ICT Radiotherapy Services, Livingston, New Jersey 07039, USA.
Med Phys. 2010 Nov;37(11):5683-90. doi: 10.1118/1.3495969.
The depth dose of a monoenergetic broad parallel proton beam has been modeled in a number of ways, but evidently not yet for oblique incidence. The purpose of this investigation is to find an accurate analytic formula for this case, which can then be used to model the depth dose of a broad beam with an initial Gaussian angular distribution.
The Bortfeld model of depth dose in a broad normally incident proton beam has been extended to the case of oblique incidence. This extension uses an empirically determined Gaussian parameter sigma(x) which (roughly) characterizes the off-axis dose of a proton pencil beam. As with Bortfeld's work, the modeling is done in terms of parabolic cylinder functions. To obtain the depth dose for an initial angular distribution, the result is integrated over the angle of incidence, weighted by a Gaussian probability function. The predictions of the theory have been compared to MCNPX Monte Carlo calculations for four phantom materials (water, bone, aluminum, and copper) and for initial proton energies of 50, 100, 150, 200, and 250 MeV.
Comparisons of the depth dose predicted by this theory with Monte Carlo calculations have established that with very good accuracy, sigma(x) can be taken to be independent both of the depth and of the angle of incidence. As a function of initial proton range or of initial proton energy, sigma(x) has been found to obey a power law to very high accuracy. Good fits to Monte Carlo calculations have also been found for an initial Gaussian angular distribution.
This investigation is the first step in the accurate modeling of a proton pencil beam with initial Gaussian angular distribution. It provides the longitudinal factor, with its Bragg peak buildup and sharp distal falloff. A transverse factor must still be incorporated into this theory and this will give the lateral penumbra of a collimated proton beam. Also, it will be necessary to model the dose of product particles from nuclear interactions of the proton beam. With the accurate modeling of a pencil beam, it will be possible to accurately take into account the effect of localized tissue inhomogeneities.
已经以多种方式对单能宽平行质子束的深度剂量进行了建模，但显然尚未对斜入射情况进行建模。本研究的目的是找到一种准确的解析公式来描述这种情况，然后可以使用该公式来模拟具有初始高斯角分布的宽束的深度剂量。
已经将宽的正常入射质子束的深度剂量的 Bortfeld 模型扩展到斜入射的情况。这种扩展使用了一个经验确定的高斯参数 sigma(x)，该参数大致描述了质子铅笔束的离轴剂量。与 Bortfeld 的工作一样，建模是根据抛物线柱面函数进行的。为了获得初始角分布的深度剂量，将结果沿入射角进行积分，并乘以高斯概率函数进行加权。该理论的预测已与用于四种模拟材料（水、骨、铝和铜）和初始质子能量为 50、100、150、200 和 250 MeV 的 MCNPX 蒙特卡罗计算进行了比较。
将该理论预测的深度剂量与蒙特卡罗计算的结果进行比较，已经确定，非常准确地，可以假定 sigma(x)既不依赖于深度也不依赖于入射角。作为初始质子射程或初始质子能量的函数，已经发现 sigma(x)非常准确地遵循幂律关系。还发现初始高斯角分布的很好的拟合。
本研究是准确模拟具有初始高斯角分布的质子铅笔束的第一步。它提供了具有布拉格峰形成和陡峭的远端下降的纵向因子。还必须将横向因子合并到该理论中，这将给出准直质子束的侧向半影。此外，还需要对质子束的核相互作用产生的产物粒子的剂量进行建模。通过对铅笔束进行准确建模，可以准确地考虑局部组织不均匀性的影响。
