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Computer simulation of microvascular exchange after thermal injury Gu, Xiaozheng 1987

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COMPUTER SIMULATION OF MICROVASCULAR EXCHANGE AFTER THERMAL INJURY by XIAOZHENG GU A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT' OF' CHEMICAL ENGINEERING We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1987 © Xiaozheng Gu, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) Abstract A computer model is developed to study the f l u i d and protein r e d i s t r i b u t i o n after thermal i n j u r i e s in rats. This model is derived by including the burned skin as a fourth compartment in the microvascular exchange model developed by Bert et a l . [6]. The pathological changes that occur after thermal in j u r i e s are introduced into the burn model as perturbations. The simulations of short-term and long-term responses were then made in t h i s four compartment (burn) model for two cases: 10% and 40% percent surface area burns. Appropriate ranges of the perturbations were estimated based on the available information in the l i t e r a t u r e . The perturbations for the 10% burn include: the plasma leak c o e f f i c i e n t in the injured skin, the tissue pressure in the injured skin, the f l u i d exchange c o e f f i c i e n t s in the injured skin, the a r t e r i a l c a p i l l a r y pressure in the injured skin and. the lymph flow c h a r a c t e r i s t i c s in the injured skin. The perturbations for the 40% burn include the perturbations for the 10% burn plus the plasma leak c o e f f i c i e n t s in the intact tissues, the f l u i d exchange c o e f f i c i e n t s in the intact tissues and the lymph flow c h a r a c t e r i s t i c s in the intact tissues. The dynamic responses of the system using these perturbations were plotted. Comparisons between the simulation predictions and the experimental data were i i characterized in terms of sum-of-squares of differences between simulation results and experimental data. Compared to the limited amount of data available in the l i t e r a t u r e , the burn model describes microvascular exchange after thermal i n j u r i e s reasonably well. The work in this thesis could e a s i l y be extended to account for f l u i d resuscitation following a thermal injury in rats and, i t is hoped that t h i s approach might eventually be applied to the resuscitation management of burn patients. Table of Contents Abstract i i L i s t of Tables iv L i s t of Figures vi 1 . Introduction 1 1.1 General Information About Burns 1 1.2 The Treatment of Burn Patients 3 1.3 The Potential Use of Computer Simulations 4 1.4 Objectives of the Present Work 7 2. Physiological Background 11 2.1 Circulatory System 11 2.1.1 Description of the Circulatory System ....11 2.1.2 The C l a s s i f i c a t i o n of the Blood Vessels ..15 2.1.3 Microvascular Exchange 17 2.1.3.1 General Description 17 2.1.3.2 The Driving Force for Microvascular Exchange 24 2.2 Physicochemical C h a r a c t e r i s t i c s of Blood 25 2.3 Tissue and P a r t i t i o n Properties 27 2.4 The Lymphatics 30 3. Physiological Changes After Burns 32 3.1 Changes in C a p i l l a r y Permeability 33 3.2 Changes within the Blood C i r c u l a t i o n 36 3.2.1 The Cardiac Output 37 3.2.2 The Main A r t e r i a l Pressure .........40 3.2.3 The Central Venous Pressure 40 3.2.4 Total Peripheral Resistance 41 3.2.5 The Blood Flow 41 iv 3.3 Changes within the Interstitium 42 3.4 Symptoms of Burn Injuries 46 3.4.1 Plasma Volume Loss (Hypovolemia) 46 3.4.2 Edema Formation and Tissue Plasma Protein Content 48 3.4.3 Lymph Flow .49 4. Brief Survey of Computer Simulations of Microvascular Exchange 51 4.1 Models of Microvascular Exchange in Normal Tissues 51 4.2 Models of Microvascular Exchange Following a Thermal Injury 54 5. The Burn Model 57 5.1 Formulation of the Burn Model 57 5.2 Mathematical Relationships 60 5.2.1 F l u i d and Protein Mass Balances 60 5.2.2 Constitutive Relationships 68 5.2.2.1 Compliance Relationships 68 5.2.2.2 C o l l o i d Osmotic Pressure Relationships 71 5. 3 Constants 73 5.3.1 Normal Steady-State Conditions 73 5.3.2 Normal Transport C o e f f i c i e n t s 76 5.4 Computer Program 79 5.4.1 Numerical Methods 79 5.4.2 Program 80 6. Results and Discussion 83 6.1 Introduction 83 6.2 The 10% Burn 87 6.2.1 Introduction 87 V 6.2.2 Transient Response to Changes in the Plasma Leak Co e f f i c i e n t and the Hydrostatic Pressure Characteristcs of the Injured Skin (Phase One) .89 6.2.3 Transient Response Including Changes to the F i l t r a t i o n and Reabsorption Coefficients in the Injured Skin (Phase Two) 96 6.2.4 Transient Response Including Changes to the A r t e r i a l C a p i l l a r y Pressure in the Injured Skin (Phase three) 102 6.2.5 Transient Response Including Changes to The Lymph Flow Charact e r i s t i c s in the Injured Skin (Phase Four) 106 6.2.6 Most Reasonable F i t for a 10% Burn 110 6.2.7 Long-term Transient Response 111 6.2.8 Implications of the 10% Burn Model 115 6.3 The 40% Burn 1 26 6.3.1 Introduction 126 6.3.2 Transient Response for Changes to the Plasma Leakage Co e f f i c i e n t and the Tissue Pressure C h a r a c t e r i s t i c s of the Injured Skin (Phase one) 127 6.3.3 Transient Response Including Changes in the Plasma Leak C o e f f i c i e n t s in the Intact Tissues (Phase Two) 132 6.3.4 Transient Response Including Changes in the F i l t r a t i o n and Reabsorption Coefficients in the Injured Skin (Phase Three) 136 6.3.5 Transient Response Including Changes in the F i l t r a t i o n and the Reabsorption Coefficients in the Intact Tissues (Phase Four) 142 6.3.6 Transient Response Including Changes in the A r t e r i a l C a p i l l a r y Pressure in the Injured Skin (Phase Five) 146 6.3.7 Transient Response Including Changes in the Lymph Flow Cha r a c t e r i s t i c s (Phase Six) 148 vi; 6.3.8 Most Reasonable F i t for a 40% Burn 153 6.3.9 Long Term Transient Response .....154 6.3.10 Implications' of the 40% Burn Model 158 7. Conclusions 173 8. Recommendations 177 NOMENCLATURE 180 SUBSCRIPTS 181 REFERENCES 182 APPENDIX A: 188 APPENDIX B: 191 APPENDIX C: 210 APPENDIX D 220 v i i L i s t of Tables Table Page 1.1 Incidence, Morbidity and Mortality of Burns Per 2 M i l l i o n People 1.2 Resuscitation Formulas to Prevent Burn Shock 4 2.1 The Protein Composition of the Blood and 26 I n t e r s t i t i a l F l u i d for Humans Under Normal Conditions 3.1 Permeability Change 34 3.2 Hemodynamic Change 38 3.3 Characteristic Effects of Experimental Burn 47 Injuries 5.1 Summary of Burn Model Equations 68 5.2 Coef f i c i e n t s in Osmotic Pressure Relationships 72 5.3 Normal Conditions: 225 g rat 75 5.4 Transport Coefficients Determined for Normal Skin 78 and Muscle 6.1 SS as a Function of K p l b and P b for 10% Burn 92 (Phase One) 6.2 SS as a Function of K p l b, K f b, and K r b for 10% 98 Burn (Upper Curve for P,, Phase Two) 6.3 SS as a Function of K p l b, K f b, K r b and P Q b for 10% 105 Burn (Upper Curve for P b, Phase Three) 6.4 SS as a Function of K f b, K r b, P a b and V l b for 10% 108 Burn (A=35, Upper Curve for P b, Phase Four) v l i i 6.5 SS as a Function of K p l b and P b for 40% Burn 129 (Phase One) 6.6 SS as a Function of K l b , K p l s and K p l m for 40% 134 Burn (Upper Curve for P b, Phase Two) 6.7 SS as a Function of K l b , K l s , K p l m, K f f a and K r f a 138 for 40% Burn (Upper Curve for Pfa, Phase Three) 6.8 SS as a Function of K p l, K f and K r for 40% Burn 144 (Upper Curve for Pfa, Phase Four) 6.9 SS as a Function of K p l, K f, K r and P a b for 40% 147 Burn (Upper Curve for P b, Phase Five) 6.10 SS as a Function of K p l, K f, Kf, P a b and Lymph 150 Flow Characteristics for 40% Burn (Upper Curve for Pb' Pab = 1 7 , 6 mrnHg, Phase Six) A.1 Compliance Relationship for Skin 189 A.2 Compliance Relationship for Muscle 190 C.1 Experimental Data for 10% Burn by Lund and Reed 215 C.2 Normalized Experimental Data for 10% Burn 215 C.3 Experimental Data for 40% Burn by Lund and Reed 216 C.4 Normalized Experimental Data for 40% Burn 216 i x L i s t of Figures Figure Page 2. 1 The Circulation 13 2.2 Cross Sectional Area and Velocity Variations in 14 the Human Circ u l a t i o n 2.3 Schematic of the Vascular System 16 2.4 The Microvascular System 19 2.5 Relationship of C a p i l l a r i e s , Tissues and 20 Lymphat ics 2.6 C l e f t s in the C a p i l l a r y Membrane 22 2.7 Farmer I r r i g a t i n g Crops - an Analog to Ca p i l l a r y 23 Perfusion 2.8 Model of Volume Exclusion, Sphere in a Random 29 Network of Rods 2.9 Relationship Between Tissue Pressure and Lymph 31 Flow 3.1 Cardiac Output, Stroke Volume and Heart Rate for 39 Guinea Pigs After a 70% Burn 3.2 The Tissue Hydrostatic Pressure of Injured Skin 44 for Rats Subjected to a 10% Burn 3.3 The Tissue Hydrostatic Pressure of Injured Skin 45 for Rats Subjected to a 40% Burn 5.1 Schematic Diagram of Burn Model 58 6.1 Simulation for Changing Kp]^ and P^ in 10% Burn 91 (A=35.0, Upper Pressure Curve for P^) x 6.2 Simulation for changing K f b and K r b in 10% 99 Burn (A=35.0, B=-0.9, Upper Curve for P b) 6.3 Simulation for Changing K l b , K f b and K r b in 10% 100 Burn (A=35.0, B=-0.5, Upper Curve for P b) 6.4 Simulation for Changing Kp-^, K j b , K r b and P a b in 103 10% Burn (A=35.0, B=-0.9, P a b = 1 0 - 0 mmHg, Upper Curve for P b) 6.5 Simulation for Changing K p^ b, Kf b , K r b and P a b in 104 10% Burn (A=35.0, B=-0.9, P fa=15.0 mmHg, Upper Curve for P b) 6.6 Simulation for Changing K f b, K r b, P a b and V^ b in 109 10% Burn (A=35.0, B=-0.9, P a b=l0.0 mmHg, Upper Curve for P b, Altered Lymph Flow Characteristic) 6.7 Steady-State Result for 10% Burn (A=35.0, B=-0.5 112 P a b=l5.0 mmHg, Upper Curve for P b, v i b = 2 * v l b o w h e n t<3 hours and v i b < 2 * v l b o ' A r t e r i a l a n c ^ Venous Capill a r y Pressure Return to Normal After Three Hours) 6.8 Steady-State Result for 10% Burn (A=35.0, B=-0.5 113 P a b=l5.0 mmHg, Upper Curve for P b, vifc> = 2* vlbo when t<3 hours and v i D < 2 ' V ^ b o , A r t e r i a l and Venous Capill a r y Pressure Return to Normal After Six Hours) 6.9 Steady-State Result for 10% Burn (A=35.0, B=-0.5 114 P a b=l5.0 mmHg, Upper Curve for P b, v i b = 2 * v l b o w h e n t<3 hours and v i b < 2 ' v l b o ' A r t e r i a l and Venous Capill a r y Pressure Return to Normal After Nine Hours) 6.10 The Changes of Compartmental F l u i d Volumes 116 Following a 10% Burn xi 6.11 The Changes of Compartmental Protein Contents 117 Following a 10% Burn 6.12 The Changes of Flu i d Flow Rates in Injured Skin 118 Following a 19% Burn 6.13 The Changes of Albumin Transport Rates in Injured 119 Skin Following a 10% Burn 6.14 The Changes of Capill a r y and Injured Skin 120 Pressures Following a 10% Burn 6.15 Simulation for Changing K p l b and P b in 40% Burn 130 (A=4.0, Upper Curve for P^) 6.16 Simulation for Changing K p l b, K p i S ' Kplm a n d P b * n 1 3 5 40% Burn (A=4.0, B=8.0, Upper Curve for Pfa) 6.17 Simulation for Changing Kp l b, Kp l s, K p l m, K f b and 139 K r b in 40% Burn (A=4.0, B=10.0, C=-0.80, Upper Curve for P b) 6.18 Simulation for Changing K p l b, K p l s, K p i m ' K f b a n d 1 4 0 K r b in 40% Burn (A=4.Q, B=10.0, C=-0.95, Upper Curve for P b) 6.19 Simulation for Changing K p l b, K p l s, K p l m, K f b, 145 K r b ' K f s ' Kfm' K r s a n d Krm i n 4 0 % B u r n (A-3.0f. B=10.0, C=-0.80, D=-0.30, Upper Curve for P b) 6.20 Simulation for Changing K p l b, K p l s, K p i m f K f b ' 1 5 1 K r b ' K f s ' Kfm' K r s ' Krm' v l b ' ' v l s a n d i n 4 0 % Burn (A=4.0, B=10.0, C=-0.95, D=-0.0, Upper Curve for P b, Second Condition for Lymph Flow Characteristics) x i i 6.21 Sim u l a t i o n f o r Changing Kp^ r K p l s , K p i m ' K f b ' 1 5 2 K r b ' K f s ' Kfm' K r s ' Krm' v l b ' ^ l s a n d ^lm i n 4 0 % Burn (A=4.0, B=10.0, C=-0.95, D=-0.0, Upper Curve for P^, T h i r d C o n d i t i o n f o r Lymph Flow C h a r a c t e r i s t i c s ) 6.22 Steady State Result f o r 40% Burn (A=4.0, B=10.0, 156 C=-0.95, D=0.0, Upper Curve for. Pfa, A r t e r i a l and Venous C a p i l l a r y Pressure Return to Normal A f t e r Three Hours) 6.23 Steady State Result f o r 40% Burn (A=4.0, B=10.0, 157 C=-0.95, D=0.0, Upper Curve f o r P^, A r t e r i a l and Venous C a p i l l a r y Pressure Return to Normal A f t e r Six Hours) 6.24 Steady State Result f o r 40% Burn (A=4.0, B=10.0, 158 C=-0.95, D=0.0, Upper Curve f o r P b, A r t e r i a l and Venous C a p i l l a r y Pressure Return to Normal A f t e r Nine Hours) 6.25 The Changes of Compartmental F l u i d Volumes 161 F o l l o w i n g a 40% Burn 6.26 The Changes of Compartmental Plasma P r o t e i n 162 Contents F o l l o w i n g a 40% Burn 6.27 The Changes of F l u i d Flow Rates i n I n t a c t Skin 163 F o l l o w i n g a 40% Burn 6.28 The Changes of Albumin T r a n s p o r t Rates i n I n t a c t 164 Skin F o l l o w i n g a 40% Burn 6.29 The Changes of C a p i l l a r y and I n t a c t Skin Pressures 165 F o l l o w i n g a 40% Burn 6.30 The Changes of F l u i d Flow Rates in In j u r e d Skin 166 F o l l o w i n g a 40% Burn x i i i 6.31 The Changes of Albumin Transport Rates in Injured 167 Skin Following a 40% Burn 6.32 The Changes of Capillary and Injured Skin 168 Pressures Following a 40% Burn A.1 Compliance Curve for Muscle and Skin of Rats 191 xiv Chapter 1 INTRODUCTION 1.1 GENERAL INFORMATION ABOUT BURNS Burns represent a major surgical trauma worldwide. Each year, thousands of people suffer from burn i n j u r i e s . To indicate the magnitude of the problem, a summary of such inju r i e s and some consequences are l i s t e d in Table 1.1. Note that, in the developing countries of the Third World (e.g. Algeria) the mortality rate i s much higher, with about 140 deaths in hospital alone per m i l l i o n population per year [1], compared to Western Nations. Burns may result from any one of the following sources [2]: 1. thermal (flame,steam, hot l i q u i d , hot metal), 2. e l e c t r i c a l (alternating current, direct current, lightning) , 3. - chemical (acid, a l k a l i , vesicant agents) or, 4. radiation (nuclear, s o l a r ) . Thermal injury accounts for 95% of hospital burn admissions and e l e c t r i c a l injury for 3% [2], 1 T a b l e 1.1 I n c i d e n c e , M o r b i d i t y and M o r t a l i t y of Burns Per M i l l i o n People [1] Burn I n j u r i e s Hospi t a i i z a t i o n Deaths Due t o Burn I n j u r i e s Burn I n j u r i e s C o u p t r i es U n i t e d S t a t e s 8333 292 38 Denmark 4140 325 10 England 250 Wales 420 S c o t l a n d 30 A l g e r i a 140 3 1.2 THE TREATMENT OF BURN PATIENTS Burns can severely damage or destroy tissue components, as well as effect a large f l u i d s h i f t from the blood to the surrounding tissues. The resulting imbalance in f l u i d d i s t r i b u t i o n causes the tissue to swell, while blood volume is reduced. This loss of blood volume can lead to shock and death. F l u i d resuscitation following a burn injury i s often the c r i t i c a l consideration with respect to saving the injured person's l i f e . Mankind has searched for soothing and healing burn medicines throughout history [3], In the f i r s t century AD, Celsus prescribed the application of honey and bran, then cork and ashes for the treatment of burns. Pliny the Elder proposed allowing burns to remain exposed to a i r rather than covering them with grease. In the second century AD, Galen prescribed vinegar and wine dressings [3]. - The current therapy for burn victims seeks restoration of health with minimal impairment and disfigurement. The main c l i n i c a l treatment of burns involves f l u i d replacement (resuscitation), which is accomplished by inj e c t i n g f l u i d s intravascularly to replace f l u i d l o s t from the c i r c u l a t i o n . Several empirical formulas have been proposed to estimate the amount of f l u i d to be replaced, based on past c l i n i c a l experience. Some examples of such formulas are l i s t e d in 4 Table 1.2. The patient's response to the burn and to resuscitation must be monitored continuously. In p a r t i c u l a r , urinary output provides a generally r e l i a b l e guide to the adequacy of resuscitation. Resuscitation f l u i d s are given to maintain a urinary output rate of 30 to 50 ml/h for adults [4]. Since the s p e c i f i c needs of individual burn patients vary, the resuscitation formulas can only serve as general guidelines. For example, burn patients with e l e c t r i c a l injury or inhalation injury usually need more f l u i d s than for other types of burns [4]. F l u i d replacement formulas risk p a r t i a l or t o t a l f a i l u r e due to both the v a r i a b i l i t y amongst patients and the approximate nature of the formulas. 1.3 THE POTENTIAL USE OF COMPUTER SIMULATIONS In the 1980's, the potential has arisen for the use of computer simulations in the management and treatment of burn patients. Mathematical relationships describing microvascular exchange in the human for normal and some pathological conditions now appear in the l i t e r a t u r e [5]. T a b l e 1.2 R e s u s c i t a t i o n Formulas to Pr e v e n t Burn Shock (131 Evans formula Brooke formula Modified Brooke formula Ariz formula Parkland formula Hypertonic formula Firtl 24 houri: Electrolyte solu-tion Normal saline Ringer's lactate Ringer's lactate Ringer's lactate Ringer's lactate Hypertonic lac-tated saline (sodium, 250 mEq/liter) ml/kg/ft burned 1.0 1.5 2.0 30 4.0 Rate based on urine output of 70 ml/hour in adults Colloid 1 ml whole blood plasma, or plasma expanders/ k g / * burn 0.3 m l / k g / * burn None None None None Free water (D,W) 2,000 ml 2.000 ml None None None None St con J 24 hours: One-half first 24-hour dose; same amount D , W One-half first 24-hour dose; same amount D , W D,W and colloid D,W and colloid Only D,W to maintain utine out-put; colloid, 0.) to 2 liters Continued at rate to maintain urine output >)0 ml . 6 The physiological changes that occur after a thermal injury can also be incorporated into mathematical models so that the results of a thermal injury can be predicted. The effectiveness of the diff e r e n t f l u i d resuscitation schemes can then be assessed. The computer simulation, i f v a l i d , can therefore be used as an aid in managing resuscitation for t h i s type of injury. As an example of the use of computer simulations in the assessment of burn i n j u r i e s , consider the model by Arturson et a l . [5]. In t h i s model, the microcirculation is divided into three compartments: plasma, burned tissue and normal tissue. Within each compartment, mass balances for the important constituents (e.g., f l u i d and albumin) can be written. The dynamic behavior of the system, i.e. the time-dependent movement of these substances into and out of the system and amongst the compartments, i s then described by the solution of a set of ordinary d i f f e r e n t i a l equations. Arturson's model has been used to describe postburn edema formation [5]. The model predicted the formation, d i s t r i b u t i o n and composition of two d i f f e r e n t kinds of edema following thermal i n j u r i e s : a l o c a l edema which i s protein-rich, and a general edema which i s protein-poor [5], 7 1.4 OBJECTIVES OF THE PRESENT WORK Despite i t s achievements, the model by Arturson et a l . provides only a limited amount of information about f l u i d and protein exchange. Its major weakness i s that i t combines intact skin and muscle together as a single compartment whereas i t is well-known that the properties of these two tissues are quite d i f f e r e n t . Additionally, other information inputs into his model are questionable. Thus, the primary objective of the present work i s to develop a four compartment (plasma, injured skin, intact skin and muscle) simulation model which w i l l adequately describe the dynamic exchanges of f l u i d and protein which take place these compartments following burn i n j u r i e s . Because of the limited amount of experimental information available for humans, the present work w i l l use the rat as the animal model. This approach has several advantages including: 1. the existence of a validated model of microvascular exchange for the normal (uninjured) rat developed by Bert et a l . [6], 2. a large amount of experimental information available in the l i t e r a t u r e for this animal under normal conditions, and 3. the a v a i l a b i l i t y in the l i t e r a t u r e of both qu a l i t a t i v e and quantitative experimental information about the systemic changes which take place in the rat following a burn. 8 In addition, experimental data for both normal and thermally injured rats is being provided to us by a group of researchers (notably R. Reed and T. Lund) from the Department of Physiology, School of Medicine, University of Bergen in Norway. The computer simulation presented here represents a part of the col l a b o r a t i v e e f f o r t between UBC and the University of Bergen. The general steps leading to the development of a non-resuscitative burn simulation for rats were as follows: 1. The three compartment (plasma, skin and muscle) model of Bert et a l . [6] for normal rats was extended to four compartments by d i v i d i n g the skin compartment into two separate portions; injured and intact skin. F l u i d and. protein mass balances were written for each compartment yi e l d i n g a set of eight ordinary d i f f e r e n t i a l equations which, along with appropriate constants as well as a u x i l i a r y and c o n s t i t u t i v e relationships, defined the burn model. 2. Assuming that normal steady-state conditions prevailed at zero time, the system was than perturbed by a l t e r i n g one or more of the c h a r a c t e r i s t i c parameters used in the model of Bert et a l [6 ] , to describe microvascular exchange in the noninjured rat. Reasonable changes to both the c o e f f i c i e n t s governing f l u i d and protein exchange as well as the hydrostatic pressures within the system were considered. •All such parametric changes 9 were permitted to decay with time to allow the system to return to normal (recuperate) after a s u f f i c i e n t l y long period. 3. The resulting set of i n i t i a l - v a l u e ordinary d i f f e r e n t i a l equations was solved numerically using the Runge-Kutta-Fehlberg. method to predict the changes in a l l compartmental variables (hydrostatic and osmotic pressures, f l u i d and protein fluxes, f l u i d volume and protein content) which occurred as a function of time following a 10% or 40% surface area burn. 4. By comparing the model predictions with the quantitative experimental results of Lund and Reed [45] for 10% and 40% burns in rats as well as with other, more qua l i t a t i v e results recorded in the l i t e r a t u r e for rats and other animals, a so-called " b e s t - f i t " model was determined for both the 10% and 40% burn cases. The "best f i t " simulation was considered to be that which was able to provide an adequate interpretation of the experimental results while requiring the simplest and " most reasonable parametric changes to the o r i g i n a l model. Obvious extensions to the present model which were not considered here include: 1. development of a burn resuscitation model for rats by adding f l u i d and protein sources and sinks such as intravascular i n j e c t i o n , wound exudation and urine 10 production to the mass balances, 2. development of a burn resuscitation model for humans along similar l i n e s . However, such developments w i l l have to wait u n t i l s u f f i c i e n t experimental data become available to allow estimates of a l l the transport parameters and constitutive relationships needed to adequately describe the extended models. Chapter 2 PHYSIOLOGICAL BACKGROUND In order to understand the f l u i d and plasma protein exchanges resulting from thermal i n j u r i e s , a review of the physiology relevant to microvascular exchange system, such as the normal function of the blood c i r c u l a t i o n , the blood plasma composition, the tissue properties and lymph flow behaviour, is required. These w i l l be discussed in the following sections. 2.1 CIRCULATORY SYSTEM 2.1.1 DESCRIPTION OF THE CIRCULATORY SYSTEM S i m p l i s t i c a l l y , the c i r c u l a t o r y system can be considered as a closed system, consisting of the heart and various kinds of blood vessels. It is a very complex, sel f - r e g u l a t i n g , feedback control system which keeps the systemic variables, such as a r t e r i a l blood pressure, within a narrow range. Except for severe upsets, adequate blood flow through a l l organs and tissues is maintained. The driving force for thi s flow i s the a r t e r i a l blood pressure, produced by ventricular contraction [ 7 ] , From the l e f t v e n t r i c l e of the heart, blood is injected into the aorta and into the coronary blood vessels which 1 1 1 2 supply the heart muscle with nutrients. The aorta branches into a r t e r i e s , a r t e r i o l e s and then c a p i l l a r i e s . The c a p i l l a r y walls allow the necessary exchange of 0 2, C0 2 and other substances between the blood and the tissues. Following this exchange process, the blood, now depleted of 0 2 and enriched in C0 2, is returned to the right v e n t r i c l e by way of the venous blood vessels. From the right v e n t r i c l e , blood i s injected into the pulmonary c i r c u l a t o r y system where the exchange of 0 2 and C0 2 takes place in the lung, such that blood returning to the l e f t atrium and the l e f t v e n t r i c l e i s oxygen-rich [7]. This c i r c u i t is i l l u s t r a t e d in Figure 2.1. The t o t a l cross-sectional area of a l l c a p i l l a r i e s , the smallest diameter elements in the c i r c u l a t o r y system, is 500 to 600 times that of the aorta. Since the volume flow rate of blood through the aorta must be i d e n t i c a l to that in the c a p i l l a r y bed, the average flow v e l o c i t y through the c a p i l l a r i e s i s much lower than that in the aorta. The area and v e l o c i t y variations throughout t h i s system are shown in Figure 2.2. 1 3 F i g u r e 2.1 The C i r c u l a t i o n [14] Capillaries Veins from head and upper extremities Superior vena cava Arteries to head and upper extremities Pulmonary artery Pulmonary veins Lung Right atrium Inferior vena cava Right ventricle Hepatic vein Liver Veins from abdomen and lower extremities Portal vein Aorta Arteries to abdomen and lower extremities Spleen Capillaries 1 4 F i g u r e 2.2 C r o s s S e c t i o n a l A r e a and V e l o c i t y V a r i a t i o n s i n t h e Human C i r c u l a t i o n [ 1 5 ] 1 5 2.1.2 THE CLASSIFICATION OF THE BLOOD VESSELS The c i r c u l a t o r y system contains a number of d i f f e r e n t types of blood vessels including Windkessel vessels, precapillary resistance vessels, followed by precapillary sphincters, c a p i l l a r y exchange vessels, and venous vessels. Differences in the microstructure of the various blood vessels r e f l e c t their p a r t i c u l a r function (see Figure 2.3). Windkessel vessels, such as the aorta and the large a r t e r i e s , offer l i t t l e resistance to flow. Their e l a s t i c walls serve as energy-storing reservoirs which enact a damping e f f e c t on the p u l s a t i l e output of the v e n t r i c l e s . Therefore, the pressure drop in the windkessel vessels i s very small [7].' Precapillary resistance vessels are responsible for most of the t o t a l resistance to flow. Compared to Windkessel vessels, the e l a s t i c components of the wall are replaced by muscle f i b r e s which function as variable r e s i s t o r s . The diameter of these vessels depends primarily on l o c a l physical and chemical factors; t h i s dependence i s p a r t i c u l a r l y acute in the resistance vessels of the heart and the brain [7]. 1 6 Figure 2.3 Schematic of the Vascular System [15] U J 2 .WINDKES'.'IEL I RESISTANCE I J .EXCHANGE , RESl SI ANCE \CARACl TANCE VENTRICLE 0-I VESSELS i VESSELS I w \CAPIL L ARIES I VfSSfLS I VESSELS m m 120 100 80 -60 40 20 c i r c u l a t i o n 17 Precapillary sphincters consist of rings of smooth muscle. They are t y p i c a l l y located near the a r t e r i a l entrance to the microvascular bed, surrounding a r t e r i a l blood vessels. Contraction of the sphincters determines the size of the c a p i l l a r y exchange area by modifying the number of c a p i l l a r i e s perfused at any one moment [8]. The c a p i l l a r y exchange vessels form the most important region of the vascular system with respect to the exchange of f l u i d and other material between the blood and tissues (microvascular exchange). The c a p i l l a r y wall acts as a selective membrane, influencing the transfer of l i f e - s u s t a i n i n g chemical compounds between the blood and tissue c e l l s [8]. This d e l i c a t e l y balanced system is obviously highly susceptible to burn i n j u r i e s in tissues such as skin. 2.1.3 MICROVASCULAR EXCHANGE The microvascular exchange system i s essential for the maintenance of the health and well-being of the individual. Since i t i s the subject of our model, a more detailed description of i t s physiology w i l l be given. 2.1.3.1 General Description The microvascular system can be divided into six parts (see Figure 2.4) based on their d i f f e r e n t location 18 and function: 1. precapillary sphincters, 2. a r t e r i a l c a p i l l a r i e s , 3. true c a p i l l a r i e s , 4. venous c a p i l l a r i e s , 5. the surrounding tissue and 6. the lymphatics. As previously mentioned, the precapillary sphincters form the connection between a r t e r i o l e s and the c a p i l l a r i e s . The c a p i l l a r i e s of the network nearest to the a r t e r i o l e s are c a l l e d a r t e r i a l c a p i l l a r i e s , and those nearest to the venules are c a l l e d venous c a p i l l a r i e s . The c a p i l l a r y network between a r t e r i o l e s and venules often contains a thoroughfare channel c a l l e d the central channel, where the blood flow i s continuous, in contrast to the branches of the c a p i l l a r y network, where the flow tends to be intermittent. The surrounding tissues include muscle and skin or other tissues. The drained substances of the surrounding tissues return to the c i r c u l a t i o n via lymphatic channels (see Figure 2.5). 1 9 Figure 2.4 The Microvascular System [ 9 ] a) Branching Network of C i r c u l a t i o n C a p i l l a r i e s A r t e r i e s A r t e r i o l e s V e n u l e s V e i n s b) A Typical Microvasular Pattern A r i e " 0 ' e Stnoll * e n u l t 20 Figure 2.5 Relationship of C a p i l l a r i e s , Tissues and Lymphatics [9] Capillary I n t e r s t i t i u m 21 The c a p i l l a r y walls consist of a single endothelial c e l l layer which separates the blood from the in t e r s t i t i u m . Ordinarily, adjacent endothelial c e l l s are separated by a narrow c l e f t (see Figure 2 .6) [ 8 ] . The c a p i l l a r i e s branch extensively without much change in c a l i b r e . The density and pattern of the c a p i l l a r y networks vary in the different tissues and organs. The tissues with the highest metabolic a c t i v i t y have more elaborate branches and c l o s e l y packed c a p i l l a r i e s than low metabolic a c t i v i t y tissues. In skin, for example, the branches are numerous [8]. Normally, a c a p i l l a r y is closed to blood flow about 60% to 95% of the time. Apparently being perfused 5% of the time is. adequate under resting circumstances. Therefore, only about 5% of c a p i l l a r i e s may be open at any.one time but this 5% i s constantly changing so that a l l c a p i l l a r i e s are perfused in turn. This may be compared to a farmer opening and closing i r r i g a t i o n head gates so as to insure that each row of crops i s i r r i g a t e d in turn. Increased metabolic a c t i v i t y increases the percentage of c a p i l l a r i e s open (Figure 2.7) [8]. 22 Figure 2.6 C l e f t s in the C a p i l l a r y Membrane h 6 ] 23 Figure 2.7 Farmer I r r i g a t i n g Crops - a n Analog to Capillary Perfusion [16] Headgate in dam — . Irrigation ditch f Fnrniei opening side ol d i lc l i Row o( crops Now being irrigated Now nol being irrigated Now not being irrigated Now not being irrigated Now being irrigated Capillary sphincter closed ^ (] Capillary closed Capillary open ..-Q C O O 24 2.1.3.2 The Driving Force for Microvascular Exchange The passage of f l u i d across the c a p i l l a r y wall is dependent on the following driving forces: 1. the blood pressure within the c a p i l l a r i e s , 2. the c o l l o i d osmotic pressure in the interstitium, 3. the c o l l o i d osmotic pressure in the blood and 4. the hydrostatic pressure in the i n t e r s t i t i u m . According to Starling's Hypothesis, the f i r s t two factors promote passage of f l u i d from the vessels to the tissues, while the last two factors favor reabsorption into the blood. The blood pressure in the a r t e r i a l c a p i l l a r i e s is normally higher than in the venous c a p i l l a r i e s , and the blood pressure on the a r t e r i a l side also normally exceeds the c o l l o i d osmotic pressure of the blood plasma. Based on these pressure relationships, f l u i d s normally pass from the vessels to the tissues in the a r t e r i a l part of the c a p i l l a r y bed and return to the vessels via the venous c a p i l l a r i e s , small venules, and lymphatic channels [8]. Pressure within the c a p i l l a r i e s can be modified at the l o c a l l e v e l by vasoconstriction and vasodilation through contraction and relaxation of smooth muscle c e l l s in the walls of a r t e r i o l e s , metarterioles and p r e c a p i l l a r y sphincters. A r t e r i a l 25 blood pressure also influences c a p i l l a r y flow. Elevation in pressure caused by an increase in cardiac output w i l l increase c a p i l l a r y flow. Elevation in blood pressure caused by a r t e r i o l a r c o n s t r i c t i o n decreases c a p i l l a r y flow. Actually, changes in a r t e r i a l pressure usually involve elements of both, so that c a p i l l a r y flow may be increased, decreased or unchanged [8], 2.2 PHYSICOCHEMICAL CHARACTERISTICS OF BLOOD Blood, which i s composed of c e l l s and plasma medium, is cir c u l a t e d through the vascular system. Almost a l l the blood c e l l s are red c e l l s . Their main function is to transport the oxygen from the lungs to the tissues and organs. White and red c e l l s e x i s t in a r a t i o of 1 : 5 0 0 in the blood. White c e l l s protect against invasion by disease organisms [ 9 ] . The plasma medium of blood i s composed of primarily of water containing proteins and small dissolved ions. The plasma proteins can be classed as albumins, globulins and fibrinogen based on their molecular weight and function. Both in blood and in i n t e r s t i t i a l f l u i d , the albumin content is dominant (see Table 2 . 1 ) . Albumin i s normally the pr i n c i p l e determinant of c o l l o i d osmotic pressure [ 1 0 ] . Table 2.1 The Prote in Composition of the Blood and I n t e r s t i t i a l F l u i d fo r Humans Under Normal Condit ions Protei n Albumi n Globul ins F i br i nogen Total Prote in Blood (g/ml) 6.042 0.027 0.003 0.073 I n t e r s t i t i a l F l u i d (g/ml) 0.025 0.011 N.A. N.A. N.A. - not ava i lab le 27 2.3 TISSUE AND PARTITION PROPERTIES "The interstitium is the connective tissue space outside the vascular and lymphatic systems and the c e l l s " [12]. There is a certain amount of water, i t s dissolved constituents, plasma proteins and glycosaminoglycans in the i n t e r s t i t i a l space [12], The basic structure of the i n t e r s t i t i u m involves collagenous f i b e r s which are made up of bundles of smaller units of collagen [12]. "The main functional e f f e c t s of the collagenous fibers are that they r e s i s t changes in tissue configuration and volume, they exclude proteins, and they immobilize the glycosaminoglycans" [12]. Other than collagenous f i b e r s , some tissues contain e l a s t i c f i b e r s which have a rubberlike consistency, hyaluronate which i s composed of unbranched high molecular weight polysaccharide, proteoglycans and c e l l s [12]. Due to the geometric shapes of the i n t e r s t i t i u m components, the extravascular space accessible to a plasma protein or other macromolecules i s limited [12]. Therefore, the e f f e c t i v e volume available to proteins i s reduced, and the e f f e c t i v e protein concentration i s increased. This exclusion phenomenon can be described by the model of a sphere in a random network of rods. The protein i s assumed to be in the shape of a sphere and the collagenous 28 fibers are assumed to be c y l i n d r i c a l rods (see Figure 2.8). The volume into which the protein cannot enter (shaded area) due to the s t e r i c hindrance of the rods is termed the excluded volume [12]. The volume accessible to the spheres can be calculated by substracting the excluded volume from the t o t a l f l u i d content of this system. The c o l l o i d osmotic pressure in a given compartment (be i t blood or tissue) is a function of the e f f e c t i v e protein concentration in that compartment; which is obtained from the protein content and the nonexcluded volume of that compartment. Another important property of the interstitium involves the relationship between i n t e r s t i t i a l f l u i d volume and tissue pressure. These are related through a- compliance rela t i o n s h i p often expressed as tissue pressure as a function of i n t e r s t i t i a l volume (see Figure A.1). In this figure note that as i n t e r s t i t i a l f l u i d volume becomes very large, only s l i g h t changes in tissue hydrostatic pressure r e s u l t . At moderate values of i n t e r s t i t i a l f l u i d volume, the-change in tissue pressure due to a change in f l u i d volume is substantially greater. F i g u r e 2.8 M o d e l of Volume E x c l u s i o n [ 1 2 ] S p h e r e i n a Random N e t w o r k of Rods 30 2.4 THE LYMPHATICS The lymphatic system is a closed vascular system [13]. It is composed of endothelial-1ined channels, which run p a r a l l e l to the a r t e r i a l and venous system. I n t e r s t i t i a l f l u i d containing, plasma protein normally drains from the i n t e r s t i t i u m and is returned to the blood c i r c u l a t i o n via the lymphatics (see Figure 2.5). The mechanism of lymph flow i s very complex but may be controlled by tissue hydrostatic pressure [13]. As the i n t e r s t i t i a l f l u i d volume increases, the tissue hydrostatic pressure increases according to the tissue compliance relationship. This in turn causes a marked increase in l o c a l lymph flow (Figure 2.9), hence preventing tissue overloading (edema) [13]. It i s hypothesized [13] that when i n t e r s t i t i a l f l u i d pressure r i s e s to a certain l e v e l , the r e l a t i v e lymph flow does not increase. If that i s true, the lymphatic system protects against edema formation only up to l i m i t i n g tissue pressure. 31 Figure 2.9 Relationship Between Tissue Pressure and Lymph Flow [13] — f 1 1 1 ' — i r -6 -4 -2 0 2 4 PT (mmHg) Chapter 3 PHYSIOLOGICAL CHANGES AFTER BURNS The experimental information in the l i t e r a t u r e concerning the c h a r a c t e r i s t i c changes of the microvascular exchange system after a burn injury w i l l be reviewed. These changes may be grouped into the following three categories: 1. changes in c a p i l l a r y permeability and ef f e c t i v e surface area, 2. changes within the blood c i r c u l a t i o n , and 3. changes within the i n t e r s t i t i u m . The results of the above physiological changes are also reviewed with respect to the following: 1. plasma volume loss, 2. edema and increased protein content in the tissues, and 3. increased lymph flow. The l i t e r a t u r e contains a number of experimental burn studies using d i f f e r e n t animal species. The l i v e animal subjects were exposed to varying extents of thermal injury. Most of the experiments were conducted without f l u i d resuscitation. Due to differences between protocols, these experiments w i l l be compared only in very general terms. 32 33 3.1 CHANGES IN CAPILLARY PERMEABILITY Following, a burn, the transfer of f l u i d and plasma proteins from the blood to the tissue increases. This was f i r s t demonstrated in 1943 by Netsky and Le i t e r [18], who measured the rate of serum protein crossing the c a p i l l a r y endothelium of dogs. The finding of an increase in tran s c a p i l l a r y macromolecular transport was confirmed later using radioactive dyes and Evans blue as test substances [18]. In fact, molecules as large as 125,000 MW, which do not normally cross the endothelial b a r r i e r , have been exchanged following a burn [19], implying an increase in e n t h o l e l i a l permeability. The investigation of change in c a p i l l a r y permeability surface area product as a result of a thermal injury has been the subject of many studies [18,20,21,22,23,24]. The results of some of these experiments are l i s t e d Table 3.1. It is d i f f i c u l t to compare these works d i r e c t l y due to differences in the animal model, degree and area of burn as well as the time interval under investigation. At best we can hope to identif y trends in the behaviour of t h i s system resultant to a thermal injury. One example of these experiments i s the work of Arturson and Mellander in 1964. In his.work, cats were subjected to a second degree burn. The c a p i l l a r y f i l t r a t i o n c o e f f i c i e n t in the injured tissue was then measured. It was found that, following an acute T A B L E 3.1 P E R M E A B I L I T Y C H A N G E [ l 8 , 20, 21, 22, 23, 24] ANIMAL BURN DEGREE PERCENT BURN AREA EXPT. TIME PARAMETERS MEASURED IN INJURED SKIN PARAMETER MEASURED IN INTACT TISSUES PARAMETERS PERCENT CHANGE DURATION OF THE CHANGE PARAMETER PERCENT CHANGE DURATION OF THE CHANGE Arturson et a l . 1964 cat 2nd one paw 0-60 minutes C a p i l l a r y F i l t r a t i o n C o e f f i c i e n t Increases ~ 300% maximum ~ 60 minutes Carvajal ec a l . 1975 guinea pig 2nd 6ctn^ The 3rd hour a f t e r burn Protein Extrava-sation Increases ~ 8.4 times maximum Brouhard et a l . 1977 rat 2nd 0.2% 0-48 hour Protein Extrava-sation Increases ~ 16 times maximum 12 hours Brouhard et a l . 1978 rat 2nd 0.2% 1/2 hour post burn 3 hour post burn 6 hour post burn Protein Extrava-sation Increases —14 times maximum Increases —2.3 times maximum Increases~l.14 times maximum Carvaj a l * et a l . 1978 rat deep 2nd 10% 20% 30% 40% 0-24 hours Protein Extrava-sation Increases ~14 times maximum Increases —12 times maximum Increases —19 times maximum Increases —12 times maximum 12 hours Protein Extrava-sation Remain norm.il. Increases —70% maxinuim Increases — 6 times maximum Increases — 2 times maximum 12 hours Lund et a l . 1986 rat 3rd 10% 40% 0-3 hours Protein Extrava-sation Increases ~54 times maximum Increases -7 times maximum Injured animals are resuscitated i n these experiments C/min/GDW: counts/minutes/gram of dry weight 35 burn, there was a very rapid transfer of f l u i d from the intravascular to the extravascular space of the injured tissue, and that the c a p i l l a r y f i l t r a t i o n c o e f f i c i e n t increased by 100 to 300 percent of i t s control value [18]. The increased c a p i l l a r y f l u i d conductivity may re f l e c t two d i s t i n c t changes in the microvascular bed: 1. Due to the application of heat, i n t e r c e l l u l a r junctions in the c a p i l l a r y membrane become wider [18]. F l u i d and proteins exude rapidly from the c a p i l l a r y to the i n t e r s t i t i a l space through these gaps in the endothelial wall. In severe cases, d i r e c t damage to the endothelial c e l l may disrupt the c a p i l l a r y wall, increasing i t s •conductivity [25]. 2. Following thermal injury to the microvascular bed thete is an associated release of histamine-1ike mediators. These mediators increase the metabolic l e v e l in the c a p i l l a r i e s , which in turn increases the number of relaxed precapillary sphincters. Hence more c a p i l l a r i e s • are perfused. The e f f e c t i v e c a p i l l a r y surface area for exchange is therefore increased [21]. However, the blood flow to the tissues may decrease which e f f e c t i v e l y decreases the c a p i l l a r y perfusion rate and the surface area for exchange. This w i l l be discussed further in Section 2.2.5. 36 In any burn, increased c a p i l l a r y leakage occurs in the injured skin. However, in the case of thermal injury involving over 25% of the t o t a l body surface, this increased leakage has been observed in nonburned tissues as well [26,27].. The information in Table 3.1 indicates that the protein exchange rate in the injured skin t y p i c a l l y increases to more than ten times i t s normal value after a burn. For rats, t h i s permeability increase l a s t s for approximately 12 hours. For the noninjured tissues, protein exchange rate remains normal for 10% burns, and increases for burns larger than 20%. However, the protein exchange rate increase in the noninjured tissues i s not as much as in the injured skin, only 1 to 5 times i t s normal value. The permeability changes in the intact tissues tend to last about 12 hours as well. Unforturnately, there i s not enough information available in the l i t e r a t u r e to allow a d i s t i n c t i o n to be made between the fraction of the c a p i l l a r y permeability surface area product change which i s due to changes in conduction across the endothelial membrane and the fraction which i s due to changes in c a p i l l a r y perfusion. 3.2 CHANGES WITHIN THE BLOOD CIRCULATION Associated with the profound changes in the c a p i l l a r y permeability after thermal i n j u r i e s , are acute effects on 37 the c i r c u l a t i o n . In par t i c u l a r blood c i r c u l a t i o n as reported invest igated: the following aspects of the in the l i t e r a t u r e have been 1. the cardiac output, 2. the main a r t e r i a l pressure, 3. the central venous pressure, 4. t o t a l peripheral resistance, and 5. the blood perfusion. The results are summarized in Table 3.2 and b r i e f l y discussed in the following sections. 3.2.1 THE CARDIAC OUTPUT There i s a marked f a l l in cardiac output soon after injury in experimental animals with extensive burns [28,29,30,31,32,33] (see Table 3.2). This large decrease in cardiac output i s due to reduced stroke volume and reduced heart rate [33]. For example, Figure 3.1 i l l u s t r a t e s the changes in the heart rate, the stroke volume, and the cardiac output following thermal injury in guinea pigs [31]. T A B L E 3.2 H E M O D Y N A M I C C H A N G E [28, 29, 30, 31, 32, 33] PRESSURE CHANGE PERFUSION PERCENT BURN AREA MAIN ARTERIAL PRESSURE CENTRE VENOUS PRESSURE BLOOD FLOW ANIMAL BURN DEGREE EXPT. TIME CARDIAC OUTPUT STROKE VOLUME TOTAL RESISTANCE INJURED TISSUE NON INJURED TISSUES Wolfe et a l . 1976 guinea pig Between 2nd and 3rd. Between 2nd and 3rd 551 70Z 0-24 hours 0-24 hours Decreases ~ 35Z maximum Decreases ~ 10Z maximum Decreases ~ 60Z maximum Decreases ~ 85Z maximum Decreases ~ 50Z maximum Decreases ~ 60Z maximum Increases ~2.0 times maximum Increases ~5.5 times maximum Ferguson et a l . 1977 guinea Pig 3rd 70Z 0-75 minutes Decreases ~ 60Z maxlmum Decreases ~ 50Z maximum Decreases ~ 95% maximum Muscle: decreases ~30-60Z maximum Intact Skin: decreases 80Z maximum Delralng* et a l . 1978 sheep 3rd 40Z 0-69 hours remains normal Decreases ~ 15Z maximum Ferguson et a l . 1980 guinea pig 3rd over 70Z 0-8 hours Decreases ~ 25Z maximum Decreases ~ 57Z maximum Decreases ~ 45Z maximum Increases ~ 2 times maximum Harms* et a l . 1981 sheep 3rd 25Z 0-72 hours Increases ~ 13Z maximum Decreases ~ 67Z maximum Lund et a l . 1986 rat 3rd 10Z 0-3 hours Decreases ~ 20Z maximum remains normal Decreases ~ 10Z maximum Decreases ~ 8Z maximum 40Z 0-3 hours Decreases ~ 35Z maximum remains normal Decreases ~ 40Z maximum Decreases ~ 25Z maximum * Injured animals are resuscitated ln these experiments 3 9 Figure 3.1 Cardiac Output, Stroke Volume and Heart Rate for Guinea Pigs After a 70% Burn [31] CARDIAC OUTPUT 2 5 0 I * 1 i 1— Prebum / 4 8 TIME (hours after burn) 40 3.2.2 THE MAIN ARTERIAL PRESSURE The main a r t e r i a l pressure (MAP) is reported to decrease following a burn [28,31,33] (see Table 3.2). The degree of reduction is dependent in part on the fraction of tot a l surface area burned [33]. For example, in the experiments of Lund and Reed [33], the f a l l in the MAP following, a 40% burn i s greater than that following a 10% burn. The MAP in the 10% burn group decreased from 110 mmHg to 97 mmHg after 15 minutes postburn, and remained at around 86-94 mmHg up to 3 hours afterwards. The maximum f a l l of the MAP in the 10% burn group was 20% of the control value. However, in the 40% burn group, MAP f e l l from 110 mmHg to 80 mmHg in the f i r s t 15 minutes after the burn was induced and remained at a 70-80 mmHg l e v e l a f t e r 30 minutes postburn up to three hours. The maximum f a l l in the 40% burn group is 30% of the control value [19]. 3.2.3 THE CENTRAL VENOUS PRESSURE Experimentally, the central venous pressure (CVP) did not appear to change s i g n i f i c a n t l y following a burn injury [24,30] (see Table 3.2). In some experiments, no change was observed [24,31]. In one experiment, the CVP decreased s l i g h t l y [32]. The experiment of Lund and Reed [33] on rats subjected to 10% and 40% burns showed that the CVP did not change s i g n i f i c a n t l y [33].. 41 3.2.4 TOTAL PERIPHERAL RESISTANCE While the cardiac output and mean a r t e r i a l pressure decreased following a burn, the t o t a l peripheral resistance increased (see Table 3.2). This finding has been supported by a number of investigators, including Wolf et a l . [28] and Ferguson et a l . [31]. Unlike other investigators, Arturson measured a decrease in the regional resistance of the venous c a p i l l a r y following a thermal injury [ 1 8 ] . 3.2.5 THE BLOOD FLOW As a consequence of the depressed cardiac output, the decrease of the mean a r t e r i a l pressure and the assumed increase of the t o t a l peripheral' resistance, a reduced c a p i l l a r y blood flow occurs both throughout the general systemic c i r c u l a t i o n and in the injured tissue [29]. In some cases, the high resistance may even halt the c a p i l l a r y flow [29]. These factors therefore tend to decrease the perfusion rate both in the injured skin and the non-injured tissues. For example, the experiments of Ferguson et a l . on guinea pigs [29] showed that blood flow in d i f f e r e n t organs and tissues was depressed tremendously. Following a third degree burn over 70% of the surface area, the blood flow decreased by 95% in the injured skin, 30% to 60% in the muscle, and 80% in the intact skin (see Table 3.2). Very l i t t l e additional quantitative information i s available 42 concerning the blood flow in tissues following a thermal injury. Personal communications with A. Haugan of The Department of Physiology, University of Bergen, Norway corroborate these findings. With respect to f l u i d exchange between the c i r c u l a t i o n and the tissue space, the tendency towards decreased perfusion opposes the effect of the histamine-1ike mediators discussed e a r l i e r (see Section 3.1). 3.3 CHANGES WITHIN THE INTERSTITIUM In 1964, Arturson noticed that after a second degree burn to cats, there was a very rapid increase of i n t e r s t i t i a l volume, but only a moderate increase of the c a p i l l a r y f i l t r a t i o n c o e f f i c i e n t [18], He suggested that some mechanism, other than the increase of c a p i l l a r y permeability, also contributed to edema formation. He suggested that the early edema may be due primarily to a temporary change in the t r a n s c a p i l l a r y osmotic pressure difference. However, t h i s has not been proved experimentally. Lund et a l . [34] investigated the importance of the tissue pressure (P b) in injured skin with respect to edema formation following a burn. P^ in rats was measured both in 10% and 40% body surface area burns up to three hours after 43 the burn. A rapid decrease in P^, uncharacteristic of the normal tissue compliance behaviour was noted. The injured tissue hydrostatic pressure following a burn as a function of time i s shown Figure 3.2 and Figure 3.3 for 10% and 4 0% burns, respectively. Though the experimental errors associated with measurement of tissue pressure in the injured skin (one standard error is shown) are large, c l e a r l y some dramatic changes are occurring within the inte r s t i t i u m . Following a 10% burn, the intradermal P^ decreased rapidly from i t s control value of -1.21 mmHg. At around 40 minutes postburn, i t reached a minimum value of -20 mmHg. In the 40% burn, P^ dropped more dramaticlly to a minimum of -24 to -30 mmHg at 30 to 40 minutes postburn [34]. In both cases, P^ rose to s l i g h t l y above i t s normal value within 180 minutes postburn [34]. Whether t h i s strongly negative pressure drop i s related to the t r a n s c a p l l l a r y osmotic pressure gradient advocated by Arturson i s not clear, but is i s believed to be a major driving force in i n t e r s t i t i a l edema formation at least in the early stage of the burn injury. 44 Figure 3.2 The Tissue Hydrostatic Pressure of Injured Skin for Rats Subjected to a 10% Burn [34] 5 to -40 I 1 1 1 1 1 0 0.5 1 1.5 2 2.5 3 T i m e ( h o u r ) 45 Figure 3.3 The Tissue Pressure Hydrostatic Pressure of Injured Skin for Rats Subjected to a 40% Burn [34] be J3 T i m e ( h o u r ) 46 3.4 SYMPTOMS OF BURN INJURIES The permeability, hemodynamic and tissue pressure changes described above cause a number of physiological responses in experimental animals following a burn injury. The available experimental data concerning these responses is summarized in Table 3.3.and discussed b r i e f l y in the sections which follow. 3.4.1 PLASMA VOLUME LOSS (HYPOVOLEMIA) Owing to the postburn physiological changes mentioned in Sections 3.1 to 3.3, f l u i d and proteins from the c i r c u l a t i o n leak rapidly to the burned tissue and, in more severe cases, even to the nonburned tissues [ l ] . This causes a large loss of plasma volume (hypovolemia) and a decreased protein content in the c i r c u l a t i o n . These results are confirmed by many investigators including Leap [37] and Reed et a l . [45] (see Table 3.3). For example, Leap measured the hematocrit change in monkeys following a 50% burn [37]. The estimated plasma volume from the hematocrit data showed a maximal plasma volume decrease of 30% to 40% in 4 hours. He also demonstrated a decrease in plasma albumin concentration of 10% to 20%. Table 3.3 C H A R A C T E R I S T I C E F F E C T S O F E X P E R I M E N T A L B U R N I N J U R I E S (18, 35, 36, 37, 38. 39, 40, 41, 42, 43, 44, 45 I ANIMAL • U R N D E C R E E PERCENT BURN AREA a r t . T I K E VOLUME OF P L A S K A COP O r PLASMA COP OP INJURED SKIN FLUID CONTENT OP INJURE!) SKIN FLUID CONTENT OF INTACT SKIN OR KUSCLE PROTEIN CONTENT 0 ' INJURED SKIN PROTEIN CONTENT Of INTACT SKIN OR HUSCLE LTHPH FLOW IN INJURED SKIN DURATION Of C D E K A Artur iot i .1 a l . 1964 cat 2nd on* pev 0 1 60 • In. Inereaira -170-1/mog t i ssue aaxlaua Arturson a t a l . 1967 dog 2nd 0 I It days *_ Increases | 40-50X J aaxlaua Increaaea 4-6 t ines aaxlaua Injured t l i i u e : longer thin 14 da?a Lcape 1968 rat 3rd sot 0 1 4 hour a A Increases T - 6JX 1 aaxlaua A Increases ~ 90X | asxlaiaa Leap* 1970 Rhesus aonhey 3rd JOT. 0 1 4 hour • Decreases 30-401 mmximum Plssaa cone, decreases 10-201 r auuLaHja A Increases T 70-BOX 1 aaxlaua • > renalna constant A Increases 1 - 5 c l i c i 1 a* a . tenia In tact ak in : } reaaln constant 1 a u s c l e : decree tea ]f 101-2 01 aaxlaua R.L. Creen 1977 •oust on* paw 0 1 40 • In-* Increase* |....... erowhard rt ml. 1978 rat 2nd 0.21 0 1 48 hour a A Increases 1 I u x l i u a » rcaalne constant Increases 24 hours Ca r v a j a l e 1 a l . 1979 rat deep 2nd IQt 701 301 40X 0 1 24 hour a Increases - 94X aaxlaua Intact s * l n . Increases 20X a u s c l s : reaslns conatsnt longer than 24 h o o t a Deallne,' et a l . 1901 sheep 3rd 251 0 1 72 houra T Increases 4.5 ttaaa | aaxloua Orel I nit F t a l . 1983 aheep 3rd 301 0 1 2 houra decreases - 20X U I I M I Dra] Inn e t a l . 1984 sheep 2nd 0 I 168 houra • Increases 100X aaxlaua I 10 t l a e s 1 aaxlaua 168 houra Artur K i n rt at . 1984 palt«nti 2nd 26X 1 it t S e e d r l A l . 1986 rat 3rd 101 40Z 0 1 1 hour* decreaeee - 201 [ ecresees - 20t as • Lama As Increaaea - 401 •a *1«ua asxlaua i, Increases 2 -5 -3 t l a e e Increaaea 501 aaxlaua d t t r * « t « t - 30X t ecreesea — 30X l*ua aaxlaua Increases —201 maxInus Increases t U nun Increases -2 ttaaa Increase* 501 aaxlaua Injured anlaale are rsausc l ts ted In thi 48 Lund and Reed [45] r e c e n t l y measured the volume and c o l l o i d osmotic pressure (COP) of plasma i n r a t s with t h i r d degree, 10% and 40% su r f a c e area burns. T h e i r r e s u l t s showed that the volume of plasma decreased by 20% to 30% w i t h i n three hours postburn. Over the same time p e r i o d , the plasma COP a l s o decreased by 20-30%. 3.4.2 EDEMA FORMATION AND TISSUE PLASMA PROTEIN CONTENT In a l l the experiments l i s t e d i n Table 3.3, the f l u i d and the p r o t e i n content of the i n j u r e d t i s s u e i n c r e a s e d a f t e r burns. For example, Leap measured the f l u i d and p r o t e i n content in the i n j u r e d s k i n of r a t s s u b j e c t e d to t h i r d degree, 50% burns. H i s r e s u l t s showed that the f l u i d content of the i n j u r e d s k i n i n c r e a s e d by 70% of i t s normal value and the p r o t e i n content by 90% [36]. Among the i n v e s t i g a t o r s l i s t e d i n Table 3.3, Reed et a l . [45] are the only ones who measured the COP of the i n j u r e d s k i n . T h e i r r e s u l t s r e v e a l e d that the burned s k i n COP i n c r e a s e d by 40% a f t e r a 10% burn, and by 20% a f t e r a 40% burn. The responses of f l u i d and p r o t e i n content i n the i n t a c t t i s s u e s from Table 3.3 are not c o n s i s t e n t . Some i n v e s t i g a t o r s found that the f l u i d content i n the i n t a c t t i s s u e s remained approximately normal a f t e r burns [37,39]. Others measured an i n c r e a s e d f l u i d content i n the i n t a c t t i s s u e s [40,45]. Thus the p r o t e i n content i n the i n t a c t 4 9 tissue could increase, decrease or remain constant according to Table 3.3. The duration of the edema also varies considerably from one investigation to the next. For example, Carvajal et a l . [24] used rats as animal models. They reported that the water content of burned tissues increases rapidly, is maximal at about 3 hours, and disappears by 24 hours postinjury. Others use di f f e r e n t animals in their experiments and claim that the edema i s present for several days to two weeks [35,43]. 3.4.3 LYMPH FLOW The lymph flow draining the various tissues has frequently been used to assess microvascular f l u i d f i l t r a t i o n rate and protein permeability c h a r a c t e r i s t i c s for the pa r t i c u l a r tissues involved. The composition of lymph is often assumed to r e f l e c t i n t e r s t i t i a l f l u i d volume and the flow rate to r e f l e c t the quantity of edema [46]. The increase of lymph flow in the injured skin postburn was reported by many investigators including Arturson et a l . [35] and Demling et a l . [41,43] (see Table 3.3). Both have estimated the lymph flow change in the injured skin. Arturson measured lymph flow in dogs after a second degree burn, and found an increase in lymph flow of 4 to 6 times 50 the control value within 6 hours postinjury [35]. Demling noticed a marked increase in lymph flow in the sheep which lasted more than four days after the burn [41,43]. However, i t is questionable whether the lymph flow measured by either group represented only that from injured skin. Chapter 4 BRIEF SURVEY OF COMPUTER SIMULATIONS OF MICROVASCULAR EXCHANGE Computer simulations have provided an additional method by which physiological systems may be studied. Models of microvascular exchange have been developed to describe the d i s t r i b u t i o n and transport of f l u i d and plasma proteins between the c i r c u l a t i o n , i n t e r s t i t i a l space and the lymphatic system. The most pertinent of these models w i l l be reviewed here. 4.1 MODELS OF MICROVASCULAR EXCHANGE IN NORMAL TISSUES Models of microvascular exchange f a l l into two categories, d i s t r i b u t e d and compartmental models. The compartmental models divide the microvascular exchange system into several well-mixed but separate compartments. The o v e r a l l mass balances, which account for f l u i d and protein transport between each compartment, are written as a set of ordinary d i f f e r e n t i a l equations. The dist r i b u t e d or position-dependent models eliminate the well-mixed assumption. Hence, a more complex description of the f l u i d and protein transport in the microvascular system i s required. The f l u i d and protein balances must now be written around a d i f f e r e n t i a l volume element in each part of the system yi e l d i n g a set of p a r t i a l d i f f e r e n t i a l equations 5 1 52 which are more d i f f i c u l t to solve. The distributed models are not s u f f i c i e n t l y well-developed at present and have yet to be extensively applied to existing experimental or c l i n i c a l data. Compartmental models, on the other hand, have been used reasonably successfully in describing the behaviour of the microvascular exchange system resultant to perturbed states. One of the most successful compartmental models of microvascular exchange for humans is the model of Wiederhielm [47]. In this two-compartment model, the dynamics of f l u i d and plasma protein transport between the c i r c u l a t i o n and the i n t e r s t i t i u m were studied. The i n t e r s t i t i a l compartment included muscle, skin and a l l the other tissues. To account for exclusion e f f e c t s , the i n t e r s t i t i u m was assumed to have two phases: a gel phase composed of mucopolysaccharides which i s accessible to f l u i d but not to proteins and a free f l u i d phase containing the plasma proteins. At any time, t h i s two-phase system i s in osmotic equilibrium. The model was used to analyse the behaviour of the microvascular exchange system following a variety of perturbations, such as changes in plasma c o l l o i d osmotic pressure, a r t e r i a l and venous pressures, i n t e r s t i t i a l mucopolysaccharide content and lymphatic obstruction [47]. This model has been shown to some extent to predict the r e d i s t r i b u t i o n of f l u i d and proteins which take place during pathological conditions. 53 Studies of f l u i d and protein transport have been furthered in the model by Bert and Pinder [48]. This model was developed from Wiederhielm's compartmental model, but an i n t e r s t i t i a l "excluded volume" was used in place of the gel phase of Wiederhielm's model. The model of Wiederhielm was modified by Bert and Pinder based on the assumption and experimental observations that the volume exclusion in a given tissue i s a function of i t s collagenous fiber content and remains constant even i f the tissue swells. Using the same input parameters as Wiederhielm, the "volume exclusion" model of Bert and Pinder predicted results for both normal and perturbed conditions which were in good agreement with those of the e a r l i e r model. The introduction of constant volume exclusion eliminated several uncertain assumptions required by the two-phase model of Wiederhielm and, as a consequence, resulted in a s i g n i f i c a n t s i m p l i f i c a t i o n to the overall model. Based on the model of Wiederhielm [47] and the model of Bert and Pinder [48], a dynamic model for i n t e r s t i t i a l f l u i d exchange in rats, the Plasma Leak model, has been developed by Bert et a l . [6], This model divided the interstitium into two separate compartments: muscle and skin. The reason for subdividing the i n t e r s t i t i u m was that the c o l l o i d osmotic pressure relationship, the compliance curve and the normal steady-state conditions for muscle and skin are very d i f f e r e n t . The Plasma Leak model was able to provide a much 54 more elaborate description of the f l u i d and protein exchanges taking place between the c i r c u l a t i o n , the i n t e r s t i t i a l space of skin and muscle and the lymphatics. The model was used to predict the behaviour of a l l microvascular exchange parameters in rats for several perturbations, such as for changes in venous pressure and plasma c o l l o i d osmotic pressure as well as for over- and dehydration. The predictions of the model compare well with the large amount of experimental data available in the l i t e r a t u r e for this experimental animal. 4.2 MODELS OF MICROVASCULAR EXCHANGE FOLLOWING A THERMAL  INJURY Recently, compartmental models have been applied to study the f l u i d and protein r e d i s t r i b u t i o n that occurs following a thermal injury. For example, Arturson [6] has recently modified the model of Wiederhielm to make i t applicable to thermal i n j u r i e s in humans. In his burn model, Arturson divided the microvascular exchange system into three compartments: the plasma, the nonburned tissues (including both muscle and intact skin) and the burned tissue (assumed to be skin). It was also assumed that each compartment was well-mixed, that there was no exchange of protein or f l u i d between the burned and nonburned tissues and that the f l u i d and protein were lost from the system only by exudation and evaporation in the injured skin [5]. 55 Arturson's mathematical model p a r a l l e l s that of Wielderhielm, except for the additional equations required to account for the existence of the burned tissue and the additional input/output. The t o t a l protein loss from the wound, was taken to be the product of f l u i d loss by exudation and the protein concentration in the injured skin [5]. The model was used to describe the rate of l o c a l and general edema formation. Its pot e n t i a l use would be in the evaluation of the f l u i d r e s u s c i t a t i o n requirements for patients who suffer from thermal i n j u r i e s . This s i m p l i s t i c model predicted the d i s t r i b u t i o n of overall edema and the rate of the edema formation. Arturson's model which considers only overa l l "tissue" f a i l s to dis t i n g u i s h between muscle and skin and their d i f f e r e n t effects on microvascular exchange. The model needs, to be improved in order to estimate f l u i d r e s u s c i t a t i o n requirements more r e a l i s t i c a l l y . Another model which describes the f l u i d s h i f t s which take place throughout the whole body during a burn is the model of Wachtel et a l . [49], This model does not consider microvascular exchange. Here, the plasma water input, urinary output, burn water l o s s , and insensible water losses via the unburned skin, lung, and g a s t r o i n t e s t i n a l tract were simulated. This model can be applied to acute burn patients. Given the patient's body size and percentage of surface area burned, the model can be used to anticipate the 56 loss of water from the wound, and calculate appropriate levels and time of resuscitation treatment. However, the model does not include the i n t e r s t i t i a l space which is an important location for f l u i d and protein exchange. Hence the structure of the model did not contain several necessary components. This gave rise to many c l i n i c a l l y u n r e a l i s t i c features. Thus, in order to find application in a c l i n i c a l setting, present day burn models need to simulate more accurately not only the systems involved but also the changes which occur as a result of thermal injury. Chapter 5 THE BURN MODEL 5.1 FORMULATION OF THE BURN MODEL The burn model described here is an extension of the compartment model developed by Bert et a l . [6] for microvascular exchange. The major difference between the present model and the previous model i s that the former has an extra compartment. As was discussed in Chapter 3, the pathological responses in burned and nonburned skin immediately after a thermal injury can be very d i f f e r e n t . It i s therefore necessary to divide the skin compartment into two parts: the burned and nonburned skin. As a consequence, the new model consists of four compartments: blood, muscle, injured skin and noninjured skin. The primary objective of the simulation i s to determine how f l u i d and plasma proteins are redistributed amongst the four compartments following a burn injury. For si m p l i c i t y , i t i s assumed that the behaviour of a l l plasma proteins can be referenced to a single protein species, albumin. A schematic drawing of the four compartment model showing the various exchange paths for f l u i d ( s o l i d lines) and albumin (dashed lines) i s given in Figure 5.1. As in the model of Bert et a l . , complete mixing i s assumed to exist in each compartment except for the plasma compartment where a 57 gure 5.1 Schematic Diagram of Burn Model « — Ci r c u l a t ion 5 9 hydrostatic pressure difference is maintained between the a r t e r i a l and venous ends of each c a p i l l a r y . Furthermore, in some numerical f i t t i n g experiments, the a r t e r i a l and venous c a p i l l a r y pressures are allowed to have dif f e r e n t values for injured skin than for noninjured skin and muscle. It i s also assumed that there is no direct interchange of f l u i d or proteins between the tissue compartments; dire c t exchanges occur only between the c i r c u l a t i o n and each tissue separately. At the c a p i l l a r y l e v e l , mass exchange is hypothesized to take place through a variety of mechanisms. F l u i d i s transfered by f i l t r a t i o n (Vj) from plasma to tissue at the a r t e r i a l end of the c a p i l l a r y , by plasma leak (V ]_) in the same di r e c t i o n but at the venous end of the c a p i l l a r y and by reabsorption (v"r) which normally takes place from tissue to plasma also at the venous end. F i l t r a t i o n and reabsorption occur through small completely-sieving pores which are assumed to exist at both ends of the c a p i l l a r y , while plasma leak takes place through large non-sieving channels in the venous c a p i l l a r y only. F l u i d in the tissue i s drained by a unidir e c t i o n a l lymph flow (V-^ ) and returned to the c i r c u l a t i o n . Albumin is exchanged between the c a p i l l a r y and the tissue space either by convection with the plasma leak ( O p i)or by d i f f u s i o n across the c a p i l l a r y membrane Plasma Proteins can only leave the tissue with the lymph (0 X ) . 60 It should also be noted that in our model, attention is focussed only on the transport of f l u i d and plasma proteins (albumin) in the microvascular exchange system. The transport of small ions between compartments and between the tissues and their c e l l s is ignored. It is assumed therefore that the small ions in the blood, i n t e r s t i t i u m and lymphatics are always in equilibrium with one another and that there is no change to the c e l l s before, during and after burns. 5.2 MATHEMATICAL RELATIONSHIPS 5.2.1 FLUID AND PROTEIN MASS BALANCES The equations used in the burn model to describe the various f l u i d exchanges between the plasma and each tissue compartment are based on a S t a r l i n g ' s type equation [46]. In these equations, the volumetric rate (V) of f l u i d movement across the vascular endothelium i s assumed to be proportional to the hydrostatic pressure (P) and the protein c o l l o i d osmotic pressure (II) differences on either side of the membrane: v = K [ ( p p - P i ) - a ( n p - n i ) ] 5.1 61 In Equation 5.1, K is a transport c o e f f i c i e n t which depends on the f l u i d exchange area and the size of the f l u i d exchange channels within the endothelial barrier [46]. The second parameter o is the protein r e f l e c t i o n c o e f f i c i e n t which usually has a value between zero and unity. If the vascular membrane is impermeable to protein, a is 1. If the protein can pass through the membrane fre e l y , a is 0. The subscripts "p" and " i " denote the plasma and i n t e r s t i t i a l compartments, respectively. F l u i d moves into the inte r s t i t i u m when the net dr i v i n g pressure difference is positive (e.g. normal f i l t r a t i o n ) ; but moves out of the inte r s t i t i u m when the net pressure difference is negative (e.g. normal reabsorption). The equations which describe the r e d i s t r i b u t i o n of f l u i d and proteins in a compartment following a burn are similar to those derived previously by Bert et a l . for their "Plasma Leak Model" (submitted to the American Journal of Physiology). They are obtained by carrying out a mass balance of f l u i d or albumin around each compartment. For example, consider the f l u i d balance around a general tissue compartment. At any instant in time, the rate of accumulation of f l u i d within the compartment must be equal to the net rate of transfer of f l u i d to that compartment, or: 62 dVj/dt = v f + v p l - v r - v 1 5.2 where is the volume of f l u i d in the tissue compartment and t is the time. Since f l u i d f i l t r a t i o n and reabsorption occur through pores which are impermeable to protein, the r e f l e c t i o n c o e f f i c i e n t in both cases i s equal to unity. Thus when Equation 5.1 is applied, tff = K f[p a-p i - ( n p - n i ) ] 5.3 and V r = KpfPi-P^mi-IIp)-] 5.4 where and K r are referred to as the f i l t r a t i o n and reabsorption c o e f f i c i e n t s , respectively, and the subscripts "a" and "v" d i f f e r e n t i a t e the a r t e r i a l and venous ends, respectively, of the plasma c a p i l l a r y . Note that f i l t r a t i o n i s considered to be positive when i t results in a net transfer of f l u i d into the tis s u e . The reverse i s true for reabsorption. 63 In the case of the plasma leak, the large pores in the venous c a p i l l a r y allow the free passage of proteins and the r e f l e c t i o n c o e f f i c i e n t is zero. Thus, the plasma leak, v p i ' is proportional to the hydrostatic pressure difference only, and Equation 5.1 becomes V = Kpl ( Pv" Pi ) 5'5 where Kp-^  is the plasma leak c o e f f i c i e n t . For overhydration of the tis s u e , the lymph flow, V-^ , is assumed to be l i n e a r l y - r e l a t e d to the change in i n t e r s t i t i a l f l u i d pressure, i . e . v l = V l o + S L ( P r P i o ) , P i > P i o 5.6a where V^ Q and P^Q are the lymph flow and tissue pressure, respectively, at normal steady-state conditions. The slope of the lymph flow relationship, SL, is termed the "lymph flow s e n s i t i v i t y " ; For underhydration, a li n e a r lymph flow re l a t i o n s h i p is also assumed, but in this case the lymph flow i s taken to be zero at the tissue hydrostatic pressure, P^e, corresponding to i t s excluded volume (see Section 2.3), 64 ^ l o ( p i " p i e ) / ( p i o - p i e ) ' 10 5.6b Furthermore, if the lymph flow calculated from Equation 5.6b is less than zero, the lymph flow is set equal to zero. It is therefore assumed the lymph flow can only occur in one direction; from the tissues to the lymphatics. Similarly, a mass balance of albumin around the same general tissue compartment yields: i n t e r s t i t i a l compartment at any given time. The mass flow rate of protein transported into the tissue along with the plasma leak, Opi, is given as the product of the plasma leak rate, V ,, and the plasma protein concentration, C , or: dQi/dt Oi 5 . 7 where Qi represents the total mass of protein in the V - V C P 5 . 8 65 The plasma protein (albumin) concentration in blood plasma is defined as: C p = Q p / V p 5.9 where Q p and V p are the mass of protein and the . t o t a l volume, respectively, of the plasma compartment. Similarly, the protein flow in lymph, , i s the product of the lymph flow rate, v"-^ , and the tissue concentration, C£, i . e . 0 X = V-L-C^ 5.10 where Cj = Qi/Vj 5.11 The mass rate of protein d i f f u s i o n across the c a p i l l a r y wall, Q ^ , i s assumed to be proportional to a concentration difference between the plasma and the tissue. In this case, i t i s more appropriate to use a tissue concentration based on the available fraction of the tissue volume. Thus, 0 d = PS(C p-CA i) 5.12 66 where P S is another transport c o e f f i c i e n t which represents the product of the protein permeability and the surface area of the c a p i l l a r y wall. CA^ i s the e f f e c t i v e tissue protein concentration based on the available volume, which is given by: C A i = Q i/(V i-VE i) 5.13 In the l a s t equation, VE^ i s the excluded volume of the tissue, i . e . , the f l u i d volume which cannot be penetrated by the protein (see Section 2.3). Equations having the form of 5.2 - 5.13 have been derived for a l l three tissue compartments and are compiled in Table 5.1. D i f f e r e n t i a l equations describing the f l u i d and protein behaviour in the plasma could be obtained by carrying out similar mass balances around the plasma compartment. However, since we assume there are no f l u i d or protein losses, no c e l l u l a r changes and no f l u i d r esuscitation in the present model, the amounts of f l u i d and protein in the system are constants and i t is easy to show that: dV p/dt = -(dV m/dt+dV s/dt+dV b/dt) 5.14 Table 5.1 Summary of Burn Model Equations Muscle C =Q /V m m m C Am= Qm/ ( vm - V E J m m m m nm= F " n ( C A J m m P =FM(V ) m m fm fm am m p m v =K [ P - P - ( n - n )) rm rm m vm m p V =K , (P - P ) pirn pirn vm m V, =V, +SL (P - P ) lm lmo m m mo Q =V C plm plm p Q . =PS (C -CA ) dm m p m Q, =V, C lm lm m dV /dt=V, + V , -V -V, m fm plm rm lm dQ /dt=Q , +Q . -Q, m plm dm lm Intact Skin C =Q /V S S S CA s=Q s/(V s-VE s). n s =F n (CA s ) P s =F n (V s ) fs fs as s p s v =K [ P - P - ( n - n )] rs rs s vs s p V =K , (P -P ) p i s p i s vs s V, =V, +SL (P -P ) is 1 so s s so Q , =V , C p i s p is p Q . =PS (C -CA ) ds s p s Is Is s dV /dt=V r t-V , -V -V , s fs p i s rs Is dQ /d t = Q , +Q . -Q, s p i s ds Is Injured Skin c V V ( V V E b > n b =F n (CA b ) v p n ( v fb fb ab b p b rb rb b vb b p VP l b = K P l b ( P v b - P b ) V1 b = V l b o + S L b ( P b - P b o ) Q =V C plb p lb p Q d b = P S b ( C p - C A b ) Q l b = V l b C b d v b / d t = v f b + v p l b - v r b - v l b d Q b / d t = Q p l b + Q d b - Q l b PIasma C =Q /V P P P n = F n ( c ) p p P =F°(V ).P =Fn(V ) a P v p dV /dt=-dtv +V +V >/dt p m s b dQ /dt=-d(Q +Q + Q, ) /d t p m s b Ch 68 and dQ p/dt = -(dQ m/dt+dQ s/dt+dQ b/dt) 5.15 where the subscripts p,m,s and b denote the plasma, muscle, noninjured skin and injured skin compartments, respectively. The necessary equations for the plasma compartment are also shown in Table 5.1. The table also l i s t s other relationships needed to complete the description of the burn model including the compliance and the c o l l o i d osmotic pressure relationships which w i l l be discussed in the next section. 5.2.2 CONSTITUTIVE RELATIONSHIPS 5.2.2.1 Compliance Relationships The compliance relationships between the hydrostatic pressure and the f l u i d volume for the muscle, intact skin and plasma compartments are taken to be the same as those used in the e a r l i e r model of Bert et a l . [6]. For skeletal muscle and normal skin, the compliance curves are based on the experimental data of Reed and Wiig [50] and Wiig and Reed [51], respectively, and are tabulated and plotted in Appendix A. To allow interpolation in the computer program, the tabulated curves were divided into three segments and f i t t e d with linear relationships at the extremes of high and low 69 f l u i d volume and with a set of cubic spline equations [52] over the central portion. Note that in order to use this compliance rel a t i o n s h i p c o r r e c t l y for intact skin when the fraction of t o t a l skin area which is burned (BSA) exceeds zero, the intact skin f l u i d volume must be divided by (1-BSA). The compliance relationships for the plasma compartment were obtained by Bert et a l . [6] by using their microvascular exchange model to f i t the dehydration and overhydration experimental results of Reed and Wiig [53]. Note that separate relationships are required for the a r t e r i a l and venous ends of the c a p i l l a r y . For convenience, Bert et a l . [6] assumed simple linear compliance relationships of the form where the "o" subscripts indicate normal steady-state values and A, and A 2 are constants whose "best f i t " values were found to be 0.00 and 5.12 mmHg/ml, respectively. In some of the burn simulations described P = P + A,(V -V ) a ao 1 v p po' 5.16 and P = P + A,(V -V ) v vo 2 v p po' 5.17 in the next Chapter, P ao and P. vo are assumed to be 70 dif f e r e n t for c a p i l l a r i e s replenishing the injured tissue compared to those interacting with intact skin or muscle. However, in a l l cases, A, and A 2 were taken to be constant at the values given above. There is no measured compliance curve available for burned tissue. However, i t is clear from the large negative pressure measured by Lund and Reed [34] (See Section 3.3, Figure 3.2 and Figure 3.3) that during the period immediately following a burn, the injured skin compliance i s very d i f f e r e n t from that of intact skin. Thus, for the f i r s t three hours postburn, the hydrostatic pressure of the injured skin compartment was unlinked from the f l u i d volume and was assumed to follow the data plotted in Figure 3.2 for 10% burn and Figure 3.3 for 40% burn. For computational purposes, the pressures at intermediate times were obtained by l i n e a r l y interpolating the discrete values. Since the measured- tissue pressures returned approximately to the intact skin compliance values after three hours, the normal skin compliance r e l a t i o n s h i p was followed for a l l times subsequent to three hours. Once again, in order to use the l a t t e r r e l a t i o n s h i p successfully, the f l u i d volume in the injured skin compartment must be normalized by dividing i t s value by BSA. 7 1 5.2.2.2 C o l l o i d Osmotic Pressure Relationships The c o l l o i d osmotic pressure associated with the presence of proteins in the f l u i d is related to the protein concentration. Bert et a l . [6] obtained expressions for the c o l l o i d osmotic pressure in the plasma, muscle and intact skin compartments by f i t t i n g experimental data for rats available in the l i t e r a t u r e . In a l l three cases, the data were f i t to cubic relationships having the general form: n = B,C + B 2C 2 + B 3C 3 5.18 where C is CA for muscle and skin and B 1 f B 2 and B 3 are constants whose b e s t - f i t values are tabulated in Table 5.2. Note that since C and CA are intensive properties of the compartments, the equations can be used without correction when BSA exceeds zero. For lack of better information, the c o l l o i d osmotic pressure relationship for injured skin was assumed to be i d e n t i c a l to that of intact skin. 72 Table 5 .2 Coefficients in Osmotic Pressure Relationships Ti ssue B B- B-Plasma Intact Skin Muscle 0.39723 0.42894 0.40891 0.00516 •0.00929 •0.00024 0.00027 0.00038 0.00040 When these c o e f f i c i e n t s are used in Equation 5.18, the units of c o l l o i d osmotic pressure must be mmHg and the units of concentration, mg/ml. 73 5.3 CONSTANTS 5.3.1 NORMAL STEADY-STATE CONDITIONS The normal steady-state conditions are those that are assumed to exist just prior to a burn injury. The normal conditions descriptive of the plasma, muscle and intact skin compartments of a 225g "standard" rat are the same as those used by Bert et a l . [6] and are l i s t e d in Table 5.3. The excluded volumes of both tissue compartments were estimated by assuming that albumin i s r e s t r i c t e d from entering one-quarter of the normal f l u i d volume of skin or muscle. When these excluded volumes are substituted into appropriate compliance relationship (see Appendix A), i t i s found that the excluded volume tissue pressures needed for Equation 5.6b are P m e= -6.78 mmHg and P g e = -7.84 mmHg. The i n i t i a l normal a r t e r i a l and venous c a p i l l a r y pressures were calculated using the relationships: Pao " p v v + R a ( p a a - p v v > 5 ' 1 9 and Pvo = Pvv^v'Paa-Pvv' 5 - 2 0 respectively, where P and P are the pressures in the Table 5.3 Normal Condi t ions: 225 g rat [6] V (ml ) VE (ml) CA or C (mg/ml) Q (mg) P (mmHg) P^ (mmHg) P (mmHg) n (mmHg) Weight (g) Ci r c u l a t i o n 6.12 0.00 35.9 219.71 24.54 5.92 20.0 12.38 Muscle Skin 10.41 2.60 17.2 134.30 -0.51 9.0 102.2 16. 88 4.22 22.0 278.50 -1.21 9.0 40.5 75 large a r t e r i e s and large veins, respectively, while Ra and Rv are constants representing the fra c t i o n a l pressure drop from the a r t e r i a l c a p i l l a r y and the venous c a p i l l a r y , respectively, to the large veins. In rats under normal conditions the assumption is made that P a a=1u0 mmHg and P v v=2 mmHg [6]. The f r a c t i o n a l resistances are taken to be the same as those for humans [6] and hence, R =0.23 and Rv=0.04. It i s assumed that the conditions s p e c i f i e d in Table 5.3 are the i n i t i a l conditions which exist at the start of each burn simulation with the exception of the following modifications. F i r s t l y , the i n i t i a l intensive properties C, CA, P and n of the injured skin compartment are presumed to be i d e n t i c a l to the corresponding normal properties of the intact skin. Secondly, the i n i t i a l extensive properties for the burned skin compartment are obtained by multiplying the values of V"s, VE g and Q s l i s t e d in Table 5.3 by BSA. Simila r l y , the i n i t i a l intact skin values are obtained by multiplying V s, VE S and Q s by (1-BSA). F i n a l l y , because the main a r t e r i a l pressure f a l l s after a burn (see Section 3.2.2) and because the d i s t r i b u t i o n of flow resistance may also change (see Section 3.2.4), the i n i t i a l values for the a r t e r i a l and venous c a p i l l a r y pressures also depend on BSA and in fact, may d i f f e r from one tissue to another. To accommodate these changes, the values of P a a and in some cases, R a appearing in Equations 5.18 and 5.19 were altered. 76 These modifications to and P,T^  are discussed in more ao vo de t a i l in Chapter 6. 5.3.2 NORMAL TRANSPORT COEFFICIENTS An examination of the burn model equations in Table 5.1 reveals that there are six transport c o e f f i c i e n t s for each tissue compartment which must be specified before the simulation can get underway. For any general tissue compartment, these six c o e f f i c i e n t s are the f i l t r a t i o n c o e f f i c i e n t Kf, the reabsorption c o e f f i c i e n t K r, the plasma leak c o e f f i c i e n t Kp-^ , the protein d i f f u s i o n c o e f f i c i e n t PS, the i n i t i a l lymph flow V^ 0 and the lymph flow s e n s i t i v i t y SL. Following a burn injury, many or perhaps a l l of these c o e f f i c i e n t s change. In some cases, these changes occur in a time-dependent way. The major emphasis in the next chapter is to try to determine what al t e r a t i o n s to the normal transport c o e f f i c i e n t s are reasonable in order to simulate the burn s i t u a t i o n . As a basis for making these al t e r a t i o n s , a set of normal transport c o e f f i c i e n t s must be defined for each of the muscle, intact skin and injured skin compartments. These normal values were derived from the transport c o e f f i c i e n t s established by Bert et a l . [6] for noninjured 77 rats. These values are l i s t e d in Table 5.4 for the two tissue compartments, intact skin and muscle, considered by those authors. The tabulated values were estimated independently for the two tissues by s t a t i s t i c a l l y f i t t i n g the simulation predictions to experimentally measured i n t e r s t i t i a l f l u i d volume and c o l l o i d osmotic pressure data as a function of venous pressure for muscle [54] and c o l l o i d osmotic pressure versus venous pressure for skin [55] for a series of perturbed states and also for normal conditions. Note that for both tissues, the f i t t i n g results predict that d i f f u s i o n plays a negligible role in protein transport. The normal transport c o e f f i c i e n t s used for muscle in the burn model are those given in Table 5.4. Because a l l of the c o e f f i c i e n t s are proportional to the f r a c t i o n a l surface area burned, BSA, the normal values for injured and intact skin were determined by multiplying the skin values shown in Table 5.4 by BSA and (1-BSA), respectively. The various alterations imposed on this set of normal c o e f f i c i e n t s are discussed further in Chapter 6. 78 Table 5.4 Transport C o e f f i c i e n t s Determined f o r Normal Skin and Muscle [6] Coef f i c i e n t s Muscle Skin Kf(ml/mmHg«h) Kr(ml/mmHg-h) KpT_(ml/mmHg«h) PS(ml/h) V l o(ml/h) SL(ml/mmHg»h) 0.0634 0. 1585 0.0145 0.0 0.2603 1.2065 0.0927 0.2943 0.0272 0.0 0.4220 1.2651 79 5.4 COMPUTER PROGRAM 5.4.1 NUMERICAL METHODS As can be seen in Table 5.1, the burn simulation consists e s s e n t i a l l y of a set of eight f i r s t - o r d e r ordinary d i f f e r e n t i a l equations obtained by carrying out f l u i d and protein (albumin) balances around the four separate compartments which make up the model. This set of eight d i f f e r e n t i a l equations (along with their corresponding a u x i l i a r y and constitutive relationships) must be solved simultaneously subject to a s e t of imposed i n i t i a l conditions. Because of the nonlinear nature of some of the a u x i l i a r y and constitutive relationships, the d i f f e r e n t i a l equations cannot be solved a n a l y t i c a l l y . The numerical method chosen to carry out t h i s solution is the Runge-Kutta-Fehlberg method with error control [ 5 6 ] . Like a l l Runge-Kutta solvers, Fehlberg's technique uses only f i r s t - o r d e r derivative evalutions ( i . e . evalutions of the right-hand-side functions of the d i f f e r e n t i a l equations) in order to obtain the simultaneous solutions of a l l dependent variables from one time step to the next. However, i t produces results which are equivalent in accuracy to a fourth-order Taylor's series and hence, permits the use of f a i r l y l i b e r a l time increments. Furthermore, Fehlberg's method includes estimates of each 80 dependent variable which are f i f t h - o r d e r accurate. Thus, by taking the difference between the fourth- and f i f t h - o r d e r estimates, i t becomes possible to predict the error corresponding to the chosen time interval or conversly, to adjust that interval to achieve a specified l o c a l and global error. In the present simulations, the maximum possible absolute error allowed for any of the eight dependent variables (V p, V m, V s, Vfa, Qp,Qm, Q s or Qfa) was 10"3 ml of f l u i d or 10~2 mg of protein. As mentioned e a r l i e r , the discrete compliance data available for skin and muscle was interpolated in the program (at least over the middle range of f l u i d volume) by using the method of cubic splines [56]. The cubic splines are a set of cubic polynomials, one such equation for each pair of pressure-volume points, whose c o e f f i c i e n t s are determined by forcing each polynomial to pass through i t s associated data points and making the f i r s t and second derivatives of neighbouring splines match at each point. The s p l i n e - f i t t i n g procedure requires, at one stage, the solution of a tridiagonal set of linear algebraic equations. The l a t t e r solution i s accomplished by using the tridiagonal matrix algorithm developed by Thomas [56]. 5.4.2 PROGRAM 81 A complete l i s t i n g of the FORTRAN program used to carry out the burn model simulations i s given in Appendix B. The program consists of a main or driver subprogram which di r e c t s the overall computation plus a myriad of function and subroutine subprograms which handle the more subordinate tasks. The main subprogram i n i t i a l i z e s the dependent variables, increments time and c a l l s the Runge-Kutta-Fehlberg subroutine to obtain a solution to the d i f f e r e n t i a l equations at each time step. It also outputs a l l of the dependent and supplemental variables in tabular form for each compartment and c a l l s p l o t t i n g subroutines which compare predicted responses to experimental data for selected variables. The main subprogram also calculates and outputs the sum-of-squares of differences between the simulation and experimental ' r e s u l t s for these selected variables. Separate subroutines and function subprograms control the different numerical procedures required, evaluate right-hand-side functions, a u x i l i a r y relationships and constitutive relationships, and perform a l l of the pl o t t i n g duties. A BLOCK DATA subprogram is used to specify the starting data as well as the discrete data needed by some of the constitutive r e l a t i o n s h i p s . The purpose and main features of each subprogram are documented within the program l i s t i n g contained in Appendix B. In general, the program i s used to predict how the proteins and f l u i d redistribute themselves with time because 82 of the transient alterations of transport c o e f f i c i e n t s and/or f l u i d pressures that occur after a burn injury having an area fraction BSA. Because of the a v a i l a b i l i t y of experimental results, only the cases of BSA = 10% and BSA 40% were studied extensively. Chapter 6 RESULTS AND DISCUSSION 6.1 INTRODUCTION It i s well-known that the physiological response of any l i v i n g organism to thermal i n j u r i e s is very complex. Furthermore, only a limited amount of experimental information is available on the pathological changes to the microvascular exchange system which occur after burns in rats or in other experimental animals. As a consequence, the approach followed in t h i s work is to hypothesize a l i s t of reasonable pathological changes which could be associated with burn i n j u r i e s in rats and input these as perturbations into the burn model for t h i s animal. Predictions based on the model are then compared with the available experimental information. In t h i s chapter, the perturbations and their resultant short- and long-term responses for 10% and 40% burns are simulated and discussed. The primary objective of t h i s investigation i s to i d e n t i f y those changes which not only give a reasonable f i t when compared with a l l of the available experimental data but also have a reasonable physiological basis. These "best" parametric changes for the two types of burns are considered to be the basis of a "satisfactory" burn model whose predictions w i l l then be examined in d e t a i l to understand better the underlying mechanisms of the physiological changes that take place 83 84 following a burn. The experimental information used as the primary comparison with the model predictions in t h i s work is the rat data of Lund and Reed [45]. The reasons for selecting their results are the following: 1. Their experimental data for the rat are far more complete and wide-ranging than those available from any other experiments. These data include measured changes in f l u i d content, protein content and c o l l o i d osmotic pressure in various compartments after 10% and 40% surface area burns. 2. Their experimental information contains necessary a u x i l i a r y information such as burn-induced changes in main a r t e r i a l and injured tissue hydrostatic pressures. 3. The normal steady-state conditions for the rat are well-characterized by the e a r l i e r microvascular exchange model of Bert et a l . [6], From the data of Lund and Reed, six variables were chosen as a basis of comparison between the model predictions and the experimental r e s u l t s . These variables were selected not only because they are important physiological variables in the tissue and plasma compartments but also because the experimental information concerning their behaviour i s reasonably complete and accurate. The six variables are the volume of plasma, the 85 i n t e r s t i t i a l f l u i d content of injured skin, the i n t e r s t i t i a l f l u i d content of intact skin, the c o l l o i d osmotic pressure in plasma, the c o l l o i d osmotic pressure in the injured skin and the albumin content of the injured skin. It should be pointed out that these data, considered to be the most complete, also have l i m i t a t i o n s . F i r s t l y , the time interval between each pair of experimental points i s one hour which i s quite large, and the t o t a l measurement time is r e s t r i c t e d to 3 hours postburn. Secondly, in the case of some experimental points, the estimated errors are f a i r l y s i g n i f i c a n t . The average normal values for some model variables used by Bert et a l . were d i f f e r e n t than those measured by Lund and Reed prior to the i n i t i a t i o n of burns. For example, in the case of the 10% burn, the normal value of COP in the plasma i s 20.0 mmHg in our simulation, while the average normal value of the same parameter in their burn experiments is 16.5 mmHg (see Tables C.1 and C.2 in Appendix C). Thus, some of the experimental data used to compare with the model required normalization which was carri e d out by multiplying the o r i g i n a l measured result by the r a t i o of the normal value of the model to that of the experiment. Sample calculations i l l u s t r a t i n g this normalization procedure are given in Appendix C. 86 It should be noted that in the experiments of Lund and Reed, the to t a l water in the tissue was measured, including both the water inside the c e l l s as well as the e x t r a c e l l u l a r water. However, only e x t r a c e l l u l a r water was used to represent tissue f l u i d volumes in our model. Thus, a more accurate normalization procedure was required to estimate the true tissue volume for intact and injured skin. To calculate the e x t r a c e l l u l a r volume, i t is necessary to know what fraction of the t o t a l tissue water i s i n t r a c e l l u l a r and, in the case of injured skin, what fraction of the c e l l s are destroyed during a burn. Since neither fraction i s well-known, sample calculations were carried out for the extreme assumptions that 20% of skin f l u i d i s i n t r a c e l l u l a r and, for burned skin, 0% or 100% of the c e l l s are destroyed following a thermal injury. The results of these calculations are also shown in Appendix C. In some cases, the tissue volumes calculated in these alternate ways are substantially larger than those obtained from the simple normalization procedure described above. However, because these estimates require a d d i t i o n a l assumptions and because the calculated differences for the extreme cases assumed are s t i l l small compared to the experimental error, i t was concluded that f l u i d volumes obtained by ignoring the correction for i n t r a c e l l u l a r water were acceptable as a basis'of comparison with the model simulations. 87 In addition to the six selected variables, the model can predict the transient behaviour of a l l the variables in each compartment including the f l u i d and plasma protein transfer rates, the f l u i d and c o l l o i d osmotic pressures, and so on. The implications of these other variables in helping form a mechanistic understanding of the various physiological events that take place following a burn w i l l be discussed in later sections. F i r s t i t i s necessary to determine the most reasonable set of parametric changes required to explain the 10% and 40% surface area burns. 6.2 THE 10% BURN 6.2.1 INTRODUCTION Based on the information presented in Chapter 3, the ch a r a c t e r i s t i c perturbations that seem l i k e l y to influence the microvascular exchange system aft e r a 10% burn injury are the following: 1. an increase in the plasma leak c o e f f i c i e n t in the injured skin ( K p l b ) , 2. an a l t e r a t i o n in the hydrostatic pressure in the injured skin ( P b ) , 3. changes in the f i l t r a t i o n and reabsorption c o e f f i c i e n t s in the injured skin ( K ^ and K ^ ) , 4. a l o c a l l y induced change which results in a decrease in the a r t e r i a l c a p i l l a r y pressure in the injured skin 88 5. an al t e r a t i o n of the lymph flow in the injured skin These are l i s t e d in what i s assumed to be the approximate order of importance with respect to the available information in the l i t e r a t u r e . The simulations employing the above l i s t e d perturbations were examined in phases. Phase One of the simulations started with the f i r s t and the second perturbations together, then in each subsequent phase, the remaining perturbations were added and examined one at time. Following a 10% burn, the main a r t e r i a l pressure has been measured [33] to decrease to 80% of i t s normal value. Substituting this new value of P Q a into equation 5.19 and 5.20 yi e l d s an a r t e r i a l c a p i l l a r y pressure of P a = 19.94 mmHg and a venous c a p i l l a r y pressure of P y = 5.12 mmHg for the entire plasma compartment. It i s these altered values of P Q and P v which were used (at least during the f i r s t three hours) for a l l simulations in the case of the 10% burn. 6.2.2 TRANSIENT RESPONSE TO CHANGES IN THE PLASMA LEAK COEFFICIENT AND THE HYDROSTATIC PRESSURE CHARACTERISTCS OF THE INJURED SKIN (PHASE ONE) 89 The plasma leak c o e f f i c i e n t in the injured skin, K p i b ' is the product of the c a p i l l a r y permeability and the available exchange surface area in that tissue. Burns cause an increased c a p i l l a r y permeability but the e f f e c t i v e surface area for exchange of f l u i d and protein might decrease due to a lower perfusion rate [29] (personal communication with A.Haugan, Bergen, 1987). According to Table 3.1, Kp^ ]-, increases after a burn, but i t appears to return to i t s normal value within about 12 hours postburn. Based on the work of Arturson et a l . [5], i t w i l l be assumed that the transient behaviour of K p i b can be expressed as an exponential function having the form K p l b = { k e ~ a t + ] ) k p l b 6 ' 1 In the above equation, the i t a l i c i z e d k [ ^ indicates the normal plasma leak c o e f f i c i e n t in the injured skin (see Section 5.3). "A" i s the magnitude of the Kp]_b change due to the burn injury. Several values of A ranging from 15 to 40 were tested in the 10% burn model. These A values were chosen because they are in the same range as values estimated experimentally (see Table 3.3). The constant a determines the time needed for the c o e f f i c i e n t to return to i t s normal value. The value of a was chosen to be 0.231 h" 1 90 so that f a l l s to half of i t s maximum value after three hours postburn, and i t returns approximately to i t s i n i t i a l value after about 12 hours. The changes in injured tissue hydrostatic pressure which occur within the f i r s t three hours following a burn were discussed in Chapter 3. Lund and Reed [34] measured P^ experimentally in rats subjected to 10% and 40% surface area burns. The measured tissue pressures, as shown in Figures 3.2 and 3.3, are characterized by "mean" values and standard errors. Through personal communication with T.Lund, i t was suggested that both the "mean" and the "mean+standard error" tissue pressure values should be t r i e d in the simulations due to the large error in the measurements. The tissue pressure variation with time (up to 3 hours) given by Lund and Reed's "mean" values i s referred to hereafter as the "middle" pressure curve. The d i s t r i b u t i o n obtained by adding the standard errors to the "mean" value for each point is c a l l e d the "upper" pressure curve. An example simulation showing the time-dependent behaviour of the six important variables obtained by perturbing only K p ^ and P^ i s plotted in Figure 6.1. The various input perturbations attempted along with a measure of the overall f i t in each case are given in Table 6.1. 91 Table 6.1 SS as a Function of K and P,_ fo r 10% Burn (Phase One) plb D — A pw \ ss D \ 15.0 20.0 25.0 30.0 35.0 40.0 Upper Curve — 1.91 — 0. 94 0. 90 1 .03 Middle Curve 2. 12 1 .99 2.52 3.51 — — N5 93 In Table 6.1 as well as in the remaining tables in this chapter, SS represents the t o t a l sum-of-squares of the normalized v e r t i c a l differences between the experimental points and the simulation for a l l six variables. The data were normalized by dividing each difference by the i n i t i a l value of that variable so that a l l the differences had equal weight. SS therefore represents how close the simulation predicts the experimental data, i . e . , the smaller the value of SS, the smaller the overall differences between the experimental and simulation r e s u l t s . For example, in Table 6.1, when A is changed from 20 to 30 using the upper pressure curve, SS f a l l s from 1.91 to 0.94 indicating a s i g n i f i c a n t improvement in the ov e r a l l f i t . The plots shown in the Figure 6.1 are those obtained for the "best" f i t of the various p o s s i b i l i t i e s attempted, i . e . , using the upper pressure curve with A=35. Since the injured tissue pressure i s very negative following a burn, both f i l t r a t i o n and reabsorption act in the same di r e c t i o n causing a greatly increased input of f l u i d to the injured tissue over the f i r s t few hours. The changes made in both K p ^ and also magnify the plasma leak rate to the injured skin. Thus both the f l u i d and the protein content of the injured skin compartment increase dramatically. Since most of the f l u i d transported to the injured skin comes from the plasma compartment, the plasma volume tends to decrease. Also, because the relat i v e 94 increase in the plasma leak rate is many times that of the f i l t r a t i o n and reabsorption rates and because the protein concentration in the plasma i s i n i t i a l l y about twice that in the injured skin, the protein concentration and consequently the c o l l o i d osmotic pressure (COP) of the l a t t e r tissue increase. Conversely, since the plasma now loses protein at a r e l a t i v e l y faster rate than f l u i d , both the plasma concentration and the COP of the plasma decrease. The massive f l u i d and protein exchanges between the plasma and the injured skin have only an indirect e f f e c t on the f l u i d content of the noninjured t i s s u e s . Consequently, as shown in Figure 6.1, the volume of intact skin tends to decrease only s l i g h t l y . When the upper curve i s used to represent P^, the f i t of the model predictions to the experimental data improves as A is increased from 15, passes through a minimum at about A = 35 and then begins to deteriorate with further increases in A. Therefore, A = 35 is assumed to be the best f i t at this stage. The magnitude A has a d i r e c t effect on the f l u i d and protein contents of the injured skin and also on the plasma volume. Raising the value of A in the simulation increases the amount of the f l u i d and protein in the injured skin and decreases the plasma volume. When a low value of A is employed, e.g., A = 15, the predicted injured skin f l u i d and protein contents are too low and the plasma volume is too high compared to the experimental data. When a high 95 value of A is used, such as A=40, the f l u i d and protein contents are higher and the plasma volume lower than the experimental data. Primarily because of these factors, SS goes through a minimum at the intermediate value of A=35. For a l l values of A, the upper pressure curve always yields a better f i t than does the middle curve. This i s because f i r s t l y , the middle pressure curve tends to cause substantial increases in f i l t r a t i o n , reabsorption and plasma leak leading to a gross overprediction of the injured skin f l u i d volume. Secondly, because the f i l t r a t i o n and (negative) reabsorption rates are so much higher, the protein content does not increase as rapidly as the f l u i d content of the injured skin. Hence, the injured skin COP tends to decrease with time while the experimental data shows that i t should increase. As a consequence, in the simulations which follow, only the upper curve w i l l be used to estimate the injured tissue pressure P^ for the f i r s t three hours postburn. The predicted results obtained by varying only K and P^ are generally representative of those measured experimentally except for one variable, the volume of intact skin. The experimental data show that t h i s volume should increase s l i g h t l y after a 10% burn, while the simulation results predict a small decrease (see Figure 6.1). Other investigators [37,39] found there was no change in intact 96 skin volume after burns (see Table 3.3). Additionally, the predicted injured skin volume for the best case in Phase One is somewhat too high and the protein content somewhat too low. As Figure 6.1 shows, this y i e l d s low values of COP in this compartment. 6.2.3 TRANSIENT RESPONSE INCLUDING CHANGES TO THE FILTRATION  AND REABSORPTION COEFFICIENTS IN THE INJURED SKIN  (PHASE TWO) The f i l t r a t i o n and reabsorption mechanisms are important causes of f l u i d interchange between the plasma and each tissue. The magnitudes of the changes in both the f i l t r a t i o n and reabsorption c o e f f i c i e n t s are subject to two competing phenomena. These c o e f f i c i e n t s could increase due to an r i s e in the permeability of vessel wall or they could decrease due to a reduction in blood flow (or perfusion rate) following a burn injury. Both types of changes have been suggested in the l i t e r a t u r e (see Chapter 3). Thus, both increases and decreases in K ^ and K r b were examined in the model. K ^ and K r b are assumed to undergo a step change at zero time (onset of burn) and to return to normal according to the following exponential relationships: K f b = (Be~ a t+1>* / 6 6.2 97 and = (Be •at + 1 ) k rb 6.3 Here, the i t a l i c i z e d kyb and k r b denote the normal values (see Section 5 . 3 ) , B r e f l e c t s the magnitude of the change (assumed to be i d e n t i c a l for both and K^) and a is the time constant. B could be p o s i t i v e (indicating an increase in and K r^), negative (i n d i c a t i n g a decrease in and Kj.^) or zero (indicating that K f b and K^^ remain at their normal values). For reasons of s i m p l i c i t y , i t i s assumed in Equation 6.2 and 6.3 that the changes in these parameters occur synchronically with Kp-j^, ( i . e . a=0.23l h" 1 ) . Simulations using d i f f e r e n t values of Kpifc>' K f b a n c^ K r b were generated using the upper pressure curve. The input perturbations and the sum-of-squares of differences, SS, are l i s t e d in Table 6.2. Examples of the Phase Two simulations are shown in Figures 6.3 and 6.4 for and decreased by 90% and 50% of their normal values, respectively. In both cases, K p i b * s assumed to increase i n i t i a l l y by 36 times i t s normal value. As can be seen in Table 6.2, when K f b and K r b are given larger than normal values (B > 0 ) , the agreement between the model predictions and the experimental data degenerates. Table 6.2 SS as a Function of K K and K ^ fo r 10% Burn (Upper Curve fo r P . , Phase Two) plb fb rb b B \ . SS 20.0 30.0 35.0 40.0 2.0 2.43 1 .64 1.61 1.74 0.0 1.91 0.94 0. 90 1 .03 -0.9 1.70 0.62 0.56 0.68 -0 .5 1.77 0.75 0.69 0.82 B = O.O corresponds to the simulat ions in Phase One 99 1 00 F i g u r e 6.3 S i m u l a t i o n f o r C h a n g i n g K l b , K f b and K r b i n 10% B u r n (A=35.0, B=-0.5, Upper C u r v e f o r P,) 0 2 i 6 0 2 * 6 TIME (hour) TIME (hour) 101 However, i f i t i s assumed that a r e d u c t i o n i n i n j u r e d t i s s u e p e r f u s i o n dominates the p e r m e a b i l i t y i n c r e a s e f o r f i l t r a t i o n and r e a b s o r p t i o n such that B < 0, then the r e s u l t s show a s i g n i f i c a n t improvement. No dramatic d i f f e r e n c e s were found in the p l o t t e d r e s u l t s obtained by s e t t i n g B = -0.5 and -0.9 although the l a t t e r y i e l d s a somewhat b e t t e r o v e r a l l f i t . However there i s not enough experimental information a v a i l a b l e to c l e a r l y j u s t i f y adopting one value of B over the other. The r e d u c t i o n s i n and Kr^ are reasonable compared to Ferguson's [29] estimate that i n a 70% burn, the blood flow i s reduced to only 5% of i t s normal value (see Table 3.2). The manner by which each of K p i b ' K f b a n ^ K r b change i n response to a burn i n j u r y i s complex and r e q u i r e s f u r t h e r experimental i n v e s t i g a t i o n . The trends p r e d i c t e d i n Phase Two are s i m i l a r to those of Phase One. However, as might be expected, the Phase Two parameter changes r e s u l t in b e t t e r f i t s of the experimental data f o r COP i n i n j u r e d s k i n (compare F i g u r e s 6.1 and 6.2). One"way of g a i n i n g separate c o n t r o l over the f l u i d and p r o t e i n i n f l u x to the i n j u r e d s k i n i s allow f o r changes i n the f i l t r a t i o n and r e a b s o r p t i o n c o e f f i c i e n t s . As mentioned before, the f i l t r a t i o n and r e a b s o r p t i o n have the same d i r e c t i o n i n the f i r s t few hours postburn, i . e . , from plasma to i n j u r e d s k i n . Thus changing and K r b together causes a much gr e a t e r change in the f l u i d exchange than j u s t changing one of these c o e f f i c i e n t s . Because of the 102 decreases in f i l t r a t i o n and reabsorption rates, the amount of f l u i d entering the injured tissue is reduced, while the amount of protein, which depends only on the plasma leak rate, is the about same as in Phase One. The net result of these changes is that the protein concentration in the injured tissue rises and hence . i t s COP increases to values which are in better accord with the experimental r e s u l t s . 6.2.4 TRANSIENT RESPONSE INCLUDING CHANGES TO THE ARTERIAL  CAPILLARY PRESSURE IN THE INJURED SKIN (PHASE THREE) As was mentioned e a r l i e r , the a r t e r i a l c a p i l l a r y pressure for a l l tissues was reduced from i t s normal value of 25.54 mmHg to 19.94 mmHg because the main a r t e r i a l pressure i s known to drop by about 20% after a 10% burn. The a r t e r i a l c a p i l l a r y pressure in the burned skin, Pa^, may be even lower due to an increase in the l o c a l peripheral resistance (see discussion in Chapter 3). Since there is no experimental information about how much P a b might decrease after a 10% burn, two additional values of P a b, 10 mmHg and 15 mmHg, were examined in the simulation. Figure 6.4 and Figure 6.5 show the simulation results for the Phase Three predictions using several of the better f i t t i n g parameter changes from Phase Two as a starting point. Table 6.3 reveals how the sum-of-squares of differences i s affected by these two alt e r a t i o n s in P a b. 1 03 F i g u r e 6.4 S i m u l a t i o n f o r C h a n g i n g K l b , K f b , K r b and P a b i n 10% B u r n (A=35.0, B=-0.9, P-, h=1 n mmHg, Uppe r C u r v e f o r P.) 0 2 * 6 0 2 4 6 TIME (hour) TIME (hour) Figure 6.5 Simulation for Changing K K f b ' K r b a n c^ P 10% Burn (A=35.0, B = -0.9, P\,h=15 rnmHg, Upper Curve for Table 6.3 SS as a funct ion of K K , K and P . (mmHg) for 10% Burn (Upper curve fo r P. , Phase Three) plb fb rf ab 3 b A.B P \ SS ab \ 30 .0 , -0 .5 35 .0 , -0 .5 40 .0 , -0 .5 3 0 . 0 , - 0 . ? 35 .0 , -0 .9 40 .0 , -0 .9 19.9 0.75 0.69 0.82 0.62 0.56 0.68 15.0 0.69 0.63 0.75 0.60 0.54 0.66 10.0 0.64 0.58 0.70 0.59 0.52 0.64 P = 19.9 mmHg corresponds to the simulations in Phase Two 1 06 Although the lower values of appear to improve the overall f i t s l i g h t l y (Table 6.3), the effects on the plotted results shown in Figures 6.4 and 6.5 are almost indiscernable from the P ^ = 19.94 mmHg results in Figure 6.3. Thus, i t appears that the predictions are much less sensitive to Pa^ changes than to the parameter alte r a t i o n s investigated in the previous phases. 6.2.5 TRANSIENT RESPONSE INCLUDING CHANGES TO THE LYMPH FLOW CHARACTERISTICS IN THE INJURED SKIN (PHASE FOUR) If the normal lymph flow rel a t i o n s h i p is used (see Equations 5.6a and 5.6b) for burned tissue, lymph flow in the injured skin is predicted to be less than zero in the f i r s t hour postburn because of the strongly negative tissue pressure P^. Since there has never been a reported measurement of a lymph flow reversal, the simulation program takes to be zero during t h i s period. From one to two hours after the burn, the injured skin lymph flow is positive but s t i l l s i g n i f i c a n t l y less than the normal lymph flow, again as a consequence of the low values of P^. As revealed in Table 3.3, some investigators have measured lymph flow subsequent to a burn injury and found that i t consistently increased. However, i t i s unclear i f the lymph flow they obtained represents only that of the injured tissue. More l i k e l y i t represents f l u i d from intact tissues as well. In the Phase Four perturbations, the lymph flow in 1 07 the injured skin for the f i r s t three hours postburn i s hypothesized to be twice i t s normal value i f the calculated from Equation 5.6a or 5.6b is less than 2 * v i b o ' After three hours, i t ' s assumed to once again follow the normal relationship, Equation 5.6a or 5.6b. The model predictions for this altered lymph flow behaviour were generated based on several of the better f i t t i n g cases from Phase Three. Table 6.4 shows the sum-of-squares f i t t i n g r e sults and Figure 6.6, the plot of the best f i t to the experimental data for Phase Four. The results l i s t e d in Table 6.4 indicate that the overall f i t between simulations and experimental data i s only s l i g h t l y improved by r a i s i n g the lymph flow to double i t s normal value during the f i r s t three hours postburn. However, there i s once again no discernable difference in the plotted results shown in Figure 6.6 compared to those of Phases Two and Three. Thus, there i s no clear indication from the model that lymph flow actually increases following a burn injury and i t appears that i t i s unnecessary to include al t e r a t i o n s in the lymph flow c h a r a c t e r i s t i c s in order to explain the results measured for a 10% burn. Table 6.4 SS as a Function of K , . , K . , P . (mmHg) and V,. fo r 10% Burn (A=35, Upper Curve fo r P , Phase fb rb ab I D D Four) -0 .9 ,10 .0 -0 .9 .15 .0 -0 .5 ,10 .0 -0 .5 ,15 .0 Normal 0.52 0.54 0.58 0.63 2 V Ibo (When t<3.0 hours a n d v l b < v l b o ) 0.49 0.50 0.53 0.58 V,. = Normal corresponds to the simulat ions in Phases Two and Three 1 b o CD 1 09 F i g u r e 6.6 S i m u l a t i o n f o r C h a n g i n g K j b , K r b , P a b and i n 10% B u r n (A=35.0, B=-0.9, P a b = 1 0 . 0 mmHg, Uppe r C u r v e f o r P^, A l t e r e d Lymph F l o w C h a r a c t e r i s t i c s ) 0 2 4 6 0 2 4 6 TIME (hour) TIME (hour) 110 6.2.6 MOST REASONABLE FIT FOR A 10% BURN Based on the four phases of f i t t i n g described above,-the i n f l u e n t i a l parameter changes which lead to the most reasonable results can be summarized as follows: 1. The plasma leak c o e f f i c i e n t in the injured skin increases, and the magnitude of the change (A in Equation 6.1) is 35. K p ^ decays to i t s normal value in about 12 hours following an exponential relationship with time. 2. The tissue pressure in the injured skin f a l l s to very negative values for a limi t e d period postburn. The use of the "mean plus one standard error" pressure curve of Lund and Reed (shown in Figure 3.2) res u l t s in a better s t a t i s t i c a l f i t than does the "mean" curve alone. 3. The perfusion rate in the injured skin decreases leading to a reduction of the i n i t i a l f i l t r a t i o n and reabsorption c o e f f i c i e n t s in the injured skin by at least 50% of their normal values. A perfusion rate decrease of 50% implies an increase in the protein permeability due to plasma leakage of about 70. The l a t t e r increase is in l i n e with the increased protein permeability of about 54 observed in the extravasation experiments of Lund and Reed [25] (see Table 3.1). The other perturbations examined, changes to P a b and V-^, have only a ne g l i g i b l e effect on the sum-of-squares f i t and on 111 the predicted results and hence, were not included as the most reasonable changes associated with the 10% burn. 6.2.7 LONG-TERM TRANSIENT RESPONSE There is a small amount of information in the li t e r a t u r e concerning the time required for the 10% burned rat to return to i t s normal state (see Table 3 . 3 ) . In the simulation, the a r t e r i a l and venous c a p i l l a r y pressures were altered from their normal values because of changes to the large artery pressure following a 10% burn. A l l of the transport c o e f f i c i e n t changes were specified in such a way that they returned to their normal values within about 12 hours (the compliance r e l a t i o n s h i p and the lymph flow c h a r a c t e r i s t i c s in the injured skin returned to normal at three hours postburn). In order to allow the entire microvascular system to eventually recuperate, both the a r t e r i a l and venous c a p i l l a r y pressures were a r b i t r a r i l y set to their normal values (Pa=24.54 mmHg and Py=5.92 mmHg) after either 3, 6 or 9 hours postburn. The steady-state results obtained for these three cases using one of the most reasonable sets of parameter changes are shown in Figures 6.7 - 6.9. In these figures, the dashed l i n e s indicate the normal values for each variable. Figures 6.7, 6.8 and 6.9 are not s i g n i f i c a n t l y d i f f e r e n t aside from the location of the discontinuity 1 1 2 Figure 6.7 Steady State Result for 10% Burn (A=35.0, B=-0.5, ab = 15.0 mmHg, Upper curve for P h, -^^ =^2 •V1 h n when t<3 hours lbo and v i b < ^ ' ^ l b o ' A r t e r i a l and Venous Capil l a r y Pressure Return to Normal After Three Hours) 12 24 TIME (hour) 36 12 24 TIME (hour) 1 1 3 Figure 6.8 Steady State Result for 10% Burn (A=35.0, B=-0.5, P ak=l5.0 mmHg, Upper Curve for P^, v i b = 2 * v i b o when t<3 hours and ^ i b < 2 ' ^ l b o ' A r t e r i a l and Venous Ca p i l l a r y Pressure Return to Normal After Six Hours) 0 12 24 36 0 12 24 36 TIME (hour) TIME (hour) 1 1 4 Figure 6.9 Steady State Result for 10% Burn (A=35.0, B=-0.5, Pab=15.0, Upper Curve for P b, v i b = 2 ' ^ i b o w h e n t < 3 n o u r s a n d V l b<2-V l b o, A r t e r i a l and Venous C a p i l l a r y Pressure Return to Normal After Nine Hours) 0 12 24 36 0 12 24 36 TIME (hour) TIME (hour) 1 1 5 caused by the s h i f t in a r t e r i a l and venous c a p i l l a r y pressures. The long-term responses, recognized to be very speculative, show that i t takes approximately 24 to 36 hours for the various parts of the system to return to normal. This prediction w i l l be discussed in greater d e t a i l in the section following. 6.2.8 IMPLICATIONS OF THE 10% BURN MODEL In this section, the model simulations are used to obtain a more mechanistic understanding of the various events which take place following a 10% surface area burn. The simulations were obtained by using the most "reasonable" set of parameter changes summarized in Section 6.2.6 and by allowing the c a p i l l a r y pressures to return to their normal values after 9 hours. A complete l i s t i n g of the transient responses of a l l variables in a l l four compartments due a 10% burn perturbation i s given in Appendix D. The time-dependent behaviour of some of the more important variables is plotted in Figures 6.10-6.14. Figures 6.10 and 6.11 show how the f l u i d volumes and protein contents, respectively, of the plasma, muscle, intact skin and injured skin compartments change as a function of time following a 10% burn. As i s indicated by the plots, immediately after the burn i s i n i t i a t e d , there is a very large transfer of both f l u i d and protein to the 1 1 6 F i g u r e 6.10 The C h a n g e s of C o m p a r t m e n t a l F l u i d V olumes F o l l o w i n g a 10% B u r n 10 15 2 0 25 Time (hour) 1 1 7 F i g u r e 6.11 The Changes of C o m p a r t m e n t a l P r o t e i n C o n t e n t s F o l l o w i n g a 10% B u r n 3 0 0 n 1 1 1 1 1 I 1 1 1 1 0 5 10 15 20 25 3 0 35 4 0 Time (hour) F i g u r e 6.12 The C h anges of F l u i d F l o w R a t e s i n I n j u r e d S k i n F o l l o w i n g a 10% B u r n 1.6 . Time (hour) 1 20 F i g u r e 6.14 The Changes o f C a p i l l a r y and I n j u r e d S k i n P r e s s u r e s F o l l o w i n g a 10% B u r n A a a tt Wi 9 CO 0) tt hi - 1 0 - 1 5 r 0 i 10 7 15 20 Time (hour) 4 0 121 injured tissue. The injured skin becomes edematous, with i t s volume approximately doubling while i t s protein content increases by an even greater extent. Although the intact tissue compartments contribute somewhat to this large exchange of f l u i d and protein, i t is the plasma compartment, which is the only compartment in d i r e c t communication with the injured skin compartment, that undergoes the severest losses. In order to understand the mechanisms behind these f l u i d and protein s h i f t s , an analysis of the f l u i d and protein fluxes into and out of the injured skin compartment is necessary. Figures 6.12 and 6.13 i l l u s t r a t e the transient behaviour of the relevant f l u i d and protein fluxes, respectively. The d r i v i n g forces behind the f l u i d and protein exchanges are functions of the hydrostatic and c o l l o i d osmotic pressures that exist in that compartment and in the c i r c u l a t i o n at any given time. These are shown for the plasma and injured tissue compartments in Figure 6.14. The most important parameter changes a f f e c t i n g the flow of . f l u i d and protein to the injured skin are the strongly-negative tissue pressure, P^, which occurs in the f i r s t few hours postburn (see Figure 6.14) and the i n i t i a l large increase in the plasma leak c o e f f i c i e n t , Kpi D« T n e combined effect of these two changes causes a very large increase in the plasma leak rate and i t s associated protein flux. Despite the fact that the f i l t r a t i o n and reabsorption c o e f f i c i e n t s are i n i t i a l l y reduced to half of their normal 1 22 values, the negative tissue pressure results in a substantial increase in the magnitudes of both of these flows. Furthermore, because of the i n i t i a l large change in P^-Pv, the reabsorption flow i s actually negative ( i . e . , into the tissue) for the f i r s t two hours postburn. Matters are further exascerbated by the fact that during much of this same period, the lymph flow i s shut down en t i r e l y due to the low values of P^. Thus, because there are only inflows of f l u i d and protein to the injured skin and no outflows, both the volume and plasm protein content of this compartment r i s e dramatically during this period. Furthermore, as was explained e a r l i e r (see Section 6.2.2), because the plasma leak rate far exceeds the combined f i l t r a t i o n and reabsorption rates and because the protein concentration i s greater in plasma than that of the tissue, protein enters the injured skin r e l a t i v e l y more quickly than does f l u i d . As a consequence, CA^ and increase while C p and lip f a l l . In addition, since the volume of the plasma compartment declines, the compliance relationship for this compartment requires that the venous c a p i l l a r y pressure, P v, must decrease. Between 2 and 3 hours postburn, the injured tissue pressure approaches values much closer to normal (-0.51 mmHg) and the plasma leak c o e f f i c i e n t f a l l s to about half of 1 23 i t s i n i t i a l value. As a result, the plasma leak f l u i d and protein fluxes f a l l rapidly from their maximum values. The plasma leakage is further reduced by the drop in P v. The increase in P^ causes a decline in the f i l t r a t i o n rate which is somewhat offset by the reduction in r i p - n ^ . More importantly, the very small difference between P y and P^ which occurs during this period reverses the sign of the reabsorption rate. Furthermore, because P^ ri s e s above i t s excluded volume l i m i t of -7.84 mmHg, the lymph flow i s turned on again. Thus, because both the reabsorption and lymph flows are positive, there are now two mechanisms for reli e v i n g the f l u i d buildup in the injured skin and both and Q_k begin to f a l l . The extrema in both curves occur when the input and output fluxes are exactly equal, e.g., at 2.3 hours in the case of Q^. Interestingly, because there are now two paths for f l u i d return and only one for protein, the protein concentration in the injured tissue continues to ris e s l i g h t l y as exhibited by the further small increase in n ^ . Note also how the lymph flow begins to increase rapidly after about 2 hours when P^ ri s e s above -0.51 mmHg and there is a switchover from Equations 5.6b to 5.6a. At 3 hours, P^ is returned to i t s compliance relationship producing the d i s c o n t i n u i t i e s apparent in Figure 6.14 and manifested in a l l of the fluxes shown in Figures 6.12 and 6.13. A second set- of noticeable d i s c o n t i n u i t i e s occurs at 9 hours when the c a p i l l a r y 1 24 pressures, P a and P y are restored to normal. Here, the larger increase in P g causes an increased f i l t r a t i o n to a l l tissues, reducing the plasma volume and consequently, ra i s i n g both C D and n . The maximum in Q at about 16 hours (Figure 6.11) i s also probably associated with this change. Even though a l l of the c o e f f i c i e n t perturbations have died out after approximately 12 hours and the imposed hydrostatic pressure changes have been relaxed at even e a r l i e r times, i t is of interest to note that a l l of the f l u i d volumes and protein contents only return to normal after an elapsed time of 24-36 hours postburn. It is suspected that the main cause of t h i s lag i s the i n a b i l i t y of the lymph flow to relieve the excessive f l u i d and plasma protein accumulation in the injured tissue quickly enough. Figure 6.12 demonstrates that V-j^ remains at a r e l a t i v e l y constant, but high value (approximately 6 times greater than i t s normal value) from about 3 to 13 hours postburn. Despite the fact that the tissue volume i s abnormally elevated during much of t h i s period, the lymph flow i s severely limited by compliance considerations. The volume (when divided through by BSA) places the operating condition far out on the f l a t portion of the compliance relationship where P^ i s almost independent of V^. It's not u n t i l the volume is lowered s u f f i c i e n t l y (after about 13 hours) that the tissue pressure begins to follow the more strongly curved part of the relationship, and P^ (Figure 6.14) and 1 2 5 v"lb (Figure 6 . 1 2 ) f a l l slowly back to their normal values. Because only gradually a l l e v i a t e s the protein buildup in the injured skin, remains unusually high u n t i l after about 15 hours. Beyond this time, the increase in ^ - n ^ results in a corresponding r i s e in the reabsorption rate which further extends the time taken to reach normal. These results are in excellent agreement with the experimental measurements of Carvajal et a l . [40] who found that, even though the permeability of thermally injured rat skin returned to normal within 12 hours, the resulting edema lasted longer than 24 hours. Figures 6.10 and 6.11 indicate that the water and protein contents of the intact skin and muscle compartments decrease only s l i g h t l y following a 10% burn injury. The i n i t i a l cause of the f l u i d volume decrease can be associated with the imposed drop in P a which decreases the f i l t r a t i o n flow to the intact tissues. The decline in the plasma protein content i s explained by the large reduction in the plasma leak to muscle and intact skin due to the i n i t i a l rapid drop in P v observed in Figure 6.14. The minima in V m, V s, Q m and Q s occur at times which lag considerably behind the extrema noted e a r l i e r for V , V^, Q p and Qfa. This is because the most important perturbations to the system (Kp]_b and P^) changes only i n d i r e c t l y affect the intact skin and muscle compartments; these changes made to the injured tissue are eventually communicated to the intact tissues 1 26 through the intervening plasma compartment. In essence, for the 10% burn, the intact tissue compartments act as adit i o n a l f l u i d and protein reservoirs which mediate the changes which take place in the plasma. 6.3 THE 40% BURN 6.3.1 INTRODUCTION A similar simulation procedure was applied to rats with 40% surface area burns. As mentioned previously (see Chapter 3), the c a p i l l a r y permeability increases not only in the injured skin, but also in the intact skin and muscle for burns exceeding 25% of the surface area. It i s therefore anticipated that the simulation of a 40% burn w i l l be more complex than that of a 10% burn. The following l i s t includes the parameter perturbations which would l i k e l y occur in a 40% burn: 1. an increase in the plasma leak c o e f f i c i e n t in the injured skin ( K p l b ) , 2. an a l t e r a t i o n in the hydrostatic pressure in the injured skin ( P B ) , 3. an increase in the plasma leak c o e f f i c i e n t s in the intact tissues (K p^ s and K p-^ m), 4. a change in the f i l t r a t i o n and reabsorption c o e f f i c i e n t s in the injured skin ( K ^ and K r b) , 127 in the injured skin (K f^ and K^) , 5. a change in the f i l t r a t i o n and reabsorption c o e f f i c i e n t s in the intact tissues (Kf s, Kf m, K r s and K r m ) , 6. a decrease in the a r t e r i a l c a p i l l a r y pressure in the injured skin ( P a b ) , and 7. an a l t e r a t i o n the lymph flow in a l l tissues ( V ^ , and V l s ) . These perturbations are l i s t e d in what i s assumed to be their order of importance with respect to causing changes in the microvascular exchange system. According to Lund and Reed [33], the . main a r t e r i a l pressure, P a a, dropped to an average of 70% of i t s normal value following a 40% burn. If this P__ i s used to recalculate the a r t e r i a l and venous c a p i l l a r y pressure from Equations 5.18 and 5.19, i t is found that Pa=17.64 mmHg and Pv=4.72 mmHg, respectively. These P a and P v values are used in a l l of the simulations for the 40% burn which follow. 6.3.2 TRANSIENT RESPONSE FOR CHANGES TO THE PLASMA LEAKAGE  COEFFICIENT AND THE TISSUE PRESSURE CHARACTERISTICS OF  THE INJURED SKIN (PHASE ONE) An increased is known to occur in 40% surface area burns (see Table 3.1). As in the 10% burn, a time-dependent exponential function of the form: 1 28 K = (Ae - a t + 1 )k 6.4 p l b pi b i s assumed. In Equation 6.4, i t a l i c i z e d k pi b is the normal plasma leak c o e f f i c i e n t of the injured skin for a 40% burn and the time constant a has the same value as for a 10% burn. follows the experimentally measured relationship shown in Figure 4.3. Again, both the "mean" (middle) and the "mean+standard error" (upper) pressure curves are used in the simulation. The sum-of-squares of differences between the simulation predictions and the experimental data obtained by Lund and Reed for a 40% burn were calculated as before and the same six dependent variables were normalized with respect to their i n i t i a l values. The SS predictions obtained in Phase One of the 40% burn simulation can be seen in Table 6.5 for a range of changes in K p ^ and for the two P^ versus time curves. The best f i t ( i . e . , lowest SS) results are shown in graphical form as Figure 6.15. As was the case for the 10% burn, the use of the middle pressure curve gave poorer f i t s for a l l A values tested. Thus, i t was decided that only the upper pressure curve would be employed in a l l of the subsequent simulations. Table 6.5 SS as a funct ion of K and P fo r 40% Burn (Phase One) plb b P b N \ ss 2.0 3.0 4.0 5.0 6.0 8.0 Upper Curve — 1.02 0.98 0.99 1 .03 1 . 26 Middle Curve 1.30 — 1.38 — 1 .82 — 131 When the upper curve for was used, the SS value went through a minimum at about A =4 although very l i t t l e difference in the f i t s was noted for the range of 3<A<6. The optimum value of A for a 10% burn was found to be 35, i.e., an order-of-magnitude greater than that determined for a 40% burn. This apparent discrepancy can be rationalized i f i t i s recalled that K p i b ' l i k e a l l transport c o e f f i c i e n t s , i s the product of the vascular surface area and the c a p i l l a r y permeability. Thus, i f the perfusion of the injured tissue in a 40% burn is substantially less than for a 10% burn, then the actual permeability could be of the same order in both cases. As w i l l be discussed in a later section (Section 6.3.4), there is other corroborating evidence for this hypothesis. As can be seen from Figure 6.15, predicted results are not in close quantitative agreement with the experimental measurements. However, even though some of the experimental trends are not p a r t i c u l a r l y obvious, fiv e out of the six predicted trends appear to be in the right d i r e c t i o n . As was the case for the 10% burn, the only variable for which the trends do not match c l o s e l y i s the volume of intact skin. The model predicts a s l i g h t decrease in V g, while the experimental data show an increase. The reasons why an increase in Kp-^ and a reduction in P^ lead to the predicted responses i l l u s t r a t e d in Figure 1 32 6.15 are s i m i l a r to those a l r e a d y d i s c u s s e d in the Phase One s i m u l a t i o n s for the 10% burn. The p r e d i c t e d edema in the i n j u r e d s k i n i s too high compared to the experimental data l e a d i n g to COP values f o r t h i s t i s s u e which are somewhat too low. A l s o , the plasma output i s i n s u f f i c i e n t and, as a consequence, the volume of plasma in the model p r e d i c t i o n i s f a r too l a r g e . To compensate for these problems of not having enough f l u i d i n the i n t a c t s k i n and too much in the plasma, the Phase Two s i m u l a t i o n s were i n t r o d u c e d . 6.3.3 TRANSIENT RESPONSE INCLUDING CHANGES IN THE PLASMA  LEAK COEFFICIENTS IN THE INTACT TISSUES (PHASE TWO) The j u s t i f i c a t i o n f o r i n c r e a s i n g the plasma leak c o e f f i c i e n t s i n the n o n i n j u r e d t i s s u e s f o r the 40% burn case has a l r e a d y been d i s c u s s e d . The perturbed plasma leak c o e f f i c i e n t s f o r the i n t a c t s k i n and muscle (Kp-j_s and K p i m ) are assumed to behave as time-dependent e x p o n e n t i a l f u n c t i o n s having the f o l l o w i n g form: K = (Be -at + 1 )k 6.5 p i s pi s and K = (Be -at + 1 ) * 6.6 plm plm 1 33 where the i t a l i c i z e d k , and k , denote normal values, B pis plm r e f l e c t s the magnitude of the change (assumed equal for both skin and muscle), and a is the time constant (again equal to 0.231 h - 1 ) . Several combinations of d i f f e r e n t K p i D ' ^pls a n d ^plm were investigated. The input perturbations and the SS values obtained in each t r i a l are l i s t e d in Table 6.6. The best f i t results for a l l six measured variables are shown in Figure 6.16. The SS values reported in Table 6.6 are not dramatically d i f f e r e n t from, but a l l represent a s l i g h t improvement over the Phase One results given in Table 6.5, In other words, when Kp^ s and ^-g\m are increased, the model provides better predictions than i f K p^ s and Kpim remain at their normal values. Since the changes in the intact tissue permeabilities after burns are expected to be s i g n i f i c a n t l y less than those of injured skin, only values of B < 10.0 were examined in this investigation. According to Table 6.6, the best results are obtained when K p i s an<3 Kpim a r e allowed to increase to about 8 - 10 times normal. Although the optimum value of B is about twice that of A, i t is expected that the actual permeability increase in the injured skin should be considerably greater than in the intact tissues. Table 6.6 SS as a K K and K . for 40% Burn (Upper Curve for P . , Phase Two) p lb p i s pirn b \ — — - - ^ B A N . SS 1 .0 3.0 5.0 8.0 10.0 2.0 — — 0.89 — 0.77 3.0 0.97 0.89 0.81 0.75 — 4.0 0.94 0.85 0.79 0.74 — 6.0 — — 0.89 0.87 — 1 36 Again, the f i t t e d results can be rati o n a l i z e d i f i t is assumed that there is a strong depression of the perfusion rate in the injured tissue following a large surface area burn. When the results of Figure 6.16 are compared to those of Figure 6.15, the two most apparent differences are that the intact skin volume f a l l s less d r a s t i c a l l y and the plasma volume more d r a s t i c a l l y than before. Both trends are in better agreement with the experimental data. The changes are consistent with increases in K p^ s and Kp-^m, which allow for a greater f l u i d transfer from the plasma to the intact tissues. However, the Phase Two perturbations have no direct effect on the f l u i d and protein contents of the injured skin. As a consequence, in this tissue, the edema remains overpredicted and the COP underpredicted. 6.3.4 TRANSIENT RESPONSE INCLUDING CHANGES IN THE FILTRATION  AND REABSORPTION COEFFICIENTS IN THE INJURED SKIN  (PHASE THREE) The f i l t r a t i o n and the reabsorption c o e f f i e i n t s in the injured skin ( K f b a n <^ K r b ^ a r e products of c a p i l l a r y permeability and surface area for transport. Thus, K ^ and Kj.^ could increase due to an augmentation of the permeability or decrease due to a drop in the perfusion rate. It is assumed that the c o e f f i c i e n t perturbations 1 37 follow the relationships: K f b = (Ce- a t-HH / f c 6.7 and K r b - ( C e ~ a t ^ ) k r b 6.8 where i t a l i c i z e d kj-^ and kr^ are the normal values, C r e f l e c t s the magnitude of the change (which could be positive or negative) and the time constant a has the same value as before ( i . e . , <Z=0.231 h ~ 1 ) . Table 6.7 l i s t s the sum-of-squares results for the Phase Three simulations obtained by using some of the better combinations of A and B from Phase Two. Figures 6.17 and 6.18 compare predicted and experimental values for the two " b e s t - f i t " cases where A=4.0, B=10.0, and C=-0.80 or C=-0.95, respectively. For the 40% burn, increasing K^b and K r b above their normal values results in a poorer f i t of the data. This induces an even higher edema in the injured skin compare to the results of Phase Two. If K j b and K r b are decreased by 50%, the SS results are improved moderately. However, i f Table 6.7 SS as a Function of K , K , , K , , K and K for 40% Burn (Upper Curve for P . Phase Three) plb p i s plm fb rb b ^ A.B 2 .0 ,8 .0 3.0,1.0 3 .0 ,8 .0 3.0,10.0 4 .0 ,3 .0 4 .0 ,5 .0 4 .0 ,10.0 2.0 1 .03 — 0.99 — — — — 0.0 — 0.97 0.75 — 0.85 0. 79 — -0.50 — 0.86 — — 0.74 — — -0.80 — 0.79 0.58 0.56 0.68 — 0.53 -0.95 — 0.77 0.58 0.55 0.66 0.67 0.52 C = 0.0 corresponds to the simulations in Phase Two 141 these c o e f f i c i e n t s are decreased by 80% or even 95%, the SS results show a s i g n i f i c a n t improvement (see Table 6.7). This required reduction in and K r b is additional confirmation that there may be much lower perfusion in the injured skin following a 40% burn than after a 10% burn. There is no discernable differences between the two cases plotted in Figures 6.17 and 6.18. Thus, i t appears that the f i l t r a t i o n and reabsorption c o e f f i c i e n t s in injured skin may decline to about 5-20% of their normal values. Since i t is expected that the permeability towards f l u i d transfer through the c a p i l l a r y wall w i l l actually increase after a burn, the inescapable conclusion i s that the perfusion rate in a 40% injury must f a l l to very low values. This prediction is consistent with the published measurements of Ferguson et a l . [24] (see Table 3.2) and with the recent experimental observations of A. Haugan in Bergen, Norway (personal communication). The Phase Three model predictions represent a s i g n i f i c a n t improvement in the f i t of the experimental data as compared to the Phase Two perturbations. Additionally, a l l six predicted variables in Figures 6.17 and 6.18 now have the same trends as the experimental data. The main effe c t of the Phase Three perturbations i s to reduce the edema in the injured skin, making i t more consistent with the experimental data. By decreasing and K r b the magnitudes of both the f i l t r a t i o n and reabsorption rates are 142 decreased. Since reabsorption i s actually reversed during the i n i t i a l stages of the burn (see Section 6.3.10), these changes lead to a lower input of f l u i d into the injured skin which tends to adjust conditions in the proper d i r e c t i o n . However these changes do not influence the protein content in the injured skin, which remains approximately the same as in Phase Two. As a consequence, the albumin concentration is higher and the COP results are in better correspondence with the measured data. 6.3.5 TRANSIENT RESPONSE INCLUDING CHANGES IN THE FILTRATION  AND THE REABSORPTION COEFFICIENTS IN THE INTACT  TISSUES (PHASE FOUR) An increased c a p i l l a r y permeability and decreased perfusion rate could occur in the intact tissues as well as in injured tissue in the case of a 40% burn (see Chapter 3). These changes could increase or reduce the values of K f s r K r s, K f m and K r m, which are assumed to follow time-dependent exponential functions of the form and K f s • = (De a t + y ) k f s 6.9 K r s = • ( D e ~ a t + 1 H rs 6.10 Kfm = • < De- a t +1>*/m 6.11 Krm = : ( D e - a t + l H r m 6.12 143 In the above equations, i t a l i c i z e d ^fs> ^ r s , ^fm an<^ ^rm represent normal values, D r e f l e c t s the magnitude of the i n i t i a l changes (assumed equal for a l l four c o e f f i c i e n t s ) and the time constant a remains 0.231 h~ 1 . Based on several of the better f i t t i n g cases from Phase Three, several model predictions for t h i s phase were generated and the sum-of-squares values are given in Table 6.8. The plotted results for the best f i t to the experimental data in t h i s phase are shown in Figure 6.19. Changing K f s, K r s, K f m and K r m for a given set of A,B, and C magnitudes yie l d s the same SS values as the Phase Three predictions (compare Table 6.8 with Table 6.7). This i s primarily because reabsorption and f i l t r a t i o n in the intact tissues transfer f l u i d in opposing d i r e c t i o n s . Thus, i f both f l u i d fluxes change by the same magnitude, their e f f e c t s tend to cancel one another. However, changing K f b and in injured skin has a much stronger influence. Because reabsorption is actually reversed (due to the very negative tissue pressures in burned skin) during the f i r s t few hours postburn, and V ^ are in the same di r e c t i o n . Hence, increases or decreases in the value of and K r b during t h i s time has twice the e f f e c t of just increasing one of the c o e f f i c i e n t s . Table 6.8 SS as a Function of K , , K„ , K fo r 40% Burn (Upper Curve fo r P . , Phase Four) pi f r b \ ^ \ A ' B D \ ^ SS -0.50 -0.80 -0.95 3.0,1.0 3 .0 ,5 .0 3.0,10.0 3 .0 ,1 .0 3 .0 ,5 .0 3.0,10.0 3 .0 ,1 .0 3 .0 ,5 .0 3.0,10.0 Norma 1 0.86 — — 0.79 — 0.56 0.77 — 0.55 -0.30 0.86 0.71 0.61 0.79 0.66 0.56 0.78 0.65 0.55 -0 .5 0.86 0.71 0.61 0.80 0.66 0.56 0.78 0.65 0.56 D - Normal corresponds to the simulat ions in Phase Three 1 46 Since the trends predicted in Phase Four are v i r t u a l l y indistinguishable from those obtained in Phase Three, i t is concluded that changing the f i l t r a t i o n and reabsorption c o e f f i c i e n t s in the intact tissues does not improve the f i t t i n g results and hence, can be l e f t out of a l l subsequent model predictions. 6.3.6 TRANSIENT RESPONSE INCLUDING CHANGES IN THE ARTERIAL  CAPILLARY PRESSURE IN THE INJURED SKIN (PHASE FIVE) The main a r t e r i a l pressure in the 40% burn i s depressed more than in the 10% burn [33] (see Table 3.2). This factor was assumed to decrease the a r t e r i a l and venous c a p i l l a r y pressures for a l l tissues (see Section 6.3.1). It is also expected that, because of a l o c a l increase in peripheral resistance in thermally injured tissue, Pa^ may be even lower than the value estimated in Section 6.3.1. Thus, Phase Five involved the examination of the e f f e c t of two lower values of P a b; namely, 10 mmHg and 15 mmHg. Based on several of the better f i t t i n g cases from Phase Three (since Phase Four does not have much influence on SS), the model predictions for these decreased P a b values were generated. Table 6.9 compares the sum-of-squares f i t t i n g , values for the resultant simulations. Table 6.9 SS as a Function of K , K , K and P (mmHg) fo r 40% Burn (Upper Curve for P^, Phase Five) pi f r ab 3 b \ ^ \ A , B p \ SS ab \ -0.8 -0 . 95 3 .0 ,1 .0 3 .0 ,8 .0 4.0,10.0 3 .0 ,1 .0 3 .0 ,8 .0 4 .0 .10.0 10.0 0.80 0.60 0.54 — 0.60 0. 54 15.0 0.79 0.59 0.54 — 0.58 0. 53 17.6 0.79 0.58 0.53 0.77 0.58 0.52 P = 17.6 mmHg corresponds to the simulations in Phase Three 1 48 No dramatic differences can been seen in Table 6.9 as Pab * s decreased from 17.6mmHg to 15mmHg or to lOmmHg. In fact, i f anything, the f i t becomes s l i g h t l y worse as P is lowered. Thus, as was the case for the 10% burn, P-^ changes do not seem to have a s i g n i f i c a n t effect on the behaviour of the system and hence do not need to be consided in future simulations. The transient results predicted in th i s phase were so similar to those obtained in Phase Three and Phase Four that no plots w i l l be presented. 6.3.7 TRANSIENT RESPONSE INCLUDING CHANGES IN THE LYMPH FLOW  CHARACTERISTICS (PHASE SIX) Not much information about lymph flow after burns is available in the l i t e r a t u r e . Unlike the 10% burn, the lymph flow of intact tissues in the 40% burn might change as well. Four d i f f e r e n t cases of lymph flow change were simulated. These cases are: when t<3 hours and < when t<3 hours and < S - V - ^ ^ Q 1- ."*lb = 2 ^ l b o ' 2- * l b = 5 ^ l b o ' 1 49 * l b = 2 ^ l b o ' When t < 3 hours and < 2-V ^ ^Q * l s • (0.5e~ a t+1) •*lso ^Im = (0.5e" a t+1) •*lmo * l b = 5 ^ l b o ' When t < 3 hours and V - ^ < S ^ V ^ o * l s = (2.5e~ a t+1) ^lrn = (2.5e" a t+1) •*lmo ' ^ Ibo ' ^lmo a n d ^Iso a r e the i n i t i a l lymph flows in injured skin, muscle and intact skin, respectively. The parameter a i s a time constant which determines how long i t takes for the intact tissue lymph flows to return to normal. It i s assumed to be the same ( i . e . , a=0.231 h~ 1) as the time constant used to relax the transport c o e f f i c i e n t changes. After three hours, the lymph flow c h a r a c t e r i s t i c s in the injured skin are assumed to return to the normal rel a t i o n s h i p given by Equations 5.6a or 5.6b. The input perturbations and sum-of-squares values for the f i t s are l i s t e d in the Table 6.10. Two plots i l l u s t r a t i n g the consequences of lymph flow changes 2 and 3 above are shown in Figures 6.20 and 6.21, respectively. Table 6.10 SS as a Function of K , , K. , K and Lymph Flow C h a r a c t e r i s t i c s fo r 40% Burn (Upper Curve for P pi f r P = 17.6 mmHg, Phase Six) SS Normal F i r s t case Second case t h i r d case Fourth case A = 4.0 B = 10.0 C = -0.95 D = 0.0 0.52 0.60 0.96 0.62 0.97 A = 3.0 B = 8.0 C = -0.80 D = 0.0 0.58 0.69 1.05 0.72 1 .05 V = Normal corresponds the simulations in Phase Three 151 F i g u r e 6.21 S i m u l a t i o n f o r C h a n g i n g K p i D ' K p l s ' K p l m ' K f b ' K r D ' K f s ' K f m ' K r s ' K r m ' * l b ' * l s a n d X'lm i n 4 0 % B u r n (A=4.0, B=10.0, C=-0.95, D=0.0, Uppe r C u r v e f o r P., Se c o n d C o n d i t i o n f o r Lymph F l o w C h a r a c t e r i s t i c ) 0 2 4 6 0 2 4 6 TIME (hour) TIME (hour) 152 Figure 6.20 Simulation for Changing Kp^, K p i S ' Kplm' K f b ' K r b ' K f s K4 K. v l b ' v l s and Vlm i n 40? Burn fm' rs' "rm1 (A=4.0, B=10.0, C=-0.95, D=0.0, Upper Curve for Pfa, Third Condition for Lymph Flow Characteristic) Z t3 oo < u. 9 O o > 5 15 J? V) 11 o UJ C L O O 5 8 J, 6 UJ 2 2< < C O < 2 -T T — • 1 1 — 1 i 1 • i : ! i i i 1 -i i i . 2 I TIME (hour) 350 O E, Z 280 2t> OH ZD uo z z o a. 70 0 15 2 13 00 3 11 .26 < lo 2 2 < _1 C L O ^ u 10 2 * TIME (hour) 1 53 According, to Table 6 . 1 0 , the more the lymph flow in the injured skin increases, the worse is the f i t of the predicted versus experimental r e s u l t s . For example, Figure 6 . 2 0 shows that, when the lymph flow in the injured skin is 5 times i t s normal value in the f i r s t three hours postburn, the plasma and injured skin volumes predicted both change much less s i g n i f i c a n t l y from their i n i t i a l values. This is because the rate of return of the f l u i d from the injured skin to the plasma through the lymphatics i s too rapid such that neither the volume of the injured skin or the volume of plasma demonstrate the expected changes. Table 6 . 1 0 also indicates that changing the lymph flow rate in the intact tissue has l i t t l e e f fect on the o v e r a l l f i t s . For example, the SS results for the second case (where only changes) are almost i d e n t i c a l to the fourth case (where V-^, and V^ m a l l change). It therefore seems reasonable to continue using the o r i g i n a l lymph flow c h a r a c t e r i s t i c s for the 40% burn. 6 . 3 i 8 MOST REASONABLE FIT FOR A 40% BURN The parameter changes which appear to y i e l d the most reasonable results for a 40% surface area burn are the following: 1. The plasma leak c o e f f i c i e n t in the injured skin increases. The i n i t i a l magnitude of the change A = 4 . 0 leads to the "best" f i t . 1 54 2. The tissue pressure in the injured skin declines dramatically for 3 hours postburn. The upper curve yi e l d s a more reasonable f i t than does the middle curve. 3. The plasma leak c o e f f i c i e n t in the two intact tissues increases to an even greater extent than in the injured skin. The magnitude of the change for both intact tissues is taken to be B = 10.0. 4. The f i l t r a t i o n and reabsorption c o e f f i c i e n t s in the injured skin are reduced to 5% of their normal values (C = -0.95). 6.3.9 LONG TERM TRANSIENT RESPONSE In the simulations of the 40% burn discussed thus far, the a r t e r i a l and venous c a p i l l a r y pressures have been changed from their normal values. A l l of the other changes have been designed so that they w i l l return to normal within about 12 hours postburn. In order to find the time required for the system to restore i t s e l f , P a and P v were set to their normal values after either 3, 6 or 9 hours postburn. The long term transient responses based on one of the best sets of conditions from Phase Three was then obtained. Figures 6.22 - 6.24 show how the injured system returns to normal a f t e r a 40% burn using these reasonable parameter changes from the e a r l i e r investigation and c a p i l l a r y pressure changes at 3, 6 and 9 hours, respectively. In 1 55 Figure 6.22 Steady State Result for 40% Burn (A=4.0, B=10.0, C=-0.95, D=0.0, Upper Curve for P^, A r t e r i a l and Venous Cap i l l a r y Pressure Return to Normal After Three Hours) 0 12 24 36 0 12 24 36 TIME (hour) TIME (hour) 1 56 Figure 6.23 Steady State Result for 40% Burn (A=4.0, B=10.0, C=-0.95, D=0.0, Upper Curve for P^, A r t e r i a l and Venous Capillary Pressure Return to Normal After Six Hours) E 6 ill < 2 CO I 12 24 TIME (hour) 36 350 o» E Z 280 CO 3 210 — 140 y 70 -o Q_ 0 15 Z 13 to Q DC =5 ~ T o > 5 30 .26 < to < _ l CL Z « Q_ O <-> u 10 12 24 TIME (hour) 36 1 57 Figure 6.24 Steady State Result for 40% Burn (A=4.0, B=10.0, C=-0.95, D=0.0, Upper Curve for P b, A r t e r i a l and Venous Capillary Pressure Return to Normal After Nine Hours) Z 13 to o 11 < U . 9 Mr o £ 6 2 Z) _ l < to < 2 -T _L T 1 r 12 24 TIME (hour) 350 36 11 L_ 9 o 2 _J O > 5 30 .26 < to < 2 1 8 O. O <-> U 10 12 24 TIME (hour) 36 1 58 these figures, the dashed l i n e s indicate the normal values for each parameter. In burn experiments such as those by Lund and Reed [45], rats subjected to a 40% injury are not expected to survive beyond 4-6 hours without f l u i d resuscitation. In the simulations shown in Figures 6.22 - 6.24, since no resuscitation is provided, i t is recognized that the longer-term results are not only very speculative but also highly u n r e a l i s t i c . Nonetheless, these longer-term transients can be used to better understand the workings of the model. For example, perhaps the most interesting feature of these figures when compared to the corresponding 10% burn results, i s that most of the system variables return to normal in a shorter period, from 12 - 24 hours postburn. These results are consistent with the explanation given e a r l i e r for the the 24 - 36 hour recovery times predicted for the 10% burn. In the present case, because the injured skin edema i s s i g n i f i c a n t l y less, the tissue compliance is able to increase the lymph flow s u f f i c i e n t l y to more rapidly relieve the f l u i d and protein buildup. 6.3.10 IMPLICATIONS OF THE 40% BURN MODEL Based on the most reasonable f i t for the 40% burn summarized in Section 6.3.8, the simulation results obtained for a l l dependent parameters are l i s t e d in Appendix D for a 1 59 period of 0-6 hours postburn. The transient responses of several of the more important model variables are plotted in Figures 6.25-6.32. Only the short-term behaviour of rats subjected to 40% burns is discussed in d e t a i l since, without resuscitation, a l l such animals are known to succumb at longer times. Figures 6.25 and 6.26 reveal how f l u i d and plasma proteins, respectively, r e d i s t r i b u t e themselves amongst the four compartments immediately following a 40% burn injury. The two figures indicate that a l l of the tissue compartments i n i t i a l l y accumulate both f l u i d and proteins. This behaviour i s quite d i f f e r e n t than for a 10% burn where only the injured skin compartment acts as a sink for both f l u i d and plasma proteins, while the two intact tissue compartments act as sources to make up for the losses from the plasma compartment. Consequently, in the 40% burn, the plasma volume and protein content are far more depressed than in the 10% case. However, the f l u i d and protein accumulation in the injured skin i s s t i l l much more prevalent than in the intact tissues. Also, as was the case for the 10% burn, the changes in muscle and intact skin occur in p a r a l l e l . Thus, only the behaviour of one of these intact tissue compartments, the noninjured skin, w i l l be examined in d e t a i l . 1 60 Figure 6.25 The Changes of Compartmental F l u i d Volumes Following a 40% Burn Time (hour) J 163 F i g u r e 6.28 The Changes, of A l b u m i n T r a n s p o r t R a t e s i n I n t a c t S k i n F o l l o w i n g a 40% Burn 40-, , 164 F i g u r e 6.29 The Changes of C a p i l l a r y and I n t a c t S k i n P r e s s u r e s F o l l o w i n g a 4C% Burn 1 6-5 F i g u r e 6.30 The Changes of F l u i d Flow Rates in Injured Skin F o l l o w i n g a 40% Burn 1.6 - i 1 1.4-1 r 1 1 :—r 0 1 2 3 4 5 Time (hour) 166 Figure 6.31 The Changes of Albumin Transport Rates in Injured Skin Following a 40% Burn 45 0 1 2 3 4 5 6 Time (hour) Figure 6.32 The Changes of C a p i l l a r y and Injured Sk Pressures following a 40% Burn 168 Figure 6.27 and 6.28 portray the time-dependent changes in the intact skin f l u i d and protein fluxes, respectively, which occur after the i n i t i a t i o n of a 40% burn. Figure 6.29 shows the behaviour of the c a p i l l a r y and intact skin hydrostatic and c o l l o i d osmotic pressures during the same period. The i n i t i a l changes to the intact skin are due to the large increase of the plasma leak c o e f f i c i e n t , Kpi s» which is 11 times i t s normal value at t=0. As a consequence of t h i s increase, both the f l u i d and protein fluxes associated with the plasma leak r i s e sharply. The increased protein influx raises the protein content of the intact skin. However, the plasma leak f l u i d flux i s tempered to some extent by an i n i t i a l drop in the f i l t r a t i o n rate due to the step decline in P a. Partly because of the l a t t e r effect and partly because the protein concentration in plasma i s much higher than that of any tissue, C g and n s also begin to r i s e . Similar changes occur in muscle (where Kp^ m i s also raised by a factor of 11) and in injured skin (where Kpib increases 5 times). Consequently, the plasma loses s i g n i f i c a n t amounts of f l u i d and proteins (the l a t t e r more quickly than the former) and both C D and n decline. The rapid loss of plasma f l u i d also causes a precipitous f a l l in P... Largely because of the behaviour of P , the plasma leak f l u i d and protein fluxes in intact skin decrease rapidly for the f i r s t 30 minutes. After t h i s time, since P y is not 1 69 allowed to descend below 2 mmHg (the large vein pressure), the plasma leak fluxes f a l l off more slowly due to the exponential decrease in K p i s ' Simultaneously, the f i l t r a t i o n rate r i s e s from an i n i t i a l l y depressed value and the reabsorption rate, after a short i n i t i a l increase, begins to f a l l . Both of these changes can be attributed to a decrease in the c o l l o i d osmotic pressure difference, I l p - n s . The i n i t i a l r i s e in V r s is associated with the i n i t i a l rapid f a l l in P v. Over the f i r s t half hour, because Vp- s^ f a l l s more rapidly than the changes in and V r s can compensate for, the intact skin f l u i d volume actually goes through a maximum and then decreases s l i g h t l y . However, after 30 minutes, when ^ s decreases at a slower rate, the ri s e in and the f a l l in V r s begin to dominate, and the volume of the tissue begins to increase once again. As the f l u i d content of the intact tissue compartment becomes large, the i n t e r s t i t i a l pressure P g begins to r i s e causing a corresponding increase in the lymph flow f l u i d and protein fluxes. Similar lymph flow changes in the other two tissues causes a restoration of proteins in the plasma compartment which, after about 3 hours has elapsed postburn, begins to raise i t s concentration and consequently i t s c o l l o i d osmotic pressure. It is clear from Figure 6.26, that the protein return flow from the injured skin compartment is primarily responsible for this phenomenon. Once n begins to increase, V"£S declines and V r s rises and, 1 70 since both changes reduce f l u i d input to the intact tissue, the compartmental volume begins to f a l l . S i m i l a r l y , because the protein outflux due to the lymph flow overwhelms the protein influx due to the plasma leak at just over 3 hours, the compartmental protein content also begins to decline. If the intact skin transients were obtained for a longer period of time, the l a t t e r changes would eventually return the tissue to i t s normal conditions. Figures 6.30, 6.31 and 6.32 show the volumetric flow rates, albumin transport rates and hydrostatic and c o l l o i d osmotic pressures, respectively, associated with the injured tissue compartment following a 40% burn. The explanation for the events which take place after this more extensive injury i s very similar to that already described in d e t a i l for the 10% burn (see Section 6.2.8). In essence, the combination of the augmented plasma leak c o e f f i c i e n t and the i n i t i a l l y lowered tissue hydrostatic pressure increases the plasma leak and f i l t r a t i o n rates, reverses the reabsorption rate and shuts down the returning lymph flow. Because there are e s s e n t i a l l y only inflows of f l u i d and proteins during the f i r s t two hours postburn, the f l u i d volume and protein content of the injured skin build up much more rapidly than either of the two intact tissues, accounting for the major losses from the plasma. However, once the reduced tissue pressure i s relieved and the a l t e r a t i o n s in the transport c o e f f i c i e n t s decay, the plasma leak f a l l s , reabsorption 171 again becomes positive, lymph of the increase in the tissue begins to return to normal. flow r i s e s steadily (because volume) and the injured tissue The major differences in the behaviour of the injured skin after a 40% burn as compared to i t s responses to a 10% burn are the following: 1. In order to adjust the protein concentration in the injured tissue to suit the experimental data, the i n i t i a l f i l t r a t i o n and reabsorption c o e f f i c i e n t s , and K^, were depressed even more strongly in the present case. As a consequence, the f i l t r a t i o n and reabsorption rates were affected less by the very low tissue pressures which occurred over the f i r s t 2 hours postburn and these two mechanisms therefore played a much less important role in potential edema formation. 2. Primarily because the magnitude of the plasma leak c o e f f i c i e n t change was much smaller than in the 10% case, the maximum percentage increase in the injured skin volume and protein content was considerably less than before. 3. Much larger changes in the plasma volume and protein content occurred over the f i r s t 3 hours postburn. These responses were primarily due to the fact that the injured skin compartment is 4 times larger in the present case and hence, is able to absorb greater 1 72 quantities of both f l u i d and protein. The losses from the plasma compartment were aggravated further because, during the f i r s t few hours of the simulation, there was a net transfer of both materials to the two intact tissue compartments as well as to the injured tissue compartment. These large decreases in plasma volume and protein concentration produced large changes in the plasma hydrostatic and c o l l o i d osmotic pressures, both in directions which helped restore the microvascular exchange system more quickly to normal. 4. For the various reasons described above, the injured skin does not become as edematous as in the 10% burn case. As a consequence, the f l u i d volume in th i s compartment is always s u f f i c i e n t l y low that the lymph flow can react through tissue compliance to quickly relieve the accumulation of f l u i d . Thus, as was discussed in Section 6.3.9, the entire system takes less time to return to normal than in the 10% burn case. Chapter 7 CONCLUSIONS Based on a s t a t i s t i c a l f i t t i n g of the model simulations to the experimental data on thermally injured rats by Lund and Reed [45], the model predictions can be summarized as follows for 10% and 40% burns, respectively. 1. For the 10% burn, the two most important perturbations to the system are an increase in the plasma leak c o e f f i c i e n t and a reduction in the tissue pressure of the injured skin compartment. By changing only these two parameters, the model predicts reasonable short-term and long-term transient results compared to the available experimental data. A decrease in the injured skin f i l t r a t i o n and reabsorption c o e f f i c i e n t s (attributed to a decrease in the tissue perfusion rate) allows for an adjustment of the protein concentration but otherwise has only a moderate effect on the res u l t s . The other parameters investigated such as the the a r t e r i a l c a p i l l a r y pressure and the lymph flow in the burned tissue, do not appear to have an important influence on the simulations. A l l the changes and predictions made in Phase One to Phase Four of the 10% burn simulation are in reasonable agreement with the experimental data presented in Tables 3.1, 3.2 and 3.3. The best results from Phase Two, Three and Four not only 1 73 174 produce very good f i t s to the experimental data of Lund and Reed [45], but also y i e l d the correct trends as compared to the measurements of other investigators compiled in Table 3.3. The simulation results reveal that following a 10% burn, the large negative tissue pressure causes a reversal in reabsorption and a zero lymph flow during f i r s t few hours postburn. Because of thi s lack of drainage and also because of a very large increase in plasma leakage during t h i s period, there i s an i n i t i a l massive influx of f l u i d (causing edema) and proteins to the injured tissue. Due to compliance considerations, the tissue edema tends to l i m i t the longer-term lymph flow. As a consequence, i t takes a r e l a t i v e l y long time (24-36 hours) to reli e v e the f l u i d buildup and restore the system to normal. 2. For the 40% burn, the experimental results indicate that the system i s more complex. Hence, to adequately describe i t s behaviour, more changes had to be made to the model. The four important perturbations required to obtain a reasonable f i t of the data were: an increase in K p l b ' a n i n c r e a s e i - n K p l s a n <^ Kplm' a decrease in and and an a l t e r a t i o n in injured tissue hydrostatic pressure. When these four parameters were changed, the model yielded a reasonable f i t of the short-term 1 75 experimental data of Lund and Reed [45] for a 40% burn. The other parameter changes, including the a r t e r i a l c a p i l l a r y pressure, the intact tissue f i l t r a t i o n and reabsorption c o e f f i c i e n t s and the lymph flow in a l l three tissues did not s i g n i f i c a n t l y influence the f i t . As was the case for 10% burn, the changes made and the results simulated are also in general agreement with the trends indicated by other investigators and summarized in Tables 3.1-3.3. Compared to the 10% burn, the plasma and intact tissue compartments no longer act as inexhaustable sources of f l u i d and protein in t h i s case. In fact, the plasma loses material to muscle and intact skin as well as injured skin. As a consequence of the limited volume of the plasma compartment and the large changes i t is forced to undergo, there i s considerably less accumulation of f l u i d and protein (on a re l a t i v e basis) in the injured skin compartment. Because of th i s lower volume (and i t s location on the compliance curve), lymph flow more e a s i l y a l l e v i a t e s the f l u i d accumulation leading to a much faster speculated return to normal. It i s recognized that the experimental data with which the simulation results were compared are very sparse and have large associated errors. Thus, the present model for 10% and 40% burns in rats should be considered as being very 176 preliminary. When more experimental information, p a r t i c u l a r l y about the effects of burns on tissue properties, become available, the model can be more extensively tested and fine-tuned. In the meantime, i t is believed that the present model provides a more rigorous representation of the microvascular changes following a burn injury than does any simulation model previously reported in the l i t e r a t u r e . Chapter 8 RECOMMENDATIONS In order to better quantify the s p e c i f i c perturbations needed for the burn model, v e r i f y the model predictions and analyze the pathological changes that occur after burns, the following additional experimental data are required: a. more complete and longer-term data on the f l u i d , albumin content and c o l l o i d osmotic pressure in each compartment after thermal i n j u r i e s , b. independent measurements of the changes in transport c o e f f i c i e n t s which take place after burns, c. more accurate measurement of the injured tissue hydrostatic pressure d i s t r i b u t i o n , d. measurement of the plasma perfusion rate to each tissue compartment before and after burns, and e. determination of the lymph flow changes in each tissue due to burn i n j u r i e s . The simulation for postburn f l u i d resuscitation could be ea s i l y incorporated into the present model by adding appropriate sink and source terms to each compartment. Experimental information for resuscitation in thermally injured rats is anticipated in the near future (personal communication with R. Reed, 1987). One future development of the model would be to subdivide the injured skin compartment in the present 1 77 1 78 model into two compartments: a central and an outer burn compartment. The central burn compartment incorporates the area of actual tissue necrosis whose size would depend on the severity of the injury. This compartment would receive no d i r e c t perfusion of blood but would communicate with the rest of the system through the outer burn compartment. The l a t t e r compartment surrounds the ce n t r a l compartment and consists of less extensively-injured skin which receives some di r e c t perfusion. It i s expected that such a model could provide a more r e a l i s t i c representation of the true nature of injured t i s s u e . Another p o t e n t i a l l y f r u i t f u l development of model would be to include a description of the small ion transport in a l l four compartments. There exists hypothetical evidence that large amount of ionic material i s released from the damaged c e l l s after thermal i n j u r i e s [18]. If this i s true, the assumption of small ion equilibrium in our model no longer holds and there may be a large osmotic pressure gradients caused by non-equilibrium differences in small ion concentrations between compartments. Inclusion of ionic transport in the present model requires four additional mass balances (one for each compartment) for each ionic species. Since c e l l u l a r water makes up a s i g n i f i c a n t fraction of rat skin, another possible extension to the model would be to include a c e l l u l a r subcompartment in the injured 1 79 and intact skin compartments and allow for f l u i d , protein and small ion exchange between this subcompartment and the interstitium. NOMENCLATURE A constant in Equations 6.1 and 6.4 B constant in Equations 6.2, 6.3, 6.5 and 6.6 BSA f r a c t i o n a l burn area C constant in Equations 6.7 and 6.8 C concentration of albumin (mg/ml) CA available concentration of albumin (mg/ml) COP c o l l o i d osmotic pressure (mmHg) CVP central venous pressure (mmHg) D constant in Equations 6.9, 6.10, 6.11 and 6.12 F n function of K transport c o e f f i c i e n t (ml/mmHg-h) MAP main a r t e r i a l pressure (mmHg) P hydrostatic pressure (mmHg) PS permeability times surface area (ml/h) Q albumin content (mg) 0 albumin transport rate (mg/h) R f r a c t i o n a l pressure drop of the blood vessel SL lymph flow s e n s i t i v i t y (ml/mmHg«h) SS sum-of-squares of differences between predicted and normalized experimental results V volume V volumetric flow rate of f l u i d (ml/hr) VE volume exclusion (ml) a time constant in Equations 6.1 to 6.12 (h" 1) n c o l l o i d osmotic pressure (mmHg) a protein r e f l e c t i o n c o e f f i c i e n t 180 SUBSCRIPTS a a r t e r i a l c a p i l l a r y aa artery b injured skin d d i f f u s i v e e excluded volume f f i l t r a t i o n i i n t e r s t i t i u m 1 lymph m muscle o normal steady-state value p plasma pi plasma leak r reabsorption s intact skin v venous c a p i l l a r y vv large vein 181 REFERENCES 1. 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Brouhard, B.H., Carvajal, H.F. & Linares, H.A., "Burn Edema and Protein Leakage in the Rat", Microvascular Research 15: 221-228, 1978. 40. Carvajal, H.F., Linares, H.A. & Brouhard B.H., "Relationship of Burn Size to Vascular Permeability Changes in Rats", Surgery, Gynecology & Obstetrics 149: 193-202, 1979. 1 86 41. Delming, R.H., Mazess, R.B. & Wolberg, W. , "The Effect of Immediate and Delayed Cold Immersion on Burn Edema Formation and Resorption", The Journal of Trauma 19(1): 56-60, 1979. 42. Delming, R.H., Trunkey, D. & Way, L., "Burns and Other Thermal Injuries", in Way, L. (ed): Current Surgical  Diagnosis and Treatment, p 233, Lange Medical Publications, Los Altos, C a l i f o r n i a (1983). 43. Demling, R.H., Kramer, G. & Harms, B., "Role of Thermal Injury-Induced Hypoproteinemia on Fluid Flux and Protein Permeability in Burned and Nonburned Tissue", Surgery 95(2): 136-144, 1984. 44. Arturson, G. & Jakobsson, O.P., "Edema Measurements in a Standard Burn Model", Burns 12: 1-7, 1985. 45. Lund, T. & Reed, R.K., "Microvascular F l u i d Exchange Following Thermal Skin Injury in the Rat: Changes in Extravascular C o l l o i d Osmotic Pressure, Albumin Mass, and Water Content", Circulatory Shock 20: 91-104, 1986. 46. Gabel, J.C. & Drake, R.E., "Plasma Proteins and Protein Osmotic Pressure", in Staub, N.C. & Taylor, A.E. (ed): Edema, pp 371-382, Raven Press, New York (1984). 47. Wielderhielm, C.A., "Dynamics of Capil l a r y F l u i d Exchange: A Nonlinear Computer Simulation", Microvascular Research 18: 48-82, 1979. 48. Bert, J.L. & Pinder, K.L., "An Anolog Computer Simulation Showing the Effect of Volume Exclusion on C a p i l l a r y F l u i d Exchange", Microvascular Research 24: 94-103, 1982. 49. Bush, J.W., Schneider, A.M., Wachtel, T.L. & Brimm J.E., "A Simulation Analysis of Plasma Water Dynamics and Treatment in Acute Burn Resuscitation", The Journal of Burn Care 7(2): 86-94, 1986. 50. Reed, R.K. & Wiig, H., "Compliance of the I n t e r s t i t i a l Space in Rats: Studies on Hindlimb Skeletal Muscle", Acta Physiology Scandinavian 113: 307-315, 1981. 187 51. Wiig, H. & Reed, R.K., "Compliance of the I n t e r s t i t i a l Space in Rats: Studies on Skin", Acta Physiology Scaninavian 307: 315, 1981. 52. Bowen, B.D., U.B.C., Department of Chemical Engineering' Internal Report, 1986. 53. Reed, R.K. & Wiig, H., "Compliance of the I n t e r s t i t i a l Space in Rats: Contribution of Skin and Skeletal Muscle I n t e r s t i t i a l F l u i d Volume to Changes in Total E x t r a c e l l u l a r F l u i d Volume", Acta Physiology Scandinavian 121: 57-63, 1984. 54. Reed, R.K., " I n t e r s t i t i a l F l u i d Volume, C o l l o i d Osmotic and Hydrostatic Pressures in Rat Skeletal Muscle: Effect of Venous Stasis and Muscle A c t i v i t y " , Acta Physiology Scandinavian 112: 7-17, 1981. 55. Fadnes, H.O, "Effect of Increased Venous Pressure on the Hydrostatic and C o l l o i d Osmotic Pressure in Subcutaneous I n t e r s t i t i a l F l u i d in Rats: Edema-Preventing Mechanisms", Scandinavian Journal of C l i n i c a l Laboratory Investigation 36: 371-377, 1976. 56. Carnahan, B., Luther, H.A. & Wilkes J.O., Applied  Numerical Methods, John Wiley & Sons Inc. (1969). APPENDIX A: COMPLIANCE CURVES FOR RAT MUSCLE AND SKIN 1. Table A.1 Compliance Relationship for Skin [6] V s(ml) Ps(mmHg) 5.92 -6.95 8.88 -5.40 1 1 .85 -3.85 14.81 -2.30 16.88 -1.21 17.77 -0.75 18.51 -0.38 19.25 -0.07 19.99 0.19 20.73 0.35 21 .47 0.44 22.21 0.50 22.95 0.53 23.69 0.55 26.65 0.60 29.61 0.63 32.58 0.65 51 .82 0.69 81.44 0.73 106.62 0.77 188 Table A.2 Compliance Relationship for Muscle [6] V m(ml) m Pm(mmHg) 2.60 -6.78 5.21 -4.69 7.81 -2.60 10.41 -0.51 11.06 -0.13 11.71 0.07 1 2.36 0.18 13.01 0.26 1 3. 66 0.33 14.31 0.38 1 4. 96 0.42 1 5.62 0.45 1 6. 92 0.49 18.22 0.50 20.82 0.51 26.03 0.54 31.23 0.57 36. 44 0.60 46.82 0.65 3. Figure A.1 Compliance Curves for Muscle and Skin of Rats [6] 2 I i i i i | i i i i | i i i ' | i i i i CD zn. _g I > ' ' ' ' ' ' • • I • • • ' 1 i ; 0 10 20 30 40 INTERSTITIAL FLUID VOLUME (ml) APPENDIX B: COMPUTER PROGRAM The computer program used to predict the transient responses of the four compartment microvascular exchange system after various burn perturbations is l i s t e d on the following pages. Documentation about the purpose of and procedures used in each routine are presented throughout the program l i s t i n g in the form of comment statments. 191 1 92 C The main program: C T h i s program i s used to p r e d i c t how the p r o t e i n C and f l u i d r e d i s t r i b u t e themselves w i t h time f o l l o w i n g C a burn h a v i n g an a r e a f r a c t i o n BSA. C IMPLICIT REAL*8(A-H,K,L.0-Z) REAL *4 RK1 ,RK2,RK3.RK4,RK5.RK6.RK7 INTEGER FLAG COMMON/BLKA/PAO,PVO,VES,VEM,VESI C0MM0N/BLKB/VPLO.QPLO.VSO.VSOI.QSO.QSOI.VMO.QMO C0MM0N/BLKC/APL1,APL2,APL3.AS1,AS2,AS3.AS1I,AS2I,AS3I,AM1,AM2, 1 AM3 COMMON/BLKL/KFSI,KPLSI,KRSI,LSOI,SLSI,PSOI,PSSI COMMON/BLKM/FSI,VPLSI.RSI.LSI,QPLSI,ODSI.OLSI COMMON/BLKG/FS,VPLS,RS,LS.OPLS,QDS,OLS COMMON/BLKH/FM,VPLM,RM,LM.OPLM,ODM,OLM COMMON/BLKI/KFS,KPLS,KRS,LSO,SLS,PSO,PSS COMMON/BLKJ/KFM,KPLM,KRM,LMO,SLM,PMO,PSM COMMON/BLKK/PA,PV,CPL.PI PL,CS,CAVS,PS,PIS.CM,CAVM,PM.PIM, 1 CSI.CAVSI.PSI,PISI COMMON/BLKZ/DEGREE COMMON/BLKS/AA,BB COMMON/BLKW/K1,K2,K3,K4,K5.K6.K7 DIMENSION Y0LD(8),YNEW(8),VT(501),VPL(501),VS(501),VM(501) . 1 0PL(501),OS(501),0M(501),T(501),VSI(501),OSI(501) DIMENSION S0LN(5O,5O1),SOL(50),DIS(3,6),DIST(6) EXTERNAL RHSF DATA NV,NP.DT.EPS,EPSR,ALPHA/8,51,0.2500,1.D-5,1.D-7,.500/ DATA RDLS.RFRS.SLPS/.ODO,.314900,1.2651D0/ DATA RDLM,RFRM,SLPM/.ODO,.4001D0,1.20S5D0/ DATA RDLSI,RFRSI,SLPSI/.ODO..3149DO,1.2S51DO/ DATA TKFM,TKRM,TKPLM,TSLM,TLMO,TKFS,TKRS,TKPLS,TSLS,TLSO/ 1 0.0634DO.O.1585D0.0.0145D0,1.2065D0.0.2603DO, 2 0.0927DO.O.2943D0.0.0272DO,1.2651D0.0.422DO/ TSTART = 1 .D-1*DT TMIN=1.D-4*DT TMAX = DT NPM=NP-1 C C D e t e r m i n a t i o n of the i n i t i a l v a l u e s of volume, e x c l u d e d volume C p r o t e i n c o n t e n t i n the s k i n compartments C VSOI=DEGREE*VSO VSO=(1.DO-DEGREE)*VSO 0SOI=DEGREE*0SO 0SO=(1.D0-DEGREE)*OSO VESI=DEGREE*VES VES=(1.DO-DEGREE)*VES C CALL SPLINS CALL SPLINM C C E s t i m a t i o n of the t r a n s p o r t c o e f f i c i e n t s i n each t i s s u e C compartment C C 1) i n t a c t s k i n compartment C CPLO=QPLO/VPLO PIPLO=CPLO*(APL1+CPL0*(APL2+CPL0*APL3)) 193 cso=oso/vso CAVSO=QSO/(VSO-VES) PSO=FC0MPS(VSO/(1.DO-DEGREE ) ) PISO=CAVSO*(AS1+CAVSO*(AS2+CAVSO*AS3)) LSO=(0.02500*VSO*(CPLO-CAVSO) )/(RDLS*CS0+CPL0-CAVSO) PSS=0.025DO*VSO-LSO VPLS0=LS0*CS0*(1.D0-RDLS)/CPL0 KPLS=VPLSO/(PVO-PSO) KRS=(LS0-VPLS0)/(RFRS*(PA0-PS0-PIPL0+PIS0)-(PS0-PV0-PIS0+ 1 PIPLO)) KFS=RFRS*KRS SLS=(1-DEGREE)*SLPS C C 2) i n j u r e d s k i n compartment C CPLOI=0PLO/VPLO PIPLO=CPLOI*(APL1+CPLOI*(APL2+CPL0I* APL3)) CSOI=0SOI/VSOI CAVSOI=0SOI/(VSOI-VESI ) PSOI=FC0MPS(VSOI/DEGREE) PISOI=CAVSOI*(AS11+CAVSOI*(AS21+CAVSOI*AS31 )) LSOI = (0.025DO*VSOI*(CPLOI-CAVSOI ) )/(RDLSI*CSOI+CPLOI-CAVSOI) PSS1=0.02500*VSOI-LSOI VPLSOI=LSOI*CSOI*(1.DO-RDLSI)/CPLOI KPLSI=VPLSOI/(PVO-PSOI) KRSI = (LSOI-VPLSOI)/(RFRSI*(PAO-PSOI-PIPLO+PISOI )-1 (PSOI-PVO-PISOI+PIPLO)) KFSI=RFRSI*KRSI SLSI=DEGREE*SLPSI C C 3) muscle compartment C CMO=0MO/VMO CAVMO=0MO/(VMO-VEM) PMO=FCOMPM(VMO) PIMO=CAVMO*(AM 1+CAVMO*(AM2+CAVM0*AM3)) LM0=(0.025D0*VM0*(CPL0-CAVM0))/(RDLM*CMO+CPLO-CAVMO) PSM=0.025DO*VMO-LMO VPLMO=LMO*CMO*(1.DO-RDLM)/CPLO KPLM=VPLMO/(PVO-PMO) KRM=(LMO-VPLMO)/(RFRM*(PAO-PMO-PIPLO+PIMO)-(PMO-PvO-PIMO+ 1 PIPLO)) KFM=RFRM*KRM SLM=SLPM C C Input the i n i t i a l v a l u e s of dependent v a r a i a b l e s f o r the e i g h t C d i f f e r e n t i a l e q u a t i o n s C YNEW(1)=VPL0 YNEW(2)=QPL0 YNEW(3)=VS0 YNEW(4)=QS0 YNEW(5)=VM0 YNEW(6)=0MO YNEW(7)=VS0I YNEW(8)=0SOI C C S o l v e the d i f f e r e n t i a l e q u a t i o n s u s i n g C R u n g e - K u t t a - F e h l b e r g method C DO 30 1=1,NP T ( I ) = ( I - 1 ) * D T A v o i d t he d i s c o n t i n u o u s p o i n t s IF (I.E0.9) T(I)=1.999DO IF (I.E0.10) T(1-1)=2.001D0 IF (I.EO.13) T(I)=2.999D0 IF (I.EQ.14) T(I-1)=3.001D0 I F ( I . E O . I ) GO TO 16 IM=I-1 CALL DESOLV(RHSF,NV,T(IM),T(I),YOLD,EPS,TSTART,TMIN,TMAX, YNEW,FLAG) IF (FLAG.EO.0.0) GO TO 80 GO TO 16 CALL R00T(RHSF,NV,0.DO.YOLD.EPSR.ALPHA,YNEW,FLAG) IF(FLAG.EO.O) GO TO 60 CALL AUXS(T(I),YNEW) CALL AUXM(T(I),YNEW) CALL AUXSI(T(I),YNEW) S t o r e the i n t e r m e d i a t e r e s u l t s SOLN(1 . 1 ) =T(I) S0LN(2,I ) = CM S0LN(3,I) =CAVM S0LN(4,I) = PIM S0LN(5,I) = PM S0LN(6,I) = FM SOLN(7,I) = VPLM S0LN(8,I) = RM S0LN(9,I) = LM SOLN(10 , 1 )=0PLM S0LN(11 , 1 )=OLM SOLN(12,I )=ODM SOLN(13 , 1 )=CS SOLN(14,I )=CAVS SOLN(15,I )=PIS SOLN(16,I )=PS SOLN(17,I ) = FS SOLN(18,I )=VPLS SOLN(19 , 1 )=RS S0LN(20,I ) = LS S0LN(21,I )=OPLS S0LN(22,I )=OLS S0LN(23,I )=0DS S0LN(24,I )=PA S0LN(25,I )=PV S0LN(26,I )=PIPL S0LN(27,I )=CPL S0LN(28,I )=CSI S0LN(29,I )=CAVSI SOLN(30.I )=PISI S0LN(31,I )=PSI S0LN(32,I ) = FSI S0LN(33,I )=VPLSI S0LN(34,I )=RSI S0LN(35.I ) = LSI S0LN(36,I )=0PLSI 195 20 30 92 91 C C C C 93 94 95 40 S0LN(37.1)= QLSI S0LN(38,I)=0DSI VPL(I )=YNEW(1) QPL(I)=YNEW(2) VS(I)=YNEW(3) 0S(I)=YNEW(4) VM(I)=YNEW(5) 0M(I)=YNEW(6) VSI(I)=YNEW(7) QSI(I)=YNEW(8) C a l c u l a t e the sum of square s of d i f f e r e n c e s between the s i m u l a t i o n and e x p e r i m e n t a l r e s u l t s IF(T(I).EQ.1.0D0) IF(T(I).EO.1.0D0) I F ( T ( I ) . EQ.1.ODO) I F ( T ( I ) . I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) I F ( T ( I ) EO EO EO EO EO EO. 1 EO. 1 DIS(1, DIS( 1 DIS(1, DIS( 1 , DIS( 1 , 1) =((VPL(I)-4.17D0)/6.12DO)**2 2) =((S0LN(26,I)-13.4D0)/20.0D0)**2 3) =((S0LN(30.I)-9.25D0)/9.0D0)**2 4) =((VSI(I)-7.852DO)/e.75200)**2 5) =((VS(I)-10.400)/10.128D0)**2 1.ODO) 1.ODO) 1.ODO) DISC 1,6)=((QSI(I)-120.37D0)/111.4D0)**2 1.999D0) 1.999D0) 999D0) 99900) DIS(2. 1 ) DIS(2.2) DIS(2,3) DIS(2.4) DIS(2,5) DIS(2.6) DIS(3 DIS(3 DIS(3 DIS(3 DIS(3 1 ) = 2) = 3) = 4) = 5) = . EO- 1.999D0) .EO.1.999D0) •E0.2.999DO) .E0.2.999DO) IF(T(I).EO.2.999D0) IF ( T (I ) .EO.2.999D0) IF(T(I).EQ.2.999DO) IF(T(I).E0.2.999D0) DIS(3,6) DO 20 0=1,NV YOLD(d)=YNEW(J) CONTINUE CONTINUE DISTA=O.ODO DO 91 1=1,6 DIST(I)=0.ODO DO 92 d=1 ,3 DIST(I)=DIST(I)+DIS(J,I ) CONTINUE DISTA=DISTA+DIST( I ) CONTINUE (VPL(I)-4.17DO)/6.12DO)**2 (S0LN(26,I)-17.6D0)/20.0D0)**2 (SOLN(30,I)-10.47DO)/9.ODO)**2 (VSKD-7 .43DO)/6.752DO)**2 (VS(I)-11.35DO)/10.128DO)**2 (OSI(I)-204.1D0)/111.4DO)**2 (VPL(I)-4.13DO)/6.12DO)**2 (S0LN(26,I)-14.20O)/2O.ODO)* *2 (SOLN(30.I)-8.D0)/9.0D0)**2 (VSI(I)-7.93D0)/6.752D0)**2 (VS(I)-1 1 . 15D0)/10. 128D0)**2 (0SI(I)-1O5.4DO)/111,4D0)**2 P r i n t the var i a b l e s output r e s u l t s : dependent and supplemental f o r each compartment and the sum of squares WRITE(6,93) F0RMAT(/,8X.'1'.10X. WRITE(G,94) ( D I S T ( I ) FORMAT(//,2X,6F10.5) WRITE(6.95) DI STA FORMAT*/,2X, 'DI STANCE = WRITE(4,40) F0RMAT(//1X,'Transient 5X,'Time',6X,'VPL',8X, 8X, 'OM'.8X, 'OSI ' ,/) WRITE(4,50) ( T ( I ) . V P L ( I ) OSI(I).1=1,NP) 2' .8X, ' 1=1,6) 10X , . 9X , .8X, '6' ) . F 10 . 5 ) re s p o n s e f o r 40% burn',///, VS',8X,'VM',7X.'VSI',8X.'OPL' ,8X,'OS' VS(I ) ,VM(I),VSI(I),OPL(I),0S( I ) ,0M( I ) . 1 96 50 FORMAT(1X,5F10.5,4F10.4) WRITE(4, 55) VPL(NP),VS(NP),VM(NP) ,VSI(NP),QPl(NP),QS(NP),QM(NP) 1 ,OSI(NP) 55 FORMAT(1X,' I n f i n i t y ',4F10.5,4F10.4) WRITE(4,2G) 26 FORMAT(//,2X , 'TIME' .7X, 'CM' , 1 1X , 'CAVM' ,10X, ' PIM' , 12X, ' PM') WRITE(4,32)((SOLN(J,I),d=1,5),1=1,NP) 32 FORMAT(1X.F5.2.4E14.5) WRITE(4.17) 17 F0RMAT(///,2X, 'TI ME'.7X, 'FM', 12X, 'VPLM'. 11X. 'RM' . 12X . 'LM', 1IX, 1 'OPLM',1OX,'QLM',11X,'ODM') WRITE(4,13)(S0LN(1.1).(SOLN(J,I).J=6,12),I=1,NP) 13 FORMAT(1X.F5.2.7E14.5) WRITE(4.19) 19 FORMAT(//,2X , 'TIME' .7X, 'CS ' . 1 IX, 'CAVS', 10X, 'PIS', 12X, 'PS') WRITE(4.14)(S0LN(1,1),(SOLN(J,I).J=13,1G ),I=1.NP) WRITE(4.18) 18 FORMAT(///,2X,'TIME',7X,'FS' , 12X,'VPLS', 1 1X, 'RS',12X. 'LS',11X, 1 ' QPLS ' , 10X , ' QLS ' , 1 1X , ' ODM' ) 14 FORMAT( 1X.F5.2.4E14.5) WRI T E ( 4 . 2 5 ) ( S 0 L N ( 1 , 1 ) . ( S 0 L N ( J , I ) , J =17.23),I=1.NP) 25 FORMAT( 1X.F5.2, 7E14..5) WRITE(4,63) 63 FORMAT(//,2X, 'TIME' ,7X, 'CSI ' ,10X, 'CAVSI',9X, 'PISI', 11X, 'PSI ' ) WRITE(4,64) (SOLN( 1 ,I ) ,(SOLN(J,I ) ,J=28,31),1 = 1,NP) 64 FORMAT(1X,F5.2,4F14.5) WRITE(4.61) 61 FORMAT(//,2X, 'TIME',7X, 'FSI', 1 1X, 'VPLSI', 10X, 'RSI' , 11X, 'LSI', 1 10X, 'QPLSI',9X, 'QLSI' , 10X, 'QDSI' ) WRITE( 4 , 6 2 ) ( S 0 L N ( 1 , I ) , ( S O L N ( J , I ) , J = 32,38),I = 1.NP ) 62 FORMAT(1X,F5.2,7E14.5) WRITE(4,21) 21 FORMAT(///,2X, 'TIME',7X, 'PA ' , 12X.'PV',12X, 'PI PL'.11X,'CPL') WRITE(4,22)(SOLN(1,1),(SOLN(d,I ) , J = 24,27),I = 1,NP ) 22 FORMAT(1X.F5.2.4E14.5) C C P l o t the r e s u l t s of s i x v a r i a b l e s : volume of the i n t a c t s k i n , C p r o t e i n c o n t e n t i n the i n j u r e d s k i n , COP i n the i n j u r e d s k i n , C volume of the i n j u r e d s k i n , volume of plasma and C COP i n plasma C RK1=K1 RK2=K2 RK3=K3 RK4=K4 RK5=K5 RK6=K6 RK7=K7 CALL CAP(RK1,RK2.RK3,RK4,RK5,RK6,RK7) CALL PVPL(T.VPL.NP) DO 234 1=1,51 SOL(I)=S0LN(26.I) 234 CONTINUE CALL PCOPP(T,SOL,NP) DO 235 1=1,51 SOL(I)=SOLN(30,I) 235 CONTINUE CALL PCOPII(T,SOL,NP) DO 236 1=1,51 SOL(I ) =SOLN( 15 . 1 ) 1 97 236 CONTINUE CALL PTTWI(T,VSI,NP) CALL PTTW(T,VS.NP) CALL PQI(T,QSI,NP) CALL PLOTND STOP 60 WRITE(5,70) 70 FORMAT(1X,'Root-solver f a i l s ! ' ) 80 WRITE(5,90) T ( I ) 90 FORMAT(1X,'ODE-solver f a i l s a t T =',F8.4) STOP END C C E v a l u a t e the r i g n t - h a n d - s i z e f u n c t i o n s of the e i g h t C d i f f e r e n t i a l e q u a t i o n s C DOUBLE PRECISION FUNCTION RHSF(I,X,Y) IMPLICIT REAL*8(A-H,L,0-Z) COMMON/BLKG/FS,VPLS,RS,LS,QPLS,QDS,OLS COMMON/BLKH/FM,VPLM,RM,LM,OPLM,QDM,QLM COMMON/BLKM/FSI,VPLSI,RSI,LSI.OPLSI.QDSI.OLSI DIMENSION Y(8) GO TO (10,20,30.40.50,60,70.80),I 10 CALL AUXS(X,Y) CALL AUXM(X.Y) CALL AUXSI(X.Y) RHSF3=FS+VPLS-RS-LS RHSF5=FM+VPLM-RM-LM RHSF7=FSI+VPLSI-RSI-LSI RHSF=-RHSF3-RHSF5-RHSF7 RETURN 20 RHSF4=QPLS+QDS-0LS RHSF6=QPLM+QDM-QLM RHSF8=QPLSI+QDSI-0LSI RHSF=-RHSF4-RHSF6-RHSF8 RETURN 30 RHSF=RHSF3 RETURN 40 RHSF=RHSF4 RETURN 50 RHSF=RHSF5 RETURN 60 RHSF = RHSF6 RETURN 70 RHSF=RHSF7 RETURN 80 RHSF = RHSF8 RETURN END C C C a l c u l a t e the a u x i l i a r y r e l a t i o n s h i p s f o r s k i n C SUBROUTINE AUXS(X.Y) IMPLICIT REAL*8(A-H,K,L,0-Z) COMMON/BLKA/PAO,PVO,VES,VEM,VESI COMMON/BLKC/APL1.APL2,APL3,AS 1 ,AS2.ASS,AS 11,AS2I.AS3I. 1 AM 1,AM2,AM3 COMMON/BLKM/FSI,VPLSI,RSI.LSI,QPLSI,QDSI,QLSI COMMON/BLKG/FS,VPLS,RS,LS.QPLS.QDS,QLS COMMON/BLKI/KFS,KPLS.KRS,LSO,SLS,PSO.PSS 1 9 8 COMMON/BLKK/PA,PV.CPL,PI PL,CS.CAVS,PS,PIS.CM,CAVM.PM,PIM, 1 CSI,CAVSI ,PSI ,PISI COMMON/8LKZ/DEGREE DIMENSION Y(8) COMMON/BLKS/AA,BB COMMON/BLKW/K1,K2.K3,K4,K5.K6,K7 KFS=K5*0.0556150*DEXP(-AA*X)+0.055615D0 KPLS=K3*0.01632DO*DEXP(-AA*X)+0.O1632D0 KRS=K5*0.176613D0*DEXP(-AA*X)+0.176613D0 LSO=0.2532DO*(1 .DO+0.ODO*DEXP(-AA*X ) ) SLS=0.759OO0O*(1 .DO+0.ODO*DEXP(-AA*X )) PA=FCOMPA(Y( 1 ) ) PV = FCOMPV(Y( 1 ) ) CPL=Y(2)/Y( 1 ) PIPL = CPL*(APL1+CPL*(APL2+CPL*APL3) ) CS=Y(4)/Y(3) CAVS=Y(4)/(Y(3)-VES) PS=FC0MPS(Y(3)/(1.DO-DEGREE)) PIS=CAVS*(AS1+CAVS*(AS2+CAVS*AS3) ) FS=KFS*(PA-PS-(PI PL-PIS ) ) VPLS=KPLS*(PV-PS) RS=KRS*(PS-PV-(PIS-PI PL)) LS=LS0+SLS*(PS-PS0) IF(LS.LT.LSO) LS=LS0*(PS+7.84D0)/(PS0+7.84D0) IF(LS.LT.O.DO) LS=0.D0 0PLS=VPLS*CPL ODS=PSS*(CPL-CAVS) QLS=LS*CS RETURN END C C C a l c u l a t e the a u x i l i a r y r e l a t i o n s h i p s f o r i n j u r e d s k i n C SUBROUTINE AUXSI(X.Y) IMPLICIT REAL*8(A-H.K,L.O-Z) COMMON/BLKA/PAO,PVO.VES.VEM,VESI C0MM0N/BLKC/APL1,APL2,APL3,AS 1.AS2,AS3,AS 11.AS2I.AS3I, 1 AM 1,AM2,AM3 COMMON/BLKL/KFSI.KPLSI,KRSI,LSOI,SLSI,PSOI,PSSI COMMON/BLKM/FSI,VPLSI.RSI,LSI.QPLSI,ODSI.OLSI CDMMON/BLKG/FS,VPLS,RS,LS,OPLS,ODS,OLS COMMON/BLKI/KFS,KPLS.KRS,LSO,SLS,PSO,PSS COMMON/BLKK/PA,PV.CPL.PI PL,CS.CAVS.PS.PIS,CM,CAVM.PM,PIM. 1 CSI,CAVSI.PSI,PISI COMMON/BLKZ/DEGREE COMMON/BLKT/PSIT(10),TT(10).NPSIT COMMON/BLKS/AA,BB COMMON/BLKW/K1.K2,K3,K4,K5,K6,K7 DIMENSION Y(8) KFSI=K4*0.037077D0*DEXP(-AA*X)+0.03707700 KPLSI=K1*0.01088D0*DEXP(-AA*X)+0.01088D0 KRSI-K4*0.117742DO*DEXP(-AA*X)+0.117742D0 LSOI=0.1688D0*(1.D0+K7*DEXP(-BB*X)) SLSI=O.50604ODO*(1.D0+K7*DEXP(-BB*X)) PA = FPAI(Y(1) ) PV=FPVI(Y(1)) CPL=Y(2)/Y(1) 'PIPL=CPL*(APL1+CPL*(APL2+CPL*APL3)) CSI=Y(8)/Y(7) CAVSI=Y(8)/(Y(7)-VESI) D e t e r m i n a t i o n of the t i s s u e p r e s s u r e i n the f i r s t t h r e e hours in the i n j u r e d s k i n DO 10 I=2.NPSIT IF ( T T ( I ) . G T . X ) GOTO 20 CONTINUE PSI=PSIT(NPSIT) GOTO 30 IM=I-1 P S I = P S I T ( I M ) + ( P S I T ( I ) - P S I T ( I M ) ) * ( X - T T ( I M ) ) / ( T T ( I ) - T T ( I M ) ) IF (X.GE.3.D0) PSI=FC0MPS(Y(7)/DEGREE) PISI=CAVSI*(AS1I+CAVSI*(AS2I+CAVSI*AS3I)) FSI=KFSI*(PA-PSI-(PIPL-PISI ) ) VPLSI=KPLSI*(PV-PSI ) RSI=KRSI*(PSI-PV-(PISI-PIPl) ) LSI=LS0I+SLSI*(PSI-PS0I) I F ( L S I . L T . L S O I ) LSI=LS0I*(PSI+7.84D0)/(PS0I+7.84DO) IF(LSI.LE.O.DO) LSI=O.DO IF(LSI.LE.K7*LS0I.AND.X.LT.3.DO) LSI=K7*LSOI OPLSI=VPLSI*CPL QDSI=PSSI*(CPL-CAVSI ) OLSI=LSI*CSI RETURN END C a l c u l a t i o n of the a u x i l i a r y r e l a t i o n s h i p s f o r muscle SUBROUTINE AUXM(X.Y) IMPLICIT REAL*8(A-H.K,L.O-Z) COMMON/BLKA/PAO,PVO,VES,VEM.VESI COMMON/BLKC/APL 1 .APL2,APL3,AS 1.AS2,AS3.AS 11,AS21.AS3I, AM 1 ,AM2.AM3 COMMON/BLKH/FM,VPLM,RM,LM,QPLM,QDM,QLM COMMON/BLKJ/KFM,KPLM,KRM,LMO,SLM,PMO.PSM COMMON/BLKK/PA,PV,CPL,PI PL,CS.CAVS.PS,PIS,CM,CAVM.PM,PIM. CSI.CAVSI,PSI.PISI COMMON/BLKZ/DEGREE COMMON/BLKS/AA,BB COMMON/BLKW/K1,K2,K3.K4,K5,K6,K7 DIMENSION Y(8) KFM=K5*0.063446D0*DEXP(-AA*X)+0.063446D0 KPLM=K3*O.O14545D0*0EXP(-AA*X)+0.O14545DO KRM=K5*0.158575D0*DEXP(-AA*X)+0.158575DO LMO=0.26025DO*(1.DO+0.ODO*DEXP(-AA*X)) SLM=1.2065D0*(1.DO+O.0D0*DEXP(-AA*X)) PA=FC0MPA(Y(1)) PV=FCOMPV(Y(1)) CPL=Y(2)/Y(1) PIPL=CPL*(APL1+CPL*(APL2+CPL*APL3)) CM=Y(6)/Y(5) CAVM=Y(6)/(Y(5)-VEM) PM=FC0MPM(Y(5)) PIM=CAVM*(AM1+CAVM*(AM2+CAVM*AM3)) FM=KFM*(PA-PM-(PIPL-PIM)) VPLM=KPLM*(PV-PM) RM=KRM*(PM-PV-(PIM-PIPL)) LM=LM0+SLM*(PM-PM0) IF(LM.LT.LMO) LM=LM0*(PM+G.78D0)/(PM0+6.78D0) IF(LM.LT.0.DO) LM=0.D0 200 QPLM=VPLM*CPL QDM=PSM*(CPL-CAVM) OLM=LM*CM RETURN END C C E s t i m a t i o n of the compliance r e l a t i o n s h i p f o r s k i n C SUBROUTINE SPLINS IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKD/X(20),Y(20),A 1,BN,N,NM COMMON/BLKO/0(100),R(101).S(100) DIMENSION H(100),A(101),B(101).C(101),D(101) DO 10 1=1,NM 10 H ( I ) = X ( I + 1 ) - X ( I ) B( 1 )=2.DO*H(1) C(1)=H(1) D(1)=3.D0*((Y(2)-Y(1))/H(1)-A1) DO 20 I=2,NM IP=I+1 IM=I-1 A(I)=H(IM) B(I ) = 2.DO*(H(IM)+H(I)) C ( I ) = H ( I ) 20 D ( I ) = 3 . D 0 * ( ( Y ( I P ) - Y ( I ) ) / H ( I ) - ( Y ( I ) - Y ( I M ) ) / H ( I M ) ) A(N)=H(NM) B(N)=2.D0*H(NM) D(N)=-3.DO*((Y(N)-Y(NM))/H(NM)-BN) CALL TDMA(A.B,C,D.R.N) DO 30 1=1,NM IP=I+1 0 ( I ) = ( Y ( I P ) - Y ( I ) ) / H ( I ) - H ( I ) * ( 2 . D O * R ( I )+R(IP) )/3.DO 30 S ( I ) = ( R ( I P ) - R ( I ) ) / ( 3 . D 0 * H ( I ) ) RETURN END C C E s t i m a t i o n of the compliance r e l a t i o n s h i p f o r muscle C SUBROUTINE SPLINM IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKE/X(19),Y(19).A 1.BN.N.NM C0MM0N/BLKP/Q(100),R(101),S(100) DIMENSION H(100),A(101).B(101).C(101),D(101) DO 10 I=1,NM 10 H ( I ) = X ( I + 1 ) - X ( I ) B(1)=2.D0*H( 1 ) C(1)=H(1) D( 1)=3.DO*((Y(2)-Y(1))/H(1)-A1) DO 20 I=2,NM IP=I+1 IM=I-1 A(I)=H(IM) B(I)=2.DO*(H(IM ) +H(I) ) C ( I ) = H ( I ) 20 D( I ) = 3 . D O * ( ( Y ( I P ) - Y ( I ) ) / H ( I ) - ( Y ( I ) - Y ( I M ) ) / H ( I M ) ) A(N)=H(NM) . B(N)=2.DO*H(NM) D(N)=-3.D0*((Y(N)-Y(NM))/H(NM)-BN) CALL TDMA(A.B.C.D.R.N) DO 30 1=1,NM 201 I P = I + I 0 ( I ) = ( Y ( I P ) - Y ( I ) ) / H ( I ) - H ( I ) * ( 2 . D O * R ( I ) + R ( I P ) ) / 3 . D O 30 S ( I ) = ( R ( I P ) - R ( I ) ) / ( 3 . D 0 * H ( I ) ) RETURN END C C SUBROUTINE TDMA(A,B,C,D,X,N) C C Thomas a l g o r i t h m C IMPLICIT REAL*8(A - H,0-Z) DIMENSION A(N),B(N),C(N),D(N),X(N),P(1O1).0(1O1) NM=N-1 P(1)=-C(1)/B(1) 0 ( D = D ( 1 ) / B ( 1 ) DO 10 1=2.N IM=I-1 DEN=A(I)*P(IM)+B(I) P(I)=-C(I)/DEN 10 Q( I ) = (D( I ) - A ( I ) * Q ( I M ) ) / D E N X(N)=0(N) DO 20 11 = 1,NM I=N-II 20 X ( I ) = P ( I ) * X ( I + 1 ) + 0 ( I ) RETURN END C C C a l c u l a t i o n of the a r t e r i a l c a p i l l a r y p r e s s u r e i n the C normal t i s s u e s C DOUBLE PRECISION FUNCTION FCOMPA(V) IMPLICIT REAL*8(A - H,K,0-Z) COMMON/BLKA/PAO.PVO,VES,VEM,VESI COMMON/BLKB/VPLO.QPLO,VSO,VSOI.QSO.QSOI,VMO.QMO COMMON/8LKW/K1,K2,K3,K4,K5,K6,K7 FC0MPA=17.64D0 RETURN END C C C a l c u l a t i o n of the a r t e r i a l c a p i l l a r y p r e s s u r e i n the C i n j u r e d s k i n C DOUBLE PRECISION FUNCTION FPAI(V) IMPLICIT REAL*8(A - H,K,0-Z) COMMON/BLKA/PAO.PVO,VES,VEM.VESI COMMON/BLKB/VPLO.QPLO,VSO,VSOI.QSO.QSOI.VMO.QMO COMMON/BLKW/K1,K2.K3,K4.K5,K6.K7 FPAI=K6 RETURN END C C C a l c u l a t o r ! of the venous c a p i l l a r y p r e s s u r e i n the C normal t i s s u e s C DOUBLE PRECISION FUNCTION FCOMPV(V) IMPLICIT REAL*8(A - H.O-Z) COMMON/BLKA/PAO,PVO.VES.VEM,VESI COMMON/BLKB/VPLO,QPLO,VSO.VSOI,QSOI,QSO.VMO.QMO FC0MPV=4.72D0+5.1193D0*(V-VPLO) IF(FC0MPV.LT.2.DO) FC0MPV=2.D0 RETURN END C a l c u l a t i o n of the venous c a p i l l a r y p r e s s u r e i n the i n j u r e d s k i n DOUBLE PRECISION FUNCTION FPVI(V) IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKA/PAO,PVO,VES.VEM,VESI COMMON/BLKB/VPLO.QPLO,VSO,VSOI,0SOI,OSO.VMO.QMO FPVI=4.7200+5.1193D0*(V-VPL0) IF(FPVI.LT.2.D0) FPVI=2.D0 RETURN END E v a l u a t i o n of the compliance r e l a t i o n s h i p f o r s k i n : the s k i n f l u i d p r e s s u r e as a f u n c t i o n of s k i n volume DOUBLE PRECISION FUNCTION FCOMPS(V) IMPLICIT REAL*8(A-H,0-Z) IF(V.LE.16.88D0) GO TO 10 IF(V.GE.32.58D0) GO TO 20 FCOMPS=FS(V) RETURN FC0MPS=-1.21DO+0.523723DO*(V-16.88DO) RETURN FC0MPS=0.65D0+0.001621D0*(V-32.58D0) RETURN END E v a l u a t i o n of the com p l i a n c e r e l a t i o n s h i p f o r muscle: the muscle f l u i d p r e s s u r e as a f u n c t i o n of muscle volume DOUBLE PRECISION FUNCTION FCOMPM(V) IMPLICIT REAL*8(A-H.0-Z) IF(V.LE.10.41D0) GO TO 10 IF(V.GE.18.22D0) GO TO 20 FC0MPM=FM(V) RETURN FC0MPM=-O.51D0+0.802817D0*(V-10.41DO) RETURN FCOMPM=0.50DO+0.005239DO*(V-18.22DO) RETURN END DOUBLE PRECISION FUNCTION FS(Z) IMPLICIT REAL*8(A-H,0-Z) C0MM0N/BLKD/X(2O),Y(20),A 1,BN,N,NM COMMON/BLKO/Q(100),R(101).S(100) 1 = 1 I F ( Z .LT,X(1)) GO TO 30 IF(Z.GE.X(NM)) GO TO 20 d = NM K=(I+J)/2 I F ( Z . L T . X ( K ) ) d=K IF(Z.GE.X(K)) I=K IF(J.EQ.1+1)' GO TO 30 203 GO TO 10 20 I=NM 30 DX=Z-X(I) FS = Y(I )+DX*(0(I)+DX*(R(I )+DX*S( I ) ) ) RETURN END C C DOUBLE PRECISION FUNCTION FM(Z) IMPLICIT REAL*8(A-H.O-Z) COMMON/BLKE/X( 19),Y(19),A 1,BN,N,NM COMMON/BLKP/Q(100),R(101).S(100) 1 = 1 I F ( Z . L T . X ( 1 ) ) GO TO 30 IF(Z . GE.X(NM)) GO TO 20 d = NM 10 K=(I+J)/2 I F ( Z . L T . X ( K ) ) J = K IF(Z.GE.X(K)) I=K IF(O.EQ.1+1) GO TO 30 GO TO 10 20 I=NM 30 DX=Z-X(I) FM=Y(I)+DX*(0(I) + DX*(R(I)+DX*S(I))) RETURN END C C T h i s s u b r o u t i n e i s f o r s o l v i n g the o r d i n a r y d i f f e r e n t i a l C e q u a t i o n s u s i n g R u n g e - K u t t a - F e h l b e r g method C SUBROUTINE DESOLV(F,M,A,B,YO,EPS.HSTART,HMIN,HMAX,YB.FLAG) IMPLICIT REAL*8(A-H,0-Z) DIMENSION YO(M),YA(10),YB(M) EXTERNAL F INTEGER FLAG BMA=B-A HOLD=HSTART X = A DO 10 1 = 1 ,M 10 YA(I)=YO(I) 20 CALL RKF(F,M,X,YA,HOLD,YB,YDIF) GAMMA=.800*(EPS*H0LD/(BMA*YDIF))**O.2500 HNEW=GAMMA*HOLD IF(GAMMA.GE.1.DO) GO TO 30 IF(HNEW.LT.HOLD/10.DO) HNEW=HOLD/10. IF(HNEW.LT.HMIN) GO TO 50 HOLD=HNEW GO TO 20 30 IF(HNEW.GT.5.D0*H0LD) HNEW=5.DO*H0LD IF(HNEW.GT.HMAX ) HNEW=HMAX IF(X+HOLD.GE.B) GO TO 70 X=X+H0LD H0LD=HNEW DO 40 1=1,M 40 YA(I)=YB(I) GO TO 20 50 . FLAG=0 B = X DO 60 1=1,M 60 YB(I)=YA(I) 204 RETURN 70 FLAG= 1 HSTART=HNEW H0LD=B-X CALL RKF(F,M,X,YA,HOLD,YB,YDIF) RETURN END C C C SUBROUTINE RKF(F,M,X,YOLD.H.YNEW,YDIFM) IMPLICIT REAL*8(A-H,0-Z) DIMENSION YOLD(M) ,YNEW(M),YARG1(10),YARG2(10) REAL*8 K1(10),K2(10),K3(10).K4(10),K5(10),K6(10) DATA C21,C31, C32,C33/.25D0,.375D0,.0937500,.2812500/ DATA C41,C42/.923076923076923073DO,0.879380974055530257D0/ DATA C43,C44/-3.27719617660446061DO,3.32089212562585323DO/ DATA C51,C52/2.0324O74O74O74O7220O,-8.DO/ DATA C53.C54/7.17348927875243647D0,-.20589668615984405D0/ DATA C61.C62/.5DO.-.296296296296296294DO/ DATA C63,C64/2.00,- 1.381676413255360500/ DATA C65,C66/.452972709551656916D0.-.275DO/ DATA C71,C72/.118518518518518509D0..518986354775828454D0/ DATA C73.C74/.506131490342016654D0,-.18D0/ DATA C75.C81/.0363636363636363636D0,.00277777777777777778DO/ DATA C82.C83/-.0299415204678362568D0,-.0291998936735778838D0/ DATA C84,C85/.02DO,.0363636363636363636DO/ DO 10 1=1,M K1(I )=H*F(I,X,YOLD) 10 YARG1(I )=YOLD(I)+C21*K1(I) XARG=X+C21*H DO 20 1=1,M K2(I)=H*F(I,XARG,YARG1) 20 YARG2(I)=YOLD(I) +C32*K1(I)+C33*K2(I ) XARG=X+C31*H DO 30 I=1,M K3(I)=H*F(I,XARG.YARG2) 30 YARG1(I)=YOLD(I)+C42*K1(I)+C43*K2(I )+C44*K3(I ) XARG=X+C41*H DO 40 1=1,M K4(I)=H*F(I,XARG,YARG1) 40 YARG2(I) = YOLD(I)+C51*K1(I)+C52*K2(I )+C53*K3(I) + C54*K4(I) XARG=X+H DO 50 I=1,M K5(I)=H*F(I.XARG.YARG2) 50 YARG1(I)=YOLD(I) + C62*K1(I)+C63 *K2(I)+C64 *K3(I)+C65 *K4(I) 1 +C66*K5(I) XARG=X+C61*H DO 60 1=1,M 60 K6(I)=H*F(I,XARG,YARG1) YDIFM=O.DO DO 70 1 = 1 ,M YNEW(I)=Y0LD(I)+C71*K1(I)+C72*K3(I)+C73*K4(I)+C74*K5(I)+ 1 C75*K6(I) YDIF = DABS(C81*K1(I) + C82*K3(I)+C83*K4(I )+C84*K5(I )+C85*K6(I)) 70 IF(YDIF.GT.YDIFM) YDIFM=YDIF RETURN END C C P l o t t i n g of plasma volume 205 SUBROUTINE PVPL(X,Y,NP) IMPLICIT REAL*8(A-H,0-Z) REAL*4 RX(51),RY(51),XE(3),YE(3),DE1(3).BAR 1,BAR2.BAR3.BAR4 REAL*4 TX,TY,TDE1 ,TDE2,DE2(3) DIMENSION X(51 ) . Y(51) DATA XE/1.0.2.0.3.0/ DATA YE/4.17,4.17.4.13/ DATA DE1/0.564,0.156,0.639/ DATA DE2/0.505.0.154,0.57/ CALL PLCTRL('METRIC',1) CALL PALPHA('SANSERIF.2 ',0) CALL FTNCMD('ASSIGN 7=-SCR;') CALL AXES(4.7,7.2,6.,1.0.2,0. ,2.0,- 1.'TIME (hour)',11,6.. 1 0.75,2,0.0,2.0.-1.'PLASMA VOLUME (ml)',18,O.25) DO 10 1=1,NP RX(I)=X(I) IF (RX(I).GT.6.0) 11=1-1 IF (RX(I).GT.6.0) GOTO 11 RY(I)=Y(I) 10 CONTINUE 11 CALL DLINE1(RX.RY,II,0.0.4.7,1.0,0.0,7.2,1.3333,0.1,0.1. 1 0.0,0.005,-12.0,18.0,1.0,'40 PER BURN',11,0.3,100.) DO 20 1=1,3 TX=XE(I)+4.7 TY=YE(I)/1.3333+7.2 TDE1=DE1(I)/1 .3333 TDE2=DE2(I)/1.3333 CALL SYMBOL(TX.TY,0.1,4,1) BAR 1=TY+TDE 1 BAR2=TY-TDE2 BAR3=TX+0.1 BAR4=TX-0.1 CALL PL0T(BAR4.BAR1,3) CALL PL0T(BAR3,BAR1,2) CALL PL0T(TX,BAR1 ,3) CALL PL0T(TX,BAR2,2) CALL PL0T(6AR4,BAR2,3) CALL PL0T(BAR3.BAR2.2) 20 CONTINUE RETURN END C C P l o t t i n g of COP i n plasma C SUBROUTINE PCOPP(X,Y.NP) IMPLICIT REAL*8(A-H,0-Z) REAL*4 RX(51),RY(51).XE(3).YE(3),DE(3),BAR 1,BAR2,BAR3,BAR4 REAL*4 TX,TY,TOE DATA XE/1:0,2.0.3.0/ DATA YE/13.4,17.6,14.2/ DATA DE/2.27,0.595,1.495/ DIMENSION X(51 ) ,Y(51 ) CALL AXES(13.5,7.2,6. ,1.0,2,0. ,2.0,-1. 'TIME ( h o u r ) ' , 1 1 .6 . , 1 0.6,2,10.0,4.0,-1,'COP IN PLASMA ( m l ) ' 2 ,18,0.25) DO 10 1=1,NP RX(I)=X(I) IF (RX(I).GT.6.0) 11=1-1 IF (RX(I).GT.6.0) GOTO 11 RY(I )=Y( I ) CONTINUE CALL DLINE1(RX, RY,II,0.0,13.5,1.0,10.0,7.2,3.333,0.1,0.1, 1 0.0,0.005.-12.0,19.0.1.0.'40 PER BURN',11.0.3.100.) DO 20 1=1,3 TX = XE(I ) + 13.5 TY=(YE(I)-10.0)/3.333+7.2 TDE=DE(I)/3.333 CALL SYMBOL(TX.TY,0.1,4,1) BAR 1 =TY + TDE BAR2=TY-TDE BAR3=TX+0.1 BAR4=TX-0.1 CALL PL0T(BAR4,BAR 1,3) CALL PL0T(BAR3,BAR1,2) CALL PL0T(TX,BAR1,3) CALL PL0T(TX,BAR2,2) CALL PL0T(BAR4,BAR2,3) CALL PL0T(BAR3,BAR2,2) CONTINUE RETURN END P l o t t i n g of COP i n the i n j u r e d s k i n SUBROUTINE PCOP11(X,Y,NP) IMPLICIT REAL*8(A-H.O-Z) REALM RX(51),RY(51 ) ,XE(3),YE(3),DE(3) REAL*4 BAR 1 ,BAR2 . BAR3,BAR4.TX.TY,YDE DATA XE/1.0,2.0,3.0/ DATA YE/9.248,10.47,8.0/ DATA DE/2.064,1.653,0.917/ DIMENSION X(51 ) .Y(51 ) CALL XAXIS(4. 7. 13.5,6.0,0.01, 1.0,0.15,2.2,0.0,2.0,0.0, 1 0.0.-1,'.',1,0.0,0.0,2,6.0) CALL YAXIS(4.7,13.5,6.0,0.01,0.6,0.15,2,2,5.0.2.0,0.1875, 1 0.1875,-1.'COP IN INJURED SKIN (mmhg)',26,0.25,0.1875,2,6.0) DO 10 1=1,NP RX(I)=X(I) IF (RX(I).GT.6.0) 11=1-1 IF (RX(I).GT.6.0) GOTO 11 RY(I)=Y(I) CONTINUE CALL DLINE1(RX,RY,II.0.0,4.7,1.0,5.0,13.5,1.6667,0.1.0.1, 1 0.0,0.005,- 12.0, 18.0. 1.0, '40 PER BURN' , 11,0.3. 100.) DO 20 1=1,3 TX = XE(I ) + 4 . 7 TY=(YE(I)-5,0)/1.6667+13.5 TDE=DE(I)/1.6667 CALL SYMBOL(TX,TY,0.1,4,1) BAR 1 =TY + TDE BAR2=TY-TOE BAR3=TX+0.1 BAR4=TX-0.1 CALL PL0T(BAR4.BAR1.3) CALL PL0T(BAR3,BAR1,2) CALL PL0T(TX,BAR1 ,3) CALL PL0T(TX,BAR2,2) CALL PL0T(BAR4,BAR2,3) CALL PL0T(BAR3,BAR2.2) 207 20 10 1 1 20 CONTINUE RETURN END P l o t t i n g of water c o n t e n t i n the i n j u r e d t i s s u e YE(3) ,DE(3) TX.TY.TDE 01.1.0.0.15,2.2.0.0.2.0,0.0, 11=1-1 GOTO 11 II,0.0.13.5.1.,5.0,13.5,1.667,0.1,0.1, 10 PER BURN',11,0.3,100.) SUBROUTINE PTTWI(X,Y,NP) IMPLICIT REAL*8(A-H.O-Z) REAL*4 RX(51 )'.RY.(51 ),XE(3) REALM BAR1 , BAR2 . BAR3 , BAR4 DIMENSION X(51 ) ,Y(51) DATA XE/1.0,2.0,3.0/ DATA YE/7.852.7.43.7.93/ DATA DE/O.95,0.7587.0.683/ CALL XAXIS( 13.5, 13.5,6.0,0 0.0,-1,'.',1,0.0,0.0.2,6.0) CALL YAXIS(13.5,13.5,6.0,0.01,0.6,0.15,2,2,5 O. 1875,-1 , 'VOLUME OF INJURED SKIN (ml)',27.0 DO 10 1=1,NP RX(I)=X(I) IF (RX(I).GT.6.0) IF (RX(I).GT.6.0) RY(I)=Y(I) CONTINUE CALL DLINE1(RX,RY, 0.0,0.005,-12.0,18.0,1.0,' DO 20 1=1,3 TX = XE(I ) + 13 . 5 TY=(YE(I)-5.0)/1.6667+13.5 TDE=DE(I)/1.6667 CALL SYMBOL(TX,TY.O BAR 1=TY + TDE BAR2=TY-TDE BAR3=TX+0.1 BAR4=TX-0.1 CALL PLOT(BAR4,BAR 1 CALL PLOT(BAR3,BAR 1 CALL PLOT(TX,BAR 1 ,3) CALL PL0T(TX,BAR2,2) CALL PL0T(BAR4,BAR2,3) CALL PL0T(BAR3,BAR2,2) CONTINUE RETURN END P l o t t i n g of water c o n t e n t i n the i n t a c t s k i n SUBROUTINE PTTW(X.Y.NP) IMPLICIT REAL*8(A-H,0-Z) REAL*4 RX(51),RY(51),XE(3),YE(3),DE(3) REAL*4 BAR1 .BAR2.BAR3,TX.TY.TDE DATA XE/1.0,2.0,3.0/ DATA YE/10.4,11.4.11.15/ DATA DE/1 .56, 1 .22,0.68/ DIMENSION X(51 ) ,Y(51 ) CALL XAXIS(4 . 7, 19 . 8,6.0.0.01 0.0,-1,'.',1.0.0,0.0,2,6.0) CALL YAXIS(4.7.19.8.6.0,0.01 O.1875,-1,'VOLUME OF INTACT DO 10 1=1,NP 0,2.0,0.1875. 25,O.1875,2,6.0) 1.4,1) .3) ,2) 1.0.0.15.2,2,0.0,2.0,0.0, ,0.6,0.15, SKIN ( m l ) ' 2,2, .26, 0,2.0,0.1875, 25.0.1875,2,6.0) 20& RX(I)=X(I) IF (RX(I).GT.6.0) 11=1-1 IF (RX(I).GT.6.O) GOTO 11 RY(I)=Y(I) 10 CONTINUE 11 CALL DLINE1(RX,RY,11.0.0,4 . 7. 1 .0.5.0. 19.8, 1 .6666667,0. 1.0.1. 1 0.0,0.005.-12.0,18.0.1.0,'10 PER BURN',11,0.3,100.) DO 20 1=1,3 TX=XE(I)+4.7 TY=(YE(I)-5.0)/1.6667+19.8 TDE=DE(I)/1.6667 CALL SYMBOL(TX,TY.O.1,4,1) BAR 1=TY+TDE BAR2=TY-TDE BAR3=TX+0.1 BAR4=TX-0.1 CALL PL0T(BAR4,BAR1 ,3) CALL PL0T(BAR3,BAR1,2) CALL PL0T(TX,BAR1,3) CALL PL0T(TX,BAR2,2) CALL PL0T(BAR4.BAR2.3) CALL PL0T(BAR3,BAR2.2) 20 CONTINUE RETURN END C C P l o t t i n g of p r o t e i n c o n t e n t i n i n j u r e d s k i n C SUBROUTINE POI(X,Y,NP) IMPLICIT REAL*8(A-H,0-Z) REAL*4 RX(51 ) ,RY(51),XE(3),YE(3),DE(3) REAL BAR 1,BAR2,BAR3.BAR4,TX,TY,TDE DATA XE/1.0,2.0,3.0/ DATA YE/120.37,204.1,105.4/ DATA DE/44.86.1 1 1.4,5.98/ DIMENSION X(51),Y(51) CALL XAXIS(13.5,19.8,6.0,0.01,1.0,0.15,2,2,0.0,2.0,0.0, 1 0.0.-1,'.',1,0.0,0.0.2,6.0) CALL YAXIS(13.5,19.8,6.0,0.01.0.6,0.15,2.2,0.0,70.0,0.1875, 1 O. 1875,-1 , 'PROTEIN IN INJURED SKIN (mg)',28.0.25,0. 1875.2,6.0) DO 10 I=1,NP RX(I)=X(I ) IF (RX(I).GT.6.0) 11=1-1 IF (RX( I ).GT.6.0) GOTO 11 RY(I)=Y(I) 10 CONTINUE 11 CALL DLINE1(RX,RY.11,0.0.13.5, 1.0,0.0.19.8.54.0000.0. 1 .0. 1 , 1 0.0,0.005,-12.0,18.0,1.0, '10 PER BURN', 1 1 ,0.3,100. ) DO 20 1=1.3 TX = XE(I ) + 13.5 TY=YE(I)/54.000+19.8 TDE=DE(I)/54.OOOO CALL SYMBOL(TX.TY,0.1,4,1) BAR 1 =TY + TDE BAR2=TY-TDE BAR3=TX+0.1 BAR4=TX-0.1 -CALL PL0T(BAR4,BAR1,3) CALL PL0T(BAR3,BAR1.2) CALL PL0T(TX,BAR1,3) 209 CALL PLOT(TX,BAR2,2) CALL PLOT(BAR4,BAR2,3) CALL PLOT(BAR3.BAR2.2) 20 CONTINUE C CALL PL0T(25.,0.0.-3) RETURN END C CALL DLINE1(RX.RY.NP.0.0.4.0.6.667.0.0,7.0,0.3333333.0. 1,0. 1 , C 1 0.0.0.02.12.0,18.0,1.0,'40 PER BURN'.11,0.3,100.) C C P l o t t i n g of the c a p t i o n i n the o u t p u t graphs C SUBROUTINE CAP(A,B.C,0,E,F,G) IMPLICIT REAL*4(A-H,0-Z) L0GICAL*1 CHAR(20) CALL PLCTRL('METRIC,1) CALL PALPHA('SANSERIF.2 ',0) CALL FTNCMD('ASSIGN 7=-SCR;') CALL PSYM(4.7.5.0.0.25,'KPLSI=',0.0.6) CALL CONVT(A,1,CHAR,NC) CALL PSYM(7.7,5.0.O.25,CHAR,0.0,NC) CALL PSYM(13.5,5.0,0.25.'Pif=',0.0,4) CALL PSYM(15.5,5.0.0.25,'top',0.0,3) CALL PSYM(4.7,4.2,0.25.'KPLS,KPLM=',0.0.10) CALL CONVT(C,1,CHAR,NC) CALL PSYM(7.7,4.2,0.25,CHAR.O.O,NC) CALL PSYM(13.5,4.2,0.25, 'KFSI,KRSI = ' ,0.0,10) CALL C0NVT(D,2,CHAR,NC) CALL PSYM(15.5,4.2.0.25,CHAR.O.O,NC) CALL PSYM(4.7,3.4,0.25,'KFM(S),KRM(S) = ',0.0,14) CALL CONVT(E,1.CHAR,NC) CALL PSYM(7.7,3.4,0.25,CHAR,0.0,NC) CALL PSYM(13.5,3.4,0.25,'PAI=',0.0,4) CALL CONVT(F,1,CHAR,NC) CALL PSYM(15.5,3.4,0.25,CHAR,0.0,NC) CALL PSYM(4.7,2.6,0.25.'LSI=',0.0,4) CALL CONVT(G,1,CHAR,NC) CALL PSYM(7.7,2.6.0.25,CHAR.O.O,NC) RETURN END APPENDIX C: SAMPLE CALCULATION For NORMALIZING EXPERIMENTAL DATA The experimental data used to compare with the simulation predictions are from the work of Lund and Reed [45]. These experimental results with th e i r standard deviations are tabulated in Tables C.1 and C .2 for 10% and 40% burn, respectively. The normal values measured in the experiments of Lund and Reed [45] and the normal values assumed by Bert et a l . (and also used as sta r t i n g values in the present simulation) are somewhat d i f f e r e n t . Therefore i t was necessary to normalize the experimental data before comparisons could be made. The c a l c u l a t i o n a l procedure used for t h i s purpose i s based on d i r e c t proportionality, i . e . , i f the normal values of the experiment and the simulation are U and V, respectively, and i f the experimental value i s W at any given time, then the normalized value X i s : X = VW/U C.1 Table C .2 and C.4 l i s t the normalized experimental data which are used as a basis of comparison with the simulation re s u l t s and which are also plotted in the output response 210 T a b l e C.1 Experimental Data f o r 10% Burn by Lund and Reed [45] I n i t i a l 60 min po s t b u r n •V (ml) 1.49±0.18 1.62±0.18 Q b (mg) 14.9+3.0 42.9+21.6 n b (mmhg) 11.2±2.2 15.1+3.4 ^ 1.78±0.22 3.06±0.75 b H e m a t o c r i t ( % ) ' 49.9+2.0 52.4+1.8 np(mmhg) 16.8±2.4 12.5+3.3 T a b l e C.2 Norm a l i z e d E x p e r i m e n t a l Data f o r 10% Burn I n i t i a l 60 min po s t burn V g ( m l ) 15. 19 16.52+1.84 Q. (mg) b 27.85 80.19+40.37 n b (mmhg) 9.00 10.79+2.45 V b 1 .69 2.90±0.71 V (ml ) P n (mmhg) 6. 12 5.41+0.40 20.00 15.43+4.07 p 120 min p o s t b u r n 180 min p o s t burn 1.57+0.21 1.53±0.04 41.7+.15.3 40.6±12.6 14.8+2.9 15.3+2.1 2.91+0.69 2.99+0.62 55.5+2.1 55.4±1.7 14.6±3.4 13. 9 +.3.0 120 min p o s t burn 180 min p o s t burn 16.00+2.14 15.60+0.41 77.94+28.60 75.89+23.55 12.69+2.49 12.87+1.77 2.76±0.65 2.84+0.59 4.87±0.43 4.73±0.34 17.8 + 4.15 15.53+.3.35 Table C.3 Experimental Data fo r 40% Burn by Lund and Reed [45] I n i t i a l Value 60 min post burn 120 min post burn 180 min post burn V (ml ) s C- (mg) b n b (mmhg) V (ml ) b H e m a t o c r i t s ) np(mmhg) 1.49*0.18 14.9+3.0 10.5+2.0 1.78+0.22 4 9 . 6 ± 1 . 8 18.9+3.0 1 .53±0.23 16.1+6.0 11.2+2.5 2.07+0.25 5 8 . 5 ± 3 . 1 12.4+3.4 1 .67±0 .18 27.3+14.9 11.4+1.8 1.96+0.20 5 9 . 0 ± 0 . 9 14.8+0.50 1 .64±0 .10 14.1+0.8 9.6+1.1 2.09+.0. 18 6 0 . 0 ± 3 . 5 15.2+1.6 Table C 4 Normalized Experimental Data f o r 40% Burn V (ml ) s Q. (mg) b n b (mmhg) V (ml ) b n (mmhg) P I n i t i a l Value 10.13 111.4 9.00 6.75 6. 12 20.00 60 min post burn 10.40+1.56 120.4+44.9 9.25+2.06 7 . 8 5 ± 0 . 9 5 4.17+0.564 13 .40±2 .27 120 min post burn 11.35+1.22 204. 1 + 111.4 10.47+1.65 7.43+0.76 4.17+0.156 17.62+.0.60 180 min post burn 11.15+0.68 105.4+6.0 8.00+0.92 7 . 9 3 ± 0 . 6 8 4.13+0.639 14.20+.1 . 50 213 graphs. These data are calculated from Equation C.1, except for the plasma volume ( V D) which is derived from the hematocrit data. A t y p i c a l example calculation i l l u s t r a t i n g the use of Equation C.1 is given below. Protein content in the injured skin one hour after a 40% burn: i n i t i a l experimental value: 14.9 mg (Table C.3), i n i t i a l simulation value: 111.4 mg (Table C.4), experimental value: 16.1 mg (Table C.3) normalized experimental value: (111.4)•(16.1)/(14.9) = 120.37 (mg) The plasma volumes were calculated from the measured hematocrit. By d e f i n i t i o n , the hematocrit i s the volume percentage of erythrocytes (red c e l l s ) in whole blood. Assume the red c e l l volume i s Y and that i t i s constant. Assume the i n i t i a l experimental hematocrit value and the value at a certain measuring time postburn are U and W, respectively, and the i n i t i a l volume of plasma in the simulation is V. Therefore, i f X i s now the normalized plasma volume, then by d e f i n i t i o n : U = Y/(Y+V) C.2 and W = Y/(Y+X) 214 C.3 Thus, when Y is eliminated by combining Equations C.2 and C.3, the following expression for the plasma volume is obtained, namely X = V-(1-W)-U/[W-(1-U)] C.4 For example, at two hours afte r the i n i t i a t i o n of a 40% burn: i n i t i a l experimental value of hematocrit: 49.5% (Table C.3), i n i t i a l simulation value of plasma: 6.12 ml (Table C.4), experimental value of hematocrit: 59% (Table C.3), normalized experimental value: 6.12-(1-0.59)-0.495/[0.59-(1-0.495)] = 4.17 (ml) A l l the normalized values of plasma volume are calculated in this manner. The normalized experimental value of the water content of the tissues cannot be derived d i r e c t l y from Equation C.1, since the t o t a l tissue water measured includes the i n t r a c e l l u l a r water. For example, l e t ' s look at the water 215 content in the intact tissue. After a 10% burn, assuming that there are no c e l l s damaged in the intact skin, the normalized values for tissue water content can be calculated in two ways: 1. Assume the i n t r a c e l l u l a r water can be ignored. The normalized values for tissue content in t h i s case are calculated from Equation C.1. These values are shown in Table C.3. 2. Assume the i n t r a c e l l u l a r water is 20% of the t o t a l tissue water, and there is no c e l l u l a r damage after the burn. The normal value of i n t e r s t i t i a l f l u i d volume from the experiment i s therefore (0.8)•(1.49ml) = 1.19(ml). If i t i s assumed also that the change in tissue water equals the change of t o t a l tissue water ( i . e . , c e l l u l a r water volume remains constant), then the value for the intact skin f l u i d volume after one hour becomes: 1.19ml+(1.62-1.49)ml = 1.32 (ml) S i m i l a r l y , after two hours the experimental result becomes: 1.i9ml+(1.57-1.49)ml =1.27 (ml) and after three hours the experimental result becomes: 1.19ml+(1.53-1.49)ml = 1.23 (ml). The normal value of skin i n t e r s t i t i a l f l u i d volume used in the simulations: 15.2 (ml). 216 Therefore, the normalized experimental value after one hour: (15.2)•(1.32)/(1.19) = 16.9 (ml), after two hours: (15.2)•(1.27)/(l.19) = 16.2 (ml), and after three hours: (15.2)•(1.23)/(1.19) =15.7 (ml) There is only a very small difference between the normalized experimental values in these two cases. Additionally, these differences are i n s i g n i f i c a n t compared to the experimental error. Because of these i n s i g n i f i c a n t differences and because the f r a c t i o n of i n t r a c e l l u l a r water in the tissue i s not precisely known for rat skin, the normalized experimental data for intact tissue volumes used for comparison with the simulation predictions are taken from Table C.4 which i s based on the assumption that the i n t r a c e l l u l a r water can be ignored. The analysis of the water content of the injured tissue could be more complicated than that for the intact skin since a f r a c t i o n of the c e l l s in the injured skin may be damaged afte r the burn. The normalized values of water content in the injured skin are calculated for three cases based: on three d i f f e r e n t assumptions. Using the 40% burn as an example: 1. Assume the i n t r a c e l l u l a r water can be ignored. The 2 1 7 normalized values in t h i s case can be calculated d i r e c t l y from Equation C.1 and are l i s t e d in Table C.4. 2. Assume the i n t r a c e l l u l a r water is 20% of the t o t a l tissue water and none of the c e l l s are damaged after the burn. The normal experimental value for skin i n t e r s t i t i a l water in the experiment: (0.8)•(1.78ml) = 1.42 ml. Assume that the change of tissue water equals the change of t o t a l tissue water, then the experimental result becomes after one hour: 1.42ml+(2.07-1.78)ml=1.71 ml, after two hours: 1.42ml+(1.96-1.78)ml=1.60 ml and after three hours: 1.42ml+(2.09-1.78)ml=1.73 ml. The i n i t i a l value used in the simulation i s 6.75 ml for intact skin. The normalized experimental values therefore become after one hour: (6.75)•(1.71)/(1.42) = 8.13 (ml), after two hours: (6.75)•(1.60)/(1.42) = 7.6 (ml) and after three hours: (6.75)•(1.73)/(1.42) = 8.22 (ml). 3. Assume that 20% of the t o t a l tissue water i s c e l l u l a r water, that a l l the c e l l s are damaged a f t e r the burn and the c e l l u l a r water disappears after the c e l l s are 2 1 8 damaged ( p o s s i b l y through exudation and e v a p o r a t i o n ) . In t h i s case, the experimental data a f t e r the burn only r e f l e c t the i n t e r s t i t i a l t i s s u e water content. The normal value of s k i n i n t e r s t i t i a l water of the experiment: (0.8)•(1.78ml) = 1.42 (ml). The experimental values are 2.07 ml a f t e r one hour, 1.96 ml a f t e r two hours and 2.09 ml a f t e r three hours. The s t a r t i n g v alue f o r i n j u r e d s k i n f l u i d volume used i n the s i m u l a t i o n f o r a 40% burn i n j u r y i s 6.75 ml. The normalized experimental values t h e r e f o r e become a f t e r one hour: (6.75)•(2.07)/(1.42) = 9.84 (ml), a f t e r two hours: (6.75)•(1.96)/(1.42) = 9.32 (ml) and a f t e r three hours: (6.75)•(2.09)/(1.42) = 9.93 (ml). The d i f f e r e n c e s between the normalized values c a l c u l a t e d i n Cases 1, 2 and 3 are small and i n s i g n i f i c a n t compared to the e r r o r s i n the experiment. I t i s l i k e l y that not a l l the c e l l s i n the i n j u r e d s k i n are damaged and f u r t h e r , the estimate of 20% i n t r a c e l l u l a r water i s b e l i e v e d to be high. Hence, Case 3 i s an extreme case and i t r e f l e c t s the l a r g e s t p o s s i b l e d i f f e r e n c e s i n i n j u r e d s k i n f l u i d volume obtained through the n o r m a l i z a t i o n procedure. Not enough i n f o r m a t i o n i s p r e s e n t l y a v a i l a b l e regarding the 219 proportion of the c e l l s damaged and the percentage of i n t r a c e l l u l a r water. Thus to simplify the problem and to eliminate questionable assumptions, the normalized values of the injured skin are calculated as shown in Case 1. These values for 10% and 40% burns are l i s t e d in Table C.2 and C.4, respectively. APPENDIX D Appendix D c o n t a i n s a complete l i s t i n g of the t r a n s i e n t responses of a l l v a r i a b l e s i n the burn model obtained using the most reasonable parameter changes f o r the 10% and 40% burns. For the 10% burn, the p e r t u r b a t i o n s a r e : A = 35.0, P b = Upper Curve, B=-0.5 and P a and P y r e t u r n to normal at 9 hours postburn. For the 40% burn, the p e r t u r b a t i o n s are: A = 4.0, P h = Upper Curve, B=10.0 and C=-0.95. 220 221 Transient response for 10% burn r i me VPL VS VM VSI OPL OS OM OS I 0 .0 6 .12000 15 .19200 10 .4 1000 1 . 68800 2 19 . 7080 250 .6500 134 . 3000 27 .8500 0 . 2S00G 6 .00146 15 .08145 10 .3 30G8 1 .99G4 1 2 IO . 1 180 250 .4275 134 . 1G90 37 . 7935 0 .50000 5 .798 13 14 .97750 10 .2585G 2 .37581 199 . 4926 249. . 9929 133 .9134 49 . 109 1 0 .75100 5 .G0125 14 .8G457 10 .18335 2 .76083 189 .63 t5 249 , . 3480 133 . 533G 59 . 9950 1 .00000 5 .49779 14 .75638 10 .112G2 3 .04320 183 .2112 248. .5482 133 .0620 67 .G86G 1 .25000 5 .42652 14 .68170 10 .06297 3 .23882 179 .3557 247 . .72 17 132 . 5748 72 .8558 t .50000 5 .3G123 14 .62802 10 .02627 3 .39448 176 .6358 246 . 9063 132 .094 1 76 .87 18 t .75000 5 .32722 14 .58496 9 .99606 3 .50176 175 .17 17 246. . 1016 13 1 .6197 79 . 6 149 2 .00000 5 .33391 14 . 55002 9 .97087 3 .55520 t75 .0289 245 . 3064 13 1 .1510 8 1 .02 17 2 .25000 5 .37287 11 .52241 9. .95028 3 .56445 175 . 9284 244 . 5 193 130 . 6873 8 1 . 3730 2 . 50100 5 .42882 14 .50048 9 .93331 3 .54738 177 .4350 243 . 7425 130 . 2299 8 1 . 1007 2 .7 5000 5 .50187 14 .48290 9 .91915 3 .50609 179 . 4992 242 . 972 1 129 .7765 80 . 2602 3 .OOIOO 5 .577 12 14 .47794 9 9 1269 3 .44225 182 .0269 242 . 2386 129 . 3454 78 . 897 1 3 .25000 5 .61444 14 48273 9 . 9 1 199 3 .40084 183 .9680 24 1 . .5670 128 .9513 78 .02 17 3 . 50O00 5 .65051 14 .48739 9 .91107 3 .36103 185 .8108 240. .9368 128 .58 19 77 . 1785 3 .750O0 5 .G84S t 14 49223 9 .91020 3 .32276 187 . 5586 240. . 3489 128 . 2379 76 . 3625 4 00000 5 .7 1745 14 .49732 9 .90946 3 .28577 189 . 2225 239 8012 127 .9181 75 . 5662 4 .25000 5 .74858 14 .50267 9 .90890 3 .24986 190 .81 16 239 . 29 18 127 .62 13 74 . 7833 4 . 50000 5 .77832 14 .50826 9 .90854 3 .21487 192 .3331 238 ,Bt91 127 . 3465 74 .0092 4 .75000 5 .80682 14 .51410 9 .9084 1 3 .18068 193 .7937 238 .3814 127 .0928 73 .2401 5 .00000 5 .83419 1 4 .52017 9 .90852 3 . 14712 195 . 1992 237 . .9773 126 .8594 72 .4721 5 .25000 5 .86055 14 .52649 9 .90889 3 .11407 196 . 5549 237 . .6053 126 .6454 7 1 . 7023 5 .50000 5 .88598 14 .53306 9 .90953 3 .08143 197 .8651 237 . .2644 126 .4501 70 .9283 S .75000 5 .91055 14 .53989 9 .91045 3 .04910 199 . 1337 23G . . 9534 12G . 2729 70 . 1480 6 .00100 5 .93425 14 .54695 9 .91167 3 .01713 200 . 3589 236 .6722 126 . 1 136 69 . 3632 6 .25000 5 .95723 14 .55429 9 .91319 2 .98529 201 .5490 236. .4186 125 .97 10 68. .5693 6 .50000 5 .979G3 14 5G193 9 .91503 2 .95342 202 . 7 107 236 . 1909 125 . 8440 67 . 7624 6 .75000 6 .00138 14 .56985 9 .91718 2 .92159 203 .84 10 235. .9891 125 .7327 66 .9452 7 .00000 6 .02254 14 .57807 9 .91965 2 .88974 204 . 94 18 235. .8 124 125 . 6366 66. .117 1 7 .25000 6 .04313 14 .58G58 9 .92245 2 .85784 20G .0145 235 . 6601 125 . 5552 G5. . 278 1 7 .50000 6 .06319 14 59539 9. .92558 2 .82585 207 .0606 235. 53 13 125 .4880 64 . 428 1 7 .7 5O00 6 .08273 14 60450 9 .92903 2 .79374 208 .0810 235 . 4253 125 .4346 63 .567 1 8 .OOOOO 6 .10178 14 .6 1392 9. .93281 2 .76 149 209 .0768 235 . 34 14 125 . 3945 62 .6953 8 .2 5000 6 .12035 14 .62364 9 93692 2 . 72909 210 .0488 235. 2789 125 . 3672 61 . 8130 8 .50000 6 .13847 14 .63367 9. .94135 2 .69651 2 10 . 9978 235 . 237 1 125. . 3525 60. .9207 8 .75000 e .15615 14 .64401 9. 946 10 2 .66374 2 1 1 . 9242 235 . 2 153 125. 3499 60 .0186 9 .00100 e .17333 14 .65461 9. 95 1 t5 2 .63091 212 . 8253 235. 2129 125 . 3588 59 . .1110 9 .25000 6 .05222 14 .73798 10. 01296 2 .60G84 213 .5121 235 . 3 1G9 125 .4 300 58 . . 2490 9 .50000 G .04 200 14 .75848 10. .03447 2 .57505 2 14 .3196 235 . 376 1 125 .4730 57 , . 3393 9. .75000 6 .05277 14 , .76778 10. 04775 2 .54169 2 15 . 1330 235. 44 13 125. .5183 56. .4 153 10 .00000 G .0G713 14 , .77582 10. 059 13 2 .50792 2 15 .93 14 235 . 52 1 8 125 . 57 17 55 . 483 1 10 .25000 G .08183 14 784 3 3 10. 0G986 2 .47397 2 IG . 7 107 235 . 6 18G 125 .634 1 54 . 5447 10 .50000 G .09628 14 . 79357 10. 08024 2 .43990 217 .4 703 235. 73 14 125. 7053 53. 6010 10 .7 5000 G . 11037 14 8035 1 10. 09038 2 .40574 2 18 . 2 100 235 . 8596 125. . 785 1 52 G533 11 .00000 G . 12407 14 , .81408 10. 1003 3 2 .37152 2 18 . 929G 236 . 0025 125 . 8732 5 1 . . 7026 11 .25000 G . 137 38 14 82520 10. 1 1016 2 .33726 2 19 .6289 236 . 1596 125 .9694 50. . 7501 11 . 50000 G .15028 14 . 83682 10. 1 1989 2 .30301 220 . 307G 236 . 3302 126. .0733 49 . 7969 1 1 .75000 G .1G278 14 , 84887 IO. 12955 2 .26880 220 . 9G5G 236 . 5 136 126. 1845 48 . 8443 12 .00000 6 . 17483 14 . 86129 10. 139 17 2 .2347 1 22 1 .6017 236. 7092 126. 3028 47 . 8943 1 2 .2U0OO (', . \HC,:i? 1 -1 87 398 10 . 1487.1 2 .2009G 222 . 2 1 2G 23G . 9 1G2 1 2G . 4278 .16 . 95 13 2 . 50000 G 197 1 6 1 4 8HG82 10. 1582 1 2 .16782 222 . 7943 237 . 1340 12G. 5590 4G . 0207 12 .7 5000 G .20729 14 , 899G8 10. 16755 2 .13547 223 . 3439 237 . 3615 126. 6960 45. 1067 13 .00000 6 .21073 14 . 91248 10. 17674 2 .10405 223 . 8599 237 . 5979 126 8382 44 . 2121 13 25000 G 22539 14 92513 10 18575 2 07374 224 3397 237 8422 126 9850 43 34 1 1 13 50000 G 233 15 14 93750 10 19452 2 04483 224 7 79 1 238 0936 127 1360 42 4993 13 75000 6 23990 14 94946 10 20299 2 01765 225 1734 238 3508 127 2904 4 1 6935 14 00000 0 24556 14 96085 10 2 1 107 1 99252 225 5 180 238 6 126 127 4475 40 9299 14 25000 0 25009 1 4 97 154 10 2 1870 1 96967 225 8 100 238 8778 127 G066 40 2 1 35 14 50000 G 25359 14 98 145 10 22585 1 949 1 1 226 0504 239 1452 127 7669 39 5455 14 75000 6 2562 1 14 99060 10 23252 1 93068 226 242B 239 4 13G 127 9279 38 9237 15 00000 6 25809 14 99903 10 23875 1 9 14 13 226 3926 239 682 1 128 08 B 8 38 3445 15 25000 6 25936 15 00684 10 2446 1 1 89919 226 5050 239 9501 128 2493 37 8036 lb ooooo 0 26008 15 014 10 10 25015 1 88567 226 5846 240 2 167 128 4090 37 297 7 15 75000 6 2603 1 15 02088 10 25540 1 87340 226 635 1 240 48 14 128 5676 36 8240 16 OOOOO G 260 1 1 15 02724 10 2604 1 1 86224 22G 6598 240 7437 128 724G 36 3799 1G 25000 G 25953 15 0332 1 10 265 19 1 85207 226 66 18 24 1 003 1 128 8798 35 9632 16 50000 6 258G 1 15 03883 10 26976 1 84280 226 6435 241 2593 129 0331 35 572 1 16 75000 6 25740 15 044 14 10 274 15 1 83431 226 6075 24 1 51 19 129 184 1 35 2044 17 OOOOO 6 25593 15 049 15 10 27838 1 82654 226 556 1 24 1 7606 129 3328 34 8585 17 25000 G 25426 15 0539 1 10 28245 1 8 1938 226 49 14 242 0052 129 4789 34 5325 17 50000 C 25240 15 05844 10 28638 1 8 1277 226 4 153 242 2455 129 6224 34 2249 17 75000 G 25040 15 0G277 10 29019 1 80664 226 3295 242 48 12 129 763 1 33 9342 18 OOOOO G 24827 15 06690 10 29389 1 80093 226 2355 242 7 124 129 9010 33 659 1 18 25000 6 24605 15 07087 10 2974B 1 79560 226 1348 242 9389 130 0360 33 3983 IS 50000 6 24374 15 07468 10 30097 1 7906 1 226 0284 243 1606 130 168 1 33 1509 18 75000 6 24 137 IS 07835 10 30437 1 7859 1 225 9174 243 3775 130 2973 32 9 159 19 OOOOO G 23894 15 08190 10 30768 1 78 149 22S 8026 243 5895 130 4235 32 6924 19 25000 6 23G4B 15 0853 1 10 3 1090 1 77730 225 6849 243 79G7 130 5467 32 4797 19 50000 G 2 3400 15 08862 10 31405 1 77334 225 5G49 243 9990 130 GG70 32 277 1 19 75000 6 23 149 IS 09182 10 3 17 11 1 76957 225 4433 244 1965 130 7843 32 0839 20 OOOOO . 6 22B98 15 09492 10 32010 1 76600 225 3205 244 3892 130 8986 31 8997 20 2 5000 6 22G47 15 09792 10 32302 1 762G0 225 1969 244 577 1 131 0101 3 1 7239 20 50000 6 22396 15 10083 10 32586 1 75935 225 0731 244 7603 131 1 186 31 5560 20 750OO G 22 146 15 10364 10 32863 1 75626 224 9493 244 9388 13 1 2243 3 1 3957 2 1 OOOOO 6 2 1898 15 10G38 10 33133 1 7533 1 224 8257 24S 1 126 13 1 3272 3 1 2425 2 1 25000 6 2 1G52 15 10903 10 33396 1 75049 224 7028 245 2819 131 4272 3 1 0960 2 1 50000 G 2 1408 15 1 1 160 10 33652 1 74780 224 5807 245 4467 13 1 5246 30 956 1 2 1 7 5000 G 2 1 167 15 1 1409 10 33901 1 74523 224 4595 245 607 1 131 6192 30 8222 22 OOOOO 6 20928 15 1 1650 10 34 144 1 74277 224 3396 245 763 1 13 1 7 112 30 6942 22 250O0 G 20693 15 1 1885 10 34380 1 74042 224 2209 245 9 148 13 1 8005 30 57 1 7 22 50000 G 204G2 15 12 112 10 34609 1 738 18 224 1038 246 0624 131 8873 30 4546 22 7SOOO G 20233 15 12332 10 34832 1 73603 223 9882 246 2058 13 1 97 16 30 3425 23 OOOOO G 20009 15 12546 10 35049 1 73397 223 8743 24G 345 1 132 OS 3 4 30 2352 23 25000 6 19788 15 12752 10 35259 1 73200 223 762 1 246 4805 132 1328 30 1326 23 50000 6 1957 1 15 12953 10 35464 1 73012 223 65 18 246 6 120 132 2099 30 0343 23 75000 6 19359 15 13 147 10 35663 1 7283 1 223 5434 246 7398 132 2846 29 9403 24 OOOOO 6 19150 15 1333G 10 35855 1 72659 223 4369 246 8638 132 357 1 29 8503 24 25000 0 I894G IS 13518 10 3G042 1 72494 223 3324 246 984 1 132 4273 29 764 1 24 50000 6 1874G 15 13695 10 3622 1 1 72335 223 2300 247 1010 132 4954 29 68 16 24 75000 6 18550 15 13867 10 36400 1 72 184 223 1296 2-47 2 144 132 5614 29 6026 25 OOOOO 6 18358 15 14033 10 3657 1 1 72039 223 0312 247 3243 132 6254 29 5270 25 25000 6 18170 15 14 193 10 3G73G 1 7 1900 222 9350 247 4310 132 6873 29 4547 25 50000 6 17987 15 14349 10 36896 1 7 1767 222 8408 247 5345 132 7473 29 3854 25 75000 G 17808 15 14 500 10 37052 1 7 1640 222 7487 247 6348 132 8054 29 3 190 26 OOOOO 6 17633 15 14646 10 37202 1 71518 222 6587 247 732 1 132 86 16 29 2555 26 25000 6 17463 15 14788 10 37348 1 71402 222 5708 247 8264 132 9161 29 1947 26 50000 6 17296 15 14925 10 37489 1 7 1290 222 4850 247 9178 132 9687 29 1365 26 75000 6 17 134 15 15057 10 37625 1 71 183 222 4012 248 0064 133 0197 29 0808 27 OOOOO G 16976 15 15 I8G 10 37758 1 7 108 1 222 3 194 248 0922 133 0690 29 0274 27 2 5000 G 1682 1 1 S 153 10 10 37BBG 1 70983 222 2397 248 1753 133 1 167 28 97G3 27 50000 6 IG67 1 15 1543 1 10 38009 1 70889 222 16 19 248 2559 133 1628 28 9274 27 7 5000 6 16525 15 15547 10 38 129 1 70799 222 0862 248 3339 133 207 3 28 8B06 2B OOOOO 6 16382 15 15660 10 38245 1 707 13 222 0123 248 4094 133 2504 28 8358 28 2 5 000 6 . 162-14 1 5 . 157G9 10 38356 1.706 31 22 1 94 04 248 482G 133 202 1 28 7929 28 50OO0 G . 16 109 5 . 15875 10 38465 1.70552 22 1 8704 248 5534 133 3323 28 75 18 28 7 50O0 6 . 15977 1 5 . 15977 10 38569 1.70477 22 1 8023 248 6220 133 37 12 28 7 125 29 00000 6 . 15850 15 . 16076 10 38670 1.70404 22 1 7360 248 6884 133 4087 28 6749 29 2 5000 6 . 15726 15 . 16172 10 38768 1.70335 22 1 67 15 248 7526 133 4450 28 6389 29 5OO00 6 15605 15 . 16264 10 38862 1.70269 22 1 G088 248 8 148 133 4 800 28 604 4 29 7 5000 6 . 15488 15. 16354 10 38953 1.70205 22 1 5478 248 8749 133 5 138 28 57 15 30 00000 G . 15374 15 . 1644 1 10 3904 1 1 70145 22 1 4885 248 333 1 133 5465 28 5399 30 2 5000 6 15263 1 5 . 16525 10 39 125 1.70086 22 1 4 309 248 9894 133 5780 28 5097 30 50000 6 - 15156 5 . 16606 10 39207 1.70031 22 1 3750 249 0439 133 6084 28 4808 30 7 5000 6 . 15052 15 . 16685 10 39286 1.69977 22 1 3 206 249 0966 133 6377 28 4531 3 1 00000 G . 14 950 15 . 16761 10 39362 1.69926 22 1 2678 249 1476 133 6660 28 4266 3 1 25000 6 . 14852 15 . 16834 10 39436 1.69878 22 1 2 166 249 1969 133 6933 28 4013 3 1 50000 G . 1 1757 1 5 . 16905 10 39507 1.69831 22 1 1668 249 2445 133 7 19G 28 377 1 3 1 75000 6 . 14GG4 15 . 16974 10 39575 1.69786 22 1 1 185 249 2906 133 7450 28 3539 32 00000 6 . 14575 15 . 1704 1 10 3964 1 1.69743 22 1 07 17 249 3352 133 7695 28 3317 32 2 5000 6 . 14488 1 5 . 17 105 10 39705 1.69702 22 1 0262 249 3783 133 793 1 28 3 105 32 S00G0 6 14 404 5 . 17 167 10 39766 t.69663 220 982 1 249 4 199 133 8 158 28 2902 32 7 5000 6 . 14322 IS. 17228 10 39825 1.69626 220 9394 249 4602 133 8377 28 2707 33 00000 G . 14243 1 5 . 17286 10 39882 1.69590 220 8979 249 4991 133 8588 28 2522 33 25000 G - 14 1G6 5 . 17342 10 39936 1.69555 220 8578 249 5367 t33 879 1 28 2344 33 50000 6 . 14092 15 . 17397 10 39989 1.69522 220 8 188 249 573 1 133 8987 28 2 174 33 7 5000 6 . 14020 15. 17450 10 40040 1.69491 220 78 1 1 249 6082 133 9176 28 201 1 34 00000 6 . 13S50 15 . 17501 10 40089 t.69461 220 7445 249 642 1 133 9357 28 1856 34 25000 6 . 1 3882 15. 17550 10 40136 1.69432 220 7091 249 6749 133 9532 28 1707 34 50000 6 . 1 38 1 7 15. 17598 10 40181 t.69404 220 6748 249 7066 133 9700 28 1565 34 75000 6 . 13754 15. 17644 10 40224 1.69378 220 64 16 249 7373 133 9862 28 1429 35 00000 C . 13G93 15 . 17G89 10 40266 t.69353 220 6094 249 7668 134 0018 28 1299 35 25000 6 . 13633 15 . 17732 10 40306 t.69329 220 5783 249 7954 134 0168 28 1 175 35 50000 6 . 13576 15. 17773 10 40345 1.69305 220 548 1 249 8230 134 0312 28 1056 35 75000 6 . 13520 15 . 178 14 10 40382 1.69283 220 5 190 249 8497 134 0451 28 094 2 36 00000 6 . 13467 15. 17853 10 404 18 1.69262 220 4907 249 8755 134 0584 28 0834 36 25000 6 . 134 15 1 5 . 1789 1 10 40453 1.69242 220 4634 249 90O4 134 07 12 28 07 30 36 50000 G . I33G5 5 . 17927 10 4048G 1.69223 220 4370 249 9244 134 0836 28 063 1 36 7 5000 G . 133 16 15 . 17963 10 405 17 1.69204 220 4 114 249 9476 134 0954 28 0536 37 00000 6 . 13269 IS . 17997 10 40548 1.69186 220 3867 249 9700 134 1068 28 0445 37 25000 6 . 13224 1 5 . 18030 10 40577 1.69169 220 3628 249 9917 134 1 177 28 0358 37 50000 6 . 13 180 5. 18062 10 40605 1.69153 220 3397 250 0126 134 1282 28 0275 37 7 5000 6 . 13 13 7 15 . 18093 to 40632 1.69138 220 3173 250 0328 134 1383 28 0196 38 OOOOO 6 . 13096 1 5 . 18 123 10 40G58 1.69123 220 2957 2 50 0523 134 1480 28 0120 38 25000 6 . 13057 5. 18 152 10 40683 1.69109 220 2748 250 071 1 134 1573 28 0048 38 50000 6 . 13019 15. 18 180 10 40706 1.69095 220 2546 250 0893 134 1662 27 9979 38 75000 G . 12982 15 . 18207 10 40729 1.69082 220 2351 250 1069 134 1747 27 9913 39 OOOOO 6 . 12946 1 5 . 18234 10 40751 1.69070 220 2 163 250 1238 134 1829 27 9849 39 25000 G . 129 12 15. 18259 10 40772 1.69058 220 198 1 250 1402 134 1908 27 9789 39 50000 G 12878 15 . 18284 10 40792 1 .G9046 220 1805 ? 50 15G0 134 1983 27 973 1 39 7 5000 6 . 12846 1 5 . 18307 10 408 1 1 1.69036 220 1635 250 17 13 134 2056 27 9676 40 OOOOO 6 . 128 IS 1 5 . 18330 10 40829 1.69025 220 147 1 250 1860 134 2 125 27 9624 40 25000 G . 12785 1 5 . 18353 10 40847 1.69015 220 1313 250 2003 134 2 192 27 9573 40 50000 G . 12756 5. 18374 10 40864 1.69006 220 1 160 250 2140 134 2255 27 9525 40 7 5000 G . 12728 1 5 . 18395 10 40880 1.68997 220 1012 250 2273 134 2316 27 9479 4 1 OOOOO 6 . 12701 1 5 . 184 16 10 40895 1.68988 220 0869 250 2401 134 2375 27 9435 4 1 25000 6 . 12675 5. 18435 10 409 IO 1.68980 220 0732 250 2525 134 2430 27 9393 4 1 50000 6 . 12650 1 5 . 18454 10 40924 1.68972 220 0599 250 2644 134 2484 27 9353 4 1 7 5000 6 . 1262G 1 5 . 18473 10 40937 1.68964 220 047 1 250 2759 134 2535 27 93 15 42 OOOOO 6 . 12603 1 5 . 18490 10 40950 1.68957 220 0347 250 287 1 134 2584 27 9278 42 2 5000 6 . 1 2580 1 5 . 18508 10 40962 1.68950 220 0227 2 50 2978 134 263 1 27 9244 42 50000 6 . 12559 5 . 18524 10 40974 1.68943 220 0112 2 50 3082 134 2675 27 92 10 42 7 5000 6 . 12538 1 5 . 1854 1 10 40985 1.68937 220 0001 2 50 3182 134 27 18 27 9178 43 OOOOO G . 125 17 1 5 . 1B556 10 40996 1 .68931 2 19 9894 250 3279 134 2759 27 9148 O O O O i ^ O ^ O u 1 0 ^ 0 ( J i O U i O L n O O l O U i O O i O u 1 O u i O U , O U i rn o c o o o o o o o o o o o o o o o o o o o o o o o o o o o fo ro ro o fO rj rj ro ro ro ro ro ro to to to u u u u u u u u U fO ro cn 0"> cn -j -0 -j 03 CD CO 03 to to to O O O O to U> O CT. 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I 2 1 8 7 E • 01 0 . 1 2 1 9 1 E • 01 0 . 1 2 I 9 4 E • 0 1 0 . 1 2 1 9 8 E '01 0 . I 2 2 0 I C •01 0 . 1 2 2 0 5 E •01 0 . 1 2 2 0 8 E * 0 1 0 . 1 2 2 1 2 E * 0 1 0. 12215E •01 0 . 1 2 2 1 8 E • O l 0 . 1222 IE • O l 0 . 1 2 2 2.1U • O l 0 . 12226E*01 0 . 1 2 2 2 9 E •01 0 . 1 2 2 3 2 E •01 0 . 1 2 2 3 4 E •01 0 . 1 9 I 3 7 E * 0 0 0 . 1 9 1 0 2 E * O 0 0 . 1 9 0 6 6 E ^ 0 0 0 . 1 9 0 3 0 E + 0 0 0 . 1 8 9 9 4 E ^ 0 0 0 . I 8 9 5 8 E ^ 0 0 0 . 1 8 9 2 3 E ^ 0 0 0 . I 8 8 8 7 E ^ 0 0 0 . 1 8 8 5 2 E * 0 0 0 . 1 8 8 1 7 E ^ O O 0 . 1 8 7 8 2 E ^ O O 0 . 1 8 7 4 8 E * 0 0 0 . 1 8 7 1 4 E ^ 0 0 0 . 1 8 6 8 1 E * O 0 0 . 1 8 6 4 8 E + 0 0 0 . 1 8 6 1 6 E ^ 0 0 0 . 1 8 5 8 4 E * 0 0 0 . 1 8 5 5 3 E * 0 0 0 . 1 8 5 2 2 E * 0 0 0 . 1 8 4 9 2 E * 0 0 0 . 1 8 4 6 3 E * O 0 0 . 1 8 4 3 4 E * 0 0 0 . 1 B 4 0 6 E * 0 0 0 . 1 8 3 7 8 E * 0 0 0 . 1 8 3 5 1 E * 0 0 0 . 1 8 3 2 5 E * 0 0 0 . 1 8 2 9 9 E + 0 0 0 . 1 8 2 7 4 E + 0 0 0 . 1 8 2 4 9 E + 0 0 0 . I 8 2 2 5 E * 0 0 0 . 1 8 2 0 2 £ * 0 0 0 . 1 8 1 7 9 E + 0 O 0 . 1 8 1 5 7 E + 0 0 0 . 1 8 1 3 5 E - * 0 0 0 . 1 8 1 1 4 E * 0 0 0 . 1 8 0 9 3 E * 0 0 0 . 1 8 0 7 3 E * 0 0 0 . 1 8 0 5 4 E » 0 0 0 . 1 8 O 3 5 E ^ 0 0 0 . 1 8 0 1 7 E + 0 0 0 . 1 7 9 9 9 E ^ O O 0 . 1 7 9 8 1 E ^ 0 0 O . 1 7 9 6 4 E * 0 0 O . 1 7 9 4 8 E ^ 0 0 O . 1 7 9 3 2 E + 0 0 O . 1 7 9 1 6 E * 0 O O . 1 7 9 0 1 E + 0 0 O . 1 7 8 B 7 E * 0 0 O . 1 7 8 7 3 E * 0 O O . 1 7 8 5 9 E ^ 0 0 O . 1 7 B 4 5 E * 0 O 0 . 1 7 8 3 2 E + 0 0 O . 1 7 8 2 0 E ^ 0 0 O . 1 7 8 0 8 E + 0 0 O . 1 7 7 9 6 E * 0 0 O . 1 7 7 8 4 E > 0 0 0 . 1 7 7 7 3 E » 0 0 0 . 1 7 7 6 3 E * 0 0 0 . 1 7 7 5 2 E ^ O O O . 1 7 7 4 2 E ^ O O 0 . 9 9 0 5 6 E + 0 0 0 . 9 9 1 3 7 E ^ 0 0 O . 9 9 2 1 7 E + 0 0 0 . 9 9 2 9 5 E ^ 0 0 0 . 9 9 3 7 3 E * O O O . 9 9 4 5 0 E ' 0 0 0 . 9 9 5 2 S E ^ O O 0 . 9 9 6 0 0 E ^ 0 0 0 . 9 9 6 7 3 E ^ O O O . 9 9 7 4 6 E + 0 0 O . 9 9 8 1 8 E * 0 0 0 . 9 9 8 8 9 E « 0 0 0 . 9 9 9 5 8 E » 0 0 0 . 1 0 0 0 3 E + 0 1 O . 1 0 0 0 9 E + 0 1 0 . 1 0 0 1 6 E * 0 I 0 . 1 0 0 2 3 E + 0 1 0 . 1 0 0 2 9 E + 0 1 O . 1 0 0 3 S E * 0 1 O . 1 0 0 4 1 E * 0 1 O . 1 0 0 4 7 E + 0 1 0 . 1 0 0 5 3 E * 0 1 0 . 1 0 0 5 9 E + 0 1 0 . I 0 0 6 S E * 0 1 O . 1 0 0 7 0 E * 0 1 0 . 1 O 0 7 6 E + 0 1 0 . 1 0 0 8 1 E + 0 1 O . 1 0 0 8 6 E + 0 1 0 . 1 0 0 9 1 E * 0 1 O . 1 0 0 9 6 E < 0 1 O . 1 0 1 0 I E + 0 1 O . 1 0 1 0 6 E * 0 1 O . 1 0 1 1 0 E + 0 1 O . 1 0 1 1 5 E + 0 1 O . 1 0 1 1 9 E + 0 1 O . 1 0 I 2 3 E * 0 1 0 . 1 0 1 2 7 E * 0 1 O . 1 0 1 3 1 E + 0 1 O . 1 0 1 3 5 E * 0 1 0 . 1 0 1 3 9 E + 0 1 O . 1 0 1 4 2 E + 0 1 O . 1 0 1 4 6 E « 0 1 O . 1 0 1 4 9 E * 0 1 O . 1 0 1 S 3 E + 0 1 O . 1 0 1 5 6 E * 0 1 O . 1 0 1 5 9 E + 0 1 O . 1 0 I 6 2 E * 0 1 O . 1 0 1 6 5 E * 0 1 O . I 0 I 6 8 E * 0 1 O . 1 0 1 7 1 E + 0 1 O . 1 0 1 7 4 E + 0 1 O . 1 0 1 7 6 E + 0 1 O . 1 0 1 7 9 E ^ 0 1 O . 1 0 1 8 1 E + 0 1 O . 1 0 I 8 4 E ^ 0 1 O . I 0 1 8 6 E ^ 0 1 O . 1 0 1 8 8 E + 0 1 0 . 1 0 1 9 1 E + 0 1 O . 1 0 1 9 3 E + 0 1 0 . 1 0 1 9 5 E + 0 1 0 . 3 7 6 0 1 E + 0 0 0 . 3 7 6 1 3 E + 0 0 0 . 3 7 G 2 4 E + 0 0 O . 3 7 6 3 5 E ^ O O O . 3 7 G 4 G E + 0 0 O . 3 7 6 5 6 E + 0 0 O . 3 7 6 6 G E ^ O O O . 3 7 6 7 6 E • O O 0 . 3 7 6 8 5 E + 0 0 0 . 3 7 6 9 5 E ' 0 0 O . 3 7 7 0 3 E ^ 0 0 0 . 3 7 7 1 2 E + 0 0 0 . 3 7 7 2 0 E * 0 0 0 . 3 7 7 2 8 E + 0 0 0 . 3 7 7 3 6 E + O O 0 . 3 7 7 4 4 E ^ 0 0 0 . 3 7 7 5 1 E + 0 0 0 . 3 7 7 5 8 E * 0 0 0 . 3 7 7 6 5 E + 0 0 0 . 3 7 7 7 2 E + 0 0 0 . 3 7 7 7 8 E + 0 0 0 . 3 7 7 8 5 E + 0 0 0 . 3 7 7 9 1 E * 0 0 0 . 3 7 7 9 6 E + 0 0 0 . 3 7 8 0 2 E * 0 0 0 . 3 7 8 0 8 E + 0 0 0 . 3 7 8 1 3 E + 0 0 0 . 3 7 8 1 8 E + 0 0 0 . 3 7 8 2 3 E + 0 0 O . 3 7 8 2 8 E * 0 0 0 . 3 7 8 3 3 E * 0 0 0 . 3 7 8 3 7 E + 0 0 O . 3 7 8 4 2 E + 0 0 0 . 3 7 8 4 6 E + 0 0 0 . 3 7 8 5 0 E + 0 0 O . 3 7 B 5 4 E * 0 0 0 . 3 7 8 5 8 E + 0 0 O . 3 7 8 6 2 E + 0 0 0 . 3 7 8 G G E * 0 0 0 . 3 7 8 6 9 E + 0 0 0 . 3 7 8 7 3 E + 0 0 O . 3 7 B 7 G E + 0 0 0 . 3 7 8 7 9 E * 0 0 0 . 3 7 8 B 2 E * 0 0 O . 3 7 8 8 5 E * 0 0 0 . 3 7 8 8 8 E + 0 0 0 . 3 7 8 9 1 E * 0 0 O . 3 7 8 9 4 E * 0 0 0 . 3 7 8 9 6 E ^ 0 0 0 . 3 7 8 9 9 E ^ 0 0 O . 3 7 9 0 1 E * 0 0 O . 3 7 9 0 4 E + 0 0 O . 3 7 9 0 6 E + 0 0 0 . 3 7 9 0 8 E * 0 O O . 3 7 9 l O E ^ O O 0 . 3 7 9 1 2 E + 0 0 0 . 3 7 9 1 4 E * 0 0 O . 3 7 9 1 6 E ^ 0 0 0 . 3 7 9 1 8 E + 0 0 O . 3 7 9 2 0 E + 0 0 0 . 6 9 2 7 0 E ^ 0 1 0 . 6 9 1 3 4 E ^ 0 1 0 . 6 8 9 9 6 E ' 0 1 0 . 6 8 8 5 7 E ' 0 1 0 . 6 8 7 1 7 C 0 1 O . 6 8 5 7 8 E * 0 I 0 . G 8 4 3 8 E - 0 1 O . G 8 3 0 0 E ' 0 1 0 . 6 8 I 6 2 E ^ 0 1 0 . 6 8 0 2 5 E ^ 0 I 0 . 6 7 8 9 0 E * 0 1 0 . 6 7 7 5 6 E ^ 0 1 0 . 6 7 6 2 3 E * 0 1 0 . 6 7 4 9 3 E ^ 0 1 0 . 6 7 3 6 4 E » 0 1 0 . 6 7 2 3 8 E ^ 0 1 O . 6 7 1 1 3 E + 0 1 0 . 6 6 9 9 1 E » 0 1 0 . 6 6 8 7 0 E * 0 1 0 . 6 6 7 5 2 £ * 0 1 0 . 6 6 6 3 7 E + 0 1 0 . 6 6 5 2 3 E ^ 0 1 0 . 6 6 4 1 2 E ^ 0 1 0 . 6 6 3 0 4 E ^ 0 1 0 . 6 6 1 9 8 E * 0 1 0 . 6 6 0 9 4 E + 0 1 0 . 6 5 9 9 2 E ^ 0 1 0 . 6 5 B 9 3 E ^ 0 1 0 . 6 5 7 9 6 E * 0 1 0 . 6 5 7 0 2 E * 0 1 0 . 6 5 6 1 0 E ^ 0 1 0 . 6 5 5 2 0 £ * 0 1 0 . 6 5 4 3 2 E + 0 1 0 . 6 5 3 4 7 E > 0 1 0 . 6 5 2 6 4 E + 0 1 O . 6 5 1 8 3 E ^ 0 I 0 . 6 5 1 0 4 E * 0 1 0 . 6 5 0 2 7 E ^ 0 1 O . 6 4 9 S 3 E * 0 1 0 . 6 4 8 8 0 E + 0 1 0 . 6 4 8 l O E ^ O I O . 6 4 7 4 1 E + 0 1 O . G 4 6 7 4 E + 0 I O . 6 4 6 1 0 E ' 0 I O . G 4 5 4 7 E * 0 1 0 . 6 4 4 8 5 E + 0 1 0 . 6 4 4 2 6 E ^ 0 1 O . 6 4 3 6 8 E • O 1 0 . 6 4 3 1 3 E * 0 1 O . G 4 2 S 8 E * 0 1 0 . 6 4 2 0 6 E ^ 0 1 0 . 6 4 1 5 4 E + 0 1 0 . 6 4 I O S E ' 0 1 O . G 4 0 5 7 E * 0 1 0 . 6 4 0 1 0 E ^ 0 1 O . G 3 9 G S E ^ O I O . 6 3 9 2 1 E * 0 1 0 . 6 3 8 7 9 E * 0 1 0 . 6 3 8 3 8 E ' 0 1 0 . 6 3 7 9 8 E ^ 0 1 0 . 6 0 G 9 1 E * 0 1 0 . 6 0 7 4 9 E * 0 1 0 . 6 0 8 0 5 E ^ 0 1 0 . 6 0 8 6 0 E ^ 0 1 0 . 6 0 9 1 4 E ^ 0 1 O . 6 0 9 6 6 E ^ 0 1 0 . 6 1 0 1 7 E « 0 1 0 . 6 1 0 G 7 E ^ 0 1 0 . 6 1 1 1 5 E * 0 1 0 . 6 1 I G 2 E ^ 0 1 6 . G 1 2 0 8 E ^ 0 1 0 . 6 1 2 5 3 E ^ 0 1 0 . 6 1 2 9 6 E ^ 0 1 O . G 1 3 3 8 E + 0 1 0 . 6 1 3 8 0 E - t 0 1 0 . 6 1 4 1 9 E * 0 1 0 . 6 1 4 5 B E ^ 0 1 0 . 6 1 4 9 6 E ^ 0 1 0 . 6 1 5 3 3 E + 0 1 0 . 6 1 5 G 8 E * 0 1 0 . 6 1 6 0 3 E + 0 1 0 . 6 1 6 3 6 E + 0 1 0 . 6 1 6 6 9 E + 0 1 0 . 6 1 7 0 0 E + 0 1 0 . 6 1 7 3 1 E * 0 1 0 . 6 1 7 6 1 E + 0 1 0 . 6 1 7 9 0 E + O 1 0 . 6 1 B 1 8 E + 0 1 0 . 6 1 8 4 5 E + 0 1 0 . 6 1 8 7 1 E * 0 1 0 . 6 I 8 9 6 E * 0 1 O . 6 1 9 2 1 E + 0 1 0 . 6 1 9 4 5 E ^ 0 1 0 . 6 1 9 6 8 E + 0 1 0 . 6 1 9 9 1 E + 0 1 O . 6 2 0 1 3 E * O 1 0 . 6 2 0 3 4 E « 0 1 0 . 6 2 0 5 4 E * 0 1 0 . 6 2 0 7 4 E + 0 1 0 . 6 2 0 9 3 E + 0 1 0 . 6 2 1 M E + 0 1 0 . 6 2 1 2 9 E + 0 1 0 . 6 2 I 4 7 E * 0 1 0 . 6 2 1 6 4 E + 0 I 0 . 6 2 1 8 0 E * 0 1 0 . 6 2 1 9 G E * 0 1 0 . 6 2 2 1 I E + 0 1 0 . 6 2 2 2 5 E + 0 1 O . 6 2 2 4 0 E ' 0 1 0 . 6 2 2 5 3 E ' 0 1 0 . 6 2 2 6 7 E * 0 1 0 . 6 2 2 8 0 E + 0 1 0 . 6 2 2 9 2 E ^ 6 l 0 . 6 2 3 0 4 E + 0 1 0 . 6 2 3 1 6 E ^ 0 1 O . G 2 3 2 7 E ' 0 1 0 G 2 3 3 8 E ^ 0 1 0 . 6 2 3 4 9 E ^ 0 1 O . 6 2 3 5 9 E * 0 1 0 . 6 2 3 6 9 E + 0 1 0 4 0 5 8 3 E - 15 0 4 0 5 3 7 E - 15 0 4 0 4 9 2 E - 15 0 4 0 4 4 6 E - 15 0 4 0 4 0 1 E - 15 0 4 0 3 5 6 E - 15 0 4 0 3 1 1 E - 5 0 4 0 2 6 7 E - 5 0 4 0 2 2 3 E - 5 0 4 0 1 8 0 E - 5 0 4 0 1 3 7 E - 15 0 4 0 0 9 5 E - 15 0 4 0 0 5 4 E - 5 0 4 0 0 1 3 E - 5 0 3 9 9 7 3 E - 5 0 3 9 9 3 4 E - 15 0 3 9 8 9 6 E - 15 0 3 9 8 5 9 E - 15 0 3 9 8 2 2 E - 15 0 3 9 7 8 G E - 15 0 3 9 7 5 1 E - 15 0 3 9 7 1 7 E - 15 0 3 9 6 8 3 E - 15 0 3 9 6 5 1 E - 15 0 3 9 6 1 9 E - 5 0 3 9 5 8 7 E - 5 0 3 9 5 5 7 E - 5 0 3 9 5 2 7 E - 5 0 3 9 4 9 9 E - 5 0 3 9 4 7 0 E - 15 0 3 9 4 4 3 E - 5 0 3 9 4 1 6 E - 5 0 3 9 3 9 0 E - 5 0 3 9 3 G 5 E - 5 0 3 9 3 4 1 E - 15 0 3 9 3 1 7 E - 15 0 3 9 2 9 4 E - 15 0 3 9 2 7 1 E - 5 0 3 9 2 4 9 E - 15 0 3 9 2 2 8 E - 15 0 3 9 2 0 7 E - 15 0 3 9 1 8 7 E - 15 0 3 9 I G 8 E - 15 0 3 9 1 4 9 E - 5 0 3 9 1 3 0 E - 5 0 3 9 1 1 2 E - 5 0 3 9 0 9 5 E - 5 0 3 9 0 7 8 E - 5 0 3 9 0 G 2 E - 5 0 3 9 0 4 6 E - 5 0 3 9 0 3 1 E - 5 0 3 9 0 1 6 E - 5 0 3 9 0 0 2 E - 5 0 3 8 9 8 8 E - 15 0 3 8 9 7 5 E - 15 0 3 B 9 G I E - 5 0 3 8 9 4 9 E - 5 0 3 8 9 3 7 E - 15 0 3 8 9 2 5 E - 15 0 3 8 9 1 3 E - 15 237 33 7b 0. 1 2 2 3 7 1 ' 0 1 0. I 7 7 3 2 E ' 0 0 0 . 1 0 1 9 7 E * 0 t 0 3 7 9 2 2 E*00 0 6 3 7 5 9 E ' 0 1 0 6 2 3 7 8 E*01 0 3 8 9 0 2 E • 15 3J 00 0. 1 2 2 3 9 E » 0 1 0 . 1 7 7 2 3 E » 0 0 0 . 1 0 1 9 9 E * 0 1 0 3 7 9 2 3 E + 0 0 0 6 3 7 2 2 E * 0 1 0 6 2 3 8 7 E*01 0 3 8 8 9 IE - 15 34 2 5 0. 1 2 2 4 I E ' 0 1 0. 1 7 7 1 4 E * 0 0 0 . 1 0 2 0 1 E * 0 1 0 37925E+00 0 6 3 6 8 6 E*01 0 6 2 3 9 6 E*01 0 3 8 8 8 IE - 15 3-1 5 0 0. 1 2 2 4 J L • 0 1 0. 17 705E*00 0 . 1 0 2 0 2 E » 0 1 O 3 7 9 2 7 E « 0 0 0 63651E+01 0 6 2 4 0 5 E'01 0 3 8 8 7 IE - 15 3 1 75 0. I22-10E -0 1 0 - 1 7 6 9 G E ' 0 0 0 . 1 0 2 0 4 E * 0 1 0 37928E+00 0 G 3 6 1 7 E'0t 0 G2413E'01 0 3 8 8 G I C - 15 3 5 O O O . I2248L"'01 0. 1 7 6 8 8 E ' 0 0 0 . 1 0 2 0 6 E * 0 1 O 3 7 9 3 0 E + 0 0 0 6 3 5 8 4 E*01 0 6 2 4 21E + 01 0 3 8 8 5 2 E - 15 3 5 2 5 0. 1 2 2 5 0 E ' O I 0 . 1 7 6 8 0 E - > 0 0 0 . 1 0 2 0 8 E * 0 1 0 3 7 9 3 1 E * 0 0 O 6 3 5 5 2 E + 0 1 0 6 2 4 2 9 E * 0 1 0 3 8 8 4 3 E - 15 35 5 0 0. I 2 2 5 2 E * 0 1 0 . 1 7 6 7 2 E * 0 0 0 . 1 0 2 0 9 E + 0 1 0 3 7 9 3 2 E + 0 0 0 6 3 5 2 2 E + 0 1 0 6 2 4 3 6 E » 0 1 0 3 8 8 3 4 E - 15 3 5 7 5 0 . 1 2 2 5 4 E * 0 1 0 . 1 7 6 6 5 E + 0 0 0 . 1 0 2 1 1 E * 0 1 0 3 7 9 3 4 E + 0 0 0 6 3 4 9 2 E + 0 1 0 6 2 4 4 3 E + 0 1 0 3 8 8 2 5 E - 15 3 6 0 0 0 . 1 2 2 5 5 E * 0 1 0 . 1 7 6 5 7 E * 0 0 0 . 1 0 2 1 2 E * 0 1 0 3 7 9 3 5 E + 0 0 0 6 3 4 6 3 E + 0 1 0 6 2 4 5 0 E ' 0 1 0 3 8 8 I 7 E - 15 3 6 2 5 0. 1 2 2 5 7 E » 0 1 0 . 1 7 6 5 0 E * 0 0 0 . 1 0 2 1 4 E * 0 1 0 3 7 9 3 6 E * 0 0 0 6 3 4 3 5 E + 0 1 0 6 2 4 5 7 E * 0 1 0 3 8 8 0 9 E - 15 3 6 5 0 0. 1 2 2 5 9 E ' 0 1 0 . 1 7 6 4 3 E * 0 O 0 . 1 0 2 1 5 E * 0 1 0 3 7 9 3 8 E + 0 0 0 6 3 4 0 9 E + 0 1 0 6 2 4 6 4 E * 0 I 0 3 8 8 0 I E - 15 3 6 7 5 0. 1 2 2 6 0 E ' 0 1 0. 1 7 6 3 7 E » 0 0 0 . 1 0 2 1 G E * 0 1 O 3 7 9 3 9 E + 0 0 0 6 3 3 8 2 E - ' 0 1 0 6 2 4 7 0 E - ' 0 1 0 3 8 7 9 4 E - 15 3 7 0 0 0 . I 2 2 6 2 E ' 0 1 0 . 1 7 6 3 0 E + 0 0 0 . 1 0 2 1 8 E * 0 1 0 3 7 9 4 0 E + 0 0 0 6 3 3 5 7 E + 0 1 0 6 2 4 7 6 E + 0 1 0 3 8 7 8 7 E - 15 3 7 2 5 0 . 1 2 2 6 4 E + 0 1 0 . 1 7 6 2 4 E * 0 0 0 . 1 0 2 1 9 E * 0 1 0 37941E*00 0 6 3 3 3 3 E + 0 1 0 6 2 4 8 2 E * 0 1 0 3 8 7 B 0 E - 15 37 5 0 0. 1 2 2 6 5 E ' 0 I 0 . 1 7 6 1 8 E + 0 0 0 . 1 0 2 2 0 E * 0 1 0 3 7 9 4 2 E * 0 0 0 6 3 3 1 0 E + 0 1 0 6 2 4 8 8 E * 0 1 0 3 8 7 7 3 E - 15 3 7 7 5 0. 1 2 2 6 7 E * 0 1 0 . 1 7 6 1 3 E * 0 0 0 . 1 0 2 2 1 E * 0 1 0 3 7 9 4 3 E * 0 0 0 6 3 2 8 7 E + 0 1 0 6 2 4 9 3 E * 0 1 0 3 8 7 6 7 E - 15 3 8 0 0 0 . 1 2 2 6 8 E * 0 1 0 . 1 7 6 0 7 E * 0 0 0 . 1 0 2 2 2 E - > 0 1 0 3 7 9 4 4 E + 0 0 0 6 3 2 6 5 E + 0 1 0 6 2 4 9 8 E * 0 1 0 3 8 7 6 0 E - 15 3 8 2 5 0 . I 2 2 6 9 E ' 0 1 0 . 1 7 6 0 2 E » 0 0 0 . 1 0 2 2 4 E * 0 1 0 3 7 9 4 5 E * 0 0 o 6 3 2 4 4 E * 0 1 O 6 2 5 O 3 E * 0 1 0 3 8 7 5 4 E - 1 5 3 8 5 0 0 . 1 2 2 7 1 E * 0 1 0 . 1 7 5 9 6 E * 0 0 0 . 1 0 2 2 S E + 0 1 0 3 7 9 4 6 E + 0 0 0 6 3 2 2 3 E + 0 1 0 6 2 5 0 8 E + 0 1 0 3 8 7 4 9 E - 15 3 8 7 5 0 . 1 2 2 7 2 E » 0 1 0 . 1 7 5 9 1 E * 0 0 0 . 1 0 2 2 6 E * 0 1 0 3 7 9 4 7 E + O 0 0 6 3 2 0 3 E + 0 1 0 6 2 5 1 3 E + 0 1 0 3 8 7 4 3 E - 15 3 9 0 0 0 . 1 2 2 7 3 E ' 0 1 0 . 1 7 5 8 7 E * 0 0 0 . 1 0 2 2 7 E + 0 1 0 3 7 9 4 8 E + 0 0 0 6 3 1 8 4 E + 0 1 0 6 2 5 1 8 E + 0 1 0 3 8 7 3 7 E - 15 3 9 2 5 0 . I 2 2 7 4 E ' 0 1 0 . 1 7 5 8 2 E + 0 0 0 . 1 0 2 2 8 E + 0 1 0 3 7 9 4 9 E + O 0 0 6 3 1 6 6 E + 0 1 0 6 2 5 2 2 E + 0 1 0 3 8 7 3 2 E - 15 3 9 5 0 0. 1 2 2 7 5 £ ' 0 1 0 . 1 7 5 7 7 E + 0 0 0 . 1 0 2 2 8 E + 0 1 0 3 7 9 4 9 E + 0 0 0 6 3 1 4 8 E * O I 0 6 2 5 2 6 E + 0 1 0 3 8 7 2 7 E - 15 39 7 5 0 . I 2 2 7 6 E * 0 1 0 . 1 7 5 7 3 E * 0 0 0 . 1 0 2 2 9 E * 0 1 0 3 7 9 5 0 E + O 0 0 6 3 1 3 1 E + 0 1 0 6 2 5 3 1 E + 0 1 0 3 8 7 2 2 E - 15 4 0 0 0 0 . I 2 2 7 7 E * 0 1 0 . 1 7 5 6 9 E + 0 0 0 . 1 O 2 3 0 E + 0 1 0 3 7 9 5 1 E + 0 0 0 6 3 1 1 4 E + 0 1 0 6 2 5 3 5 E * 0 1 0 3 8 7 1 7 E - 15 4 0 2 5 0 . 1 2 2 7 9 E * 0 1 0 . 1 7 5 6 5 E * 0 0 0 . 1 0 2 3 1 E » 0 1 0 3 7 9 5 2 E + 0 0 0 6 3 0 9 8 E + 0 1 0 6 2 5 3 8 E * 0 1 0 3 8 7 1 3 E - 15 4 0 5 0 0 . 1 2 2 7 9 E ' 0 1 0 . 17561E*00 0 . 1 0 2 3 2 E * 0 1 0 3 7 9 5 2 E + 0 0 0 6 3 0 8 2 E + 0 1 0 6 2 5 4 2 E + 0 1 0 3 8 7 0 8 E - 15 4 0 7 5 0 . 1 2 2 8 0 E * 0 1 0 . 1 7 5 5 7 E * 0 0 0 . 1 0 2 3 3 E + 0 1 0 3 7 9 5 3 E + 0 0 0 6 3 0 6 7 E + 0 1 0 6 2 5 4 6 E * 0 1 0 3 8 7 0 4 E - 15 4 1 O O 0 . 1 2 2 8 1 1 » 0 1 0 . 1 7 5 S 3 E * 0 0 0 . 1 0 2 3 3 E + 0 1 0 3 7 9 S 4 E * 0 0 0 6 3 0 5 3 E + 0 1 0 6 2 5 4 9 £ ' 0 1 0 3 8 7 0 0 E - 15 4 1 2 5 0. I 2 2 8 2 E > 0 1 0 . 1 7 5 5 0 E + 0 0 0 . 1 0 2 3 4 E . 0 1 o 3 7 9 5 5 E * 0 0 0 6 3 0 3 9 E + 0 1 0 6 2 5 5 3 E - ' 0 1 o 3 8 G 9 6 E - 15 4 1 5 0 0. 1 2 2 8 3 E » 0 1 0 . 1 7 5 4 6 E * 0 0 0 . 1 0 2 3 5 E * 0 1 0 3 7 9 5 5 E + 0 0 0 6 3 0 2 5 E + 0 1 0 6 2 5 5 6 E + 0 1 0 3 8 6 9 2 E - 15 4 1 75 0. 1 228-1E ' 0 1 0 . 1 7 5 4 3 E + 0 0 0 . 1 0 2 3 5 E + 0 1 o 3 7 9 5 6 E + 0 0 0 6 3 0 1 2 E + 0 1 0 6 2 5 5 9 E + 0 1 0 3 8 6 8 9 E - 15 4 3 OO 0. 1 2 2 8 5 E ' 0 1 0 . 1 7 5 4 0 E » 0 0 0 . 1 0 2 3 6 E * O 1 o 3 7 9 5 6 E * 0 O 0 6 3 0 0 0 E + 0 1 0 6 2 5 G 2 E * 0 1 0 3 8 6 8 5 E - 15 4 2 2 5 0 . 1 2 2 8 5 E * 0 1 0 . 1 7 5 3 7 E * 0 0 0 . 1 0 2 3 7 E + 0 1 0 3 7 9 5 7 E + 0 0 0 6 2 9 8 8 E + 0 1 0 6 2 5 6 5 E + 0 1 0 3 8 6 8 2 E - 15 4 2 5 0 0 . I 2 2 8 6 E * 0 1 0 . 1 7 5 3 4 E + 0 0 0 . 1 0 2 3 7 E * 0 1 0 3 7 9 5 7 E * 0 0 0 6 2 9 7 6 E + 0 1 0 6 2 5 6 8 E * 0 1 0 3 8 6 7 8 E - 15 •12 7 5 0. 1 2 2 8 7 E ' 0 1 0 . 1 7 5 3 1 E * 0 0 0 . 1 0 2 3 8 E + 0 1 0 3 7 9 5 8 E + 0 O 0 6 2 9 6 5 E + 0 1 0 6 2 5 7 O E * 0 1 0 3 8 6 7 5 E - 15 -13 0 0 0 . 1 2 2 8 8 E ' 0 1 0 . 1 7 5 2 8 E + 0 0 0 . 1 0 2 3 8 E * O I 0 3 7 9 5 9 E * 0 0 0 6 2 9 5 4 E + 0 1 0 6 2 5 7 3 E + 0 1 0 3 8 6 7 2 E - 15 4 3 2 5 0 . 1 2 2 8 8 E * 0 1 0 . 1 7 5 2 6 E + 0 0 0 . 1 0 2 3 9 E * 0 1 0 3 7 9 5 9 E * 0 0 0 6 2 9 4 3 E + 0 1 0 6 2 5 7 6 E * 0 1 0 3 8 6 6 9 E - 15 4 3 5 0 0 . 1 2 2 8 9 E - 0 1 0 . 1 7 5 2 3 E + 0 0 0 . 1 0 2 3 9 E + O 1 0 3 7 9 6 0 E * 0 0 0 6 2 9 3 4 E + 0 1 0 6 2 5 7 8 E * 0 1 0 3 8 6 6 6 E - 15 4 3 7 5 0 . 1 2 2 9 0 E * 0 1 0 . 1 7 5 2 1 E + 0 0 0 . 1 0 2 4 0 E + 0 1 0 3 7 9 6 0 E + 0 0 0 6 2 9 2 4 E + 0 1 0 G 2 5 8 1 E + 0 1 0 3 8 6 6 3 E - 15 4 4 0 0 0 . I 2 2 9 0 E * 0 1 0 . 1 7 5 1 9 E + 0 0 0 . 1 0 2 4 0 E » O 1 0 3 7 9 6 1 E + 0 0 0 6 2 9 1 6 E + 0 1 0 6 2 5 8 3 E * 0 1 0 3 8 6 6 1 E - 15 4 4 2 5 0 . 1 2 2 9 1 E * 0 1 0 . 1 7 5 1 7 E + O 0 0 . 1 0 2 4 0 E + 0 1 0 3 7 9 6 1 E + 0 0 0 6 2 9 0 7 E + 0 1 0 6 2 5 8 5 E * 0 1 0 3 8 6 5 8 E - 15 4 4 5 0 0 . 1 2 2 9 I E ' 0 1 0. 1 7 5 1 4 E * 0 0 0 . 1 0 2 4 1 E * 0 I 0 3 7 9 6 2 E + 0 0 0 6 2 8 9 9 E + 0 1 0 6 2 5 8 7 E * 0 1 0 3 8 6 5 6 E - 15 44 75 0. 1 2 2 9 2 E ' 0 1 0. 1 7 S 1 3 E < 0 0 0 . 1 0 2 4 I E » 0 1 0 3 7 9 G 2 E + 0 0 0 6 2 8 9 1 E + 0 1 0 6 2 5 8 9 E * 0 1 0 3 8 G 5 4 E - 15 4 5 0 0 0 . 1 2 2 9 2 E * 0 1 0 . 1 7 5 1 1 E + 0 0 0 . 1 0 2 4 1 E * 0 1 0 3 7 9 6 3 E + 0 0 0 6 2 8 8 4 E + 0 1 0 6 2 5 9 1 E + 0 1 0 3 8 6 5 1 E - 15 4 5 2 5 0 . 1 2 2 9 3 E ' 0 1 0 . 1 7 5 0 9 E * 0 0 0 . 1 0 2 4 2 E * 0 1 0 3 7 9 6 3 E * 0 0 0 6 2 8 7 7 E + 0 1 0 6 2 5 9 3 E + 0 1 0 3 8 6 4 9 E - 15 4 5 5 0 0 . 1 2 2 9 3 E - 0 1 0 . 1 7 5 0 7 E * 0 0 0 . 1 0 2 4 2 E * 0 1 0 3 7 9 6 3 E * 0 0 0 6 2 8 7 0 E + 0 1 0 6 2 5 9 5 E * 0 1 0 3 8 6 4 7 E - 15 4 5 7 5 0 . 1 2 2 9 3 E » 0 1 0 . 1 7 5 O 5 E * 0 O 0 . 1 0 2 4 2 E + 0 1 0 3 7 9 6 4 E + 0 0 0 6 2 8 6 3 E * 0 1 0 6 2 5 9 7 E + 0 1 0 3 8 6 4 6 E - 15 4 6 0 0 0. I 2 2 9 4 E ' 0 1 0 . 1 7 5 0 4 E + 0 O 0 . 1 0 2 4 2 E * O t 0 3 7 9 6 4 E * 0 0 0 6 2 8 5 7 E + 0 1 0 6 2 5 9 9 E + 0 1 0 3 8 6 4 4 E - 15 4 6 2 5 0. I 2 2 9 4 E » 0 1 0 . 1 7 5 0 2 E * 0 0 0 . 1 0 2 4 3 E * 0 1 0 3 7 9 6 5 E * 0 0 0 6 2 8 5 1 E + 0 1 c 6 2 6 0 1 E - > O I 0 3 8 6 4 2 E - 15 4 6 5 0 0. 1 2 2 9 5 E + 0 1 0 . 1 7 5 0 1 E * 0 0 0 . 1 0 2 4 3 E + 0 1 0 3 7 9 6 5 E + 0 O 0 6 2 8 4 5 E * 0 1 0 6 2 6 0 3 E + 0 1 0 3 8 6 4 0 E - 15 4 6 7 5 ' 0 . 1 2 2 9 5 E ' 0 1 0 . 1 7 4 9 9 E + 0 0 0 . 1 0 2 4 3 E * 0 1 0 3 7 9 6 6 E + 0 0 0 6 2 8 3 9 E + 0 1 0 6 2 6 0 4 E + 0 1 0 3 8 6 3 9 E - 15 47 0 0 0 . 1 2 2 9 5 E + 0 1 0 . 1 7 4 9 8 E * 0 0 0 . 1 0 2 4 4 E * 0 1 0 3 7 9 6 6 E * 0 0 0 6 2 8 3 4 E + 0 1 0 6 2 6 0 6 E + 0 1 0 3 8 6 3 7 E - 15 4 7 2 5 0. 1 2 2 9 5 E + 0 1 0 . 1 7 4 9 7 E - » 0 0 0 . 1 0 2 4 4 E * O 1 0 3 7 9 6 6 E + 0 0 0 6 2 8 2 9 E + 0 1 0 6 2 6 0 7 E + 0 1 0 3 8 6 3 6 E - 15 4 7 S O 0. I 2 2 9 6 E ' 0 1 0 . I 7 4 9 5 E ' 0 0 0 . 1 0 2 4 4 E * 0 1 0 3 7 9 6 7 E + 0 0 0 6 2 8 2 4 E > 0 1 0 6 2 6 0 9 E - ' 0 1 0 3 8 6 3 4 E - 15 4 7 7 5 0 . 1 2 2 9 G E » 0 1 0 . 1 7 4 9 4 E * 0 0 0 . 1 0 2 4 4 E » 0 1 0 3 7 9 G 7 E + 0 0 0 6 2 8 1 9 E + 0 1 0 6 2 6 1 0 E + 0 1 0 3 8 6 3 3 E - 15 4 8 0 0 0 . 1 2 2 9 6 E « 0 1 0 . 1 7 4 9 3 E < - 0 0 0 . 1 0 2 4 5 E + O 1 0 3 7 9 6 8 E + 0 0 0 6 2 8 1 4 E + 0 1 0 6 2 6 1 2 E + 0 1 0 3 8 6 3 1 E - 15 4 8 2 5 0 . 1 2 2 9 7 E ' 0 1 0 . 1 7 4 9 2 E * 0 0 0 . 1 0 2 4 5 E + 0 1 0 3 7 9 6 8 E + 0 0 0 6 2 8 0 9 E * 0 1 0 6 2 6 1 3 E » 0 1 0 3 8 6 3 0 E - 15 4 8 5 0 0 . 1 2 2 9 7 E * 0 1 0 . 1 7 4 9 1 E + 0 0 0. 1 0 2 4 5 E + 0 1 0 3 7 9 6 8 E + 0 0 0 6 2 8 0 5 E + 0 1 0 6 2 6 1 5 E + 0 1 0 3 8 6 2 9 E - 15 238 48 75 0 . 12297E 01 0 17490E*00 0 10245E->01 0 37969E<00 0 62801E+01 0 62616E+01 0 3B628E- 15 J<J OO 0. 1 2207E 01 0 17489E'00 0 I0246E<0I 0 379G9E<00 0 62797E »01 0 6 2 6 1 7 E » 0 1 0 38G27E- 15 49 25 0. 12298E 01 0 1748BE<00 0 1024GE'O1 0 3 7 9 6 9 E » 0 0 0 G2793E*01 0 62618E*01 0 38625E- 15 49 50 0. 12298E 01 0 17487E*00 0 10246E*01 0 37970E*00 0 62789E*01 0 62620E+01 0 38G24E- 15 49 75 0. 122U8E 01 0 1748GE'00 0 10246E*01 0 37970E*O0 0 G2785E+01 0 62621E+01 0 38623E- 15 50 OO 0. 12298E 01 0 1 7 4 8 5 E » 0 0 0 10246E+01 0 37970E*00 0 627B2E*01 0 62622E*01 0 38622E- 15 TIME CSI CAVS! 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S O 12 . 75 13 . 0 0 13 . 2 5 1 3 . 5 0 13 .75 14 . 0 0 14 . 2 5 14 . 5 0 14 .75 15 . 0 0 1 5 . 2 5 0 1 9 9 4 0 E >02 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 I 9 9 4 0 E • 0 2 0 I 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E * 0 2 0 I 9 9 4 0 E • 0 2 0 1 9 9 4 0 E * 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E * 0 2 0 1 9 9 4 0 E * 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 I 9 9 4 0 E * 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E * 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E * 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 4 0 E • 0 2 0 1 9 9 J 0 E • 0 2 0 1 9 9 4 0 E * 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 S 4 0 E * 0 2 0 2 4 5 4 0 E « 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E * 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 S 4 0 E • 0 2 0 2 4 S 4 0 E ' 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 S 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 S 4 0 E ' 0 2 0 2 4 5 4 0 E • 0 2 0 2 4 5 4 0 E • 0 2 0 . 3 4 7 2 2 E - 0 0 . 2 4 6 4 4 E » 0 O . 1 9 3 4 7 E * 0 0 . 1 5 6 9 8 E » 0 O . 1 2 3 5 6 E * 0 O . 1 0 6 1 5 E + 0 0 . 1 0 9 5 7 E * 0 O . 1 2 9 5 2 E + 0 O . 158 I 7 E + 0 O . I 9 5 5 6 E * 0 0 . 2 3 4 0 8 E » 0 O . 2 5 3 I 9 E » 0 0 . 2 7 1 6 5 E + 0 0 . 2 8 9 2 1 E + 0 O . 3 O 5 9 2 E + 0 O . 3 2 I 8 6 E * 0 0 . 3 3 7 0 9 E * 0 O . 3 5 1 6 7 E » 0 0 . 3 6 S 6 9 E * 0 0 . 3 7 9 1 8 E * 0 O 3 9 2 2 0 E < 0 0 . 4 O 4 7 8 E * 0 0 . 4 1 6 9 I E + 0 O . 4 2 8 6 7 E * 0 0 . 4 4 0 1 4 E + 0 0 . 4 5 1 2 8 E » 0 0 . 4 6 2 1 1 E * 0 0 . 4 7 2 6 5 E » 0 O . 4 8 2 9 2 E + 0 O . 4 9 2 9 2 E * 0 O . 5 0 2 6 7 E » 0 O . S 1 2 1 8 E * 0 0 . 5 2 1 4 6 E + 0 0 . 5 3 0 S 1 E « 0 0 . 5 3 9 3 0 E * 0 0 . 5 5 7 3 0 E * 0 0 . 5 5 2 0 7 E + 0 0 . 5 5 7 5 8 E + 0 0 . 5 G 4 9 3 E * 0 O . 5 7 2 4 6 E ^ 0 O . 5 7 9 8 6 E + 0 0 . 5 8 7 0 7 E + 0 O . 5 9 4 0 8 E + 0 0 . 6 0 0 9 0 E + 0 0 . 6 0 7 5 0 E » 0 0 . 6 1 3 9 0 E + 0 O . 6 2 O O 7 E + 0 0 . 6 2 5 9 5 E + 0 0 . G 3 1 5 O E + 0 C . G 3 G 6 9 E + 0 0 . 6 4 1 5 2 E * 0 0 . 6 4 5 9 5 E » 0 0 . 6 4 9 9 3 E * 0 0 . 6 5 3 3 8 E * 0 0 . G S G 2 8 E - 0 0 . G 5 8 6 0 E + 0 0 . G 6 0 3 9 E + 0 0 . 6 6 1 7 3 E + 0 0 . 6 6 2 6 9 E + 0 0 . 6 6 3 3 4 E + 0 O . 1 8 4 6 6 E + 0 2 0 . 1 7 9 2 5 E * 0 2 O . 1 7 4 1 7 E « 0 2 0 . 1 7 1 6 1 E + 0 2 0 . I 7 0 6 3 E * 0 2 0 . 1 7 0 0 4 E + 0 2 0 . 1 6 9 4 1 E * 0 2 O . I 6 8 7 6 E + 0 2 O . 1 6 8 2 1 E + 0 2 O . 1 6 7 6 7 E * 0 2 O . 1 6 7 7 9 E * 0 2 O . 1 6 8 9 7 E + 0 2 0 . 1 7 0 0 5 E * 0 2 0 . 1 7 1 0 6 E * O 2 O . 1 7 2 0 2 E * 0 2 0 . 1 7 2 9 3 E + 0 2 O . 1 7 3 8 0 E + O 2 O . 17464E-I -02 0 . 1 7 5 4 4 E + 0 2 O . 1 7 6 2 1 E * 0 2 O . I 7 6 9 5 E + 0 2 0 . 1 7 7 6 7 E + 0 2 O . 1 7 8 3 6 E * 0 2 O . 1 7 9 0 3 E + 0 2 0 . 1 7 9 6 9 E + 0 2 O . 1 8 0 3 2 E + 0 2 0 . 1 8 0 9 4 E + 0 2 O . 1 8 1 5 4 E + 0 2 O . 1 8 2 1 3 E + 0 2 0 . 1 8 2 7 0 E + 0 2 O . 1 8 3 2 6 E + 0 2 O . 1 8 3 8 0 E * 0 2 0 . 1 8 4 3 3 E + 0 2 O . 1 8 4 8 4 E + 0 2 O . 1 8 5 3 4 E + 0 2 O . 1 9 3 4 9 E + 0 2 O . 1 9 5 4 9 E * 0 2 O . 1 9 G 2 4 E + 0 2 0 . 1 9 6 7 3 E + 0 2 0 . 1 9 7 1 8 E + 0 2 O . 1 9 7 6 0 E + 0 2 0 . 1 9 8 0 1 E + 0 2 O . I 9 8 4 0 E + 0 2 O . 1 9 8 7 9 E + 0 2 0 . 1 9 9 1 6 E + 0 2 O . 1 9 9 5 2 E + 0 2 O . 1 9 9 8 7 E + 0 2 O . 2 0 0 2 1 E + 0 2 O . 2 0 0 5 4 E + 0 2 O . 2 0 0 8 G E + 0 2 0 . 2 0 1 1 6 E » 0 2 0 . 2 0 1 4 5 E + 0 2 O . 2 0 1 7 3 E + 0 2 0 . 2 0 1 9 8 E « 0 2 O . 2 0 2 2 2 E + 0 2 O . 2 0 2 4 5 E * 0 2 0 . 2 0 2 6 4 E + 0 2 0 . 2 0 2 8 1 E * 0 2 0 . 2 0 2 9 5 E * 0 2 O . 2 O 3 0 6 E + O 2 O . 3 4 4 0 6 E + 0 2 O . 3 3 8 5 5 E * 0 2 O . 3 3 3 2 4 E « 0 2 O . 3 3 0 5 2 E + 0 2 0 . 3 2 9 4 7 E + 0 2 O . 3 2 8 8 2 E + 0 2 0 . 3 2 8 1 4 E + 0 2 O . 3 2 7 4 4 E * 0 2 O . 3 2 6 B 4 E + 0 2 0 . 3 2 6 2 5 E * 0 2 0 . 3 2 6 3 8 E + 0 2 O . 3 2 7 6 7 E * 0 2 0 . 3 2 8 8 4 E + 0 2 O . 3 2 9 9 3 E + 0 2 0 . 3 3 0 9 6 E * 0 2 0 . 3 3 1 9 3 E + 0 2 0 . 3 3 2 8 5 E + 0 2 0 . 3 3 3 7 3 E * 0 2 0 . 3 3 4 5 8 E + 0 2 O . 3 3 5 3 9 E + 0 2 O . 3 3 G I 6 E * 0 2 0 . 3 3 6 9 1 E + 0 2 O . 3 3 7 6 3 E + 0 2 0 . 3 3 8 3 3 E * 0 2 0 . 3 3 9 0 0 E + 0 2 0 . 3 3 9 6 6 E * 0 2 0 . 3 4 0 2 9 E + 0 2 0 . 3 4 0 9 1 E + 0 2 0 . 3 4 1 5 0 E + O 2 0 . 3 4 2 0 8 E + 0 2 0 . 3 4 2 6 5 E + 0 2 O . 3 4 3 2 0 E + 0 2 0 . 3 4 3 7 3 E + 0 2 O . 3 4 4 2 5 E + 0 2 O . 3 4 4 7 5 E + 0 2 0 . 3 5 2 7 8 E + 0 2 0 . 3 5 4 7 2 E + 0 2 0 . 3 5 5 4 3 E + 0 2 0 . 3 5 5 9 0 E + 0 2 0 . 3 5 6 3 2 E + 0 2 O . 3 5 6 7 3 E + 0 2 0 . 3 5 7 1 1 E * 0 2 O . 3 5 7 4 9 E * 0 2 0 . 3 5 7 8 5 E + 0 2 0 . 3 5 8 2 1 E * 0 2 0 . 3 5 8 5 5 £ » 0 2 O . 3 5 8 8 8 E * 0 2 0 . 3 5 9 2 0 E * 0 2 0 . 3 5 9 5 1 E + 0 2 0 . 3 5 9 8 1 E + 0 2 0 . 3 6 0 0 9 E + 0 2 0 . 3 6 0 3 6 E + 0 2 O . 3 6 0 6 2 E + 0 2 0 . 3 6 0 8 6 E + O 2 O . 3 6 1 0 9 E + 0 2 O . 3 6 1 2 9 E + 0 2 0 . 3 6 1 4 7 E + 0 2 0 . 3 6 1 6 3 E * 0 2 0 . 3 6 1 7 6 E * 0 2 0 . 3 6 1 B 7 E + 0 2 C M o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o S S S S S S S S S S S S S B S S S S S S S S S S S ^ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o SSSSSSSSSSSSSS'SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS iiiiiiiiliiiiiiiiiliiiiiiliiM r. 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O O 4 2 . 2 5 4 2 . 5 0 4 2 . 7 5 4 3 . 0 0 4 3 . 2 5 4 3 . 5 0 4 3 . 7 5 4 4 . 0 0 4 4 . 2 5 4 4 . 5 0 4 4 . 7 5 4 5 . 0 0 4 5 . 2 5 0 . 245.10L" - 0 2 0 . 2 4 5 4 0 E « 0 2 0 . 2 4 5 4 0 E +02 0 . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + O 2 0 . 2 4 5 1 0 E - 0 2 O . 2 4 5 1 0 E + O 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E +02 0 . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + O 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 0 4 O E +02 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 S 4 0 E + 0 2 O . 2 4 5 4 0 E + O 2 O . 2 4 5 4 0 E <02 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E > 0 2 0 . 2 4 5 4 0 E + O 2 O . 2 4 5 4 0 E >02 O . 2 4 S 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 G E . 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E --02 0 . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 0 . 2 4 5 4 0 E + 0 2 O . 2 4 S 4 0 E + 0 2 O . 2 4 5 4 0 E +02 O . 2 4 5 4 0 E + C 2 O . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E >02 0 . 2 4 5 4 0 E + 0 2 O . 2 4 5 4 0 E + 0 2 O . 6 0 8 1 6 E + 0 1 O . G 0 7 6 2 E + 0 1 O . 6 0 7 1 0 E + 0 1 0 . 6 0 6 6 0 E + 0 1 0 . 6 0 6 I 1E +01 0 . 6 0 S 6 4 E + O I 0 . 6 0 5 1 8 E + 0 1 O . 6 0 4 7 4 E +0 1 0 . 6 0 4 3 0 E + 0 1 0 . 6 0 3 8 9 E + 0 1 0 . 6 0 3 4 8 E + 0 1 0 . 6 0 3 0 9 E + 0 1 0 . 6 0 2 7 1 E + 0 1 0 . 6 0 2 3 4 E + 0 1 0 . 6 0 1 9 8 E » 0 1 0 . 6 0 1 6 4 E + 0 1 0 . 6 0 1 3 0 E + 0 1 0 . 6 0 0 9 8 E + 0 1 0 . 6 0 0 6 6 E + 0 1 0 . 6 0 0 3 6 E + 0 1 0 . 6 0 0 0 7 E + 0 1 0 . 5 9 9 7 8 E + 0 1 0 . 5 9 9 5 1 E + 0 1 0 . 5 9 9 2 4 E + 0 1 0 . 5 9 8 9 9 E + 0 1 O . 5 9 8 7 4 E + 0 1 O . 5 9 8 5 0 E + 0 I 0 . 5 9 8 2 6 E + O I O . 5 9 8 0 4 E + 0 1 0 . 5 9 7 8 2 E + 0 1 O . 5 9 7 6 1 E + 0 1 O . 5 9 7 4 1 E + 0 1 O . 5 9 7 2 1 E + 0 1 O . 5 9 7 0 3 E + 0 1 O . S 9 6 8 4 E + 0 1 0 . 5 9 6 6 7 E + 0 1 0 . 5 9 6 5 0 E + 0 1 0 . 5 9 6 3 3 E + 0 1 0 . 5 9 6 1 7 E + 0 1 O . 5 9 6 0 2 E + 0 1 0 . 5 9 5 8 7 E + 0 1 0 . 5 9 5 7 3 E + 0 1 0 . 5 9 5 5 9 E + 0 1 O . 5 9 5 4 6 E + 0 1 O . 5 9 5 3 3 E + 0 1 O 5 9 5 2 1 E + 0 1 O . 5 9 5 0 9 E + 0 1 O . 5 9 4 9 7 E + 0 1 O . 5 9 4 8 6 E + 0 1 O . 5 9 4 7 5 E + 0 1 0 . 5 9 4 6 5 E + 0 1 0 . 5 9 4 5 5 E + 0 1 0 . 5 9 4 4 6 E + 0 1 0 . 5 9 4 3 7 E + 0 1 O . 5 9 4 2 9 E + 0 1 0 . 5 9 4 2 1 E + 0 1 0 . 5 9 4 1 4 E + 0 1 0 . 5 9 4 0 7 E + 0 1 O . 5 9 4 O O E + 0 1 O . 5 9 3 9 4 E + 0 I 0 . 2 0 0 9 2 E + 0 2 0 . 2 0 0 8 9 E + 0 2 0 . 2 0 0 8 7 E + 0 2 O . 2 0 0 8 4 E + 0 2 0 . 2 0 0 8 I E + 0 2 O . 2 0 0 7 9 E + 0 2 0 . 2 0 0 7 6 E + 0 2 O . 2 0 0 7 4 E + 0 2 0 . 2 0 0 7 1 E + 0 2 O . 2 0 0 6 9 E + 0 2 0 . 2 0 0 6 7 E + 0 2 0 . 2 0 0 6 4 E + 0 2 O . 2 0 O 6 2 E + O 2 0 . 2 0 0 6 0 E + 0 2 O . 2 0 0 S 8 E + 0 2 0 . 2 0 O 5 6 E + 0 2 0 . 2 0 0 5 4 E + 0 2 0 . 2 0 0 S 3 E + 0 2 0 . 2 0 0 5 I E + 0 2 0 . 2 0 0 4 9 E + 0 2 0 . 2 0 O 4 8 E + 0 2 0 . 2 0 0 4 6 E + 0 2 0 . 2 0 0 4 4 E +02 0 . 2 0 0 4 3 E + 0 2 0 . 2 0 0 4 1 E + 0 2 0 . 2 0 0 4 0 E + 0 2 O . 2 0 O 3 9 E + O 2 O . 2 0 0 3 7 E + 0 2 O . 2 0 0 3 6 E + 0 2 O 2 0 0 3 5 E + 0 2 O . 2 0 0 3 4 E + 0 2 0 . 2 O 0 3 2 E + O 2 0 . 2 0 0 3 I E + 0 2 0 . 2 0 O 3 0 E + 0 2 O . 2 0 0 2 9 E + 0 2 O . 2 O O 2 8 E + 0 2 0 . 2 O O 2 7 E + 0 2 O . 2 0 0 2 6 E + 0 2 0 . 2 0 0 2 5 E + 0 2 0 . 2 O O 2 5 E + O 2 O . 2 0 0 2 4 E + 0 2 0 . 2 0 O 2 3 E + 0 2 O . 2 0 0 2 2 E + 0 2 O . 2 0 0 2 1 E + 0 2 O . 2 0 0 2 1 E + 0 2 O . 2 0 0 2 0 E + 0 2 O . 2 0 0 I 9 E + 0 2 O . 2 0 0 1 8 E + 0 2 0 . 2 0 0 1 8 E + 0 2 0 . 2 0 0 1 7 E + 0 2 0 . 2 0 0 1 7 E + 0 2 O . 2 0 0 1 6 E + 0 2 0 . 2 0 0 1 5 E + 0 2 O . 2 0 0 1 5 E + 0 2 O . 2 0 0 1 4 E + 0 2 0 . 2 0 0 1 4 E + 0 2 O . 2 O O 1 3 E + 0 2 O . 2 0 0 1 3 E + 0 2 0 . 2 0 0 1 2 E + 0 2 O . 2 O 0 1 2 E + 0 2 O . 3 5 9 8 7 E + 0 2 0 . 3 5 9 8 4 E + 0 2 0 . 3 5 9 8 1 E + 0 2 0 . 3 5 9 7 9 E + 0 2 0 . 3 5 9 7 6 E + 0 2 O . 3 5 9 7 4 E + 0 2 O . 3 5 9 7 1 E + 0 2 O . 3 5 9 6 9 E + 0 2 O . 3 5 9 6 7 E + 0 2 O . 3 5 9 6 5 E + 0 2 0 . 3 5 9 6 3 E + O 2 0 . 3 5 9 6 1 E + 0 2 0 . 3 5 9 5 9 E + 0 2 0 . 3 5 9 5 7 E + 0 2 0 . 3 5 9 5 5 E + 0 2 0 . 3 5 9 5 3 E + 0 2 0 . 3 5 9 5 1 E + 0 2 O . 3 5 9 5 0 E + 0 2 0 . 3 5 9 4 8 E + 0 2 0 . 3 5 9 4 6 E + 0 2 O . 3 5 9 4 5 E + 0 2 0 . 3 5 9 4 3 E + 0 2 0 . 3 5 9 4 2 E + 0 2 O . 3 5 9 4 0 E + O 2 0 . 3 5 9 3 9 E + 0 2 O . 3 5 9 3 8 E + 0 2 O . 3 5 9 3 6 E + 0 2 0 . 3 5 9 3 5 E + O 2 O . 3 5 9 3 4 E + 0 2 0 . 3 5 9 3 3 E + O 2 O . 3 5 9 3 2 E + 0 2 0 . 3 S 9 3 1 E + 0 2 O . 3 5 9 3 0 E + 0 2 0 . 3 5 9 2 9 E + 0 2 O . 3 5 9 2 8 E + 0 2 0 . 3 5 9 2 7 E + 0 2 0 . 3 5 9 2 6 E + 0 2 O . 3 5 9 2 5 E + 0 2 0 . 3 5 9 2 4 E + 0 2 O . 3 5 9 2 3 E + 0 2 O . 3 5 9 2 2 E + 0 2 0 . 3 5 9 2 1 E + O 2 O . 3 5 9 2 1 E + 0 2 O . 3 S 9 2 0 E + 0 2 O . 3 5 9 1 9 E + 0 2 O . 3 5 9 1 9 E + 0 2 O . 3 5 9 1 8 E + 0 2 O . 3 5 9 1 7 E + 0 2 0 . 3 5 9 1 7 E + 0 2 0 . 3 5 9 1 6 E + 0 2 0 . 3 5 9 1 6 E + 0 2 0 . 3 5 9 1 5 E + 0 2 0 . 3 5 9 1 5 E + 0 2 0 . 3 5 9 1 4 E + 0 2 O . 3 5 9 1 3 E + 0 2 O . 3 5 9 1 3 E + 0 2 0 . 3 5 9 1 3 E + 0 2 O . 3 5 9 1 2 E + 0 2 0 . 3 5 9 1 2 E + 0 2 O . 3 5 9 1 1 E + 0 2 8 S S S 8 S l S S 8 S S S 8 S S S 8 i s ! S o o o o o o o o o o c o o o o o o o o H i i i i i i i S i i i i S i l i S sssssssssss-ssssssss o o o o o o o o o o o c o o o o o o o s i - L s | - U H 0 M > l - L n O U l O C i O < n - - J m m m m m m m m m m r n m m m m m 0 6 0 0 6 0 0 0 0 6 0 0 0 0 6 0 6 0 6 O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o IIIIISIIIIIIIIIIIII T r a n s l e n t r e s p o n s e f o r J.0% b u r n T i m e V P L V S 0 . 0 6 . 1 2 0 0 0 10 . 1 2 8 0 0 0 . 2 5 0 0 0 5 . 7 6 7 3 2 10 1 8 4 6 8 0 . 5 0 0 0 0 5 . 4 5 8 2 3 10 1 7 0 6 4 0 . 7 5 0 0 0 5 . 1 5 1 4 6 1 0 1 8 0 2 0 1 OOOOO 4 . 9 0 8 7 0 1 0 . 2 0 4 3 8 1 . 2 5 0 0 0 4 . 7 0 7 8 9 1 0 . 2 3 4 2 3 1 . 5 0 0 0 0 4 .50719 1 0 . 2 6 0 3 8 1 . 7 5 0 0 0 4 . 3 5 6 4 1 1 0 . 2 8 1 6 3 2 . 0 0 1 0 0 4 . 2 8 6 10 1 0 . 3 0 6 0 4 o . 2 5 0 0 O 4 . 2 9 3 16 1 0 . 3 3 8 3 2 2 . 5 0 0 0 0 4 . 3 5 9 6 4 1 0 . 3 7 6 5 9 2 . 7 5 0 0 0 4 . 4 8 9 5 6 1 0 . 4 1 8 2 6 3 . 0 0 1 0 0 4 . 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