# Chapter 4Integrated Analysis of Cardiac Output Control

## 4.1. Graphical Analysis of Cardiac Output Regulation Based on Combined Venous Return and Cardiac Function Curves

Regulation of cardiac output requires interaction of factors affecting the return of blood to the heart from the peripheral tissues and factors affecting the pumping ability of the heart itself. The previous two chapters described ways to develop quantitative descriptions of venous return and cardiac output as functions of right atrial pressure. Both function curves are graphical expressions of complex algebraic equations, each having the same two variables, flow and right atrial pressure. While neither alone can be solved for either flow or right atrial pressure, solving the two expressions by plotting both simultaneously on the same axes can yield solutions for both. Plotting the two relationships on the same right atrial pressure versus flow coordinates reveals that they intersect at the equilibrium point that is the cardiac output and right atrial pressure for the cardiovascular system described by those venous return and cardiac output functions. Figure 4.1 presents the combination of a venous return curve and cardiac function curve plotted together, intersecting at the equilibrium point [31].

#### FIGURE 4.1

Presented in the figure are a normal venous return curve intersecting a normal cardiac function curve at the equilibrium point.

Essentially, all factors that may affect the regulation of cardiac output alter the venous return and cardiac output relationships in the limited number of ways that were described in the previous chapters. Strengthening or weakening the heart’s pumping ability can only increase or decrease the slope or plateau of the cardiac function curve, and all effectors of venous return can only alter the plateau, the slope, or the intercept of the venous return curve. Therefore, each action or stimulus affecting the cardiovascular system can be understood in terms of its effects on the two function curves, and if the effects are known, the resulting cardiac output can be determined.

Cardiac output responses to changes in mean systemic pressure and resistance to venous return can be determined graphically. Increasing or decreasing mean systemic pressure will shift the venous return curve to the left or right without affecting its slope. For example, if the cardiovascular system described by the function curves shown in Figure 4.1 received an input such as a rapid transfusion of blood that raised mean systemic pressure from 7 to 10 mm Hg, the venous return curve would be shifted parallel to the right, as shown in Figure 4.2 as the blue line. Consequently, the new venous return curve would intersect the normal cardiac function curve at a new equilibrium point, approximately 7.5 L/min and 0.7 mm Hg right atrial pressure.

#### FIGURE 4.2

A normal cardiac function curve intersecting the normal venous return curve and venous return curves obtained following events that altered resistance to venous return and/or mean systemic pressure.

If resistance to venous return was increased by 20%, as a result of reduction in metabolic demand by the body, the slope of the venous return curve would be reduced but the x-axis intercept would remain unchanged, as shown in the yellow line in Figure 4.2. The solution for the values of cardiac output and right atrial pressure of the cardiovascular system under these conditions would be the intersection of the normal cardiac output curve and the new venous return curve, where cardiac output is 4.4 L/min and right atrial pressure is –0.5 mm Hg.

In exercise, increased metabolic demand of the body may decrease resistance to venous return, and elevated sympathetic nervous system activity may raise the mean systemic pressure. If resistance to venous return were decreased by 50% and mean systemic pressure were increased to 12 mm Hg, the slope of the venous return curve would be increased, and the x-axis intercept would be shifted 5 mm Hg to the right, yielding the venous return curve illustrated in Figure 4.2 as green line. The cardiac output and right atrial pressure under these conditions would be the intersection of the new venous return curve and the normal cardiac output function curve, at which cardiac output is 14.4 L/min and right atrial pressure is 2 mm Hg.

Cardiac output responses to changes in the pumping ability of the heart can be determined graphically. Nearly all effectors of cardiac pumping ability shift the function curve and/or the plateau. Raising sympathetic nervous system activity, for example, increases the slope and raises the plateau, while sympathetic blockade and parasympathetic stimulation have the opposite effect on both properties of the function curve.

If emotional excitement increased sympathetic stimulation by 50%, the new cardiac function curve would be shifted to the left, and the plateau would be increased compared to the normal curve. In Figure 4.3, the normal cardiac function curve and the normal venous return curve are plotted as dark blue lines intersecting at 5.0 L/min cardiac output and 0 mm Hg right atrial pressure, and the new cardiac curve is plotted as the yellow line, which intersects the normal venous return curve at a point where cardiac output is only slightly greater than normal and right atrial pressure is slightly below normal.

#### FIGURE 4.3

Normal venous return curve intersecting the normal cardiac function curve and cardiac function curves obtained after increased or decreased sympathetic stimulation of the heart.

Conversely, if a pharmacological antagonist of the sympathetic nervous system transmitters were administered reducing the sympathetic nervous system effect on the heart by 50%, the pumping ability of the heart would be limited, yielding the function curve shown as the red line in Figure 4.3. It intersects the normal venous return curve where cardiac output is only slightly reduced from normal, and right atrial pressure is slightly greater than 0 mm Hg.

Cardiac output changes can be determined graphically when multiple factors affect simultaneously both venous return and cardiac pumping ability. Most challenges faced by the cardiovascular system entail simultaneous actions on venous return and cardiac pumping ability. The effect of such complex changes can also be solved using graphical methods.

If a motor vehicle accident results in lacerations causing great pain and hemorrhage of several hundred milliliters, mean systemic pressure may fall to 5 mm Hg, resistance to venous return may increase 33% greater than normal, and cardiac pumping ability may increase to 125% of normal. What would be the new levels of cardiac output and right atrial pressure in this circumstance? The answer is not intuitively obvious, but it can be estimated by analyzing the effects of the perturbations on the venous return and cardiac function curves and plotting the functions together graphically to obtain the new equilibrium point. In Figure 4.4, the normal function curves are plotted as dark blue lines (venous return as solid lines and cardiac output as dashed), and the new relationships are presented as red lines. The venous return curve can be plotted knowing the mean systemic pressure, which is the x-axis intercept, 5 mm Hg, and the slope (1/resistance to venous return = 1/1.33), which is decreased to 75% of normal, approximately 0.5 L/min/mm Hg. The cardiac pumping ability, which is 125% of normal, may be represented as a function curve having a plateau value 22% higher than normal (22 L/min) but achieved at the normal plateau value of right atrial pressure, approximately 4 mm Hg. The intersection of the new curves occurs at a point where cardiac out is approximately 2.8 L/min and right atrial pressure is –0.7 mm Hg.

#### FIGURE 4.4

Cardiac output and venous return curves under normal conditions and following trauma and hemorrhage.

## 4.2. Algebraic Analysis of Cardiac Output Regulation

Investigators attempted in the past 100 years to derive algebraic expressions useful in analyzing cardiac output regulation. Some are more useful than others.

Cardiac output can be expressed as a function of heart rate and stroke volume. This is intuitively obvious and is frequently written as an equation:

4.1

The equation is useful in some circumstances in predicting cardiac output when only heart rate changes. However, most responses of the cardiovascular system involve changes in variables not included in the equation, especially those that determine stroke volume. Furthermore, the equation has led to some misunderstanding of regulation of cardiac output.

Ohm’s law can be adapted to expressions of cardiac output regulation. The most straightforward adaptation is the following:

4.2

where Pa is the systemic arterial pressure and Rs is the systemic resistance. This equation assumes that right atrial pressure is 0. The expression can be made more useful by including it as a variable:

4.3

where Pra is the right atrial pressure. The formula can be rearranged to find systemic resistance, but its usefulness in determining cardiac output is limited by simultaneous changes occurring in more than one variable in most situations.

The quantitative effects of changes in venous resistance on venous return are much greater than proportionally similar changes in arterial resistance. Ohm’s law can be useful in understanding the flow of fluid through rigid tubes, but because the circulatory system is made up of compliant vessels, Guyton found it was necessary to modify Ohm’s law in order for it to be useful in analysis of the cardiovascular system. Instead of using one simple term for systemic resistance, arterial and venous resistances had to be weighted separately according to the capacitances of the two segments of the vascular system [32]. The need for the modification is due to the arrangement of arterial resistance, Ra, and venous resistance, Rv, in series and each being positioned following capacitance segments, whose volumes are functions of the arterial pressure and capacitance and venous pressure and capacitance, respectively. Extra volume in each segment, EVa and EVv, is the contained volume greater than the unstressed volume. The series arrangement of the arterial and venous resistances and capacitance sections is presented schematically in Figure 4.5.

The derivation of the expression incorporating the capacitance-weighted arterial and venous resistances begins by restating arterial and venous pressures separately. The pressure in the arteries, Pa, is equal to right atrial pressure, Pra, plus the pressure drop from the root of the aorta to the right atrium; the pressure in the veins is equal to venous pressure, Pv, minus the right atrial pressure. Pressures in the arteries and veins are defined by the product of cardiac output and the separate arterial and venous resistances, Ra and Rv:

4.4
4.5

The extra volume, EV, in the arterial and venous segments is equal to the pressure in that segment times its capacitance, Ca or Cv:

4.6
4.7

The mean systemic pressure is equal to the extra volume greater than the unstressed volume of the total systemic vascular system, EVsyst, divided by the systemic capacitance, Cs. The extra volume EVs is equal to the sum of Eva and EVv , and Cs is the sum of Ca and Cv:

4.8

Substituting the expressions for EVv and EVa derived in Equations (4.6) and (4.7) into Equation (4.8) and rearranging to solve for cardiac output yields the following:

4.9

Equation (4.9) is very helpful in understanding complex cardiovascular phenomena, and many of these will be analyzed in subsequent chapters of this presentation. But in the context of analyzing factors affecting venous return, the denominator makes clear the potentially preponderant effect of venous resistance on venous return and cardiac output. The importance of venous resistance is even more striking when viewed in light of the experimental data, indicating that the ratio of venous to arterial capacitances may be as much as 18:1 [3]. By substituting these values into Equation (4.9), we can appreciate the magnitude of effects on venous resistance changes:

4.10

Simplification yields:

4.11

Therefore, at least under some conditions, a change in venous resistance may have a 19-fold greater effect on cardiac output than the same percentage change in arterial resistance. Normally, arterial resistance is approximately seven times greater than venous resistance. Consider the consequences of a 20% increase in total systemic resistance, with all of it occurring in arterial resistance. Calculated venous return/cardiac output would decrease 6%. However, if the 20% increase in total systemic resistance is confined to the venous portion of the system, the calculated decrease in venous return would be 53%.

While the quantitative superiority of changes in venous resistance over changes in arterial resistance in affecting cardiac output and venous return is impressive, it should be considered in the context of the usual operation of the cardiovascular control system. In normal conditions of health, arterial resistance is much more dynamic than venous resistance; arterial resistance is strongly affected by the sympathetic nervous system, circulating vasoactive hormones, such as epinephrine, angiotensin II, and PGE2, as well as by local tissue autoregulatory mediators, especially those sensitive to reduction in tissue pO2. Venous resistance, on the other hand, is comparatively less sensitive to the effects of these factors. Therefore, changes in arterial resistance frequently may contribute to the greater share of changes in total systemic resistance.

Cardiac output can be expressed as a function of capillary pressure, right atrial pressure, and venous resistance. Guyton proposed that cardiac output could also be expressed as a function of the pressure gradient from the mid-point of the capillaries to the right atrium divided by the vascular resistance from the capillary mid-point to the right atrium [32]:

4.12

The terms of Starling’s law of the capillary, which states that capillary pressure is equal to tissue pressure plus plasma colloid osmotic pressure minus tissue colloid osmotic pressure, can be substituted in Equation (4.12) to yield:

4.13

in which Pt is the tissue fluid pressure, Pcop and Pcot are plasma and tissue colloid osmotic pressure, respectively, and Pra is the right atrial pressure. This expression of cardiac output, like Starling’s law of the capillary, is only valid in steady-state conditions; if one determinant of capillary pressure changes, several hours may be required for equilibrium to be reestablished.

The value of Equation (4.13) is derived from the independence of the determinants of capillary pressure from all other circulatory variables and from the other elements of the equation; colloid osmotic pressure in the plasma and extracellular space are functions of protein concentrations in the two compartments, tissue pressure is a function of the physical characteristics of the extracellular space and the volume of fluid it contains, right atrial pressure is primarily a function of the pumping ability of the heart, and resistance to venous return is determined by the physical characteristics of the venous vessels and the viscosity of the blood. This independence of the variables improves the probability that the equation is a valid and useful algebraic expression of cardiac output.

Equation (4.13) has several interesting and important implications. For example, it implies that cardiac output varies as a direct function of both tissue fluid pressure and plasma colloid osmotic pressure, with cardiac output increasing with increases in either variable. Additionally, the equation states that cardiac output varies inversely with right atrial pressure and resistance to venous return and that it is independent of arterial pressure except in conditions where afterload significantly affects right atrial pressure.

The equation can also be helpful in understanding regulation of cardiac output in heart failure. Frequently, patients with signs of severe congestive failure will have normal cardiac output at rest while having high right atrial pressure. These patients are also known to have high levels of tissue fluid pressure, which would contribute to elevated capillary pressure. Consequently, even with high right atrial pressure, the patients may have, according to Equation (4.13), a pressure gradient for venous sufficiently elevated to maintain normal cardiac output.

## 4.3. Summary

Because regulation of cardiac output requires interaction of factors affecting return of blood to the heart from the peripheral tissues and factors affecting the pumping ability of the heart itself, both must be considered in attempts to analyze cardiac output control. Plotting venous return curves simultaneously with cardiac function curves is an effective means of considering the numerous important variables’ roles in regulation of output. Most algebraic expressions of cardiac output deal explicitly with only factors affecting venous return, while the pumping function of the heart contributes indirectly via its effect of right atrial pressure. The most significant equations are based on Ohm’s law, in which cardiac output is equated to the pressure gradient for venous return divided by the resistance to venous return. Differences between equations result from differences in expressing the pressure gradient and the resistance.