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A computer simulation of the pulmonary microvascular exchange system - alveolar flooding Heijmans, Franciscus R. C. 1985

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A COMPUTER SIMULATION OF THE PULMONARY MICROVASCULAR EXCHANGE SYSTEM - ALVEOLAR FLOODING By FRANCISCUS R. C. HEIJMANS B . E . S c , U n i v e r s i t y of Western O n t a r i o , London, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Chemical E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1985 © F r a n c i s c u s R.C. Heijmans, 1985 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f G>^P,\MA.tCjtld ETrY^V^tQ^^iynOj The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall ..'•*• Vancouver. Canada V6T 1Y3 Date \?J^I&g DE-6 (3/81) - i i -ABSTRACT Previous models of the pulmonary microvascular exchange system (28,29) have been r e s t r i c t e d to the study of f l u i d and solute exchange between the pulmonary m i c r o c i r c u l a t i o n , i n t e r s t i t i a l tissue space, and lymphatics. In severe pulmonary edema the capacities of the lymphatics and tissue space are exceeded. The f l u i d and solutes entering the i n t e r s t i t i u m from the c i r c u l a t i o n w i l l , then, be transported Into the a i r space. The accumulation of f l u i d i n the a i r space impairs the d i f f u s i o n of gas (oxygen and carbon dioxide) between the a i r space and blood c i r c u l a t i o n ; i f th i s f l u i d accumulation i s excessive a patient's health may be compromised. In this thesis severe pulmonary edema i s studied by including the a i r space as a fourth compartment into the i n t e r s t i t i a l model developed by Bert and Pinder (29). A computer simulation of the four compartment (alveolar) model was developed on a d i g i t a l computer. Tests of the model were made to study the e f f e c t of the parameters which were introduced into the alveolar model. These parameters include: a f i l t r a t i o n c o e f f i c i e n t that describes the alveolar membrane f l u i d conductivity, an extravascular f l u i d volume that represents the point at which f l u i d enters the a i r space, the alveolar f l u i d pressure at the onset of f l u i d flow into the a i r space, and the rate of alveolar f l u i d pressure - i i i -change r e l a t i v e to an alveolar f l u i d volume change. For each case the dynamic response of the exchange system was recorded. In addition, two types of pulmonary edema were simulated: 1) h y d r o s t a t i c a l l y induced edema, and 2) edema induced by changes to the f l u i d and solute permeability of the porous membrane separating the c i r c u l a t o r y and i n t e r s t i t i a l compartments. Due to the l i m i t e d data a v a i l a b l e on the i n t e r a c t i o n of the a i r space with the other three compartments of the pulmonary microvascular exchange system, only p a r t i a l v e r i f i c a t i o n of the appropriate range of values of the alveolar model parameters and the predictions of the simulations was possible. The alveolar model developed i n this thesis i s an i n i t i a l approximation but appears to provide a s a t i s f a c t o r y approach for the i n c l u s i o n of the a i r space i n the pulmonary microvascular exchange system. - i v -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENT . xiv 1. BACKGROUND 1 1.1 Introduction 1 1.2 The Circulation 3 1.2.1 The Pulmonary and Systemic Circulatory Systems 3 1.2.2 Physical Characteristics of Blood 5 1.2.3 The Pulmonary Circulation 7 1.3 The Vascular Membrane 13 1.3.1 Structure of the Vascular Membrane.... 13 1.3.2 Physical Properties of the Vascular Membrane 16 1.4 The Interstitium. 17 1.4.1 Structure and Composition of the Interstitium 17 1.4.2 Volume Exclusion 19 1.4.3 The Alveolar and Extra-alveolar Tissue Sub-compartments 21 1.5 The Lymphatics 26 1.5.1 Structure of the Lymphatics 26 1.5.2 Contractility and Pumping in Lymphatic Vessels 28 1.6 The Air Space 29 1.6.1 Arrangement of the Air Space 29 1.7 Barriers Between the I n t e r s t i t i a l Space and Air Space 32 1.7.1 Alveolar Membrane 32 1.7.2 Surfactant Lining the Wall of the Air Space 32 1.8 The Normal Fluid Pathways in the Pulmonary Microvascular Exchange System 36 1.9 Pulmonary Edema 36 1.9.1 Definition of Pulmonary Edema 36 1.9.2 Cli n i c a l Causes of Pulmonary Edema 38 1.9.2.1 Hydrostatic Pulmonary Edema 38 1.9.2.2 Permeability Pulmonary Edema 39 -v-TABLE OF CONTENTS (Cont.d) Page 2. INTERSTITIAL MODEL 42 2.1 Introduction 42 2.2 Modelling of the Pulmonary M i c r o c i r c u l a t i o n 43 2.3 Modelling of the I n t e r s t i t i u m 46 2.4 Modelling the Vascular Membrane 51 2.5 St a r l i n g ' s Hypothesis: Transendothelial F l u i d Flow 51 2.6 Kedem-Katchalsky Solute Flux Equation: Transendothelial Solute Flow 56 2.7 Modelling of the Lymphatics 60 2.8 F l u i d and Solute Material Balances 61 2.8.1 F l u i d Material Balance 61 2.8.2 Solute Material Balance 64 3. ALVEOLAR MODEL 65 3.1 Introduction 65 3.2 Modelling of the A i r Space 65 3.3 Modelling of the Alveolar Membrane 67 3.4 The Onset of Alveolar Flooding 68 3.5 Modelling of F l u i d and Solute Transport Across the Alveolar Membrane 71 3.6 Pressure-volume Relationship of the Alveolar F l u i d . .. 73 3.6.1 F l u i d Pressure-volume Relationship of an Individual Alveolus 73 3.6.2 F l u i d Pressure-volume Relationship for the A i r Space Compartment 77 3.7 Representation of the E p i t h e l i a l F i l t r a t i o n C o e f f i c i e n t , KAS 78 3.7.1 Representation of KAS as a Variable 78 3.7.2 Representation of KAS as a Constant 79 3.8 Integration of the A i r Space Compartment with the Pulmonary Microvascular Exchange Model... 80 4. COMPUTER SIMULATION OF ALVEOLAR MODEL 83 4.1 Introduction 83 4.2 The Computer Program 84 4.2.1 Input Data F i l e s EDA and PDA 84 4.2.2 The Main Program UBCEDEMA 89 4.2.3 Tabulated and Graphical Output of Variables from the Computer Program 92 4.3 C h a r a c t e r i s t i c Points Along the Transient Response 92 4.3.1 The Onset of Alveolar Flooding 92 4.3.2 The Point of Maximum T r a n s e p i t h e l i a l Flow.. 95 v i -TABLE OF CONTENTS (Cont.d) Page 4.3.3 The Point at which the Total Extravascular Fluid Volume equals 1000 ml.. 95 4.4 Outline of Simulations for the Alveolar Model 96 4.4.1 Simulations to study the Effect of the Parameters Introduced with the Alveolar Model: KAS, SL, VTONS and B 96 4.4.2 Simulations to Study the Effect of a Maximum Lymph Flow........ 97 4.4.3 Simulations to Study the Effect of the Endothelial Permeability Parameters, Endothelial F i l t r a t i o n Coefficient and the Circulatory Hydrostatic Pressure on the Alveolar Model 99 5. RESULTS AND DISCUSSION 102 5.1 Transient Responses of the PMVES for Constant KAS 102 5.2 Transient Responses of the PMVES for Variable KAS 109 5.2.1 Transient Response of the Pulmonary Microvascular Exchange System to changes in NK 117 5.3 The Response of the PMVES to changes in the Parameter VTONS 123 5.4 Transient Responses of the PMVES to changes in the Parameter SL 131 5.5 Responses of the PMVES to changes in the Parameter B 139 5.6 The Response of the PMVES to a Maximum Lymph Flow 151 5.7 The Responses of the PMVES to changes in PMV 157 5.8 The Response of the PMVES to changes in KF 162 5.9 The Response of the PMVES to changes inAPERM 169 5.10 Control of Error in the Numerical Solutions 177 SUMMARY AND CONCLUSIONS 179 RECOMMENDATIONS FOR FURTHER WORK 181 NOMENCLATURE 182 REFERENCES 185 - v i i -TABLE OF CONTENTS (Cont.d) Page APPENDIX A THE COMPUTER PROGRAM 190 A l . l Input Files EDA and PDA 190 A1.2 The Main Program UBCEDEMA 193 A1.3 Plotting Section of Main Program UBCEDEMA 211 Computer Program of Subroutine DLINE 219 Computer Program of Subroutine BBLINE 224 A1.4 Operation of Computer Program 227 - v i i i -L I S T OF TABLES Table Description Page 1. The Protein Composition of the Blood and Int e r s t i t i a l Fluid 8 2. Major Causes of Cardiogenic Pulmonary Edema 40 3. PPMV versus VIS1 for the Transition Region of the In t e r s t i t i a l Compliance Curve 52 4. Input Parameters to the In t e r s t i t i a l Computer Simulation for Normal Conditions 59 5. Content of Input Fil e EDA 85 6. Variables in Input Fil e PDA 87 7. Equations used in the Alveolar Model 90 8. Variables Tabulated and/or Plotted by Main Program UBCEDEMA 93 9. Characteristic Points along Transient Response and Variables Recorded 94 10. Simulations Conducted to Study the Effect of Parameters Introduced with the Alveolar Model 98 11. Simulations Conducted to Study the Effect of the Endothelial Permeability Parameters and Filtr a t i o n Coefficient and the Circulatory Hydrostatic Pressure on the Alveolar Model 101 12. Conditions of PMVES Simulations Conducted to Study Changes in KAS 102 13. Conditions of the PMVES Simulations for a Variable Epithelial F i l t r a t i o n Coefficient 110 14. I n i t i a l Conditions of PMVES/Simulation at a PMV of 50 mmHg I l l 15. Conditions of the PMVES Simulations Conducted to Study Changes in NK 119 16. Conditions of the PMVES Simulations to Study Changes in VTONS 124 - i x -L I S T OF TABLES (Cont'd.) Table Description Page 17. Conditions of the PMVES Simulations to Study Changes i n SL 133 18. Conditions of the PMVES Simulations Conducted to Study the Changes i n B 140 19. Conditions of the PMVES Simulations Conducted to Study the Changes i n the Maximum Lymph Flow 152 20. Conditions of the PMVES Simulations Conducted to Study Changes i n PMV 158 21. Conditions of the PMVES Simulations Conducted to Study Changes In KF 163 22. Conditions of the PMVES Simulations Conducted to Study Changes i n APERM 171 (Tables i n Appendix) 23. Example of Numerical Values Assigned to Parameters i n F i l e EDA 191 24. Example of Numerical Values Assigned to the Parameters i n F i l e PDA 192 25. Explanation of Subroutine DRKC 208 26. Explanation of Subroutine CMPLNC 210 27. Explanation of Subroutine AXIS 212 28. Explanation of Subroutine PLOT 213 29. Explanation of Subroutine PLOTND 214 30. Explanation of Subroutine LINE 215 31. Explanation of Subroutine DASHLN 216 32. Explanation of Subroutine PSYM 217 33. Explanation of Subroutine DLINE 218 34. Explanation of Subroutine BBLINE 223 35. Explanation of Subroutine NUMBER 225 36. Explanation of Subroutine LEGEND 226 -x-LIST OF FIGURES Figure Description Page 1. Schematic of the Pulmonary Microvascular Exchange System 2 2. Illustration of Pulmonary and Systemic Circulatory Systems 4 3. Composition of Blood Plasma and I n t e r s t i t i a l Fluid 6 4a. Branching Network of Circulation 9 4b. Pulmonary Circulation at Level of Alveolus 9 5a,b,c. Illustration of Zone Model as Developed by West, Dollery and Naimark 11 6. Illustration of Blood Flow at Different Heights of the Lung 14 7. Illustration of Vascular Membrane Separating Interstitium and Circulation 15 8. Cross-sectional View of Interstitium and Contents and Epithelial and Endothelial Membranes 18 9. Models of Volume Exclusion in the I n t e r s t i t i a l Space.... 20 10. Schematic of Pulmonary Microvascular Exchange System showing Sub-compartments of Interstitium and Circulation, and Surfactant Layer Lining Air Space 22 11. Compliance Curves of Alveolar and Extra-alveolar Tissue Sub-compartments 24 12. Structural Features of a Terminal Lymphatic 27 13a. Branching Network of Air Space 30 13b. Lobes, Lobules and Bronchopulmonary Segments of the Lungs 31 14a. Illustration of Gas Bubble Surrounded by Liquid 34 14b. Illustration of an Ideal Alveolus with Surfactant Lining 34 15. Tissue Compliance Curve used in Models 50 - x i -LIST OF FIGURES (Cont.d) Figure Description Page 16a. Schematic of the Three Compartment Interstial Model Illustrating Fluid Flows and Accumulation 63 16b. Schematic of the Three Compartment I n t e r s t i t i a l Model Illustrating Solute Flows and Accumulation 63 17a,b,c,d. Schematic Representation of Fluid F i l l i n g an Alveolus During Acute Pulmonary Edema 75 18a. Schematic of Four Compartment Alveolar Model Illustrating Fluid Flows 82 18b. Schematic of Four Compartment Alveolar Model Illustrating Solute Flows 82 19. Transient Responses of VIS1 for Constant KAS 104 20. Transient Responses of JAS for Constant KAS 105 21. Transient Responses of PAS for Constant KAS 107 22a. Transient Responses of QA, QG, QNETA, and QNETG for Variable KAS 112 22b. Transient Responses of JV, JL, JNET1, VIS1, PPMV, and PIPMV for Variable KAS 113 22c. Transient Responses of QA, QG, VIS1, CTA and CTG for Variable KAS 115 22d. Transient Responses of JAS, PAS, KAS and PPMV for Variable KAS 116 22e. Tranient Responses of JV, JL, JNET1 and JAS for Variable KAS 118 23a. Transient Responses of JAS for Changes in NK 120 23b. Transient Responses of VIS1 for Changes in NK 122 24. Steady State Values of VIS1 for given PMV for In t e r s t i t i a l Edema 125 25a. Transient Responses of (JV-JL) and JAS for VTONS of 420 ml and 500 ml 126 - x i i -LIST OF FIGURES (Cont.d) Figure Description Page 25b. Time to Reach a VTOT of 1000 ml for Different VTONS 128 25c. The Maximum Transepithelial Flow and the JAS at a VTOT of 1000 ml for Different VTONS 129 25d. The Fluid Volume VIS1 at a VTOT of 1000 ml for Different VTONS 130 26a. Transient Responses of JAS for Different SL and NK = 0.05 h r - 1 mmHg-1 134 26b. Transient Responses of PAS for Different SL and NK = 0.05 h r - 1 mmHg-1 135 26c. Transient Responses of VIS1 for Different SL and NK = 0.05 h r - 1 mmHg-1 136 27a. Transient Respones of PAS for Different SL and NK = 50.0 h r - 1 mmHg-1 137 27b. Transient Responses of JAS for Different SL and NK = 50.0 h r - 1 mmHg-1 138 28a. The Maximum Transepithelial Flow and JAS at a VTOT of 1000 ml for Different B 141 28b. Transient Responses of (PPMV-PAS) for B = -10 mmHg and B = -6 mmHg 143 28c. Transient Responses of KAS for B = -10 mmHg and B = -6 mmHg 144 28d. Time to Reach a VTOT of 1000 ml for Different B 145 28e. Transient Responses of (PPMV-PAS) for B = -3 mmHg and B = 0.5 mmHg 146 28f. Transient Responses of KAS for B = -3 mmHg and B = 0.5 mmHg 147 28g. Fluid Volume VIS1 at a VTOT of 1000 ml for Different B 149 29a. Transient Responses of JL for Different JL(max) 153 - x i i i -L I S T OF FIGURES (Cont.d) Figure Description ' Page 29b. Transient Responses of (JV-JL) for Different JL(max) 155 29c. Time to Reach a VTOT of 1000 ml for Different JL(max) 156 30a. Transient Response of (JV-JL) and JAS for Different PMV 159 30b. Time to Reach the Onset of Alveolar Flooding for Different PMV 160 30c. Time to Reach a VTOT of 1000 ml for Different PMV 161 31a. Transient Responses of (JV-JL) and JAS for Different KF 165 31b. Time to Reach a VTOT of 1000 ml for Different KF 166 31c. Time to Reach the Onset of Alveolar Flooding for Different KF 167 31d. Fluid Volume VIS1 at a VTOT of 1000 ml for Different KF 168 32a. Transient Responses of Albumin Protein Concentration CAVA for different APERM 172 32b. Transient Responses of Albumin Protein Weight QA for Different APERM 174 32c. Transient Responses of Fluid Volume VIS1 for Different APERM 175 32d. Time to Reach a VTOT of 1000 ml for Different APERM.... 178 - x i v -ACKNOWLEDGEMENTS I would l i k e to thank my supervisors, Dr. J.L. Bert and Dr. K.L. Pinder, for t h e i r guidance throughout the course of th i s work. Thanks are also due to Dr. P. Pare* and Dr. P. Dodek of the Pulmonary Research Unit at St. Paul's Hospital, who provided the medical perspective i n th i s work. The manuscript was typed by Mrs. M. Woschee and Mrs. K. L e s l i e , whose work i s much appreciated. The computer subroutines DLINE and BBLINE were generously provided by Dr. B. Bowen. F i n a l l y , I would l i k e to thank the Natural Sciences and Engineering Research Council of Canada for t h e i r f i n a n c i a l support. -I -BACKGROUND 1.1 Introduction Biological systems, because of their complex interactions and non-linear nature, are sometimes studied by a combination of experimental models and computer simulations. In this thesis the behaviour of the f lu id and solute exchange system of the lungs is investigated by means of a computer simulation. Figure 1 shows a schematic of the pulmonary microvascular exchange system (PMVES). It involves the exchange of f lu id and solute between the main compartments of the lung, that i s , the blood c i rculat ion, air space, tissue space (or interst i t ium), and lymphatics. The term microvascular refers to the circulat ing system of the small blood vessels. The i n t e r s t i t i a l compartment interacts direct ly with the other three compartments, which are assumed not to interact direct ly with each other. Separating the compartments are the vascular membrane - between the circulat ion and interstit ium - the alveolar barriers - between the interstit ium and air space - and the lymphatic capi l lary membrane - between the interstit ium and lymph channel. The pulmonary microvascular exchange system operates in conjunction with other physiological and biochemical systems to carry out the functions of the lung. The primary function of the lung is that of gas exchange, involving the countercurrent transfer of oxygen and carbon dioxide between the blood and air space. -2-Figure 1: Schematic of the Pulmonary Microvascular Exchange System: I l l u s t r a t e s Fluid-flow pathways (50) - 3 -The following sections discuss the lung compartments, t h e i r components, and the bar r i e r s to flow of the pulmonary microvascular exchange system. A section o u t l i n i n g the exchange of f l u i d and solute between the compartments and the c l i n i c a l importance of studying pulmonary f l u i d and solute exchange w i l l also be presented. 1.2 The Circulation 1.2.1 The Pulmonary and Systemic Circulatory Systems C i r c u l a t i o n of blood i n the body takes place i n the pulmonary and systemic vascular systems. Figure 2 i l l u s t r a t e s these two systems. The systemic c i r c u l a t i o n refers to the vascular network of the whole body where the blood i s deoxygenated, while the pulmonary c i r c u l a t i o n i s confined to the lungs where the blood i s oxygenated. The heart combines the two c i r c u l a t o r y systems to form a loop of continuous blood flow. The heart i s composed of two halves with each half containing two chambers - an atrium and a v e n t r i c l e . The r i g h t atrium of the right heart receives blood from the systemic c i r c u l a t i o n . The right v e n t r i c l e then pumps the blood Into the pulmonary vasculature. The oxygenated blood of the pulmonary system enters the l e f t atrium of the l e f t heart and i s subsequently pumped through the systemic c i r c u l a t i o n by the l e f t v e n t r i c l e . The primary function of the systemic c i r c u l a t i o n i s to transport nutrients and oxygen to the body tissues, and remove the wastes, such as carbon dioxide. -4-Figure 2: I l l u s t r a t i o n of Pulmonary and Systemic C i r c u l a t o r y Systems (54) SYSTEMIC CIRCULATION - 5 -The pulmonary microvasculature (a network of small blood v e s s e l s ) , while supplying nutrients to the surrounding tissue, i s also in intimate contact with the a i r spaces of the lungs. One of the purposes of this unique arrangement i s to exchange the carbon dioxide of the incoming (oxygen d e f i c i e n t ) blood for the oxygen present i n the a i r space. This thesis i s concerned with the exchange of f l u i d and solute i n the pulmonary microvasculature. 1.2.2 Physical Characteristics of Blood C i r c u l a t i n g through the vascular systems i s blood: "a viscous f l u i d composed of c e l l s and plasma" ( 1 ) . The blood c e l l s may be divided Into two classes: the red blood c e l l s which make up 99% of the c e l l s , and the white blood c e l l s . The primary function of these red blood c e l l s Is to transport hemoglobin, which i n turn c a r r i e s oxygen from the lungs to the tissues ( 1 ) . The plasma medium of the blood i s almost I d e n t i c a l i n e l e c t r o l y t e composition to the lung i n t e r s t i t i a l f l u i d . The watery f l u i d contains e l e c t r o l y t e s , nonelectrolytes and proteins. Figure 3 i l l u s t r a t e s the composition of plasma and i n t e r s t i t i a l f l u i d . The e l e c t r o l y t e s Include the cations - sodium, potassium, calcium, and magnesium - and the anions - chloride, bicarbonate, phosphate, s u l f a t e , and organic acid ions. Blood plasma and lung i n t e r s t i t i a l f l u i d d i f f e r from each other i n t h e i r protein content which i s greater i n the blood plasma. Three major classes of proteins e x i s t , based on - 6 -Figure 3 : Composition of Blood Plasma and I n t e r s t i t i a l F l u i d (1) EXTRACELLULAR FLUID I N T E R S T I T I A L F L U I D -7-their molecular weights and functions; these classes are albumins, globulins and fibrinogen. The protein composition of blood plasma and lung i n t e r s t i t i a l f l u i d are given in Table 1. The content of the globulins and albumin exceeds that of fibrinogen. The primary function of albumin is to cause a colloid osmotic pressure at c e l l membranes (1). 1.2.3 The Pulmonary Circulation The pulmonary circulation is composed of a branching network of blood vessels, as shown in Figure 4. Blood that enters the right heart is pumped through the pulmonary artery. The pulmonary artery branches Into vessels of smaller diameter, eventually reaching the size of the arterioles - less than 100 u in diameter - and then the capillaries - approximately 7 u in diameter. At the level of the capillaries oxygen and carbon dioxide are exchanged between the blood in the capillaries and the gas of the air space. The capillary surface area available for exchange is 50 to 70 m2, while that of the air space Is about 20% larger (2). The capillaries then unite to form the venules - less than 200 u in diameter. The venules combine further until they form the pulmonary vein which delivers the blood to the atrium of the le f t heart. The vessels of the microvasculature are thought to be composed of a r t e r i a l , capillary, and venous segments. As blood flows from the arte r i a l to the capillary and then to the venous segments of the vessel, the fluid hydrostatic pressure decreases. The longitudinal - 8 -Table 1 Table 1: The Protein Composition of the Blood and In t e r s t i t i a l Fluid (29) Protein Blood I n t e r s t i t i a l Fluid (g/ml) (g/ml) Albumin .042 .025 Globulins .027 .011 Fibrinogen(55) 0.003 N.A. Total Protein 0.073 N.A. - not available - 9 -F i g u r e 4 a) B r a n c h i n g Network of C i r c u l a t i o n (54) C a p i l l a r i e s A r t e r i e s A r t e r i o l e s V e n u l e s V e i n s -10-pressure p r o f i l e of the blood vessel may be simply i l l u s t r a t e d by assigning a hydrostatic pressure to the f l u i d of each vascular segment: PA for the a r t e r i a l end, PMV for the c a p i l l a r y , and PV for the venous end. The pressures i n descending order of value are PA > PMV > PV. The c a p i l l a r y segment i s i n close contact with the a i r space, while the a r t e r i a l and venous ends are less so; the diagrams of Figure 5a show the lo c a t i o n of the three segments i n r e l a t i o n to the a i r space. West, Dollery, and Naimark ( 3 ) proposed a Zone Model for the lungs, based on measurements of the r e l a t i v e values of PA, PV and the hydrostatic pressure of the gas in the a i r space PALV. In zone I, the a i r space pressure i s greater than the a r t e r i a l and venous pressures: PALV > PA > PV (1) Figure 5a i l l u s t r a t e s zone I. Under these conditions the c a p i l l a r y segment of the blood vessel w i l l be collapsed; the c a p i l l a r y f l u i d pressure, PMV, w i l l assume a value approximately equal to PALV. Therefore blood w i l l not flow through the c a p i l l a r y since the pressure gradient that causes flow (PA - PALV) i s negative. In zone II the a i r space pressure Is between the a r t e r i a l and venous pressures: PA > PALV > PV ( 2 ) Figure 5b ( i ) shows the case when PALV i s much larger than PV. At the downstream end of the c a p i l l a r y the c a p i l l a r y pressure w i l l assume a value approximately equal to PALV. The pressure gradient causing -11-Figure 5 : I l l u s t r a t i o n of Zone Model as Developed by West Dollery and Naimark ( 3 ) (PA=arterial pressure PV=venous pressure; PALV=alveolar pressure) a) Zone I: PALV > PA > PV • c a p i l l a r y collapsed Arteriole V e n i c e b) Zone I I : PA > PALV > PV •c o n s t r i c t i o n at downstream end of vessel PA(1) p v d : PA(1)< PA(2) PVC 1)( PV(2) PA(2) c) Zone I I I : PA > PV > PALV •vessel held open -12-blood flow through the c a p i l l a r y becomes (PA - PALV). The c a p i l l a r y w i l l be constricted at the downstream end. If PA and PV are elevated, such that PV approaches PALV, the condition shown i n Figure 5b ( i i ) r e s u l t s . The pressure gradient causing blood flow w i l l remain (PA -PALV) since PMV i s approximately equal to PALV, but the c o n s t r i c t i o n moves further downstream. The f i n a l zone, zone I I I , has the following r e l a t i o n s h i p : PA > PV > PALV ( 3 ) The pressure gradient causing blood flow now becomes (PA - PV). The c a p i l l a r y f l u i d pressure, PMV, w i l l be between PA and PV, and therefore exceed PALV. The greater PVM i s above PALV the more the vessel w i l l distend. Figure 5c shows the vessel under zone III conditions. West, Dollery and Naimark ( 3 ) have suggested that the human lung may be h o r i z o n t a l l y divided into zones. The blood c i r c u l a t i o n i s affected by g r a v i t y . In the upper section of the lung under normal conditions zone I conditions p r e v a i l ; there i s no blood flow through the vessels. Zone II conditions p r e v a i l i n the middle section of the lung. At the junction of zone I and zone II PALV equals PA. At h o r i z o n t a l positions of the lung lower than this junction of zones I and II the hydrostatic pressure of the blood r i s e s due to grav i t y . Therefore, at v e r t i c a l positions lower than the junction of zones I and II PA and PV increase; (PA - PALV) increases and (PALV - PV) decreases. The blood flow increases as the d r i v i n g force causing flow -13-(PA - PALV) increases. When PV equals PALV there i s no c i r c u l a t o r y c o n s t r i c t i o n and the pressure gradient causing flow becomes (PA - PV). Zone III conditions p r e v a i l i n the lower section of the lung. At lower positions i n zone III the blood flow r i s e s due to the distension of the vessels. Figure 6 i l l u s t r a t e s the Zone Model, showing how blood flow changes at d i f f e r e n t positions i n the lung. 1.3 The Vascular Membrane 1.3.1 Structure of the Vascular Membrane The vascular membrane, shown i n Figure 7 , separates the blood c i r c u l a t i o n from the i n t e r s t i t i u m . The membrane i s composed of an endothelial c e l l layer that i s i n contact with the c i r c u l a t i o n , and an underlying basement membrane. "Whenever the endothelial c e l l s come i n t o contact, there are junctions that are continuous along the l i n e of contact. These vary from " t i g h t " junctions to "gap" junctions, which themselves vary from the tight to the "leaky" type, the p a r t i c u l a r condition depending on the width of the junction." (4) Wissig and Charonis ( 5 ) suggest that the basement membrane functions as a "supportive role for the endothelial l i n i n g of the c a p i l l a r i e s " . The basement membrane may " s t a b i l i z e the wall, a s s i s t i n g i t to withstand the hydrostatic pressure of the perfusing blood". - 1 4 -F i g u r e 6: I l l u s t r a t i o n of B l o o d Flow a t D i f f e r e n t H e i g h t s o f t h e Lung (Pa = a r t e r i a l p r e s s u r e , P = A i r P r e s s u r e , P = venous p r e s s u r e ) ( 3 ) > zone I PA > Po ) P» zone 2 PQ >P A> Py Distance zone 3 D ) D > D 'a 'Y ' A 3iood flow -15-Figure 7: I l l u s t r a t i o n of vascular membrane separating I n t e r s t i t i u m and C i r c u l a t i o n : Values of hydrostatic (PMV, PPMV) and oncotic (PIMV, PIPMV) pressures at normal conditions. . - B A S E M E N T M E M B R A N E C A P I L L A R Y W A L L K F = 1 . 1 2 m l / h r / m m H g S I G D = 0 . 7 5 P S A = 3.0 P S G = 1 . 0 m l / h r S I G F A = 0.4 S I G F G =0.6 -16-1.3*2 Physical Properties of the Vascular Membrane Studies of the permeability of the vascular membrane to fluid and solute rarely differentiate between the permeabilities of the endothelial c e l l layer and the basement membrane. In the remainder of this thesis the c e l l layer and basement membrane w i l l be lumped together, and referred to as either the vascular or endothelial membrane. The vascular membrane of the pulmonary microvasculature is classified as being "leaky" (6); fluid and solute are transported across the membrane, generally from the circulation to the interstitium. Among the factors that determine the a b i l i t y of a solute to cross the endothelium are the solute size, electrical charge of the solute (2), distribution of the size of the membrane openings and the population of these openings. In the case of globulins and albumin - with an average radius of about 5.2 and 3.7 nm, respectively - a larger fraction of albumins cross the vascular membrane as compared to globulins. The fraction of solute crossing the vascular membrane may be obtained experimentally by comparing the solute concentration in the interstitium to that of the plasma. The solute concentration of the interstitium is generally approximated by the solute concentration in lung lymph, since i n t e r s t i t i a l samples are d i f f i c u l t to obtain experimentally. Yoffey and Courtice (7) have obtained estimates of the concentration ratios: the ratio of the i n t e r s t i t i a l concentration of albumin to the plasma albumin concentration was 0.80, while the -17-i n t e r s t i t i a l - t o - p l a s m a concentration r a t i o for glo b u l i n was 0.55. These re s u l t s indicate that a greater f r a c t i o n of plasma albumin crosses the vascular membrane than plasma gl o b u l i n s . 1.4 The Interstitium 1.4.1 Structure and Composition of the Interstitium The i n t e r s t i a l space of the lung i s that space which i s between the vascular membrane and the alveolar membrane. The components of the i n t e r s t i t i u m are numerous. Figure 8 shows a section of the i n t e r s t i t i u m . The basic structure involves a "skeleton of collagen f i b r e s " ; the space between the f i b r e s i s an i n t e r s t i t i a l matrix of water, s a l t s , plasma proteins and glycosaminoglycans (8). _ The collagen f i b r e s range i n diameter from 0.5 u to 1.0 u. The actual arrangement of the f i b r e s i n tissue i s not well known. Bert and Pearce (9) state that the f i b r e s i n many tissues are organized into bundles of p a r a l l e l f i b r e s ; these "bundles may be assembled into highly ordered arrays, as i n cornea, or Into f e l t - l i k e mats randomly oriented p a r a l l e l to the surface, as i n dermis" (10). "The main fu n c t i o n a l e f f e c t s of the collagen f i b r e s are that they r e s i s t changes i n tissue configuration and volume, they exclude proteins, and they immobilize the glycosaminoglycans (see below) of the i n t e r s t i t i a l matrix" (8). The i n t e r s t i t i a l matrix may be described with two major phases i n the i n t e r s t i t i u m : one a free f l u i d and the other a complex gel structure. The free f l u i d phase i s comparable to the plasma f l u i d of the blood, except that the i n t e r s t i t i a l protein concentration d i f f e r s . -18-Figure 8: Cross-sectional View of In t e r s t i t i u m and Contents, E p i t h e l i a l and Endothelial Membranes (RBC-red blood c e l l ; WBC-white blood cell) ( 5 5 ) AIR -19-The gel phase is composed of macromolecules called glycosaminoglycans or GAGs. The primary GAG is hyaluronic acid. Hyaluronic acid is an unbranched, random c o i l macromolecule. In tissue, hyaluronic acid occupies a volume 1000 times that of its own unhydrated structure (8). Other GAGs, such as chondroitin sulphate, are present in smaller quantity; they are attached as side branches to a protein backbone, forming proteoglycans. The GAGs show noticeable "internal entanglement" (8). The collagen fibres are entangled with the GAGs, and immobilize the mass. Thus, a gel-like structure results which exhibits solid-like behaviour (9). 1.4.2 Volume Exclusion "Volume exclusion refers to a property of matter that prevents two materials from occupying the same space at the same time" (10). The interwoven latticework of the fibres, primarily collagen, in conjunction with hyaluronate, contribute to the exclusion phenomenon by limiting the "extravascular-extracellular space accessible to a plasma protein" (9). Therefore, the volume of the i n t e r s t i t i a l space available to proteins is reduced, which results in an increase in the effective protein concentration. The exclusion phenomenon may be explained by: 1) the rod and sphere model, and 2) the sphere in a random network of rods model. In the former model, shown in Figure 9a, the rod represents an isolated collagen bundle in the tissue while the protein is approximated by the sphere. The protein w i l l be excluded from the volume occupied by the rod, and the volume that is one protein radius from the rod - shown by -20-Figure 9: Models of Volume Exclusion i n the I n t e r s t i t i a l Space (9) a) Rod and Sphere Model (thatched area i s excluded space) b) Sphere i n a Random Network of Rods Model (thatched area excluded space) -21-the shaded area in the side and front view given in Figure 9a. The collagen bundles may be arranged into f e l t - l i k e mats, with the bundles randomly oriented. In this case the bundles may be represented as a random network of rods, hence, the sphere in a random network of rods model. Figure 9b shows the sphere in a random network of rods model -the centre of the sphere (protein) i s excluded from the shaded area. In both models the excluded space (volume) is dependent on the size of the protein. Thus In either model, globulins, with an average diameter of 5.2 nm w i l l be excluded from a larger volume than albumins having an average diameter of 3.7 nm. The effective concentrations of the proteins may be calculated from the i n t e r s t i t i a l protein weights and the volumes available to the proteins. The colloid osmotic pressure calculated from the effective protein concentration w i l l be greater than that determined from the protein concentration that does not account for the excluded volume of the proteins, i.e., in the latter case, the concentration uses the i n t e r s t i t i a l volume, while in the former case the volume Is the i n t e r s t i t i a l volume less the excluded volume. 1.4.3 The Alveolar and Extra-alveolar Tissue Sub-compartments The interstitium is subdivided into alveolar and extra-alveolar sub-compartments, as shown in Figure 10. The sub-compartments are distinguished by at least two factors: 1) the effect of lung inflation (or a rise in the hydrostatic pressure of the air space) on the tissue hydrostatic pressure, and 2) the tissue compliance curves of the two sub-compartments. - 2 2 -Figure 10: Schematic of Pulmonary Microvascular Exchange System showing sub-compartments of I n t e r s t i t i u m and C i r c u l a t i o n and surfactant layer l i n i n g a i r space (PALV-alveolar (gas) pressure; PAS=alveolar f l u i d pressure; PPMV=pressure of al v e o l a r i n t e r s t i t i a l f l u i d ; PEA=pressure of extra-alveolar i n t e r s t i t i a l pressure; PMV=microvascular pressure) (2) J V J V. Ly mphat ic u Ex t ra -a l veo la r I PAS A l v e o l i / / PALV UV. Lymphat ic PEA t ^ J A lveo la r J Ext ra-a lveo lar I n t e r s t i t i u m i Interst i t i u m P P M V i I n t e r s t i t i u m ! i PMV E x t r a - a l v e o l a r A r t e r i o l e s A l v e o l a r W a l C a p i l i G r i e s , E x t r a - a l v e o l a r i V e n u l e s -23-Fi r s t , the alveolar tissue sub-compartment lies ln close proximity to the alveoli (or air sacs) of the air space. The extra-alveolar interstitium is not necessarily adjacent to the air space. The result of this arrangement is the different responses of the hydrostatic pressures of the sub-compartments to an increase in the air pressure of the air space. Lai-Fook (11), and Parker, Guyton and Taylor (12) have suggested that the alveolar tissue hydrostatic pressure (PPMV) assumes a value related to the air pressure (PALV). The air space is lined with a surfactant fl u i d that produces an alveolar tissue pressure slightly less than PALV (see section 1.7.2). The hydrostatic pressure of the extra-alevolar tissue (PEA) is less than the pressure of the alveolar tissue (13). Howell, et a l . (14), Hida, et a l . (15) and Goldberg (16) have observed experimentally that PEA decreases with a rise in PALV; reasons for this decrease have been related to the configuration of the air space and blood vessels (14), but generally remain unclear. The effect of a rise in the air pressure, PALV, i s , then, an equivalent rise in the alveolar tissue hydrostatic pressure, PPMV, and a drop in the extra-alveolar hydrostatic pressure, PEA. Second, the tissue compliance curves of the two subcompartments differ in shape and value. Tissue compliance is defined as the change in extravascular-extracellular (EVEA) flu i d volume relative to the change in tissue fl u i d pressure, i.e. (AV/AP). Prichard (17) gives hypothetical complicance curves for the alveolar and extra-alveolar tissue spaces, as shown in Figure 11. Figure 11 illustrates that for any tissue fl u i d volume less than point D the alveolar tissue - 2 4 -Figure 11: Compliance Curves of Alveolar and Extra-alveolar Tissue Sub-compartments - Alveolar Flooding Occurs When Point C' i s reached. (17) V O L U M E - 2 5 -hydrostatic pressure (PPMV) i s greater than the extra-alveolar tissue hydrostatic pressure (PEA). The pressure gradient that r e s u l t s w i l l cause f l u i d to drain p r e f e r e n t i a l l y from the alveolar tissue to the extra-alveolar t i s s u e . The tissue compliance for each sub-compartment varies as f l u i d accumulates i n the t i s s u e . Both hypothetical curves i n Figure 11 i l l u s t r a t e a low compliance (low AV/AP) for volumes less than that at point A on the extra-alveolar tissue compliance curve and point B^on the a l v e o l a r tissue compliance curve. In this low compliance region i t i s postulated (18) that the gel phase - of glycosaminoglycans (GAGs) - i s compressed, such that as the f l u i d volume r i s e s i n the low compliance region the gel i s returning to i t s neutral states. As f l u i d volume increases from points A and B^he compliance curves of the extra-alveolar and alveolar tissues, respectively, undergo a t r a n s i t i o n to a higher compliance. In the region of high compliance i t i s hypothesized (18) that the GAGs fragment as the tissue expands; the f l u i d volume r i s e s sharply with l i t t l e change i n the tissue f l u i d pressure. In the extra-alveolar tissue the high compliance proceeds u n t i l the f l u i d volume reaches point C, whereupon maximum expansion of tissue elements has occurred. The extra-alveolar tissue returns to a low compliance, and the r i s e i n tissue f l u i d pressure i s high for further increases i n f l u i d volume. The alveolar tissue continues to expand under a high compliance for f l u i d volumes beyond point B*. When the tissue elements are stretched to t h e i r upper l i m i t the f l u i d empties into the a i r space which has a large capacity; thus enabling the alveolar tissue compliance to remain high. -26-1.5 The Lymphatics 1.5.1 Structure of the Lymphatics The lymphatics provide drainage channels for the fl u i d that enters the interstitium. Figure 10 illustrates that the terminal lymphatic capillaries are situated in the extra-alveolar interstitium, but not the alveolar interstitium. The fluid and solute that is transported across the vascular membrane into the alveolar tissue space must, therefore, flow through the tissue space to reach the lymphatics; as noted in the previous section a pressure gradient from the alveolar to extra-alveolar interstitium causes this flow of f l u i d . The lymphatic capillary membrane is assumed to be extremely permeable to fluid and solute (19). The membrane is composed of an endothelial c e l l layer and an underlying basement membrane. The basement membrane is "poorly developed, and is commonly absent, or at best, discontinuous" (19) and therefore, does not provide significant resistance to solute and fluid flow. Figure 12 shows that anchoring filaments are attached to the cellular wall, and embedded in the surrounding tissue. When the perilymphatic tissue expands, due to f l u i d accumulation, the filaments maintain the cellular junctions open, and the vessel does not collapse (17,18). The tissue fluid is then able to continue flowing into the lymphatic vessel. As the fluid moves downstream through the lymph vessel i t is prevented from returning to the terminal lymphatic end by lymphatic valves (Figure 12). -27-Figure 12: S t r u c t r u a l Features of a Terminal Lymphatic (18) -28-1.5*2 Contractility and Pumping in Lymphatic Vessels To further f a c i l i t a t e the drainage of fluid and solutes there is a contractile and pumping motion of the lymphatic capillaries produced by muscular contractions, respiration, tissue movements, and an intrinsic pumping mechanism (17,19). The operation of the intrinsic pumping mechanism is not clearly understood. Guyton (18) suggested the following explanation of the mechanism: "As fluid accumulates in the tissue space i t expands the tissue. The anchoring filaments of the lymphatic capillaries, embedded in the tissue, become expanded along with the expansion of the tissues, thereby creating a suction Inside the lymphatic capillaries. This suction pulls fluid inward through the (openings) of the lymphatic capillary, the inward movement of flu i d causing the flap valves of the lymphatic membrane cells (see Figure 12) to open inward". Guyton then postulates that the f l u i d - f i l l e d lymphatic capillary w i l l be compressed by the fluid in the surrounding tissue, raising the pressure at the capillaries end, and forcing the fluid forward into the collecting lymphatic. The force that propels the fluid forward into the collecting lymphatic may alternatively be the respiration movements or muscular contractions• -29-1.6 The Air Space 1.6.1 Arrangement of the Air Space Figure 13a illustrates that the air space is composed of a branching network of air ducts (or airways). Air enters via the mouth or nose and passes through the trachea. The trachea continues to the hilum of the chest, whereupon the duct bifurcates into the l e f t and right main bronchial ducts (bronchi); the main bronchi lead to the l e f t and right lung units. The le f t and right lung units are composed of individual lung segments or lobes (Figure 13b). Lobar bronchi branch from the l e f t and right main bronchi to each lobe. Each lobar bronchi branches into terminal bronchioles and then respiratory bronchioles (Figure 13a). The respiratory bronchioles divide into alveolar ducts. At the periphery of the alveolar ducts are air sacs or alveoli. There are about 300 million alveoli, each having an equivalent diameter of 25 to 30 u (17). It is at the level of the alveolus that the gas exchange occurs between the air space and blood circulation. The major sites for fluid and solute exchange are located at the level of the respiratory units, that i s , the respiratory bronchiole, alveolar ducts and alveoli. - 3 0 -Figure 13b: Lobes, lobules and bronchopulmonary segments of the Lungs (54) -32-1.7 Barriers Between the Interstitial Space and Air Space 1.7.1 Alveolar Membrane The alveolar membrane is composed of a cellular layer and a basement membrane. The cells of the membrane, primarily of squamous type I, normally form continuous tight junctions. The membrane behaves functionally as though i t contained water-filled pores having a radius between 0.6 and 1.0 nm (6,20). Therefore the transport of water across the alveolar membrane is possible. Although water crosses the epithelium, i t does not accumulate in excess in the air space; there must be mechanisms to remove the water. The action of the surfactant lining the epithelium functions as one possible means to prevent fluid accumulation (see sections 1.7.2 and 3.6). Another mechanism to remove the water is evaporation to the inspired a i r . However, the inspired air becomes completely humidified in the upper airways; therefore, only a small amount of water w i l l be removed by evaporation (17). 1.7.2 Surfactant Lining the Wall of the Air Space Two secretions line the walls of the respiratory unit: the alveolar wall (of the alveoli) is covered with a surfactant, while th walls of the respiratory bronchiole are lined with the secretions of the mucous glands, goblet cel l s , and Clara cells (17). It is the surfactant lining of the alveolar wall that is believed to play an important role in the mechanics of the lung and i pulmonary microvascular exchange (21). The surfactant is a surface -33-active lipoprotein - primarily composed of dipalmitoly lecithin (DPL) - that has a low surface tension of 24-34 dynes/cm (21); water characteristically has a surface tension of 68 dynes/cm. The advantage of the lower surface tension is illustrated by the Laplace equation developed for a spherical gas bubble AP = PG - PL = — (4) r where PG = hydrostatic pressure of gas in bubble PL = liquid hydrostatic pressure x = surface tension r = radius of curvature Figure 14a shows the location of the pressures, PG and PL, and the radius of curvature, r. In the adjacent diagram, Figure 14b, a schematic of the alveolus is shown and the variables of the Laplace equation indicated. Assuming a constant radius of curvature, one can see from equation (4) that the lower the surface tension (x), the smaller the pressure differential (PG - PL). Therefore, in comparing the lung surfactant to water, the lower surface tension of DPL reduces the pressure differential, (PG - PL), needed in the alveolus to maintain a stable air sac. DPL is a cationic surfactant which has two structural features that aid in maintaining the alveoli virtually dry. F i r s t , DPL has a positive charge from the quaternary nitrogen ion that may firmly attach to the negative charges that abound In the alveolar membrane. These bonds may be compared to those between cationic surfactants and -34-Figure 14a: I l l u s t r a t i o n of Gas Bubble Surrounded by L i q u i d : To I l l u s t r a t e Variables of Laplace's Equation (PG=gas pressure; PL=liquid pressure; r=radius of curvature) Figure 14b: I l l u s t r a t i o n of an Ideal Alveolus with Surfactant Lining Alveolus A lveo lus Ep i the l i a l / M e m b r a n e . S u r f a c t a n t / / F l u i d ' P A S ( P L ) / ^ u r ^ a c < 2 t e n s i o n r - 3 5 -textiles. The surfactants applied to textiles have been shown to raise the "water entry pressure" to over 760 mm Hg, which is the pressure needed on the convex side of the liquid surface to break through the barrier. Although the DPL surfactant may only have a fraction of one percent efficiency in comparison to the textile surfactants, i t does serve as an "appreciable protection" against acute fluid accumulation in the air space (21). Second, DPL has "long hydrocarbon chains which are oriented outwards to provide a hydrophobic surface with a consequent lack of wettability" (21). Wettability is the abil i t y of a liquid to adhere to a surface - the greater the surface area a given volume of liquid covers, the greater the wettability. The lung surfactant has a low wettability; water is repelled by the surfactant. Thus liquid water is prevented from crossing the surfactant lining. The surfactant is produced by type II epithelial cells that are interspersed throughout the predominantly type I epithelial cellular membrane. Any minor breaks in the surfactant layer may be quickly eliminated by replenishing the surface with the newly-secreted surfactant. -36-1.8 The Normal Fluid Pathways In the Pulmonary Microvascular  Exchange System Figure 1 shows the arrangement of the pulmonary microvascular exchange system, and the normal pathways for fluid and solute. The endothelium is permeable to fluid and solutes. A net flow of fl u i d and solutes is generally observed to occur from the circulation to the interstitium. Once in the alveolar interstitium the fluid and solute flow along a pressure gradient to the extra-alveolar interstitium. The lymphatics drain the extra-alveolar tissue space. If steady state is achieved the flow of fl u i d and solutes across the endothelium is equivalent to the flow of fluid and solutes leaving the tissue space through the lymphatics. Under normal conditions fl u i d may traverse the epithelium between the interstitium and air space. However fluid entering the air space may be evaporated (a minor amount) or return to the interstitium. Therefore the net flow is negligible, and the air space is assumed to be virtually dry. 1.9 Pulmonary Edema 1.9.1 Definition of Pulmonary Edema Edema is defined as "a swelling due to an effusion of watery fluid into the intercellular spaces of connective tissue" (17). Prichard (17) presented a further definition of pulmonary edema: "a pathologic state in which there is abnormal water storage in the lungs". -37-Pulmonary edema may be i n i t i a t e d from the c i r c u l a t i o n compartment by elevated c i r c u l a t o r y f l u i d pressure and/or changes i n the permeability of the endothelium. A l t e r n a t i v e l y , edema may be induced by changes to the permeability of the epithelium. The following discussion on the sequence of events during pulmonary edema w i l l be for edema i n i t i a t e d from the c i r c u l a t i o n compartment. A r i s e i n the c i r c u l a t o r y hydrostatic pressure, or a change i n the permeability of the vascular membrane w i l l lead to increased f l u i d f i l t r a t i o n from the c i r c u l a t i o n to the i n t e r s t i t i u m . The blood c a p i l l a r y segment because of i t s larger surface area and higher f l u i d and solute permeability, i n comparison to the a r t e r i a l and venous segments, i s the major s i t e of f l u i d and solute transudation (22,23). The f l u i d and solute enter the alveolar i n t e r s t i t i u m , and flow towards the extra-alveolar tissue space along the pressure gradient between the two tissue sub-compartments. The lymphatics drain the f l u i d that flows into the extra-alveolar tissue space. As the f l u i d volume r i s e s i n the tissue space the lymph flow also r i s e s . If the edema i s moderate the lymph flow may equalize with the transendothelial flow, and a new steady state w i l l be established. However, i n severe pulmonary edema the transendothelial flow exceeds the flow capacity of the lymphatics. The f l u i d pressure i n the extra-alveolar tissue space r i s e s as f l u i d accumulates i n t h i s sub-compartment. This r e s u l t s i n a decreased alveolar to extra-alveolar tissue f l u i d flow, and an increase i n alveolar tissue f l u i d volume. Eventually the capacity of the i n t e r s t i t i u m i s exceeded and -38-breaks in the epithelium develop. Fluid may then flow from the circulation into the air space, as well as into the lymphatics. The location of the epithelial breaks is not well-known. Staub and Gee (24), and Zumsteg et a l . (23) suggest that the breaks are present at the level of the terminal bronchioles; these epithelial breaks are possibly present under normal conditions, but the pressure gradient does not normally favour i n t e r s t i t i a l to alveolar fl u i d transport. Egan (25) proposes that the breaks may develop between the normally tight junctions between the epithelial c e l l s . In both cases, the openings are not considered to be uniformly spread over the epithelium, but occur at a relatively few discrete sites. The protein flow into the air space occurs through these openings. The fl u i d accumulation in the air space is spotty and the alveoli f i l l under an a l l or nothing phenomena. This individual f i l l i n g is described in more detail in section 3.6. 1.9.2 Clinical Causes of Pulmonary Edema "Pulmonary edema may arise from a number of causes and in a wide variety of diseases" (17). In most cases the cause of edema w i l l be an increase in the pulmonary microvascular hydrostatic pressure and/or an increase in the fluid and solute permeability of the vascular and/or epithelial membranes. 1.9.2.1 Hydrostatic pulmonary edema. The elevation of the microvascular hydrostatic pressure raises the driving force that causes f l u i d flow from the circulation to the interstitium. If the high transmembrane flow of fluid overwhelms the -39-safety factors of the exchange system - that are trying to regulate the flow - then the progression of edema may proceed through the i n t e r s t i t i a l and alveolar stages. Hydrostatic edema may r e s u l t from upsets to the cardiac system. For example, i f the heart malfunctions, and the rate of f l u i d pumped from the right v e n t r i c a l Is greater than the rate of f l u i d pumped from the l e f t v e n t r i c a l , f l u i d w i l l accumulate i n the pulmonary c i r c u l a t i o n . As the blood volume of the c i r c u l a t i o n r i s e s , the vascular hydrostatic pressure w i l l also r i s e . The r e s u l t w i l l be an increased transendothelial flow of f l u i d and solutes. The major causes of cardiogenic pulmonary edema are l i s t e d i n Table 2 . 1.9.2.2 Permeability pulmonary edema. Permeability changes to the endothelial and/or e p i t h e l i a l membranes w i l l also r e s u l t i n the formation of edema. E n d o t h e l i a l permeability to f l u i d and solute may be altered by intravenous i n j e c t i o n s of excessive doses of drugs, such as heroin (diamorphine) or methadone. Moderate edema w i l l r e s u l t i n the accumulation of f l u i d I n t e r s t i t i a l l y . In severe pulmonary edema alveolar flooding w i l l occur as the i n t e r s t i t i a l and lymphatic f l u i d c a p acities are exceeded. The permeability of the epithelium to f l u i d and solute may be affected by the i n h a l a t i o n of smoke or v o l a t i l e toxins, such as nitrogen oxides. Increasing the e p i t h e l i a l permeability w i l l i n i t i a t e a l veolar edema; the f l u i d leaving the i n t e r s t i t i u m may then enter e i t h e r the a i r space or the lymphatics. -40-Table 2 Table 2: Major Causes of Cardiogenic (Hydrostatic) Pulmonary Edema (17) Left v e t r i c u l a r f a i l u r e Cardiomyopathy M i t r a l stenosis M i t r a l r e g u r g i t a t i o n Left a t r i a l thrombi Lef t a t r i a l myxoma Cor t r i a t r i a t u m Loculated c o n s t r i c t i v e p e r i c a r d i t i s -41-In either of the above cases, the other membrane may also be injured by the i n s u l t . Therefore the permeability of the endothelium and the epithelium w i l l be changed, r e s u l t i n g i n a more severe case of pulmonary edema. -42-INTERSTITIAL MODEL 2.1 Introduction Modelling of the complex, nonlinear interactions of the pulmonary microvascular exchange enables the study of the effect of perturbations to this system. Pulmonary edema is the pathophysiological disease that is most commonly studied (13,17). Moderate pulmonary edema, induced by perturbations to the circulation and/or permeability of the vascular membrane, w i l l result in the accumulation of fl u i d in the interstitium. The tissue between the circulation and air space w i l l expand, thereby impairing the diffusion of oxygen and carbon dioxide between the two compartments. The two classes of computer models used to study the i n t e r s t i t i a l phase of moderate pulmonary edema are the Pore Models (Blake and Staub (26); Harris and Roselli (27); Roselli, Parker and Harris (28)) and the Lumped Compartment Models (Bert and Pinder (29); Prichard, Rajagopalan and Lee (30)). Both types of models have been developed to provide transient and steady state responses. The pore model assumes that the vascular membrane is composed of a population of pores; the pores are of uniform or different sizes. The fluid and solute are assumed to be transported across the membrane through these pores. The parameters employed to define the membrane transport properties to fluid and solute are expressed in terms of the pore dimensions. To simulate the transient responses of the pulmonary microvascular exchange with the Pore Model, the pore structure should be established I n i t i a l l y , with the help of experimental data obtained -43-under steady state conditions (28). The lumped compartment model, on the other hand, does not interpret the physical structure of the membrane. The parameters used to define the membrane transport properties are obtained from experimental data. The lumped compartment model developed by Bert and Pinder (29) to study i n t e r s t i t i a l edema forms the basis for the work done In this thesis on alveolar edema. Their three compartment model, t i t l e d the I n t e r s t i t i a l Model, w i l l be discussed f i r s t . Incorporation of the air space w i l l lead to the development of the Alveolar Model. 2.2 Modelling of the Pulmonary Microcirculation The circulating compartment interacts directly with the i n t e r s t i t i a l compartment In the pulmonary microvascular exchange system. To develop a lumped compartment for the circulation, assumptions and simplifications are needed in defining i t s macroscopic properties - the hydrostatic and colloid osmotic pressures, and the protein concentrations. At different levels of the lung's circulation the value of the hydrostatic pressure changes. West, Dollery and Naimark (3) have identified three zones along the height of the lung - see section 1.2.3. The third zone was identified as a region where PA > PV > PALV. In this region a l l the blood capillaries are recruited, and thus, involved in fluid and solute exchange. The In t e r s t i t i a l model assumes zone III conditions prevail. - 4 4 -Th e hydrostatic pressure of the blood c i r c u l a t i o n also varies a x i a l l y along the length of the blood vessel from the a r t e r i a l to venous ends. The blood vessel was divided into a r t e r i a l , venous and c a p i l l a r y segments. If the c i r c u l a t i o n i s divided into three sub-compartments, then hydrostatic pressures would be assigned to each segment. Ad d i t i o n a l information would be needed because of the vessel sub-division. (1) The surface area of the wall of each segment would be needed i n order to determine the a v a i l a b l e area for f l u i d f i l t r a t i o n . (2) The permeability c h a r a c t e r i s t i c s of the membrane of each segment to f l u i d and solute would be required. And (3) there would be a preference to divide the i n t e r s t i t i u m into an alveolar sub-compartment - i n t e r a c t i n g with the c a p i l l a r y segment - and an extra-a l v e o l a r sub-compartment - i n t e r a c t i n g with the venous and a r t e r i a l segments. The physical properties of these i n t e r s t i t i a l sub-compartments would then be required. However, s u f f i c i e n t information i s not a v a i l a b l e to define the i n t e r s t i t i a l sub-compartments and vascular segments. Therefore, the d i v i s i o n of the c i r c u l a t i o n into segments was not ca r r i e d out. In the i n t e r s t i t i a l model, the pulmonary c i r c u l a t i o n was represented as a lumped compartment with one c i r c u l a t o r y hydrostatic pressure (PMV). The microvascular hydrostatic pressure representing the c i r c u l a t o r y compartment, i n zone III conditions, may be calculated from the l e f t a t r i a l and pulmonary a r t e r i a l hydrostatic pressures (3): -45-PMV = PLA +0.4 (PPA - PLA) (5) where PMV, PLA, PPA = the capillary, left a t r i a l , and pulmonary arte r i a l hydrostatic pressures, respectively. The proteins of the microvascular exchange system were assumed to be represented by albumin and globulins. Albumin is the most abundant protein in the exchange system, and the most osmotically active macromolecule. Globulins represent the class of larger proteins, and are the second most abundant proteins. The contents of the circulation compartment were assumed to be well-mixed. As a result the concentrations of albumin (CMVA) and the globulins (CMVG) are uniform. Intracompartmental solute diffusive flow is therefore eliminated. The presence of proteins in the circulation generates a compartmental colloid osmotic pressure, PIMV. Combining the concentrations of albumin and the globulins provides the total protein concentration CPMV: CPMV = CMVA + CMVG (6) The colloid osmotic or oncotic pressure may then be calculated from the total protein concentration by the Landis and Pappenhiemer Equation (7), with concentration changed to units of g/ml: PI = 210 CP + 1600 CP 2 + 9000 CP3 (7) - 4 6 -where PI = oncotic pressure, CP = t o t a l protein concentration. In the case of the c i r c u l a t i o n compartment CP i s equated to CPMV and PI to PIMV. The macroscopic properties of the lumped c i r c u l a t i o n compartment are: the hydrostatic pressure (PMV), the c o l l o i d osmotic pressure (PIMV), and the protein concentrations for albumin (CMVA) and globulins (CMVG). 2.3 Model l ing of the I n t e r s t i t i u m Previous discussion of the i n t e r s t i t i u m suggested the presence of an alveolar sub-compartment and an extra-alveolar sub-compartment. Each sub-compartment would have Its own macroscopic properties. However, values of the properties for each sub-compartment are d i f f i c u l t to obtain. The approach taken i n the I n t e r s t i t i a l model was to assume one i n t e r s t i t i a l compartment. The i n t e r s t i t i a l compartment was composed of two phases - a free f l u i d phase and a gel phase. The f l u i d present i n both phases would be of a volume VIS - the i n t e r s t i t i a l f l u i d volume. However, solutes would have access to only a f r a c t i o n of the i n t e r s t i t i a l f l u i d volume, or conversely would be excluded from a f r a c t i o n of the I n t e r s t i t i a l f l u i d volume. This "excluded" volume Is generally found i n the gel phase, and the f r a c t i o n of the i n t e r s t i t i a l volume from which the solute i s excluded Is dependent on the diameter of the solute. Albumin representing the smaller c l a s s - s i z e of proteins i s excluded from a f l u i d volume (VEXA) of 75.5 ml (29), while gl o b u l i n , - 4 7 -representing the larger class-size of proteins is excluded from a fl u i d volume (VEXG) of 150 ml (29). In the i n t e r s t i t i a l model neither of these excluded volumes are assumed to rise as flu i d accumulates in the interstitium (29). An available volume accessible to albumin and globulin results: VAVA = VIS - VEXA (8) VAVG = VIS - VEXA (9) where VAVA, VAVG = the fluid volume available to albumin and globulin, respectively. VEXA, VEXG = the fluid volume excluded from albumin and globulin, respectively. VIS = i n t e r s t i t i a l fluid volume. The available volumes for albumin and globulin are used in the evaluation of the protein concentration of the available spaces. These protein concentrations are the effective concentrations of the proteins: CAVA=^A_ ( 1 0 ) C A V G = v i c < n ) where CAVA, CAVG = the effective i n t e r s t i t i a l concentration of albumin and globulin, respectively. QA, QG = the weight of albumin and globulin, respectively, in the interstitium. -48-The oncotic pressure a c t u a l l y exerted i n the i n t e r s t i t i u m i s calc u l a t e d from the summation of the e f f e c t i v e i n t e r s t i t i a l concentrations: CPPMV = CAVA + CAVG (12) where CPPMV = t o t a l e f f e c t i v e protein concentration i n the Int e r s t i t i u m The i n t e r s t i t i a l oncotic pressure PIPMV may then be calculated by using the Landis and Pappenhiemer expression by set t i n g CP i n equation ( 7 ) equal to CPPMV. The i n t e r s t i t i u m i s assumed to be drained by nonsieving lymph ducts. These ducts c o l l e c t the f l u i d that passes through the i n t e r s t i t i a l space, that i s , both the space which excludes the proteins and the space which i s accessible to the proteins. The volume of this f l u i d space i s VIS. Albumin and glo b u l i n have protein weights i n the i n t e r s t i t i u m of QA and QG, r e s p e c t i v e l y . Assuming the lymphatics are well mixed, the protein concentrations of the f l u i d entering the lymphatics are given by: CTA = 2A_ ( 1 3) CTG = ( U ) where CTA, CTG = the tissue concentration of albumin and gl o b u l i n , r e s p e c t i v e l y . -49-Under normal steady state conditions the lymph flow i s equivalent to the transendothelial flowrate; no f l u i d accumulates i n the i n t e r s t i t i u m . However, a perturbation to the f l u i d exchange system where the transendothelial flow exceeds the lymph f l u i d flow w i l l lead to an accumulation of f l u i d i n the i n t e r s t i t i u m . In the i n t e r s t i t i u m the hydrostatic pressure r i s e s with the r i s e i n f l u i d volume according to the re l a t i o n s h i p provided by the tissue compliance curve. Although a compliance curve should exist for each of the tissu e sub-compartments - alveolar and extra-alveolar - uniting these sub-compartments into one compartment allows for the assumption of one tissue compliance curve. Bert and Pinder (29) have recalculated the compliance curve of Parker et a l . (31) to account for the dimensions of a human lung; the extravascular-extra-alveolar (EVEA) f l u i d volume, VIS1, i s used instead of the i n t e r s t i t i a l volume, VIS. These two volumes are related by the expression: The c e l l u l a r volume, VCELL, i s assumed to be constant. The hydrostatic pressure - EVEA f l u i d volume r e l a t i o n s h i p i s shown i n Figure 15. At f l u i d volumes less than 380 ml the tissue compliance i s low and the r e l a t i o n s h i p i s given by: VIS1 = VIS + VCELL ( 1 5 ) PPMV = 0.227 VIS1 - 89.0 (16) where PPMV = i n t e r s t i t i a l hydrostatic pressure. -51-If the EVEA fluid volume exceeds 460 ml a high tissue compliance is present, the expression is given by: PPMV = .017 VIS1 - 6.73 (17) The smooth transition zone between the low and high compliance curves is represented by a series of points approximating the smooth curve -see Table 3. An interpolation routine is used to relate i n t e r s t i t i a l hydrostatic pressure to the EVEA fluid volume i n this region. 2.4 Modelling the Vascular Membrane The circulation and i n t e r s t i t i a l compartments are separated by the vascular membrane. Fluid and solute exchange occurs between these two compartments across this barrier. The barrier is composed of a cellular layer and basement lamina, each component having a specific resistance to fluid and solute flow. The model combines these resistances to fluid and solute flow, and assumes that the vascular membrane may be represented by a resistance to fl u i d and solute that is uniform throughout the membrane. 2.5 Starling's Hypothesis: Transendothelial Fluid Flow Starling's Hypothesis has been used to represent the transendothelial flow from the circulation compartment to the i n t e r s t i t i a l compartment (8,29,32). -52-Table 3 Table 3: PPMV versus VIS1 for transition region of i n t e r s t i t i a l compliance curve. VIS1 PPMV (ml) (mmHg) 380 -2.74 390 -2.4 400 -1.9 410 -1.3 420 -0.8 430 -0.3 440 0.15 450 0.6 460 1.09 -53-JV = KF[(PMV-PPMV) - SIGD(PIMV-PIPMV)] (18) where JV = transendothelial fluid flowrate (ml/hr) KF = endothelial f i l t r a t i o n coefficient (ml/hr/mm Hg) SIGD = solute reflection coefficient (a^) PMV, PPMV = circulation and i n t e r s t i t i a l hydrostatic pressures, respectively (mm Hg) PIMV, PIPMV = circulatory and effective i n t e r s t i t i a l oncotic pressures, respectively (mm Hg). This hypothesis assumes that the flow of flu i d across the endothelium may be defined by two driving forces: 1) the hydrostatic pressure gradient between the i n t e r s t i t i a l and circulation compartments (PMV - PPMV), and 2) the colloid osmotic pressure gradient (PIMV - PIPMV). The parameter KF characterizes the permeability of the endothelial membrane to fluid transport, while SIGD characterizes an effective oncotic pressure gradient across the endothelium. Figure 7 shows a schematic of the circulation and tissue space separated by the endothelium; the normal values of the circulation and i n t e r s t i t i a l hydrostatic (PVM, PPMV) and oncotic (PIMV, PIPMV) pressures, and of the parameters KF and SIGD are indicated. These are the values assumed by Bert and Pinder (29) to be representative of existing experimental data. The hydrostatic pressure difference (PMV - PPMV) of 12.3 mm Hg w i l l force fl u i d from the circulation to the interstitium. -54-Equation (7) showed that oncotic pressure may be calculated from the total protein concentration of a compartment, the total protein concentration of the circulation compartment is 69 g/litre -assuming an albumin concentration of 42 g/1 and a globulin concentration of 27 g/1. The total protein concentration of the available i n t e r s t i t i a l space was 36 g/1. The albumin concentration (25 g/1) and the globulin concentration (11 g/1) of the available i n t e r s t i t i a l space were calculated by equations (10) and (11), respectively - VAVA and VAVG were evaluated from known (or assumed) values of VTS1, VCELL, VEXA and VEXG. The resultant oncotic pressures are 25 mm Hg for the circulation and 18.9 mm Hg for the interstitium, which gives an oncotic pressure difference (PIMV-PIPMV) of 6.1 mm Hg. Contrary to the effect of a hydrostatic pressure gradient - causing fluid movement from a high to a low hydrostatic pressure - an oncotic pressure gradient w i l l force fluid from the lower oncotic pressure to the higher oncotic pressure. The transport of fluid to the compartment of higher protein concentration results in the dilution of the receiving compartment's proteins, and concentration of the proteins of the compartment losing f l u i d . Therefore the oncotic pressure gradient forces fl u i d from the interstitium to the circulation. If the membrane separating the compartments is permeable to proteins, the proteins w i l l also be able to move from the circulation to the interstitium; the magnitude of the oncotic pressure driving force w i l l be reduced. The solute reflection coefficient (SIGD) measures the ab i l i t y of the proteins to pass through a membrane. If the membrane is semi-permeable, only -55-water can flow across the membrane and the solutes are restricted to the compartments on either side of the membrane. In this case the solute reflection coefficient equals one (SIGD = 1) and the oncotic pressure driving force is (PIMV-PIPMV). However, i f the solutes are able to freely pass across the membrane then the solute reflection coefficient would equal zero (SIGD = 0) and the oncotic pressure driving force is non-existent. In the case of the plasma proteins -albumin and globulins - a fraction of the proteins may cross the endothelial membrane, while the remaining fraction does not - the membrane sieves the proteins. For the i n t e r s t i t i a l model the solute reflection coefficient for the endothelium is 0.75 (29). Therefore, the effective oncotic pressure driving force between the circulation and interstitium is 0.75 (PIMV-PIPMV). The flu i d f i l t r a t i o n coefficient of the endothelium, KF, represents the permeability of the membrane to f l u i d . KF is the product of the membrane's flu i d conductivity (the inverse of the membrane's resistance to flu i d flow) and the surface area of the membrane (33): KF = LF(SA) (19) where LF = the fluid conductivity coefficient of the membrane SA = the surface area of the membrane By maintaining the circulation pressure above the alveolar (gas) pressure, i.e. zone III conditions, a l l the capillaries are assumed to be recruited. In zone III conditions the membrane surface area is assumed to remain relatively constant, so that the increase in the -56-fluid f i l t r a t i o n coefficient is primarily due to an increase in the endothelial fluid conductivity ( a l l capillaries are recruited and the surface area increases only slightly because of vessel distension). Bert and Pinder (29) reviewed the data and made appropriate selection of the values for KF (1.12 ml/hr) and SIGD (0.75). The driving force for flu i d flow across the endothelium is the difference [(PMV-PPMV) - SIGD(PIMV-PIPMV)]. The flu i d f i l t r a t i o n rate across the membrane, JV, is the product of the flu i d f i l t r a t i o n coefficient, KF, and the driving force noted above, i.e., JV is calculated by equation (18), Starling's Hypothesis. 2.6 Kedem-Katchalsky Solute Flux Equation; Transendothelial  Solute Flow The Kedem-Katchalsky (K-K) solute flux equation has been used to describe solute flow across the endothelial membrane: (JS)i = (PS)i((CMV)i-(CAV)i) + ( l - ( S I G F ) i ) [ ( C M V ) 1 * ( C A V ) 1 ] J V (20) diffusive term convective term (JS)i = the flowrate of the solute ' i ' across the membrane (g/hr) (PS)i = the permeability - surface area product of the membrane to solute ' i ' (ml/hr) (SIGF)i = the solvent drag reflection coefficient of the membrane to solute ' i ' - 5 7 -(CMV)i,(CAV)i = the concentration of solute ' i ' i n the c i r c u l a t i o n and av a i l a b l e tissue space, respectively (g/ml) JV = the f l u i d f i l t r a t i o n rate (ml/hr) The K-K solute f l u x equation i s separated into a d i f f u s i v e term and a convective term. The d i f f u s i v e term i s the product of a concentration gradient ((CMV)i-(CAV)i) for solute * i ' - albumin or globulins - and the permeability-surface area product ( P S ) i . The d i r e c t i o n of solute flow i s from the higher concentration to the lower concentration. In the i n t e r s t i t i a l model the d i f f u s i v e flow of albumin and g l o b u l i n across the endothelium i s from the c i r c u l a t i o n to the i n t e r s t i t i a l free f l u i d phase that i s av a i l a b l e to proteins. The protein concentration of th i s i n t e r s t i t i a l f l u i d space accessible to proteins i s CAVA (albumin) and CAVG ( g l o b u l i n ) . It i s these a v a i l a b l e concentrations that represent the i n t e r s t i t i a l protein concentrations used i n the K-K solute flux equation. The permeability surface area products, PSA and PSG, i l l u s t r a t e the permeability of the endothelial membrane to the s p e c i f i e d solute. Bert and Pinder (29) reviewed the data on these parameters and made appropriate s e l e c t i o n s . They have assumed a PSA for albumin of 3.0 ml/hr; a value within the range of PSA derived for canine and sheep models. The permeability-surface area product for glob u l i n , PSG, was set at one-third the value for albumin, i . e . , PSG was assumed to be 1.0 ml/hr (29). - 5 8 -The convective term of the K-K solute flux equation is assumed to represent the flow of solute that i s coupled with the flu i d flow. The arithmetic mean of the solute concentration Is a simplified expression of the mean logarithmic solute concentration that results from the analytical derivation of the convective term of the K-K solute flux equation. The product of the arithmetic mean of the solute concentration and the transendothelial fluid f i l t r a t i o n rate yields the rate of convective solute flow that crosses an endothelial membrane that is highly permeable to solute ' i ' . A solvent drag reflection coefficient for solute ' i ' of zero, (SIGF)i = 0, is assumed to represent an endothelium highly permeable to the solute. If the endothelium is semi-permeable to solute ' i ' , then solute ' i ' is not transported across the membrane and the solvent drag reflection coefficient for solute ' i ' equals one, (SIGF)i = 1. Therefore the solvent drag reflection coefficient indicates the ab i l i t y of the f l u i d to 'drag' solute ' i ' across the membrane. After reviewing the data in literature, Bert and Pinder (29) selected the value of the solvent drag reflection coefficient for albumin, SIGFA, to be approximately 0.4; near the midrange of values determined in dogs as JV varied. The solvent drag reflection coefficient for globulin, SIGFG, would be expected to be greater than SIGFA, since its diameter is larger. Bert and Pinder (29) selected a value of 0.6 for SIGFG. Table 4 l i s t s a l l the input parameters discussed in the i n t e r s t i t i a l model. The estimates used by Bert and Pinder (29) are shown. -59-Table 4 Table 4: Input Parameters to the I n t e r s t i t i a l Computer Simulation for Normal Conditions (29) Bert & Pinder PMV 9 mmHg PIMV 25 mmHg KF 1.12 ml/hr mmHg SIGD 0.75 VCELL 150 ml VIS1 379. ml Albumin PSA 3.0 ml/hr SIGFA 0.4 VEXA 73.5 ml QA 5.38 g CMVA 0.042 g/ml Globulin PSG 1.0 ml/hr SIGFG 0.6 VEXG 115.5 ml QG 2.48 g CMVG 0.0271 g/ml - 6 0 -2.7 Modelling of the Lymphatics The lymphatics function, in part, as drainage channels for f l u i d and solute leaving the interstitium. Although an intrinsic pumping mechanism within the lymphatics is thought to actively drain the tissue (17,18), in the i n t e r s t i t i a l model the lymphatics assume a passive role. The lymph fluid flow has been expressed as a function of total extravascular fluid volume (VTOT), i n t e r s t i t i a l pressure (PPMV), and also circulation hydrostatic pressure (PMV) (34,35,36, 37). An increase in either of the variables - VTOT, or PPMV, - or parameter PMV - is equivalent to a rise in the i n t e r s t i t i a l (+ cellular) f l u i d volume, VIS1 (= VIS + VCELL). Bert and Pinder (29) have replotted and recalculated the data of Erdmann et a l . (35) to develop a relationship for the lymph fluid flow, JL as a function of the i n t e r s t i t i a l (+ cellular) fl u i d volume, VIS1, for the dimensions of human lungs. Accounting for experimental and c l i n i c a l trends, the relationship i s : JL = 0.17 VIS1 - 55.6 (21) where JL = lymph flu i d flow (ml/hr) VIS1 = i n t e r s t i t i a l (+ cellular) or extravascular -extra-alveolar (EVEA) flu i d volume (ml) Therefore, any rise in i n t e r s t i t i a l f l u i d volume w i l l cause JL to increase. - 6 1 -The solutes are also assumed to passively leave the i n t e r s t i t i u m and enter the lymphatics. The lymphatic c a p i l l a r y membrane i s assumed to be highly permeable to the proteins. The protein concentration of the lymph i s thus assumed to be equivalent to the tissue protein concentration: ( C L ) ± = (CT) i (22) where (CL)^ = concentration of protein ' i ' i n lymph (CT)^ = tissue concentration of protein ' i ' The solute flow into the lymphatics i s assumed to be completely convective, that i s : (Solute Flowrate i n t o _ , . the Lymphatics) " J L < C T > i 2.8 Fluid and Solute Material Balances 2.8.1 Fluid Material Balance The f l u i d flows previously described i n section 2.5 and 2.7 were of the f l u i d entering and e x i t i n g the i n t e r s t i t i u m ; f l u i d enters from the c i r c u l a t i o n and e x i t s through the lymphatics. In steady state conditions the f l u i d flows into and out of the i n t e r s t i t i u m , JV and JL r e s p e c t i v e l y , are assumed equal. If unsteady state conditions p r e v a i l i n the three-compartment model, and JV exceeds JL, there i s an accumulation of f l u i d i n the i n t e r s t i t i u m . A statement of the f l u i d material balance around the i n t e r s t i t i u m i s : F l u i d Flowrate into the In t e r s t i t i u m Rate of F l u i d Accumulation i n the I n t e r s t i t i u m + F l u i d Flowrate out of the In t e r s t i t i u m (24) Using the appropriate nomenclature r e s u l t s i n : JV = JNET1 + JL (25) where JNET1 = rate of f l u i d accumulation i n the i n t e r s t i t i u m The material balance of equation (25) applies to the three-compartment i n t e r s t i t i a l model. Figure 16a i l l u s t r a t e s the three compartment model and the f l u i d flows. Equation (25) i l l u s t r a t e s that i f the f l u i d flow into the i n t e r s t i t i u m , JV, and out of the i n t e r s t i t i u m , JL, are known, then the rate of accumulation of f l u i d i n the i n t e r s t i t i u m i s simply the difference (JV-JL). The i n t e g r a l of JNET1 with time w i l l y i e l d the difference between the f l u i d volume In i n t e r s t i t i a l (+ c e l l u l a r ) space at time zero and time t : In the i n t e r s t i t i a l model f l u i d does not accumulate i n the a i r space; the EVEA or i n t e r s t i t i a l (+ c e l l u l a r ) f l u i d volume, VIS1, w i l l be equivalent to the t o t a l extravascular f l u i d volume, VTOT. VIS1 - VIS10 = f tJNETl dt o (26) -63-Figure 16a: Schematic of the Three Compartment I n t e r s t i t i a l Model I l l u s t r a t i n g F l u i d Flows and Accumulation Ci rculation J V Air Space . Interstitium Accumulat ion J N E T 1 -> Lymph Flow JL Figure 16b: Schematic of the Three Compartment I n t e r s t i t i a l Model I l l u s t r a t i n g Solute Flows and Accumulation Circulation Concentration CMV; JS; Air Space In te rs : ! t i u m A c c u m u l a t i o n ' Q N E T j C o n c e n t r a t i o n CT; ^Lymph Flow J L C T ; r -64-2.8.2 Solute Material Balance The material balance for the proteins - albumin and globulin -around the interstitium may be stated as follows: Flowrate of Rate of Accumulation Flowrate of Solute Solute ' i ' into = of Solute ' i ' in + ' i ' out of the (27) the Interstitium the Interstitium Interstitium The solute material balance may be rewritten, with the solute flow variables inserted: (JS) ± = (QNET)± + JL(CT) 1 (28) where (QNET)^ = rate of accumulation of solute ' i ' in the interstitium The solute flows are illustrated in Figure 16. Equation (28) illustrates that i f the solute flow Into the interstitium, (JS)^, and out of the interstitium, (QNET)^, are known, then the rate of accumulation of solute in the interstitium is simply the difference ((JS) 1-JL(CT) 1). The integral of (QNET)^ with time w i l l yield the difference between the solute weight in the interstitium at time zero and time t: Q± " (Q0) ± = Jl (QNET)i dt (23) where Q^»(QO)^ = the weight of solute ' i ' at time t and zero, respectively (gm). - 6 5 -ALVEOLAR MODEL 3.1 Introduction Previous models (28,29) of pulmonary microvascular exchange were developed to study the exchange of fluid and solutes between the i n t e r s t i t i a l , circulation, and lymphatic compartments. The air space was not included as a fourth compartment since the experimental data needed to include i t were not available. This thesis reports one way of integrating the air space into the i n t e r s t i t i a l model developed by Bert and Pinder (29), and previously described in section 2. The four compartment model that results is referred to as the alveolar model. The limited data which are available on the involvement of the air space In the pulmonary microvascular exchange were combined with assumptions - in areas of the model lacking experimental evidence - to integrate the air space with the lumped compartment model. 3.2 Modelling of the Air Space The major regions for fl u i d and solute exchange are at the level of the respiratory units, that i s , the respiratory bronchiole, alveolar ducts, and alveoli. Each lung lobe is composed of many respiratory units. In experimental studies of the pulmonary microvascular exchange i t was found that the air space of a lobe or segment of a lobe may be f i l l e d with fl u i d (12,38). The effects of f i l l i n g these air spaces with fluid remain localized, i.e., other lung sections would continue - 6 6 -to operate normally. In severe pulmonary edema the accumulation of fl u i d in the air space may also be confined to isolated sections of the lung (39). During severe pulmonary edema Staub et a l . (39) proposed that the accumulation of fluid in the air space was not uniform. They proposed that an alveolus would not remain partially f i l l e d , but when flui d started to accumulate in the alveolus i t would f i l l rapidly and independently of the neighbouring alveoli. Ideally, modelling of the air space should incorporate the concept of individual alveoli f i l l i n g independently. The air space would then be modelled as a compartment composed of millions of sub-compartments that represent the alveoli. As the flu i d accumulates in the air space the number of small sub-compartments f i l l e d would increase. However, as a f i r s t approximation in the development of the alveolar model i t is unreasonable to incorporate this degree of complexity. Consequently in this model the air space compartment is represented as a reservoir with a capacity of several l i t r e s . During normal conditions the base of the compartment is lined with a thin layer of surface active f l u i d . Macklin (40) estimated that the total fluid volume in the air space under normal conditions is 20 ml; a volume that is negligible In comparison to the large capacity of the air space. At normal conditions the compartment w i l l be f i l l e d primarily with gas (or a i r ) . The fluid that accumulates in the air space w i l l be assumed to be well-mixed. Therefore, the compartment w i l l have a uniform protein concentration. -67-3.3 Modelling of the Alveolar Membrane Figure 8 i l l u s t r a t e s that the a i r space i s separated from the other compartments of the pulmonary microvascular exchange by the alveolar wall composed of a c e l l u l a r and a basement membrane. For s i m p l i c i t y , the i n d i v i d u a l resistances of each membrane to f l u i d and solute flow are represented i n the alveolar model by a single r e s i s t -ance. Hence the alveolar wall w i l l be taken as being uniformly perm-eable to f l u i d and solutes throughout i t s thickness and surface area. In the alveolar model the exchange of f l u i d and solutes across the alveolar membrane w i l l be assumed to occur between only the i n t e r s t i t i a l and alveolar compartments. S t r u c t u r a l l y , the alveolar wall separates the a i r space and i n t e r s t i t i u m . At other s i t e s the basement membrane of the alveolar wall i s fused to the basement membrane of the vascular wall (Figure 8 ) . F l u i d and solute may also be transported across this vascular-alveolar wall, between the c i r c u l a - t i o n and a i r space. However, i f f l u i d and solutes could leave the c i r c u l a t i o n across the vascular wall (to the i n t e r s t i t i u m ) and the vascular-alveolar membrane (to the a i r space) a factor would be required to indicate the proportion of f l u i d and solutes leaving the c i r c u l a t i o n for the i n t e r s t i t i u m , and for the a i r space. Furthermore, the permeability to f l u i d and solute of the vascular-a l v e o l a r membrane would be needed. Neither the p r o p o r t i o n a l i t y factor nor the vascular-alveolar permeability to f l u i d and solute are ava i l a b l e i n l i t e r a t u r e . Therefore, to simplify the matter, the c i r c u l a t i o n was assumed to int e r a c t with only the i n t e r s t i t i u m , which then interacts,, with the a i r space. - 6 8 -3.A The Onset of Alveolar Flooding During i n t e r s t i t i a l pulmonary edema, fluid and protein are exchanged between the circulation, i n t e r s t i t i a l , and lymphatic compartments. The air space remains virtually dry, and is assumed t not participate in the pulmonary microvascular exchange. Following the onset of alveolar flooding the air space receives fluid and protein from the i n t e r s t i t i a l compartment. The onset of alveolar flooding Is assumed to occur at a specific extravascular-extra-alveolar (EVEA) flu i d volume, VTONS. Preceding alveolar flooding the extravascular-extra-alveolar fluid volume (VIS1) is equivalent to the total extravascular fluid volume (VTOT). In the alveolar model, when the flu i d volume VIS1 (or VTOT) reaches VTONS alveolar flooding begins. The subsequent progression the alveolar edema w i l l depend on other factors introduced in later sections (3.5, 3.6). The value of VTONS is dependent on the cause of the edema. Edema may be caused by a perturbation that originates from the vascular compartment, and augments the transendothelial flow - throu an elevated PMV, or change in endothelial permeability. Fluid w i l l accumulate in the i n t e r s t i t i a l space until the EVEA flu i d volume (VIS1) exceeds VTONS, at which point alveolar edema begins. A numbe of values of VTONS have been suggested under these circumstances: I l i f f (41) has suggested that alveolar flooding occurs after a 30% increase in VTOT from the normal total extravascular fluid volume; i.e., for a VTOT of approximately 379 ml under normal conditions VTONS would be 490 ml. -69-Prichard (17) has stated that alveolar flooding occurs at a total extravascular fluid volume (VTOT) equivalent to twice the i n t e r s t i t i a l f l u i d volume (VIS = VTOT - VCELL) at normal conditions; for an i n t e r s t i t i a l fluid volume of approximately 229 ml (= 379 - 150 ml) under normal conditions, VTONS would be 458 ml. Finally, Staub (13) has stated that alveolar flooding occurs after a 30% increase in lung weight from normal conditions; for a wet lung weight of approximately 672 ml under normal conditions, VTONS would be 585 ml (VTONS = (1.3(672) - 195 - 94): Blood content = 195 g; Blood-free dry lung weight = 94 g; Density of flu i d = 1 g/ml). Therefore, in alveolar edema caused by a transendothelial flow greater than normal, the range of VTONS lies between 450 ml and 600 ml. Severe pulmonary edema may also arise from permeability changes to the epithelial membrane induced by an irrit a n t introduced intravenously, or from the air side. The parameter VTONS may s t i l l be used to indicate the onset of alveolar flooding. If the irritant is introduced from the air side VTONS w i l l assume the extravascular fluid volume at normal conditions, 379 ml, since no i n t e r s t i t i a l accumulation of flu i d w i l l have occurred. If the ir r i t a n t is introduced intravenously i t may alter the epithelial permeability to flu i d and solute after i t changes the endothelial permeability; the length of time before the epithelium Is injured may depend on the dosage of the i r r i t a n t . Introduction of a large dose may cause injury to the epithelium immediately after the irrit a n t damages the endothelium; in this case VTONS w i l l be just above 379 ml (normal extravascular fluid volume). On the other hand, the epithelial -70-permeability may change at a longer period of time after the endothelium is injured by a small dosage of the i r r i t a n t . During this period of time, before the epithelium is injured, the EVEA fluid volume (VIS1) w i l l rise, and the value of VTONS w i l l then be above 379 ml - the upper value of VTONS w i l l be that EVEA fluid volume at which edema is induced by an elevated circulation hydrostatic pressure or a permeability change to only the endothelium. The estimates provided for VTONS were determined by experimental measurements on the lungs of animals (41,42), and then recalculated for the dimensions of human lungs. Using these estimates to approximate the value of VTONS for human lungs may provide erroneous results. Acquiring the measurements for the animal lungs in vitro w i l l also affect the values of VTONS (P. Dodek, personal communication). The expansion of lungs in vivo w i l l be restricted by the chest wall, while in vitro expansion of the lungs w i l l not be restricted. Therefore the estimates obtained with the lungs in vitro (41,42) may be unusually high. A series of computer simulations of the alveolar model were performed to study the effect of different values of VTONS. In other simulations where the purpose was to study the effect of different values of other parameters VTONS was set at approximately 460 ml. This value of VTONS is close to the fluid volume equal to twice the i n t e r s t i t i a l fluid volume, 458 ml. - 7 1 -3 .5 Modelling of Fluid and Solute Transport Across the Alveolar Membrane During alveolar flooding f l u i d and solute are transported across the alveolar membrane from the i n t e r s t i t i u m into the a i r space. Modelling the f l u i d flow across the alveolar membrane may be approached i n a manner s i m i l a r to that used i n the modelling of f l u i d movement across the endothelium. The dr i v i n g forces are the hydrostatic and c o l l o i d osmotic pressure gradients, while the e p i t h e l i a l conductivity to f l u i d flow i s described by an e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t . The form of the expression i s s i m i l a r to S t a r l i n g ' s Hypothesis: JAS = KAS[(PPMV-PAS) - SIGDAS(PIPMV-PIAS) (24) where JAS = t r a n s e p i t h e l i a l f l u i d f i l t r a t i o n rate KAS = e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t PPMV,PAS = f l u i d hydrostatic pressure of i n t e r s t i t i a l space and alveolar f l u i d space, respectively PIPMV,PIAS = c o l l o i d osmotic pressure of i n t e r s t i t i a l space and alveolar f l u i d space, respectively SIGDAS = solute r e f l e c t i o n c o e f f i c i e n t of epithelium To insure that the t r a n s e p i t h e l i a l f l u i d flow i s zero before the onset of alveolar flooding the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t -representing the f l u i d conductivity of the epithelium - w i l l be set equal to zero. Following the onset of alveolar flooding KAS w i l l be greater than zero. The actual value depends on the expression used for KAS. -72-Vreim et a l . (43,44) have illustrated with dog lung experiments exposed to conditions that caused severe pulmonary edema with alveolar flooding (induced hydrostatically or by permeability changes to the vascular and alveolar membranes), that the protein concentrations in the alveolar fluid, i n t e r s t i t i a l f l u i d and lymph were similar - with no s t a t i s t i c a l l y significant differences. The protein concentration of the i n t e r s t i t i a l f l u i d refers to the tissue concentration (CTA, CTG). In hydrostatically induced edema the protein concentration of the tissue fluid was less than in the plasma. If the edema was induced by permeability changes to both membranes the protein concentration of the alveolar, i n t e r s t i t i a l , lymphatic and circulation compartments was similar. The similarity in protein concentration between the tissue and alveolar fluid suggests that the alveolar membrane was highly permeable to proteins. This situation would permit the assumption that during alveolar flooding the solute reflection coefficient of the epithelium (SIGDAS) is equal to zero. Therefore, during alveolar flooding, the expression for transepithelial fluid flow reduces to: JAS = KAS(PPMV-PAS) (25) Preceeding alveolar flooding KAS is set equal to zero, so that there is no need to define the solute reflection coefficient of the epithelium and the colloid osmotic pressure of the alveolar f l u i d . The variables PAS and KAS w i l l be defined in sections 3.6 and 3.7, respectively. -73-The observation by Vreim et a l . (43,44) that solute concentration in the tissue space and alveolar fluid were similar during alveolar flooding supports a convective non-sieving flow of solutes across the epithelium: Flowrate of Solute ' i ' across = JAS(CT)i (26) the epithelium The fluid that enters the air space w i l l be the fluid that passes through the free-fluid channels of the interstitium which are accessible to proteins and also the gel phase of the i n t e r s t i t i a l space which is not available to the proteins. As a result the protein concentration of fluid entering the air space Is assumed to be the tissue concentration (CTA,CTG). 3.6 Pressure-volume Relationship of the Alveolar Fluid 3.6.1 Fluid Pressure-volume Relationship of an Individual Alveolus (i) Normal Conditions A schematic of an alveolus and i t s surrounding tissue is shown in Figure 14b. Lining the alveolus is the surfactant f l u i d . The pressures assigned to this schematic are the air pressure (PG), the alveolar fl u i d pressure (PL), and the i n t e r s t i t i a l hydrostatic pressure (PPMV). The average radius of curvature of the air-liquid interface is r. As mentioned in section 1.7.2, Laplace's equation, equation (4), has been applied to the alveolus. The pressure -74-d i f f e r e n t i a l (PG-PL) i s equal to 2T/T ( T = surface tension of the sur f a c t a n t ) . To ensure a stable alveolus, Laplace's expression should hold. The e p i t h e l i a l c e l l u l a r membrane i s water porous. If the f l u i d s on both sides of the membrane reach equilibrium under normal conditions then the hydrostatic pressures of f l u i d on each side on the membrane should be equal, i . e . , PL = PPMV (45). Under normal conditions the net exchange of f l u i d across the epithelium should be e s s e n t i a l l y zero. A schematic of the alveolus under normal conditions i s shown i n Figure 17. ( i i ) F l u i d F i l l i n g of the Alveolus The f i l l i n g of an alveolus with f l u i d may be divided into three phases (17,39). F i r s t , i f the i n t e r s t i t i a l hydrostatic pressure i s elevated s l i g h t l y (because of a perturbation to the microvascular exchange system) a hydrostatic pressure gradient from the i n t e r s t i t i u m to the a i r space develops. F l u i d w i l l flow along t h i s gradient into the alveolus. I f a steady state i s achieved at th i s new i n t e r s t i t i a l pressure, the alveolar f l u i d pressure w i l l have r i s e n to equal the i n t e r s t i t i a l pressure. To maintain Laplace's equation (and a stable alveolus) the extra f l u i d i n the alveolus must increase the radius of curvature (Figure 17). The a f f e c t of the f l u i d entering the alveolus on the surface tension at the a i r - l i q u i d i n t e r f a c e i s unclear. If the surfactant i s d i l u t e d , the surface tension w i l l r i s e ; Laplace's equation w i l l be maintained by a greater r i s e i n the radius of -75-F i g u r e 17: Schematic R e p r e s e n t a t i o n o f F l u i d F i l l i n g an A l v e o l u s D u r i n g Acute Pulmonary Edema. Phase 1: r a d i u s of c u r v a t u r e ( r ) i n c r e a s e s d u r i n g i n i t i a l phase of f i l l i n g . Phase 2: r a d i u s of c u r v a t u r e ( r ) d e c r e a s e s d u r i n g f i l l i n g . Phase 3: r a d i u s o f c u r v a t u r e i n c r e a s e s d u r i n g f i n a l phase of f i l l i n g . ( M o d i f i e d from (39)) a) N o r m a l c)Phas<2 2 d ) P h a s e 3 x -76-curvature. Tierney and Johnson (46) have performed experiments in which plasma fluid was slowly introduced under a surfactant lining; the surface tension at the air-liquid interface remained at its normal value. Nonetheless, in this f i r s t phase the alveolar fl u i d pressure rises with alveolar fluid volume. Guyton and co-workers (45,47) have suggested that this f i r s t phase of f i l l i n g may continue up to a slightly positive i n t e r s t i t i a l hydrostatic pressure (+1 or +2 mm Hg). Secondly, i f the i n t e r s t i t i a l hydrostatic pressure increases beyond this slightly positive i n t e r s t i t i a l pressure the alveolus w i l l become unstable and fluid w i l l rush into the air sac. During this second phase the radius of curvature, r, decreases (Figure 17). Laplace's equation (4) illustrates that i f r is decreasing then (PG-PL) must increase; assuming that PG remains constant, PL must decrease. The transepithelial fluid pressure gradient that is formed causes further fluid flow into the alveolus. The surface tension at the air-liquid interface is assumed to rise to that of water and remain constant (17). Therefore, in this second phase the alveolar flu i d pressure is inversely proportional to the alveolar fl u i d volume. Thirdly, the radius of curvature of the air - l i q u i d interface reaches a second c r i t i c a l value whereafter further increase in alveolar f l u i d volume causes the radius of curvature to increase (Figure 17). Applying Laplace's equation to the alveolus indicates that (PG-PL) w i l l begin to decrease once again; the alveolar fl u i d -77-pressure PL w i l l r i s e . In this t h i r d phase the alveolar f l u i d pressure r i s e s , once again, with the alveolar f l u i d volume. 3.6.2 Fluid Pressure-volume Relationship for the Air Space Compartment In the alveolar model a f l u i d pressure-volume r e l a t i o n s h i p w i l l be developed for the alveolar f l u i d of the a i r space compartment. As f l u i d enters the a i r space a pressure head develops i n the compartment. As a f i r s t approximation for the compartment a s t r a i g h t -l i n e r e l a t i o n s h i p of pressure as a function of volume i s assumed. The equation w i l l take the following form: PAS = SL VAS + B (27) where PAS = alveolar f l u i d hydrostatic pressure VAS = f l u i d volume of a i r space SL = slope of PAS-VAS curve B = PAS at the onset of alveolar flooding The pressure PAS w i l l be the pressure r e s i s t i n g f l u i d transport into the a i r space. Preceding the onset of alveolar flooding t r a n s e p i t h e l i a l flow i s assumed to be equal to zero. The value of the alveolar f l u i d pressure, PAS, i s not needed during the pre-alveolar edema phase. At the onset of alveolar flooding the value of the alveolar f l u i d pressure i s required; i t w i l l be given by B. No estimates of B have been documented. As a r e s u l t , a range of estimates of B w i l l be investigated i n t h i s t h e s i s . For an i n i t i a l estimate B s h a l l be taken -78-as the value of the alveolar f l u i d pressure at normal conditions, i . e . B w i l l be equivalent to the i n t e r s t i t i a l hydrostatic pressure at normal conditions. The slope of the PAS-VAS r e l a t i o n s h i p i s the parameter SL. As with B, no estimates of SL have been documented. In this thesis the e f f e c t of the parameter SL on the microvascular exchange system w i l l also be investigated. An i n i t i a l estimate of SL s h a l l be a value of .25 x 10" 2 mm Hg/ml. 3.7 Representation of the Epithelial Filtration Coefficient. KAS The e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t represents the f l u i d c onductivity of the alveolar b a r r i e r . Preceding the onset of alveolar flooding the parameter i s set equal to zero. Following the onset of a l v e o l a r flooding the value of KAS may be a function of time or some other v a r i a b l e , or KAS may assume a constant value. In this thesis two proposals for the value of KAS w i l l be investigated. 3.7.1 Representation of KAS as a Variable: KAS = NK(VISl-VTONS) for VIS1 > VTONS (28) In t h i s circumstance the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t i s proportional to the EVEA f l u i d volume, VIS1, less the f l u i d volume at the onset of alevolar flooding VTONS. The parameter NK may be considered a s e n s i t i v i t y parameter: as NK r i s e s , the e p i t h e l i a l -79-f i l t r a t i o n c o e f f i c i e n t w i l l be more se n s i t i v e to a given (VIS1-VT0NS). The f l u i d volume difference (VIS1-VT0NS) gives an i n d i c a t i o n of the ef f e c t of the expansion of the extra-vascular-extraalveolar f l u i d space above i t s f l u i d volume at the onset of alveolar flooding, on KAS. A range of magnitude of NK w i l l be examined i n t h i s t h e s i s . 3.7.2 Representation of KAS as a Constant: KAS = KASO for VTOT > VTONS (29) Preceding alveolar flooding the t o t a l extravascular f l u i d volume (VTOT) i s equivalent to the extravascular-extra-alveolar f l u i d volume (VIS1). When VTOT reaches the f l u i d volume VTONS the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t r i s e s i n s t a n t l y to a constant value of KASO. The f l u i d volume VTOT i s then composed of VIS1 and the alveolar f l u i d volume, VAS; provided VTOT exceeds VTONS KAS w i l l remain at a value of KASO. The EVEA f l u i d volume VIS1 may become greater or less than the f l u i d volume VTONS. In t h i s study simulations were conducted to determine the response of the pulmonary microvascular exchange system to d i f f e r e n t values of KASO. -80-3.8 Integration of the Air Space Compartment with Pulmonary  Microvascular Exchange Model In the i n t e r s t i t i a l model the material balances for the i n t e r s t i t i u m involved an i n f l u x of f l u i d and solute from the c i r c u l a t i o n and an e f f l u x of f l u i d and solute through the lymphatics. The addition of the a i r space compartment to the i n t e r s t i t i a l model presents few d i f f i c u l t i e s . During alveolar flooding, f l u i d and solute cross the epithelium from the tissue space into the a i r space. The f l u i d material balance denoted by equation (24) i s s t i l l applicable, but i n t h i s case: Rate of F l u i d Flow out of = JL + JAS (30) the I n t e r s t i t i u m S i m i l a r l y , the flow rate of solute ' i 1 out of the i n t e r s t i t i u m i s : Flow rate of solute 'I* out = JL(CT)i + JAS(CT)i (31) of the I n t e r s t i t i u m The material balances of the alveolar model, around the i n t e r s t i t i u m , are: F l u i d Material Balance JNET1 = JV - JL - JAS (32) M a t e r i a l Balance for Solute ' i ' (QNET)i = ( J S ) i - JL(CT)i - JAS(CT)i (33) where I = albumin or g l o b u l i n s . -81-Figure 18 Illustrates the material balances of the alveolar model. The material balances shown in equations (32) and (33) are applicable before and after the onset of alveolar flooding. Preceeding the onset of alveolar flooding there is no transepithelial fluid or solute flow, since KAS, and therefore JAS, in equation (25) are equal to zero. The time integrals of JNET1, QNETA, QNETG, and JAS yield the variables VIS1, QA, QG, and VAS, respectively, which are needed in the evaluation of the fluid and solute flows of the material balances shown in equations (32) and (33). JV, JL and JAS of the flu i d material balance are evaluated by equations (18), (21) and (25), while (JS)i of the solute material balance is determined by equation (20). The solute concentrations CTA and CTG are evaluated by equations (13) and (14). -82 -Figure 18a: Schematic of Four Compartment A l v e o l a r Model I l l u s t r a t i n g F l u i d Flows Circulation J V Interstit ium Accumu la t i on J NET 1 A i r Space -> Lymphat ics J L Figure 18b : Schematic of Four Compartment A l v e o l a r Model I l l u s t r a t i n g S o l ute Flows C i r c u l c t ion U S ) ; I n t e r s t i t i u m A c c u m u l a t i o n (Q N E T ) . A i r Space O A S - ( C T ) . -> Lymphatics JL-(CT); -83-COMPUTER SIMULATION OF ALVEOLAR MODEL 4.1 Introduction T r a d i t i o n a l l y , the study of b i o l o g i c a l models involved the use of human and/or animal experiments. Recently, computer simulations have provided an a d d i t i o n a l method for te s t i n g the b i o l o g i c a l models. Due to the complexities that a r i s e from the int e r a c t i o n s of a p h y s i o l o g i c a l system, numerous s i m p l i f i c a t i o n s are necessary when developing a computer model. In spite of t h i s , a s i m p l i f i e d model of a b i o l o g i c a l system examined with a computer may provide: 1) b i o l o g i c a l s c i e n t i s t s and health care professionals with an estimate of the progress of a pathophysiological state and the e f f e c t s of c l i n i c a l management, and 2) pulmonary researchers with an alternate t o o l with which to test t h e i r experimental f i n d i n g s . The model may provide suggestions on the e s s e n t i a l variables and parameters that should be measured to va l i d a t e the model. Bert and Pinder (29) f i r s t developed the i n t e r s t i t i a l model on an AD-80 analog computer. This i n i t i a l step i n the modelling allowed a clear v i s u a l i z a t i o n of the i n t e r r e l a t i o n s h i p s between v a r i a b l e s . Thus, the analog computer was a useful t o o l while developing the model and i t s s o l u t i o n . However, due to problems i n s c a l i n g , amplifer i n s t a b i l i t y , and the li m i t e d capacity of the AD-80 analog computer the i n t e r s t i t i a l model was translated into d i g i t a l form, using FORTRAN. The sol u t i o n for the alveolar model was also i n i t i a t e d on the AD-80 analog computer. However, due to the l i m i t e d capacity of the -84-analog computer, the solution to the Alveolar Model was completed on a di g i t a l computer using FORTRAN. The dig i t a l computer system employed for this task was the AMDAHL 4TOV-12 of the University of British Columbia. A benefit of using the university f a c i l i t i e s was the access to a variety of library subroutines. The following sub-sections describe the computer program developed for the solution of the alveolar model, and the simulations that were performed to study the operation of the alveolar model. 4.2 The Computer Program The computer program of the Alveolar Model may be divided into three sections: 1) the input data f i l e s EDA and PDA, 2) the main program UBCEDEMA, and 3) the output of results - tabulated and/or plotted. In Appendix A some additional Information on the computer program w i l l be discussed. 4.2.1 Input Data Files EDA and PDA The parameters listed in the two data f i l e s are defined in Tables 5 and 6. File EDA (listed in Table 5) contains parameters that were introduced with the i n t e r s t i t i a l and alveolar models, such as the plasma protein concentrations (CMVA,CMVG) and the cellular fluid volume (VCELL). Fi l e PDA (listed in Table 6) includes four parameters that were introduced with the i n t e r s t i t i a l and alveolar models - PMV, SL, NK, and KASO. The remaining parameters of f i l e PDA pertain to the operational aspect of the simulation. The f i r s t two input parameters indicate -85-Table 5: Content of Input F i l e EDA VTOTO VASO QAO QGO PIMV VCELL VEXA VEXG VTONS VLMPH B CMVA CMVG I n i t i a l value of t o t a l extravascular f l u i d volume (ml) I n i t i a l value of alveolar f l u i d volume (ml) I n i t i a l value of i n t e r s t i t i a l albumin weight (g) I n i t i a l value of i n t e r s t i t i a l g l o b u l i n weight (g) Microvascular c o l l o i d osmotic pressure (mmHg) C e l l u l a r f l u i d volume (a constant)(ml) Excluded volume for albumins (ml) Excluded volume for globulins (ml) (EVEA) Extravascular extraalveolar (EVEA) f l u i d volume at the onset of alveolar flooding (ml) EVEA f l u i d volume that corresponds to the maximum lymph flow (ml) Alveolar f l u i d pressure at the onset of alveo l a r flooding (mmHg) Microvascular albumin concentration (g/ml) Microvascular g l o b u l i n concentration (g/ml) FPPMV(l)-FPPMV(9) Interpolation values for tissue compliance curve (mmHg) Continued.... - 8 6 -Table 5: Content of Input F i l e EDA (Continued) KF : Endothelial f l u i d f i l t r a t i o n c o e f f i c i e n t (ml/hr/mmHg) SIGD : Endothelial osmotic r e f l e c t i o n c o e f f i c i e n t for f l u i d flow PSA : Permeability times surface area "of vascular membrane for albumins (ml/hr) PSG : Permeability times surface area of vascular membrane for globulins (ml/hr) SIGFA : Endothelial r e f l e c t i o n c o e f f i c i e n t f o r albumins SIGFG : Endothelial r e f l e c t i o n c o e f f i c i e n t f o r globulins -87-Table 6 : Variables i n Input F i l e PDA PLOTS (Y=1,N=2) TABLES 1 (Y=1,N=2) Ca l l s plot routine (Yes or No) sets NN Ca l l s table routine (Yes or No) sets NNN TAUMX1 SUBNT1 TAUMX2 SUBNT2 STEPSZ HMIN TOL Maximum time for output of p r i n t i n g i n t e r v a l SUBNT1 (hr) Time i n t e r v a l for p r i n t i n g output up to TAUMX1. Must divide evenly into TAUMX1. (hr) Maximum time for cal c u l a t i o n s (hr) Time i n t e r v a l for p r i n t i n g output between TAUMX1 and TAUMX2. Must divide evenly into TAUMX1 and TAUMX2. (hr) I n i t i a l stepsize to be used i n c a l c u l a t i o n s i n v o l v i n g UBC Computing Centre subroutine "DRKC" (hr) Minimum value of stepsize to be used i n subroutine "DRKC" (hr) Maximum tolerance l e v e l allowed i n ca l c u l a t i o n s of subroutine "DRKC" KASO NK SL PMV XS(1),XS(2) Constant value of e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t (ml/hr/mmHg) S e n s i t i v i t y parameter for va r i a b l e e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t (hr/mmhg) Slope of alveolar f l u i d pressure-volume curve (mmHg/ml) Cir c u l a t o r y hydrostatic pressure (mmHg) X coordinates of legend curve representations ( l e f t and ri g h t values) for plots (inches) Continued.... - 8 8 -Table 6: Variables i n Input F i l e PDA (Continued) YS(1)-YS(4) Y coordinates of legend curve representations for plots (inches) SFT Value of increments on abscissa (time)(hr/inch) SFX Value of Increments on ordinate for f l u i d flow ((ml/hr)/inch) SFY Increments on ordinate for f l u i d volume (ml/inch) ZMN Minimum value of hydrostatic and c o l l o i d osmotic pressure for ordinate (mmHg) SFZ Increments on ordinate for hydrostatic and c o l l o i d osmotic pressure (mmHg/inch) SFK Increments on ordinate for e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t ((ml/hr/mmHg)/inch) SFR Increments on ordinate for solute weights (g/inch) SFU Increments on ordinate for solute concentrations ((g/ml)/inch) - 8 9 -whether plots and/or tables are desired. The parameters from TAUMX1 through to TOL apply to the time i n t e g r a t i o n steps and the time i n t e r v a l s for p r i n t i n g . The parameters from XS(1) through to SFU • pe r t a i n to the p l o t t i n g of the transient responses. Preceding each simulation i t w i l l generally be necessary to adjust the parameters i n f i l e s EDA and PDA to the desired values. 4.2.2 The Main Program UBCEDEMA The main program may also be divided into three sections. In the f i r s t section the parameter values l i s t e d i n Tables 5 and 6 are read into the computer. In addition, a number of variables and counters are i n i t i a l i z e d . The second section of UBCEDEMA involves the solu t i o n of the equations over the time i n t e r v a l s s p e c i f i e d i n f i l e PDA. The equations solved are l i s t e d i n Table 7. The stepout i n time for the integrations i s made using the Runge-Kutta-Merson technique as packaged by the UBC Computing Centre; the technique i s provided through the subroutine "DRKC". The beginning and end values of a time i n t e r v a l must be sp e c i f i e d for the subroutine. DRKC uses an extern a l l y declared subroutine FUNC to cal c u l a t e the values of the d i f f e r e n t i a l s JNET1, JAS, QNETA and QNETG; FUNC contains the equations l i s t e d i n Table 7. After i n t e g r a t i o n over the time i n t e r v a l subroutine "DRKC" provides the values of the int e g r a l s VIS1, VAS, QA and QG (whose d i f f e r e n t i a l s are JNET1, JAS, QNETA and QNETG r e s p e c t i v e l y ) . The series of equations are then solved with the new values of the int e g r a l s to y i e l d the desired variables; when a -90-Table 7: Equations used i n the Alveolar Model 1) JV = KF[(PMV-PPMV)-SIGD(PIMV-PIPMV)] 2) a) JSA = PSA(CMVA-CAVA) + (1-SIGFA)(CMVA + CAVA)JV/2. b) JSG - PSG(CMVG-CAVG) +. (1-SIGFG)(CMVG + CAVG)JV/2. 3) JL = 0.17 VIS1-55.6 4) JAS = KAS(PPMV-PAS) 5) JNET1 = JV-JL-JAS 6) VIS1 =Jt JNET1 6t + VIS10 7) VAS = 0/ t JAS 6t + VASO 8) a) QNETA =JSA-JL CTA-JAS CTA b) QNETG = JSG-JL CTG-JAS CTG 9) a) QA = Q/ t QNETA 6t + QAO b) QG = Q/ t QNETG 6t + QGO 10) VTOT = VIS1 + VAS 11) VIS = VIS1 - VCELL 12) a) VAVA = VIS-VEXA b) VAVG = VIS-VEXG 13) a) CTA = QA/VIS b) CTG = QG/VIS 14) a) CAVA = QA/VAVA b) CAVG = QG/VAVG 15) PIPMV = 210 CP + 1600 CP 2 + 9000 CP 3 Continued.... Table 7: Equations used i n the Alveolar Model (Continued) 16) PPMV = fn(VISl) (by in t e r p o l a t i o n ) 17) PAS = SL VAS + B 18) a) KAS = 0 for VIS1 < VTONS KAS = NK(VISl-VTONS) for VIS1 > VTONS or b) KAS = 0 for VTOT < VTONS KAS = KASO for VTOT > VTONS 19) CP = CAVA + CAVG -92-p r i n t i n g i n t e r v a l i s reached the variables are stored In array form f o r use i n the table and p l o t t i n g sections. When TAUMX2, the maximum time for cal c u l a t i o n s i s reached, tabulation begins as described i n the t h i r d section. 4.2.3 Tabulated and graphical output of variables from the computer program Table 8 l i s t s the variables that are tabulated and/or graphically I l l u s t r a t e d . The transient response of these variables i s recorded. The conditions of the simulation are l i s t e d before the output of the tables; the conditions l i s t e d consists of most of the variables i n the data f i l e s EDA and PDA. The value of variables at several c h a r a c t e r i s t i c points along the transient response were also recorded. The c h a r a c t e r i s t i c points and the variables recorded at these points are shown i n Table 9. 4.3 Characteristic Points Along the Transient Response of Variables The c h a r a c t e r i s t i c points chosen to represent the transient responses of the PMVES simulation are l i s t e d i n Table 9. 4.3.1 The onset of alvaeolar flooding This f i r s t c h a r a c t e r i s t i c point represents the point of t r a n s i t i o n i n a simulation from i n t e r s t i t i a l edema to alveolar edema. At the onset of alve o l a r flooding the variables recorded w i l l be: 1) the time to reach the onset of alveolar flooding ( t ( o n s e t ) ) , and 2) the r a t i o of the tissue albumin concentration to the plasma albumin concentration - CTA/CMVA. Table 8 : Variables Tabulated and/or Plotted by Main Program UBCEDEMA Variable Tabulated Plotted JV • / JL • • JAS • JNET1 • / VTOT / / VIS1 / / VAS / / VIS • / PPMV • • PAS / / •PAS) or (PDIF) / PIPMV • / KAS / • QA / QG / • VAVA / VAVG • CTA • CTG • • CAVA • CAVG QNETA QNETG - 9 4 -Table 9 : C h a r a c t e r i s t i c Points along Transient Response and Variables Recorded C h a r a c t e r i s t i c Point 1) Onset of alveolar flooding 2) Maximum T r a n s e p i t h e l i a l Flow 3) Total Extravascular F l u i d Volume of 1000 ml. Variables Recorded time CTA/CMVA JAS time VIS1 VTOT CTA/CMVA JAS time VIS1 CTA/CMVA - 9 5 -4.3.2 The point of maximum transepithelial flow In the simulations of the PMVES discussed i n this thesis the transient response of the t r a n s e p i t h e l i a l flow (JAS) has a maximum value. The variables recorded at th i s c h a r a c t e r i s t i c point are: 1) the maximum value of the t r a n s e p t h e l i a l flow JAS(max), 2) the time to reach a maximum JAS, 3) the EVEA and t o t a l extravascular f l u i d volumes, VIS1 and VTOT, res p e c t i v e l y , and 4) the r a t i o of the tissue albumin concentration to the plasma albumin concentration, CTA/CMVA(max). 4.3.3 The point at which the total extravascular fluid volume (VTOT) equals 1000 ml. Staub (13) and Gump et a l (48) have found that i n most cases where human lungs were subjected to acute edema the t o t a l extravascular f l u i d volume generally reached approximately 1100 ml., approximately one-half of the fun c t i o n a l r e s i d u a l capacity of the lung. The patients investigated died under intensive care a f t e r severe i n j u r y i n which i t was thought that respiratory f a i l u r e was the major contributing factor.(48) In th i s thesis a c h a r a c t e r i s t i c point was established below the t o t a l extravascular f l u i d volume of 1100 ml, at a VTOT of 1000 ml. The variables recorded at th i s c h a r a c t e r i s t i c point are: 1) the time to reach a VTOT of 1000 ml, 2) the t r a n s e p i t h e l i a l flow, JAS(1000), 3) the EVEA f l u i d volume, VIS1(1000), and the r a t i o of the tissue albumin concentration to the plasma albumin concentration, CTA/CMVA(1000). The behaviour of the exchange system to a perturbation may be -96-analyzed by comparing the variables between these c h a r a c t e r i s t i c points. The change i n the rate of alveolar flooding over the course of alveolar edema may be i l l u s t r a t e d by comparing the t r a n s e p i t h e l i a l flows JAS(max) and JAS(IOOO). The r a p i d i t y with which the f l u i d accumulates i n the extravascular space can be shown by comparing the time to reach the onset of alveolar flooding and the time to reach a VTOT of 1000 ml. Comparison of the EVEA f l u i d volume, VIS1, at JAS(max) and at a VTOT of 1000 ml indicates the amount of I n t e r s t i t i a l edema that formed In excess of the EVEA f l u i d volume at the onset of al v e o l a r flooding given by VTONS. The difference (VT0T-VIS1) at these two c h a r a c t e r i s t i c points y i e l d s the f l u i d volume of the a i r space. Also, comparison of the albumin concentration r a t i o s at the c h a r a c t e r i s t i c points i l l u s t r a t e s the transient behaviour of the protein concentrations. 4.4 Outline of Simulations for the Alveolar Model 4.4.1 Simulations to study the effect of the parameters introduced with the alveolar model: KAS, SL, VTONS, and B . The addition of the a i r space to the lumped compartment model developed by Bert and Pinder (29) introduced several new parameters: KAS, SL, VTONS and B. The e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t , KAS, was represented as: 1) KAS=KAS0, i f VTOT exceeded VTONS, or 2) KAS=NK(VIS1-VT0NS), i f VIS1 exceeded VTONS. If VTOT i n (1) and VIS1 i n (2) were less than VTONS, KAS was set equal to zero. Therefore, the expressions for KAS introduced the parameters KASO (for KAS as a -97-constant value) and NK (for KAS as a v a r i a b l e ) . Simulations were performed to study the e f f e c t of these parameters on the responses of the microvascular exchange system. The exchange system was subjected to a perturbation that increased the transendothelial flow of f l u i d and solute. For the simulations conducted for this thesis the perturbation was an elevated c i r c u l a t o r y hydrostatic pressure, usually to 50 mmHg. In Table 10 are l i s t e d the simulations that were ca r r i e d out. For example, when a set of simulations were conducted to study the parameter VTONS, the range of values of VTONS was 414 to 500 ml; the other parameters assumed values as shown i n row 4: NK = 0.5 hr~1mmHg"1, KASO = 0, SL = 0.25xl0" 2 mmHg/ml and B = -3.03 mmHg. 4.4.2 Simulations to study the effect of a maximum lymph flow In the alveolar model the lymph flow rate i s assumed to be given by equation (21), as a function of the EVEA f l u i d volume, VIS1. Observations by Drake et a l (34) and others, (17,18,49) have shown that the lymph flow reaches a maximum value: 1) during pulmonary edema induced by an elevated c i r c u l a t o r y hydrostatic pressure (PMV), 2) during pulmonary edema induced by changes i n the endothelial permeability to f l u i d and solutes. In th i s thesis simulations w i l l be performed to study the e f f e c t of a maximum lymph flow on the responses of the exchange system. A maximum lymph flow w i l l be represented as JL(max). An EVEA f l u i d volume, defined as VLMPH, w i l l correspond to JL(max). The point - 9 8 -Table 10: Simulations conducted to study the e f f e c t of parameters introduced with the alveolar model* Parameter value Varied Parameter KAS NK (hr^mmHg - 1) KASO (ml/hr/mmHg) SL (mmHg/ml) VTONS (ml) -B (mmHg) KAS NK 0.05 0.5 5.0 50.0 0 .25xl0- 2 460 -3.03 KASO 0 1.0 2.5 5.0 10.0 .25x10-2 460 -3.03 SL 0.5 50.0 0 .25x20"l+ .25xlO- 3 .25x10-2 -75x10-2 460 -3.03 VTONS 0.5 0 .25x10-2 414 to 500 -3.03 8 0.05 0 .25xl0- 2 460 -10.0 to 1.0 * Parameters from Table 11 are PMV = 50 mmHg KF = KF(normal) = 1.12 ml/hr/mmHg SIGD = .75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFG = .40 SIGFA = .60 - 9 9 -at which the lymph flow becomes a maximum may be s p e c i f i e d i n reference to the onset of alveolar flooding (17,34); since VTONS i s the f l u i d volume at the onset of alveolar flooding, representing JL(max) by VLMPH w i l l permit easy comparison between these two points. The value of the maximum lymph flow, JL(max), or i t s corresponding f l u i d volume, VLMPH, i s not c l e a r l y defined. Drake et a l (49) performed lymph flow experiments on dog lungs i n vivo. Their r e s u l t s showed that maximum lymph flow occurred at a PMV between 20 and 25 mmHg. The corresponding f l u i d volume for a PMV of 25 mmHg i s 414 ml. or 90% of VTONS. Prichard (17) suggested that alveolar flooding occurs shortly a f t e r the attainment of a maximum lymph flow, i . e . , VLMPH < VTONS. The range of values of VLMPH considered i n th i s study i s from approximately 414 ml. (JL(max)=14.8 ml/hr) to an extremely high value (e.g. 5000 ml) which indicates that there i s no maximum lymph flow. For VLMPH equal to the f l u i d volume VTONS(460 ml) the value of JL(max) i s 22.6 ml/hr. 4.4.3 Simulations to study the effect of the endothelial permeability parameters, endothelial filtration coefficient, and the circulatory hydrostatic pressure on the alveolar model. In the i n t e r s t i t i a l model Bert and Pinder (1984) investigated the e f f e c t on the pulmonary microcascular exchange system of changing the endothelial permeability parameters-SIGD, PSA, PSG, SIGFA and SIGFG - the endothelial f i l t r a t i o n c o e f f i c i e n t , KF, and the c i r c u l a t o r y hydrostatic pressure, PMV. One of the objectives of the -100-current work i s to observe the e f f e c t of the same parameters on the predictions of the PMVES simulation using the alveolar model. Table 11 l i s t s the simulations performed. In a l l three sets of simulations the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t , KAS, i s represented as a function of VIS1 (equation (28)). The c i r c u l a t o r y hydrostatic pressure (PMV) was raised to 50 mmHg i n the simulations i n v e s t i g a t i n g KF, and the permeability parameters. -101-Table 11: Simulations conducted to study the ef f e c t of the endo t h e l i a l permeability parameters and f i l t r a t i o n c o e f f i c i e n t , and the c i r c u l a t o r y hydrostatic pressure on the alveolar model* Parameter value Varied Parameter PMV (mmHg) KF/KF(norraal) APERM (%) * PMV 35 to 60 1 0 KF KF(normal) 50 1 to 10 0 APERM 50 1 0 to +100 * The permeability parameters are changed by a s p e c i f i c percent (APERM). I f APERM i s +50%, then: Normal Value Value a f t e r APERM of + 50% SIGD 0.75 0.375 PSA 3.00 4.50 PSG 1.00 1.50 SIGFA 0.40 .2 SIGFG 0.60 .3 Note that SIGD, SIGFA and SIGFG decrease and PSA and PSG increase. * Parameters from Table 10 were set at: NK = 0.5 h r - 1 mmHg-1 SL = .25xl0~ 2 mmHg/ml VTONS = 460 ml B = -3.03 mmHg -102-RESDLTS AND DISCUSSION 5.1 Transient Responses of the Pulmonary Microvascular Exchange  System for Constant Epithelial F i l t r a t i o n Coefficient, KAS The e f f e c t of changes i n KAS (defined by equation (29)) on the predictions of the PMVES simulation.was studied under the conditions l i s t e d i n Table 12. In the following section (5.2) the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t was represented by equation (28) as a function of VIS1, which resulted i n values for KAS i n the range of 0 to 10 ml/hr/mm Hg. Therefore, for the case of a constant e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t the values of KAS studied ranged from 1.0 to 10.0 ml/hr/mm Hg. At time zero the PMVES was subjected to a step change i n PMV from a normal value of 9.0 mm Hg. to 50 mm Hg. The time for the onset of alveolar flooding i n the simulations conducted for a constant KAS was 2 hrs. Preceding the onset of al v e o l a r flooding the f l u i d volume, VIS1, r i s e s to a value of VTONS (460 ml), as shown i n Figure 19. The t r a n s e p i t h e l i a l flow i s determined as the product of KAS and (PPMV-PAS). At the onset of alveolar flooding the pressure difference (PPMV-PAS) w i l l be s i m i l a r f o r a l l KAS. Therefore, the value of JAS depends on the value of KAS; Figure 20 shows that as KAS increases, JAS at the onset of alveolar flooding increases. This value of JAS w i l l influence the subsequent responses of variables - such as PAS or VIS1 - predicted by the PMVES simulation. The alveolar f l u i d pressure, PAS, i s determined as a function of VAS by equation (27). The d i f f e r e n t i a l of PAS with respect to time gives: -103-Table 12 Conditions of PMVES Simulations Conducted to Study Changes i n KAS* KAS = KASO for VTOT > VTONS KASO = 1.0, 2.5, 5.0 and 10.0 ml/hr/mm Hg PMV = 5 0 mm Hg KF = 1.12 ml/hr/mm Hg APERM = 0 SIGD = 0.75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFA = 0.4 SIGFG = 0.6 VTONS = 460. ml SL = 0.25x10"2 mm Hg/ml B = -3.03 mm Hg **VLMPH = 5000. ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 19,20,21. * * I f VLMPH = 5000 ml, then there i s no maximum lumph flow, i . e . JL(max) = N.A. -104-Figure 19 Transient Responses of VIS1 for Constant KAS (Conditions as i n Table 12) - Response continued to time when VTOT = 1000 ml. 550 £ 500 CO 450 O > 400-350- I 0 i 20 40 60 80 TIME (hrs) Legend KAS= to KAS= 2.5 KAS= 5.0 KAS = 10.0 100 120 -105-Figure 20 Transient Responses of JAS for Constant KAS (Conditions as i n Table 12) - Response continued to time when VTOT = 1000 ml. 50 40 H 00 < Q 30 H 20 H O H r 0 i 20 40 60 80 TIME (hrs) Legend KAS= to KAS = 2.5 KAS= 5.0 KAS = 10.0 100 120 -106-dPAS dVAS = SL . - 3 — (34) dt dt for constant SL and B. The de r i v a t i v e (dVAS/dt) i s equivalent to JAS, so that: 2^2- = SL .JAS (35) dt Equation (35) i l l u s t r a t e s that the slope of the PAS curve from the onset of alveolar flooding depends on the value of JAS, hence KAS. Figure 21 shows that as KAS increases, the slope of the PAS curve increases. The i n t e r s t i t i a l f l u i d pressure (PPMV) i s determined as a function of VIS1 through the tissue compliance curve. The d e r i v a t i v e of VIS1 i s equal to JNET1 by d e f i n i t i o n , so that: JNET1 = d V ^ S 1 = JV - JL - JAS (36) at The value of (dVISl/dt) depends on the values of JAS, JV and JL. The response of VIS1 immediately a f t e r the onset of alveolar flooding i s shown i n Figure 19. For the case where KAS = 10 ml/hr/mm Hg VIS1 decreases a f t e r onset, as JAS i s greater than (JV-JL) which causes JNET1 to be negative. Reducing KAS to 5.0 ml/hr/mm Hg and below res u l t s i n p o s i t i v e values for JNET 1, and a r i s e i n VIS1 following the onset of alveolar flooding. Following the onset of alveolar flooding the predictions of the PMVES simulation for KAS = 10 ml/hr/mm Hg are as follows: The v a r i a b l e PPMV decreases as VIS1 decreases; the r i s e i n PAS and the decline i n PPMV causes a rapid decrease i n JAS (Figure 20). As would be expected the lymph flow, JL, decreases as VIS1 decreases. -107-Figure 21 Transient Responses of PAS for Constant KAS (Conditions as i n Table 12) - Response continued to time when VTOT = 1000 ml. -1.5 - i rj) E CO Ld o: CO CO Ld CL O -2H -2.5 H -3H o Q -4.5 H r 0 20 60" 40  80 TIME (hrs) Legend KAS= 10 KAS= 2.5 KAS= 5.0 KAS = 10.0 —T— 100 120 -108-Re c a l l i n g equation (36), the decrease i n JAS and JL leads to an increase i n JNET1 (or a decrease i n the ne g a t i v i t y i n JNET1). As the simulation proceeds VIS1 w i l l eventually decrease to a value where JNET1 equals zero; Figure 19 shows that for KAS = 10 ml/hr/mm Hg the f l u i d volume VIS1 reaches a minimum value. JNET1 then becomes p o s i t i v e and VIS1 increases (Figure 19). PPMV and JL w i l l also increase again. However, the pressure difference (PPMV-PAS) continues to decrease, and as a re s u l t JAS decreases (Figure 20). In Figures 19, 20, and 21 the res u l t s of the simulation were recorded u n t i l a VTOT of 1000 ml. was attained; steady state was not achieved at t h i s point. For KAS = 10 ml/hr/mm Hg the rate of f l u i d leaving the i n t e r s t i t i u m exceeds the rate of f l u i d entering. The f l u i d d e f i c i t , i n t h i s case, was made up by depleting the i n t e r s t i t i u m of i t s f l u i d ; the f l u i d volume VIS1 decreased from i t s value at the onset of al v e o l a r flooding (Figure 21). For the case of KAS equal to 5, 2.5 and 1.0 ml/hr/mm Hg, following the onset of alveolar flooding VIS1 continued to r i s e (Figure 21). -109-5.2 Transient Responses of the Pulmonary Microvascular Exchange System for Variable Epithelial Filtration Coefficient (KAS) The variable e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t , KAS, i s calculated as: KAS = NK(VIS1 - VTONS) for VIS1 > VTONS (28) The e f f e c t of this representation of KAS on the PMVES was investigated under the conditions given i n Table 13 - which were chosen according to the c r i t e r i a given i n the model d e s c r i p t i o n s . At time zero the PMVES was subjected to a step change i n the c i r c u l a t o r y hydrostatic pressure from a normal value of 9 mm Hg to 50 mm Hg. In th i s section the transient responses of the PMVES w i l l be discussed using Figures 22a to 22e. During the simulation the variables l i s t e d i n Table 14 w i l l increase from t h e i r steady state values at a normal PMV of 9 mm Hg to the i n i t i a l values given for PMV = 50 mm Hg. The response of the PMVES proceeds through a phase of i n t e r s t i t i a l edema followed by alveolar edema. During the transient response a number of variables are integrated over time. These are: the rate of f l u i d accumulation i n the i n t e r s t i t i a l (and c e l l u l a r ) space (JNET1) and i n the a i r space (JAS), and the rate of accumulation of solutes i n the i n t e r s t i t i a l space (QNETA, QNETG). The integrated variables y i e l d VIS1, VAS, QA and QG, r e s p e c t i v e l y . Figures 22a and 22b show that during the i n t e r s t i t i a l edema phase (time zero to 2.3 hrs) QA, QG (Figure 22a) and VIS1 (Figure 22b) r i s e as a re s u l t of the p o s i t i v e values of QNETA, QNETG, and JNET1; JAS i s zero. The r i s e i n -110-Table 13 Conditions of the PMVES Simulations for a Variable E p i t h e l i a l F i l t r a t i o n C o e f f i c i e n t * KAS = NK(VIS1-VTONS) for VIS1 > VTONS NK = 0.5 h r - 1 mmHg-1 PMV = 50 mmHg KF = 1.12 ml/hr/mmHg APERM = 0 SIGD = 0.75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFA = 0.40 SIGFG = 0.60 VTONS = 460. ml SL = 0.25x10"2 mmHg/ml B = -3.03 mmHg VLMPH = 5000. ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 22a-22e. Table 14 In i t i a l Conditions of PMVES Simulation at a PMV of 50 mmHg Variable Steady State I n i t i a l Value Value at a at a PMV=9 mmHg PMV=50 mmHg JV (ml/hr) 8.82 54.7. JNET1 (ml/hr) 0. 46.0 QNETA (g/hr) 0. 0.66 QNETG (g/hr) 0. 0.70 -112-Figure 22a Transient Responses of QA, QG, QNETA and QNETG for Variable KAS with NK = 0.5 hr 1 mmHg 1 (Conditions as i n Table 13) 1.5 v2 o _ i L i _ Ld O tn 0 . 5 • 0 . 5 / — Q A ' V— QNETA \ 1—QNETG \ \ 1 i 0 T 5 -6 8 O U J UJ I— o (71 10 15 TIME (hrs) 2 0 25 Figure 22b Transient Responses of JV, JL, JNETl, VIS1, PPMV and PIPMV for Variable KAS with NK = 0.5 h r - 1 mmHg-1 (Conditions as in Table 13) 2 - | r 4 8 0 CD X E an ZD ui ui Ld cn Q_ ( J Ul o Q >-0 -- 1 -- 2 -- 3 - 4 - J Ul o Q ZD r 2 0 10 15 TIME (hrs) - 1 8 cn E J , LU C£ LO (/) Ld cc: Q_ o 14 O o 16 L 1 2 I -114-VIS1 r e s u l t s i n an Increase i n PPMV according to the tissue compliance curve, and also an increase i n JL according to equation (21). Figure 22b shows these i n t e r r e l a t i o n s h i p s . The r a t i o of the solute weights (QA and QG) to the i n t e r s t i t i a l volume (VIS=VIS1-VCELL) gives the tissue protein concentrations CTA and CTG. The responses of CTA and CTG are dependent on the rates of change i n QA or QG and VIS. As shown i n Figure 22c, during the i n t e r s t i t i a l edema phase CTA and CTG decli n e . The e f f e c t i v e protein concentrations, CAVA and CAVG, also decrease. The tissue oncotic pressure PIPMV, decreases since i t i s related to the t o t a l e f f e c t i v e protein concentration (CAVA + CAVG). The decrease i n PIPMV and r i s e i n PPMV results i n a decrease i n JV (Figure 22b). The decrease i n JV and increase i n JL cause JNET1 to decrease (Figure 22b). Eventually VIS1 reaches the f l u i d volume corresponding to the onset of alveolar flooding, VTONS. For the simulation conducted under the conditions given i n Table 13 the onset of alveolar flooding occurs at 2.5 hrs. The t r a n s e p i t h e l i a l flow, JAS, that r e s u l t s i s calculated by equation (25). Whether JAS r i s e s or f a l l s i s dependent on the rates of change of PAS, KAS and PPMV: *Um = KAS + (PPMV-PAS) -KAS d ^ A S > (37) dt at at at Immediately following the onset of alveolar flooding JAS increases -Figure 22d shows the responses of PPMV, KAS, PAS and JAS. The r i s e i n JAS re s u l t s i n a further decline i n JNET1 (Figure 22e), -115-Figure 22c Transient Responses of QA, QG, VIS1, CTA and CTG for Variable KAS with NK = 0.5 h r - 1 mmHg-1 (Conditions as i n Table 13) 480-1 0.030 460-*j= 440 H Ld ZD 420 H __l O > Q 400-j ZD _ J 1_L_ 380 H 360 J 0.025 H 8 E o LY. O 0.015-j ZZ. o o Ld o 0.005 IT cn -6 ^—^ 00 1 ~c o Ld Ld 1 -4 ZD 1 — l O (/) 10 15 TIME! (hrs) -116-Figure 22d Transient Responses of JAS, PAS, KAS and PPMV for Variable KAS with NK = 0.5 h r - 1 mmHg-1 (Conditions as i n Table 13) TIME (hrs) -117-which i n turn reduces the rate of increase of VIS1. The reduction i n the rate of change of VIS1 diminishes the rate of change of PPMV and KAS. JAS reaches a maximum value (Figure 22d) when JAS equals (JV-JL) or (d(JAS)/dt) equals zero. JAS then declines since the rate of change of PAS exceeds the rates of change of PPMV and KAS (Figure 22d). As JAS declines, JNET1 increases, which raises VIS1. The r i s e i n VIS1 increases PPMV and JL (Figure 22b). If PPMV increases then JV w i l l decrease (Figure 22b). Figure 22e i l l u s t r a t e s the transient responses of JV, JL, JAS and JNET1. Steady state i s achieved when JV equals JL, I.e., JNET1 and JAS equal zero. 5.2.1 Transient Response of the Pulmonary Microvascular Exchange System to changes in NK In the case of a variable e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t , NK represents the pr o p o r t i o n a l i t y between KAS and the f l u i d volume difference (VIS1-VT0NS). The conditions i n the PMVES under which NK was studied are l i s t e d i n Table 15. At time zero the PMVES was subjected to a step change i n PMV to 50 mm Hg. Different orders of magnitude of NK were studied, ranging from 0.05 h r - 1 mmHg-1 to 50.0 h r - 1 mmHg-1. The transient responses of JAS for d i f f e r e n t NK are i l l u s t r a t e d i n Figure 23a. The time for the onset of alveolar flooding (JAS > 0) i s appoximately 2.5 hrs. A f t e r the onset of alveolar flooding JAS i n -118-Figure 22e Transient Responses of JV, JL, JNET1 and JAS for Variable KAS with NK = 0.5 h r " 1 mmHg-1 (Conditions as i n Table 13): Insert i l l u s t r a t e s responses up to steady state (approximately 300 h r s ) . -119-Table 15 Conditions of the PMVES Simulations Conducted to Study Changes i n NK* KAS =NK(VIS1-VT0NS) for VIS1 > VTONS NK = 0.05, 0.5, 5.0 and 50.0 h r - 1 mmHg-PMV = 50 mmHg KF = 1.12 ml/hr/mmHg APERM = 0 SIGD = 0.75 PSA = 3.00 ml/hr PSG = 1.00 ml/hr SIGFA = 0.40 SIGFG = 0.60 VTONS = 460. ml SL = 0.25xl0 - 2 mmHg/ml B -3.03 mmHg VLMPH = 5000. ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 23a and 23b. -120-Figure 23a Transient Responses of JAS for Changes i n NK (Conditions as i n Table 15) - Response continued to time when VTOT = 1000 ml. -121-each case r i s e s to a maximum; as NK increases the maximum JAS also increases. The t r a n s e p i t h e l i a l flow i s determined by the product of KAS and (PPMV-PAS), as shown i n equation (25). For the conditions of Table 15 the value of (PPMV-PAS) i s 4.12 mm Hg at the onset of alveolar flooding for a l l cases of NK. Therefore JAS i s dependent on the value of KAS which i s composed of the product of NK and (VIS1-VTONS). Figure 23b shows that as NK increases (VIS1-VT0NS) decreases (VTONS=460 ml). When NK i s increased by a factor of 10, from 0.05 to 0.5 h r - 1 mmHg-1, the reduction i n (VIS1-VT0NS) i s less than a factor of 10; JAS increases as NK i s raised from 0.05 to 0.5 h r - 1 mmHg-1 (Figure 23a). However, r a i s i n g NK by a factor of 10, from 5.0 to 50.0 h r - 1 mmHg-1 reduces (VIS1-VT0NS) by a factor of almost 10. The benefit of the increase i n NK i s eliminated and the r e s u l t i n g JAS for NK=5.0 and 50.0 h r - 1 mmHg-1 are almost equal (Figure 23a). Therefore, as NK i s increased the s e n s i t i v i t y of the c o e f f i c i e n t KAS to (VIS1-VT0NS) increases. It may be surmised that f o r low values of NK (e.g. NK=.05 h r - 1 mmHg-1) a large increase i n VIS1 i s required to induce the movement of f l u i d and solutes across the alveolar membrane. If NK i s large (e.g. NK=5.0 or 50.0 h r - 1 mmHg-1) then very l i t t l e increase i n VIS1 i s required to induce the movement of f l u i d and solute across the alveolar membrane. A value of NK=0.5 h r - 1 mmHg-1 was used i n the remaining simulations since i t allowed some increase i n VIS1. -122-Figure 23b Transient Responses of VIS1 for Changes i n NK (Conditions as i n Table 15) - Response continued to time when VTOT = 1000 ml. -123-5.3 The Response of the Pulmonary Microvascular Exchange System to  changes i n the parameter VTONS The parameter VTONS represents the EVEA f l u i d volume at the onset of alveolar flooding. The e f f e c t of changes i n VTONS on the PMVES were studied under the conditions l i s t e d i n Table 16. At time zero the PMVES was subjected to a step change i n PMV from 9 to 50 mmHg. Since a variable e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t with an NK of 0.5 h r - 1 mmHg-1 was employed, the shape of the transient responses of the variables i s s i m i l a r to those discussed i n Section 5.2. For VIS1 less than VTONS f l u i d and solute exchange take place between the c i r c u l a t o r y , i n t e r s t i t i a l and lymphatic compartments only ( i . e . , i n t e r s t i t i a l edema i s simulated). A simulation of i n t e r s t i t i a l edema w i l l i l l u s t r a t e that steady state conditions may be achieved for a given perturbation to the PMVES; steady state i s established when the transendothelial and lymph flows for both f l u i d and solutes are equal. If the perturbation i s an elevated PMV, then the i n t e r s t i t i a l (and c e l l u l a r ) f l u i d volume at steady state (VISl(ss)) may be obtained for each PMV (Figure 24). The other conditions of the simulation are l i s t e d with Figure 24. In the alveolar model, r a i s i n g VTONS to a very large value, i n th i s case 5000 ml w i l l r e s u l t i n the modelling of i n t e r s t i t i a l edema. Figure 24 i l l u s t r a t e s that at a PMV of 50 mmHg the f l u i d volume VISl(ss) i s 566 ml. In the alveolar model a VTONS greater than 566 ml w i l l result i n i n t e r s t i t i a l edema only. The PMVES w i l l reach steady state when VIS1 reaches 566 ml. However, i f VTONS i s set below a value of 566 ml, -124-Table 16 Conditions of the PMVES Simulations to study the changes i n VTONS* VTONS: varied from 410 to 500 ml KAS = NK(VIS1 - VTONS) f o r VIS1 > VTONS Nk = 0.5 h r - 1 mmHg-1 PMV = 50 mmHg KF = 1 . 1 2 ml/hr APERM = 0 SIGD = 0.75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFA = 0.4 SIGFG = 0.6 SL = 0.25xl0 - 2 mmHg/ml B = -3.03 mmHg VLMPH = 5000. ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 25a to 25d. -125-Figure 24 Steady state values of VIS1 for given PMV for I n t e r s t i t i a l Edema (Conditions: VTONS = 5000 ml, VLMPH = 5000 ml, KF = 1.12 ml/hr/mmHg, SIGD = 0.75, PSA =3.0 ml/hr, PSG =1.0 ml/hr, SIGFA = 0.4, SIGFG =0.6) 750-1 0 1 400-1 , , —, , , 1 20 30 40 50 60 70 80 CIRCULATION HYDROSTATIC PRESSURE PMV (mm -126-Figure 25a Transient Responses of (JV-JL) and JAS for VTONS of 420 ml and 500 ml (Conditions as i n Table 16) - Response continued to time when VTOT = 1000 ml. Legend (JV-JL), VTONS=420  JAS, VTONS=420 (JV-JL), VTOHS=500 JAS, VTONS=5O0 I I I I I 10 20 30 40 50 TIME (hrs) -127-the f l u i d volume VIS1 w i l l reach a value equal to VTONS and alve o l a r flooding w i l l occur. As VTONS i s reduced from 566 ml the difference between JV and JL at the onset of alveolar flooding Increases. Figure 25a shows the transient respone of (JV-JL) up to VT0T=1000 ml for a VTONS of 420 ml and 500 ml; the time for the onset of alveolar flooding i s approximately 1.0 and 4.5 hrs., r e s p e c t i v e l y . The value of (JV-JL) i s greater for VT0NS=420 ml than a VT0NS=500 ml. Since (JV-JL) i s equivalent to the rate of f l u i d accumulation i n the t o t a l extravascular space ( i n t e r s t i t i a l + c e l l u l a r + alveolar) the time to reach a VTOT of 1000 ml (Figure 25b) and to the onset of flooding are less for the smaller VTONS. Figure 25a also i l l u s t r a t e s that following the onset of alveolar flooding JAS r i s e s to a maximum very r a p i d l y - the shape of the JAS curve was discussed i n section 5.2. As (JV-JL) at the onset of alveolar flooding increases, JAS also increases. Therefore, as VTONS i s reduced JAS increases (Figure 25c). The transient response of JAS i s i l l u s t r a t e d i n Figure 25c by showing the maximum JAS and the JAS at a VTOT of 1000 ml. The simulations of the PMVES for changes i n VTONS also I l l u s t r a t e that the EVEA f l u i d volume at a VTOT of 1000 ml increases as VTONS r i s e s (Figure 25d); due to the increase i n i n t e r s t i t i a l f l u i d accumulation before VIS1 reaches VTONS. In section 3.4 the upper l i m i t of VTONS experimentally observed was approximately 600 ml. For the simulations conducted i n the study of VTONS the upper value of VTONS was only 500 ml. The lower value of -128-Figure 25b Time to Reach a VTOT of 1000 ml for Di f f e r e n t VTONS (Conditions as i n Table 16) -129-Figure 25c The Maximum T r a n s e p i t h e l i a l Flow (JAS(max)) and the JAS at a VTOT of 1000 ml for Dif f e r e n t VTONS (Conditions as i n Table 16) 30 i r i i i i r 400 420 440 460 480 500 520 VTONS (ml) -130-Figure 25d The F l u i d Volume VIS1 at a VTOT of 1000 ml for Different VTONS (Conditions as i n Table 16) 520 visi(iooo) 400 420 440 460 480 500 520 VTONS (ml) -131-VTONS var i e s , depending on the o r i g i n of the perturbation. As VTONS i s lowered the rate of alveolar flooding increases. An i n t e r s t i t i a l hydrostatic pressure, PPMV (onset), corresponds to the EVEA f l u i d volume VTONS, as determined by the tissue compliance curve. One may speculate that the alveolar b a r r i e r s cannot withstand i n t e r s t i t i a l hydrostatic pressures above PPMV (onset). Guyton et al.(47) suggested that the alveolar b a r r i e r s cannot withstand p o s i t i v e i n t e r s t i t i a l pressure. In the remaining simulations VTONS was set at 460 ml, which corresponds to an i n t e r s t i t i a l hydrostatic pressure of +1.1 mmHg. 5.4 Transient Responses of the Pulmonary Microvascular Exchange  System to changes i n the Parameter SL The parameter SL i s defined as the slope of the expression r e l a t i n g PAS to VAS - i n equation (27). The e f f e c t of SL on the PMVES was studied under the conditions shown i n Table 17. At time zero the PMVES was subjected to a step change i n PMV from a normal value of 9 mmHg to 50 mmHg. Tests were made with values of SL between 0.25xl0 - 1 + mmHg/ml and 0.75xlO~ 2 mmHg/ml. In the simulations conducted to study changes i n SL the alveo l a r f l u i d volume at a VTOT of 1000 ml was approximately 500 ml. Using equation (27), the change i n the alveolar f l u i d pressure that corresponds to a change i n VAS of 500 ml i s 0.0125 mmHg for SL=0.25xlO _ l t mmHg/ml and 3.75 mmHg for SL=0.75xl0 - 2 mmHg/ml. For the case of SL=0.25xl0 - 1 + mmHg/ml the change i n PAS i s i n s i g n i f i c a n t as compared to the change for SL=0.75xlO - 2 mmHg/ml. The variable PAS -132-i s used i n evaluating the pressure gradient that causes f l u i d flow from the i n t e r s t i t i u m to the a i r space ( i . e . (PPMV-PAS)); for a given PPMV at a VTOT of 1000 ml and the conditions of Table 17, (PPMV-PAS) decreases as SL increases. The e f f e c t of SL on the PMVES was studied for NK=0.05 hr _ 1mm Hg - 1 and NK=50.0 h r " 1 mm Hg - 1. ( i ) Simulations with NK=0.05 h r - 1 mm Hg - 1. Figure 26a shows that following the maximum JAS, the rate of decrease i n JAS r i s e s as SL i s increased. This i s caused by the r i s e i n PAS as SL i s increased (Figure 26b). The f l u i d material balance of equation (32) i l l u s t r a t e s that (JNET1 + JAS) equals (JV-JL). As the simulation of the PMVES progresses to steady state conditions (JV-JL) decreases. Following the onset of alveolar flooding the change i n (JV-JL) i s generally small during non-steady state conditions to a VTOT of 1000 ml. Therefore, as JAS decreases i n the period just before a VTOT of 1000 ml, JNET1 w i l l increase. The increase i n JNET1 may be r e f l e c t e d by the change i n the transient response of VIS1. Since the rate of decline i n JAS increases as SL i s raised, JNET1 would increase with SL. Figure 26c shows that the increase i n VIS1 i s more pronounced as SL increases. ( i i ) Simulations with NK=50.0 h r - 1 mm Hg - 1. Following the time of the onset of alveolar flooding (2.5 hr s ) : for NK=50 h r - 1 mm Hg - 1 an Increase i n SL also increases the alveolar f l u i d pressure, as seen by Figure 27a. However, the transient responses of JAS, up to the time that VTOT=1000 ml, i s not -133-Table 17 Conditions of the PMVES Simulation Conducted to Study Changes i n SL* SL = 0.25 x I O - 4 , 0.25 x 10 - 3, 0.25 x 1 0 - 2 and 0.75 x I O - 2 mmHg/ml KAS = NK(VISl-VTONS) for VISL > VTONS NK = 0.5, 50. h r - 1 mm Hg - 1 PMV = 5 0 mm Hg KF = 1.12 ml/hr/mm Hg APERM = 0 SIGD = 0.75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFA = 0.40 SIGFG = 0.60 VTONS = 460 ml B = -3.03 mm Hg VLMPH = 5000. ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 26a,b,c and 27 a,b. -134-Figure 26a Transient Responses of JAS for Different SL and NK = 0.05 hr mmHg"1 (Conditions as i n Table 17) - Responses continued to time when VTOT = 1000 ml 15 TIME (hrs) -135-Figure 26b Transient Responses of PAS for Dif f e r e n t SL and NK = 0.05 hr mmHg-1 (Conditions as i n Table 17) - Responses continued to time when VTOT = 1000 ml TIME (hrs) -136-Figure 26c Transient Responses of VIS1 for Dif f e r e n t SL and NK = 0.05 hr mmHg-1 (Conditions as i n Table 17)-Responses continued to time when VTOT = 1000 ml 550-i £ 500-> 450 O > 400-350-Legend SL = 0.25E-4 SL = 0.25E-3 SL = O^SE-2 SL = 0.75E-2 20 40 60 80 TIME (hrs) -137-F i g u r e 27a T r a n s i e n t Responses o f PAS f o r D i f f e r e n t SL and NK = 50.0 hr mmHg-1 ( C o n d i t i o n s as i n T a b l e 17) - Responses c o n t i n u e d t o time when VTOT = 1000 ml TIME (hrs) -138-Figure 27b Transient Responses of JAS for Different SL and NK = 50.0 hr mmHg-1 (Conditions as in Table 17) - Responses continued to time when VTOT = 1000 ml 0 10 TIME (hrs) 20 Legend SL = 0.25E-4 SL = 0.25E-3 SL = 0.25E-2 SL = 0.75E-2 30 -139-affected by the d i f f e r e n t values of SL - see Figure 27b. In the determination of JAS the variables VIS1 and PAS exhibit competing e f f e c t s ; the r i s e i n PAS attempts to suppress JAS, while a r i s e i n VIS1 attempts to augment JAS by increasing KAS and PPMV. In the s i t u a t i o n where NK=50 h r - 1 mm Hg - 1, the difference (VIS1-VT0NS) need only r i s e by 0.1 ml for a doubling of KAS; JAS i s , thus, more responsive to KAS than to PAS. 5.5 Responses of the Pulmonary Microvascular Exchange System to  changes in the parameter B. The parameter B represents the value of the alveolar f l u i d pressure at the onset of alveolar flooding. The e f f e c t of B on the predictions of the PMVES simulation was studied under the conditions l i s t e d i n Table 18. At time zero the PMVES was subjected to a step change in PMV from 9 mm Hg to 50 mm Hg. For a l l values of B studied, the time for onset of alveolar flooding i s about 2.5 hrs. Following the onset of alveolar flooding the d r i v i n g force that causes t r a n s e p i t h e l i a l flow i s (PPMV-PAS). As B decreases, the i n i t i a l value of the pressure gradient (PPMV-PAS) w i l l increase. The transient response of JAS may be I l l u s t r a t e d by recording the maximum JAS (JAS (max)) and the JAS at a VTOT of 1000 ml (JAS (1000)) for each value of B (Figure 28a); the transient response of JAS between these two points i s a gradual decrease from JAS (max) to JAS (1000). Figure 28a i l l u s t r a t e s that JAS (max) and JAS (1000) increase as B i s reduced from approximately 1.0 mm Hg to -4 mm Hg, but does not change -140-Table 18 Conditions of the PMVES Simulations Conducted to Study the Changes i n B* B: varied from -10 to 1.0 mm Hg KAS = NK(VISl-VTONS) for VIS1 > VTONS NK = 0.5 h r - 1 mm Hg - 1 PMV = 5 0 mm Hg KF = 1.12 ml/hr/mm Hg APERM = 0 SIGD = 0.75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFA = 0.40 SIGFG = 0.60 VTONS = 460 ml SL = 0.25xlO" 2 mm Hg/ml VLMPH = 5 0 0 0 ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 28a to 28g. -141-Figure 28a The maximum t r a n s e p i t h e l i a l flow (JAS(max)) and JAS at a VTOT of 1000 ml for Different B (Conditions as i n Table 18) 1 I I I I I [ -10 - 8 - 6 - 4 - 2 0 2 PAS @ ONSET OF ALVEOLAR FLOODING, B (mm -142-s i g n i f I c a n t l y for B less than -4 mm Hg. The following discussion w i l l treat these regions separately. ( i ) The region of B less than -4 mm Hg. JAS i s determined by the product of KAS and (PPMV-PAS). Figure 28b i l l u s t r a t e s the transient response of (PPMV-PAS) for B=-10 mm Hg and B= -6 mm Hg; the area of i n t e r e s t i s for time greater than the time at the onset of alveolar flooding (2.5 h r s ) . As B i s decreased from -6 mm Hg to -10 mm Hg, (PPMV-PAS) Increases. Figure 28c shows the transient response of KAS for B= -10 mm Hg and B= -6 mm Hg; as B i s decreased from -6 mm Hg to -10 mm Hg, KAS decreases. The factor by which (PPMV-PAS) increases as B i s lowered from -6 mm Hg to -10 mm Hg i s equivalent to the factor by which KAS decreases as B i s lowered from -6 mm Hg to -10 mm Hg. Therefore, the product of KAS and (PPMV-PAS) for B= -6 mm Hg y i e l d s a value approximately equal to the product of KAS and (PPMV-PAS) for B= -10 mm Hg. Figure 28a i l l u s t r a t e s that JAS(max) and JAS(1000) for B= -6 mm Hg and -10 mm Hg are approximately equal. Since JAS(1000) i s the same for values of B less than -4 mm Hg, the time to reach a VTOT of 1000 ml i s also s i m i l a r (Figure 28d). ( i i ) The region of B greater than -4 mm Hg For the region of B greater than -4 mm Hg the transient responses of (PPMV-PAS) and KAS are i l l u s t r a t e d for B= -3 mm Hg and B= 0.5 mm Hg i n Figures 28e and 28f r e s p e c t i v e l y . Figure 28e i l l u s t r a t e s that following the time of the onset of alveolar flooding (2.5 hrs) - 1 4 3 -Figure 28b Transient Responses of (PPMV-PAS) for B = -10 mmHg and B = -6 mmHg (Conditions as in Table 18) - Responses continued to time when VTOT = 1000 ml -144-Figure 28c Transient Responses of KAS for B = -10 mmHg and B = -6 mmHg (Conditions as i n Table 18) - Responses continued to time when VTOT = 1000 ml Legend B = -10 B = - 6 0 10 20 30 -145-Figure 28d Time to reach a VTOT of 1000 ml for D i f f e r e n t B (Conditions as i n Table 18) 55 -10 - 8 -6 - 4 - 2 0 2 PAS @ ONSET OF ALVEOLAR FLOODING, B (mm -146-Figure 28e Transient Responses of (PPMV-PAS) for B = -3 mmHg and B = 0.5 mmHg (Conditions as i n Table 18) - Responses continued to time when VTOT = 1000 ml -147-Figure 28f Transient Responses of KAS for B = -3 mmHg and B = 0.5 mmHg (Conditions as i n Table 18) - Responses continued to time when VTOT = 1000 ml -148-the pressure difference (PPMV-PAS) i s less for B = 0.5 mm Hg than for B = -3 mm Hg, while i n Figure 28f KAS increases as B i s increased from -3 mm Hg to 0.5 mm Hg. After a time of about 7.5 hrs. the slope of the KAS-time curve for B = 0.5 mm Hg i s greater than for B = -3 mmHg. However, the product of KAS and (PPMV-PAS) at JAS(max) and at JAS(IOOO) i s less f or B = 0.5 mm Hg than for B = -3 mm Hg (Figure 28a). As B i s raised from -4 mm Hg, the factor by which (PPMV-PAS) decreases i s higher than the factor by which KAS increases. The decline i n JAS as B i s raised above -4 mm Hg leads to an increase i n the time to reach a VTOT of 1000 ml (Figure 28d). An increase i n B also causes a r i s e i n the f l u i d volume VIS1 at a VTOT of 1000 ml, as shown i n Figure 28g. At a PMV of 50 mm Hg and the conditions of Table 18 the steady state VIS1 i s 566 ml (as stated i n section 5.3). At an EVEA f l u i d volume of 460 ml (VTONS) the i n t e r s t i t i a l hydrostatic pressure, PPMV, i s 1.09 mm Hg. As B i s increased towards 1.09 mm Hg the value of KAS increases; Figure 28f shows that KAS increases when B i s raised from -3 mm Hg to 0.5 mm Hg. Corresponding to the increase i n KAS i s a r i s e i n VIS1. If VIS1 r i s e s to 566 ml, then steady state w i l l be achieved and alveolar flooding terminated. As B approaches 1.09 mm Hg the t o t a l extravascular f l u i d volume (VIS1+VAS) at steady state w i l l decrease; the f l u i d volume VAS, i n p a r t i c u l a r w i l l decrease. The bulk of the edema f l u i d i s then contained i n the i n t e r s t i t i a l compartment. If B i s elevated to values above 1.09 mm Hg, and conditions remain at those stated i n Table 18, then the pressure gradient -149-Figure 28g F l u i d Volume VIS1 at a VTOT of 1000 ml for D i f f e r e n t B (Conditions as i n Table 18) -150-(PPMV-PAS) at the onset of a l v e o l a r flooding w i l l become negative, and the t r a n s e p i t h e l i a l flow w i l l be i n i t i a l l y from the alveolar space to the i n t e r s t i t i a l space. However, the f l u i d volume of the a i r space at the onset of alveolar flooding i s zero; a t r a n s e p i t h e l i a l flow from the a i r space to the i n t e r s t i t i a l space i n i t i a l l y w i l l reduce VAS to negative values. Since negative values i n VAS are u n r e a l i s t i c , the upper l i m i t of B i s 1.09 mm Hg or the i n t e r s t i t i a l pressure corresponding to a f l u i d volume of VTONS. The alveolar f l u i d pressure (PL=PAS) i s related to the gas pressure (PG) according to Laplace's equation, equation (4). Elevation of PG w i l l raise PL. At the onset of alveolar flooding PL i s equivalent to B. If B i s increased, then the t o t a l extravascular f l u i d volume (VIS1+VAS) at steady state conditions w i l l decrease, i . e . there w i l l be less f l u i d accumulation i n the a i r space. C l i n i c a l l y , the parameter B may be elevated by applying a p o s i t i v e pressure to the gas space, thus increasing PG, which i n turn increases B. C a l d i n i et a l . (50) subjected i s o l a t e d dog lung lobes to continuous p o s i t i v e -pressure v e n t i l a t i o n during an increase i n the pulmonary microvascular pressure (PMV), i . e . , hydrostatic pulmonary edema was induced. If the a i r space pressure (PG) was maintained above a c e r t a i n value no "appreciable amount of l i q u i d accumulated i n the airways". For PG l e s s than this value "frothy bloody l i q u i d flowed continuously from the endotracheal cannula" during acute pulmonary edema. This set of observations by C a l d i n i et a l . (50) supports the s i g n i f i c a n c e of the elevation i n B. -151-5.6 The Response of the Pulmonary Microvascular Exchange System  to a Maximum Lymph Flow Lymph flow Is related to the f l u i d volume VIS1 through equation (21). A maximum capacity of the lymphatics may be defined by either JL(max) or equivalently VLMPH. In the computer program the maximum capacity of the lymphatics i s related to the EVEA f l u i d volume VLMPH. JL w i l l be determined by equation (21) u n t i l VIS1 reaches VLMPH; further increases i n VIS1 w i l l not rai s e JL. The ef f e c t of changes i n JL(max) (or VLMPH) on the predictions of the PMVES simulation was studied under the conditions l i s t e d i n Table 19. The e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t was, again, considered to be var i a b l e with NK = 0.5 h r - 1 mm Hg - 1. At time zero the PMVES was subjected to a step-change i n PMV from 9 mm Hg to 50 mm Hg. A f i r s t estimate of JL(max) as suggested i n section 4.4.2, could be the lymph flow corresponding to a f l u i d volume VTONS (460 ml). Using equation (21) to calculate the lymph flow, JL(max) becomes 22.6 ml/hr. The value of JL(max) was varied above and below 22.6 ml/hr. Figure 29a shows the lymph flow responses, up to a time when VTOT = 1000 ml, for JL(max) equal to 14.8, 22.6 and 24.3 ml/hr, and when no maximum lymph flow e x i s t s . When JL(max) i s approximately 7% greater than 22.6 ml/hr ( i . e . , 24.3 ml/hr), Figure 29a i l l u s t r a t e s that the response of JL i s very close to the response of JL when no maximum lymph flow i s defined. Decreasing JL(max) from 22.6 ml/hr by approximately 35% ( i . e . , 14.8 ml/hr) produces a noticeable change i n the lymph flow response, i n comparison to the response of JL when no -152-Table 19 Conditions of the PMVES Simulations Conducted to Study Changes i n the Maximum Lymph Flow* JL(max) varied from 14 to 38 ml/hr corresponds to a VLMPH of 410 to 550 ml. KAS = NK(VISl-VTONS) for VIS1 > VTONS NK = 0.5 h r - 1 mm Hg - 1 PMV = 5 0 mm Hg KF = 1.12 ml/hr/mm Hg APERM = 0 SIGFA SIGFG VTONS SL B SIGD PSA PSG 0.75 3.0 1.0 0.40 0.60 ml/hr ml/hr 460 0.25xl0 - 2 -3.03 ml mmHg/ml mmHg *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 29a to 29c. -153-Figure 29a Transient Responses of JL for Different JL(max) (or VLMPH) (Conditions as i n Table 19) - Responses continued to time when VTOT = 1000 ml o —I L L . Q T 10 Legend JL(max) = N.A. JL(max) = 22.6  JL(max) = 14.8 TIME (hrs) 20 I 30 -154-maxiraum lymph flow i s defined. Figure 29b i l l u s t r a t e s the transient responses of (JV-JL) for three maximum lymph flows: 14.8, 22.6 and 29.4 ml/hr. Increasing JL(max) from 22.6 ml/hr to 29.4 ml/hr (a 30% increase) produces a small drop i n (JV-JL) that averages 2 ml/hr, while a decrease i n JL(max) from 22.6 ml/hr to 14.8 ml/hr (a 35% decrease) produces an average r i s e i n (JV-JL) of about 7.5 ml/hr. Since (JV-JL) represents the rate of f l u i d accumulation i n the t o t a l extravascular space, increasing (JV-JL) w i l l lead to a decrease i n the time to reach a VTOT of 1000 ml (t(1000)) as shown i n Figure 29c. It can be seen that decreasing JL(max) below 22.6 ml/hr y i e l d s a rapid decrease i n t(1000). For JL(max) above 22.6 ml/hr the increase i s less rapid u n t i l JL(max) reaches approximately 30 ml/hr, a f t e r which t(1000) becomes constant. . From the above discussion one may conclude that i f the capacity of the lymphatics i s exceeded before the EVEA f l u i d volume reaches a value of VTONS, then the rate at which f l u i d accumulates to a VTOT of 1000 ml i s accelerated. Prichard (17) discusses pulmonary edema and lymphatic i n s u f f i c i e n c y : "Occlusion or p a r t i a l occlusion of the lymphatic system may be responsible for pulmonary edema i n patients with s i l i c o s i s , . . . , and possibly shock lung". He suggests that the lymphatic capacity may be reduced by phys i c a l a l t e r a t i o n s of the lymphatic channels or lung t i s s u e . In the alveolar model the lymphatic capacity may be reduced by changes i n the parameter VLMPH -155-Figure 29b Transient Responses of (JV-JL) for Dif f e r e n t JL(max) (Conditions as i n Table 19) - Responses continued to time when VTOT = 1000 ml 50-f Legend JL(max) = 14.8 JL(max}_= 22.6 JL(max) a 29A 10-o -10 TIME (hrs) T 20 30 -156-Figure 29c Time to Reach a VTOT of 1000 ml for Dif f e r e n t JL(max) (Conditions as in Table 19) 30-| 28 H 26 A ~ 22-1 20-1 10 15 20 25 30 MAXIMUM JL (ml/hr) 35 40 -157-( f l u l d volume corresponding to JL(max)). The response of the predictions of the PMVES simulation may then be observed. 5.7 The Responses of the Pulmonary Microvascular Exchange  System to changes in the value to which PMV is elevated. In the preceding simulations pulmonary edema was simulated by elevat i n g PMV from 9 mm Hg to 50 mm Hg at time zero. The following discussion w i l l consider step changes i n PMV to d i f f e r e n t values of PMV under the conditions l i s t e d i n Table 20. The e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t was represented as a variable with NK equal to 0.5 h r - 1 mm Hg - 1. The parameter PMV i s introduced i n the alve o l a r model with St a r l i n g ' s Hypothesis, equation (18). From equation (18) one can see that elevating PMV increases the transendothelial flow (JV). The transendothelial flow less the lymph flow equals the rate of f l u i d accumulation i n the t o t a l extravascular space. Figure 30a i l l u s t r a t e s that increasing PMV from 40 to 60 mm Hg increases (JV-JL). In the i n t e r s t i t i a l phase VTOT i s equivalent to VIS1; therefore as PMV i s increased the time for VIS1 to reach the onset of alveolar flooding (t(onset)) decreases (Figure 30b). For PMV=40 mm Hg the value of t(onset) i s 3.0 hrs, and for PMV=60 mm Hg the value of t(onset) i s 1.5 hrs. Following the onset of alveolar flooding (JV-JL) decreases very slowly ( i n comparison to before alveolar flooding), as shown i n Figure 30a; (JV-JL) for PMV=60 mm Hg i s s t i l l greater than (JV-JL) for PMV=40 mm Hg. Under the conditions of Table 20 JAS approaches (JV-JL) -158-Conditions of the PMVES Simulations Conducted to Table 20 ti Study Changes i n PMV* PMV: varied from 35 to 60 mm Hg KAS = NK(VIS1-VTONS) for VIS1 > VTONS NK = 0.5 h r - 1 mm Hg - 1 KF = 1.12 ml/hr/mm Hg APERM = 0 SIGD = 0.75 PSA = 3.0 ml/hr PSG = 1.0 ml/hr SIGFA = 0.40 SIGFG = 0.60 VTONS = 4 6 0 ml SL = 0.25xlO - 2 mmHg/ml B = -3.03 mmHg VLMPH = 5000 ml *These are the conditions of the simulations that produced the r e s u l t s I l l u s t r a t e d i n Figures 30a to 30c. -159-Figure 30a Transient Response of (JV-JL) and JAS for Dif f e r e n t PMV (Conditions as i n Table 20) - Responses continued to a VTOT = 1000 ml Legend (JV-JL), PMV = 40 JAS, PMV=40 TIME (hrs) -160-Figure 30b Time to Reach the Onset of Alveolar Flooding for Different PMV (Conditions as i n Table 20) V 61 Ld 1-1 1 1 1 1 1 35 40 45 50 55 60 HYDROSTATIC PRESSURE PMV (mm Hg) -161-Figure 30c Time to Reach a VTOT of 1000 ml for D i f f e r e n t PMV (Conditions as i n Table 20) 120 CIRCULATION HYDROSTATIC PRESSURE PMV(mm -162-(Figure 30a). The time to reach a VTOT (=VIS1+VAS) of 1000 ml decreases as PMV increases (Figure 30c). In both Figure 30b and Figure 30c i t can be seen that the times, t(onset) and t(1000), r e s p e c t i v e l y , are most s e n s i t i v e to small Increases i n PMV above 35 mm Hg and le v e l s o f f at higher PMV values. If PMV i s reduced to 30 mm Hg, no alveolar flooding would occur since at a PMV of 30 mm Hg and for the conditions of Table 20, the steady state VIS1 i s 460 ml, which i s also equivalent to the assumed VTONS (460 ml). Therefore at a PMV of 30 mm Hg t(onset) and t(1000) would be i n f i n i t e . Increasing PMV above 30 mm Hg would induce alveolar f l o o d i n g . In cardiogenic pulmonary edema the c i r c u l a t o r y hydrostatic pressure, PMV, i s elevated above i t s normal value causing an increased transendothelial flow. For the conditions l i s t e d i n Table 20 Increases i n PMV up to 30 mm Hg produce only i n t e r s t i t i a l edema. Raising PMV above 30 mm Hg causes severe pulmonary edema with alveolar flooding. 5.8 The Response of the Pulmonary Microvascular Exchange System  to changes in the Endothelial Filtration Coefficient, KF The e f f e c t of changes i n KF on the predictions of the PMVES simulation was studied under the conditions l i s t e d i n Table 21. The perturbation to the exchange system consisted of a step change i n the c i r c u l a t o r y hydrostatic pressure, at time zero, from 9 mm Hg to 50 mm Hg and changes to the f l u i d conductivity of the endothelial membrane. -163-Table 21 Conditions of the PMVES Simulations Conducted to Study Changes i n KF* KF: varied from normal (KF(norm)) to 10 KF(norm) KF(norm) =1.12 ml/hr/mm Hg KAS = NK(VISl-VTONS) for VIS1 > VTONS NK = 0.5 h r - 1 mm Hg - 1 PMV = 5 0 mm Hg APERM = 0 SIGFA SIGFG VTONS SL B VLMPH SIGD PSA PSG 0.75 3.0 1.0 0.4 0.6 ml/hr ml/hr 460 0.25xl0- 2 -3.03 5000 ml mmHg/ml mmHg ml *These are the conditions of the simulations that produced the r e s u l t s i l l u s t r a t e d i n Figures 31a to 31d. -164-The endothelial f i l t r a t i o n c o e f f i c i e n t was introduced into the model with S t a r l i n g ' s Hypothesis, equation (18); elevation of KF w i l l r e s u l t i n an increase i n the transendothelial flow, JV. The f l u i d flow difference (JV-JL) i s the rate of f l u i d accumulation i n the extravascular f l u i d space. Figure 31a i l l u s t r a t e s that an Increase i n KF re s u l t s i n an increase i n (JV-JL). Therefore the time to reach a t o t a l extravascular f l u i d volume of 1000 ml decreases with an increase i n KF (Figure 31b). The time to reach a VTOT of 1000 ml (t(1000)) decreases by approximately two-thirds (29.4 hrs to 9.9 hrs) as KF i s doubled to 2KF(normal) and by half (9.9 hrs to 4.3 hrs) when KF i s doubled from 2KF(normal) to 4KF(normal). Therefore, t(1000) i s most s e n s i t i v e to the range of KF close to KF(normal). Increases i n the endothelial f i l t r a t i o n c o e f f i c i e n t also reduces the time to reach the onset of alveolar flooding (Figure 31c). As with t(1000), the time to reach the onset of al v e o l a r flooding i s most sens i t i v e to changes i n KF near KF(normal). Following the onset of alveolar flooding the t r a n s e p i t h e l i a l flow (JAS) r i s e s to a value close to (JV-JL) (Figure 31a). As KF increases JAS also increases. The rate of f l u i d accumulation i n the i n t e r s t i t i a l (and c e l l u l a r ) space (JNET1) also increases with KF. The i n t e g r a l over time of JNET1 i s the f l u i d volume VIS1. In Figure 31d the f l u i d volume VIS1 at a VTOT of 1000 ml i s seen to increase with KF. The i n i t i a l value of (JV-JL) at a KF/KF(normal) of 10 i s approximately 200 ml/hr, as compared to 45 ml/hr for KF=KF(norm). - 1 6 5 -Figure 31a Transient Responses of (JV-JL) and JAS for Dif f e r e n t KF (Conditions as i n Table 21) - Responses continued to time when VTOT = 1000 ml 250 200 H (/) 3 o 150 100 A Legend (JV-JL), 1KF(norm) J^SJKF^orm)_ (JV-JL), 2KF(norm) Jj^^FfcormJ 50-1 20 10 TIME (hrs) 30 -166-Figure 31b Time to Reach a VTOT of 1000 ml for Di f f e r e n t KF (Conditions as i n Table 21) 30 i i i i 1 0 2 4 6 8 KI/KF(normal) : ENDOTHELIUM -167-Figure 31c Time to Reach the Onset of Alveolar Flooding f or Different KF (Conditions as i n Table 21) -168-Figure 31d 450 H 1 1 1 - | 0 2 4 6 8 10 KI/KF(normal) : ENDOTHELIUM -169-The magnitude of the f l u i d flow for the condition with KF=10 KF(normal) i s large. Experimental r e s u l t s should be obtained to v e r i f y this magnitude of transendothelial flow. Guyton et a l . (51) studied hydrostatically-induced pulmonary edema on dog lungs. In cases where alveolar flooding occurred the dogs did not survive past one or two hours af t e r the onset of edema. Pare" and Dodek. (personal communication,52) stated that alveolar flooding begins within a f r a c t i o n of an hour following the onset of acute pulmonary edema. Under the conditions shown i n Table 21 and with a normal KF the times to reach the onset of alveolar flooding and a VTOT of 1000 ml are 2.3 hrs and 29.4 hrs, r e s p e c t i v e l y . Doubling the value of KF reduces the time to reach the onset of alveolar flooding to 0.9 hrs and the time to reach a VTOT of 1000 ml to 9.9 hrs - much closer to the range of the suggested times. 5.9 The Response of the Pulmonary Microvascular Exchange System  to changes i n APERM The plasma protein permeability parameters of the endothelial membrane are: 1) the solute r e f l e c t i o n c o e f f i c i e n t - SIGD 2) the permeability-surface area product for albumin and globulin-PSA, PSG 3) the solvent drag r e f l e c t i o n c o e f f i c i e n t f o r albumin and globulin-SIGFA, SIGFG -170-Changes i n the solute permeability of the endothelium are defined by APERM - this change i s expressed as a percent of normal. Increases i n APERM refer s to increases i n PSA and PSG and decreases i n SIGD, SIGFA and SIGFG. The e f f e c t of changes i n APERM on the predictions of the PMVES simulation was studied under the conditions l i s t e d i n Table 22. The e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t was represented as a function of (VIS1-VT0NS) with NK equal to 0.5 h r - 1 mmHg-1. At time zero the PMVES was subjected to a step change i n PMV to 50 mmHg. In the cases where APERM was greater than zero the perturbation to the PMVES was an increase i n PMV, and an increase i n the permeability of the endothelium to solutes. Solute flow from the c i r c u l a t i o n to the i n t e r s t i t i u m i s expressed mathematically by the Kedem-Katchalsky (K-K) solute f l u x equation (equation (20)); the d i f f u s i v e term includes (PS)^, while the convective term includes (SIGF)^. The i n t e r s t i t i a l protein concentration used i n equation (20) i s (CAV)^ - the e f f e c t i v e protein concentration. Figure 32a i l l u s t r a t e s the transient response of CAVA for changes i n APERM from 0 to 100%. For the case of APERM=0 CAVA decreases from 0.038 g/ml at time zero to approximately 0.029 g/ml at which time VTOT = 1000 ml; CAVA i s always less than CMVA (.042 g/ml), as shown i n Figure 32a. Therefore, for APERM = 0 the values of both the d i f f u s i v e term and the convective term of equation (20) are p o s i t i v e . In Figure 32b the albumin content i n the i n t e r s t i t i u m (QA) r i s e s to a maximum for APERM = 0 and then declines slowly; i n i t i a l l y -171-Table 22 Conditions for the PMVES Simulations Conducted to Study Changes i n APERM* APERM: 0% 50% 90% 100% SIGD 0.75 0.375 0.075 0 PSA(ml/hr) 3.0 4.5 5.7 6.0 PSG(ml/hr) 1.0 1.5 1.9 2.0 SIGFA 0.4 0.2 0.04 0 SIGFG 0.6 0.3 0.06 0 KAS = NK(VIS1 -VTONS) for VISI > VTONS NK = 0.5 hr - 1 mmHg"1 PMV 50 mmHg KF 1.12 ml/hr/mmHg VTONS = 460 ml SL 0.25xl0- 2 mmHg/ml B -3.03 mmHg VLMH = 5000 ml * These are the conditions of the simulations that produced the r e s u l t s of Figures 32a to 32c. -172-Figure 32a: Transient Responses of Albumin Protein Concentration CAVA for d i f f e r e n t APERM (Conditions as in Table 22 -Responses continued to time when VTOT = 1000 ml 0.08 0.06-0.04-Legend PERM = o % PERM = 50 % PERM = 90 % 0.02-0.00 10 TIME (hrs) 20 30 -173-there Is a net accumulation of albumin i n the i n t e r s t i t i u m , followed by a net depletion. An increase i n APERM by 50% re s u l t s i n the response of CAVA as shown i n Figure 32a; CAVA drops to a minimum and then r i s e s to a value above CMVA by the time VTOT reaches 1000 ml (approximately 21 h r s ) . S i m i l a r l y , for the case of APERM = 90% CAVA increases from 0.038 g/ml at time zero to a value greater than CMVA by the time VTOT = 1000 ml (approximately 21 h r s ) . The time at which CAVA=CMVA i s 14.5 hrs. for APERM = 50% and 3.6 hrs. for APERM=90%. Solute concentration i s determined by the r a t i o of solute weight to f l u i d volume. Following the onset of alveolar flooding (2.5 hrs) the albumin weight i n the i n t e r s t i t i a l f l u i d r i s e s (Figure 32b); In addition, as APERM increases QA increases. However, the f l u i d volume VIS1, as shown i n Figure 32c, increases very l i t t l e following the onset of alv e o l a r flooding; as APERM increases the change i n VIS1 i s also n e g l i g i b l e . Therefore, the r a t i o of QA to VIS1 w i l l r i s e following the onset of alveolar flooding and as APERM increases. Whether CAVA exceeds CMVA, or not, i s dependent on the size of the increase i n QA. QA i s the i n t e g r a l of QNETA over time, which i s determined by equation (33): QNETA = JSA-JL.CTA-JAS.CTA (33) As JSA i s increased QA w i l l also increase. For values of CAVA near CMVA, the convective term of equation (20) i s dominant i n the evaluation of JSA; as SIGFA i s reduced ( i . e . APERM increased) the value of the convective term increases, and QA r i s e s . -174-Figure 32b: Transient Responses of Albumin Protein Weight QA for d i f f e r e n t APERM (Conditions as i n Table 22) -Responses continued to time when VTOT = 10Q0 ml 20 Legend PERM = o % E^RK4 = 50%_ PERM = 90 % PERM = 100 % 0 10 20 30 TIME (hrs) -175-Figure 32c: Transient Responses of F l u i d Volume VIS1 for d i f f e r e n t APERM (Conditions as i n Table 22) - Responses continued to time when VTOT = 1000 ml 500 > 400-o > 350 300 Legend PERM = 0 % PERM = 5 0 % PERM = 9 0 % PERM = 100% 10 20 TIME (hrs) 30 -176-Figure 32d i l l u s t r a t e s the time to reach a VTOT of 1000 ml (t(1000)) for changes i n APERM; as APERM increases from 0 to approximately 80%, t(1000) decreases from 29 hrs. to 19 hrs. The parameter SIGD i s introduced with S t a r l i n g ' s Hypothesis and combined with (PIMV-PIPMV) to y i e l d the e f f e c t i v e oncotic pressure difference between the c i r c u l a t i o n and i n t e r s t i t i u m . As APERM increases SIGD decreases, r e s u l t i n g i n a decreasing e f f e c t i v e oncotic pressure d i f f e r e n c e . Therefore the transendothelial flow rate increases. However, the increase i n JV i s small since the hydrostatic pressure diff e r e n c e (PPMV-PPMV) i s dominant when PMV i s elevated to 50 mmHg. The primary e f f e c t of changes i n APERM i s on the transmembrane protein movement. Sh i r l e y et a l . (53) have c a r r i e d out tracer studies on the lymph and plasma of dog lungs. They observed that following a large increase i n blood volume (which corresponds to a large r i s e i n PMV) the r a t i o of lymph to plasma solute concentrations rose. From t h i s observation Shirley et a l . (53) suggested that the endothelial membrane became more porous during the high PMV - this was c l a s s i f i e d as the stretched pore model. Figure 32a shows that at an elevated PMV of 50 mmHg the r a t i o CAVA/CMVA (comparable to the lymph-plasma r a t i o ) may r i s e i f APERM i s increased to values such as 50%. However, i n the res u l t s of Sh i r l e y et a l . (53) the r i s e i n lymph to plasma solute concentration i s primarily due to a drop i n the plasma concentration following the infusion of an albumin s o l u t i o n , and not due to a s i g n i f i c a n t r i s e i n lymph solute concentration. In the case of the -177-simulations of the PMVES where APERM i s changed i t i s the lymph solute concentration that i s increased. 5.10 Control of Error in the Numerical Solutions The numerical integration technique used i n the computer program of the alveolar model was the Runge-Kutta-Merson method, supplied as a subroutine by the UBC Computing Centre. In the evaluation of the variable s , such as VIS1, because of the approximations used In the c a l c u l a t i o n there i s an error associated with the solu t i o n i n each int e g r a t i o n step. This error was minimized i n the computer program by employing a variable time step and s e t t i n g a tolerance l e v e l on the size of the error that was acceptable. If the error i n the integrated variables exceeded a tolerance l e v e l of 1 x 1 0 - 9 , the time step was reduced and the int e g r a t i o n was repeated using a smaller time step. A s t r a i g h t - l i n e i n t e r p o l a t i o n method was used to evaluate PPMV (from VIS1) and the values of variables at the c h a r a c t e r i s t i c point such as the time to reach a VTOT of 1000 ml. In th i s technique the error i n the solutions i s dependent on the difference i n value of the independent variable at the end-points of the i n t e r v a l . Generally, as the time i n t e r v a l of the cal c u l a t i o n s decreased, the error i n the dependent variable decreased. For the ca l c u l a t i o n s of the variables at the c h a r a c t e r i s t i c points, this time i n t e r v a l was 0.25 hrs. -178-Figure 32d: Time to Reach a VTOT of 1000 ml for Dif f e r e n t APERM (Conditions as i n Table 22) 30 O O O > o < o 28 A 26-24 H 22-1 20-1 18-| 1 , , , 0 20 40 60 80 100 PERM(%) : ENDOTHELIUM -179-SUMMARY AND CONCLUSIONS 1. The alveolar model describes one way to integrate the a i r space with the lumped compartment model ( i n t e r s t i t i a l model) of the PMVES as developed by Bert and Pinder (29). 2. Estimates for the parameters introduced with the alveolar model were obtained: (a) E p i t h e l i a l F i l t r a t i o n c o e f f i c i e n t , KAS: The c o e f f i c i e n t may be represented as a constant or a v a r i a b l e . A varia b l e c o e f f i c i e n t appears to be more compatible with l i t e r a t u r e (33). We suggest a value for the s e n s i t i v i t y parameter NK of 0.5 h r - 1 mmHg-1. (b) VTONS: For cases of hydrostatic edema and edema caused by changes to the endothelial permeability, a value of 460 ml i s a good estimate. A VTONS of 460 ml corresponds to a PPMV of +1.1 mmHg - a value within the range suggested by Guyton(47). (c) SL: No value of SL was found i n the l i t e r a t u r e survey. We suggest a value of 0.25 x I O - 2 mmHg/ml. (d) B: No value of B was found i n the l i t e r a t u r e survey. We suggest a value of -3.03 mmHg, which corresponds to the expected alveolar f l u i d pressure at normal conditions. 3. The presence of a maximum lymph capacity may be e a s i l y introduced i n t o the model through parameter VLMPH. In the simulations of the PMVES car r i e d out for this thesis a f l u i d volume VLMPH less than VTONS w i l l accelerate the progression of edema. Values of VLMPH -180-greater than VTONS have very l i t t l e e f f e c t on the progression of edema (up to the time when VTOT = 1000 ml). 4. In h y d r o s t a t i c a l l y induced pulmonary edema PMV must be raised above a minimum pressure for alveolar flooding to occur. The time to reach the onset of alveolar flooding i s most se n s i t i v e to small r i s e s i n PMV above this minimum value. 5. If PMV i s raised above the minimum value s p e c i f i e d i n (4), then increases i n KF w i l l accelerate the progress of edema. The time to reach the onset of alveolar flooding i s most sens i t i v e to small increases i n KF above i t s normal value. 6. The i n t e r s t i t i a l to plasma protein concentration r a t i o may increase or decrease during the progression of pulmonary edema depending on the size of the change i n the endothelial solute permeability parameters. - 1 8 1 -RECOMMENDATIONS FOR FURTHER WORK 1. To further u t i l i z e the computer simulation of the alveolar model experimental data must be obtained to v e r i f y the model's predictions, or to suggest a l t e r n a t i v e ways to integrate the a i r space with the e x i s t i n g i n t e r s t i t i a l models of the PMVES. 2. The computer program of the alveolar model should be improved so that i t may be more "user f r i e n d l y " . In t h i s way the simulation may be used by the health care profession for teaching purposes. 3 . Future development of the alveolar model should incorporate: (a) the ef f e c t s of lung i n f l a t i o n on the PMVES (b) the Zone Model of the lung proposed by West, Dollery and Naimark ( 3 ) (c) the e f f e c t of l o c a l i z e d pulmonary edema on the PMVES (d) a recovery phase following the progression of pulmonary edema. -182-NOMENCLATURE CAVA,CAVG (CL)i CMVA,CMVG CP,CPMV,CPPMV CTA,CTG JAS,JL,JV JL(max) JNET1 JSA.JSG KAS,KF KASO LF NK PA PALV.PG PAS,PL Alveolar f l u i d pressure at the onset of alveolar flooding (mmHg) The e f f e c t i v e i n t e r s t i t i a l concentration of albumin and g l o b u l i n respectively (g/ml) Lymph protein concentration for solute ' i ' (g/ml) The blood plasma concentration of albumin and glob u l i n , r e s p e c t i v e l y (g/ml) The t o t a l protein concentration for blood plasma (CPPMV) and i n t e r s t i t i a l f l u i d (CPPMV) (g/ml) The tissue concentration of albumin and gl o b u l i n , r e s p e c t i v e l y (g/ml) The t r a n s e p i t h e l i a l , lymph, and transendothelial f l u i d flows, respectively (ml/hr) The parameter representing the value of the maximum lymph flow (ml/lhr) The rate of f l u i d accumulation i n the extravascular-extraalveolar space (ml/hr) The transendothelial solute flows for albumin and globu l i n , respectively (g/hr) The f l u i d f i l t r a t i o n c o e f f i c i e n t of the epithelium and endothelium, res p e c t i v e l y (ml/hr/mmHg) The parameter representing a constant e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t (ml/hr/mmHg) The f l u i d conductivity c o e f f i c i e n t of the endothelium (cm/hr/mmHg) The s e n s i t i v i t y parameter for a var i a b l e e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t ( h r - 1 mmHg-1) The hydrostatic pressure of the a r t e r i a l segment of a blood vessel (mmHg) The hydrostatic pressure of the gas i n the a i r space (mmHg) The alveolar f l u i d pressure (mmHg) Continued.... - 1 8 3 -NOMENCLATURE (Cont.d) PI,PIAS,PIMV, PLA PMV PPA PPMV,PEA PSA,PSG PV QA.QG QNETA,QNETG (QO)i SA SIGD.SIGDAS SIGFA,SIGFG SL VAS VAVA.VAVG VCELL VEXA.VEXG The c o l l o i d osmotic (or oncotic) pressure of the alveolar f l u i d (PIAS), blood plasma (PIMV), and i n t e r s t i t i a l f l u i d (PIPMV) (mmHg) The l e f t a t r i a l hydrostatic pressure (mmHg) The pulmonary microvascular hydrostatic pressure (mmHg) The pulmonary a r t e r i a l hydrostatic pressure (mmHg) The hydrostatic pressure of the alveolar and extra-alveolar i n t e r s t i t i a l f l u i d , r e spectively (mmHg) The endothelial permeability-surface area products for albumin and glob u l i n , respectively (ml/hr) The hydrostatic pressure of the venous segment of a blood vessel (mmHg) The i n t e r s t i t i a l solute content of albumin and glob u l i n , respectively (gm) The rate of solute accumulation i n the i n t e r s t i t i u m for albumin and gl o b u l i n respectively (g/hr) The solute weight i n the i n t e r s t i t i u m for solute ' i ' at time zero (gm) Surface area of pulmonary c a p i l l a r y membrane (cm 2) The solute r e f l e c t i o n c o e f f i c i e n t for f l u i d of the endothelial and e p i t h e l i a l membranes, re s p e c t i v e l y . The solvent drag r e f l e c t i o n c o e f f i c i e n t of the endothelial membrane to albumin and glo b u l i n , r e s p e c t i v e l y The slope of the PAS-VAS curve for the alveolar f l u i d (mmHg/ml) The alveolar f l u i d volume (ml) The a v a i l a b l e i n t e r s t i t i a l f l u i d volume for albumin and g l o b u l i n , respectively (ml) The c e l l u l a r f l u i d volume (ml) The excluded f l u i d volume to albumin and g l o b u l i n , r e s p e c t i v e l y (ml) Continued.... -184-VIS VIS1 VIS10 VLMPH VTONS VTOT NOMENCLATURE (Cont.d) The i n t e r s t i t i a l f l u i d volume (VIS=VTOT-VAS-VCELL)(ml) The f l u i d volume of the i n t e r s t i t i a l ( + c e l l u l a r ) space or the extravascular-extraalveolar (EVEA) space (VIS1=VT0T-VAS)(ml) The i n t e r s t i t i a l ( + cellular) f l u i d volume at time zero (ml) The i n t e r s t i t i a l ( + cellular) f l u i d volume corresponding to the maximum lymph flow (ml) The i n t e r s t i t i a l ( + cellular) f l u i d volume at the onset of alveolar flooding (ml) The t o t a l extravascular f l u i d volume (ml) The radius of curvature: used i n Laplace's Equation (cm) The surface tension of a f l u i d (dynes/cm) -185-REFERENCES 1. 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Macklin, C C , The pulmonary alveolar mucoid f i l m and the pneumonocytes, Lancet 1_: 1099-1104, 1954. 41. I l i f f , L.D., Extra-alveolar vessels and edema development i n excised dog lungs, C i r c u a l t i o n Research 2iB: 524-532, 1971. -188-42. Hughes, J.M.B., J.B. G l a z i e r , J.E. Maloney and J.B. West, E f f e c t of extra-alveolar vessels on d i s t r i b u t i o n of blood flow In the dog lung. Journal of Applied Physiology 2_5: 701-712, 1968. 43. Vreim, C.E. & N.C. Staub, Protein composition of lung f l u i d s i n acute alloxan edema i n dogs, American Journal of Physiology 230(2): 376-379, 1976. 44. Vreim, C.E., P.D. Snashall, & N.C. Staub, Protein composition of lung f l u i d s i n anesthetized dogs with acute cardiogenic edema, American Journal of Physiology 231(5): 1466-1469, 1976. 45. Guyton, A.C, A.E. Taylor, R.E. Drake & J.C. Parker, Dynamics of subatmospheric pressure i n the pulmonary i n t e r s t i t i a l f l u i d , In: Lung Liquids, American E l s e v i e r , New York (1976). 46. Tierney, D.F. & R.P. Johnson, Altered surface tension of lung extracts and lung mechanics, Journal of Applied Physiology 20(6): 1253-1260, 1965. 47. Guyton, A.C, J.C. Parker, A.E. Taylor, T.E. Jackson, and D.S. Moffatt, Forces governing water movement i n the lung, In: Pulmonary Edema, Edited by A.F. Fishman and E.M. Renkin, American P h y s i o l o g i c a l Society, Baltimore, 1979. 48. Gump, F.E., Y. Mashima, A. Ferenczy, & J.M. Kinney, Pre-and postmortem studies of lung f l u i d s and e l e c t r o l y t e s , The Journal of Trauma, 11/6): 474-482, 1971. 49. Drake, R.E., J.H. Smith & J.C. Gabel, Estimation of the f i l t r a t i o n c o e f f i c i e n t i n i n t a c t dog lungs. American Journal of Physiology 238(7): H430-H438, 1980. 50. C a l d i n i , P., J.D. L e i t h , & M.J. Brennan, E f f e c t of continuous positive-pressure v e n t i l a t i o n (CPPV) on edema formation i n dog lung, Journal of Applied Physiology 39/4): 672-679, 1975. 51. Guyton, A.C, & A.W. Lindsey, E f f e c t of elevated l e f t a t r i a l pressure and decreased plasma protein concentration on the development of pulmonary edema, C i r c u l a t i o n Research 7: 649-657, 1959. ~ 52. P a r i , P. and P. Dodek, Personal communication. 53. S h i r l e y J r . H.H., C.G. Wolfram C.G., K. Wasserman and H.S. Mayerson, C a p i l l a r y Permeability to Macromolecules: stretched pore phenomenon. American Journal of Physiology 190(2): 189-193, 1957. 54. Jacob, S.W. C.A. Francone, and W.J. Lossow, Structure and Function i n Man, F i f t h E d i t i o n , W.B. Saunders Company, Toronto, 1982. -189-55. Prockop, D.J. Collagen, E l a s t i n and Proteoglycans: Matrix for F l u i d Accumulation i n the Lung, In: Pulmonary Edema, Edited by A.P. Fishman and E.M. Renkin, American P h y s i o l o g i c a l Society, Baltimore, 1979. 56. Moore, C. UBC RKC: Runge Kutta with Error Control, Computing Centre, The University of B r i t i s h Columbia, 1983. 57. Mair, S.G., UBC PLOT: The UBC plot subroutines and programs. Computing Centre, The University of B r i t i s h Columbia, 1984. -190-APPENDIX A: The Computer Program A l . l Input Files EDA and PDA The parameters of the input f i l e s EDA and PDA were defined i n Table 5 and 6, res p e c t i v e l y . An example of the numerical values assigned to these parameters i s shown i n Table 23 for EDA and Table 24 for PDA. (The numerical values must s t a r t i n column 11 for EDA and column 21 for PDA). The parameters introduced with the alveolar model - VTONS, SL, B, and VLMPH - assume values as stated i n section 4.4. In this example the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t i s represented as a variab l e ; therefore NK i s assigned a value, i n th i s case 0.5 h r - 1 mmHg-1, and KASO i s set equal to zero (see Table 24). If KAS i s represented as a constant, then NK i s set equal to zero and KASO assigned a constant value. The permeability parameters of the endothelium and the endothelial f i l t r a t i o n c o e f f i c i e n t assume the normal values as selected by Bert and Pinder (29). As shown i n f i l e PDA the hydrostatic pressure PMV i s set at 50 mmHg, raised from i t s normal value of 9 mmHg. The meaning of the parameters TAUMX1, SUBNT1, TAUMX2 and SUBNT2 may be explained as follows: Up to a time of 25 hrs (TAUMX1) the values of the variables l i s t e d i n Table 9 w i l l be stored every hour (SUBNT1); SUBNT1 must divide evenly into TAUMX1. From a time of 25 hrs (TAUMX1) to 100 hrs (TAUMX2) the values of the variables w i l l be stored every 5 hrs. (SUBNT2); SUBNT2 must divide evenly into TAUMX1 and TAUMX2. A det a i l e d account of the transient response of the PMVES - 1 9 1 -Table 23: Example of Numerical Values Assigned to Parameters i n F i l e EDA 1 VTOTO 378 .7D0 2 VASO 0 . ODO 3 QAO 5 .81D0 4 QGO 2 . 36D0 5 PIMV 25 .0D0 6 VCELL 150 . ODO 7 VEXA 73 . 5D0 8 VEXG 1 1 5 . 5D0 9 VTONS 459 .9D0 10 VLMPH 5000 . ODO 1 1 B -3 .03D0 1 2 CMVA 0 .042D0 1 3 CMVG 0 .0271D0 1 4 FPPMV(1) = -2 .74D0 1 5 FPPMV(2) = -2 . 40D0 1 6 FPPMV(3) = - 1 . 9D0 i 7 FPPMV(4) = -1 . 3D0 18 FPPMV(5) = -0 • 8D0 1 9 FPPMV(6) = -0 . 3D0 20 FPPMV(7) = 0 . 1 5D0 2 1 FPPMV(8) = 0 . 60D0 22 FPPMV(9) = 1 .09D0 23 KF 1 . 1 2D0 24 SIGD 0 .750D0 25 PSA 3 . 00D0 26 PSG 1 . 00D0 27 SIGFA 0 .400D0 28 SIGFG 0 .600D0 -192-Table 24: Example of Numerical Values Assigned to the Parameters i n F i l e PDA 1 P L 0 T S ( Y = 1 , N = 2 ) = 2 2 T A B L E S 1 ( Y = 1 , N = 2 ) = 1 3 T A B L E S 2 ( Y = 1 , N = 2 ) = 1 4 T A U M X 1 = 2 5 . O D O 5 S U B N T 1 = 1 . 0 D 0 6 T A U M X 2 = 1 0 0 . O D O 7 S U B N T 2 = 5 . ODO 8 S T E P S Z = 0 . 0 1 D 0 9 H M I N = 0 . 0 0 1 D 0 1 0 T O L = 0 . 0 0 0 0 0 0 0 0 0 1 D 0 11 K A S O = 0 . 0 0 12 NK = 0 . 5 0 13 S L = 0 . 0 0 2 5 14 P M V = 5 0 . 0 0 . 15 X S ( 1 ) = 5 . 8 7 5 16 X S ( 2 ) = 7 . 0 0 17 Y S 1 ( 1 ) = 7 . 5 6 2 5 18 Y S 2 ( 1 ) = 7 . 1 8 7 5 19 Y S 3 0 ) = 6 . 8 1 2 5 2 0 Y S 4 ( 1 ) = 6 . 4 3 7 5 21 S F T = 1 0 . 0 0 2 2 S F X = 1 5 . 0 2 3 S F Y = 6 0 0 . 0 2 4 ZMN = - 5 . 0 2 5 S F Z = 5 . 0 2 6 S F K = 2 0 . 0 2 7 S F R = 2 . 5 2 8 S F U = 0 . 0 1 0 -193-slmulation i s provided from time zero to TAUMX1, as values are recorded every hour. During the i n t e r v a l from TAUMX1 to TAUMX2 the response w i l l be recorded every 5 hrs., and hence less d e t a i l e d . TAUMX1 was chosen from preliminary tests as the time when a l l the variables were not changing r a p i d l y with time. A1.2 The Main Program UBCEDEMA Following i s a copy of the main program UBCEDEMA. Explanations of the contents are presented throughout the program by comment statments. The numerical integrations of JNET1, JAS, QNETA and QNETG over time to y i e l d VIS1, VAS, QA and QG, resp e c t i v e l y , are conducted with the subroutine "DRKC", provided by the University of B r i t i s h Columbia Computing Centre. The d e f i n i t i o n s of the terms are presented i n Table 25. The tissue compliance curve i s required to determine the i n t e r s t i t i a l hydrostatic pressure PPMV corresponding to the f l u i d volume VIS1. A subroutine e n t i t l e d CMPLNC i s used to rela t e PPMV to VIS1 (statement 166 and 683): the terms of CMPLNC are defined i n Table 26. -194-COMPUTER PROGRAM UBCEDEMA L i s t i n g of UBCEDEMA at 11:55:23 on MAR 25, 1985 f o r CCid=HEIJ Page 1 1 IMPLICIT REAL*8(A -Z) 2 DIMENSION Y l0(5),F10(5),FPPMV(10 ),G(5),S(5),T(5) 3 COMMON/BLCA/SIGD,PSA,PSG,SIGFA,SIGFG 4 COMMON/BLCB/PIMV,VCELL,VEXA,VEXG,CMVA,CMVG 5 COMMON/BLCC/FPPMV,VTONS,B,VLMPH,KF,KASO,NK,SL,PMV 5.5 COMMON/BLCD/PMV1,PMV2,PTAU1,PTAU2 6 REALM X1(201),X2(201),X3(20l),X4(201),Y1(201),Y2(201), 7 *Y3(201),Y4(201),21(201),Z2(201) ,Z3(201),R1(201),R2(201), 8 *S1(201),S2(201),U1(201),U2(201),V1(201),V2(201),W1(201), 9 *W2(20l),TI(201),KA(201),PI(201),XS(2),YS1(2),YS2(2), 10 *YS3(2),YS4(2) 11 REAL*4 TMN,SFT,XMN,SFX,YMN,SFY,ZMN,A 1,AA1 , 12 *SFZ,RMN,SFR,UMN,SFU,KMN,SFK 13 INTEGER L,LL,LLL,LLLL,M,N,NN,NNN,NNNN,1,11,1 CHECK,I FLAG, 14 * I I , J , J J , J J J J 15 EXTERNAL FUNC 16 C 17 C INPUT DATA FROM FILE EDA 18 C 19 RSAD(4,1)VTOTO,VASO,QAO,QGO,PIMV 20 1 FORMAT(4(T11., D1 3 . 5 , /),T11,D13.5) 21 READ(4,3)VCELL,VEXA,VEXG,VTONS,VLMPH, 22 *B 23 3 FORMAT(5(T11,D13.5,/),T11,D13.5) 2 4 READ(4,4)CMVA,CMVG 25 4 FORMAT(T11,D13.5,/,T11,D13.5) 26 READ(4,5)(FPPMV(J),J=1,9) 27 5 FORMAT(8 (T11 ,D13.5,/),T11,D13.5) 28 READ(4 ,6)KF,SIGD,PSA,PSG,SIGFA,SIGFG 29 6 FORMAT(5(T11,D13.5,/),T11,D13.5) 30 C 3 1 C INPUT DATA FROM FILE PDA 32 C 3 3 READ(5,12)NN 3 4 READ(5 , i 2)NNN 3 5 READ(5,12)NNNN 3 6 12 FORMAT(T21 ,I 1 ) 3 7 READ(5,16)TAUMX1 3£ READ(5,16)SUBNT1 3S READ(5,16 JTAUMX2 4 0 READ(5,16)SUBNT2 41 READ(5,16)STEPSZ 4 2 READ(5,17)HMIN 4 3 RSAD(5, 17JTOL 4 4 READ(5,16)KASO 4 5 READ(5,16)NK 46 READ(5,16)SL 4 7 READ(5,16)PMV 46 16 FORMAT(T21,D13.6) 4 9 17 FORMAT(T21,D19.12) 50 C 5 1 C PRINTING OF INPUT DATA 52 C 53 WRITE(7,401) 54 401 FORMAT(1H1,1 OX,'INPUT PARAMETERS FOR RUN',///) 55 WRITE(7,402)CMVA,CMVG 56 402 FORMAT(IX,'PLASMA CONCENTRATION : ALBUMIN',8X,'CMVA 57 *D13.5,/,24X,'GLOBULIN',7X,'CMVG = ',D13.5,/) -195-L i s t i n g o f UBCEDEMA a t 11:55:23 on MAR 25, 1985 f o r CCid=HEIJ Page 2 53 WRI TE(7 ,403 )VCELL , VEXA,VEXG 59 403 FORMAT(IX,'CELLULAR VOLUME' ,23X , 'VCELL = ' , D 1 3 . 5 , / / , 1 X , 60 * 'EXCLUDED VOLUME : AL3UMIN' ,13X, 'VEXA = ' , D 1 3 . 5 , / , 1 9 X , 6 1 * 'GLOBULIN' , 12X, 'VEXG = ' ,D t 3 . 5 , / ) 62 WRITE(7,404)SIGD 63 404 FORMAT(IX,'REFLECTION C O E F F I C I E N T ' , 1 6 X , ' S I G D ' , D 1 3 . 5 , / ) 64 WRITE(7,405)PSA,PSG 65 405 FORMAT(IX,'PERMEABILITY SURFACE AREA : AL3UMIN',3X, 6 6 * ' P S A = ' , D 1 3 . 5 , / , 2 9 X , ' G L O B U L I N PSG = ' , 67 * D 1 3 . 5 , / ) 68 WRITE(7,406)SIGFA,SIGFG 69 406 FORMAT(1X,'REFLECTION COEFFICIENT : AL3UMIN' ,6X, 70 * ' S I G F A = ' , D i 3 . 5 , / , 2 6 X , ' G L O B U L I N ' , 5 X , ' S I G F G = ' , 7 1 * D 1 3 . 5 , / ) 72 WRITE(7,407)KAS0,NK,SL 73 407 FORMAT(IX,'ALVEOLAR FLOODING CONSTANTS : ' , 9 X , ' K A S 0 = ' , 74 * D 1 3 . 5 , / , 3 9X, 'NK = ' , D 1 3 . 5 , / , 3 9 X , ' S L = ' , D 1 3 . 5 , / ) 75 WRITE(7,4 08)VTONS,VLMPH 76 408 FORMAT(IX,'ONSET VOLUME FOR FLOODING' ,13X,"VTONS = ' , 77 * D 1 3 . 5 , / / , 1X,'VOLUME OF MAXIMUM LYMPH FLOW' ,1 OX,'VLMPH = ' , 78 * D 1 3 . 5 , / ) 79 WRITEO, 409 ) KF , P M V 80 409 FORMAT(1X,'ENDOTHELIUM FILTRATION COEFFICIENT KF = ' , 81 * D13. 5 , / / , I X , ' P L A S M A HYDROSTATIC PRESSURE' ,11X, 'PMV = ' , 82 * D 1 3 . 5 , / ) 83 WRITE(7,410)TAUMX1 ,SUBNT1 ,TAUMX2,SUBNT2,STEPS Z,HMIN,TOL 84 4 1 0 FORMAT(IX,'MAXIMUM TIME OF PRINTING INTERVAL 1 ' , 3 X , 85 *'TAUMX1 = ' , D 1 3 . 5 , / , I X , ' P R I N T I N G OUTPUT INTERVAL l ' , 1 2 X , 86 * ' SUBNT 1 ' , D 1 3 . 5 , / / , 1 X , ' M A X I M U M TIME OF RUN',19X, 87 88 *'TAUMX2 = ' , D 1 3 . 5 , / , 1 X , ' P R I N T I N G OUTPUT INTERVAL 2 ' , 12X , *'SUBNT2 = ' , D 1 3 . 5 , / / , 1 X , ' I N I T I A L STEPSIZE F O R CALCULATIONS', 89 * 5X, 'STEPS Z = ' , D 1 3 . 5 , / / , 1 X , ' M I N I M U M STEPS IZE F O R CALCULATIONS 90 * 5 X , ' KM IN = ' , D 1 3 . 5 , . / / , IX, ' MAXIMUM TOLERANCE ' , 2 1 X , 9 1 * ' T C L = ',313 . 5 ) 92 c 93 c I N I T I A L !ZiNG VARIABLES & COUNTERS 94 c 95 1=0 b c : C H E C K = : 9 7 I?LAG= ' 98 L = \ 99 M = 2 00 C = 0.0 0 \ TAu = 0 . 0 02 H S T E ? = S T E ? S Z 03 SU3:NT=SU3N?1 04 Q A = Q A O 05 Q G = Q G O 06 VAS=VASO 107 V T O T = V T O T O 105 V I S i 0=V70T0- V A S C 109 VIS1=VIS10 1 1 0 1 1 1 c c ASSIGNING INITIAL VALUES TO ARRAY YIO : NEEDED 1 1 2 c FOR UBC SUBROUTINE "DRKC" 1 1 3 c 1 1 4 YIO(1)=VTOTO 1 1 5 YIO(2)=QAO -196-L i s t i n g of UBCEDEMA at 11:55 :23 on MAR 25, 1985 for CCid=HEIJ Page 3 1 1 6 YIO(3)=QGO I 1 7 1 18 YIO(4)=VASO YI0(5)=VIS10 1 1 9 C 120 C ASSIGNING INITIAL VALUES TO VARIABLES THAT ARE TO 121 C BE DETERMINED AT THE "CHARACTERISTIC POINTS" 122 C 123 JASMAX=0.DO 124 JASLO=0.DO 125 TIMAX=TAU 126 TILO=TAU 127 TAUON=0.D0 1 28 VTOTLO=VTOT 1 29 VTOTMX=VTOT 130 VISlLO=VIS1 131 VIS1MX=VIS1 1 32 RCAMAX=0.D0 133 CTAON=QA/VISl 134 CTALO=0.DO 135 C 136 C DETERMINATION OF THE NUMBER OF INTERVALS (II) OF 137 C SIZE SU3NT2 IN THE FIRST TIME RANGE OF MAXIMUM 1 38 C VALUE TAUMX1 139 c 140 A1=TAUMX1 141 AA1=SUBNT2 142 I I= IF IX (A1) / !F IX (AAl ) 143 c 144 195 CONTINUE 1 45 c 146 c CALCULATION OF INTERSTITIAL VOLUME 147 c 148 c VIS1=VTOT-VAS 149 VIS=VIS1-VCELL 1 50 c 151 c CALCULATION OF AL3UMIN PARAMETERS 1 52 c 1 53 CTA = C'A/VI S 1 54 VAVA=VIS-VEXA 1 55 C A V A = Q A / V A. V A 1 56 r 1 57 c CALCULATION OF GLOBULIN PARAMETERS 1 58 C 1 59 CTG=QG/VIS 1 60 VAVG=VIS-VEXG 1 6 1 CAVG=QG/VAVG 162 C 1 63 C CALCULATION OF CENTRAL PARAMETERS 1 64 c CALCULATION OF INTERSTITIAL PRESSURE 165 c 166 CALL CMPLNC (VIS ' ,FPPMV,PPMV) 167 c 1 68 c CALCULATION OF INTERSTITIAL ONCOTIC PRESSURE 169 c 170 CP=CAVA+CAVG 171 PIPMV=210.D0*CP+!600.DO*CP*CP+9000.DO*CP*CP*CP 172 c 173 c CALCULATION OF CAPILLARY FILTRATION RATE -197-L i s t i n g of U3CEDEMA at 11: 55 :23 cn MAR 25, 1985 f o r CCicl=HEIJ Page 4 174 C 17 5 JV=XF*((PMV-PPMV)-SIGD*(PIMV-PIPMV)) 176 C 177 C 173 C CALCULATION OF LYMPH FLOW 179 C 180 IF(VIS1 .LT.VLMPH)GO TO 203 13! C CALCULATION OF MAXIMUM LYMPH FLOW 132 JL=0.17D0*VLM?K-5.56D1 183 GO TO 204 184 203 JL=0 .17D0*V IS1 -5 .56D1 185 C 186 C CALCULATION OF RATE OF EXTRAVASCULAR FLUID 137 C ACCUMULATION 188 C 189 204 JNET=JV-JL 190 FIO( l )=JNET 1 9 1 C 192 C DETERMINATION IF ALVEOLAR FLOODING OCCURS 193 C 194 IF(VTOT.LT.VTONS)GO TO 207 195 C 196 C CALCULATION OF FILTRATION COEFFICIENT OF 197 C EPITHELIAL MEM3RANS 1 98 C 199 KAS=KAS0+NK*(VISI-VTONS) 200 GO TO 208 201 207 KAS=0.D0 202 C 203 C CALCULATION OF HYDROSTATIC PRESSURE OF FLUID 204 C IN THE AIR SPACE 205 C 206 203 PAS=SL*VAS^3 207 C 208 C CALCULATION CF HYDROSTATIC PRESSURE DIFFERENCE • 209 C (??MV-?AS' BETWEEN" INTERSTITIUM & AIR SPACE 2 10 C 2 1 ! ? n r =- = ? ? M V - ? A S 2 12 C 213 C CALCULATION CF TRANSEPITHELIAL FLUID FLOW RATE 2 14 C 215 jAS=KAS* IPPMV-PAS) 216 F I C U ) = J A S 2:7 C 2 18 C CALCULATION Cr RATE OF ACCUMULATION OF 220 C 22 1 JN'ET i = J V - J L - J A S 222 FIC';3)=JNET1 223 C 224 C CALCULATION OF TRAMSENDOTHELIUM ALBUMIN 22 5 C FLOW RATE 226 C 227 JSA=?SA*(CMVA-CAVA)+(I.DO-SIGFA)*(CMVA+CAVA)*JV/2.DO 227.3 DIFFUS=?SA*(CMVA-CAVA) 227.6 CONVEC=(1.D0-SIGFA)*(CMVA+CAVA)*JV/2 .D0 228 C 229 C CALCULATION OF RATE OF INTERSTITIAL ALBUMIN -198-L i s t i n g of UBCEDEMA at 11:55:23 on MAR 25, 1985 for CCid=HEIJ Page 5 230 C ACCUMULATION 231 C 232 QNETA=JSA-CTA*(JL+JAS) 233 FIO(2)=QNETA 234 C 235 C CALCULATION OF TRANSENDOTHELIUM GLOBULIN 236 C FLOW RATE 237 C 238 JSG=PSG*(CMVG-CAVG)+(1.D0"SIGFG)*(CMVG+CAVG)*JV/2.DO 239 C 240 C CALCULATION OF RATE OF INTERSTITIAL GLOBULIN 241 C ACCUMULATION 242 C 243 QNETG=JSG-CTG*(JL+JAS) 244 FIO(3)=QNETG 245 C 246 C SETTING UP OF TABLES (2) FOR FURTHER CALCULATIONS 247 C USING OTHER PROGRAMS 248 C 249 IF(NNNN.EQ.2)GO TO 399 250 IF(TAU.GT.99.0)GO TO 399 251 WRITE(1 ,310)TAU,JV,JL,JNET1 ,JAS F0RMAT(5(E12.5,2X)) 252 310 253 C WRITE(2,320)TAU,VIS1,PPMV,KAS,PAS 254 C 320 FORMAT(5(E12.5,2X)) 255 C WRITE(3,330)TAU,QA,QG,PIPMV 256 C 330 FORMAT(4(E12.5,2X)) 257 c CTCMV=CTA/CMVA 253 c WRITE(6,34 0)TAU,QNETA,QNETG,CTA,CTG 259 c 340 FORMAT(5(E12.5,2X)) 260 c WRITE(8,350)TAU,DIFFUS,CONVEC,JSA,QNETA 261 c 350 F0RMAT(5(E12.5,2X)) 262 399 CONTINUE 263 c 264 c DETERMINATION OF THE TIME OF ONSET OF ALVEOLAR 265 c FLOODING (TION) & THE CORRESPONDING ALBUMIN CONCENTR-266 c -ATION (RCAON) 267 c 26S I F ( I FLAG.GT.1 )GO TO 261 269 IF(JAS.EQ.0.D0)GO TO 261 27C TION=(TAU+TAUON)/2.D0 27 1 RCAON=((CTA+CTAON)/2.D0)/CMVA 272 IFLAG=2 273 c 274 c DETERMINATION OF MAXIMUM JAS (JASMAX) & CORRESPONDING 275 c TIME (TIMAX) ,ALBUMIN CONCENTRA I ON RATIO (RCAMAX) , 276 c TOTAL EXTRAVASCULAR FLUID VOLUME (VTOTMX), & 277 c EXTRAVASCULAR-EXTRAALVEOLAR FLUID VOLUME (VIS1 MX) 273 c 279 261 IF(JAS.LT.JASMAX)GO TO 262 280 JASMAX=JAS 28 1 TIMAX=TAU 282 RCAMAX=CTA/CMVA 283 VTOTMX=VTOT 284 VIS1MX=VIS1 285 c 286 c DETERMINATION OF THE TIME (TIOT) TO REACH A 287 c VTOT OF 1000 ML & THE CORRESPONDING -199-L i s t i n g of UBCEDEMA at 11:55 :23 on MAR 25, 1985 for CCid=HEIJ Page 6 288 C EXTRAVASCULAR-EXTRAALVEOLAR FLUID VOLUME (VISIOT) , 289 C TRANSEPITHELIAL FLUID FLOWRATE (JASCT) , SLOPE (JASSL?) 290 C OF THE TRANSEPITHELIAL FLUID FLOWRATE-TIME CURVE AT 291 C VTOT=1000 ML, AND ALBUMIN CONCENTRATION RATIO (RCAOT) 292 C 293 262 IF(VTOT.LT.1000.DO)GO TC 263 294 IF (I CHECK.GT.1)GO TO 264 295 FACT=(1000.D0-VTOTLO)/(VTOT-VTOTLO) 296 VI S10T=VIS1LO+FACT*(VIS1-VIS1LO) 297 JASOT=JASLO+FACT*(JAS-JASLO) 298 JASSLP=(JASLO-JAS) / (T ILO-TAU) 299 RCAOT=(CTALO+FACT*(CTA-CTALO))/CMVA 300 TIOT=TILO+FACT*(TAU-TILO) 301 ICHECK=2 302 GO TO 264 303 C REPLACING OLD VALUES OF DESIGNATED V ^ ' A B L E S WITH NEW 304 C VALUES IF VTOT IS LESS THAN 1000 ML 305 263 IF(VTOT.LT.VTOTLO)GO TO 264 306 VTOTLO=VTOT 307 VIS1L0=VIS1 308 JASLO=JAS 309 CTALO=CTA 310 TILO=TAU 31 1 C 312 C DETERMINATION OF WHETHER A NSW TIME INTERVAL 313 C (I1*SUBINT) FOR PRINTING OF VARIA3LES HAS BEEN REACHED 314 C IF SO, THE VARIA3LES ARE STORED IN ARRAY FORM. 315 C IF NOT, TIME ITERATIONS ARE CONTINUED. 3 1 6 C 317 264 IF (C .GE . (FLOAT( I 1 )*SU3IMT)) M=M+1 318 IF(M.GT.1)GO TC 2 60 319 c 320 c DETERMINATION OF BEGINNING (A) AND END (C) OF 32 1 c CALCULATION INTERVAL IN UNITS OF TIME 322 c 323 2 1 0 A = TAU 324 C=TAU+0.2 5D0 325 HSTE?=.1ODO 326 CTAON=CTA 327 TAUON=A 328 c 329 c UTILIZATION OF U3C SUBROUTINE "DRKC" TO EVALUATE 330 c TIME INTEGRAL CF VARIABLES QA.QG.VAS, « VTOT BETWEEN" 33 1 c CALCULATION INTERVAL BEGINNING AT ' A' & ENDING AT 332 c ' C OF WIDTH 0 .2 5 333 c 334 CALL D R K C ( 5 , A , C , Y l O , F I C , H S T E ? , H M I N , T C L , F U N C , G , S , T j 335 VTOT=YIO(1) 336 QA=YIO(2) 337 QG=YIO(3) 333 VAS=YIO(4) 339 VIS 1=YIO(5) 340 TAU = C 34 1 GO TO 195 342 c 343 c ARRANGEMENT OF VARIABLES FOR PRINTING AND PLOTTING 344 c 345 260 1=1 + 1 -200-L i s t i n g o f U3CEDEMA a t 1 1 : 5 5 : 2 3 on MAR 2 5 , 1985 f o r C C i d = H E I J P a g e 7 346 M= 1 347 C 3 43 C SET - U P OF ARRAYS FOR STORAGE OF D I F F E R E N T V A R I A B L E S 349 C 350 T l ( I ) = T A U 351 X I ( I ) = J V 352 X 2 ( I ) = J L 353 X 3 ( I ) = J N E T I 354 X 4 ( I ) = J A S 355 Y 1 ( I ) = V T O T 356 Y 2 ( I ) = V A S 357 Y 3 ( I ) = V I S 1 353 Y 4 ( I ) = V I S 359 Z 1 ( I ) =PPMV 360 Z 2 ( I ) = P A S 36 i 2 3 ( 1 ) = P D I F 362 K A ( I ) = K A S 363 RI ( I ) = Q A 364 R 2 ( I ) = Q G 365 S 1 ( I ) = V A V A 366 S 2 ( I ) = V A V G 367 U l ( I ) = C T A 363 U 2 ( I ) = C T G 369 V I ( I ) = C A V A 370 V 2 ( I ) = C A V G 37 1 W l ( I ) = Q N E T A 372 W2( I )=QNETG 373 P I ( I ) = P I P M V 374 I F ( C . L T . T A U M X ! ) I 1 = 1 375 I F ( C . L T . T A U M X 1 ) G O TO 210 376 SU3INT=SUBNT2 3 77 1 1 = 1 1 + 1 3 7 S 1 1 = 1 1 379 I F ( C . G E . T A U M X 2 ) G O TO 267 33 0 GO TO 210 3 £ '. C 3 £ 2 C c TERMINATION OF C A L C U L A T I O N S J C J 3 c 4 2 5 " CONTINUE — c — f I F ( N N N . E Q . 2 ) G O TO 900 3 c o 3c 7 L c {-PRINT ING OF T A 3 L E S J C £ 3 3 9 W R I T E ( 7 , 4 1 4 ) T I O N , R C A O N 3 9 0 * : i - i ^ FORMAT( 1H1 , 15X , ' O N S E T OF A L V E O L A R F L O O D I N G ' , / / / , ** ^ 1 * IX, ' T I M E OF ONSET IS ' , D 1 2 . 5 , / / , IX, ' A L B U M I N ' , c ~ * ' CONCENTRATION RATIO - C T A / C M V A - IS ' , D 1 2 . 5 , / / / / ) - C ~ W R I T E ( 7 , 4 1 5 ) 3S4 4 1 5 F O R M A T ( 5 X , ' V A R I A B L E S AT THE POINT OF T H E ' , 395 * ' MAXIMUM ALVEOLAR FLOODING R A T E ' , / / ) 396 W R I T E ( 7 , 4 2 0 ) J A S M A X , T l M A X , R C A M A X 397 420 F O R M A T ( I X , ' T H E MAXIMUM F L U I D FLOWRATE I N T O ' , 393 * ' THE AIR S P A C E IS ' , D 1 2 . 5 , / / , 1 X , ' T H E C O R R E S P O N D I N G ' , 399 * ' T IME I S ' , 2 4 X , D 1 2 . 5 , / / , I X , ' T H E ALBUMIN CONCENTRATION 400 * ' RAT IO - C T A / C M V A - IS ' , D 1 2 . 5 , / ) 401 W R I T E ( 7 , 4 2 1 ) V T O T M X , V I S 1 M X 402 421 F O R M A T ( I X , ' T H E E X T R A V A S C U L A R F L U I D VOLUME IS ' , 1 5 X , 403 * D 1 2 . 5 , / / , I X , ' T H E I N T E R S T I T I A L AND C E L L U L A R VOLUME IS - 2 0 1 -L i s t i n g of UBCEDEMA at 11:55:23 on MAR 25, 1985 f o r CCid=KEIJ Page 8 404 *9X,D-12.5,////) 405 WRITE(7,425) 406 425 FORMAT(IX,'THE MAGNITUDE OF VARIA3LES AT A TOTAL', 407 *' EXTRAVASCULAR FLUID VOLUME OF 1000 ML',//) 408 WRITE(7, 4 30)TIOT,VIS10T,JASOT,JASSL?,RCAOT 409 430 FORMAT(IX,'TIME',32X,'TAU = ',D12.5,//,IX, 410 *'INTERSTITIAL AND CELLULAR VOLUME',4X,'VIS1 = ',Dl2.5, 411 *//,IX,'FLUID FLOWRATE INTO THE AIR SPACE JAS = ', 4 12 *D12.5,//,1X,'RATE OF CHANGS OF JAS' ,15X,'DJAS/DT = ', 413 *D12.5,//,IX,'ALBUMIN CONCENTRATION RATIO',92,'CTA/CMVA = ' 414 *,D12.5) 415 WRITE(7,434) 416 4 34 FORMAT(1H1,18X,'TABLE 1 : OUTPUT OF FLUID FLOWS',//) 417 WRITE(7,435) 418 435 FORMAT(2X,'TAU(HRS) ' , 6X, 'JV(ML/HR)' ,5X,'JL(ML/HR)' , 419 *3X,'JNET1(ML/HR)',4X,'JAS(ML/HR)',/) 4 20 WRITE(7, 4 4 0 ) ( T I ( J ) ,X1 ( J ) , X 2 ( J ) , X 3 ( J ) , X 4 ( J ) , J = 1 ,I) 421 440. FORMAT(5(E12.5,2X) ) 422 WRITE(7,444) 423 444 FORMAT(1H1,17X,'TABLE 2 : OUTPUT CF FLUID VOLUMES',//) 424 WRITE(7,445) 4 2 5 4 4 5 FORMAT(2X,'TAU(HRS)',6X,'VTOT(ML)',6X,'VAS(ML)', 426 *7X,'VI SI (ML)' ,6X,'VIS(ML)' ,/) 427 W R I T E ( 7 , 4 5 0 ) ( T I ( J ) , Y 1 ( J ) , Y 2 ( J ) , Y 3 ( J ) , Y 4 ( J ) , J = l , 1 ) 428 450 FORMAT(5(E12.5,2X)) 429 WRITE(7,454) 430 454 FORMAT(1H1,IX,'TABLE 3 : OUTPUT OF HYDROSTATIC PRESSURES -', 431 *' PPMV,PAS,(PPMV-PAS) -',//,12X,'& EPITHELIUM FILTRATION", 432 *' COEFFICIENT - KAS',//) 433 WRITE(7,455) 434 455 FORMAT(2X,'TAU(HRS)',4X,'PPMV(MM H G ) ' , 4 X , 435 *'PAS(MM HG)',4X,'PDIF(MM HG)',7X,'KAS'/) 4 36 WRITE(7,460)(TI(J),Z1 ( J ) , Z 2 ( J ) , Z 3 ( J ) , K A ( J ) , J = 1 ,1) 4 3 7 4 60 FORMAT(E12.5,2X,E12.5,2X,E12.5,3X,E!2.5,!X,E12.5) 438 WRITE (7 ,464) 43S 464 FORMAT(1H1,5X,'TABLE 4 : OUTPUT OF INTERSTITIAL SOLUTE', 4 4 0 *'WEIGHTS - Q A , Q G -',//, 16X,' & INTERSTITIAL AVAILABLE', 4 4 1 *' VOLUMES - VAVA,VAVG',//) 4-=2 WRITE ( 7 , 4 65) 4 4 3 4 6 5 FORMAT ( 2 X,'TAU(HRS)',7X, ' Q A ( G M ) ' , S X , ' Q G i G M ; ' , 7 X , 4 4 4 * ' V A V A ( M L ) ' , 6X, 'VAVG(ML) ' , / ) 4 4 5 WRITE( 7,4 50 )(TI(J),R1 (J) ,R2(J) ,S1 (J) , S 2 ( J ) , j = ' , I ) 446 4 70 FORMAT(5(El 2.5,2X)) 447 WRITE(7 ,474) 448 4 74 FORMAT(1H1 ,5X,'TABLE 5 : OUTPUT OF INTERSTITIAL - C T A , C T G 449 * ' 5. AVAILABLE' ,//, 1 6X , ' INTERSTITIAL - CAVA,CAVG - ', 450 *'CONCENTRATIONS',//) 451 WRITE(7,4 75) 4 52 4 7 5 FORMAT(2X,'TAU(HRS)',5X,'CTA(GM/ML) ' , 4X , 453 *'CTG(GM/ML)',4X,'CAVA(GM/ML)',3X,'CAVG(GM/ML)',/) 4 54 WRITE( 7,4 80)(TI(J),U 1 ( J ) , U 2 ( J ) , V I ( J ) , V 2 ( J ) , J = 1 , I ) 455 480 FORMAT(5(E12.5,2X)) 456 WRITE(7,484) 457 484 FORMAT(1H1,'TABLE 6 : OUTPUT OF INTERSTITIAL SOLUTE', 4 58 *' ACCUMULATION RATES - QNETA,',//,11X,'QNETG - ', 459 * ' ( , INTERSTITIAL ONCOTIC PRESSURE - PIPMV',//) 460 WRITE(7,485) 461 485 FORMAT(2X,'TAU(HRS)*,4X,'QNETA(GM/HR)',2X, -202-L i s - i n g of UBCEDEMA a t 11: 5 5 : 23 on MAP. 2 5 , 1 9 8 5 for CCid=HEIJ Page 9 462 * ' Q N E T G ( G M / H R ) ' , 3 X , ' P I P M V ( M M H G ) ' , / ) 462 W R I T E ( 7 ,490! (TI (J) ,W1 (J) ,W2(J) , P I ( J ) ,J=1 , I ) 464 490 F O R M . A T ( i ( E ! 2 . 5 , 2 X ) ) 465 C 466 C T E R M I N A T I O N OF P R I N T I N G S E C T I O N 467 C 468 900 C O N T I N U E 469 I F ( N N . E Q . 2 ) G O TO 1000 470 C 47 1 C P L O T T I N G OF G R A P H S 472 c 473 c I N P U T D A T A F R O M F I L E P D A F O R S Y M B O L L E G E N D 474 c -475 R E A D (5,901 )XS(1) ,XS(2) 476 90 ! F O R M A T ( T 2 i , E 1 3 . 5 , / , T 2 1 , S13 . 5 ) 477 R E A D ( 5,90 2)YS1 ( 1 ) ,YS2( 1 ) , YS3(1 ),YS4(1 ) 478 902 F O R M A T ( 3 ( T 2 1 , S 1 2 . 5 , / ) , T 2 1 , E 1 3 . 5 ) 479 YS;(2)=?S1(1) 480 YS2(2)=YS2(1) 43 ; YS 3(2)=YS 3(1 ) 432 YS4(2)=YS4(i) 4S3 c 4S4 r P L O T T I N G OF F L U I D F L O W R A T E S 435 c 486 T M N = 0 . 0 487 X M N = 0 . 0 433 c I N P U T D A T A F R O M F I L E P D A 489 R E A D ( 5 , 9 0 3 ) S E T 490 R E A D ( 5 , 9 0 3)SFX 49; 90 3 F O R M A T ( T 2 i , E l 3.5) 492 C A L L A X I S ( 2 . 2 5 , 3 . , ' T I M E ( H R S ) ' , - 1 0 , 5 . , 0 . , T M N , S F T ) 493 C A L L A X I S ( 2 . 2 5 , 3 . , ' F L U I D F L O W S ( M L / H R ) ' , 1 8 , 5 . , 9 0 . , X M N , S F X ) 494 CALL D L I N E ( T I , X i , I , T M N , 2 . 2 5 , S F T , X M N , 3 . , S F X , . 1 2 5 , . 1 2 5 , . 0 6 2 5 ) 495 CALL D L I N E ( 7 1 , X 2 , I , T M N , 2 . 2 3 , S F T , X M N , 3 . , S F X , . 2 5 , . 2 5 , . 0 7 5 ) 496 CALL D C : M E - : T : , XO , I ,TMN , 2 . 2 3 , S F T , X M N , 3 . , S F X , . 1 5 , . 0 6 2 5 , . 0 5 ) 497 CALL = B L Z N E ( T : , X 4 , I , T M N , 2 . 2 3 , S F T , X M N , 3 . , S F X , 2 , 1 , 0 . 1 , 2 ) 493 CALL L E G E N D ( P M V , K F , K A S 0 , N K , S L , S I G D , P S A , P S G , S I G F A , S I G F G ) 499 SCO CALL PLOT X S ' : } , " S 1 ( : } , 3 ) 50 i CALL DA3HLM •: .125, .0525 , .125 , .0625) 502 CALL ? L O T i X S ; 2 ) , V S 1 ( 2 ) , 4 ) 303 CALL ?SYM(5'.0,7. ! 25 , . i 25 , ' J L = ' , . 0 , 8 ) 304 CALL ?107 (XS- ' !) ,YS2! i ) , 3 ) 50 3 CALL DASHLNi .23, .075, . 23 , .075 ) 30 6 CA" L ?' ZZ; SS •. 1 : , : S3 '.' 2 ) , 4 ) 50 7 CA" " PS7M'. 3 . 0 . £ . 730 , . ' 25 , ' JM'TT i = ' , . 0 , 8 ) 50 3 CALL ? L 0 T ( X S ' • : ) , ? S 3 ( i ) , 3 ) 50 9 CALL DASHLN• . 13, .03 , .0625, .03) 5 : 0 CALL PLOT;XS•2} ,YS3 i 2) ,4) 5 : ! CALL PS YM ( '3.0,6.275, . 125, ' J A S = ' , . 0 , 8 ) 3 i 2 CALL L I M E ; X S , Y S 4 , 2 , 1 ) 5; 3 C A L L PLOT!12 . 0 , 0 . , - 3 ) 5i4 c 51 5 c P L O T T I N G OF F L U I D V O L U M E S 516 c 517 Y M N = 0.0 518 c I N P U T D A T A F R O M F I L E PDA 519 R E A D ( 5 , 9 0 3 ) S F Y - 2 0 3 -L i s t i n g of UBCEDEMA at 11:55:23 on MAR 25, 1985 f o r CCid=HEIJ Page 10 520 CALL AXIS(2.25,3.,'TIME (HRS)' ,- 10,5.,0.,TMN,SFT) 521 CALL AXIS(2.25,3.,'FLUID VOLUMES(ML)',17,5.,90.,YMN,SFY) 522 CALL B3LINE(TI,Y1 ,1,TMN,2.25,SFT,YMN,3.,SFY,2,1,0.1,2) 52 3 CALL DLINE(TI,Y2,1,TMN,2.25,SFT,YMN,3.,SFY, .125, .125, .0625) 524 CALL DLINE(TI,Y3,1,TMN,2.25,SFT,YMN,3.,SFY,.15, .,0625,.05) 525 CALL LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) 526 CALL PSYM(5.0,7.50,.125,'VTOT = ',.0,7) 527 CALL LINE(XS,YS1,2,1) 528 CALL PSYM(5.0,7.125,.125,'VIS1 = ',.0,7) 529 CALL PLOT(XS(1),YS2(1),3) 530 CALL DASHLN(.125,.0625,.125,.0625) 531 CALL PLOT(XS(2),YS2(2),4) 532 CALL PSYM(5.0,6.750,.125,'VAS = ',.0,7) 533 CALL PLOT(XS(1),YS3(1),3) 534 CALL DASHLN(.15,.05,.0625,.05) 535 CALL PLOT(XS(2),YS3(2),4) 536 CALL PLOT(12.0,0.,-3) 537 C 5 38 C PLOTTING OF HYDROSTATIC PRESSURES 539 C 540 C INPUT DATA FROM FILE PDA 541 READ(5,903)ZMN 542 READ(5,903)SFZ 543 CALL AXIS(2.25,3.,'TIME (HRS)' ,-10,5.,0 . ,TMN,SFT) 544 CALL AXIS(2.25,3.,'HYDROSTATIC & OSMOTIC PRESSURES(MM HG)', 545 *38,5.,90.,ZMN,SFZ) 546 CALL BBLINE(TI,Z1 ,1,TMN,2.25,SFT,ZMN,3.,SFZ,2,1,0.1,2) 547 CALL DLINE(TI,PI,1,TMN,2.25,SFT,ZMN,3. ,SFZ, .15, .0625, .05) 54 8 CALL DLINE(Tl,Z2,I,TMN,2.25,SFT,ZMN,3.,SFZ,.125,.125,.0625) 549 CALL LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) 550 CALL PSYM(5.0,7.50,.125,'PIPMV = ',.0,8) 55 1 CALL PLOT(XS(1),YS1(1),3) 552 CALL DASHLN(.15, .05, .0625, .05) 553 CALL PLOT(XS(2),YS1(2),4) 554 CALL PSYM(5.0,7.125,.125,'PPMV = ',.0,8) 555 CALL LINE(XS,YS2,2,1) 556 CALL PSYM(5.0,6.750,.125,'PAS = ',.0,8) 557 CALL PLOT(XS(1),YS3(1),3) 558 CALL DASHLN(.125, .0625, . 125, .0625 ) 559 CALL PLOT(XS(2),YS3(2),4) 560 CALL PLOT(12.0,0.,-3) 561 C 562 C PLOTTING OF EPITHELIUM FILTRATION COEFFICIENT 563 C 564 KMN=0.0 56 5 C INPUT DATA FROM FILE PDA 566 READ(5,903)SFK 567 CALL'AXIS(2.25,3.,'TIME (HRS)' ,- 10,5 . , 0 . ,TMN,SFT) 568 CALL AXIS(2.25,3.,'KAS (ML/HR/MM HG) ' , 17,5. , 90. ,KMN,SFK) 569 CALL BBLINE(TI ,KA,I,TMN,2.25,SFT,KMN,3.0,SFK,2, 1,0.1,2) 570 CALL LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG, SIGFA,SIGFG) 571 CALL PLOT(12.0,0.,-3) 572 C 573 C PLOTTING OF SOLUTE WEIGHTS 574 C 575 RMN=0.0 57 6 C INPUT DATA FROM FILE PDA 577 READ(5,903)SFR - 2 0 4 -L i s t i n g o f UBCEDEMA a t 1 1 : 5 5 : 2 2 on MAR 2 5 , 1985 f o r C C i d = K E I J Page 11 578 C A L L A X I S ( 2 . 2 5 , 3 . , ' T I M E ( HRS ) ' , - 1 0 , 5 . , 0 . , TMN , S F T ) 579 C A L L A X I S ( 2 . 2 5 , 3 . , ' S O L U T E W E I G H T S ( G M ) ' , 1 8 , 5 . , 9 0 . , R M N , S F R ) 580 C A L L 3 3 L I N E { T l , R 1 , I , T M N , 2 . 2 5 , S F T , R M N , 3 . , S F R , 2 , 1,0. 1 , 2 ) 531 C A L L D L I N E ( T l , R 2 , I , T M N , 2 . 2 5 , S F T , R M N , 3 . , S F R , . 1 2 5 , . 1 2 5 , . 0 6 2 5 ) 582 C A L L L E G E N D ( P M V , K F , K A S O , N K , S L , S I G D , P S A , P S G , S I G F A , S I G F G ) 583 C A L L ? S Y M ( 5 . 0 , 7 . 5 0 , . 1 2 5 , ' Q A = ' , . 0 , 5 ) 584 C A L L L I N E ( X S , Y S 1 , 2 , 1 ) 585 C A L L ?SYM( 5 . 0 , 7 . 1 2 5 , . 1 2 5 , ' QG = ' , . 0 , 5 ) 586 C A L L P L O T ( X S ( 1 ) , Y S 2 ( I ) , 3 ) 587 C A L L D A S H L N C . 1 2 5 , . 0 6 2 5 , . 1 2 5 , . 0 6 2 5 ) 588 C A L L P L O T ( X S ( 2 ) , Y S 2 ( 2 ) , 4 ) 589 C A L L ? L O T ( 1 2 . 0 , 0 . , - 3 ) 590 C 591 C P L O T T I N G OF SOLUTE CONCENTRATIONS 592 C 593 UMN=0.0 594 C INPUT DATA FROM F I L E PDA 595 R E A D ( 5 , 9 0 3 ) SFU 596 C A L L A X I S ( 2 . 2 5 , 3 . , ' T I M E ( H R S ) ' , - 1 0 , 5 . , 0 . , T M N , S F T ) 597 C A L L A X I S ( 2 . 2 5 , 3 . , ' S O L U T E C O N C E N T R A T I O N S ( G M / M L ) ' , 2 8 , 5 . , 598 * 9 0 . , U M N , S F U ) 599 C A L L 3 3 L I N S ( T l , U 1 , 1 , T M N , 2 . 2 5 , S F T , U M N , 3 . , S F U , 2 , 1 , 0 . 1 , 2 ) 600 C A L L D L I N E ( T I , U 2 , I , T M N , 2 . 2 5 , S F T , U M N , 3 . , S F U , . 1 2 5 , . 1 2 5 , . 0 6 2 5 ) 601 C A L L L E G E N D ( P M V , K F , K A S O , N K , S L , S I G D , P S A , P S G , S I G F A , S I G F G ) 602 C A L L P S Y M ( 5 . 0 , 7 . 5 0 , . 1 2 5 , ' C T A = ' , . 0 , 6 ) 603 C A L L L I N E ( X S , Y S 1 , 2 , 1 ) 604 C A L L P S Y M ( 5 . 0 , 7 . 1 2 5 , . 1 2 5 , ' C T G = ' , . 0 , 6 ) 60 5 C A L L P L O T ( X S ( 1 ) , Y S 2 ( 1 ) , 3 ) 606 C A L L D A S H L N ( . 1 2 5 , . 0 6 2 5 , . 1 2 5 , . 0625 ) 60 7 C A L L ? L O T ( X S ( 2 ) , 7 3 2 ( 2 ) , 4 ) 605 C 609 C TERMINATION OF PLOTTING S E C T I O N 610 C 6 1 1 C A L L PLOTN'D 612 1000 STOP 613 END 614 C 615 C 616 C SU3R0U7INE CMPLNC : FOR THE C A L C U L A T I O N CF 613 C COMPLIANCE CURVE 619 C 620 SUBROUTINE C M P L N C ' V i 3 1 FPPMV PPM 1/) 62 1 C 622 I M P L I C I T = E A 1 * 3 ( A - : • 623 DIMENSION FPPMV(iO) 624 R E A L * 4 S V I S l 62 5 INTEGER MLG.MH* 626 I F ( V I S 1 . L T . 4 6 0 . D 0 ) G O TO 610 627 P ? M V=0 . 0 1 7 D 0 * V I S :-6 . 7 3 D C 626 GO TO 650 629 610 I F ( V I S 1 . G E . 3 3 0 . 0 D 0 ) G O TO 620 630 P P M V = 0 . 2 2 7 D 0 * V I S i - 3 9 . 0 D 0 631 GO TO 650 632 C I N T E R P O L A T I O N OVER CURVED PORTION OF COMPLIANCE 6 3 3 620 D V I S 1 = ( V I S 1 - 3 8 0 . D 0 ) / 1 0 . D 0 6 3 4 SVIS1=DVIS1 635 M L O = I N T ( S V I S1)+1 - 2 0 5 -L i s t i n g o f UBCEDEMA a t 11:55:23 on MAR 2 5 , 1985 f o r C C i d = H E I J Page 12 636 MHI=.MLO+l 637 P PMVLO= F P PMV(MLO) 638 PPMVHI=FPPMV(MKI) 639 P P M V = (1 . D 0 + D V I S l - D F L O A T ( M L O ) ) * ( P P M V H I - P P M V L O ) + P P M V L 0 640 650 CONTINUE 64 1 RETURN 642 END 643 C 644 C 645 C SUBROUTINE FUNC : TO BE USED IN CONJUNCTION WITH 646 C SUBROUTINE DRKC FOR T H E E V A L U A T I O N OF THE D E R I V A T I V E S 647 C ( F I O ) OF DEPENDENT V A R I A B L E S (Y IO) 648 c 649 SUBROUTINE F U N C ( T I M E , Y l O , F I O ) 650 c 651 I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) 652 R E A L * 8 J A S , J L , J N E T , J N E T 1 , J S A , J S G , J V , K F , K A S , N K , K A S O 653 D IMENSION YIO(5) ,F IO(5) 654 C O M M O N / B L C A / S I G D , P S A , P S G , S I G F A , S I G F G 655 C 0 M M 0 N / 3 L C B / P I M V , V C E L L , V E X A , V E X G , C M V A , C M V G 656 C O M M O N / B L C C / F P P M V ( 1 0 ) , V T O N S , B , V L M P H , K F , K A S 0 , N K , S L , P M V 656.5 C O M M O N / B L C D / P M V1 , P M V 2 , P T A U 1 , P T A U 2 657 VTOT=YIO(1) 658 Q A = Y I O ( 2 ) 659 Q G = Y I O ( 3 ) 660 V A S = Y I O ( 4 ) 661 VIS1=YIO(5) 662 .5 c 663 c C A L C U L A T I O N OF I N T E R S T I T I A L VOLUME 664 c 665 c V I S l = V T O T - V A S 666 V I S = V I S 1 - V C E L L 667 c 668 c C A L C U L A T I O N OF ALBUMIN PARAMETERS 669 c 670 CTA=QA/VIS 671 VAVA=VIS - V S X A 672 CAVA=QA/VAVA 67 3 c 6~4 c C A L C U L A T I O N OF G L 0 3 U L I N PARAMETERS 675 c 676 CTG=QG/VIS 6 7 7 VAVG=VIS - V E X G S~3 CAVG=QG/VAVG 679 c 630 c C A L C U L A T I O N OF CENTRAL PARAMETERS 6c ! £ £ 9 c C A L C U L A T I O N OF I N T E R S T I T I A L P R E S S U R E O O i 6 5 3 C A L L C M P L N C ( V I S ! , F P P M V , P P M V ) 634 c 63 5 c C A L C U L A T I O N OF I N T E R S T I T I A L ONCOTIC P R E S S U R E 686 c 687 CP=CAVA+CAVG 688 - PIPMV=210 . D 0 * C P + 1 6 0 0 . D O * C P * C P + 9 0 0 0 . D 0 * C P * C P * C P 689 c 690 c C A L C U L A T I O N OF C A P I L L A R Y F I L T R A T I O N R A T E 691 c 692 J V = K F * ( ( P M V - P P M V ) - S I G D * ( P I M V - P I P M V ) ) -206-L i s t i n g of UBCEDEMA at 11:55:23 on MAR 25, 1985 f o r CCid=HEIJ Page 13 693 C 694 C 695 C CALCULATION OF LYMPH FLOW 696 C 697 IF(VIS 1 .LT.VLMPH)GO TO 203 698 JL=0.17D0*VLMPH-5.56D1 699 GO TO 204 700 203 JL=0.17D0*VIS1-5.56D1 701 C 702 C CALCULATION OF RATE OF EXTRAVASCULAR FLUID 703 C ACCUMULATION 704 C 705 204 JNET=JV-JL 706 FIO(1)=JNET 707 C 708 C DETERMINATION IF ALVEOLAR FLOODING OCCURS 709 C 710 IF(VTOT.LT.VTONS)GO TO 207 7 1 1 c 712 c CALCULATION OF FILTRATION COEFFICIENT OF 713 c EPITHELIAL MEMBRANE 714 c 715 KAS = KAS0+NK*(VIS 1-VTONS) 716 GO TO 208 717 207 KAS=0.D0 718 c 719 c CALCULATION OF HYDROSTATIC PRESSURE OF FLUID 720 c IN THE AIR SPACE 72 1 c 722 208 PAS=SL*VAS+3 723 c 724 c CALCULATION OF TRANSEPITHELIAL FLUID FLOW RA1 FT ,TT 725 c 726 JAS=KAS*(PPMV-PAS) 727 FIO(4)=JAS 728 c 729 c CALCULATION OF RATE OF ACCUMULATION OF 730 c INTERSTITIAL FLUID 73 1 c 732 JNET1=JV-JL-JAS 733 FIO(5)=JNST1 7 3 4 c "7 3 5 c CALCULATION OF TRANSENDOTKELI UM A.L SUM I N 736 c FLOW RATE 7 37 c 738 JSA=PSA*(CMVA-CAVA) + ( 1 .DO-SIGFA)*(CMVA-CAVA) *JV/2 739 c 740 c CALCULATION OF RATE OF INTERSTITIAL ALBUMIN 74 1 c ACCUMULATION 742 c 743 QNETA=JSA-CTA*(JL+JAS) 744 FIO(2)=QNETA 745 c 746 c CALCULATION OF TRANSENDOTHELIUM GLOBULIN 747 c FLOW RATE 748 c 749 JSG=PSG*(CMVG-CAVG)+(1.DO-SIGFG)*(CMVG+CAVG) *JV/2 750 c -207-L i s t i n g of UBCEDEMA a t 11:55:23 on MAR 2 5 , 1985 f o r C C i d = H E I J Page 14 751 C C A L C U L A T I O N OF RATE OF I N T E R S T I T I A L G L O B U L I N 752 C ACCUMULATION 753 C 754 Q N E T G = J S G - C T G * ( J L + J A S ) 755 F I O ( 3 ) = Q N E T G 756 RETURN 757 END 758 C 759 C 760 C SUBROUTINE LEGEND : FOR THE PRINTING OF CONSTANTS 761 C ON P L O T S 762 C 763 SUBROUTINE L E G E N D ( P M V , K F , K A S O , N K , S L , S I G D , P S A , P S G , S I G F A , S I G F G ) 764 C 765 R E A L * 3 P M V , K F , K A S O , N K , S L , S I G D , P S A , P S G , S I G F A , S I G F G 766 C A L L PSYM(2.00,1.875, . 1 2 5 , ' P M V (MM HG) = ' , . 0 , 2 1 ) 7 67 C A L L N U M 3 E R ( 4 . 2 5 , 1 . 8 7 5 , . 1 2 5 , P M V , . 0 , 2 ) 768 C A L L P S Y M ( 2 . 0 0 , 1 . 6 0 , . 1 2 5 , ' K F (ML /HR/MM HG) = ' , . 0 , 2 1 ) 769 C A L L N U M 3 E R U . 2 5 , 1 . 6 0 , . 1 2 5 , K F , . 0 , 4 ) 770 C A L L P S Y M ( 2 . 0 0 , 1 . 3 2 3 , . 1 2 3 , ' K A S O (ML /HR/MM HG) = ' , . 0 , 2 1 ) 771 C A L L N U M B E R ( 4 . 2 5 , 1 . 3 2 5 , . 1 2 5 , K A S O , . 0 , 5 ) 772 C A L L P S Y M ( 2 . 0 0 , 1 . 0 5 0 , . 125 , 'NK ( /HR/MM HG) = ' , . 0 , 2 1 ) 773- C A L L N U M 3 E R ( 4 . 2 5 , 1 . 0 5 0 , . 1 2 5 , N K , . 0 , 5 ) 774 C A L L PSYM( 2 . 0 0 , 0 . 7 7 5 , . 1 2 5 , ' S L (MM H G / M L ) = ' , . 0 , 2 1 ) 77 5 C A L L N U M B E R ! 4 . 2 5 , 0 . 7 7 5 , . 1 2 5 , S L , . 0 , 5 ) 776 C A L L P S Y M ( 5 . 2 5 , 1 . 3 7 5 , . 1 2 5 , ' S I G D = ' , . 0 , 1 4 ) 77 7 C A L L NUMBER( 6 . 7 5 , 1 . 8 7 3 , . 1 2 5 , S I G D , . 0 , 5 ) 776 C A L L P S Y M ( 5 . 2 5 , 1 . 6 , . 1 2 5 , ' P S A (ML/HR) = ' , . 0 , 1 5 ) 779 C A L L N U M 3 E R ( 6 . 7 5 , 1 . 6 , . 1 2 5 , P S A , . 0 , 4 ) 780 C A L L P S Y M ( 5 . 2 3 , 1 . 3 2 3 , . 1 2 5 , ' P S G (ML /HR) = ' , . 0 , 1 5 ) 781 C A L L N U M 3 E R ( 6 . 7 5 , 1 . 3 2 5 , . 1 2 5 , P S G , . 0 , 4 ) 782 C A L L P S Y M ( 5 . 2 5 , 1 . 0 5 , . 1 2 5 , ' S I G F A = ' , . 0 , 1 4 ) 733 C A L L N U M 3 E R ' 6 . 7 5 , 1 . 0 5 , . 1 2 5 , S I G F A , . 0 , 5 ) 784 C A L L P S Y M ( 5 . 2 3 , . 7 7 5 , . 123 , ' S I G F G = ' , . 0 , 1 4 ) 7 £ 5 C A L L N U M B E R ( £ . 7 5 , . 7 7 5 , . 1 2 5 , S I G F G , . 0 , 5 ) 786 RETURN 757 END -208-Table 25: Explanation of Subroutine DRKC (56) C A L L D R K C ( N , X , Z , Y , F , H , H M I N , E , F U N C , G , S , T ) w h e r e : N i s a n I N T E G E R v a r i a b l e o r c o n s t a n t . O n e n t r y , i t c o n t a i n s t h e n u m b e r o f d i f f e r e n t i a l e q u a t i o n s t o b e s o l v e d . X i s a R E A L * 8 v a r i a b l e . O n e n t r y , i t c o n t a i n s t h e i n p u t v a l u e o f t h e i n d e p e n d e n t v a r i a b l e . O n e x i t , i t c o n t a i n s t h e f i n a l v a l u e o f t h e i n d e p e n d e n t v a r i a b l e . Z i s a R E A L * 8 v a r i a b l e o r c o n s t a n t . O n e n t r y , i t c o n t a i n s t h e f i n a l v a l u e o f t h e i n d e p e n d e n t v a r i a b l e a t t h e e n d p o i n t o f i n t e g r a t i o n . Y i s a R E A L * 8 , o n e - d i m e n s i o n a l a r r a y , o f d i m e n s i o n > N . O n e n t r y , Y ( I ) I = 1 , . . . , N c o n t a i n t h e i n i t i a l v a l u e s o f t h e d e p e n d e n t v a r i a b l e s . O n e x i t , Y c o n t a i n s t h e v a l u e s o f t h e d e p e n d e n t v a r i a b l e s a t t h e e n d p o i n t o f i n t e g r a t i o n . F i s a R E A L * 8 , o n e - d i m e n s i o n a l a r r a y , o f d i m e n s i o n >N. On e x i t , i t c o n t a i n s t h e o u t p u t v e c t o r o f d e r i v a t i v e s Y ' ( X ) a t X = Z . H i s a R E A L * 8 v a r i a b l e . O n e n t r y , i t c o n t a i n s t h e i n p u t s t e p - s i z e . H i s c h a n g e d b y D R K C t o c o n t a i n t h e s t e p - s i z e u s e d a t t h e c u r r e n t i n t e g r a t i o n s t e p . F o r m a x i m u m a c c u r a c y i n s i n g l e p r e c i s i o n , . H = ( Z - X ) / 6 4 . H M I N i s a R E A L * 8 v a r i a b l e o r c o n s t a n t . O n e n t r y , i t c o n t a i n s a l o w e r b o u n d f o r t h e s t e p - s i z e i n o r d e r t o p r e v e n t i n f i n i t e c y c l i n g . F o r d i f f i c u l t p r o b l e m s ( e . g . t h e o r b i t p r o b l e m — s e e t h e s a m p l e p r o b l e m i n U B C D I F S Y ) D R K C m a y r e q u i r e H M I N ^ l 0 " 8 * H . U s u a l l y H M I N = 1 0 " 2 * H . i s a d e q u a t e . . S e e s u b s e c t i o n ( c ) , b e l o w , f o r a m e t h o d t o d e t e c t w h e n H M I N h a s b e e n r e a c h e d . - 2 0 9 -Table 25: (cont'd.) E i s a REAL*8 v a r i a b l e or c o n s t a n t . On e n t r y , i t c o n t a i n s an e r r o r tolerance-. If ES I , the o r i g i n a l s t e p - l e n g t h ' w i l l be m a i n t a i n e d throughout the i n t e g r a t i o n . In s i n g l e p r e c i s i o n , E £ . 5 x l 0 " s . FUNC i s the name of a SUBROUTINE subprogram which i s coded by the user to e v a l u a t e the d e r i v a t i v e s . FUNC must be d e c l a r e d EXTERNAL i n the u s e r ' s c a l l i n g program. T y p i c a l l y t h i s s u b r o u t i n e would look l i k e : SUBROUTINE FUNC (X,Y,F) IMPLICIT REAL*8(A~H,O -Z) DIMENSION Y ( 1 ) , F ( 1 ) F( 1)= f , ( X , Y ( 1 ) , . . . , Y ( N ) ) F(N)= f N ( X , Y ( 1 ) , . . . , Y ( N ) ) RETURN END whe r e: X i s a REAL*8 v a r i a b l e which c o n t a i n s the c u r r e n t v a l u e . o f the independent v a r i a b l e . Y i s a REAL*8, on e - d i m e n s i o n a l a r r a y , which c o n t a i n s the c u r r e n t v a l u e s of the dependent v a r i a b l e s . F i s a REAL * 3, one-dimensional a r r a y . On exit; from FUNC i t c o n t a i n s the v a l u e s of the d e r i v a t i v e s . S, T are REAL*8, one-dimensional a r r a y s , dimensioned They are work areas r e q u i r e d i n t e r n a l l y . G, >N . -210-Table 26: Explanation of Subroutine CMPLNC CALL CMPLNC (VIS1, FPPMV, PPMV) VIS1 i s a REAL*8 v a r i a b l e , on entry VIS1 i s the EVEA f l u i d volume to which the corresponding i n t e r s t i t i a l pressure i s required FPPMV i s a REAL*8 array, containing the i n t e r s t i t i a l hydrostatic pressures l i s t e d i n Table 3 - the curved portion of the compliance curve. PPMV is a REAL*8 va r i a b l e , on exit i t contains the i n t e r s t s i t i a l pressure corresponding to VIS1. -211-A1.3 Plotting Section of Main Program UBCEDEMA The p l o t t i n g section of the simulation i s executed from within the program UBCEDEMA, s t a r t i n g on l i n e 466. Plots w i l l be drawn i f parameter NN i n f i l e PDA [PLOTS(Y=1,N=2)] i s set equal to 1. The plots drawn are of the time v a r i a t i o n of: 1) the f l u i d flows JV, JL, JNET1 and JAS, 2) the f l u i d volumes VTOT, V.IS1 and VAS, 3) the e p i t h e l i a l f i l t r a t i o n c o e f f i c i e n t KAS, 4) the protein content (albumin and globulin) i n the i n t e r s t i t i u m QA and QG and 5) the tissue protein concentration (albumin and globulin) CTA and CTG. For each set of plots a number of subroutines are c a l l e d : AXIS, PLOT, PLOTND, LINE, DASHLN, PSYM, DLINE, BBLINE, NUMBER and LEGEND. These subroutines are explained i n Tables 27 through 36. - 2 1 2 -Table 27: Explanation of subroutine AXIS (57) AXIS Purpose AXIS draws an a x i s wi th t i c k marks every i n c h , p l a c e s s c a l e va lues near the t i c k marks, and i d e n t i f i e s the a x i s with a l a b e l . T h i s s u b r o u t i n e was not des igned fo r use in the m e t r i c system and does not g ive p l e a s i n g r e s u l t s i f met r ic u n i t s are u s e d . Users who p r e f e r to work in met r i c are a d v i s e d to use s u b r o u t i n e AXPLOT, d e s c r i b e d on a f o l l o w i n g page. AXPLOT a l s o g i v e s the user more c o n t r o l over the p l o t t i n g of an a x i s . How To Use CALL AXIS(X ,Y ,3CD , N , S ,THETA,XMIN,DX) (X,Y) are the c o o r d i n a t e s of the s t a r t of the a x i s . BCD i s a H o l l e r i t h l i t e r a l ( n H x x x . . . or ' x x x . . . ' ) or an a r r a v c o n t a i n i n g BCD i n f o r m a t i o n to be used as the a x i s l a b e l . N i s p o s i t i v e i f the l a b e l i s to be w r i t t e n on the c o u n t e r c l o c k w i s e s i d e of the a x i s , and n e g a t i v e i f the l a b e l i s to be . w r i t t e n on the c l o c k w i s e s i d e . The a b s o l u t e va lue of N i s the number of c h a r a c t e r s in BCD. S i s the l eng th of the a x i s in f l o a t i n g - p o i n t u n i t s . T h i s va lue w i l l be rounded up to the neares t whole u n i t . THETA i s the d i r e c t i o n of the a x i s in f l o a t i n g - p o i n t d e g r e e s . For a.~ x a x i s , THETA i s u s u a l i v 0.0, and for a v a x i s , 90.0. XMIN i s the va lue to be a s s i c n e d to the o r i c i n ( i . e . the v c _ u e o : a x i s ) . ';.-) c ta t: or. o c o c s i t e trie t i r s t tic:-; cr. DX i s the s c a l e increment (number of u n i t s per inch or c e n t i m e t r e ) . The s c a l e a n n o t a t i o n s at each s u c c e s s i v e t i c k c : the a x i s w i l l be XMIN, XMIN+DX, XMIN+2*DX, e t c . Note: XMIN and DX may be c a l c u l a t e d us ing s u b r o u t i n e SCALE. -213-Table 28: Explanation of subroutine PLOT (57) PLOT P u r p o s e T h i s subprogram i s the basic p l o t s u b r o u t i n e . It generates the pen movements r e q u i r e d to move the per. in a s t r a i g h t l i n e from i t s present p o s i t i o n to the p o s i t i o n i n d i c a t e d in the c a l l . It i s a l s o used to r e l o c a t e the o r i g i n of the p l o t t e r c o o r d i n a t e system in the x d i r e c t i o n . -low To Use CALL PLOT (X , Y , I PEN') w e r e : are the c o o r d i n a t e s of the p o i n t to which the pen i s to move. Y must be between - 0 . 5 and 34.5 inches (-1.27 and 87.63 c e n t i m e t r e s ) . X must be between - 1 . 0 and 200 .0 inches (-2.54 and 508.0 cm) . If the y coord ina t - ; i s l a r g e r than 10.5 inches (26.67 cm), the p l o t wi}\ automat i c a l l y c'ueued. be d i r e c t e d to l a rge paper when i t i s IPEN s p e c i f i e s whether the pen i s to be up or down d u r i n g i t s move to ( X , Y ) . The v a l u e s fo r IPEN a r e : + I fo r no change . If the pen i s down be fore the c a l l , i t w i l l remain down. +2 for pen down. If the pen i s up before the c a l l , the pen w i l l be lowered b e f o r e the movement to (X,Y) i s made. If down, i t w i l l remain down. pen up. If the pen i s down be fore the c a l l , i t .1 be r a i s e d before i t i s moved to ( X , Y ) . If up, w i l l remain UD. for pen dashed , wh i le the pen i s moved to (X,Y) i t w i l l be r a i s e d and lowered to produce dashed l i n e s i". accordance with the l a s t c a l l to the s u b r o u t i n e i TOO r - = Tne s u b r o u t i n e PLOT i s a l s o used to r e l o c a t e the o r i g i n on the graph paper in the x d i r e c t i o n . If I PEN = - 1 , - 2 , or - 3 , the pen i s moved to the ( X , 0 . 0 ) p o s i t i o n on the paper ( r e g a r d l e s s cf the y c o o r d i n a t e s p e c i f i e d ) , and t h i s p o s i t i o n then becomes the new o r i g i n ( 0 . 0 , 0 . 0 ) . The o r i g i n of the y c o o r d i n a t e may not be r e l o c a t e d . The o r i g i n of the x c o o r d i n a t e i s r e l o c a t e d when a s e r i e s of graphs i s to be drawn. -214-Table 29: Explanation of subroutine PLOTND (57) PLOTND Purpose To terminate p l o t t i n g . How To Use CALL PLOTND Commen t s To ensure that a complete p l o t i s r e c e i v e d , t e rmina te your p l o t t i n g with a c a l l to PLOTND, or a c a l l to PLOT with a n e g a t i v e IPEN v a l u e . If PLOTND i s u s e d , i t must be the l e s t p l o t t e r subprogram c a l l e d . No o ther c a l l s to the p l o t r o u t i n e s are a l l o w e d a f t e r PLOTND. PLOTND w i l l p r i n t the f o l l o w i n g message: *** U3C P l o t S u b r o u t i n e s - End of P l o t t i n g * * * Number of p l o t frames generated = n I f t h i s p l o t i s queued f o r p l o t t i n g i t w i l l take a p p r o x i m a t e l y m minutes to p l o t at an approximate c o s t of d . d d d o l l a r s ( U n i v e r s i t y r a t e s ) and use I i n c h e s of paper . Approx imate ly p% of the time w i l l be spent p l o t t i n g with the pen r a i s e d . The numbers g i v e n are approximate and are in tended to g ive the user some idea of how much i t w i l l cos t to produce the p l o t . "n" i s the number of frames in the p l o t f i l e , "m" i s the number of minutes to draw the p l o t on the p l o t t e r , " d . d d " i s the approximate c o s t in d o l l a r s at academic r a t e s , and " i " i s the number of inches of paper r e q u i r e d . " p " , the pen-up t i m e , i s i n c l u d e d as a measure of e f f i c i e n c y of the p l o t . When the p l o t i s cop ied to *PLOT* the time a c t u a l l y used in drawing the p l o t w i l l be c a l c u l a t e d and your account char ged . A user w i l l r e c e i v e on ly part of h i s p l o t i f the l a s t c a l l to a p l o t t e r subprogram is not to PLOTND, or to PLOT with a n e g a t i v e IPEN v a l u e . He may r e c e i v e an e r r o r message at the time the p l o t i s queued i n d i c a t i n g a m i s s i n g PEND r e c o r d . -215-Table 30: Explanation of subroutine LINE (57) LINE Purpose To draw a l i n e th rough a s e r i e s of p o i n t s . How To Use CALL L IKE(ARKAYX,ARRAYY,N,J ) where: ARRAYX c o n t a i n s the >: c o o r d i n a t e s of N data p o i n t s . ARRAY? c o n t a i n s the y c o o r d i n a t e s of the p o i n t s . N i s the number of p o i n t s th rough which the l i n e i s to be drawn. j shou ld be set to -1 i f the pen i s t o . b e down when moving t o the f i r s t p o i n t ; o t h e r w i s e , i t shou ld be set to +1 . If J i s +1 , the l i n e may be drawn backwards i f t h i s i s more e f f i c i e n t . T h i s means that the pen may not a lways f i n i s h at the c o o r d i n a t e s of the l a s t p o i n t in the l i n e ; i t w i l l sometimes f i n i s h at the c o o r d i n a t e s of the f i r s t p o i n t . I f 3 i s set to - 1 , the l i n e i s always drawn f o r w a r d . - 2 1 6 -Table 31: Explanation of subroutine DASHLN (57) DA5KIN Purr-cse jvemen i s (I PEN = 4 ) To s e : a c a t t e r n ror c a s ne c l i n e ^^n m o v e me < T c Use CALL DA SHLNtDASH 1,SPAC1 ,DASH2,S?AC2) where: DASH! i s the l e n g t h of the f i r s t da s h . . SPACl i s the l e n g t h of the f i r s t s p a c e . DASH 2 i s l V of the second d a s h . SPAC2 i s the l e n g t h of the second s p a c e . These i n d i c a t a 1lowed v a l u e s sh ing the 1 .ould be engths entered as p o s i t i v e in user u n i t s . Zero If DASHLN has been c a l l e d , a c a l l to PLOT wi th an IPEN va lue of 4 w i l l r e s u l t in a dashed l i n e be ing drawn; The f i r s t dash w i l l be DASHl inches or c e n t i m e t r e s in l e n g t h , the f i r s t space w i l l be SPAC1 inches or c e n t i m e t r e s in l e n g t h , and s i m i l a r l y for the second dash and s p a c e . Then the p a t t e r n w i l l repeat i t s e l f . By us ing a v a r i e t y of p a r a m e t e r s , many types of dashed l i n e s may be c r e a t e d . If dash ing i s d i s c o n t i n u e d , for example , in order to draw a symbol , and the pen i s then r e t u r n e d to the same p o s i t i o n where dash ing was d i s c o n t i n u e d , the dashes w i l l be c o r r e c t l y c o n n e c t e d . Comments The l e n g t h s c : dashes and spaces at a n g l e s of 45 degrees w i l l be a p p r o x i ^ a t e l y 1.4 times t h e i r l e n g t h as s p e c i f i e d in the c a l l . T h i s d i f f e r e n c e i s r a r e l y n o t i c e a b l e in a f i n i s h e d -217-Table 32: Explanation of subroutine PSYM (57) PSYM Purpose PSYM produces text on a p l o t . T h i s r o u t i n e : c h a r a c t e r s e t s . See the d e s c r i p t i o n of i n f o r m a t i o n on a l t e r n a t i v e c h a r a c t e r s e t s . s used f o r a l l PALPHA for How To Use CALL PSYM(X,Y,HEIGHT,STRING,ANGLE,LENGTH,*RC4) where: X ,Y are the f l o a t i n g - p o i n t (REAL*4) c o o r d i n a t e s of the f i r s t c h a r a c t e r to be drawn. For most c h a r a c t e r s e t s , i n c l u d i n g the s tandard one, t h i s i s the l o w e r - l e f t c o r n e r of the f i r s t c h a r a c t e r . If e i t h e r c o o r d i n a t e i s - 0 . 0 (hexadecimal 80000000), PSYM c o n t i n u e s from the end of the l a s t c h a r a c t e r s t r i n g drawn. HEIGHT i s the f l o a t i n g - p o i n t (REAL*4) he ight in which the s t r i n g i s drawn. STRING i s the c h a r a c t e r s t r i n g to be drawn. ANGLE i s the f l o a t i n g - p o i n t (REAL*4) angle in the c h a r a c t e r s t r i n g (us ing a c o u n t e r c l o c k w i s e c o n v e n t i o n ) . i nche s degrees of pos i t i ve LENGTH i s the f u l l w o r d i n t e g e r c h a r a c t e r s in STRING. (INTEGER*4) number of iRC4 i s the e x i t taken for an u n s u c c e s s f u l r e t u r n ; STRING i s not drawn. E i t h e r the parameters are in e r r o r ( f o r i n s t a n c e , c a l l i n g PSYM for the f i r s t time wi th X or Y equa l to - 0 . 0 ) , or there i s an e r r o r in a u s e r - d e f i n e d c h a r a c t e r s e t . -218-Table 33: Explanation of subroutine DLINE CALL DLINE(XY,N,X0,SX,SFX,Y0,YS,SFY,D1,D2,DB) where: X r e a l array of Independent x-values Y r e a l array of dependent y-values N integer number of pairs of (x,y) points XO r e a l s t a r t i n g value of x-axis (units) XS r e a l s t a r t i n g l o c a t i o n of x-axis (inches) SFX r e a l x-axis scale factor (units/inch) YO r e a l s t a r t i n g value of y-axis (units) YS r e a l s t a r t i n g l o c a t i o n of y-axis (inches) SFY r e a l y-axis scale factor ( u n i t s / inch) DI length of f i r s t dash (inches) D2 length of second dash (inches) D3 length of space between dashes (inches) Subroutine plots a smooth dashed curve. - 2 1 9 -COMPUTER PROGRAM OF SUBROUTINE DLINE L i s t i n g of DLINE at 11:55:38 on MAR 25, 1985 for CCid=HEIJ Page 1 SUBROUTINE D L I N E ( X , Y , N , X 0 , X S , S c X , Y 0 , Y S , S E Y , D i ,D2,DB) 2 DIMENSION X(N),Y(N) 3 COMMON/BLKA/XP(101 ) ,YP(101 ) ,M, MM 4 M=N 5 MM=M-1 6 DO 10 1=1 , N XP( I )=XS+(X( I ) -X0) /SFX 7 8 YP( I )=YS+(Y( I ) -Y0) /SFY 9 10 CONTINUE 10 1 1 CALL SPLINE XI=XP(l) 12 YI=YP(1) 1 3 CALL P L O T ( X I , Y l , 3 ) 1 4 IFLAG=1 1 5 20 ITER=1 1 6 G O T O ( 3 0 , 4 0 , 5 0 , 6 0 ) , I FLAG 1 7 30 D=D1 18 IPEN=2 19 GO TO 7 0 20 40 D=DB 21 IPEN=3 22 GO TO 70 23 50 D=D2 24 IPEN=2 25 GO TO 7 0 26 60 D=DB 27 IPEN=3 28 70 XF1=XI+D/2. 29 80 XF = XF 1 -FX(XF 1 ,XI ,YI ,D)/DFX(XF1 ,XI ,YI ,D) 30 I F ( X F . L E . X I ) GO TO 8 5 31 I F ( A 3 S ( ( X F 1 - X F ) / X F ) . L T . 0 . 0 0 0 0 1 ) GO TO 90 32 ITER=ITER+1 33 IF ( ITER.GT.20 ) GO TO 90 34 XF1=XF 35 GO TO 8 0 36 85 XI =XI 37 FX1=FX(X1,XI ,YI ,D) 38 DX=D/2. 39 86 XF=X1+DX 40 I F ( D X . L T . 0 . C 0 0 0 i ) GO TO 90 4 1 DX=DX/2. 42 F X F = F X ( X F , X I , X I , D » 43 IF(FX 1 * F X F . L T . 0 . ) GO TO 8 6 44 X1=XF 45 F X ' = F X F 46 GO TO 6 6 47 90 IF (XF .GT .X?(N) ) XF=XP(N) 48 YF=F(XF) 49 CALL PLOT(XF,YF , I PEN) 50 IF (XF .EQ.XP(N) ) RETURN 51 XI =XF 52 Yl = YF 53 IFLAG=IFLAG+1 54 IF(I FLAG.EQ.5 ) IFLAG=1 55 GO TO 20 56 END 57 C 58 SUBROUTINE SPLINE -220-L i s t i n g of DLINE at 11:55:38 on MAR 25, 1985 for CCid=HEIJ Page 2 59 C 60 C I n t e r p o l a t i o n us ing c u b i c s p l i n e s wi th f i t t e d end p o i n t s 6 1 c Input : X Array of independent x - v a l u e s 62 c Y Array of dependent y - v a l u e s 63 c N Number of data p o i n t s 64 c NM N-1 65 c Output : Q ,R,S C o e f f i c i e n t s of cub ic s p l i n e equat ions 66 c 67 COMMON/ELKA/X(101 ) ,Y(101 ) ,N ,NM 68 CCMMCN/BLK3/Q(100),R(101),S(100) 69 DIMENSION H(100) ,A(101) ,B(101) ,C(101) ,D(101),COEFF(4,5) 70 c 71 c C o e f f i c i e n t mat r i ces f o r end po in t c u b i c s 72 c 73 INTEGER FLAG 74 M=4 75 IS = 0 76 FLAG=0 7 7 MF=M-i-1 78 MM=M~1 79 1 0 DO 20 1=1,M 80 I I = I - I S 8 1 C O E F ? ( I , M ? ) = Y ( I I ) 82 COEFF(1,1)=!. 83 DO 20 J=2,M 84 20 C O E F F ( I , J ) = C O E F F ( I , J - 1 ) * X ( I I ) 35 c 86 c Gauss e l i m i n a t i o n to f i n d A4 and B4 87 c 88 DO 3 0 K=1,MM 89 K?=K*1 90 DO 3 0 I=K?,M 9 i DO 3 0 J = X ? , M ? 92 30 C C E F F ; I , J ) = C O E F F ( I , J ) - C O E F F ( I , K ) * C O E F F ( K , 3 ) / C O E F F ( K , K ) 9 3 I F ( F L A G . N E . 0 ) GO TO 40 94 A 4 = C O E F F ( M , M P ) / C O E F F ( M , M ) 95 ?LAG= ; 9 6 I S = N - M 9 - G C 70 -0 9c -= L 3 - = C C E F F ' M MP) 'COEF?(M M) 99 C 1 00 c C a l c u l a t e H(I) • n i i '•J • c 10 2 D C 5 0 1=1,MM ' C 2 3 0 r. > I ' = X ' I - ' ) - X ' - ! i 04 c i 0 3 c Coef f i c i e n t s of t r i d i a c o n a l equa t ions 1 06 c 1 07 A i 1 1=0.0 1 OS 3( i :•=-:-;( i) 1 09 c. I )=:-:( i ) ! 1 0 D(. 1 ) = 3.*H( 1 )*H( 1 )*A4 1 ! 1 DO 60 I=2,NM 1 1 2 I P = I + 1 1 1 3 IM=I-1 1 1 4 A(I)=H(IM) 1 1 5 B(I ) = 2 . * ( K(IM ) + H(I)) 1 1 6 C( I )=H(I) -221-L i s t i n g of DLINE at 11:55:38 on MAR 25, 1985 for CCid=HEIJ Page 3 17 60 D ( I ) = 3 . * ( ( Y ( I P ) - Y ( I ) ) / H ( l ) - ( Y ( I ) - Y ( I M ) ) / H ( I M ) ) 18 A(N)=H(NM) 19 B(N)=-H(NM) 20 C(N)=0.0 21 D(N)=-3.*H(NM)*H(NM)*B4 22 C 23 c C a l l Thomas a l g o r i t h m to s o l v e t r i d i a g o n a l set 24 c 25 CALL TDMA(A,B,C,D,R,N) 26 • c 27 c Determine Q(I) and S(I) 28 c 29 DO 70 1=1,NM 30 IP=I+1 31 Q ( I ) = ( Y ( I P ) - Y ( I ) ) / H ( I ) - H ( I ) * ( 2 . * R ( I ) + R ( l P ) ) / 3 . 32 70 S( I ) = (R ( IP ) -R ( I ' ) ) / ( 3 . *H ( I ) ) 33 RETURN . 34 END 35 C 36 SUBROUTINE TDMA{A,B,C,D,X,N) 37 C 38 c Thomas a l g o r i t h m 39 c 40 DIMENS I ON A ( N ) , B ( N ) , C ( N ) , D ( N ) , X ( N ) , P ( 1 0 1 ) , Q ( 1 0 1 ) 4 1 NM=N-1 42 P(1 )= -C(1 ) /3 (1 ) 43 Q(1)=D(1) /B(1) 44 DO 10 1=2,N 45 IM=I-1 46 DEN=A(I)*P(IM)+3(I) 47 P( I )= -C( I ) /DEN 48 10 Q(I )=(D(I ) -A( I )*Q(IM)) /DEN 4 9 X(N)=Q(N) 50 DO 20 11=1,NM 51 I=N-II 52 20 X(I)=?(!)*X(I+1)-Q(I) 53 RETURN 54 END 55 C 56 FUNCTION F(Z) 57 C 58 C S p l i n e i n t e r p o l a t i o n f u n c t i o n . I n t e r p o l a t i o n i n t e r v a l 59 C by b i s e c t i o n . 60 c 61 COMMON/3LKA./X ( 1 0 1 ) , Y ( 1 0 1 ) , N , NM 62 COMMON/BLK3/Q(100),R(101),S(100) 63 1 = 1 164 I F ( Z . L T . X (1 ) ) GO TO 3 0 65 IF(Z .GE.X(NM)) GO TO 2 0 66 J = NM 167 10 K=(I+J)/2 1 68 I F ( Z . L T . X ( K ) ) J=K 169 I F ( Z . G E . X ( K ) ) I=K 170 I F ( J . E Q . I + 1 ) GO TO 3 0 171 GO TO 10 172 20 I=NM 173 30 DX=Z-X(I) 174 F=Y(I)+DX*(Q(I )+DX*(R(I )+DX*S(I ) ) ) - 2 2 2 -L i s t i n g of DLINE at 11:55:38 on MAR 25, 1985 f o r CCid=HEIJ Page 4 17 5 RETURN 176 END 177 C 178 FUNCTION FX(Z,XO,YO,D) 17 9 C0MM0N/3LKA/X(101 ) ,Y(101) ,N,NM 180 COMMON/BLK3/Q(100),R(101),S(100) 181 1=1 182 I F ( Z . L T . X ( 1 ) ) GO TO 30 183 IF(Z.GE.X(NM)) GO TO 20 184 J=NM 185 10 K=(I+J)/2 186 I F ( Z . L T . X ( K ) ) J=K 187 I F ( Z . G E . X ( K ) ) I=K 188 I F ( J . E Q . I + 1 ) GO TO 30 189 GO TO 10 190 20 I=NM 191 30 DX=Z-X(I) 192 FX=(Z-X0)**2+(Y( I ) -Y0+DX*(Q( I )+DX*(R( I )+DX*S( I ) ) ) ) * *2-D*D 1 93 RETURN 194 END 195 C 196 FUNCTION DFX(Z,X0,Y0,D) 1 97 COMMON/BLK.A/X ( 1 0 1 ) , Y ( 1 0 1 ) , N , NM 198 COMMON/BLK3/Q(100),R(101),S(100) 199 1=1 200 I F ( Z . L T . X ( 1 ) ) GO TO 30 201 I F ( Z . G E . X ( N M ) ) GO TO 2 0 202 J=NM 203 10 K=(I+J)/2 204 I F ( Z . L T . X ( K ) ) J=K 205 I F ( Z . G E . X ( K ) ) I=K 206 I F ( J . E Q . I + ! ) GO TO 30 207 GO TO 10 208 20 I=NM 209 30 DX=Z-X(I) 210 DFX=2 . * (Z -X0 ) -2 . * (Y ( I ) -Y0+DX* (Q( l )+DX* (R( I )+DX*S( I ) ) ) ) * 211 1 (Q(I )+DX*(2.*R(I )+DX* 3 . *S( I ) ) ) 2 1 2 RETURN 2 1 3 END -223-Table 34: Explanation of subroutine BBLINE SUBROUTINE BBLINE (X,Y,N,XO,XS,SFX,YO,YS,SFY,L,J,H,K) where: X r e a l array of independent x-values Y r e a l array of dependent y-values N integer number of pairs of (x,y) points XO r e a l s t a r t i n g value of x-axis (units) XS r e a l s t a r t i n g l o c a t i o n of x-axis (inches) SFX r e a l x-axis scale factor (units/Inch) YO r e a l s t a r t i n g value of y-axis (units) YS r e a l s t a r t i n g l o c a t i o n of y-axis (inches) SFY r e a l y-axis scale factor (units/inch) L integer l i n e type control parameter L=0 symbols plotted only L=l curved l i n e plotted with symbols L=2 curved l i n e plotted without symbols L=-l s t r a i g h t l i n e plotted with symbols L=-2 straight l i n e plotted without symbols J integer symbol control parameter H r e a l symbol height K integer o r i e n t a t i o n parameter K>0 y i s single-valued function of x K<0 x i s single-valued function of y outine plots a smooth curve -224-COMPUTER PROGRAM OF SUBROUTINE BBLINE l i s t i n g of B3LINE at 11:55:46 on MAR 25, 1985 for CCid=HEIJ Page 1 0 1 1 1 2 1 3 1 4 1 5 l 6 ! 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 36 E2 53 54 55 56 57 53 59 60 61 62 63 64 C C C C C C C C C c c c c c c c c c c c-c c c SUBROUTINE B 3 L I N E ( X , Y , N , X O , X S , S F X , Y 0 , Y S , S F Y , L , J , H , K ) 1 0 30 40 50 PARAME X v N X0 XS SFX Y0 YS SFY L J H K ERS: r e a l array of independent x - v a l u e s r e a l a r ray of dependent y - v a l u e s in teger number of p a i r s of (x ,y ) p o i n t s r e a l s t a r t i n g va lue of x - a x i s ( u n i t s ) s t a r t i n g l o c a t i o n of x - a x i s ( inches) x - a x i s s c a l e f a c t o r ( u n i t s / i n c h ) s t a r t i n g v a l u e of y - a x i s ( u n i t s ) s t a r t i n g l o c a t i o n of y - a x i s ( inches ) y - a x i s s c a l e f a c t o r ( u n i t s / i n c h ) in teger l i n e type c o n t r o l parameter L=0 symbols p l o t t e d on ly curved l i n e p l o t t e d wi th symbols curved l i n e p l o t t e d wi thout symbols s t r a i g h t l i n e p l o t t e d wi th symbols s t r a i g h t l i n e p l o t t e d wi thout symbols in teger symbol c o n t r o l parameter r e a l symbol h e i g h t in teger o r i e n t a t i o n parameter K>0 y i s s i n g l e - v a l u e d f u n c t i o n of x K<0 x i s s i n g l e - v a l u e d f u n c t i o n of y r e a l r e a l r e a l r e a l r e a l L=1 L = 2 L=-1 L=-2 DIMENSION X(N),Y(N) COMMON/BLKA/XP(101),YP(101),M,MM M=N MM=M-1 DO 10 I=1,N X?( I )=XS+(X( I ) -X0) /SFX YP( I )=YS+(Y( I ) -Y0) /SFY I F ( K . G T . O . O R . L . L E . O ) GO TO 10 TEM?=XP(I) XP(I)=YP(I) Y?(I)=TEMP CONTINUE I F ( L . L E . O ) GO TO 4 0 CAE IF (X, I F U . I F ( K . IF (K. DG 3 0 SPLINE GT.0) CALL LT .0 ) CALL G T . 0 . A N D . L . L T . 0 . A N D . L . 1=2,N PLOT(XP(1),YP(1 PLOT(YP(1),XP(1) 3) 3) E Q . l ) CALL SYM30L(XP(1) EQ.1) CALL SYM30L(YP(1) Y P (1 ) , K ,J X P ( 1 ) , H , J ,0.,-1) 0 . , - 1 ) PLOT(XX,YY, PLOT(YY,XX, CALL CALL SYMBOL(XP(I ) ,YP( I ) SYMBOL(YP( I ) ,XP( I ) , IM=I- 1 DX=(X?( I ) -XP( IM)) /2 0 . DO 20 JJ=1,20 XX=X?(IM)+JJ*DX YY=F(XX) IF (K .GT .O) CALL I F ( K . L T . O ) CALL CONTINUE I F ( K . G T . 0 . A N D . L . E Q . 1 ) I F ( X . L T . 0 . A N D . L . E Q . 1 ) CONTINUE RETURN CALL PLOT(XP(1) ,YP(1) I F ( L . E Q . 0 . O R . L . E Q . - 1 ) DO 50 I=2,N I F ( L . L T . O ) CALL P L O T ( X P ( I ) , Y P ( I ) , 2 ) I F ( L . E Q . O ) CALL P L O T ( X P ( I ) , Y P ( I ) , 3 ) I F ( L . E Q . 0 . O R . L . E Q . - 1 ) CALL SYMBOL(XP( I ) ,YP( I ) ,H ,J CONTINUE RETURN END , 0 . ,-1 ) ,0.,-D ,3) CALL SYMBOL( XP ( 1 ) , YP ( 1 ) , H , J , 0 . , - 1 ) , 0 . ,-1 ) -225-Table 35: Explanation of subroutine NUMBER (57) NUMBER Purpose T h i s s u b r o u t i n e w i l l p l o t a f l o a t i n g - p o i n t number. How To Use CALL NUM3E?( X , Y ,HT,FLOAT,THETA,N) where: (X,Y) are the c o o r d i n a t e s of the lower l e f t - h a n d corner of the number. HT i s the he igh t of the number. (For more i n f o r m a t i o n , r e f e r to the SYM30L r o u t i n e d e s c r i p t i o n . ) FLOAT i s the f l o a t i n g - p o i n t number to be drawn.-THETA i s the a n g l e . N s p e c i f i e s the number of dec ima l d i g i t s to the r i g h t of the dec ima l p o i n t . N=0 puts a d e c i m a l p o i n t at the end of the number, N=-1 suppresses the dec ima l p o i n t . N shou ld not be l a r g e r than 3. The number i s t r u n c a t e d , not rounded. For example, i f FLOAT=-17 .795 and N= 3, the c h a r a c t e r s -17 .795 are drawn; N= 0, the c h a r a c t e r s - 1 7 . are drawn; N = - i , the c h a r a c t e r s -17 are drawn. Examole T h i s example would p l o t the number 12.3 at ( 5 . 3 , 6 . 2 ) : X=12.3 CALL N U M B E R ( 5 . 3 , 6 . 2 , 0 . 2 1 , X , 0 . 0 , 1 ) R e s t r i c t ions 1. The i n t e g e r p o r t i o n of the number to be. p l o t t e d must not exceed 7 c h a r a c t e r s . 2. NUMBER may net be used to p l o t i n t e g e r s d i r e c t l y . The i n t e g e r must be conver ted to f l o a t i n g - p o i n t form and rj 1 o 11 e d w i t h N = - 1 . -226-Table 36: Explanation of subroutine LEGEND LEGEND (PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) - terms are parameters used i n the computer program UBCEDEMA -227-A1.4 Operation of Computer Program 1) The computer programs UBCEDEMA, BBLINE and DLINE are compiled as follows: //RUN *FTN SCARDS=UBCEDEMA+BBLINE+DLINE The compiled programs are then stored under the temporary name - LOAD. 2) Execution of the programs using the data f i l e s EDA and PDA i s as follows: //RUN -LOAD 4=EDA 5=PDA 7=-TABLES 9=-PL0TS The tabulated and graphical output Is stored i n the temporary f i l e s -TABLES and - PLOTS, r e s p e c t i v e l y . 

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