Including robustness in multi-criteria optimization for intensity-modulated proton therapy.

We present a method to include robustness in a multi-criteria optimization (MCO) framework for intensity-modulated proton therapy (IMPT). The approach allows one to simultaneously explore the trade-off between different objectives as well as the trade-off between robustness and nominal plan quality. In MCO, a database of plans each emphasizing different treatment planning objectives, is pre-computed to approximate the Pareto surface. An IMPT treatment plan that strikes the best balance between the different objectives can be selected by navigating on the Pareto surface. In our approach, robustness is integrated into MCO by adding robustified objectives and constraints to the MCO problem. Uncertainties (or errors) of the robust problem are modeled by pre-calculated dose-influence matrices for a nominal scenario and a number of pre-defined error scenarios (shifted patient positions, proton beam undershoot and overshoot). Objectives and constraints can be defined for the nominal scenario, thus characterizing nominal plan quality. A robustified objective represents the worst objective function value that can be realized for any of the error scenarios and thus provides a measure of plan robustness. The optimization method is based on a linear projection solver and is capable of handling large problem sizes resulting from a fine dose grid resolution, many scenarios, and a large number of proton pencil beams. A base-of-skull case is used to demonstrate the robust optimization method. It is demonstrated that the robust optimization method reduces the sensitivity of the treatment plan to setup and range errors to a degree that is not achieved by a safety margin approach. A chordoma case is analyzed in more detail to demonstrate the involved trade-offs between target underdose and brainstem sparing as well as robustness and nominal plan quality. The latter illustrates the advantage of MCO in the context of robust planning. For all cases examined, the robust optimization for each Pareto optimal plan takes less than 5 min on a standard computer, making a computationally friendly interface possible to the planner. In conclusion, the uncertainty pertinent to the IMPT procedure can be reduced during treatment planning by optimizing plans that emphasize different treatment objectives, including robustness, and then interactively seeking for a most-preferred one from the solution Pareto surface.