In proton therapy treatment planning, the aim is to ensure tumor control while sparing the various surrounding risk structures. The biological effect of the irradiation depends on both physical dose and linear energy transfer (LET). In order to include LET alongside physical dose in plan creation, we propose to formulate the proton treatment planning problem as a particularly structured multi-criteria bi-level optimization problem, which we call hierarchical.
We show that the hierarchical multi-criteria bi-level problem can be reduced to a standard multi-criteria optimization (MCO) problem employing a specific domination cone. As the unfavorable properties of this cone prohibit a direct application of standard MCO solution methods, we further illustrate how a more convenient approximate cone can be constructed. Based on the found reduction to a standard MCO problem, we then describe a novel approach to calculate a Pareto front representation for the hierarchical multi-criteria bi-level problem. As a point of reference, we also discuss a second, more brute-force approach. We apply both approaches to a prostate and a head and neck case, and compare the results.