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LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAIGN 
 
 510.64 
 
 no. 613-^17 
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 December 1973 
 
 A TWO COMPARTMENTAL MODEL OF THE 
 RESPIRATORY SYSTEM 
 
 by 
 
 Paul Robert Haskitt 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
 THE LIBRARY OF THE 
 
 FEB 15 1974 
 
 ' 1 OF ILLINOIS 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/twocompartmental616hask 
 
UIUCDCS-R-73-6l6 
 
 A TWO COMPARTMENTAL MODEL OF THE 
 RESPIRATORY SYSTEM 
 
 by 
 
 Paul Robert Haskitt 
 
 December 1973 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 
 UNIVERSITY OF ILLINOIS AT URBANA- CHAMPAIGN 
 
 URBANA, ILLINOIS 6l801 
 
 This work was supported in part by the Department of Computer Science 
 and submitted in partial fulfillment for the Master of Science degree 
 in Computer Science, 1973. 
 
1X1 
 
 ACKNOWLEDGMENT 
 
 I would like to thank the following people for all their help 
 in making this thesis possible. To my advisor, Professor D. Watanabe, 
 I owe a starting point and also the acquisition of financial backing 
 which made the testing of this model possible. Thanks also goes to 
 Mr. Al Whaley for his help in the computer implementation of this model 
 using the GPASS system now being developed at the University of Illinois. 
 
IV 
 
 PREFACE 
 
 This thesis describes a two c 01153 art mental model of the human 
 
 respiratory control system. The goal of this model is a balance 
 
 between simplicity and accuracy; more detailed models can be found in 
 the references. 
 
TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION 1 
 
 2. BACKGROUND 3 
 
 3. GENERAL DESCRIPTION OF THE SYSTEM 6 
 
 3.1 Physical Description 6 
 
 3.2 System Assumptions 7 
 
 k. DERIVATION OF STEADY STATE SYSTEM EQUATIONS 10 
 
 k.l Gas Balance Equations 12 
 
 k.2 Concentration Equations 13 
 
 k.3 Data Fitting Equations lk 
 
 4.^4 Steady State System Equations 17 
 
 5. RESULTS OF THE STEADY STATE SYSTEM 18 
 
 5.1 Hypoxia at Altitude 19 
 
 5.2 Hypoxia at Sea Level 20 
 
 5.3 Carbon Dioxide Inhalation 22 
 
 5.U Metabolic Disturbances 23 
 
 6. OVERALL EVALUATION OF THE STEADY STATE MODEL 27 
 
 7. THE DYNAMIC SYSTEM 29 
 
 7.1 Background 29 
 
 7-2 Evaluation of the Dynamic Model 33 
 
 8. CONCLUSIONS ABOUT THE TOTAL MODEL 37 
 
 LIST OF REFERENCES 38 
 
 APPENDICES 
 
 A. SYSTEMS AND UNITS FOR EQUATIONS OF SYSTEM 39 
 
 B. NORMAL PARAMETER VALUES 1+1 
 
VI 
 
 LIST OF TABLES 
 
 Table Page 
 
 1. Partial Pressures of Body Gases 11 
 
 2. Comparison of Model Hypoxia Results 21 
 
 3. Comparison of Model CO Inspiration Results 2 3 
 
 k. Classification of Work Intensity 2k 
 
 5. Accumulated Steady State Results 28 
 
 6. Body Gas Stores 31 
 
 7. Transient System Response to MR = 1.00 35 
 
 2 
 
 8. Transient System Response to MR = 2.00 36 
 
 2 
 
 A-l. Table of Symbols and Units kO 
 
 B-l. "Normal" Parameter Values U2 
 
VI 1 
 
 LIST OF FIGURES 
 
 Figure Page 
 
 1. Two Compartment System 6 
 
 2. Dissociation Curve lU 
 
 3. Altitude Hypoxia 19 
 
 k. Hypoxia at Sea Level 20 
 
 5. System Response to CO Inspiration 22 
 
 6. System Respiratory Response to Consumption 25 
 
 7. System Blood Flow Response to Consumption 2 6 
 
 8. Cardiac Output Transient Response to Exercise 3^ 
 
1. INTRODUCTION 
 
 The "closed- loop" characteristic of the respiratory control 
 system was first recognized, at least implicitly, in the works of 
 Haldane and Priestly [l] around 1905. Gray [2, 3] was responsible for 
 the first explicit algebraic model of the respiratory control system 
 in 19^-5 • Gray's model was restricted to steady -state responses. 
 However, the model was capable of handling such forcings as CO 
 inhalation, arterial anoxemia, and metabolic disturbances in the acid- 
 base balance. In 195^ a model based on Gray's model was proposed [h]. 
 Although able to handle dynamic simulation studies, this model was 
 severely restricted in that it only considered the forcing of carbon 
 dioxide inhalation. These early models had to be kept fairly simple 
 because the computational difficulty of actually running a model 
 increased considerably with even slight increases in the complexity of 
 the model itself. 
 
 As larger, more powerful computing facilities were realized, 
 interest in biological systems simulation began to increase proportion- 
 ately. Since Defares, Derkson, and Duyff [5] began refining the 
 original dynamic model of the respiratory control system in i960, many 
 improved models, both analog and digital, have appeared [6, 7]. 
 Milhorn et al. [7] in 1965 and later Grodins et al. [8] in 1967 have 
 produced two of the more recent of these models. 
 
 Although both of these latter models, especially Grodin's 
 model, are extremely accurate in their representations, they are also 
 extremely complex. For instance, Grodin's model requires the 
 simultaneous solution of a system of 12 nonlinear differential equations, 
 which is not an easy or inexpensive task, even with the aid of 
 computer technology. 
 
The purpose of this model of the respiratory system is to 
 simplify the horrendous calculations needed for a solution of previous 
 detailed models, while preserving the capability of handling such 
 forcings as CO inhalation, hypoxia at sea level or altitude, metabolic 
 disturbances, or any combination of these in the steady state, and 
 metabolic disturbances in a transient analysis. A number of assumptions 
 and approximations are made to keep the model as simple and easy to 
 handle as possible. Although these modifications are, of course, 
 accompanied by small errors, it is believed that the ratio of accuracy 
 to complexity of this model is at least as good as those of other 
 current models . 
 
2 . BACKGROUND 
 
 Respiratory gases which are located near the alveolar capillary 
 walls are called alveolar gases. It is this gas that supplies oxygen 
 molecules to the "blood and receives carbon dioxide from the blood through 
 the porous membrane of the alveoli. 
 
 The two most plausible definitions for the composition of mean 
 alveolar gas are l) a mixture of the different gases leaving the various 
 alveoli during expiration, and 2) a mixture of the different gases 
 contained in the various alveoli (an imaginary mixture). 
 
 In either case, the very slight differences in these seeminly 
 acceptable definitions raise some doubt as to the interpretation and 
 actual importance of the small differences between the alveolar and 
 arterial partial pressures of and CCL. 
 
 The factor generally used to relate carbon dioxide output rate 
 to oxygen uptake rate by the body is their ratio, which is called the 
 respiratory quotient, RQ. In the actual gas exchange of the tissue cells, 
 the amount of carbon dioxide output is less than the oxygen uptake. 
 Thus, the respiratory quotient is less than one. 
 
 Another factor, closely related to the respiratory quotient, 
 is the respiratory gas exchange ration, R. This is the ratio of the 
 amount of expired gases to inspired gases. The respiratory quotient 
 is the metabolic gas exchange ratio between the blood and the tissue. 
 
 During a steady state condition, the respiratory quotient 
 remains constant and equal to the gas exchange ratio. In this state 
 the amount of oxygen inspired is sufficient to supply the body's 
 metabolic needs without affecting the oxygen stores in the body. 
 
In an unsteady state, however, some of the gas used to satisfy 
 the metabolic needs may come from oxygen stores, thus causing the gas 
 exchange ratio to become larger than the respiratory quotient because 
 the amount of inspired gas (the denominator) becomes smaller. As the 
 body gas stores are replenished, the gas exchange ratio begins to 
 approach the respiratory quotient as a steady state is reached. As 
 long as the body gas stores are undergoing readjustments, the respiratory 
 gas exchange ratio differs from the respiratory quotient. This is not 
 to imply that in the steady state the actual body gas stores are 
 constant, but rather that their amounts are nearly constant. The gas 
 in the stores is, of course, constantly being used and replaced with 
 "fresh" gas. 
 
 The control of alveolar ventilation can be affected by many 
 factors as suggested by Gray [2]. Based on laboratory experiments, 
 there are three main factors that can increase ventilation: l) the 
 administration of a fixed acid, 2) the inhalation of a carbon dioxide 
 rich atmosphere, and 3) the inhalation of an oxygen poor atmosphere. 
 There are, of course, other factors that may also increase ventilation, 
 but the three mentioned above seem to be the most important. The three 
 factors then must be based on the amount or concentration of acid, 
 carbon dioxide, and oxygen at some point in the body or atmosphere. 
 
 The factors are considered to be measured in arterial blood. 
 The che mo receptors of the carotid and aortic bodies are exposed to 
 arterial blood. Changes in external respiratory controls seem to affect 
 the composition of venous blood only indirectly. It is, therefore, 
 logical and consistent that arterial concentrations of the three chemical 
 
agents control the respiration rate. For the concentrations of the two 
 gases, carbon dioxide and oxygen, the partial pressures of the gases is 
 an appropriate substitute. The partial pressure of a gas which is in a 
 gas mixture occupying a certain volume is the pressure that gas would 
 exert in the same volume in the absence of the other gases in the mixture. 
 The concentration of acid in the blood is represented by the hydrogen 
 ion concentration in arterial blood. 
 
 Laboratory experiments have shown that these three chemical 
 factors affect each other. The Haldane effect is one instance of the 
 influence of one factor on another. This is the phenomenon where at 
 any partial pressure of carbon dioxide, the CO content decreases as 
 the partial pressure of oxygen increases. In other words the carbon 
 dioxide partial pressure required to establish a given CO content 
 increases with the partial pressure of . 
 
 The Bohr effect is the counterpart of the Haldane effect for 
 the partial pressure of carbon dioxide. At a given partial pressure 
 of oxygen, the saturation is decreased by an elevation in the partial 
 pressure of carbon dioxide. That is, an increase in the partial pressure 
 of CO decreases the affinity of hemoglobin for oxygen. 
 
 Variations in pH (H ion concentration) have the same effect as 
 the partial pressure of CO in the Bohr effect. An increase in the 
 partial pressure of CO causes the pH to be lowered. Therefore, it is 
 not exactly clear whether the Bohr effect is a result of the partial 
 pressure of CO or a result of a change in pH. The most widely accepted 
 theory is that the partial pressure of CO rather than the pH effect is 
 directly responsible for the Bohr effect. 
 
3. GENERAL DESCRIPTION OF THE SYSTEM 
 
 3. 1 Physical Description 
 
 The model consists of a lung reservoir and a lumped tissue 
 reservoir. The lungs are regarded as a rigid box of constant volume, 
 uniform content, and zero dead space which is aerated by a continuous 
 flow of gas • 
 
 
 V TP 
 
 
 LUNG F A 
 
 
 *V TP 
 
 
 V l' r i 
 
 
 
 * V E^E 
 
 
 ^ ->n n 
 
 
 
 V 
 
 ' V >y u 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 TISSUE MR 
 
 
 
 
 
 
 
 
 
 
 Figure 1. Two Compartment System 
 
7 
 
 The TISSUE box is a lumped representative of the body's tissues. It is 
 characterized by the tissue metabolic rate, which is the rate in liters 
 per minute that is used or CO is produced, depending on the subscript 
 (see appendix). 
 
 The LUNG and TISSUE boxes are joined by a lumped model of the 
 circulatory system. 
 
 An uninterrupted stream of gas flows into and out of the LUNG 
 box. In the box a certain amount of oxygen passes through the thin 
 membrane of the alveoli and enters the blood. At this same interface 
 carbon dioxide passes from the venous blood into the LUNG box and flows 
 out as part of the exhaled flow of gas. This passage of gases through 
 the alveolar membrane is caused by the diffusion mechanism. Since the 
 concentration gradient causes gas molecules to traverse the porous 
 membrane, the input atmosphere has a great affect on how much gas is 
 added to or removed from the blood. The arterial blood then carries 
 these gases to the TISSUE box. In the TISSUE box oxygen is removed and 
 carbon dioxide is added to the blood at a rate dictated by the metabolic 
 rate. Then these gases are carried by the venous blood back to the 
 lung box. 
 
 3.2 System Assumptions 
 
 A number of assumptions were made when this system was designed: 
 
 1) The lungs are a box of constant volume, uniform content, and 
 zero dead space ventilated by a continuous unidirectional stream of gas. 
 
 2) Rapid phasic changes in alveolar and blood gas concentrations 
 with each respiratory cycle are ignored. Variations caused by cyclic 
 changes in respiration are not of primary interest here and will be ignored. 
 
8 
 
 3) Arterial gas pressures, alveolar gas pressures, and gas 
 pressures in expired air are all equal at all times. 
 
 h) Nitrogen, the third major component in air, does not take 
 part in the respiration process. Although nitrogen does enter the blood 
 in dissolved form it does not, for the most part, leave the blood until 
 the gas is expired. 
 
 5) Alveolar ventilation is a function of hydrogen ion 
 concentration, and alveolar carbon dioxide and oxygen partial pressures. 
 This assumption comes from Gray's multiple factor theory [2] and has 
 been used and supported by many researchers since Gray. Based on the 
 multiple factor theory, the total effect of the three chemical agents 
 
 is taken to be the algebraic sum of the separate effects of each of 
 the three agents. Factors other than the three mentioned previously 
 which affect ventilation are disregarded; one such factor is temperature. 
 Since a rise in body temperature increases the respiration rate, for this 
 model the body temperature is assumed to be 37° C. (98.6° F. ) , which is 
 nearly exact for the cases considered, except when physical exercise 
 is involved. 
 
 6) Transport delays in blood flow, blood sensing delays, and 
 reaction delays are ignored. Obviously, for the real case this 
 assumption is not really correct. However, these delays seem only to 
 retard slightly the actual response of the system. Therefore, these 
 delays are ignored with the admission that they are real and important, 
 but do not drastically discredit the results. These delays could be 
 added later if their effect appears necessary. 
 
7) The respiratory quotient is constant and equal to unity. 
 This means that the rate at which carbon dioxide is placed into the 
 blood by the tissues is exactly equal to the rate at which oxygen is 
 removed from the blood by the tissues. In the actual case the 
 respiratory quotient is somewhere between .80 and 1.00. The reason 
 for this is that the oxidation reaction that takes place in the cells 
 themselves uses more oxygen than the carbon dioxide it produces. The 
 average respiratory quotient in a resting position is about . 8U. 
 
 8) The cases considered and the data used are for healthy, 
 non-diseased subjects. Diseased reactions could conceivably be studied 
 with some minor changes in this model. 
 
 This is a list of the major assumptions. Obviously many 
 details of the real-life system which were not listed must be left 
 out, but a list of the minor assumptions would not only be lengthy and 
 boring, but also would not serve any useful purpose. 
 
10 
 k. DERIVATION OF STEADY STATE SYSTEM EQUATIONS 
 
 Dalton's law of partial pressures is very important throughout 
 this model. A description of this law follows. 
 
 As was mentioned earlier, if a gas is contained in a gas 
 mixture occupying a volume, V, the partial pressure, P, of the gas 
 component is the pressure that this gas would exert if it were the only 
 gas contained in this same volume V. Each of the components of a gas 
 mixture exerts a partial pressure which is proportional to its volume 
 percent of the total. Obviously, the sum of all of the partial 
 pressures is the total gas pressure exerted "by the mixture. 
 
 In this model, the gas mixture considered is ambient air, 
 and therefore, the partial pressure of a gas is the product of the 
 total pressure (barometric pressure) and the fraction, F, of the gas 
 in the mixture. Table 1 shows a typical breakdown of dry air at a 
 barometric pressure, B, of 760 mm Hg. 
 
 When this air enters the lungs, it is moistened by the upper 
 respiratory system. Alveolar gas is generally assumed to be saturated 
 with water vapor. At a body temperature of 37° (98.6° F. ) water vapor 
 exerts a partial pressure of ^7 mm Hg. This water vapor pressure must 
 be considered when the partial pressures of gases in this wet mixture 
 are calculated (Table l) . The total pressure exerted by the wet gas 
 mixture is still equal to the barometric pressure. The partial pressure 
 of each gas in the wet mixture could be calculated by multiplying the 
 barometric pressure and the volumetric fraction of each gas including 
 water vapor. Since the wet volumetric fractions of the gases are more 
 difficult to obtain than the dry volumetric fractions, it is easier to 
 
11 
 
 consider the total pressure as the "barometric pressure of dry air minus 
 the pressure due to the added water vapor. The partial pressures of 
 the gases can then be calculated using their fractions in dry air. Wet 
 gas is always used as the gas mixture when physiological respiratory 
 models are studied. 
 
 Dalton's equation for the partial pressure of a gas in a wet 
 mixture at 37° C. becomes 
 
 P = (B-U7) F. 
 
 
 °2 
 
 co 2 
 
 N 2 
 
 H 2 
 
 SUM 
 
 Fraction , F ( dry gas ) 
 
 .lk6 
 
 .055 
 
 • 799 
 
 
 
 1.00 
 
 Partial pressure, P, mm Hg. 
 (dry gas) , P = 760 F. 
 
 111 
 
 k2 
 
 607 
 
 
 
 760 
 
 Partial pressure mm Hg. 
 (wet gas) , P = (B-U7) F. 
 
 10 k 
 
 39 
 
 570 
 
 hi 
 
 760 
 
 Table 1. Partial Pressures of Body Gases 
 When a gas and a liquid are in contact as they are in the 
 lungs across the porous alveolar membranes , gas molecules are exchanged. 
 When the number of gas molecules entering the liquid equals the number 
 of molecules escaping from the liquid over an interval of time, i.e. 
 there is no net change in the number of molecules in the liquid, a 
 condition of equilibrium exists. Gas molecules exert a partial pressure 
 in the liquid phase as well as in the gas phase. If the partial 
 pressures of a gas in the gas and liquid phases are equal, then an 
 equilibrium condition exists and the liquid is said to be saturated with 
 the gas. If in this saturated condition the partial pressure of the gas 
 in the gaseous phase is suddenly decreased or increased, then the liquid 
 
12 
 
 will give off or absorb gas until another equilibrium condition is 
 reached. This effect is very important when considering a step change 
 in the input atmosphere. 
 
 k.l Gas Balance Equations 
 
 It is obvious that the amount of oxygen used by the body is 
 the difference between the amount of oxygen inspired and the amount of 
 oxygen expired. In the steady state this difference would correspond 
 to the metabolic rate of oxygen consumption, but in an unsteady state 
 this oxygen consumption corresponds not only to the metabolic rate, but 
 also to the replenishment of oxygen stores . Thus , 
 
 V 2 I0 2 A0 2 ' 
 
 Since 
 
 V = V F 
 
 io 2 v i io 2 
 
 and 
 
 V = V F 
 A0 2 E A0 2 ' 
 
 this equation becomes 
 
 V = V F - V F 
 V 2 V I I0 2 V E A0 2 * 
 
 With the assumption that the inspired ventilation rate equals the 
 expiratory respiration rate, the equation becomes 
 
 V = VENT(F - F ) 
 V 2 V ^ M I0 2 A0 2 ; 
 
 VENT is usually measured in l./min., BTPS , because these are the 
 
 conditions of gas in the lungs. BTPS means Body Temperature and 
 
 Pressure Saturated with water vapor. V and V pn are measured in 
 
 U 2 LU 2 
 
13 
 
 STPD units (Standard Temperature and Pressure Dry). To convert from 
 
 dry air to moist air, the multiplier (B-U7)/T60 is used. Therefore, 
 
 VENT = (863/(B-UT))V /(F - F ) (U.l) 
 
 U 2 1U 2 AU 2 
 
 Another obvious observation is that the total amount of 
 
 CO produced by metabolism is equal to the difference between the 
 
 expired CO amount and the inspired CO amount, 
 
 V = V - V 
 
 C0 2 AC0 2 IC0 2 
 
 Proceeding as above, 
 
 V = VENTfF - F ) 
 
 CO V ACO ICO ; 
 
 Since the respiratory quotient was assumed to be unity, 
 
 V = V 
 
 co 2 o 2 
 
 It then follows that 
 
 F = F - F + F (U.2) 
 
 A0 2 I0 2 AC0 2 IC0 2 
 
 k.2 Concentration Equations 
 
 The concentration of oxygen in arterial blood is the sum of 
 the concentration of dissolved and combined oxygen. Combined oxygen 
 is in the form of oxyhemoglobin, HbO . 
 
 The concentration of dissolved oxygen can easily be 
 
 calculated using 
 
 C (dissolved) = a (B-UT) F.. /760 , 
 
 a0 2 2 A0 2 
 
 where a is the Bunsen solubility coefficient for oxygen. 
 2 
 
Ik 
 
 100 -- 
 
 
 90 
 
 
 80 
 
 CM 
 
 o 
 
 TO 
 
 o 
 
 60 
 
 is; 
 o 
 
 H 
 E-j 
 
 50 
 
 n 
 
 Uo 
 
 EH 
 
 <! 
 CO 
 
 Qn 
 
 vs. 
 
 20-. 
 
 10 + 
 
 
 
 ifc 2*0 §6 $) 50 - 6fe 7^ 80" 90" lfe 
 
 PARTIAL PRESSURE OF 
 
 Figure 2. Dissociation Curve 
 
 The concentration of combined oxygen (oxyhemoglobin) can be 
 
 expressed by the oxyhemoglobin dissociation curve (Figure 2). A 
 
 mathematical expression that fits this curve was found and can be 
 expressed as 
 
 C aHb0 2 = *> (l-exp(-.05(B-U T ) F^f, (U.3) 
 
 where Hb is the blood oxygen capacity in liters (STPD) /liter blood. 
 
 k. 3 Data Fitting Equations 
 
 An equation is now needed for the regulation of blood flow. 
 An equation taken from Grodin's work will be used. Blood flow is equal 
 to the normal flow of blood plus any change due to oxygen plus any 
 change due to carbon dioxide. Thus, 
 
15 
 
 FLOW = FLOW + AFLOW + AFLOW . (k.h) 
 
 The changes in flow due to and CO were measured by varying one 
 gas pressure while keeping the other constant. It was determined that 
 AFLOW Q = 9.6551 - .2885P AQ + .00292UlP| - .000010033P| Q , 
 
 P A0 2 * 1Q 5. 
 
 = otherwise, 
 and 
 
 AFL0W C0 2 ■ - 3(P AC0 2 " to) > k0< - P AC0 2 i 6 °> 
 
 = otherwise. 
 
 Equation (U.U), therefore, controls flow given the values of 
 
 F and F . The normal value of flow is found in numerous places 
 
 A0_ AuJ- 
 
 in the literature and is, although slightly variable, about 6.1 liters 
 per minute. 
 
 Gray [k] developed an equation for ventilation based on the 
 
 multiple factor theory. The equation expresses the dependence of 
 
 + 
 ventilation on the three factors H , P _ , P and is 
 
 LU 2 U 2 
 VENT = VENT D (.22H + + .262P n _ - 18.0 + (105-P~ ) 2 /P^ ). 
 
 R co 2 o 2 o 2 
 
 Here VENT is the value of VENT corresponding to the following normal 
 n 
 
 system variable values, F = .21, F = 0, B = 760 , and V = .250, 
 
 i0 2 2 °2 
 
 and is 5.^03 l./min. (BTPS). Solving for H we find 
 
 H + = VENT/1.1909 - 1.1909P ACO + 81.8181 - ((105-P A0 ) 2 /P A0 )/-22 (U.5) 
 
16 
 
 Hydrogen ion is produced in the blood according to the 
 following reaction, 
 
 C0 2 + H 2 2 H 2 C0 3 2 H + + HCO" 
 
 Since blood contains almost 1,000 times as much dissolved CO as H CO , 
 CO is usually treated as if it were the proton donor. 
 
 When the law of mass action [9] is applied to the above 
 reaction, the result is 
 
 H + HC0~ C0 2 = K A 
 
 where H , HCO , and CO are the concentrations of hydrogen ion, 
 bicarbonate ion, and carbon dioxide, and K. is the ionization or 
 
 og 
 
 -6 
 
 dissociation constant of the weak acid H_C0 . The value -log K. 
 
 is called the pK and equals 6.1. Therefore, K = .795 x 10 
 Solving the above equation for CO yields 
 
 C0 2 = H + HCO~/(795 x 10~ 9 ) 
 If H is measured in nanomoles /liter rather than moles /liter, the 
 
 equation becomes 
 
 Now. 
 
 C0 2 = H + HC0 3 /T95 
 
 C0 2 = ka C02 (B-U T ) F AC02 
 
 where k, the conversion factor from atmospheres to mm. of mercury, 
 
 4 co 
 
 equals .00132 and a , the solubility coefficient for gas in blood, 
 
 2 
 equals .510 liters (STPD) gas/liter blood/atm. at 37° C. Therefore, 
 
 P ACQ = H + HCO"/. 5352 (U.6) 
 
IT 
 
 The bicarbonate content HCO of blood can be expressed by 
 the following equation, derived by Gray [3] and utilized by Grodins [8], 
 
 HCO" = BHC0 o ^ + .375(Hb-HbO n ) - (.16+2. 3Hb) 
 3 3b 2 
 
 (log(HC0~)/(.01P AC0 )-.lk) 
 
 where BHC0„, the standard bicarbonate content of blood in liters 
 3b 
 
 CO (STPD) /liter blood at 37° C. , equals .5^7 s and Hb equals .21. 
 Therefore , 
 
 HCO" = .6338+ .375(.2-Hb0 2 ) - .2692 ln(HCO~/( .01P ACQ )) (U.7) 
 
 k.k Steady State System Equations 
 
 All of the equations needed to describe the steady state 
 
 have now been derived and a list of the seven equations follows: 
 
 VENT = 863 V /((B-U7)(F -F )) 
 U 2 iU 2 AU 2 
 
 F = F - F + F 
 
 A0 2 I0 2 AC0 2 IC0 2 
 
 C aHb0 2 = -2(1 - exp(-.05(B-U T ) F^)) 2 
 
 FLOW = 6.1 + A FLOW + A FLOW 
 
 °2 C °2 
 
 H + = VENT/1.1909 - 1.1909P ACO + 81.8181 - ((105-P A0 ) 2 /P A0 2 )/-22 (U.5) 
 
 P ACO = H+HC0 3/-5352 (U.6) 
 
 HCO" = .6338+ .375(.2-Hb Q ) - .2692 ln(HC0^/(.01P AC0 )) (U.7) 
 
 In these equations, VENT is measured in l./min. (BTPS) ; 
 
 V and V , which are the oxygen uptake and carbon dioxide production, 
 
 2 UU 2 
 
 are measured in l./min (STPD) ; P AQ and P Aro are in mm. Hg. 
 
 (U.1) 
 
 (U.2) 
 
 (*.3) 
 
 (k.k) 
 
18 
 
 5. RESULTS OF THE STEADY STATE SYSTEM 
 
 The solution of the system in the steady state is accomplished 
 
 by solving the seven system equations given F , F , B, and V . 
 
 2 p 2 
 
 For example, to find the system variables for altitude hypoxia at 
 
 15,000 feet, B is set equal to U30 , its value at 15,000 feet, and 
 
 V , F , and F are set equal to their "normal" values. 
 
 The solution of the system is determined as follows. First, 
 
 a value for F is guessed. F is then found using equation (k.2). 
 
 ALU p 
 
 Values of VENT and C are then calculated from equations (U.l, U.3). 
 
 3x1 D U 
 
 Equation (U.7) is then solved for HC0~ by guessing a value for HC0~ 
 
 calculating the right hand side of the equation, taking the difference 
 
 between the left and right hand sides to determine a measure of the 
 
 error, and then averaging the two values to find the new estimate of 
 
 HCO . If the difference is less than .001, an acceptable value for 
 
 HC0 has been found; otherwise, this is repeated until a suitable value 
 
 is found. The next step is to calculate H and P flpn using equations 
 
 ALU 
 
 (k.5, h.6) . This value of P. m readily gives a value for F , which 
 
 ALU ALU 
 
 can be compared to the guess for F . If the difference is greater 
 
 than .001, the two values of F are averaged to yield a new estimate 
 
 of F , and the entire procedure is repeated; otherwise, acceptable 
 
 ALU 
 
 values for F and the other variables have been found and the value 
 
 for FLOW can be found from equation (U.U), completing the solution. 
 
 All other conditions can be analyzed in an analogous manner. 
 
19 
 
 5.1 Hypoxia at Altitude 
 
 Hypoxia is a condition in which the body cells lack oxygen. 
 The cause of a hypoxic state can be anything that causes an abnormal 
 resistance to the flow of oxygen to the cells. Hypoxic conditions can 
 only be defined in reference to a normoxic condition. A body, after 
 spending an extended length of time at altitude, can adapt to the 
 hypoxic condition and thus enter a normoxic condition for that body. 
 Now if this system descended to sea level after being in normoxia at 
 altitude, a condition of hyperoxia would result. 
 
 This study is concerned with subjects who have spent a very 
 short time at high altitude. The relationship between alveolar 
 ventilation and altitude in feet is given in Figure 3. The graph was 
 formed from the combined data of De jours [10] and Milhorn [7]. 
 
 -p 
 
 Pn 
 
 EH 
 
 3 
 
 o— 
 
 o Experimental 
 x Computed 
 
 5 TO 15 20 25 
 
 ALVEOLAR VENTILATION ( liters /min. ) 
 Figure 3. Altitude Hypoxia 
 It can be seen from the figure that the values of ventilation 
 predicted by the model are very close to the actual experimental results 
 The other system variables are also reasonably close to other accepted 
 values. Table 2 gives a comparison of system variable values predicted 
 by this model and values accepted by Grodins [8] in his model. 
 
20 
 
 5.2 Hypoxia at Sea Level 
 
 A condition of hypoxia can be caused at sea level by changing 
 the inspired atmosphere to an oxygen poor mixture. As would seem 
 likely, the lower the percentage of oxygen in inspired air, the higher 
 will be the ventilation and blood flow. There are many other 
 situations which might cause a condition of hypoxia at sea level. These 
 include such afflictions as emphysema, obesity, and partial paralysis. 
 
 Figure k gives the relationship between alveolar ventilation 
 in the steady state and the percent of oxygen inspired. The data 
 graphed in Figure k is taken from Milhorn [7]. Although the values of 
 ventilation predicted by this model seem to be low for inspired 
 atmospheres containing very little oxygen, the predicted values compare 
 very favorably with values calculated by the very involved model given 
 by Grodins [8]. Table 2 gives a comparison of these values. 
 
 s 
 
 
 
 
 
 H 
 
 
 EH 
 
 
 H < 
 
 
 Ej J 
 
 
 SB 
 
 a 
 
 CQ £j 
 
 ■H 
 
 >h£ 
 
 ^ 
 
 Q 
 
 • 
 
 <t K 
 
 H 
 
 gs 
 
 
 CO 
 
 
 o Experimental 
 x Computed 
 
 PERCENT INSPIRED 
 Figure k. Hypoxia at Sea Level 
 
21 
 
 cq 
 
 H 
 
 n 
 
 o 
 ce 
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 rH 
 < 
 CO 
 
 CM 
 
 O 
 
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 (h 
 
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22 
 
 5. 3 Carbon Dioxide Inhalation 
 
 The condition of hypercapnia is an increase of carbon dioxide 
 in the blood. When a mixture rich in CO is inspired, the rate of 
 breathing increases and reaches a new, nearly stable value. The 
 hyperventilation due to the inspiration of a hypercapnic mixture lasts 
 until the mixture is changed and decreases gradually when the gas 
 mixture is replaced by air. Hypercapnia can occur in many situations, 
 such as any form of asphyxia or suffocation. 
 
 Figure 5 shows the curve of alveolar ventilation versus the 
 percent of carbon dioxide in the inspired air for data from this model 
 and from that of Milhorn [7]. It can be seen that the values predicted 
 by this model compare quite satisfactorily with the values given 
 in Milhorn. 
 
 o 
 
 H 
 EH 
 
 <! H • 
 
 EH EH S 
 
 CO S| 
 
 Q 
 
 < K H 
 
 w o 
 
 h3 
 
 30+ 
 
 25- 
 20- 
 
 15-h 
 
 10- 
 
 &^-o 
 
 o Milhorn 
 x Present 
 A Grodins 
 
 ' i ' i ' 
 
 PERCENT CO INSPIRED 
 Figure 5« System Response to CO Inspiration 
 
 Although Grodins' model follows Milhorn's experimental results 
 a little better than the model proposed here, the proposed model does 
 
23 
 
 predict ventilation values fairly well. A comparison between the system 
 variables for the proposed model and Grodins 1 model is given in Table 3. 
 
 VARIABLE PROPOSED GRODINS 
 
 B 
 
 T6o 
 
 760 
 
 760 
 
 760 
 
 760 
 
 760 
 
 \ 
 
 .250 
 
 .250 
 
 .250 
 
 .215 
 
 .215 
 
 .215 
 
 H 
 
 .21 
 
 .21 
 
 .21 
 
 .21 
 
 .21 
 
 .21 
 
 F 
 
 ico 2 
 
 .01 
 
 .03 
 
 .05 
 
 .01 
 
 .03 
 
 .05 
 
 F 
 A0 2 
 
 .1633 
 
 .1809 
 
 .19 1 +8 
 
 .1585 
 
 .1783 
 
 .1913 
 
 F 
 AC0 2 
 
 .0567 
 
 .0591 
 
 .0652 
 
 .0563 
 
 .0585 
 
 .0665 
 
 s 
 
 116. k 
 
 129.0 
 
 138.9 
 
 113.0 
 
 127.1 
 
 136. h 
 
 P AC0 2 
 
 kO.k 
 
 U2.1 
 
 U6.5 
 
 1*0.1 
 
 Ul.7 
 
 kl.h 
 
 VENT 
 
 6.hQ 
 
 10.1+0 
 
 19.91 
 
 6.10 
 
 9.88 
 
 16.67 
 
 FLOW 
 
 6.23 
 
 6.7^ 
 
 8.05 
 
 6.04 
 
 6.52 
 
 8.22 
 
 Hb0 2 
 
 .1988 
 
 .1993 
 
 .1996 
 
 .1983 
 
 • 1991 
 
 .1993 
 
 H co 3 
 
 .5513 
 
 .5587 
 
 • 5763 
 
 NOT 
 GIVEN 
 
 NOT 
 GIVEN 
 
 NOT 
 GIVEN 
 
 Table 3. Comparison of Model CO Inspiration Results 
 
 5. h Metabolic Disturbances 
 
 Metabolic increases due to exercise are the only type of 
 metabolic disturbance considered in this model. During the first few 
 seconds after the onset of exercise, the ventilation rate increases 
 very quickly. Then for the next few minutes of exercise, the 
 ventilation rate increases gradually until a steady state is reached. 
 
2k 
 
 At this point of this study, the only concern is the level 
 of the steady state reached and not the details of the transient 
 behavior. This aspect will be considered later. It is assumed that 
 the ventilation rate is adequate for normal saturation of blood 
 flowing through the lungs. The exercise in this model is applied 
 instantaneously from a state of rest and continued indefinitely until 
 a steady state is reached. 
 
 A metabolic rate of .250 liters per minute corresponds 
 to a resting state. The exact value of the resting metabolism is not 
 agreed upon in the literature , but all opinions are rather close , 
 and .250 liters per minute seems reasonable. A metabolic rate of 
 1.000 liters per minute is associated with moderate exercise, such 
 as walking at a speed of about three miles per hour. A metabolic rate 
 of 2.000 liters per minute can support a system performing heavy to 
 very heavy work. See Table k for a classification of work intensity. 
 
 
 METS 
 
 O2 ( liter/min. ) 
 
 Cal. /min. 
 
 VERY LIGHT WORK 
 
 1-2 
 
 be low . 5 
 
 below 2.5 
 
 LIGHT WORK 
 
 2-3 
 
 0.5-1.0 
 
 2.5-5.0 
 
 MODERATE WORK 
 
 3-U 
 
 1.0-1.5 
 
 5.0-7.5 
 
 HEAVY WORK 
 
 k-6 
 
 1.5-2.0 
 
 7. 5-10.0 
 
 VERY HEAVY WORK 
 
 6-8 
 
 2.0-2.5 
 
 10.0-12.5 
 
 MAXIMAL WORK 
 
 over 8 
 
 over 2.5 
 
 over 12.5 
 
 Table k. Classification of Work Intensity 
 
25 
 
 The relationship between oxygen consumption and ventilation 
 rate is generally agreed to be linear. Figure 6 gives a comparison of 
 the results calculated by this model and the results of data from 
 Morehouse and Miller [ll]. 
 
 35- 
 
 304 
 
 25 
 
 20 
 
 15-- 
 
 10-- 
 
 5-- 
 
 ■h 
 
 o 
 
 H 
 
 H 
 
 E-i 
 
 o Experimental 
 x Computed 
 
 
 
 ~% 1^5 tH r! 
 
 fe 
 
 1.'25 1.50 1.75 2.00 
 OXYGEN CONSUMPTION ( 1. /Win. ) 
 
 Figure 6. System Respiratory Response to Consumption 
 
 It can be seen from the figure that the results predicted by 
 this model are, although consistently low, reasonable when compared to 
 experimental data. According to the experimental data, resting 
 ventilation is about nine liters per minute, which is much higher than 
 the resting ventilation generally agreed upon in the literature. If 
 the resting ventilation is adjusted lower, the two curves become 
 very comparable. 
 
26 
 
 The only other metabolic dependent variable for which data 
 is readily available and for which this model is appropriate is blood 
 flow. Figure 7 shows the experimental and predicted values for blood 
 flow versus the metabolic rate. The experimental data is due to 
 Chapman [12] . 
 
 25 
 20 
 
 •h 15 -- 
 s 
 
 H 
 
 o 
 
 10-- 
 
 5 -- 
 
 x 
 
 o Experimental 
 x Computed 
 
 ^25 ^0 .75 1.00 1.25 1.50 i!t5 s'.OO 2 J .25 
 2 CONSUMPTION (l./Min.) 
 
 Figure 7. System Blood Flow Response to Consumption 
 
 From the figure it can be seen that the predicted values do 
 not compare with experimental values very favorably. This discrepancy 
 is due to the fact that this model does not consider the efficiency of 
 ventilation. At high ventilation rates the increased ventilation does 
 not serve any purpose. The delivery of oxygen to the tissues is limited 
 by the blood flow rate. This model assumes high ventilation is 
 perfectly efficient, thus keeping the blood flow rate lower than the 
 actual experimental values. 
 
27 
 6. OVERALL EVALUATION OF THE STEADY STATE MODEL 
 
 A summary of the results for the steady state model is 
 given in Table 5. The steady state model behaves acceptably in 
 general. Using this model, system variables for hypoxias and carbon 
 dioxide inhalation can be predicted with reasonable accuracy. The 
 model, under metabolic control, predicts ventilation values con- 
 sistently below the experimental data gathered by Morehouse and 
 Miller [ll]. However, blood flow values, under metabolic control, 
 are quite low for the model because this model does not account for 
 the lack of efficiency of ventilation at high ventilation rates. 
 
 Overall, this model predicts experimental system 
 variables quite well. Considering the simplicity of this model, it 
 looks even better when compared to similar but more complicated models. 
 
28 
 
 
 
 
 
 
 
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 -=f 
 
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 o 
 
 
 
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 -=t- 
 
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 -=r 
 
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 t— 
 
 
 
 
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 t— 
 
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 & 
 
 « pq 
 
 
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 r <; 
 
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 > H 
 
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 en 
 
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 -p 
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 -P 
 
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 CD 
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29 
 
 7. THE DYNAMIC SYSTEM 
 
 Now that the steady state model has "been designed and 
 evaluated, a model 'which can predict system variables in the unsteady 
 or dynamic state will he formulated. 
 
 7. 1 Background 
 
 First of all it must be decided just which variables can 
 change instantaneously when the system is forced into an unsteady 
 state. It is unrealistic to allow all system variables to change 
 instantaneously because this would lead to a system that adjusts to 
 disturbances instantaneously, which conflicts with real life. It 
 seems reasonable to assume that the system reaction is limited either 
 by how fast ventilation can change or by how fast the blood flow rate 
 can change. 
 
 If exercise resulting in a step change in metabolic rate 
 is considered, the ventilation rate jumps very sharply during the 
 first few seconds and then gradually levels off to a steady state. 
 At the step completion of exercise ventilation decreases in a similar 
 manner. The ventilation rate suddenly drops in the first few seconds 
 after the completion of exercise and then gradually decreases to a 
 steady state. 
 
 The time required for ventilation to rise to a steady state 
 depends upon the severity of the exercise undertaken. The time required 
 for ventilation rate to decrease to a steady state after exercise has 
 ceased depends both on the severity of the exercise and also on the 
 duration of the exercise. If the preceding exercise was extremely 
 intense, recovery of ventilation rate might take more than an hour. 
 
30 
 
 Since the ventilation rate can change very rapidly for short 
 periods of time, it seems more likely that the system is limited "by 
 how fast blood flow is able to change. Wow if it is assumed that 
 blood flow controls system change, then an equation describing the 
 variation of flow with time must be developed. 
 
 In order to develop the equation, first the difference 
 
 between V and MR must be discussed for this model. V is the 
 
 2 2 2 
 
 oxygen uptake that the system is able to support in a steady state. 
 
 MR is the oxygen uptake demanded by the system because of exercise. 
 
 In other words, for any value of flow, V is the calculated value 
 
 of oxygen uptake that the system can support. The difference between 
 
 the oxygen uptake the system can support and the oxygen uptake 
 
 demanded by the conditions must be supplied from the oxygen stores 
 
 in the body. 
 
 Table 6 shows the volumes of the various gases stored in the 
 
 different compartments of a 155 pound man at rest. The stores given 
 
 in the figure represent only those that can be utilized fairly quickly, 
 
 that is, within a few minutes or a few hours. Other stores such as 
 
 bone stores do exist, but they cannot be brought into play rapidly. 
 
 The size of these stores is changed during muscular exercise when the 
 
 tissue partial pressure of CO increases and the partial pressure of 
 
 decreases. The partial pressures of the gases in venous blood vary 
 
 in the same direction as those in the tissues. The partial pressures 
 
 of gases in arterial blood actually vary quite a bit less except 
 
 during heavy exercise. This model, however, is not concerned with 
 
 which stores have been affected, but rather only that the body stores as 
 
 a whole have changed. 
 
31 
 
 BLOOD TOTAL ^^ 
 
 GAS 
 
 TISSUES VENOUS ARTERIAL Blood + (2.51 BTPS) 
 
 Tissues B=760mmHg 
 ml. STPD ml. STPD ml. STPD ml. STPD ml. STPD 
 
 co 2 
 
 3330 
 
 2120 
 
 650 
 
 6100 
 
 116 
 
 °2 
 
 200 
 
 610 
 
 270 
 
 1080 
 
 300 
 
 N 2 
 
 fat-free i+90 
 fat k$0 
 
 U2 
 
 Ik 
 
 996 
 
 1560 
 
 Table 6. Body Gas Stores 
 
 The amount of gas that has "been borrowed from these gas 
 
 stores during exercise must be restored. If the gas stores of oxygen 
 
 are used up and there is still an additional need for oxygen, then 
 
 some of the body cells will start to die. For this reason it is 
 
 very important that the oxygen stores be replenished as soon as 
 
 reasonably possible. The amount of gas which is owed to the stores 
 
 will be referred to as the accumulated debt and will be denoted by 
 
 PDEBT because, at least for this model, the accumulated debt will 
 
 always be positive. Once the stores are full again, they do not 
 
 take on additional gases since this would lead to a negative 
 
 accumulated debt . 
 
 The symbol DEBT' is reserved for the instantaneous debt. 
 
 This is the instantaneous difference between V and MR . The 
 
 2 2 
 value of DEBT' may be either positive or negative depending on 
 
 whether PDEBT is being built up or paid off. 
 
 The rate of change of flow must depend both upon PDEBT 
 
 and on DEBT'. The value of DEBT' will govern the change in flow 
 
 at the outset of exercise, but as PDEBT gets larger it also 
 
 influences flow. Then when the value of flow finally rises to a 
 
32 
 
 value large enough to make DEBT' equal to zero, PDEBT alone controls 
 
 FLOW. As FLOW gets larger than that required for the MR , DEBT' 
 
 2 
 affects FLOW negatively and PDEBT affects FLOW positively until 
 
 PDEBT is zero. Then FLOW decreases to a steady state. 
 
 The instantaneous debt may now be defined algebraically as 
 
 DEBT' = MR - V . 
 2 2 
 
 Since FLOW depends both on DEBT' and also on PDEBT, the 
 simplest dependence is assumed. Thus to this point 
 
 FLOW = A(DEBT') + B(PDEBT) + C, 
 where the coefficients A, B, C are functions of FLOW. The coefficient 
 C in this equation must be zero since when DEBT* and PDEBT are both 
 zero, FLOW is also zero. 
 
 The rate of blood flow must be limited above and below. 
 According to the literature an appropriate upper limit on flow is 
 30 liters per minute and an appropriate lower limit is h.G2 liters 
 per minute [13]. It seems reasonable that as the flow approaches 
 either limit, the rate of change of the flow must decrease. Therefore, 
 FLOW must also depend on the difference between the FLOW and each of 
 the limits, normalized by the value of FLOW itself, leading to the 
 following equation 
 
 FLOW = (D(DEBT') + E( PDEBT) )(30-FLOW)(FLOW-U.62)/FLOW 2 . 
 The values for D and E can be chosen to best fit the experimental data. 
 
 equation 
 
 The dependence of V on FLOW can be expressed by the 
 
 V = (FLOW-U.62)/5.9 
 
 This equation is taken from the literature [12] and is based on 
 experimental data. 
 
33 
 
 This experimentally derived equation shows that if blood flow 
 were to drop to as low as h.62 liters per minute, there would no longer 
 be any oxygen uptake by the body, and the system would begin to die. 
 
 Now all of the equations necessary for a transient solution 
 have been developed. The system of equations follows: 
 
 FLOW = (D(DEBT') + E(PDEBT))(30-FL0W)(FL0W-U.62)/FL0W 2 . 
 
 DEBT' = MR - V 
 
 2 2 
 
 V n = (FLOW-U.62)/5.9 
 
 U 2 
 
 PDEBT = H( DEBT) DEBT, 
 
 where H is the Heaviside step function. MR is the only variable 
 
 2 
 that must be specified by the user. The remaining variables of the 
 
 system can be calculated using the steady state equations given 
 
 V and FLOW. 
 
 7-2 Evaluation of the Dynamic Model 
 
 The flow equation with D = 13 and E - .5 was integrated for 
 step metabolic rates of 1 and 2 liters per minute using Euler's method 
 with a time step of 1 second. The values of the system variables durin; 
 the first two minutes of exercise are given in Tables 7 and 8. The 
 predicted flows are compared with data adapted from Guyton [13] in 
 Figure 8. Data on the other system variables during exercise is not 
 readily available in the literature. Given the agreement between the 
 predicted and experimental cardiac outputs, it is reasonable to hope 
 that the other system variables are predicted equally well. 
 
3k 
 
 •iH 
 o 
 Jh 
 0) 
 X 
 
 H 
 
 O 
 -P 
 
 <D 
 
 en 
 
 O 
 ft 
 
 CQ 
 
 (1) 
 ffi 
 
 •P 
 
 ft 
 
 •P 
 
 O 
 u 
 
 O 
 
 CO 
 CD 
 
 •H 
 
 1 
 
 o 
 
 1 
 IJA 
 
 1 
 
 O 
 
 <M 
 
 H 
 
 H 
 
 (•qtw/'t) inamo oviaavo 
 
35 
 
 T 
 
 FLOW 
 
 FLOW 
 
 PDEBT 
 
 DEBT' 
 
 m o 2 
 
 \ 
 
 V 
 
 0.0 
 
 6.10 
 
 9.26 
 
 .000 
 
 .749 
 
 1.00 
 
 .250 
 
 5.42 
 
 0.1 
 
 7.02 
 
 8.67 
 
 .068 
 
 .593 
 
 1.00 
 
 .407 
 
 8.50 
 
 0.2 
 
 7.82 
 
 6.98 
 
 .122 
 
 .458 
 
 1.00 
 
 .542 
 
 11.12 
 
 0.3 
 
 8.45 
 
 5.38 
 
 .163 
 
 .351 
 
 1.00 
 
 .649 
 
 13.00 
 
 o.i* 
 
 8.93 
 
 4.10 
 
 .195 
 
 .270 
 
 1.00 
 
 .730 
 
 14.36 
 
 0.5 
 
 9-30 
 
 3.14 
 
 .219 
 
 .207 
 
 1.00 
 
 .793 
 
 15.44 
 
 0.6 
 
 9.5 8 
 
 2.42 
 
 .237 
 
 .160 
 
 1.00 
 
 .841 
 
 16.23 
 
 0.7 
 
 9.80 
 
 1.87 
 
 .2 52 
 
 .123 
 
 1.00 
 
 .877 
 
 16.79 
 
 0.8 
 
 9.97 
 
 1.46 
 
 .263 
 
 .094 
 
 1.00 
 
 .906 
 
 17.34 
 
 0.9 
 
 10.10 
 
 1.14 
 
 .271 
 
 .072 
 
 1.00 
 
 .928 
 
 17.62 
 
 1.0 
 
 10.20 
 
 .89 
 
 .278 
 
 .0 54 
 
 1.00 
 
 .946 
 
 17.95 
 
 i.i 
 
 10.28 
 
 • 70 
 
 .282 
 
 .040 
 
 1.00 
 
 .960 
 
 18.21 
 
 1.2 
 
 10.35 
 
 .55 
 
 .286 
 
 .030 
 
 1.00 
 
 .970 
 
 18.41 
 
 1.3 
 
 10. ko 
 
 .44 
 
 .289 
 
 .021 
 
 1.00 
 
 .979 
 
 18.43 
 
 1.4 
 
 10. 44 
 
 .35 
 
 .290 
 
 .014 
 
 1.00 
 
 .986 
 
 18.56 
 
 1.5 
 
 10.47 
 
 .27 
 
 .292 
 
 .009 
 
 1.00 
 
 .991 
 
 18.66 
 
 1.6 
 
 10.49 
 
 .22 
 
 .292 
 
 .005 
 
 1.00 
 
 .995 
 
 18.74 
 
 1.7 
 
 10.51 
 
 .17 
 
 .293 
 
 .001 
 
 1.00 
 
 .999 
 
 18.80 
 
 1.8 
 
 10.53 
 
 .14 
 
 .293 
 
 -.001 
 
 1.00 
 
 1.001 
 
 18.85 
 
 1.9 
 
 10.5*4 
 
 .11 
 
 .292 
 
 -.003 
 
 1.00 
 
 1.003 
 
 18.89 
 
 2.0 
 
 10.55 
 
 .08 
 
 .292 
 
 -.005 
 
 1.00 
 
 1.005 
 
 18.92 
 
 Table 7. Transient System Response to MR = 1.00 
 
 2 
 
36 
 
 T 
 
 FLOW 
 
 FLOW 
 
 PDEBT 
 
 DEBT' 
 
 "V 
 
 \ 
 
 V 
 
 0.0 
 
 6.10 
 
 21.62 
 
 .000 
 
 1.7^9 
 
 2.00 
 
 .250 
 
 5.U2 
 
 0.1 
 
 8.35 
 
 20.69 
 
 .159 
 
 1.368 
 
 2.00 
 
 .632 
 
 12.63 
 
 0.2 
 
 10.16 
 
 11+.83 
 
 .282 
 
 1.06l 
 
 2.00 
 
 • 939 
 
 17.78 
 
 0.3 
 
 11.1+6 
 
 10.73 
 
 .379 
 
 .81+1 
 
 2.00 
 
 1.195 
 
 21.22 
 
 0.1+ 
 
 12.1+1 
 
 8.05 
 
 .U55 
 
 .679 
 
 2.00 
 
 1.321 
 
 23.78 
 
 0.5 
 
 13.11+ 
 
 6.23 
 
 .518 
 
 .556 
 
 2.00 
 
 1.1+1+1+ 
 
 25.51 
 
 0.6 
 
 13.70 
 
 1+.9U 
 
 .569 
 
 .1+60 
 
 2.00 
 
 I.5I4O 
 
 26.77 
 
 0.7 
 
 11+.16 
 
 3.99 
 
 .612 
 
 .381+ 
 
 2.00 
 
 1.616 
 
 27.86 
 
 0.8 
 
 lU.52 
 
 3.27 
 
 .6U8 
 
 .321 
 
 2.00 
 
 1.679 
 
 28.69 
 
 0.9 
 
 11+.83 
 
 2.71 
 
 .678 
 
 .270 
 
 2.00 
 
 1.730 
 
 29.31+ 
 
 1.0 
 
 15.08 
 
 2.27 
 
 .703 
 
 .227 
 
 2.00 
 
 1.773 
 
 30.07 
 
 1.1 
 
 15.29 
 
 1.91 
 
 .721+ 
 
 .192 
 
 2.00 
 
 1.808 
 
 30.1+U 
 
 1.2 
 
 15.1+7 
 
 1.62 
 
 .7^2 
 
 .161 
 
 2.00 
 
 1.839 
 
 30.95 
 
 1.3 
 
 15.62 
 
 1.39 
 
 .757 
 
 .135 
 
 2.00 
 
 I.865 
 
 31.85 
 
 1.1+ 
 
 15-75 
 
 1.19 
 
 • 770 
 
 .113 
 
 2.00 
 
 I.887 
 
 31.52 
 
 1.5 
 
 15.86 
 
 1.02 
 
 .780 
 
 .09I+ 
 
 2.00 
 
 1.906 
 
 31.81+ 
 
 1.6 
 
 15.96 
 
 .88 
 
 .789 
 
 .078 
 
 2.00 
 
 1.922 
 
 31.89 
 
 1.7 
 
 16.01+ 
 
 .76 
 
 .796 
 
 .06^ 
 
 2.00 
 
 1.936 
 
 32.12 
 
 1.8 
 
 l6.ll 
 
 .66 
 
 .802 
 
 .052 
 
 2.00 
 
 I.9I+8 
 
 32.32 
 
 1.9 
 
 16.18 
 
 .57 
 
 .807 
 
 .01+1 
 
 2.00 
 
 1.959 
 
 32.50 
 
 2.0 
 
 16.23 
 
 • 50 
 
 .810 
 
 .032 
 
 2.00 
 
 1.968 
 
 32.68 
 
 
 Table 8. Transient System Response to MR =2.00 
 
37 
 
 8. CONCLUSIONS ABOUT THE TOTAL MODEL 
 
 Some obvious shortcomings of this model are that it only 
 considers chemical control in the tissues, blood flow constants are 
 ignored, and the Bohr, Haldane and pH effects are not taken into 
 account. A more detailed model might include a brain reservoir with 
 some sort of a sensing capability possible on the blood flowing 
 through the brain reservoir. The blood flow transport delays could 
 be taken into account, making the location of the blood chemoreceptors 
 very important. It must be remembered, however, that every bit of 
 complexity that is added serves to defeat the purpose of this model. 
 
 In its present state, this model is applicable to many 
 conditions including the results of hypoxic, hypercapnic and 
 asphyxiatic conditions. A variety of disease situations could be 
 studied by altering the appropriate normal control equations. 
 Biological conditions that can be simulated include carbon monoxide 
 poisoning, infections with anaerobic bacteria, crisis in sickle cell 
 anemia, cyanosis in "blue babies" who have large shunts of venous 
 blood, and any other condition that might be treated by the 
 inspiration of an oxygen rich atmosphere at normal or even increased 
 barometric pressures. Situations such as voluntary hyperventilation, 
 as done by many athletes, can be studied. Conditions associated 
 with scuba diving, high altitude flying, and mountain climbing could 
 be investigated. 
 
 In spite of the errors in this model, it does serve a very 
 useful purpose; it shows that a very simple respiratory simulation 
 model can be conceived and run with satisfactory results. 
 
38 
 
 LIST OF REFERENCES 
 
 [l] Haldane, J. S. and Priestley, J. G. , "The Regulation of the Lung 
 Ventilation," J. Physiol. , 32 , 225-266, 1905. 
 
 [2] Gray, J. S., "The Multiple Factor Theory of Respiratory 
 
 Regulation," AAFSAM Project Report No. 386 (l), 19 U5. 
 
 [3] Gray, J. S. , "The Multiple Factor Theory of Respiratory 
 
 Regulation," MFSAM Project Report No. 386 (2), 19 U5. 
 
 [k] Grodins, F. S., Gray, J. S., Shroeder, K. R. , Norins , A. L. 
 and Jones, R. W. , "Respiratory Responses to CO 
 Inhalation: A Theoretical Study of a Nonlinear 
 Biological Regulator," J. Appl. Physiol. , J_, 283-308, 195U. 
 
 [5] Defares, J. G. , Derksen, H. E. and Duyff, J. ¥. , "Cerebral Blood 
 Flow in the Regulation of Respiration," Acta Physiol. 
 Pharmacol. Neerlandica , 9, 327-360, i960. 
 
 [6] Defares, J. G. , "Principles of Feedback Control and Their 
 Application to the Respiratory Control System," 
 Handbook of Physiology, Vol. 1, Section 3: Respiration , 
 ed. W. 0. Fenn and H. Rahn, Washington, D. C. , I96U. 
 
 [7] Milhorn, H. T. , Jr., Benton, R., Ross, R. and Guyton, A. C. , 
 "A Mathematical Model of the Human Respiratory Control 
 System," Biophysical J. , 5_> 27-^6, 1965. 
 
 [8] Grodins, F. S. , Buell, J. and Bart, A. J., Mathematical Analysis 
 and Digital Simulation of the Respiratory Control System , 
 The RAND Corporation, RM-52U1+-PR, 1967. 
 
 [9] Comroe , J. H. , Physiology of Respiration , Year Book Medical 
 Publishers Incorporated, Chicago, 1965. 
 
 [10] De Jours , P., Respiration , Oxford Press , New York, 1966. 
 
 [ll] Morehouse, L. E. and Miller, A. T. , Jr., Physiology of Exercise , 
 C. V. Mosby Co., Saint Louis, 1971. 
 
 [12] Chapman, C. B., Physiology of Muscular Exercise , American Heart 
 Association Monograph No. 15, American Heart Association, 
 Inc. , New York, 19.67- 
 
 [13] Guyton, A. C. , Circulatory Physiology: Cardiac Output and its 
 
 Regulation , W. B. Saunders Co., Philadelphia and London, 19 63. 
 
39 
 
 APPENDIX A 
 SYSTEMS AND UNITS FOR EQUATIONS OF SYSTEM 
 
ko 
 
 For this paper symbols have, in general, two subscripts. 
 The first subscript specifies the location of the variable and the 
 second identifies the chemical species involved. One or both of 
 these subscripts may be omitted. 
 
 SYMBOL 
 
 SUBSCRIPTS 
 LOCATION SPECIES 
 
 DEFINITION 
 
 UNITS 
 
 a 
 
 B 
 BHCO 
 
 3 
 
 F 
 
 FLOW 
 A FLOW 
 
 Hb 
 MR 
 
 VENT 
 
 a 
 v 
 
 A 
 E 
 I 
 
 A 
 a 
 
 I 
 
 E 
 I 
 
 gas 
 HbO 
 K 
 
 + 2 
 
 liter (STPDP /liter 
 
 blood) /atm, 37° C. 
 
 mmHg 
 
 liters CO (STPD)/ 
 
 liter blood, 37° C. 
 
 liters (STPD) 
 liters (STPD) 
 nanomoles 
 liter blood 
 liter blood 
 dimension less 
 
 gas solubility coefficient 
 
 for gas in blood 
 
 barometric pressure 
 
 standard bicarbonate 
 
 content of blood 
 
 concentration 
 
 of gas 
 
 of oxyhemoglobin 
 
 of hydrogen ion 
 
 in blood at lung exit 
 
 in blood at lung entrance 
 
 volumetric fraction 
 gas of gas 
 
 in dry alveolar gas 
 
 in dry expired gas 
 
 in dry inspired gas 
 
 blood flow, cardiac output liters /min. 
 
 normal (resting) 
 
 change in blood flow 
 CO due to carbon dioxide 
 
 due to oxygen 
 
 blood oxygen capacity 
 
 metabolic rate 
 CO of carbon dioxide production 
 
 of oxygen consumption 
 
 tension or partial 
 
 pressure 
 gas of gas 
 
 in alveoli 
 
 in blood at lung exit 
 
 in inspired air 
 
 gas flow rate 
 
 expiratory 
 
 inspiratory 
 
 liters /min. 
 
 liters (STPD) /liter blood 
 liters (STPD) /min. 
 
 mmHg 
 
 liters (BTPD)/min. 
 
 Table A-l. Table of Symbols and Units 
 
kl 
 
 APPENDIX B 
 NORMAL PARAMETER VALUES 
 
1+2 
 
 a co 2 
 
 — 
 
 .510 
 
 % 
 
 = 
 
 .024 
 
 B 
 
 = 
 
 760 
 
 F 
 
 ico 2 
 
 = 
 
 .000 
 
 F 
 
 io 2 
 
 = 
 
 .210 
 
 F 
 
 iw 2 
 
 = 
 
 .790 
 
 Hb 
 
 = 
 
 .200 
 
 MR co 2 
 
 = 
 
 .250 
 
 m o 2 
 
 = 
 
 .250 
 
 FLO¥ N 
 
 = 
 
 6.100 
 
 Table B-l. "Normal" Parameter Values 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 port Mo. 
 UIUCDCS-R-73-6I6 
 
 3. Recipient's Accession No. 
 
 4. Title and Subtitle 
 
 A TWO COMPARTMENTAL MODEL OF THE RESPIRATORY SYSTEM 
 
 5. Report Date 
 
 December 1973 
 
 6. 
 
 7. Author(s) 
 
 Paul Robert Haskitt 
 
 8. Performing Organization Rept. 
 No. 
 
 ?. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 12. Sponsoring Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 13. Type of Report & Period 
 Covered 
 
 Thesis Research 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 A simplified two compartmental model of the human respiratory system 
 is presented. The model is used to study the steady state responses to 
 hypoxia at altitude and sea level, carbon dioxide inhalation, and 
 metabolic disturbances, and the transient response to metabolic 
 disturbances. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 respiration 
 simulation 
 
 7b. Identifiers/Open-Ended Terms 
 
 7c. COSATI Field/Group 
 
 8. Availability Statement 
 
 unlimited 
 
 ORM NTIS-3B ( 10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 
 Page 
 UNCLASSIFIED 
 
 21. Nu. M Pages 
 
 22. Price 
 
 USCOMM-DC 40329-P7 1 
 
.** 
 
 <$>' 
 
*»*"» 
 
 «ft 
 
UNIVERSITY OF ILLINOIS-URBAN A 
 510.84 IL6R no. C002 no. 61 3-617(1 973 
 Internal report / 
 
 3 01 
 
 2 088401036 
 
 m 
 
 
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