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

Bo Jiang,Xianliang Wang,Xiaodong Zhang,Yang Zhang,Xiaochun Wang,Fada Guan,Ronald X. Zhu,Yupeng Li

doi:10.1038/s41598-018-28683-5

Abstract

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.

Highlights

Radiation therapy plays an important role in the treatment of cancer; approximately 50% of all cancer patients receive radiation therapy during the course of their illness, and radiation therapy contributes to 40% of curative treatment for cancer[1]
Because most dose distributions have been described on the basis of the Molière theory[16,17], which leads to Gaussian distribution, the long-tailed portion of the distribution created by the halo has been fitted with empirical models such as two Gaussians with Cauchy-Lorentz function[9] or three Gaussians[18]
We sought to determine whether the long-tailed proton lateral dose distribution follows the power-law, and if so, whether this phenomenon can be explained by similar mechanisms leading to long-tailed distributions in other research areas

Summary

Introduction

Radiation therapy plays an important role in the treatment of cancer; approximately 50% of all cancer patients receive radiation therapy during the course of their illness, and radiation therapy contributes to 40% of curative treatment for cancer[1]. Owing to the physical characteristics of proton, the maximum dose can be located in the tumor and negligible dose downstream of the Bragg peak For these reasons, proton radiotherapy has more powerful capabilities to spare the normal tissues and reduce the burden of treatment-related complications compared to photon radiotherapy. We sought to determine whether the long-tailed proton lateral dose distribution follows the power-law, and if so, whether this phenomenon can be explained by similar mechanisms leading to long-tailed distributions in other research areas. Determining whether the long-tailed proton dose distribution follows the power law will help elucidate the mechanisms of halo formation and improve dose calculation

Methods

Results

Conclusion