Combinatorial optimisation in radiotherapy treatment planning

The goal of radiotherapy is to cover a target area with a desired radiation dose while keeping the exposition of non-target areas as low as possible in order to reduce radiation side effects. In the case of Intensity Modulated Proton Therapy (IMPT), the dose distribution is typically designed via a treatment planning optimisation process based on classical optimisation algorithms on some objective functions. We investigate the planning optimisation problem under the point of view of the Theory of Complexity in general and, in particular, of the Combinatorial Optimisation Theory. We firstly give a formal definition of a simplified version of the problem that is in the complexity class NPO. We prove that above version is computationally hard, i.e. it belongs to the class NPO\PTAS if NP not equal P. We show how Combinatorial Optimisation Theory can give valuable tools, both conceptual and practical, in treatment plan definition, opening the way for new deterministic algorithms with bounded time complexity which have to support the technological evolution up to adaptive plans exploiting near real time solutions.