as well as their confidence intervals for individual datasets 855.2 Contour plots of OBJ during early and late responses for Hubbard’s data(1984) 88Schematic of the circulatory system in human bodyThe general patten of the capillary networkStructure and types of capillaries found in the human bodyScanning electron micrographs of collagenous and elastic fibres in humanskinRelationship between interstitial fluid volume and interstitial fluid pressureSchematic diagram depicting the exclusion phenomenonSchematic of the lymphatic systemMicrostructure of the terminal lymphatics3.13.23.33.43.53438464850x5.3 Simulations of a 100 mL saline or albumin infusion with 1.4 L fluid intakeduring waking hours 925.4 Simulations of a 200 mL saline or albumin infusion with 1.4 L fluid intakeduring waking hours 945.5 Simulations of acute saline infusion in selected patients 965.6 Simulation of 2 L normal saline infusion within 2 hours 975.7 Simulations of steady-state Vj vs. 11PL and V1 vs. P in heart failurepatients 1005.8 Simulations of V1 vs. 11PL and Hi vs. 11PL in nephrotic syndrome patients 1025.9 Steady-state effects of graded reduction of plasma oncotic pressure on fluidand protein exchangeSensitivity analysis for LSSesitivityanlysis for aSensitivity analysis for P0Residual plot for the best-fit parameters of compliance relationship #3Plot of.JL vs. P1Effect of Koomans’ (1985) albumin infusion on select microvascular exchange variables 1191501521531545.105.115.125.135.145.15103104106107109112D.1 Surface and contour plots of the objective functionE.1 Simulations of a 100 mL saline or albumin infusion using compliance relationship #1E.2 Simulations of a 100 mL saline or albumin infusion using compliance relationship #2E.3 Simulations of a 200 mL saline or albumin infusion using compliance relationship #1xiE.4 Simulations of a 200 mL saline or albumin infusion using compliance relationship #2 155E.5 Simulations of acute saline infusion in selected patients using compliancerelationship #1 156E.6 Simulations of acute saline infusion in selected patients using compliancerelationship #2 157E. 7 Simulation of 2 L normal saline infusion within 2 hours using compliancerelationship #1 158E.8 Simulation of 2 L normal saline infusion within 2 hours using compliancerelationship #2 159E.9 Simulations of steady-state V1 vs. 11PL and V1 vs. P in heart failurepatients using compliance relationship #1 160E.10 Simulations of steady-state V1 vs. 11PL and V1 vs. P in heart failurepatients using compliance relationship #2 161E.11 Simulations of Vj vs. 11PL and Hi vs. 11PL in nephrotic syndrome patientsusing compliance relationship #1 162E.12 Simulations of Vj vs. 11FL and H1 vs. 11PL in nephrotic syndrome patientsusing compliance relationship #2 163E.13 Effect of Koomans’s (1985) albumin infusion on select microvascular exchange variables using compliance relationship #1 164E.14 Effect of Koomans’s (1985) albumin infusion on select microvascular exchange variables using compliance relationship #2 165xiiAcknowledgementFirst 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 graciousness when I became a new mother.Second, I am indebted to Drs. P. Englezos, G. Wong, N. Lee and T. Nicol, for theirhelpfully 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 NaturalSciences and Engineering Research Council of Canada and the Norwegian Council forScience.Finally, I would like to dedicate this work to my husband Xiao Ping Yang, for hislove and encouragement.xli’Chapter 1IntroductionBy virtue of the unique properties of water as a solvent and as medium in which a widerange of exchanges and metabolic reactions are possible, it plays a very important role inmaintaining homeostasis. Water, as the major component of all living tissues, typicallymakes 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 totalbody water, and the extracellular compartment, which retains about 40% of the totalbody water. The extracellular water, in turn, exists as two compartments, namely, theplasma 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 interstitialwater about 30% [95].The word fluid, which will be used throughout this dissertation, identifies the mixtureof water and non-protein components dissolved in the water. The maintenance of thecomposition and volume of this aqueous phase is critical to the vital functions of thehuman body. The regulation of composition and volume, to a large extent, can bedescribed by system and control theory. Feedback control is the typical control strategyused for regulating the microvascular exchange system, which consists of the capillaries,the interstitium and the lymphatic system. Using the feedback signals, the microvascularexchange system is capable of reacting to changes in its environment by passively alteringits own properties so that the pre-perturbation state can be achieved again. Thus, undernormal conditions, the composition and volume of the plasma and interstitial fluid remain1Chapter 1. Introduction 2in a very narrow range and the body functions normally.The utilization of mathematical models in the description of the biological behaviourof systems is now common practice. The advantages of mathematical models are self-evident. The human body is well recognized to be a complex control system. Whenphysicians attempt to diagnose and correct the malfunctioning of the system, they arefaced with the problem that most physiological and biochemical state variables are influenced by too many factors to be grasped by the unaided human mind. Therefore,reliable mathematical models as a complementary approach to physiological experimentation is of potential importance to aid the clinician in gaining insight into these intricateinteractions.Several models of microvascular exchange have been developed [7, 32, 106]. Thesemodels 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 microvascularexchange system after perturbations. The current model uses the same general compartmental approach and mechanistic descriptions as many of these earlier models. However,it differs from the latter in one important respect: whereas the previous models assumethe values of transport parameters a priori, in the current model these are determined onthe basis of a statistical fit of the model predictions to selected experimental data. Thepresent work is a continuation of an earlier study carried out by Chapple [13]. Chapplemodelled the microvascular exchange system by using the uncoupled Starling model andthe plasma leak model to represent the transcapillary exchange of fluid and proteins. Thecoupled Starling model is used in the current study. The coupled Starling model wasfound to yield the best statistical fit between the model predictions and the experimentaldata for nephrotic patients [14]. Furthermore, it is generally thought that the coupledStarling model provides the most accurate description of the transcapillary protein fluxChapter 1. Introduction 3[73].The objectives of the current study are:1. to formulate a coupled Starling model for describing the microvasdillar exchangesin humans;2. to assess and assemble all of the available literature data for the response of themicrovascular exchange system to both diseaseinduced and artificially applied perturbations;3. to design a reliable parameter estimation procedure;4. to obtain a set of best-fit transport parameters so that the model is fully describedand 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 thatthese kinds of models could be used for pretesting therapeutic procedures on individualpatients in certain disease states, e.g. for the on-line simulation of trial fluid therapy forburn 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 fluidand plasma proteins between the circulation, interstitial space of skin and muscle, andthe lymphatics in the rat under burn condition.The thesis is organized in the following manner. Chapter 2 reviews the basic physiology relevant to the microvascular exchange system. This chapter provides a physicalbasis for better understanding the transport mechanisms of fluid and proteins within thesystem. Chapter 3 develops a compartmental model based on the coupled Starling representation of the transcapillary fluid and protein transfers. The necessary constitutiveChapter 1. Introduction 4relationships and the normal steady-state conditions of the system are also presentedin this chapter. Chapter 4 describes the optimization procedure designed to obtain thebest-fit transport parameters. The experimental data used in statistical fitting are assessed and the numerical techniques used in the current work are reviewed. Chapter 5presents the results of the parameter estimation procedure and the simulations obtainedusing these best-fit parameters. Statistical procedures are used to evaluate the reliabilityof the best-fit parameters. Finally, the parameters and model are validated from a physiological point of view. Chapter 6 summarizes the conclusions drawn from the currentstudy and Chapter 7 makes recommendations for future work. For the convenience ofthe reader, all mathematical symbols used throughout the thesis are listed and definedin a special Nomenclature section following the text.Chapter 2Physiological Principles Of The Microvascular Exchange SystemMany physiological systems involve the maintenance of the milieu interieur, or homeostasis, so that optimal function can be fulifihled. Milieu interieur is defined by ClaudeBernard [84] as the pervasive extravascular and extracellular space in which all cellsbathe. The cardiovascular system, which consists of the heart and a series of bloodvessels forming a closed network, plays an important role in enabling homeostasis to beachieved. It supplies oxygen and nutrients necessary for life to the tissues and carriesaway metabolic waste products from the tissues. Oxygenated and nutrient rich bloodis 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. oxygen, plasma proteins, vitamins, hormones, heat, etc., are transported across the capillarywalls to the surrounding tissue space called the interstitium. Meanwhile, metabolic-end-products, e.g. carbon dioxide, water and heat, are collected from the tissue. As a result ofthese exchanges, blood becomes oxygen depleted and is collected by venules before beingtransported, via large veins, back to the right atrium. Rhythmic muscle contractions andone-way valves force the blood to flow in one direction. When the muscles in the wall ofthe right ventricle contract, blood is pumped into the pulmonary artery and hence passesto the lungs where the transfer of gasses with the inspired air occurs. Oxygenated bloodthen returns to the left atrium by the pulmonary veins, finally emptying into the leftventricle. On the other hand, interstitial fluid containing plasma proteins drains fromthe interstitium and is returned to the blood circulation via the lymphatic system. The5Chapter 2. Physiological Principles Of The Microvascular Exchange System 6entire circulatory system is illustrated in Fig. 2.1.The current investigation is concerned with the microvascular exchange system. Itconsists of three components: capillaries, interstitium, and the lymphatic system. Thestructure and functions of these components will be discussed in this chapter.2.1 Capillaries2.1.1 Arrangement of the CapillariesThe microcirculation consists of arterioles, capillaries, and venules. Capillaries are tubesof 5 to 10 m in diameter and their walls consist of a single layer of flattened endothelialcells, 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 ofa tissue per square millimeter of area) has been measured in red skeletal muscle in dog,and has been found to be about 1000 capillaries/mm2.Thus the distance between bloodcapillaries 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 complexmeshwork which has the general pattern shown in Fig. 2.2. Most physiologists agree thatthe local circulation rate should be in accordance with the needs of every tissue as well asthe variant physiological states of the subject. Therefore the capillaries are able to openor close depending on the local requirements of the system. Whether the capillary is off oron is controlled by the precapillary sphincter. When the sphincter muscle contracts, thechannel closes and vice versa. The dilation and constriction of precapillary sphinctersproduce a continuously changing pattern of flow through the capillary network. In aparticular segment of the capillary bed, the blood may flow rapidly through one channelfor a period of time, then cease to flow or even flow in the opposite direction, dependingon which sphincters are open. Normally, only around 5 — 10% of the total availableChapter 2. Physiological Principles Of The Microvascular Exchange System 7CapillariesArlercs to head andupper exItemiliesCapilttiesFigure 2.1: Schematic of the circulation systems in human body [17].Chapter 2. Physiological Principles Of The Microvascular Exchange System 8Figure 2.2: Schematic diagram showing the general pattern of the capillarynetwork [53].MerioleArteriovenousanastomosisTruecapillariesThoroughfarechannelVenulechapter 2. Physiological Principles Of The Microvascular Exchange System 9capillaries in the human adult at rest are open [93]. But during exercise, the number ofopen 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 thecirculation and tissues, while the other part of the circulation, e.g., the small arteriolesand venules, only account for a small amount of the duty.2.1.2 The Composition and Properties of the BloodBlood is a transport medium for nutrients and wastes. A constancy of blood compositionis vital for survival and the mechanisms that operate to keep it constant are vital processesof homeostasis.Blood is a suspension of erythrocytes, leukocytes, platelets, and other particulatesiii a. complex solution of dissolved gases, salts, proteins, carbohydrates, and lipids. Thespecific 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 arterialblood pressure is prevented by constriction of arteries and veins. However, the heart ratewill 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 testtube. Heparin is added to prevent blood clotting. Then the tube is placed in a centrifugeto 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 totalvolume. The straw-colored supernatant liquid, called plasma, is a cell free fluid. Thespecific gravity of plasma varies between 1.028 and 1.032 [85]. The ratio of blood cells toChapter 2. Physiological Principles Of The Microvascular Exchange System 10total 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 chemicalcomposition of plasma typically yields the results depicted in Table 2.1. Despite thefact that chemical substances are constantly entering and leaving the blood stream, itsgeneral composition is relatively uniform in the higher-order animals. As can be seenfrom the table, about 90% (by weight) of the plasma is water. Proteins account forabout 7- 9% of the total weight. Some of these proteins are also found elsewhere in thebody, but when they occur in blood, they are called plasma proteins. Plasma proteinsexert an osmotic pressure (around 25 mmHg) which influences the passage of water andother solutes through capillaries. The globulins, classified as a1, a2, and ‘y globulins,have a wide range of molecular weights varying from 100,000 to 450,000 [53]. Albuminsconstitute the majority of the plasma proteins (55 - 64%) and possess &relatively lowmolecular weight of the order of 68,000. Also, albumins are not transported freely acrossthe intact vascular endothelium. Therefore, they provide a significant colloid osmoticpressure gradient between the blood and the interstitium which greatly influences thetransmembrane exchanges of water and other solutes.2.1.3 Structure of the Transport Barrier and the Transport MechanismsThe transport rates of water and solutes across the capillary wall are determined bythree 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 WallCapillaries of different tissues vary considerably, both anatomically and functionally. Forexample, in the skin, there are arteriovenous anastomoses (AVAs) which are wide-bore,Chapter 2. Physiological Principles Of The Microvascular Exchange System 11Constituent Concentration (g/100 mL)Proteins 6.0 - 8.0Albumin 3.4- 5.0Total globulin 2.2- 4.0Transferrin 0.25Haptoglobin 0.03 - 0.2Hemopexin 0.05 - 0.1Ceruloplasmin 0.03- 0.05Ferritin 0.015- 0.3NonproteinsWaterCholesterol 0.14- 0.25Glucose 0.07 - 0.11Urea nitrogen < 0.02Uric acid < 0.008Creatinine < 0.002Iron < 0.0002Table 2.1: Chemical compositions of plasma in the human [74].Chapter 2. Physiological Principles Of The Microvascular Exchange System 12direct channels between the arterioles and venules (see Fig. 2.2). The AVAs are underneurogenic control; their shunting capabilities enable them to reduce heat loss throughthe skin during exposure to cold.The capillary wall is made up of a number of microscopic structures, some of whichaffect the transcapillary exchange. Those that have been observed and are thoughtby most physiologists to play a role in transcapillary exchange are intercellular clefts,fenestrae, and pinocytotic vesicles (Fig. 2.3). Intercellular clefts are the junctions betweenendothelial cells. They can be loosely joined to form large pores or tightly joined to formsmall 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 aminimally restrictive pathway through the endothelium, while others are closed with aslight central thickening or knob. Inside the cytoplasm of the eridothelium, there arernaiy 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. Thesevesicles may be important in protein transfer. Sometimes the vesicles fuse together andform a transitory open channel.The above structures determine the transport properties of the capillary wall. According to the degree of continuity, capillaries can be separated into three types: continuous,discontinuous and fenestrated capillaries (see Fig. 2.3). Continuous capillaries are widelydistributed; characteristically, they are found in lung, brain, kidney as well as skeletaland smooth muscles. There are no recognizable intercellular openings in this type ofcapillary. Fenestrated capillaries have small gaps which are either closed as in endocrineglands and intestinal villi or open as in renal glomeruli. These gaps usually range from40— 60 nm in radius [100]. Discontinuous capillaries have large nonselective gaps, ranging from 100 — 1000 nm in radius [100]. These gaps are sufficiently wide to allow largeproteins and even cells to pass through. This type of capillary is typically found in liver,Chapter 2. Physiological Principles Of The Microvascular Exchange System 13CONTINUOUSCAPILLARYTISSUEBASEMENTMEMBRANEF EN E STR AT EDCAPILLARY5—11DISCONTINUOUSCAPILLARY1 5.PLASMATISSUE100— 1000Figure 2.3: Structure and types of capillaries found in the human body. 1,plasmalemmal vesicles; 2, intracellular clefts; 3, fused plasmalemmal vesicles;4, fenestrae; 5, nonselective gaps. Unit of the radius is nm [100].1.PLASMA2.503.10—4050 2—6TISSUE 40— 60Chapter 2. Physiological Principles Of The Microvascular Exchange System 14bone marrow and spleen.2.1.3.2 Basement MembraneThe basement membrane is composed of several matrix-specific components, includingthe structural protein collagen, the glycoproteins laminin and entactin, and a large heparm sulfate proteoglycan, which covers the external surface of capillary endothelial cells,penetrated here and there in discontinuous capillaries but not in the other types. Itprovides surfaces on which epithelial cells adhere and assumes a polarized orientation.The thickness of the membrane averages 30— 80 nm. The transport properties of thebasement membrane are attributed to the presence of pores. The membrane is negativelycharged, 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 inseries. 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 Interstitial Compartment2.2.1 Structure and Composition of the InterstitiumThe word interstitium is borrowed from Latin; it generally refers to the connective tissuespace situated outside the vascular and lymphatic systems and the parenchymal cells.Half a decade ago, interstitium was perceived as an inert, metabolically inactive substance, which only provided support for cells and held tissues together. But this concepthas been extensively revised. It is now appreciated that the interstitium not only provides a framework for parenchymal cells and a space for distribution of blood vesselsChapter 2. Physiological Principles Of The Microvascular Exchange System 15Type ExamplesMyxoid Nucleus polposus, synovium, areolar connective tissueFibrous Tendon, ligament, fascia, dermisElastic Ligamentum nuchae, arteryAdipose tissue White fat, brown fatMuscle Striated muscle, smooth muscleVascular tissue Artery, capillary, veinCartilage Hyaline cartilage, fibrocartilage, elastic cartilageBone Compact bone, trabecular boneTable 2.2: Classification of connective tissue.and nerve fibers, but it also provides a suitable transport medium for nutrient and endmetabolic waste products between cells and capillary blood. In physiological situations,various exchanges occur between the interstitium and the circulatory compartment andlymph, but these exchanges are in a dynamic equilibrium.According to their anatomical structure and physiological function, connective tissuescan be classified into many types (Table 2.2). These tissues are extremely heterogeneousboth in their cellular population and composition. The cell component may occupyfrom 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 interstitial components (e.g. bone cells), while the others maintain a specialized internalfunction (e.g. muscle cells have active contractile function). The compositon of interstitia varies not only from one tissue to another, but also from domain to domain withinthe tissue. Most interstitia are intimate composites of two phases: the structural macromolecules and interstitial fluid. The structural macromolecules include collagens, elastin,glycosaminoglycans (GAGs) and proteoglycans. Interstitial fluid consists of water, salts,plasma proteins, 02, C02, hormones, vitamins, etc.. The basic structure of interstitiais similar. It can be described as an elastic, three-dimensional gel-like structure whichchapter 2. Physiological Principles Of The Microvascular Exchange System 16is produced by the complex aggregation of collagenous and non-collagenous componentsby covalent, electrostatic, and hydrogen bonds. The following discussion will emphasisethe 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 FibresCollagenous fibres are the soft, flexible, white fibres which are a characteristic constituentof all types of connective tissues. In normal adult skin, they account for 70— 75% ofthe dry weight [104]. They are composed of bundles of fibrils which can be visualizedwith the scanning electron microscope (Pig. 2.4 a). These bundles range from one toseveral hundred microns in diameter, but their texture may be loose enough to admitmolecules as large as plasma proteins between the individual fibrils. These fibrils intersperse with GAGs and can glide smoothly past each other. The collagen molecule is thebasic structllral unit of collagenous fibres and the most common protein in the body. Itis 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 approximately1000 amino acids and has a molecular weight of around 100 kDa [29].The collagen molecule is biosynthesized from a precursor known as procollagen. Procollagen is enzymatically trimmed of its nonhelical ends and spontaneously assemblesinto fibrils extracellularly. These fibrils interact with each other in the direction parallelto the axis of each molecule and form the well organized collagen fibers.Collagenous fibres determine the strength and elasticity of the tissue. They demonstrate poor elasticity but excellent mechanical strength.Chapter 2. Physiological Principles Of The Microvascular Exchange System(a)(c)(b)17Figure 2.4: Electron micrographs of collagenous and elastic fibres. (a) collage-.nous fibres have a regular banding patern(x45,000) [103]; (b) triple-helicalcollagen molecule; (c) elastic fibres in human skin after removal of otherextracellular matrix components. The fibres are randomly oriented and ofvarying size [90].C 87. 1Chapter 2. Physiological Principles Of The Microvascular Exchange System 182.2.1.2 Elastic FibresCompared to collagenous fibres, elastic fibres account for only a small fraction of connective tissue, 2— 4% of the dry weight in normal adult skin [104]. Nonetheless, they playan 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%); thesecondary one has a distinct appearance and each unit is referred to as a microfibril (notshown on Fig. 2.4 c). Elastin is a unique protein which is composed of hydrophobicamino acid sequences. The individal hydrophobic chains crosslink and form a randomconfiguration. This results in a very insoluble protein that possesses properties analogousto rubber. The microfibrils are composed of several glycoproteins that vary in size from25k to 34k daltons. They can exist in fibrillar arrangements or as aggregates independentof their association with elastin.2.2.1.3 Glycosaminoglycans and ProteoglycansGlycosarninoglycans (GAGs), formerly called mucopolysaccharides, distribute widely inmost tissues. However, except for hyaluronate, GAGs do not exist in vivo as free polymers. Instead, these GAGs normally covalently link to a core protein at the terminal endand form compounds which are called proteoglycans.The general structure of GAGs consists of three levels. The first level is an unbranchedchain of repeating disaccharide units in which one of the monosaccharides is an aminosugar (hexosamine) and the other is usually a hexuronic acid. The type and numberof disaccharide units vary in different GAGs. Four basic types of GAGs are recognizedand defined according to the type of hexosamine: hyaluronic acid, chondroitin/dermatansulfate, heparin/heparan sulfate, and keratin sulfate. The second level is the chargedChapter 2. Physiological Principles Of The Microvascular Exchange System 19side 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 backboneand the side groups. The backbone and the side groups form fibrils up to several hundredmicrons in length. The fibrils are folded every few hundred angstroms so that instead ofbeing long linear molecules they are actually jumbled, folded, springlike coils occupyinga space having a diameter of several hundred millimicrons [49]. GAGs interact withproteins, collagens and proteoglycans and form a continuous gel-like substance. GAGsdemonstrate a highly hydrophilic character.Knowledge of the structure of the proteoglycans in tissues other than cartilage isincomplete. In cartilage, the average proteoglycan molecule consists of a core protein towhich 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 proteinportion constitutes only 5— 10% of the entire molecule, and the rest of the moleculeconsists of complex carbohydrates. In the presence of hyaluronate, protein cores bind tothe hyaluronate chain and form aggregates [16]. The high charge density permits thesemolecules to attract a large volume of water and contribute to the ability of proteoglycansto absorb compressive loads.2.2.1.4 Interstitial FluidInterstitial fluid is a filtrate of blood. It consists of the same nonparticulate componentsas plasma except it has lower macromolecule concentrations. Interstitial fluid is thecontinuous bathing media through which nutrients diffuse to the parenchymal cells andmetabolic-end-products arrive at the lymphatics.Chapter 2. Physiological Principles Of The Microvascular Exchange System 202.2.2 Physicochemical Properties of the InterstitiumThe physicochemical properties of the interstitium are determined by the structure, composition, and physicochemical conditions of the compartment. In this section, the compliance of the interstitial space, exclusion caused by the macromolecules, and colloidosmotic behaviour will be discussed.2.2.2.1 ComplianceFrom the previous section, we know that collagen molecules can glide a short distancealong the axis of the fibre, but the amount of movement is limited by the interactionbetween 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 bestretched out. When the external force is removed, the fibre will be drawn back andregain its original state. This specific property is called elasticity. Moreover, elasticfibres like GAGs and proteoglycans have spring-like configurations; therefore they allpossess an elastic behaviour similar to springs.The compliance of human tissues is a poorly understood property, but one whichis very important to microvascular exchange. It involves the interaction between thefluid and the solid components of the interstitium, the state of charge of the structuralcomponents, the fibrillar organization, composition, etc.. For reasons of convenience,compliance (FCOMPI) in current study is defined byFCOMPI = AV1//P (2.1)where 1.V is the change in interstitial volume when the interstitium is subjected to ahydrostatic pressure change, AP1. To some extent, it determines the degree of elasticityof the interstitial space. The interstitium has high compliance if small changes of interstitial pressure can induce large changes in the interstitial fluid volume. When pressureChapter 2. Physiological Principles Of The Microvascular Exchange System 21Figure 2.5: Relationship of interstitial fluid pressure to change in leg weightduring progressive increase in interstitial fluid volume. Weight is correlatedwith 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 ofthe interstitial space at a certain volume. Even though knowledge of human tissue compliance is incomplete, studies on other animal tissues suggest that the pressure-volumecurve should be sigmoidal [34, 2], as exemplified by Fig. 2.5. Low compliance occurs atlow and normal interstitial volume. This is one of the tissue’s passive mechanisms forpreventing dehydration and edema. When the volume decreases, P1 decreases significantly. Thus the driving force for filtration to the interstitium increases and preventsCwzCl,4—eCHANGE IN LEG WEIGHT (per cent)+110 +120 4430Chapter 2. Physiological Principles Of The Microvascular Exthange System 22further dehydration. When the volume increases, Pj increases significantly to decreasefluid filtration and hence prevent edema. High compliance occurs at high V1. This makesthe 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 thetissue which therefore exhibits, once again, a low compliance. At this stage, the plasmavolume will increase due to the elevation in the driving force for absorption.2.2.2.2 ExclusionExclusion refers to fact that two or more objects of any volume can not overlap andoccupy the same space at the same time. This phenomenon exists in the interstitiumand was first described by Ogston and Phelps in 1961 [65].To depict the exclusion phenomenon more clearly, two models are introduced. Oneis the sphere-and-rod model (Fig. 2.6 b). The spheres represent plasma proteins andthe rods represent structural components of the interstitia. In this model, the center ofthe sphere is not accessible to a volume of 7rl(r3 + rr)2 surrounding the rod, where rris the radius of the rod and 1 is its length; r3 is the radius of the spere. This volumeis called the excluded-volume (VEX). The other model is closer to the real situation inthe interstitium. It is a sphere located inside a random network of rods. The excluded-volume fraction for this model has been calculated to be 1 — e_ (rS+T)2,where L is thetotal length of the rods per unit volume (Fig. 2.6 c). The above models can be usedin hypothetical descriptions of the exclusion phenomenon, but because the organizationwithin the interstitium is not well defined, they can be used only to estimate the excludedvolume of the interstitium. VEX is determined by many factors. There are different sizesof collagen, elastin, GAGs and proteoglycans which contribute to limiting the volumeavailable to the plasma proteins. The spatial arrangement of the fibres greatly affectsthe value of VEX. For example, a larger number of smaller fibres arranged randomly1;‘ I I‘I I4Chapter 2. Physiological Principles Of The Microvascular Exchange System 23A 8——---FFigure 2.6: Schematic diagram depicting the exclusion phenomenon. (a) twospheres can’t occupy the same space at the same time; (b) sphere-and-rodmodel; (c) a sphere located inside a random network of rods [16].Chapter 2. Physiological Principles Of The Microvasculai Exchange System 24causes much greater exclusion than an equal weight of a smaller number of larger fibresin an orderly lineup [6]. In addition, exclusion is also affected by the nature of the chargeon the proteins and the various structural components they interact with. For example,both GAGs and albumin are negatively charged. Because similarly charged entities repelone another, albumin is excluded by GAGs to a greater extent than other fibrils whichhave a similar size but are uncharged. Since the structural components differ from tissueto tissue, the exclusion effect is tissue dependent.The best way to study exclusion inside interstitium is through in vivo experimentation. Unfortunately, the results of only a few in vivo studies are available. One way toestimate VEX is to use a multiple-indicator technique. One indicator has a very smallmolecular size (e.g. sucrose, Cr-EDTA) and its exclusion other than by solid structurescan 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 animportant role in excluding plasma proteins. Other materials might also play an important role. This issue is not yet clarified. In vivo studies show that the excluded-volumefraction by collagenous fibres ranges from 25% to 53% of the volume of the interstitialmatrix [6, 5, 86].Due to exclusion, the interstitial space available to the plasma proteins is less thanthe interstitial fluid volume; consequently, the effective protein concentration and chemical activity of the protein is greater than the mass of plasma proteins divided by theinterstitial fluid volume. The colloid osmotic pressure, which depends on the proteinconcentration, is therefore sensitive to the interstitial exclusion phenomenon.Chapter 2. Physiological Principles Of The Microvascular Exchange System 252.2.2.3 Colloid Osmotic BehaviourBecause capillary walls restrict protein movement, changes in the concentration of bodyfluids will affect the movement of water between fluid compartments due to the phenomenon of osmosis. Osmosis is the movement of water across a semipermeable membrane 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 smallerdegree, osmosis also affects the movement of solutes. The forces of friction cause somesolutes 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 sidesof the membrane. Colloid osmotic pressure is caused by a relative deficit of permeatingmolecules on one side of the membrane versus the other side. It should be mentioned thatcolloid 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, exertsoncotic pressure within the blood vessels and helps maintain the water content of theblood in the intravascular space.The normal plasma colloid osmotic pressure is approximately 25— 30 mmHg, whichconsists of approximately 19 mmllg due to plasma proteins and 9 mmHg due to cationsheld by the Donnan equilibrium effect ‘.The direction of the colloid osmotic pressure is opposite to that of the hydrostaticpressure within the interstitium. Together they counteract the oncotic and hydrostaticforces within the capillary and maintain a dynamic equilibrium between the two compartments.There is a good correlation between protein concentration and colloid osmotic pressure. This issue will be discussed in more detail in the next chapter. Experiments show1Negatively charged proteins attract positive ions and create a high ion density environment, consequently causing osmotic pressure. This phenomenon is called the Donnan equilibrium effect [35].Chapter 2. Physiological Principles Of The Microvascular Exchange System 26that about 68% of the total colloid osmotic pressure is due to albumin, 6% is due toglobulin, and the rest is due to the other proteins [53].2.3 Lymphatic SystemThe 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 theinterstitial fluid back to the blood circulation (Fig. 2.7). The drainage is important influid and plasma protein distribution and transport in the body. The cells bathed by thelymph, namely, lymphocytes, are primarily responsible for the specificity of immunological responses.2.3.1 Terminal LymphaticsThe terminal (or initial) lymphatic is the part of the lymphatic system that is involvedwith 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 endothelialcells. The differences between a terminal lymphatic and a blood capillary are that theformer has a much thinner wall but its diameter is much larger (15— 20 pm) and itsbasement membrane is highly attenuated and frequently absent completely.It is well recognized that terminal lymphatics are discontinuous, with open junctionsbetween endothelial cells (except those in brain, spinal cord and ocular space). Someworkers have even identified some ultrastructures around these junctions. For example,Leak [51, 50] observed anchoring filaments that attach the basement membrane to theadjacent collagenous and elastic fibres. The unattached sites around the junctions formflaps 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 anchoringChapter 2. Physiological Principles Of The Microv.ascular Exchange System 27Figure 2.7: Schematic of the lymphatic vessels in an area of bat wing at abifurcation of two large vascular channels: terminal lymphatics (irregularblack bulbs), collecting lymphatics (black bold lines), blood vessels (shaded),and capillaries (thin lines) [15j.Chapter 2. Physiological Principles Of The Microvascular Exchange System 28IFigure 2.8: Microstruture of the terminal lymphatics. When the anchoringfilaments are tightened, the microvalves are open and the lymphatics arefilled; when the filaments are loosened, the microvalves are closed and thelymphatics are emptied [90, 12].-filaments and the microvalves are of potential importance in supporting the postulatedtransport mechanism for lymph which will be disscussed later. Collecting lymphaticsinto, which initial lymphatic fluids derived from the interstitium continuously drain aredistinguished from terminal lymphatics by the appearance of macroscopic bileaflet valvesand smooth muscle intima, and by the fact that they contract spontaneously [37]. Thebileaflet valves are one-way valves and hence prevent backflow. The collecting lyinhaticsfinally converge at the thoracic duct and, through the duct, lymph is returned to the bloodcirculation.2.3.2 Mechanism of Lymph FormationThe question concerning the mechanism by which the interstitial fluid is transported intothe terminal lymphatics has long been disscussed but still remains controversial.The first postulated mechanism is that lymph is formed by the periodic contractionIChapter 2. Physiological Principles Of The Microvascular Exchange System 29of the terminal lymphatics which is caused by its own smooth muscle. This mechanism issupported by research on bat wing lymphatic endings [38, 62]. These endings have theirown 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 anyother tissue. A detailed review of the microanatomy of terminal lymphatics in differentorgans has been presented [90].The second postulated mechanism is considered to be more common. In this postulation, lymph is formed by the contraction of the tissues surrounding the terminal endings.Muscle contraction, intestine motilities, skin tension, vasomotion, etc., stretch the anchoring filaments and expand the terminal lymphatics. The unattached microvalves areopened, 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 themicrovalves 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 fluiddue to the large intercellular junctions and the incomplete basement membrane, bothof which result in a nonsieving in the terminals. Some investigators have observedhyaluronate present in the prenodal lymph [77]. After the lymph passes through thelymph nodes, some components are degraded and the composition of the lymph changes.But whether the assumption that compositions of initial lymph and interstitial fluid arethe same is correct or not still needs to be proven [2].Chapter 3Model Formulation3.1 IntroductionWith the development of computer sciences, a new interdisciplinary approach for studyingbiological systems has appeared. Computer modelling has provided researchers with apowerful tool for better understanding intricate systems, which are generally influencedby too many factors to be grasped by the unaided human mind.In studies of the microvascular exchange system (MVES), many different mathematical models have been developed on the basis of the Starling’s hypothesis. To elucidatehis hypothesis, Starling wrote [96]:Although the osmotic pressure of the pro teids of the plasma is so insignificant it is of an order of magnitude comparable to that of the capillary pressures; and whereas capillary pressure determines transudations the osmoticpressure of the pro teids determines absorption. Moreover, if we leave thefunctional resistance of the capillary wall to the fluid through it out of account, the osmotic attraction of the serum for the extravascular fluid will beproportional to the force expended in the production of the latter, so thatat any given time, there must be a balance between the hydrostatic pressureof the blood in the capillaries and the osmotic attraction of the blood for thesurrounding fluids. With increased capillary pressure there must be increasedtransudation until equilibrium is established at a somewhat higher point, when30Chapter 3. Model Formulation 31there is a more dilute fluid in the tissue spaces and therefore a higher absorbingforce to balance the increased capillary pressure. With diminished capillarypressure there will be an osmotic absorption of salt solution from the extravascular 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 thediminished capillary pressure.The center of the hypothesis is that the fluid and protein exchanges across the capillarywall are governed by the hydrostatic pressures and colloid osmotic pressures on bothsides 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 wereused 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], weremore interested in the overall regulation of body fluid. Hence, in their models, not onlytransport mechanisms across the capillary wall were considered, but also the influence ofhormone release and renal function.According to the spatial distribution properties of the parameters which describe thesystem, MVES models are divided into two types: lumped parameter or compartmentalmodels and distributed parameter models. In compartmental models, all the variablesunder consideration are spatially invariant, i.e. any changes in those variables are considered equal and simultaneous throughout the volume for which the laws of conservation(mass, energy, and momentum) have been established. These models can be representedin terms of algebraic equations (static systems) or ordinary differential equations (dynamic systems). In the distributed parameter model, however, some or all of the variablesare not the same throughout the whole system volume at any given time, i.e. the valuesChapter 3. Model Formulation 32of the variables are spatially varying. Ordinary differential eqilations (one-dimensionalstatic systems) or partial differential equations (multi-dimensional and/or dynamic systems) are required to describe these types of models.Currently, our interest is in compartmental models of the MVES. A distributed parameter model, developed by other members of the research group [101, 102], was usedto investigate the fluid and protein transfers inside an idealized interstitium. For cornparmental models, based on different assumptions about the structure and transportmechanisms of the capillary-interstitium interface, different mathematical descriptions ofthe 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 plasmaleak model, and used these in an overall MVES model to simulate experimental data fromnephrotic patients. In the uncoupled Starling model, albumin is assumed to pass throughthe capillary wall by convection along with the filtrate from the ircu1ation to the interstitium, and by diffusion which takes place via a separate pathway. These two mechanismsare non-interacting and hence the name uncoupled Starling model. In the plasma leakmodel, it is assumed that two types of pores exist in the capillary wall, namely, theso-called small and large pores. Albumin is hypothesized to pass through the capillarywall by convection through the large pores and by diffusion through other parts of thewall, but is completely rejected by the small pores. In the current study, the coupledStarling hypothesis (or the Patlak formulation) will be used. This macromolecule transport 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 333.2 General Assumptions of Compartmental MVES ModelsOne of the goals of this work is to fomulate a mathematical model which describes thenormal behaviour of the MVES. As we know, all computer models involve varying degreesof simplification of the real system. By making appropriate assumptions, a complexsystem 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 approachfor 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, andterminal lymphatics. It is a complex biological system. Some structures and properties ofthe system are still unknown or controversial. Therefore, simplifying assumptions mustbe 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 bedivided into?In the present model, the MVES is divided into two compartments, the circulationand a general tissue compartment (Fig. 3.1). The capillary-interstitium interface, whichconsists of the capillary wall and its basement membrane, lies between these two compartments. It is obvious that lymph and blood can each be treated as a single compartmentdue 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 intoa general interstitial compartment due to their heterogeneities in properties, form andfunction (see Table 2.2). Iii previous modelling studies of rats [41, the interstitium wasseparated into individual skin and muscle compartments, since, on a weight basis, theyare the two largest tissues in the body. However, this separation is not reasonable inhuman studies because of the lack of information on human tissues. The most completeChapter 3. Model Formulation 34Figure 3.1: Schematic diagram of the compartmental model of the MVES.Chapter 3. Model Formulation 35set of information on humans is available for subcutaneous tissue and dermis. Fortunately, 50— 70% of the total interstitial fluid is stored in loose connective tissue such asskin and as little as 10% in muscle [2]. Also. results from the rat model indicate thatskin and muscle compartments behave similarly [76]. Therefore, the properties of thegeneral interstitial compartment will be assumed to be approximately equal to those ofsubcutaneous tissue.The microscopic geometry of both the capillary and the interstitium, which make upan exchange unit, also supports the compartmental assumption. Here, an exchange unitis defined as the cross-sectional area through which fluid and proteins must travel fromthe capillary to the nearest terminal lymphatic. The exchange unit for subcutaneoustissue has been estimated to be less than 8.1x104im2in area [113]. Assuming theexchange unit is a square, the distance between the capillary and the nearest terminallymphatic will be around 0.03 1um. The time required for fluid and solute molecules todiffuse 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, afew seconds are instantaneous. Therefore, the compartmental method is an appropriateapproach 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 tobe homogeneous, i.e. whatever changes happen in one place occur throughout the wholecompartment volume simultaneously. Other important assumptions include:• All plasma proteins are referenced to a single protein species, albumin. This isjustified by the fact that albumin accounts for more than 50% of the total plasmaprotein in humans and, by virtue of its smaller molecular weight and greater netChapter 3. Model Formulation 36charge, it accounts for approximately 65% of the total plasma colloid osmotic pressure.• 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 thesystem perturbations investigated in the dllrrent study, the volume and compositions of intracellular fluid are unchanged. Parenchymal cells are always in dynamicequilibrium with extracellular fluid.• Physicochemical properties of macromolecules, such as GAGs, proteoglycans, collagenous and elastic fibres, remain unchanged throughout the perturbations. Compliance of the interstitial compartment is assumed to follow the “most likelyhuman 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 dynamicsassociated with the MVES.The MVES is not an isolated system. Thus the third question concerns the assumptions about the surrounding environment. The effects of nervous system and hormonerelease on the MVES is considered only indirectly in the current study.In summary, the MVES is simplified to two homogeneous compartments, the circulation and the interstitium. Fluid and albumin are the only significant species withinthe system. Mass exchanges take place at the capillary-interstitium interface and at theterminal lymphatics due to passive driving forces.Chapter 3. Model Formulation 373.3 Coupled Starling ModelWhen the principle of mass conservation is applied, the following equation applicable inboth the circulation and the interstitium can be derived:Accumulation of S = Inflow of S — Outflow of S + Generation of S — Consumption of Swhere S denotes fluid or albumin content. If the generation and consumption terms equalzero, the above equation simplifies toAccumulation of S Inflow of S — Outflow of S (3.1)This form of the conservation of mass equation will be used to simulate perturbationsinvolving 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 inor out of the system, MVES models can be classified depending on the mechanism oftranscapillary exchange. Several mechanisms, such as those described by the Patlak [10]as well as uncoupled Starling and plasma leak models, have been investigated in ourresearch [13]. Because the Patlak formulation requires fewer fitting parameters thanthe plasma leak model and provides a more reasonable coupling of protein diffusion andconvection 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 homoporous model. In this model, the pores in the capillary membrane are assumed tobe a single size and this size is characterized by the value of the albumin reflection coefficient (o). Fluid is transported from the capillary to the interstitium by filtrationaccording to the Starling hypothesis described earlier. Solutes (i.e., albumin) are transported passively by diffusion and convection through the same channels that carry thefluid and thereby coupled. Interstitial fluid is drained back to the circulation by lymph.Chapter 3. Model Formulation 38BLOOD CAPiLLARY INTERSTITIUMLWALJPFigure 3.2: Schematic of the coupled Starling model.The hydrostatic pressure within the capillary is represented by a single value termed thecapillary hydrostatic pressure, Pc (see Fig. 3.2).The quantitative analysis of the fluid filtration flow across a membrane was firstintroduced 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 osmoticpressures, respectively; and subscripts C, PL, and I denote capillary, plasma, and interstitium, respectively. KF is the filtration coefficient (mL.mmHg1.h’) and its value isdetermined by the hydraulic conductivity and total area available for fluid transport. a isthe albumin reflection coefficient and its value is determined by the properties (e.g. size,charge) of the solute molecule and the channel (i.e. pore) through which solute moleculesChapter 3. Model Formulation 39pass. If ci = 1, the membrane is perfectly impermeable to solute and the full osmoticpressure of the solution opposes filtration; if ci = 0, the membrane allows solute moleculesto pass through freely and the osmotic pressure of the solution offers no resistance tofiltration. The filtration rate is proportional to the net Starling driving force (bracketedterm in Eq. 3.2).Albumin is transported across the membrane by diffusion with the superimpositionof convection. Bresler and Groome [10] solved the one- dimensional convective-diffusionequation for a uniform cross-section pore to derive the following equation for transmembrane albumin transport:/ CFL — CI,Av exp(—Pe)Qs=JF1—ci). 331 — exp(—Fe)where Fe is the modified Péclet number given byFe = (1 — ci). JF/PS (3.4)Here Qs 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 albuminconcentration and is calculated as the interstitial albumin content divided by the interstitial volume available to albumin (Vr,Av), which will be discussed later. PS is the productof membrane permeability to albumin and the membrane surface area (mL/h) and usedto 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 thesolute.By rearranging Eq. 3.3, the albumin transport rate can be partitioned into two components, i.e.,Qs = JF (1 — ci) . CPL + i_e(x1p(Pe . (CPL — CI,Av). (3.5)Chapter 3. Model Formulation 40where the first term is referred to as the convective component and the second term asthe diffusive component. Equation 3.5 is not equivalent to the sum of the uncoupledconvective and diffusive solute transports through a single channel [13], i.e..Qs = JF (1 — a). [(CpL + CI,Av)/2]+ FS (CPL — CI,Av). (3.6)Thus it can be seen that the convective and diffusive transfers of albumin in the Patlakmodel mutually influence each other.Interstitial fluid is drained back to the circulation compartment by lymph flow. Asdiscussed in Chapter 2, bileaflet valves on terminal and collecting lymphatics ensureunidirectional lymph flow. Under normal conditions, the lymph flowrate, JL, is alwaysassumed to be positive, i.e. lymph always flows from the interstitium to the terminallymphatics. It is also assumed that the composition of the initial lymph is the same asthat of the interstitial fluid. Therefore, there is no difference in colloid osmotic pressurebetween the lymph and interstitial fluids. Hydrostatic pressure in the interstitium isassumed to be the only driving force for lymph flow deviations from the normal baselinelevel. A linear relationship between the lymph flowrate and the tissue hydrostatic pressureof the type developed by Bert et al. [4] for their rat model is also employed in thecurrent human model. During tissue overhydration, the lymph flowrate, JL, is assumedto increase proportionally to the change in interstitial hydrostatic pressure by a factorLS, which is called the lymph flow sensitivity (mL.mmHg1.h’). Thus,= JL,O + L5 (Pi — F1,0) when P1 Pi,o (3.7)where JL,O is the lymph flowrate at normal steady-state conditions (mL/h). As discussedin Chapter 2, this normal lymph flow is formed by the contraction of the tissues surrounding the terminal lymphatics, as well as by the spontaneous contraction of the collectinglymphatics.Chapter 3. Model Formulation 41During tissue dehydration, another linear relationship is used, namelyJL JL,O (P1 — PJ,Ex)/(PI,o — PI,Ex) when P1 <P1,0 (3.8)FI,EX is the interstitial hydrostatic pressure when the tissue dehydrates till the interstitial fluid volume is equal to the excluded volume (VI,Ex). This is considered to be thelimit to which the tissue can be dehydrated. At this condition, the lymph flow, accordingto Eq. 3.8, ceases completely, i.e. JL = 0. Further dehydration is beyond the scope ofthe current discussion, because it would be accompanied by drastic changes in the structure and properties of the interstitium, e.g. cell dehydration, fiber and macromoleculebreakdown, etc.. Such changes conflict with the assumptions made at the beginning ofthis chapter.Because of the non-sieving character of the terminal lymphatic walls (o is assumed tobe zero), plasma proteins within the intersititium are assumed to be transported acrossthe wall solely by convection with lymph flow. Therefore, the albumin exchange rateacross the lymphatic wall, Qr, is given as the product of the lymph flowrate and thealbumin concentration in the interstitium:QL = Jr C1 (3.9)Equation 3.9 assumes that, once the plasma protein solution from the available volumeenters the lymphatics, it mixes with the interstitial fluid leaving the remaining portions ofthe mobile fluid volume according to their relative volumes; therefore, C1 is used insteadof CI,Av.Now the mass balances for the circulatory and interstitial compartment can be written more specifically. The fluid and protein balances, respectively, for the circulatorycompartment becomedVp/dt = JL — JF + D1 (3.10)Chapter 3. Model Formulation 42anddQpL/dt = QL — Qs + D2 (3.11)where D denotes external perturbations to the system, e.g. fluid infusion (D positive),protein infusion (D2 positive), urine output (D1 and D2 negative), etc.. For the interstitialcompartment, the two balances becomedV1/dt= JF — JL (3.12)andclQj/dt= Q — (3.13)If both compartments are at steady-state, then the left hand sides of Eqs. 3.10— 3.13 areset to zero, i.e. the fluid volumes (VPL and V1) and albumin contents (QPL and Qi) inboth compartments remain constant.3.4 Constitutive RelationshipsWhen the dynamic characteristics of the microvascular exchange system are studied,some variables are subject to variations with time until a new balance is obtained. Thesevariations involve the changes in volume, concentration, hydrostatic and colloid osmoticpressures, etc.. Thus, in order to complete the description of the model, it is necessary toestablish additional, constitutive relationships between these variables. In this section,three such relationships will be discussed, i.e. the circulatory compliance, the interstitialcompliance, and the relationship between albumin concentration and colloid osmoticpressure.3.4.1 Circulatory ComplianceDue to the microscopic nature of the capillaries, the techniques for measuring capillaryhydrostatic pressures, P0, are not yet reliable. Therefore, it becomes necessary to relateChapter 3. Model Formulation 43Pc to variables which can he measured directly. It is believed that Pc is dependent onthe arterial pressure, FAA, and venous pressure, PVV according to [72]:Pc = [(Rvv/RAA) . FAA + Pvv]/[1 + (Rvv/RAA)} (3.14)where RAA and Rv are the precapillary and postcapillary resistances, respectively. Thephysical meaning of the above equation is obvious. However, the use of Eq. 3.14 isproblematic, 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 inarterial and venous pressures result in changes to another measurable output, the plasmavolume, VPL. Circulatory compliance, FCOMPC, is defined as the ratio of the changein 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 hypothesis, we know that there are other factors (e.g. colloid osmotic pressures, transportproperties of the membrane) besides the capillary hydrostatic pressure which determinethe plasma volume. Other active mechanisms associated with hormonal, neural or myological behaviour which will not be considered directly in this study may also affect thecirculatory compliance. In fact, for humans, there are insufficient data even to establisha quantitative relationship between Pc and VPL. Thus for the sake of simplicity, a linearrelationship is assumed to apply, i.e.FCOMPC = LVPL//PC = constantorPc = Pc,o + PC,GRAD• (VPL — VpLO) (3.16)Chapter 3. Model Formulation 44where PC,GRAD is the reciprocal of the circulatory compliance.Since, to our knowledge, PC,GRAD has never been measured in humans, therefore, anestimate of its value is needed. Bert et al. [4] estimated PC,GRAD = 5.05 mmHg/mL inthe rat microcirculation by statistically fitting a set of fluid volume distribution data asa function of total fluid volume. This value is scaled for use in the current human modelby ensuring that equal fractional increases in plasma volume for both the rat and humangive rise to equal elevations in capillary hydrostatic pressure. In the rat, PC,GRAD = 5.05mmHg/mL corresponds to an elevation in P of 30.91 mmHg when VPL is raised to twiceof its normal level, (from 6.12 mL to 12.24 mL), i.e. PC,GRAD = 30.91 mmHg/6.12 mL.In human, with a normal plasma volume of 3200 mL (the normal values for human willbe discussed in next section), to satisfy the scaling criterion, PC,GRAD should be equal to30.91 mmHg/3200 mL = 0.009659 mmHg/mL.3.4.2 Interstitial ComplianceThe interstitial compartment is considered to be a storage reservoir for body fluid. Interstitial compliance, FCOMPI, therefore, is an important property which can markedlyaffect the distribution of body fluid. It is defined by Eq. 2.1. Although there is a generalawareness of the importance of the interstitial compliance, very little actual informationabout it is available in the literature. Reed and Wiig [109, 79, 110, 111] have conducted aseries of experiments to study the interstitial compliance relationship in mammals otherthan humans. After studying the compliance characteristics of skin and skeletal muscle in rat, cat and dog, they concluded that the compliance of both skin and skeletalmuscle follows a similar trend: low compliance during severe tissue dehydration, moderate compliance between moderate dehydration and moderate overhydration, and highcompliance during severe tissue overhydration. Experimental data from the rat [109, 79]were fitted and used in compliance relationships for the skin and muscle compartmentsChapter 3. Model Formulation 45in the rat microvascular model of Bert et al. [4].The only experimental data on human tissue compliance known to the author is fromStranden and Myhre [99]. They studied 46 patients with unilateral leg edema. The tissuehydrostatic pressure, F1, was measured in subcutaneous tissue by the wick-in-needletechnique. The subcutaneous tissue volume increase was calculated as the differencein volume between the edematous and the contralateral leg. The results are shownin 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 tothem. However, it was found that the compliance varied significantly at different levelsof interstitial tissue hydration: P1 increased markedly with increasing subcutaneous tissuevolume in patients with moderate edema (0— 100% subcutaneous tissue volume increase),but insignificant further increase in P1 was observed with additional edema (100- 600Under these circumstances, a so-called “most-likely” human interstitial compliancerelationship is constructed on the basis of the information from Stranden and Myhre aswell 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 theinterstitial hydrostatic pressure of rat according to:PI,HUMAN — PI,HUMAN,O = (PI,RAT — PJ,RAT,o). (3.17)and by scaling the interstitial volume according to:VI,HUMAN.OVI,HUMAN = VI,RAT ( ) (3.18)VI,J?AT,IJwhere 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 aFigure 3.3: The interstitial fluid pressure-volume relationship in patients following arterial reconstruction for femoropopliteal atherosclerosis. • represents patients with postoperative edema. represents patients without postoperative edema (mean±SD). represents healthy controls. Number of subjects investigated is bracketed below [99]. The solid line is the “most-likely”human interstitial compliance relationship.46Chapter 3. Model FormulationE 5-E- 4-3-.‘ 2-3 0-0,I.G-3-C. •.. . . . a..C• ..•••I I I I I I I I0 200 400 600 800Subcutaneous tissue volume increase (%)Chapter 3. Model Formulation 47Dehydration: Pi = —0.7 + 1.96154 x 103(V — 8.4 x i0)Intermediate: cubic spline interpolation (see subroutine SPLINS)Overhydration:Compliance #1 P1 = 1.88 + 1.8 x 10—5(Vi — 1.26 x iO)Compliance #2 Pi = 1.88 + 5.0 x 10—5(V — 1.26 x 10)Compliance #3 Pi = 1.88 + 1.05 x 10—4(V — 1.26 x i0)Table 3.1: Mathematical descriptions of the interstitial compliance relationship.series of discrete data points obtained from the skin compliance relationship in the ratmicrovascular model [4]. The “most-likely” human interstitial compliance so generatedis plotted as the solid line (compliance #1) in Figs. 3.3 and 3.4. Figure 3.3 shows thatthe selected curve is a reasonable representation of the experimental data points.The relationship is artificially separated into three regions: the “dehydration segment”(V1 8.4 L), the “intermediate segment” (8.4 L < V1 < 12.6 L), and the “overhydrationsegment” (V1 12.6 L). The confidence of the overhydration region is considered to bethe lowest, therefore, two other overhydration compliances (compliance #2 and #3) aregenerated by arbitrarily increasing the slope in this region, while the other two regionsremain the same as that in compliance relationship #1. The mathematical descriptionsof the three compliance relationships are summarized in Table 3.1.3.4.3 Colloid Osmotic Pressure RelationshipSince colloid osmotic pressure is caused by the fact that solute molecules can not readilydiffuse through the semipermeable capillary membrane, it is natural to assume that thereis some kind of correlation between protein concentration and colloid osmotic pressure. In1963, Landis and Pappenheimer [47] derived three empirical equations to relate albumin,globulin, and total protein concentrations to colloid osmotic pressure. However, in theChapter 3. Model Formulation 485.00.0-tzOCompliance curve 1-5.0- PJ’.P Curvej2 -cQmpU..cc.w’x.-.0.0 tOM 20.0 30.0 40.0v1(L)Figure 3.4: The “most-likely” human interstitial compliance relationship [13].Chapter 3. Model Formulation 49current model, the single protein species albumin has been chosen to be respresentativeof 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 smallermolecular weight and greater net charge; and3. the measurements of its content and concentration are the most widely reported.Therefore, we chose to determine the relationship between the albumin concentrationand the total colloid osmotic pressure so that, in case one of them is given, the othercan be calculated. Chapple [13] has developed such a relationship by a least squaresfitting of the data collected from the circulatory compartment of patients with nephroticsyndrome [31, 63, 241. Nephrosis is a kidney disease which causes a lowering of the bloodprotein level. The fitted relationship was forced to pass through the point 11PL = 0 wheuCFL = 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 thefirst and second order relationships and because the higher order polynomials producednon-monotonic behaviour within the range of interest, the linear relationshipin—3 TT‘—‘FL — 1.522 x ,u “FL 3.19was selected as the best choice. In Eq. 3.19, the units of CPL and 11FL are g/mL andmmHg, respectively. This relationship as well as the experimentally measured data pointsare plotted on Fig. 3.5.Landis and Pappenheimer [47] had earlier determined the following relationship between total protein concentration (C) and total colloid osmotic pressure (H):fl = 0.21 x C + 1.6 x i0 x C2 + 9.0 x 10-6 x C3. (3.20)Chapter 3. Model FormulationC)CC)50Figure 3.5: Relationship between albumin concentration and total colloid osmotic pressure [31, 63, 24].50.10.0 20.0II (mmHg)Chapter 3. Model Formulation 51The good agreement between Eq. 3.19 and Eq. 3.20 over the concentration range ofinterest shows that neglecting protein differences does not affect the LI values significantly[14].Because of the difficulties in collecting interstitial fluid, very little information aboutthe interstitial compartment is available. If it is assumed that the relative osmotic activityof 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 fromsome of the tissue volume.3.5 Normal Steady-State ConditionsIn order to study the transient responses of the MVES after the system is disturbed, aninitial point from which these transient responses deviate must be specified. This initialpoint is taken to be the normal steady-state conditions in the current study.As we know, different individuals possess different body weights, fluid volumes, pressures, 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 andhydrostatic pressures are taken at heart level of the thorax, if possible.To make comparison convenient, the normal steady-state conditions of the “referenceman” in the current model are the same as those used by Chapple [13], and they areChapter 3. Model Formulation 52Variable Circulation Tissue ReferencesV0, mL 3200 8400 [21, 106, 94, 56, 26]VEX, mL — 2100 [76](assumed value)ll, mmHg 25.9 14.7 [23, 46, 64, 27]C0, g/L 39.4 16.8 calculated from Eq. 3.19 and 3.23CAV,O, g/L — 22.4 calculated from Eq. 3.21Qo, g 126.1 141.1 calculatedPA,o, mmHg 5.92 — [106]Pv,o, mmHg 24.54 — [106]F1,o, mmHg— -0.7 [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 emphasizedhere. First, the normal fluid volumes in both compartments are confidently known. Thevalues have been further verified by more updated information. Fauchald [25] measuredthe 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 calculationof the interstitial fluid volume as Vi,o = ECV— VPL,O. In this manner, Fauchald andNoddeland et al. found that the normal range of interstitial fluid volume lay between6.0 L— 11.3 L and 7.1 L — 12.2 L, respectively. These measurements made us feel moreconfident about the fluid volume values listed in the table.The second point concerns the excluded-volume, VEX. The degree of exclusion isexpressed in various ways. Sometimes, it is based on the matrix space instead of the“free-fluid” space (i.e. V1), while sometimes, it is based on the excluding material’svolume or weight alone. These different bases make it difficult to use the informationavailable in the literature. Moreover, there is little information concerning exclusion inChapter 3. Model Formulation 53human 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 doghindpaw 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 equalsthe 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 compartment, CI,Av. It is defined as the ratio of interstitial albumin content to the interstitialfluid volume available as distribution space for that protein, i.e.CI,Av = Q1/(V— VEX) (3.22)It is the effective interstitial albumin concentration rather than the actual interstitial albumin concentration that-determines the interstitial colloid osmotic pressure. Thus, given111,0 (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 interstitial albumin concentration required in the lymph drainage expression (Eq. 3.9) isobtained fromC1 = Q1/Vj (3.23)A similar equation applies to the plasma albumin concentration, i.e.CPL = QPL/VPL (3.24)3.6 SummaryA summary of the complete set of equations required by the coupled Starling model islisted in Table 3.3.HHHHIIIIIIII1’9CDIII_—CDqI—-__-I-‘lJCl)C)c_—i2II0‘-jIC)__CDCDI‘-(i)IC)ICD—IHI—.lIc)—Le.DICl’_-:(IC Cl) CC)Cl)0I.‘—S—Cl) I- — -. 0IICDHqe-l ICD -.+C3---.Chapter 4Parameter Estimation And Data AnalysisIn 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 howto use these data to obtain the unknown parameters for the coupled Starling model.4.1 Parameters to be DeterminedA 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 physicochemical properties of the system. These constants are called the internal parametersof the MVES model. Their values used in the model affect the fluid and protein distribution within the MVES significantly. For example, the fluid filtration coefficient, KF,the diffusive permeability coefficient, PS, the solute reflection coefficient, a, the normallymph flow rate, JL,O, and the lymph flow sensitivity, LS, are all internal parameterswhich characterize the transport properties of the capillary wall and the terminal lymphatics; 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 characterizesthe efficiency of the lymphatic system in removing accumulated interstitial fluid. Theseparameters are assumed to remain unchanged at the constant value assigned to themthroughout the duration of whatever perturbation is applied to the MVES. One of theobjectives of the current study is to determine the optimal values for these parameters.55Chapter 4. Parameter Estimation And Data Analysis 56Unknown DescriptionLS lymph flow sensitivity (mL.mmHg’.h1)a albumin reflection coefficientPc,o capillary hydrostatic pressure at normal steady-state (mmHg)KF the fluid filtration coefficient (mL.mmHg’.h1)PS permeability-surface area product (mL.h’)JL,O 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 compartmental 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 modelpredictions to available measurements from the literature.Because some of the aforementioned parameters have not been measured directly inhumans or even in animals, and some have only been measured inaccurately (these detailswill 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 interstitial fluid volume, P1 F(V1), can only be estimated (see Table 3.1). Thus, thecapillary pressure, Pc,o, is added to the list of unknown model parameters listed in Table 4.1, while the three interstitial compliance relationships discussed in Section 3.4.2are introduced into the model one at a time to test how they influence the results of theparameter estimation procedure.Chapter 4. Parameter Estimation And Data Analysis 574.2 Additional Relationships between the UnknownsOf the six unknowns parameters listed in Table 4.1, three can be eliminated via additionalrelationships which exist between these parameters. These relationships will be discussedin the sections that immediately follow.4.2.1 Steady-state Balances at Normal ConditionsTwo equations can be derived from the fluid and protein mass balances which must existin the intersititial or circulatory compartments under normal steady-state conditions.At normal steady-state, the fluid balance can be written asJF,O — JL,O = 0 (4.1)and the protein balance asQs,o — QL,O = 0 (4.2)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, PS, JL,O, a and Pc,o:KF [Pco P1,0 — a (HPL,o — L[’,o)] — JL,O = 0 (4.3)andJF,O . (1 — a) . [CPLO CIAvoexp(PeO)1— JL,O C1,0 = 0 (4.4)The degrees of freedom for a model is equal to the number of unknowns minus the numberof 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 bespecified to completely characterize the model.Chapter 4. Parameter Estimation And Data Analysis 584.2.2 Albumin Clearance RelationshipIn addition, several workers have investigated the clearance rate of125-radiolabelled (or‘311-radiolabelled) albumin from the subcutaneous tissue in humans. It is assumed thatinjecting the tracer does not alter the normal steady-state conditions of the system.The interstitial albumin disappearance half-time, T112 is defined as the time required toeliminate 50% of the total introduced labelled albumin from interstitium to the lymphaticsystem 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 Geigercounter at the injection site at various time intervals [22]. The percentage of the initialradioactivity is plotted as a function of time on a semilogarithmic graph allowing T172 tobe determined as the slope of the disappearance curve [48]. The rate of disappearanceof albumin from the interstitial compartment is defined as the albumin turnover rate,A1bTQ (fraction lost pr hdur). Thus, A1bTO and T112 are related by(1 — A1bTQ)T1/2 = 0.5 (4.5)The average value of T112 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—C,0— CZAV,o exp(—Peo) 4 7— (—F,0 1 — exp(—Peo) — L,0 ( . )According to the definition of albumin turnover rate, at t = 0,dQ*I AlL 17 fY*= iWTQ VI,OChapter 4. Parameter Estimation And Data Analysis 59Also, at t = 0, there is no labelled albumin in plasma, so that CLO = 0. Furthermore,since CZ0 = Q,0/Vj,o and CZAv,0 = Q,0/(V1,o— VEX), substituting into Eqs. 4.7 and 4.8(at t = 0) and then setting them equal yieldsJF,O . (1 — a) . exp(—Peo) JL,O —+ — — AibTo — 0 (4.9)[1— exp(—Peo)1(Vi,o — VJ,Ex) V1,0where— (1 — a). JF,OFe0—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 consequence, 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 OptimizedThe albumin reflection coefficient (a), the lymph flow sensitivity (LS) and the capillaryhydrostatic pressure at normal steady-state (Pc,o) were selected as the three parametersto 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 clearancerelationship; 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 to1. Hence, because of its convenient bounds, a was chosen as another parameter to beChapter 4. Parameter Estimation And Data Analysis 60optimized continuously within these bounds. Finally, Pc,o was chosen to be the thirdparameter investigated because its bounds can also be estimated, as is discussed in thenext section. Because the optimization program (see Section 4.5.3) is robust enough tosolve a three-dimensional problem and since a single set of the parameters is requiredfor validation purposes, we chose to optimize LS, a and Pc,o continuously, instead oftreating 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 manipulating Eqs. 4.3, 4.4 and 4.9:andA1bTQ(1—cr).ecp(—Peo)[1—exp(—Peo)J(VJ,o—VJ,Ex) +(1 — a). JL,OP5=CJ,o—(1—o-)CJ,Av,Q]ln[ CI,o—(—)CpL,oJL,OKF = (4.12)Pc,o — Pi,o — a. (IIPL,O—At this point, the model is completely defined. Thus, knowing the initial conditionsof the system and the perturbation, we can solve the four coupled ordinary differentialequations, Eqs. 3.10, 3.11, 3.12 and 3.13, to find the transient or steady-state responses ofvolumes and protein contents in both the interstitial and the circulatory compartments.In the process, the protein concentrations, osmotic pressures and hydrostatic pressuresof the system are also calculated.JL,O = (4.10)(4.11)Chapter 4. Parameter Estimation And Data Analysis 614.2.4 Parameter Search RangesTo save computational costs, bounds that encompass the region within which physiologically feasible parameter values must lie were calculated.The upper and lower bounds on Pc,o are the same as those used by Chapple [13]. Theupper limit was obtained by assuming that a trariscapillary pressure gradient of 1 mmHgis maintained under normal steady-state conditions. Thus, since Pi,o = —0.7 mmHg,fli,o = 14.7 mmllg and 11PL,O = 25.9 mmHg, and since the maximum Pc,o is obtainedwhen o 1, then Pc,o < 11.5 mmHg. The lower limit was obtained on the basis ofexperimental 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 pc,o 11.0 mmHg.- As was discussed in Section LB is independent of the steady-state mass balance.Thus there is no restriction imposed on LS from these mass balance equations and LSmay range from zero to infinity. To reduce computational costs, LS is assigned thebounds:0 LS 500 mL . mmHg’ . hr’In most cases, the value of LB which produces the best fit is significantly less than theartificially imposed upper limit.Because KF, PS, and JL,O must be positive, the possible values of a must be restrictedaccordingly. Since JF,O is positive (which is equal to JL,O at normal steady-state), thento ensure that KF remains positive in Eq. 3.2,Pc,o — Pi,o — 0-. (H,o—> 0 (4.13)or,P,0 — Pi,o (4.14)L[PL,O — 111,0Chapter 4. Parameter Estimation And Data Analysis 62Since a < 1, therefore, 1 — a 0. And normal lymph flow is assumed to be positive, i.e.,JL,O > 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 existsin the evaluation of logarithms, givesCPL,0 — I,0<a 1. (4.16)CPL,0To ensure that JL,O remains positive, then according to Eq. 4.10(1— a) . exp(—Peo)0 4 17[1 — exp(—Peo)](Vj,o— VI,Ex) + V1,0>Rearranging Eq. 4.11 gives,(1— a) JL,O—1—(1—a). CI,Av,oj (418)PS n— (1 — a) CPL,0Taking exponentials on both side of the above equation gives,e(l_JL0 Ci,o — (1 — a) CI,Av,o (4.19)C1,0 — (1 — a). CPL,0or, according to the definition of the Péclet number (Eq. 3.4),e_Pe0C1,0— (1 — a) CpL,o (4.20)C1,o — (1 — a) CI,Av,oUpon substituting Eq. 4.20, Eq. 4.17 can be simplified asa>1[CPLO — C1,o(VIO— V1Ex). (CpL,o— CJ,Av)]. (4.21)CPL,0 V1,oAlso, since CJ,Av,o = Q1,0/(V1,0— VI,Ex) and Cj,o =Q1,0/V then= (VIO_EX). CI,Av,o (4.22)Chapter 4. Parameter Estimation And Data Analysis 63P,o (mmHg) clmjn amax7 0.574 0.6888 0.574 0.7779 0.574 0.86610 0.574 0.95511 0.574 1.000Table 4.2: Bounds on a for Pc,o equal to 7 to 11 mmflg.Substituting Eq. 4.22 into Eq. 4.21 yieldsCI,Avo —a> . (4.23)CI,Av,oThe range of a values which simultaneously satisfies inequalities 4.14, 4.16 and 4.23 isfound to beCPL,0 — Ci,0.— }CFL,0 11PL,O — 111,0The upper bound is chosen to be either 1 or (Pc,0— PI,o)/(llpL,o— ui,o), whichever issmaller. The latter expression, as can be seen, depends on P,0. The actual bounds ona for Pc,o values ranging from 7 to 11 mmHg are listed in Table 4.2.4.3 Analysis of DataEver since the general principles of the Starling hypothesis for transcapillary fluid exchange were widely accepted, measurements of the Starling forces and other relatedfactors in the MVES have interested many investigators. After years of exploration,reliable approaches for measuring many of these factors have been developed. Generalknowledge of these measuring techniques will assist us in evaluating the quality of theavailable experimental data.For example, interstitial colloid osmotic pressure data can be obtained by:Chapter 4. Parameter Estimation And Data Analysis 641. direct sampling by micropipettes and catheters. The drawback of the method isthat the applied suction could increase the net capillary filtration and therebyinfluence the interstitial protein concentration [86, 87];2. implanted capsules or wicks based on the assumption that a fluid compartment incontact with the interstitium will finally attain the same protein concentration andhydrostatic pressure as the free-fluid phase of the interstitium [23];3. noninvasive blister suction method [80]. This method is subject to the same criticism as per # 1 above.The wick technique is the most widely used method and is generally thought to be themost representative and accurate. Once the interstitial fluid is collected, the interstitialcolloid osmotic pressure can be measured directly by an osmometer, and protein concentrationcan be determined by using a number of methods. Blood samples are easy tocollect; therefore, plasma osmotic pressures and protein concentrations can be determinedimmediately using these same techniques.Alternative methods for the measurement of hydrostatic pressures in interstitiuminclude [2]:1. the use of chronically implanted perforated plastic capsules (with diameters of 1cm 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 manometerthrough a fluid-filled tube. However, it is impossible to place a micropipette or needle inside the capillary because the capillaries have internal diameters of 5 — 10 m only.Chapter 4. Parameter Estimation And Data Analysis 65Thus, to the knowledge of the author, no one has so far directly measured the capillaryhydrostatic pressure.The methods for the measurement of plasma volume and interstitial volume are basedon the dilution principle. A measured amount (Q) of a suitable tracer substance isadministered by injection. After equilibrium throughout the whole body is attainedand corrections are made for metabolic losses of the tracer, the concentration (C) ofthe substance is measured in a suitable sample of the body water. The volume of thecompartment (V) is then given by the relation, V = Q/C. The dilution agents used inthe measurement of extracellular fluid volume (ECV) include insulin, sucrose, mannitol,sodium thiocyanate and the radioactive ions, 35SO, 82Br and 24Na+. Tracers usedin the measurement of plasma volume include Evans blue dye,‘311-labelled albumin or51Cr-labelled erythrocytes. The interstitial volume is then calculated as Vi = ECV—VPL.In this section, we begin by preseiiting all of the available literature dataon humanswhich we hope to include in the parameter estimation procedure. We then analyze eachdata set in detail to determine whether it really provides information about the problemunder study and whether the measured results can reflect the relationships between thedependent variables and the parameters we investigated. Finally, we discuss how thesedifferent data sets can be combined and used in the optimization procedure.4.3.1 Experimental DataSix sets of useful data were found by an extensive search of the literature. The quantitiesreported in these independent studies include colloid osmotic pressures, fluid volumes,protein contents and protein concentrations, which were measured by applying the varioustechniques 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 thesedifferent measurants comparable, we have chosen to convert all of them into percentageChapter 4. Parameter Estimation And Data Analysis 66changes, 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 xlOOxowhere X denotes the quantity measured in the experiment and subscripts t and 0 indicatequantities 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- XNQRMx 100XNORMwhere X and XNORM represent the same variables measured in a patient and in a normalsubject, respectively.This normalization procedure has several advantages. First, as was mentioned, itmakes all the measured quantities unitless so that they can be compared. Second, itconverts the magnitude of all the quantities into a similar range, again so that they aremore comparable. For example, before conversion, Vj has a value of around 8400 mL andF1, a value of about —0.7 mmHg. It is difficult to compare 8400 mL with —0.7 mmHg.But, after conversion, their percentage changes are usually less than 100% as can be seenin sections which follow. Third, using percentage changes simplifies the normalizationprocess because it eliminates the differences in baseline values of variables in differentexperiments. For example, consider the variable VPL. In one experiment, the initialplasma volume may be 2547 mL, while in another it may be 3227 mL. If percentagechanges are used, the differences in starting values are eliminated automatically.Only the modified data, based on the normalization procedure discussed above, arepresented in the section which follow. The raw experimental data are listed in AppendixA. For each of these six sets of results, the objectives, perturbations, quantities measuredand sample sizes of the experiments are described in detail below.Chapter 4. Parameter Estimation And Data Analysis 67Time (hr) -1.5 1 3 6 9 12Saline, 100 mL Mean 0.00 -3.37 -0.41 -2.86 -4.09 -7.36N=4 SD + 0.00 5.85 8.43 7.09 6.63 10.16Albumin, 100 mL Mean 0.00 2.04 -0.41 -2.45 -1.64 -4.50N=4 SD + 0.00 11.11 10.19 7.64 9.73 6.33Saline, 200 mL Mean 0.00 -3.73 -0.83 -1.66 2.07 -0.41N=4 SD ± 0.00 16.94 10.94 11.71 19.88 14.71Albumin, 200 mL Mean 0.00 13.28 12.86 6.22 8.30 6.64N=4 SD ± 0.00 12.51 20.36 14.87 17.93 13.24Table 4.3: Percentage change in plasma colloid osmotic pressure at room temperature. t = 0 designates the end of the infusion period [41].4.3.1.1 Set A: Saline and Albumin InfusionTo study albumin-induced plasma volume expansion, Hubbard et al. [41] infused eighthealthy male volunteers with solutions of saline or saline plus albumin over a 1.5 hourperiod in a thermoneutral enviroment. Four of them were raiadomly assigned to a lowdosage treatment, i.e. intravenous infusion of 100 mL saline or 100 mL saline with 25 galbumin, while the other four were assigned to a high dosage treatment, i.e. 200 mL salineor 200 mL saline with 50 g albumin. Just before the infusion and 1, 3, 6, 9 and 12 hourspost-infusion, plasma volume was measured by dye dilution using indocyanine green,plasma osmotic pressure was measured using an oncometer, and total plasma proteincontent was determined by commercial tests. The results are tabulated in Tables 4.3,4.4 and 4.5 (for 11FL, VPL and QFL, respectively) in terms of percentage changes frompre-infusion values. Both average changes and standard deviations are provided. Duringthe observation period (0-13.5 hours), an average net weight gain of 1.4 kg due to fluidintake occurred. This will be considered as part of the perturbation. Therefore, the totalfluid 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 68Time (hr) -1.5 1 3 6 9 12Saline, 100 mL Mean 0.00 1.80 0.59 3.52 7.41 6.56N=4 SD + 0.00 38.10 38.25 39.17 39.99 39.35Albumin, 100 mL Mean 0.00 11.04 10.30 11.66 10.82 10.52N=4 SD ± 0.00 41.82 45.62 45.10 43.14 43.34Saline, 200 mL Mean 0.00 5.00 4.50 5.00 3.38 5.24N=4 SD ± 0.00 19.75 17.37 16.25 18.41 17.67Albumin, 200 mL Mean 0.00 13.25 11.88 7.85 8.72 9.74N=4 SD ± 0.00 18.41 14.79 15.61 14.84 16.36Table 4.4: Percentage change in plasma volume at room temperature. t = 0designates the end of the infusion period [41].Time (hr) -1.5 1 3 6 9 12Saline, 100 mL Mean 0.00 0.04 1.14 0.81 3.56 3.64N=4 SD ± 0.00 6.60 7.27 7.55 6.16 6.29Albumin, 100 mL Mean 0.00 9.41 8.31 7.25 7.67 6.57N=4 SD ± 0.00 5.54 5.42 7.21 5.52 6.12Saline, 200 mL Mean 0.00 0.77 3.08 4.10 4.32 5.51N=4 SD ± 0.00 6.19 7.48 7.60 9.15 9.88Albumin, 200 mL Mean 0.00 19.06 15.60 12.56 10.38 11.58N=4 SD ± 0.00 10.44 9.32 7.99 8.29 8.25Table 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 69However, because QPL VPL x CPL, the results presented in the three tables are notindependent. Only VFL and CPL, the measured quantities, are selected to be comparedwith the model predictions.4.3.1.2 Set B: Acute Saline InfusionTo study the effect of rapid intravenous infusion of physiologic saline solution on thepulmonary arterial and capillary pressure of otherwise normal human subjects, Doyleet al. [18] studied twelve adult men convalescing from noncardiac ailments. In eachcase, 900— 1000 mL of normal saline solution were injected intravenously in 6.5 — 13mm. Blood volumes (BV) and hematocrits (HCT) were measured both pre-infusionand post-infusion. Plasma volumes were then calculated from VPL = BV. (1 — HCT). Insome patients, e.g. Patients #4, #8 and #10, the plasma volume kept increasing after theinfusion, (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 fromfurther consideration, leaving only five sets to be used in the parameter estimation. Thesedata, 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 experimentalerrors are used. The estimate of the standard error (SE) is based on the assumptionsthat an error of ±316 mL/m2of body surface area was associated with the measurementsof blood volume [18] and that the hematocrit readings were accurate. The calculation ofthe estimated standard error is illustrated in Appendix B.Chapter 4. Parameter Estimation And Data Analysis 70Patient # Remarks VPL AVPL SE(rnL) %Control 2547.4+331.3 0.00 26.011 l000mL/9.5mins 3182.5±361.8 24.93 30.4538 mm. after infusion 2766.0±342.8 8.58 27.58Control 3227.9±342.3 0.00 21.212 l000mL/llmins 4031.2±366.1 24.89 24.5960 mm. after infusion 3726.9+358.0 15.46 23.33Control2382.2±273.7 0.00 22.986 l000mL/l3mins 2927.2+300.3 22.88 26.7215 mm. after infusion 2711.2±290.4 13.81 25.27Control 2292.8±303.1 0.00 26.447 l000mL/llmins 2956.9±336.1 28.96 31.7130 mm. after infusion 2668.9±323.1 16.40 29.48Control 2606.0+286.9 0.00 22.0211 900mL/9rnins 3122.3+311.2 19.81 25.1340 mm. after infusion 2817.2+297.7 8.10 23.33Table 4.6: Percentage change in plasma volume after rapid saline infusion [18].Chapter 4. Parameter Estimation And Data Analysis 71VFL (mL) CPL (g/L) QPL (g)t=0 hr 3200 34 108.8- ±5 ±16t=2.5 hr 3541 30 106.2±1177 +5 ±53Percentage 10.66 -11.76 -2.39change (%) +36.78 ±27.68 +63.07Table 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 InfusionIn 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 atwo-hour period. Only 111 patients tolerated the entire dosage. Hemoglobin, hematocrit,total protein and albumin were measured before and after the infusion. Hemoglobin content is assumed to remain constant throughout the study. Hence, the percentage changeof albumin concentration (ACpL%) and plasma volume (AVpL%) can be calculated (seeAppendix B) and compared with the results predicted by the model using the same perturbation. All data in this set (Table 4.7) are quantitatively significant because eachdata point represents 111 repeat measurements. Also, since QFL VPL x CPL, only VPLand CPL were selected for use in parameter estimation.4.3.1.4 Set D: Heart FailurePatients 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 whomhad diuretic-resistant fluid retention with anasarca. Noddeland et al. [63] examined22 patients who had angina pectoris in another independent experiment. 11FL and HChapter 4. Parameter Estimation And Data Analysis 72were measured in both experiments. Noddeland et al. also measured the interstitialhydrostatic pressure F1, and Pc was then estimated as the isogravimet’ric capillary pressure, Pc,iso . From the filtration equation JF = KF• [Pc — Pi — u (HpL — fl)j, weknow that only when JF = 0, Pc = Pc,iso. Because the filtration rate is relativelylow 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, thisrelation is used to estimate the capillary hydrostatic pressure Pc in his patients. VPLand ECV were also measured; accordingly, Vj was calculated from the difference betweenECV and VPL. The results are listed in Table 4.8. The mean value of each group isused, so these data points are also quantitatively significant. Note that the pressures arenormalized 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 dataset undergoing normalization, and H represents the normal colloid osmotic pressure ofreference man. Also the fluid volumes are proportionally normalized with respect to thevalue 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 behaves normally [105]. The most pathological condition studied in heart failure as part ofthis 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 thecapillaries, i.e. Pc,iso = P1 + o (IIPL— Hi). It is not a measure of the actual capillary pressure, inprinciple.Chapter 4. Parameter Estimation And Data Analysis 73Anasarca Heart Failure Angina Pectoris Normal(N=7) (N=13) (N=22)11PL mmHg 20.3 23.4 24.0 25.9Hi mmHg 8.1 10.4 12.2 14.7Pc mmllg 20.1 16.91 11.45 10.0V1 mL 13120 12040 7450 8400SD 17340 15420 6750—/V?7o 56.2 43.3 -11.4—SD 206.4 183.6 80.4—Table 4.8: Experimental data at steady-state for patients with heart failure.patients begin losing dry tissue. Consequently, the protein content in the interstitiumwill be severely lowered. As well, the tissue structure will change [57]. Patients in thiscondition are not considered to have normal MVES parameters and are therefore notconsidered in this study. Thus, the corresponding data point (see Table 4.8) has beeneliminated from further consideration.In the parameter estimation procedure, 11PL, H, and Pc are fixed at the valuestabulated in Table 4.8, because these values are thought to characterize the differentextents of heart failure. Thus, the only Starling force component which can be alteredis the interstitial hydrostatic pressure. Parameter estimation in this case is based on thedifferences between experimental and predicted /V1 values.4.3.1.5 Set E: Nephrotic SyndromeThe data from chronic nephrotic patients which were used in Chapple’s [13] study arealso included in the current investigation. The nephrotic syndrome is characterized byproteinuria (i.e. excessive urinary excretion of plasma proteins) sufficient to induce hypoalbuminemia (i.e., a plasma albumin content below that of normal) and edema. Experiments show that patients with nephrotic syndrome exhibit a normal microvascularChapter 4. Parameter Estimation And Data Analysis 74exchange 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 transcapillary forces and fluid distributions in chronic nephrotic patients. All of them reporteddata relating H to 11FL (see Fig. 4.1, lower panel), but only Fadnes et al. [24] measuredVj at different 11PL (see Fig. 4.1, upper panel). All of the data are normalized followingthe same procedure as that used in Chapple’s study [13], i.e., pressures (LI1 and IIFL)are normalized arithmetically with respect to the normal steady-state values of referenceman (Eq. 4.24); while the fluid volumes are normalized proportionally (Eq. 4.25).In the parameter estimation procedure, the circulatory compartment is assumed tobehave as an infinite source/sink of both fluid and proteins and the interstitium is allowedto change its fluid and albumin contents to attain the steady-state corresponding to eachperturbed 11PL value.4.3.1.6 Set F: Saline infusion before extracorporeal circulationBefore 13 patients with coronary artery disease were operated on using extracorporealcirculation, Rein et al. [80] injected intravenously 1500 - 2000 mL of Ringer’s acetate.was measured subcutaneously on the chest using the blister suction method andll represents the average colloid osmotic pressure during the suction period (1.5 hrs).11PL was measured in a blood sample collected from a cubital vein. P1 was measuredas well, but was not used in the parameter estimation, because, after investigation, wefound that the error in P1 controlled the fit. The reason for this is that Pj has the lownegative baseline value of —0.7 mmHg in the model. Using the best-fit parameters oftissue compliance relationship #3 (which will be discussed in Chapter 5), the injectionof 1750 mL Ringer’s acetate will cause an elevation in P1 of around 1.63 mmHg. Thiscorresponds to a AP1% = 1.63/(—0.7) = —233%. According to Eq. 4.26, this point willChapter 4. Parameter Estimation And Data Analysis.7550.0-37.5-25.0-•.12.5t •• .‘0.00.0 5.0 10.0 15.0 20.0 25.0 3(1T(mmHg)- 20.0-16.0-12.0-..$8.0-. .a a4.0-•3.•a• •. ••.I I I I0.0 5.0 10.0 15.0 20.0 25.0 30.0(mmHg)Figure 4.1: Normalized data for patients with nephrotic syndrome [13]. Theupper panel has 18 points and the lower panel has 66 points, all of which areincluded in the parameter estimation procedure.Chapter 4. Parameter Estimation And Data Analysis 76“FL (mmHg) ll (mmHg)t=0 hr 24.1 14.7±1.1 ±0.9t=3 hr 17.5 13.7±1.2 ±1.2Percentage -27.4 -6.8change (%) ±8.3 ±13.9Table 4.9: Experimental data from Rein et al. [80]. Blister suction methodwas used to measure Hi, hence, All1 represents the average osmotic pressureduring the suction period (1.5 hrs).contribute to the objective function by 5843 units. Compared with the OBJmjn of 74.57units (from Table 5.1), we can see that this point will dominate the shape of the surfaceplot if it is included in the parameter estimation procedure. Hence, it was eliminated inthe current study. The useful results are tabulated in Table 4.9. Although the MVESapparently 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 StrategyFirst, we analyse the suitability of the experimental data for the parameter estimationtask at hand. Except for Set D and Set E, all the other experimental data sets use at leasta 900 mL fluid infusion within a short period. Now, let us discuss how the system willlikely be affected by 1000 mL fluid infusion. Assume that the 1000 mL of fluid is quicklyredistributed to the interstitium after it is injected intravenously, and that the plasmavolume is therefore essentially unchanged. This is the typical physiological response ofthe MVES to fluid infusion. According to the model, an extra 1000 mL of fluid in theinterstitium will increase Pj from -0.7 to 0.86 mmHg, i.e. AP 1.5 mmHg. With LSChapter 4. Parameter Estimation And Data Analysis 77ranging from 0 to 300 mL.mmHg’•h’, the lymph flow, therefore, could increase upto 450 mL•h1 above the normal level. Thus, the lymph flow sensitivity will affect theMVES lymph flow return rate moderately when the system is subjected to a perturbationat this value of fluid infusion.For an albumin infusion, 25 g of albumin in plasma will increase the plasma colloidosmotic pressure by approximately 5 mmHg, while 50 g will cause 11PL to increase byapproximately 10 mmHg. Since JF = KF• (P — Pj — g. (IIPL — Hi)), we know thata will influence the MVES filtration rate and, consequently, the plasma volume, plasmacolloid osmotic pressure, etc., which are the quantities being compared between the modelpredictions 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 concludethat the experimental data do provide the information about the parameters under studyandean 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 parameter estimation procedure. We have to keep in mind that, when these different data setsare combined, it is done so on the assumption that the transport parameters for thesedifferent 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 tomake 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 withineach set. To solve this problem, each point was assigned a weight factor, W. If thepoint represents an individual value, then W = 1/SD. This means that points whichhave 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 ntimes, then W = n/SD, i.e., the standard error of the mean (SE = SD//) is usedChapter 4. Parameter Estimation And Data Analysis 78to calculate the weight factor. This implies that if an average value is used, that point isquantitatively 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 modelpredictions must be compared with the experimental data point at that specific time. Asa convenience, the vertical distance instead of the shortest distance (which was used inChapple’s studies) between the prediction and measurement was selected for parameterestimation purposes.4.4 Parameter Estimation ProcedureTo obtain the best-fit values of u, LS, and Pc,o, the weighted least squares fitting criterionis selected. Thus, the parameter estimation procedure is based on finding the values ofthe unknown parameters which will minimize an objective function, OBJ, formed bysumming, over all data points, the squares of the vertical distances between the measuredand predicted percent changes, i.e.OBJ = wi,j(Ax%epij — X%simjj)2© given LS, a, Pc,o (4.26)i=1 j=1where N denotes the number of data sets and M denotes the number of data points inthe ith data set. The weight for each data point reflects the importance or accuracy ofthat 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) arethe best-fit parameters.Chapter 4. Parameter Estimation And Data Analysis 794.5 Numerical Methods and Computer Programs4.5.1 Transient SolutionsTo find the transient response of the MVES after a perturbation, the set of four simultaneous first—order nonlinear ordinary differential equations listed in Table 3.3 must besolved. These equations represent mass balances of fluid and proteins in both the interstitial and the circulatory compartments. They are integrated over time using theRunge-Kutta-Fehlberg method with error control [40]. The advantage of employing thismethod is that the local time step size is adjusted (i.e. in sharply curved regions the stepsize will be smaller, while in slowly-changing regions it will be larger) so that the cumulative 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 allowableerror is set to 0.01 mL and 0.01 g for fluid volume and protein content, respectively.4.5.2 Steady-state SolutionsThe steady-state solutions are obtained by solving the set of simultaneous nonlinearalgebraic equations that result when the accumulation terms of the ordinary differential equations are dropped. The numerical technique employed to accomplish this taskis Newton’s iterative method [11]. In this method, the non-linear equations are transformed to a set of linearized equations. The required partial differentials (to calculate thecoefficients of the Jacobian matrix) are approximated by finite differences. The resultinglinear algebraic equations are solved by Gauss elimination incorporating full pivot selection. The maximum allowable error is also set to 0.01 mL and 0.01 g for fluid volumeand protein content, respectively.Chapter 4. Parameter Estimation And Data Analysis 804.5.3 Computer ProgramsThe optimization routine used in current study is the UBC NLP, which is a nonlinearfunction optimization program. The parameter values which yield a global minimumin the objective function are obtained by invoking the subroutine GRG (in which thegeneralized reduced gradient method is employed) or subroutine NLPQL (in which aslightly modified version of the quadratic approximation method of Wilson et al. [89]is employed) from the interactive monitor program, NLMON. The monitor programprovides an interface to the nonlinear optimizing routines. To use the monitor program,a FORTRAN function subprogram named XDFUNC must be supplied to evaluate theobjective function. This function subprogram is listed in Appendix F. Since the currentstudy deals with a constrained optimization problem, a subprogram named XDCONSmust 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 derivativesusing central differencing.The monitor is invoked by issuing the MTS command$RUN FUNC.0 + NA:NLMONwhere FUNC.0 is a file containing the compiled versions of the aforementioned subprograms (i.e. XDFUNC and XDCONS). The monitor takes care of the difference in thecalling sequences among the different routines. The global optimum is found by rerunning the program using several different starting points. If all runs return the same pointas the optimum, it is assumed that the optimum solution is the global one. Upper andlower bounds are set as the search ranges of the parameters. The step size used in thecalculation of numerical derivative (DELTA) is set to l.D—3.Besides the minimum of the objective function, the monitor program can also provideChapter 4. Parameter Estimation And Data Analysis 81important statistical information about the best-fit parameters, such as covariance andconfidence interval. Some basic concepts related to the statistical analysis are explainedin Appendix C. For detailed information on how to use the monitor program, refer tothe manual of UBC NLP.Appendix F also includes the routine that simulates the transient response of theuncoupled Starling model to a specified perturbation according to Koomans’ experimentand also determines the final steady-state conditions of the system following the disturbance.Chapter 5Results And Discussion5.1 IntroductionNow that the mathematical model has been formulated (Chapter 3) and the parameterestimation procedure has been established (Chapter 4), in this chapter, the best-fit parameters for each independent data set will be presented separately. Following that, thepossibility of combining these different data sets will be investigated. Then the best-fitparameters for the combined data set (including Sets A — E) for the coupled Starlingmodel determined by employing the least squares method will be presented. With thesestatistically determined parameters, the model is fully described and is used to simulatevariolls experimental situations. The simulations are compared with the experimentalresults in terms of both the trends and the fit.Because statistical fitting between the experimental data and the model predictionsduring parameter estimation is one of the major characteristics of the current work, aconsiderable amount of attention will be devoted to the parameter estimation proceduresin the following discussion. Statistical analyses are included in the discussion to evaluatethe reliability of the estimated parameters. Sensitivity analyses are carried out to investigate how the transport parameters influence the objective function. The correctnessof the computer program and its ability to converge to a set of known parameters arealso investigated (see Section 5.2.6). Residual analysis is carried out to investigate thedistribution of the errors between the predicted and the experimental data.82Chapter 5. Results And Discussion 83Finally, the model is validated by comparing the best-fit transport parameters withavailable literature values, and by comparing simulation predictions with the measureddynamic behaviour of nephrotics following an albumin infusion [45], a set of data whichhas not been used for and hence is independent of the parameter estimate procedure.5.2 Results5.2.1 Best-fit Parameters for the Coupled Starling ModelOne of the major goals of the current study was to obtain a set of best-fit mass exchangeparameters for humans so that the model becomes fully described and can then be appliedto various clinical and experimental situations.Using the estimation procedure described in Chapter 4 and applying it to all 138data points, three sets of best-fit parameters corresponding to the three different tissuecompliance relationships were obtained and are listed in Table 5.1. From the table, we cansee that all three compliance relationships give a similar fit to the experimental data, witha maximum difference in OBJmjn of less than 6 units, or about 7%. Clearly, compliancerelationship #3 produces the lowest OBJmin and hence will receive the highest degreeof interest in the following sections. The best-fit Pc,o values were relatively consistentat about 11.0 mmHg. The best-fit u values increase slightly as the tissue compliancedecreases during overhydration, i.e., as the compliance changes from relationship #1to #3. The best-fit LS values decrease when the tissue compliance decreases duringoverhydration.5.2.2 The Feasibility of Combining Different Data SetsAs was discussed in Section 4.3.2, the validity of combining data from individual patientshaving different diseases to obtain a single, comprehensive set of parameters is based onChapter 5. Results And Discussion 84Compliance # 1 2 # 3LS (rnL.mmHg1.h’) 53.04±0.31 52.57+5.27 43.08±4.62C.I. (L,U) (52.43,53.64) (42.24,62.90) (34.03,52.13)U 0.9560±0.005 0.9761±0.018 0.9888±0.002C.I. (L,U) (0.9453,0.9666) (0.9399,1.0000) (0.9840,0.9936)pc,o (mmHg) 10.95±0.05 10.99±0.17 11.00±0.03C.I. (L,TJ) (10.86,11.04) (10.66,11.33) (10.95,11.05)PS (mL.h’) 71.68 72.52 73.01KF (mL.mmHg’.h’) 83.99 101.08 121.05JL,O (mL.h’) 77.26 77.06 75.74Fe 0.0487 0.0254 0.0116OBJmin 79.67 77.68 74.57Table 5.1: Best-fit parameters and their confidence intervals, as well as theassociated transport coefficients corresponding to the three tissue compliance relationships. C.I. denotes confidence interval (see Appendix C for itscalculation); L and U denote lower and upper limits respectively.the assumption that all the subjects involved in those independent studies have similarmicrovascular exchange transport coefficients.To test the possibility of combining the data collected for the current study with thosefrom nephrotic patients used by Chapple [13], each data set was analysed separately andthe 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. BecauseP00 was treated as a discrete parameter during the parameter estimation procedurefor nephrotic patients and because Pc,o = 11 mmHg is believed to produce the mostreasonable fit to the experimental data [13], therefore, for the sake of convenience, in allcases shown in Table 5.2, tissue compliance relationship #3 with Pc,o = 11 mmHg wasemployed.In order to visualize the fits corresponding to each set of data, the confidence limits aswell as the optimum values for LS and cr listed in Table 5.2 are plotted on Fig. 5.1. In theChapter 5. Results And Discussion 85000.0500.00 Set A Hubbard et al.)XSet B Doyle et al.)V Set C Mullin.s et al.)@Set D Heart failure400.0 • Set E Nephrotic syndrome)= Combined data300.0--200.0-100.0’0.0- I I • L:J V0.5 0.6 0.7 0.6 0.9 1.0UFigure 5.1: Best-fit LS and a’, as well as their confidence intervals for individualdata sets. The results correspond to tissue compliance relationship #3 withPc,o = 11 mmflg.Chapter 5. Results And Discussion 86Set # LS Confidence Interval ci Confidence IntervalA 0.0 0.0 LS 55.4 0.9630 0.9614 ci 0.9646B 232.5 0.0< LS 600.0 1.0000 0.8332< ci 1.0000C 0.0 0.0 LS 172.4 0.9855 0.9517 ci <1.0000D 79.0 0.0< LS 600.0 0.9886 0.5744< ci 1.0000E 42.7 31.1 LS 54.3 0.9892 0.9579 ci 1.0000Combined 39.9 39.5< LS 40.3 0.9829 0.9765< ci <0.9893Table 5.2: Best-fit LS and ci, as well as their confidence intervals for individualdata sets. The results correspond to tissue compliance relationship #3 withPc,o = 11. mmHg. “Combined” represents the best- fit parameters obtainedwhen data Sets A, B, C and D are combined.ci direction (i.e., x-direction in the figure), the confidence intervals for these independentdata sets all overlap, with the narrowest interval being that of Set A, ranging from 0.9614to 0.9646, and widest being Set D, ranging from 0.5744 to 1.0. The best-fit ci valuesranged from 0.9630 (the best-fit ci of Set A) to 1.0 (the best-fit ci of Set B); the majoritywere around 0.98. It can be said with certainty that the different subjects investigatedin 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 from0 to 232.5 mL.mmHg’.h’. Once again, all the confidence intervals for LS overlap eachother to a great extent. Best-fit values for LS for Sets B and D have particularly largeconfidence intervals, covering the entire range we investigated (0 — 600 mL.mmHg1.h’).The reason why some data sets are not sensitive to LS may be explained as follows. In themodel, the lymph flow sensitivity affects the MVES by altering the lymph flow rate: i.e.,JL = JL,O + LS (P1 —F1,0). That the objective function is not sensitive to LS may resultfrom the fact that, in Sets B and D, AP1 is always close to zero. That is, the experimentaldata depict very small changes in V1. These result in very small changes in P1 so thatJL does not change significantly. However, this is not the situation in most experimentsChapter 5. Results And Discussion 87as discussed in Section 4.3.2, i.e., by approximation, 1 L of infused saline may cause anincrease ill interstitial hydrostatic pressure of about 1.5 mmHg. In real systems, the lackof sensitivity of the LS parameter estimation may also relate to the time dependenceof lymph flow (personal communication with R.K. Reed.). Unfortunately, there is noquantitative information about such effects nor have we included delays in lymph flowin the present model. This is perhaps one of its defects. For Sets B and C, data arecollected in less than 2.5 hours and we suspect that the lymphatic system had not beenfully triggerred during the period investigated. To prove this hypothesis, we separatedthe data of Set A into two parts; one is the so-called early response corresponding to t3 hrs post-infusion; the other is the late response corresponding to 9 t < 12 hrspost-infusion. From Fig. 5.2, we found that, in the early response, the objective functionis not sensitive to LS, i.e., the contour lines are vertical; while in the late response, theobjective function is sensitive to LS. Even though the predicted best-fit LS of the lateresponse equals to zero, Fig. 5.2 B provides much more information about the behaviourof the lymphatic system than does Fig. 5.2 A, suggesting that the lymphatic responsehas not been fully triggerred at the earlier time. The procedure used for generating thecontour plots is detailed in Appendix D.When data Sets A, B, C and D are combined, we found that the best-fit LS and aare 39.9 mL.mmHg1.hand 0.9829, respectively, with 95% confidence intervals of 39.5LS 40.3 mLmmHg’.h’ and 0.9765 a 0.9893. It can be seen that the best-fita and LS values and the confidence intervals for Set E (see Table 5.2) are very similarto those of the combined data set, and the 95% confidence intervals overlap to a largeextent. Thus, we conclude that the MVES in uephrotic patients behaves normally andthat the nephrotic data can be combined with the other data collected for parameterestimation purposes in the current study.Chapter 5. Results And Discussion 88Figure 5.2: Contour plots of OBJ during early and late responses for Hubbard’sdata [41]. A represents the early response, t3 hrs; B represents the lateresponse, 9 hrst12 hrs.0.80 0.85 0.90SIGChapter 5. Results And Discussion 89It should also be mentioned that the best-fit parameters for the nephrotic data obtained for the coupled Starling model by applying the shortest distance least squarescriterion were [14]:a = 0.996LS 39.8 mL.mmHg1.hThese parameters are similar to the best-fit parameters obtained by the current parameterestimation procedure. This agreement adds further to our confidence in the reliability ofthe current parameter estimates presented in Table s.2.In summary, there is a total number of 138 experimental data points which canbe used for parameter estimation purposes. The subjects include healthy males, adultmen convalescing from non-cardiac ailments, patients with some degree of heart failure,-- -patients-with non-cardiac multiple-system diseases, patients with nephrotic syndrome andnormals. All of these subjects are considered to exhibit normal microvascular exchangebehaviour. We can therefore say that the current study represents a general investigationof the human microvascular exchange system under normal conditions.5.2.3 Simulations Using Best-fit ParametersBy just looking at the OBJmjn value, it is difficult to tell how well the experimental dataare fitted. Does the model predict the correct response trends after the microvascularexchange 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 answerthese questions is to compare each set of experimental data with the corresponding modelsimulation results obtained using the best-fit parameters.Chapter 5. Results And Discussion 905.2.3.1 Transient Responses of llpj, CPL and VPL after Saline or AlbuminInfusions (Sets A, B and C)To properly compare the model predictions with the experimental data, we must alsounderstand 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 tomake the circulation system self-regulating. There are many delicate pressure receptorsand chemoreceptors on the walls of the blood vessels. If these receptors detect anychanges in blood pressure and/or chemical composition, the nervous system responds tothese changes directly (i.e. by stimulating nerve fibers to control cardiac functions andvessel contraction or relaxation); or indirectly (i.e. by stimulating the endocrine systemto 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 conditionsin the capillaries and in the tissues in the immediate vicinity of the capillaries. Theseconditions are generally referred to as the Starling forces.Suppose a certain amount of isotonic solution is injected intravenously. After theinjection, the plasma volume expands, and its concentration of high molecular weightsolute is diluted. Lundvafl et al. [54] suggested that, in man, there may be particularlyefficient mechanisms for maintaining constant plasma volume via fluid transfer betweenthe bloodstream and the large fluid reservoir of skeletal muscle and skin. The increasedhydrostatic pressure caused by volume expansion and the decreased osmotic pressure dueto solute dilution are considered to be the driving forces for eliminating extra fluid fromthe bloodstream and consequently maintaining constant plasma volume.After a hyperoncotic solution is injected intravenously, the Starling forces will alter inChapter 5. Results And Discussion 91the following manner. The capilliary hydrostatic pressure increases due to the increase inthe plasma volume in addition to the capillary colloid osmotic pressure, therefore addingan additional driving force for fluid exchange. As we mentioned before, the circulationsystem has been shown to be self-regulating. If the capilllary wall is permeable to thesolute, the solute molecules will penetrate to the interstitiurn until a balanced osmoticpressures on both sides of the wall is attained. If the capillary wall is not permeable tothe solute, water is drawn into the bloodstream, thereby diluting the plasma to achievethe same result. Because the capillary wall is semi-permeable to albumin, both of theaforementioned 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 capillaryhydrostatic pressure. ‘Whether the plasma volume is increased or decreased depends onwhich effect dominates.Figures 5.3 — 5.6 demonstrate the model predictions of the transient responses of thecirculatory system oncotic pressure and volume following saline infusions with or withoutalbumin. The experimental protocols were described in Section 4.3.1. Since the lowestOBJmin value was obtained for compliance relationship #3, the best-fit parameters associated 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 pressure 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 systern and are in good quantitative agreement with the experimental data. These curvesshow that the circulatory compartment remains relatively stable after the perturbation.Changes in 11PL and VPL are relatively small compared with the large experimental error associated with the data. For a closed circulatory system, a 1500 mL fluid input, ifChapter 5. Results And Discussion,J,J.iJ -15.0-(A)92I I I I I I I I1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.03.0 4.5 6.0Time(hrs)I I I7.5 9.0 10.5 12.0Figure 5.3: (A) Simulation of a 100 mLsaline infusion. (B) Simulation of a 100mL 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 Hubbardet al. [41]); solid line: model simulation. For the best-fit parameters of tissuecompliance #3.—5.0—25.CI— I-I-‘? I I I I I I I I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 160.035.0-10.0—15.0——40.0V.11. HI I—1.5 0.0 1.5 3.0I I4.5 6.0Time(hrs)I I7.5 9.0 10.5(B)tz I—1.5 0.015.0-—25.0-60.035.0-10.0.;—15.0-—40.0-_______•1’ II I—1.5 0.0 1.5FChapter 5. Results And Discussion 93all 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 mechanismsof the circulatory system, the plasma volume increases by only 11.5% at 12 hours post-infusion, corresponding to an actual volume increase of 367 mL. Thus, most of the excessfluid is shifted to the interstitium. Plasma oncotic pressure decreases slightly (6.27%) dueto the dilution effect of the infused saline. Figure 5.3 (B) represents the response of thesubjects to a hyperoncotic albumin infusion. During the infusion period (-1.5— 0 hr), theplasma 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 after0.64 hr. The steeper slope is due to the rapid loss of injected albumin from the vascularspace [41]. The return of 11PL to normal afterwards is mainly attributable to the dilutioneffect 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 capillaryhydrostatic pressure; therefore, the plasma volume increases.Figure 5.4 presents the percentage changes in 11PL and VPL with time when the volunteers (N=4) are subjected to higher dose infusions, i.e. a 200 mL saline infusion ora 200 mL saline infusion with 50 g albumin (data Set A). The transient responses aresimilar to those obtained with lower dose infusions. There is no reason for experimentally measured 11PL 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 themeasurement. Overall, the simulation results fit the experimental data reasonably wellwithin the error of measurement.Figure 5.5 (data Set B) illustrates the percentage change in plasma volume whichoccurs after 900— 1000 mL of normal saline were injected intravenously within 6.5 — 13minutes (specific information corresponding to individual patients is listed in Table 4.6).The model predictions are in good agreement with the experimental measurements. ForChapter 5. Results And Discussion 94I I I1.5 3.0 4.5 6.0Time(hrs)I I I I7.5 9.0 10.5 12.0I I I—1.5 00 1.5I I I I I I I3.0 4.5 6.0 7.5 9.0 10.5 12.0Time(hrs)Figure 5.4: (A) Simulation of a 200 mL saline infusion. (B) Simulation of a 200mL 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 Hubbardet al. [41]); solid line: model simulation. For the best- fit parameters of tissuecompliance #3.(A)35.015.0—5.0—25.01;.4—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.060.035.0-10.0—15.0-Afl flI I—1.5 0.0T(B)60.035.0-10.0-i—15.0-—40.0Chapter 5. Results And Discussion 95patient #1, 1000 mL of normal saline was infused within 9.5 minutes. At the end of theinfusion (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 theinfusion period (9.5 minutes). Compared with the experimental measurement, i.e., a24.9% increase, the agreement is good. The model predicts the average filtration rateduring this period is 947 mL/h, which is much higher than the normal filtration rate. At38 minutes post-infusion, the model predicts a 10.9% increase in plasma volume while a8.5% increase was measured experimentally. At t=2 hour, the VPL increase is only 224mL (a 7% increase) and the system appears to be close to steady-state. The 7% increasein VFL is due to the assumption that within these 2 hours, water lost from skin andduring respiration, as well urine output are negligible because no such information waspresented by the investigators.-- Figure 5.6- (data Set C) shows the percentage changes in plasma volume and plasmaconcentration following a 2 L normal saline infusion within 2 hours. During the infusionperiod (0 — 2 hour), VPL is predicted to increase steadily to a maximum of 3929 mL (a 23%increase), while CPL decreases steadily to a minimum of 32.4 g/L. Accordingly, 11FL isexpected to decrease because of the linear relationship between CFL and 11PL (Eq. 3.19).Afterwards, both VpL and CPL tend to return to normal. All these trends match thephysiological responses of the MVES after an isotonic solution infusion. The modelpredictions at half an hour after the infusion match the experimental measurements verywell within the large experimental errors in terms of both absolute values and percentagechanges. The model predicts that VPL increases to 3659 mL (a 14.36% increase), andCPL decreases to 34.89 g/L ( a 11.45% decrease); experimentally it was found that 11PLat 0.5 h post-infusion was 3541±1177 mL and CPL was 30±5 g/L.Chapter 5. Results And Discussion 96Patient #120.010.0u.00.0 30.0 60.0 90.0 120.0Patient #230.020.00.0 30.0 60.0 90.0 120.0Patient #630.020.0- 10.00o0.0 30.0 60.0 90.0 120.0Patient #730.00.00.0 30.0 60.0 90.0 120.0Patient #1130.020.010.0nfl.________________________________0.0 30.0 60.0 120.0Time(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 parameters of tissue compliance #.Chapter 5. Results And Discussion 9770.0-40.0-10.0-—20.0-—50.0-0.0 1.0 2.0 3.0 4.0 5.030.0-10.0--10.0- F—30.0-—50.()-0.0 10 2.0 3.0 4.0 5.0Time(hrs)Figure 5.6: Transient responses of plasma volume and plasma albumin concentration after a 2 L of normal saline infusion within 2 hours [60j. Filledcircle: experimental data point; solid line: model simulation. For the best-fitparameters of tissue compliance #3-Chapter 5. Results And Discussion 985.2.3.2 Simulations of Heart Failure (Set D)Chronic heart failure is the pathophysiological state in which an abnormality in cardiacfunction is responsible for the failure of the heart to pump at a rate commensuratewith the requirements of the metabolizing tissues. It is frequently caused by a defectin myocardial contraction. As a result of the deficiency of contraction, the volume ofblood delivered into the systemic vascular bed is chronically reduced, and a complexsequence of adjustments occurs that ultimately results in the abnormal accumulation offluid in interstitia in the body. These adjustments include a rise in venous pressure. Theincrement in venous pressure is transmitted to the capillary level and therefore increasesthe transcapillary pressure. This will accelerate the transcapillary filtration rate (Eq. 3.2),and water will accumulate in the interstitium. As V1 increases, Pi will increase and Hwill decrease. In turn, the rise in Pi and the reduction in llj both act to oppose furtherincrease in JF• In addition, lymph flow increases with the elevation in P1. As the fallingfiltration 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 V1 = 8.3 L, and increasesto 2.35 mmHg for patients with heart failure, corresponding to Vj = 17.0 L for tissuecompliance #3. In the experiments, P1 was observed to decrease to 0.6 mmHg lower thancontrol level and Vj was observed to decrease to 7.5 L with a measurement error of +6.8L 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 withour predicted value which is around the normal range of V1. For patients with heartfailure, P1 was not measured and V1 was observed to be 12.0±15.4 L [25], which containsthe range of the predicted value. These basically match the adjustments which occur inheart failure as discussed above.Chapter 5. Results And Discussion 99In addition, several studies on animal [19, 36] have shown that there exists a critical 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 sucha critical pressure. But it is certain that when Pc exceeds 17 mmllg, massive edemaoccurs (see Fig. 5.7).5.2.3.3 Simulations of V1 vs. 11PL and Hj vs. 11PL in Nephrotic Syndrome(Set E)Interstitial edema in patients with chronic nephrotic syndrome is caused by the pathological removal by the kidneys of plasma protein from the blood stream which eventuallyresults in hypoproteinemia.Thern current model assumes that the lowered plasma colloid osmotic pressure- is theonly cause of edema formation in nephrotic patients. Figure 5.8 was constructed bydecreasing 11PL step by step at a constant plasma volume, and predicting the steady-state interstitial volumes and interstitial colloid osmotic pressures which occur for eachnew 11PL value. From this figure, we can see that both Vj and H are well fitted as 11PLincreases. Figure 5.9 shows the steady-state fluid fluxes (i.e. JF and JL) and proteinfluxes (i.e. Qs and QL), as well as the albumin contents in both the interstitial and thecirculatory 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 obviousslope change around 11PL equal to 25.9 mmHg. The predictions shown in Figs. 5.8 and 5.9are helpful in investigating edema formation and its mechanisms. In the upper panel ofFig. 5.8, it can be seen that as 11PL decreases from 28 to 12 mmHg, V1 increases from 7.7to 11.5 L. In other words, V1 increases 3.8 L as HPL decreases 16 mmHg. As 11PL falls, thetranscapillary fluid flux and lymph flow both rise to about double their normal levels (seeChapter 5. Results And Discussion 10025.0I I10.0 12.5 15.0P(mmHg)Figure 5.7: Simulations of steady-state Vi vs. 11PL and V1 vs. Pc in heart failurepatients [63, 25]. For the best-fit parameters of tissue compliance #3.22.5 25.0rrP(mmHg)20.015.010.05.0.7.5 17.5 20.0Chapter 5. Results And Discussion 101Fig. 5.9). Due to the loss of plasma proteins from the bloodstream (at HPL = 12 mmHg,QFL = 58.46 g), the interstitial protein content drops dramatically (at 11PL = 12 mmHg,Qi = 58.51 g). When 11PL decreases to 12 mmHg, Pi1 decreases to 4.1 mmHg. Thus itcan be seen that protein washout (i.e., a net reduction in Qi) is the predominant edemapreventing mechanism during this period. If HPL decreases further, protein washout cannot prevent the further expansion of interstitial fluid volume. According to Fig. 5.8, as11PL decreases from 12 to 4 mmHg, the interstitial volume increases from 11.5 L to 43.5L, which is a very severe state of tissue edema. At 11PL = 4 mmllg, Qi = 19.5 g and P1= 5.1 mmllg. This shows that protein washout does not occur as extensively as when11PL is decreased from 28 to 12 mmHg because most of the protein has already beenwashed out, and the increase in tissue hydrostatic pressure plays a more important rolein preventing further edema formation. However, due to the high tissue compliance, thebcease in P1 also fails to prevent the edema formation effectively. Thus, massive edemaformation takes place.A plasma colloid osmotic pressure of 12 mmHg can be regarded as the critical levelbelow which severe edema occurs. To prevent edema formation, 11PL must be maintainedabove this critical value.5.2.3.4 Summary of the Simulations using Best-fit ParametersBased on the above discussion, it is concluded that the model predicts the correct trendsand values after the MVES is perturbed. Each independent data set has been well fittedby using the best-fit parameters.5.2.4 Sensitivity AnalysisThough we have obtained three sets of best-fit parameters, we do not know how thevariables (LS, a, and Pc,o) influence the objective function. If a slight deviation from20.0-18.0-12.08.0-4.0-0.00.0 5.00.0I --I10.0 15.0 20.0 25.0 30.0PL(mmllg)I I I I5.0 10.0 15.0 20.0 25.0 30.0PL(mmHg)102Chapter 5. Results And Discussion50.0-37.5.25.0-12.50.0fI.....aI•.Figure 5.8: Simulations of Vi vs. IIPL and fl1 vs. 1IPL in nephrotic syndromepatients [24, 46, 64, 27]. For the best-fit parameters of tissue compliance #3.Figure 5.9: Steady-state effects of graded reduction of plasma oncotic pressureon fluid and protein exchange. For the best-fit parameters of tissue compliance #3. The solid line in the last panel is the tissue protein content, thedashed line is the protein content of the plasma compartment.103Chapter 5. Results And Discussion350.0280.0210.0140.070.00.0.0 5.0 10.0 15.0 20.0 26.0 31.51.2O .60.30.0150.0120.090.060.030.00.0I— I5.0 10.0 15.0 20.0 Z50 3I.03.030.00.0 .0 1Ô.O 15.01r(mmHg)20.0 26.0Chapter 5. Results And Discussion 104CFigure 5.10: Sensitivity analysis for LS. The analysis is conducted for tissuecompliance #3, Pc,o 11.00 rnmHg and a = 0.9888.the best-fit value affects the objective function significantly, then the location of thisvalue must be chosen very accurately. The study on whether the optimum fit is sensitiveto changes in the fitting variables is called a sensitivity analysis. A sensitivity analysis ishelpful in evaluating the degree of reliability of the best-fit parameters.5.2.4.1 Lymph Flow Sensitivity— LSTo analyse the sensitivity of the objective function to LS, Pc,o and a are fixed at theirbest-fit values, LS is varied on either side of its optimum point, and the correspondingOBJ value is calculated. The results obtained for compliance relationship #3 (i.e. Pc,o11.00 mmHg, a = 0.9888) with LS varying from 10— 110 mL.mrnllg’.lr’ are shownin 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 43.08250.50.0 70.0LS (mi/rnmHg.h)Chapter 5. Results And Discussion 105mL•mmHg’•h1.If LS increases to 60 mL•mmHg’•h1,the OBJ rises only to 80.83units, 4.33 greater than OBJmin. If LS decreases to 20 rnL•mmHg’.h’, the OBJincreases to 115.01 units, 38.51 greater than OBJmim. Thus, it is clear that, if LS = 43.08mL.mmHg1.his not the “true” value of the lymph flow sensitivity, then the “true”value is “more likely” to be found at a greater value than at a lesser value. This analysisreveals an asymmetry in the objective function and suggests that regions on one side of thebest-fit value are more likely than regions oi the other side of to yield the “true” value ofthe parameter. Such a conclusion can not be reached by simply examining the calculatedconfidence interval on LS (Table 5.1), which are symmetrically positioned about the best-fit value. ÔOBJ/8LS is fiat for LS ranging from 35 to 50 mL.mmHg1.hThis intervalcoincides with the 95% confidence interval on LS (Table 5.1), which is interpretated asthe probability of OBJmin falling inside it is 95%.5.2.4.2 Albumin Reflection Coefficient — aThe sensitivity analysis for a is carried out for tissue compliance relationship #3, Pc,o= 11.00 mmHg, LS = 43.08 mL•mmllg’.h1,and a varying from 0.8 — 1.0. The resultsare plotted on Fig. 5.11. The objective function decreases from 172.46 to 80.14 as a isincreased 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 varyby less than 3.64 units. The optimum a does not appear to be biased towards eitherdirection in the plateau region.5.2.4.3 Capillary hydrostatic pressure at normal steady-state—Pc,oAs it was mentioned in Section 3.3.1, the minimum value of Pc,0, below which the modelbreaks down, is restricted by the choice of a. With a set to 0.9888, the interval overwhich the capillary hydrostatic pressure could be investigated was limited to F,üChapter 5. Results And Discussion 10601.00Figure 5.11: Sensitivity analysis for o. The analysis is conducted for tissue.compliance #3, Pc,o = 10.8431 mmHg, and LS = 49.7848 mL•rnmHg’h’.10.37 mmHg. Therefore, the interval for the sensitivity analysis was selected to be10.5 Pc,o 15.0 mmHg. From Fig. 5.12, we note that the optimum Fc,o lies between10.75 to 11.25 mmHg; the objective function increases dramatically when Pc,o is lessthan 10.75 mmHg or greater than 11.25 mmHg. Thus, even though it is still reasonablephysiologically for Pc,o to exceed the upper limit of 15 mmflg, a worse fit to the currentexperimental data is definitely obtained when Pc,o 11.5 mmHg.5.2.5 Péclet NumberThe Péclet number is a dimensionless mass transfer parameter which describes the ratioof convective to diffusive exchange. Here, the Péclet number is defined (according toEq. 3.4) as Pe (1—o}JF/PS, and it represents the importance of convection comparedto diffusion as a mechanism for transcapillary albumin transfer. From Table 5.1, we note0.80 0.85 0.90 0.950•Chapter 5. Results And Discussion 107275.235.0195.0C155.0115.U75.’Figure 5.12: Sensitivity analysis for F,o. The analysis is conducted for tissuecompliance #3, LS = 43.08 mL-rnn11g’•h’ and o- = 0.9888.that the Péclet numbers at steady-state for all three compliance relationships are verylow. Therefore, we conclude that, at steady-state, albumin is mainly transported bydiffusion.5.2.6 Verification of FitThis test is used to check the correctness of the computer program and its ability toconverge to a set of known parameters [112]. In this test, 138 “error-free” data points aregenerated by solving the model equations using the best-fit parameter values when theMVES is subjected to the same perturbations as those stated in Section 4.3.1. The datagenerated 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 transport parameters areassumed to be unknown and the same optimization procedure as that used with the real12.0 12.5 13.0 13.5P.0 (mmHg)Chapter 5. Results And Discussion 108experimental data is followed to find a set of new parameter estimates for these “error-free” data. Theoretically, if the program is working properly, a set of parameters identicalto 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 resultscorresponding to all of the quantities which were measured experimentally were predictedby the mathematical model and substituted back into the optimization program. Thebest-fit parameters obtained in this way were:LS = 43.126 mL.mmHg’.h’a = 0.98877Pc,o = 11.000 mmHgwith OBJmjn equal to 0.014885. As can be seen by comparing the above set of parameterswith those listed in Table 5.1 for this case, the best-fit values are all reproducibleto atleast three significant figures. Therefore, we are confident that the parameter estimationprocedure is reliable.5.2.7 Residual AnalysisA check of the normal distribution of the errors can be made by constructing a residualplot. Here, the errors refer to the differences between the predicted and the experimentaldata, 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 obviouspattern. Although the fitting parameters obtained by the least squares procedure do notdepend on a normal distribution of the errors, the calculation of the confidence intervalwill depend on this assumption. The true confidence intervals may differ greatly from thecalculated values if the normality assumption is not satisfied. Therefore, it is worthwhilechecking to determine whether the distribution of the errors is normal.Chapter 5. Results And Discussion 1093Figure 5.13: Residual plot for the best-fit parameters of compliance relationship X and X refer to the simulation value and the experimentalmeasurement, 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., thesimulation 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 and78 are positive. The residuals scatter at random around the zero line and no trends areobserved. Thus, it appears that the residuals are normally distributed. The residual plotis also useful for highlighting major departures in the observed values of the data fromanticipated patterns. Here, no pattern of systematic departure of the points around thezero line is evident. Thus, the residual plot in Fig. 5.13 suggests that each experimentaldata point is properly weighted.Xsim/SDChapter 5. Results And Discussion 110Fi (mmHg) (mL/h+2.0 2.8+1.0 1.4+1.0 0.6+0.5 0.6Table 5.3: Experimental data of interstitial hydrostatic pressures vs. lymphflow in the leg superficial lymphatics [98]. L denotes the lymph flowrate inthe leg superficial lymphatics.5.3 Validation of the Best-fit ParametersThe best-fit parameters listed in Table 5.1 are calculated on the basis of a statistical fittingbetween the model predictions and the experimental measurements. In this section,we try to compare these best-fit parameters with estimated values available from theliterature. Additionally, a comparison between model predictions and experimental datawhich have not been used in the optimization would also be very helpful to validate thesebest-fit parameters. Unfortunately, no other new data have been found by the author sofar. 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 — LSTo the knowledge of the author, no one has experimentally determined the lymph flowsensitivity for humans which is defined as aforementioned. Stranden et al. [98] foundthat there was a significant correlation between lymph flow and interstitial hydrostaticpressure. The data from their experiments on patients with local leg edema is proposedby the author for a comparison with LS as estimated in this work. The data are listed inTable 5.3. Note that the lymph flow rate measured in the experiment (denoted by jL) isJL )Chapter 5. Results And Discussion 111Lymph flowrate (mL/h) SourceLSL 0.25 [67]LSL 0.34 [68]TD 84.0 [20]TD 70.8 [33]Table 5.4: Lymph flow rates in the leg superficial lymphatics and in the thoracicduct. LSL denotes the leg superficial lymphatics; TD denotes the thoracicduct.that in the leg superficial lymphatics, which have diameters in the range of 0.1 0.4 mm.But lymph flow rate (JL) in Eq. 3.7 refers to the lumped whole body flow rate. Therefore.JL must be converted to a total lymph flow in the thoracic duct which has a diameter of2— 3 mm in the neck region (see later discussion). Table 5.4 shows some measurementsof 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 26077.4/0.295) times higher than that in the leg superficial lymphatics. Scaling L by 260,LS is found to be 400 mL.mmHg’.h’ by linear fitting between JL and (P1 — F1,0). Ifthe scale factor is arbitrarily altered to 100 (because of varying topography and caliber ofthe cannulated vessels), LS is found to be 154 mL.mmHg.h’ (see Fig. 5.14). In bothcases, normal lymph flows are found to be negative, which is unreasonable and conflictswith the assumption in the current model and with physiological evidence. This mightbe due to a discrepancy in P1,0 between the reference man and the patients involved inStranden’s experiments. However, mathematically, this discrepancy does not affect theslope of the straight line, i.e. LS.Compared with the LS value estimated by the above procedure, the lymph flowsensitivity predicted by the model (which ranges from 43.08 to 53.04 mL.mmHg’.h1)tends to be underestimated. The extent of underestimation is difficult to evaluate becauseChapter 5. Results And Discussion 112800.0-600.0-400.0-200.0-0.0-—200.0-—400.0 I I I—1.0—0.5 0.0 0.5 1.0 1.5 2.0 2.5Pi(mmHg)Figure 5.14: Plot of 1L vs. F1. Filled circles: experimental data obtained byscaling iL (Table 5.3) by 260; open circles: obtained using scale factor of 100;solid line: 1L —378 + 400 x (Fi — P1,o); dashed line: JL = —145 + 154 x (F1—F1,0).the experimental data are too few and too scattered.Another quantitative criterion was also applied to validate the value of LS. Thiscriterion is the ratio of maximal to basal lymph flowrate, JLRMB (i.e., JL,MAX/JL,0).The maximum lymph flowrate is assumed to occur at a cutoff of Vj = 20 L [23]; thus,according to the tissue compliance relationships, F1 is calculated to be between 2.01 and2.66 mmHg. If the estimated basal lymph flowrate and lymph flow sensitivity are assumedto be correct (see Table 5.1), according to JL = JL,O + LS(F1—F1,o), JL,MAX is estimatedto 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 observedvalues of JLRMB in animal are 5 to 10 [2]. Comparison between the values of predicted andexperimental JLRMB suggest that the model predicted LS values are close to the lowerbound of the experimentally determined LS.0-oChapter 5. Results And Discussion 1135.3.2 Albumin Reflection Coefficient — aExperimental estimations of the reflection coefficient (a) have been based either on thepore estimation method [42] or on steady-state lymph flow analyses. Renkin [82] estimated that small pores having an inner diameter of 40 A have a reflection coefficientfor albumin of 0.95, and large pores have a a value of 0.45. Reed et al. [78] estimatedthat 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 indicatesmall pore populations with radii 40— 50 A in subcutaneous tissue and 60 A in skeletalmuscle; large pore populations have a a radius of around 200 A [73]. The number ratioof large to small pores is around 1:3500. Therefore, the whole body capillary membranea 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 proteinflux equations (Eq. 3.6) or the nonlinear equation (Eq. 3.5). Lymph ff6w as well asprotein concentrations in lymph and plasma are measured at two or more steady lymphflow states produced by elevating venous pressure. The estimation of a is in reality aproblem 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 usingdifferent mathematical formulations. Renkin et al. [81] reported a values ranging from0.98 to >0.99 based on their recent study on rat skin and muscle.Reflection coefficients estimated by both methods are in good agreement with thevalues obtained by statistical fitting in current study. It is concluded that the albuminreflection coefficient is close to unity.Chapter 5. Results And Discussion 114Species Method PS (mL/h/lOOg wet tissue) PS (mL/h) SourceDog LA 0.707 269.36 [82]Dog LA 0.593 225.93 [88]Cat ID 4.530 1725.93 [92]Rabbit ID 4.434 1689.35 [69]Human ID 0.467 177.93 [33]Human ID 2.760 1051.56 [70]Table 5.5: Experimental estimates of PS values. The PS values were normalized per 100 g wet tissue for comparison. LA denotes lymphatic analysis;ID denotes indicator dilution. Values in the fourth column are calculated byassuming that skin and skeletal muscle for the whole body weigh 10.1 kg and28.0 kg, respectively.5.3.3 Permeability-Surface Area Product— PSThe permeability-surface area product (PS) is the most widely used index to describethe diffusive characteristics across capillary wall. The penneabilities vary for differentsolutes. The PS value used in the present study refers to that for albumin.Experimental estimates of the permeability-surface area product are typically obtained by one of two methods. One is the steady-state protein flux analysis which wasmentioned in the previous section. The other is the indicator dilution method, or single injection, residue detection method. In the latter method, a single bolus of labelledalbumin is injected intra-arterially and blood samples are collected at time intervals toanalyse their radioactivity. An indicator dilution curve is then constructed to estimatethe 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 estimatedskeletal muscle and skin for the whole body, weighing 28.0 kg and 10.1 kg respectively, thetotal whole body permeability-surface area product ranges from 178 to 1725.93 mL/h.Chapter 5. Results And Discussion 115Paaske et al. [70, 92] pointed out in their reports that the indicator dilution PS values forboth human and animal tissues were 3 to 10 times higher than those obtained by othermethods [88, 82]. However, no explanation for this overestimation was given. Comparedto the values listed in Table 5.1, which are close to 70 rnL/h, it seems that the PSestimated in the current study is about two times lower than expected, if human tissueshave permeability coefficients similar to animal tissues. But, it should be kept in mindthat the physiological measurements of PS are highly uncertain.5.3.4 Fluid Filtration Coefficient—Experimental determinations of the capillary filtration coefficient (KF) in human havebeen limited to two methods: the venous occlusion method [57, 91, 54, 52] and theosmotic transient method [71, 44]. Both methods are developed based on the Starlingequation, he.,-JF=KF[Pc—PI—o(HPL—f11)].The venous occlusion method is based on the assumption that a change in Pc is theonly source for the change in fluid filtration rate (JF) when venous pressure is raisedor lowered, at least at times near the perturbation. The osmotic transient method isbased on the assumption that a change in 11PL is the change in driving force whichdominates JF variations after the injection of a hyperoncotic albumin solution. Thetranscapillary filtration rate is determined with the aid of a well-balanced volume pistonrecorder connected to the plethysmograph which can record the changes in tissue volume.Plasma volume is assumed to remain constant. The capillary filtration coefficient iscalculated as the filtration rate divided by the concomitant pressure change.Table 5.6 presents the experimentally determined KF values for mixed tissue (skin andskeletal muscle) in human. The table contains a wide range of values. Most investigatorsChapter 5. Results And Discussion 116KF (mL/(min.mmHg.100g soft tissue)) KF (mL/(mmllg.h)) Reference0.0012 27.43 [91]0.0036* 82.30 [71]0.0046k 105.16 [44]0.0058 132.59 [52]0.0077 176.02 [57]0.054 1234.44 [54]Table 5.6: Experimentally determined KF values. All these measurements weremade in human by the venous occlusion method except * were measured bythe osmotic transient method. Values in the second column are calculated byassuming that skin and skeletal muscle for the whole body weigh 10.1 kg and28.0 kg, respectively.report a KF value of around 0.005 mL/(min.mmHg•lOOg tissue) by using the venousocclusion method with increases in venous pressure of 30 — 60 mmHg. However, Lundvallet al. [54] suggested that such high increases of venous pressure may lead to closureof the precapillary sphincters, thereby reducing the capillary surface area available fortranscapillary exchanges. Also, it may cause large decreases in regional blood flow. Ifthese postulates are tenable, they will cause an underestimation of the experimental KFvalues. Lundvall et al. reported a ten times higher KF value (0.054 mL/(min.mmHg.100g tissue)) by increasing venous pressure by only 1.6 mmHg. A similar magnitude of KFwas 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 KFvalues 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,ODirect measurements of lymph flow in human have been made mostly either in the leglymphatics or the thoracic duct. Lymph flow rate in the leg superficial lymphatics isChapter 5. Results And Discussion 117simply measured by determining the volume of lymph collected in a calibrated syringeconnected to the cannula during a measured period of time. Lymph flow is found tooccur only simultaneously with the lymph pulse [67].The normal lymph flow rate (JL,o) in the current model refers to the lumped wholebody flow rate. Therefore, comparisons should only be made between the estimated JL,Oand the lymph flowrate measured in the thoracic duct. Measured thoracic duct flowsare of the order of 1— 3 mL/min, or 60 — 180 mL/h [75]. The normal lymph flowratespredicted in the current study are within the lower side of the normal range at about 78mL/h for all tissue compliance relationships.5.4 Simulations of a Single Intravenous Infusion of Human AlbuminIn 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 nephroticsyndrome were presented. All these data have been used in the parameter estimationprocedure; therefore, it is not surprising that these data are well represented by the modelpredictions. To test further that the parameters obtained based on the aforementioneddata are generally applicable to simulations of a microvascular exchange system whichexhibits normal behavior, a comparison was made between the simulation predictionsand the dynamic results from an experimental study by Koomans et al. [45]. The datafrom 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 estimatedtransport parameters.Koomans’ experiment was designed to study the fate of a single intravenous infusionof human albumin in 10 patients with nephrotic syndrome. In their experiment, 60 gof human albumin in 300 mL of solution were infused continuously over a period of 1.5Chapter 5. Results And Discussion 118hours. Plasma volume, and colloid osmotic pressure in both interstitium and plasma weremeasured before and immediately after the albumin infusion, as well as at 1 and 24 hourpost-infusion. Urinary albumin loss was observed to be 10.5 g/day before infusion and26.4 g/day during the first day after infusion. Accordingly, the excess albumin excretionrate was 15.9 g/day. The 300 mL of infused fluid was observed to be excreted within thefirst 24 hours post-infusion. In addition, patients were reported to possess an averagepre-infusion interstitial fluid volume of 18.25 L. Plasma volume, measured experimentally, was very close to normal. Plasma and interstitial colloid osmotic pressures weredetermined from the steady-state simulations of nephrotic patients (see Fig. 5.8), whichwere 10.574 and 3.304 mmHg, respectively. Thus, the corresponding albumin contentsin both circulatory and interstitial compartments could be calculated (albumin content= volume x albumin concentration, and albumin concentration can be determined fromthe colloid osmotic pressure relationship, see Eqs. 3.19 and 3.21). Starting with theseinitial conditions, the model predictions of the transient responses when the system wassubjected to the same perturbations as the experimental counterpart, are presented inFig. 5.15. Figure 5.15 shows that the model predictions are in good agreement withthe experimental data in terms of both absolute values and trends. Both 11PL and VpLfollow a uniform pattern of initial increase and subsequent decrease which starts immediately after infusion. The pattern bf change in the interstitial fluid evidently differs fromthat in the plasma: after a slight initial increase, Il rises further during the 24 hourspost-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. Thecontinuous accumulation of interstitial albumin mass during the 24 hours post-infusionproves that part of the infused albumin disappears into the interstitium where it causesan increase in tissue fluid oncotic pressure [45].Finally, the good agreement between the model predictions and the experimentalFigure 5.15: Transient responses of UPL, [1, VPL as well as QPL and Qi aftersingle infusion of 60 g of human albumin solution [45]. Solid lines representthe dynamic responses of the model. Dotted lines represent the steady-statevalues obtained by running the program for a long time. Experimental dataare shown as filled circles with error bars.Chapter 5. Results And DiscussionS425.020.0-15.010.0-5.0 - I I I12.0 24.0 36.0 48.06.0-5.0-4.0-3.0-2.0-1.0-D.CI24.0 36.0 48.012.0119QPLQI150.0aC’100.0-50.0-0.0- I I I0.0 12.0 24.0 36.0 43.0Time(hrs)Chapter 5. Results And Discussion 120data 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 thesum of squares of the differences between model predictions and experimental data in theform of percentage changes, are also applicable for predicting the absolute values. This, inturn, shows that the initial values of the reference man (see Table 3.2) are representativeof those of the majority of human beings.Chapter 6ConclusionsBased on the discussion in the previous chapter, it is concluded that the coupled Starlingmodel is capable of providing good descriptions of microvascular exchange and fluid andalbumin distribution in the human. According to statistical fitting results, the best-fit tothe available experimental information is obtained with tissue compliance relationship #3and the transport parameters LS = 43.08 + 4.62 mL•mmHg1h’, a = 0.9888 ± 0.002,Pc,o = 11.00±0.03 mmHg, PS = 73.01 mL•h’, KF = 121.05 mL• mmHg’.h1and JL,O= 75.74 mLh1. Simulations of the available experimental data using these parametersgave a reasonable fit in terms of both trends and absolute values. All of the best-fitparameter values so obtained were in reasonable agreement with estimated values basedon experimental measurements where comparisons with literature data were possible.The fact that the albumin reflection coefficient is near unity indicates that transcapillary 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 thatdiffusion is the dominant mechanism of transcapillary protein transport under normalconditions.The good agreement between the best-fit parameters obtained by studying normalsubjects and those obtained for chronic nephrotic patients suggests that the microvascular exchange system behaves close to normal in chronic nephrotic patients. For patients121Chapter 6. Conclusions 122suffering from mild nephrotic syndrome, protein washout (i.e. a net reduction in interstitial protein content) is the predominant edema preventing mechanism. For thosesuffering from severe nephrotic syndrome, all edema preventing mechanisms fail, leadingto 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 parametersobtained by statistically fitting between the model predictions and other experimentaldata for humans, further proved that the coupled Starling model can successfully simulatethe transport mechanisms of the microvascular exchange system in the human.Chapter 7Future WorkThe 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 experimentally determined tissue compliance relationship when clinical measurements of this property become available. One can then investigate how closethe -“most-likely” compliance relationship is to the real one. -(b) Dividing the generic tissue compartment into two compartments, i.e., a skeletalmuscle compartment and a skin compartment, when the properties of thesetwo separate compartments become available.(c) Considering the influence of cellular component.2. Develop a new model to study the behavior of the microvascular exchange systemafter a burn injury. Because of the limited amount of information available forhumans, the microvascular exchange system in the burn model should probably bedivided into three compartments as the first step in developing a more comprehensive model. The three compartments would be the circulatory, normal tissue andburned tissue compartments. The transport parameters obtained in current studycould he used to describe the transport characteristics of the normal tissue. Someof the transport properties will be subject to a transient adjustment due to the123Chapter 7. Future Work 124burn injury [8, 9]. Using available data from burn patients a similar optimizationprocedure is recommended for finding the optimal parameters which describe thesetransient adjustments. Such a model is of potential use as an aid in monitoringfluid resuscitation in burn injured patients.Bibliography[1] Arturson, G., Groth, T., Hedlund, A., and Zaar,B. Computer simulation of fluidresuscitation in trauma: First pragmatic validation in thermal injury. J. Burn CareRehabil. 10:292-299,1989.[2] Aukiand, K., and Nicolaysen, G. Interstitial fluid volume: Local regulatory mechanisms. Physiol. Rev. 61(3) :556-638,1981.[3] Bell, D.R. Watson, P.D., and Renkin, E.M. Exclusion of plasma proteins in theinterstitium of tissues from dog hindpaw. Am. J. Physiol. (Heart Circ. Physiol. 8)239:H532-H538, 1980.[4] Bert, J.L., Bowen, B.D., and Reed, R.K. Microvascular exchange and interstitial volume regulation in the rat: model validation. Am. J. 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Handbook of Physiology.Vol. IV, sect.2, pp.251-307,1984.NomenclatureSymbol Meaning UnitsAib AlbuminBV Blood volume (ml)C Concentration of albumin (g.1’)D Perturbation (g.h’ or ml.h1)ECV Extracellular volume (ml)F ( ) Function ofFCOMFI Tissue compliance relationship (ml.mmHg’)HCT HematocritJ Fluid transport rate (ml.h)K Fluid transport coefficient (ml.mmHg1.h’)Length (m)L Length or lower boundLS Lymph flow sensitivity (ml.mmHg.h1)OBJ Objective function valueF Hydrostatic pressure (mmllg)FCOMPC Capillary compliance (ml.mmHg’)Fe Péclet numberPS Permeability-surface area product of (ml.h1)capillary with respect to albuminQ Albumin content (g)134Nomenclature 135Q Albumin transport rate (g.h1)r Radius (m)RAP Right atrial pressure (mmHg)SD Standard deviationSE Standard errorT Time (h)U Upper boundV Volume (ml)W Weight factorH Colloid osmotic pressure (mmHg)a Albumin reflection coefficientChangeSuperscripts and SubscriptsAV Interstitial volume available to albuminC CapillaryD DiffusionEX Excludedexp Experimental valueF FiltrationGRAD GradientHUMAN HumanI Interstitial compartmentNomenclature 136iso IsogravimetricL Lymphmax Maximum valuemm Minimum valueNORM Normal steady-stateFL PlasmaRAT RatRMB Ratio fo maximal to basal levelS Solutesim Simulation valueTO Turnover rateo Normal steady-state1/2 Half-time* Tracer propertiesAppendix ARaw Experimental Data137Appendix A. Raw Experimental Data 138Time (hr) -1.5 1 3 6 9 12Saline, 100 mL Mean 24.1 -0.8 -0.1 -0.7 -1.0 -1.8N=4 SD ±0.4 ±1.0 ±0.7 ±0.6 ±1.5Albumin, 100 mL Mean 24.8 0.5 -0.1 -0.6 -0.4 -1.1N=4 SD ±1.6 +1.4 ±0.8 +1.3 +0.5Saline, 200 mL Mean 24.0 -0.9 -0.2 -0.4 0.5 -0.1N=4 SD +2.3 ±0.8 +1.0 ±2.9 ±1.7Albumin, 200 mL Mean 24.2 3.2 3.1 1.5 2.0 1.6N=4 SD +0.9 ±2.8 +1.6 +2.3 ±1.2Table A.1: Change in plasma colloid osmotic pressure (mmHg) at room temperature. t = 0 designates the end of infusion.Time (hr) -1.5 1 3 6 9 12Saline, 100 mL Mean 2,727 49 16 96 202 179N=4 SD ±883 ±140 ±155 ±154 ±142 ±132Albumin, 100 mL Mean 301 281 318 295 287N=4 SD +160 ±270 +244 ±198 ±206Saline, 200 mL Mean 3223 161 145 161 109 169N=4 SD ±394 ±223 ±148 ±110 ±186 +155Albumin, 200 mL Mean 427 383 353 281 314N=4 SD +147 +36 ±66 ±50 ±95Table A.2: Change in plasma volume (mL) at room temperature. t = 0 designates the end of infusion.Appendix A. Raw Experimental Data 139Time (hr) 4.5 1 3 6 9 12Saline, 100 mL Mean 236 0.1 2.7 1.9 8.4 8.6N=4 SD +8.2 +9.7 ±10.4 ±6.9 ±7.2Albumin, 100 mL Mean 22.2 19.6 17.1 18.1 15.5N=4 SD ±5.0 ±4.8 ±9.1 +5.1 +6.6Saline, 200 mL Meái 234 1.8 7.2 9.6 f0T 12.9N=4 SD ±4.8 ±7.6 ±7.8 ±11.4 ±13.0Albumin, 200 mL Mean 44.6 36.5 29.4 24.3 27.1N=4 SD +13.0 +10.7 +7.9 +8.8 +8.6Table A.3: Change in total plasma albumin content (g) at room temperature.t = 0 designates the end of infusion.Appendix A. Raw Experimental Data 140Patient Surface Remarks HCT General Blood Plasma# Area (m2) (%) Volume (mL) Volume (mL)Control 42.4 4422.6 2547.41 1.82 l000mL/9.5mins 37.1 50.59.6 3182.538 mm. after infusion 40.4 4641.0 2766.0Control 32.3 4768.0 3227.92 1.60 l000mL/llmins 27.6 5568.0 4031.260 mm. after infusion 29.2 5264.0 3726.9Control 37.7 4116.0 2564.33 1.68 l000mL/l0mins 31.6 4905.6 3355.4Control 40.0 4273.5 2564.14 1.65 l000mL/llmins 34.3 4983.0 3273.835 mm. after infusion 33.2 5148.0 3438.9Control 34.7 4026.1 2629.05 1.63 l000mL/llmins 28.4 4922.6 3524.6Control 47.5 4537.5 2382.26 1.65 l000mL/l3mins 42.4 5082.O 2927.2-15 mm. after infusion 44.3 4867.5 2711.2Control 43.9 4086.9 2292.87 1.71 l000mL/llmins 37.8 4753.8 2956.930 mi after infusion 40.2 4463.1 2668.9Control 52.3 5635.5 2688.18 1.95 950mL/6.5mins 45.2 6396.0 3505.030 mm. after infusion 44.4 6493.5 3610.4Control 44.1 5056.6 2826.69 1.93 950mL/7.5mins 38.1 5867.2 3631.8Control 45.7 5632.0 3058.210 1.76 950mL/8.5mins 44.8 5737.6 3167.230 mm. after infusion 43.1 5966.4 3394.9Control 46.9 4907.7 2606.011 1.71 900mL/9mins 42.4 5420.7 3122.340 mm. after infusion 44.9 5112.9 2817.2Table A.4: Hemodynamic effect of rapid intravenous infusion of physiologicsaline infusion.Appendix A. Raw Experimental Data 141Baseline 1/2 hour post-infusionHGB (g.100mL’) 12.5+1.7 11.7+1.7HCT () 38.4±5.2 36.2+5.10PL (g.lOOmL’) 3.4±0.5 3.0+0.5Table A..5: Mean concentration of hemoglobin and albumin, and mean hematocrits before and after saline infusion,Patients with Patients with NormalHeart Failure Anasarca Subjects11PL (mmHg) 26.1+4.2 23.0+2.6 28.6±3.4Hi (mmHg) 11.5±3.4 9.2+2.6 15.8±2.7RAPT (mmHg) 11.15 16.29 0Pct (mmHg) 13.71 16.90 6.8VPL (mL/cm) 19.2±4.5 21.7±4.4 17.3±1.5Body Height (cm) 171.9+10.7 171.9±10.7 172.0±9.2VpL/VI 0.27±0.07 0.28±0.09 0.36±0.06V/ (mL) 12224±6795 13322±7945 8266± 2536Table A.6: 11PL, H and VPL in patients with heart failure (N=13), patients with anasarca (N=7) and n normal subjects. calculated fromH —0.4 x RAP + 15.8; t calculated from Pc = 0.62 x RAP + 6.8; * the calculation of SD is the same as that described in APPENDIX B.Angina pectoris Controls11FL (mmHg) 24.9±2.1 26.8±3.7U1 (mmllg) 10.65±2.35 13.15±2.5RAP (rnmHg) 5.1±1.7 1.5±2.25Pc (rnmHg) 12.6±2.9 11.15±2.7VPL (rnL/cm) 18.8±1.8 19.9±1.35Body Height (cm) 175±6 181±5VPL/VI 0.37±0.08 0.36±0.08Vj (mL) 8892±3079 10005+3179Table A.7: 11PL, H1 and VFL in patients with angina pectoris and in controls. *the calculation of SD is the same as that described in APPENDIX B.Appendix A. Raw Experimental Data 1421IPL (mrnHg) Vi (L)9.2 19.0110.7 16.0211.7 14.4012.7 12.9813.7 7.9813.7 6.7114.2 10.7014.2 8.2718.7 9.9418.7 9.1120.7 7.7920.7 6.7321.7 6.6124.7 8.4425.7 8.3425.7 7.6926.7 8.2526.7 9.38Table A.8: 11FL vs. V1 for patients with nephrotic syndrome.Appendix A. Raw Experimental Data 14311PL (mmHg) Hi (mmHg) 11PL (mmHg) fT1 (mmHg)2.3 2.3 15.5 6.45.3 5.9 15.5 6.45.8 1.9 15.7 3.76.6 2.6 17.3 5.97.0 3.4 17.4 7.77.0 2.9 17.5 5.97.8 0.9 17.5 3.98.3 2.7 17.9 8.08.9 1.7 18.0 8.69.0 4.9 18.2 8.29.0 4.9 18.4 8.49.2 3.2 18.7 5.29.5 4.4 18.7 6.79.5 3.9 19.0 6.99.6 0.7 19.0 9.99.9 1.6 19.4 7.710.3 29 19.9 I&.710.7 5.7 20.7 7.711.3 5.4 21.0 10.411.5 3.9 21.0 10.911.5 3.9 21.7 8.711.7 4.7 22.9 14.711.9 4.0 23.6 11.712.3 1.9 23.6 13.212.4 4.5 23.6 14.012.9 4.3 24.0 12.213.0 3.4 24.4 11.713.2 7.0 24.8 12.713.7 5.7 24.2 15.213.9 7.2 24.4 14.414.2 5.7 24.9 13.214.2 4.7 26.4 13.714.5 4.9 26.9 14.7Table A.9: ll vs. 11PL for patients with nephrotic syndrome.Appendix BCalculation Of Error PropagationCalculation of the propagation of experimental error is necessary in the following situations:1. When the compared quantity is not directly measured in experiment. For example,blood volume and hemotocrits are measured experimentally, then plasma volumeis calculated from their prodllct. The uncertainty associated with the calculatedplasma 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 fromabsolute 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 quantitiesx1, x2,•.•, x, by the general equationU=f(xi,x2, (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 themeasured x’s is then given bydU =-1—dx + +... + -‘—dx (B.2)ax1 ax2144Appendix B. Calculation Of Error Propagation 145where -dx is the partial derivative off with respect to x taken with all the remainingx’s held constant.When Ax1, Ax2, ..., Ax are sufficiently small that higher order terms of the Taylorexpansion can be neglected, the differentials dU, dx1, dx2, ..., dx can he replaced bythe finite increments AU, Ax1, Ax2, AxAU = LA1+ LA2+... +1—Ax (B.3)ax2The maximum uncertainty of quantity U is calculated according to Eq. B.3.ExamplesIn 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) isHGBBBV= HGBx BVBSubstituting the numbers into the above equation givesBV = (0.125/0.117) x 5195 = 5550.2 mLApply Eq. B.3ABVP = AHGBB x BVB + HGBB x BVB x AHGBPP P= 1561.3 mLSince plasma volume (VPL) equals to BVPL x (1 — HCT), givenbaseline hemotocrit: HCTB = 0.384±0.052;post-infusion hernotocrit: HCT = 0.362±0.051, we getAppendix B. Calculation Of Error Propagation 146baseline plasma volume: VPL,B = 3200 mL;post-infusion plasma volume: VPL,F = 3541±1177 (SD1) mL. Therefore,VPL%= [VPLF—1] x 100 = 10.65%VPL,Band standard deviation for AVPL% isSDVPL%= SD1x 100 = 36.78VPL,BSince QPL = CPL x VPL, therefore,baseline albumin content: QPL,B = 34 x 3.2 = 108.8 g. with a standard deviation ofSD2= /-GPL,B x VPL,B = 16 g;post-infusion albumin content: QPL,P = 30 x 3.541 = 106.2 g, with a standard deviationof SD3 = /CPL,P x VPL,P + CPL,P x SD1 = 5 x 3.541 + 30 x 1.117 = 53.0 g.Thus, the percentage change of albumin content isQPL% = [QPLP—1] x 100 = —2.39Q PL,BApply Eq. B.3, the standard deviation for QFL% isSDQPL%= QPL,B + x SD2— 53.0 + 106.2 x 16 — 63 07— 108.8 108.82 —All these values correspond to those listed in Table 4.7.Appendix CBasic Concepts Related To Statistical AnalysisTo permit a better understanding of the statistical analysis presented in chapter 5, severalrelevant basic concepts will be clarified here.Covariance MatrixThe covariance matrix is defined as the inverse of the matrix of second partial derivatives (i.e. the Hessian matrix) of the objective function, expressed as:u2{X} a{X,Y}u{X,Y} a2{Y}where a2 { } is called the variance operator (read “variance of”); a{ , } is called thecovariance operator (read “covariance of”); X and Y are two random variables. Thevariance 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 areindependent, a{X,Y} = 0, however, the converse is not necessarily so.Confidence IntervalThe probability that a correct interval estimate of an unknown parameter X isobtained is called the confidence coefficient and is denoted by 1—c. The interval,147Appendix C. Basic Concepts Related To Statistical Analysis 148L <X < U, within which the value of the parameter in question would be expected tolie is called a 100(1 — c) percent confidence interval for the parameter X. The interpretation of this interval is that, if in repeated random samplings, a large number of suchintervals 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 samplesize is reasonably large, a variable with a sample mean X and a standard deviation SDhas the 95% confidence interval [61]X—1.96xSD<XX+1.96xSDResidual PlotA residual (e) is the difference between an observed value (X) and the correspondinganticipated value (X), i.e. e = X — X. A staudardized residual (é) is the residual dividedby the standard deviation, i.e. é = e/SD. A residual plot is a plot of standardizedresiduals (or residuals) plotted against the fitted value X. It often provides useful cluesfor the evaluation of the aptness of the model (see Chapter 5).Appendix DSurface Plot And Contour PlotThe objective function (Eq. 4.19) is an unknown function of LS, a, and the interstitial compliance relationship. It is determined by the differences between the simulationresults and the experimental data. It is non-linear and can not be expressed as an analytical equation. Therefore, analytical methods can not be applied to find the location ofOBJmin. To allow a visual estimate of the minimum region, surface and contour plots ofthe objective function are generated first. Then, starting with an initial point located inside the minimum region, a numerical optimization program is applied to find the best-fitparameters.The surface plot has three dimensions. In this case, the X—axis is a, the Y—axis isLS, and the Z—axis is OBJ. It is generated by increasing a and LS in incrementalsteps (a 20 x 20 grid is chosen) between their lower and upper bounds, and calculatingOBJ at the given values of a, LS, Pc,o and interstitial compliance relationship by usingEq. 4.14. Figure D.1 shows three surface plots viewed from different angles. The figurewas constructed by using 138 experimental data points.A contour plot is a projection drawing of the surface plot (Fig. D.1). Contour linesare drawn by connecting points having the same OBJ value on the projection. From thecontour plot, it is easy to locate the minimum region of the objective function (e.g. theshaded area on Fig. D.1).149Appendix D. Surface Plot And Contour Plot1.0150Figure 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 areais the minimum region of OBJ. For compliance relationship #3.9OD0’00.5 0.6 0.7 0.8 0.9aAppendix ESimulations At Best-Fit Parameters15160.040.020.00.0—20.0—40.0Time(hrs)Figure E.1: (A) Simulation of 100 mL saline infusion; (B) Simulation of 100mE 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(A)15230.0-15.0:0.0-115.0——30.0-—1.5 0.0I I I I I I I1.5 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 12.0Time(hrs)(B)G.liz30.0-15.0-0.0-I—15.0-—30.0--rT—1.5 0.0I I I I I I I1.5 3.0 4.5 6.0 7.5 9.0 10.5p1—1.5 0.0 1.5 3.0 4.5 6.0Appendix E. Simulations At Best-Fit Parameters60.C -40.0-20.0-—20.0-—40.C—1.560.040.020.00.0—20.0—40.0I I I I I I I I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0Time(hrs)Figure E.2: (A) Simulation of 100 mL saline infusion; (B) Simulation of 100mL albumin infusion; Fluid intake during waking hours (0— 13.5 hr) is 1.4 L(N=4). For tissue compliance relationship #2.(A)15330.0-15.0—0.0 -—15.0-—30.CI—._T I I I I I I I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 1 . 0-tffEH0.0 1.5 3.0 9.0 10.5 12.0I I I4.5 6.0 7.5Time(hrs)(B)30.0-15.0-—15.0-30.0-—1.5 0.0 1.5 3.0 4.5Appendix E. Simulations At Best-Fit Parameters 154(A)30.0-15.0— III I I I I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 160.0-40.020:0—20.0—40.0-—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0Time(hrs)(B)30.0I I I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.060.0-IL—-—20.0-—40.0— I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0Time(hrs)Figure E.3: (A) Simulation of 200 mL saline infusion; (B) Simulation of 200mL albumin infusion; Fluid intake during waking hours (0— 13.5 hr) is 1.4 L(N=4). For tissue compliance relationship #1.6 0.0-400-20.0-0.0-i—20.0-—40.0-Figure E.4: (A) Simulation of 200 mL saline infusion; (B) Simulation of 200mL albumin infusion; Fluid intake during waking hours (0— 13.5 hr) is 1.4 L(N=4). For tissue compliance relationship #2.155Appendix E. Simulations At Best-Fit Parameters(A)3 0.0-15.00.0____________________—15.0 I—30.0-—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 1_40.0:C.—1.5 0.0I I I I1.5 3.0 4.5 6.0 7.5Time(hrs)I I I9.0 10.5 12.0(B)30.015.00.0—15.0—30.—1.5 0.0 1.5I I I I I—1.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5Time(hrs)12.0Appendix F. Simulations At Best-Fit Parameters 156Patient #130.020.0120.010.00.00.0 30.0 60.0Patient #230.00.0 I0.0 30.0 60.0Patient #630.00.00.0 30.0 60.0 90.0Patient #730.020.0-0.00.0 30.0 60.0 90.0 120.0Patient #11- ____________________________________________________20.010.0_0.00.0 30.0 60.0 90.0 120.0Time(nilns)Figure E.5: Simulations of acute saline infusion in selected patients. For tissuecompliance relationship #1.Appendix E. Simulations At Best-Fit Parameters 157Patient #130.0K 200I 000.00030.0 60.0Patient #230.020010•0-0.000_________________________________________________________0.0 30.0 60.0Patient #630 020.00.00.0 30.0 60.0 90.0 0.0Patient #730.020.010.0____0.00.0 30.0 60.0 90.0 120.0Patient #1130.0_________________20.010.00.00.0 30.0 60.0 90.0 120.0Time(mlns)Figure E.6: Simulations of acute saline infusion in selected patients. For tissuecompliance relationship #2.Appendix E. Simulations At Best-Fit Parameters 15875.0-.0 .0 L 5.050.0-25.0-Time(hrs)Figure E.7: Transient responses in plasma volume and plasma albumin concentration after 2 L of normal saline infusion within 2 hours. For tissuecompliance relationship #1.Appendix F. Simulations At Best-Fit Parameters 15975.0-oLo.o.oLo5.o50.O-25.0-0 0 0 0Time(hrs)Figure E.8: Transient responses in plasma volume and plasma albumin concentration after 2 L of normal saline infusion within 2 hours. For tissuecompliance relationship #2.Appendix E. Simulations At Best-Fit Parameters 16025.020.015.0i:.:20.0 22.5 25.0 27.5 30.0PLmmHg25.0-20.0-15.0-10.0-5.0-7.5 10.0 12.5 15.0 17.5 20.0P(mmHg)Figure E.9: Simulations of steady-state V1 vs. HPL and V1 vs. Pc in heart failurepatients. For tissue compliance relationship #1.Appendix E. Simulations At Best-Fit Parameters 16125.020.015.0-10.0-5.0-25.0-20.015.0I I20.0 22.5 25.0PL(mmHg)30.010.0-5.0 I I I7.5 10.0 12.5 15.0P(mmHg)17.5 20.0I27.5Figure E.1O: Simulations of steady-state Vj vs. IIPL and V1 vs. Pc in heartfailure patients. For tissue compliance relationship #2.Appendix E. Simulations At Best-Fit Parameters 162100.080.060.040.020.00.0 I— I I I0.0 5.0 10.0 15.0 20.0 25.0 31PL(mmHg)20.0-16.0-12.0-8.0-4.0-0.0 I I I I0.0 5.0 10.0 15.0 200‘TrP(mmHg)25.0 30.0Figure E.11: Simulations of V1 vs. IIPL and H vs. ‘IPL in nephrotic syndromepatients. For the best-fit parameters of tissue compliance relationship #1.S1;.II.I.a...•. I•.Appendix E. Simulations At Best-Fit Parameters 163Figure E.12: Simulations of V1 vs. 1IPL and H vs. HPL in nephrotic syndromepatients. For the best-fit parameters of tissue compliance relationship #2.100.0-80.0-60.0-40.0-20.0-0.020.0I I0.0 5.0 10.0 150 20.0 25.0 37rL(mmHg)).0.16.0-12.0-8.0-4.0-0.0...a•0.0 5.0I I I10.0 15.0 20.0P1. (mmHg)25.0 30.0Time(hrs)Figure E.13: Transient responses of colloid osmotic pressures (IIPL and Hi),plasma volume (Vpjj, and protein contents (QPL and Qz) after single infusionof 60 g of human albumin solution. For the best-fit parameters of tissuecompliance relationship #1.Appendix E. Simulations At Best-Fit Parameters25.0-20.0-15.0-10.0-5.0-6.0-5,0-4.0-3.0-2.0-S4$‘-aI I I12.0 24.0 36.0 4 I.01.0-0.0 -.L12.0164QPLQI24.0 36.0 48.0150.0-0.0 12.0 24.0 36.0 48.0004C,100.0-50.0-0.0--..--.-.0.0 12.0 24.0 36.0 48.0Appendix E. Simulations At Best-Fit ParametersTime(hrs)165Figure E.14: Transient responses of colloid osmotic pressures (HPL and Hi),plasma volume (Vpjj, and protein contents (QPL and Qj) after single infusionof 60 g of human albumin solution. For the best-fit parameters of tissuecompliance relationship #2.25.020.0-15.0-10.0-5.06.05.0-4.0-3.02.0’1.0-0.0-— I I I0.0 12.0 24.0 36.0 4 [.0S4S040J-I I I0.0 12.0 24.0 36.0 48.04.0-3.5.3.0-2.5-150.0-100.0-50.0-0.0 -).0 12.0 24.0 36.0 4 I.0-,.:-0.0 12.0QPLQI24.0 36.0 48.0Appendix FList Of Computer ProgramsF.1 Parameter List of Steady-State and Transient SimulatorsCC A list of program variables of all programs listedC ****************************************************CC Variable key:CC ACC = Accuracy limitC ALPHA = Scaling factor for ROOTNV = Number of nonlinearequation - -C PIPL = Plasma colloid osmotic pressureC PISKIN Tissue colloid osmotic pressureC P15 = Tissue colloid osmotic pressureC PSNRN = Normal tissue hydrostatic pressureC QPLS = Plasma: albumin content_calculatedC QSS = Tissue: albumin content_calculatedC QPLNRM Plasma: normal albumin contentC QSNRN = Tissue: normal albumin contentC QTNRM = Normal total albumin contentC QTOT = Whole body: albumin contentC RHSF = Nonlinear equations to be solvedC VPLNRN Plasma: normal volumeC VIFNRM Tissue: normal volumeC VTNRN = Normal total extra-cellular volumeC VTOT = Whole body: volumeC VEXS = Tissue excluded volumeC YNEW = Initial solution estimatesC YOLO = Final solution valuesC CAVS = Protein concentration in available volume166Appendix F. List Of Computer Programs 167C CPL = Protein concentration in plasmaC CS = Protein concentration in tissueC iFS = Fluid filtration flowrateC JLS = Fluid lymph flowrateC PC = Capillary hydrostatic pressureC PCNRN = Normal PCC QPS = Protein flowrate across the capillary wallC QLS = Lymphatic protein flowrateC AS = Initial gradient of tissue compliance curveC BS = Final gradient of tissue compliance curveC KFS = Tissue filtration transport coefficientC LSNRM = Tissue basal lymph flowrateC LSS = Lymph flow sensitivityC P55 = Permeability-surface area productC SIGS = Albumin reflection coefficientC NPS = Number of points on tissue compliance curveC NPSN1NPS- 1F.2 Listing of FORTRAN function XDFUNC1CFUNCTION XOFUNC(X,N)CCC This function is used to calculate the value of the objectiveC function at specific values of SIGS, LSS end PCNRM.CIMPLICIT REAL*8CA-H,K,L,3,O-Z)CCONMON/BLKA2/CPLNRM , CSNRM , CASNRMCONMON/BLKB/JFS , ASCOMMON/ELKO/QPS , QLSCONMON/BLKG/CPL ,CS, CAVSCOMMON/BLKI/PIPL ,PISCOMMON/BLKF/PC ,PCNRM,PCGRADCONNON/BLKH/LSNRMCONMON/BLKJ/VTNRN, VIFNRN, VPLNRNCONNON/BLKK/VEXSCONNON/ELKL/LSS ,KFS , SIGS , P55Appendix F. List Of Computer Programs 168COMMON/BLKT/QTNRM , QSNNM , QPLNRMCOMMON/BLKU/PSNRM ,PSCOMMON/BLKBB/PIPNRN ,PISNRMCOMMON/BLKZB/QTOT, VTOTCOMMON/BLKO/VSP(14) ,PSP(14) ,AS,BS,NPS,NPSM1COMMON/A/VPINIT , QPINIT , VSINIT , QSINITCDIMENSION X(N) ,G(6)CLBS = x(i)*40.D0SIGS = X(2)PCNRN = X(3)*10.D0CPIPNRM=25 .900PISNRM=14 .700CALL SPLINSCALL COEFFCPLINIT = 25.900PIINIT = 14.7D0VIFINT = 8400.000CVPINIT = VPLNRMQPINIT = VPLNRM * FALB(PLINIT)CVSINIT = VIFINTQSINIT = FALB(PIINIT) * (VIFINT- VEXS)CVTINIT = VPINIT + VSINITQTINIT = QPINIT + QSINIICC Each data set is progrsned separately so that it can be calledC 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)CAppendix F. List Of Computer Programs 169SUM = 0.00CALL HUB(i.D-2,4,SUM1,SUM)0(1) = SUMSUM = 0.00CALL DOYLE(i.D-2,4,SUM)0(2) = SUMSUM = 0.00CALL MULLIMS(i .D-2,4,SUM)0(3) = SUMSUM = 0.00CALL HF(1.D-2,.SD0,i.D-2,4,SUM)0(4) = SUMSUM = 0.00CALL MEPHRO(SUM)0(5) = SUM0(6) = 0(1)+G(2)+0(3)+G(4)+0(S)XDFUMC = 0(6)CRETURNENDCSUBROUTINE XDCONS(X,N,M,ME,MMAX,G)CCC This subroutine is used to evaluate the constraints imposed onC the parameters to be estimated.CIMPLICIT REAL*6(A-H,K,L,0-Z)INTEGER N,M,ME,MMAXDIMENSION X(N) ,G(MMAX)CCOMMON/BLKA2/CPLMRM , CSNRM , CASNRMCOMMON/BLKF/PC,PCNRM,PCGRADCOMMON/BLKU/PSNRM ,PSCOMMOM/BLKBB/PIPNRM ,PISNRMCIF(M.LE.0) GO TO 1000(1) = (x(3)*io.Do- PSNRM)/(PIPNRN— PISMRM) — x(2)/lo.DoAppendix F. List Of Computer Programs 170RETURN100 CONTINUEJUMP = -MGO TO (1),JUMPRETURN1 0(1) = (X(3)*10.OO — PSNRM)/(PIPNRN- PISNEM)— X(2)/10.OORETURNENDCSUBROUTINE HUB (EPS ,NV, SUM1 ,SUM)CCC This subroutine is used to evalute the sum of square of the errorC contributed by Hubbard’s data only.CIMPLICIT REAL*8(A-H,K,L,J,O-Z)INTEGER FLAGCCOMMON/BLKI/PIPL,PIS--COMMON/A/VPINIT , QPINIT ,VSINIT ,QSINITCOMMON/BLKB/JFS , JLSCOMMON/BLKO/QPS ,QLSCOMMON/SLKG/CPL ,CS, CAVSCOMMON/BLKF/PC ,PCNRM,PCGRADCOMMON/BLKH/LSNRMCOMMON/BLKL/LSS ,KFS ,SIGS ,PSSCOMMON/BLKU/PSNRM ,PSCOMMON/BLKZB/QTOT ,VTOTCDIMENSION YOLD(4) ,YNEW(4)DIMENSION SOLN(6,2),EXP1(6,2,4),T1(6),SD1(6,2,4)EXTERNAL RHSFBCC Experimental data.CDATA 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 1712 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/CDO 60 IPET1,4CC Set initial conditions.CYNEW(1) = VPINITYNEW(2) = QPINITYNEW(3) = VSINITYNEW(4) = QSIMITCC Solve differential equations.CDO 30 I = 0,5IP=I+1TI = T1(IP)IF(I,EQ.0) GOTO 10TIM T1(I)DT=TI-TIMTSTART=1 .D-1*DTTMIN1 . D-4*DTTMAX=DTCALL RK4C(RHSFB,NV,TIM,TI,YOLD,EPS,YNEW,NFTJNC,FLAG,IPET)IF(FLAG.EQ.0) GOTO 7010 CALL AUXSAM(YNEW)SOLN(IP,1) PIPLAppendix F. List Of Computer Programs 172SOLN(IP,2) = YNEW(1)DO 20 J = l,NVYOLO(J) = YNEW(J)20 CONTINUE30 CONTINUECC Calculate the sum of square of error.C00 50 .1=1,2SOLNOSOLN(1,J)00 40 1=2,6SOLN(I ,J)=(SOLN(I,J)-SOLN0)/SOLNOERROR’SOLNCI , .1) -EXP1 (I, .1, IPET)SUN SUM +4.00*(ERR0R**2)/(5D1(I,J,IpET)**2)40 CONTINUE50 CONTINUE60 CONTINUERETURNC70 WRITE(6,80)80 FORMAT(1X,’ODE SOLVER FAILS 1 ‘)STOPENDCSUBROUTINE DOYLE (EPS ,NV,SUM)CCC This subroutine is used to evaluate the sum of square of errorC contributed by Doyle’s data only.CIMPLICIT REAL*8(A-H,K,L,J,O-Z)INTEGER FLAGCOMMON/B/T2(3,S)CONNON/BLKI/PIPL ,PISCOMMON/A/VPINIT,QPINIT,VSINIT,QSINITCOMMON/BLKB/JFS ,J1SCOMMON/BLKD/QPS ,QLSAppendix F. List Of Computer Programs 173COMMON/BLKG/CPL ,CS, CAVSCOMMON/SLKF/PC ,PCNRM,PCGRADCOMMON/BLKH/LSNRMCOMNON/BLKL/LSS ,KFS , 5105 ,PSSCOMMON/BLKI3/PSNRN ,PSCOMMON/8LKZ8/QTOT ,VTOTCDIMENSION YOLO(4) ,YNEW(4)DIMENSION SOLN(3,S) ,EXP2(3,S) ,502(3,S)EXTERNAL RHSFCCC Experimental data.CDATA 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/C00 60 IPET1,SCC Set initial conditions.CYNEW(1) = VPINITYNEW(2) = QPINITYNEW(3) = VSINITYNEW(4) = QSINITCC Solve the differential equations.CDO 30 I 0,21P1+1TI = T2(IP,IPET)/60.D0IF(I.EQ.0) GOTO 10Appendix F. List Of Computer Programs 174TIM = T2(I,IPET)/60.D0DTTI-TIMTSTART1 .O-1*DTTMIN1 .D-4*DTThAXDTCALL RK4C(RHSFC,NV,TIM,TI,YOLD,EPS,YNEW,NFUNC,FLAG,IPET)IF(FLAG,EQ.O) GOTO 7010 CALL AUXSAM(YNEW)SOLN(IP,IPET) = YNEW(1)DO 20 .3 = 1,NVYOLO(J) = YNEWCJ)20 CONTINUE30 CONTINUECC Calculate the sum of square of error.CSOLNOSOLN(1 ,IPET)DO 50 1=2,3SOLN(I , IPET)(SOLN(I ,IPET)-SOLNO)/SOLNOERROR=SOLN(I ,IPET)-EXP2(I,IPET)sUMsUM+(ERROR*ERR0R)/(5D2C.I ,IPET)*SD2(I,IPET))50 CONTINUE60 CONTINUERETURNC70 WRITE(6,80)80 FORMAT(1X,’ODE SOLVER FAILS 2’)STOPENDCSUBROUTINE MULLINSCEPS ,NV,SUM)CCC This subroutine is used to evaluate the sum of square of errorC contributed by Mullins’ data only.CIMPLICIT REAL*8(A-H,K,L,J,O-Z)INTEGER FLAGAppendix F. List Of Computer Programs 175CCOMMON/BLKI/PIPL , PISCOMMON/A/VPINIT , QPINIT ,VSINIT , QSINITCOMMON/BLKB/JFS , JLSCOMMON/BLKD/QPS , QLSCOMMON/BLKG/CPL ,Cs, CAVSCOMMON/BLKF/PC ,PCNRM,PCGRADCOMMON/BLKH/LSNRMCOMMON/BLKL/LSS , KFS , SIGS , PSSCOMMON/BLKU/PSNRN ,PSCOMMON/BLKZB/QTOT ,VTOTCDIMENSION YOLD(4) ,YNEW(4)DIMENSION SOLN(2,2) ,EXP4(2,2) ,T4(2) ,SD4(2,2)EXTERNAL RHSFECC Experimental data.CDATA 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/CC Set initial conditions.CYNEW(1) = VPINITYNEW(2) QPINITYNEW(3) VSINITYNEW(4) QSINITCC Solve the differential equations.CIPET2DO 30 I = 0,1IP=I+1TI = T4(IP)IF(I.EQ.0) GOTO 10TIM = T4(I)DT=TI-TIMTSTART=l .D-1*DTAppendix F. List Of Computer Pro,gTams 176TMIN=1 .D-4*DTTMAX=DTCALL RK4C(RHSFE,NV,TIM,TI,YOLD,EPS,YNEW,NFUNC,FLAG,IPET)IF(FLAG.EQ.0) GOTO 7010 CALL AUXSAM(YNEW)SOLN(IP,i) = YNEW(1)SOLN(IP,2) = CPLDO 20 3 = 1,NVYOLD(J) = YNEW(3)20 CONTINUE30 CONTINUECC Calculate the sum of square of error.CDO 60 3=1,2SOLNOSOLN(1 ,J)DO 50 1=2,2SOLN(I , J)’(SOLN(I ,J)—SDLND)/SOLNOERROR”SOLN(I !4U,3)SUM=SUM+i ii. D0* (ERROR*ERRDR) / (SD4(I, 3) **2)50 CONTINUE60 CONTINUERETURNC70 WRITE(6,80)80 FORMAT(iX,’DDE SOLVER FAILS 3’)STOPENDC *************************************SUBROUTINE HF(ACC,ALPHA,EPS,NV,SUM)C *************************************CC This subroutine is used to evaluate the sum of square of eorrorC contributed by the data from heart failure patients.CIMPLICIT REAL*8(A-N,K,L,J,D-Z)INTEGER FLAG,FLAGGCAppendix F. List Of Computer Programs 177COMMON/BLKA2/CPLNRM , CSNRM , CASNRMCOMMON/BLKB/JFS , JLSCOMMON/BLKD/QPS , QLSCOMMON/BLKG/CPL ,Cs, CAVSCOMMON/BLKI/PIPL ,PISCOMMON/BLKF/PC ,PCNRM,PCGRADCOMMON/BLKH/LSNRJ4COMM0N/BLKJ/VTNR4 ,VIFNRM,VPLNRNCOMMON/ELKK/VEXSCOMMON/BLKL/LSS , KFS ,SIGS ,PSSCOMMONJBLKT/QTNRM , QSNRN , PLNRMCOMMON/BLKU/PSNRM ,P5COMMON/BLKBB/PIPNRN ,PISNRMCOMMON/BLKDD/ALBSTOCOMNON/BLKZB/QTOT ,VTOTCOMMON/VQPL/VPL ,QPLCDIMENSION 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 RHSFACC Experimental data.CDATA 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/CTSTART1 . 0-1*0 .00500TMIN1 .0-4*0.00500TMAX=0 .0100CC Set initial conditions.VSG = VIFNRN+3000.D0DO 40 IPET2,3Appendix F. List Of Computer Programs 178PIPL = A(IPET)P15 = B(IPET)PC = C(IPET)CPL = FALB(PIPL)PCC = PCVPL = FVPL(PCC)QPL = VPL*FALB(PIPL)QSG = VSG*FALB(PIS)*3.DO/4.OOYOLD(1) = VSGCC Solve the differential equation.CCALL ROOT(RHSFA,l,O.DO,YOLD,ACC,ALPHA,YNEW,FLAG)IF(FLAG.EQ,O) THENPRINT*,’----ROOT FAILED----’STOPENDIFIF(VSG.LE.O.OO.OR.QSG.LE.O.OO) THENPRINT*, ‘----NEGATIVE VOL/Q GENERATED----’STOPENDIFCC Calculate the sum of square of error.CVSG = YNEW(1)CIS = FALB(PIS)QS = YNEW(l)*FALB(PIS)QTOT = QS + QPLPS = FCOMPS(YNEW(1))Y1CIPET) = YNEW(1)/1000.DOCHANGE = (YNEW(l)-vIFNRM)/VIFNRNERROR = CHANGE-XPVI(IPET)= SUN + p0INT(IPET)*ERR0R**2/SD3(IPET)**240 CONTINUERETURNENDC ************************SUBROUTINE NEPHRO (SUM)Appendix F. List Of Computer Programs 179CCC This subroutine is used to evaluate the sum of square of errorC contributed by the data from nephrotic patients.CIMPLICIT REAL*8(A-H,K,L,J,O-2)INTEGER FLAGCCOMMON/BLKA2/CPLNRM , CSNRM , CASNRMCOMMON/BLKB/JFS JLSCOMMON/BLKD/QPS , QLSCOMNON/BLKG/CPL ,CS,CAVSCOMMONIBLKI/PIPL ,PISCOMMON/BLKF/PC , PCNRM , PCGRADCOMMON/BLKH/LSNRMCOMMON/BLKJ/VTNRM ,VIFNRM,VPLNRMCOMMON/BLKK/VEXSCOMMON/BLKL/LSS ,KFS,SIGS,PSSçoMMoN/T0T/vToT , QTOTCOMNON/BLKU/PSNRM ,PSCOMMON/BLKY/PIPPS(100) ,PISKIN (100) • NA ,NAM1COMMON/BLKBB/PIPNRM , PISNRMCOMNON/BLKDD/ALBSTOCOMMON/BLKZ/DIFV(19) ,DPIPL(19) ,NNCOMMON/BLKHH/PIPLPI (66) ,PII (66)CDIMENSION YOLD(2),YNEW(2),EXP6(66),SD6(66),EXP7(19) ,SD7(19)EXTERNAL RHSFCDATA ACC,ALPHA,NV/1.D-2,0.5D0,2/CC Experimental data.CDATA 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,Appendix F. List Of Computer Programs 180+ -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/CNA = 97NAM1 = NA - 1PIPLUL = 28.00CC GUESS THE INITIAL VALUES OF QPL, QS, VPL, VSCQPLG = VPLNRM * FALB(PIPNRM)Appendix F. List Of Computer Programs 181QSGO= (VIFNRN — 400.00) * FALB(PISNRM) * 3.00/4.00VPLG = VPLNRMVSG0 = VIFNRM - 400.00DIV = 0.2500CSUMPIO .00VSG=VSG0QSGQSGOCC Solve the differential equations.C00 20 IN = 1,66PIPP = PIPLPI(IN)CPL = FALB(PIPP)YOLD(1) = VSGYOLD(2) = QSGCALL ROOT(RHSF,NV,0.O0,YOLO,ACC,ALPHA,YNEW,FLAG)IF(FLAG.EQ.0)THENPRINT*, ‘*****RQQT FAILED! ****‘STOPENOIFVSG = YNEW(1)QSG = YNEW(2)IF(VSG.LE.O.O0.OR.QSG.LE.D.O0)ThENPRINT*,’ IN = ‘,IN,’VSG = ‘,VSG,’QSG = ‘,QSGPRINT*,’** NEGATIVE VOL/Q GENERATED *****‘STOPENOIFCC Calculate the sum of square of error contributed by PISKIN.CVPLG = VTOT - VSGQPLG = QTOT - QSGCAVS = QSG / (VSG - VEXS)PISKIN(IN) = FPI(CAVS)CHANGE = (PISKIN(IN)-PISNRM) /PISNRHERROR CHANGE-EXP6 (IN)SUMPISUNPI+ERROR**2/506 (IN) **220 CONTINUEAppendix F. List Of Computer Programs 182CSUMIFV = 0.00VSGVSG0QSG=QSGODO 40 1=1,19PIPPD?IPL(I)CPL=FALB (PIPP)YOLO(i)VSGYOLO (2) =QSGCALL ROOT(RHSF,NV,0.O0,YOLD,ACC,ALPHA,YNEW,FLAG)IF(FLAG.EQ.0) THENPRINT*, ‘*****ROOT FAILEIH*****’STOPENOIFVSG = YNEW(1)QSG = YNEW(2)IFQVSG.LE.O.O0.OR.QSG.LE.0.O0) THENPRINT*,’ I = ‘,I,’VSG= ‘,VSG,’QSG= ‘,QSGPRINT*, ‘*****NEGATIVE VOL/Q GENERATEDSTOPENOIFCC Calculate the sum of square of error contributed by VIF.CIF(PIPP.EQ. 17.400) THENERROR = 0.00GO TO 35END IFCHANGE= (YNEW (1) -VIFNRN) /VIFNRNERROR= CHANGE-EXP7 (I)35 SUNIFVSUNIFV+ERROR**2/5O7 (I) **250 FORNAT(IX,SF1O.4)40 CONTINUESUN = SUN+SIJNPI+SUNIFVRETURNENDCSUBROUTINE COEFFAppendix F. List Of Computer Programs 183CCC This subroutine is used to calculate the values of the transportC parameters: LSNRN, P55, KFS.CIMPLICIT REAL*8(A-H,J,K,L,0-Z)COMMON/BLKA2/CPLNRN , CSNRM , CASNRMCOMMON/BLKT/QTNRM QSNRM , QPLNRMCOMMON/ELKDO/ALBSTOCOMMON/BLKH/LSNRMCOMMON/ELKF/PC , PCNRM , PCGRADCOMMON/BLKU/PSNRM ,PSCOMMON/BLKK/VEXSCOMMON/BLKBS/PIPNRM ,PISNRMCOMMON/BLKL/LSS , KFS ,SIGS ,PSSCOMMON/BLKJ/VTNRN,VIFNRM , VPLNRNCC Set the normal steady-state conditions.CPIPNRM = 25.900PISNRM = 14.700PSNRM = FCOMPS(VIFNRM)CPLNRM = FALB(PIPNRM)CASNEM = FALB(PISNRM)CSNRM = CASNEM * (1.00 - VEXS / VIFNRM)QPLNRN = VPLNRM * CPLNRMQSNRM = VIFNRM * CSNRMQTNRM = QPLNRM + QSNRMVPLNRM = VPLNRMVSNRM = VIFNRMVTNRM = VPLNRM + VSNRMCC Calculate LSNRM.EPNRM = (CSNRM-(1 .00-SIGS)*CPLNRM)/(CSNRM-(1 .D0-SIGS)*CASNRM)Ml = VIFNRM - VEXSIF(ABS(1.00-SIGS) .LT.1.0-8) THENLSNRM = ALBSTO/ (CSNRM/ (CPLNRM-CASNRN) /W1+i . 00/VIFNRM)ELSEAppendix F. List Of Computer Programs 184LSNRM = ALBSTO/((i .OO-SIGS)*EPNRM/ (1 .OO-EPNRM)/W1+1 .OO/VIFNRM)ENDIFCC Calculate P55.CIF(ABS(i.DO-SIGS).LT.l.O-8) THENPSS=CSNRM*LSNRM/ (CPLNRN-CASNRM)ELSEP55 = Ci .DO-SIGS)*LSNRM/OLOGC1 .DO/EPNRM)ENDIFCC Calculate KFS.CKFS = LSNRM / (PCNRM - PSNRM - 5105 * CPIPNRM - PISNRM))PE = -DLOGCEPNRM)RETURNENDCSUBROUTINE AUXSAMCY)C **********************CC This subroutine is used to calculate all system variable values atC given compartmental protein and fluid contents.CINPLICIT REAL*8(A-H,J,K,L,O—Z)CCOMMON/BLKB/JFS , ASCOMNON/BLKO/QPS , QLSCONNON/BLKF/PC,PCNRM,PCGRADCOMNON/BLKG/CPL ,CS, CAVSCONNON/BLKH/JLSNRMCOMMON/BLKI/PIPL ,PISCOMMON/BLKK/VEXSCOMNON/BLKL/LSS ,KFS, SIGS, P55COMMON/BLKT/QTNRM , QSNRN , QPLNRMCOMMON/BLKU/PSNRN ,PSCONNON/BLKZA/PRESLOCOMNON/BLKZB/QTOT ,VTOTAppendix F. List Of Computer Programs 185CDIMENSION Y(4)CCPL = Y(2) I Y(1)PC = FCOMPC(Y(1))Cs = Y(4)/ Y(3)CAVS = Y(4)/ (Y(3) - VEXS)PIPL = FPI(CPL)PIS = FPI(CAVS)PS = FCOMPS(Y(3))JFS = KFS * (PC — PS — SIGS* (PIPL- PIS))JLS = .JLSNRM + LSS * (PS - PSNRM)IF(JLS .LT. JLSNRM) JLS=JLSNRM*(PS-PRESLO)/(PSNRM-PRESLO)ULJLS1O .OO*JLSNRMIF(JLS . GT.ULJLS) JLSUL.JLSEP= (1.00 — SIGS) * JFS / P55IF(EP.GT.5O.00) THENQPS=JFS* (1 . DO-SIGS) *CPLELSEIF(EP.LT.-5O.OO) THENQPSJFS*(1.OO—SIGS)*CAVSELSEIF(ABS(O.D0-EP) .LT.1.D-8) THENQPS=PSS* (CPL-CAVS)ELSEQPS = JFS *(1.oo-sIG5)*(CpL-CAv5*Exp(-Ep))/(1.oo-Exp(-Ep))ENDIFQLS = JLS*CSRETURNENDCSUBROUTINE AUXNEP (Y)CCIMPLICIT REAL*B(A-H,J,K,L,O—Z)CCOMMON/BLKB/JFS , AS,QLS,PCNRM , PCGRADCOMMON/BLKG/CPL ,CS, CAVSAppendix F. List Of Computer Programs 186COMMON/BLKH/JLSNRMCOMMON/BLKI/PIPL,PISCOMMON/BLKK/VEXSCOMMON/BLKL/LSS , KFS , SIGS ,PSSCOMMON/TOT/VTOT , QTOTCOMMON/ELKU/PSNRM ,PSCOMMON/BLKZA/PRESLOCDIMENSION Y(2)CVDLPLA = 3200.00VTOT = VOLPLA + Y(1)QPLAS = VOLPLA * CPLQTDT = QPLAS + Y(2)PC = PCNRMCS = Y(2) / Y(1)CAVS = Y(2) / (Y(i) — VEXS)PIPL = FPI(CPL)PIS = FPI(CAVS)PS = FCDMPS(Y(1))JFS = KFS * (PC — PS — SIGS * (PIPL - PIS))JLS = JLSNRN + LSS * (PS - PSNRM)IF(JLS .LT. JLSNRM) JLSJLSNRM*(PS-PRESLO)/(PSNRM-PRESLD)ULJLS=10 . D0*JLSNRMIF(JLS.GT.ULJLS) .JLS=ULJLSEP= (1.00 - SIGS) * iFS / p55IF(EP.GT.50.DC) THENQPS=JFS* (1. DO—SIGS) *CPLELSEIF(EP.LT.-50.DD) THENQPS=—JFS* (1.00-5105) *CAV5ELSEIF(ABS(0.D0-EP).LT.1.D-8) THENQp5=PSS* (CPL-CAVS)ELSEQPS = iFS *(1.DQ-5I05)*(CPL—CAV5*EXP(—Ep))/(1.DQ-EXp(—EP))ENDIFQLS = .JLS * CSRETURNENDAppendix F. List Of Computer Programs 187C ******************************SUBROUTINE TOMACA,B,C,O,X,N)C ******************************CC This subroutine uses the Thomas algorithm to find the solutionC of a tridiagonal matrix.CIMPLICIT REAL*8CA-H,O-Z)CC Thomas algorithmCDIMENSION ACN),B(N),C(N),O(N),X(N),PC1OI),QCiD1)CNM = N- 1PU) = —CU) / BU)QC1) = OC1) / BC.)00 10 I = 2,NIM = I—iDEN = AU) * PCIM) + B(I)PCI) = -CCI) / OENQCI) = COCI)— ACI) * QCIM)) / OEN10 CONTINUEXCN) = QCN)DO 20 II = 1,NMI = N- IIXCI) = PCI) * XCI+1) + QCI)20 CONTINUERETURNENDC ************************************DOUBLE PRECISION PUNCTION FALBCPI)C ************************************CC This function calculates Albumin concentration- using a fittedC CAlb) vs. C.0.Ppl relationship.CIMPLICIT REAL*8CA-H,O-Z)CAppendix F. List Of Computer Programs 188CONMON/BLKCC/CALB1 ,CALB2CFALB = (CALB1 + CALB2 * P1) / 1.0+3RETURNENDC **********************************DOUBLE PRECISION FUNCTION FPI(C)CCC This function calculates colloid osmotic pressure - using aC fitted C.O.P vs. (Alb) relationship.CIMPLICIT REAL*8(A-H,O-Z)CCOMMON/BLKN/CPI1 ,CPI2CR = C * 1.0+3FPI = CPI1 + CPI2 * RRETURNENDC ****************************************DOUBLE PRECISION FUNCTION RHSFA(IJX,Y)C ****************************************CC This function evaluates the non-linear equations for a givenC set of Y values. Same as RHSFB,RHSFC,RBSFD,RHSFE,RHSFF.CIMPLICIT REAL*8(A—H,J,K,L,O—2)CCOMMON/BLKB/JFS , JLSCDMMON/BLKD/QPS , QLSDIMENSION Y(i)CCALL AUXALT(Y)RHSF3 = JFS— JLSRHSF4 = QPS - QLSGOTO(1O,20),I10 RI-ISFA = RHSF3Appendix F. List Of Computer Programs 189RETURN20 RHSFA = RHSF4RETURNENDCDOUBLE PRECISION FUNCTION RHSFB(I,T,Y,IPET)CCIMPLICIT REAL*8(A-H,.J,K,L,O-Z)CCONMON/BLKB/JFS , JLSCONMON/BLKD/QPS , QLSDIMENSION Y(4) ,SAL1(4) ,PRO(4)CDATA SAL1/100.DD,100.OD,200.DD,200.DD/DATA PRO? 0.00, 25.00, 0.00, 50.00?CCALL AUXSAM(Y)IF(T.LT.0.000)THENRHSFI = .JLS — iFS + SAL1(IPETV1.500RHSF2 = QLS - QPS + PRO(IPET)*.6SDO/1.SDORHSF3 = iFS— ASRHSF4 QPS — QLSELSEIF(T.GT.D.DDO.AND.T.LT. 13.SDD)TNENRNSFI. = AS — iFS + 1400.DD?13.SDORHSF2 = QLS — QPSRHSF3 = iFS— ASRHSF4 = QPS— QLSELSERHSF1 = JLS - iFSRHSF2 = QLS — QPSRHSF3 = iFS - JLSRHSF4 = QPS - QLSENDIFGDTO(10,2D,3D,40) ,I10 RHSFB = RUSF1RETURN20 RHSFB = RHSF2Appendix F. List Of Computer Programs 190RETURN30 RHSFB = RHSF3RETURN40 RHSFB = RHSF4RETURNENDCDOUBLE PRECISION FUNCTION RHSFCCI,T,Y,IPET)CCIMPLICIT REAL*8(A-H,J,K,L,0—Z)CCOMMON/BLKB/JFS,JLSCOMMON/BLKD/QPS QLSCOMMON/B/T2(3,S)DIMENSION Y(4) ,SAL2(5)CDATA SAL2/4*1000.D0,900.D0/CCALL AUXSAM(Y)IF(T.LE.T2(2,IPET)/60.D0)THENRHSF1 itS— JFS + SAL2(IPET)/T2(2,IPET)*60.O0RHSF2 = QLS — QPSRHSF3 = IFS— ESRHSF4 = QPS- QLSELSERHSF1 = itS- JFSRHSF2 = QLS— QPSRHSF3 = .TFS — JLSRHSF4 = QPS- QLSENOIFGDTO(10,20,30,40),I10 RHSFC = RHSF1RETURN20 RHSFC = RHSF2RETURN30 RHSFC = RHSF3RETURNAppendix F. List Of Computer Programs 19140 RHSFC = RHSF4RETURNENDC *********************************************DOUBLE PRECISION FUNCTION RHSFO(I,T,Y,IPET)C *********************************************CIMPLICIT REAL*8(A-H,J,K,L,O-Z)CCOMMON/BLKB/JFS ,JLSCOMMON/BLKO/QPS, QLSCOMMON/B/T2(3,5)DIMENSION Y(4) ,SAL3(2) ,PRO3(2)CDATA SAL3/4420 .0500,5996. 83D0/DATA PRO3/29.3500,9.6500/CCALL AUXSAM(Y)IF(T.LT.1:soo) THENRHSF1 = .JLS - JFS + SAL3(IPET)/1.SDORHSF2 = QLS- QPS + PRO3(IPET)*.6500/1.SDORHSF3 = JFS— JLSRHSF4 = QPS- QLSELSERHSF1 = JLS— JFSRHSF2 = QLS- QPSRHSF3 = JFS - JLSRHSF4 = QPS - QLSENDIFGOTO(10,20,30,40),I10 RHSFD = RHSF1RETURN20 RHSFD = RHSF2RETURN30 RHSFD = RHSF3RETURN40 RHSFD = RHSF4RETURNAppendix F. List Of Computer Programs 192ENDC *********************************************DOUBLE PRECISION FUNCTION RHSFE(I,T,Y,IPET)CCINPLICIT REAL*8(A—H,J,K,L,O-Z)CCOMNON/BLKB/JFS , JLSCOMMON/BLKO/QPS, QLSDIMENSION Y(4) ,SAL4(2) ,URI(2)CDATA SAL4/-2800 .00,2000.00/OATA URI/O .000,1500.00/CCALL AUXSAN(Y)IF(T.LT.2,000)THENRHSF1 = ES — .JFS + SAL4(IPET)/2.OORBSF2 = QLS - QPSRRSF3=JFS— JLSRHSF4 = QPS- QLSELSEIF(T.GE.2,000.ANO.T.LT.24.000)THENRHSF1 = JLS— JFS- URI(IPET)/22.DORHSF2 = QLS— QPSRHSF3 = JFS- JLSRHSF4 = QPS- QLSELSERHSF1 = JLS - JFSRNSF2 = QLS- QPSRHSF3 = iTS- ASRHSF4 = QPS- QLSENOIFGOTO(1O,20,30,40) ,I10 RUSFE = RHSF1RETURN20 RESFE = RHSF2RETURN30 RHSFE = RHSF3RETURNAppendix F. List Of Computer Programs 19340 RHSFE = RHSF4RETURNENDC ****************************************DOUBLE PRECISION FUNCTION RHSFFCI,T,Y)CCIMPLICIT REAL*8(A-H,J,K,L,O-Z)CCDMMON/BLKB/JFS , ASCOMMON/BLKD/QPS , QLSDIMENSION Y(4)CCALL AUXSAM(Y)RHSF1 = JLS - JFS + 1750.D0/2,SDORHSF2 = QLS - QPSRHSF3 = JFS - ASRHSF4 = QPS- QLSGO-TD(10,2&,30,40),I10 RHSFF = RHSF1RETURN20 RHSFF = RHSF2RETURN30 RHSFF = RHSF3RETURN40 RHSFF = RHSF4RETURNENDC ****************************************DOUBLE PRECISION FUNCTION RHSF(I,X,Y)C ****************************************CIMPLICIT REAL*8(A-H,J,K,L,D—Z)CCOMMON/BLKB/JFS , ASCOMMON/BLKD/QPS , QLSDIMENSION Y(2)CAppendix F. List Of Computer Programs 194CALL AUXNEP(Y)RHSF3 = JFS - JLSRHSF4 = QPS- QLSGOTO(10,20),I10 RHSF = RHSF3RETURN20 RHSF = RHSF4RETURNENDCBLOCK DATACCIMPLICIT REAL*8(A-H,K,L,O-Z)CCOMMON/B/T2(3,S)COMNON/BLKF/PC ,PCNRM,PCGRADCONNON/BLKJ/VTNRN,VIFNRN,VPLNR14COMMON/BLKK/VEXSCOMMON/BLKN/CPI 1, CPI2CONNON/BLKO/VSP(14) ,PSP(14) ,AS,BS,NPS,NPSN1CONNON/BLKZ/DIFV(19) ,DPIPL(19) ,NNCOMMON/BLKHH/PIPLPI (66) ,PII (66)CONNDN/BLKCC/CALB1 , CALB2COMMON/BLKDD/ALBSTOCOMMDN/BLKZA/PRESLOCDATA 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,Appendix F. List Of Computer Programs 195+ 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/Appendix F. List Of Computer Programs 196ENDC *************************************DOUBLE PRECISION FUNCTION FCOMPS(V)C *************************************CC This function calculates the skinC compartments hydrostatic pressure.CIMPLICIT REAL*8(A-H,O-Z)CCOMMON/BLKO/VSP(i4),PSP(14),AS,BS,NPS,NPSM1CIF(V.LE.VSP(1)) GO TO 10IF(V.GE.VSP(9)) GO TO 20FCOMPS = FS(V)RETURNiO FCONPS = PSP(i) + AS * C V — VSPC1))RETURN20 FGOMPS PSPCP) + BS * KV — VSPC9))RETURNENDC *******************SUBROUTINE SPLINSCCC This subroutine SPLINES the experimentlyC obtained PS vs. VS data set.CIMPLICIT REAL*B(A-H,O-Z)CCOMMON/BLKO/XCi4) ,YC14) ,A1,BN,N,NMCOMMON/BLKQ/QC1Oi) ,RC1O1) ,S(100)CDIMENSION HC100) ,AC1O1) ,BC10I) ,C(101) ,OC1O1)CDO 10 I = 1,NMH) = XCI+i)— XCI)10 CONTINUEAppendix F. List Of Computer Programs 197AC1) = 0,00BU) = 2.00 * HO)CU) = MCi)0(1) = 3.00 * CCYC2) — Y(1)) / MU) — Al)DO 20 I = 2,NMIP = 1+1IM = I-iAU) = MCDI)BCI) = 2.00 * (MCDI) + MCI))CCI) = MCI)DCI) = 3.00 * CCYCIP)— YCI)) / MCI) — CYCI) — Y(IM))/MCIM))20 CONTINUEACN) = MCNM)BCN) = 2.00 * MCNM)CCN) = 0.00OCN) = -3.00 * CCYCN)- YCNM)) / MCNM) - BN)CALL TOMACA,B,C,O,R,N)00 30 I = l,NMIP = I + 1QCI) = CYCIP)— YCI)) / MCI) — MCI) * (2.00 *RCI)+R(IP))/3.00SCI) = CRCIP)— RCI)) / C3.00 * MCI))30 CONTINUERETURNENDC **************************+******DOUBLE PRECISION FUNCTION FSCZ)C *********************************CC This function evaluates skin compartment hydrostatic pressureC - using a splined set of data.CIMPLICIT REAL*BCA-M,O-Z)CCOMMON/BLKO/X(14) ,Y(l4) ,A1,BN,N,NMCOMMON/BLI{Q/QClOl) ,R(101) ,S000)C11IF(Z.LT.X(1)) GO TO 30Appendix F. List Of Computer Programs 198IF(Z.GE.X(NM)) GO TO 203 = NM10 K(I+J)/2IF(Z.LT.X(K)) 3 = KIF(Z.GE.X(K)) I = KIF(J.EQ.I+1) GO TO 30GO TO 1020 INM30 OX=Z—x(I)FS = Y(I) + OX * (Q(I) + OX * (RU) + ox * 5(I)))RETURNENOC *************************************DOUBLE PRECISION FUNCTION FCOMPC(V)C *************************************CC This function is used to calculate capillary hydrostaticC pressure at given volume.C-IMPLICIT REAL*B(A-H,O-Z)CCOMMON/BLKF/PC,PCNRM , PCGRAOCOMMON/BLKJ/VTNRM ,VIFNRM,VPLNRMCFCOMPC = PCNRN + PCGRAO * (V - VPLNRM)IF(FCOMPC.LT.2.O0) FCOMPC = 2.00RETURNENDC ***************************************************SUBROUTINE RK4C(F,M,A,B ,YA,EPS,YB ,NFUN,FLAG, IPET)C ***************************************************CIMPLICIT REAL*B(A-H,K,O-Z)INTEGER FLAGDIMENSION YA(M) ,YI(lO),YB(N),YBOLO(iO),YARG1(10),yARG2(10)DIMENSION K1(1G) ,K2(10) ,K3(iO) ,K4(10)CFLAG1Appendix F. List Of Computer Programs 199NINT=1NFUN=010 NFUN=NFUN+4*NINT*MDfl (B-A) /NINTXIADO 20 11,M20 YI(I)=YA(I)00 70 INT1,NINT00 30 11,MK1(I)=Dx*Fa ,XI ,YI,IPET)30 YARG1(I)=YI(I)+K1(I)/2.O0XARG=XI+0X/2 .0000 40 11,MK2(I)=DX*F(I ,XARG,YARG1 ,IPET)40 YARG2(I)=YI(I)+K2(I)/2.D000 80 11,MK3 (I) 0X*F (I ,XARG,YARG2 ,IPET)50 YARG1(I)=YI(I)+K3(I)xI=xI+0x0060 11,MK4w=ox*Fa ,XI ,YARG1 ,IPET)60 YI(I)YI(I)+(K1(I)+2.O0*(K2(I)+K3(I+K4(I))/6.oo70 CONTINUE00 80 11,M80 YB(I)=YI(I)IF(NINT.EQ.1) GO TO 100YBOIFM=0 .0000 90 1=1,1190 YBOIFM=OMAX1(DABS((YB(I)—YBOJ.D(I))/YB(I)),yBDIfl4)IF (YNOIFM . LT. EPS) RETURNIF(NINT.GT.10000) GO TO 120100 DO 110 I=1,M110 YBOLO(I)=YB(I)NINT2*NINTGO TO 10120 FLAG=0RETURNENOAppendix F. List Of Computer Programs 200C **********************************************SUBROUTINE GAUSS(A,N,NOR,NOC,X,RNORM, IERROR)CCC Purpose:C Uses Gauss elimination with partial pivot selection toC solve simultaneous linear equations of form JA(*;X:;C:.CC Arguments:C A Augmented coefficient matrix containing all coefficientsC 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 Solution vector.C RNORM Measure of size of residual vector ;C:-]A(*;X:.C IERROR Error flag.C 1 Successful Gauss elimination.C =2 Zero diagonal entry after pivot selection.- -CIMPLICIT REAL*B(A-H ,O-Z)OIMENSION A(NDR,NOC) ,x(N) ,B(SO,51)CNM = N - 1NP = N + iCC Set up working matrix BC00 20 I = 1,N00 10 .3 = 1,NPB(I.3) = A(I,J)10 CONTINUE20 CONTINUECC Carry out elimination process N-i timesC00 80 IC = i,NNKP = K + 1CAppendix F. List Of Computer Programs 201C Search for largest coefficient in column K, rows K through NC IPIVOT is the row index of the largest coefficientCBIG = 0.00DO 30 I = K,NSF = OABS(B(I,K))DO 26 3 = KP,NPSF = OMAX1(SF,OABS(BCJ,K)))26 CONTINUEAB = OABS(B(I,K) / SF)IF(AB.LE.BIG) GOTO 30BIG = ABIPIVOT = I30 CONTINUECC INTERCHANGE ROWS K AND IPIVOT IF IPIVOT.NE.KCIF(IPIVOT.EQ.K) GOTO 5000 40 3 = K,NPTEMP = B(IPIVOT,J)B(IPIVOT,J) = B(K,J)B(K,J) = TEMP40 CONTINUESO IF(B(K,K).EQ.O.DO) GOTO 130CC Eliminate BCI,K) from rows K+i through NC00 70 I = KP,NQUOT = B(I,K) / B(K,K)B(I,K) = 0.00DO 60 3 = KP,NP8(1,3) = 6(1,3) — QUOT * B(K,J)60 CONTINUE70 CONTINUE80 CONTINUEIF(B(N,N).EQ.0.O0) GOTO 130CC Back substitute to find solution vectorCAppendix F. List Of Computer Programs 202XCN) = BCN,NP) / B(N,N)DO 100 II = 1,NMSUM = 0.00I = N - IIIP = I + 1DO 90 3 = IP,NSUM = SUM + B(I,J) * XCJ)90 CONTINUEXCI) = (B(I,NP) — SUM) / BCI,I)100 CONTINUECC Calculate norm of residual vector, C-A*XC Normal return with IERROR = 1CRSQ = 0.0000 120 I = l,NSUN = 0.00DO 110 .1 = 1,NSUM = SUM + ACI,J) * XCJ)110 dONTIUERSQ = RSQ + CDABS(ACI,NP)- SUM)) ** 2120 CONTINUERNORM = OSQRTCRSQ)IERROR = 1RETURNCC Abnormal return because of zero entry on diagonal, IERROR2C130 IERROR = 2CCC find the solution of a set of simultaneous RETURN RETURNENDC **********************************************SUBROUTINE ROOTCF,M,X,YOLD,EPS,ALPHA,Y,FLAG)C **********************************************CC This subroutine uses Newton’s method toAppendix F. List Of Computer Programs 203C find the solution of a set of simultaneousC non-linear equations.CIMPLICIT REAL*8(A-H,O-Z)INTEGER FLAGDIMENSION YOLD(M) ,Y(M),DY(1O),DELY(10),A(lo,11)CNP = N + 1DO 10 I 1,MY(I) = YOLD(I)DELY(I) = 1.0—6 * YOLO(I)10 CONTINUEDO 60 ITER = 1,100FLAG = 0DO 40 I 1,MDO 30 J = iMPIF(J.EQ.MP) GOTO 20y(J) = y(j) + DELY(3)FUP = F(I,X,Y)Y(J) = Y(J) - 2.00 * DELY(J)FDOWN = F(I,X,Y)y(j) = y(3) + DELY(J)A(I,J) = (FUP— FDOWN) / (2.00 * DELY(J))GOTO 3020 A(I,J) = —F(I,X,Y)30 CONTINUE40 CONTINUECALL GAUSS(A,M,10,11,DY,RNORM,IERROR)CFLAG 1DO 50 I = 1,MY(I) = YCI) + ALPHA * DY(I)IF(DABS(DY(I)).GT.EPS) FLAG = 050 CONTINUEIF(FLAG.EQ,1) RETURN60 CONTINUERETURNENDAppendix F. List Of Computer Programs 204C **********************SUBROUTINE AUXALT(Y)CCC This subroutine is used to calculate the system variables fromC compartmental fluid and protein contents.CIMPLICIT REAL*8(A-H,J ,K,L,O-Z)COMMON/BLKB/JFS , .315COMMON/BLKO/QPS , QLSCOMMON/BLKF/PC , PCNRM,PCGRAOCOMMON/B LKG/CPL ,CS, CAVSCOMNON/BLKH/JLSNRMCOMMON/BLKI/PIPL ,PISCOMMON/BLKK/VEXSCOMMON/BLKL/LSS ,KFS ,SIGS ,PSSCOMMON/BLKZB/QTOT ,VTOTCOMMON/BLKU/PSNRM ,PSCOMMON/BLKZA/PRESLOCOMMON/VQPL/VPL , QPLDIMENSION Y(i)Cqis = Y(l)*FALB(PIS)CS = QIS / Y(i)CAVS = QIS / (Y(i) - vExS)PS = FCOMPS(Y(Ij)JFS = KFS * (PC — PS - SIGS * (PIPL - PIS))JLS = JLSNRM + LSS * (PS— PSNRM)IF (.315. LT. JLSNRM) THENJLSJLSNBN* (P5-PRESLO) / (PSNRM—PRESLO)ENOIFULJLS1O .OO*JLSNRMIF(JLS . GT.ULJLS) JLSULJLSEP= (1.00— SIGS) * iFS / P55IF(EP.OT.SO.DO) THENQPS.TFS* Ci .00—5105) *CPLELSEIF (EP . LT. -50.00) THENQPS=JFS*(1 .D0-SIGS)*CAVSELSEIF(ABS(0.D0-EP).LT.l.O-B) THENAppendix F. List Of Computer Programs 205QPSPSS* (CPL-CAVS)ELSEQPS = JFS *(1.O0—SIGS)*(CPL—CAVS*EXP(—Ep))/(j.D0-EXP(-Ep))ENDIFQLS = .JLS * CSRETURNENDC *****************************************************************SUBROUTINE OESOLV(F,M,A,B,YO,EPS,HSTART,HMIN,HMAX,YB,FLAG,IPET)CCIMPLICIT REAL*B (A-H, O-Z)DIMENSION YO(M) ,YA(10) ,YB(M)EXTERNAL FINTEGER FLAGBNAB-AHOLD=HSTARTDO 10 I=1,M10 YA(I)=Y0(I)20 CALL RKF(F,M,X,YA,HOLD,YB,YOIF,IPET)GAZIMM . BD0* (EPS*HOLD/ (BMA*YDIF) ) **O .2500HNEW=GAMMA*MOLDIF(GAMMA.GE.1.D0) GO TO 30IF(HNEW.LT.HOLD/10.OO) HNEW=HOLD/10.IF(BNEW.LT.HMIN) GO TO 50HOLDHNEWGO TO 2030 IF (HNEW . GT .5. D0*HOLO) HNEW5 . O0*HOLDIF(NNEW . GT .HMAX) HNEWHMAXIF(X+HOLD.GE.B) GO TO 70XX+H OLDHOLDHNEWDO 40 11,M40 YA(I)=YB(I)GO TO 2050 FLAG=0BXAppendix F. List Of Computer Programs 206DO 60 11,M60 YB(I)YA(I)RETURN70 FLAG=1HSTART=HNEWHDLDB-XCALL RKF(F,M,X,YA,HOLD,YB,YDIF,IPET)RETURNENDCSUBROUTINE RKF(F,M,X,YOLD,H,YNEW,YDIFM,IPET)CCIMPLICIT 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)CDATA 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,MK1(I)3*F(I,X,YDLD,IPET)10 YARG1(I)YDLD(I)+C21*K1(I)XARG=X+C21*HDO 20 I=1,NK2(I)=H*F(I,XARG,YARG1, IPET)20 YARG2 (I) YDLD(I)+C32*K1 CI)+C33*K2 (I)XARG=X+C31*HAppendix F. List Of Computer Programs 20700 30 11,MK3(I)H*F(I,XARG,YARG2,IPET)30 YARG1(I)Y0L0(I)+C42*Ki(I)+C43*K2(I)+C44*K3(I)XARGX+C41*H00 40 I=1,MK4(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+H00 50 11,MK5(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*H00 60 11,M60 K6(I)=H*F(I,XARG,YARG1,IPET)YDIFMO .0000 70 11,MYNEW(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=YDIFRETURNENDC *************************************DOUBLE PRECISION FUNCTION FVPL(PCC)C *************************************CC This function is used to calculate the plasma volume at givenC capillary hydrostatic pressure.CIMPLICIT REAL*8(A-H,O-Z)COMMON/BLKF/PC,PCNRM,PCGRADCOMMON/BLKJ/VTNRN , VIFNRM, VPLNRMCFVPL = VPLNRM + (PCC - PCNRM)/PCGRAOIF(FVPL.LT.2372.O0) FVPL = 2372.00RETURNENDAppendix F. List Of Computer Programs 208F.3 Listing of program PATDYNC Filename: PATDYNC This progam is used to simulate the transient responses of theC microvascular exchsnge system after a single intravenous infusionC of human albumin. Unless specify, subroutines and functions calledC in this program are the same as those listed in Section F.2 withC the same names.CIMPLICIT REAL*8(A-H,K,L,J,O—Z)INTEGER FLAG,FLAG6,FLAG7,FLAGGCOMMON/BLKA2/CPLNRM ,CSNRM , CASNRMCOMMON/BLKB/JFS , JLSCCMMON/BLKD/QPS ,QLSCOMMON/3LKG/CPL,CS,CAVSCOMMON/5LKI/PIPL ,PISCOMMON/BLKF/PC ,PCNRM,PCGRADCOMMON/BLKH/LSNRNCOMMON/BLKJ/VTNRN ,VIFNRM , VPLNRMCOMMON/BLKK/VEXSCOMMON/BLKL/LSS , KFS , SIGS ,PSSCOMMON/BLKT/QTNRM, QSNRM , QPLNRMCOMMON/BLKU/PSNRM ,PSCOMMON/BLKBB/PIPNRM ,PISNRMCOMMON/BLKDD/ALBSTOCOMMON/BLKZB/QTOT , VTOTCOMMON/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,RHSFACDATA ACC,ALPHA,EPS,DT,NP,NV/i,O-6,.500,i.000,O.OSOO,769,4/FLAG7 = 0FLAGS = 0TSTART1 .0-1*0 .00500TMIN=l .0-4*0.00500Appendix F. List Of Computer Programs 209TMAXDTCC Set LSS, SIGS, and PCNRM paremeter values.CLSS = 43.08086800SIGS = 0.9887763700PCNRM = 11.000CALL SPLINSCALL COEFFCC Set initial fluid volumes end protein contents.CPLINIT = 10.57400PIINIT = 3.30400VIFINT = 18250.00VPINIT = VPLNRMQPINIT = VPLNRM * FALB(PLINIT)VSINIT = VIFINTQSINIT = FALB(PIINIT) * (VIFINT— VEXS)VTINIT = VPINIT + VSINITQTINIT = QPINIT + QSINITYNEW(1) = VPINITYNEW(2) = QPINITYNEW(3) = VSINITYNEW(4) = QSINITCC Solve differential balances.C00 30 I = 1,1940TIM = (I-1)*OTTI = I * DTIF(I.EQ,1) GOTO 14CALL DESOLV(RHSF,NV,TIM,TI,YOL0,Ep5,TSTP.RT,TMIN,TMAX,+ YNEW,FLAG)IF(FLAG.EQ,0) GOTO 33CALL AUXSAM(YNEW)CC Store results at select time intervals.CAppendix F. List Of Computer Programs 21014 IF(TI.LT.3.DO.OR.TI.EQ.3.000)THENIK=(I—1)/4 +1REALI = IRI = (REALI— 1.00)14.00 + 1.00RIK 1KCHK = RI- RIKELSEIK=(I-i)120+ 13REALI = IRI = (REALI—l.O0)/20.O0 + 13,00RIK = 1KCHK = RI - RIKENOIFIF(CHK.NE.0.O0) GOTO 5615 CALL AUXSAN(YNEW)T(IK)0T*(I—1)SOLN(1,IK) = YNEW(1)SOLN(2,IK) = YNEW(2)SDLN(3,IK) = PC -SOLN(4,IK) = PIPLSOLN(5,IK) = CPLSOLN(6,IK) = PISSOLN(T,IK) = CSSOLN(8,IK) = CAVSSOLN(9,IK) = PSSOLN(1O,IK) = .TFSSOLN(11,IK) = JLSSOLN(12,IK) = QPSSOLN(14,IK) = QLSSOLN(15,IK) = YNEW(3)SOLN(16,IK) = YNEW(4)56 DO 20 J = 1,NVYOLO(J) = YNEW(J)20 CONTINUE30 CONTINUECC Calculate final steady-state variable values and store them.Appendix F. List Of Computer Programs 211VTOT = 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- YFINAL(1)SOLF(2) = QTOT- YFINAL(2)SOLF(3) = PCSCLF(4) = PIPLSOLF(S) = CPLSOLF(6) = PISSCLF(7) = CSSOLF(8) = CAVSSOLF(9) PSSOLF(10) = JFSSOLF(11) = .TLSSOLF(12) = QPSSOLF(14) = QLSSOLF(1S) = YFINAL(1)SCLF(16) = YFINAL(2)CC Output variables stored and associated times.C00 112 1=1,109WRITE(6,91)T(I) ,SOLN(4,I),SOLN(6,I) ,SOLNC3,I) ,SOLNC9,I)91 FORMAT(6X,F6.3,SX,F7.4,5X,F7.4,SX,F7.4,5X,F7.4)112 CONTINUEWRITE(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=1,109WRITE(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 CONTINUEWRITE(6,45)SOLF(15) ,SOLF(1) ,SOLF(16) ,SOLF(2)45 FORNAT(17X,F10.3,5X,F7.2,SX,F7.3,5X,F7,3)00 113 1=1,109WRITE(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)Appendix F. List Of Computer Programs 212113 CONTINUEWRITE(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=1,109WRITE(6,58)T(I),SOLN(5,I),SOLN(5,I),5OLN(7,I)68 FORMAT(6X,F6.3,5X,F7.S,5X,F7.5,5X,F7.5)117 CONTINUEWRITE(6,24)SOLF(6) ,SOLF(8) ,SOLF(7)24 FORMAT(17X,F7.5,5X,F7.5,6X,F7.5)STOPC33 WRITE(6,40)40 FORNAT(1X,’OOE SOLVER FAILS! ‘)STOPENDC ***************************************DOUBLE PRECISION FUNCTION RHSF(I,T,Y)C ***************************************C -- -C This function evaluates the non-linearC equations for a given set of Y values.CIMPLICIT REAL*8CA-H,J,K,L,O-Z)CONMON/BLKB/JFS ,ASCOMNON/BLKO/QPS , QLSDIMENSION YC4)CCALL AUXSAN(Y)IF(T.LT. 1.600)THENRHSF1 = JLS- .JFS + 200.00RHSF2 = QLS - QPS + 40.00*0.6500RHSF3 = JFS — JLSRHSF4 = QPS - QLSELSEIFCT.GE. 1.500.AND.T.LE.26.500)THENRHSF1 = JLS— JFS- (300.00/24.00)RHSF2 = QLS - QPS - (15.aoo/24.Do)*o.65D0RHSF3 = JFS- JLSRHSF4 = QPS - QLSAppendix F. List Of Computer Programs 213ELSEIF(T.GT.25.500.AND.T.LE.92.639D0)THENRHSF1 = JLS — iTSRUSF2 = QLS — QPS — (15.8D0/24.D0)*o.e500RHSF3 JFS— JLSRHSF4 = QPS — QLSELSERHSF1 = AS- JFSRHSF2 = QLS - QPSRHSF3 = IFS - ASRHSF4 = QPS - QLSENDIFGOTO(10,20,30,40),I10 RHSF = RHSF1RETURN20 RBSF = RHSF2RETURN30 RHSF = RHSF3RETURN40 RHSF = RHSF4RETURN -END

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