An Approximation of the $$M_2$$ M 2 Closure: Application to Radiotherapy Dose Simulation

Pichard, Teddy; Alldredge, Graham W.; Brull, Stéphane; Dubroca, Bruno; Frank, Martin

doi:10.1007/s10915-016-0292-8

Cited by 14 publications

(24 citation statements)

References 41 publications

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“…Instead, the method of moments is often used as it requires lower computational power (see some comparisons e.g. in the previous work [49,48,46]). In the following, the construction of the M 1 model associated to the kinetic model (1-2) is recalled.…”

Section: A Moment Modelmentioning

confidence: 99%

“…In order to use this scheme, one needs to solve (22) which is a nonlinear equation on the vector (ψ n ) l=1,...,lmax . For this purpose, we propose an iterative solver inspired of [20], which was tested in [48,47,50,6,45].…”

Section: An Iterative Solver For the 1d Schemementioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Pichard¹,

Brull²,

Dubroca³

2019

CICP

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Numerical schemes for the systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider here a system modeling the steady transport of photons and electrons in the field of radiotherapy, which leads to very stiff CFL conditions at the discrete level. We circumvent this issue by constructing an implicit scheme based on a relaxation approach. The physics is modeled by an entropy-based moment system, namely the M 1 model. This model is non-linear, possesses potentially no hyperbolic operator. It is furthermore only valid under a condition called realizability, which corresponds to the positivity of an underlying kinetic distribution function. The present numerical approach is applicable to non-linear systems which possess potentially no hyperbolic operator, and it preserves the realizability property. However the discrete equations are non-linear and we propose a numerical method to solve such non-linear systems. Our approach is tested on academic and practical cases in 1D, 2D and 3D and it is shown to require significantly less computational power than reference methods.

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Section: A Moment Modelmentioning

confidence: 99%

Section: An Iterative Solver For the 1d Schemementioning

confidence: 99%

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

Pichard¹,

Brull²,

Dubroca³

2019

CICP

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“…An iterative solver was proposed in [15] to compute Ψ n l,m at each iteration and a complete analysis of this scheme is postponed to a futur paper.…”

Section: A Numerical Scheme For 2d Equationsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

Pichard

Dubroca

Brull

et al. 2018

Theory, Numerics and Applications of Hyperbolic Problems II

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Typical external radiotherapy treatments consist in emitting beams of energetic photons targeting the tumor cells. Those photons are transported through the medium and interact with it. Such interactions affect the motion of the photons but they are typically weakly deflected which is not well modeled by standard numerical methods.The present work deals with the transport of photons in water. The motion of those particles is modeled by an entropy-based moment model, i.e. the M 1 model. The main difficulty when constructing numerical approaches for photon beam modelling emerges from the significant difference of magnitude between the diffusion effects in the forward and transverse directions. A numerical method for the M 1 equations is proposed with a special focus on the numerical diffusion effects.

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