Minimizing Setup and Beam-On Times in Radiation Therapy

Abstract

Radiation therapy is one of the commonly used cancer therapies. The radiation treatment poses a tuning problem: it needs to be effective enough to destroy the tumor, but it should maintain the functionality of the organs close to the tumor. Towards this goal the design of a radiation treatment has to be customized for each patient. Part of this design are intensity matrices that define the radiation dosage in a discretization of the beam head. To minimize the treatment time of a patient the beam-on time and the setup time need to be minimized. For a given row of the intensity matrix, the minimum beam-on time is equivalent to the minimum number of binary vectors with the consecutive "1"s property that sum to this row, and the minimum setup time is equivalent to the minimum number of distinct vectors in a set of binary vectors with the consecutive "1"s property that sum to this row. We give a simple linear time algorithm to compute the minimum beam-on time. We prove that the minimum setup time problem is APX-hard and give approximation algorithms for it using a duality property. For the general case, we give a 24 13 approximation algorithm. For unimodal rows, we give a 9 7 approximation algorithm. We also consider other variants for which better approximation ratios exist.

Approximation algorithms for minimizing segments in radiation therapy

“…The literature contains several leaf-sequencing algorithms which attempt to reduce the number segments through the use of various heuristic techniques [11,6,7,10]. Bansal et al [2] show that the single row version of the problem is APX-complete. They provide a 24/13-approximation algorithm for the single row problem and give some better approximations for more constrained versions.…”

Nonnegative integral subset representations of integer sets

“…replace the second appearance with 2t. Somewhat similar optimization problems arising from radiation therapy are considered in [1,3,4]. The case in which the elements of T can be negative or fractional is considered in [2,5,6].…”

Faster Optimal Algorithms for Segment Minimization with Small Maximal Value

“…The segmentation problem is known to be NP-complete in the strong sense, even for a single row [9,2,3], as well as APX-complete [4]. Bansal et al [4] provide a ¾ ½¿-approximation algorithm for the single-row problem and give better approximations for more constrained versions.…”

“…Bansal et al [4] provide a ¾ ½¿-approximation algorithm for the single-row problem and give better approximations for more constrained versions. Work by Collins et al [10] shows that the single-column version of the problem is NP-complete and provides some nontrivial lower bounds given certain constraints.…”

“…4 As the segmentation problem focuses on the time incurred for establishing leaf positions, optimizing the beam-on-time also has implications for making procedures more efficient by reducing the time spent administering the treatment corresponding to the segments themselves.…”