Combinatorial Benders cuts for decomposing IMRT fluence maps using rectangular apertures

We consider the problem of decomposing Intensity Modulated Radiation Therapy (IMRT) fluence maps using rectangular apertures. A fluence map can be represented as an integer matrix, which denotes the intensity profile to be delivered to a patient through a given beam angle. We consider IMRT treatment machinery that can form rectangular apertures using conventional jaws, and hence, do not need sophisticated multi-leaf collimator (MLC) devices. The number of apertures used to deliver the fluence map needs to be minimized in order to treat the patient efficiently. From a mathematical point of view, the problem is equivalent to a minimum cardinality matrix decomposition problem. We propose a combinatorial Benders decomposition approach to solve this problem to optimality. We demonstrate the efficacy of our approach on a set of test instances derived from actual clinical data. We also compare our results with the literature and solutions obtained by solving a mixed-integer programming formulation of the problem.