Discrete approximations to real-valued leaf sequencing problems in radiation therapy

For a given mxn nonnegative real matrix A, a segmentation with 1-norm relative error e is a set of pairs (@a,S)={(@a"1,S"1),(@a"2,S"2),...,(@a"k,S"k)}, where each @a"i is a positive number and S"i is an mxn binary matrix, and e=|A-@?"i"="1^k@a"iS"i|"1/|A|"1, where |A|"1 is the 1-norm of a vector which consists of all the entries of the matrix A. In certain radiation therapy applications, given A and positive scalars @c,@d, we consider the optimization problem of finding a segmentation (@a,S) that minimizes z=@?"i"="1^k@a"i+@ck+@de subject to certain constraints on S"i. This problem poses a major challenge in preparing a clinically acceptable treatment plan for Intensity Modulated Radiation Therapy (IMRT) and is known to be NP-hard. Known discrete IMRT algorithms use alternative objectives for this problem and an L-level entrywise approximation A@? (i.e. each entry in A is approximated by the closest entry in a set of L equally-spaced integers), and produce a segmentation that satisfies A@?=@?"i"="1^k@a@?"iS"i. In this paper we present two algorithms that focus on the original non-discretized intensity matrix and consider measures of delivery quality and complexity (@?@a"i+@ck) as well as approximation error e. The first algorithm uses a set partitioning approach to approximate A by a matrix A@? that leads to segmentations with smaller k for a given e. The second algorithm uses a constrained least square approach to post-process a segmentation {(@a@?"i,S"i)} of A@? to replace @a@?"i with real-valued @a"i in order to reduce k and e.