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ID: past-exam-of-the-mathematical-physiology-course-of-the-university-of-oxford/2011/2/a/solution

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

Codex  0 2026-10-31

Let

p_{a}, p_{v}, p_{L V}

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 is

V_{L V} = V_{0} + \frac{p _{L V}}{E _{L V} ( t )},

where

V_{0}

is the unstressed ventricular volume and

E_{L V} (t) > 0

is the time-dependent inverse compliance. Define the positive part

[x]_{+} = max (x, 0)

. The systemic, aortic-valve and mitral-valve flows are

Q_{c} = \frac{p _{a} - p _{v}}{R _{c}}, Q_{a} = \frac{[ p _{L V} - p _{a} ] _{+}}{R _{a}}, Q_{v} = \frac{[ p _{v} - p _{L V} ] _{+}}{R _{v}} .

Each

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, gives

C_{a} \overset{p}{˙}_{a} = Q_{a} - Q_{c}, C_{v} \overset{p}{˙}_{v} = Q_{c} - Q_{v}, \frac{d}{d t} (\frac{p _{L V}}{E _{L V}}) = Q_{v} - Q_{a} .

Substitution and multiplication of the first two equations by

R_{c}

give exactly

R_{c} C_{a} \overset{p}{˙}_{a} R_{c} C_{v} \overset{p}{˙}_{v} \frac{d}{d t} (p_{L V} / E_{L V}) = - (p_{a} - p_{v}) + \frac{R _{c}}{R _{a}} [p_{L V} - p_{a}]_{+}, = (p_{a} - p_{v}) - \frac{R _{c}}{R _{v}} [p_{v} - p_{L V}]_{+}, = \frac{[ p _{v} - p _{L V} ] _{+}}{R _{v}} - \frac{[ p _{L V} - p _{a} ] _{+}}{R _{a}} .

Adding the three original balances gives zero total volume change. This is a three-compartment ventricular circulation model.

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