Relaxation Schemes for the M₁ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation

Abstract

Because of stability constraints, most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar. This problem emerges with the M1 system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the M1 system is non-linear, and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability. In this paper, we propose a numerical method that overcomes the stability constraint and preserves the realizability property. For this purpose, we relax the M1 system to obtain a linear flux term. Then we extend the stencil of the difference quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation example.

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

“…The computational time is although higher when using the new relaxation parameters (15) than when using the one in (11). This is actually due to the method used to compute Ψ n l,m from (14) or (16). The conditioning of problem (14) is simply better than the one of (16) which explains the difference of computational times.…”

“…In order to handle the non-linearity in (8), the relaxation approach proposed in [16] and based on the previous work of [14,1] is used.…”

“…The present work is a follow-up to [5,16,15] which is devoted to adapt the numerical scheme presented in [15] to accurately model beams of photons. Such beams travel almost straight in a human-sized medium.…”

A Numerical Approach for a System of Transport Equations in the Field of Radiotherapy

“…This work aims to construct a numerical solver for systems of steady transport equations emerging in the field of radiotherapy. It is a follow-up to ( [21,5,49]) and it analyses the numerical method used [46,50,48,47,14,6,45].…”

“…Such a condition becomes very restrictive in the presence of low density media. This typically corresponds to require a step size (see [5,49] or Section 3 below) proportional to the minimum density in the medium. This problem was first studied for application in radiotherapy in [5] and it was circumvented by the use of a clever change of variables.…”