Rival approaches to mathematical modelling in immunology
Journal of Computational and Applied Mathematics
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The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin (Nat. Immunol. 1 (2000) 239-244) have proposed to normalize CFSE data by the 2-fold expansion that is associated with each division, and have argued that the mean of the normalized data increases linearly with time, t, with a slope reflecting the division rate p. We develop a number of mathematical models for the clonal expansion of quiescent cells after stimulation and show, within the context of these models, under which conditions this approach is valid. We compare three means of the distribution of cells over the CFSE profile at time t: the mean, µ(t), the mean of the normalized distribution, µ2 (t), and the mean of the normalized distribution excluding nondivided cells, µ2°(t).In the simplest models, which deal with homogeneous populations of cells with constant division and death rates, the normalized frequency distribution of the cells over the respective division numbers is a Poisson distribution with mean µ2(t) = pt, where p is the division rate. The fact that in the data these distributions seem Gaussian is therefore insufficient to establish that the times at which cells are recruited into the first division have a Gaussian variation because the Poisson distribution approaches the Gaussian distribution for large pt. Excluding nondivided cells complicates the data analysis because µ2°(t) ≠ pt, and only approaches a slope p after an initial transient.In models where the first division of the quiescent cells takes longer than later divisions, all three means have an initial transient before they approach an asymptotic regime, which is the expected µ(t) = 2pt and µ2(t) = µ2° (t) = pt. Such a transient markedly complicates the data analysis. After the same initial transients, the normalized cell numbers tend to decrease at a rate e-dt, where d is the death rate.Nonlinear parameter fitting of CFSE data obtained from Gett and Hodgkin to ordinary differential equation (ODE) models with first-order terms for cell proliferation and death gave poor fits to the data. The Smith-Martin model with an explicit time delay for the deterministic phase of the cell cycle performed much better. Nevertheless, the insights gained from analysis of the ODEs proved useful as we showed by generating virtual CFSE data with a simulation model, where cell cylce times were drawn from various distributions, and then computing the various mean division numbers.