BACKGROUND: While minimizing plan delivery time is beneficial for proton therapy in terms of motion management, patient comfort, and treatment throughput, it often poses a tradeoff with optimizing plan quality. A key component of plan delivery time is the energy switching time, which is approximately proportional to the number of energy layers, that is, the cardinality.
OBJECTIVE: This work aims to develop a novel optimization method that can efficiently compute the pareto surface between plan quality and energy layer cardinality, for the planner to navigate through this quality-and-efficiency tradeoff and select the appropriate plan of a balanced tradeoff.
METHODS: A new IMPT method CARD is proposed that (1) explicitly incorporates the minimization of energy layer cardinality as an optimization objective, and (2) automatically generates a set of plans sequentially with a descending order in number of energy layers. The energy layer cardinality is penalized through the l1,0-norm regularization with an upper bound, and the upper bound is monotonically decreased to compute a series of treatment plans with gradually decreased energy layer cardinality on the quality-and-efficiency pareto surface. For any given treatment plan, the plan optimality is enforced using dose-volume planning objectives and the plan deliverability is imposed through minimum-monitor-unit (MMU) constraints, with optimization solution algorithm based on iterative convex relaxation.
RESULTS: The new method CARD was validated in comparison with the benchmark plan of all energy layers (P0), and a state-of-the-art method called MMSEL, using prostate, head-and-neck (HN), lung, pancreas, liver and brain cases. While labor-intensive and time-consuming manual parameter tuning was needed for MMSEL to generate plans of predefined energy layer cardinality, CARD automatically and efficiently computed all plans with sequentially decreasing predefined energy layer cardinality all at once. With the acceptable plan quality (i.e., no more than 110% of total optimization objective value from P0), CARD achieved the reduction of number of energy layers to 52% (from 77 to 40), 48% (from 135 to 65), 59% (from 85 to 50), 67% (from 52 to 35), 80% (from 50 to 40), and 30% (from 66 to 20), for prostate, HN, lung, pancreas, liver, and brain cases, respectively, compared to P0, with overall better plan quality than MMSEL. Moreover, due to the nonconvexity of the MMU constraint, CARD provided the similar or even smaller optimization objective than P0, at the same time with fewer number of energy layers, that is, 55 versus 77, 85 versus 135, 45 versus 52, and 25 versus 66 for prostate, HN, pancreas, and brain cases, respectively.
CONCLUSIONS: We have developed a novel optimization algorithm CARD that can efficiently and automatically compute a series of treatment plans of any given energy layer sequentially, which allows the planner to navigate through the plan-quality and energy-layer-cardinality tradeoff and select the appropriate plan of a balanced tradeoff.

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 intoy1; (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.

OBJECTIVE: Spot-scanning arc therapy (SPArc) is an emerging proton modality that can potentially offer a combination of advantages in plan quality and delivery efficiency, compared with traditional IMPT of a few beam angles. Unlike IMPT, frequent low-to-high energy layer switching (so called switch-up (SU)) can degrade delivery efficiency for SPArc. However, it is a tradeoff between the minimization of SU times and the optimization of plan quality. This work will consider the energy layer optimization (ELO) problem for SPArc and develop a new ELO method via energy matrix (EM) regularization to improve plan quality and delivery efficiency.
METHODS: The major innovation of EM method for ELO is to design an EM that encourages desirable energy-layer map with minimal SU during SPArc, and then incorporate this EM into the SPArc treatment planning to simultaneously minimize the number of SU and optimize plan quality. The EM method is solved by the fast iterative shrinkage-thresholding algorithm and validated in comparison with a state-of-the-art method, so-called energy sequencing (ES).
RESULTS: EM is validated and compared with ES using representative clinical cases. In terms of delivery efficiency, EM had fewer SU than ES with an average of 35% reduction of SU. In terms of plan quality, compared with ES, EM had smaller optimization objective values and better target dose conformality, and generally lower dose to organs-at-risk and lower integral dose to body. In terms of computational efficiency, EM was substantially more efficient than ES by at least 10-fold.
CONCLUSIONS: We have developed a new ELO method for SPArc using EM regularization and shown that this new method EM can improve both delivery efficiency and plan quality, with substantially reduced computational time, compared with ES.

OBJECTIVE: When treating lung cancer patients with intensity-modulated proton therapy (IMPT), target coverage can only be guaranteed when utilizing motion mitigation. The three motion mitigation techniques, gating, breath-hold, and dose repainting, all benefit from a more rapid application of the treatment plan. A lower limit for the ungated treatment time is defined by the number of energy layers in the IMPT plan. By limiting this number during treatment planning, IMPT could become more viable for lung cancer patients. We investigate to what extend the number of layers can be reduced in single-field optimization (SFO) and multifield optimization (MFO) plans and which implications it has on the plan quality and robustness.
METHODS: We have implemented three distinct layer-reducing strategies in the treatment planning system Hyperion; constant energy steps, exponential energy steps, and an adaptive strategy, where the spot weights are exposed to a group sparsity penalty in combination with layer exclusion during optimization. Four levels of increasing layer removal are planned for each strategy. SFO and MFO plans with three treatment fields are created for eleven locally advanced NSCLC patients on the midventilation 4DCT phase to simulate a breath-hold. A minimum dose to the target is ensured for each degree of layer reduction, reflecting the plan quality in the homogeneity index (HI). Plan quality was also assessed by a robustness evaluation, where the patient setup was shifted 2 mm or 4 mm in six directions.
RESULTS: The three strategies result in very similar target coverages and robustness levels as a function of removed layers. The HI increases unacceptably for all the SFO plans after 50% layer removal as compared to the reference plan, while all the MFO plans are clinically acceptable with up to a highest removed percentage of 75%. The robustness level is constant as a function of removed layers. The SFO plans are significantly more robust than the MFO plans with all P-values below 0.001 (Wilcoxon signed-rank). The overall mean D98% CTV dose difference is at 2-mm setup error amplitude: 0.7 Gy (SFO) and 1.9 Gy (MFO), and at 4 mm: 3.2 Gy (SFO) and 5.4 Gy (MFO), respectively.
CONCLUSIONS: The number of layers in MFO plans can be reduced substantially more than in SFO plans without compromising plan quality. Furthermore, as the robustness is independent of the number of layers, it follows that if the level of robustness is acceptable or enforced via robust optimization, MFO plans could be candidates for treatment time reductions via energy layer reductions.