# Optimal Treatment Planning in Radiotherapy based on Boltzmann Transport Equations

### From DFG-SPP 1253

## Project leaders

MATHCCES

Department of Mathematics

RWTH Aachen University

phone: +49 (0)241 80 98661

email: frank@mathcces.rwth-aachen.de

homepage: www.mathcces.rwth-aachen.de/doku.php/staff/frank

Lehrstuhl C für Mathematik

RWTH Aachen

Tel:

Fax:

Email: herty@mathc.rwth-aachen.de

Homepage:

## DFG funded assistants

Richard Barnard (RWTH Aachen)

barnard@mathcces.rwth-aachen.de

## Description

We plan to investigate optimal control problems used in clinical external beam radiotherapy. The starting point is the formulation of the treatment planning problem as an optimal control problem for the Boltzmann Continuous Slowing Down model. Mathematically, the model consists of an integro--partial differential equation in energy, spatial and angular variables. Besides analytical investigations on existence and regularity of solutions and optimal controls we study moment approximations to the high--dimensional phase space in order to devise efficient numerical schemes. The relation between optimization and moment approximation will also be investigated. We use the analytical and numerical results on real $CT-$patient data and compare the optimal treatment plans obtain by solving the control problem with standard Monte--Carlo methods.

Besides surgery and chemotherapy, the use of ionizing radiation is one of the main tools in the therapy of cancer today. According to WHO data, in the year 2007 there were about 11.3 million new cancer cases. More than half of the patients that are treated receive radiation therapy at one point during their treatment. Since the early days of radiation treatment high energy photons have been the most important type of radiation. Other types of radiation include high energy electrons and heavy charged particles like protons and ions. The latter type of radiation is of growing importance, but has not reached the widespread use of photons and electrons, yet. The aim of radiation treatment is to deposit enough energy in cancer cells so that they are destroyed. On the other hand, healthy tissue around the cancer cells should be harmed as little as possible. Furthermore, some regions at risk, like the spinal chord, should receive almost no radiation at all.

It is still current practice that treatment plans involve several fixed beam directions which are selected by an experienced physician by hand. Radiation facilities where the beam head rotates around the patient and where the beam is shaped by multileaf collimators are entering clinical practice. These methods have become known as Intensity-Modulated Radiation Therapy (IMRT). Patient motion during treatment is also one of the future challenges in the field of external beam radiotherapy. For instance, tumors near the lung move due to breathing. Techniques addressing this problem have become known as 4D radiotherapy (4DRT) \cite{BucBevRoa05}, meaning that time, the fourth dimension, also has to be taken into account. A further technique, named Image-Guided Radiotherapy (IGRT) is currently being developed. In this method, the radiation is used to create patient images during treatment. All of these novel techniques require mathematical modeling and optimization techniques.

Before the treatment of the patient can be started, the expected dose distribution, i.e.\ the distribution of absorbed radiative energy in the patient, has to be calculated. Most dose calculation algorithms in clinical use rely on the Fermi--Eyges theory of radiation. In recent work \cite{KriSau05}, however, it has been shown that these can produce errors of up to 12\% near inhomogeneities.

This project is based on dose calculation using a Boltzmann transport equation. Similar to Monte Carlo simulations it relies on a rigorous model of the physical interactions in human tissue that can in principle be solved exactly. Monte Carlo simulations are widely used, but it has been argued that a grid-based Boltzmann solution should have the same computational complexity \cite{Bor98}. Furthermore, Monte Carlo can only be used in derivative--free methods for optimal dose distributions. In constrast, when optimizing using Boltzmann's equations it is possible to exploit structural information for numerical and analytical purposes of the optimization problem.