# Mathematical modelling for dose deposition in photontherapy

Pichard, Teddy; Frank, Martin (Thesis advisor); Herty, Michael Matthias (Thesis advisor)

*Aachen (2016, 2018)* [Dissertation / PhD Thesis]

*Page(s): 216 Seiten*

Abstract

Radiation therapy is one of the most common types of cancertreatments. It consists in irradiating the patient with beams of energetic par-ticles (typically photons). Such particles are transported through the mediumand interact with it. Especially, during such interactions, a part of the en-ergy of the transported is deposited in the medium, this is the so-called dose,responsible for the biological effect of the radiation.The aim of the present thesis is to develop numerical method for dosecomputation that are competitive in terms of computational cost and accuracycompared to reference method such as the statistical Monte Carlo methods orthe empirical superposition-convolution methods.The motion of such particles is studied through a system of linear transportequations at the kinetic level with a special consideration on the conservationof mass, momentum and energy.Computational costs required to solve directly such systems is typicallyhigher than available in medical center. In order to reduce those costs, themoment method is used. This consists in averaging the transport equationover one of the variables. However, such a method leads to a system of equa-tions with more unknowns than equation. An entropy minimization procedureis used to close this system, leading to the so-called M N models. The mo-ments extraction preserves the major properties of the kinetic system such ashyperbolicity, entropy decay and realizability (existence of a positive solution).However, computing numerically the M N closure may also be computationallycostly for the application in medical physics, and furthermore it is valid onlyunder condition, called realizability condition, on the unknowns. The realiz-ability domain, i.e. the domain of validity of the M N model, is studied. Basedon these results, approximations of the first order entropy-based closures, i.e.the M 1 and the M 2 closures, are developped for 3D problems which requirelower computational costs to compute.The resulting moment equation are non-linear and valid under realizabilitycondition. Standard numerical schemes for moment equations are constrainedby stability conditions which happen to be very restrictive when the mediumcontains low density regions. Numerical approaches adapted to moment equa-tions are developped. The non-linearity is treated by using a relaxation methodoriginally developped for hyperbolic systems of equations. Then incondition-ally stable schemes are proposed to treat the problem of restrictive stabilityconditions. A first explicit scheme based on the method of characteristics isproposed for hyperbolic equations. A second numerical scheme with implicitnon-linear flux terms is proposed. Those schemes preserve the realizabilityproperty and they are competitive in terms of computational costs comparedto reference approaches.

### Identifier

- REPORT NUMBER: RWTH-2016-11929