It has been recently shown that an additional therapeutic gain may be achieved if a radiotherapy plan is altered over the treatment course using a new treatment paradigm referred to in the literature as spatiotemporal fractionation. Because of the nonconvex and large-scale nature of the corresponding treatment plan optimization problem, the extent of the potential therapeutic gain that may be achieved from spatiotemporal fractionation has been investigated using stylized cancer cases to circumvent the arising computational challenges. This research aims at developing scalable optimization methods to obtain high-quality spatiotemporally fractionated plans with optimality bounds for clinical cancer cases. In particular, the treatment-planning problem is formulated as a quadratically constrained quadratic program and is solved to local optimality using a constraint-generation approach, in which each subproblem is solved using sequential linear/quadratic programming methods. To obtain optimality bounds, cutting-plane and column-generation methods are combined to solve the Lagrangian relaxation of the formulation. The performance of the developed methods are tested on deidentified clinical liver and prostate cancer cases. Results show that the proposed method is capable of achieving local-optimal spatiotemporally fractionated plans with an optimality gap of around 10%–12% for cancer cases tested in this study. Summary of Contribution: The design of spatiotemporally fractionated radiotherapy plans for clinical cancer cases gives rise to a class of nonconvex and large-scale quadratically constrained quadratic programming (QCQP) problems, the solution of which requires the development of efficient models and solution methods. To address the computational challenges posed by the large-scale and nonconvex nature of the problem, we employ large-scale optimization techniques to develop scalable solution methods that find local-optimal solutions along with optimality bounds. We test the performance of the proposed methods on deidentified clinical cancer cases. The proposed methods in this study can, in principle, be applied to solve other QCQP formulations, which commonly arise in several application domains, including graph theory, power systems, and signal processing.
On subspace properties of the quadratically constrained quadratic program
