Computational analysis of CFSE proliferation assay. (English) Zbl 1113.92021
Summary: Carboxy fluorescin diacetate succinimidyl ester (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 undergo. Interpretation and understanding of such population data can be greatly improved through the use of mathematical modelling. We apply a heterogeneous linear compartmental model, described by a system of ordinary differential equations similar to those proposed by D. G. Kendall [Biometrika 35, 316–330 (1948; Zbl 0032.04703)]. 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 Phytohemagglutinin (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 cytotoxic T lymphocyte (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.
The experimental data set that we specifically analyze specifies the following characteristics of the kinetics of Phytohemagglutinin (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 cytotoxic T lymphocyte (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.
MSC:
| 92C37 | Cell biology |
| 92C40 | Biochemistry, molecular biology |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 93A30 | Mathematical modelling of systems (MSC2010) |
| 92C30 | Physiology (general) |
Keywords:
Modelling T-cell kinetics; CFSE assay; Heterogenous compartmental model; Parameter estimation; Computation of confidence intervalsCitations:
Zbl 0032.04703Software:
bootstrapReferences:
| [1] | Abu-Absi N.R., Zamamiri A., Kacmar J., Balogh S.J., Srienc F. (2003) Automated flow cytometry for acquisition of time-dependent population data. Cytometry A 51, 87–96 · doi:10.1002/cyto.a.10016 |
| [2] | Baker C.T.H., Bocharov G.A., Ford J.M., Lumb P.M., Norton S.J., Paul C.A.H., Junt T., Ludewig B., Krebs P. (2005) Computational approaches to parameter estimation and model selection in immunology. J. Comput. Appl. Math. 184, 50–76 · Zbl 1072.92020 · doi:10.1016/j.cam.2005.02.003 |
| [3] | Baker C.T.H., Bocharov G.A., Paul C.A.H., Rihan F.A. (2005) Computational modelling with functional differential equations: identification, selection and sensitivity. Appl. Numer. Math. 53, 107–129 · Zbl 1069.65082 · doi:10.1016/j.apnum.2004.08.014 |
| [4] | Bard Y. (1974) Nonlinear Parameter Estimation. Academic, New York · Zbl 0345.62045 |
| [5] | Bernard S., Pujo-Menjouet L., Mackey M.C. (2003) Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys. J. 84, 3414–3424 · doi:10.1016/S0006-3495(03)70063-0 |
| [6] | Burnham K.P., Anderson D.R., (2002) Model selection and multimodel inference–a practical information-theoretic approach, 2nd ed. Springer, Berlin Heidelberg New York · Zbl 1005.62007 |
| [7] | De Boer R.J., Perelson A.S. (2005) Estimating division and death rates from CFSE data. J. Comput. Appl. Math. 184(1): 140–164 · Zbl 1074.92016 · doi:10.1016/j.cam.2004.08.020 |
| [8] | Efron B., Tibshirani R. (1986) Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat. Sci. 1(1): 54–77 · Zbl 0587.62082 · doi:10.1214/ss/1177013815 |
| [9] | Efron B., Tibshirani R. (1993) Introduction to the bootstrap. Chapman and Hall, New York · Zbl 0835.62038 |
| [10] | Eisen M. (1979) Mathematical models in cell biology and cancer chemotherapy. Springer, Berlin Heidelberg New York · Zbl 0414.92005 |
| [11] | Ganusov V.V., Pilyugin S.S., De Boer R.J., Murali-Krishna K., Ahmed R., Antia R. (2005) Quantifying cell turnover using CFSE data. J. Immunol. Methods 298, 183–200 · doi:10.1016/j.jim.2005.01.011 |
| [12] | Gett A.V., Hodgkin P.D. (2000) A cellular calculus for signal integration by T cells. Nat. Immunol. 1, 239–244 · doi:10.1038/79782 |
| [13] | Gudmundsdottir H., Wells A.D., Turka L.A. (1999) Dynamics and requirements of T cell clonal expansion in vivo at the single-cell level: effector function is linked to proliferative capacity. J. Immunol. 162: 5212–5223 |
| [14] | Kendall D.G. (1948) On the role of variable generation time in the development of a stochastic birth process. Biometrika 35, 316–330 · Zbl 0032.04703 |
| [15] | Murali-Krishna K., Ahmed R. (2000) Cutting edge: naive T cells masquerading as memory cells. J. Immunol. 165: 1733–1737 |
| [16] | Muyng I.J., Balasubramanian V., Pitt M.A. (2000) Counting probability distributions: differential geometry and model selection. PNAS USA 97: 11170–11175 · Zbl 0997.62099 · doi:10.1073/pnas.170283897 |
| [17] | Pilyugin S.S., Ganusov V.V., Murali-Krishna K., Ahmed R., Antia R. (2003) The rescaling method for quantifying the turnover of cell populations. J. Theoret. Biol. 225(2): 275–283 · doi:10.1016/S0022-5193(03)00245-5 |
| [18] | Rubinov S.I. (1980) Cell kinetics. In: Segel L.A. (eds) Mathematical Models in Molecular and Cellular Biology. Cambridge University Press, Cambridge, pp 502–522 |
| [19] | Smith J.A., Martin L.(1973) Do cell cycle? Proc Nat. Acad. Sci. USA 70(4): 1263–1267 · doi:10.1073/pnas.70.4.1263 |
| [20] | Veiga-Fernandes H., Walter U., Bourgeois C., McLean A., Rocha B. (2000) Response of naive and memory CD8+ T cells to antigen stimulation in vivo. Nat. Immunol. 1, 47–53 · doi:10.1038/76907 |
| [21] | Venzon D.J., Moolgavkar S.H. (1988) A method for computing profile-likelihood-based confidence intervals. Appl. Statist. 37(1): 87–94 · doi:10.2307/2347496 |
| [22] | Wells A.D., Gudmundsdottir H., Turka L.A. (1997) Following the fate of individual T cells throughout activation and clonal expansion. Signals from T cell receptor and CD28 differentially regulate the induction and duration of a proliferative response. J. Clin. Invest. 100, 3173–3183 · doi:10.1172/JCI119873 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
