HPC Tier Module
Arterial Windkessel Hemodynamics
Resolve aortic pressure from a lumped Windkessel model with closed-form benchmarks - exact f64, in-browser.
This tool runs inside GDBS HPC. What it computes and the authorities it applies are below.
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).
Get an email when tools like this one change
Leave your email and we will tell you when we add or update Healthcare & Biomedical tools like this one, at most once a month.
We email you once to confirm. After that, unsubscribe from any issue in one click. Your address goes to VaultSync only (privacy).
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.