Explicit solutions of the radiative transport equation in the P3 approximation

Explicit solutions of the monoenergetic radiative transport equation in the P3 approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light.

A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P3 equations in closed form.

The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P3 approximation is close to the exact solution of the radiative transport equation.

The authors derived exact analytical solutions of the P3 equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.