An analytical model for the upper bound estimation of respiratory motion–induced dose uncertainty in spot-scanning proton beam therapy

Purpose: We developed an analytical model of a spot-scanning beam delivery system to estimate the upper bound of respiratory motion–induced dose uncertainty for a given treatment plan. Methods: The effective delivery time for each spot position in the treatment plan was calculated on the basis of the parameters of the delivery system. The upper bound of the dose uncertainty was then calculated as a function of the effective delivery time. Two-dimensional (2D) measurements with a detector array on a one-dimensional moving platform were obtained to validate the model. Results: We performed 351 two-dimensional measurements on a moving platform for different delivery sequences of a single-layer uniform pattern and patient treatment field. The measured dose uncertainty was a strong function of the effective delivery time: The shortest effective delivery time resulted in a maximum absolute dose error of >90%, while the longest ones resulted in a maximum absolute dose error of 4.9% for a single layer and 9.7% for a patient field with heterogeneity. The relationship of the effective delivery time and the measured dose uncertainty followed the analytical formula. Conclusions: With our analytical model, the upper bound of the dose uncertainty due to motion can be estimated in spot-scanning proton therapy without four-dimensional dynamic dose calculation.