Li Haisen S, Romeijn H Edwin, Dempsey James F
Department of Radiation Oncology, University of Florida College of Medicine, Gainesville, Florida 32610-0385, USA.
Med Phys. 2006 Sep;33(9):3508-18. doi: 10.1118/1.2241996.
We developed an analytical method for determining the maximum acceptable grid size for discrete dose calculation in proton therapy treatment plan optimization, so that the accuracy of the optimized dose distribution is guaranteed in the phase of dose sampling and the superfluous computational work is avoided. The accuracy of dose sampling was judged by the criterion that the continuous dose distribution could be reconstructed from the discrete dose within a 2% error limit. To keep the error caused by the discrete dose sampling under a 2% limit, the dose grid size cannot exceed a maximum acceptable value. The method was based on Fourier analysis and the Shannon-Nyquist sampling theorem as an extension of our previous analysis for photon beam intensity modulated radiation therapy [J. F. Dempsey, H. E. Romeijn, J. G. Li, D. A. Low, and J. R. Palta, Med. Phys. 32, 380-388 (2005)]. The proton beam model used for the analysis was a near monoenergetic (of width about 1% the incident energy) and monodirectional infinitesimal (nonintegrated) pencil beam in water medium. By monodirection, we mean that the proton particles are in the same direction before entering the water medium and the various scattering prior to entrance to water is not taken into account. In intensity modulated proton therapy, the elementary intensity modulation entity for proton therapy is either an infinitesimal or finite sized beamlet. Since a finite sized beamlet is the superposition of infinitesimal pencil beams, the result of the maximum acceptable grid size obtained with infinitesimal pencil beam also applies to finite sized beamlet. The analytic Bragg curve function proposed by Bortfeld [T. Bortfeld, Med. Phys. 24, 2024-2033 (1997)] was employed. The lateral profile was approximated by a depth dependent Gaussian distribution. The model included the spreads of the Bragg peak and the lateral profiles due to multiple Coulomb scattering. The dependence of the maximum acceptable dose grid size on the orientation of the beam with respect to the dose grid was also investigated. The maximum acceptable dose grid size depends on the gradient of dose profile and in turn the range of proton beam. In the case that only the phantom scattering was considered and that the beam was aligned with the dose grid, grid sizes from 0.4 to 6.8 mm were required for proton beams with ranges from 2 to 30 cm for 2% error limit at the Bragg peak point. A near linear relation between the maximum acceptable grid size and beam range was observed. For this analysis model, the resolution requirement was not significantly related to the orientation of the beam with respect to the grid.
我们开发了一种分析方法，用于确定质子治疗计划优化中离散剂量计算的最大可接受网格尺寸，从而在剂量采样阶段保证优化剂量分布的准确性，并避免多余的计算工作。剂量采样的准确性通过以下标准判断：连续剂量分布可从离散剂量中重建，误差限制在2%以内。为了将离散剂量采样引起的误差控制在2%以内，剂量网格尺寸不能超过最大可接受值。该方法基于傅里叶分析和香农 - 奈奎斯特采样定理，是我们之前对光子束调强放射治疗分析的扩展[J. F. Dempsey, H. E. Romeijn, J. G. Li, D. A. Low, and J. R. Palta, Med. Phys. 32, 380 - 388 (2005)]。用于分析的质子束模型是水介质中近单能（宽度约为入射能量的1%）且单向的无限小（非积分）笔形束。所谓单向，是指质子粒子在进入水介质之前处于同一方向，且不考虑进入水之前的各种散射。在调强质子治疗中，质子治疗的基本强度调制实体是无限小或有限尺寸的子束。由于有限尺寸子束是无限小笔形束的叠加，用无限小笔形束得到的最大可接受网格尺寸结果也适用于有限尺寸子束。采用了Bortfeld提出的解析布拉格曲线函数[T. Bortfeld, Med. Phys. 24, 2024 - 2033 (1997)]。横向轮廓用与深度相关的高斯分布近似。该模型包括布拉格峰的展宽以及多次库仑散射引起的横向轮廓展宽。还研究了最大可接受剂量网格尺寸与束相对于剂量网格方向的依赖性。最大可接受剂量网格尺寸取决于剂量轮廓的梯度，进而取决于质子束的射程。在仅考虑模体散射且束与剂量网格对齐的情况下，对于射程为2至30 cm的质子束，在布拉格峰点2%误差限制下，所需网格尺寸为0.4至6.8 mm。观察到最大可接受网格尺寸与束射程之间存在近似线性关系。对于该分析模型，分辨率要求与束相对于网格的方向没有显著关系。
