HPC Tier Module
Arterial Windkessel Hemodynamics
Resolve aortic pressure from a lumped Windkessel model with closed-form benchmarks - exact f64, in-browser.
See it run - a worked example, 100% in this browser tab
The problem
Cardiovascular-systems teaching and model exploration need a Windkessel solver that is reproducible and self-validating, not a black box whose pressure waveform cannot be checked against the physics it claims to follow.
The local-first solution
This plugin integrates the cited 2- and 4-element Windkessel ODEs in exact f64 RK4 and compares the result to two scheme-independent closed forms - the diastolic decay constant and the cycle-mean Ohm analog - entirely in your browser with no upload or API key.
What it does
RK4 integration of the WK2 (R-C) and WK4 (R-C-Zc-L) lumped arterial models
Half-sine systolic ejection forcing scaled exactly to the prescribed stroke volume
Fits the diastolic decay time constant tau and checks it equals R*C
Verifies the cycle-mean Ohm relation mean(P) = CO * R
Reduces exactly to WK3 and WK2 as Zc and L are varied
Reports a GeoNum conditioning signal with a Richardson residual fallback
Honest scope
Exact: the WK2/WK4 ODEs, the half-sine forcing, RK4 integration, and the two closed-form benchmarks (tau = R*C and cycle-mean = CO*R). It is a lumped 0-D model with no spatial information - no wave travel, reflection, taper, or location-specific waveforms - and the half-sine ejection is idealized; R, C, Zc, L, HR, SV are surfaced inputs with cited physiological defaults. It is a research and education tool, not a medical device, and not for clinical, diagnostic, or treatment decisions.
Authorities cited
- Frank, O. (1899). Die Grundform des arteriellen Pulses (Erste Abhandlung: mathematische Analyse). Zeitschrift fuer Biologie 37, 483-526. - The original two-element (Windkessel) arterial model: C dP/dt + P/R = Q.
- Westerhof, N., Lankhaar, J.-W., & Westerhof, B. E. (2009). The arterial Windkessel. Medical & Biological Engineering & Computing 47(2), 131-141. DOI 10.1007/s11517-008-0359-2. - 2/3/4-element Windkessel review; governing equation, diastolic decay tau = R*C (eqn 2), mean pressure = CO*R, half-sine inflow (Fig. 2).
- Stergiopulos, N., Westerhof, B. E., & Westerhof, N. (1999). Total arterial inertance as the fourth element of the Windkessel model. American Journal of Physiology - Heart and Circulatory Physiology 276(1), H81-H88. DOI 10.1152/ajpheart.1999.276.1.H81. - The 4-element (parallel) Windkessel: characteristic impedance Z_c and total arterial inertance L.
- Westerhof, N., Elzinga, G., & Sipkema, P. (1971). An artificial arterial system for pumping hearts. Journal of Applied Physiology 31(5), 776-781. DOI 10.1152/jappl.1971.31.5.776. - The 3-element (R-C-Z_c) Windkessel adding the characteristic impedance of the proximal aorta.
- Westerhof, N., Stergiopulos, N., Noble, M. I. M., & Westerhof, B. E. (2019). Snapshots of Hemodynamics, 3rd ed. Springer. DOI 10.1007/978-3-319-91932-4. - Windkessel models, arterial input impedance, characteristic impedance, and physiological parameter ranges (R, C, Z_c, L).
- Press, W. H., et al. (2007). Numerical Recipes, 3rd ed., Sec. 17.1, CUP. - classical fourth-order Runge-Kutta (RK4); local truncation error O(h^5) / global O(h^4).
Run the Windkessel solve
Compute the pressure trace and benchmark errors entirely in your browser - nothing uploaded to anyone's cloud. Save the run to Sandbox, attach it to a Worklog case, or share the parameter set through a Gate portal.