Lexicographic extension of the reference point method applied in radiation therapy treatment planning

In radiation therapy treatment planning, generating a treatment plan is a multi-objective optimisation problem. The decision-making strategy is uniform for each group of cancer patients, e.g. prostate cancer, and can thus be automated. Predefined priorities and aspiration levels are assigned to each objective, and the strategy is to attain these levels in order of priority. Therefore, a straightforward lexicographic approach is sequential ϵ-constraint programming where objectives are sequentially optimised and constrained according to predefined rules, mimicking human decision-making. The clinically applied 2-phase ϵ-constraint (2pϵc) method captures this approach and generates clinically acceptable treatment plans. However, the number of optimisation problems to be solved for the 2pϵc method, and hence the computation time, scales linearly with the number of objectives. To improve the daily planning workload and to further enhance radiation therapy, it is extremely important to reduce this time. Therefore, we developed the lexicographic reference point method (LRPM), a lexicographic extension of the reference point method, for generating a treatment plan by solving a single optimisation problem. The LRPM processes multiple a priori defined reference points into modified partial achievement functions. In addition, a priori bounds on a subset of the partial trade-offs can be imposed using a weighted sum component. The LRPM was validated for 30 randomly selected prostate cancer patients. While the treatment plans generated using the LRPM were of similar clinical quality to those generated using the 2pϵc method, the LRPM decreased the average computation time from 12.4 to 1.2 minutes, a speed-up factor of 10.