current beat, older beats fainter
end-systolic P–V relation (Emax)
end-diastolic (passive) P–V relation
The pressure–volume loop needs the elastance ventricle. Switch the heart model to “Elastance ventricle”.
Time-varying elastance [5, 6]: PLV = e(t)·Emax·(V − V0) + (1 − e(t))·Ppassive(V) − RV·e(t)·Q. The activation e(t) is a double-Hill curve [6] scaled by the systole duration. The passive filling curve is Ppassive = 0.26·(eB·(V − 15 ml) − 1) mmHg; its stiffness B is adjustable (default 0.03; lower B: compliant, dilated ventricle; higher B: stiff, hypertrophied ventricle). The ventricle fills from the filling pressure through the mitral valve. Stroke volume, ejection time and flow are therefore results of contractility, filling, valve and afterload.
The valve gradient grows with the square of flow, ΔP = K·Q². By Bernoulli, ΔP = ½ρv² ≈ 4v² (v in m/s) [8], so K corresponds to a valve area of about √(4/K) cm² (K in 10−4 mmHg·s²/ml²). The blood in the outflow tract has an inertia L = ρ·l/A, from the pressure needed to accelerate the blood column, ΔP = ρ·l·dv/dt with v = Q/A (an inertance term as in the four-element windkessel [4], but placed in series here). With ρ = 1.06 g/ml, l = 2 cm and a diameter of 2 cm, L ≈ 0.0005 mmHg·s²/ml. Because L is small, ventricular and aortic root pressure differ only by L·dQ/dt and the valve gradient during ejection. Late in systole the flow continues briefly after the ventricular pressure has fallen below the aortic pressure, then reverses and closes the valve. This short backflow produces the dicrotic notch. In aortic regurgitation the closed valve leaks through a regurgitant orifice: blood flows back from the aorta into the ventricle throughout diastole, with the same Bernoulli relation. The ejection fraction is computed from the total ejected volume; with regurgitation the net forward stroke volume is smaller.
The “AV GRADIENT” block shows the pressure drop across the aortic valve, from the left ventricle to the aortic root, during ejection. In each time step the model computes it from the valve flow Q as ΔP = K·Q². This is the valve part only: the pressure needed to accelerate the blood (L·dQ/dt) and the drop across Zc are not included, so it corresponds to the gradient measured by Doppler echocardiography [8].
For orientation, guidelines grade aortic stenosis as severe at a mean gradient of at least 40 mmHg, vmax of at least 4 m/s or a valve area below 1.0 cm², and aortic regurgitation as severe at a regurgitant volume of at least 60 ml or a regurgitant fraction of at least 50 % [16, 17]. A normal valve in this model gives a peak gradient of a few mmHg and vmax below 1 m/s; the “Aortic stenosis” condition gives about 72/44 mmHg and vmax about 4.2 m/s.
Three-element windkessel [2, 3]: the compliant aorta stores part of the stroke volume and drains through the peripheral resistance Rp = R − Zc, so pressure falls exponentially in diastole with τ = Rp·C. The characteristic impedance Zc of the proximal aorta adds a pressure component proportional to flow. Its default of 0.04 mmHg·s/ml lies in the range measured in adults without heart disease [9–11]; higher values are found in hypertension, especially isolated systolic hypertension [9, 12]. The difference between aortic root and Pc corresponds to the “excess pressure” of the reservoir–wave concept, which was found to be proportional to aortic flow [13]. As a guide, Zc and the pulse wave velocity are also calculated from the compliance: Bramwell–Hill, c = √(V / (ρ·C)) [7], for an aorta 50 cm long and 3.3 cm wide, and Zc = ρ·c/A. This guide value is approximate, because C stands for the whole arterial tree, of which the thoracic aorta holds only about half [13]. Compliance falls with pressure, C(P) = C0·e−0.012·(P − 100), with C0 set at 100 mmHg.
The red trace and all pressure readouts show Pc; the dashed line shows the aortic root pressure P. In the flow chart, the optional dashed line shows the outflow at the end of the aorta into the periphery, Pc/Rp: it is nearly constant, because the compliant aorta turns the pulsatile ejection into a steady flow (windkessel function). The ECG is an illustration timed to the simulated beats and does not feed back into the model. With the heart model set to “Flow source”, ejection is imposed as in the original BPSIM [1]: a half-sine or skewed flow profile, constant compliance, and a notch produced by a brief change in compliance; the LV pressure outside ejection is then an illustration as well.
| Constant | Value | Unit | Meaning | Source |
|---|---|---|---|---|
| Blood density ρ | 1.06 | g/ml | Inertance and Bernoulli gradient | standard value |
| Unloaded LV volume V0 | 15 | ml | Volume at zero pressure | assumption ([6] uses 20 ml) |
| Passive filling curve | A = 0.26, B = slider (default 0.03) | mmHg, 1/ml | Ppassive = A·(eB·(V−V0) − 1) | assumption |
| Activation exponents | 1.32, 21.9 | – | Double-Hill function e(t) | [6] |
| Activation time constants | 0.75·Ts, 1.25·Ts | s | Ts = systole duration slider; ratio 1.67 as in [6] (1.68); e(t) normalised to a peak of 1 | adapted from [6] |
| Internal resistance of the ventricle RV | 0.3 | mmHg·s/ml | PLV reduced by RV·e(t)·Q during ejection | assumption |
| Mitral resistance | 0.005 | mmHg·s/ml | Filling from the filling pressure, one-way | assumption |
| Outflow tract | l = 2, d = 2 | cm | Inertance L = ρ·l/A ≈ 0.0005 mmHg·s²/ml, in series | assumption; concept [4] |
| Aortic valve gradient | ΔP = K·Q² | mmHg | K = 4/A²·10−4 (A = valve area in cm²) | [8] |
| Aortic regurgitation | Q = −f(ΔP), R = 0.05 | mmHg·s/ml | Backflow in diastole through the regurgitant orifice: K·Q² + R·Q = Pc − PLV, K = 4/AR²·10−4 | Bernoulli [8]; R assumed |
| Valve closure | 1 ml backflow, +0.01 | ml, mmHg·s²/ml² | Closing volume and extra resistance of the closing leaflets | assumption |
| Pressure dependence of compliance | 0.012, reference 100 mmHg | 1/mmHg | C(P) = C0·e−0.012·(P−100) (elastance model only) | assumption |
| Aorta for the PWV and Zc guide | length 50, diameter 3.3 | cm | c = √(V/(ρ·C)), Zc = ρ·c/A | [7]; dimensions assumed |
| Unit conversion | 1 mmHg = 1333.2 | dyn/cm² | Converts ρ·l/A and ρ·c/A to mmHg·s²/ml and mmHg·s/ml | physical constant |
| Initial state | VLV 120, Pc 90 (flow model: aortic volume 125) | ml, mmHg, ml | Start values; the model reaches steady state within a few beats | assumption |
| Numerical integration | explicit Euler, 1/3000 | s | Time step of the simulation | – |
| Flow model: ejection profiles | x1.5(1−x)3.5; half sine; x1.2(1−x)1.2 | – | Skewed (peak at 30 %), original BPSIM, aortic stenosis (peak at 50 %) | [1] for the half sine; others assumed |
| Flow model: incisura | +8 % of C for 0.1 s | – | Half-sine compliance change at the end of ejection (original: +0.1 ml/mmHg) | adapted from [1] |
| Flow model: LV pressure outside ejection | τ 35 ms, 50 ms, 4 mmHg | – | Relaxation time constant, isovolumic contraction, atrial kick; filling from 3 mmHg towards the filling pressure within about 0.4 s (illustration) | assumption |
| ECG | QTc 0.40 s | s | Shapes are illustrations timed to the simulated beats; QT scales with the RR interval (Bazett, QT = 0.40·√RR), PR and wave widths shorten moderately at high rates | assumption; Bazett formula |
Values chosen for teaching (see “Model assumptions”). Emax was calibrated so that the elastance model reaches the stroke volume listed in the SV column; SV is set directly only in the flow model.
Orange: parameters that represent the primary cause. Blue: parameters that represent the physiological (regulatory or structural) compensation.
HR heart rate, Ts systole duration, R total peripheral resistance, C0 aortic compliance at 100 mmHg, Zc characteristic impedance, Emax contractility, Pfill filling pressure, B LV diastolic stiffness, K aortic valve resistance, AR regurgitant orifice area, SV stroke volume (calibration target).
Program by Ralf Mrowka, © 1991–2026. Model parameters of the original BPSIM from James E. Randall, Microcomputers and Physiological Simulation, Addison Wesley, 1981, ISBN 0-201-06128-7. Educational simulation only. The values are model output and are not measurements of any person.