Bortfeld T
Deutsches Krebsforschungszentrum (DKFZ), Abteilung Medizinische Physik and Universität Heidelberg, Fakultät für Physik und Astronomie, Heidelberg, Germany.
Med Phys. 1997 Dec;24(12):2024-33. doi: 10.1118/1.598116.
The knowledge of proton depth-dose curves, or "Bragg curves," is a fundamental prerequisite for dose calculations in radiotherapy planning, among other applications. In various cases it is desirable to have an analytical representation of the Bragg curve, rather than using measured or numerically calculated data. This work provides an analytical approximation of the Bragg curve in closed form. The underlying model is valid for proton energies between about 10 and 200 MeV. Its main four constituents are: (i) a power-law relationship describing the range-energy dependency; (ii) a linear model for the fluence reduction due to nonelastic nuclear interactions, assuming local deposition of a fraction of the released energy; (iii) a Gaussian approximation of the range straggling distribution; and (iv) a representation of the energy spectrum of poly-energetic beams by a Gaussian with a linear "tail." Based on these assumptions the Bragg curve can be described in closed form using a simple combination of Gaussians and parabolic cylinder functions. The resulting expression can be fitted to measurements within the measurement error. Very good agreement is also found with numerically calculated Bragg curves.
质子深度剂量曲线，即“布拉格曲线”的知识，是放射治疗计划剂量计算及其他应用的基本前提。在各种情况下，希望有布拉格曲线的解析表达式，而不是使用测量数据或数值计算数据。这项工作提供了一种封闭形式的布拉格曲线解析近似。基础模型适用于约10至200 MeV的质子能量。其主要的四个组成部分为：(i) 描述射程 - 能量依赖性的幂律关系；(ii) 由于非弹性核相互作用导致注量降低的线性模型，假设释放能量的一部分进行局部沉积；(iii) 射程离散分布的高斯近似；(iv) 用具有线性“尾部”的高斯函数表示多能束的能谱。基于这些假设，布拉格曲线可用高斯函数和抛物柱面函数的简单组合以封闭形式描述。所得表达式在测量误差范围内可拟合测量值。与数值计算的布拉格曲线也有很好的一致性。
