A circular matrix-merging algorithm with application in Volumetric Intensity-Modulated Arc Therapy

In this paper, we study an optimization problem, called the circular matrix-merging (CMM) problem: Given a cyclic sequence of n non-negative matrices that are associated with some locations on a circle and an operation that merges two consecutive matrices into one matrix at the expense of some merging error, merge the n matrices into the minimum number of matrices under the constraint of a threshold of the total merging error. This problem arises in Volumetric Intensity-Modulated Arc Therapy (VMAT) for radiation cancer treatment, where the circular structure represents a delivery arc path of 360° around the patient and the matrices represent the radiation fluence maps at selected delivery angles on that path. By computing the minimum k matrices, the CMM algorithm produces a most efficient delivery plan that meets clinical requirements. Based on dynamic programming and computing a set of k-link shortest paths, we present a polynomial time algorithm for the CMM problem, improving the quality and efficiency of VMAT delivery.