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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 B.E.Sc,  A  University  THESIS  R. C. HEIJMANS  of Western  Ontario,  London,  SUBMITTED IN P A R T I A L F U L F I L L M E N T OF  THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D  SCIENCE  in  THE  FACULTY OF GRADUATE  (Department  We  accept to  THE  of Chemical  this  thesis  the r e q u i r e d  ©  Engineering)  as c o n f o r m i n g standard  UNIVERSITY OF B R I T I S H April Franciscus  STUDIES  COLUMBIA  1985  R.C. H e i j m a n s , 1985  1982  In presenting  this thesis i n p a r t i a l fulfilment of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y of B r i t i s h Columbia, I agree that it  freely  t h e L i b r a r y s h a l l make  a v a i l a b l e f o r r e f e r e n c e and study.  I further  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 copying o f t h i s  thesis  f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d 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 . understood that for  financial  copying o r p u b l i c a t i o n o f t h i s  D e p a r t m e n t 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 C o l u m b i a 1956 Main M a l l ..'•*• V a n c o u v e r . Canada V6T 1Y3  DE-6  (3/81)  thesis  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  permission.  Date  It i s  \?J^I&g  written  -ii-  ABSTRACT  P r e v i o u s models of the pulmonary m i c r o v a s c u l a r 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  between the pulmonary m i c r o c i r c u l a t i o n , lymphatics. lymphatics  and s o l u t e exchange  interstitial  t i s s u e space, and  I n severe pulmonary edema the c a p a c i t i e s of the and t i s s u e space a r e exceeded.  The f l u i d  e n t e r i n g 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 , t r a n s p o r t e d Into the a i r space. space i m p a i r s  The accumulation  i f this  fluid  then, be  of f l u i d  the d i f f u s i o n of gas (oxygen and carbon  the a i r space and blood c i r c u l a t i o n ;  and s o l u t e s  i n the a i r  d i o x i d e ) between  accumulation i s  e x c e s s i v e a p a t i e n t ' s h e a l t h may be compromised. In t h i s  t h e s i s severe  pulmonary edema i s s t u d i e d by i n c l u d i n g  the a i r space as a f o u r t h compartment i n t o the i n t e r s t i t i a l developed  by Bert and P i n d e r ( 2 9 ) .  A computer s i m u l a t i o n of the f o u r compartment was developed study  on a d i g i t a l  the e f f e c t  a l v e o l a r model.  computer.  ( a l v e o l a r ) model  T e s t s of the model were made t o  of the parameters which were i n t r o d u c e d i n t o the These parameters i n c l u d e : a f i l t r a t i o n  t h a t d e s c r i b e s the a l v e o l a r membrane f l u i d extravascular f l u i d  coefficient  c o n d u c t i v i t y , an  volume that r e p r e s e n t s the p o i n t a t which  e n t e r s the a i r space, fluid  model  the a l v e o l a r f l u i d  flow i n t o the a i r space,  fluid  p r e s s u r e a t the onset of  and the r a t e of a l v e o l a r f l u i d  pressure  -iii-  change r e l a t i v e  to an a l v e o l a r f l u i d  volume change.  dynamic response of the exchange system was r e c o r d e d .  F o r each case the In a d d i t i o n ,  two types of pulmonary edema were s i m u l a t e d : 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 s o l u t e  p e r m e a b i l i t y of the porous membrane s e p a r a t i n g the c i r c u l a t o r y and interstitial  compartments.  Due t o 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 o t h e r three compartments  of the pulmonary m i c r o v a s c u l a r  exchange system, only p a r t i a l v e r i f i c a t i o n of the a p p r o p r i a t e range of v a l u e s of the a l v e o l a r model parameters and the p r e d i c t i o n s of the s i m u l a t i o n s was p o s s i b l e . i s an i n i t i a l  The a l v e o l a r model developed i n t h i s  thesis  a p p r o x i m a t i o n but appears to p r o v i d e a s a t i s f a c t o r y  approach f o r the i n c l u s i o n o f the a i r space i n the pulmonary m i c r o v a s c u l a r exchange system.  -ivTABLE OF CONTENTS Page i i  ABSTRACT TABLE OF CONTENTS  iv  LIST OF TABLES  viii  LIST OF FIGURES  x  ACKNOWLEDGEMENT  1.  BACKGROUND 1.1 1.2  Introduction The C i r c u l a t i o n 1.2.1 The Pulmonary and Systemic Circulatory Systems 1.2.2 Physical Characteristics of Blood 1.2.3 The Pulmonary C i r c u l a t i o n 1.3 The Vascular Membrane 1.3.1 Structure of the Vascular Membrane.... 1.3.2 Physical Properties of the Vascular Membrane 1.4 The I n t e r s t i t i u m . 1.4.1 Structure and Composition of the Interstitium 1.4.2 Volume Exclusion 1.4.3 The Alveolar and Extra-alveolar Tissue Sub-compartments 1.5 The Lymphatics 1.5.1 Structure of the Lymphatics 1.5.2 C o n t r a c t i l i t y and Pumping i n Lymphatic Vessels 1.6 The A i r Space 1.6.1 Arrangement of the A i r Space 1.7 Barriers Between the I n t e r s t i t i a l Space and A i r Space 1.7.1 Alveolar Membrane 1.7.2 Surfactant Lining the Wall of the Air Space 1.8 The Normal F l u i d Pathways i n the Pulmonary Microvascular Exchange System 1.9 Pulmonary Edema 1.9.1 D e f i n i t i o n of Pulmonary Edema 1.9.2 C l i n i c a l Causes of Pulmonary Edema 1.9.2.1 Hydrostatic Pulmonary Edema 1.9.2.2 Permeability Pulmonary Edema  .  xiv  1 1 3 3 5 7 13 13 16 17 17 19 21 26 26 28 29 29 32 32 32 36 36 36 38 38 39  -vTABLE OF CONTENTS (Cont.d)  2.  INTERSTITIAL MODEL 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8  3.  3.6  3.7  3.8  4.  4.3  42 43 46 51 51 56 60 61 61 64  65  Introduction M o d e l l i n g of the A i r Space M o d e l l i n g of the A l v e o l a r Membrane The Onset of A l v e o l a r F l o o d i n g M o d e l l i n g of F l u i d and S o l u t e T r a n s p o r t A c r o s s the A l v e o l a r Membrane Pressure-volume R e l a t i o n s h i p of the A l v e o l a r F l u i d . .. 3.6.1 F l u i d Pressure-volume R e l a t i o n s h i p of an I n d i v i d u a l A l v e o l u s 3.6.2 F l u i d Pressure-volume R e l a t i o n s h i p f o r the A i r Space Compartment R e p r e s e n t a t i o n 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 3.7.1 R e p r e s e n t a t i o n of KAS as a V a r i a b l e 3.7.2 R e p r e s e n t a t i o n of KAS as a Constant I n t e g r a t i o n of the A i r Space Compartment w i t h the Pulmonary M i c r o v a s c u l a r Exchange Model...  COMPUTER SIMULATION OF ALVEOLAR MODEL 4.1 4.2  42  Introduction M o d e l l i n g of the Pulmonary M i c r o c i r c u l a t i o n M o d e l l i n g of the I n t e r s t i t i u m M o d e l l i n g the V a s c u l a r Membrane S t a r l i n g ' s Hypothesis: T r a n s e n d o t h e l i a l F l u i d Flow Kedem-Katchalsky S o l u t e F l u x E q u a t i o n : T r a n s e n d o t h e l i a l S o l u t e Flow M o d e l l i n g of the Lymphatics F l u i d and S o l u t e M a t e r i a l Balances 2.8.1 F l u i d M a t e r i a l Balance 2.8.2 S o l u t e M a t e r i a l Balance  ALVEOLAR MODEL 3.1 3.2 3.3 3.4 3.5  Page  Introduction The Computer Program 4.2.1 Input Data F i l e s EDA and PDA 4.2.2 The Main Program UBCEDEMA 4.2.3 T a b u l a t e d and G r a p h i c a l Output of V a r i a b l e s from the Computer Program C h a r a c t e r i s t i c P o i n t s Along the T r a n s i e n t Response 4.3.1 The Onset of A l v e o l a r F l o o d i n g 4.3.2 The P o i n t of Maximum T r a n s e p i t h e l i a l Flow..  65 65 67 68 71 73 73 77 78 78 79 80  83 83 84 84 89 92 92 92 95  vi-  TABLE OF CONTENTS (Cont.d) Page 4.3.3  4.4  5.  The Point at which the Total Extravascular F l u i d Volume equals 1000 ml.. 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 4.4.2 Simulations to Study the Effect of a Maximum Lymph Flow........ 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  RESULTS AND DISCUSSION Transient Responses of the PMVES f o r Constant KAS 5.2 Transient Responses of the PMVES f o r Variable KAS 5.2.1 Transient Response of the Pulmonary Microvascular Exchange System to changes in NK 5.3 The Response of the PMVES to changes i n the Parameter VTONS 5.4 Transient Responses of the PMVES to changes i n the Parameter SL 5.5 Responses of the PMVES to changes i n the Parameter B 5.6 The Response of the PMVES to a Maximum Lymph Flow 5.7 The Responses of the PMVES to changes i n PMV 5.8 The Response of the PMVES to changes i n KF 5.9 The Response of the PMVES to changes inAPERM 5.10 Control of Error i n the Numerical Solutions  95 96  96 97  99 102  5.1  SUMMARY AND CONCLUSIONS  102 109 117 123 131 139 151 157 162 169 177 179  RECOMMENDATIONS FOR FURTHER WORK  181  NOMENCLATURE  182  REFERENCES  185  -viiTABLE OF CONTENTS (Cont.d) Page APPENDIX A Al.l A1.2 A1.3 A1.4  THE COMPUTER PROGRAM Input F i l e s EDA and PDA The Main Program UBCEDEMA Plotting Section of Main Program UBCEDEMA Computer Program of Subroutine DLINE Computer Program of Subroutine BBLINE Operation of Computer Program  190 190 193 211 219 224 227  -viiiL I S T OF TABLES  Table 1.  Description The Protein Composition of the Blood and I n t e r s t i t i a l Fluid  Page  8  2.  Major Causes of Cardiogenic Pulmonary Edema  40  3.  PPMV versus VIS1 for the Transition Region of the I n t e r s t i t i a l Compliance Curve  52  4.  Input Parameters to the I n t e r s t i t i a l Computer Simulation for Normal Conditions  59  5.  Content of Input F i l e EDA  85  6.  Variables i n Input F i l e PDA  87  7. 8.  Equations used i n the Alveolar Model Variables Tabulated and/or Plotted by Main Program UBCEDEMA  90 93  Characteristic Points along Transient Response and Variables Recorded  94  Simulations Conducted to Study the Effect of Parameters Introduced with the Alveolar Model  98  9. 10. 11.  12. 13. 14. 15. 16.  Simulations Conducted to Study the Effect of the Endothelial Permeability Parameters and F i l t r a t i o n Coefficient and the Circulatory Hydrostatic Pressure on the Alveolar Model  101  Conditions of PMVES Simulations Conducted to Study Changes i n KAS  102  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 Coefficient  110  I n i t i a l Conditions of PMVES/Simulation at a PMV of 50 mmHg  Ill  Conditions of the PMVES Simulations Conducted to Study Changes i n NK  119  Conditions of the PMVES Simulations to Study Changes i n VTONS  124  -ix-  Table 17.  18.  19.  20.  21.  22.  23.  24.  L I S T OF TABLES (Cont'd.) Description C o n d i t i o n s of the PMVES S i m u l a t i o n s Changes i n SL  Page  to Study 133  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted to Study the Changes i n B  140  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted to Study the Changes i n the Maximum Lymph Flow  152  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted t o Study Changes i n PMV  158  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted t o Study Changes In KF  163  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted to Study Changes i n APERM  171  ( T a b l e s i n Appendix) Example of Numerical Values A s s i g n e d i n F i l e EDA  to Parameters 191  Example of Numerical  t o the  Values  Assigned  Parameters i n F i l e PDA  192  25.  E x p l a n a t i o n of Subroutine  DRKC  208  26.  E x p l a n a t i o n of Subroutine  CMPLNC  210  27.  E x p l a n a t i o n of Subroutine  AXIS  212  28.  E x p l a n a t i o n of Subroutine  PLOT  213  29.  E x p l a n a t i o n of Subroutine  PLOTND  214  30.  E x p l a n a t i o n of Subroutine  LINE  215  31.  E x p l a n a t i o n of Subroutine  DASHLN  216  32.  E x p l a n a t i o n of Subroutine  PSYM  217  33.  E x p l a n a t i o n of Subroutine  DLINE  218  34.  E x p l a n a t i o n of Subroutine  BBLINE  223  35.  E x p l a n a t i o n of Subroutine  NUMBER  225  36.  E x p l a n a t i o n of Subroutine  LEGEND  226  -xLIST OF FIGURES Figure 1. 2.  Description  Schematic of the Pulmonary Microvascular Exchange System  Page 2  I l l u s t r a t i o n 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 F l u i d  6  4a.  Branching Network of C i r c u l a t i o n  9  4b.  Pulmonary C i r c u l a t i o n at Level of Alveolus  9  5a,b,c. 6. 7.  I l l u s t r a t i o n of Zone Model as Developed by West, Dollery and Naimark I l l u s t r a t i o n of Blood Flow at Different Heights of the Lung  11 14  I l l u s t r a t i o n of Vascular Membrane Separating Interstitium and C i r c u l a t i o n  15  Cross-sectional View of Interstitium and Contents and E p i t h e l i a l and Endothelial Membranes  18  9.  Models of Volume Exclusion i n 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 C i r c u l a t i o n , and Surfactant Layer Lining A i r Space  22  8.  11.  12.  Compliance Curves of Alveolar and Extra-alveolar Tissue Sub-compartments  24  Structural Features of a Terminal Lymphatic  27  13a. Branching Network of A i r Space  30  13b. Lobes, Lobules and Bronchopulmonary Segments of the Lungs  31  14a. I l l u s t r a t i o n of Gas Bubble Surrounded by Liquid  34  14b. I l l u s t r a t i o n of an Ideal Alveolus with Surfactant Lining 15. Tissue Compliance Curve used i n Models  34 50  -xi-  LIST OF FIGURES (Cont.d) Figure  Description  Page  16a. Schematic of the Three Compartment I n t e r s 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  63  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  63  17a,b,c,d.  Schematic Representation of F l u i d F i l l i n g an Alveolus During Acute Pulmonary Edema  75  18a. Schematic of Four Compartment Alveolar Model I l l u s t r a t i n g Fluid Flows  82  18b. Schematic of Four Compartment Alveolar Model I l l u s t r a t i n g 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 f o r Variable KAS  112  22b. Transient Responses of JV, JL, JNET1, VIS1, PPMV, and PIPMV f o r Variable KAS  113  22c. Transient Responses of QA, QG, VIS1, CTA and CTG f o r Variable KAS  115  22d. Transient Responses of JAS, PAS, KAS and PPMV f o r Variable KAS  116  22e. Tranient Responses of JV, JL, JNET1 and JAS f o r Variable KAS  118  23a. Transient Responses of JAS for Changes i n NK  120  23b. Transient Responses of VIS1 f o r Changes i n NK  122  24.  Steady State Values of VIS1 for given PMV f o r I n t e r s t i t i a l Edema 25a. Transient Responses of (JV-JL) and JAS for VTONS of 420 ml and 500 ml  125 126  -xiiLIST 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 f o r Different VTONS  130  26a. Transient Responses of JAS for Different SL and NK = 0.05 h r mmHg  134  26b. Transient Responses of PAS for Different SL and NK = 0.05 h r mmHg  135  26c. Transient Responses of VIS1 for Different SL and NK = 0.05 h r mmHg  136  27a. Transient Respones of PAS for Different SL and NK = 50.0 h r mmHg  137  27b. Transient Responses of JAS for Different SL and NK = 50.0 h r mmHg  138  28a. The Maximum T r a n s e p i t h e l i a l Flow and JAS at a VTOT of 1000 ml for Different B  141  28b. Transient Responses of (PPMV-PAS) f o r B = -10 mmHg and B = -6 mmHg  143  - 1  -1  - 1  - 1  - 1  - 1  28c. Transient  -1  -1  -1  -1  Responses of KAS for  B = -10 mmHg and B = -6 mmHg 28d. Time to Reach a VTOT of 1000 ml for Different B 28e. Transient Responses B = -3 mmHg and B = 28f. Transient Responses B = -3 mmHg and B =  of (PPMV-PAS) f o r 0.5 mmHg of KAS f o r 0.5 mmHg  144 145  146 147  28g. Fluid Volume VIS1 at a VTOT of 1000 ml f o r Different B  149  29a. Transient  153  Responses of JL for Different JL(max)  -xiiiL I S T OF FIGURES  Figure  (Cont.d)  Description  '  Page  29b. Transient Responses of (JV-JL) f o r Different JL(max)  155  29c. Time to Reach a VTOT of 1000 ml f o r Different JL(max)  156  30a. Transient Response of (JV-JL) and JAS f o r 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 f o r Different PMV  161  31a. Transient Responses of (JV-JL) and JAS f o r 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 f o r 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 f o r Different APERM  175  32d. Time to Reach a VTOT of 1000 ml for Different APERM....  178  -xiv-  ACKNOWLEDGEMENTS  I would l i k e K.L.  Pinder,  to thank my s u p e r v i s o r s , Dr. J . L . B e r t and Dr.  f o r t h e i r guidance throughout the course  of t h i s work.  Thanks are a l s o due to Dr. P. Pare* and Dr. P. Dodek of the Pulmonary Research U n i t at S t . Paul's medical  H o s p i t a l , who p r o v i d e d the  p e r s p e c t i v e i n t h 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 a p p r e c i a t e d . The provided  computer s u b r o u t i n e s  DLINE and BBLINE were  generously  by Dr. B. Bowen.  Finally,  I would l i k e  to thank the N a t u r a l S c i e n c e s and  E n g i n e e r i n g Research C o u n c i l of Canada f o r t h e i r  financial  support.  -I BACKGROUND  1.1  Introduction B i o l o g i c a l 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 l u i d and solute exchange system of the lungs i s 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 l u i d and solute  between the main compartments of the lung, that i s , the blood c i r c u l a t i o n , a i r space, tissue space (or i n t e r s t i t i u m ) , and lymphatics.  The term microvascular refers to the c i r c u l a t i n g system  of the small blood v e s s e l s .  The i n t e r s t i t i a l compartment  interacts  d i r e c t l y with the other three compartments, which are assumed not to interact  d i r e c t l y with each other.  Separating the compartments are  the vascular membrane - between the c i r c u l a t i o n and interstitium alveolar barriers - between the i n t e r s t i t i u m  -  the  and a i r space - and the  lymphatic c a p i l l a r y membrane - between the interstitium  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 i s  that of gas exchange, involving the countercurrent and carbon dioxide between the blood and a i r space.  transfer of oxygen  -2-  F i g u r e 1:  Schematic of t h e Pulmonary M i c r o v a s c u l a r Exchange System: I l l u s t r a t e s F l u i d - f l o w pathways (50)  -3-  The  f o l l o w i n g s e c t i o n s d i s c u s s the lung compartments,  components, and the b a r r i e r s exchange system.  to flow o f the pulmonary m i c r o v a s c u l a r  A section outlining  the exchange of f l u i d  between the compartments and the c l i n i c a l pulmonary f l u i d  1.2  and s o l u t e exchange w i l l  importance  and s o l u t e  of s t u d y i n g  a l s o be p r e s e n t e d .  The Circulation  1.2.1  The Pulmonary and Systemic Circulatory Systems Circulation  and  their  of blood i n the body takes p l a c e i n the pulmonary  systemic v a s c u l a r systems.  systems.  F i g u r e 2 i l l u s t r a t e s these two  The systemic c i r c u l a t i o n r e f e r s  to the v a s c u l a r network of  the whole body where the blood i s deoxygenated, w h i l e the pulmonary circulation The  h e a r t combines the two c i r c u l a t o r y  continuous half  i s c o n f i n e d to the lungs where the blood i s oxygenated.  blood f l o w .  systems to form a loop o f  The heart i s composed of two h a l v e s w i t h each  c o n t a i n i n g two chambers - an a t r i u m and a v e n t r i c l e .  a t r i u m of the r i g h t h e a r t r e c e i v e s blood from circulation.  The r i g h t v e n t r i c l e  pulmonary v a s c u l a t u r e . e n t e r s the l e f t through  transport  the s y s t e m i c  then pumps the blood Into the  The oxygenated blood o f the pulmonary system  a t r i u m of the l e f t h e a r t and i s subsequently  the systemic c i r c u l a t i o n by the l e f t The  The r i g h t  primary  pumped  ventricle.  f u n c t i o n of the systemic c i r c u l a t i o n i s t o  n u t r i e n t s and oxygen to the body t i s s u e s ,  wastes, such as carbon d i o x i d e .  and remove the  -4-  F i g u r e 2:  I l l u s t r a t i o n o f Pulmonary and Systemic C i r c u l a t o r y Systems ( 5 4 )  SYSTEMIC CIRCULATION  -5-  The  pulmonary m i c r o v a s c u l a t u r e  (a network of s m a l l blood  v e s s e l s ) , w h i l e s u p p l y i n g n u t r i e n t s to the surrounding  tissue, i s also  in  One  i n t i m a t e c o n t a c t with the a i r spaces  of the l u n g s .  of  the  purposes of t h i s unique arrangement i s to exchange the carbon of  the incoming  air  (oxygen d e f i c i e n t ) blood  i n the  with the exchange of f l u i d  and  solute  the pulmonary m i c r o v a s c u l a t u r e .  1.2.2  Physical Characteristics of Blood C i r c u l a t i n g through  fluid  composed of c e l l s  divided  Into two  the c e l l s , red  f o r the oxygen present  space. T h i s t h e s i s i s concerned  in  dioxide  and  blood c e l l s  plasma" ( 1 ) .  The  blood c e l l s  the red blood c e l l s  the white blood c e l l s .  The  primary  to the t i s s u e s  f u n c t i o n of  of plasma and  fluid.  interstitial  the anions - c h l o r i d e , b i c a r b o n a t e ,  watery Figure 3  fluid.  The  calcium,  and  phosphate,  organic acid  fluid  from each other i n t h e i r p r o t e i n content which i s g r e a t e r  in  the blood plasma.  plasma and  The  s u l f a t e , and differ  Blood  carries  proteins.  e l e c t r o l y t e s Include the c a t i o n s - sodium, potassium,  ions.  these  Identical in  to the lung i n t e r s t i t i a l  the composition  be  (1).  c o n t a i n s e l e c t r o l y t e s , n o n e l e c t r o l y t e s and  magnesium - and  may  which make up 99% of  plasma medium of the blood i s almost  illustrates  "a v i s c o u s  Is to t r a n s p o r t hemoglobin, which i n t u r n  e l e c t r o l y t e composition fluid  and  classes:  oxygen from the lungs The  the v a s c u l a r systems i s b l o o d :  lung  interstitial  Three major c l a s s e s of p r o t e i n s e x i s t , based  on  -6-  Figure 3:  Composition o f 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 i n Table 1.  The content of the  globulins and albumin exceeds that of fibrinogen.  The primary  function of albumin i s to cause a c o l l o i d osmotic pressure at c e l l membranes (1).  1.2.3  The Pulmonary C i r c u l a t i o n The pulmonary c i r c u l a t i o n i s composed of a branching network of  blood vessels, as shown i n Figure 4.  Blood that enters the right  heart i s pumped through the pulmonary artery. branches  The pulmonary artery  Into vessels of smaller diameter, eventually reaching the  size of the a r t e r i o l e s - less than 100 u i n diameter - and then the c a p i l l a r i e s - approximately 7 u i n diameter.  At the l e v e l of the  c a p i l l a r i e s oxygen and carbon dioxide are exchanged between the blood in the c a p i l l a r i e s and the gas of the a i r space.  The c a p i l l a r y  surface area available for exchange i s 50 to 70 m , while that of the 2  a i r space Is about 20% larger (2).  The c a p i l l a r i e s then unite to form  the venules - less than 200 u i n diameter.  The venules combine  further u n t i l they form the pulmonary vein which delivers the blood to the atrium of the l e f t heart. The vessels of the microvasculature are thought to be composed of a r t e r i a l , c a p i l l a r y , and venous segments.  As blood flows from the  a r t e r i a l to the c a p i l l a r y and then to the venous segments of the vessel, the f l u i d hydrostatic pressure decreases.  The longitudinal  -8-  Table 1  Table 1: The Protein Composition of the Blood and I n t e r s t i t i a l F l u i d (29) Protein  Blood (g/ml)  I n t e r s t i t i a l Fluid (g/ml)  Albumin Globulins Fibrinogen(55)  .042 .027 0.003  .025 .011 N.A.  Total Protein  0.073  N.A. - not available  -9-  Figure  a)  Branching  Network of  4  Circulation  (54)  Capillaries Arteries  Arterioles  Venules  Veins  -10-  p r e s s u r e p r o f i l e of the blood v e s s e l may a s s i g n i n g a h y d r o s t a t i c pressure segment:  PA > PMV  The p r e s s u r e s  5a  f o r the c a p i l l a r y ,  i n descending  order of value are  segment  i s i n c l o s e contact with  the a i r space,  the a r t e r i a l and venous ends are l e s s so; the diagrams of F i g u r e  show the l o c a t i o n of the three segments i n r e l a t i o n  space.  to the a i r  West, D o l l e r y , and Naimark ( 3 ) proposed a Zone Model f o r the  l u n g s , based on measurements  of the r e l a t i v e  v a l u e s o f PA, PV and the  h y d r o s t a t i c p r e s s u r e of the gas i n the a i r space PALV. air  and PV f o r  > PV.  The c a p i l l a r y while  i l l u s t r a t e d by  to the f l u i d of each v a s c u l a r  PA f o r the a r t e r i a l end, PMV  the venous end.  be simply  space p r e s s u r e  In zone I, the  i s g r e a t e r than the a r t e r i a l and venous  pressures:  (1)  PALV > PA > PV  F i g u r e 5a i l l u s t r a t e s zone I . segment  Under these c o n d i t i o n s the c a p i l l a r y  of the blood v e s s e l w i l l be c o l l a p s e d ; the c a p i l l a r y  p r e s s u r e , PMV,  w i l l assume a value approximately  T h e r e f o r e blood w i l l not flow through  equal to PALV.  the c a p i l l a r y  g r a d i e n t t h a t causes flow (PA - PALV) i s n e g a t i v e . space p r e s s u r e  fluid  s i n c e the p r e s s u r e In zone I I the a i r  Is between the a r t e r i a l and venous p r e s s u r e s :  (2)  PA > PALV > PV  F i g u r e 5b ( i ) shows the case when PALV i s much l a r g e r than PV. downstream  end of the c a p i l l a r y  v a l u e approximately  the c a p i l l a r y  equal to PALV.  At the  p r e s s u r e w i l l assume a  The p r e s s u r e g r a d i e n t c a u s i n g  -11-  Figure  a)  5:  I l l u s t r a t i o n of Zone Model as Developed by West D o l l e r y and Naimark ( 3 ) ( P A = a r t e r i a l p r e s s u r e PV=venous p r e s s u r e ; PALV=alveolar p r e s s u r e )  Zone I : PALV > PA > PV •capillary collapsed  Venice  Arteriole  b)  Zone I I : PA > PALV > PV • c o n s t r i c t i o n a t downstream end of v e s s e l  PA(1)  pvd:  PA(2)  PA(1)< PA(2) PVC 1)( PV(2)  c)  Zone I I I : PA > PV > PALV • v e s s e l h e l d open  -12-  blood  flow through  the c a p i l l a r y becomes (PA - PALV).  The  capillary  will  be c o n s t r i c t e d at the downstream end.  such  that PV approaches PALV, the c o n d i t i o n shown i n F i g u r e 5b ( i i )  results.  The  I f PA and PV are e l e v a t e d ,  p r e s s u r e g r a d i e n t c a u s i n g blood flow w i l l  PALV) s i n c e PMV  i s approximately  moves f u r t h e r downstream.  The  equal to PALV, but  final  remain (PA -  the  constriction  zone, zone I I I , has  the f o l l o w i n g  relationship:  (3)  PA > PV > PALV  The  pressure  capillary  g r a d i e n t causing blood  fluid  p r e s s u r e , PMV,  t h e r e f o r e exceed PALV. vessel w i l l  distend.  The  will  flow now  be between PA  g r e a t e r PVM  F i g u r e 5c  becomes (PA - PV). and  PV,  The  and  i s above PALV the more the  shows the v e s s e l under zone I I I  conditions. West, D o l l e r y and Naimark ( 3 ) lung may  be h o r i z o n t a l l y  a f f e c t e d by g r a v i t y .  have suggested  d i v i d e d i n t o zones.  lung.  there i s no blood  Zone I I c o n d i t i o n s p r e v a i l i n the middle  At the j u n c t i o n of zone I and  than t h i s  I I the h y d r o s t a t i c p r e s s u r e of the blood  T h e r e f o r e , at v e r t i c a l p o s i t i o n s lower and  I I PA and  decreases.  PV  The  flow  than  through  s e c t i o n of  zone I I PALV equals PA.  h o r i z o n t a l p o s i t i o n s of the lung lower and  blood c i r c u l a t i o n i s  In the upper s e c t i o n of the lung under normal  c o n d i t i o n s zone I c o n d i t i o n s p r e v a i l ; the v e s s e l s .  The  t h a t the human  the  At  j u n c t i o n of zones I  r i s e s due  to g r a v i t y .  the j u n c t i o n of zones I  i n c r e a s e ; (PA - PALV) i n c r e a s e s and blood flow i n c r e a s e s as the d r i v i n g  (PALV -  PV)  force causing  flow  -13-  (PA - PALV) i n c r e a s e s . c o n s t r i c t i o n and  When PV  equals PALV there i s no  the p r e s s u r e g r a d i e n t c a u s i n g flow becomes (PA -  Zone I I I c o n d i t i o n s p r e v a i l i n the lower lower  p o s i t i o n s i n zone I I I the blood  of the v e s s e l s .  Figure 6 i l l u s t r a t e s  b l o o d flow changes at d i f f e r e n t  1.3  circulatory  s e c t i o n of the l u n g .  flow r i s e s due  At  to the d i s t e n s i o n  the Zone Model, showing  p o s i t i o n s i n the  PV).  how  lung.  The Vascular Membrane  1.3.1  Structure of the Vascular Membrane The  v a s c u l a r membrane, shown i n F i g u r e 7,  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 . endothelial c e l l  The  separates  the blood  membrane i s composed of  l a y e r t h a t i s i n c o n t a c t w i t h the c i r c u l a t i o n ,  an and  an  u n d e r l y i n g basement membrane. "Whenever the e n d o t h e l i a l c e l l s j u n c t i o n s that are continuous from " t i g h t "  come i n t o c o n t a c t , there are  along the l i n e of c o n t a c t .  j u n c t i o n s to "gap"  These vary  j u n c t i o n s , which themselves vary  the t i g h t  to the " l e a k y " type, the p a r t i c u l a r c o n d i t i o n depending  the width  of the  junction."  W i s s i g and  Charonis  assisting blood".  The  (5)  suggest  that the basement membrane  basement membrane may  i t to w i t h s t a n d  on  (4)  f u n c t i o n s as a " s u p p o r t i v e r o l e f o r the e n d o t h e l i a l l i n i n g of capillaries".  from  "stabilize  the  the w a l l ,  the h y d r o s t a t i c p r e s s u r e of the p e r f u s i n g  -14-  Figure  6:  I l l u s t r a t i o n of Blood 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 Pressure, P = venous pressure)(3)  zone I  >  P > Po ) P» A  zone 2 Py  PQ > P > A  Distance  zone 3 D  'a  )  D  'Y  >  D 'A  3iood flow  -15-  Figure  7:  I l l u s t r a t i o n o f v a s c u l a r membrane s e p a r a t i n g 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 o f h y d r o s t a t i c (PMV, PPMV) and o n c o t i c (PIMV, PIPMV) p r e s s u r e s a t normal c o n d i t i o n s .  .-BASEMENT MEMBRANE  CAPILLARY WALL K F = 1.12ml/hr/mmHg SIGD = 0.75 P S A = 3.0 PSG=1.0ml/hr 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 f l u i d  and solute rarely d i f f e r e n t i a t e 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 i s c l a s s i f i e d as being "leaky" (6); f l u i d and solute are transported across the membrane, generally from the c i r c u l a t i o n 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, e l e c t r i c a l charge of the solute (2), d i s t r i b u t i o n 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 f r a c t i o n of albumins cross the vascular membrane as compared to globulins. The f r a c t i o n of solute crossing the vascular membrane may be obtained experimentally by comparing the solute concentration i n the i n t e r s t i t i u m to that of the plasma.  The solute concentration of the  i n t e r s t i t i u m i s generally approximated by the solute concentration i n 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 r a t i o s :  the r a t i o 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-  interstitial-to-plasma  concentration  r a t i o f o r g l o b u l i n was 0.55.  These r e s u l t s i n d i c a t e that a g r e a t e r crosses  f r a c t i o n of plasma albumin  the v a s c u l a r membrane than plasma g l o b u l i n s .  1.4 The Interstitium 1.4.1 Structure and Composition of the Interstitium The  interstial  space of the lung  i s that space which i s between  the v a s c u l a r membrane and the a l v e o l a r membrane. the  i n t e r s t i t i u m a r e numerous.  interstitium. fibres";  8 shows a s e c t i o n of the  The b a s i c s t r u c t u r e i n v o l v e s a " s k e l e t o n  of c o l l a g e n  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 m a t r i x of  water, s a l t s , plasma p r o t e i n s _  Figure  The components of  The c o l l a g e n  and glycosaminoglycans ( 8 ) .  f i b r e s range i n diameter from 0.5 u t o 1.0 u.  a c t u a l arrangement of the f i b r e s i n t i s s u e i s not w e l l known. and  Bert  Pearce (9) s t a t e that the f i b r e s i n many t i s s u e s are organized  i n t o bundles o f p a r a l l e l f i b r e s ; highly  ordered a r r a y s ,  oriented  these "bundles may be assembled  as i n cornea, or Into  felt-like  p a r a l l e l to the s u r f a c e , as i n dermis" ( 1 0 ) .  into  mats randomly "The main  f u n c t i o n a l e f f e c t s of the c o l l a g e n f i b r e s a r e that they r e s i s t in  The  changes  t i s s u e c o n f i g u r a t i o n and volume, they exclude p r o t e i n s , and they  i m m o b i l i z e the g l y c o s a m i n o g l y c a n s (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 m a t r i x may be d e s c r i b e d  i n the i n t e r s t i t i u m : structure. the blood,  one a f r e e f l u i d  The f r e e f l u i d except that  w i t h two major phases  and the other  a complex g e l  phase i s comparable t o the plasma f l u i d of  the i n t e r s t i t i a l  protein concentration  differs.  -18-  Figure  8:  C r o s s - s e c t i o n a l View o f I n t e r s t i t i u m and Contents, E p i t h e l i a l and E n d o t h e l i a l Membranes (RBC-red blood c e l l ; WBC-white blood c e l l ) ( 5 5 )  AIR  -19-  The gel phase i s composed of macromolecules c a l l e d glycosaminoglycans or GAGs. Hyaluronic  The primary GAG  i s hyaluronic acid.  acid i s an unbranched, random c o i l macromolecule.  tissue, hyaluronic acid occupies a volume 1000 unhydrated structure (8). are present  In  times that of i t s own  Other GAGs, such as chondroitin  sulphate,  i n smaller quantity; they are attached as side branches to  a protein backbone, forming proteoglycans. " i n t e r n a l entanglement" (8).  The GAGs show noticeable  The collagen f i b r e s are entangled with  the GAGs, and immobilize the mass.  Thus, a g e l - l i k e structure results  which exhibits s o l i d - l i k e 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 f i b r e s , primarily collagen, i n  conjunction with hyaluronate,  contribute to the exclusion phenomenon  by l i m i t i n g 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 i s reduced, which results i n an increase i n the e f f e c t i v e protein The  concentration.  exclusion phenomenon may  be explained by:  1) the rod and  sphere model, and 2) the sphere in a random network of rods model. the former model, shown i n Figure 9a, the rod represents  an isolated  collagen bundle i n the tissue while the protein i s approximated by sphere.  The  In  protein w i l l be excluded from the volume occupied  the  by the  rod, and the volume that i s one protein radius from the rod - shown by  -20-  F i g u r e 9:  Models of Volume E x c l u s i o n i n the Space ( 9 )  a)  Rod and Sphere Model  b)  Sphere i n a Random Network of Rods Model excluded space)  Interstitial  (thatched area i s excluded  space)  (thatched area  -21-  the shaded area i n the side and front view given i n 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 i n a random network of rods model.  Figure 9b shows the sphere i n 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) i s dependent the protein.  on the size of  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 c o l l o i d osmotic pressure calculated from the e f f e c t i v e  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 . , i n the l a t t e r case, the concentration uses the i n t e r s t i t i a l volume, while i n 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 i n t e r s t i t i u m i s subdivided into alveolar and extra-alveolar  sub-compartments, as shown i n Figure 10. distinguished by at least two factors:  The sub-compartments are 1) the effect of lung  i n f l a t i o n (or a r i s e i n the hydrostatic pressure of the a i r space) on the tissue hydrostatic pressure, and 2) the tissue compliance curves of the two sub-compartments.  -22-  F i g u r e 10:  Schematic o f Pulmonary M i c r o v a s c u l a r Exchange System showing sub-compartments o f 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 s u r f a c t a n t l a y e r l i n i n g a i r space ( P A L V - a l v e o l a r (gas) p r e s s u r e ; PAS=alveolar f l u i d p r e s s u r e ; PPMV=pressure o f a l 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 o f e x t r a - a l v e o l a r i n t e r s t i t i a l p r e s s u r e ; PMV=microvascular p r e s s u r e ) ( 2 )  Alveoli  JV  /  /  PALV  UV.  J V.  Ly mphat ic  Lymphatic PAS  u Extra-alveolar Interstitium  I i  Alveolar Interst i t i u m  !  PEA  P P M V  t^J  J Extra-alveolar i Interstitium i  PMV  Extra-alveolar  Alveolar Wal  Arterioles  Capil iGries  , Extra-alveolar i  Venules  -23-  F i r s t , the alveolar tissue sub-compartment l i e s l n close proximity  to the a l v e o l i (or a i r sacs) of the a i r space.  The  extra-  alveolar i n t e r s t i t i u m i s not necessarily adjacent to the a i r space. The  result of this arrangement i s the d i f f e r e n t responses of the  hydrostatic pressures of the sub-compartments to an increase i n the a i r pressure of the a i r 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 a i r pressure (PALV). The a i r space i s lined with a surfactant f l u i d that produces an alveolar tissue pressure s l i g h t l y less than PALV (see section 1.7.2). The hydrostatic pressure of the extra-alevolar tissue (PEA) than the pressure of the alveolar tissue (13).  i s less  Howell, et a l . (14),  Hida, et a l . (15) and Goldberg (16) have observed experimentally PEA  that  decreases with a r i s e in PALV; reasons for this decrease have been  related to the configuration of the a i r space and blood vessels but generally remain unclear. pressure,  The effect of a r i s e i n the a i r  PALV, i s , then, an equivalent  hydrostatic pressure, hydrostatic pressure,  (14),  r i s e i n the alveolar tissue  PPMV, and a drop i n the  extra-alveolar  PEA.  Second, the tissue compliance curves of the two d i f f e r in shape and value.  subcompartments  Tissue compliance i s defined as the change  i n extravascular-extracellular (EVEA) f l u i d volume r e l a t i v e to the change in tissue f l u i d pressure,  i . e . (AV/AP).  Prichard (17)  hypothetical complicance curves for the alveolar and tissue spaces, as shown in Figure 11. any  gives  extra-alveolar  Figure 11 i l l u s t r a t e s that for  tissue f l u i d volume less than point D the alveolar tissue  -24-  F i g u r e 11:  Compliance Curves o f A l v e o l a r and E x t r a - a l v e o l a r T i s s u e Sub-compartments - A l v e o l a r F l o o d i n g Occurs When P o i n t C' i s r e a c h e d . (17)  VOLUME  -25-  hydrostatic  p r e s s u r e (PPMV) i s g r e a t e r  hydrostatic  p r e s s u r e (PEA).  cause f l u i d  to d r a i n p r e f e r e n t i a l l y from the a l v e o l a r t i s s u e to the  extra-alveolar The  than the e x t r a - a l v e o l a r  The p r e s s u r e g r a d i e n t  that  tissue  results  will  tissue.  t i s s u e compliance f o r each sub-compartment v a r i e s as f l u i d  accumulates i n the t i s s u e .  Both h y p o t h e t i c a l  illustrate  (low AV/AP) f o r volumes l e s s than that at  a low compliance  p o i n t A on the e x t r a - a l v e o l a r  i s postulated  (18) that  In t h i s low compliance  fluid  the g e l i s r e t u r n i n g  volume i n c r e a s e s  the e x t r a - a l v e o l a r  from p o i n t s  the  pressure. until  (18) that  In the r e g i o n  In the e x t r a - a l v e o l a r  the f l u i d  in fluid  of high  compliance  The e x t r a - a l v e o l a r  volume.  compliance  fluid  proceeds  C, whereupon maximum expansion of  compliance, and the r i s e i n t i s s u e f l u i d  expand under a h i g h  t i s s u e returns  to a  p r e s s u r e i s high f o r  The a l v e o l a r t i s s u e c o n t i n u e s t o  compliance f o r f l u i d  When the t i s s u e elements a r e s t r e t c h e d  volumes beyond p o i n t  to t h e i r upper l i m i t  empties i n t o the a i r space which has a l a r g e c a p a c i t y ; the  As  change i n the t i s s u e  t i s s u e the high  volume reaches p o i n t  further increases  states.  the GAGs fragment as the t i s s u e expands;  volume r i s e s s h a r p l y w i t h l i t t l e  t i s s u e elements has o c c u r r e d . low  to i t s n e u t r a l  and a l v e o l a r t i s s u e s , r e s p e c t i v e l y , undergo a  i s hypothesized fluid  volume r i s e s i n the low  A and B ^ h e compliance curves of  t r a n s i t i o n t o a h i g h e r compliance. it  region  the g e l phase - of g l y c o s a m i n o g l y c a n s  (GAGs) - i s compressed, such that as the f l u i d compliance r e g i o n  11  t i s s u e compliance curve and p o i n t B^on  the a l v e o l a r t i s s u e compliance c u r v e . it  curves i n F i g u r e  a l v e o l a r t i s s u e compliance to remain  high.  thus  B*.  the f l u i d enabling  -26-  1.5  The Lymphatics  1.5.1  Structure of the Lymphatics The  lymphatics provide drainage channels for the f l u i d that  enters the i n t e r s t i t i u m . Figure 10 i l l u s t r a t e s that the  terminal  lymphatic c a p i l l a r i e s are situated i n the extra-alveolar i n t e r s t i t i u m , but not the alveolar i n t e r s t i t i u m . The f l u i d and solute that i s transported  across the vascular membrane into the alveolar tissue  space must, therefore, flow through the tissue space to reach the lymphatics;  as noted i n the previous  section a pressure  gradient from  the alveolar to extra-alveolar i n t e r s t i t i u m causes this flow of fluid. The lymphatic c a p i l l a r y membrane i s assumed to be extremely permeable to f l u i d and solute (19).  The membrane i s composed of an  endothelial c e l l layer and an underlying basement membrane.  The  basement membrane i s "poorly developed, and i s commonly absent, or at best, discontinuous"  (19) and therefore, does not provide  resistance to solute and f l u i d flow. filaments are attached surrounding t i s s u e .  significant  Figure 12 shows that anchoring  to the c e l l u l a r wall, and embedded i n the  When the perilymphatic  tissue expands, due  f l u i d accumulation, the filaments maintain the c e l l u l a r open, and the vessel does not collapse (17,18).  The  to  junctions  tissue f l u i d i s  then able to continue flowing into the lymphatic vessel.  As the  fluid  moves downstream through the lymph vessel i t i s prevented from returning to the terminal lymphatic end by lymphatic valves 12).  (Figure  -27-  F i g u r e 12:  S t r u c t r u a l F e a t u r e s of a T e r m i n a l Lymphatic  (18)  -28-  1.5*2  C o n t r a c t i l i t y and Pumping i n Lymphatic Vessels To further f a c i l i t a t e the drainage of f l u i d and solutes there  i s a c o n t r a c t i l e and pumping motion of the lymphatic c a p i l l a r i e s produced by muscular contractions, respiration, tissue movements, and an i n t r i n s i c pumping mechanism (17,19). The operation of the i n t r i n s i c pumping mechanism i s not c l e a r l y understood. mechanism: tissue.  Guyton (18) suggested the following explanation of the "As f l u i d accumulates  i n the tissue space i t expands the  The anchoring filaments of the lymphatic c a p i l l a r i e s ,  embedded i n the tissue, become expanded along with the expansion of the tissues, thereby creating a suction Inside the lymphatic capillaries.  This suction pulls f l u i d inward through the (openings)  of the lymphatic c a p i l l a r y , the inward movement of f l u i d causing the flap valves of the lymphatic membrane c e l l s (see Figure 12) to open inward".  Guyton then postulates that the f l u i d - f i l l e d  lymphatic  c a p i l l a r y w i l l be compressed by the f l u i d i n the surrounding tissue, r a i s i n g the pressure at the c a p i l l a r i e s end, and forcing the f l u i d forward into the c o l l e c t i n g lymphatic. The force that propels the f l u i d forward into the c o l l e c t i n g lymphatic may contractions•  a l t e r n a t i v e l y be the respiration movements or muscular  -29-  1.6 1.6.1  The A i r Space Arrangement of the A i r Space Figure 13a i l l u s t r a t e s that the a i r space i s composed of a  branching network of a i r ducts (or airways). or nose and passes through the trachea.  A i r enters v i a the mouth  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 l e 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). alveolar ducts. or a l v e o l i .  The respiratory bronchioles divide into  At the periphery of the alveolar ducts are a i r sacs  There are about 300 m i l l i o n a l v e o l i , each having an  equivalent diameter of 25 to 30 u (17).  It i s at the l e v e l of the  alveolus that the gas exchange occurs between the a i r space and blood circulation. The major s i t e s for f l u i d and solute exchange are located at the l e v e l of the respiratory units, that i s , the respiratory bronchiole, alveolar ducts and  alveoli.  -30-  Figure 13b:  Lobes, lobules and bronchopulmonary segments of the Lungs ( 5 4 )  -32-  1.7  Barriers Between the Interstitial Space and Air Space  1.7.1  Alveolar Membrane The  alveolar membrane i s composed of a c e l l u l a r  basement membrane.  The c e l l s of the membrane, primarily of squamous  type I, normally form continuous tight junctions. behaves functionally as though i t contained a radius between 0.6  layer and a  and  1.0 nm  (6,20).  The membrane  w a t e r - f i l l e d pores having  Therefore  the transport of  water across the alveolar membrane i s possible. Although water crosses the epithelium, i t does not accumulate in excess i n the a i r space; there must be mechanisms to remove the water.  The action of the surfactant l i n i n g the epithelium  functions  as one possible means to prevent f l u i d accumulation (see sections 1.7.2  and 3.6).  Another mechanism to remove the water i s evaporation  to the inspired a i r . However, the inspired a i r becomes completely humidified  i n the upper airways; therefore, only a small amount of  water w i l l be removed by evaporation  1.7.2  (17).  Surfactant Lining the Wall of the Air Space Two  secretions l i n e the walls of the respiratory unit:  the  alveolar wall (of the a l v e o l i ) i s covered with a surfactant, while th walls of the respiratory bronchiole are lined with the secretions of the mucous glands, goblet c e l l s , and Clara c e l l s  (17).  It i s the surfactant l i n i n g of the alveolar wall that i s believed to play an important role i n the mechanics of the lung and i pulmonary microvascular  exchange (21).  The surfactant i s a surface  -33-  active lipoprotein - primarily composed of dipalmitoly l e c i t h i n  (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 i s i l l u s t r a t e d by the Laplace equation developed for a spherical gas bubble  AP = PG - PL = — r where  (4)  PG = hydrostatic pressure of gas i n bubble PL = l i q u i d hydrostatic pressure x  = surface tension  r  = radius of curvature  Figure 14a shows the location of the pressures, PG and PL, and radius of curvature, r.  In the adjacent diagram, Figure 14b,  the a  schematic of the alveolus i s 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 d i f f e r e n t i a l  (PG - PL).  Therefore, i n comparing  the lung surfactant to water, the lower surface tension of DPL the pressure d i f f e r e n t i a l ,  reduces  (PG - PL), needed i n the alveolus to  maintain a stable a i r sac. DPL  i s a cationic surfactant which has two s t r u c t u r a l features  that aid i n maintaining the a l v e o l i v i r t u a l l y dry. positive charge from the quaternary  F i r s t , DPL has a  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-  F i g u r e 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 V a r i a b l e s of L a p l a c e ' s E q u a t i o n (PG=gas pressure; PL=liquid pressure; r=radius of curvature)  F i g u r e 14b:  I l l u s t r a t i o n of an I d e a l A l v e o l u s w i t h S u r f a c t a n t  Lining  Alveolus  Alveolus  Epithelial Membrane Surfactant 'PAS(PL) ^ /  u r  ^  a c < 2  /  .  //Fluid  tension  r  -35-  textiles.  The surfactants applied to t e x t i l e s have been shown to  raise the "water entry pressure" pressure  to over 760 mm Hg, which i s the  needed on the convex side of the l i q u i d surface to break  through the b a r r i e r .  Although the DPL surfactant may only have a  f r a c t i o n of one percent  e f f i c i e n c y i n comparison to the t e x t i l e  surfactants, i t does serve as an "appreciable  protection" against  acute f l u i d accumulation i n the a i r space (21). Second, DPL has "long hydrocarbon chains which are oriented outwards to provide a hydrophobic surface with a consequent lack of w e t t a b i l i t y " (21).  Wettability i s the a b i l i t y of a l i q u i d to adhere  to a surface - the greater the surface area a given volume of l i q u i d covers, the greater the w e t t a b i l i t y .  The lung surfactant has a low  w e t t a b i l i t y ; water i s repelled by the surfactant. i s prevented from crossing the surfactant The  Thus l i q u i d water  lining.  surfactant i s produced by type II e p i t h e l i a l c e l l s that are  interspersed throughout the predominantly type I e p i t h e l i a l membrane.  cellular  Any minor breaks i n 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 f l u i d and solute. endothelium  i s permeable to f l u i d and solutes.  and solutes i s generally observed interstitium.  The  A net flow of f l u i d  to occur from the c i r c u l a t i o n to the  Once i n the alveolar i n t e r s t i t i u m the f l u i d and solute  flow along a pressure gradient to the extra-alveolar i n t e r s t i t i u m . The lymphatics drain the extra-alveolar tissue space.  If steady state  is achieved the flow of f l u i d and solutes across the endothelium i s equivalent to the flow of f l u i d and solutes leaving the tissue space through the lymphatics. Under normal conditions f l u i d may between the i n t e r s t i t i u m and a i r space. a i r space may interstitium.  be evaporated  traverse the epithelium However f l u i d entering the  (a minor amount) or return to the  Therefore the net flow i s n e g l i g i b l e , and the a i r space  i s assumed to be v i r t u a l l y dry.  1.9 1.9.1  Pulmonary Edema Definition of Pulmonary Edema Edema i s defined as "a swelling due to an effusion of watery  f l u i d into the i n t e r c e l l u l a r spaces of connective tissue" (17). Prichard (17) presented a further d e f i n i t i o n of pulmonary edema: pathologic state i n which there i s abnormal water storage i n the lungs".  "a  -37-  Pulmonary edema may be i n i t i a t e d compartment by e l e v a t e d the  permeability  circulatory fluid  of the endothelium.  will  p r e s s u r e and/or changes i n  A l t e r n a t i v e l y , edema may be  induced by changes to the p e r m e a b i l i t y following discussion  from the c i r c u l a t i o n  of the e p i t h e l i u m .  on the sequence of events d u r i n g  be f o r edema i n i t i a t e d  permeability  filtration capillary and  of the v a s c u l a r  membrane w i l l  fluid  the  area and h i g h e r  and s o l u t e  transudation  t i s s u e space along the p r e s s u r e g r a d i e n t  two t i s s u e sub-compartments.  t i s s u e space.  As the f l u i d  the t i s s u e space the lymph flow a l s o r i s e s .  a new steady s t a t e w i l l  (22,23).  between  the e x t r a - a l v e o l a r sub-compartment.  I f the edema i s  w i t h the t r a n s e n d o t h e l i a l  alveolar  tissue f l u i d  volume.  Eventually  flow,  be e s t a b l i s h e d .  of the l y m p h a t i c s .  The f l u i d  t i s s u e space r i s e s as f l u i d This  that  volume r i s e s  However, i n severe pulmonary edema the t r a n s e n d o t h e l i a l exceeds the flow c a p a c i t y  fluid  The lymphatics d r a i n the f l u i d  moderate the lymph flow may e q u a l i z e and  The blood  the a l v e o l a r i n t e r s t i t i u m , and flow towards  flows i n t o the e x t r a - a l v e o l a r in  fluid  i n comparison to the a r t e r i a l and venous  and s o l u t e enter  the e x t r a - a l v e o l a r  or a change i n  lead to i n c r e a s e d  segment because of i t s l a r g e r s u r f a c e  segments, i s the major s i t e of f l u i d The  pressure,  from the c i r c u l a t i o n t o the i n t e r s t i t i u m .  solute permeability,  pulmonary edema  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 h y d r o s t a t i c the  The  flow  pressure i n  accumulates i n t h i s  r e s u l t s i n a decreased a l v e o l a r t o e x t r a flow, and an i n c r e a s e  the c a p a c i t y  i n alveolar tissue  fluid  of the i n t e r s t i t i u m i s exceeded and  -38breaks i n the epithelium develop.  F l u i d may then flow from the  c i r c u l a t i o n into the a i r space, as well as into the lymphatics. The location of the e p i t h e l i a l breaks i s not well-known.  Staub  and Gee (24), and Zumsteg et a l . (23) suggest that the breaks are present at the l e v e l of the terminal bronchioles; these e p i t h e l i a l 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 f l u i d transport.  Egan (25) proposes that the breaks may develop between the  normally tight junctions between the e p i t h e l i a l c e l l s .  In both cases,  the openings are not considered to be uniformly spread over the epithelium, but occur at a r e l a t i v e l y few discrete s i t e s . flow into the a i r space occurs through these The f l u i d accumulation  The protein  openings.  i n the a i r space i s spotty and the  a l v e o l i f i l l under an a l l or nothing phenomena.  This individual  f i l l i n g i s described i n more d e t a i l i n section 3.6.  1.9.2  Clinical Causes of Pulmonary Edema "Pulmonary edema may arise from a number of causes and i n a  wide variety of diseases" (17).  In most cases the cause of edema w i l l  be an increase i n the pulmonary microvascular hydrostatic pressure and/or an increase i n the f l u i d and solute permeability of the vascular and/or e p i t h e l i a l 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 c i r c u l a t i o n to the interstitium.  If the high transmembrane flow of f l u i d overwhelms the  -39-  s a f e t y f a c t o r s of the exchange system - t h a t a r e t r y i n g t o r e g u l a t e the  flow - then the p r o g r e s s i o n  interstitial  and a l v e o l a r  Hydrostatic For  of edema may proceed through the  stages.  edema may r e s u l t from upsets to the c a r d i a c  example, i f the heart  m a l f u n c t i o n s , and the r a t e of f l u i d  from the r i g h t v e n t r i c a l Is g r e a t e r the  than the r a t e of f l u i d  system. pumped  pumped from  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  circulation. vascular  As the blood volume of the c i r c u l a t i o n r i s e s , the  hydrostatic  increased  pressure w i l l also r i s e .  transendothelial  causes of c a r d i o g e n i c  1.9.2.2  The r e s u l t w i l l be an  flow o f f l u i d and s o l u t e s .  pulmonary edema a r e l i s t e d  i n Table 2 .  Permeability pulmonary edema.  Permeability  changes to the e n d o t h e l i a l and/or  membranes w i l l a l s o r e s u l t i n the f o r m a t i o n of edema. permeability injections  The major  epithelial Endothelial  t o f l u i d and s o l u t e may be a l t e r e d by i n t r a v e n o u s  of e x c e s s i v e  o r methadone.  doses of drugs, such as h e r o i n  Moderate edema w i l l  Interstitially.  (diamorphine)  r e s u l t i n the accumulation of f l u i d  I n severe pulmonary edema a l v e o l a r f l o o d i n g  will  o c c u r as the i n t e r s t i t i a l and lymphatic f l u i d c a p a c i t i e s a r e exceeded. The  permeability  of the e p i t h e l i u m  to f l u i d  a f f e c t e d by the i n h a l a t i o n of smoke or v o l a t i l e nitrogen  oxides.  Increasing  a l v e o l a r edema; the f l u i d  and s o l u t e may be  t o x i n s , such as  the e p i t h e l i a l p e r m e a b i l i t y  will  l e a v i n g the i n t e r s t i t i u m may then  e i t h e r the a i r space or the l y m p h a t i c s .  initiate  enter  -40Table 2  T a b l e 2:  Major Causes of C a r d i o g e n i c ( H y d r o s t a t i c ) 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 Mitral regurgitation L e f t a t r i a l thrombi L e f 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 e i t h e r of the above cases, the other membrane may a l s o be i n j u r e d by the i n s u l t .  T h e r e f o r e the p e r m e a b i l i t y of the endothelium  and the e p i t h e l i u m w i l l be changed, r e s u l t i n g pulmonary edema.  i n a more severe  case of  -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 i s the  pathophysiological disease that i s most commonly studied (13,17). Moderate pulmonary edema, induced by perturbations to the c i r c u l a t i o n and/or permeability of the vascular membrane, w i l l result i n the accumulation of f l u i d i n the i n t e r s t i t i u m .  The tissue between the  c i r c u l a t i o n and a i r space w i l l expand, thereby impairing the d i f f u s i o n 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 R o s e l l i (27); R o s e l l i , 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 i s composed of a population of pores; the pores are of uniform or different s i z e s .  The f l u i d  and  solute are assumed to be transported across the membrane through these pores.  The parameters  employed to define the membrane transport  properties to f l u i d and solute are expressed i n 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 a i r  space w i l l lead to the development of the Alveolar Model.  2.2  Modelling of the Pulmonary M i c r o c i r c u l a t i o n The c i r c u l a t i n g compartment interacts d i r e c t l y 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 c i r c u l a t i o n ,  assumptions and simplifications are needed i n defining i t s macroscopic properties - the hydrostatic and c o l l o i d osmotic pressures, and the protein concentrations. At different levels of the lung's c i r c u l a t i o n the value of the hydrostatic pressure changes. West, Dollery and Naimark (3) have i d e n t i f i e d three zones along the height of the lung - see section 1.2.3. PALV.  The t h i r d zone was i d e n t i f i e d as a region where PA > PV > In this region a l l the blood c a p i l l a r i e s are recruited, and  thus, involved i n f l u i d and solute exchange. assumes zone I I I conditions p r e v a i l .  The I n t e r s t i t i a l model  -44-  Th e h y d r o s t a t i c pressure a x i a l l y along  the l e n g t h  venous ends.  The blood  capillary  segments.  compartments, segment.  of the blood v e s s e l was  (1) The s u r f a c e  divided into a r t e r i a l ,  (2)  each segment  to f l u i d  area  would  for f l u i d  c h a r a c t e r i s t i c s of the membrane of  and s o l u t e would be r e q u i r e d .  And  (3)  there  to d i v i d e the i n t e r s t i t i u m i n t o an a l v e o l a r sub-  a l v e o l a r sub-compartment  i s not a v a i l a b l e to d e f i n e  segments was not c a r r i e d o u t .  information  sub-compartments  and  the d i v i s i o n of the c i r c u l a t i o n  into  In the i n t e r s t i t i a l model, the  represented  c i r c u l a t o r y h y d r o s t a t i c pressure  c i r c u l a t o r y compartment,  arterial  sub-  However, s u f f i c i e n t  the i n t e r s t i t i a l  Therefore,  The m i c r o v a s c u l a r  - and an e x t r a -  of these i n t e r s t i t i a l  compartments would then be r e q u i r e d .  pulmonary c i r c u l a t i o n was  segment  - i n t e r a c t i n g w i t h the venous and  The p h y s i c a l p r o p e r t i e s  from the l e f t a t r i a l  to each  of the w a l l of each segment  - i n t e r a c t i n g with the c a p i l l a r y  segments.  sub-  would be needed because of the v e s s e l  The p e r m e a b i l i t y  would be a p r e f e r e n c e  vascular  venous and  would be assigned  to determine the a v a i l a b l e area  filtration.  compartment  v e s s e l from the a r t e r i a l to  then h y d r o s t a t i c p r e s s u r e s  be needed i n order  c i r c u l a t i o n also varies  I f the c i r c u l a t i o n i s d i v i d e d i n t o three  A d d i t i o n a l information  sub-division.  segments.  of the blood  as a lumped compartment  w i t h one  (PMV).  h y d r o s t a t i c pressure  representing  i n zone I I I c o n d i t i o n s , may  the  be c a l c u l a t e d  and pulmonary a r t e r i a l h y d r o s t a t i c p r e s s u r e s  (3):  -45-  PMV = PLA +0.4 (PPA - PLA) where  (5)  PMV, PLA, PPA = the c a p i l l a r y , l e f t a t r i a l , a r t e r i a l hydrostatic pressures,  The proteins of the microvascular  and pulmonary  respectively.  exchange system were assumed  to be represented by albumin and globulins.  Albumin i s the most  abundant protein i n 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 c i r c u l a t i o n compartment were assumed to be  well-mixed.  As a result the concentrations  globulins (CMVG) are uniform. flow is therefore The  of albumin (CMVA) and the  Intracompartmental solute d i f f u s i v e  eliminated.  presence of proteins i n the c i r c u l a t i o n generates a  compartmental c o l l o i d osmotic pressure, concentrations concentration  PIMV.  Combining the  of albumin and the globulins provides the t o t a l protein CPMV:  CPMV = CMVA + CMVG The  (6)  c o l l o i d osmotic or oncotic pressure may then be calculated from  the t o t a l protein concentration Equation (7), with concentration  by the Landis and Pappenhiemer changed to units of g/ml:  PI = 210 CP + 1600 CP + 9000 CP 2  3  (7)  -46-  where  PI = o n c o t i c  pressure,  CP = t o t a l p r o t e i n  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  t o PIMV. The macroscopic p r o p e r t i e s of the lumped  compartment pressure  a r e : the h y d r o s t a t i c p r e s s u r e (PMV), the c o l l o i d  osmotic  (PIMV), and the p r o t e i n c o n c e n t r a t i o n s f o r albumin (CMVA) and  globulins  2.3  circulation  (CMVG).  M o d e l l i n g o f the  Interstitium  P r e v i o u s d i s c u s s i o n of the i n t e r s t i t i u m suggested the presence of  an a l v e o l a r sub-compartment and an e x t r a - a l v e o l a r  Each sub-compartment would have I t s own macroscopic  sub-compartment. properties.  However, v a l u e s of the p r o p e r t i e s f o r each sub-compartment are difficult to  to o b t a i n .  The approach taken i n the I n t e r s t i t i a l  assume one i n t e r s t i t i a l The i n t e r s t i t i a l  free f l u i d  model was  compartment.  compartment was composed  phase and a g e l phase.  The f l u i d  would be of a volume VIS - the i n t e r s t i t i a l  of two phases - a  p r e s e n t i n both phases fluid  volume.  However,  s o l u t e s 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  fluid  volume, or c o n v e r s e l y would be excluded from a f r a c t i o n of the Interstitial in  fluid  volume.  T h i s "excluded" volume Is g e n e r a l l y  the g e l 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  found  volume from  which the s o l u t e i s excluded I s dependent on the diameter of the solute.  Albumin r e p r e s e n t i n g the s m a l l e r c l a s s - s i z e of p r o t e i n s i s  e x c l u d e d from a f l u i d  volume (VEXA) of 75.5 ml (29), w h i l e  globulin,  -47-  representing  the larger c l a s s - s i z e of proteins  i s excluded from a  f l 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 r i s e as f l u i d accumulates i n the i n t e r s t i t i u m (29). An available volume accessible to albumin and globulin r e s u l t s :  where  VAVA = VIS - VEXA  (8)  VAVG = VIS - VEXA  (9)  VAVA, VAVG = the f l u i d volume available to albumin and globulin,  respectively.  VEXA, VEXG = the f l u i d volume excluded from albumin and globulin,  respectively.  VIS = i n t e r s t i t i a l f l u i d volume. The available volumes f o r albumin and globulin are used i n the evaluation  of the protein concentration of the available spaces.  These protein concentrations are the e f f e c t i v e concentrations of the proteins: CAVA=^A_  C A V G  where  (  = vic  1  0  <  )  n)  CAVA, CAVG = 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 globulin,  respectively.  QA, QG = the weight of albumin and globulin, in the i n t e r s t i t i u m .  respectively,  -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  c a l c u l a t e d from the summation of the e f f e c t i v e  interstitial  concentrations:  CPPMV = CAVA + CAVG  where  (12)  CPPMV = t o t a l e f f e c t i v e p r o t e i n c o n c e n t r a t i o n i n the Interstitium  The  interstitial  u s i n g the L a n d i s (7)  equal  PIPMV may then be c a l c u l a t e d by  and Pappenhiemer e x p r e s s i o n by s e t t i n g CP i n equation  to CPPMV.  The ducts.  oncotic pressure  i n t e r s t i t i u m i s assumed to be d r a i n e d by n o n s i e v i n g  These ducts  interstitial  collect  the f l u i d  space, that i s , both  that passes through the  the space which excludes the  p r o t e i n s and the space which i s a c c e s s i b l e to the p r o t e i n s . volume of t h i s f l u i d  entering  where  a r e w e l l mixed, the p r o t e i n c o n c e n t r a t i o n s  the lymphatics  CTA  =  CTG  =  The  space i s VIS. Albumin and g l o b u l i n have p r o t e i n  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 . lymphatics  lymph  Assuming the  of the f l u i d  a r e g i v e n by:  2A_  CTA, CTG = the t i s s u e c o n c e n t r a t i o n of albumin and globulin, respectively.  (13  )  (U)  -49-  Under normal steady s t a t e c o n d i t i o n s equivalent the  t o the t r a n s e n d o t h e l i a l  interstitium.  However,  will  flowrate;  a perturbation  system where the t r a n s e n d o t h e l i a l  the h y d r o s t a t i c  volume a c c o r d i n g curve.  no f l u i d  accumulates i n  to the f l u i d  exchange  flow exceeds the lymph f l u i d  l e a d to an accumulation of f l u i d  interstitium  the lymph flow i s  i n the i n t e r s t i t i u m .  p r e s s u r e r i s e s w i t h the r i s e  to the r e l a t i o n s h i p p r o v i d e d  flow  In the  in fluid  by the t i s s u e  compliance  Although a compliance curve should e x i s t f o r each of the  t i s s u e sub-compartments - a l v e o l a r and e x t r a - a l v e o l a r sub-compartments i n t o one compartment t i s s u e compliance c u r v e .  Bert  - uniting  these  a l l o w s f o r the assumption of one  and P i n d e r  (29) have r e c a l c u l a t e d the  compliance curve of Parker et a l . (31) to account f o r the dimensions of a human lung; VIS1,  the e x t r a v a s c u l a r - e x t r a - a l v e o l a r  i s used i n s t e a d  of the i n t e r s t i t i a l  (EVEA) f l u i d  volume,  volume, V I S . These two  volumes a r e r e l a t e d by the e x p r e s s i o n :  (15)  VIS1 = VIS + VCELL  The c e l l u l a r volume, VCELL, i s assumed hydrostatic  p r e s s u r e - EVEA f l u i d  Figure  At f l u i d  low  15.  t o be c o n s t a n t .  The  volume r e l a t i o n s h i p i s shown i n  volumes l e s s than 380 ml the t i s s u e compliance i s  and the r e l a t i o n s h i p i s g i v e n by:  (16)  PPMV = 0.227 VIS1 - 89.0  where  PPMV = i n t e r s t i t i a l  hydrostatic  pressure.  -51-  If the EVEA f l u i d volume exceeds 460 ml a high tissue compliance i s present, the expression i s given by: PPMV = .017 VIS1 - 6.73  (17)  The smooth t r a n s i t i o n zone between the low and high compliance i s represented by a series of points approximating see Table 3.  curves  the smooth curve -  An interpolation routine i s used to relate  interstitial  hydrostatic pressure to the EVEA f l u i d volume i n this region.  2.4 Modelling the Vascular Membrane The c i r c u l a t i o n and i n t e r s t i t i a l compartments are separated by the vascular membrane.  F l u i d and solute exchange occurs between these  two compartments across this b a r r i e r .  The barrier i s composed of a  c e l l u l a r layer and basement lamina, each component having a s p e c i f i c resistance to f l u i d and solute flow.  The model combines these  resistances to f l u i d and solute flow, and assumes that the vascular membrane may be represented by a resistance to f l u i d and solute that i s uniform throughout  2.5  the membrane.  Starling's Hypothesis:  Transendothelial Fluid Flow  Starling's Hypothesis has been used to represent the transendothelial flow from the c i r c u l a t i o n compartment to the i n t e r s t i t i a l compartment (8,29,32).  -52Table 3  Table 3:  PPMV versus VIS1 for t r a n s i t i o n region of i n t e r s t i t i a l compliance  VIS1 (ml)  PPMV (mmHg)  380 390 400 410 420 430 440 450 460  -2.74 -2.4 -1.9 -1.3 -0.8 -0.3 0.15 0.6 1.09  curve.  -53-  JV = KF[(PMV-PPMV) - SIGD(PIMV-PIPMV)] where  JV = transendothelial f l u i d flowrate  (18)  (ml/hr)  KF = endothelial f i l t r a t i o n c o e f f i c i e n t  (ml/hr/mm Hg)  SIGD = solute r e f l e c t i o n c o e f f i c i e n t (a^) PMV, PPMV = c i r c u l a t i o n and i n t e r s t i t i a l hydrostatic pressures,  respectively (mm Hg)  PIMV, PIPMV = circulatory oncotic pressures,  and effective  interstitial  respectively (mm Hg).  This hypothesis assumes that the flow of f l u 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 c i r c u l a t i o n  compartments (PMV - PPMV), and 2) the c o l l o i d osmotic gradient  (PIMV - PIPMV).  pressure  The parameter KF characterizes the  permeability of the endothelial membrane to f l u i d transport, while SIGD characterizes an effective oncotic pressure endothelium.  gradient across the  Figure 7 shows a schematic of the c i r c u l a t i o n and tissue  space separated  by the endothelium; the normal values of the  c i r c u l a t i o n and i n t e r s t i t i a l hydrostatic (PVM, PPMV) and oncotic (PIMV, PIPMV) pressures, indicated.  and of the parameters KF and SIGD are  These are the values assumed by Bert and Pinder  representative of existing experimental data. pressure  (29) to be  The hydrostatic  difference (PMV - PPMV) of 12.3 mm Hg w i l l force f l u i d 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 .  -54-  Equation (7) showed that oncotic pressure may be calculated from the t o t a l protein concentration of a compartment, the t o t a l protein concentration of the c i r c u l a t i o n compartment i s 69 g / l i t r e assuming an albumin concentration of 42 g/1 and a globulin concentration of 27 g/1.  The t o t a l 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 c i r c u l a t i o n and 18.9 mm Hg for the i n t e r s t i t i u m , which gives an oncotic pressure difference (PIMV-PIPMV) of 6.1 mm Hg. Contrary to the effect of a hydrostatic pressure gradient - causing f l u i d movement from a high to a low hydrostatic pressure - an oncotic pressure gradient w i l l force f l u i d from the lower oncotic pressure to the higher oncotic pressure.  The transport of f l u i d to the  compartment of higher protein concentration results i n the d i l u t i o n 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 f l u i d from the i n t e r s t i t i u m to the circulation.  If the membrane separating the compartments i s permeable  to proteins, the proteins w i l l also be able to move from the c i r c u l a t i o n to the interstitium; the magnitude of the oncotic pressure driving force w i l l be reduced.  The solute r e f l e c t i o n c o e f f i c i e n t  (SIGD) measures the a b i l i t y of the proteins to pass through a membrane.  I f the membrane i s 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 r e f l e c t i o n c o e f f i c i e n t equals one (SIGD = 1) and the oncotic pressure  driving force i s (PIMV-PIPMV).  However, i f the solutes are  able to freely pass across the membrane then the solute r e f l e c t i o n c o e f f i c i e n t would equal zero (SIGD = 0) and the oncotic driving force i s non-existent.  pressure  In the case of the plasma proteins -  albumin and globulins - a f r a c t i o n of the proteins may cross the endothelial membrane, while the remaining f r a c t i o n does not - the membrane sieves the proteins.  For the i n t e r s t i t i a l model the solute  r e f l e c t i o n c o e f f i c i e n t for the endothelium i s 0.75 (29). the e f f e c t i v e oncotic pressure  Therefore,  driving force 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 i s 0.75 (PIMV-PIPMV). 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 endothelium, KF, represents  the permeability of the membrane to f l u i d .  KF i s the  product of the membrane's f l u i d conductivity (the inverse of the membrane's resistance to f l u i d flow) and the surface area of the membrane (33): KF = LF(SA) where  (19)  LF = the f l u i d conductivity c o e f f i c i e n t of the membrane SA = the surface area of the membrane  By maintaining pressure,  the c i r c u l a t i o n pressure  above the alveolar (gas)  i . e . zone III conditions, a l l the c a p i l l a r i e s are assumed to  be r e c r u i t e d .  In zone I I I conditions the membrane surface area i s  assumed to remain r e l a t i v e l y constant,  so that the increase i n the  -56-  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 i s primarily due to an increase i n the endothelial f l u i d conductivity ( a l l c a p i l l a r i e s are recruited and the surface area increases only s l i g h t l y 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 f l u i d flow across the endothelium difference [(PMV-PPMV) - SIGD(PIMV-PIPMV)].  i s the  The f l u i d f i l t r a t i o n rate  across the membrane, JV, i s the product of the f l u i d  filtration  c o e f f i c i e n t , KF, and the driving force noted above, i . e . , JV i s calculated by equation (18), Starling's  2.6  Hypothesis.  Kedem-Katchalsky Solute Flux Equation; Solute Flow  Transendothelial  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 ) [  d i f f u s i v e term  (JS)i  ( C M V ) 1  *  ( C A V ) 1  ]JV  (20)  convective term  = 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 r e f l e c t i o n c o e f f i c i e n t of the membrane to solute ' i '  -57-  (CMV)i,(CAV)i  = the c o n c e n t r a t i o n of s o l u t e ' i '  i n the  c i r c u l a t i o n and a v a i l a b l e t i s s u e space, r e s p e c t i v e l y (g/ml) JV = the f l u i d  The  filtration  K-K s o l u t e f l u x e q u a t i o n  convective gradient  term.  rate  i s separated  f o r solute  *i'  the p e r m e a b i l i t y - s u r f a c e area product flow i s from the h i g h e r  across  of a c o n c e n t r a t i o n  - albumin or g l o b u l i n s - and  (PS)i.  The d i r e c t i o n of s o l u t e  c o n c e n t r a t i o n t o the lower c o n c e n t r a t i o n .  model the d i f f u s i v e  In  flow of albumin and g l o b u l i n  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 a v a i l a b l e to p r o t e i n s .  concentration is  i n t o a d i f f u s i v e term and a  The d i f f u s i v e term i s the product  ((CMV)i-(CAV)i)  the i n t e r s t i t i a l  (ml/hr)  of t h i s  interstitial  fluid  CAVA (albumin) and CAVG ( g l o b u l i n ) .  concentrations  that r e p r e s e n t  products,  PSA and PSG, i l l u s t r a t e  membrane t o the s p e c i f i e d data on these  space a c c e s s i b l e to p r o t e i n s I t i s these a v a i l a b l e  the i n t e r s t i t i a l  used i n the K-K s o l u t e f l u x e q u a t i o n .  solute.  The p r o t e i n  protein  concentrations  The p e r m e a b i l i t y s u r f a c e  area  the p e r m e a b i l i t y of the e n d o t h e l i a l B e r t and P i n d e r  parameters and made a p p r o p r i a t e  (29) reviewed the  selections.  They have  assumed a PSA f o r albumin of 3.0 ml/hr; a v a l u e w i t h i n the range of PSA area  d e r i v e d f o r canine product  and sheep models.  The p e r m e a b i l i t y - s u r f a c e  f o r g l o b u l i n , PSG, was s e t a t o n e - t h i r d the v a l u e f o r  albumin, i . e . , PSG was assumed to be 1.0 ml/hr ( 2 9 ) .  -58-  The  convective  to represent  term of the K-K solute flux equation i s assumed  the flow of solute that i s coupled with the f l u i d  The arithmetic mean of the solute concentration expression  Is a simplified  of the mean logarithmic solute concentration  from the a n a l y t i c a l derivation of the convective solute flux equation.  flow.  that  results  term of the K-K  The product of the arithmetic mean of the  solute concentration and the transendothelial f l u i d f i l t r a t i o n rate yields  the rate of convective  solute flow that crosses an endothelial  membrane that i s highly permeable to solute ' i ' . r e f l e c t i o n c o e f f i c i e n t for solute ' i ' assumed to represent  A solvent drag  of zero, (SIGF)i = 0, i s  an endothelium highly permeable to the solute.  If the endothelium i s semi-permeable to solute ' i ' ,  then solute ' i ' i s  not transported across the membrane and 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 solute ' i '  equals one, (SIGF)i = 1.  solvent drag r e f l e c t i o n c o e f f i c i e n t to 'drag' solute ' i '  indicates the a b i l i t y  across the membrane.  l i t e r a t u r e , Bert and Pinder  Therefore the  After reviewing  of the f l u i d the data i n  (29) selected the value of the solvent  drag r e f l e c t i o n c o e f f i c i e n t for albumin, SIGFA, to be approximately 0.4;  near the midrange of values determined i n dogs as JV varied.  The  solvent drag r e f l e c t i o n c o e f f i c i e n t for globulin, SIGFG, would be expected to be greater than SIGFA, since i t s diameter i s larger. and Pinder  (29) selected a value of 0.6 f o r SIGFG.  Table 4 l i s t s a l l the input parameters discussed i n t e r s t i t i a l model. shown.  Bert  i n the  The estimates used by Bert and Pinder  (29) are  -59Table 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 PIMV KF SIGD VCELL VIS1 Albumin PSA SIGFA VEXA QA CMVA Globulin PSG SIGFG VEXG QG CMVG  9 mmHg 25 mmHg 1.12 ml/hr mmHg 0.75 150 ml 379. ml  3.0 ml/hr 0.4 73.5 ml 5.38 g 0.042 g/ml 1.0 ml/hr 0.6 115.5 ml 2.48 g 0.0271 g/ml  -60-  2.7  Modelling  of the Lymphatics  The lymphatics function, i n part, as drainage channels f o r f l u i d and solute leaving the i n t e r s t i t i u m .  Although an i n t r i n s i c  pumping mechanism within the lymphatics i s thought to a c t i v e l y drain the tissue (17,18), i n the i n t e r s t i t i a l model the lymphatics assume a passive r o l e .  The lymph f l u i d flow has been expressed as a function  of t o t a l extravascular  f l u i d volume (VTOT), i n t e r s t i t i a l pressure  (PPMV), and also c i r c u l a t i o n hydrostatic pressure (PMV) (34,35,36, 37).  An increase i n either of the variables - VTOT, or PPMV, - or  parameter PMV - i s equivalent  to a r i s e i n the 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 (= 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 f l u i d flow, JL as a function of the 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, f o r 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 where  (21)  JL = lymph f l u i d flow (ml/hr) VIS1 = i n t e r s t i t i a l (+ c e l l u l a r ) or extravascular -extra-alveolar  Therefore, increase.  (EVEA) f l u i d volume (ml)  any r i s e i n 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  -61-  The  s o l u t e s are a l s o assumed to p a s s i v e l y l e a v e  i n t e r s t i t i u m and  enter the l y m p h a t i c s .  The  lymphatic  the capillary  membrane i s assumed to be h i g h l y permeable to the p r o t e i n s .  The  p r o t e i n c o n c e n t r a t i o n of the lymph i s thus assumed to be e q u i v a l e n t the  to  tissue protein concentration:  (CL)  where  ±  = (CT)  (22)  i  (CL)^ = c o n c e n t r a t i o n of p r o t e i n ' i '  i n lymph  ( C T ) ^ = t i s s u e c o n c e n t r a t i o n of p r o t e i n ' i '  The  s o l u t e flow i n t o the lymphatics  convective,  _ "  , J L  <  C T  . >i  Fluid and Solute Material Balances  2.8.1  Fluid Material Balance The  fluid  were of the f l u i d  flows  p r e v i o u s l y d e s c r i b e d i n s e c t i o n 2.5  e n t e r i n g and  from the c i r c u l a t i o n and  exiting  the i n t e r s t i t i u m ;  e x i t s through the l y m p h a t i c s .  s t a t e c o n d i t i o n s the f l u i d and  completely  that i s : ( S o l u t e Flowrate i n t o the Lymphatics)  2.8  i s assumed to be  flows  i n t o and  J L r e s p e c t i v e l y , are assumed e q u a l .  and  fluid In  2.7 enters  steady  out of the i n t e r s t i t i u m ,  JV  I f unsteady s t a t e c o n d i t i o n s  p r e v a i l i n the three-compartment model, and  JV exceeds J L , 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 .  m a t e r i a l balance around  F l u i d Flowrate i n t o the Interstitium  A statement  of the f l u i d  the i n t e r s t i t i u m i s :  Rate of F l u i d Accumulation i n the I n t e r s t i t i u m  U s i n g the a p p r o p r i a t e nomenclature  F l u i d Flowrate out of the Interstitium  +  results i n :  (25)  JV = JNET1 + J L  JNET1 = r a t e of f l u i d  where  (24)  accumulation  i n the i n t e r s t i t i u m  The m a t e r i a l balance of e q u a t i o n (25) a p p l i e s to the three-compartment interstitial  model.  model and the f l u i d fluid  F i g u r e 16a i l l u s t r a t e s the three compartment flows.  E q u a t i o n (25) i l l u s t r a t e s that i f the  flow i n t o 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 , J L ,  a r e known, then the r a t e of accumulation of f l u i d is  simply the d i f f e r e n c e  will  (JV-JL).  i n the i n t e r s t i t i u m  The i n t e g r a l of JNET1 with  y i e l d the d i f f e r e n c e between the f l u i d  time  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 : VIS1 - VIS10 = In  f JNETl dt  the i n t e r s t i t i a l model f l u i d  the EVEA or i n t e r s t i t i a l  (26)  t  o  does not accumulate  (+ c e l l u l a r ) f l u i d  e q u i v a l e n t to the t o t a l e x t r a v a s c u l a r f l u i d  i n the a i r space;  volume, VIS1, w i l l volume, VTOT.  be  -63-  F i g u r e 16a:  Schematic o f t h e 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  Air Space . Interstitium Ci rculation  JV  Accumulat ion J N E T 1  -> Lymph  Flow  JL  F i g u r e 16b:  Schematic o f t h e 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 S o l u t e Flows and Accumulation  Air Space Inters:! t i u m  Circulation  Accumulation JS;  Concentration CMV;  ' Q NETj Concentration CT;  ^Lymph JLCT;  Flow  r  -64-  2.8.2  Solute Material Balance The material balance for the proteins - albumin and globulin -  around the i n t e r s t i t i u m may be stated as follows: Flowrate of Solute ' i ' into the Interstitium  =  Rate of Accumulation of Solute ' i ' i n the Interstitium  +  Flowrate of Solute ' i ' out of the (27) Interstitium  The solute material balance may be rewritten, with the solute flow variables inserted: (JS) where  ±  = (QNET) + J L ( C T ) ±  (28)  1  (QNET)^ = rate of accumulation of solute ' i '  i n the  interstitium The solute flows are i l l u s t r a t e d i n Figure 16.  Equation (28)  i l l u s t r a t e s that i f the solute flow Into the i n t e r s t i t i u m , (JS)^, and out of the i n t e r s t i t i u m , (QNET)^, are known, then the rate of accumulation of solute i n the i n t e r s t i t i u m i s simply ((JS) -JL(CT) ). 1  1  the difference  The i n t e g r a l of (QNET)^ with time w i l l y i e l d the  difference between the solute weight i n the i n t e r s t i t i u m at time zero and time t:  Q  where  ±  " (Q0) = Jl ±  (QNET) dt i  Q^»(QO)^ = the weight of solute ' i ' respectively (gm).  (23)  at time t and zero,  -65-  ALVEOLAR MODEL  3.1  Introduction Previous models (28,29) of pulmonary microvascular exchange  were developed to study the exchange of f l u i d and solutes between the i n t e r s t i t i a l , c i r c u l a t i o n , and lymphatic compartments. was  The a i r space  not included as a fourth compartment since the experimental data  needed to include i t were not a v a i l a b l e . This thesis reports one way of integrating the a i r 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 i n section 2.  The four compartment model that  results i s referred to as the alveolar model. are available on the involvement  The limited data which  of the a i r space In the pulmonary  microvascular exchange were combined with assumptions - i n areas of the model lacking experimental evidence - to integrate the a i r space with the lumped compartment model.  3.2  Modelling of the A i r Space The major regions for f l u i d and solute exchange are at the  l e v e l of the respiratory units, that i s , the respiratory bronchiole, alveolar ducts, and a l v e o l i .  Each lung lobe i s composed of many  respiratory units. In experimental studies of the pulmonary microvascular exchange i t was  found that the a i r space of a lobe or segment of a lobe may  f i l l e d with f l u i d  (12,38).  be  The effects of f i l l i n g these a i r spaces  with f l u i d remain l o c a l i z e d , i . e . , other lung sections would continue  -66-  to operate normally.  In severe pulmonary edema the accumulation of  f l u i d i n the a i r 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 f l u i d i n the a i r space was not uniform.  They  proposed that an alveolus would not remain p a r t i a l l y f i l l e d , but when f l u i d started to accumulate i n the alveolus i t would f i l l rapidly and independently of the neighbouring a l v e o l i . Ideally, modelling of the a i r space should incorporate the concept of individual a l v e o l i f i l l i n g independently. The a i r space would then be modelled as a compartment composed of m i l l i o n s of subcompartments that represent the a l v e o l i . the  As the f l u i d accumulates i n  a i r space the number of small sub-compartments f i l l e d  increase.  would  However, as a f i r s t approximation i n the development  of the  alveolar model i t i s unreasonable to incorporate this degree of complexity.  Consequently i n this model the a i r space compartment i s  represented as a reservoir with a capacity of several l i t r e s . During normal conditions the base of the compartment i s lined with a thin layer of surface active f l u i d .  Macklin (40) estimated  that the t o t a l f l u i d volume i n the a i r space under normal conditions is 20 ml; a volume that i s n e g l i g i b l e In comparison to the large capacity of the a i r 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 f l u i d that accumulates i n the a i r space w i l l be assumed to be well-mixed. concentration.  Therefore, the compartment w i l l have a uniform protein  -67-  3.3  Modelling of the Alveolar Membrane 8 illustrates  Figure other  that  the a i r space i s separated  compartments of the pulmonary m i c r o v a s c u l a r  from the  exchange by the  a l v e o l a r w a l l composed of a c e l l u l a r and a basement membrane. simplicity,  the i n d i v i d u a l r e s i s t a n c e s of each membrane t o f l u i d and  s o l u t e flow a r e r e p r e s e n t e d ance. eable  i n the a l v e o l a r model by a s i n g l e  Hence the a l v e o l a r w a l l w i l l to f l u i d  and s o l u t e s  be taken as being  throughout i t s t h i c k n e s s  In the a l v e o l a r model the exchange of f l u i d the  a l v e o l a r membrane w i l l  interstitial  For  resist-  uniformly  perm-  and s u r f a c e  area.  and s o l u t e s  be assumed to occur between only the  and a l v e o l a r compartments.  S t r u c t u r a l l y , the a l v e o l a r  w a l l s e p a r a t e s the a i r space and i n t e r s t i t i u m .  At other  basement membrane of the a l v e o l a r w a l l i s fused  to the basement  membrane of the v a s c u l a r w a l l be  transported  across  and be the  the c i r c u l a t i o n a c r o s s  Fluid  and s o l u t e may a l s o  However, i f f l u i d the v a s c u l a r w a l l  and s o l u t e s  could  ( t o the i n t e r s t i t i u m )  the v a s c u l a r - a l v e o l a r membrane ( t o the a i r space) a f a c t o r would required  to i n d i c a t e the p r o p o r t i o n  of f l u i d  and s o l u t e s  leaving  c i r c u l a t i o n f o r the i n t e r s t i t i u m , and f o r the a i r space.  Furthermore, the p e r m e a b i l i t y  to f l u i d  a l v e o l a r membrane would be needed. nor  8).  (Figure  s i t e s the  t h i s v a s c u l a r - a l v e o l a r w a l l , between the  c i r c u l a - t i o n and a i r space. leave  across  Neither  the v a s c u l a r - a l v e o l a r p e r m e a b i l i t y  available i n literature.  Therefore,  and s o l u t e of the v a s c u l a r the p r o p o r t i o n a l i t y f a c t o r  to f l u i d  to s i m p l i f y the matter, the  c i r c u l a t i o n was assumed to i n t e r a c t with only then i n t e r a c t s , , w i t h the a i r space.  and s o l u t e are  the i n t e r s t i t i u m , which  -68-  3.A  The Onset of Alveolar Flooding During i n t e r s t i t i a l pulmonary edema, f l u i d and protein are  exchanged between the c i r c u l a t i o n , compartments.  i n t e r s t i t i a l , and lymphatic  The a i r space remains v i r t u a l l y dry, and i s assumed t  not p a r t i c i p a t e i n the pulmonary microvascular  exchange.  Following  the onset of alveolar flooding the a i r space receives f l u i d 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 s p e c i f i c extravascular-extra-alveolar  (EVEA) f l u i d volume, VTONS.  Preceding alveolar flooding the extravascular-extra-alveolar volume (VIS1) i s equivalent (VTOT).  to the t o t a l extravascular  fluid  f l u i d volume  In the alveolar model, when the f l u 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  i n later  sections (3.5, 3.6). The value of VTONS i s 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 i n endothelial permeability.  Fluid  will  accumulate i n the i n t e r s t i t i a l space u n t i l the EVEA f l u i d volume (VIS1) exceeds VTONS, at which point alveolar edema begins.  A numbe  of values of VTONS have been suggested under these circumstances: Iliff  (41) has suggested that alveolar flooding occurs a f t e r a 30%  increase i n VTOT from the normal t o t a l extravascular i.e.,  f l u i d volume;  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 t o t a l extravascular f l u i d 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 f l u i d volume of approximately 229 ml (= 379 - 150 ml) under normal conditions, VTONS would be 458 ml. F i n a l l y , Staub (13) has stated that alveolar flooding occurs a f t e r a 30% increase i n 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): g; Blood-free dry lung weight = 94 g; Therefore,  Blood content = 195  Density of f l u i d = 1 g/ml).  i n alveolar edema caused by a transendothelial flow greater  than normal, the range of VTONS l i e s between 450 ml and 600 ml. Severe pulmonary edema may also arise from permeability changes to the e p i t h e l i a l membrane induced  by an i r r i t a n t  intravenously, or from the a i r side.  introduced  The parameter VTONS may s t i l l be  used to indicate the onset of alveolar flooding.  If the i r r i t a n t i s  introduced from the a i r 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 f l u i d w i l l have occurred.  If the i r r i t a n t i s  introduced intravenously i t may a l t e r the e p i t h e l i a l permeability to f l u 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 i r r i t a n t  damages the  endothelium; i n this case VTONS w i l l be just above 379 ml (normal extravascular f l u i d volume).  On the other hand, the e p i t h e l i a l  -70-  permeability may change at a longer period of time after the endothelium i s injured by a small dosage of the i r r i t a n t .  During this  period of time, before the epithelium i s injured, the EVEA f l u i d volume (VIS1) w i l l r i s e , 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 f l u i d volume at which edema i s induced by an elevated c i r c u l a t i o n 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 r e s u l t s .  Acquiring the measurements for the animal lungs i n  v i t r o w i l l also affect the values of VTONS (P. Dodek, personal communication).  The expansion of lungs i n vivo w i l l be r e s t r i c t e d by  the chest wall, while i n v i t r o expansion of the lungs w i l l not be restricted.  Therefore the estimates obtained with the lungs i n v i t r o  (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 i s close to the f l u i d volume equal to twice the i n t e r s t i t i a l f l u i d volume, 458 ml.  -71-  3.5  Modelling of F l u i d and Solute Transport Across the Alveolar Membrane During  alveolar flooding fluid  and s o l u t e are t r a n s p o r t e d  a c r o s s the a l v e o l a r membrane from the i n t e r s t i t i u m M o d e l l i n g the f l u i d  i n t o the a i r space.  flow a c r o s s the a l v e o l a r membrane may be  approached i n a manner s i m i l a r t o that used i n the m o d e l l i n g o f f l u i d movement  a c r o s s the endothelium.  h y d r o s t a t i c and c o l l o i d  osmotic  e p i t h e l i a l c o n d u c t i v i t y to f l u i d filtration  coefficient.  Starling's  Hypothesis:  JAS  where  The d r i v i n g  f o r c e s are the  p r e s s u r e g r a d i e n t s , w h i l e the flow i s d e s c r i b e d by an e p i t h e l i a l  The form of the e x p r e s s i o n i s s i m i l a r t o  = KAS[(PPMV-PAS) - SIGDAS(PIPMV-PIAS)  JAS = t r a n s e p i t h e l i a l f l u i d  filtration  KAS = e p i t h e l i a l f i l t r a t i o n  coefficient  PPMV,PAS = f l u i d  PIPMV,PIAS = c o l l o i d  space,  osmotic  space and a l v e o l a r f l u i d  respectively  p r e s s u r e of i n t e r s t i t i a l  space,  respectively  SIGDAS = s o l u t e r e f l e c t i o n c o e f f i c i e n t To  i n s u r e t h a t the t r a n s e p i t h e l i a l f l u i d  of  a l v e o l a r f l o o d i n g the e p i t h e l i a l f i l t r a t i o n  equal to z e r o .  for  KAS.  of e p i t h e l i u m  flow i s zero b e f o r e the onset coefficient  -  c o n d u c t i v i t y of the e p i t h e l i u m - w i l l  F o l l o w i n g the onset  g r e a t e r than z e r o .  rate  h y d r o s t a t i c p r e s s u r e of i n t e r s t i t i a l  space and a l v e o l a r f l u i d  r e p r e s e n t i n g the f l u i d  (24)  of a l v e o l a r  be s e t  f l o o d i n g KAS w i l l  be  The a c t u a l v a l u e depends on the e x p r e s s i o n used  -72-  Vreim et a l . (43,44) have i l l u s t r a t e d 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 i n the alveolar f l u i d , 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  s i g n i f i c a n t 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 f l u i d was less than i n the plasma. induced by permeability concentration  If the edema was  changes to both membranes the protein  of the alveolar, i n t e r s t i t i a l , lymphatic and c i r c u l a t i o n  compartments was s i m i l a r .  The s i m i l a r i t y i n protein  concentration  between the tissue and alveolar f l u i d suggests that the alveolar membrane was highly permeable to proteins.  This s i t u a t i o n would  permit the assumption that during alveolar flooding the solute r e f l e c t i o n c o e f f i c i e n t of the epithelium  (SIGDAS) i s equal to zero.  Therefore, during alveolar flooding, the expression f o r t r a n s e p i t h e l i a l f l u i d flow reduces to:  JAS = KAS(PPMV-PAS)  (25)  Preceeding alveolar flooding KAS i s set equal to zero, so that there i s no need to define the solute r e f l e c t i o n c o e f f i c i e n t of the epithelium  and the c o l l o i d osmotic pressure of the alveolar  fluid.  The variables PAS and KAS w i l l be defined i n sections 3.6 and 3.7, respectively.  -73-  The observation by Vreim et a l . (43,44) that solute concentration i n the tissue space and alveolar f l u i d were similar during alveolar flooding supports a convective non-sieving flow of solutes across the epithelium:  Flowrate of Solute ' i ' across the epithelium  = JAS(CT)i  (26)  The f l u i d that enters the a i r space w i l l be the f l u i d that passes through the f r e e - f l u i d channels of the interstitium which are accessible to proteins and also the gel phase of the space which i s not available to the proteins.  interstitial  As a result the protein  concentration of f l u i d entering the a i r space Is assumed to be the tissue concentration (CTA,CTG).  3.6 3.6.1 (i)  Pressure-volume Relationship of the Alveolar Fluid Fluid Pressure-volume Relationship of an Individual Alveolus Normal Conditions A schematic of an alveolus and i t s surrounding  i n Figure 14b.  tissue i s shown  Lining the alveolus i s the surfactant f l u i d .  The  pressures assigned to this schematic are the a i r pressure (PG), alveolar f l u i d pressure pressure  (PPMV).  interface i s r. equation  the  (PL), and the i n t e r s t i t i a l hydrostatic  The average radius of curvature of the a i r - l i q u i d As mentioned i n section 1.7.2, Laplace's  (4), has been applied to the alveolus.  The  equation,  pressure  -74-  differential  (PG-PL) i s equal  surfactant).  to 2T/T ( T = s u r f a c e t e n s i o n of the  To ensure a s t a b l e a l v e o l u s , L a p l a c e ' 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.  f l u i d s on both s i d e s of the membrane reach conditions  then the h y d r o s t a t i c p r e s s u r e s  membrane should conditions  be e q u a l ,  e s s e n t i a l l y zero. is  of f l u i d  across  on each s i d e on the  Under normal  the e p i t h e l i u m  should be  A schematic of the a l v e o l u s under normal  conditions  shown i n F i g u r e 17.  (ii)  Fluid F i l l i n g The  filling  phases (17,39). elevated  of the A l v e o l u s of an a l v e o l u s w i t h  First,  slightly  fluid  i f the i n t e r s t i t i a l  t o the a i r space d e v e l o p s . the a l v e o l u s . pressure,  Fluid will  pressure.  ( F i g u r e 17).  pressure  To m a i n t a i n  a l v e o l u s ) the e x t r a f l u i d curvature  gradient  equation  i s diluted,  will  from the i n t e r s t i t i u m  flow a l o n g  will  this gradient  at t h i s new  into  interstitial  have r i s e n to equal the  Laplace's  equation  (and a s t a b l e  i n the a l v e o l u s must i n c r e a s e the r a d i u s of The a f f e c t  of the f l u i d  the s u r f a c e t e n s i o n at the a i r - l i q u i d  surfactant  three  h y d r o s t a t i c pressure i s  I f a steady s t a t e i s achieved  the a l v e o l a r f l u i d  interstitial  may be d i v i d e d i n t o  (because of a p e r t u r b a t i o n t o the m i c r o v a s c u l a r  exchange system) a h y d r o s t a t i c p r e s s u r e  on  e q u i l i b r i u m under normal  i . e . , PL = PPMV ( 4 5 ) .  the net exchange of f l u i d  I f the  the a l v e o l u s  interface i s unclear.  the s u r f a c e t e n s i o n w i l l  be maintained by a g r e a t e r  entering  rise  rise;  Laplace's  i n the r a d i u s of  I f the  -75-  Figure 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 A c u t e P u l m o n a r y Edema. P h a s e 1: r a d i u s o f c u r v a t u r e ( r ) i n c r e a s e s during i n i t i a l phase of f i l l i n g . P h a s e 2: r a d i u s o f 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 filling. P h a s e 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 p h a s e 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)Phase 3  x  -76-  curvature.  Tierney and Johnson (46) have performed experiments i n  which plasma f l u i d was  slowly introduced under a surfactant l i n i n g ;  the surface tension at the a i r - l i q u i d value.  Nonetheless,  interface remained at i t s normal  i n this f i r s t phase the alveolar f l u i d  pressure  rises with alveolar f l u i d volume. Guyton and co-workers (45,47) have suggested phase of f i l l i n g may  that this  continue up to a s l i g h t l y positive  hydrostatic pressure (+1 or +2 mm  first  interstitial  Hg).  Secondly, i f the i n t e r s t i t i a l hydrostatic pressure increases beyond this s l i g h t l y positive i n t e r s t i t i a l pressure the alveolus w i l l become unstable and f l u i d w i l l rush into the a i r sac. second phase the radius of curvature, r, decreases  During  this  (Figure 17).  Laplace's equation (4) i l l u s t r a t e s that i f r i s decreasing then (PG-PL) must increase; assuming that PG remains constant, PL must decrease.  The t r a n s e p i t h e l i a l f l u i d pressure gradient that i s formed  causes further f l u i d flow into the alveolus. the a i r - l i q u i d  The surface tension at  interface i s assumed to r i s e to that of water and  remain constant (17).  Therefore, i n this second phase the alveolar  f l u i d pressure i s inversely proportional to the alveolar f l u i d volume. Thirdly, the radius of curvature of the a i r - l i q u i d reaches a second c r i t i c a l value whereafter  interface  further increase i n  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 f l u i d  -77-  p r e s s u r e PL w i l l pressure r i s e s ,  3.6.2  In t h i s t h i r d phase the a l v e o l a r  once a g a i n , with the a l v e o l a r f l u i d  fluid  volume.  Fluid Pressure-volume Relationship for the Air Space Compartment In  the a l v e o l a r model a f l u i d  be developed fluid  rise.  f o r the a l v e o l a r f l u i d  e n t e r s the a i r space  compartment.  As a f i r s t  pressure-volume  relationship  will  of the a i r space  compartment.  As  a p r e s s u r e head develops  approximation  f o r 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 p r e s s u r e as a f u n c t i o n equation w i l l  take the f o l l o w i n g  i n the  of volume i s assumed.  form:  PAS = SL VAS + B  where  The  (27)  PAS = a l v e o l a r f l u i d VAS = f l u i d  hydrostatic  pressure  volume of a i r space  SL  = s l o p e of PAS-VAS  B  = PAS at the onset of a l v e o l a r  p r e s s u r e PAS w i l l  The  curve flooding  be the p r e s s u r e r e s i s t i n g f l u i d  transport  into  the a i r space. P r e c e d i n g the onset of a l v e o l a r f l o o d i n g t r a n s e p i t h e l i a l flow i s assumed to be equal to z e r o .  The v a l u e of the a l v e o l a r  p r e s s u r e , PAS, i s not needed d u r i n g the p r e - a l v e o l a r  edema phase.  the onset of a l v e o l a r  f l o o d i n g the v a l u e of the a l v e o l a r  pressure i s required;  i twill  been documented. investigated  be g i v e n by B.  fluid At  fluid  No e s t i m a t e s of B have  As a r e s u l t , a range of e s t i m a t e s of B w i l l be  i n this thesis.  F o r an i n i t i a l  e s t i m a t e B s h a l l be taken  -78-  as the value of the a l v e o l a r f l u i d B will normal  be e q u i v a l e n t to the i n t e r s t i t i a l  of SL have been documented.  .25 x 10"  2  mm  An  initial  As  In t h i s t h e s i s  of the parameter SL on the m i c r o v a s c u l a r exchange system  a l s o be i n v e s t i g a t e d .  the will  e s t i m a t e of SL s h a l l be a v a l u e of  Hg/ml.  Representation of the Epithelial Filtration Coefficient. KAS The  epithelial filtration  conductivity  c o e f f i c i e n t r e p r e s e n t s the  of the a l v e o l a r b a r r i e r .  Preceding  flooding  the parameter i s set equal to z e r o .  alveolar  flooding  the v a l u e of KAS  other v a r i a b l e , or KAS  may  may  the onset  will  be  fluid of  alveolar  F o l l o w i n g the onset  be a f u n c t i o n  assume a constant v a l u e .  p r o p o s a l s f o r the value of KAS  3.7.1  p r e s s u r e at  s l o p e of the PAS-VAS r e l a t i o n s h i p i s the parameter SL.  w i t h B, no e s t i m a t e s  3.7  hydrostatic  conditions. The  effect  p r e s s u r e at normal c o n d i t i o n s , i . e .  of  of time or some In t h i s t h e s i s  two  investigated.  Representation of KAS as a Variable: KAS  = NK(VISl-VTONS) f o r VIS1  In t h i s circumstance proportional the onset  the e p i t h e l i a l  to the EVEA f l u i d  parameter:  The  (28)  filtration  volume, VIS1,  of a l e v o l a r f l o o d i n g VTONS.  considered a s e n s i t i v i t y  > VTONS  coefficient is  l e s s the f l u i d  parameter NK may  as NK r i s e s ,  the  volume at be  epithelial  -79-  filtration The f l u i d effect  coefficient will  be more s e n s i t i v e to a g i v e n (VIS1-VT0NS).  volume d i f f e r e n c e (VIS1-VT0NS) g i v e s an i n d i c a t i o n of the  of the expansion  above i t s f l u i d  of the e x t r a - v a s c u l a r - e x t r a a l v e o l a r f l u i d  volume at the onset  space  of a l v e o l a r f l o o d i n g , on KAS.  A range of magnitude of NK w i l l  be examined i n t h i s  thesis.  3.7.2 Representation of KAS as a Constant: KAS = KASO f o r VTOT > VTONS  Preceding  alveolar  (29)  f l o o d i n g the t o t a l  extravascular f l u i d  (VTOT) i s e q u i v a l e n t to the e x t r a v a s c u l a r - e x t r a - a l v e o l a r f l u i d (VIS1).  When VTOT reaches  filtration The f l u i d  coefficient  the f l u i d  fluid  volume  volume VTONS the e p i t h e l i a l  rises instantly  to a constant v a l u e of KASO.  volume VTOT i s then composed of VIS1 and the a l v e o l a r  volume, VAS; p r o v i d e d VTOT exceeds VTONS KAS w i l l KASO.  The EVEA f l u i d  volume  fluid  remain a t a value of  volume VIS1 may become g r e a t e r or l e s s than the  volume VTONS. In t h i s  response  study  s i m u l a t i o n s were conducted  t o determine the  of the pulmonary m i c r o v a s c u l a r exchange system t o d i f f e r e n t  v a l u e s 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 m a t e r i a l  i n t e r s t i t i u m involved c i r c u l a t i o n and The  the  epithelium  fluid  material  and  and  solute  from  to the  the  the  s o l u t e through the  a i r space compartment  difficulties.  cross  but  i n f l u x of f l u i d  an e f f l u x of f l u i d  a d d i t i o n of the  p r e s e n t s few  an  balances f o r  lymphatics.  interstitial  During a l v e o l a r f l o o d i n g , f l u i d  from the  t i s s u e space i n t o the  balance denoted by  e q u a t i o n (24)  model  and  a i r space.  is s t i l l  solute The  applicable,  i n t h i s case:  Rate of F l u i d Flow out of the I n t e r s t i t i u m  Similarly,  the  =  JL + JAS  flow r a t e of s o l u t e  Flow r a t e  ' i  1  out  (30)  of the  interstitium i s :  of  s o l u t e ' I * out = J L ( C T ) i + JAS(CT)i of the I n t e r s t i t i u m  The  material  interstitium, Fluid Material  balances of the a l v e o l a r model, around  (31)  the  are: Balance  JNET1 = JV - JL - JAS  (32)  M a t e r i a l Balance f o r S o l u t e ' i '  (QNET)i = ( J S ) i - J L ( C T ) i - J A S ( C T ) i  where  I = albumin or  globulins.  (33)  -81-  Figure 18 I l l u s t r a t e s the material balances of the alveolar model.  The  material balances shown i n equations (32) and (33) are applicable before and after the onset of alveolar flooding.  Preceeding the onset  of alveolar flooding there i s no t r a n s e p i t h e l i a l f l u i d or solute flow, since KAS, and therefore JAS, i n equation (25) are equal to zero. The time integrals of JNET1, QNETA, QNETG, and JAS y i e l d the variables VIS1, QA, QG, and VAS, respectively, which are needed i n the evaluation of the f l u i d and solute flows of the material balances shown i n equations (32) and (33). JV, JL and JAS of the f l u i d material balance are evaluated by equations (18), (21) and (25), while ( J S ) i of the solute material balance i s determined by equation (20). The solute concentrations CTA and CTG are evaluated by equations (13) and (14).  -82-  Figure  18a: Schematic o f 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  Air  Space  Interstitium Circulation  JV  Accumulation  J NET 1  -> Lymphatics JL  F i g u r e 18b  :  Schematic o f 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 u t e Flows  Air  Space  Interstitium C i r c u l c t ion  US);  Accumulation  OAS-(CT). (Q N E T ) .  ->  Lymphatics JL-(CT);  -83-  COMPUTER SIMULATION OF ALVEOLAR MODEL  4.1  Introduction Traditionally,  the study of b i o l o g i c a l models i n v o l v e d the use  of human and/or animal experiments. have p r o v i d e d an a d d i t i o n a l method Due  R e c e n t l y , computer s i m u l a t i o n s  f o r t e s t i n g the b i o l o g i c a l models.  to the c o m p l e x i t i e s t h a t a r i s e from the i n t 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 n e c e s s a r y when d e v e l o p i n g a computer model.  In s p i t e 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 h e a l t h care p r o f e s s i o n a l s w i t h an estimate of the progress of a p a t h o p h y s i o l o g i c a l s t a t e and the e f f e c t s of c l i n i c a l management, and 2) pulmonary r e s e a r c h e r s w i t h an a l t e r n a t e t o o l w i t h which to t e s t  t h e i r experimental  findings.  The model  may  p r o v i d e s u g g e s t i o n s on the e s s e n t i a l v a r i a b l e s and parameters t h a t should be measured  to v a l i d a t e the model.  B e r t and P i n d e r (29) f i r s t an AD-80 analog computer.  developed  This i n i t i a l  the i n t e r s t i t i a l model on  step i n the m o d e l l i n g  a c l e a r 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  variables.  Thus, the analog computer was a u s e f u l t o o l w h i l e d e v e l o p i n g and  i t s solution.  allowed  the model  However, due to problems i n s c a l i n g , a m p l i f e r  i n s t a b i l i t y , and the l i m i t e d i n t e r s t i t i a l model was  c a p a c i t y of the AD-80 analog computer the  translated  i n t o d i g i t a l form, u s i n g FORTRAN.  The s o l u t i o n f o r the a l v e o l a r model was AD-80 a n a l o g computer.  also i n i t i a t e d  on the  However, due to the l i m i t e d c a p a c i t y of the  -84-  analog computer, the solution to the Alveolar Model was completed on a d i g i t a l computer using FORTRAN.  The d i g i t a l computer system employed  for this task was the AMDAHL 4TOV-12 of the University of B r i t i s h Columbia.  A benefit of using the university f a c i l i t i e s was the access  to a variety of l i b r a r y  subroutines.  The following sub-sections describe the computer program developed  f o r 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 F i l e s EDA and PDA The parameters l i s t e d i n the two data f i l e s are defined i n  Tables 5 and 6.  F i l e EDA ( l i s t e d i n 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 c e l l u l a r volume (VCELL).  fluid  F i l e PDA ( l i s t e d i n 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  I n i t i a l v a l u e of t o t a l e x t r a v a s c u l a r f l u i d volume (ml)  VASO  I n i t i a l v a l u e of a l v e o l a r (ml)  QAO  I n i t i a l v a l u e of i n t e r s t i t i a l weight (g)  albumin  QGO  I n i t i a l v a l u e of i n t e r s t i t i a l weight (g)  globulin  PIMV  Microvascular c o l l o i d (mmHg)  pressure  VCELL  Cellular fluid  VEXA  Excluded  volume f o r albumins (ml)  VEXG  Excluded  volume f o r g l o b u l i n s (ml) (EVEA)  VTONS  E x t r a v a s c u l a r e x t r a a l v e o l a r (EVEA) f l u i d volume at the onset of a l v e o l a r f l o o d i n g (ml)  VLMPH  EVEA f l u i d volume that corresponds maximum lymph flow (ml)  B  A l v e o l a r f l u i d p r e s s u r e at the onset o f a l v e o l a r f l o o d i n g (mmHg)  CMVA  M i c r o v a s c u l a r albumin  CMVG  Microvascular globulin concentration (g/ml)  fluid  osmotic  volume  volume (a c o n s t a n t ) ( m l )  to the  c o n c e n t r a t i o n (g/ml)  FPPMV(l)-  FPPMV(9)  I n t e r p o l a t i o n values f o r t i s s u e curve (mmHg)  compliance  Continued....  -86-  Table 5:  Content of Input F i l e EDA ( C o n t i n u e d )  KF  : Endothelial fluid (ml/hr/mmHg)  filtration  coefficient  SIGD  : E n d o t h e l i a l osmotic r e f l e c t i o n f o r f l u i d flow  PSA  : P e r m e a b i l i t y times s u r f a c e area "of v a s c u l a r membrane f o r albumins (ml/hr)  PSG  : P e r m e a b i l i t y times s u r f a c e area of v a s c u l a r membrane f o r g l o b u l i n s (ml/hr)  SIGFA  : Endothelial albumins  reflection  coefficient for  SIGFG  : Endothelial reflection globulins  coefficient for  coefficient  -87-  Table 6 :  V a r i a b l e s i n Input F i l e  PDA  PLOTS (Y=1,N=2)  Calls plot s e t s NN  TABLES 1 (Y=1,N=2)  C a l l s t a b l e r o u t i n e (Yes or No) s e t s NNN  TAUMX1  Maximum time f o r output i n t e r v a l SUBNT1 ( h r )  SUBNT1  Time i n t e r v a l f o r p r i n t i n g output up to TAUMX1. Must d i v i d e evenly i n t o TAUMX1. (hr)  TAUMX2  Maximum time  SUBNT2  Time i n t e r v a l f o r p r i n t i n g output between TAUMX1 and TAUMX2. Must d i v i d e evenly i n t o TAUMX1 and TAUMX2. ( h r )  STEPSZ  I n i t i a l s t e p s i z e 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 s u b r o u t i n e "DRKC" ( h r )  HMIN  Minimum v a l u e of s t e p s i z e to be used i n s u b r o u t i n e "DRKC" ( h r )  TOL  Maximum t o l e r a n c e l e v e l allowed i n c a l c u l a t i o n s of s u b r o u t i n e "DRKC"  KASO  Constant v a l u e of e p i t h e l i a l c o e f f i c i e n t (ml/hr/mmHg)  NK  S e n s i t i v i t y parameter f o r v a r i a b l e epithelial filtration coefficient (hr/mmhg)  SL  Slope of a l v e o l a r curve (mmHg/ml)  PMV  C i r c u l a t o r y h y d r o s t a t i c p r e s s u r e (mmHg)  XS(1),XS(2)  X c o o r d i n a t e s of legend curve r e p r e s e n t a t i o n s ( l e f t and r i g h t v a l u e s ) f o r plots (inches)  r o u t i n e (Yes or No)  of p r i n t i n g  for calculations (hr)  fluid  filtration  pressure-volume  Continued....  -88-  Table  6:  V a r i a b l e s i n Input F i l e PDA  (Continued)  YS(1)-YS(4)  Y c o o r d i n a t e s of legend curve representations f o r plots (inches)  SFT  Value of increments on a b s c i s s a (time)(hr/inch)  SFX  Value of Increments on o r d i n a t e f o r f l u i d flow ( ( m l / h r ) / i n c h )  SFY  Increments on o r d i n a t e f o r f l u i d (ml/inch)  ZMN  Minimum v a l u e of h y d r o s t a t i c and c o l l o i d osmotic p r e s s u r e f o r o r d i n a t e (mmHg)  SFZ  Increments on o r d i n a t e f o r h y d r o s t a t i c and c o l l o i d osmotic p r e s s u r e (mmHg/inch)  SFK  Increments on o r d i n a t e f o r e p i t h e l i a l f i l t r a t i o n coefficient ((ml/hr/mmHg)/inch)  SFR  Increments on o r d i n a t e f o r s o l u t e weights (g/inch)  SFU  Increments on o r d i n a t e f o r s o l u t e concentrations ((g/ml)/inch)  volume  -89-  whether p l o t s and/or t a b l e s are d e s i r e d . through  to TOL apply  intervals  to the time  for printing.  The parameters from TAUMX1  i n t e g r a t i o n steps and the time  The parameters from XS(1) through  • p e r t a i n to the p l o t t i n g o f the t r a n s i e n t Preceding  to SFU  responses.  each s i m u l a t i o n i t w i l l g e n e r a l l y be necessary t o  adjust  the parameters i n f i l e s EDA and PDA to the d e s i r e d v a l u e s .  4.2.2  The Main Program UBCEDEMA The  the f i r s t  main program may a l s o be d i v i d e d i n t o t h r e e s e c t i o n s . s e c t i o n the parameter v a l u e s l i s t e d  read i n t o the computer. counters  In  i n T a b l e s 5 and 6 a r e  In a d d i t i o n , a number of v a r i a b l e s and  are i n i t i a l i z e d .  The  second s e c t i o n of UBCEDEMA i n v o l v e s the s o l u t i o n of the  equations  over  the time  intervals  equations  s o l v e d are l i s t e d  specified  i n Table 7.  in file  The stepout  PDA.  The  i n time  f o r the  i n t e g r a t i o n s i s made u s i n g the Runge-Kutta-Merson technique as packaged by the UBC Computing Centre; through  the s u b r o u t i n e  "DRKC".  i n t e r v a l must be s p e c i f i e d  the technique  The b e g i n n i n g  f o r the s u b r o u t i n e .  i s provided  and end values of a time DRKC uses an  e x t e r n a l l y d e c l a r e d s u b r o u t i n e FUNC to c a l c u l a t e the v a l u e s of the d i f f e r e n t i a l s JNET1, JAS, QNETA and QNETG; FUNC c o n t a i n s the equations listed  i n Table 7.  subroutine and  A f t e r i n t e g r a t i o n over  the time  interval  "DRKC" p r o v i d e s the v a l u e s of the i n t e g r a l s VIS1, VAS, QA  QG (whose d i f f e r e n t i a l s  respectively).  are JNET1, JAS, QNETA and QNETG  The s e r i e s of equations  v a l u e s of the i n t e g r a l s  to y i e l d  a r e then s o l v e d w i t h the new  the d e s i r e d v a r i a b l e s ; when a  -90-  Table 7:  E q u a t i o n s used i n the A l v e o l a r 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)  J L = 0.17 VIS1-55.6  4)  JAS = KAS(PPMV-PAS)  5)  JNET1 = JV-JL-JAS  6)  VIS1 =J  7)  VAS = /  8)  a) QNETA =JSA-JL CTA-JAS CTA b) QNETG = JSG-JL CTG-JAS CTG  9)  a) QA = /  t  QNETA 6t + QAO  b) QG = /  t  QNETG 6t + QGO  JNET1 6t + VIS10  t  t  0  JAS 6t + VASO  Q  Q  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 C P  2  + 9000 C P  3  Continued....  Table 7:  Equations used i n the A l v e o l a r Model (Continued)  16) PPMV = f n ( V I S l )  (by i n t e r p o l a t i o n )  17) PAS = SL VAS + B 18) a) KAS = 0 f o r VIS1 < VTONS KAS = NK(VISl-VTONS) f o r VIS1 > VTONS or b)  KAS = 0 f o r VTOT < VTONS KAS = KASO f o r VTOT > VTONS  19) CP = CAVA + CAVG  -92-  printing for  i n t e r v a l i s reached the v a r i a b l e s  are s t o r e d  use i n the t a b l e and p l o t t i n g s e c t i o n s .  third  form  When TAUMX2, the maximum  time f o r c a l c u l a t i o n s i s reached, t a b u l a t i o n the  In a r r a y  begins as d e s c r i b e d i n  section.  4.2.3 Tabulated and graphical output of variables from the computer program T a b l e 8 l i s t s the v a r i a b l e s Illustrated. The  The t r a n s i e n t  conditions  tables;  listed  d a t a f i l e s EDA and PDA. c h a r a c t e r i s t i c points  are t a b u l a t e d  are l i s t e d  consists  c h a r a c t e r i s t i c points  are  shown i n T a b l e 9.  graphically  i s recorded.  b e f o r e the output of the  of most of the v a r i a b l e s  i n the  The v a l u e of v a r i a b l e s a t s e v e r a l  along the t r a n s i e n t  The  and/or  response of these v a r i a b l e s  of the s i m u l a t i o n  the c o n d i t i o n s  that  response were a l s o  and the v a r i a b l e s  recorded.  recorded at these  points  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  responses o f the PMVES s i m u l a t i o n  4.3.1  chosen to r e p r e s e n t the t r a n s i e n t are l i s t e d  i n Table 9.  The onset of alvaeolar flooding This  first  c h a r a c t e r i s t i c point  t r a n s i t i o n i n a simulation  r e p r e s e n t s the p o i n t of  from i n t e r s t i t i a l edema to a l v e o l a r edema.  At  the onset o f a l v e o l a r f l o o d i n g  1)  the time to reach the onset of a l v e o l a r  2)  the r a t i o of the t i s s u e albumin c o n c e n t r a t i o n  concentration  - CTA/CMVA.  the v a r i a b l e s  recorded w i l l be:  flooding  ( t ( o n s e t ) ) , and  to the plasma albumin  Table 8 :  Variables  T a b u l a t e d and/or P l o t t e d 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  • •  •  • •  -94-  Table 9 :  C h a r a c t e r i s t i c P o i n t s along T r a n s i e n t Response and V a r i a b l e s Recorded  Characteristic  Point  1)  Onset of a l v e o l a r  flooding  2)  Maximum Flow  3)  Total Extravascular Volume of 1000 ml.  Transepithelial  Fluid  Variables  Recorded  time CTA/CMVA JAS time VIS1 VTOT CTA/CMVA JAS time VIS1 CTA/CMVA  -95-  4.3.2  The point of maximum transepithelial flow In the s i m u l a t i o n s  transient value.  response of the t r a n s e p i t h e l i a l flow  The v a r i a b l e s recorded  1) the maximum v a l u e to  of the PMVES d i s c u s s e d  i n t h i s t h e s i s the  (JAS)  has a maximum  at t h i s c h a r a c t e r i s t i c point are:  o f the t r a n s e p t h e l i a l flow JAS(max),  reach a maximum JAS,  2) the time  3) the EVEA and t o t a l e x t r a v a s c u l a r  fluid  volumes, VIS1 and VTOT, r e s p e c t i v e l y , and 4) the r a t i o of the t i s s u e albumin c o n c e n t r a t i o n t o 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  where human lungs were s u b j e c t e d extravascular f l u i d approximately  t o acute  edema the t o t a l  volume g e n e r a l l y reached approximately  one-half  cases  1100 ml.,  of the f u n c t i o n a l r e s i d u a l c a p a c i t y of the  lung.  The p a t i e n t s i n v e s t i g a t e d d i e d under i n t e n s i v e care  severe  i n j u r y i n which i t was thought that r e s p i r a t o r y f a i l u r e was the  major c o n t r i b u t i n g f a c t o r . ( 4 8 )  In t h i s  thesis a characteristic  was  e s t a b l i s h e d below the t o t a l e x t r a v a s c u l a r f l u i d  ml,  a t a VTOT o f 1000 ml.  c h a r a c t e r i s t i c point are: 2)  after  The v a r i a b l e s recorded  volume of 1100  at this  1) the time to reach a VTOT of 1000 ml,  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 t i s s u e albumin c o n c e n t r a t i o n plasma albumin c o n c e n t r a t i o n , The  point  to t h e  CTA/CMVA(1000).  behaviour of the exchange system to a p e r t u r b a t i o n may be  -96-  analyzed  by comparing the v a r i a b l e s between these  points.  The change i n the r a t e of a l v e o l a r f l o o d i n g over  of a l v e o l a r edema may be i l l u s t r a t e d flows JAS(max) and JAS(IOOO).  characteristic the course  by comparing the t r a n s e p i t h e l i a l  The r a p i d i t y w i t h which the f l u i d  accumulates i n the e x t r a v a s c u l a r space can be shown by comparing the time  to reach  the onset  VTOT of 1000 m l .  of a l v e o l a r f l o o d i n g and the time  t o reach a  Comparison of the EVEA f l u i d volume, VIS1, at  JAS(max) and at a VTOT of 1000 ml i n d i c a t e s the amount of I n t e r s t i t i a l edema t h a t formed In excess  of the EVEA f l u i d  a l v e o l a r f l o o d i n g g i v e n by VTONS. two  c h a r a c t e r i s t i c points yields  volume at the onset of  The d i f f e r e n c e (VT0T-VIS1) at these the f l u i d  volume of the a i r space.  A l s o , comparison of the albumin c o n c e n t r a t i o n r a t i o s a t 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 t r a n s i e n t  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  developed KAS,  a d d i t i o n of the a i r space to the lumped compartment model by B e r t and P i n d e r  SL, VTONS and B.  represented  as:  (29) i n t r o d u c e d s e v e r a l new parameters:  The e p i t h e l i a l  filtration  1) KAS=KAS0, i f VTOT exceeded VTONS, or  KAS=NK(VIS1-VT0NS), i f VIS1 exceeded VTONS. in  c o e f f i c i e n t , KAS, was  I f VTOT i n (1) and VIS1  (2) were l e s s than VTONS, KAS was s e t equal to z e r o .  the e x p r e s s i o n s  2)  Therefore,  f o r KAS i n t r o d u c e d the parameters KASO ( f o r KAS as a  -97-  constant  v a l u e ) and NK ( f o r KAS as a v a r i a b l e ) .  Simulations  were performed t o study  the e f f e c t  parameters on the responses of the m i c r o v a s c u l a r exchange system was s u b j e c t e d  o f these  exchange system. The  t o a p e r t u r b a t i o n t h a t i n c r e a s e d the  t r a n s e n d o t h e l i a l flow of f l u i d  and s o l u t e .  F o r the s i m u l a t i o n s  conducted f o r t h i s t h e s i s the p e r t u r b a t i o n was an e l e v a t e d h y d r o s t a t i c pressure, In Table  circulatory  u s u a l l y t o 50 mmHg.  10 a r e l i s t e d  the s i m u l a t i o n s  that were c a r r i e d o u t .  F o r example, when a s e t of s i m u l a t i o n s were conducted t o study the parameter VTONS, the range o f v a l u e s other  parameters assumed v a l u e s  hr~ mmHg" , 1  4.4.2  1  o f VTONS was 414 t o 500 ml;  the  as shown i n row 4: NK = 0.5  KASO = 0, SL = 0 . 2 5 x l 0 "  2  mmHg/ml and B = -3.03 mmHg.  Simulations to study the effect of a maximum lymph flow In the a l v e o l a r model the lymph flow r a t e i s assumed t o be  g i v e n by equation  (21), as a f u n c t i o n of the EVEA f l u i d  Observations  by Drake et a l (34) and o t h e r s ,  shown that the lymph flow reaches a maximum v a l u e : pulmonary edema induced pressure  (PMV),  (17,18,49) have 1) d u r i n g  by an e l e v a t e d c i r c u l a t o r y h y d r o s t a t i c  2) d u r i n g pulmonary edema induced  e n d o t h e l i a l permeability to f l u i d simulations w i l l  volume, VIS1.  and s o l u t e s .  be performed t o study  by changes i n t h e  In t h i s t h e s i s  the e f f e c t  of a maximum lymph  f l o w on the responses of the exchange system. A maximum lymph flow w i l l fluid  be r e p r e s e n t e d  volume, d e f i n e d as VLMPH, w i l l  correspond  as JL(max).  An EVEA  t o JL(max).  The p o i n t  -98-  Table  10:  S i m u l a t i o n s conducted to study the e f f e c t i n t r o d u c e d w i t h the a l v e o l a r model*  Parameter value Varied Parameter  NK  of parameters  KAS NK KASO ( h r ^ m m H g ) (ml/hr/mmHg) -1  0.05 0.5 5.0 50.0  0  SL (mmHg/ml)  .25xl0-  2  VTONS (ml) -  B (mmHg)  460  -3.03  460  -3.03  460  -3.03  KAS 0 KASO  1.0 2.5 5.0 10.0  .25x10-2  .25x20" .25xlO.25x10-2 -75x10-2 l+  SL  3  0.5 50.0  0  VTONS  0.5  0  .25x10-2  414 to 500  -3.03  8  0.05  0  .25xl0-  460  -10.0 to 1.0  * Parameters from Table PMV = KF = SIGD = PSA = PSG = SIGFG = SIGFA =  2  11 a r e 50 mmHg KF(normal) = 1.12 ml/hr/mmHg .75 3.0 ml/hr 1.0 ml/hr .40 .60  -99-  at  which the lymph flow becomes a maximum may be s p e c i f i e d i n  r e f e r e n c e to the onset the f l u i d  of a l v e o l a r f l o o d i n g (17,34);  volume a t the onset  JL(max) by VLMPH w i l l  permit  s i n c e VTONS i s  of a l v e o l a r f l o o d i n g , representing easy  comparison between these two  points. The  v a l u e o f the maximum lymph flow, JL(max), o r i t s  corresponding al  fluid  volume, VLMPH, i s not c l e a r l y d e f i n e d .  (49) performed lymph flow experiments on dog lungs  results  Drake e t  i n vivo.  Their  showed t h a t maximum lymph flow o c c u r r e d at a PMV between 20  and  25 mmHg.  The c o r r e s p o n d i n g  414  ml. o r 90% o f VTONS.  f l o o d i n g occurs  volume f o r a PMV o f 25 mmHg i s  P r i c h a r d (17) suggested  s h o r t l y a f t e r the attainment  i.e.,  VLMPH < VTONS.  study  i s from approximately  extremely  fluid  high value  maximum lymph f l o w .  that a l v e o l a r  of a maximum lymph  flow,  The range of v a l u e s of VLMPH c o n s i d e r e d i n t h i s 414 ml. (JL(max)=14.8 ml/hr) t o an  ( e . g . 5000 ml) which i n d i c a t e s t h a t there i s no F o r VLMPH equal t o the f l u i d  volume VTONS(460 ml)  the v a l u e of JL(max) i s 22.6 ml/hr.  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. In the i n t e r s t i t i a l  the e f f e c t  model B e r t and P i n d e r  (1984) i n v e s t i g a t e d  on the pulmonary m i c r o c a s c u l a r exchange system of changing  the e n d o t h e l i a l p e r m e a b i l i t y parameters-SIGD, PSA, PSG, SIGFA and SIGFG - the e n d o 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 , KF, and the  c i r c u l a t o r y h y d r o s t a t i c p r e s s u r e , PMV.  One of the o b j e c t i v e s of the  -100-  c u r r e n t work i s to observe the e f f e c t  of the same parameters  on the  p r e d i c t i o n s of the PMVES s i m u l a t i o n u s i n g the a l v e o l a r model. Table 11 l i s t s simulations  the e p i t h e l i a l f i l t r a t i o n  as a f u n c t i o n of VIS1 pressure  the s i m u l a t i o n s performed.  (PMV)  was  (equation  (28)).  In a l l t h r e e s e t s of  c o e f f i c i e n t , KAS,  i s represented  The c i r c u l a t o r y h y d r o s t a t i c  r a i s e d to 50 mmHg i n the s i m u l a t i o n s i n v e s t i g a t i n g  KF, and the p e r m e a b i l i t y  parameters.  -101-  Table 11: S i m u l a t i o n s conducted t o study the e f f e c t of t h e e n d o t h e l i a l p e r m e a b i l i t y 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 h y d r o s t a t i c p r e s s u r e on the a l v e o l a r model*  Parameter value Varied Parameter  PMV  KF KF(normal)  APERM  PMV (mmHg)  KF/KF(norraal)  1  35 to 60  1 t o 10  50  50  1  APERM (%) *  0  0  0 t o +100  * The p e r m e a b i l i t y parameters a r e changed by a s p e c i f i c p e r c e n t (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 i n c r e a s e . * Parameters  from T a b l e 10 were s e t a t : NK = 0.5 h r mmHg SL = . 2 5 x l 0 ~ mmHg/ml VTONS = 460 ml B = -3.03 mmHg -  2  1  -1  -102-  RESDLTS AND DISCUSSION  5.1 Transient Responses of the Pulmonary Microvascular Exchange System f o r 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 , KAS The  effect  of changes i n KAS ( d e f i n e d by e q u a t i o n  p r e d i c t i o n s of the PMVES simulation.was listed  i n Table  filtration  12.  s t u d i e d under the c o n d i t i o n s  In the f o l l o w i n g s e c t i o n (5.2) the e p i t h e l i a l  c o e f f i c i e n t was r e p r e s e n t e d by e q u a t i o n  of VIS1, which r e s u l t e d ml/hr/mm Hg. filtration  coefficient  (28) as a f u n c t i o n  i n v a l u e s f o r KAS i n the range of 0 to 10  T h e r e f o r e , f o r the case of a constant  10.0 ml/hr/mm Hg.  (29)) on the  epithelial  the v a l u e s of KAS s t u d i e d ranged from 1.0 to  At time  zero the PMVES was s u b j e c t e d to a step  change i n PMV from a normal v a l u e of 9.0 mm Hg. to 50 mm Hg. The conducted  time  f o r the onset  of a l v e o l a r f l o o d i n g i n the s i m u l a t i o n s  f o r a constant KAS was 2 h r s . P r e c e d i n g  a l v e o l a r f l o o d i n g the f l u i d volume, VIS1, r i s e s  the onset of  to a v a l u e of VTONS  (460 m l ) , as shown i n F i g u r e 19.  The t r a n s e p i t h e l i a l flow i s  determined  and (PPMV-PAS).  as the product  of KAS  At the onset of  a l v e o l a r f l o o d i n g the p r e s s u r e d i f f e r e n c e (PPMV-PAS) w i l l be s i m i l a r f o r a l l KAS.  Therefore,  the v a l u e of JAS depends on the v a l u e of KAS;  F i g u r e 20 shows that as KAS i n c r e a s e s , JAS a t the onset flooding  increases.  responses  T h i s v a l u e of JAS w i l l  of a l v e o l a r  i n f l u e n c e the subsequent  of v a r i a b l e s - such as PAS or VIS1 - p r e d i c t e d by the PMVES  simulation. The  alveolar  of VAS by e q u a t i o n gives:  fluid (27).  p r e s s u r e , PAS, i s determined The d i f f e r e n t i a l  as a f u n c t i o n  of PAS w i t h r e s p e c t to time  -103-  Table  12  C o n d i t i o n s of PMVES S i m u l a t i o n s Conducted t o Study Changes i n KAS*  KAS = KASO f o r VTOT > VTONS KASO = 1.0, 2.5, 5.0 and 10.0 ml/hr/mm Hg PMV KF APERM SIGD PSA PSG SIGFA SIGFG VTONS SL B **VLMPH  = = = = = = = = = = = =  5  0 1.12 0 0.75 3.0 1.0 0.4 0.6 460. 0.25x10" -3.03 5000.  mm Hg ml/hr/mm Hg  ml/hr ml/hr  2  ml mm Hg/ml mm Hg ml  *These are the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 19,20,21. * * I f VLMPH = 5000 ml, JL(max) = N.A.  then there i s no maximum lumph flow, i . e .  -104-  F i g u r e 19  T r a n s i e n t Responses o f VIS1 f o r Constant KAS ( C o n d i t i o n s as i n Table 12) - Response c o n t i n u e d to time when VTOT = 1000 ml.  550  £  500  CO  450 O >  Legend  400-  KAS=  to  KAS=  2.5  KAS=  5.0  KAS = 10.0  350- I 0  i  20  40  60  TIME (hrs)  80  100  120  -105-  Figure  20  T r a n s i e n t Responses o f JAS f o r Constant KAS ( C o n d i t i o n s as i n Table 12) - Response c o n t i n u e d to time when VTOT = 1000 m l .  50  Legend KAS= to  40 H  KAS = 2.5 KAS= 5.0 KAS = 10.0  00 <  30 H  Q  20 H  OH  r  0  i  20  40  60  TIME (hrs)  80  100  120  -106-  dPAS dt for  dVAS = SL  constant SL and The  . - 3 — dt  B.  derivative  2^2dt  (34)  (dVAS/dt) i s e q u i v a l e n t to JAS,  = SL  so t h a t :  .JAS  E q u a t i o n (35) i l l u s t r a t e s  (35)  that the s l o p e of the PAS  curve from the  onset of a l v e o l a r f l o o d i n g depends on the v a l u e of JAS, F i g u r e 21 shows that as KAS  hence  i n c r e a s e s , the s l o p e of the PAS  KAS.  curve  increases. The  interstitial  f u n c t i o n of VIS1 of  VIS1  fluid  p r e s s u r e (PPMV) i s determined  through the t i s s u e compliance  i s equal to JNET1 by d e f i n i t i o n , JNET1 =  d V  ^  curve.  The  as a derivative  so t h a t :  = JV - JL - JAS  S 1  at  (36)  The v a l u e of ( d V I S l / d t ) depends on the v a l u e s of JAS, JV and J L . response of VIS1  immediately  shown i n F i g u r e 19.  the onset of a l v e o l a r  For the case where KAS  decreases a f t e r onset, as JAS JNET1 to be n e g a t i v e . results  after  flooding i s  = 10 ml/hr/mm Hg  i s g r e a t e r than (JV-JL) which  Reducing KAS  causes  following  flooding.  F o l l o w i n g the onset of a l v e o l a r PMVES s i m u l a t i o n f o r KAS  f l o o d i n g the p r e d i c t i o n s of the  = 10 ml/hr/mm Hg are as f o l l o w s :  v a r i a b l e PPMV decreases as VIS1  d e c r e a s e s ; the r i s e  d e c l i n e i n PPMV causes a r a p i d decrease i n JAS be expected  VIS1  to 5.0 ml/hr/mm Hg and below  i n p o s i t i v e v a l u e s f o r JNET 1, and a r i s e i n VIS1  the onset of a l v e o l a r  The  i n PAS  The and  ( F i g u r e 20).  the lymph flow, JL, decreases as VIS1  decreases.  the  As would  -107-  F i g u r e 21  T r a n s i e n t Responses o f PAS f o r Constant KAS ( C o n d i t i o n s as i n T a b l e 12) - Response c o n t i n u e d to time when VTOT = 1000 ml.  rj)  E  -1.5  -i  -2H  CO Ld  -2.5 H  o:  CO CO Ld CL  -3H  O  Legend  o  KAS= 10  Q  KAS= 2.5 KAS= 5.0 KAS = 10.0  -4.5 H r 0  20  40  6 0" 60  TIME (hrs)  80  —T— 100  120  -108-  R e c a l l i n g equation  ( 3 6 ) , the decrease  i n c r e a s e i n JNET1 ( o r a decrease s i m u l a t i o n proceeds JNET1 equals fluid  VIS1 w i l l  i n JAS and JL leads to an  i n the n e g a t i v i t y i n JNET1).  e v e n t u a l l y decrease  volume VIS1 reaches a minimum v a l u e .  increase again.  19,  decrease,  to a v a l u e where  zero; F i g u r e 19 shows that f o r KAS = 10 ml/hr/mm Hg the  p o s i t i v e and VIS1 i n c r e a s e s ( F i g u r e 19).  to  As the  20, and 21 the r e s u l t s  JAS decreases  becomes  PPMV and J L w i l l  However, the p r e s s u r e d i f f e r e n c e  and as a r e s u l t  VTOT of 1000 ml.  JNET1 then  also  (PPMV-PAS) c o n t i n u e s  (Figure 20).  In F i g u r e s  of the s i m u l a t i o n were r e c o r d e d u n t i l a  was a t t a i n e d ; steady  s t a t e was not a c h i e v e d a t t h i s  point. For KAS = 10 ml/hr/mm Hg the r a t e of f l u i d i n t e r s t i t i u m exceeds the r a t e of f l u i d in  entering.  l e a v i n g the The f l u i d  deficit,  t h i s case, was made up by d e p l e t i n g 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  alveolar flooding  from  i t s v a l u e a t the onset of  (Figure 21).  F o r the case of KAS equal to 5, 2.5 and 1.0 ml/hr/mm Hg, f o l l o w i n g the onset ( F i g u r e 21).  of a l v e o l a r f l o o d i n g VIS1 continued  to r i s e  -109-  5.2 Transient Responses of the Pulmonary Microvascular Exchange System for Variable Epithelial Filtration Coefficient (KAS) The  variable epithelial  filtration  c o e f f i c i e n t , KAS, i s  calculated as: KAS The  effect  = NK(VIS1 - VTONS) f o r VIS1 > VTONS  of t h i s r e p r e s e n t a t i o n of KAS on the PMVES was i n v e s t i g a t e d  under the c o n d i t i o n s g i v e n i n Table to  the c r i t e r i a  13 - which were chosen a c c o r d i n g  g i v e n i n the model d e s c r i p t i o n s .  PMVES was s u b j e c t e d pressure  At time zero the  to a step change i n the c i r c u l a t o r y h y d r o s t a t i c  from a normal value of 9 mm Hg to 50 mm Hg.  the t r a n s i e n t responses of the PMVES w i l l 22a  (28)  In t h i s s e c t i o n  be d i s c u s s e d u s i n g F i g u r e s  to 22e. During  the s i m u l a t i o n the v a r i a b l e s l i s t e d  i n c r e a s e from t h e i r steady the i n i t i a l v a l u e s The  state values  i n Table  14 w i l l  at a normal PMV of 9 mm Hg to  g i v e n f o r PMV = 50 mm Hg.  response of the PMVES proceeds through a phase o f  interstitial  edema f o l l o w e d by a l v e o l a r edema.  During  the t r a n s i e n t  response a number of v a r i a b l e s are i n t e g r a t e d over  time.  the r a t e of f l u i d  (and c e l l u l a r )  accumulation  i n the i n t e r s t i t i a l  space (JNET1) and i n the a i r space (JAS), and the r a t e of of  s o l u t e s i n the i n t e r s t i t i a l  v a r i a b l e s y i e l d VIS1, 22b  space (QNETA, QNETG).  VAS, QA and QG, r e s p e c t i v e l y .  show that d u r i n g the i n t e r s t i t i a l  These a r e :  accumulation  The i n t e g r a t e d F i g u r e s 22a and  edema phase (time zero t o 2.3  h r s ) QA, QG ( F i g u r e 22a) and VIS1 ( F i g u r e 22b) r i s e as a r e s u l t of the p o s i t i v e values  of QNETA, QNETG, and JNET1; JAS i s z e r o .  The r i s e i n  -110-  Table 13  C o n d i t i o n s of the PMVES S i m u l a t i o n s f o r a V a r i a b l e Epithelial Filtration Coefficient*  KAS = NK(VIS1-VTONS) f o r VIS1 > VTONS NK = 0.5 h r mmHg -  PMV KF APERM SIGD PSA PSG SIGFA SIGFG VTONS SL B VLMPH  = = = = = = = = = = = =  1  -1  50 1.12 0 0.75 3.0 1.0 0.40 0.60 460. 0.25x10" -3.03 5000.  mmHg ml/hr/mmHg  ml/hr ml/hr  2  ml mmHg/ml mmHg ml  *These are the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 22a-22e.  Table 14  I n i t i a l Conditions of PMVES Simulation at a PMV of 50 mmHg Variable  JV JNET1 QNETA QNETG  (ml/hr) (ml/hr) (g/hr) (g/hr)  Steady State Value at a PMV=9 mmHg 8.82 0. 0. 0.  I n i t i a l Value at a PMV=50 mmHg 54.7. 46.0 0.66 0.70  -112-  F i g u r e 22a  T r a n s i e n t Responses o f QA, QG, QNETA and QNETG f o r V a r i a b l e KAS w i t h NK = 0.5 hr mmHg ( C o n d i t i o n s as i n Table 13) 1  1  1.5  8  / — Q A  v2  o_ i  ' V—  -6  QNETA  O 0.5  UJ  Li_  \  Ld  1—QNETG  UJ  \  I— \ 1  O  o  i  tn  •0.5 0  (71  T  5  10  15  TIME (hrs)  20  25  F i g u r e 22b T r a n s i e n t Responses o f JV, J L , JNETl, VIS1, PPMV and PIPMV f o r V a r i a b l e KAS w i t h NK = 0.5 h r mmHg ( C o n d i t i o n s as i n T a b l e 13) - 1  -1  2-|  r20  r480  CD X  cn -18  E 0-  an ZD  ui ui  J, LU C£ 16 LO (/) Ld  Ul -1-  o  Ld  cc:  cn Q_  (J  - 2 -  Q_  Q  o  ZD  Ul  o Q  E  14 -3  O  o  >-4-J  L  10  15  TIME (hrs)  12  I  -114-  VIS1  r e s u l t s i n an Increase i n PPMV a c c o r d i n g  c u r v e , and a l s o an i n c r e a s e  i n JL according  to the t i s s u e compliance  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 s o l u t e weights  (QA  and QG) to the i n t e r s t i t i a l  tissue protein concentrations CTG  CTA and CTG.  The responses of CTA and  a r e dependent on the r a t e s of change i n QA or QG and VIS.  shown i n F i g u r e decline.  related  22c, d u r i n g  the i n t e r s t i t i a l  The t i s s u e o n c o t i c p r e s s u r e  CAVA and CAVG, a l s o  PIPMV, decreases s i n c e i t i s  to the t o t a l e f f e c t i v e p r o t e i n c o n c e n t r a t i o n  decrease i n PIPMV and r i s e  ( F i g u r e 22b).  As  edema phase CTA and CTG  The e f f e c t i v e p r o t e i n c o n c e n t r a t i o n s ,  decrease.  The  volume (VIS=VIS1-VCELL) g i v e s the  (CAVA + CAVG).  i n PPMV r e s u l t s i n a decrease i n JV  The decrease i n JV and i n c r e a s e  i n J L cause JNET1 to  decrease ( F i g u r e 22b). Eventually  VIS1 reaches the f l u i d  onset of a l v e o l a r f l o o d i n g , VTONS. the c o n d i t i o n s at  2.5 h r s .  by  equation  given  F o r the s i m u l a t i o n  conducted under  i n T a b l e 13 the onset of a l v e o l a r f l o o d i n g occurs  The t r a n s e p i t h e l i a l flow, (25).  volume c o r r e s p o n d i n g to the  JAS, that r e s u l t s i s c a l c u l a t e d  Whether JAS r i s e s or f a l l s  i s dependent on the  r a t e s of change of PAS, KAS and PPMV: *Um dt  = KAS  at  + (PPMV-PAS)  -KAS ^ d  at  > at A S  (37)  Immediately f o l l o w i n g the onset of a l v e o l a r f l o o d i n g JAS i n c r e a s e s Figure The  22d shows the responses of PPMV, KAS, PAS and JAS.  r i s e i n JAS r e s u l t s i n a f u r t h e r d e c l i n e i n JNET1 ( F i g u r e 22e),  -  -115-  F i g u r e 22c  T r a n s i e n t Responses o f QA, QG, VIS1, CTA and CTG f o r V a r i a b l e KAS w i t h NK = 0.5 h r mmHg ( C o n d i t i o n s as i n Table 13) - 1  480-1  460-  *j=  -1  0.030  E  8  0.025 H  IT  cn -6 ^—^  440 H  00  o  1  Ld  ZD 420 H  ~c  o  LY.  Ld  __l  O O 0.015-j > ZZ. Q 400-j ZD  Ld  o o  _J 1_L_  1  - 4 ZD 1  — l  O (/)  Ld  380 H  360  J  o  0.005 10  15  TIME! (hrs)  -116-  Figure  22d  T r a n s i e n t Responses of JAS, PAS, KAS and PPMV f o r V a r i a b l e KAS w i t h NK = 0.5 h r mmHg ( C o n d i t i o n s as i n Table 13) - 1  -1  TIME (hrs)  -117-  which i n t u r n reduces the r a t e of i n c r e a s e the  r a t e of change of VIS1 d i m i n i s h e s  KAS.  JAS reaches a maximum v a l u e  or  (d(JAS)/dt)  equals z e r o .  o f VIS1.  The r e d u c t i o n i n  the r a t e of change o f PPMV and  (Figure  22d) when JAS equals  (JV-JL)  JAS then d e c l i n e s s i n c e the r a t e of  change of PAS exceeds the r a t e s of change of PPMV and KAS ( F i g u r e 22d). As  JAS d e c l i n e s , JNET1 i n c r e a s e s , which r a i s e s VIS1.  i n VIS1 i n c r e a s e s will  PPMV and J L ( F i g u r e  decrease ( F i g u r e 22b).  Figure  22b).  The r i s e  I f PPMV i n c r e a s e s  22e i l l u s t r a t e s  the t r a n s i e n t  responses of JV, J L , JAS and JNET1.  Steady s t a t e i s achieved  e q u a l s J L , I.e.,  zero.  5.2.1  JNET1 and JAS equal  filtration  c o e f f i c i e n t , NK  represents  the p r o p o r t i o n a l i t y between KAS and the f l u i d  difference  (VIS1-VT0NS).  studied  subjected  are l i s t e d  t o a step  The c o n d i t i o n s  i n Table 15.  - 1  At time zero  the PMVES was  D i f f e r e n t orders o f  from 0.05 h r  -  1  mmHg  -1  t o 50.0  mmHg . -1  The in Figure is  volume  i n the PMVES under which NK  change i n PMV t o 50 mm Hg.  magnitude of NK were s t u d i e d , r a n g i n g hr  when JV  Transient Response of the Pulmonary Microvascular Exchange System to changes in NK In the case of a v a r i a b l e e p i t h e l i a l  was  then JV  t r a n s i e n t responses o f JAS f o r d i f f e r e n t NK a r e i l l u s t r a t e d 23a. The time f o r the onset of a l v e o l a r f l o o d i n g (JAS > 0)  appoximately 2.5 h r s .  A f t e r the onset o f a l v e o l a r f l o o d i n g JAS i n  -118-  Figure  22e  T r a n s i e n t Responses o f JV, J L , JNET1 and JAS f o r V a r i a b l e KAS w i t h NK = 0.5 h r " mmHg ( C o n d i t i o n s as i n T a b l e 13): I n s e r t i l l u s t r a t e s responses up to steady s t a t e ( a p p r o x i m a t e l y 300 h r s ) . 1  -1  -119-  T a b l e 15  Conditions  of the PMVES S i m u l a t i o n s Conducted to Study Changes i n NK*  KAS =NK(VIS1-VT0NS) f o r VIS1 > VTONS NK = 0.05, 0.5, 5.0 and 50.0 h r mmHg - 1  PMV KF APERM SIGD PSA PSG SIGFA SIGFG VTONS SL B VLMPH  = = = = = = = = = = =  50 1.12 0 0.75 3.00 1.00 0.40 0.60 460. 0.25xl0 -3.03 5000.  -  mmHg ml/hr/mmHg  ml/hr ml/hr  - 2  ml mmHg/ml mmHg ml  *These are the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 23a and 23b.  -120-  F i g u r e 23a  T r a n s i e n t Responses o f JAS f o r Changes i n NK ( C o n d i t i o n s as i n T a b l e 15) - Response c o n t i n u e d to time when VTOT = 1000 m l .  -121-  each case r i s e s increases.  to a maximum; as NK i n c r e a s e s the maximum JAS a l s o  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 Table  15 the value  o f (PPMV-PAS) i s 4.12 mm Hg a t the onset o f  a l v e o l a r f l o o d i n g f o r a l l cases the v a l u e VTONS).  Figure  -  1  of NK.  Therefore  of KAS which i s composed o f the product  (VTONS=460 m l ) . 0.5 h r  ( 2 5 ) . F o r the c o n d i t i o n s of  23b shows that as NK i n c r e a s e s  JAS i s dependent on of NK and (VIS1-  (VIS1-VT0NS) decreases  When NK i s i n c r e a s e d by a f a c t o r of 10, from 0.05 t o  mmHg , the r e d u c t i o n i n (VIS1-VT0NS) i s l e s s than a f a c t o r -1  of 10; JAS i n c r e a s e s as NK i s r a i s e d from 0.05 t o 0.5 h r (Figure hr  - 1  1  mmHg  -1  23a). However, r a i s i n g NK by a f a c t o r of 10, from 5.0 to 50.0  mmHg  benefit  -  -1  reduces (VIS1-VT0NS) by a f a c t o r o f almost  10. The  o f the i n c r e a s e i n NK i s e l i m i n a t e d and the r e s u l t i n g JAS f o r  NK=5.0 and 50.0 h r  -  as NK i s i n c r e a s e d  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 t o (VIS1-  1  mmHg  -1  a r e almost equal  (Figure 23a).  Therefore,  VT0NS) i n c r e a s e s . I t may be surmised t h a t f o r low v a l u e s o f NK ( e . g . NK=.05 h r mmHg ) a l a r g e i n c r e a s e i n VIS1 i s r e q u i r e d t o induce -1  fluid  and s o l u t e s a c r o s s  NK=5.0 or 50.0 h r r e q u i r e d t o induce a l v e o l a r membrane. remaining  -  1  the a l v e o l a r membrane.  mmHg ) then very -1  little  the movement of f l u i d A value  simulations  o f NK=0.5 h r  s i n c e i t allowed  1  I f NK i s l a r g e ( e . g .  i n c r e a s e i n VIS1 i s  mmHg  1  the movement o f  and s o l u t e a c r o s s the -  -  -1  was used i n the  some i n c r e a s e i n VIS1.  -122-  F i g u r e 23b  T r a n s i e n t Responses o f VIS1 f o r Changes i n NK ( C o n d i t i o n s as i n T a b l e 15) - Response c o n t i n u e d to time when VTOT = 1000 m l .  -123-  5.3  The Response of the Pulmonary Microvascular Exchange System to changes i n the parameter VTONS The parameter VTONS r e p r e s e n t s  onset  of a l v e o l a r f l o o d i n g .  the EVEA f l u i d  The e f f e c t o f changes i n VTONS on the  PMVES were s t u d i e d under the c o n d i t i o n s l i s t e d zero  the PMVES was s u b j e c t e d  mmHg.  mmHg  filtration  of  0.5 h r  of  the v a r i a b l e s i s s i m i l a r t o those  -1  between the c i r c u l a t o r y , interstitial  edema w i l l  discussed  interstitial  for  c o e f f i c i e n t with an NK  illustrate  that steady  i n S e c t i o n 5.2.  and lymphatic  edema i s s i m u l a t e d ) .  compartments  only  A simulation of i n t e r s t i t i a l  s t a t e i s e s t a b l i s h e d when  f o r both  fluid  and s o l u t e s a r e  I f the p e r t u r b a t i o n i s an e l e v a t e d PMV, then the i n t e r s t i t i a l  cellular) fluid  volume a t steady  each PMV ( F i g u r e 24).  l i s t e d w i t h F i g u r e 24.  i n t h i s case  interstitial  edema.  the f l u i d  s t a t e ( V I S l ( s s ) ) may be o b t a i n e d  The other c o n d i t i o n s of the s i m u l a t i o n are  I n the a l v e o l a r model, r a i s i n g VTONS to a very  large value,  5000 ml w i l l  result  F i g u r e 24 i l l u s t r a t e s  i n the m o d e l l i n g of  that at a PMV of 50 mmHg  volume V I S l ( s s ) i s 566 m l .  In the a l v e o l a r model a VTONS g r e a t e r than 566 ml w i l l in  place  s t a t e c o n d i t i o n s may be achieved f o r  the t r a n s e n d o t h e l i a l and lymph flows  (and  At time  and s o l u t e exchange take  a g i v e n p e r t u r b a t i o n t o the PMVES; steady  equal.  16.  was employed, the shape o f the t r a n s i e n t responses  For VIS1 l e s s than VTONS f l u i d  (i.e.,  i n Table  to a step change i n PMV from 9 to 50  Since a v a r i a b l e e p i t h e l i a l - 1  volume a t the  interstitial  VIS1 reaches  edema o n l y .  The PMVES w i l l  reach steady  result  s t a t e when  566 ml. However, i f VTONS i s s e t below a v a l u e of 566 ml,  -124-  Table 16  C o n d i t i o n s o f the PMVES S i m u l a t i o n s to study the changes i n VTONS*  VTONS:  v a r i e d from 410 to 500 ml  KAS = NK(VIS1 - VTONS) Nk = 0.5 h r mmHg -  PMV KF APERM SIGD PSA PSG SIGFA SIGFG SL B VLMPH  1  f o r VIS1 > VTONS  -1  = 50 = 1 . 1 2 = 0 = 0.75 = 3.0 = 1.0 = 0.4 = 0.6 = 0.25xl0 = -3.03 = 5000.  mmHg ml/hr  ml/hr ml/hr  - 2  mmHg/ml mmHg ml  *These are the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 25a to 25d.  -125-  F i g u r e 24  Steady s t a t e v a l u e s Edema ( C o n d i t i o n s : ml/hr/mmHg, SIGD = SIGFA = 0.4, SIGFG  o f VIS1 f o r g i v e n PMV f o r I n t e r s t i t i a l VTONS = 5000 ml, VLMPH = 5000 ml, KF = 1.12 0.75, PSA = 3 . 0 ml/hr, PSG = 1 . 0 ml/hr, =0.6)  750-1  0  1  400-1 20  ,  30  , —,  40  50  ,  60  , 1  70  80  CIRCULATION HYDROSTATIC PRESSURE PMV (mm  -126-  F i g u r e 25a  T r a n s i e n t Responses of (JV-JL) and JAS f o r VTONS of 420 ml and 500 ml ( C o n d i t i o n s as i n Table 16) - Response c o n t i n u e d 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  flooding w i l l  occur.  reach a v a l u e equal to VTONS and a l v e o l a r  As VTONS i s reduced  from 566 ml the d i f f e r e n c e  between JV and JL at the onset of a l v e o l a r f l o o d i n g I n c r e a s e s . 25a  shows the t r a n s i e n t  respone  Figure  of (JV-JL) up to VT0T=1000 ml f o r a  VTONS of 420 ml and 500 ml; the time f o r the onset of a l v e o l a r f l o o d i n g i s approximately of  1.0 and 4.5 h r s . , r e s p e c t i v e l y .  The v a l u e  (JV-JL) i s g r e a t e r f o r VT0NS=420 ml than a VT0NS=500 ml.  ( J V - J L ) i s e q u i v a l e n t to the r a t e of f l u i d e x t r a v a s c u l a r space  accumulation  Since  i n the t o t a l  ( i n t e r s t i t i a l + c e l l u l a r + a l v e o l a r ) the time to  reach a VTOT of 1000 ml ( F i g u r e 25b) and t o the onset of f l o o d i n g are l e s s f o r the s m a l l e r VTONS. F i g u r e 25a a l s o i l l u s t r a t e s a l v e o l a r f l o o d i n g JAS r i s e s  that f o l l o w i n g  to a maximum v e r y r a p i d l y - the shape o f  the JAS curve was d i s c u s s e d i n s e c t i o n 5.2. of  JAS  JAS i s i l l u s t r a t e d  Illustrate  The t r a n s i e n t  response  i n F i g u r e 25c by showing the maximum JAS and t h e  s i m u l a t i o n s of the PMVES f o r changes i n VTONS a l s o that the EVEA f l u i d  as VTONS r i s e s accumulation In  of  T h e r e f o r e , as  at a VTOT of 1000 ml. The  was  As (JV-JL) a t the onset  a l v e o l a r f l o o d i n g i n c r e a s e s , JAS a l s o i n c r e a s e s .  VTONS i s reduced JAS i n c r e a s e s ( F i g u r e 2 5 c ) . of  the onset of  volume at a VTOT of 1000 ml i n c r e a s e s  ( F i g u r e 25d); due t o the i n c r e a s e i n i n t e r s t i t i a l  fluid  b e f o r e VIS1 reaches VTONS.  s e c t i o n 3.4 the upper l i m i t  approximately  600 ml.  of VTONS e x p e r i m e n t a l l y observed  F o r the s i m u l a t i o n s conducted  VTONS the upper v a l u e of VTONS was only 500 ml.  i n the study  The lower v a l u e of  -128-  F i g u r e 25b  Time to Reach a VTOT of 1000 ml f o r D i f f e r e n t VTONS ( C o n d i t i o n s as i n Table 16)  -129-  F i g u r e 25c  The Maximum T r a n s e p i t h e l i a l Flow (JAS(max)) and t h e JAS a t a VTOT of 1000 ml f o r D i f f e r e n t VTONS ( C o n d i t i o n s as i n T a b l e 16)  30  i 400  r 420  i 440  i 460  i 480  VTONS (ml)  i 500  r 520  -130-  F i g u r e 25d  The F l u i d Volume VIS1 a t a VTOT o f 1000 ml f o r D i f f e r e n t VTONS ( C o n d i t i o n s as i n Table 16)  520  visi(iooo)  400  420  440  460  480  VTONS (ml)  500  520  -131-  VTONS v a r i e s , depending on the o r i g i n of the p e r t u r b a t i o n . is  lowered  the r a t e o f a l v e o l a r f l o o d i n g i n c r e a s e s .  h y d r o s t a t i c p r e s s u r e , PPMV ( o n s e t ) , corresponds volume VTONS, as determined  h y d r o s t a t i c p r e s s u r e s above PPMV ( o n s e t ) . that the a l v e o l a r b a r r i e r s cannot pressure.  In the remaining  corresponds  5.4  withstand  curve.  fluid One may  interstitial  Guyton e t a l . ( 4 7 ) positive  suggested  interstitial  s i m u l a t i o n s VTONS was s e t a t 460 ml, which  to an i n t e r s t i t i a l  h y d r o s t a t i c p r e s s u r e of +1.1 mmHg.  Transient Responses of the Pulmonary Microvascular Exchange System to changes i n the Parameter SL The  parameter SL i s d e f i n e d as the slope of the e x p r e s s i o n  r e l a t i n g PAS t o VAS - i n e q u a t i o n was  withstand  An i n t e r s t i t i a l  to the EVEA  by the t i s s u e compliance  s p e c u l a t e t h a t the a l v e o l a r b a r r i e r s cannot  As VTONS  (27).  The e f f e c t o f SL on the PMVES  s t u d i e d under the c o n d i t i o n s shown i n Table 17.  At time zero the  PMVES was s u b j e c t e d t o a step change i n PMV from a normal v a l u e o f 9 mmHg to 50 mmHg.  T e s t s were made with v a l u e s of SL between 0 . 2 5 x l 0  mmHg/ml and 0.75xlO~ In  2  mmHg/ml.  the s i m u l a t i o n s conducted  t o study changes i n SL the  alveolar fluid  volume a t a VTOT of 1000 ml was approximately  Using equation  (27), the change i n the a l v e o l a r f l u i d  corresponds SL=0.25xlO  -1+  500 m l .  p r e s s u r e that  t o a change i n VAS of 500 ml i s 0.0125 mmHg f o r _lt  mmHg/ml and 3.75 mmHg f o r S L = 0 . 7 5 x l 0  case of SL=0.25xl0  -1+  -2  mmHg/ml.  For the  mmHg/ml the change i n PAS i s i n s i g n i f i c a n t as  compared t o the change f o r S L = 0 . 7 5 x l O  -2  mmHg/ml.  The v a r i a b l e PAS  -132-  i s used i n e v a l u a t i n g  the p r e s s u r e  gradient  that causes f l u i d  from the i n t e r s t i t i u m  to the a i r space ( i . e . (PPMV-PAS)); f o r a given  PPMV at a VTOT of 1000 ml and the c o n d i t i o n s  flow  of Table 17, (PPMV-PAS)  d e c r e a s e s as SL i n c r e a s e s . The Hg  - 1  (i)  effect  of SL on the PMVES was s t u d i e d  and NK=50.0 h r " mm H g 1  Simulations Figure  - 1  -  1  mm H g  - 1  .  f o l l o w i n g the maximum JAS, the r a t e of  decrease i n JAS r i s e s as SL i s i n c r e a s e d . i n PAS as SL i s i n c r e a s e d  (Figure  e q u a t i o n (32) i l l u s t r a t e s  that  simulation decreases.  The f l u i d m a t e r i a l balance of As the  of the PMVES p r o g r e s s e s to steady s t a t e c o n d i t i o n s Following  VTOT of 1000 ml.  (JV-JL)  the onset of a l v e o l a r f l o o d i n g the change i n  Therefore,  non-steady s t a t e c o n d i t i o n s  as JAS decreases i n the p e r i o d  a VTOT of 1000 ml, JNET1 w i l l  may be r e f l e c t e d Since  26b).  T h i s i s caused by the r i s e  (JNET1 + JAS) equals ( J V - J L ) .  (JV-JL) i s g e n e r a l l y s m a l l d u r i n g  before  _1  .  w i t h NK=0.05 h r  26a shows that  f o r NK=0.05 hr mm  increase.  to a  just  The i n c r e a s e  i n JNET1  by the change i n the t r a n s i e n t response of VIS1.  the r a t e of d e c l i n e i n JAS i n c r e a s e s  would i n c r e a s e w i t h SL.  Figure  as SL i s r a i s e d , JNET1  26c shows t h a t the i n c r e a s e  i n VIS1 i s  more pronounced as SL i n c r e a s e s . (ii)  Simulations Following  hrs):  w i t h NK=50.0 h r  transient  mm H g  - 1  .  the time of the onset of a l v e o l a r f l o o d i n g (2.5  f o r NK=50 h r  alveolar f l u i d  - 1  - 1  mm H g  pressure,  - 1  an I n c r e a s e i n SL a l s o i n c r e a s e s the  as seen by F i g u r e  27a. However, the  responses of JAS, up to the time t h a t VTOT=1000 ml, i s not  -133-  Table 17  C o n d i t i o n s of the PMVES S i m u l a t i o n Conducted t o Study Changes i n SL*  SL = 0.25 x I O  - 4  , 0.25 x 1 0  - 3  , 0.25 x 1 0  - 2  and 0.75 x I O  - 2  mmHg/ml  KAS = NK(VISl-VTONS) f o r VISL > VTONS NK = 0.5, 50. h r mm H g -  PMV KF APERM SIGD PSA PSG SIGFA SIGFG VTONS B VLMPH  1  - 1  = 5 0 = 1.12 = 0 = 0.75 = 3.0 = 1.0 = 0.40 = 0.60 = 460 = -3.03 = 5000.  mm Hg ml/hr/mm Hg  ml/hr ml/hr  ml mm Hg ml  *These a r e the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 26a,b,c and 27 a,b.  -134-  F i g u r e 26a  T r a n s i e n t Responses o f JAS f o r D i f f e r e n t SL and NK = 0.05 hr mmHg" ( C o n d i t i o n s as i n Table 17) - Responses continued t o time when VTOT = 1000 ml 1  15  TIME (hrs)  -135-  F i g u r e 26b  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 = 0.05 hr mmHg ( C o n d i t i o n s as i n Table 17) - Responses c o n t i n u e d t o time when VTOT = 1000 ml -1  TIME (hrs)  -136-  F i g u r e 26c  T r a n s i e n t Responses o f VIS1 f o r D i f f e r e n t SL and NK = 0.05 hr mmHg ( 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 -1  550-i  £ 500> 450 O >  Legend  400-  SL = 0.25E-4 SL = 0.25E-3 SL = O^SE-2 SL = 0.75E-2  350-  20  40  TIME (hrs)  60  80  -137-  F i g u r e 27a  T r a n s i e n t R e s p o n s e s o f PAS f o r D i f f e r e n t S L a n d NK = 50.0 h r mmHg ( C o n d i t i o n s a s i n T a b l e 17) - R e s p o n s e s c o n t i n u e d t o t i m e when VTOT = 1 0 0 0 m l -1  TIME (hrs)  -138Figure 27b  Transient Responses of JAS for Different SL and NK = 50.0 hr mmHg (Conditions as in Table 17) - Responses continued to time when VTOT = 1000 ml -1  Legend SL = 0.25E-4 SL = 0.25E-3 SL = 0.25E-2 SL = 0.75E-2  0  10  20  TIME (hrs)  30  -139-  affected  by the d i f f e r e n t  d e t e r m i n a t i o n of JAS effects; VIS1  the r i s e  the v a r i a b l e s VIS1  i n PAS  attempts to augment JAS  o n l y r i s e by 0.1  and PAS  attempts to suppress  s i t u a t i o n where NK=50 h r  mm  - 1  Hg , - 1  than to  In  the  competing  while a r i s e i n  and PPMV.  In the  the d i f f e r e n c e (VIS1-VT0NS) need JAS  i s , thus, more  PAS.  Responses of the Pulmonary Microvascular Exchange System to changes i n the parameter B. The  parameter B r e p r e s e n t s the v a l u e of the a l v e o l a r  p r e s s u r e at the onset  of a l v e o l a r f l o o d i n g .  p r e d i c t i o n s of the PMVES s i m u l a t i o n was listed  i n Table  change i n PMV the time  18.  At time  from 9 mm  f o r onset  Hg  Hg.  response (max)) and  of JAS  may  the JAS  v a l u e of B ( F i g u r e 28a);  hrs.  f l o o d i n g the d r i v i n g As B d e c r e a s e s ,  at a VTOT of 1000  the t r a n s i e n t  p o i n t s i s a g r a d u a l decrease  28a  illustrates  from approximately  1.0  mm  (max) Hg  force that the  increase.  The  be I l l u s t r a t e d by r e c o r d i n g the maximum  two  t h a t JAS  the  For a l l values of B s t u d i e d ,  of a l v e o l a r f l o o d i n g i s about 2.5 of a l v e o l a r  of B on  s u b j e c t e d to a step  v a l u e of the p r e s s u r e g r a d i e n t (PPMV-PAS) w i l l  transient (JAS  effect  fluid  s t u d i e d under the c o n d i t i o n s  causes t r a n s e p i t h e l i a l flow i s (PPMV-PAS). initial  The  zero the PMVES was  to 50 mm  F o l l o w i n g the onset  JAS  exhibit  JAS,  by i n c r e a s i n g KAS  ml f o r a d o u b l i n g of KAS;  r e s p o n s i v e to KAS  5.5  v a l u e s of SL - see F i g u r e 27b.  response  from JAS  and  JAS  to -4 mm  ml  (max)  (JAS  (1000)) f o r each  of JAS  between these  to JAS  (1000).  (1000) i n c r e a s e as B i s Hg,  but  does not  change  Figure reduced  -140-  Table 18  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted to Study the Changes i n B*  B:  v a r i e d from -10 to 1.0 mm Hg  KAS = NK(VISl-VTONS) f o r VIS1 > VTONS NK = 0.5 h r mm H g - 1  PMV KF APERM SIGD PSA PSG SIGFA SIGFG VTONS SL VLMPH  - 1  = 5 0 = 1.12 = 0 = 0.75 = 3.0 = 1.0 = 0.40 = 0.60 = 460 = 0.25xlO" =5000  mm Hg ml/hr/mm Hg  ml/hr ml/hr  2  ml mm Hg/ml ml  *These a r e the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 28a t o 28g.  -141-  F i g u r e 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 o f 1000 ml f o r D i f f e r e n t B ( C o n d i t i o n s as i n Table 18)  1 -10  I  I  -8  -6  I  - 4 - 2  I  I  [  0  2  PAS @ ONSET OF ALVEOLAR FLOODING, B (mm  -142-  signifIcantly treat (i)  f o r B l e s s than -4 mm Hg.  The f o l l o w i n g d i s c u s s i o n  these r e g i o n s s e p a r a t e l y . The r e g i o n of B l e s s than -4 mm Hg. JAS  i s determined  by the product of KAS and (PPMV-PAS).  28b  illustrates  and  B= -6 mm Hg; the area of i n t e r e s t  the t r a n s i e n t  i s f o r time g r e a t e r than the (2.5 h r s ) .  from -6 mm Hg t o -10 mm Hg, (PPMV-PAS) I n c r e a s e s . the t r a n s i e n t  As B i s decreased F i g u r e 28c shows  response of KAS f o r B= -10 mm Hg and B= -6 mm Hg; as B  decreased from -6 mm Hg t o -10 mm Hg, KAS d e c r e a s e s .  which (PPMV-PAS) i n c r e a s e s as B i s lowered is  Figure  response of (PPMV-PAS) f o r B=-10 mm Hg  time a t the onset of a l v e o l a r f l o o d i n g  is  will  The f a c t o r by  from -6 mm Hg to -10 mm Hg  e q u i v a l e n t t o the f a c t o r by which KAS decreases as B i s lowered  from -6 mm Hg to -10 mm Hg.  T h e r e f o r e , the product of KAS and (PPMV-  PAS) f o r B= -6 mm Hg y i e l d s a v a l u e a p p r o x i m a t e l y e q u a l t o the product of  KAS and (PPMV-PAS) f o r B= -10 mm Hg.  F i g u r e 28a i l l u s t r a t e s  that  JAS(max) and JAS(1000) f o r B= -6 mm Hg and -10 mm Hg a r e a p p r o x i m a t e l y equal. S i n c e JAS(1000) i s the same f o r v a l u e s of B l e s s the time t o reach a VTOT of 1000 ml i s a l s o s i m i l a r (ii)  than -4 mm Hg,  ( F i g u r e 28d).  The r e g i o n of B g r e a t e r than -4 mm Hg For  responses  the r e g i o n of B g r e a t e r than -4 mm Hg the t r a n s i e n t of (PPMV-PAS) and KAS a r e i l l u s t r a t e d  0.5 mm Hg i n F i g u r e s 28e and 28f r e s p e c t i v e l y .  f o r B= -3 mm Hg and B= F i g u r e 28e i l l u s t r a t e s  t h a t f o l l o w i n g the time of the onset o f a l v e o l a r f l o o d i n g  (2.5 h r s )  -143-  Figure 28b  Transient Responses of (PPMV-PAS) f o r B = -10 mmHg and B = -6 mmHg (Conditions as i n Table 18) - Responses continued to time when VTOT = 1000 ml  -144-  F i g u r e 28c  T r a n s i e n t Responses o f KAS f o r B = -10 mmHg and B = -6 mmHg ( C o n d i t i o n s a s i n Table 18) - Responses c o n t i n u e d t o time when VTOT = 1000 ml  Legend B = -10 B= -6  0  10  20  30  -145-  Figure  28d  Time to r e a c h a VTOT of 1000 ( C o n d i t i o n s as i n Table 18)  ml  for Different B  55  -10  -8  -6  - 4 - 2  0  2  PAS @ ONSET OF ALVEOLAR FLOODING, B (mm  -146-  F i g u r e 28e  T r a n s i e n t Responses o f (PPMV-PAS) f o r B = -3 mmHg and B = 0.5 mmHg ( C o n d i t i o n s a s i n Table 18) - Responses continued t o time when VTOT = 1000 ml  -147-  F i g u r e 28f  T r a n s i e n t Responses o f KAS f o r B = -3 mmHg and B = 0.5 mmHg ( C o n d i t i o n s a s i n T a b l e 18) - Responses c o n t i n u e d t o time when VTOT = 1000 ml  -148-  the p r e s s u r e d i f f e r e n c e B = -3 mm -3 mm  Hg  Hg,  (PPMV-PAS) i s l e s s f o r B = 0.5  w h i l e i n F i g u r e 28f KAS  to 0.5  mm  Hg.  However, the product  of KAS  and  JAS(IOOO) i s l e s s f o r B = 0.5 28a).  mm  mm  i s g r e a t e r than f o r B = -3 mmHg.  than f o r B = -3 mm  Hg,  as B i s r a i s e d  above -4 mm  to reach a VTOT of 1000  An  ml  ml,  s e c t i o n 5.3).  interstitial increased  566 ml,  Hg  state w i l l  As B approaches 1.09  in particular will  If  the v a l u e of KAS  i n the i n t e r s t i t i a l  The  of 50 mm  mm  Hg.  Hg  and  (as s t a t e d  As B i s  i n c r e a s e s ; F i g u r e 28f  from -3 mm  Hg  to 0.5  mm  I f VIS1  be a c h i e v e d and a l v e o l a r mm  Hg  Hg. rises  flooding  the t o t a l e x t r a v a s c u l a r f l u i d decrease;  the f l u i d  bulk of the edema f l u i d  volume i s then  compartment.  B i s e l e v a t e d to v a l u e s above 1.09  remain at those s t a t e d i n T a b l e 18,  at  (VTONS) the  i s a r i s e i n VIS1.  state w i l l  decrease.  volume VIS1  i s 566 ml  volume of 460 ml  to the i n c r e a s e i n KAS  volume (VIS1+VAS) at steady  contained  At a PMV  s t a t e VIS1  i n c r e a s e s when B i s r a i s e d  then steady  terminated.  mm  The  leads to an i n c r e a s e i n  h y d r o s t a t i c p r e s s u r e , PPMV, i s 1.09  shows t h a t KAS  to  At an EVEA f l u i d  increases.  i n the f l u i d  as shown i n F i g u r e 28g.  towards 1.09  Corresponding  Hg  a rise  the c o n d i t i o n s of Table 18 the steady in  (Figure  ( F i g u r e 28d).  i n c r e a s e i n B a l s o causes  a VTOT of 1000  Hg  at  the f a c t o r by which (PPMV-PAS)  i s h i g h e r than the f a c t o r by which KAS  d e c l i n e i n JAS the time  Hg  than f o r  h r s . the s l o p e of  (PPMV-PAS) at JAS(max) and  As B i s r a i s e d from -4 mm  decreases  Hg  Hg  i n c r e a s e s as B i s i n c r e a s e d from  A f t e r a time of about 7.5  the KAS-time curve f o r B = 0.5  mm  mm  Hg,  and c o n d i t i o n s  then the p r e s s u r e g r a d i e n t  VAS,  -149-  Figure  28g  F l u i d Volume VIS1 at a VTOT of 1000 B ( C o n d i t i o n s as i n T a b l e 18)  ml  for Different  -150-  (PPMV-PAS) at the onset  of a l v e o l a r f l o o d i n g w i l l  the t r a n s e p i t h e l i a l flow w i l l the i n t e r s t i t i a l the onset  space.  be i n i t i a l l y  and  from the a l v e o l a r space to  However, the f l u i d  volume of the a i r space at  of a l v e o l a r f l o o d i n g 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 negative values. upper l i m i t  The  to a f l u i d  alveolar fluid  (PG)  space i n i t i a l l y w i l l  S i n c e n e g a t i v e v a l u e s i n VAS  of B i s 1.09  corresponding  pressure  become n e g a t i v e ,  mm  Hg  reduce VAS  are u n r e a l i s t i c ,  or the i n t e r s t i t i a l  to the  pressure  volume of VTONS. pressure  (PL=PAS) i s r e l a t e d  a c c o r d i n g to L a p l a c e ' s  equation,  gas  equation (4).  E l e v a t i o n of PG w i l l  r a i s e PL.  is  I f B i s i n c r e a s e d , then the t o t a l e x t r a v a s c u l a r  e q u i v a l e n t to B.  fluid  At the onset  to the  volume (VIS1+VAS) at steady  there w i l l  be l e s s f l u i d  the parameter B may thus  of a l v e o l a r f l o o d i n g PL  state conditions w i l l  accumulation  i n the a i r space.  be e l e v a t e d by a p p l y i n g a p o s i t i v e  gas  space,  i n c r e a s i n g PG,  al.  (50) s u b j e c t e d i s o l a t e d dog  decrease, i . e . Clinically, pressure  which i n t u r n i n c r e a s e s B. lung lobes to continuous  to the  C a l d i n i et positive-  p r e s s u r e v e n t i l a t i o n d u r i n g an i n c r e a s e i n the pulmonary m i c r o v a s c u l a r pressure air  (PMV), i . e . , h y d r o s t a t i c pulmonary edema was  space p r e s s u r e  (PG) was  maintained  " a p p r e c i a b l e amount of l i q u i d less  than  t h i s value  the e n d o t r a c h e a l  above a c e r t a i n v a l u e  accumulated i n the a i r w a y s " .  " f r o t h y bloody  liquid  c a n n u l a " d u r i n g acute  flowed  I f the  no For  PG  c o n t i n u o u s l y from  pulmonary edema.  o b s e r v a t i o n s by C a l d i n i et a l . (50) supports e l e v a t i o n i n B.  induced.  T h i s set of  the s i g n i f i c a n c e of  the  -151-  5.6  The Response of the Pulmonary Microvascular Exchange System to a Maximum Lymph Flow Lymph flow Is r e l a t e d  (21).  to the f l u i d volume VIS1 through  A maximum c a p a c i t y of the lymphatics may be d e f i n e d by e i t h e r  JL(max) or e q u i v a l e n t l y VLMPH. c a p a c i t y of the lymphatics JL  equation  w i l l be determined  In the computer program the maximum  i s related  by e q u a t i o n  to the EVEA f l u i d volume VLMPH.  (21) u n t i l VIS1 reaches VLMPH;  f u r t h e r i n c r e a s e s i n VIS1 w i l l not r a i s e J L . The e f f e c t  of changes i n  JL(max) ( o r VLMPH) on the p r e d i c t i o n s of the PMVES s i m u l a t i o n was s t u d i e d under the c o n d i t i o n s l i s t e d filtration 0.5 h r  -  1  i n Table  19.  The  epithelial  c o e f f i c i e n t was, a g a i n , c o n s i d e r e d to be v a r i a b l e w i t h NK =  mm H g  - 1  .  change i n PMV from A first  At time zero the PMVES was s u b j e c t e d to a s t e p 9 mm Hg to 50 mm Hg.  estimate of JL(max) as suggested  i n s e c t i o n 4.4.2,  c o u l d be the lymph flow c o r r e s p o n d i n g t o a f l u i d volume VTONS (460 ml).  Using e q u a t i o n  22.6 ml/hr. ml/hr.  (21) to c a l c u l a t e  the lymph flow, JL(max) becomes  The v a l u e of JL(max) was v a r i e d above and below 22.6  F i g u r e 29a shows the lymph flow responses,  up to a time when  VTOT = 1000 ml, f o r JL(max) equal t o 14.8, 22.6 and 24.3 ml/hr, and when no maximum lymph flow e x i s t s . g r e a t e r than 22.6 ml/hr ( i . e . , that the response  24.3 m l / h r ) , F i g u r e 29a  of J L i s very c l o s e to the response  maximum lymph flow i s d e f i n e d . approximately  When JL(max) i s approximately  35% ( i . e . ,  the lymph flow response,  7%  illustrates  of J L when no  D e c r e a s i n g JL(max) from 22.6 ml/hr by  14.8 ml/hr) produces a n o t i c e a b l e change i n i n comparison t o the response  o f J L when no  -152-  Table 19  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted t o Study Changes i n the Maximum Lymph Flow*  JL(max) v a r i e d from 14 to 38 ml/hr corresponds to a VLMPH of 410 to 550 ml. KAS = NK(VISl-VTONS) f o r VIS1 > VTONS NK = 0.5 h r mm H g -  PMV KF APERM SIGD PSA PSG SIGFA SIGFG VTONS SL B  1  - 1  = 5 = =  0 1.12 0 0.75 3.0 1.0 0.40 0.60 460 0.25xl0 -3.03  mm Hg ml/hr/mm Hg  ml/hr ml/hr  - 2  ml mmHg/ml mmHg  *These a r e the c o n d i t i o n s of the s i m u l a t i o n s t h a t produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 29a to 29c.  -153-  F i g u r e 29a  T r a n s i e n t Responses of J L f o r D i f f e r e n t JL(max) (or VLMPH) ( C o n d i t i o n s as i n Table 19) - Responses continued to time when VTOT = 1000 ml  o —I  Legend  LL. Q  JL(max) = N.A.  JL(max) = 22.6 JL(max) = 14.8  T 10  20  TIME (hrs)  I  30  -154-  maxiraum lymph flow i s d e f i n e d . F i g u r e 29b i l l u s t r a t e s  the t r a n s i e n t responses of (JV-JL) f o r  t h r e e maximum lymph f l o w s : 14.8, 22.6 and 29.4 ml/hr. JL(max) from 22.6 ml/hr  t o 29.4 ml/hr  Increasing  (a 30% i n c r e a s e ) produces a  s m a l l drop i n (JV-JL) that averages 2 ml/hr, w h i l e a decrease i n JL(max) from 22.6 ml/hr average  rise  i n (JV-JL) of about  the r a t e of f l u i d increasing  to 14.8 ml/hr  (a 35% decrease) produces an  7.5 ml/hr.  Since (JV-JL) r e p r e s e n t s  accumulation i n the t o t a l e x t r a v a s c u l a r  (JV-JL) w i l l  space,  lead to a decrease i n the time to reach a VTOT  of 1000 ml ( t ( 1 0 0 0 ) ) as shown i n F i g u r e 29c.  I t can be seen t h a t  d e c r e a s i n g JL(max) below 22.6 ml/hr y i e l d s a r a p i d decrease i n t(1000).  F o r JL(max) above 22.6 ml/hr  the i n c r e a s e i s l e s s  u n t i l JL(max) reaches a p p r o x i m a t e l y 30 ml/hr, a f t e r which  rapid  t(1000)  becomes c o n s t a n t . . From the above d i s c u s s i o n one may conclude that i f the c a p a c i t y of the l y m p h a t i c s i s exceeded  b e f o r e the EVEA f l u i d  v a l u e of VTONS, then the r a t e a t which 1000  fluid  volume reaches a  accumulates  to a VTOT of  ml i s a c c e l e r a t e d . Prichard  insufficiency:  (17) d i s c u s s e s pulmonary edema and lymphatic " O c c l u s i o n or p a r t i a l o c c l u s i o n of the l y m p h a t i c  system may be r e s p o n s i b l e f o r pulmonary edema i n p a t i e n t s w i t h silicosis,...,  and p o s s i b l y shock l u n g " .  He suggests that the  l y m p h a t i c c a p a c i t y may be reduced by p h y s i c a l a l t e r a t i o n s o f the lymphatic  channels or lung t i s s u e .  I n the a l v e o l a r model the  lymphatic  c a p a c i t y may be reduced by changes i n the parameter VLMPH  -155-  F i g u r e 29b  T r a n s i e n t Responses of (JV-JL) f o r D i f f e r e n t JL(max) ( C o n d i t i o n s as i n T a b l e 19) - Responses c o n t i n u e d t o time when VTOT = 1000 ml  50-f  Legend JL(max) =  14.8  JL(max}_= 22.6 JL(max) a 29A  10-  o10  T  TIME (hrs)  20  30  -156-  F i g u r e 29c  Time to Reach a VTOT of 1000 ml f o r D i f f e r e n t JL(max) ( C o n d i t i o n s as i n T a b l e 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 c o r r e s p o n d i n g  to JL(max)).  The response  of the  p r e d i c t i o n s of the PMVES s i m u l a t i o n may then be observed.  5.7 The Responses of the Pulmonary Microvascular Exchange System to changes in the value to which PMV i s elevated. In the p r e c e d i n g s i m u l a t i o n s pulmonary edema was s i m u l a t e d by e l e v a t i n g PMV from 9 mm Hg t o 50 mm Hg a t time z e r o . discussion w i l l  c o n s i d e r step changes i n PMV to d i f f e r e n t v a l u e s o f  PMV under the c o n d i t i o n s l i s t e d filtration 0.5  hr  -  1  i n Table 20.  The e p i t h e l i a l  c o e f f i c i e n t was r e p r e s e n t e d as a v a r i a b l e with NK equal t o  mm H g  The  The f o l l o w i n g  - 1  .  parameter PMV i s i n t r o d u c e d i n the a l v e o l a r model w i t h  S t a r l i n g ' s Hypothesis,  equation  (18).  From equation (18) one can see  t h a t e l e v a t i n g PMV i n c r e a s e s the t r a n s e n d o t h e l i a l flow ( J V ) . t r a n s e n d o t h e l i a l flow l e s s the lymph flow equals accumulation  i n the t o t a l e x t r a v a s c u l a r space.  that i n c r e a s i n g PMV from interstitial increased  the time f o r VIS1 to reach the onset  ( t ( o n s e t ) ) decreases  the r a t e of f l u i d F i g u r e 30a i l l u s t r a t e s  40 to 60 mm Hg i n c r e a s e s ( J V - J L ) .  phase VTOT i s e q u i v a l e n t to VIS1;  ( F i g u r e 30b).  The  I n the  t h e r e f o r e as PMV i s of a l v e o l a r  flooding  F o r PMV=40 mm Hg the v a l u e of  t ( o n s e t ) i s 3.0 h r s , and f o r PMV=60 mm Hg the v a l u e of t ( o n s e t ) i s 1.5 hrs.  F o l l o w i n g the onset of a l v e o l a r f l o o d i n g  s l o w l y ( i n comparison to b e f o r e a l v e o l a r 30a;  (JV-JL) f o r PMV=60 mm Hg i s s t i l l  mm Hg.  (JV-JL) decreases  very  f l o o d i n g ) , as shown i n F i g u r e  g r e a t e r than  (JV-JL) f o r PMV=40  Under the c o n d i t i o n s of Table 20 JAS approaches  (JV-JL)  -158-  Table 20  C o n d i t i o n s of the PMVES S i m u l a t i o n s Conducted ti to Study Changes i n PMV*  PMV:  v a r i e d from 35 to 60 mm Hg  KAS = NK(VIS1-VTONS) f o r VIS1 > VTONS NK = 0.5 h r mm H g -  KF APERM SIGD PSA PSG SIGFA SIGFG VTONS SL B VLMPH  1  - 1  = 1.12 = 0 = 0.75 = 3.0 = 1.0 = 0.40 = 0.60 = 4 6 0 = 0.25xlO = -3.03 = 5000  ml/hr/mm Hg  ml/hr ml/hr  -2  ml mmHg/ml mmHg ml  *These a r e the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s I l l u s t r a t e d i n F i g u r e s 30a to 30c.  -159-  F i g u r e 30a  T r a n s i e n t Response o f (JV-JL) and JAS f o r D i f f e r e n t PMV ( C o n d i t i o n s as i n T a b l e 20) - Responses c o n t i n u e d to a VTOT = 1000 ml  Legend (JV-JL), PMV = 4 0 JAS, PMV=40  TIME (hrs)  -160-  Figure  30b  Time to Reach the Onset of A l v e o l a r F l o o d i n g f o r D i f f e r e n t PMV ( C o n d i t i o n s as i n T a b l e 20)  V  6  1  Ld  1-1 35  1  40  1  45  1  50  1  55  1  60  HYDROSTATIC PRESSURE PMV (mm Hg)  -161-  Figure  30c  Time to Reach a VTOT o f 1000 ml f o r D i f f e r e n t PMV ( C o n d i t i o n s as i n T a b l e 20)  120  CIRCULATION HYDROSTATIC PRESSURE PMV(mm  -162-  ( F i g u r e 3 0 a ) . The time to reach a VTOT (=VIS1+VAS) of 1000 ml decreases  as PMV  increases (Figure 30c).  In both F i g u r e 30b and F i g u r e 30c i t can be seen that the times,  t ( o n s e t ) and t ( 1 0 0 0 ) ,  Increases  r e s p e c t i v e l y , a r e most  s e n s i t i v e to small  i n PMV above 35 mm Hg and l e v e l s o f f a t h i g h e r PMV  If  PMV i s reduced  at  a PMV of 30 mm Hg and f o r the c o n d i t i o n s of Table  values.  to 30 mm Hg, no a l v e o l a r f l o o d i n g would occur  since  20, the steady  s t a t e VIS1 i s 460 ml, which i s a l s o e q u i v a l e n t t o the assumed VTONS (460 m l ) . T h e r e f o r e be i n f i n i t e .  at a PMV of 30 mm Hg t ( o n s e t ) and t(1000) would  I n c r e a s i n g PMV above 30 mm Hg would induce a l v e o l a r  flooding. In c a r d i o g e n i c pulmonary edema the c i r c u l a t o r y h y d r o s t a t i c p r e s s u r e , PMV,  i s e l e v a t e d above i t s normal value  t r a n s e n d o t h e l i a l flow. Increases  For the c o n d i t i o n s l i s t e d  c a u s i n g an i n c r e a s e d i n Table 20  i n PMV up t o 30 mm Hg produce only i n t e r s t i t i a l edema.  R a i s i n g PMV above 30 mm Hg causes severe  pulmonary edema w i t h a l v e o l a r  flooding.  5.8  The Response of the Pulmonary Microvascular Exchange System to changes i n the Endothelial Filtration Coefficient, KF The e f f e c t  of changes i n KF on the p r e d i c t i o n s of the PMVES  s i m u l a t i o n was s t u d i e d under the c o n d i t i o n s l i s t e d  i n Table  21.  The  p e r t u r b a t i o n to the exchange system c o n s i s t e d of a step change i n the c i r c u l a t o r y h y d r o s t a t i c p r e s s u r e , a t time zero, from 9 mm Hg to 50 mm Hg and changes to the f l u i d  c o n d u c t i v i t y of the e n d o t h e l i a l membrane.  -163-  Table 21  Conditions  of the PMVES S i m u l a t i o n s Conducted t o Study Changes i n KF*  KF: v a r i e d from normal (KF(norm)) to 10 KF(norm) KF(norm) =1.12 ml/hr/mm Hg  KAS = NK(VISl-VTONS) f o r VIS1 > VTONS NK = 0.5 h r mm H g -  PMV APERM SIGD PSA PSG SIGFA SIGFG VTONS SL B VLMPH  1  - 1  = =  5  0 0 0.75 3.0 1.0 0.4 0.6 460 0.25xl0-3.03 5000  mm Hg  ml/hr ml/hr  2  ml mmHg/ml mmHg ml  *These a r e the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s i l l u s t r a t e d i n F i g u r e s 31a t o 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 i n t r o d u c e d i n t o the  model w i t h S t a r l i n g ' s H y p o t h e s i s , e q u a t i o n (18); e l e v a t i o n of KF w i l l result  i n an i n c r e a s e i n the t r a n s e n d o t h e l i a l flow, JV.  flow d i f f e r e n c e  (JV-JL) i s the r a t e of f l u i d  extravascular f l u i d KF  results  space.  total extravascular f l u i d  accumulation  F i g u r e 31a i l l u s t r a t e s  i n an i n c r e a s e i n ( J V - J L ) .  The f l u i d i n the  that an Increase i n  T h e r e f o r e the time t o reach a  volume of 1000 ml decreases w i t h an i n c r e a s e  i n KF ( F i g u r e 31b). The time t o reach a VTOT of 1000 ml ( t ( 1 0 0 0 ) ) decreases by approximately  t w o - t h i r d s (29.4 h r s to 9.9 h r s ) as KF i s  doubled  t o 2KF(normal) and by h a l f  doubled  from 2KF(normal) to 4KF(normal).  sensitive  (9.9 h r s t o 4.3 h r s ) when KF i s T h e r e f o r e , t(1000)  to the range o f KF c l o s e t o KF(normal).  I n c r e a s e s i n the e n d o t h e l i a l f i l t r a t i o n reduces 31c).  i s most  coefficient  also  the time t o reach the onset of a l v e o l a r f l o o d i n g ( F i g u r e As with t ( 1 0 0 0 ) , the time to reach the onset of a l v e o l a r  f l o o d i n g i s most s e n s i t i v e t o changes i n KF near Following f l o w (JAS) r i s e s  the onset of a l v e o l a r f l o o d i n g  The r a t e of f l u i d  (and c e l l u l a r ) space  As KF  accumulation  i n the  (JNET1) a l s o i n c r e a s e s w i t h KF.  i n t e g r a l over time of JNET1 i s the f l u i d the f l u i d  the t r a n s e p i t h e l i a l  to a v a l u e c l o s e t o (JV-JL) ( F i g u r e 31a).  i n c r e a s e s JAS a l s o i n c r e a s e s . interstitial  KF(normal).  volume VIS1. I n F i g u r e 31d  volume VIS1 a t a VTOT of 1000 ml i s seen  to i n c r e a s e with  KF. The  i n i t i a l v a l u e of (JV-JL) a t a KF/KF(normal) of 10 i s  approximately  200 ml/hr,  The  as compared t o 45 ml/hr f o r KF=KF(norm).  -165-  F i g u r e 31a  T r a n s i e n t Responses o f (JV-JL) and JAS f o r D i f f e r e n t KF ( C o n d i t i o n s as i n T a b l e 21) - Responses c o n t i n u e d to time when VTOT = 1000 ml  250  Legend (JV-JL), 1KF(norm)  200 H  J^SJKF^orm)_  (JV-JL), 2KF(norm) Jj^^FfcormJ  150  (/)  3 o  100 A  50-1  10  20  TIME (hrs)  30  -166-  F i g u r e 31b  Time to Reach a VTOT of 1000 ml f o r D i f f e r e n t KF ( C o n d i t i o n s as i n Table 21)  30  i  i  i  i  0  2  4  6  1 8  KI/KF(normal) : ENDOTHELIUM  -167-  F i g u r e 31c  Time to Reach t h e Onset of A l v e o l a r F l o o d i n g f o r D i f f e r e n t KF ( C o n d i t i o n s as i n T a b l e 21)  -168-  F i g u r e 31d  450  H 0  1  1  1  2  4  6  - |  8  KI/KF(normal) : ENDOTHELIUM  10  -169-  The magnitude of the f l u i d KF(normal) i s l a r g e . verify  flow f o r the c o n d i t i o n with KF=10  Experimental  r e s u l t s should be o b t a i n e d to  t h i s magnitude of t r a n s e n d o t h e l i a l flow. Guyton e t a l . (51) s t u d i e d h y d r o s t a t i c a l l y - i n d u c e d  edema on dog l u n g s .  I n cases where a l v e o l a r  pulmonary  f l o o d i n g o c c u r r e d the  dogs d i d not s u r v i v e past one or two hours a f t e r the onset  of edema.  Pare" and Dodek. ( p e r s o n a l communication,52) s t a t e d that a l v e o l a r f l o o d i n g begins w i t h i n a f r a c t i o n of an hour f o l l o w i n g the onset of acute pulmonary edema.  Under the c o n d i t i o n s shown i n Table 21 and  w i t h a normal KF the times  t o reach the onset of a l v e o l a r f l o o d i n g and  a VTOT of 1000 ml a r e 2.3 h r s and 29.4 h r s , r e s p e c t i v e l y . the v a l u e of KF reduces  the time t o reach the onset  Doubling  of a l v e o l a r  f l o o d i n g to 0.9 h r s and the time to reach a VTOT of 1000 ml to 9.9 h r s - much c l o s e r t o the range of the suggested  5.9  times.  The Response of the Pulmonary Microvascular Exchange System to changes i n APERM The  plasma p r o t e i n p e r m e a b i l i t y parameters of the e n d o t h e l i a l  membrane a r e : 1)  the s o l u t e r e f l e c t i o n c o e f f i c i e n t - SIGD  2)  the p e r m e a b i l i t y - s u r f a c e area product  f o r albumin and  g l o b u l i n - P S A , PSG 3)  the s o l v e n t drag r e f l e c t i o n c o e f f i c i e n t  globulin-SIGFA,  SIGFG  f o r albumin and  -170-  Changes i n the s o l u t e p e r m e a b i l i t y o f the endothelium APERM - t h i s change i s expressed  a r e d e f i n e d by  as a percent of normal.  APERM r e f e r s t o i n c r e a s e s i n PSA and PSG and decreases and  Increases i n  i n SIGD, SIGFA  SIGFG. The  effect  of changes i n APERM on the p r e d i c t i o n s o f the PMVES  s i m u l a t i o n was s t u d i e d under the c o n d i t i o n s l i s t e d epithelial  filtration  i n T a b l e 22. The  c o e f f i c i e n t was r e p r e s e n t e d as a f u n c t i o n o f  (VIS1-VT0NS) w i t h NK equal t o 0.5 h r  - 1  mmHg . -1  At time zero the  PMVES was s u b j e c t e d t o a s t e p change i n PMV t o 50 mmHg.  I n the cases  where APERM was g r e a t e r than zero the p e r t u r b a t i o n t o the PMVES was an i n c r e a s e i n PMV, and an i n c r e a s e i n the p e r m e a b i l i t y o f the endothelium  to s o l u t e s .  S o l u t e flow from expressed equation  mathematically  the c i r c u l a t i o n t o the i n t e r s t i t i u m i s by the Kedem-Katchalsky (K-K) s o l u t e f l u x  ( e q u a t i o n ( 2 0 ) ) ; the d i f f u s i v e  the c o n v e c t i v e term i n c l u d e s (SIGF)^. c o n c e n t r a t i o n used  i n equation  protein concentration. of  term  The i n t e r s t i t i a l  (20) i s (CAV)^  g/ml  - the e f f e c t i v e  from 0.038 g/ml a t time zero t o a p p r o x i m a t e l y 0.029  as shown i n F i g u r e 32a.  than CMVA (.042  T h e r e f o r e , f o r APERM = 0 the v a l u e s of  both the d i f f u s i v e term and the c o n v e c t i v e term o f e q u a t i o n positive. rises  response  F o r the case of APERM=0  a t which time VTOT = 1000 ml; CAVA i s always l e s s  g/ml),  protein  F i g u r e 32a i l l u s t r a t e s the t r a n s i e n t  CAVA f o r changes i n APERM from 0 t o 100%.  CAVA decreases  i n c l u d e s (PS)^, w h i l e  I n F i g u r e 32b the albumin  content  (20) a r e  i n the i n t e r s t i t i u m (QA)  to a maximum f o r APERM = 0 and then d e c l i n e s s l o w l y ;  initially  -171-  Table 22  C o n d i t i o n s f o r the PMVES S i m u l a t i o n s Conducted t o Study Changes i n APERM*  50% 0.375 4.5 1.5 0.2 0.3  0% 0.75 3.0 1.0 0.4 0.6  APERM: SIGD PSA(ml/hr) PSG(ml/hr) SIGFA SIGFG  90% 0.075 5.7 1.9 0.04 0.06  100% 0 6.0 2.0 0 0  KAS = NK(VIS1 -VTONS) f o rVISI > VTONS NK = 0.5 h r mmHg" - 1  PMV KF VTONS SL B VLMH  =  =  1  50 1.12 460 0.25xl0-3.03 5000  2  mmHg ml/hr/mmHg ml mmHg/ml mmHg ml  * These are the c o n d i t i o n s of the s i m u l a t i o n s that produced the r e s u l t s of F i g u r e s 32a t o 32c.  -172-  F i g u r e 32a:  T r a n s i e n t Responses o f Albumin P r o t e i n C o n c e n t r a t i o n CAVA f o r d i f f e r e n t APERM ( C o n d i t i o n s as i n T a b l e 22 Responses c o n t i n u e d to time when VTOT = 1000 ml  0.08 Legend  0.06-  PERM = o % PERM = 50 % PERM = 90 %  0.04-  0.02-  0.00 10  20  TIME (hrs)  30  -173-  there I s a net accumulation by  of albumin i n the i n t e r s t i t i u m ,  followed  a net d e p l e t i o n . An  i n c r e a s e i n APERM by 50% r e s u l t s  i n the response of CAVA as  shown i n F i g u r e 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 , f o r the case of APERM = 90% CAVA i n c r e a s e s from 0.038 g/ml at  time zero to a value g r e a t e r than CMVA by the time VTOT = 1000 ml  (approximately  21 h r s ) .  The time at which CAVA=CMVA i s 14.5 h r s . f o r  APERM = 50% and 3.6 h r s . f o r APERM=90%.  Solute concentration i s  determined by the r a t i o of s o l u t e weight t o f l u i d the onset  volume.  Following  of a l v e o l a r f l o o d i n g (2.5 h r s ) the albumin weight i n the  interstitial QA i n c r e a s e s .  fluid  rises  ( F i g u r e 32b); I n a d d i t i o n , as APERM i n c r e a s e s  However, the f l u i d  i n c r e a s e s very l i t t l e  volume VIS1, as shown i n F i g u r e 32c,  f o l l o w i n g the onset  of a l v e o l a r f l o o d i n g ; as  APERM i n c r e a s e s the change i n VIS1 i s a l s o n e g l i g i b l e .  T h e r e f o r e , the  r a t i o of QA to VIS1 w i l l  r i s e f o l l o w i n g the onset  and  Whether CAVA exceeds CMVA, or n o t ,  as APERM i n c r e a s e s .  dependent on the s i z e of the i n c r e a s e i n QA. QNETA over  of a l v e o l a r f l o o d i n g is  QA i s the i n t e g r a l of  time, which i s determined by e q u a t i o n ( 3 3 ) :  QNETA = JSA-JL.CTA-JAS.CTA  As  JSA i s i n c r e a s e d QA w i l l  CMVA, the c o n v e c t i v e  (33)  also increase.  term of e q u a t i o n  e v a l u a t i o n of JSA; as SIGFA i s reduced  F o r v a l u e s of CAVA near  (20) i s dominant i n the ( i . e . APERM i n c r e a s e d ) the  v a l u e o f the c o n v e c t i v e term i n c r e a s e s , and QA  rises.  -174-  F i g u r e 32b:  T r a n s i e n t Responses o f Albumin P r o t e i n Weight QA f o r d i f f e r e n t APERM ( C o n d i t i o n s as i n T a b l e 22) Responses continued t o time when VTOT = 10Q0 ml  20  Legend PERM = o % ^ERK4 = 50%_ PERM = 90 % PERM = 100 % 0  10  20  TIME (hrs)  30  -175-  F i g u r e 32c:  T r a n s i e n t Responses o f F l u i d Volume VIS1 f o r d i f f e r e n t APERM ( C o n d i t i o n s as i n T a b l e 22) - Responses c o n t i n u e d to time when VTOT = 1000 ml  500  > 400-  o >  Legend PERM = 0 %  350  PERM = 5 0 % PERM = 9 0 % PERM = 1 0 0 %  300  10  20  TIME (hrs)  30  -176-  F i g u r e 32d i l l u s t r a t e s (t(1000))  the time to reach a VTOT of 1000 ml  f o r changes i n APERM; as APERM i n c r e a s e s from 0 t o  approximately  80%, t(1000) decreases from 29 h r s . t o 19 h r s . The  parameter SIGD i s i n t r o d u c e d with with  (PIMV-PIPMV) t o y i e l d  S t a r l i n g ' s H y p o t h e s i s and combined  the e f f e c t i v e  oncotic pressure d i f f e r e n c e  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 . decreases, difference.  resulting  i n a decreasing  Therefore  As APERM i n c r e a s e s SIGD  e f f e c t i v e oncotic  pressure  the t r a n s e n d o t h e l i a l flow r a t e i n c r e a s e s .  However, the i n c r e a s e i n JV i s s m a l l s i n c e the h y d r o s t a t i c difference The  (PPMV-PPMV) i s dominant when PMV i s e l e v a t e d  primary e f f e c t  pressure  t o 50 mmHg.  of changes i n APERM i s on the transmembrane p r o t e i n  movement. S h i r l e y et a l . (53) have c a r r i e d out t r a c e r s t u d i e s on the lymph and plasma o f dog l u n g s . i n c r e a s e i n blood  They observed that f o l l o w i n g a l a r g e  volume (which corresponds t o a l a r g e r i s e i n PMV)  the r a t i o of lymph to plasma s o l u t e c o n c e n t r a t i o n s observation  rose.  From  this  S h i r l e y e t a l . (53) suggested t h a t the e n d o t h e l i a l  membrane became more porous d u r i n g as the s t r e t c h e d pore model. of 50 mmHg the r a t i o CAVA/CMVA  the high PMV - t h i s was c l a s s i f i e d  F i g u r e 32a shows t h a t a t an e l e v a t e d PMV (comparable t o the lymph-plasma r a t i o )  may r i s e i f APERM i s i n c r e a s e d t o v a l u e s r e s u l t s o f S h i r l e y e t a l . (53) the r i s e  such as 50%.  However, i n the  i n lymph t o plasma s o l u t e  c o n c e n t r a t i o n i s p r i m a r i l y due t o a drop i n the plasma  concentration  f o l l o w i n g the i n f u s i o n of an albumin s o l u t i o n , and not due t o a significant  r i s e i n lymph s o l u t e c o n c e n t r a t i o n .  In the case of the  -177-  simulations  of the PMVES where APERM i s changed i t i s the lymph  concentration  that  solute  i s increased.  Control o f Error in the Numerical Solutions  5.10  The  n u m e r i c a l i n t e g r a t i o n technique used i n the computer  program of the a l v e o l a r model was the Runge-Kutta-Merson method, supplied  as a s u b r o u t i n e by the UBC Computing C e n t r e .  evaluation  of the v a r i a b l e s ,  such as VIS1,  I n the  because of the  approximations used In the c a l c u l a t i o n there i s an e r r o r w i t h the s o l u t i o n i n each i n t e g r a t i o n s t e p .  This  i n the computer program by employing a v a r i a b l e a tolerance the  error  1 x 10 using  - 9  associated  e r r o r was minimized  time step and s e t t i n g  l e v e l on the s i z e of the e r r o r that was a c c e p t a b l e .  i n the i n t e g r a t e d  variables  exceeded a t o l e r a n c e  If  l e v e l of  , the time step was reduced and the i n t e g r a t i o n was repeated  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 e v a l u a t e PPMV (from VIS1)  and the values of v a r i a b l e s  at the c h a r a c t e r i s t i c p o i n t  such as the time to reach a VTOT of 1000 ml. e r r o r i n the s o l u t i o n s  In t h i s technique the  i s dependent on the d i f f e r e n c e  i n v a l u e of the  independent v a r i a b l e a t the end-points o f the i n t e r v a l . the  Generally,  as  time i n t e r v a l of the c a l c u l a t i o n s decreased, the e r r o r i n the  dependent v a r i a b l e decreased. at the c h a r a c t e r i s t i c p o i n t s ,  F o r the c a l c u l a t i o n s o f the v a r i a b l e s t h i s time i n t e r v a l was 0.25 h r s .  -178-  F i g u r e 32d:  Time to Reach a VTOT of 1000 ml f o r D i f f e r e n t APERM ( C o n d i t i o n s as i n Table 22)  30  O O O  28 A 26-  >  o <  24 H  o  22-1  20-1  18-| 0  1 20  , 40  , 60  , 80  PERM(%) : ENDOTHELIUM  100  -179-  SUMMARY AND CONCLUSIONS  1.  The a l v e o l a r model d e s c r i b e s one way t o i n t e g r a t e the a i r space  w i t h the lumped compartment model ( i n t e r s t i t i a l model) o f the PMVES as developed 2.  by Bert and P i n d e r  Estimates  (29).  f o r the parameters i n t r o d u c e d with the a l v e o l a r model  were o b t a i n e d : (a)  Epithelial Filtration  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 r e p r e s e n t e d as a constant or a v a r i a b l e . variable coefficient literature  (33).  appears  We suggest  parameter NK of 0.5 h r (b)  -  1  A  t o be more compatible  with  a v a l u e f o r the s e n s i t i v i t y  mmHg . -1  VTONS: For cases of h y d r o s t a t i c edema and edema caused by changes to the e n d o t h e l i a l p e r m e a b i l i t y , a value of 460 ml is  a good e s t i m a t e .  A VTONS o f 460 ml corresponds  to a  PPMV of +1.1 mmHg - a v a l u e w i t h i n the range suggested by Guyton(47). (c)  SL: No v a l u e o f SL was found suggest  (d)  i n the l i t e r a t u r e  a v a l u e o f 0.25 x I O  B: No v a l u e of B was found  - 2  survey.  mmHg/ml.  i n the l i t e r a t u r e  survey.  suggest a v a l u e o f -3.03 mmHg, which corresponds expected 3.  The presence  alveolar fluid  VTONS w i l l  We  t o the  p r e s s u r e a t normal c o n d i t i o n s .  of a maximum lymph c a p a c i t y may be e a s i l y i n t r o d u c e d  i n t o the model through PMVES c a r r i e d  We  parameter VLMPH.  out f o r t h i s t h e s i s a f l u i d  In the s i m u l a t i o n s of the volume VLMPH l e s s  a c c e l e r a t e the p r o g r e s s i o n o f edema.  Values  than  o f VLMPH  -180-  g r e a t e r than VTONS have very  little  effect  (up  to the time when VTOT = 1000 m l ) .  4.  I n h y d r o s t a t i c a l l y induced  above a minimum p r e s s u r e reach  the onset  on the p r o g r e s s i o n of edema  pulmonary edema PMV must be r a i s e d  f o r a l v e o l a r f l o o d i n g to o c c u r .  The time to  of a l v e o l a r f l o o d i n g i s most s e n s i t i v e to s m a l l  rises  i n PMV above t h i s minimum v a l u e . 5.  I f PMV i s r a i s e d above the minimum value s p e c i f i e d  i n c r e a s e s i n KF w i l l reach  the onset  a c c e l e r a t e the progress  of edema.  i n ( 4 ) , then The time to  of a l v e o l a r f l o o d i n g i s most s e n s i t i v e t o s m a l l  i n c r e a s e s i n KF above i t s normal v a l u e . 6.  The i n t e r s t i t i a l  to plasma p r o t e i n c o n c e n t r a t i o n r a t i o may  i n c r e a s e or decrease  d u r i n g the p r o g r e s s i o n of pulmonary edema  depending on the s i z e of the change i n the e n d o t h e l i a l s o l u t e permeability  parameters.  -181-  RECOMMENDATIONS FOR FURTHER WORK  1.  To f u r t h e r u t i l i z e  experimental or  data must be o b t a i n e d to v e r i f y  the model's  model  predictions,  to suggest a l t e r n a t i v e ways to i n t e g r a t e the a i r space w i t h the  existing 2.  the computer s i m u l a t i o n of the a l v e o l a r  i n t e r s t i t i a l models of the PMVES.  The computer program of the a l v e o l a r model should be improved so  t h a t i t may  be more "user f r i e n d l y " .  In t h i s way  the s i m u l a t i o n may  be used by the h e a l t h care p r o f e s s i o n f o r t e a c h i n g purposes. 3.  F u t u r e development of the a l v e o l a r model should i n c o r p o r a t e : (a)  the e f 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, D o l l e r y and Naimark  (3)  (c)  the e f f e c t  (d)  a r e c o v e r y phase f o l l o w i n g the p r o g r e s s i o n of pulmonary edema.  of l o c a l i z e d  pulmonary edema on the PMVES  -182NOMENCLATURE A l v e o l a r f l u i d p r e s s u r e a t the onset of a l v e o l a r f l o o d i n g (mmHg) CAVA,CAVG  The and  e f f e c t i v e i n t e r s t i t i a l concentration g l o b u l i n r e s p e c t i v e l y (g/ml)  (CL)i  Lymph p r o t e i n c o n c e n t r a t i o n  CMVA,CMVG  The blood plasma c o n c e n t r a t i o n g l o b u l i n , r e s p e c t i v e l y (g/ml)  CP,CPMV,CPPMV  The t o t a l p r o t e i n c o n c e n t r a t i o n f o r blood plasma (CPPMV) and i n t e r s t i t i a l f l u i d (CPPMV) (g/ml)  CTA,CTG  The t i s s u e c o n c e n t r a t i o n r e s p e c t i v e l y (g/ml)  JAS,JL,JV  The t r a n s e p i t h e l i a l , lymph, and t r a n s e n d o t h e l i a l f l o w s , r e s p e c t i v e l y (ml/hr)  JL(max)  The parameter r e p r e s e n t i n g lymph flow ( m l / l h r )  JNET1  The r a t e of f l u i d accumulation i n the e x t r a v a s c u l a r e x t r a a l v e o l a r space (ml/hr)  JSA.JSG  The t r a n s e n d o t h e l i a l s o l u t e flows f o r albumin and g l o b u l i n , r e s p e c t i v e l y (g/hr)  KAS,KF  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 e p i t h e l i u m and endothelium, r e s p e c t i v e l y (ml/hr/mmHg)  KASO  The parameter r e p r e s e n t i n g a constant f i l t r a t i o n c o e f f i c i e n t (ml/hr/mmHg)  LF  The f l u i d c o n d u c t i v i t y (cm/hr/mmHg)  NK  The s e n s i t i v i t y parameter f o r a v a r i a b l e filtration coefficient ( h r mmHg )  epithelial  PA  The h y d r o s t a t i c p r e s s u r e of the a r t e r i a l blood v e s s e l (mmHg)  segment of a  PALV.PG  The h y d r o s t a t i c (mmHg)  PAS,PL  The  f o r solute  of albumin  ' i ' (g/ml)  of albumin and  of albumin and g l o b u l i n ,  fluid  the v a l u e of the maximum  coefficient  - 1  epithelial  of the endothelium  -1  p r e s s u r e of the gas i n the a i r space  alveolar fluid  p r e s s u r e (mmHg) Continued....  -183-  NOMENCLATURE  (Cont.d)  PI,PIAS,PIMV,  The c o l l o i d osmotic ( o r o n c o t i c ) p r e s s u r e of the a l v e o l a r 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)  PLA  The  PMV  The pulmonary m i c r o v a s c u l a r h y d r o s t a t i c (mmHg)  PPA  The  PPMV,PEA  The h y d r o s t a t i c p r e s s u r e of the a l v e o l a r and e x t r a a l v e o l a r i n t e r s t i t i a l f l u i d , r e s p e c t i v e l y (mmHg)  PSA,PSG  The e n d o t h e l i a l p e r m e a b i l i t y - s u r f a c e area products f o r albumin and g l o b u l i n , r e s p e c t i v e l y (ml/hr)  PV  The h y d r o s t a t i c p r e s s u r e of the venous segment of a blood v e s s e l (mmHg)  QA.QG  The i n t e r s t i t i a l s o l u t e content of albumin and g l o b u l i n , r e s p e c t i v e l y (gm)  left  atrial  hydrostatic  pressure  pulmonary a r t e r i a l h y d r o s t a t i c  (mmHg) pressure  pressure  (mmHg)  QNETA,QNETG  The for  r a t e of s o l u t e accumulation i n the i n t e r s t i t i u m albumin and g l o b u l i n r e s p e c t i v e l y (g/hr)  (QO)i SA  The s o l u t e weight i n the i n t e r s t i t i u m f o r s o l u t e ' i ' a t time zero (gm) S u r f a c e area of pulmonary c a p i l l a r y membrane ( c m )  SIGD.SIGDAS  The s o l u t e r e f l e c t i o n c o e f f i c i e n t f o r f l u i d of t h e e n d o t h e l i a l and e p i t h e l i a l membranes, r e s p e c t i v e l y .  SIGFA,SIGFG  The s o l v e n t drag r e f l e c t i o n c o e f f i c i e n t of the e n d o t h e l i a l membrane to albumin and g l o b u l i n , respectively  2  SL  The slope of the PAS-VAS curve f o r the a l v e o l a r (mmHg/ml)  VAS  The  alveolar f l u i d  VAVA.VAVG  The and  available i n t e r s t i t i a l fluid g l o b u l i n , r e s p e c t i v e l y (ml)  VCELL  The  cellular  VEXA.VEXG  The excluded f l u i d r e s p e c t i v e l y (ml)  fluid  fluid  volume (ml) volume f o r albumin  volume (ml) volume to albumin and g l o b u l i n ,  Continued....  -184-  NOMENCLATURE  (Cont.d)  VIS  The i n t e r s t i t i a l VCELL)(ml)  VIS1  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 e x t r a v a s c u l a r - e x t r a a l v e o l a r (EVEA) space (VIS1=VT0T-VAS)(ml)  VIS10  The i n t e r s t i t i a l (ml)  VLMPH  The 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 c o r r e s p o n d i n g to the maximum lymph flow (ml)  VTONS  The 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 of a l v e o l a r f l o o d i n g (ml)  VTOT  The t o t a l e x t r a v a s c u l a r The r a d i u s (cm) The s u r f a c e  fluid  volume  (VIS=VTOT-VAS-  (+cellular) fluid  of c u r v a t u r e :  tension  fluid  volume at time zero  volume at the onset  volume (ml)  used i n L a p l a c e ' s E q u a t i o n  of a f l u i d  (dynes/cm)  -185-  REFERENCES 1.  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Brennan, E f f e c t of continuous p o s i t i v e - p r e s s u r e v e n t i l a t i o n (CPPV) on edema f o r m a t i o n i n dog lung, J o u r n a l of A p p l i e d P h y s i o l o g y 39/4): 672-679, 1975.  51.  Guyton, A . C , & A.W. L i n d s e y , E f f e c t of e l e v a t e d l e f t a t r i a l p r e s s u r e and decreased plasma p r o t e i n c o n c e n t r a t i o n 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.  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 P e r m e a b i l i t y to Macromolecules: s t r e t c h e d pore phenomenon. American J o u r n a l of P h y s i o l o g y 190(2): 189-193, 1957.  54.  Jacob, S.W. C.A. Francone, and W.J. F u n c t i o n i n Man, F i f t h E d i t i o n , W.B. 1982.  and  P. Dodek, P e r s o n a l  communication.  Lossow, S t r u c t u r e and Saunders Company, Toronto,  -189-  55.  Prockop, D.J. C o l l a g e n , E l a s t i n and P r o t e o g l y c a n s : M a t r i x f o r F l u i d Accumulation i n the Lung, I n : Pulmonary Edema, E d i t e d by A.P. Fishman and E.M. Renkin, American P h y s i o l o g i c a l S o c i e t y , B a l t i m o r e , 1979.  56.  Moore, C. UBC RKC: Runge K u t t a w i t h E r r o r C o n t r o l , Computing Centre, The U n i v e r s i t y of B r i t i s h Columbia, 1983.  57.  M a i r , S.G., UBC PLOT: The UBC p l o t s u b r o u t i n e s and programs. Computing Centre, The U n i v e r s i t y of B r i t i s h Columbia, 1984.  -190-  APPENDIX A:  Al.l  Input F i l e s EDA and PDA The  Table  parameters of the i n p u t f i l e s EDA and PDA were d e f i n e d i n  5 and 6, r e s p e c t i v e l y .  assigned for  The Computer Program  An example o f the n u m e r i c a l  to these parameters i s shown i n Table  PDA.  (The n u m e r i c a l v a l u e s must s t a r t  column 21 f o r PDA).  values  23 f o r EDA and Table 24  i n column 11 f o r EDA and  The parameters i n t r o d u c e d with  the a l v e o l a r model  - VTONS, SL, B, and VLMPH - assume v a l u e s as s t a t e d i n s e c t i o n 4.4. In t h i s 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 v a r i a b l e ; t h e r e f o r e NK i s a s s i g n e d a v a l u e , i n t h i s case 0.5 h r mmHg , and KASO i s s e t equal t o zero -1  r e p r e s e n t e d as a c o n s t a n t , a s s i g n e d a constant endothelium  value.  (see Table  - 1  2 4 ) . I f KAS i s  then NK i s s e t equal t o zero and KASO The p e r m e a b i l i t y parameters of the  and the e n d o 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  normal v a l u e s as s e l e c t e d by Bert and Pinder  (29).  assume the  As shown i n f i l e  PDA the h y d r o s t a t i c p r e s s u r e PMV i s s e t a t 50 mmHg, r a i s e d  from i t s  normal v a l u e of 9 mmHg. The  meaning of the parameters TAUMX1, SUBNT1, TAUMX2 and SUBNT2  may be e x p l a i n e d as f o l l o w s :  Up t o a time of 25 h r s (TAUMX1) the  v a l u e s o f the v a r i a b l e s l i s t e d  i n Table 9 w i l l be s t o r e d every hour  (SUBNT1); SUBNT1 must d i v i d e evenly hrs  i n t o TAUMX1.  From a time  (TAUMX1) t o 100 h r s (TAUMX2) the v a l u e s o f the v a r i a b l e s w i l l be  s t o r e d every 5 h r s . (SUBNT2); SUBNT2 must d i v i d e evenly i n t o and  of 25  TAUMX2.  A d e t a i l e d account  o f the t r a n s i e n t  response  TAUMX1  of the PMVES  -191-  T a b l e 23:  1 2 3 4 5 6 7 8 9 10 1 1 1 2 1 3 14 15 1 6 i 7 18 1 9 20 21 22 23 24 25 26 27 28  Example o f Numerical Values to Parameters i n F i l e EDA  VTOTO VASO QAO QGO PIMV VCELL VEXA VEXG VTONS VLMPH B CMVA CMVG FPPMV(1) FPPMV(2) FPPMV(3) FPPMV(4) FPPMV(5) FPPMV(6) FPPMV(7) FPPMV(8) FPPMV(9) KF SIGD PSA PSG SIGFA SIGFG  = = = = = = = = =  Assigned  378 .7D0 0 . ODO 5 .81D0 2 . 36D0 25 .0D0 150 . ODO 73 . 5D0 1 1 5. 5D0 459 .9D0 5000 . ODO -3 .03D0 0 .042D0 0 .0271D0 -2 .74D0 -2 . 40D0 - 1. 9D0 -1 . 3D0 -0 • 8D0 -0 . 3D0 0 . 1 5D0 0 . 60D0 1 .09D0 1 . 1 2D0 0 .750D0 3 . 00D0 1 . 00D0 0 .400D0 0 .600D0  -192-  T a b l e 24:  1 2 3 4 5 6 7 8 9 1  0  11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  Example o f Numerical Values Assigned Parameters i n F i l e PDA  PL0TS(Y=1,N=2) TABLES 1(Y=1,N=2) TABLES 2(Y=1,N=2) TAUMX1 SUBNT1 TAUMX2 SUBNT2 STEPSZ HMIN TOL KASO NK SL PMV XS(1) XS(2) YS1(1) YS2(1) YS30) YS4(1) SFT SFX SFY ZMN SFZ SFK SFR SFU  to the  = 2 = 1 = 1 = 25.ODO = 1.0D0 = 100.ODO = 5 . ODO = 0.01D0 = 0.001D0 = 0.0000000001D0 = 0.00 = 0 . 5 0 = 0.0025 = 50.00. = 5.875 = 7.00 = 7.5625 = 7.1875 = 6.8125 = 6.4375 = 10.00 = 15.0 = 600.0 = -5.0 = 5.0 = 20.0 = 2.5 = 0.010  -193-  s l m u l a t i o n i s p r o v i d e d from time zero to TAUMX1, as v a l u e s a r e r e c o r d e d 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 h r s . , and hence l e s s TAUMX1 was chosen  from p r e l i m i n a r y t e s t s as the time when a l l the  v a r i a b l e s were not changing  A1.2  detailed.  r a p i d l y with  time.  The Main Program UBCEDEMA F o l l o w i n g i s a copy of the main program UBCEDEMA.  of the contents a r e presented throughout  Explanations  the program by comment  statments. The  n u m e r i c a l i n t e g r a t i o n s of JNET1, JAS, QNETA and QNETG over  time to y i e l d  VIS1,  VAS, QA and QG, r e s p e c t i v e l y , a r e conducted  the s u b r o u t i n e "DRKC", p r o v i d e d by the U n i v e r s i t y of B r i t i s h Computing C e n t r e .  with  Columbia  The d e f i n i t i o n s of the terms a r e p r e s e n t e d i n T a b l e  25. The  t i s s u e compliance  curve i s r e q u i r e d  to determine the  i n t e r s t i t i a l h y d r o s t a t i c p r e s s u r e PPMV c o r r e s p o n d i n g to the f l u i d volume VIS1. VIS1 26.  A subroutine e n t i t l e d  (statement  166 and 683):  CMPLNC i s used  t o r e l a t e PPMV to  the terms of CMPLNC a r e d e f i n e d i n T a b l e  -194-  COMPUTER PROGRAM UBCEDEMA  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 1 2 3 4 5 5.5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  C C C  22  23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 3£ 3S 40 41 42 43 44 45 46 47 46 49 50 5 1 52 53 54 55 56 57  C C C  C C C  1  IMPLICIT REAL*8(A-Z) DIMENSION Y l 0 ( 5 ) , F 1 0 ( 5 ) , F P P M V ( 1 0 ) , G ( 5 ) , S ( 5 ) , T ( 5 ) COMMON/BLCA/SIGD,PSA,PSG,SIGFA,SIGFG COMMON/BLCB/PIMV,VCELL,VEXA,VEXG,CMVA,CMVG COMMON/BLCC/FPPMV,VTONS,B,VLMPH,KF,KASO,NK,SL,PMV COMMON/BLCD/PMV1,PMV2,PTAU1,PTAU2 REALM X1(201),X2(201),X3(20l),X4(201),Y1(201),Y2(201), *Y3(201),Y4(201),21(201),Z2(201),Z3(201),R1(201),R2(201), *S1(201),S2(201),U1(201),U2(201),V1(201),V2(201),W1(201), *W2(20l),TI(201),KA(201),PI(201),XS(2),YS1(2),YS2(2), *YS3(2),YS4(2) REAL*4 TMN,SFT,XMN,SFX,YMN,SFY,ZMN,A 1,AA1 , *SFZ,RMN,SFR,UMN,SFU,KMN,SFK INTEGER L,LL,LLL,LLLL,M,N,NN,NNN,NNNN,1,11,1 CHECK,I FLAG, *II,J,JJ,JJJJ EXTERNAL FUNC INPUT DATA FROM F I L E EDA RSAD(4,1)VTOTO,VASO,QAO,QGO,PIMV 1 FORMAT(4(T11., D1 3 . 5 , /),T11,D13.5) READ(4,3)VCELL,VEXA,VEXG,VTONS,VLMPH, *B 3 FORMAT(5(T11,D13.5,/),T11,D13.5) READ(4,4)CMVA,CMVG 4 FORMAT(T11,D13.5,/,T11,D13.5) READ(4,5)(FPPMV(J),J=1,9) 5 FORMAT(8(T11,D13.5,/),T11,D13.5) READ(4,6)KF,SIGD,PSA,PSG,SIGFA,SIGFG 6 FORMAT(5(T11,D13.5,/),T11,D13.5) INPUT DATA FROM F I L E PDA READ(5,12)NN READ(5 , i 2)NNN READ(5,12)NNNN 12 FORMAT(T21 ,I 1 ) READ(5,16)TAUMX1 READ(5,16)SUBNT1 READ(5,16 JTAUMX2 READ(5,16)SUBNT2 READ(5,16)STEPSZ READ(5,17)HMIN RSAD(5, 17JTOL READ(5,16)KASO READ(5,16)NK READ(5,16)SL READ(5,16)PMV 16 FORMAT(T21,D13.6) 17 FORMAT(T21,D19.12) PRINTING OF INPUT DATA WRITE(7,401) 401 FORMAT(1H1,1 OX,'INPUT PARAMETERS FOR RUN',///) WRITE(7,402)CMVA,CMVG 402 FORMAT(IX,'PLASMA CONCENTRATION : ALBUMIN',8X,'CMVA *D13.5,/,24X,'GLOBULIN',7X,'CMVG = ',D13.5,/)  -195-  Listing  of  UBCEDEMA a t  53 59 60  on MAR 2 5 ,  1985  for  CCid=HEIJ Page  62 63 64 65 66  c c  c  INITIAL!ZiNG  VARIABLES  & COUNTERS  1=0 :CHECK=:  I?LAG= ' L= \  M= 2 C = 0.0 TAu = 0 . 0 HSTE?=STE?SZ  SU3:NT=SU3N?1 QA=QAO QG=QGO  VAS=VASO VTOT = VTOTO  V I S i 0=V70T0-VASC VIS1=VIS10  c c c c  2  WRI T E ( 7 , 4 0 3 ) V C E L L , VEXA,VEXG FORMAT(IX,'CELLULAR V O L U M E ' , 2 3 X , ' V C E L L = ',D13.5,//,1X, * ' E X C L U D E D VOLUME : A L 3 U M I N ' , 1 3 X , ' V E X A = ',D13.5,/,19X, * ' G L O B U L I N ' , 12X, 'VEXG = ' ,D t 3 . 5 , / ) WRITE(7,404)SIGD ',D13.5,/) 404 FORMAT(IX,'REFLECTION C O E F F I C I E N T ' , 1 6 X , ' S I G D WRITE(7,405)PSA,PSG 405 FORMAT(IX,'PERMEABILITY SURFACE AREA : A L 3 U M I N ' , 3 X , *'PSA = ',D13.5,/,29X,'GLOBULIN PSG = ', *D13.5,/) WRITE(7,406)SIGFA,SIGFG 406 FORMAT(1X,'REFLECTION COEFFICIENT : A L 3 U M I N ' , 6 X , *'SIGFA = ',Di3.5,/,26X,'GLOBULIN',5X,'SIGFG = ', *D13.5,/) WRITE(7,407)KAS0,NK,SL 407 FORMAT(IX,'ALVEOLAR FLOODING CONSTANTS : ' , 9 X , ' K A S 0 = ', * D 1 3 . 5 , / , 3 9X,'NK = ' ,D13.5,/,39X,'SL = ',D13.5,/) WRITE(7,4 08)VTONS,VLMPH 408 FORMAT(IX,'ONSET VOLUME FOR FLOODING' , 1 3 X , " V T O N S = ', * D 1 3 . 5 , / / , 1X,'VOLUME OF MAXIMUM LYMPH FLOW' , 1 OX,'VLMPH = ', *D13.5,/) W R I T E O , 409 ) KF , P M V KF = ', 409 FORMAT(1X,'ENDOTHELIUM FILTRATION COEFFICIENT * D 1 3 . 5 , / / , I X , ' P L A S M A HYDROSTATIC P R E S S U R E ' , 1 1 X , ' P M V = ', *D13.5,/) WRITE(7,410)TAUMX1 ,SUBNT1 ,TAUMX2,SUBNT2,STEPS Z,HMIN,TOL 4 1 0 FORMAT(IX,'MAXIMUM TIME OF PRINTING INTERVAL 1 ' , 3 X , *'TAUMX1 = ' , D 1 3 . 5 , / , I X , ' P R I N T I N G OUTPUT INTERVAL l ' , 1 2 X , * ' SUBNT 1 ' , D 1 3 . 5 , / / , 1 X , ' M A X I M U M TIME OF R U N ' , 1 9 X , *'TAUMX2 = ' , D 1 3 . 5 , / , 1 X , ' P R I N T I N G OUTPUT INTERVAL 2 ' , 1 2 X , *'SUBNT2 = ' , D 1 3 . 5 , / / , 1 X , ' I N I T I A L S T E P S I Z E F O R C A L C U L A T I O N S ' , * 5X, ' S T E P S Z = ' , D 1 3 . 5 , / / , 1 X , ' M I N I M U M STEPS IZE F O R CALCULATIONS * 5 X , ' KM IN = ' , D 1 3 . 5 , . / / , IX, ' MAXIMUM TOLERANCE ' , 2 1 X , *'TCL = ',313.5)  403  61  67 68 69 70 7 1 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9 1 92 93 94 95 bc 97 98 99 00 0 \ 02 03 04 05 06 107 105 109 1 1 0 11 1 1 1 2 1 1 3 1 1 4 1 1 5  11:55:23  ASSIGNING I N I T I A L VALUES TO ARRAY YIO FOR UBC SUBROUTINE "DRKC" YIO(1)=VTOTO YIO(2)=QAO  : NEEDED  -196-  Listing 1 1 6 I 1 7 1 18 1 1 9 120 121 122 123 124 125 126 127 1 28 1 29 130 131 1 32 133 134 135 136 137 1 38 139 140 141 142 143 144 1 45 146 147 148 149 1 50 151 1 52 1 53 1 54 1 55 1 56 1 57 1 58 1 59 1 60 1 6 1 162 1 63 1 64 165 166 167 1 68 169 170 171 172 173  of UBCEDEMA a t  1 1 : 5 5 : 2 3 on MAR 2 5 ,  1 9 8 5 f o r C C i d = H E I J Page  YIO(3)=QGO YIO(4)=VASO YI0(5)=VIS10 C C C C  ASSIGNING INITIAL VALUES TO VARIABLES THAT ARE TO BE DETERMINED AT THE "CHARACTERISTIC POINTS" JASMAX=0.DO JASLO=0.DO TIMAX=TAU TILO=TAU TAUON=0.D0 VTOTLO=VTOT VTOTMX=VTOT VISlLO=VIS1 VIS1MX=VIS1 RCAMAX=0.D0 CTAON=QA/VISl CTALO=0.DO  C C C C  DETERMINATION OF THE NUMBER OF INTERVALS ( I I ) OF SIZE SU3NT2 IN THE FIRST TIME RANGE OF MAXIMUM VALUE TAUMX1  c  A1=TAUMX1 AA1=SUBNT2 II=IFIX(A1)/!FIX(AAl)  c c c c c c c c  195  CONTINUE CALCULATION OF INTERSTITIAL VOLUME VIS1=VTOT-VAS VIS=VIS1-VCELL CALCULATION OF AL3UMIN PARAMETERS CTA = C'A/VI S  VAVA=VIS-VEXA C A V A = Q A / V A. V A r  c  CALCULATION OF GLOBULIN  PARAMETERS  C CTG=QG/VIS VAVG=VIS-VEXG CAVG=QG/VAVG C C  c c  c c  CALCULATION OF CENTRAL PARAMETERS CALCULATION OF INTERSTITIAL PRESSURE CALL  CMPLNC(VIS',FPPMV,PPMV)  CALCULATION OF INTERSTITIAL ONCOTIC  PRESSURE  c CP=CAVA+CAVG PIPMV=210.D0*CP+!600.DO*CP*CP+9000.DO*CP*CP*CP c c  CALCULATION OF CAPILLARY  FILTRATION RATE  3  -197-  Listing  of U3CEDEMA a t 1 1 : 5 5 : 2 3  174 17 5 176 177 173 179 180 13! 132 183 184 185 186 137 188 189 190 1 91 192 193 194 195 196 197 1 98 199 200 201 202 203 204 205 206 207 208 209 2 10  cn MAR 2 5 ,  1 9 8 5 f o r CCicl=HEIJ P a g e  C JV=XF*((PMV-PPMV)-SIGD*(PIMV-PIPMV)) C C C C  CALCULATION OF LYMPH FLOW  I F ( V I S 1 . L T . V L M P H ) G O TO 203 C CALCULATION OF MAXIMUM LYMPH FLOW JL=0.17D0*VLM?K-5.56D1 GO TO 204 203 JL=0.17D0*VIS1-5.56D1 C C CALCULATION OF RATE OF EXTRAVASCULAR FLUID C ACCUMULATION C 204 J N E T = J V - J L FIO(l)=JNET C C DETERMINATION IF ALVEOLAR FLOODING OCCURS C I F ( V T O T . L T . V T O N S ) G O TO 207 C C CALCULATION OF FILTRATION COEFFICIENT OF C EPITHELIAL MEM3RANS C KAS=KAS0+NK*(VISI-VTONS) GO TO 208 207 KAS=0.D0 C C CALCULATION OF HYDROSTATIC PRESSURE OF FLUID C IN THE AIR SPACE C 203 PAS=SL*VAS^3 C C CALCULATION CF HYDROSTATIC PRESSURE DIFFERENCE • C ( ? ? M V - ? A S ' BETWEEN" INTERSTITIUM & AIR SPACE C  21!  ? n r =- = ? ? M V - ? A S  2 12 213 2 14 215 216 2:7  C C C  2 18  C  220 22 1 222 223 224 22 5 226 227 227.3 227.6  C  228  C  229  C  CALCULATION CF TRANSEPITHELIAL  FLUID  FLOW RATE  jAS=KAS*IPPMV-PAS) FICU)=JAS  C CALCULATION Cr  RATE OF ACCUMULATION OF  JN'ET i = J V - J L - J A S FIC';3)=JNET1 C C C C  CALCULATION OF TRAMSENDOTHELIUM FLOW RATE  ALBUMIN  JSA=?SA*(CMVA-CAVA)+(I.DO-SIGFA)*(CMVA+CAVA)*JV/2.DO DIFFUS=?SA*(CMVA-CAVA) CONVEC=(1.D0-SIGFA)*(CMVA+CAVA)*JV/2.D0 CALCULATION OF RATE OF I N T E R S T I T I A L ALBUMIN  4  -198-  L i s t i n g o f UBCEDEMA a t 11:55:23 on MAR 25, 1985 f o r C C i d = H E I J Page 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 253 259 260 261 262 263 264 265 266 267 26S 269 27C 27 1 272 273 274 275 276 277 273 279 280 28 1 282 283 284 285 286 287  ACCUMULATION  C C  QNETA=JSA-CTA*(JL+JAS) FIO(2)=QNETA  C C C C  CALCULATION OF TRANSENDOTHELIUM FLOW RATE  GLOBULIN  JSG=PSG*(CMVG-CAVG)+(1.D0"SIGFG)*(CMVG+CAVG)*JV/2.DO  C C C C  CALCULATION OF RATE OF INTERSTITIAL GLOBULIN ACCUMULATION  C C C C  SETTING UP OF TABLES (2) FOR FURTHER USING OTHER PROGRAMS  QNETG=JSG-CTG*(JL+JAS) FIO(3)=QNETG  310 C C 320 C C 330  c c c 340 c c 350 c c c c c  c c c c c c  c c c  5  399  CALCULATIONS  IF(NNNN.EQ.2)GO TO 399 IF(TAU.GT.99.0)GO TO 399 WRITE(1 ,310)TAU,JV,JL,JNET1 ,JAS F0RMAT(5(E12.5,2X)) WRITE(2,320)TAU,VIS1,PPMV,KAS,PAS FORMAT(5(E12.5,2X)) WRITE(3,330)TAU,QA,QG,PIPMV FORMAT(4(E12.5,2X)) CTCMV=CTA/CMVA WRITE(6,34 0)TAU,QNETA,QNETG,CTA,CTG FORMAT(5(E12.5,2X)) WRITE(8,350)TAU,DIFFUS,CONVEC,JSA,QNETA F0RMAT(5(E12.5,2X)) CONTINUE DETERMINATION OF THE TIME OF ONSET OF ALVEOLAR FLOODING (TION) & THE CORRESPONDING ALBUMIN CONCENTR-ATION (RCAON) I F ( I FLAG.GT.1 )GO TO 261 IF(JAS.EQ.0.D0)GO TO 261 TION=(TAU+TAUON)/2.D0 RCAON=((CTA+CTAON)/2.D0)/CMVA IFLAG=2 DETERMINATION OF MAXIMUM JAS (JASMAX) & CORRESPONDING TIME (TIMAX) ,ALBUMIN CONCENTRA I ON RATIO (RCAMAX) , TOTAL EXTRAVASCULAR F L U I D VOLUME (VTOTMX), & EXTRAVASCULAR-EXTRAALVEOLAR FLUID VOLUME (VIS1 MX)  261 IF(JAS.LT.JASMAX)GO TO 262 JASMAX=JAS TIMAX=TAU RCAMAX=CTA/CMVA VTOTMX=VTOT VIS1MX=VIS1 DETERMINATION OF THE TIME (TIOT) TO REACH A VTOT OF 1000 ML & THE CORRESPONDING  -199-  Listing 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 31 1 312 313 314 315 316 317 318 319 320 32 1 322 323 324 325 326 327 328 329 330 33 1 332 333 334 335 336 337 333 339 340 34 1 342 343 344 345  of  UBCEDEMA a t C C C C C  1 1 : 5 5 : 2 3 on MAR 25,  1985 f o r  CCid=HEIJ  Page  6  EXTRAVASCULAR-EXTRAALVEOLAR FLUID VOLUME ( V I S I O T ) , TRANSEPITHELIAL FLUID FLOWRATE ( J A S C T ) , SLOPE (JASSL?) OF THE TRANSEPITHELIAL FLUID FLOWRATE-TIME CURVE AT VTOT=1000 ML, AND ALBUMIN CONCENTRATION RATIO (RCAOT) 262  IF(VTOT.LT.1000.DO)GO TC 263 I F ( I CHECK.GT.1)GO TO 264 FACT=(1000.D0-VTOTLO)/(VTOT-VTOTLO) VI S10T=VIS1LO+FACT*(VIS1-VIS1LO) JASOT=JASLO+FACT*(JAS-JASLO) JASSLP=(JASLO-JAS)/(TILO-TAU) RCAOT=(CTALO+FACT*(CTA-CTALO))/CMVA TIOT=TILO+FACT*(TAU-TILO) ICHECK=2 GO TO 264 C REPLACING OLD VALUES OF DESIGNATED V ^ ' A B L E S WITH NEW C VALUES IF VTOT IS LESS THAN 1000 ML 263 I F ( V T O T . L T . V T O T L O ) G O TO 264 VTOTLO=VTOT VIS1L0=VIS1 JASLO=JAS CTALO=CTA TILO=TAU C C DETERMINATION OF WHETHER A NSW TIME INTERVAL C (I1*SUBINT) FOR PRINTING OF VARIA3LES HAS BEEN REACHED C IF SO, THE VARIA3LES ARE STORED IN ARRAY FORM. C IF NOT, TIME ITERATIONS ARE CONTINUED. C 264 I F ( C . G E . ( F L O A T ( I 1 )*SU3IMT)) M=M+1 I F ( M . G T . 1 ) G O TC 2 60 c c DETERMINATION OF BEGINNING (A) AND END (C) OF CALCULATION INTERVAL IN UNITS OF TIME c  c  c c c c c c  c  c c  2 1 0 A = TAU C=TAU+0.2 5D0 HSTE?=.1ODO CTAON=CTA TAUON=A UTILIZATION OF U3C SUBROUTINE "DRKC" TO EVALUATE TIME INTEGRAL CF VARIABLES Q A . Q G . V A S , « VTOT BETWEEN" CALCULATION INTERVAL BEGINNING AT ' A' & ENDING AT ' C OF WIDTH 0 . 2 5 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 VTOT=YIO(1) QA=YIO(2) QG=YIO(3) VAS=YIO(4) VIS 1=YIO(5) TAU = C GO TO 195 ARRANGEMENT OF VARIABLES FOR PRINTING AND PLOTTING 260 1=1 + 1  -200-  Listing 346 347 3 43 349 350  351  352 353 354 355 356 357 353 359 360 36 i 362 363 364 365 366 367 363 369 370 37 1 372 373  374  375 376 3 77  37S 379 33 0 3 £ '. 3£2 J C J  of  U3CEDEMA  339 390 ** ^ 1 c~  -  C ~  3S4 395 396 397 393 399 400 401 402 403  o n MAR 2 5 ,  1985  for  C C i d = H E I J Page  7  11=11+1  11=11  IF(C.GE.TAUMX2)GO GO TO 2 1 0 C C  TERMINATION  TO 2 6 7  OF C A L C U L A T I O N S  c  2 5"  —c —  J C £  11:55:23  M= 1 C C S E T - U P OF A R R A Y S FOR S T O R A G E O F D I F F E R E N T V A R I A B L E S C Tl(I)=TAU XI(I)=JV X2(I)=JL X3(I)=JNETI X4(I)=JAS Y1(I)=VTOT Y2(I)=VAS Y3(I)=VIS1 Y4(I)=VIS Z1(I)=PPMV Z2(I)=PAS 23(1)=PDIF KA(I)=KAS RI(I)=QA R2(I)=QG S1(I)=VAVA S2(I)=VAVG Ul(I)=CTA U2(I)=CTG VI(I)=CAVA V2(I)=CAVG Wl(I)=QNETA W2(I)=QNETG PI(I)=PIPMV IF(C.LT.TAUMX!)I1=1 I F ( C . L T . T A U M X 1 ) G O TO 2 1 0 SU3INT=SUBNT2  3c4  3co 3c 7  at  CONTINUE I F ( N N N . E Q . 2 ) G O TO 900  f  L  PRINTING  c {-  OF T A 3 L E S  WRITE(7,414)TION,RCAON F O R M A T ( 1H1 , 1 5 X , ' O N S E T OF A L V E O L A R F L O O D I N G ' , / / / , * I X , ' T I M E OF O N S E T I S ' , D 1 2 . 5 , / / , I X , ' A L B U M I N ' , * ' CONCENTRATION RATIO - C T A / C M V A - IS ',D12.5,////) WRITE(7,415) 4 1 5 F O R M A T ( 5 X , ' V A R I A B L E S A T T H E P O I N T OF T H E ' , * ' MAXIMUM A L V E O L A R F L O O D I N G R A T E ' , / / ) WRITE(7,420)JASMAX,TlMAX,RCAMAX 420 F O R M A T ( I X , ' T H E MAXIMUM F L U I D F L O W R A T E I N T O ' , * ' THE AIR SPACE IS ' , D 1 2 . 5 , / / , 1 X , ' T H E CORRESPONDING', * ' TIME I S ' , 2 4 X , D 1 2 . 5 , / / , I X , ' T H E ALBUMIN CONCENTRATION * ' RATIO - CTA/CMVA - IS ',D12.5,/) WRITE(7,421)VTOTMX,VIS1MX ',15X, 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 V O L U M E I S *D12.5,//,IX,'THE I N T E R S T I T I A L AND C E L L U L A R VOLUME I S -*  :i i^  -201-  L i s t i n g o f UBCEDEMA at 404 405 406 407 408 409 410 411 4 12 413 414 415 416 417 418 419 4 20 421 422 423 424 425 426 427 428 429 430 431 432 433 434  435 4 36 437 438 43S 440  44 1 4-=2 443 444 445 446 447 448 449 450 451 4 52 453 4 54 455 456 457 4 58 459 460 461  11:55:23 on MAR  25, 1985 f o r C C i d = K E I J Page  8  *9X,D-12.5,////) WRITE(7,425) 425 FORMAT(IX,'THE MAGNITUDE OF VARIA3LES AT A TOTAL', *' EXTRAVASCULAR FLUID VOLUME OF 1000 ML',//) WRITE(7, 4 30)TIOT,VIS10T,JASOT,JASSL?,RCAOT 430 FORMAT(IX,'TIME',32X,'TAU = ',D12.5,//,IX, *'INTERSTITIAL AND CELLULAR VOLUME',4X,'VIS1 = ',Dl2.5, * / / , I X , ' F L U I D FLOWRATE INTO THE AIR SPACE JAS = ', *D12.5,//,1X,'RATE OF CHANGS OF JAS',15X,'DJAS/DT = ', *D12.5,//,IX,'ALBUMIN CONCENTRATION RATIO',92,'CTA/CMVA = ' *,D12.5) WRITE(7,434) 4 34 FORMAT(1H1,18X,'TABLE 1 : OUTPUT OF FLUID FLOWS',//) WRITE(7,435) 435 FORMAT(2X,'TAU(HRS) ' , 6X, 'JV(ML/HR)' ,5X,'JL(ML/HR)' , *3X,'JNET1(ML/HR)',4X,'JAS(ML/HR)',/) WRITE(7, 4 4 0 ) ( T I ( J ) ,X1 ( J ) , X 2 ( J ) , X 3 ( J ) , X 4 ( J ) , J = 1 , I ) 440. FORMAT(5(E12.5,2X) ) WRITE(7,444) 444 FORMAT(1H1,17X,'TABLE 2 : OUTPUT CF FLUID VOLUMES',//) WRITE(7,445) 4 4 5 FORMAT(2X,'TAU(HRS)',6X,'VTOT(ML)',6X,'VAS(ML)', *7X,'VI SI (ML)' , 6 X , ' V I S ( M L ) ' ,/) WRITE(7,450)(TI(J),Y1(J),Y2(J),Y3(J),Y4(J),J=l,1) 450 FORMAT(5(E12.5,2X)) WRITE(7,454) 454 FORMAT(1H1,IX,'TABLE 3 : OUTPUT OF HYDROSTATIC PRESSURES -', *' PPMV,PAS,(PPMV-PAS) -',//,12X,'& EPITHELIUM FILTRATION", *' COEFFICIENT - KAS',//) WRITE(7,455) 455 FORMAT(2X,'TAU(HRS)',4X,'PPMV(MM H G ) ' , 4 X , *'PAS(MM HG)',4X,'PDIF(MM HG)',7X,'KAS'/) W R I T E ( 7 , 4 6 0 ) ( T I ( J ) , Z 1 ( J ) , Z 2 ( J ) , Z 3 ( J ) , K A ( J ) , J = 1 ,1) 4 60 F O R M A T ( E 1 2 . 5 , 2 X , E 1 2 . 5 , 2 X , E 1 2 . 5 , 3 X , E ! 2 . 5 , ! X , E 1 2 . 5 ) WRITE(7,464) 464 FORMAT(1H1,5X,'TABLE 4 : OUTPUT OF INTERSTITIAL SOLUTE', *'WEIGHTS - Q A , Q G -',//, 16X,' & INTERSTITIAL AVAILABLE', *' VOLUMES - VAVA,VAVG',//) WRITE ( 7 , 4 65) 4 6 5 FORMAT(2X,'TAU(HRS)',7X,'QA(GM)',SX,'QGiGM;',7X, * ' V A V A ( M L ) ' , 6 X , 'VAVG(ML) ' , / ) WRITE( 7,4 50 ) ( T I ( J ) , R 1 ( J ) ,R2(J) ,S1 ( J ) , S 2 ( J ) , j = ' , I ) 4 70 FORMAT(5(El 2.5,2X)) WRITE(7,474) 4 74 F O R M A T ( 1 H 1 , 5 X , ' T A B L E 5 : OUTPUT OF INTERSTITIAL - C T A , C T G * ' 5. AVAILABLE' ,//, 1 6 X , ' INTERSTITIAL - CAVA,CAVG - ', *'CONCENTRATIONS',//) WRITE(7,4 75) 4 7 5 F O R M A T ( 2 X , ' T A U ( H R S ) ' , 5 X , ' C T A ( G M / M L ) ' , 4X , *'CTG(GM/ML)',4X,'CAVA(GM/ML)',3X,'CAVG(GM/ML)',/) WRITE( 7,4 8 0 ) ( T I ( J ) , U 1 ( J ) , U 2 ( J ) , V I ( J ) , V 2 ( J ) , J = 1 , I ) 480 FORMAT(5(E12.5,2X)) WRITE(7,484) 484 FORMAT(1H1,'TABLE 6 : OUTPUT OF INTERSTITIAL SOLUTE', *' ACCUMULATION RATES - QNETA,',//,11X,'QNETG - ', * ' ( , INTERSTITIAL ONCOTIC PRESSURE - PIPMV',//) WRITE(7,485) 485 FORMAT(2X,'TAU(HRS)*,4X,'QNETA(GM/HR)',2X,  -202-  Lis-ing 462 462 464 465 466 467 468 469 470 47 1 472 473 474 475 476 477 478 479 480 43 ; 432 4S3 4S4 435 486 487 433 489 490 49; 492 493 494 495 496 497 493 499 SCO 50 i 502 303 304 50 3 30 6 50 7 50 3 50 9 5: 0 5:! 3i 2 5; 3 5i4 51 5  516  517 518 519  of  UBCEDEMA a t  11: 5 5 : 23 on MAP. 2 5 ,  1 9 8 5 for CCid=HEIJ  Page  9  *'QNETG(GM/HR)',3X,'PIPMV(MM H G ) ' , / ) W R I T E ( 7 , 4 9 0 ! ( T I ( J ) ,W1 ( J ) ,W2(J) , P I ( J ) ,J=1 , I ) 490 F O R M . A T ( i ( E ! 2 . 5 , 2 X ) ) C C  T E R M I N A T I O N  P R I N T I N G  OF  S E C T I O N  C  900  C O N T I N U E  IF(NN.EQ.2)GO  TO 1000  C C  P L O T T I N G  c c c  I N P U T  c r  c c  OF  D A T A  G R A P H S  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  R E A D (5,901 ) X S ( 1 ) , X S ( 2 ) 90 ! F O R M A T ( T 2 i , E 1 3 . 5 , / , T 2 1 , S 1 3 . 5 ) R E A D ( 5,90 2)YS1 ( 1 ) ,YS2( 1 ) , YS3(1 ),YS4(1 ) 902 F O R M A T ( 3 ( T 2 1 , S 1 2 . 5 , / ) , T 2 1 , E 1 3 . 5 ) YS;(2)=?S1(1) YS2(2)=YS2(1) YS 3(2)=YS 3(1 ) YS4(2)=YS4(i) P L O T T I N G  OF  F L U I D  F L O W  R A T E S  TMN=0.0 XMN=0.0 I N P U T  D A T A  F R O M  F I L E  P D A  READ(5,903)SET R E A D ( 5 , 9 0 3)SFX 90 3 F O R M A T ( T 2 i , E l 3.5) CALL AXIS(2.25,3.,'TIME ( H R S ) ' , - 10, 5 . , 0 . , T M N ,S F T ) CALL AXIS(2.25,3.,'FLUID FLOWS(ML/HR)',18,5.,90.,XMN,SFX) 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 ) 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 ) 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 ) CALL =BLZNE(T:,X4,I,TMN,2.23,SFT,XMN,3.,SFX,2,1,0.1,2) 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 ) CALL CALL CALL CALL CALL CALL CA" L CA" " CALL CALL CALL CALL CALL C A L L  c c c  T  P L O T T I N G  Y M N  c  PLOT X S ' : } , " S 1 ( : } , 3 ) DA3HLM •: . 1 2 5 , .0525 , . 1 2 5 , .0625) ?LOTiXS;2),VS1(2),4) ? S Y M ( 5 ' . 0 , 7 . ! 25 , . i 25 , ' J L = ',.0,8) ?107(XS-'!),YS2! i ) , 3 ) DASHLNi . 2 3 , . 0 7 5 , . 2 3 , .075 ) ?' ZZ; SS •. 1 : , : S3 '.' 2 ) , 4 ) PS7M'. 3 . 0 . £ . 730 , . ' 25 , ' JM' T i = ' , . 0 , 8 ) ?L0T(XS'•:),?S3(i),3) DASHLN• . 13, . 0 3 , . 0 6 2 5 , .03) PLOT;XS•2} ,YS3 i 2) ,4) PS YM ( ' 3 . 0 , 6 . 2 7 5 , . 125, ' J A S = ',.0,8) LIME;XS,YS4,2,1) PLOT!12.0,0.,-3)  I N P U T  OF  F L U I D  V O L U M E S  = 0.0  D A T A  F R O M  F I L E  READ(5,903)SFY  PDA  -203-  Listing 520 521 522 52 3 524 525 526 527 528 529 530 531 532 533 534 535 536 537 5 38 539 540 541 542 543 544 545 546 547 54 8 549 550 55 1 552 553 554 555 556 557 558 559 560 561 562 563 564 56 5 566 567 568 569 570 571 572 573 574 575 57 6 577  o f UBCEDEMA a t 11:55:23 on MAR CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL  25, 1985 f o r C C i d = H E I J Page  10  AXIS(2.25,3.,'TIME (HRS)' ,- 10,5.,0.,TMN,SFT) AXIS(2.25,3.,'FLUID VOLUMES(ML)',17,5.,90.,YMN,SFY) B3LINE(TI,Y1,1,TMN,2.25,SFT,YMN,3.,SFY,2,1,0.1,2) DLINE(TI,Y2,1,TMN,2.25,SFT,YMN,3.,SFY, .125, .125, .0625) DLINE(TI,Y3,1,TMN,2.25,SFT,YMN,3.,SFY,.15, .,0625,.05) LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) PSYM(5.0,7.50,.125,'VTOT = ',.0,7) LINE(XS,YS1,2,1) PSYM(5.0,7.125,.125,'VIS1 = ',.0,7) PLOT(XS(1),YS2(1),3) DASHLN(.125,.0625,.125,.0625) PLOT(XS(2),YS2(2),4) PSYM(5.0,6.750,.125,'VAS = ',.0,7) PLOT(XS(1),YS3(1),3) DASHLN(.15,.05,.0625,.05) PLOT(XS(2),YS3(2),4) PLOT(12.0,0.,-3)  C C PLOTTING OF HYDROSTATIC PRESSURES C C INPUT DATA FROM F I L E PDA READ(5,903)ZMN READ(5,903)SFZ CALL AXIS(2.25,3.,'TIME (HRS)' ,-10,5.,0 . ,TMN,SFT) CALL AXIS(2.25,3.,'HYDROSTATIC & OSMOTIC PRESSURES(MM HG)', *38,5.,90.,ZMN,SFZ) CALL BBLINE(TI,Z1,1,TMN,2.25,SFT,ZMN,3.,SFZ,2,1,0.1,2) CALL DLINE(TI,PI,1,TMN,2.25,SFT,ZMN,3. ,SFZ, . 1 5 , .0625, .05) CALL DLINE(Tl,Z2,I,TMN,2.25,SFT,ZMN,3.,SFZ,.125,.125,.0625) CALL LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) CALL PSYM(5.0,7.50,.125,'PIPMV = ',.0,8) CALL P L O T ( X S ( 1 ) , Y S 1 ( 1 ) , 3 ) CALL DASHLN(.15, .05, .0625, .05) CALL P L O T ( X S ( 2 ) , Y S 1 ( 2 ) , 4 ) CALL PSYM(5.0,7.125,.125,'PPMV = ',.0,8) CALL L I N E ( X S , Y S 2 , 2 , 1 ) CALL PSYM(5.0,6.750,.125,'PAS = ',.0,8) CALL P L O T ( X S ( 1 ) , Y S 3 ( 1 ) , 3 ) CALL DASHLN(.125, .0625, . 125, .0625 ) CALL P L O T ( X S ( 2 ) , Y S 3 ( 2 ) , 4 ) CALL PLOT(12.0,0.,-3) C C PLOTTING OF EPITHELIUM FILTRATION COEFFICIENT C KMN=0.0 C INPUT DATA FROM F I L E PDA READ(5,903)SFK CALL'AXIS(2.25,3.,'TIME (HRS)' ,- 10,5 . , 0 . ,TMN,SFT) CALL AXIS(2.25,3.,'KAS (ML/HR/MM HG) ' , 17,5. , 90. ,KMN,SFK) CALL BBLINE(TI ,KA,I,TMN,2.25,SFT,KMN,3.0,SFK,2, 1,0.1,2) CALL LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG, SIGFA,SIGFG) CALL PLOT(12.0,0.,-3) C C PLOTTING OF SOLUTE WEIGHTS C RMN=0.0 C INPUT DATA FROM F I L E PDA READ(5,903)SFR  -204-  Listing 578 579 580 531 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 60 5 606 60 7 605 609  610 6 1 1 612 613 614  615 616 613 619 620 62 1 622 623 624 62 5 626 627 626 629 630  631 632  633 634 635  of  UBCEDEMA  at  CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL C C C  11:55:22  on MAR 2 5 ,  1985  for  C C i d = K E I J Page  11  AXIS(2.25,3.,'TIME ( HRS ) ' , - 1 0 , 5 . , 0 . , TMN , S F T ) AXIS(2.25,3.,'SOLUTE WEIGHTS(GM)',18,5.,90.,RMN,SFR) 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 ) DLINE(Tl,R2,I,TMN,2.2 5,SFT,RMN,3.,SFR,.125,.125,.0625) LEGEND(PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) ?SYM(5.0,7.50,.125,'QA = ',.0,5) LINE(XS,YS1,2, 1 ) ? S Y M ( 5 . 0 , 7 . 1 2 5 , . 1 2 5 , ' QG = ' , . 0 , 5 ) PLOT(XS(1),YS2(I),3) DASHLNC.125,.0625,.125,.0625) PLOT(XS(2),YS2(2),4) ?LOT(12.0,0.,-3)  P L O T T I N G OF SOLUTE C O N C E N T R A T I O N S  UMN=0.0 C INPUT DATA FROM F I L E PDA R E A D ( 5 , 9 0 3 ) SFU CALL A X I S ( 2 . 2 5 , 3 . , ' T I M E (HRS)' , - 1 0 , 5 . , 0 . , T M N , S F T ) CALL AXIS ( 2 . 2 5 , 3 . , 'SOLUTE C O N C E N T R A T I O N S ( G M / M L ) ' , 2 8 , 5 . , *90.,UMN,SFU) CALL 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 ) CALL DLINE(TI,U2,I,TMN,2.25,SFT,UMN,3.,SFU,.125,.125,.0625) CALL 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 ) CALL P S Y M ( 5 . 0 , 7 . 5 0 , . 1 2 5 , ' C T A = ' , . 0 , 6 ) CALL L I N E ( X S , Y S 1 , 2 , 1 ) CALL P S Y M ( 5 . 0 , 7 . 1 2 5 , . 1 2 5 , ' C T G = ' , . 0 , 6 ) CALL P L O T ( X S ( 1 ) , Y S 2 ( 1 ) ,3) CALL DASHLN(. 1 2 5 , . 0 6 2 5 , . 1 25, .0625) CALL ? L O T ( X S ( 2 ) , 7 3 2 ( 2 ) , 4 ) C C T E R M I N A T I O N OF P L O T T I N G S E C T I O N C C A L L PLOTN'D 1000 STOP END C C C S U 3 R 0 U 7 I N E CMPLNC : FOR T H E C A L C U L A T I O N C F C C  COMPLIANCE  CURVE  S U B R O U T I N E C M P L N C ' V i 3 1 FPPMV  PPM /) 1  C IMPLICIT =EA1*3(A-:• DIMENSION FPPMV(iO) REAL* 4 SVISl INTEGER MLG.MH* IF(VIS1.LT.460.D0)GO TO 610 P?MV=0.017D0*VIS:-6.73DC GO TO 650 610 I F ( V I S 1 . G E . 3 3 0 . 0 D 0 ) G O T O 620 PPMV=0.227D0*VISi-39.0D0 GO TO 6 5 0 C I N T E R P O L A T I O N OVER CURVED P O R T I O N OF  620  DVIS1=(VIS1-380.D0)/10.D0 SVIS1=DVIS1 MLO=INT(SVIS1)+1  COMPLIANCE  -205-  Listing  of  636 637 638 639 640 64 1 642 643 644 645 646 647 648 649 650  UBCEDEMA  650  C C C C C  c  656.5  .5  666  667 668 669 670 671 672 67 3  6~4 675 676 677 S~3 679 630 6c !  CCid=HEIJ  Page  12  S U B R O U T I N E FUNC : TO B E U S E D I N C O N J U N C T I O N WITH S U B R O U T I N E DRKC FOR T H E E V A L U A T I O N OF T H E D E R I V A T I V E S (FIO) OF DEPENDENT V A R I A B L E S (YIO) FUNC(TIME,YlO,FIO)  c c c c c c c  VIS1=YIO(5) CALCULATION  OF I N T E R S T I T I A L  VOLUME  VISl=VTOT-VAS VIS=VIS1-VCELL CALCULATION  OF A L B U M I N  PARAMETERS  CTA=QA/VIS VAVA=VIS-VSXA CAVA=QA/VAVA  c c c  CALCULATION  OF G L 0 3 U L I N  PARAMETERS  CTG=QG/VIS VAVG=VIS-VEXG CAVG=QG/VAVG  c c c  CALCULATION CALCULATION  OF C E N T R A L P A R A M E T E R S OF I N T E R S T I T I A L P R E S S U R E  9  i 653 634 63 5 686 687 688 689 690 691 692 OO  for  IMPLICIT REAL*8(A-H,0-Z) REAL*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 DIMENSION YIO(5),FIO(5) COMMON/BLCA/SIGD,PSA,PSG,SIGFA,SIGFG C0MM0N/3LCB/PIMV,VCELL,VEXA,VEXG,CMVA,CMVG COMMON/BLCC/FPPMV(10),VTONS,B,VLMPH,KF,KAS 0,NK,SL,PMV COMMON/BLCD/PMV1,PMV2,PTAU1,PTAU2 VTOT=YIO(1) QA=YIO(2) QG=YIO(3) VAS=YIO(4)  657 658 659 660  £ £  1985  MHI=.MLO+l P PMVLO= F P PMV(MLO) PPMVHI=FPPMV(MKI) PPMV=(1.D0+DVISl-DFLOAT(MLO))*(PPMVHI-PPMVLO)+PPMVL0 CONTINUE RETURN END  SUBROUTINE  651  662 663 664 665  11:55:23 on MAR 2 5 ,  c  652 653 654 655 656  661  at  CALL  c c c c c c  CMPLNC(VIS!,FPPMV,PPMV)  CALCULATION  OF I N T E R S T I T I A L  ONCOTIC  PRESSURE  CP=CAVA+CAVG 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 CALCULATION  OF C A P I L L A R Y  FILTRATION  JV=KF*((PMV-PPMV)-SIGD*(PIMV-PIPMV))  RATE  -206-  Listing 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 71 1 712 713 714 715  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 C C C C  C C C C C C C  c c c c  716  717 718 719 720 72 1 722 723 724 725 726 727 728 729 730 73 1 732 733 734 "7 3 5 736 7 37 738 739 740 74 1 742 743 744 745 746 747  748 749 750  c c c c c c c c c c c c c c c c c c c c c c c c  CALCULATION OF LYMPH FLOW I F ( V I S 1 .LT.VLMPH)GO TO 203 JL=0.17D0*VLMPH-5.56D1 GO TO 204 203 JL=0.17D0*VIS1-5.56D1 CALCULATION OF RATE OF EXTRAVASCULAR FLUID ACCUMULATION 204 JNET=JV-JL FIO(1)=JNET DETERMINATION I F ALVEOLAR FLOODING OCCURS IF(VTOT.LT.VTONS)GO TO 207 CALCULATION OF FILTRATION COEFFICIENT OF E P I T H E L I A L MEMBRANE KAS = KAS0+NK*(VIS 1-VTONS) GO TO 208 207 KAS=0.D0 CALCULATION OF HYDROSTATIC PRESSURE OF FLUID IN THE AIR SPACE 208 PAS=SL*VAS+3 CALCULATION OF TRANSEPITHELIAL FLUID FLOW RA  1 T TT F  ,  JAS=KAS*(PPMV-PAS) FIO(4)=JAS CALCULATION OF RATE OF ACCUMULATION OF INTERSTITIAL FLUID JNET1=JV-JL-JAS FIO(5)=JNST1 CALCULATION OF TRANSENDOTKELI UM A.L SUM I N FLOW RATE JSA=PSA*(CMVA-CAVA) + ( 1 .DO-SIGFA)*(CMVA-CAVA) *JV/2 CALCULATION OF RATE OF INTERSTITIAL ALBUMIN ACCUMULATION QNETA=JSA-CTA*(JL+JAS) FIO(2)=QNETA CALCULATION OF TRANSENDOTHELIUM FLOW RATE  GLOBULIN  JSG=PSG*(CMVG-CAVG)+(1.DO-SIGFG)*(CMVG+CAVG) *JV/2  13  -207-  Listing 751 752  753 754 755 756 757 758  759 760 761 762 763  764  765 766 7 67 768 769 770 771 772 773774 77 5 776 77 7 776 779 780 781 782 733 784 7£5 786 757  of  UBCEDEMA C C C  at  11:55:23 on MAR 2 5 ,  C A L C U L A T I O N OF R A T E ACCUMULATION  1985  f o r C C i d = H E I J Page  14  OF I N T E R S T I T I A L G L O B U L I N  QNETG=JSG-CTG*(JL+JAS) FIO(3)=QNETG RETURN END C C C C C  S U B R O U T I N E LEGEND ON P L O T S  : FOR T H E PRINTING  OF CONSTANTS  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 ) C REAL*3 PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG C A L L P S Y M ( 2 . 0 0 , 1 . 8 7 5 , . 1 2 5 , ' P M V (MM HG) = ',.0,21) CALL N U M 3 E R ( 4 . 2 5 , 1 . 8 7 5 , . 1 2 5 , P M V , . 0 , 2 ) CALL P S Y M ( 2 . 0 0 , 1 . 6 0 , . 1 2 5 , ' K F ( M L / H R / M M HG) = ',.0,21) CALL N U M 3 E R U . 2 5 , 1. 6 0 , . 1 2 5 , K F , . 0 , 4 ) CALL P S Y M ( 2 . 0 0 , 1 . 3 2 3 , . 1 2 3 , ' K A S O ( M L / H R / M M HG) = ' , . 0 , 2 1 ) CALL N U M B E R ( 4 . 2 5 , 1 . 3 2 5 , . 1 2 5 , K A S O , . 0 , 5 ) C A L L P S Y M ( 2 . 0 0 , 1 . 0 5 0 , . 1 2 5 , ' N K ( / H R / M M HG) = ',.0,21) CALL NUM3ER(4.25,1.050,.125,NK,.0,5) C A L L PSYM( 2 . 0 0 , 0 . 7 7 5 , . 1 2 5 , ' S L (MM H G / M L ) = ',.0,21) CALL N U M B E R ! 4 . 2 5 , 0 . 7 7 5 , . 1 2 5 , S L , . 0 , 5 ) CALL P S Y M ( 5 . 2 5 , 1 . 3 7 5 , . 1 2 5 , 'SIGD = ',.0,14) C A L L NUMBER( 6 . 7 5 , 1 . 8 7 3 , . 1 2 5 , S I G D , . 0 , 5 ) CALL P S Y M ( 5 . 2 5 , 1 . 6 , . 1 2 5 , ' P S A (ML/HR) = ',.0,15) CALL NUM3ER(6.75,1.6,.125,PSA,.0,4) CALL P S Y M ( 5 . 2 3 , 1 . 3 2 3 , . 1 2 5 , ' P S G (ML/HR) = ',.0,15) CALL NUM3ER(6.75,1.325,.125,PSG,.0,4) CALL P S Y M ( 5 . 2 5 , 1 . 0 5 , . 1 2 5 , ' S I G F A = ',.0,14) CALL NUM3ER'6.75,1.05,.125,SIGFA,.0,5) CALL PSYM(5.23, . 7 7 5 , . 123, 'SIGFG = ',.0,14) CALL N U M B E R ( £ . 7 5 , . 7 7 5 , . 1 2 5 , S I G F G , . 0 , 5 ) RETURN END  -208-  Table 25:  CALL  Explanation of Subroutine DRKC (56)  DRKC(N,X,Z,Y,F,H,HMIN, E ,FUNC,G,S,T)  where: N  is an INTEGER variable or constant. On e n t r y , c o n t a i n s the number of d i f f e r e n t i a l e q u a t i o n s to solved.  X  is a REAL*8 input value of contains the va r i a b l e .  Z  is a REAL*8 variable or constant. On e n t r y , it c o n t a i n s the f i n a l v a l u e of the independent variable at the end p o i n t of integration.  Y  is a REAL*8, one-dimensional array, o f d i m e n s i o n >N. On e n t r y , Y ( I ) I = 1 , . . . , N c o n t a i n the i n i t i a l values of the dependent v a r i a b l e s . On e x i t , Y contains the v a l u e s of the dependent v a r i a b l e s at the end point of integration.  F  is a REAL*8, one-dimensional On exit, it contains d e r i v a t i v e s Y ' ( X ) at X=Z.  H  is a REAL*8 v a r i a b l e . On entry, it contains the input step-size. H i s c h a n g e d by DRKC t o c o n t a i n the step-size used at the c u r r e n t integration step. For maximum a c c u r a c y in single precision,.H=(Z-X)/64.  HMIN  is a REAL*8 variable or constant. On entry, it c o n t a i n s a lower bound for the s t e p - s i z e in order to prevent i n f i n i t e cycling. For d i f f i c u l t problems (e.g. the o r b i t p r o b l e m — s e e the sample problem in UBC DIFSY) DRKC may require HMIN^l0" *H. Usually HMIN=10" *H. i s a d e q u a t e . . See s u b s e c t i o n (c), below, f o r a m e t h o d t o d e t e c t when HMIN has been reached.  variable. On entry, it contains the t h e i n d e p e n d e n t v a r i a b l e . On e x i t , it final value of the independent  array, of dimension the output vector  8  2  it be  >N. of  -209-  Table 25:  E  (cont'd.)  is a REAL*8 variable or constant. On e n t r y , i t c o n t a i n s an e r r o r t o l e r a n c e - . I f ES I , the original step-length' will be maintained 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£.5xl0" . s  FUNC  i s t h e name o f a SUBROUTINE subprogram which is coded by t h e u s e r t o e v a l u a t e t h e 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 user's calling 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 (X,Y(1),...,Y(N)) N  RETURN END whe r e: X  is a REAL*8 v a r i a b l e w h i c h c o n t a i n s t h e c u r r e n t v a l u e . o f the independent variable.  Y  is a REAL*8, one-dimensional array, which contains the c u r r e n t values of the dependent variables .  F  i s a REAL * 3, o n e - d i m e n s i o n a l a r r a y . On exit; from FUNC i t c o n t a i n s t h e v a l u e s of t h e d e r i v a t i v e s .  G, S, T a r e REAL*8, o n e - d i m e n s i o n a l a r r a y s , >N . They a r e work a r e a s r e q u i r e d i n t e r n a l l y .  dimensioned  -210-  T a b l e 26:  E x p l a n a t i o n of Subroutine CMPLNC  CALL  CMPLNC (VIS1, FPPMV, PPMV)  VIS1  i s a REAL*8 v a r i a b l e , on e n t r y VIS1 i s the EVEA f l u i d volume t o 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 a r r a y , c o n t a i n i n g the i n t e r s t i t i a l h y d r o s t a t i c pressures l i s t e d i n T a b l e 3 - the curved p o r t i o n of the compliance c u r v e .  PPMV  i s a REAL*8 v a r i a b l e , on e x i t i t c o n t a i n s the i n t e r s t s i t i a l p r e s s u r e c o r r e s p o n d i n g to VIS1.  -211-  A1.3  Plotting Section o f Main Program UBCEDEMA The  plotting  s e c t i o n of the s i m u l a t i o n i s executed  the program UBCEDEMA, s t a r t i n g on l i n e 466. parameter NN  in file  PDA  Plots w i l l  epithelial and  2) the f l u i d  filtration  c o e f f i c i e n t KAS,  (albumin and  The  flows JV,  volumes VTOT, V.IS1 and VAS,  g l o b u l i n ) i n the i n t e r s t i t i u m QA  concentration  be drawn i f  [PLOTS(Y=1,N=2)] i s set equal to 1.  p l o t s drawn are of the time v a r i a t i o n o f : 1) the f l u i d JNET1 and JAS,  from w i t h i n  3)  the  4) the p r o t e i n content  and  g l o b u l i n ) CTA  QG and  and  5) the t i s s u e  (albumin protein  CTG.  For each set of p l o t s a number of s u b r o u t i n e s are  called:  AXIS, PLOT, PLOTND, LINE, DASHLN, PSYM, DLINE, BBLINE, NUMBER and LEGEND.  JL,  These s u b r o u t i n e s are e x p l a i n e d i n Tables 27 through  36.  -212-  Table 27:  E x p l a n a t i o n o f s u b r o u t i n e AXIS  (57)  AXIS Purpose AXIS draws an a x i s values near the label.  w i t h t i c k marks e v e r y i n c h , places scale t i c k m a r k s , and i d e n t i f i e s the a x i s w i t h a  T h i s s u b r o u t i n e was not d e s i g n e d f o r use i n the m e t r i c system and does not g i v e p l e a s i n g r e s u l t s i f m e t r i c u n i t s a r e used. Users who prefer to work in metric 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 following page. AXPLOT also gives the user more c o n t r o l o v e r the p l o t t i n g of an axis. How To Use CALL A X I S ( X , Y , 3 C D , N , S , T H E T A , X M I N , D X )  (X,Y)  are  the  c o o r d i n a t e s of  BCD  is a Hollerith l i t e r a l ( n H x x x . . . or 'xxx...') 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 u s e d as the a x i s label.  N  is positive if the label is 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 t h e a x i s , and n e g a t i v e i f the label is 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 v a l u e of N i s t h e number of c h a r a c t e r s in BCD.  S  i s the l e n g t h of This value will unit.  axis in rounded  floating-point units. up to the n e a r e s t whole  THETA  i s the d i r e c t i o n of the a x i s i n For a.~ x a x i s , THETA i s u s u a l i v  f l o a t i n g - p o i n t degrees. 0.0, and f o r a v axis,  the be  the  start  of  the  axis.  90.0.  XMIN  is  the  vc _ ue  o :  value  to be a s s i c n e d to the o r i c i n ( i . e . ';.-) c ta t : or. o c o c s i t e trie t i r s t tic:-; cr.  the  axis). DX  is the scale increment (number of u n i t s per i n c h or centimetre). The s c a l e a n n o t a t i o n s at each successive 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  using subroutine  SCALE.  -213-  Table 28:  E x p l a n a t i o n of s u b r o u t i n e PLOT (57)  PLOT Purpose  This subprogram i s the b a s i c 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. i n a s t r a i g h t line from i t s p r e s e n t p o s i t i o n to the p o s i t i o n indicated in the call. It i s a l s o u s e d to r e l o c a t e the o r i g i n of the plotter c o o r d i n a t e s y s t e m i n the x d i r e c t i o n . -low To Use CALL  PLOT (X , Y , I PEN')  we r e : are the c o o r d i n a t e s of the p o i n t t o which the pen i s t o move. Y must be between - 0 . 5 and 34.5 inches (-1.27 and 87.63 centimetres). X must be between - 1 . 0 and 200 .0 i n c h e s ( - 2 . 5 4 and 5 0 8 . 0 c m ) . If the y c o o r d i n a t - ; i s l a r g e r t h a n 10.5 i n c h e s ( 2 6 . 6 7 cm), the plot wi}\ be directed to l a r g e paper when i t i s automat i c a l l y c'ueued. IPEN  s p e c i f i e s whether the pen i s t o be up or down i t s move t o ( X , Y ) . The v a l u e s f o r IPEN a r e : + I  for no c h a n g e . If t h e i t w i l l remain down.  +2  f o r pen down. If the pen i s up b e f o r e the c a l l , pen w i l l be lowered b e f o r e t h e movement to ( X , Y ) made. If down, i t w i l l r e m a i n down. pen u p . If the pen .1 be r a i s e d b e f o r e i t w i l l r e m a i n UD.  pen  is is  is  down b e f o r e  during  down b e f o r e the moved to ( X , Y ) .  the  call, the is  call, it If up,  for pen d a s h e d , w h i l e the pen i s moved to ( X , Y ) it w i l l be r a i s e d and l o w e r e d t o p r o d u c e dashed lines i". accordance w i t h the l a s t c a l l t o the s u b r o u t i n e  i TOO  r-=  to r e l o c a t e the o r i g i n on Tne s u b r o u t i n e PLOT i s a l s o used the graph paper i n the x d i r e c t i o n . If I PEN = - 1 , - 2 , or - 3 , the pen is moved to the (X,0.0) position on the paper ( r e g a r d l e s s c f 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 origin of the y coordinate may not be relocated. The origin 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 g r a p h s is to be drawn.  -214-  T a b l e 29:  E x p l a n a t i o n o f s u b r o u t i n e PLOTND (57)  PLOTND Purpose To  terminate  plotting.  How To Use CALL PLOTND Commen t s To ensure that a c o m p l e t e p l o t i s r e c e i v e d , t e r m i n a t e your p l o t t i n g w i t h a c a l l to PLOTND, or a call to PLOT with a negative 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 other calls to the plot routines are allowed after 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  Subroutines -  End of  Plotting  ***  Number of p l o t frames g e n e r a t e d = n If this plot is queued for plotting it will take approximately m minutes to p l o t a t an a p p r o x i m a t e 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 p a p e r . A p p r o x i m a t e l y p% of the time w i l l be s p e n t plotting with the pen r a i s e d . The numbers given a r e a p p r o x i m a t e and a r e i n t e n d e d to g i v e t h e u s e r some i d e a of how much i t w i l l c o s t to produce the plot. "n" is the number of f r a m e s i n the p l o t f i l e , "m" i s t h e number of m i n u t e s to draw the p l o t on the p l o t t e r , "d.dd" i s the a p p r o x i m a t e c o s t in d o l l a r s a t a c a d e m i c r a t e s , and " i " i s the number of i n c h e s 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 t h e p l o t . When the p l o t i s c o p i e d to * P L O T * the time a c t u a l l y used i n d r a w i n g the p l o t w i l l be c a l c u l a t e d and your a c c o u n t c h a r g e d . A u s e r w i l l r e c e i v e o n l y p a r t of h i s p l o t i f t h e l a s t c a l l to a p l o t t e r s u b p r o g r a m i s not to PLOTND, or to PLOT with a negative IPEN v a l u e . He may r e c e i v e an e r r o r message at the t i m e 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-  T a b l e 30:  E x p l a n a t i o n o f s u b r o u t i n e LINE (57)  LINE Purpose To  draw  a line  through  a s e r i e s of  points.  How To Use CALL  LIKE(ARKAYX,ARRAYY,N,J)  where: ARRAYX  contains  the  >: c o o r d i n a t e s of  N data  ARRAY?  contains  the  y c o o r d i n a t e s of  the  N  i s the number be d r a w n .  j  should be set t o -1 i f t h e pen i s t o . b e down when moving t o t h e f i r s t p o i n t ; o t h e r w i s e , i t should be set to +1. If J is +1, the l i n e may be drawn backwards i f t h i s i s more efficient. This means that the pen may not always finish at the c o o r d i n a t e s of the l a s t p o i n t i n t h e l i n e ; it will sometimes finish at the c o o r d i n a t e s of the first point. If 3 i s set to - 1 , the l i n e i s always drawn forward.  of  points  through  points.  points.  w h i c h the  line  is  to  -216-  T a b l e 31:  E x p l a n a t i o n o f s u b r o u t i n e DASHLN  (57)  DA5KIN Purr-cse To  se: a  cattern ror  c a s ne c l i n e  ^^n m jvemen o v e me <i s  (I PEN = 4 )  T c Use CALL DA SHLNtDASH 1 , S P A C 1 ,DASH2,S?AC2) where: DASH!  is  the l e n g t h  of  the  first  da s h . .  SPACl  is  the  length  of  the  first  space.  DASH 2  is  of  the  second dash.  SPAC2  is  of  the  second space.  l  V  the l e n g t h  These v a l u e s sh. o u l d be e n t e r e d as p o s i t i v e Zero i n d i c a t i n g the 1 e n g t h s i n user u n i t s . a 1lowed If DASHLN has been c a l l e d , a c a l l t o PLOT w i t h an IPEN v a l u e of 4 w i l l r e s u l t i n a dashed l i n e being drawn; The first d a s h w i l l be DASHl i n c h e s or c e n t i m e t r e s i n l e n g t h , the first space will be SPAC1 inches or c e n t i m e t r e s i n l e n g t h , and s i m i l a r l y f o r the s e c o n d dash and s p a c e . Then the pattern will repeat itself. By u s i n g a v a r i e t y of p a r a m e t e r s , many t y p e s of d a s h e d l i n e s may be c r e a t e d . If d a s h i n g i s d i s c o n t i n u e d , f o r e x a m p l e , i n o r d e r t o draw a symbol, and the pen i s then r e t u r n e d t o the same p o s i t i o n where d a s h i n g was d i s c o n t i n u e d , the d a s h e s w i l l be correctly connected. Comments The l e n g t h s c : d a s h e s and s p a c e s a t a n g l e s of 45 d e g r e e s w i l l be approxi^ately 1.4 times t h e i r l e n g t h as s p e c i f i e d i n the call. This d i f f e r e n c e is rarely noticeable in a finished  -217-  T a b l e 32:  E x p l a n a t i o n o f s u b r o u t i n e PSYM (57)  PSYM Purpose PSYM p r o d u c e s t e x t on a p l o t . T h i s r o u t i n e : s used f o r character sets. See the description of PALPHA 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 .  all 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 t o be drawn. For most character sets, including the standard one, this is 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 either coordinate is -0.0 (hexadecimal 80000000), PSYM continues 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 t h e f l o a t i n g - p o i n t (REAL*4) w h i c h the s t r i n g i s d r a w n .  STRING  is  ANGLE  is the f l o a t i n g - p o i n t (REAL*4) a n g l e i n d e g r e e s of the character string (using a pos i t i ve counterclockwise convention).  LENGTH  is the fullword integer c h a r a c t e r s i n STRING.  iRC4  the  character  string  to  height  in  i nche s  be d r a w n .  (INTEGER*4)  number  of  i s the e x i t t a k e n f o r an u n s u c c e s s f u l r e t u r n ; STRING i s not d r a w n . E i t h e r the p a r a m e t e r s are in error (for i n s t a n c e , c a l l i n g PSYM f o r the f i r s t time w i t h X or Y equal to - 0 . 0 ) , or t h e r e i s an error in a user-defined character set.  -218-  Table CALL  33:  E x p l a n a t i o n of s u b r o u t i n e DLINE  DLINE(XY,N,X0,SX,SFX,Y0,YS,SFY,D1,D2,DB)  where:  X Y N XO XS SFX YO YS SFY DI D2 D3  r e a l a r r a y of Independent x - v a l u e s r e a l a r r a y of dependent y - v a l u e s i n t e g e r 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 v a l u e of x - a x i s ( u n i t s ) r e a l s t a r t i n g l o c a t i o n of x - a x i s ( i n c h e s ) r e a l x-axis scale f a c t o r ( u n i t s / i n c h ) r e a l s t a r t i n g v a l u e of y - a x i s ( u n i t s ) r e a l s t a r t i n g l o c a t i o n of y - a x i s ( i n c h e s ) r e a l y-axis scale factor ( u n i t s / inch) l e n g t h of f i r s t dash ( i n c h e s ) l e n g t h of second dash ( i n c h e s ) l e n g t h of space between dashes ( i n c h e s )  S u b r o u t i n e p l o t s a smooth  dashed  curve.  -219-  COMPUTER PROGRAM OF SUBROUTINE DLINE  Listing 1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 1 4 1 5 1 6 1 7 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  of DLINE a t  10  20 30 40  50 60 70 80  85 86  90  C  11:55:38  on MAR 2 5 ,  1985 f o r C C i d = H E I J Page  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) DIMENSION X ( N ) , Y ( N ) COMMON/BLKA/XP(101 ) ,YP(101 ) , M , MM M=N MM=M-1 DO 10 1 = 1 , N XP(I)=XS+(X(I)-X0)/SFX YP(I)=YS+(Y(I)-Y0)/SFY CONTINUE CALL SPLINE XI=XP(l) YI=YP(1) CALL P L O T ( X I , Y l , 3 ) IFLAG=1 ITER=1 G O T O ( 3 0 , 4 0 , 5 0 , 6 0 ) , I FLAG D=D1 IPEN=2 GO TO 7 0 D=DB IPEN=3 GO TO 70 D=D2 IPEN=2 GO TO 7 0 D=DB IPEN=3 XF1=XI+D/2. XF = XF 1 - F X ( X F 1 ,XI ,YI , D ) / D F X ( X F 1 ,XI ,YI ,D) I F ( X F . L E . X I ) GO TO 8 5 I F ( A 3 S ( ( X F 1 - X F ) / X F ) . L T . 0 . 0 0 0 0 1 ) GO TO 90 ITER=ITER+1 I F ( I T E R . G T . 2 0 ) GO TO 90 XF1=XF GO TO 8 0 XI =XI FX1=FX(X1,XI,YI,D) DX=D/2. XF=X1+DX I F ( D X . L T . 0 . C 0 0 0 i ) GO TO 90 DX=DX/2. FXF=FX(XF,XI,XI,D» I F ( F X 1 * F X F . L T . 0 . ) GO TO 8 6 X1=XF FX'=FXF GO TO 6 6 I F ( X F . G T . X ? ( N ) ) XF=XP(N) YF=F(XF) CALL P L O T ( X F , Y F , I PEN) I F ( X F . E Q . X P ( N ) ) RETURN XI =XF Yl = YF IFLAG=IFLAG+1 I F ( I F L A G . E Q . 5 ) IFLAG=1 GO TO 20 END SUBROUTINE SPLINE  -220-  Listing  of  59  60 6 1 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80  8  C C  88  89 90 9 i 92 93 94 95 96 9 9c 99 1 00  c c c  Coefficient  1  0  10 2 'C2 i 04 i03 1 06 1 07 1 OS 1 09 ! 1 0 1 ! 1 1 1 2 1 1 3 1 1 4  1 15 1 1 6  Page  2  matrices  for  end p o i n t  cubics  INTEGER FLAG M=4 IS = 0 FLAG=0 MF=M-i-1 MM=M~1 DO 20 1=1,M I I = I - I S  COEF?(I,M?)=Y(II )  c c c  20  COEFF(1,1)=!. DO 20 J=2,M COEFF(I,J)=COEFF(I,J-1)*X(II) Gauss  30  -  • n i i '•J •  f o r CCid=HEIJ  COMMON/ELKA/X(101),Y(101),N,NM CCMMCN/BLK3/Q(100),R(101),S(100) DIMENSION H ( 1 0 0 ) , A ( 1 0 1 ) , B ( 1 0 1 ) , C ( 1 0 1 ) , D ( 1 0 1 ) , C O E F F ( 4 , 5 )  82 83  35 86 87  1985  I n t e r p o l a t i o n u s i n g c u b i c s p l i n e s w i t h f i t t e d end p o i n t s Input: X A r r a y of i n d e p e n d e n t x - v a l u e s Y A r r a y of d e p e n d e n t y - v a l u e s N Number of d a t a p o i n t s NM N-1 O u t p u t : Q , R , S C o e f f i c i e n t s of c u b i c s p l i n e e q u a t i o n s  c c c c c c  1  84  11:55:38 on MAR 2 5 ,  DLINE at  -= L  elimination  to  find  A4 and B4  DO 3 0 K=1,MM K?=K*1 DO 3 0 I = K ? , M DO 3 0 J = X ? , M ? CCEFF;I,J)=COEFF(I,J)-COEFF(I,K)*COEFF(K,3)/COEFF(K ,K ) I F ( F L A G . N E . 0 ) GO TO 40 A4=COEFF(M,MP)/COEFF(M,M) ?LAG= ; I S = N-M GC 70 -0 3 - = C C E F F ' M MP) 'COEF?(M M)  C  c c  Calculate 30  c c c  DC  50  r.  '  > I  =  H(I)  1=1,MM X ' I - '  Coef f i c i e n t s Ai  1  3(  i :•=-:-;(  )  - X ' - !  of  tridiaconal  1=0.0 i)  c. I )=:-:( i ) D(. 1 ) = 3.*H(  DO 60  1 )*H( I=2,NM  1  )*A4  IP=I+1  IM=I-1  A(I)=H(IM) B(I)=2.*(K(IM)+H(I)) C(I)=H(I)  equations  -221-  Listing  of DLINE a t 11:55:38 on MAR 2 5 , 1985 f o r C C i d = H E I J Page  17 18  60  19  20 21 22 23 24 25 26 • 27 28 29 30 31 32 33  37  38 39  40 41 42  c c  Call  Determine  70  61  62 63 164 65 66 167 1 68  c c  DO 70 1=1,NM IP=I+1 Q(I)=(Y(IP)-Y(I))/H(I)-H(I)*(2.*R(I)+R(lP))/3. S(I) = (R(IP)-R(I'))/(3.*H(I)) RETURN . END  Thomas  10  20  algorithm  DIMENS I ON A ( N ) , B ( N ) , C ( N ) , D ( N ) , X ( N ) , P ( 1 0 1 ) , Q ( 1 0 1 ) NM=N-1 P(1)=-C(1)/3(1) Q(1)=D(1)/B(1) DO 10 1=2,N IM=I-1 DEN=A(I)*P(IM)+3(I) P(I)=-C(I)/DEN Q(I)=(D(I)-A(I)*Q(IM))/DEN X(N)=Q(N) DO 20 11=1,NM I=N-II X(I)=?(!)*X(I+1)-Q(I) RETURN END  C FUNCTION F ( Z ) C C C  Spline interpolation by b i s e c t i o n .  c  10  169  170 171 172 173 174  Q ( I ) and S ( I )  SUBROUTINE T D M A { A , B , C , D , X , N )  C  47  58 59 60  to solve t r i d i a g o n a l set  C  44 45 46  57  Thomas a l g o r i t h m  CALL T D M A ( A , B , C , D , R , N )  c c c  43  48 49 50 51 52 53 54 55 56  D(I)=3.*((Y(IP)-Y(I))/H(l)-(Y(I)-Y(IM))/H(IM)) A(N)=H(NM) B(N)=-H(NM) C(N)=0.0 D(N)=-3.*H(NM)*H(NM)*B4  C  34  35 36  3  20 30  function.  Interpolation  COMMON/3LKA./X ( 1 0 1 ) , Y ( 1 0 1 ) , N , NM COMMON/BLK3/Q(100),R(101),S(100) 1=1 I F ( Z . L T . X ( 1 ) ) GO TO 3 0 I F ( Z . G E . X ( N M ) ) GO TO 2 0 J = NM K=(I+J)/2 I F ( Z . L T . X ( K ) ) J=K I F ( Z . G E . X ( K ) ) I=K I F ( J . E Q . I + 1 ) GO TO 3 0 GO TO 10 I=NM DX=Z-X(I) F=Y(I)+DX*(Q(I)+DX*(R(I)+DX*S(I)))  interval  -222-  Listing 17 5 176 177 178 17 9 180 181 182 183 184 185 186 187 188 189 190 191 192 1 93 194 195 196 1 97 198 199 200 201  of DLINE at  1985 f o r C C i d = H E I J Page  4  RETURN END C  10  20 30  FUNCTION F X ( Z , X O , Y O , D ) C0MM0N/3LKA/X(101),Y(101),N,NM COMMON/BLK3/Q(100),R(101),S(100) 1=1 IF(Z.LT.X(1)) GO TO 30 I F ( Z . G E . X ( N M ) ) GO TO 20 J=NM K=(I+J)/2 I F ( Z . L T . X ( K ) ) J=K IF(Z.GE.X(K)) I=K I F ( J . E Q . I + 1 ) GO TO 30 GO TO 10 I=NM DX=Z-X(I) FX=(Z-X0)**2+(Y( I )-Y0+DX*(Q(I)+DX*(R(I)+DX*S(I))))**2-D*D RETURN END  C FUNCTION D F X ( Z , X 0 , Y 0 , D ) COMMON/BLK.A/X ( 1 0 1 ) , Y ( 1 0 1 ) , N , NM COMMON/BLK3/Q(100),R(101),S(100) 1=1  IF(Z.LT.X(1)) GO TO 30 I F ( Z . G E . X ( N M ) ) GO TO 2 0 J=NM  202  203 204 205 206 207 208 209 210 211 21 2 21 3  1 1 : 5 5 : 3 8 on MAR 2 5 ,  10  20 30  K=(I+J)/2 I F ( Z . L T . X ( K ) ) J=K IF(Z.GE.X(K)) I=K I F ( J . E Q . I + ! ) GO TO 30 GO TO 10 I=NM DX=Z-X(I) DFX=2.*(Z-X0)-2.*(Y(I)-Y0+DX*(Q(l)+DX*(R(I)+DX*S(I))))* 1 (Q(I ) + D X * ( 2 . * R ( I )+DX* 3 . * S ( I ) ) ) RETURN END  -223-  Table 34: SUBROUTINE  E x p l a n a t i o n of s u b r o u t i n e BBLINE BBLINE  (X,Y,N,XO,XS,SFX,YO,YS,SFY,L,J,H,K)  where: X Y N XO XS SFX YO YS SFY L  J H K  r e a l a r r a y of independent x - v a l u e s r e a l a r r a y of dependent y - v a l u e s i n t e g e r 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 v a l u e of x - a x i s ( u n i t s ) r e a l s t a r t i n g l o c a t i o n of x - a x i s ( i n c h e s ) r e a l x-axis scale factor (units/Inch) r e a l s t a r t i n g v a l u e of y - a x i s ( u n i t s ) r e a l s t a r t i n g l o c a t i o n of y - a x i s ( i n c h e s ) r e a l y-axis scale factor (units/inch) i n t e g e r l i n e type c o n t r o l parameter L=0 symbols p l o t t e d only L=l curved l i n e p l o t t e d with symbols L=2 curved l i n e p l o t t e d without symbols L=-l s t r a i g h t l i n e p l o t t e d with symbols L=-2 s t r a i g h t l i n e p l o t t e d without symbols i n t e g e r symbol c o n t r o l parameter r e a l symbol h e i g h t i n t e g e r 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  o u t i n e p l o t s a smooth curve  -224COMPUTER PROGRAM OF SUBROUTINE BBLINE  l i s t i n g of B3LINE a t  10 1 1 12 13 14 15  l 6 ! 7 18 19 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 36  SUBROUTINE  C C C C C C C C C  c c c c c c c c c c cc c c  10  E2  53 54 55 56 57 53  30 40  59 60  61 62 63 64  11:55:46 on MAR 2 5 , 1985 f o r C C i d = H E I J Page  50  B3LINE(X,Y,N,XO,XS,SFX,Y0,YS,SFY,L,J,H,K)  PARAME ERS: X r e a l a r r a y of i n d e p e n d e n t x - v a l u e s v r e a l a r r a y of dependent y - v a l u e s N i n t e g e r number of p a i r s o f ( x , y ) p o i n t s r e a l s t a r t i n g v a l u e of x - a x i s ( u n i t s ) X0 XS r e a l s t a r t i n g l o c a t i o n of x - a x i s ( i n c h e s ) SFX real x-axis scale factor (units/inch) Y0 r e a l s t a r t i n g v a l u e of y - a x i s ( u n i t s ) YS r e a l s t a r t i n g l o c a t i o n of y - a x i s ( i n c h e s ) SFY real y-axis scale factor (units/inch) L integer l i n e type c o n t r o l parameter L=0 symbols p l o t t e d o n l y L=1 c u r v e d l i n e p l o t t e d w i t h s y m b o l s L = 2 c u r v e d l i n e p l o t t e d w i t h o u t symbols L=-1 s t r a i g h t l i n e p l o t t e d w i t h symbols L=-2 s t r a i g h t l i n e p l o t t e d w i t h o u t symbols i n t e g e r symbol c o n t r o l p a r a m e t e r J H r e a l symbol h e i g h t integer o r i e n t a t i o n parameter K 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 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 SPLINE 3) IF ( X , G T . 0 ) CALL PLOT(XP(1),YP(1 I F U . L T . 0 ) CALL PLOT(YP(1),XP(1) 3) I F ( K .G T . 0 . A N D . L . E Q . l ) CALL SYM30L(XP(1) Y P ( 1 ) , K , J ,0.,-1) I F ( K . L T . 0 . A N D . L . EQ.1) CALL SYM30L(YP(1) X P ( 1 ) , H ,, 0J. , - 1 ) DG 3 0 1=2,N IM=I- 1 DX=(X?(I)-XP(IM))/2 0 . DO 20 J J = 1 , 2 0 XX=X?(IM)+JJ*DX YY=F(XX) I F ( K . G T . O ) CALL P L O T ( X X , Y Y , I F ( K . L T . O ) CALL P L O T ( Y Y , X X , CONTINUE I F ( K . G T . 0 . A N D . L . E Q . 1 ) CALL S Y M B O L ( X P ( I ) , Y P ( I ) , 0 . ,-1 ) I F ( X . L T . 0 . A N D . L . E Q . 1 ) CALL S Y M B O L ( Y P ( I ) , X P ( I ) , ,0.,-D CONTINUE RETURN CALL P L O T ( X P ( 1 ) , Y P ( 1 ) ,3) I F ( L . E Q . 0 . O R . L . E Q . - 1 ) CALL SYMBOL( XP ( 1 ) , YP ( 1 ) , H , J , 0 . , - 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 S Y M B O L ( X P ( I ) , Y P ( I ) , H , J , 0 . ,-1 ) CONTINUE RETURN END  -225-  T a b l e 35:  E x p l a n a t i o n o f s u b r o u t i n e NUMBER (57)  NUMBER Purpose This  subroutine  will  plot  a floating-point  number.  How To Use CALL NUM3E?( X , Y , H T , F L O A T , T H E T A , N ) where: (X,Y)  are the  the coordinates number.  HT  i s the h e i g h t of the number. (For more 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  is  the  floating-point  THETA  is  the  angle.  N  specifies the number of d e c i m a l d i g i t s t o t h e r i g h t of the d e c i m a l p o i n t . N=0 p u t s a d e c i m a l p o i n t a t the end of the number, N = - 1 s u p p r e s s e s the decimal point. N should not be l a r g e r than 3. The number i s t r u n c a t e d , not r o u n d e d . F o r e x a m p l e , i f F L O A T = - 1 7 . 7 9 5 and N= 3, N= 0, N=-i,  the the the  of  the  number  lower  to  left-hand  corner  of  information,  be drawn.-  c h a r a c t e r s - 1 7 . 7 9 5 are drawn; c h a r a c t e r s - 1 7 . a r e drawn; c h a r a c t e r s -17 a r e d r a w n .  Examole This  example  would p l o t  the  number  12.3  at  (5.3,6.2):  X=12.3 CALL  NUMBER(5.3,6.2,0.21,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 exceed 7 c h a r a c t e r s . 2.  the  number t o  be.  plotted  must  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 . i n t e g e r must be converted to floating-point form rj 1 o 1 1 e d w i t h N = - 1 .  not The and  -226T a b l e 36:  E x p l a n a t i o n of s u b r o u t i n e  LEGEND  LEGEND (PMV,KF,KASO,NK,SL,SIGD,PSA,PSG,SIGFA,SIGFG) - terms a r e parameters used i n the computer program UBCEDEMA  -227-  A1.4  1)  Operation of Computer Program  The computer programs UBCEDEMA, BBLINE and DLINE are compiled as  follows: //RUN  *FTN The  SCARDS=UBCEDEMA+BBLINE+DLINE compiled programs are then s t o r e d under the temporary  name  - LOAD. 2)  E x e c u t i o n of the programs u s i n g the data f i l e s  EDA and PDA i s as  follows: //RUN The  -LOAD  4=EDA  5=PDA  7=-TABLES  9=-PL0TS  t a b u l a t e d and g r a p h i c a l output Is s t o r e d  TABLES and - PLOTS,  respectively.  i n the temporary  files  -  

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