# Bioheat equation

Heat transfer in living tissue can be considered as heat conduction with internal heat generation from two sources: metabolic heat generation and blood perfusion. The former results from a series of chemical reactions that occur in the living cells, while the latter is due to energy exchange between the living tissue and blood flowing in the small capillaries. If the heat transfer in a tissue can be considered as one-dimensional and the thermal conductivity of the tissue is assumed to be independent from temperature, the energy equation for living tissue can be obtained by

$\frac{1}{{A(s)}}\frac{d}{{ds}}\left( {A(s)\frac{{dT}}{{ds}}} \right) + \frac{{{{q'''}_m} + {{q'''}_b}}}{k} = 0,{\rm{ }}{s_1} < s < {s_2} \qquad \qquad(1)$

where q'''m and q'''b are the rate of metabolic heat generation and blood perfusion, respectively. Equation (1) is generalized for Cartesian (s = x,A(s) = const), cylindrical (s = r,A(s) = r), and spherical (s = r,A(s) = r2) coordinate systems. The internal heat generation due to blood perfusion can be obtained by

${q'''_b} = \omega {\rho _b}{c_{pb}}({T_a} - T) \qquad \qquad(2)$

where ω is the blood perfusion rate – defined as volume flow rate of blood flow per unit volume of the tissue, ρb is the density of blood, cpb is the specific heat of blood under constant pressure, and Ta is the blood temperature. Depending on whether the blood temperature is higher than the tissue temperature, the internal heat generation obtained from eq. (2) can be either positive (Ta > T) or negative (Ta < T).

Substituting eq. (2) into eq. (1), the energy equation for the tissue becomes

$\frac{1}{{A(s)}}\frac{d}{{ds}}\left( {A(s)\frac{{dT}}{{ds}}} \right) + \frac{{{{q'''}_m} + \omega {\rho _b}{c_{pb}}({T_a} - T)}}{k} = 0,{\rm{ }}{s_1} < s < {s_2}$

which is nonhomogeneous due to the presence of metabolic heating and blood perfusion. The above equation can be rearranged into

$\frac{1}{{A(s)}}\frac{d}{{ds}}\left( {A(s)\frac{{dT}}{{ds}}} \right) - \frac{{\omega {\rho _b}{c_{pb}}[T - {T_a} - {{q'''}_m}/(\omega {\rho _b}{c_{pb}})]}}{k} = 0,{\rm{ }}{s_1} < s < {s_2} \qquad \qquad(3)$

If the blood temperature and the internal heat generation due to metabolic heating are constant, eq. (3) can be homogenized by introducing the following excess temperature

$\vartheta = T - {T_a} - {q'''_m}/(\omega {\rho _b}{c_{pb}}) \qquad \qquad(4)$

i.e.,

$\frac{1}{{A(s)}}\frac{d}{{ds}}\left( {A(s)\frac{{d\vartheta }}{{ds}}} \right) - {m^2}\vartheta = 0,{\rm{ }}{s_1} < s < {s_2} \qquad \qquad(5)$

where

${m^2} = \frac{{\omega {\rho _b}{c_{pb}}}}{k} \qquad \qquad(6)$

Equation (5) is identical to the energy equation for extended surface and therefore, the solution for heat conduction from an extended surface can be readily applied to the solution of the bioheat equation.

For different parts of the body, the generalized bioheat equation (5) can be changed to different forms. For the tissue with thickness much less than its curvature radius, the tissue can be treated as a plane wall and eq. (5) becomes

$\frac{{{d^2}\vartheta }}{{d{x^2}}} - {m^2}\vartheta = 0,{\rm{ }}0 < x < L \qquad \qquad(7)$

If the effects of curvature need to be considered, eq. (5) can be rewritten for cylindrical or spherical coordinate systems, i.e.,

$\frac{{{d^2}\vartheta }}{{d{r^2}}} + \frac{1}{r}\frac{{d\vartheta }}{{dr}} - {m^2}\vartheta = 0,\qquad{\rm{ }}{r_1} < r < {r_2}\qquad{\rm{ (Cylindrical) }} \qquad \qquad(8)$
$\frac{{{d^2}\vartheta }}{{d{r^2}}} + \frac{2}{r}\frac{{d\vartheta }}{{dr}} - {m^2}\vartheta = 0,\qquad{\rm{ }}{r_1} < r < {r_2}\qquad{\rm{ (Spherical) }} \qquad \qquad(9)$

The bioheat equation in different coordinate systems can be solved with appropriate boundary conditions.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.