Trade-off bounds for the Pareto surface approximation in multi-criteria IMRT planning

One approach to multi-criteria IMRT planning is to automatically calculate a data set of Pareto-optimal plans for a given planning problem in a first phase, and then interactively explore the solution space and decide on the clinically best treatment plan in a second phase. The challenge of computing the plan data set is to ensure that all clinically meaningful plans are covered and that as many clinically irrelevant plans as possible are excluded to keep computation times within reasonable limits. In this work, we focus on the approximation of the clinically relevant part of the Pareto surface, the process that constitutes the first phase. It is possible that two plans on the Pareto surface have a small, clinically insignificant difference in one criterion and a significant difference in another criterion. For such cases, only the plan that is clinically clearly superior should be included into the data set. To achieve this during the Pareto surface approximation, we propose to introduce bounds that restrict the relative quality between plans, the so-called trade-off bounds. We show how to integrate these trade-off bounds into the approximation scheme and study their effects. The proposed scheme is applied to two artificial cases and one clinical case of a paraspinal tumor. For all cases, the quality of the Pareto surface approximation is measured with respect to the number of computed plans, and the range of values occurring in the approximation for different criteria is compared. Through enforcing trade-off bounds, the scheme disregards clinically irrelevant plans during the approximation. Thereby, the number of plans necessary to achieve a good approximation quality can be significantly reduced. Thus, trade-off bounds are an effective tool to focus the planning and to reduce computation time.