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Young DB. Control of Cardiac Output. San Rafael (CA): Morgan & Claypool Life Sciences; 2010.

## 5.1. Rationale For Building and Using Mathematical Models of the Cardiovascular System

Models are developed beginning with hypotheses concerning the organization of the system and the function of its components. For example, to develop a cardiovascular model useful for analyzing hypotheses concerning the importance of factors affecting venous return in cardiac output regulation, the model could be organized with emphasis on components related to venous return. The components’ functions could be described mathematically from data collected experimentally or from clinical observations. The function curves described in earlier chapters that were derived from experimental data could be expressed by one or more simple algebraic equations solvable by a digital computer. The components’ equations could then be arranged in series with the solution of the first, the dependent variable, being the independent variable in the next, and the solution to the final equation being the independent variable in the first equation, forming a loop that could be solved iteratively by the computer. Each iteration may be considered a unit of time chosen by the investigator. The inclusion of time as a component of the simulation gives this technique an advantage over other methods in that the function of the system can be studied at more than one point in time after a change in conditions.

To begin a simulation, variables are assigned initial values, often normal values. The simulation can then be started with the series of equations solved iteratively with the value of selected variables listed numerically or plotted graphically versus time. The investigator may choose to have the variables’ values expressed after each iteration or at longer intervals. After an initial period of operation, the investigator may introduce a change in the system designed to represent an experimental manipulation or a pathological condition. For example, to simulate a hemorrhage, a function could be introduced into the series of equations that would subtract a specified amount of blood in each iteration, representing a blood loss at a rate such as 10 mL/min. The responses of the variables of the simulated system over the subsequent period (number of iterations) could then be observed.

The construction of a model requires the investigator to consider his hypotheses in careful detail, eliminating “fuzzy thinking” or “hand-waving” logic. To write down the steps in the logic as a series of equations, the investigator must be able to think clearly through the details of a proposed hypothesis, frequently with greater rigor than when considering an idea mentally or when writing in prose.

Simulations enable the investigator to test the validity of hypotheses. A finding of disagreement of simulation solutions with actual system responses suggests that the hypotheses inherent in the model may not be valid. Often, such a situation is highly instructive, and by carefully studying the source of the model’s error, the investigator may gain insight into the function of the system.

If the results of the model simulation agree with known responses of the actual system under consideration, the agreement supports, but does not prove, the validity of the hypotheses inherent in the model. To return to the hemorrhage example, if the simulated responses of significant variables, such as mean systemic pressure, right atrial pressure, arterial pressure, and cardiac output, to a hemorrhage of 10 mL/min agreed with known responses of the actual cardiovascular system to a similar rate of blood loss, the agreement could be viewed as favoring the tenets of the hypotheses. Agreement of many simulated responses of the model to responses of the actual system provides even greater support for the hypotheses’ validity.

Model simulations permit careful observation of the components’ responses with time. Once the soundness of the model has been amply supported and the investigator has reasonable confidence in the strength of the hypotheses, simulations can be used to analyze in great detail the temporal responses of any and all system components to challenges of the investigator’s choice. Furthermore, the investigator can study simulations of events or situations that may be impossible to study experimentally. These processes provide an opportunity for heightened understanding of the workings of complex systems.

The inclusion of time as a variable in model simulations forces the investigator to recognize the significance of temporal responses of the system to challenges. Throughout the history of the study of cardiovascular physiology, nearly all experimental investigations were conducted over the course of a few hours. The data collected, usually at the conclusion of the study, were used to construct hypotheses assumed to be applicable to the long-term or steady-state operation of the cardiovascular system. But when constructing a mathematical model of the system, the investigator must consider how all variables change with time, and frequently, one may realize that cardiovascular responses require far longer to complete than was assumed previously. This realization led Guyton’s laboratory to undertake studies of the cardiovascular system that extended over many days and weeks, requiring grueling effort but yielding valuable results.

## 5.2. Cardiac Output Analysis Using A Simplified Cardiovascular Model

Beginning more than 40 years ago, Guyton and coworkers recognized the potential importance of using mathematical models of the circulatory system to analyze the control of cardiovascular function. They began their work before digital computers were generally available, doing initial simulations using analog machines. A most basic model they published from work with an early digital computer was based on the hypotheses that the heart pumped blood at the rate it returned from the circulation, without an appreciable change in right atrial pressure [33], hypotheses closely related to the Frank–Starling law of the heart [16]. Included were functions expressing the relationships between cardiac output, arterial pressure, total peripheral resistance, blood volume, and extracellular fluid volume. The model comprised the eight functions illustrated as block diagrams in Figure 5.1, arranged in a loop.

The first function, Block 1, represents the relationship between arterial pressure and urinary excretion; in the simplified model, excretion is of extracellular fluid. As arterial pressure increases, urinary output increases, with small increases in pressure causing large increases in urinary excretion. The quantization of the relationship was based on Selkurt’s studies on kidneys perfused at controlled pressures [35]. The slope of the relationship is approximately a 6% increase in output for each 1 mm Hg increase in arterial pressure between 100 and 200 mm Hg.

In this simplified scheme, the rate of fluid intake is set as a parameter, a value that is fixed as a constant value that is not affected by the functions of the model, but can be changed by intervention of the investigator. The fluid intake rate is summated with the solution of the first function, the rate of urinary output, in the second function, Block 2. The output of this function is the rate of change of extracellular fluid volume. Initially, the rate of intake and excretion are equal to each other, and the subsequent variables in the feedback loop are unchanging, in a steady-state condition.

The rate of change of extracellular fluid volume is integrated over time in Block 3, the output being the extracellular fluid volume at a specific time point.

Block 4 expresses the relationship between changes in extracellular fluid volume and changes in blood volume. The function is complex, based on observations that increases in extracellular fluid volume produce increases in blood volume up to an extracellular fluid volume of approximately 22 L in a healthy man, but greater elevations in fluid volume are associated with edema formation with little further elevation of blood volume. The output of Block 4 is blood volume.

Block 5 depicts the function relating changes in blood volume to changes in mean systemic pressure, a function described from experimental data by Richardson et al. [8].

Right atrial pressure is considered in this simplified case to be a parameter. In Block 6, right atrial pressure is subtracted from mean systemic pressure, the result being the pressure gradient for venous return.

Resistance to venous return is a parameter that in Block 7 is set as the denominator, mean systemic pressure being the numerator. The output of this function is venous return or cardiac output.

Block 8 is a multiplication function, cardiac output multiplied by another parameter, total peripheral resistance. The result is arterial pressure, the independent variable of Block 1.

Introducing changes in the model’s functions or parameters in ways that simulate experimental manipulations or pathological conditions can test the model’s hypotheses. For example, the rate of fluid intake, the relationship between extracellular fluid volume and blood volume, resistance to venous return, right atrial pressure, total peripheral resistance, and the kidneys’ fluid excretory function in relation to arterial pressure all can be altered by adjusting the parameter values or the equations used to express the functions. The resulting simulated responses then can be compared to experimental or clinical data. This process has been carried out repeatedly over the past decades, and in all cases, the tests have supported the model’s validity and increased confidence in its accuracy. Consequently, this simple series of functions has been used as a basis for many, much more complicated cardiovascular models. It has also been used to gain understanding of basic principles of cardiovascular control.

The model describes a negative feedback control system that is fundamental to cardiovascular physiology. Although very few functions are included, along with several significant implications of complex physiological phenomena, the model illustrates key concepts in cardiovascular physiology. The importance of the pressure gradient for venous return as a determinant of cardiac output can be clearly appreciated in the model; changes in either mean systemic pressure or right atrial pressure resulting in reduction in the pressure gradient for venous return will reduce cardiac output. Reduction in blood volume from hemorrhage or dehydration reduces pressure gradient for venous return and consequently cardiac output in simulations and in experimental and clinical studies. Elevation of right atrial pressure also reduces pressure gradient for venous return, which is a prominent factor in the diminished cardiac output seen in cases of acute and chronic heart failure. The immediate effects of changes in total peripheral resistance on arterial pressure are to alter it in proportion to the change in total peripheral resistance, but the long-term simulated effects are more complicated, as discussed below.

Inclusion of renal excretory function in the model permits testing of the importance of changes in renal function on long-term circulatory control. The model predicts that if arterial pressure rises above the normal level, even by only a few mm Hg, for example, due to an infusion of a large amount of fluid, the rate of fluid excretion will rise to a level significantly greater than normal. If fluid intake remains at the normal level, the system will be in a state of negative fluid balance as long as arterial pressure is greater than normal, leading to a progressive reduction in extracellular fluid volume, blood volume, mean systemic pressure, pressure gradient for venous return, cardiac output, and finally arterial pressure. The effect will continue until arterial pressure falls back to the normal level, at which renal excretion will equal the rate of fluid intake and the system will again be in a stable condition with all variables at their normal values. Significantly, the simulations suggest that the correction of such a blood pressure rise would require several hours or more to complete, depending on the magnitude of the initial blood pressure rise. The prediction has been verified by experiments in which blood pressure has been raised or lowered by several different means, all confirming the operation of the negative feedback as proposed in the model.

The operation of this simple negative feedback loop has several implications that are important to understanding hypertension and blood pressure regulation in general. First, any factor that acutely increases blood pressure but does not affect the kidneys’ ability to excrete fluid will not cause sustained hypertension. As long as renal function is normal with respect to fluid excretion, any increase in blood pressure will elicit operation of the negative feedback loop just described and, consequently, return blood pressure to the normal level. Therefore, although as discussed above, the immediate effect of an increase in total peripheral resistance on blood pressure is immediately apparent, the long-term response is more subtle if the vascular resistance change excludes renal resistance. Second, by analogous reasoning, sustained impairment in renal ability to excrete fluid will cause sustained hypertension. If the kidneys’ function changes in a way that prevents fluid excretion at the normal rate at the normal blood pressure and fluid intake continues at a rate greater than excretion, a positive fluid balance will continue until arterial pressure rises to a level that increase renal fluid excretion to equal the rate of intake. Arterial pressure will remain at that level for as long as renal excretory function is impaired, requiring higher than normal perfusion pressure to balance excretion with intake of fluid. Similarly, augmenting renal ability to excrete fluid, which is the mechanism of action of diuretics and other antihypertensive medications, will reduce steady-state blood pressure.

## 5.3. Cardiac Output Analysis Using An Expanded Cardiovascular Model

The model described above is very useful as a conceptual tool for studying basic cardiovascular function and the processes of modeling physiological processes. However, it is admittedly limited by its simplicity. Guyton and Coleman developed a more complex scheme built around the same fundamental core with additional components derived from the effects on the circulatory system of changes in heart strength, autonomic nervous system activity, and local vascular autoregulatory mechanisms. The block diagram of the more complex model is illustrated in Figure 5.2 [33].

### FIGURE 5.2

The model contains 29 functions that reduce complex physiological relationships to simplified mathematical expressions. It illustrates the modeling process of incorporating additional complexity by incrementally adding functions around a central core. This is about the largest model that can be studied in block diagram form; illustrations of more complex models begin to resemble a maze. However, construction of this one can be followed graphically with some brief explanation.

Blocks 1–8 are the same functions used in the previous model. Its operation is affected by input from three negative feedback control loops: the local tissue autoregulatory system, Blocks 9–14; the effects of changes in heart strength on cardiovascular system, Blocks 16–19 and 27; and the effects of changes in autonomic nervous system function on circulatory control, Blocks 20–29.

### 5.3.1. Local Tissue Autoregulation

Previously, local tissue autoregulation was discussed with regard to its effect on vascular resistance (Chapter 2). Most tissues have the inherent capacity to regulate their flow of blood to meet metabolic needs. If flow is inadequate for their current requirements, local mechanisms within the resistance vessels of the tissue respond by causing vasodilation of the vessels regulating flow to the capillaries of the tissue, thereby increasing tissue blood flow and nutrient delivery. If flow is greater than required, the same control mechanisms act to increase vascular resistance and decrease flow. Block 9 depicts the function relating cardiac output as the independent variable to the rate of change in tissue vascularity, representing vascularity of all tissues. As cardiac output decreases, the reduced flow through the tissues of the body increases the value of the vascularity variable. Blocks 10–12 are a modeling technique used to integrate the change in the vascularity variable and provide a time delay in the response determined by a time constant, parameter K2, in Block 12. The product of Block 11, vascularity, divides the value of parameter K1, representing the effect of metabolic rate on vascular resistance, in Block 13, yielding arterial resistance. In Block 14, arterial resistance is multiplied by a factor from the autonomic nervous system functions, the product of which is summated in Block 14 with the parameter representing venous resistance, *V*_{res}, yielding total peripheral resistance. The arterial resistance variable is also used along with *V*_{res} to determine resistance to venous return, in Block 15, which is an equation derived from the ratio of arterial to venous resistance and the relative contribution of each to resistance to venous return (Chapter 2). In the model, the effects of a value of whole body tissue flow rate (CO) that is inadequate to meet the metabolic demands of the tissues is an increase in the vascularity factor resulting in decreases in arterial resistance, total peripheral resistance, and resistance to venous return, leading to iterative increases in cardiac output that ultimately meet the blood flow demands of the tissues.

The local autoregulatory system has been studied experimentally and can produce marked reductions in resistance to venous return and increases in cardiac output within seconds of the start of increased metabolic demand in exercise. Over longer periods, the magnitude of the effect can be as great as a several hundred percent increase in whole body flow. The system’s temporal responses are both very rapid and can be sustained indefinitely; the rapid response is caused by vasodilation of existing vessels, whereas the long-term response is due to growth of additional blood vessels.

### 5.3.2. Cardiac Function Effects on Regulation of Cardiac Output

A greatly simplified scheme to include the effects of changes in cardiac function on the cardiovascular system is represented in Blocks 16–19 and 27. Block 16 is a cardiac function curve plotted with venous return (or cardiac output) plotted as the independent variable and right atrial pressure as the dependent variable, so for this modeling situation, venous return determines right atrial pressure, with an increase in venous return resulting in an increase in right atrial pressure. In the normal range, increases in venous return cause only small increases in right atrial pressure. Block 17 is an adjustable parameter representing heart strength. Block 18 is the function relating the effect of arterial pressure, or afterload, on cardiac output; as arterial pressure increases to high levels, cardiac output is impaired. The effect is combined in Block 27 with the autonomic effect on the heart, the product of which divides the heart strength factor in Block 19. The product of Block 19 is used to modify the cardiac the function curve in Block 16.

### 5.3.3. Autonomic Nervous System Effects on Cardiac Output Regulation

Blocks 20–29 represent the functions of the autonomic nervous system that interact with circulatory control and cardiac output regulation. Block 20 represents the relationship between arterial pressure as the independent variable and the autonomic factor as the dependent variable. Blocks 21 and 22 apportion the factor, with one-fourth going to the chemoreceptors and three-fourths to the baroreceptors. Baroreceptor drive to the autonomic system is known to adapt over time. A change in arterial pressure from the initial level causes a change in baroreceptor drive, but this adapts within a few days back to the level associated with the initial pressure. Blocks 23–25 model this adaptation phenomenon. Chemoreceptor drive does not adapt, and it is summated with the adapted baroreceptor output in Block 26, yielding the autonomic multiplier. The autonomic multiplier interacts with the other components of the model system in three areas: in Block 27, it multiplies the effect of arterial pressure on cardiac strength (Block 18); in Block 28, it multiplies the effect of the vascularity factor on arterial resistance; and in Block 29, the autonomic factor multiplies the effect of blood volume on mean systemic pressure (Block 5).

This more complex model, while still an extremely simplistic representation of cardiovascular function, enables testing and exploration of many more hypotheses than the first eight-function loop. Comparisons of experimental data and clinical observations with simulations in many varied circumstances have verified the model’s validity for several decades.

Very significantly, this model, which is still at a basic level of complexity, clearly predicts the independence of long-term arterial blood pressure control from all cardiovascular functions except the relationship between renal perfusion pressure and renal excretory function. Manipulation of functions such as resistance to venous return, peripheral vascular resistance (excluding renal vascular resistance), vascular capacitance (compliance), heart strength (within normal limits that preclude heart failure), and autonomic nervous system function (excluding effects on renal function) will not affect the predicted long-term regulation of arterial pressure. These predictions have been tested in experimental studies in many animal models and in clinical studies for nearly half a century; data from each have confirmed the prediction and clearly demonstrated the dominance of the renal perfusion pressure–renal sodium and water excretion rate–body fluid volume negative feedback system in long-term blood pressure control. Alterations in other portions of the cardiovascular system have significant short-term effects on blood pressure, but those are overridden by the long-term ability of the renal body fluid volume regulatory mechanism to maintain arterial pressure at its control level.

## 5.4. Digital Human

Modeling of physiological systems has developed for several decades since Guyton and Coleman made their first cardiovascular efforts. Coleman’s work has progressed to his current model, *Digital Human*, which is much more complex than earlier contributions, and includes many more physiological systems and greater detail. However, its cardiovascular functions closely resemble the core of the first models, with added units representing peripheral feedback control systems interacting with the central functions. Because the same fundamental cardiovascular functions persist in the current model, its simulations can be used to illustrate and test basic cardiovascular concepts and regulation of cardiac output.

Coleman designed *Digital Human* to be an open source publication, easily used by those interested in cardiovascular and broadly related areas of physiology. It can be downloaded from http://digitalhuman.org and used on office computers using Windows operating systems (a Macintosh version may be available in the near future). The reader is encouraged to make use of this valuable resource.

## 5.5. Summary

Several aspects of modeling the cardiovascular system can be helpful in gaining an understanding of cardiac output regulation and circulatory control in general. The process of building the model requires the investigator to carefully think through the proposed hypotheses in sufficient detail to assemble a series of equations, each representing a component of the proposed system. Generally, it is very difficult to assemble mentally the details of complex proposals, and nearly impossible when several parallel and in-series components with differing time constants interact. Often, just the attempt to write down the details of hypotheses in preparation for the mathematical expressions is sufficient to disprove untenable ideas.

Having a solvable mathematical model representing a hypothetical system provides a means to test the proposed system by comparing the results of model simulations with experimental data and clinical observations. In many instances, such comparisons are the only way to test complex proposals. One case of nonconformity of simulated results with data from actual experiments is sufficient to reject a hypothesis and force the investigator to seek the errors of his thinking, “back to the drawing boards.” Probably, most useful models are completed atop a heap of corrected errors. Model simulations can never definitively prove the validity of a hypothesis, but if model simulations are found to agree with data from a wide variety of experiments and clinical situations, investigators and other students of the subject can have some measures of confidence in the proposed hypothetical system. With each additional finding of agreement, the confidence increases.

Once a model’s validity is firmly supported and the investigator has confidence that the model is an accurate representation of the actual system, its simulations can be studied to investigate the detailed functions and relationships of the cardiovascular system in ways that may not be possible in experimental physiological or clinical analyses. The investigator and students of the subject both can share this aspect of mathematical modeling.

- Analysis of Cardiac Output Regulation by Computer Simulation - Control of Cardia...Analysis of Cardiac Output Regulation by Computer Simulation - Control of Cardiac Output

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