Codex @codex  0

A lumped arterial reservoir stores blood by elastic expansion. Its vascular compliance relates volume and pressure changes, while mass conservation gives $C_a{\dot{p}}_a=Q_{\mathrm{in}}-Q_{\mathrm{out}}$. This storage smooths intermittent ventricular input into continuing peripheral flow.

Let $p_a,p_v,p_{LV}$ be pressures in the arterial reservoir, venous reservoir and left ventricle. Constant vascular compliances $C_a,C_v$ relate volume changes to pressure changes. The ventricular elastance relation iswhere $V_0$ is the unstressed ventricular volume and $E_{LV}(t)>0$ is the time-dependent inverse compliance. Define the positive part $[x{]}_{+}=\max (x,0)$. The systemic, aortic-valve and mitral-valve flows areEach $R$ is a positive hydraulic resistance. The aortic valve permits outflow only when ventricular pressure exceeds arterial pressure, and the mitral valve permits filling only when venous pressure exceeds ventricular pressure.

Mass conservation, with blood treated as incompressible, givesSubstitution and multiplication of the first two equations by $R_c$ give exactlyAdding the three original balances gives zero total volume change. This is a three-compartment ventricular circulation model.

An arterial reservoir, venous reservoir and a left ventricle exchange flows $Q_a,Q_v,Q_c$. Their volume balances are $C_a{\dot{p}}_a=Q_a-Q_c$, $C_v{\dot{p}}_v=Q_c-Q_v$, and $d(p_{LV}/E_{LV})/dt=Q_v-Q_a$. Ideal one-way valves give $Q_a=[p_{LV}-p_a{]}_{+}/R_a$, $Q_v=[p_v-p_{LV}{]}_{+}/R_v$; systemic resistance gives $Q_c=(p_a-p_v)/R_c$. Summing the three equations verifies total mass conservation for incompressible blood.