HPC Tier Module
Pennes Bioheat Equation (Ablation / Hyperthermia)
March transient tissue heating with the Pennes bioheat equation and a cited thermal dose - exact, in-browser.
See it run - a worked example, 100% in this browser tab
The problem
Thermal-therapy modeling and engineering verification need a bioheat solver that reports temperature and thermal dose against exact benchmarks, flags numerical instability, and surfaces every tissue constant rather than hiding them.
The local-first solution
This plugin integrates the cited Pennes bioheat PDE by explicit or implicit finite differences in exact f64, computes the Sapareto-Dewey CEM43 dose, and checks the solution against two closed-form benchmarks - all deterministic and in your browser with no upload.
What it does
Centred 2nd-order Laplacian with FTCS explicit or backward-Euler implicit time marching
Pennes perfusion heat-sink term in both the explicit update and implicit diagonal
Von Neumann stability number computed and hard-flagged when violated
Sapareto-Dewey CEM43 cumulative-equivalent-minutes thermal dose by trapezoidal integration
Steady-state perfusion benchmark with the exact penetration depth sqrt(k/(w_b c_b))
Pure-diffusion sine-mode decay benchmark when perfusion and sources are off
Honest scope
Exact: the Laplacian stencil, the explicit and implicit updates, the Thomas solve, the stability number, the trapezoidal CEM43 integral, and the two closed-form benchmarks. The Pennes perfusion term is an approximate model treating blood as a uniform, instantly-equilibrating sink at fixed arterial temperature, with temperature-independent properties; it does not model discrete-vessel cooling, nonlinear perfusion, the Arrhenius damage integral, phase change, or 3D anatomy, and CEM43 is an isoeffect dose reported, not a clinical determination. Research, education, and engineering verification only - not a medical device and not for clinical or treatment-planning decisions.
Authorities cited
- Pennes, H. H. (1948). Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of Applied Physiology 1(2), 93-122. - The bioheat equation rho c dT/dt = k lap T + w_b c_b (T_a - T) + Q_met and the perfusion heat-sink term. DOI 10.1152/jappl.1948.1.2.93.
- Sapareto, S. A., & Dewey, W. C. (1984). Thermal dose determination in cancer therapy. International Journal of Radiation Oncology, Biology, Physics 10(6), 787-800. - The CEM43 = integral R^(43-T) dt cumulative-equivalent-minutes thermal-dose model (R=0.5 above, 0.25 below 43 C). DOI 10.1016/0360-3016(84)90379-1.
- Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids, 2nd ed. Oxford. - Separable sine-mode solutions and their exponential decay for the heat equation (the w_b=0 pure-diffusion benchmark).
- LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM. - Centred 2nd-order Laplacian (sec 3.2), forward/backward Euler time stepping, Dirichlet BCs. DOI 10.1137/1.9780898717839.
- Strikwerda, J. C. (1989/2004). Finite Difference Schemes and Partial Differential Equations, 2nd ed. SIAM. - von Neumann stability of explicit reaction-diffusion schemes (the perfusion term tightens the FTCS limit). DOI 10.1137/1.9780898717938.
- Press, W. H., et al. (2007). Numerical Recipes, 3rd ed. CUP, sec 2.4. - The Thomas algorithm for tridiagonal systems (the exact backward-Euler line solve).
- Hasgall, P. A., et al. (2022). IT'IS Database for thermal and electromagnetic parameters of biological tissues, v4.1. IT'IS Foundation. DOI 10.13099/VIP21000-04-1. - Cited default tissue/blood properties (k, rho, c, w_b, c_b, Q_met).
- Duck, F. A. (1990). Physical Properties of Tissue: A Comprehensive Reference Book. Academic Press. - Thermal conductivity, perfusion, and metabolic-heat ranges used for the cited defaults.
- Weinbaum, S., & Jiji, L. M. (1985). A new simplified bioheat equation for the effect of blood flow on local average tissue temperature. ASME J. Biomech. Eng. 107(2), 131-139. - The countercurrent correction that Pennes omits (cited as a NOT-MODELED limitation). DOI 10.1115/1.3138533.
Run the bioheat solve
Compute the temperature field and thermal dose in your browser, with nothing uploaded to anyone's cloud. Save the run to Sandbox, attach it to a Worklog case, or share parameters through a Gate portal.