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

For a given image nonnegative real matrix image, a segmentation with 1-norm relative error image is a set of pairs image, where each image is a positive number and image is an image binary matrix, and image, where image is the 1-norm of a vector which consists of all the entries of the matrix image. In certain radiation therapy applications, given image and positive scalars image, we consider the optimization problem of finding a segmentation image that minimizes image subject to certain constraints on image. 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 image-level entrywise approximation image (i.e. each entry in image is approximated by the closest entry in a set of image equally-spaced integers), and produce a segmentation that satisfies image. In this paper we present two algorithms that focus on the original non-discretized intensity matrix and consider measures of delivery quality and complexity image as well as approximation error image. The first algorithm uses a set partitioning approach to approximate image by a matrix image that leads to segmentations with smaller image for a given image. The second algorithm uses a constrained least square approach to post-process a segmentation image of image to replace image with real-valued image in order to reduce image and image.