Mathematical Optimization of HDR Brachytherapy

Abstract: One out of eight deaths throughout the world is due to cancer. Developing new treatments and improving existing treatments is hence of major importance. In this thesis we have studied how mathematical optimization can be used to improve an existing treatment method: high-dose-rate (HDR) brachytherapy.HDR brachytherapy is a radiation modality used to treat tumours of for example the cervix, prostate, breasts, and skin. In HDR brachytherapy catheters are implanted into or close to the tumour volume. A radioactive source is moved through the catheters, and by adjusting where the catheters are placed, called catheter positioning, and how the source is moved through the catheters, called the dwelling time pattern, the dose distribution can be controlled.By constructing an individualized catheter positioning and dwelling time pattern, called dose plan, based on each patient's anatomy, it is possible to improve the treatment result. Mathematical optimization has during the last decade been used to aid in creating individualized dose plans. The dominating optimization model for this purpose is a linear penalty model. This model only considers the dwelling time pattern within already implanted catheters, and minimizes a weighted deviation from dose intervals prescribed by a physician.In this thesis we show that the distribution of the basic variables in the linear penalty model implies that only dwelling time patterns that have certain characteristics can be optimal. These characteristics cause troublesome inhomogeneities in the plans, and although various measures for mitigating these are already available, it is of fundamental interest to understand their cause.We have also shown that the relationship between the objective function of the linear penalty model and the measures commonly used for evaluating the quality of the dose distribution is weak. This implies that even if the model is solved to optimality there is no guarantee that the generated plan is optimal with respect to clinically relevant objectives, or even near-optimal. We have therefore constructed a new model for optimizing the dwelling time pattern. This model approximates the quality measures by the concept conditional value-at-risk, and we show that the relationship between our new model and the quality measures is strong. Furthermore, the new model generates dwelling time patterns that yield high-quality dose distributions.Combining optimization of the dwelling time pattern with optimization of the catheter positioning yields a problem for which it is rarely possible to find a proven optimal solution within a reasonable time frame. We have therefore developed a variable neighbourhood search heuristic that outperforms a state-of-the-art optimization software (CPLEX). We have also developed a tailored branch-and-bound algorithm that is better at improving the dual bound than a general branch-and-bound algorithm. This is a step towards the development of a method that can find proven optimal solutions to the combined problem within a reasonable time frame.