## Simple Circulatory System

Systemic resistance R

Arterial pressure Figure 15.6 Schematic diagram of the simplest circulation model, with a single-chambered heart and a single loop.

Arterial pressure

Figure 15.6 Schematic diagram of the simplest circulation model, with a single-chambered heart and a single loop.

To illustrate how all the above pieces fit together to give a model of the circulatory system, consider a simple circulatory system with one loop and a single-chambered heart (Fig. 15.6). To begin with, suppose we have only a heart and a resistive closed loop. For the resistive closed loop, we suppose that the total flux is related to the pressure drop through

so that in steady state the flux through the loop must match the cardiac output, yielding

where Vh = Vmax — Vmin. Equation (15.35) gives a relationship between arterial and venous pressure that must be maintained in a steady-state condition. Unfortunately, these are not uniquely determined by this equation. The reason the solution is not completely determined is primarily because we have not allowed the circulation loop to be a compliance vessel. If we allow the loop to be a compliance vessel, then there is an additional relationship between pressure and total volume that must be satisfied.

To see how this works for a simple system, suppose that the circulatory loop consists of a compliance vessel with cross-sectional area given by (15.13). It follows from (15.16) that

3Ry and the total volume of the vessel is given by

Vo 4 1(1 + YPa)3 - (1 + YPv)3 These two equations, together with

give a system of three equations in terms of the four unknowns Q, Pa, Pv, and V. (Of course, it is also possible to regard Pa and Pv as known, i.e., measured, quantities, and then view y and R as unknowns. This would determine the resistance and compliance corresponding to a given pressure difference.) This is too many unknowns for the number of equations, and so we must find another equation before the solution is uniquely determined. The final equation comes from conservation of blood. Because blood is assumed to be incompressible, and because the heart chambers are assumed to have a fixed volume (as cardiac output is expressed in terms of the average output), it follows that V must be constant. The system is then completely determined. However, because it is nonlinear, a closed-form solution is not apparent, and the easiest way to obtain a solution is to solve the equations numerically. compliances i Vh=0

Compliance vessels

Heart

Heart Figure 15.7 Schematic diagram of the two-compartment model of the circulation. The heart and pulmonary system are combined into a single vessel, and the systemic capillaries are modeled as a resistance vessel. The larger arteries and veins are modeled as compliance vessels.