Research Article Local Search Algorithms for the Beam Angles’ Selection Problem in Radiotherapy Guillermo Cabrera-Guerrero , 1 Nibaldo Rodriguez, 1 Carolina Lagos , 1 Enrique Cabrera, 2 and Franklin Johnson 3 1 Escuela de Ingenier´ıa Inform´ atica, Pontificia Universidad Cat´ olica de Valpara´ıso, Valparaiso, Chile 2 CIMFAV, Universidad de Valpara´ıso, Valparaiso, Chile 3 Departamento de Computaci´ on e Inform´ atica, Universidad de Playa Ancha, Valparaiso, Chile Correspondence should be addressed to Guillermo Cabrera-Guerrero; guillermo.cabrera@pucv.cl Received 7 December 2017; Accepted 28 March 2018; Published 6 May 2018 Academic Editor: Josefa Mula Copyright © 2018 Guillermo Cabrera-Guerrero et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. One important problem in radiation therapy for cancer treatment is the selection of the set of beam angles radiation will be delivered from. A primary goal in this problem is to find a beam angle configuration (BAC) that leads to a clinically acceptable treatment plan. Further, this process must be done within clinically acceptable times. Since the problem of selecting beam angles in radiation therapy is known to be extremely hard to solve as well as time-consuming, both exact algorithms and population-based heuristics might not be suitable to solve this problem. In this paper, we compare two matheuristic methods based on local search algorithms, to approximately solve the beam angle optimisation problem (BAO). Although the steepest descent algorithm is able to find locally optimal BACs for the BAO problem, it takes too long before convergence, which is not acceptable in clinical practice. us, we propose to use a next descent algorithm that converges quickly to good quality solutions although no (local) optimality guarantee is given. We apply our two matheuristic methods on a prostate case which considers two organs at risk, namely, the rectum and the bladder. Results show that the matheuristic algorithm based on the next descent local search is able to quickly find solutions as good as the ones found by the steepest descent algorithm. 1. Introduction Radiation is one of the most common therapies used to treat patients suffering from cancer. e purpose of radiation therapy is to deliver a dose of radiation to a tumour in order to sterilize all cancer cells and to minimize the collateral effects on the surrounding healthy organs and tissues. Intensity modulated radiation therapy (IMRT) is the most common technique within radiation therapy. We can separate IMRT problem into three sequen- tial optimisation (sub-)problems: the beam angle optimisa- tion (BAO) problem, the fluence map optimisation (FMO) problem, and the multileaf collimator sequencing problem, Ehrgott et al. [1]. In the BAO problem, we determine the num- ber and directions of the beam angles we shall use to produce a treatment plan. e set of beams used to treat a patient is called beam angle configuration (BAC). en, in the FMO problem, we determine the radiation intensities that will be delivered from each beam angle. e solution to this problem is a vector of intensities that is called fluence map. Finally, a sequence of movements of a physical device called multileaf collimator is computed in order to efficiently deliver the fluence map computed during the previous phases. It is clear from here that the selection of beam angles in the BAO phase has a big impact on the quality of the fluence map computed in the FMO phase. at is, a good combination of beam angles will lead to a good quality fluence map and, consequently, will produce a good quality treatment plan. In this paper the problem of selecting a good quality BAC is addressed. To measure the quality of a BAC we need to solve the associated FMO, that is, we solve the FMO problem for each evaluated BAC. Computing the optimal fluence map for a Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4978703, 9 pages https://doi.org/10.1155/2018/4978703