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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.

A lumped venous reservoir stores blood at a pressure determined by its volume and vascular compliance. If its inflow and outflow are $Q_{\mathrm{in}}$, $Q_{\mathrm{out}}$, the constant-compliance approximation gives $C_v{\dot{p}}_v=Q_{\mathrm{in}}-Q_{\mathrm{out}}$. It forms the filling side of a three-compartment ventricular circulation model.