The highly conformal planned dose distribution achievable in intensity modulated proton therapy (IMPT) can severely be compromised by uncertainties in patient setup and proton range. While several robust optimization approaches have been presented to address this issue, appropriate methods to accurately estimate the robustness of treatment plans are still lacking. To fill this gap we present Polynomial Chaos Expansion (PCE) techniques which are easily applicable and create a meta-model of the dose engine by approximating the dose in every voxel with multidimensional polynomials. This Polynomial Chaos (PC) model can be built in an automated fashion relatively cheaply and subsequently it can be used to perform comprehensive robustness analysis. We adapted PC to provide among others the expected dose, the dose variance, accurate probability distribution of dose-volume histogram (DVH) metrics (e.g. minimum tumor or maximum organ dose), exact bandwidths of DVHs, and to separate the effects of random and systematic errors. We present the outcome of our verification experiments based on 6 head-and-neck (HN) patients, and exemplify the usefulness of PCE by comparing a robust and a non-robust treatment plan for a selected HN case. The results suggest that PCE is highly valuable for both research and clinical applications.

Purpose We aimed to derive a "robustness recipe" giving the range robustness (RR) and setup robustness (SR) settings (ie, the error values) that ensure adequate clinical target volume (CTV) coverage in oropharyngeal cancer patients for given Gaussian distributions of systematic setup, random setup, and range errors (characterized by standard deviations of Σ, σ, and ρ, respectively) when used in minimax worst-case robust intensity modulated proton therapy (IMPT) optimization. Methods and Materials For the analysis, contoured computed tomography (CT) scans of 9 unilateral and 9 bilateral patients were used. An IMPT plan was considered robust if, for at least 98% of the simulated fractionated treatments, 98% of the CTV received 95% or more of the prescribed dose. For fast assessment of the CTV coverage for given error distributions (ie, different values of Σ, σ, and ρ), polynomial chaos methods were used. Separate recipes were derived for the unilateral and bilateral cases using one patient from each group, and all 18 patients were included in the validation of the recipes. Results Treatment plans for bilateral cases are intrinsically more robust than those for unilateral cases. The required RR only depends on the ρ, and SR can be fitted by second-order polynomials in Σ and σ. The formulas for the derived robustness recipes are as follows: Unilateral patients need SR = -0.15Σ² + 0.27σ² + 1.85Σ - 0.06σ + 1.22 and RR=3% for ρ = 1% and ρ = 2%; bilateral patients need SR = -0.07Σ² + 0.19σ² + 1.34Σ - 0.07σ + 1.17 and RR=3% and 4% for ρ = 1% and 2%, respectively. For the recipe validation, 2 plans were generated for each of the 18 patients corresponding to Σ = σ = 1.5 mm and ρ = 0% and 2%. Thirty-four plans had adequate CTV coverage in 98% or more of the simulated fractionated treatments; the remaining 2 had adequate coverage in 97.8% and 97.9%. Conclusions Robustness recipes were derived that can be used in minimax robust optimization of IMPT treatment plans to ensure adequate CTV coverage for oropharyngeal cancer patients.

Purpose To quantify the impact of the degree of robustness against setup errors and range errors on organ-at-risk (OAR) dose and normal tissue complication probabilities (NTCPs) in intensity-modulated proton therapy for oropharyngeal cancer patients. Material and methods For 20 oropharyngeal cases (10 unilateral and 10 bilateral), robust treatment plans were generated using ‘minimax’ worst-case optimization. We varied the robustness against setup errors (‘setup robustness’) from 1 to 7 mm and the robustness against range errors (‘range robustness’) from 1% to 7% (+1 mm). We evaluated OAR doses and NTCP-values for xerostomia, dysphagia and larynx edema. Results Varying the degree of setup robustness was found to have a considerably larger impact than varying the range robustness. Increasing setup robustness from 1 mm to 3, 5, and 7 mm resulted in average NTCP-values to increase by 1.9, 4.4 and 7.5 percentage point, whereas they increased by only 0.4, 0.8 and 1.2 percentage point when increasing range robustness from 1% to 3%, 5% and 7%. The degree of setup robustness was observed to have a clinically significant impact in bilateral cases in particular. Conclusions For oropharyngeal cancer patients, minimizing setup errors should be given a higher priority than minimizing range errors.