### Mathematisch-Naturwissenschaftliche Fakultät

## Institut für Mathematik

### Fachgebiet: Optimierung

Betreuer: Prof. Dr. Konrad Engel

**Dipl.Math.** **Antje** **Kiesel**

(e-mail: antje.kiesel@uni-rostock.de )

*Approximated multileaf collimator field segmentation*

The dissertation deals with optimization problems arising in radiation therapy. Prescribed fluence distributions have to be realized as good as possible by a superposition of several differently shaped fields. Multileaf collimators are used for field shaping. This task can be modeled as discrete optimization problem, where given nonnegative integer matrices have to be decomposed into a nonnegative integer linear combination of certain 0-1-matrices (segments). Depending on the technical and dosimetric constraints one wants to consider, exact or approximate decomposition problems arise. These problems are solved using methods from combinatorial and discrete optimization as well as graph theory. Furthermore, in the second part of the thesis, a continuous fluence model is introduced, where the target fluences are real-valued nonnegative functions of two variables. The aim of this approach is the improved modeling of the characteristics of radiation, especially penumbra effects at the border of the radiation fields. A quadratic optimization problem is deduced. Whereas parts of the thesis are just of mathematical interest, other parts explicitly deal with clinically applicable algorithms. These are tested for a clinical case and the numerical results are displayed.