A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays

Abstract

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. The intracellular dye CFSE is a powerful experimental tool for the analysis of a population of dividing cells, and numerous mathematical treatments have been aimed at using CFSE data to describe an immune response [30,31,32,37,38,42,48,49]. Recently, partial differential equation structured population models, with measured CFSE fluorescence intensity as the structure variable, have been shown to accurately fit histogram data obtained from CFSE flow cytometry experiments [18,19,52,54]. In this report, the population of cells is mathematically organized into compartments, with all cells in a single compartment having undergone the same number of divisions. A system of structured partial differential equations is derived which can be fit directly to CFSE histogram data. From such a model, cell counts (in terms of the number of divisions undergone) can be directly computed and thus key biological parameters such as population doubling time and precursor viability can be determined. Mathematical aspects of this compartmental model are discussed, and the model is fit to a data set. As in [18,19], we find temporal and division dependence in the rates of proliferation and death to be essential features of a structured population model for CFSE data. Variability in cellular autofluorescence is found to play a significant role in the data, as well. Finally, the compartmental model is compared to previous work, and statistical aspects of the experimental data are discussed.

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

“…It is assumed that all cells are undivided at t = 0, with some initial FI n 0 (0, x) = Φ(x). Several parameterizations of the division and death rate functions α i (t) and β i (t) have been proposed [23], with best-fit results obtained when the division rates are piecewise-linear functions of time and the death rates are constants, with both division and death rates depending upon the number of divisions undergone. It has also been shown [23,79] that variability (among all cells in the population) in the autofluorescence parameter x a is a significant physical feature of an accurate mathematical model and can be accurately modeled with a lognormal distribution [23,46,79].…”

“…Several parameterizations of the division and death rate functions α i (t) and β i (t) have been proposed [23], with best-fit results obtained when the division rates are piecewise-linear functions of time and the death rates are constants, with both division and death rates depending upon the number of divisions undergone. It has also been shown [23,79] that variability (among all cells in the population) in the autofluorescence parameter x a is a significant physical feature of an accurate mathematical model and can be accurately modeled with a lognormal distribution [23,46,79]. The best-fit solution to model (3.16) for a particular data set (data originally from [56]) is shown in Figure 8 and demonstrates the suitability of the given model in describing data from CFSE-based proliferation assays.…”

“…Though the reconstruction of CFSE profiles from computed cell numbers has been suggested [47,SI Text], a label-structured model is a more fundamental and more complex effort, as one must account for the intracellular dynamics of label dilution and turnover while simultaneously estimating proliferation and death dynamics at the population level. In return for this added difficulty, one can potentially avoid some issues of biased cell counts, particularly as this method should be less reliant on distinct peak separations in the CFSE histogram data [23,79]. Figure 7.…”

“…Also, though the structural dependence of the proliferation and death rate functions can be used to infer division-dependent characteristics [21,79], this interpretation is neither straightforward nor intuitive [74]. In order to alleviate these difficulties, a simple reformulation [23,46,74,79] of the model (3.15) can be used to describe the structured densities of subpopulations of cells corresponding to distinct generations, which interact through division. Let n i (t, x) be the structured density of a subpopulation of cells at time t and measured FI x and having undergone i divisions.…”

“…The result of [46] is a more intuitive model formulation which can be viewed as a unifying framework linking models of cell numbers (described in Sections 3.1-3.4) and models of label dynamics (Section 3.5). Ideas from [23,46] have recently been combined with the cyton formulation [33,47] (see also Section 3.4) to yield a conservation-based probabilistic model that provides an excellent fit to histogram data [24].…”