A Dual Algorithm to Obtain Highly Practical Solutions in Static Multileaf Collimation

Abstract

An intensity map x is a m× n matrix with nonnegative integral entries corresponding to beamlets. We will assume for this work that the MLC has m rows called channels, and n columns. One ”step” in a sequence can then be coded as a m×n 0-1 (read ”zero one”) shape matrix with entries corresponding to the entries in the intensity map. A 0 in the shape matrix codes that a leaf from the MLC blocks the region corresponding to the entry, and a 1 marks this region open for this step. The positive entries in a channel must be consecutive, as the opening is coherent. The k shape in a sequence is denoted by Sk and the index set of all allowable shape matrices is K. This set is influenced by hardware-specific constraints concerning the separation of leaves. In general, there is a minimum distance that must separate the leaf ends in one channel at all times (∆intra ≥ 0), and the distance from one leaf to an opposing leaf in a neighboring channel must be at least ∆inter . Every shape matrix has a corresponding opening time αk, called the monitor unit of this shape. The general sequencing problem is to express the intensity map as a sum of shapes and monitor units to keep the total treatment time as low as possible to minimize the stress endured by the patient. A major component of the treatment time is the sum of the monitor units, called the beam-on time (BOT). Boland et al [1] have formulated the problem to minimize the beamon time as a network flow problem with side constraints. We will only give the formal definition of a network structure GT = (VT , AT ) that is equivalent to the formulations in [1] and the problem statement here. For more details, refer to the works [1] and [2]. The set of nodes and the set of arcs are given by