# Optimizations of Approximation Models for solving Kinetic Equations and their Applications

## Motivation

Radiation therapy is commonly applied to the cancerous tumor because of its ability to control cell growth. In radiation therapy treatment planning, the aim is to deposit enough energy in cancer cells in order to destroy them while at the same time healthy tissue should not be harmed. Ionizing radiation works by damaging the DNA of exposed tissue leading to cellular death. To spare normal tissues (such as skin or organs which radiation must pass through to treat the tumor), shaped radiation beams are aimed from several angles of exposure to intersect at the tumor, providing a much larger absorbed dose there than in the surrounding, healthy tissue.

Recently several algorithm and models have been developed to control the radiation beams with high energy particles. Many of these models are based on the numerical solution of the Boltzmann Transport Equations. These models are still inecient and heavy mech run when used with high spatial density of nodes with the objective of solving processes with small spatial scales.

## Introduction

A model for high energy particles such as photons, electrons and positrons traveling in tissues and interacting with tissues atoms via several types of interactions such as Coulomb interactions, bremsstrahlung and pair production has been recently introduced and used for optimal treatment planning [2]. The arising equations are Boltzmann type equations and it has been argued that a gridbased solution should have the same computational complexity as MonteCarlo simulations. However, an approach based on kinetic equations has the advantage of exploiting structural information during optimization.

This project is devoted to derive ecient computational methods for the high dimensional Boltzmann type equation for use in radiotherapy, which enable dierent approaches based on reduced basis and proper orthogonal decomposition will be employed to break the computational complexity.

## Research Plan

The development of ecient (in terms of computational time and memory requirements) gridbased numerical schemes for the resolution of transport equations in [2] and the associated optimal control problem by used reduced order models is the focus of this project. The work should include at least some of the following topic:

- The structured review of proper orthogonal decomposition POD) and reduced order models (ROM) for kinetic equations. Proper Orthogonal Decomposition (POD), which finds applications in computa tionally processing large amounts of high-dimensional data with the aim of obtaining low-dimensional descriptions that capture much of the phenomena of interest.
- The application and implementation of reduced order models for the equation of radiotherapy [2] and their validation by MonteCarlo methods.
- Extension of the results for the forward equation [2] to the optimal control problem.
- Various approximations (spherical harmonics, moment methods) exist for transport equation [2]. Review the existing results and approximation properties. Then, transfer the method developed in (2) to the spherical harmonics approximation of the transport equation and the M1 moment model for the transport equation [2].
- The results of (4) should be applied to the approximation of the optimal control problem. In particular, we again try to treat the spherical harmonics approxi- mation and at least the M1 moment model.

## References

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- Frank M., M. Herty, and M. Schäfer, Optimal treatment planning in radiotherapy based on Boltzmann transport calculations, Math. Models Methods Appl. Sci., 18 (2008), pp. 573592.
- Herty M., R. Pinnau, and G. Thömmes, Asymptotic and discrete concepts for optimal control in radiative transfer, ZAMM Z. Angew. Math. Mech., 87 (2007), pp. 333347.
- Shepard D. M., M. C. Ferris, G. H. Olivera, and T. R. Mackie, Optimizing the delivery of radiation therapy to cancer patients, SIAM Rev., 41 (1999), pp. 721744.
- J. Tervo, M. Vauhkonen, and E. Boman, Optimal control model for radiation therapy inverse planning applying the Boltzmann transport equation, Linear Algebra Appl., 428 (2008), pp. 12301249.
- Benjamin seibold and Martin Frank, StaRMAP - a sechond order staggered grid method for spherical harmonics moment equtions of radiative transfer, 2012
- Volkwein S., Model Reduction using Proper Orthogonal Decomposition, Lecture Notes, 2012