A treatment plan optimization method with direct minimization of number of energy jumps for proton arc therapy

Objective. The optimization of energy layer distributions is crucial to proton arc therapy: on one hand, a sufficient number of energy layers is needed to ensure the plan quality; on the other hand, an excess number of energy jumps (NEJ) can substantially slow down the treatment delivery. This work will develop a new treatment plan optimization method with direct minimization of (NEJ), which will be shown to outperform state-of-the-art methods in both plan quality and delivery efficiency.Approach. The proposed method jointly optimizes the plan quality and minimizes the NEJ. To minimize NEJ, (1) the proton spotsxis summed per energy layer to form the energy vectory; (2)yis binarized via sigmoid transform intoy; (3)y₁is multiplied with a predefined energy order vector via dot product intoy₂; (4)y₂is filtered through the finite-differencing kernel intoy₃in order to identify NEJ; (5) only the NEJ ofy₃is penalized, whilexis optimized for plan quality. The solution algorithm to this new method is based on iterative convex relaxation.Main results. The new method is validated in comparison with state-of-the-art methods called energy sequencing (ES) method and energy matrix (EM) method. In terms of delivery efficiency, the new method had fewer NEJ, less energy switching time, and generally less total delivery time. In terms of plan quality, the new method had smaller optimization objective values, lower normal tissue dose, and generally better target coverage.Significance. We have developed a new treatment plan optimization method with direct minimization of NEJ, and demonstrated that this new method outperformed state-of-the-art methods (ES and EM) in both plan quality and delivery efficiency.