Technical Note: On the analytical proton dose evaluation in compounds and mixtures

By combining the physical processes occurring due to the interaction of protons with matter, analytical theories published so far have provided acceptable models for calculating depth-dose distributions in homogeneous media. As a well-defined and comprehensive theory, the formula derived by Bortfeld models the dose transferred to the target in terms of the parabolic cylinder function. The model also includes three parameters with values specified for an initial proton energy and for the target material. These parameters are obtainable through the data gathered in nuclear data tables. The analytical studies using this interesting model are therefore restricted to those materials for which the data have been provided in these tables. This study aims to find general solutions for calculation of these parameters for a compound or mixture composed of an arbitrary choice of constituent elements.

Inspired by formulas dedicated for calculating the range and the probability of undergoing nonelastic nuclear interactions for protons in desired compounds, the analytical methods for finding the three mentioned parameters are investigated. The accuracy of the methods suggested is examined through comparison of the results with those which are calculated using the data taken from nuclear data tables. By employing the calculated parameters using the derived formulas in the Bortfeld model, the dose distribution at depth in a chosen target is calculated.

For an arbitrary selection of compounds, the predictions of the analytical depth-dose model using these parameters have been found to closely match the results employing the parameters calculated using the data reported in nuclear data tables.

The formulas presented are general, mathematically easy to handle, and valid for almost every compound or mixture including materials of interest for proton radiotherapy purposes, making the Bortfeld model more practical and advantageous.