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

Published online by Cambridge University Press:  20 August 2015

Christophe Berthon ,
Martin Frank ,
Céline Sarazin  and
Rodolphe Turpault
Christophe Berthon*
Affiliation:
Université de Nantes, Laboratoire de Mathématiques Jean Leray, UMR6629, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
Martin Frank*
Affiliation:
RWTH Aachen University, Mathematics, Center for Computational Engineering Science, Schinkelstrasse 2, 52062 Aachen, Germany
Céline Sarazin*
Affiliation:
Université de Nantes, Laboratoire de Mathématiques Jean Leray, UMR6629, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
Rodolphe Turpault*
Affiliation:
Université de Nantes, Laboratoire de Mathématiques Jean Leray, UMR6629, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
*
Email:christophe.berthon@math.univ-nantes.fr
Corresponding author.Email:frank@mathcces.rwth-aachen.de
Email:celine.sarazin@univ-nantes.fr
Email:rodolphe.turpault@univ-nantes.fr
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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.

Keywords

35L6565N0892C99Radiotherapyhyperbolic system of conservation lawssource term approximationsfinite volume methods
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1184 - 1210
Copyright
Copyright © Global Science Press Limited 2011

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