A Quantum Computing Approach for Minimum Monitor Unit Optimization In Proton Pencil Beam Scanning
Abstract
Purpose
For proton spots to be deliverable, the intensity of each spot must either be zero or exceed a specific minimum monitor unit (MMU) threshold, which is a nonconvex problem. This study develops a quantum computing (QC)-based optimization framework to address the MMU constraint and balance delivery efficiency with plan quality.
Methods
The MMU optimization problem is formulated as a mixed-integer programming (MIP) model with binary variables for spot selection and continuous variables for intensity modulation. An iterative convex relaxation strategy is first applied to decouple dose–volume constraints. The resulting problem is solved using the alternating direction method of multipliers (ADMM), which separates the optimization of continuous and binary variables. The binary subproblem is further reformulated as a quadratic unconstrained binary optimization (QUBO) model and solved using the Quantum Approximate Optimization Algorithm (QAOA).
Results
The proposed QC-based method was evaluated on representative prostate and head-and-neck cases and compared with conventional intensity-modulated proton therapy (IMPT) and a classical MIP-based approach. Compared with standard IMPT, the QC method achieved comparable plan quality while substantially reducing the number of active spots, leading to a reduction in delivery time of up to 12.2%. When compared with the classical MIP approach using the same number of non-zero spots, the QC method showed superior plan quality, with improved target dose conformity and normal tissue sparing. In a head-and-neck case, the conform index for IMPT, QC, and classical MIP were 0.90, 0.90, and 0.58, respectively. Notably, the QC method achieved this performance using only 500 non-zero spots, compared with 2,442 spots for IMPT.
Conclusion
A novel QC-based framework for MMU optimization is presented, demonstrating the potential to substantially reduce delivery time while achieving plan quality comparable to or better than classical optimization approaches.