Power-law relationship in the long-tailed sections of proton dose distributions

The halo portion of a proton therapy dose creates a long tail in proton dose distributions, but so far study of this phenomenon has been limited. We used statistical methods and mathematical models to confirm that the long-tailed portion of proton dose distributions exhibits a power-law relationship. By analyzing 299 measured dose profiles, we found that all proton lateral dose distributions had a significant power-law scaling correlation with a high correlation coefficient in the tail. We set up a dual-mechanism model, containing both direct and indirect impact mechanisms. In the direct impact mechanism, the proximal dose deposition is mainly due to the direct impact of a proton on a particle. In the indirect mechanism, the impact of a proton on a given particle is considered in terms of the proton's impact on a neighboring particle that then impacts the given particle. We found that the indirect impact mechanism led to a tail in the distribution exhibiting a power-law relationship because the probability of the indirect impacts was proportional to the distance; i.e., the longer the distance, the larger the indirect impact probability. Upon analyzing the experimental data, we observed that the power-law exponent increased with proton energy.