On the Global Optimization of the Beam Angle Optimization Problem in Intensity-Modulated Radiation Therapy

The beam angle optimization (BAO) problem remains an important and challenging problem in intensitymodulated radiation therapy (IMRT) treatment planning. BAO consists on the selection of appropriate radiation incidence directions and may influence the quality of the IMRT plans, both to enhance organs sparing and to improve tumor coverage. This is a very difficult global optimization problem since it is a highly non-convex continuous optimization problem with many local minima.Many conventional BAO approaches are based on single-beam metrics to solve a relaxed combinatorial formulation of the BAO problem. Typically, the quality of the solutions obtained is not simply related to the final value of an objective function but rather judged by dose-volume histograms or considering a set of physical dose metrics. For that reason, and also due to the fact that the global optimum value is unknown, it is difficult to perceive, in medical physics point of view, how good a solution is or how much could it be improved. In a mathematical point of view, it is difficult to acknowledge how far a solution is from the global optimum. The objective of this paper is to present the difficulties in obtaining near global optimumsolutions for the BAO problem, particularly when using single-beam approaches considering discrete subsets of all possible beam angles. The benefits of using a derivative-free approach for a continuous formulation of the BAO problem are discussed using a retrospective treated case of head-and-neck tumor at the Portuguese Institute of Oncology of Coimbra.