Past exam of the mathematical physiology course of the University of Oxford / 2011 / 2

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 isovolumetric contraction, ventricular pressure lies between the venous and arterial pressures, $p_v<p_{LV}<p_a$. Both valves are closed, so $Q_v=Q_a=0$ and ventricular volume remains at its end-filling value $V^{+}$. ConsequentlyHere $V^{+}$ is end-diastolic ventricular volume and $V_0$ is the zero-pressure/unstressed volume.

The arterial and venous reservoirs still exchange systemic blood:If contraction is short compared with both $R_c{C}_a$ and $R_c{C}_v$, the fractional changes are small andwhere the superscript values refer to the beginning of contraction. Thus pressure can rise rapidly in the left ventricle through increasing elastance while the other pressures are approximately fixed. This stage ends when $p_{LV}$ reaches $p_a$ and the aortic valve opens.

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.