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

Abstract

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.

Keywords: L-BFGS-B; deliverable robustness; intensity-modulated proton therapy (IMPT); minimum MU constraint; quadratic optimization.

Figure 1

Algorithm diagrams for the conventional robust optimization (left) and the proposed deliverable robust optimization (right). L‐BFGS indicates limited memory Broyden–Fletcher–Goldfarb–Shanno algorithm. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 2

Dose–volume histograms (DVHs) for all three methods in patient 1 with prostate cancer (first column), patient 2 with head–neck cancer (second column), and patient 5 with lung cancer (third column). The first row indicates DVHs for CTVs; the second row indicates DVHs for organ at risk. Part (a) is the test with realistic minimum MU constraint and part (b) is with artificially large (10 times larger than that in part (a)) minimum MU constraint. Please note that figures in the top row are zoomed in. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 3

Comparison of CTV D_95% (CTV, clinical target volume; D_95%, the minimum normalized dose that covers 95% of the region of interest with the highest dose of plans A, B, and C with realistic (small) minimum MU constraint and artificially 10 times larger (large) minimum MU constraint, in all seven patients. Error bars indicate the DVH family bandwidth at D_95%, which indicates plan robustness. [Color figure can be viewed at wileyonlinelibrary.com]