The article by Curran-Everett, “A classic learning opportunity from Arthur Guyton and colleagues (1955): circuit analysis of venous return,” (3) is meant to be an example of teaching with classic papers. I have argued (1, 2) that the findings of the Guyton et al. study (4) that Curran-Everett discusses have been misinterpreted, particularly in the profound misconcept that mean circulatory pressure (P_{mc}) is somehow the driving force for steady-state flow (Q) through the peripheral vasculature. Nonetheless, I agree that students stand to benefit from study of the paper's findings, both for comprehension of the physics of the peripheral vasculature and for appreciation of the history of development of cardiovascular physiology. Unfortunately, the Curran-Everett presentation includes a serious error. I also have comments about other aspects of Curran-Everett's development that I think interfere with the purpose of exploiting this learning opportunity.

The error is in Fig. 2, which shows physical models for discussion of venous return and analogous electrical circuits. In Fig. 2, the two circuit diagrams show the pressures at the inflow and outflow ports as P_{mc} and right atrial pressure (P_{ra}), respectively. The inflow port pressure should be arterial pressure (P_{a}), not P_{mc}. With (P_{mc} − P_{ra}) as the pressure differential between the input and output ports, Q would not then be described by the *Eqs. 1* and *2* of the article, but simply as (P_{mc} − P_{ra}) divided by the sum of the resistances.

If the schematics of Fig. 2 were to show a constant current generator or flow source, respectively, connected between the input and output ports, and if pressures at the ports were P_{a} and P_{ra}, then they would represent the circuit that Guyton et al. actually worked with and would be consistent with equations in Curran-Everett's development. Q would relate to pressures at the ports as follows: (1)

Through algebraic manipulation, we can replace this simple description of the behavior of the circuit with a complicated one, Curran-Everett's *Eq. 2*. The manipulation eliminates P_{a} in the flow-to-volume relationship by equating two expressions for the total volume contained within the vasculature: one expressed in terms of the pressures in the arterial and venous compartments, P_{a}*C*_{a} + P_{v}*C*_{v} (where *C*_{a} is arterial capacitance and *C*_{v} is venous capacitance); and one based on the definition of P_{mc}, P_{mc}(*C*_{a} + *C*_{v}). The latter has physical meaning in terms of measureable quantities only in the situation of zero flow, with P_{mc} being the pressure observed throughout the vasculature when flow stops and the blood volume distributes passively according to the elastic properties of the vascular compartments.

In the equality P_{mc}(*C*_{a} + *C*_{v}) = P_{a}*C*_{a} + P_{v}*C*_{v}, we can use the details of the pressure-flow relationships of the circuit described in *Eq. 1* above to replace P_{v} with P_{ra} + Q*R*_{v} and P_{a} with P_{ra} + Q*R*_{v} + Q*R*_{a} (where *R*_{a} is arterial resistance and *R*_{v} is venous resistance) to obtain the following equation: (2)

*Equation 2* connects Q, P_{mc}, and P_{ra}. By rearranging it, i.e., solving for Q (see the appendix), one obtains Curran-Everett's *Eq. 2*. But why do that? It distracts from what is really driving flow and establishing the pressures, that is, the energy of the current or flow source manifested in driving pressure (P_{a} − P_{ra}). (P_{mc} − P_{ra}) simply isn't a “driving pressure.” The intent to show this differential as a driving pressure is perhaps what led to the error of placing P_{mc} at the input ports in Curran-Everett's Fig. 2.

Beyond the problem with this error, I see another with the emphasis on vascular impedance. The more a student knows about impedance, the more confusing this treatment must be. Curran-Everett, as did Guyton and coworkers, refers to the denominator in his *Eq. 2* as an impedance. Impedance is the appropriate concept for understanding voltage and current (or, by analogy, pressure and flow) in a circuit containing both resistive and capacitive elements, especially when driven sinusoidally. In such circuits, the voltage-to-current ratio depends on the frequency at which the power source cycles, and the quantitative expression for the voltage-to-current ratio typically includes terms with resistances and capacitances multiplied together, such as *R*_{v}*C*_{v} in Curran-Everett's *Eq. 2*.

But quantitative expressions of circuit impedance necessarily include the driving frequency. The denominator of Curran-Everett's *Eq. 2* is not about impedance for the obvious reason that it is not frequency dependent. Curran-Everett's *Eq. 2* describes a steady-state relationship between flow and pressures, equivalent in electrical terms to a voltage-to-current relationship when voltages and currents are not changing, i.e., a direct current as opposed to alternating current situation. The superficial similarity to an expression for impedance, i.e., the *RC* products, is the consequence of the algebraic manipulation necessary to replace P_{a} in the simple relationship for flow, Q = (P_{a} − P_{ra})/(R_{a} + *R*_{v}).

I also see potential for confusion in the complicated expressions for the partial derivatives of Curran-Everett's *Eq. 2*. Students disinclined toward calculus will surely be daunted by the series beginning with Curran-Everett's *Eq. 5*. But those who look closely at the physical models and *Eq. 1* above will see that the partial derivatives of Q with respect to *R*_{a} and *R*_{v}, with the driving pressure (P_{a} − P_{ra}) held constant, are identical. In terms of the electrical circuits in Curran-Everett's Fig. 2, a 1-Ω resistance change has exactly the same effect on flow regardless of which resistance changes.

Curran-Everett's partial derivatives come from the expression with the artificial driving pressure (P_{mc} − P_{ra}). They reveal a magnified effect of changes in *R*_{v} because the constraint imposed in forming the total derivative is that (P_{mc} − P_{ra}) be constant. That constraint brings in the effect on volume distribution of the large ratio of *C*_{v} to *C*_{a} via the complicated way of expressing P_{a} in terms of P_{mc}. Curran-Everett's thought experiment that concentrates on the effects of altered pressure gradients on the distribution of a fixed total volume between the two compartments seems to me a much better direction to pursue, both toward the goal of helping students understand the physics of the peripheral vasculature and toward appreciation of the classic experiment (4).

In summary, it seems to me that an approach to helping students to learn from the classic paper of Guyton et al. (4) should begin with focus on the advances made possible by their innovation of holding total volume constant while flow was varied and should tackle the difficult problem of understanding how the Starling resistor functioned as a negative feedback control element for establishing the flow that would result in a specific P_{ra} (2). That approach and technology revealed the characteristics of the isovolumic “venous return curves” that would have been concealed had they forced varied rates of flow through the vasculature without regard to total circulating volume.

With the aid of simple models such as those illustrated in Curran-Everett's Fig. 2 (with the correction that P_{a} be put in the place of P_{mc}), students can be encouraged to think about how a fixed total volume is distributed among the capacitive elements of the vasculature at different levels of flow when flow changes, along the direction that Curran-Everett does take in the thought experiment described. With this concept grasped, students can move on to understanding of the built-in negative feedback mechanism: increasing Q through the peripheral vasculature obligates a generally upstream-ward transfer of volume, thus reducing P_{ra} and in turn acting to reduce cardiac output through the influence of P_{ra} on stroke volume (1, 2). They can then advance to thinking about how transient differences between venous return and cardiac output occur during intercompartment transfers of volume.

Inveigling P_{mc} into these conceptual views is not necessary. It seems to me that any conceptual legitimacy of P_{mc} lies in its proxy representation of total volume. But that representation is complicated by the coinvolvement of tone within the smooth muscle of the vasculature, i.e., P_{mc} reflects the volume within the vasculature but also the state of vascular smooth muscle. Rather than P_{mc}, why not speak directly in terms of the volume contained within the vasculature?

## APPENDIX

Begin by rearranging *Eq. 2* above as follows:

Collect the terms as follows: Collect the terms and rearrange them as follows: Then,

All that remains to make this identical to Curran-Everett's *Eq. 2* is to replace (*C*_{a} + *C*_{v}) with total capacitance and divide both sides by [*R*_{v}*C*_{v} + (*R*_{a} + *R*_{v})(*C*_{a})]. In showing how P_{ra} falls, relative to P_{mc}, with increases in flow, it reflects the redistribution of volume that occurs as Q, P_{a}, and P_{v} change under the constraint of fixed total volume.

- © 2008 American Physiological Society