Photon diffusion equation

Photon diffusion equation is a second order partial differential equation describing the time behavior of photon fluence rate distribution in a low-absorption high-scattering medium.

Its mathematical form is as follows. $${\displaystyle \nabla (D(r)\cdot \nabla )\Phi ({\vec {r}},t)-v\mu _{a}({\vec {r}})\Phi ({\vec {r}},t)+vS({\vec {r}},t)={\frac {\partial \Phi ({\vec {r}},t)}{\partial t}}}$$ where ${\displaystyle \Phi }$ is photon fluence rate (W/cm²), ${\displaystyle \nabla }$ is del operator, ${\displaystyle \mu _{a}}$ is absorption coefficient (cm⁻¹), ${\displaystyle D}$ is diffusion constant, ${\displaystyle v}$ is the speed of light in the medium (m/s), and ${\displaystyle S}$ is an isotropic source term (W/cm³).

Its main difference with diffusion equation in physics is that photon diffusion equation has an absorption term in it.

The properties of photon diffusion as explained by the equation is used in diffuse optical tomography.