Analytical solutions of Pennes equation

Feb 7, 2017 by Hugo Milan

The Pennes bio-heat equation in time-domain is defined as:

\begin{equation} \rho c_{p}\frac{\partial T}{\partial t} = k\nabla^2 T + S + Q_m + \omega_b\rho_b c_{b} (T_b - T) \end{equation}

with the following definition of flux

\begin{equation} \bar{q} = -k\nabla T \end{equation}

where \(T\) is temperature, \(t\) is time, \(\rho\) is density, \(c_{p}\) is specific heat, \(k\) is heat conductivity, \(S\) is source, \(Q_m\) is metabolic heat generation, \(\omega_b\) is blood perfusion, \(\rho_b\) is blodd density, \(c_{b}\) is blood specific heat, and \(T_b\) is blood temperature.

1) One-dimension

Click here to see how to solve the problem with the following initial condition and boundary conditions:

$$\begin{matrix} T(y, t = 0) = T_{c}\\ T(y = 0, t) = T_{c}\\ T(y = H, t) = T_{s} \end{matrix}$$

2) Two-dimensions

Click here to see how to solve the problem with the following initial condition and boundary conditions:

$$\begin{matrix} T(x,y, t = 0) = T_{c}\\ T(x,y = 0, t) = T_{c}\\ T(x,y = H, t) = T_{s}\\ \dfrac{\partial T}{\partial x}(x = 0, y, t) = 0\\ \dfrac{\partial T}{\partial x}(x = L, y, t) = \dfrac{q_x}{k} \end{matrix}$$

3) Three-dimensions

Click here to see how to solve the problem with the following initial condition and boundary conditions:

$$\begin{matrix} T(x,y,z, t = 0) = T_{c}\\ T(x,y = 0,z, t) = T_{c}\\ T(x,y = H, z,t) = T_{s}\\ \dfrac{\partial T}{\partial x}(x = 0, y, z,t) = 0\\ \dfrac{\partial T}{\partial x}(x = L, y, z,t) = \dfrac{q_x}{k}\\ \dfrac{\partial T}{\partial z}(x, y, z=0,t) = 0\\ \dfrac{\partial T}{\partial z}(x, y, z = T_z,t) = \dfrac{q_z}{k} \end{matrix}$$

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