The 'worst case scenario' (also known as the minimax approach in optimization terms) is a common approach to model the effect of delivery uncertainties in proton treatment planning. Using the 'dose-error-bar distribution' previously reported by our group as an example, we have investigated in more detail one of the underlying assumptions of this method. That is, the dose distributions calculated for a limited number of worst case patient positioning scenarios (i.e. limited number of shifts sampled on a spherical surface) represent the worst dose distributions that can occur during the patient treatment under setup uncertainties. By uniformly sampling patient shifts from anywhere within a spherical error-space, a number of treatment scenarios have been simulated and dose deviations from the nominal dose distribution have been computed. The dose errors from these simulations (comprehensive approach) have then been compared to the dose-error-bar approach previously reported (surface approximation) using both point-by-point and dose- and error-volume-histogram analysis (DVH/EVHs). This comparison has been performed for two different clinical cases treated using intensity modulated proton therapy (IMPT): a skull-base and a spinal-axis tumor. Point-by-point evaluation shows that the surface approximation leads to a correct estimation (95% accuracy) of the potential dose errors for the 96% and 85% of the irradiated voxels, for the two investigated cases respectively. We also found that the voxels for which the surface approximation fails are generally localized close to sharp soft tissue-bone interfaces and air cavities. Moreover, analysis of EVHs and DVHs for the two cases shows that the percentage of voxels of a given volume of interest potentially affected by a certain maximum dose error is correctly estimated using the surface approximation and that this approach also accurately predicts the upper and lower bounds of the DVH curves that can occur under positioning uncertainties. In conclusion, the assumption that the larger the patient shift the worse the dose error does not always hold on a point-by-point basis. Nevertheless, when performing a volumetric analysis, a limited set of worst case error scenarios correctly represents the worst quality of the plan in presence of setup errors. As a consequence of these results, we believe that the worst case scenario approach can be used in the IMPT planning procedure for estimating plan robustness provided that the possible limitations of this approach are known.
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