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. 2011 Jan;73(1):116-50.
doi: 10.1007/s11538-010-9524-5. Epub 2010 Mar 3.

Estimation of cell proliferation dynamics using CFSE data

H T Banks  1 , Karyn L Sutton, W Clayton Thompson, Gennady Bocharov, Dirk Roose, Tim Schenkel, Andreas Meyerhans
Affiliations

Estimation of cell proliferation dynamics using CFSE data

H T Banks et al. Bull Math Biol. 2011 Jan.

Abstract

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.

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Figures

Fig. 1
Fig. 1
Original CFSE histogram data.
Fig. 2
Fig. 2
Left: OLS Residuals vs. Model for observations with constant variance. Right: OLS Residuals vs. Model for observations with nonconstant variance.
Fig. 3
Fig. 3
Modified residuals for observations with nonconstant variance.
Fig. 4
Fig. 4
Graphical presentation of estimated (OLS) birth and death rate functions α(z) (left) and β(z) (right).
Fig. 5
Fig. 5
OLS best fit model solution to original PDE formulation with Eq. (2) in comparison to the data: t = 0, 24, 48 hrs.
Fig. 6
Fig. 6
OLS best fit model solution to original PDE formulation with Eq. (2) in comparison to the data: t = 72, 96, 120 hrs.
Fig. 7
Fig. 7
Original data sets shown in translated log intensity s = z + ct, with c = ĉ = .0032888, as estimated with the OLS procedure for the modified model (14). Note that subsequent division peaks are now strongly correlated with specific regions in the state variable, unlike the original log intensity variable z (see Fig. 1).
Fig. 8
Fig. 8
Graphical representation of the best fit (OLS) death rate β(s) (top) and proliferation rate α(t, s) (bottom). Numerical values are given in Tables 4 and 5.
Fig. 9
Fig. 9
Improved model solution evaluated at the best fit (OLS) parameters in comparison to the original data: t = 0, 24, 48 hrs.
Fig. 10
Fig. 10
Improved model solution evaluated at the best fit (OLS) parameters in comparison to the original data: t = 72, 96, 120 hrs.
Fig. 11
Fig. 11
Best fit (OLS) model solution shown in terms of the translated coordinate s = z + ct.
Fig. 12
Fig. 12
Estimated probability distribution of label loss rates c within the population.
Fig. 13
Fig. 13
OLS residuals as a function of model value for each time measurement.
Fig. 14
Fig. 14
GLS residuals as a function of model value for each time measurement.
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References

    1. Banks HT, Davis JL. A comparison of approximation methods for the estimation of probability distributions on parameters. Appl Numer Math. 2007;57:753–777.
    1. Banks HT, Fitzpatrick BG. Inverse problems for distributed systems: statistical tests and ANOVA. Proc. International Symposium on Math. Approaches to Envir. and Ecol. Problems, Springer Lecture Note in Biomath; July, 1988; Berlin: Springer; Brown University; 1989. pp. 262–273. LCDS/CCS Rep. 88-16.
    1. Banks HT, Fitzpatrick BG. J Math Biol. Vol. 28. University of Southern California; 1990. Statistical methods for model comparison in parameter estimation problems for distributed systems; pp. 501–527. CAMS Tech. Rep. 89-4, September, 1989.
    1. Banks HT, Fitzpatrick BG. Quart Appl Math. Vol. 49. University of Southern California; 1991. Estimation of growth rate distributions in size-structured population models; pp. 215–235. CAMS Tech. Rep. 90-2, January, 1990.
    1. Banks HT, Iles DW. On compactness of admissible parameter sets: convergence and stability in inverse problems for distributed parameter systems. Proc. Conf. on Control Systems Governed by PDE’s; February, 1986; Gainesville, FL. 1987.
    2. Science. Vol. 97. Springer; Berlin: NASA Langley Res. Ctr; Hampton VA: 1986. Springer Lecture Notes in Control & Inf; pp. 130–142. ICASE Report #86-38.

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