A Tailored Branch-and-Bound Method for Optimizing the Dwelling Time Pattern and Catheter Positioning in HDR Brachytherapy

High dose-rate (HDR) brachytherapy is one type of treatment for prostate cancer, in which a radioactive source is moved through catheters implanted into the prostate. For each patient, a unique treatment plan is constructed. This plan determines for example the catheter positioning and the dwelling time pattern, that is, where and for how long the source should stop. Mathematical optimization methods are frequently used to find high-quality dwelling time patterns. However, choosing the catheter positioning is usually done without any aid of mathematical optimization methods. Researchers have recently suggested some optimization models for catheter positioning, and also heuristics for solving them. However, there are no available methods for finding the optimal solution of these models within a clinically acceptable time frame. In this paper we present the foundation for a branch-and-bound method that has been tailored to the catheter positioning problem. Tests show that this tailored branch-and-bound method has some promising features, for example that the dual bound is improved faster than when using a standard branch-and-bound method. But the tests also show that further research is required to develop it into a method that can find the optimal solution fast enough