The minimum entropy approximation to the radiative transfer equation

To ensure the optimal treatment of cancer using radiotherapy, the administered dose must be accurate to within a few percent. However, the models used in clinical dose calculation today fail to achieve this accuracy in the inhomogeneities encountered in tissues and organs such as bone, lungs and sinus passages. With the aim of improving upon the currently available heuristic models, we consider a deterministic method: the radiative transfer equation solved with the method of moments, closed using the minimum entropy principle. It can be shown, that the minimum entropy closure ensures the hyperbolicity of the moment system and non negativity of the distribution function [APS91, DF99]. The first order minimum entropy system, however, produces unphysical shocks in numerical experiments involving two photon beams [BH01, FHK07]. We close the second order system by numerically inverting the resulting non-linear system. By avoiding the use of iterative procedures, we ensure the closure is calculated to extremely high precision. We investigate the properties of the resulting system and the highest order moment N-3 (N-1, N-2). We show that the underlying distribution function on the boundary of the moments' admissible domain becomes a linear combination of two Diracs, a fact that bodes well for overcoming the shortcomings of the first order minimum entropy system.