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Frontiers in Mathematical Biology

Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based Proliferation Assay

William C. Thompson

North Carolina State University


The mammalian immune system is comprised of a complex network of cells which interact with each other as well as with external stimuli, and an immune response is characterized by the rapid proliferation (via division) of lymphocytes following exposure to some stimulating agent. Flow cytometric analysis of a proliferating cell population is a powerful and popular tool for the study of cell division and division-linked changes in cell behavior, as it permits the quick assessment of the phenotypic properties of a culture of proliferating cells. In particular, the development of the intracellular dye carboxyfluorescein succinimidyl ester (CFSE) for the fluorescent labeling of cells has led to the need for quantitative models of division dynamics. Some key features of a mathematical description of an immune response are an estimate of the number of responding cells and the manner in which those cells divide, differentiate, and die. Numerous mathematical treatments of CFSE flow cytometry data have been proposed to describe an immune response, and each is motivated by the desire to relate the estimated numbers of cells in the population to average rates of division and death. Alternatively, a structured partial differential equation (PDE) model (with CFSE fluorescence intensity as the structure variable) can be fit directly to flow cytometry data. After reviewing the data collection process and describing previous mathematical work, we focus on the application of such structured PDE models to CFSE histogram data. Several extensions and modifications of previous models are discussed and suggestions are presented to improve the agreement between model solutions and experimental data as well as to improve the physiological understanding of the model parameters. Next, the resulting structured PDE model is generalized into a system of PDE models representing the compartmentalization of the population of cells in terms of the number of divisions undergone since the beginning of the experiment. Mathematical aspects of this compartmental model are discussed, and the model is fit to a data set. It is shown that the compartmental model permits the quantification of cell counts in terms of the number of divisions undergone, so that key biological parameters such as population doubling time and precursor viability can be determined. Finally, statistical models for the observed variability/noise in CFSE histogram data are discussed with implications for uncertainty quantification. It is revealed that several commonly held assumptions regarding the data collection procedure are not accurately reflected in the actual data. Using several additional data sets, experimental, intra-individual, and inter-individual variability in CFSE histogram data is qualitatively analyzed. The data collection procedure is then reexamined and a new statistical model of the data is hypothesized. The models presented produce meaningful quantitative descriptions of the behavior of a dynamic population of cells and are sufficiently general to describe a wide array of proliferative behavior. Several generalizations of these models are also discussed with an eye toward experimental application. This work constitutes a significant first step toward the meaningful analysis of an immune response, and could provide a useful complement in experimental or diagnostic studies of the immune system.