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The science
Time-varying elastance
Each ventricle is a chamber whose stiffness changes during the beat: P = E(t) (V - V0) with E(t) = Ed + (Emax - Ed) e(t) and activation e(t) = sin^2(pi t / Ts) during systole of length Ts = 0.324 sqrt(T) (capped at 0.6 T), zero in diastole. The pressure-volume loop is the trajectory of (V, P): filling along the soft diastolic line, isovolumic contraction, ejection, isovolumic relaxation. The loop area is the stroke work.
Valves are diodes
Flow through a valve is Q = (P_up - P_down) / R_valve when positive and zero otherwise. Aortic stenosis multiplies R_aortic, so a larger pressure gradient is needed for the same flow; regurgitation adds a leak path Q = dP / R_leak backwards through the closed valve.
Vessels
The aorta is a windkessel, C dP/dt = Q_in - Q_out, draining through the systemic resistance R_s into the compliant veins. Between beats the aortic pressure falls as exp(-t / (R_s C_ao)). The lungs are the same idea with a much lower resistance. Six volumes are integrated with fourth-order Runge-Kutta at 0.5 ms; total blood volume is conserved exactly.
Emergent results
Mean arterial pressure follows MAP - CVP = CO x SVR; the stroke volume rises with filling (Frank-Starling); the left and right outputs match in steady state. None of these is programmed in.
Honest simplifications: passive atria (no atrial kick), no blood inertia, no baroreflex, no coronary flow, one fixed body size; the numbers are model values in the physiological range, not patient data. This page runs the same physics code as the 3D page.
What to try
- Choose Aortic stenosis and watch the left ventricular pressure peak far above the aortic pressure.
- Raise the heart rate: filling time shrinks and the stroke volume falls while the output usually rises.
- Raise the systemic resistance: the blood pressure climbs while the output falls a little.
- Add leak: the diastolic pressure collapses because blood runs back into the ventricle.