Minimum-monitor-unit optimization via a stochastic coordinate descent method

Objective: Deliverable proton spots are subject to the minimum monitor-unit (MMU) constraint. The MMU optimization problem with relatively large MMU threshold remains mathematically challenging due to its strong nonconvexity. However, the MMU optimization is fundamental to proton radiotherapy (RT), including efficient IMPT, proton arc delivery (ARC), and FLASH-RT. This work aims to develop a new optimization algorithm that is effective in solving the MMU problem. Approach: Our new algorithm is primarily based on stochastic coordinate decent (SCD) method. It involves three major steps: first to decouple the determination of active sets for dose-volume-histogram (DVH) planning constraints from the MMU problem via iterative convex relaxation method; second to handle the nonconvexity of the MMU constraint via SCD to localize the index set of nonzero spots; third to solve convex subproblems projected to this convex set of nonzero spots via projected gradient descent method. Main results: Our new method SCD is validated and compared with alternating direction method of multipliers (ADMM) for IMPT and ARC. The results suggest SCD had better plan quality than ADMM, e.g., the improvement of conformal index (CI) from 0.51 to 0.71 during IMPT, and from 0.22 to 0.86 during ARC for the lung case. Moreover, SCD successfully handled the nonconvexity from large MMU threshold that ADMM failed to handle, in the sense that (1) the plan quality from ARC was worse than IMPT (e.g., CI was 0.51 with IMPT and 0.22 with ARC for the lung case), when ADMM was used; (2) in contrast, with SCD, ARC achieved better plan quality than IMPT (e.g., CI was 0.71 with IMPT and 0.86 with ARC for the lung case), which is compatible with more optimization degrees of freedom from ARC compared to IMPT. Significance: To the best of our knowledge, our new MMU optimization method via SCD can effectively handle the nonconvexity from large MMU threshold that none of the current methods can solve. Therefore, we have developed a unique MMU optimization algorithm via SCD that can be used for efficient IMPT, proton arc delivery (ARC), FLASH-RT, and other particle RT applications where large MMU threshold is desirable (e.g., for the delivery of high dose rates or/and a large number of spots).