Sensitivity analysis for dose deposition in radiotherapy via a   Fokker-Planck model
  In this paper we study the sensitivities of electron dose calculations with respect to the stopping power and the transport coefficients. We focus on the application to radiotherapy simulations. We use a Fokker-Planck approximation to the Boltzmann transport equation. Equations for the sensitivities are derived by the adjoint method. The Fokker-Planck equation and its adjoint are solved numerically in slab geometry using the spherical harmonics expansion ($P_N$) and an HLL finite volume method. Our method is verified by comparison to finite difference approximations of the sensitivities. Finally, we present numerical results of the sensitivities for the normalized average dose deposition depth with respect to the stopping power and transport coefficients, demonstrating the increasing relative sensitivities as beam energy decreases. 
Richard C. Barnard, Martin Frank, Kai Krycki
math.NA