Cardinality‐constrained plan‐quality and delivery‐time optimization method for proton therapy
Authors: Lin Bowen , Li Yuliang , Liu Bin , Fu Shujun , Lin Yuting , Gao Hao 
Published: June 2024

AbstractBackgroundWhile 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.PurposeThis 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.MethodsA 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.ResultsThe 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.ConclusionsWe 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.