Pressure–Volume Loop

What is simulated

The left ventricle is a chamber whose stiffness rises and falls each beat (time-varying elastance). Blood enters through the mitral valve from an atrium held at a fixed pressure, leaves through the aortic valve into an arterial tree that behaves like a leaky balloon (a two-element Windkessel), and drains from there into the body through a resistance.

Equations

T  = 60 / HR
Ts = min(0.16 + 0.3·T, 0.45·T)
τ  = t mod T
e(τ) = sin(π·τ / Ts) for 0 ≤ τ < Ts, else 0

Pes(V) = Ees·(V − V0)              ESPVR
Ped(V) = A·(exp(λ·(V − V0)) − 1)    EDPVR
Plv    = e·Pes(V) + (1 − e)·Ped(V)

Qmv = max(0, (Pla − Plv) / Rmv)     mitral
Qav = max(0, (Plv − Pao) / Rav)     aortic

dV/dt   = Qmv − Qav
C·dPao/dt = Qav − Pao / Rs

Integrated with fixed-step fourth-order Runge–Kutta, dt = 0.5 ms. Fixed values: V0 = 10 mL, A = 0.18 mmHg, Rmv = 0.005 and Rav = 0.01 mmHg·s/mL. The run starts from V = 120 mL and Pao = 80 mmHg.

Reading the numbers

• EDV is the volume at the start of each cycle; ESV is the smallest volume in the beat.
• SV = EDV − ESV, EF = SV / EDV, output = SV × HR.
• Stroke work is the area inside the loop, by the shoelace formula over every sampled (V, Plv) point.
• E_a, effective arterial elastance, is end-systolic pressure divided by SV. On the plot it is the steepness of the dashed line from (EDV, 0) up to the end-systolic corner: the magnitude of its slope, since the line itself falls as volume rises.
• Valve events come from the flows: a valve is open while blood moves through it. The loop counts as settled once EDV and ESV each move by less than 0.1 mL from one beat to the next.

Simplifications

• Left atrial pressure is held constant and is the preload control. A real atrium fills, contracts and empties; here it is a fixed reservoir.
• Valves are ideal one-way diodes with a small linear resistance: no regurgitation, no inertia of blood, no valve dynamics.
• No pericardium, no right heart, no interaction between the ventricles, and no reflexes: changing one slider changes nothing else.
• Because the mitral resistance is small, filling usually comes close to finishing before the next beat, so this model understates how a short diastole limits filling. It does fall short at high heart rates, and with a very compliant ventricle (low λ) at low preload even at normal rates.
• The end-systolic corner lands on the ESPVR only if the aortic valve shuts near peak activation. That holds near baseline (within about 1 mmHg), but at extreme combinations, such as very low resistance and compliance with high preload, ejection runs on after the peak and the corner can fall tens of mmHg below the line. The model checks table flags this.
• The default parameters were tuned towards the typical adult values in the Clinical values tab. Not every baseline value can sit inside its range (for example, at baseline end-diastolic pressure is approximately equal to atrial pressure, whose typical range differs), and the tab lists each place where the baseline differs. The parameters are not fitted to any person, and the model's own outputs are not reference values or thresholds.
• When filling reaches equilibrium through the low-resistance mitral valve, as it does at baseline, the ventricle's end-diastolic pressure is approximately equal to the fixed atrial pressure. When filling is incomplete (a short diastole at high heart rates, or a very compliant ventricle at low preload), end-diastolic pressure stays below atrial pressure.

Sources

• Suga H, Sagawa K, Shoukas AA. Load independence of the instantaneous pressure-volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio. Circ Res. 1973;32:314–322.
• Suga H, Sagawa K. Instantaneous pressure-volume relationships and their ratio in the excised, supported canine left ventricle. Circ Res. 1974;35:117–126.
• Sunagawa K, Maughan WL, Burkhoff D, Sagawa K. Left ventricular interaction with arterial load studied in isolated canine ventricle. Am J Physiol. 1983;245:H773–H780.
• Frank O. Die Grundform des arteriellen Pulses. Z Biol. 1899;37:483–526.