Département de Physique, de Génie Physique et D'optique et Centre de Recherche sur le Cancer, Université Laval, Quebéc, QC, G1R 0A6, Canada. Département de Radio-oncologie et Centre de Recherche du CHU de Québec, Université Laval, Quebéc, QC, G1R 2J6, Canada.
Phys Med Biol. 2017 Dec 19;63(1):015019. doi: 10.1088/1361-6560/aa9166.
The equivalent restricted stopping power formalism is introduced for proton mean energy loss calculations under the continuous slowing down approximation. The objective is the acceleration of Monte Carlo dose calculations by allowing larger steps while preserving accuracy. The fractional energy loss per step length ϵ was obtained with a secant method and a Gauss-Kronrod quadrature estimation of the integral equation relating the mean energy loss to the step length. The midpoint rule of the Newton-Cotes formulae was then used to solve this equation, allowing the creation of a lookup table linking ϵ to the equivalent restricted stopping power L , used here as a key physical quantity. The mean energy loss for any step length was simply defined as the product of the step length with L . Proton inelastic collisions with electrons were added to GPUMCD, a GPU-based Monte Carlo dose calculation code. The proton continuous slowing-down was modelled with the L formalism. GPUMCD was compared to Geant4 in a validation study where ionization processes alone were activated and a voxelized geometry was used. The energy straggling was first switched off to validate the L formalism alone. Dose differences between Geant4 and GPUMCD were smaller than 0.31% for the L formalism. The mean error and the standard deviation were below 0.035% and 0.038% respectively. 99.4 to 100% of GPUMCD dose points were consistent with a 0.3% dose tolerance. GPUMCD 80% falloff positions ([Formula: see text]) matched Geant's [Formula: see text] within 1 μm. With the energy straggling, dose differences were below 2.7% in the Bragg peak falloff and smaller than 0.83% elsewhere. The [Formula: see text] positions matched within 100 μm. The overall computation times to transport one million protons with GPUMCD were 31-173 ms. Under similar conditions, Geant4 computation times were 1.4-20 h. The L formalism led to an intrinsic efficiency gain factor ranging between 30-630, increasing with the prescribed accuracy of simulations. The L formalism allows larger steps leading to a [Formula: see text] algorithmic time complexity. It significantly accelerates Monte Carlo proton transport while preserving accuracy. It therefore constitutes a promising variance reduction technique for computing proton dose distributions in a clinical context.
在连续慢化近似下，引入了等效限制阻止本领形式，用于质子平均能量损失计算。目的是通过允许更大的步长来加速蒙特卡罗剂量计算，同时保持准确性。通过割线法和高斯-克朗罗积分方程的积分估计，获得了每步长的分数能量损失 ϵ，该积分方程将平均能量损失与步长相联系。然后，牛顿-科特斯公式的中点规则用于求解该方程，从而创建一个将 ϵ与等效限制阻止本领 L 相关联的查找表，此处 L 用作关键物理量。任何步长的平均能量损失都简单地定义为步长与 L 的乘积。质子与电子的非弹性碰撞被添加到基于 GPU 的蒙特卡罗剂量计算代码 GPUMCD 中。质子连续慢化用 L 形式进行建模。在验证研究中，单独激活电离过程并使用体素化几何形状，将 GPUMCD 与 Geant4 进行了比较。首先关闭能量散裂以单独验证 L 形式。对于 L 形式，Geant4 和 GPUMCD 之间的剂量差异小于 0.31%。平均误差和标准偏差分别低于 0.035%和 0.038%。99.4%至 100%的 GPUMCD 剂量点与 0.3%剂量容限一致。GPUMCD 的 80%下降位置 ([Formula: see text])与 Geant 的 [Formula: see text]在 1 μm 内匹配。有能量散裂时，在布拉格峰下降处的剂量差异小于 2.7%，在其他地方小于 0.83%。[Formula: see text]位置在 100 μm 内匹配。使用 GPUMCD 传输一百万个质子的总计算时间为 31-173 ms。在类似条件下，Geant4 的计算时间为 1.4-20 h。L 形式导致固有效率增益因子在 30-630 之间，随着模拟精度的提高而增加。L 形式允许更大的步长，从而导致 [Formula: see text]算法时间复杂度。它显著加速了蒙特卡罗质子传输，同时保持了准确性。因此，它构成了在临床环境中计算质子剂量分布的有前途的方差减少技术。
