Robust optimization in IMPT using quadratic objective functions to account for the minimum MU constraint

Purpose: Currently, in clinical practice of intensity-modulated proton therapy (IMPT), the influence of the minimum monitor unit (MU) constraint is taken into account through postprocessing after the optimization is completed. This may degrade the plan quality and plan robustness. This study aims to mitigate the impact of the minimum MU constraint directly during the plan robust optimization. Methods and materials: Cao et al. have demonstrated a two-stage method to account for the minimum MU constraint using linear programming without the impact of uncertainties considered. In this study, we took the minimum MU constraint into consideration using quadratic optimization and simultaneously had the impact of uncertainties considered using robust optimization. We evaluated our method using seven cancer patients with different machine settings. Result: The new method achieved better plan quality than the conventional method. The D_95% of the clinical target volume (CTV) normalized to the prescription dose was (mean [min-max]): (99.4% [99.2%-99.6%]) vs. (99.2% [98.6%-99.6%]). Plan robustness derived from these two methods was comparable. For all seven patients, the CTV dose-volume histogram band gap (narrower band gap means more robust plans) at D_95% normalized to the prescription dose was (mean [min-max]): (1.5% [0.5%-4.3%]) vs. (1.2% [0.6%-3.8%]). Conclusion: Our new method of incorporating the minimum MU constraint directly into the plan robust optimization can produce machine-deliverable plans with better tumor coverage while maintaining high-plan robustness.