School of Mathematics and Statistics, The University of Melbourne, Australia.
Department of Microbiology and Immunology, The University of Melbourne at the Peter Doherty Institute for Infection and Immunity, Australia.
Math Biosci. 2022 Dec;354:108928. doi: 10.1016/j.mbs.2022.108928. Epub 2022 Nov 2.
Nanoparticles are increasingly employed as a vehicle for the targeted delivery of therapeutics to specific cell types. However, much remains to be discovered about the fundamental biology that dictates the interactions between nanoparticles and cells. Accordingly, few nanoparticle-based targeted therapeutics have succeeded in clinical trials. One element that hinders our understanding of nanoparticle-cell interactions is the presence of heterogeneity in nanoparticle dosage data obtained from standard experiments. It is difficult to distinguish between heterogeneity that arises from stochasticity in nanoparticle-cell interactions, and that which arises from heterogeneity in the cell population. Mathematical investigations have revealed that both sources of heterogeneity contribute meaningfully to the heterogeneity in nanoparticle dosage. However, these investigations have relied on simplified models of nanoparticle internalisation. Here we present a stochastic mathematical model of nanoparticle internalisation that incorporates a suite of relevant biological phenomena such as multistage internalisation, cell division, asymmetric nanoparticle inheritance and nanoparticle saturation. Critically, our model provides information about nanoparticle dosage at an individual cell level. We perform model simulations to examine the influence of specific biological phenomena on the heterogeneity in nanoparticle dosage in the absence of heterogeneity in the cell population. Under certain modelling assumptions, we derive analytic approximations of the nanoparticle dosage distribution. We demonstrate that the analytic approximations are accurate, and show that nanoparticle dosage can be described by a Poisson mixture distribution with rate parameters that are a function of Beta-distributed random variables. We discuss the implications of the analytic results with respect to parameter estimation and model identifiability from standard experimental data. Finally, we highlight extensions and directions for future research.
纳米粒子越来越多地被用作将治疗剂靶向递送到特定细胞类型的载体。然而，对于决定纳米粒子与细胞之间相互作用的基本生物学，仍有许多需要探索。因此，很少有基于纳米粒子的靶向治疗在临床试验中取得成功。阻碍我们理解纳米粒子-细胞相互作用的一个因素是，从标准实验中获得的纳米粒子剂量数据存在异质性。很难区分纳米粒子-细胞相互作用中的随机性引起的异质性，以及细胞群体中的异质性引起的异质性。数学研究表明，这两种异质来源都对纳米粒子剂量的异质性有意义的贡献。然而，这些研究依赖于纳米粒子内化的简化模型。在这里，我们提出了一个包含一系列相关生物学现象的纳米粒子内化的随机数学模型，如多阶段内化、细胞分裂、不对称的纳米粒子遗传和纳米粒子饱和。关键的是，我们的模型提供了关于单个细胞水平纳米粒子剂量的信息。我们进行模型模拟，以检查在细胞群体中不存在异质性的情况下，特定生物学现象对纳米粒子剂量异质性的影响。在某些建模假设下，我们推导出纳米粒子剂量分布的解析近似。我们证明了这些解析近似是准确的，并表明纳米粒子剂量可以用一个泊松混合分布来描述，其率参数是 Beta 分布随机变量的函数。我们讨论了解析结果在标准实验数据的参数估计和模型可识别性方面的意义。最后，我们强调了未来研究的扩展和方向。
