Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T12:34:38.748Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011, pp. 1184 - 1210
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Ahnesjö, A., Saxner, M., and Trepp, A., A pencil beam model for photon dose calculation, Med. Phys., 19(2) (1992), 263–273.Google Scholar
[2]Andreo, P., Monte Carlo techniques in medical radiation physics, Phys. Med. Biol., 36(7) (1991), 861–920.CrossRefGoogle ScholarPubMed
[3]Berthon, C., Stability of the MUSCL schemes for the Euler equations, Commun. Math. Sci., 3 (2005), 133–157.Google Scholar
[4]Berthon, C., Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys., 218 (2006), 495–509.Google Scholar
[5]Berthon, C., Charrier, P., and Dubroca, B., An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions, J. Sci. Comput., 31 (2007), 347–389.Google Scholar
[6]Berthon, C., Dubois, J., and Turpault, R., Numerical approximation of the M1 model, SMF Publication, Panoramas et Synthèses, 28 (2009), 55–86.Google Scholar
[7]Berthon, C. and Turpault, R., Asymptotic preserving HLL schemes, to appear in NMPDE, 2010.Google Scholar
[8]Boman, E., Tervo, J., and Vauhkonen, M., Modelling the transport of ionizing radiation using the finite element method, Phys. Med. Biol., 50 (2005), 265–280.Google Scholar
[9]Börgers, C., Complexity of Monte Carlo and deterministic dose-calculation methods, Phys. Med. Biol., 43 (1998), 517–528.Google Scholar
[10]Bouchut, F., Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Frontiers in Mathematics Series, Birkhäuser, 2004.Google Scholar
[11]Buet, C. and Cordier, S., An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer, Numer. Math., 108 (2007), 199–221.Google Scholar
[12]Buet, C. and Després, B., Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Trans., 85 (2004), 385–418.Google Scholar
[13]Buet, C. and Després, B., Asymptotic preserving and positive schemes for radiation hydrodynamics, J. Comput. Phys., 215 (2006), 717–740.Google Scholar
[14]Cygler, J. E.et al., Clinical use of a commercial monte carlo treatment planning system for electron beams, Phys. Med. Biol., 50 (2005), 1029–1034.Google Scholar
[15]Dubroca, B. and Feugeas, J. L., Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I., 329 (1999), 915–920.Google Scholar
[16]Dubroca, B. and Frank, M., An iterative method for transport equations in radiotherapy, submitted to Proceedings of ECMI 08, 2009.Google Scholar
[17]Duclous, R., Dubroca, B., and Frank, M., Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843–3857.Google Scholar
[18]Eyges, L., Multiple scattering with energy loss, Phys. Rev., 74 (1948), 1534–1535.Google Scholar
[19]Frank, M., Dubroca, B., and Klar, A., Partial moment entropy approximationtoradiative transfer, J. Comput. Phys., 218 (2006), 1–18.Google Scholar
[20]Frank, M., Hensel, H., and Klar, A., A fast and accurate moment method for dose calculation in electron radiotherapy, SIAM J. Appl. Math., 67 (2007), 582–603.Google Scholar
[21]Gosse, L. and Toscani, G., Asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337–342.Google Scholar
[22]Gosse, L. and Toscani, G., Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641–658.CrossRefGoogle Scholar
[23]Goudon, T., Coulombel, J. F., and Golse, F., Diffusion approximation and entropy-based moment closure for kinetic equations, Asymp. Anal., 45(1-2) (2005), 1–39.Google Scholar
[24]Grad, H., On kinetic theory of rarified gases, Commun. Pure Appl. Math., 2 (1949), 331–407.Google Scholar
[25]Harten, A., Lax, P. D., and Leer, B. van, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35–61.Google Scholar
[26]Hogstrom, K. R., Mills, M. D., and Almond, P. R., Electron beam dose calculations, Phys. Med. Biol., 26(3) (1981), 445–459.Google Scholar
[27]Jette, D., Electron dose calculation using multiple-scattering theory, A. Gaussian multiple-scattering theory, Med. Phys., 15(2) (1988), 123–137.Google ScholarPubMed
[28]Jette, D. and Bielajew, A., Electron dose calculation using multiple-scattering theory: second-order multiple-scattering theory, Med. Phys., 16(5) (1989), 698–711.Google Scholar
[29]Krieger, T. and Sauer, O. A., Monte Carlo- versus pencil-beam-/collapsed-cone- dose calculation in a heterogeneous multi-layer phantom, Phys. Med. Biol., 50 (2005), 859–868.Google Scholar
[30]Kudchadker, R. J., Antolak, J. A., Morrison, W. H., Wong, P. F., and Hogstrom, K. R., Utilization of custom electron bolus in head and neck radiotherapy, J. Appl. Clin. Med. Phys., 4 (2003), 321–333.Google Scholar
[31]Le Veque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.Google Scholar
[32]Levermore, C. D., Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021–1065.Google Scholar
[33]Mihalas, D. and Weibel-Mihalas, B., Foundations of Radiation Hydrodynamics, Dover, 1999.Google Scholar
[34]Minerbo, G. N., Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541–545.Google Scholar
[35]Pomraning, G. C., The Equations of Radiation Hydrodynamics, Pergamon Press, 1973.Google Scholar
[36]Salvat, F., Fernandez-Varea, J. M., and Sempau, J., PENELOPE-2008,A Code System for Monte Carlo Simulation of Electron and Photon Transport, OECD, 2008, ISBN 978-92-64-99066-1.Google Scholar
[37]Shiu, A. S. and Hogstrom, K. R., Pencil-beam redefinition algorithm for electron dose distributions, Med. Phys., 18(1) (1991), 7–18.Google Scholar
[38]Siantar, C. L. H.et al., Description and dosimetric verification of the peregrine Monte Carlo dose calculation system for photon beams incident on a water phantom, Med. Phys., 28(7) (2001), 1322–1337.Google Scholar
[39]Spezi, E. and Lewis, G., An overview of Monte Carlo treatment planning for radiotherapy, Radiat. Prot. Dos., 131(1) (2008), 123–129.Google Scholar
[40]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Second edition, Springer-Verlag, Berlin, 1999.Google Scholar
[41]Turpault, R., A consistent multigroup model for radiative transfer and its underlying mean opacities, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357–371.Google Scholar
[42]Turpault, R., Frank, M., Dubroca, B., and Klar, A., Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363–371.Google Scholar
[43]Leer, B. van, Towards the ultimate conservative difference scheme, V, a second-order sequel to Godunov’s method, J. Comput. Phys., 32 (1979), 101–136.Google Scholar
[44]Vassiliev, O. N.et al., Feasibility of a multigroup deterministic solution method for 3d radiotherapy dose calculations, Int. J. Radiat. Oncol. Biol. Phys., 72(1) (2008), 220–227.Google Scholar
[45]Wen, X. and Jin, S., Convergence of an immersed interface upwind scheme for linear advec-tion equations with piecewise constant coefficients i: L 1-error estimates, J. Comput. Math., 26 (2008), 1–22.Google Scholar