Energy Layer Optimization for IMPT Based on a Mixed-Integer Model and Variational Quantum Computing
Abstract
Purpose
Intensity-modulated proton therapy (IMPT) provides superior dose conformity and sparing of healthy tissues compared with photon radiotherapy. Improving delivery efficiency is clinically important for reducing motion-induced uncertainties, enhancing plan robustness, and improving patient comfort, particularly in breath-hold treatments. In pencil beam scanning systems, long energy switching times remain a major delivery bottleneck, making energy-layer optimization a critical strategy for accelerating IMPT delivery. To address this challenge, we propose a quantum computing–based optimization framework for IMPT energy-layer reduction, which integrates mixed-integer modeling with variational quantum algorithms to efficiently handle the associated combinatorial complexity.
Methods
The energy-layer optimization problem is formulated as a mixed-integer optimization model, where continuous variables represent beam intensities and binary variables indicate energy-layer selection. An iterative convex relaxation strategy is first employed to decouple dose–volume constraints. The resulting problem is solved using the alternating direction method of multipliers (ADMM), which separates beam intensity optimization under minimum monitor unit (MMU) constraints from discrete energy-layer selection. The MMU-constrained beam intensity subproblem either admits a closed-form solution or is efficiently solved using the conjugate gradient method. Meanwhile, the energy-layer selection subproblem is reformulated as a quadratic unconstrained binary optimization (QUBO) problem and solved using variational quantum algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA), enabling efficient exploration of the associated combinatorial search space.
Results
The proposed framework achieves plan quality comparable to conventional IMPT planning and CARD while substantially reducing the number of energy layers. For head-and-neck and lung cases, the number of energy layers is reduced to 35 from 61/41 and 56/40, respectively. This reduction leads to shorter beam delivery times, decreasing from 100.6 s and 232.0 s to 90.7 s and 215.5 s.
Conclusion
A quantum computing–based method for IMPT energy-layer optimization is presented, demonstrating improved delivery efficiency without compromising plan quality.