Department of Radiology and Medical Informatics, University Hospital, Geneva, 1211, Switzerland.
Université Paris-Saclay, CEA, Service d'Études des Réacteurs et de Mathématiques Appliquées, 91191, Gif-sur-Yvette, France.
Sci Rep. 2023 Jun 19;13(1):9887. doi: 10.1038/s41598-023-36919-2.
Monte Carlo (MC) simulations allowing to describe photons propagation in statistical mixtures represent an interest that goes way beyond the domain of optics, and can cover, e.g., nuclear reactor physics, image analysis or life science just to name a few. MC simulations are considered a "gold standard" because they give exact solutions (in the statistical sense), however, in the case of statistical mixtures their implementation is often extremely complex. For this reason, the aim of the present contribution is to propose a new approach that should allow us in the future to simplify the MC approach. This is done through an explanatory example, i.e.; by deriving the 'exact' analytical expression for the probability density function of photons' random steps (single step function, SSF) propagating in a medium represented as a binary (isotropic-Poisson) statistical mixture. The use of the SSF reduces the problem to an 'equivalent' homogeneous medium behaving exactly as the original binary statistical mixture. This will reduce hundreds MC simulations, allowing to obtain one set of wanted parameters, to only one equivalent simple MC simulation. To the best of our knowledge the analytically 'exact' SSF for a binary (isotropic-Poisson) statistical mixture has never been derived before.
蒙特卡罗（MC）模拟允许描述光子在统计混合物中的传播，其应用领域不仅限于光学，还可以涵盖核反应堆物理、图像分析或生命科学等领域。MC 模拟被认为是一种“黄金标准”，因为它们给出了精确的解（在统计学意义上），然而，在统计混合物的情况下，其实现通常非常复杂。出于这个原因，本研究的目的是提出一种新的方法，该方法应能使我们在未来简化 MC 方法。这是通过一个解释性的例子来实现的，即通过推导出在介质中传播的光子随机步（单步函数，SSF）的概率密度函数的“精确”解析表达式，该介质表示为二进制（各向同性-泊松）统计混合物。使用 SSF 将问题简化为具有与原始二进制统计混合物完全相同行为的“等效”均匀介质。这将减少数百次 MC 模拟，允许仅通过一次等效的简单 MC 模拟获得一组所需的参数。据我们所知，对于二进制（各向同性-泊松）统计混合物，以前从未推导出过解析上“精确”的 SSF。
