American Institute of Mathematical Sciences

Advanced
2012, 9(4): 699-736. doi: 10.3934/mbe.2012.9.699

A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays

H. Thomas Banks 1, , W. Clayton Thompson 2, , Cristina Peligero 3, , Sandra Giest 3, , Jordi Argilaguet 3, and  Andreas Meyerhans 3,

1. 

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8212
2. 

Center for Research in Scientific Computation, and Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8212, United States
3. 

ICREA Infection Biology Lab, Department of Experimental and Health Sciences, Univ. Pompeu Fabra, 08003 Barcelona, Spain, Spain, Spain, Spain

Received  February 2012 Revised  July 2012 Published  October 2012

Some key features of a mathematical description of an immune response are an estimate of the number of responding cells and the manner in which those cells divide, differentiate, and die. The intracellular dye CFSE is a powerful experimental tool for the analysis of a population of dividing cells, and numerous mathematical treatments have been aimed at using CFSE data to describe an immune response [30,31,32,37,38,42,48,49]. Recently, partial differential equation structured population models, with measured CFSE fluorescence intensity as the structure variable, have been shown to accurately fit histogram data obtained from CFSE flow cytometry experiments [18,19,52,54]. In this report, the population of cells is mathematically organized into compartments, with all cells in a single compartment having undergone the same number of divisions. A system of structured partial differential equations is derived which can be fit directly to CFSE histogram data. From such a model, cell counts (in terms of the number of divisions undergone) can be directly computed and thus key biological parameters such as population doubling time and precursor viability can be determined. Mathematical aspects of this compartmental model are discussed, and the model is fit to a data set. As in [18,19], we find temporal and division dependence in the rates of proliferation and death to be essential features of a structured population model for CFSE data. Variability in cellular autofluorescence is found to play a significant role in the data, as well. Finally, the compartmental model is compared to previous work, and statistical aspects of the experimental data are discussed.
Keywords: partial differential equations, cell division number, inverse problems., Cell proliferation, CFSE, label structured population dynamics.
Mathematics Subject Classification: Primary: 49N45, 92C37, 97M10; Secondary: 35F16, 35L4.
Citation: H. Thomas Banks, W. Clayton Thompson, Cristina Peligero, Sandra Giest, Jordi Argilaguet, Andreas Meyerhans. A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays. Mathematical Biosciences & Engineering, 2012, 9 (4) : 699-736. doi: 10.3934/mbe.2012.9.699
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show all references

References:
[1]

H. T. Banks, "A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering,", CRC Press/Taylor-Francis, (2012). Google Scholar

[2]

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[3]

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H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics,, Math. Biosciences, 183 (2003), 63. doi: 10.1016/S0025-5564(02)00218-3. Google Scholar

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H. T. Banks, D. M. Bortz, G. A. Pinter and L. K. Potter, Modeling and imaging techniques with potential for application in bioterrorism,, CRSC-TR03-02, FR28 (2003), 03. Google Scholar

[6]

H. T. Banks, L. W. Botsford, F. Kappel and C. Wang, Modeling and estimation in size structured population models,, LCDS/CSS Report 87-13, (1987), 87. Google Scholar

[7]

H. T. Banks, Frederique Charles, Marie Doumic, Karyn L. Sutton and W. Clayton Thompson, Label structured cell proliferation models,, Appl. Math. Letters, 23 (2010), 1412. doi: 10.1016/j.aml.2010.07.009. Google Scholar

[8]

H. T. Banks, M. Davidian, J. Samuels and K. L. Sutton, An inverse problem statistical methodology summary,, CRSC-TR08-01, (2008), 08. Google Scholar

[9]

H. T. Banks and J. L. Davis, A comparison of approximation methods for the estimation of probability distributions on parameters,, Appl. Num. Math., 57 (2007), 753. doi: 10.1016/j.apnum.2006.07.016. Google Scholar

[10]

H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich, A. K. Dhar and C. L. Browdy, A comparison of probabilistic and stochastic formulations in modeling growth uncertainty and variability,, CRSC-TR08-03, 3 (2009), 08. Google Scholar

[11]

H. T. Banks and B. G. Fitzpatrick, Inverse problems for distributed systems: statistical tests and ANOVA,, LCDS/CSS Report 88-16, 81 (1989), 88. Google Scholar

[12]

H. T. Banks and B. F. Fitzpatrick, Estimation of growth rate distributions in size-structured population models,, CAMS Tech. Rep. 90-2, 49 (1991), 90. Google Scholar

[13]

H. T. Banks and N. L. Gibson, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters,, CRSC-TR04-01, 18 (2005), 04. Google Scholar

[14]

H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters,, CRSC-TR05-29, 64 (2006), 05. Google Scholar

[15]

H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems,", Birkhauser, (1989). Google Scholar

[16]

H. T. Banks and G. A. Pinter, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue,, CRSC-TR04-03, 3 (2005), 04. Google Scholar

[17]

H. T. Banks and L. K. Potter, Probabilistic methods for addressing uncertainty and variability in biological models: Application to a toxicokinetic model,, CRSC-TR02-27, 192 (2004), 02. Google Scholar

[18]

H. T. Banks, Karyn L. Sutton, W. Clayton Thompson, G. Bocharov, Marie Doumic, Tim Schenkel, Jordi Argilaguet, Sandra Giest, Cristina Peligero and Andreas Meyerhans, A new model for the estimation of cell proliferation dynamics using CFSE data,, CRSC-TR11-05, 373 (2011), 11. doi: 10.1016/j.jim.2011.08.014. Google Scholar

[19]

H. T. Banks, Karyn L. Sutton, W. Clayton Thompson, Gennady Bocharov, Dirk Roose, Tim Schenkel and Andreas Meyerhans, Estimation of cell proliferation dynamics using CFSE data,, CRSC-TR09-17, 70 (2011), 09. doi: 10.1007/s11538-010-9524-5. Google Scholar

[20]

H. T. Banks, W. C. Thompson, C. Peligero, S. Giest, J. Argilaguet and A. Meyerhans, A compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays,, Technical Report CRSC-TR12-03, (2012), 12. Google Scholar

[21]

H. T. Banks and H. T. Tran, "Mathematical and Experimental Modeling of Physical and Biological Processes,", CRC Press, (2009). Google Scholar

[22]

H. T. Banks, B. G. Fitzpatrick, Laura K. Potter and Yue Zhang, Estimation of probability distributions for individual parameters using aggregate population observations,, CRSC-TR98-06, (1998), 98. Google Scholar

[23]

G. Bell and E. Anderson, Cell growth and division I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures,, Biophysical Journal, 7 (1967), 329. Google Scholar

[24]

K. P. Burnham and D. R. Anderson, "Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach,", (2nd Edition), (2002). Google Scholar

[25]

Nigel J. Burroughs and P. Anton van der Merwe, Stochasticity and spatial heterogeneity in T-cell activation,, Immunological Reviews, 216 (2007), 69. Google Scholar

[26]

R. Callard and P. D. Hodgkin, Modeling T- and B-cell growth and differentiation,, Immunological Reviews, 216 (2007), 119. Google Scholar

[27]

Robin E. Callard, Jaroslav Stark and Andrew J. Yates, Fratricide: a mechanism for T memory-cell homeostasis,, Trends in Immunology, 24 (2003), 370. doi: 10.1016/S1471-4906(03)00164-9. Google Scholar

[28]

R. J. Carroll and D. Ruppert, "Transformation and Weighting in Regression,", Chapman Hall, (2000). Google Scholar

[29]

M. Davidian and D. M. Giltinan, "Nonlinear Models for Repeated Measurement Data,", Chapman and Hall, (2000). Google Scholar

[30]

R. J. DeBoer, V. V. Ganusov, D. Milutinovic, P. D. Hodgkin and A. S. Perelson, Estimating lymphocyte division and death rates from CFSE data,, Bull. Math. Biol., 68 (2006), 1011. doi: 10.1007/s11538-006-9094-8. Google Scholar

[31]

R. J. DeBoer and Alan S. Perelson, Estimating division and death rates from CFSE data,, J. Comp. and Appl. Mathematics, 184 (2005), 140. doi: 10.1016/j.cam.2004.08.020. Google Scholar

[32]

E. K. Deenick, A. V. Gett and P. D. Hodgkin, Stochastic model of T cell proliferation: a calculus revealing IL-2 regulation of precursor frequencies, cell cycle time, and survival,, J. Immunology, 170 (2003), 4963. Google Scholar

[33]

Mark R Dowling, Dejan Milutinovic and Philip D Hodgkin, Modelling cell lifespan and proliferation: is likelihood to die or to divide independent of age?,, J. R. Soc. Interface, 2 (2005), 517. Google Scholar

[34]

K. Duffy and V. Subramanian, On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics,, J. Math. Biol., 59 (2009), 255. doi: 10.1007/s00285-008-0231-x. Google Scholar

[35]

D. A. Fulcher and S. W. J. Wong, Carboxyfluorescein diacetate succinimidyl ester-based assays for assessment of T cell function in the diagnostic laboratory,, Immunology and Cell Biology, 77 (1999), 559. doi: 10.1046/j.1440-1711.1999.00870.x. Google Scholar

[36]

Vitaly V. Ganusov, Dejan Milutinovi and Rob J. De Boer, IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: Insights from modeling CFSE data,, J. Immunology, 179 (2007), 950. Google Scholar

[37]

V. V. Ganusov, S. S. Pilyugin, R. J. De Boer, K. Murali-Krishna, R. Ahmed and R. Antia, Quantifying cell turnover using CFSE data,, J. Immunological Methods, 298 (2005), 183. doi: 10.1016/j.jim.2005.01.011. Google Scholar

[38]

A. V. Gett and P. D. Hodgkin, A cellular calculus for signal integration by T cells,, Nature Immunology, 1 (2000), 239. Google Scholar

[39]

M. Kot, "Elements of Mathematical Ecology,", Cambridge University Press, (2001). Google Scholar

[40]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. Math. Biol., 28 (1990), 671. doi: 10.1007/BF00160231. Google Scholar

[41]

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