Estimation of cell proliferation dynamics using CFSE data.

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57-89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1-26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data.

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log(10)-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate alpha(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize alpha(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.

Numerical modelling of label-structured cell population growth using CFSE distribution data.

BACKGROUND: The flow cytometry analysis of CFSE-labelled cells is currently one of the most informative experimental techniques for studying cell proliferation in immunology. The quantitative interpretation and understanding of such heterogenous cell population data requires the development of distributed parameter mathematical models and computational techniques for data assimilation. METHODS AND RESULTS: The mathematical modelling of label-structured cell population dynamics leads to a hyperbolic partial differential equation in one space variable. The model contains fundamental parameters of cell turnover and label dilution that need to be estimated from the flow cytometry data on the kinetics of the CFSE label distribution. To this end a maximum likelihood approach is used. The Lax-Wendroff method is used to solve the corresponding initial-boundary value problem for the model equation. By fitting two original experimental data sets with the model we show its biological consistency and potential for quantitative characterization of the cell division and death rates, treated as continuous functions of the CFSE expression level. CONCLUSION: Once the initial distribution of the proliferating cell population with respect to the CFSE intensity is given, the distributed parameter modelling allows one to work directly with the histograms of the CFSE fluorescence without the need to specify the marker ranges. The label-structured model and the elaborated computational approach establish a quantitative basis for more informative interpretation of the flow cytometry CFSE systems.

Numerical stability analysis of a large-scale delay system modeling a lateral semiconductor laser subject to optical feedback.

This paper highlights the use of advanced numerical tools to study the stability of large-scale systems of delay differential equations (DDEs). Specifically, we consider a model describing a semiconductor laser subject to conventional optical feedback and lateral carrier diffusion. The symmetry of the governing rate equations allows external cavity mode solutions (ECMs) to be computed as steady state solutions. Using the software package DDE-BIFTOOL, branches of ECMs are computed as a function of varying feedback strength. The stability along these branches is computed by solving eigenvalue problems, the size of which is governed by a step-length heuristic. In this paper, we employ an improved heuristic which substantially reduces the size of these eigenvalue problems. This approach makes the stability analysis of large-scale systems of DDEs computationally feasible.

Computational analysis of CFSE proliferation assay.

CFSE based tracking of the lymphocyte proliferation using flow cytometry is a powerful experimental technique in immunology allowing for the tracing of labelled cell populations over time in terms of the number of divisions cells undergone. Interpretation and understanding of such population data can be greatly improved through the use of mathematical modelling. We apply a heterogenous linear compartmental model, described by a system of ordinary differential equations similar to those proposed by Kendall. This model allows division number-dependent rates of cell proliferation and death and describes the rate of changes in the numbers of cells having undergone j divisions. The experimental data set that we specifically analyze specifies the following characteristics of the kinetics of PHA-induced human T lymphocyte proliferation assay in vitro: (1) the total number of live cells, (2) the total number of dead but not disintegrated cells and (3) the number of cells divided j times. Following the maximum likelihood approach for data fitting, we estimate the model parameters which, in particular, present the CTL birth- and death rate "functions". It is the first study of CFSE labelling data which convincingly shows that the lymphocyte proliferation and death both in vitro and in vivo are division number dependent. For the first time, the confidence in the estimated parameter values is analyzed by comparing three major methods: the technique based on the variance-covariance matrix, the profile-likelihood-based approach and the bootstrap technique. We compare results and performance of these methods with respect to their robustness and computational cost. We show that for evaluating mathematical models of differing complexity the information-theoretic approach, based upon indicators measuring the information loss for a particular model (Kullback-Leibler information), provides a consistent basis. We specifically discuss methodological and computational difficulties in parameter identification with CFSE data, e.g. the loss of confidence in the parameter estimates starting around the sixth division. Overall, our study suggests that the heterogeneity inherent in cell kinetics should be explicitly incorporated into the structure of mathematical models.