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Objectives  The beam output of a double scattering proton system varies for each combination of beam option, range, and modulation and therefore is difficult to be accurately modeled by the treatment planning system (TPS). This study aims to design an empirical method using the analytical and machine learning (ML) models to estimate proton output in a double scattering proton system.    Materials and Methods  Three analytical models using polynomial, linear, and logarithm–polynomial equations were generated on a training dataset consisting of 1,544 clinical measurements to estimate proton output for each option. Meanwhile, three ML models using Gaussian process regression (GPR) with exponential kernel, squared exponential kernel, and rational quadratic kernel were also created for all options combined. The accuracy of each model was validated against 241 additional clinical measurements as the testing dataset. Two most robust models were selected, and the minimum number of samples needed for either model to achieve sufficient accuracy ( ± 3%) was determined by evaluating the mean average percentage error (MAPE) with increasing sample number. The differences between the estimated outputs using the two models were also compared for 1,000 proton beams with a randomly generated range, and modulation for each option.    Results  The polynomial model and the ML GPR model with exponential kernel yielded the most accurate estimations with less than 3% deviation from the measured outputs. At least 20 samples of each option were needed to build the polynomial model with less than 1% MAPE, whereas at least a total of 400 samples were needed for all beam options to build the ML GPR model with exponential kernel to achieve comparable accuracy. The two independent models agreed with less than 2% deviation using the testing dataset.    Conclusion  The polynomial model and the ML GPR model with exponential kernel were built for proton output estimation with less than 3% deviations from the measurements. They can be used as an independent output prediction tool for a double scattering proton beam and a secondary output check tool for a cross check between themselves.

Comprehensive Output Estimation of Double Scattering Proton System With Analytical and Machine Learning Models