Numerical methods for Boltzmann transport equations in radiotherapy treatment planning

• Numerische Verfahren zum Lösen von Boltzmann-Transportgleichungen in der Strahlentherapie-Planung

Jörres, Christian; Herty, Michael (Thesis advisor); Frank, Martin (Thesis advisor); Pareschi, Lorenzo (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2015)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2015

Abstract

A key element in radiotherapy treatment planning is the dose optimization. Computer supported radiation treatment and diagnostic imaging technics are constantly evolving and demand for innovative algorithms to take advantage of this improvements. We give an extended overview about recent considered deterministic Boltzmann transport models used in dose calculation. Further we introduce optimal control problems governed by these transport equations, modelling physical objectives to obtain optimal treatment plans. We proof, that for an optimal boundary control problem governed by the Fokker-Planck equation in slab geometry the approach first discretize then optimize is equivalent to first optimize then discretize. Therefore we discretize the Fokker-Planck equation applying a PN approximation with Mark-boundary conditions. Further we analyse the asymptotic behavior of an optimal control problem governed by the relaxed Boltzmann equation that we derive from the Boltzmann Continuous Slowing Down approximation by applying a diffusive scaling. We proof, that the optimal control problem is asymptotic preserving, if the relaxed Boltzmann model is discretized by a globally stiffly accurate implicit explicit Runge-Kutta method of type A or ARS. Numerical tests in slab geometry and an order analysis verify the theoretical results. Finally, we present a new approach to discretize optimal control problems for radiative transport equations. The Reduced Velocity Method is exemplarily discussed on the example of the steady state Boltzmann equation. Here, we treat the velocity space as parameter space and discretize the steady state equation on the basis of a low and finite dimensional set of velocity grid points. The decisive aspect is the determination of these points by a Greedy algorithm, taking into account the optimal control problem in order to approximate the continuous optimal control.

Institutions

• Lehr- und Forschungsgebiet Mathematik [114620]
• Chair of Applied and Computational Mathematics [115010]
• Department of Mathematics [110000]