Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation

Abstract

The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.

Convergence of filtered spherical harmonic equations for radiation transport

“…where ψ N is either ψ FP N or ψ P N . 4 To estimate E N and R N we use the trapezoidal rule for the integrals and we approximate ψ by a P Ntrue solution, with N true N sufficiently large. Thus the reference solution has a sharply higher angular resolution than ψ FP N and ψ P N but the same spatial resolution.…”

“…(2,4) 1.3188 0.6213 (4,8) 1.8212 0.8161 (8,16 ) 1.5208 0.8293 (16,32) 1.5782 0.8679 (a) even order moments…”

“…There are a variety of nonlinear approximations that ensure positivity. For example, entropy-based methods [13,30] yield, among other things, positive approximations for low-order expansions and have produced promising results in several applications [4,14,16,25,32,39]. However, the implementation of high-order expansions is computationally expensive because of the complicated relationship between the coefficients and the moments of the expansion [1,2].…”

Optimization and large scale computation of an entropy-based moment closure

“…In this section, we present an overview of several implementation details including (i) quadrature approximation in angle, (ii) finite volume discretization in space, (iii) solving the M N minimization problem (6), and (iv) time discretization method. Since many indices and variables are used, a glossary is available in Appendix D for reference.…”

“…The resulting method, colloquially termed M N in the radiation transport community, yields a nonlinear hyperbolic balance law that, unlike the P N method, formally captures the correct streaming limit [13], but at the same time, is accurate in scattering dominated regimes [9,17]. The M N method has been shown to be more accurate in several test cases [8,14,16,20] and in several different applications [6,15,18,33,36,45]. However, it requires the solution of a minimization problem at every spatial cell of the discretized domain.…”