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k-Wave Toolbox


Compute exact solution to Pennes' bioheat equation in homogeneous media.


T = bioheatExact(T0, S, material, dx, t)
T = bioheatExact(T0, S, [D, P, Ta], dx, t)


bioheatExact calculates the exact solution to Pennes' bioheat equation in a homogeneous medium on a uniform Cartesian grid using a Fourier-based Green's function solution assuming a periodic boundary condition [1]. The function supports inputs in 1D, 2D, and 3D. The exact equation solved is given by

dT/dt = D * d^2T/dx^2 - P * (T - Ta) + S

where the coefficients are defined below. Pennes' bioheat equation is often given in the alternative form

P0 * C0 * dT/dt =  Kt * d^2T/dx^2 - Pb * Wb * Cb * (T - Ta) + Q

T:  temperature                     [degC]
C0: tissue specific heat capacity   [J/(kg.K)]
P0: tissue density                  [kg/m^3] 
Kt: tissue thermal conductivity     [W/(m.K)]
Pb: blood density                   [kg/m^3] 
Wb: blood perfusion rate            [1/s]
Ta: blood arterial temperature      [degC]
Cb: blood specific heat capacity    [J/(kg.K)]
Q:  volume rate of heat deposition  [W/m^3]

In this case, the function inputs are calculated by

D = Kt / (P0 * C0);
P = Pb * Wb * Cb / (P0 * C0);
S = Q / (P0 * C0);

If the perfusion coefficient P is set to zero, bioheatExact calculates the exact solution to the heat equation in a homogeneous medium.

[1] Gao, B., Langer, S., & Corry, P. M. (1995). Application of the time-dependent Green's function and Fourier transforms to the solution of the bioheat equation. International Journal of Hyperthermia, 11(2), 267-285.


T0 matrix of the initial temperature distribution at each grid point [degC]
S matrix of the heat source at each grid point [K/s]
material material coefficients given as a three element vector in the form: material = [D, P, Ta], where
D diffusion coefficient [m^2/s]
P perfusion coefficient [1/s]
Ta arterial temperature [degC]
dx grid point spacing [m]
t time at which to calculate the temperature field [s]


T temperature field at time t [degC]

See Also