Journal of Mathematical Biology

, Volume 69, Issue 6–7, pp 1547–1583

# Mathematical models for CFSE labelled lymphocyte dynamics: asymmetry and time-lag in division

• Tatyana Luzyanina
• Jovana Cupovic
• Burkhard Ludewig
Article

## Abstract

Since their invention in 1994, fluorescent dyes such as carboxyfluorescein diacetate succinimidyl ester (CFSE) are used for cell proliferation analysis in flow cytometry. Importantly, the interpretation of such assays relies on the assumption that the label is divided equally between the daughter cells upon cell division. However, recent experimental studies indicate that division of cells is not perfectly symmetric and there is unequal distribution of protein between sister cell pairs. The uneven partition of protein or mass to daughter cells can lead to an overlap in the generations of CFSE-labelled cells with straightforward consequences for the resolution of individual generations. Numerous mathematical models developed so far for the analysis of CFSE proliferation assay incorporate the premise that the CFSE fluorescence intensity is halved in the two daughter cells. Here, we propose a novel modelling approach for the analysis of the CFSE cell proliferation assays which are characterized by poorly resolved peaks of cell generations in flow cytometric histograms. We formulate a mathematical model in the form of a system of delay hyperbolic partial differential equations which provides a good agreement with the CFSE histograms time-series data and allows an analytical treatment. The model is a further generalization of the recently proposed class of division- and label-structured models as it considers an asymmetric cell division. In addition, the basic structure of the cell cycle, i.e. the resting and cycling cell compartments, is taken into account. The model is used to estimate fundamental parameters such as activation rate, duration of the cell cycle, apoptosis rate, CFSE decay rate and asymmetry factor in cell division of monoclonal T cells during cognate interaction with dendritic cells.

## Keywords

Division- and label-structured cell population dynamics  Delay hyperbolic PDE model Asymmetric cell division Inverse problem

## Mathematics Subject Classification (2000)

35R30 92-08 92C37

## Notes

### Acknowledgments

The authors acknowledge the support of this work provided by the Swiss National Science Foundation, the Von-Tobel Foundation (Zurich), the Russian Foundation of Basic Research (11-01-00117a), the Programme of the Russian Academy of Sciences (Basic research for Medicine) and by the Swedish Institute, Visby Program.

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## Authors and Affiliations

• Tatyana Luzyanina
• 1
Email author
• Jovana Cupovic
• 2
• Burkhard Ludewig
• 2