A new penalty/stochastic approach to an application of the covering problem: The gamma knife treatment

¹ COPPE – PESC, Federal University of Rio de Janeiro, Cidade Universitária, Centro de Tecnologia, Bloco H, 21941-972 Rio de Janeiro, RJ, Brazil.
² Colégio Pedro II, Campo de São Cristóvão, 177, São Cristóvão, 20921-903 Rio de Janeiro, RJ, Brazil.
³ CEFET/RJ – Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Av. Maracanã, 229, Maracanã, 20271-110 Rio de Janeiro, RJ, Brazil.
⁴ Universidade Federal Rural do Rio de Janeiro, BR-465, Km 7, 23897-000 Seropédica, RJ, Brazil.
⁵ LIX, École Polytechnique, 91128 Palaiseau, France.

^* Corresponding author: heldermv@cos.ufrj.br, helder.venceslau@cefet-rj.br

Received: 2 January 2018
Accepted: 19 February 2020

Abstract

The covering problem of a three dimensional body using different radii spheres is considered. The motivating application – the treatment planning of Gamma Knife radiosurgery – is briefly discussed. We approach the problem only by the geometric covering point of view, that is, given a set of spheres and a body, the objective is to cover the body using the smallest possible number of spheres, regardless of the dosage issue. In order to solve this mathematical programming problem, we consider an approach based on the application of penalty and stochastic local search techniques. Finally, some illustrative results and comparisons are presented.