Mathematical Problems in Engineering

Volume 2018, Article ID 4978703, 9 pages

https://doi.org/10.1155/2018/4978703

## Local Search Algorithms for the Beam Angles’ Selection Problem in Radiotherapy

^{1}Escuela de Ingeniería Informática, Pontificia Universidad Católica de Valparaíso, Valparaiso, Chile^{2}CIMFAV, Universidad de Valparaíso, Valparaiso, Chile^{3}Departamento de Computación e Informática, Universidad de Playa Ancha, Valparaiso, Chile

Correspondence should be addressed to Guillermo Cabrera-Guerrero; lc.vcup@arerbac.omrelliug

Received 7 December 2017; Accepted 28 March 2018; Published 6 May 2018

Academic Editor: Josefa Mula

Copyright © 2018 Guillermo Cabrera-Guerrero et al. This 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.

#### Abstract

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. Thus, 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. The 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 sequential optimisation (sub-)problems: the beam angle optimisation (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 number and directions of the beam angles we shall use to produce a treatment plan. The set of beams used to treat a patient is called beam angle configuration (BAC). Then, in the FMO problem, we determine the radiation intensities that will be delivered from each beam angle. The 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. That 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 BAC is a time-consuming process. Further, one practical constraint when solving the BAO problem is that we need to obtain results within clinically acceptable times. In our experience, “clinically acceptable times” are around 12 hours (i.e., running algorithms overnight) so treatment planners can decide among alternative treatment plans during the day after patients images have been obtained. Thus, we need to find efficient strategies that produce good quality treatment plans within these time limits; that is, not too many BACs can be evaluated during the optimisation process. For this reason, sophisticated (meta-)heuristic algorithms such as population-based algorithms might not be suitable for solving this problem.

The aim of this paper is to compare two different local search strategies to capture the trade-off between solutions quality and the required time before convergence. Using the next descent algorithm, we aim to accelerate the search without any major impairment to treatment plans’ quality.

The remainder of this paper is organised as follows. In Section 2 we introduce the BAO problem we aim to solve in this paper and a brief review on different approaches dealing with the BAO problem is presented. In Section 3 both the steepest descent and the next descent algorithms are presented and their main differences are highlighted. In Section 4 the set of instances considered in our experiments are presented. Obtained results are also discussed in this section. Finally, in Section 5 some conclusions are drawn and the future work is outlined.

#### 2. Intensity Modulated Radiation Therapy: An Overview

During the last three decades or so, many researchers have worked on the problem of finding efficient treatment plans for radiation therapy for cancer treatment. Most of their efforts have been focused on the problem of finding the optimal fluence map that can be delivered to the patients given a predefined BAC (see, e.g., [2–5]). Unfortunately, significantly less attention has been paid to the problem of finding the best BAC. This is, in part, because the BAO problem is, from a mathematical point of view, much harder to solve than the FMO problem. Further, treatment planners rely on their experience to choose BACs that are good enough to produce clinically acceptable treatment plans. Figure 1 shows the traditional try-and-error process followed by treatment planners in clinical practice.