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The mitral valve separates the left atrium from the left ventricle. In a reservoir model with the atrium incorporated into the venous filling compartment, ideal one-way flow is $Q_v=[p_v-p_{LV}{]}_{+}/R_v$, where $[x{]}_{+}=\max (x,0)$ and $R_v$ is open-valve hydraulic resistance. The positive-part rule prevents reverse flow in this idealization.

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.

During ejection the mitral valve is closed, so $Q_v=0$. Adding the arterial and ventricular balances cancels aortic flow:where the last approximation assumes $p_v\ll p_a$.

Model the rapid contraction as a switch to approximately constant systolic elastance $E_S$, with $C_s=1/E_S$ the systolic ventricular compliance. A sufficiently small aortic-valve resistance makes $p_{LV}\simeq p_a$ after a short pressure-equalization transient. The systolic pressure must be high enough to open the valve, $E_S(V^{+}-V_0)>p_a^{+}$. During that transient neglect systemic runoff. Conservation of combined arterial and stressed ventricular volume givessoThis is an approximation across a rapid transient, not a discontinuity at finite valve resistance. The subsequent common pressure obeys $R_c(C_a+C_s){\dot{p}}_a\simeq -p_a$, givingThe source's $R_C$ in this expression denotes the same systemic resistance $R_c$.

At the end of firing, the remaining ventricular volume is $V_{\mathrm{end}}\simeq V_0+C_s{p}_a(\Delta t_F)$. The stroke volume is $V^{+}-V_{\mathrm{end}}$, henceThe assumptions are a closed mitral valve throughout ejection, negligible venous pressure, rapid valve equalization relative to systemic runoff, and nearly constant systolic elastance during the interval measured by $\Delta t_F$. These give the ventricular ejection pressure and stroke-volume approximation.

After a rapid switch to systolic compliance $C_s$, negligible valve resistance equalizes arterial and ventricular pressures. Neglecting runoff during this short stage conserves combined stored volume and gives the displayed pressure. Thereafter, with closed mitral valve, negligible venous pressure and constant $C_s$, the pressure decays as ${\hat{p}}_a{e}^{-t/[R_c(C_a+C_s)]}$. At firing duration $\Delta t_F$, the stroke volume is $V^{+}-V_0-C_s{\hat{p}}_a{e}^{-\Delta t_F/[R_c(C_a+C_s)]}$. Each approximation follows from an explicit valve or time-scale separation condition.