Rasouli Fatemeh S, Masoudi S Farhad, Jette David
Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran 15418-49611, Iran.
ICT Radiotherapy Services, Livingston, New Jersey 07039.
Med Phys. 2016 Jan;43(1):303. doi: 10.1118/1.4938575.
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.
通过结合质子与物质相互作用所产生的物理过程，迄今为止发表的分析理论已为计算均匀介质中的深度剂量分布提供了可接受的模型。作为一种定义明确且全面的理论，博尔特费尔德推导的公式以抛物柱面函数来模拟传递到靶区的剂量。该模型还包括三个参数，其值针对初始质子能量和靶物质进行了指定。这些参数可通过核数据表中收集的数据获得。因此，使用这个有趣模型的分析研究仅限于那些在这些表中已提供数据的材料。本研究旨在找到一种通用解法，用于计算由任意选择的组成元素构成的化合物或混合物的这些参数。
受专门用于计算质子在所需化合物中的射程和发生非弹性核相互作用概率的公式启发，研究了用于确定上述三个参数的分析方法。通过将所得结果与使用核数据表中的数据计算得出的结果进行比较，检验了所建议方法的准确性。利用在博尔特费尔德模型中通过推导公式计算出的参数，计算了选定靶区深度处的剂量分布。
对于任意选择的化合物，已发现使用这些参数的分析深度剂量模型的预测结果与使用核数据表中报告的数据计算得出的参数的结果非常匹配。
所提出的公式具有通用性，在数学上易于处理，并且对几乎每种化合物或混合物均有效，包括质子放射治疗中感兴趣的材料，这使得博尔特费尔德模型更具实用性和优势。
