  Sector duration optimization (SDO) is a problem arising in treatment planning
for stereotactic radiosurgery on Gamma Knife. Given a set of isocenter
locations, SDO aims to select collimator size configurations and irradiation
times thereof such that target tissues receive prescribed doses in a reasonable
amount of treatment time, while healthy tissues nearby are spared. We present a
multiobjective linear programming model for SDO with a state-of-the-art
delivery system, Leksell Gamma Knife Icon, to generate a diverse collection of
solutions so that one that best suits patient-specific needs can be chosen by
clinicians. We develop a generic two-phase solution strategy based on the
epsilon-constraint method for solving multiobjective optimization models, which
aims to systematically increase the number of high-quality solutions obtained,
instead of conducting a traditional uniform search. To improve solution quality
further and accelerate the procedure, we incorporate some general and
problem-specific enhancements. Moreover, we propose an alternative version of
our two-phase strategy, which makes use of machine learning tools to devote the
computational effort rather to solving epsilon-constraint models that are
predicted to yield clinically desirable solutions. In our computational study
on eight previously treated real test cases, a significant portion of obtained
solutions outperforms clinical results and those from a single-objective model
from the literature. In addition to significant benefits of the algorithmic
enhancements, our experiments illustrate the usefulness of machine learning
strategies, reducing the overall run times nearly by half while still capturing
a sufficient selection of clinically desirable solutions.
