Battelle Center for Mathematical Medicine, The Abigail Wexner Research Institute, 575 Children's Crossroad, Columbus, OH, 43215, USA.
Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA.
Bull Math Biol. 2022 Aug 17;84(10):103. doi: 10.1007/s11538-022-01063-x.
Mathematical modeling provides a rigorous way to quantify immunological processes and discriminate between alternative mechanisms driving specific biological phenomena. It is typical that mathematical models of immunological phenomena are developed by modelers to explain specific sets of experimental data after the data have been collected by experimental collaborators. Whether the available data are sufficient to accurately estimate model parameters or to discriminate between alternative models is not typically investigated. While previously collected data may be sufficient to guide development of alternative models and help estimating model parameters, such data often do not allow to discriminate between alternative models. As a case study, we develop a series of power analyses to determine optimal sample sizes that allow for accurate estimation of model parameters and for discrimination between alternative models describing clustering of CD8 T cells around Plasmodium liver stages. In our typical experiments, mice are infected intravenously with Plasmodium sporozoites that invade hepatocytes (liver cells), and then activated CD8 T cells are transferred into the infected mice. The number of T cells found in the vicinity of individual infected hepatocytes at different times after T cell transfer is counted using intravital microscopy. We previously developed a series of mathematical models aimed to explain highly variable number of T cells per parasite; one of such models, the density-dependent recruitment (DDR) model, fitted the data from preliminary experiments better than the alternative models, such as the density-independent exit (DIE) model. Here, we show that the ability to discriminate between these alternative models depends on the number of parasites imaged in the analysis; analysis of about [Formula: see text] parasites at 2, 4, and 8 h after T cell transfer will allow for over 95% probability to select the correct model. The type of data collected also has an impact; following T cell clustering around individual parasites over time (called as longitudinal (LT) data) allows for a more precise and less biased estimates of the parameters of the DDR model than that generated from a more traditional way of imaging individual parasites in different liver areas/mice (cross-sectional (CS) data). However, LT imaging comes at a cost of a need to keep the mice alive under the microscope for hours which may be ethically unacceptable. We finally show that the number of time points at which the measurements are taken also impacts the precision of estimation of DDR model parameters; in particular, measuring T cell clustering at one time point does not allow accurately estimating all parameters of the DDR model. Using our case study, we propose a general framework on how mathematical modeling can be used to guide experimental designs and power analyses of complex biological processes.
数学建模为量化免疫学过程并区分驱动特定生物学现象的替代机制提供了一种严谨的方法。通常情况下，免疫现象的数学模型是由建模人员在收集到实验合作者收集的数据后开发的，用于解释特定的数据集。是否有足够的数据来准确估计模型参数或区分替代模型通常不会被调查。虽然之前收集的数据可能足以指导替代模型的开发并帮助估计模型参数，但这些数据通常无法区分替代模型。作为一个案例研究，我们开发了一系列功效分析来确定最佳样本量，以便准确估计模型参数并区分描述 CD8 T 细胞围绕疟原虫肝脏阶段聚类的替代模型。在我们的典型实验中，将疟原虫孢子虫经静脉感染小鼠，然后将激活的 CD8 T 细胞转移到感染的小鼠中。在 T 细胞转移后不同时间，使用活体显微镜计算在单个感染肝细胞附近发现的 T 细胞数量。我们之前开发了一系列旨在解释寄生虫数量高度可变的数学模型；其中一种模型，即密度依赖招募（DDR）模型，比替代模型（如密度独立退出（DIE）模型）更好地拟合了初步实验数据。在这里，我们表明区分这些替代模型的能力取决于分析中成像的寄生虫数量；在 T 细胞转移后 2、4 和 8 小时分析约 [公式：见正文] 个寄生虫将允许超过 95%的概率选择正确的模型。收集的数据类型也有影响；随着时间的推移跟踪单个寄生虫周围的 T 细胞聚类（称为纵向（LT）数据）比从更传统的方式成像不同肝区/小鼠中的单个寄生虫（横断面（CS）数据）生成的 DDR 模型参数的估计更精确且偏差更小。然而，LT 成像需要将小鼠在显微镜下存活数小时，这在伦理上可能是不可接受的。我们最后表明，测量的时间点数量也会影响 DDR 模型参数估计的精度；特别是，在一个时间点测量 T 细胞聚类不能准确估计 DDR 模型的所有参数。使用我们的案例研究，我们提出了一个一般框架，说明如何使用数学建模来指导复杂生物过程的实验设计和功效分析。
