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Models and Equations

Models and Equations

An elementary model of the vasculature with two capacitors connected by a resistor, exhibits properties analogous to the relationship between venous return and central venous pressure. This was the model Levy used for illustration in his 1979 essay {??} and has been used elsewhere, including the present day textbook of Berne and Levy {textbook}.

In their textbook chapter, Berne and Levy offer a quantitative description of venous return curves, or "vascular function curves" as they prefer to refer to them, based on this model. Their model is simply a single resistance, R, representing the total peripheral resistance, with a capacitor on either end. The capacitor at the upstream end of the resistor corresponds to the capacitance of the arterial system; Ca, the one at the downstream end, the capacitance of the venous system Cv.

With zero flow through the system the volume within it distributes itself between the two capacitors, with pressure the same in each -- this is the mean circulatory pressure (Pmc). When flow is pumped through the system, a pressure difference develops between the upstream and downstream ends of the resistance. This represents the difference between arterial pressure (Pa) and the central venous pressure (Pcv).

For this pressure difference to occur, the volumes within the capacitors have to change. They have to change reciprocally because there is no other place for volume to go. Since pressure at the arterial end moves upward, the arterial capacitance takes up volume, the exact volume donated as a result of pressure moving downward at the venous end. Arterial capacitance is approximated as 20 times greater than the venous capacitance, so the increase in Pa due to the volume that is moved is 20 times greater than the decrease in Pv due to loss of that volume.

In this model, when flow is zero, and pressure in both compartments at mean circulatory pressure, total volume (V) equals

When flow is forced through this model of the vasculature, total volume is unchanged, and distributed in proportion to the individual compliances and pressures:

Berne and Levy put these relationships together with the relationship between the pressures and the flow and expressed venous pressure in terms of flow (Q), mean circulatory pressure, and the parameters of the system

In this relationship, Pmc is a constant determined by total volume and total capacitance. Also, the slope of the relationship between Pv, expressed as the dependent variable, and Q is negative. Thus, the model predicts the tendency of central venous pressure to fall as cardiac output is driven upward.

What the apparent resistance in the slope of the "venous return curve" reveals. return to top

Note that the slope of the relationship as expressed in (1), though related to the total resistance is certainly not the resistance of the venous portion of the total peripheral resistance. Neither is the slope equal to total peripheral resistance, R. Yet the slope of vascular function curves is commonly interpreted as one or the other of these.

One way of looking at the relationship in (1) is to imagine Pmc as a fixed pressure source that drives flow through a resistance. The pressure drop across this resistance is (Pmc - Pv). Does this resistance have any physical meaning? Quantitatively, it is the total peripheral resistance multiplied by the ratio of the arterial capacitance to the total capacitance

Emphatically, this construct has no physical meaning as an actual resistance in the vascular system across which the cardiac output or venous return develops a pressure gradient between mean circulatory pressure and central venous pressure. It is a virtual resistance, so to speak, like the Thevenin equivalent in engineering theory of passive networks. Yet this mathematical abstraction has influenced people to think that venous return is driven by the pressure head represented by the difference between mean circulatory and central venous pressure. The resistance through which this pressure head drives flow is taken to correspond to an anatomically identifiable resistance component in the series combination that makes up total peripheral resistance.

Although the elementary model of the vasculature with a single resistance representing total peripheral resistance and total system capacitance divided into only an arterial and venous compartment does illustrate the phenomenon that central venous pressure falls when cardiac output (venous return) is driven upward (with fixed peripheral resistance and capacitances), it cannot possibly show how central venous pressure relates to venous return in terms of mean circulatory pressure and venous resistance. It contains no representation of venous resistance.

Confusion arises because the apparent resistance in the formalism of equation (1), above, is about one-twentieth of total peripheral resistance if the ratio of Ca and Cv is set at 1:20. Numerically, this is not an unreasonable estimate to assign to the resistance of the venous system. But in no way is it physically meaningful to describe venous resistance as determined by the product of total peripheral resistance and a compliance ratio.

A three-capacitor, two resistance model that specifically includes representation of venous system resistance. return to top

To show relationships that reveal the influence of venous resistance, venous resistance must be included in the model. The model illustrated in Figure ???, has one more resistor and capacitor than the model above. The total vascular resistance is divided among individual resistances representing, respectively, the resistance of the arterial and venous segments of the peripheral vasculature.

Pressure (Ppv, peripheral venous pressure) at the junction of the two resistances is equivalent to the pressure at the upstream end of the venous vasculature, i.e. pressure at the downstream end of the capillaries.

Pressure at the downstream end of the venous resistance is labeled Pcv, i.e., the central venous pressure. The additional capacitance, Cpv, at the junction of the resistances, represents peripheral venous capacitance. The capacitance at the downstream end of the venous resistance represents capacitance of the central veins, Ccv. In this model, then, total venous volume is distributed between a peripheral and central compartment.

This model is a specific instance of a more general treatment described by Guyton et al {Guyton, Lindsey, Kaufman, Effect of mean circulatory filling pressure …AJP 180, '55}. It comes much closer to the model implied in the way authors describe dynamics of system volume. For example, Mohrman and Heller speak in terms of a peripheral and central venous pool {M&H text}. Rowell {monograph} describes volume dynamics that accompany changes in cardiac output similarly.

In this model the volume changes that accompany changes in the pressure profile around the vasculature due to altered cardiac output are seen, primarily, as movement of volume between the two venous compartments rather than the physical absurdity of stating that the volume shift reflected in altered central venous pressure is compensated by an equal change in arterial system volume. Furthermore, the model is amenable to analysis of changes in which arterial pressure remains constant when cardiac output changes through adjustment of the arterial resistance. The following analysis, however, shows that this model specifically does not support a description of venous return as driven by mean circulatory pressure through the venous resistance.

In this model, the zero-flow mean circulatory pressure also, of course, reflects the total volume (V) within the system distributed among the three compartments.

Total volume remains fixed so, at any time, Pmc can be expressed in terms of the instantaneous pressures in the compliant compartments, as in (2), below. The other two expressions, (3) and (4) below, are the relationships between steady-state flow and pressures. Steady state flow is Q, equal to both cardiac output and venous return, of which the analog is the constant-current generator in Figure ???. The pressures are: Pa, arterial; Ppv, the pressure in the peripheral compliant compartment; and Pcv, the pressure in the central compliant compartment, i.e., central venous pressure):

These relationships can be manipulated with the same approach used by, e.g., Berne and Levy, to develop an expression that appears to relate flow to mean circulatory pressure and parameters of the model. First, rewrite (2), i.e. state it in the form that expresses that mean circulatory pressure multiplied by the sum of the capacitances equals the total volume which also equals the sum of the volume in the three compartments.

Replace Pa and Ppv in terms of Pcv, using (3) and (4):

Collect all the terms with Pcv

The apparent resistance in the slope of the venous return curve. return to top

Rearrange and divide by the sum of the three capacitances:

How does this compare with equation 1 (above, equation ?? in the main article) ? It is similar in having the appearance of the property of a flow driven by a gradient between Pmc and Pcv across a resistor related to the arterial and venous resistances with weighting factors based on the three capacitances. As a check, note that setting both Cpv and Rv to zero is equivalent to reducing the model to the model behind equation (1), i.e., a single resistance with a capacitance at either end. We can make the relationship look easier to interpret by asserting that arterial system compliance is negligible, so that Ca can be set to zero.

Again, the virtual resistance represents no single component of the model. return to top

But either way, it is absurd to interpret the slope of this relationship between Pcv and Q as venous resistance or to give any meaning to Pmc as a fixed pressure on the upstream end of this venous resistance. We know that the system modeled is driven by flow (a current source), not by an internal pressure gradient (voltage difference.) The formulations in equations 5 and 5a are simply mathematical constructs with a convenient relationship between a constant Pmc, set by the system volume and the total capacitance, and the other parameters of the system. In engineering terms, this is like the Thevenin equivalent in terms of a virtual voltage source for a network powered by a current source.