Simple circuit model for the heart

The right side of the human heart circulates blood to the lungs to receive oxygen. The left side, which circulates oxygenated blood throughout the body, is considered here. Figure 1 shows a simple circuit model, closely based on problem 1.15 from Siebert’s Circuits, Signals and Systems.

Fig. 1: "Electrical analog of the heart and circulatory system" (Siebert)
Fig. 1: “Electrical analog of the heart and circulatory system” (Siebert)

Electric charge models blood, current models blood flow, and voltage models blood pressure. Diodes \(D_m\) and \(D_a\) are the input (mitral) and output (aortic) valves. Currents \(i_v\) and \(i_a\), the currents through these diodes, are the venous and arterial blood flows. A time-varying capacitance, \(C(t)\) models the pumping of the heart, and circulates the charge/blood through the body (\(R\)) as current \(i_b\).

When the heart contracts (systole) \(C(t)\) suddenly reduces to \(C_a\) increasing the (heart) capacitor voltage, \(u\), and driving blood through the aortic valve \(D_a\). (Capacitance, charge and voltage are related by  \(q=C(t)u\)—so instantaneously reducing \(C\) increases \(u\)). Some of the blood is forced through the body (resistor \(R\)), and some is absorbed by the elastic capacities of the arteries (\(C_a\)) and veins (\(C_v\)).

When the heart relaxes (diastole) \(C(t)\) suddenly increases (to \(C_d\)), lowering voltage \(u\), and allowing blood to return via the mitral valve, \(D_m\). (The total amount or blood/charge in the system is constant.)

We have simulated the circuit here and embedded the  main results, interactively, below (as with Siebert, we do not trouble ourselves with units, and only consider the broad messages of the model).  In the figure the heart alternately contracts (beginning at odd \(t\)), and relaxes (beginning at even \(t\)). The black trace shows the pressure in the heart (voltage \(u\)); the red trace shows the arterial pressure (\(u_a\)); the blue trace shows the venous pressure (\(u_v\)).



The slider controls the body’s resistance to circulation, \(R\). Increasing \(R\) increases the arterial pressure and reduces the venous pressure. We can also edit the table to experiment with \(C_a\) and \(C_v\) (the arterial and venous elastic capacities) and \(C_s\) and \(C_d\) (the degree of contraction and relaxation of the heart). For instance, increasing \(C_s\), reduces the contraction, and so the hearts ability to pump.

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