An energy optimization method based on mixed-integer model and variational quantum computing algorithm for faster IMPT

Intensity-modulated proton therapy (IMPT) offers superior dose conformity with reduced exposure to surrounding healthy tissues compared to conventional photon therapy. Improving IMPT delivery efficiency reduces motion-related uncertainties, enhances plan robustness, and benefits breath-hold techniques by shortening treatment time. Among various factors, energy switching time plays a critical role, making energy layer optimization (ELO) essential. This work develops an energy layer optimization method based on mixed integer model and variational quantum computing algorithm to enhance the efficiency of IMPT. The energy layer optimization problem is modeled as a mixed-integer program, where continuous variables optimize the dose distribution and binary variables indicate energy layer selection. To solve it, iterative convex relaxation decouples the dose-volume constraints, followed by the alternating direction method of multipliers (ADMM) to separate mixed-variable optimization and the minimum monitor unit (MMU) constraint. The resulting beam intensity subproblem, subject to MMU, either admits a closed-form solution or is efficiently solvable via conjugate gradient. The binary subproblem is cast as a quadratic unconstrained binary optimization (QUBO) problem, solvable using variational quantum computing algorithms. With nearly the same plan quality, the proposed method noticeable reduces the number of the used energies. For example, compared to conventional IMPT, QC can reduce the number of energy layers from 61 to 35 in HN case, from 56 to 35 in lung case, and from 59 to 32 to abdomen case. The reduced number of energies also results in fewer delivery time, e.g., the delivery time is reduced from 100.6, 232.0, 185.3 seconds to 90.7, 215.4, 154.0 seconds, respectively.