## Abstract

The circuit analysis of an electric analog of the systemic circulation, the focus of a classic paper by Guyton, Lindsey, and Kaufmann, provides a framework for understanding the factors that impact venous return and for appreciating the value of modeling physiological systems. The classic 1955 paper by Guyton, Lindsey, and Kaufmann gives your students an opportunity to learn about modeling from the physiologist who pioneered it (Guyton) and demonstrates that mathematics and data graphics are fundamental tools with which to learn about the regulation of the cardiovascular system. In this essay, I outline avenues of discovery by which your students can explore the factors that impact venous return.

- cardiac output
- cardiovascular physiology
- electric analog
- systems analysis
- teaching

the cardiovascular system lends itself to scientific study with quantitative tools such as electric analog models. Why? Because general characteristics and specific components of the cardiovascular system can be represented by elements of an electronic circuit. The elements of an electronic circuit (resistors, capacitors, and inductors) can be described using mathematical equations, and the behavior of the entire circuit can be reduced to a single equation and depicted using data graphics.

By 1955, data graphics were familiar players in physiology: Fenn, Rahn, and Otis had used data graphics to great advantage in their pivotal study of pulmonary gas exchange (4). It was not until the classic paper by Guyton, Lindsey, and Kaufmann (6), however, that electric analogs and circuit analysis demonstrated their promise as research tools in physiology.^{1}

Why is this classic paper by Guyton, Lindsey, and Kaufmann a valuable educational resource? It is valuable because it demonstrates the use of circuit analysis in the study of the cardiovascular system. It is valuable because it illustrates that mathematics and data graphics are fundamental tools with which to learn about the regulation of the cardiovascular system. And last, it is valuable because it gives your students an opportunity to learn about modeling from the physiologist who pioneered its application to the cardiovascular system: Arthur Guyton.

### Arthur Guyton: Some Background

In retrospect, we might have expected that Arthur Guyton would pioneer the use of circuit analysis in an effort to better understand the cardiovascular system.^{2} His father, Dr. Billy Guyton, was an ophthalmologist and Dean of the School of Medicine at the University of Mississippi. And, for 5 years before she married Dr. Billy, his mother, Kate Smallwood, taught mathematics, chemistry, and physics in China. The Guytons must have provided a fertile intellectual environment for young Ott.^{3}

Like many of us, Guyton constructed some of his childhood creations using an Erector Set. Unlike many of us, however, the teenage Guyton also constructed radio transmitters and receivers, oscilloscopes, and other electronic devices. One summer, Guyton read textbooks on the mathematical analysis of electronic circuits and on the physics of electronics. He would say later that he learned electronics more thoroughly than any course he took in high school or college.

In his senior year of high school, Guyton juggled the career choices of engineering, a PhD in physics, and medicine. Because he realized he would have more job opportunities, he opted for medicine. Guyton was delighted, however, that medicine provided endless opportunities for him to satisfy his love of electronics, physics, and construction.^{4}

As much as Guyton loved electronics, physics, and construction, he loved his wife Ruth (Fig. 1) and their family even more: Ruth was the greatest thing that ever happened to me in my life. It was her willingness to have ten children that made our family what it is.

It is perhaps remarkable that Guyton could work at home in the midst of his family: Working at home at night, he would dictate into a tape recorder the lectures, textbook revisions, and research papers that occupy a busy scholar's life. Transcribing the tapes to the typewriter, [his long-time secretary] Mrs. Howard heard the work of one of this century's greatest scientists over a steady background noise of kick-ball games, hide-and-seek and leapfrog.

If Guyton worked on this classic 1955 paper at home, there was background noise from just 6 of their 10 children.^{5}

### Circuit Analysis: the Early Models

In textbooks of general (Refs. 5 and 11) and cardiovascular (Ref. 15) physiology, you are likely to find data graphics that depict the theoretical impact of changes in some aspect of the systemic circulation. You are unlikely to find the mathematical equations behind those simulations. In this classic paper by Guyton, Lindsey, and Kaufmann, you will find both.

In 1955, Guyton, Lindsey, and Kaufmann (6) modeled venous return using three anatomic compartments: arteries and arterioles, capillaries and venules, and major veins (Fig. 2). From the electric analog of this model, Guyton, Lindsey, and Kaufmann derived venous return Q (in ml/min) as (1) where P_{mc} and P_{ra} are mean circulatory pressure^{6} and right atrial pressure; *C*_{3} and *R*_{3} are the capacitance of and resistance in the arteries and arterioles; *C*_{2} and *R*_{2} are the capacitance of and resistance in the capillaries and venules; *C*_{1} and *R*_{1} are the capacitance of and resistance in the major veins; and *C* is the total capacitance of the systemic circulation, where *C* = *C*_{1} + *C*_{2} + *C*_{3}. In *Eq. 1*, the symbols for mean circulatory pressure and right atrial pressure are the ones used by Guyton in 1959 (7).

In 1959, Guyton (7) modeled venous return using a two-compartment model (see Fig. 2). In this simplified model, venous return is (2) where ΔP is the pressure gradient for venous return; *C*_{a} and *R*_{a} are the capacitance of and resistance in arteries, arterioles, and capillaries; *C*_{v} and *R*_{v} are the capacitance of and resistance in venules and veins; and where *C* = C_{v} + *C*_{a}.

### Strategies for Students

Graduate students and undergraduate students who know some cardiovascular physiology can benefit from reading this paper by Guyton, Lindsey, and Kaufmann.^{7} As with other learning activities, your students can answer the questions individually or in small groups. Bear in mind that the process of developing a reasonable answer to each question is more important than the answer itself. This emphasis is something Guyton would have appreciated: he advocated discovery learning long before the phrase existed (3, 14). Svinicki (16) summarizes the pedagogy behind discovery learning.

### Opportunities for Discovery Learning

In this section, I list questions that your students can use to think about the classic 1955 paper by Guyton, Lindsey, and Kaufmann, and I provide answers you can use as a foundation for discussion. *Question 1* addresses a general experimental issue: control groups. *Questions 2* and *3* address quantitative issues specific to Guyton's model of the systemic circulation. *Questions 4* and *5* probe the physiology behind Guyton's model. *Question 6* addresses another general experimental issue: analog models.

This paper by Guyton, Lindsey, and Kaufmann provides learning opportunities that can be quite mathematical. Although some of your students may enjoy mathematics, many will not. Only *question 2* relies on mathematics that many of your students will want to avoid. To circumvent the matter of mathematics, I provide three solutions to *question 2*: a theoretical solution that involves a basic rearrangement of *Eq. 2*, an empirical solution that involves substituting numbers into *Eq. 2*, and a comprehensive solution based on partial differential equations. Because quantitative approaches to physiology research have been useful historically–they continue to be useful–I wanted to include *question 2*. It goes without saying that you can choose the solution most appropriate for your students.

### Question 1. Control groups are integral components of most research studies. Guyton, Lindsey, and Kaufmann did not use an explicit control group. Why not?

Guyton, Lindsey, and Kaufmann did not use an explicit control group because they assessed first the adequacy and then the predictions of their electric analog. There was no need to control for other factors that could have affected venous return. Nevertheless, to better understand the impact of the peripheral circulation on venous return, Guyton, Lindsey, and Kaufmann did use all-important comparisons in their experiments [see Fig. 3 and original Fig. 4 from Guyton, Lindsey, and Kaufmann (6)].

### Question 2. Consider the simplified 1959 model of the systemic circulation (Fig. 2 and Eq. 2). To which of the four model elements (arterial capacitance, arterial resistance, venous capacitance, and venous resistance) is venous return most responsive?

On p. 465–466, Guyton, Lindsey, and Kaufmann provide an answer that addresses the relative importance of venous and arterial resistance to changes in venous return: Though the impedance to venous return, as expressed by the denominator in the formulae, is the most difficult factor to test experimentally, a few general principles have been observed. If one observes the denominator of [

When I studied the denominator of *Eq. 2*], it becomes obvious that the resistances in the venous circuit [*R*_{v}] are highly important in determining the overall impedance to venous return, whereas the resistance in the arterioles and arteries [*R*_{a}] is relatively unimportant in the determination of the overall impedance to venous return.*Eq. 2*, it was not at all obvious to me why this should be so.

The first of three solutions to this question involves rearranging the basic equation for venous return (*Eq. 2*). Before doing this, your students can simplify their lives if they assume that arterial and venous resistance impact venous return more than do arterial and venous capacitance. How can your students justify this simplification? The model for venous return (*Eq. 2*) describes venous return during steady-state conditions: this means the volumes of the arterial and venous compartments remain constant. As a consequence, during steady-state conditions, it is resistance that has the prevailing effect on venous return. Your students can determine only the resistance–arterial or venous–to which venous return is more responsive.

This is how your students can rearrange the equation for venous return to arrive at an answer.

step 1. Begin with the equation for venous return (*Eq. 2*), and multiply out terms in the denominator

step 2. Group terms in the denominator by resistance: (3) Because *C* = *C*_{v} + *C*_{a}, you can simplify *Eq. 3* to (4) Venous return is more responsive to changes in venous resistance. Intuitively, this makes sense: changes in venous resistance impact the entire blood volume that is stored upstream of venous resistance; changes in arterial resistance impact just a small fraction, *C*_{a}/*C*, of the blood volume that is stored upstream of arterial resistance.

Your students can also arrive at the answer to this question using algebraic calculations or differential equations. No matter which path your students elect to follow, they must first decide whether to change the value of each element by an absolute or relative amount. Given that resistance and capacitance have different measurement scales, relative changes are more useful. All that remains is to choose initial values for the pressure gradient for venous return and for each of the four elements. These are the initial values I used:

The pressure gradient for venous return, ΔP, is 7 mmHg.

Total systemic resistance,

*R*_{a}+*R*_{v}, is 0.02 mmHg·ml^{−1}·min.This is the quintessential value reported for total systemic resistance (15).

*R*_{a}is nine times greater than R_{v}:*R*_{a}:*R*_{v}= 9:1.I based this assumption on the estimated fractional drop in pressure from the aorta to distal capillaries (11).

Total capacitance of the systemic circulation,

*C*_{v}+*C*_{a}, is 285 ml/mmHg.*C*_{v}is 19 times greater than*C*_{a}:*C*_{v}:*C*_{a}= 19:1.

With these initial values, venous return is 2414 ml/min (Table 1).

Table 1 lists values for venous return that will enable your students to answer this question. Figure 4 depicts the values. It is clear that venous return is most responsive to changes in venous resistance. There is a problem: this conclusion, albeit correct, can change if the initial conditions–the initial values of the elements–change.

If your students use partial differential equations to answer this question, they will arrive at one solution regardless of the initial values of the elements. To do this, your students must calculate the rate of change of venous return with respect to element *x*, δQ/δ*x*, while holding constant the values of the three remaining elements. A few of your students may want to calculate the partial differential equations by hand. Most will prefer to use a software package like Mathematica (17) to simplify the differentiation.

These are the partial differential equations that describe the rate of change of venous return with respect to each of the four elements: (5) (6) (7) (8) As *Eqs. 5*–*8* demonstrate, if the initial values of the four elements change, so too will the magnitudes of the partial derivatives. The ranking of the magnitudes of the partial derivatives, however, will remain constant (see Fig. 5).

To see this, the simplest approach is to compare resistances and capacitances separately. To compare δQ/δ*R*_{v} to δQ/δ*R*_{a}, divide *Eq. 5* by *Eq. 6* and simplify: (9) That is, δQ/δ*R*_{v} exceeds δQ/δ*R*_{a} by a factor of 1 + (*C*_{v}/*C*_{a}). To compare δQ/δ*C*_{a} to δQ/δ*C*_{v}, divide *Eq. 8* by *Eq. 7* and simplify (10) In other words, the magnitude of δQ/δ*C*_{a} exceeds the magnitude of δQ/δ*C*_{v} by a factor of *C*_{v}/*C*_{a}. At this point, we have these paired rankings of the magnitudes of the partial derivatives:

To complete the sequential ranking, we must compare δQ/δ*R*_{a} to δQ/δ*C*_{a} by dividing *Eq. 6* by *Eq. 8*: The last step is to decide if It is: C_{a}(*C*_{v} + *C*_{a})/*C*_{v} > 1, and 1/*R*_{a} > 1. Therefore, the ranking of the magnitudes of the partial derivatives is Just as Guyton, Lindsey, and Kaufmann stated, venous return is indeed most responsive to changes in venous resistance.

If your students get queasy just thinking about differential equations, you can still use this approach to help them estimate the relative responsiveness of venous return to the four elements of the simplified 1959 model of the systemic circulation. How? Present *Eqs. 5*–*8* and ask your students how they can compare the four rates of change of venous return. That question can help lead them to the strategy that they can divide one equation by another to assess the relative responsiveness of venous return to each of the four elements.

### Question 3. Guyton, Lindsey, and Kaufmann define capacitance as dV/dP, the change in volume per unit change in pressure. If the total capacitance of the systemic circulation changes by a factor of k, what will be the impact on venous return?

There are two approaches your students can take to answer this question: the definition of capacitance and the simplified equation for venous return (*Eq. 2*).

If total capacitance *C* changes by a factor of *k*, then the new total capacitance *C** is *C** = *kC* = *k*(dV/dP). This means that every unit change in pressure now produces a change in volume that differs from the original change in volume by a factor of *k*. But the model for venous return (*Eq. 2*) describes venous return during steady-state conditions only: there are no provisions for the parameters to change as functions of time. Because mean circulatory pressure remains constant, the volumes of the arterial and venous compartments remain constant despite the change in total capacitance. As a result, venous return remains constant.

This is how your students can arrive at the same conclusion using *Eq. 2*. The first step in this approach is to derive the following rearranged version (*Eq. 4*) of the simplified equation for venous return: If total capacitance *C* changes by a factor of *k*, then the new total capacitance is *kC*. This means that arterial capacitance *C*_{a} also changes by a factor of *k*: *kC*_{a}. As a result, the weighting factor associated with arterial resistance, *kC*_{a}/*kC* = *C*_{a}/*C*, is unchanged. Venous return remains constant despite a *k*-factor change in the total capacitance of the systemic circulation.

### Question 4. When Guyton, Lindsey, and Kaufmann increased mean circulatory pressure, they found that the actual increases in venous return exceeded the theoretical increases predicted from their model (Fig. 6). Why did this happen? What is the implication of the steepest portion of the actual venous return response?

At a given right atrial pressure, venous return is a function of mean circulatory pressure (*Eq. 2*). The theoretical relationship between venous return and mean circulatory pressure is a straight line, the slope of which is (11)

The actual relationship is curvilinear: as mean circulatory pressure increases, the slope of the relationship between venous return and mean circulatory pressure continues to increase until it reaches some maximum value (Fig. 6). How can this happen? To simplify matters, assume venous and arterial capacitance remain constant. From a pure mathematical perspective, if the denominator in *Eq. 11* decreases, then the slope will increase. The real question is how can your students explain this from a physiological perspective? Blood vessels are distensible. Therefore, when mean circulatory pressure increases, blood vessels distend, resistance decreases, the denominator in *Eq. 11* decreases, and a given change in mean circulatory pressure produces a greater change in venous return.

The implication of the steepest portion of the actual venous return response–the slope increases no more–is that mean circulatory pressure has decreased vascular resistance to some minimum value.

### Question 5. In a typical person, mean arterial blood pressure is 90–100 mmHg higher than mean circulatory pressure, but central venous pressure is just 5–10 mmHg lower than mean circulatory pressure. How do you explain this? If this person begins to develop congestive heart failure, what will happen to mean arterial and central venous pressures?

Your students can provide a qualitative answer to this question if they consider the definition of capacitance: dV/dP, the change in volume per unit change in pressure. Suppose that the heart of this typical person is arrested instantaneously and mean circulatory pressure is measured to be 7 mmHg. The sinus rhythm reestablishes itself, and some volume of blood is translocated from the venous side to the arterial side of the heart: the arterial side gains the volume of blood that the venous side loses. In each vascular compartment, the resulting change in pressure is inversely proportional to capacitance: arterial pressure increases more than venous pressure falls because arterial capacitance is less than venous capacitance.

Your students can answer this question more quantitatively if they do a basic thought experiment, the foundation of which is a physical model of the systemic circulation. In the model, suppose total resistance is maintained at 0.02 mmHg·ml^{−1}·min and venous capacitance is 19 times greater than arterial capacitance. A pump generates flow (cardiac output) through the model.

Before the pump is turned on, there is no flow through the model, and pressure throughout the model is mean circulatory pressure. Suppose the characteristics of the model are such that mean circulatory pressure is 7 mmHg.

Imagine now that the pump is turned on and generates instantaneously a steady flow of 1000 ml/min through the model. What happens to arterial and venous pressures? By virtue of the relationship of ΔP = Q*R*, we know that ΔP, the difference between arterial and venous pressures, must be 20 mmHg to drive a flow of 1000 ml/min through a resistance of 0.02 mmHg·ml^{−1}·min: All that remains is to determine arterial and venous pressures. When the pump is turned on, it translocates some volume of fluid from the venous side to the arterial side of the model. This means that venous pressure will fall, and arterial pressure will rise. Because venous capacitance is 19 times greater than arterial capacitance, the translocation of fluid will generate a decrease in venous pressure that is 1/19 the increase in arterial pressure. Therefore, at a flow of 1 l/min, venous pressure will decrease from 7 to 6 mmHg and arterial pressure will increase from 7 to 26 mmHg (Fig. 7).

If the pump generates a steady flow of 5000 ml/min through the model, the difference between arterial and venous pressures must be 100 mmHg. Venous pressure will decrease to 2 mmHg and arterial pressure will increase to 102 mmHg (see Fig. 7).

In congestive heart failure, cardiac contractility is compromised. As a result, mean arterial pressure decreases and central venous pressure increases. When central venous pressure is sufficiently elevated, the external jugular veins become distended: this visible distention is a classic sign of congestive heart failure.

### Question 6. In their circuit analysis, Guyton, Lindsey, and Kaufmann used an electric analog that was a gross representation of the systemic circulation. Discuss the value of a model that merely approximates an actual physiological system.

Some scientists discount or even dismiss results obtained from an approximate model of an actual biological system. This is silly. Other scientists consider as gospel results obtained from an approximate model of an actual biological system. This too is silly.

In essence, a model is a mathematical description of some system. Because most physiological systems are complex, most models are simple representations of the actual system. This simplification facilitates not only the development of the model but also the interpretation of the predictions generated by the model. Scientists who use models recognize that the predictions generated by the approximate model are only approximate.

An approximate model is valuable for two reasons: it provides a cogent tool with which to think about a complex system, and it generates ideas that can be tested. Guyton, Lindsey, and Kaufmann capitalized on both.

### Summary

The application of circuit analysis to the study of venous return, the essence of this paper by Guyton, Lindsey, and Kaufmann (6), became a cornerstone for research into the regulation of the cardiovascular system. From the three-compartment model of venous return he published in 1955, Guyton developed an analog model of the entire cardiovascular system (12).^{9}

This classic paper by Guyton, Lindsey, and Kaufmann gives your students an opportunity to learn about modeling of the cardiovascular system from the physiologist who pioneered it. The paper also demonstrates that mathematics and data graphics are fundamental tools with which to learn cardiovascular physiology. And the paper is an opportunity for your students to learn about Arthur Guyton: a husband and father who lived a most remarkable life, a physiologist who is treasured by his former students (14), and a person who is likely to continue to inspire future generations of children (1).

## Footnotes

↵1 John E. Hall (13) has written about the impact of Arthur Guyton and his research on cardiovascular physiology.

↵2 This background comes from a biography written by Brinson and Quinn (3).

↵3 Guyton's nickname.

↵4 Among his other projects, Guyton built an electronic apparatus to measure pressure, a device to detect aerosol particles, leg braces, a hoist to lift patients, and a motorized wheelchair that was controlled by a joystick.

↵5 David, Robert, John, Steven, Catherine, and Jean were born between 1944 and 1954. Douglas, James, Thomas, and Gregory were born between 1956 and 1967. All 10 are now physicians.

↵6 Mean circulatory [filling] pressure is the theoretical pressure that results when cardiac output stops immediately and pressures in the circulation equilibrate instantaneously (9).

↵7

*Circulatory Physiology: Cardiac Output and Its Regulation*(8) is an outstanding supplemental resource.↵8 The unstressed blood volume is the volume that can be added to a vasculature without increasing its mean circulatory pressure.

- © 2007 American Physiological Society