Approximation and Segment Count Reduction in Intensity Modulated Radiation Therapy

For a given mtimesn nonnegative real matrix A, a binary segmentation with 1-norm relative error e is a set of pairs (alpha,S)={(alpha₁, S₁), (alpha₂, S₂), ..., (alpha_k, S_k)}, where each alpha_i is a positive number and S_i is an mtimesn binary matrix, and e=|A-Sigma_i ^k=alpha_iS_i |1/|A|₁ where |A|₁ is the 1-norm of a vector which consists of exactly all the entries of the matrix A. Given A and positive scalars gamma, delta, we consider the optimization problem of finding a binary segmentation alphaS that minimizes z=Sigma_i ^k=alpha_i+gammak+deltae subject to certain constraints on S_i. This problem provides a major challenge in preparing a clinically acceptable treatment plan for intensity modulated radiation therapy (IMRT) and is known to be NP-hard. In this paper, we present an effective heuristic algorithm for the above problem with an additional constraint klesK, where K is a given positive integer. Reducing k subject to achieving an acceptable value of e is very desirable in both IMRT and intensity modulated arc therapy (IMAT). Our algorithm can be used as a valuable tool to find a clinically deliverable solution with a tradeoff between these two competing metrics