Pulsatile Heart and Circulation

The lumped parameter model of the intact pulsatile heart and circulation is illustrated in Figure 1 in terms of its electrical circuit analog. Here, charge is analogous to blood volume ($Q$, ml), current, to blood flow rate ($\dot{q}$, ml/s), and voltage, to pressure ($P$, mmHg). The model consists of six compartments which represent the left and right ventricles ($l, r$), systemic arteries and veins ($a, v$), and pulmonary arteries and veins ($pa, pv$). Each compartment consists of a conduit for viscous blood flow with resistance ($R$) and a volume storage element with compliance ($C$) and unstressed volume ($Q^0$). Two of the resistances and two of the compliances are nonlinear. The systemic venous resistance is represented by a Starling resistor (with chamber pressure set to atmospheric pressure), while the pulmonary arterial resistance is represented by an infinite number of parallel Starling resistors (with chamber pressure equal to alveolar ($alv$) pressure), arranged vertically, one on top of the other. The pressure-volume relationships of the left and right ventricles consist of an essentially linear regime (characterized by compliance and unstressed volume), a diastolic volume limit ($Q^{max}$), and a systolic pressure limit ($P^{max}$). The compliances of the linear regime of the ventricular pressure-volume relationship vary periodically over time (time evolution is precisely determined by the end-diastolic compliance ($ed$), the end-systolic ($es$) compliance, and the heart rate ($F$)) and are responsible for driving the flow of blood. The four ideal diodes represent the ventricular inflow and outflow valves and ensure uni-directional blood flow. Finally, the reference pressure is set to intrathoracic ($th$) pressure for the ventricular and pulmonary compartments.

  

Figure 2: Electrical circuit analog of the human heart-lung unit preparation designed for measuring cardiac function curves. Each box encompassing a circuit element denotes a nonlinear element.

Figure 2 illustrates the electrical circuit analog of the lumped parameter model of the human heart-lung unit preparation. The input pressure to the heart-lung unit here is defined to be the node labelled $P_{\text{\lq\lq $ra$''}}(t)$ - the location of where the right atrium would be if it were explicitly included in the model. Cardiac function curves may be obtained from this preparation by varying the independent voltage sources, $P_a$ and $P_v$, and time-averaging the resulting $\dot{q}_l(t)$ and $P_{\text{\lq\lq $ra$''}}(t)$.

Figure 3 illustrates the electrical circuit analog of the lumped parameter model of the human systemic circulation preparation. Venous return curves may be measured from this preparation by adjusting the value of $C_r(t)$ at end-diastole ($C_r^{ed}$) in order to vary $P_{\text{\lq\lq $ra$''}}(t)$ - the pressure that impedes flow into the right ventricle - and time-averaging the resulting $\dot{q}_v(t)$ and $P_{\text{\lq\lq $ra$''}}(t)$. Note that the independent current source here ( $\dot{q}_v(t)$) keeps the mean systemic ($ms$) pressure precisely constant throughout the measurement period by pumping into the systemic circulation whatever is pumped out.

  

Figure 3: Electrical circuit analog of the human systemic circulation preparation designed for measuring venous return curves. Each box encompassing a circuit element denotes a nonlinear element.