Some problems and solutions for modeling overall cardiovascular regulation

Some Problems and Solutions for Modeling Overall Cardiovascular Regulation* ARTHUR C. GUYTON. THOMAS G. COLEMAN, R. DAVIS MANNING, JR., AND JOHN E. HALL Department of PIyxiolugy and Biophysics, U~tit~ersi{v of Mississippi Medicul Center, Juckson, Mississippi39,716

ABSTRACT A brief history of the development of mathematical models of the cardiovascular system is presented. Until the advent of computers, very little modeling of transient physiological phenomena was done, but this is now commonplace. The problem of stability in complex physiological models fortunately is averted by the fact that the physiological controls are themselves highly stable. The reason for this is that evolution has eliminated unstable feedback loops because they are lethal. Indeed, enough safety fackvr has been provided in the design of the body so that even poor mathematical models are often quite stable. An especially important use of complex cardiovascular models has been to derive new concepts of cardiovascular function. One such concept is the “princ.ple of infinite gain” for long-term control of arterial pressure, which states that the long-term >:vel of ‘arterial pressure is controlled by a balance between the fluid intake and the output of fluid by the kidneys, not by the level of total peripheral resistance as has been a long-standing misconception based on acute rather than chronic animal experiments.

INTRODUCTION The goal of this paper is to recall some of the history in the development of mathematical modeling of cardiovascular regulation and especially to point out some important concepts that have come from this effort. The entire discipline of physiology attempts to explain the mechtisms by which the body functions. Therefore, from the beginning, physiologists have been inveterate modelers. Unfortunately, though, most of this modeling has been nonquantitative, most being done in the mind rather than in a formal way. Not that we object to mental modeling, indeed far from this, but when the models become complex, the mind, with all of its preconccc:ions, often

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plays horrendous tricks and leads to very false understanding of even some of the most basic principles of physiology. In the 1940s and 195Os, algebraic and graphical analysis of simple physiological mechanisms began to appear. These used mainly algebraic equations or multiple intersecting graphical curves to analyze steady-state function, but the number of equations or graphical curves was usually limited to a maximum of 8 to 10. On rare occasions, differential equations were used; a prime example was the development of a system of equations by Hodgkin and Huxley that described the generation of an action potential [ 111. for which these two authors received the Nobel prize. The solution of these equations was a task that would have deterred all but a very few physiologists from completing the project. Yet, with the advent of analog and digital computers, such systems and equations are now commonplace, and the solution times are typically a fraction of a second. Our own work in mathematical analysis of circulatory function anti regulation began in the early 1950s with the use of complex graphical analysis [2]. This proved to be an excellent method for studying steady-state conditions or the change from one steady state to another. However, as one would expect, such methodology was almost useless for ana.l_yzingsequential, rapidly changing events in circulatory function. When analog computers became available in the 1950s and digital computers in the 1960s. our models proceeded rapidly through progressive stages of sophistication, as shown in Figures 1, 2, and 3 [3. 41. Figure 1 illustrates the relationship of the kidney output of fluid to the overall fluid balance of the body as well as to arterial pressure and cardiac output. The negative feedback loop of this figure has bccomc fundamental in essentially all mathematical analyses of long-term circulatory function. Therefore. let us list the functions of the successive blocks: (1) effect of increasing arteriJ1 pressure (AP) on urinary output of fluid $JO); (2) subtraction of output of fluid from intake of fluid to the body (Intake) to give rate of change of eutracellular fluid volume @K/L/~): (3) integration of rate of change of fluid volume to give actual extracellular fluid volume (ECFV); (4) effect of changing extracellular fluid volume cn blood volume (RV); (5) effect of changing the blood volume on the mean systemic pressure (MSP) that forces blood from the systemic circulation back toward the heart; (6) subtraction of right atria1 pressure (RAP) from mean systemic pressure to determine pressure gradient from the systemic circulation back to the j’ieart (MS P-RAP); (7) division of pressure gradient by resistance to venous return (RVR) to give venous return (VR) to the heart. which is also equal to c;lirdiac output JO); and (8) multiplication of cardiac output by total peripheral resistaLtice(TPR) to give arterial pressure (AP), which was the starting point in block 1. Thus, this simple mathematical loop describes a negative feedback operation that will eventually stabilize (a) the arterial

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RAP

FIG. 1. A simple model showing ;1s the r& of the kidney in long-term and Coleman

[ 3) with permission

the basic components of circulatory function ~1s\vclI control of the circulation. (Reprinted from (iuyton

from W. B. Saunders

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FIG. 2. involving (Rcprintcd

An expansion local vascular from <;uyton

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of the moel in Figure 1, with several additional feedback loops function of the hcrlrt. control, nenous control, and pumping and C&man [3j with permission from W. R. Saunders C‘o.)

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pressure, (b) the cardiac output, (c) the extracellularskid vahune, and (d) the blood volume. The model of Figure 2 [3J added still other important ,factors in the control of the circulatian, including: (a) two nervaus fkedba& mechanisms, the baroreceptor and chemoreceptor mechtisms for feedbaclc control of the , arterial pressure; (b) the role of local control of blood flow by the tissues in affecting the resistance to blood flow through the s$stemic circulation; 2nd (c) the role of cardiac function in circulatory control. With this model, it was now possible to simulate a considerable share of the operational charactcristics of the circulatory system. But the mare complex mechanisms still required much more information, which led to the model in Figure 3 [4]. This made1 is hardly different from that of Figure 2 except that. multiple inp it variables have been added so that almost any change in any major condition of circulatory function can be studied. The model of Figure 3 has about 30 separate negative feedback loops, and models developed after this one have many mare laaps. Let us pause for a few moments and discuss some of the problems in the use of such models. THE PROBLEM OF ESTABLISHING FUNCTIONS AND PARAMETERS

I

THE INDIVIDUAL

Mast physiologists are deterred at the very start from building even simple models, to say nothing of large ones, because they say that sufficient detailed information is not available in the literature. However, a special characteristic of physiological simulation makes it possible t<) succeed even though ori first thought one might not believe it possible. Namely, through evolution the animal body has developed an operating system that is remarkably stable against wide variations in almost any single input parameter. The principle of biological variability is well known. No two persons have exactly the same quantitative functions far operation of the separate mechanisms, and no two will have nearly the same values far the different input variables. vet, each human being is a successful operating system. To give a simple example, the normal go-year-old person has a heart capable of pumping ktnly about one-half as much blood as a 20-year-old person. Yet, even the W-year-old normally is not in heart failure and can function quite adequately as long as he does not exercise beyond the limit of tis heart’s pumping capacity. Therefore, even if a model builder should fail to choose exact parameters for the average human being, almost any chosen parameter within a reasonable from + 300 per cent to -75 per cent of the mean range -sometimes value-will be that of some individual who is operating reasonably normally. The reason for this amazing functional capability is the vast number of negative feedback control loops in the functional systems of the body. such as the one illustrated in Figure 1. Evolution has developed a setof powerful.

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no-nonsense control loops, usually highly damped, that will provide appropriate compensation for large variability of input variables. THE PROBLEM OF STABILITY The next question that we must ask is why a system such as that in Figure 3, with hundreds of functional interactions, does not decay immediately into uncontrollable instability. Here again, the answer is that evolution has eliminated essentially all positive feedback loops from the functioning body. The only instances in which po&tive feedback loops are present are to perform special tasks (such as trans&ssion of nerve impulses) that function in a negative-feedback way within some other mechanistic loop. Therefore, despite the imprecision of functional data for biological systems, nature has fortunately provided the animal body with such an inherently stable operating system that even very inaccurate attempts at simulation nevertheless can often give rise to very helpful concepts. THE PROBLEM OF RAPID SOLUTION OF THE EQUATIONS If one needs a model that can simulate rapid transients as well as very long-term transients, the mathematical problem of achieving continuous solutions without using inordinate computer time often becomes very difficult, because the model usually will have many short-time-constant loops interdigitated with very long-time-constant loops, leading to extremely stiff equations. For instance, the model of Figure 3 has time constants that range from as short as a few thousandths of a minute up to more than a month, more than a ten-million-fold ratio. Obviously, the strategy for continuously solving equations of this type can be quite complex. One way to achieve a solution is to segment the model so that the short-time-constant loops can be brought to steady states, and then long iteration intervals are used to solve the long-term loops. Unfortunately, though, this type of solution requires knowledgeable development of a specific strategy for each specific model. Recently, we have been attempting to develop a method that is semiautomatic for shifticg the time frame of the solutions. The basis of this method is to keep account, in an appropriate array, of the degree of oscillation of the differential input to each separate integrator in the model. Then this inforsnation is used to adjust automatreally the time constant of each integrator as necdcd for different time frames of solution. Under most conditions, this procedure can smooth out the variables in the short-term loops, preventing them from oscillating, and allowing their use in the continuous solution of the Bong-term variables when input variables are changed. However, an appropriate method is also needed to allow shift from one time frame of solution to another. In experimental tests of such procedures, we have been able to obtain reasonable solutions for reasonably simple models when the

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time frame has been varied as much as 10,000-fold. However, with more complex models, calculations increase approximately as the square of the number of feedback loops, and progress beyond our present experimental approach must be hoped for. SOME REPRESENTATIVE CARDIOVASCULAR THAT HAVE COME FROM MODELING

CONCEPTS

Simple models, or even intermediate models such as that in Figure 2, tie valuable mainly to prove that the modeler can put his preconceived ideas into mathematical notation. But when one proceeds to more complex models, such as the one in Figure 3, the world suddenly changes, because again and again solutions provided by the model fail to fit as expected. In most instances, it is the preconceived concepts that are wrong, in which case the model serves beautifully to correct one’s understanding, thus leading to a better concept. Let us review briefly several such concepts that have come from mathematical modeling of the cardiovascular system.

THE INFINITE-GAIN PRINCIPLE PRESSURE CONTROL

FOR LONG-TERM

ARTERIAL

One of our earlier models of renal function gave the solution in Figure 4 for the effect of progressively increasing a.rte
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Note the initia! large increases in arterial pnzssure, cardiac output, and urinary output. However, the great increase in urinary output caused progressive loss of fluid from the body until the arterial pressure returned to exactly the same level from which it started. Figure 7 illustrates another test of this principle (4). Referring again to Figure 5, one can see that the long-term level of arterial pressure could be increased to a higher value if one shouilld increase the level aE sjlt md water

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intake. And another way to increase the pressure would be to shift the renal-function curve to the ight in the diagram. In effect, these two changes would increase the ‘kset point” for arterial pressure control to a level much higher than the 100 mm Hg indicated by the equilibrium point in Figure 5. Then :he pressure should be controlled, again with infinite gain, around this new set point. Figure 7 3~ trates a simulation of such charrges, using the -model of Figure 3. That is, the renal1 output curve of Figure 5 ‘cKasshifted to the right by removing 70 per cent of the kidney mass, and the water and salt intake to the animal was increased severalfold (4). Looking at the curve second from the bottom, one sees that the simulated arterial pressure rises over a few days from 100 mm Hg to a new steady-state value of 140 mm )-Ig and holds exactly at that level thereafter, because the set .>oint of the infin+gain control mechanism has been raised to this new level But observe also in this figure the many other changes that take place in the circulation in order to achieve the final result,. There are important successive changes in extracellular fluid volume, blood volume, degree of aMonomic nervous stimulation. cardiac output, total peripheral resistance, aQd urinary output. Next, to show that the simulation predicts the changes actually observed in animal experiments, Figure 8 illustrates average curves, obtained from studies in dogs in which experimental tests of the simulation iHI h;igurc 8 were

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DAYS FIG. 7. Circulation: overall regulation. Simulation of the kculatory changes caused by suddenI> increasing the set point of the kidney-body-fluid pressure control system to a higher Ievcl. (Reproduced from Guyton, Coleman. and Grringer [4] uvith ptxmission from .d/lrltd Rtww qf Pl~~w~icq~. Vol. 34 ?‘ 1972 by Annual Reviews Inc.)

performed [7]. Note the almost exact correspondence between the curves in Figures 7 and 8.

The a!!)ove studies suggest that there are only two primary determinants of the long-term arterial pressure: (1) the level of salt and water intake and (2) the pressure level of the renal-function curve for urine output. Yet, a vast majority of physiologists and clinicians alike have insisted for years that arterial pressure is controlled either entirely or almost entirely by changes in the resistance to blood flow through the systemic vascular system, ctiIed the

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“ total peripheral resistance.” The reason for this widespread belief is that the short-term level of arterial pressure is indeed controlled to a major extent by changes in total peripheral resistance. However, in the short-term situation, the kidney does not have time to readjust tihe blood volume and therefore cannot establish its overriding effect in controlling arterial pressure. Yet, when one puts both the short-term arterial pressure regulating mechanisms and the long-term renal mechanism together in the same mathematical model, it then is easy to show that changes in total peripheral resistance should have no effect on the long-term level of the artetial pressure. This is illustrated by the simulation in Figure 9 [9]. Note that the time scale of this simulation is in days. The solid curves in this figure represent the sequential changes in arterial pressure, blood volume, and extracellular fluid volume following a threefold increase in total peripheral resistance at time zero. The instantaneous effect is an increase in arterial pressure, as shown by the uppermost curve, but this elevation of the arterial pressure causes rapid loss of fluid through the kidneys, with depletion of both the extracellular fluid volume and the blood volume until the arterial pressure returns exactly to the original starting level. This is the effect that is also predicted by the principle

DAYS

Frc. 8. Average data from dog expzrimcqtts verifying the simulated results illustrated in Figure 7. (Modified from Guyton [7\ with permission frorl W. B. Saunders CO.)

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Thercforc, it appears that changes in total peripheral resi *tance arc not the mt’;m~ bv which the long-term arterial pressure Icv& IS controlled. To test this still further, Figure 10 illustrates the effect in multiple clinical conditions of abnormal total peripheral resistances on arterial pressure and cardiac output 171.Note that the arterial pressure is normal in all of these conditions. On the other hand. the cardiac output is reciprocally proportional to the total peripheral resistance. This is mathematically exactly what one would expect, because the arterial pressure is regulated by ‘an independent mechanism, and cardiac output is equal to arterial pressure divided by total peripheral rG?;t ancc. S-l-t: DY OF PHYSIOLOGICAL MECHANISMS WHEN DIRECT EXPERIMENTS CANKOT BE PERFORMED In physiological research we are now reaching the point where most mechanisms that can bt; studied easily :lave been studied. Yet, many still

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TOTAL PERIPHERAL RESISTAWCE (pet cent of normal) FIG. 10. Data from the clinical literature showing th,at changes in total peripheral resistance do not affect the long-term arterial pressure level. Instead, the cardiac output changes reciprocally with the changes in total peripheral resistance. (Rcprintcd from Guyton [7] with permission from W. B. Saunders Co. Note: the cardiac-output value for the loss of all four limbs is a calculated value.)

unstudied but very important mechanisms are beyond atiack by presently available direct experimental methods. Therefore, probably the most important role that can be played by the use of complex mathematical models is to analyze and to understand these mechanisms, because indirect information from the interactions of these mechanisms with other, already weil-studied mechanisms can be used to work out with reasonable precision their functional attributes. As an example, Figure 11 illustrates the computer simulated effects on arterial pressure and renin secretion that would be caused by constriction of either the afferent arterioles or efferent arterioles of the kidneys or both. There is no method available for selectively constricting these vessels in an experimental animal. Yet the effects that such constriction can have on circulatory function can be computed. There is good evidence that the most common type of hypertension, called essential hypertension, may result at least partly from constriction of the efferent arterioles, probably along with constriction of the afferent arterioles as well. Therefore, note in the figure the predicted results when afferent and efferent arterioles are constricted together. The predictions are (1) that hypertension will occur, and (2) that the rate of renin secretion will fall to very low values. These are the exact events that are known to occur in the early stages of essential hypertension before pathological changes develop in the circulation. Other predictions from this same simulation, but not shown

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in

the figure, also match up with essential hypertension. Therefore, even though it is generally stated that the cause of essential hypertension is unknown, the mathematical model at least suggests a plausible cause. Thus, the value of mathematical models is not nzrely to describe what we already know, but also to give insight into mechanisms and concepts that are not accessible to direct experimental attack. REFERENCES 1 2 3

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A. L. Ho&kin and A. F. Huxley, Quantitative description of mcmbranc current and its applicatic rn to conduction and excitation in nerve. J. Ph.~w’ol. (Zmui. ) 117:S()o (1952). A. C. Guyton. Determination of cardiac output by equating venous return curves with cardiac respond curves, Phvsiol. Rev. 35:123 (1955). A. C C;uvton and T. G. Coleman, Long-term regulation of the circulation; intcrrclationship.4 witIt hod) fluid volumes. in Pl~~rwtul BUSCY o/’ C’ircuhrto~~~ Trtrmport, Rqplhw umi E.YCIwg~c (E. B. Reeve and A. C. Ciuyton, Ed,.). W. B. Saunders, Philadelphia, 1967. p. i 79. A. C. (iuvton. T. G. Colernan, an.1 II. J. Grctnger. Circulation: CSvt~allregulation, .~rm. Rw. PJqwol. 34:13 (1972). A. C. Guyton, J. B. Langston. and G. Navor. Theory for renal autoregulation by feedback at the juxtagl~omerular apparatus, <‘in*. Res. 14:1-187 (1964).

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A. C. Guyton, T. G. Coleman, A. W. Cow-Icy,Jr., R. D. Manning, Jr., R. A. Norman,

Jr., and J. D. Ferguson, A systems analysis approach to understanding long-range, arterial blood pressure control and hypertension, Circ. Res. 35:159 (1974). A. C. Guyton, Circu!utoy Ph_vsio!ogvIII: Arteriui Pressure und Hjpertension, W. B, Saunders, Philadelphia, 1980. W. A. Dobbs, Jr., J. W. Prat?ler, and A. C. Guyton, Relative importance of nervous control of cardiac output and arterial pressure, Amer. J. Curdiol. 24507 (1971). A. C. Guyton and T. G. Coleman, Quantitative analysis of the pathophysiology of hypertension, Circ. Res. 24(Suppi. 1): I-l (1969).