# Mathematical Models of Dividing Cell Populations: Application to CFSE Data

H.T. Banks; W. Clayton Thompson

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 5, page 24-52
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topBanks, H.T., and Clayton Thompson, W.. "Mathematical Models of Dividing Cell Populations: Application to CFSE Data." Mathematical Modelling of Natural Phenomena 7.5 (2012): 24-52. <http://eudml.org/doc/222192>.

@article{Banks2012,

abstract = {Flow cytometric analysis using intracellular dyes such as CFSE is a powerful experimental
tool which can be used in conjunction with mathematical modeling to quantify the dynamic
behavior of a population of lymphocytes. In this survey we begin by providing an overview
of the mathematically relevant aspects of the data collection procedure. We then present
an overview of the large body of mathematical models, along with their assumptions and
uses, which have been proposed to describe the dynamics of proliferating cell populations.
While much of this body of work has been aimed at modeling the generation structure (cells
per generation) of the proliferating population, several recent models have considered the
more fundamental task of modeling CFSE histogram data directly. Such models are analyzed
and recent results are discussed. Finally, directions for future research are
suggested.},

author = {Banks, H.T., Clayton Thompson, W.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {cell proliferation; cell division number; CFSE; ordinary differential equations; cytons; label structured population dynamics; partial differential equations; inverse problems},

language = {eng},

month = {10},

number = {5},

pages = {24-52},

publisher = {EDP Sciences},

title = {Mathematical Models of Dividing Cell Populations: Application to CFSE Data},

url = {http://eudml.org/doc/222192},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Banks, H.T.

AU - Clayton Thompson, W.

TI - Mathematical Models of Dividing Cell Populations: Application to CFSE Data

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/10//

PB - EDP Sciences

VL - 7

IS - 5

SP - 24

EP - 52

AB - Flow cytometric analysis using intracellular dyes such as CFSE is a powerful experimental
tool which can be used in conjunction with mathematical modeling to quantify the dynamic
behavior of a population of lymphocytes. In this survey we begin by providing an overview
of the mathematically relevant aspects of the data collection procedure. We then present
an overview of the large body of mathematical models, along with their assumptions and
uses, which have been proposed to describe the dynamics of proliferating cell populations.
While much of this body of work has been aimed at modeling the generation structure (cells
per generation) of the proliferating population, several recent models have considered the
more fundamental task of modeling CFSE histogram data directly. Such models are analyzed
and recent results are discussed. Finally, directions for future research are
suggested.

LA - eng

KW - cell proliferation; cell division number; CFSE; ordinary differential equations; cytons; label structured population dynamics; partial differential equations; inverse problems

UR - http://eudml.org/doc/222192

ER -

## References

top- O. Arino, E. Sanchez, G.F. Webb. Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence. Mathematical Analysis and Applications, 215 (1997), 499–513. Zbl0886.92020
- B. Asquith, C. Debacq, A. Florins, N. Gillet, T. Sanchez-Alcaraz, A. Mosley, L. Willems. Quantifying lymphocyte kinetics in vivo using carboxyfluorein diacetate succinimidyl ester. Proc. R. Soc. B, 273 (2006), 1165–1171.
- H.T. Banks. A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering. CRC Press/Taylor-Francis, Boca Raton London New York, 2012.
- H.T. Banks, V. A. Bokil, S. Hu, F.C.T. Allnutt, R. Bullis, A.K. Dhar, C.L. Browdy, Shrimp biomass and viral infection for production of biological countermeasures, CRSC-TR05-45. North Carolina State University, December 2005 ; Mathematical Biosciences and Engineering, 3 (2006), 635–660. Zbl1113.92023
- H.T. Banks, D.M. Bortz, S.E. Holte, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics, Math. Biosciences.183 (2003), 63–91. Zbl1011.92037
- H.T. Banks, D. M. Bortz, G.A. Pinter, L.K. Potter. Modeling and imaging techniques with potential for application in bioterrorism. CRSC-TR03-02, North Carolina State University, January 2003 ; Chapter 6 in Bioterrorism : Mathematical Modeling Applications in Homeland Security, (H.T. Banks and C. Castillo-Chavez, eds.), Frontiers in Applied Math, FR28, SIAM, Philadelphia, PA, 2003, 129–154.
- H.T. Banks, L.W. Botsford, F. Kappel, C. Wang, Modeling and estimation in size structured population models. LCDS/CSS Report 87-13, Brown University, March 1987 ; Proc. 2nd Course on Math. Ecology (Trieste, December 8-12, 1986), World Scientific Press, Singapore, 1988, 521–541.
- H.T. Banks, F. Charles, M. Doumic, K. L. Sutton, W. C. Thompson. Label structured cell proliferation models. CRSC-TR10-10, North Carolina State University, June 2010 ; Appl. Math. Letters, 23 (2010), 1412–1415. Zbl1205.35321
- H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich, A.K. Dhar, C.L. Browdy. A comparison of probabilistic and stochastic differential equations in modeling growth uncertainty and variability. CRSC-TR08-03, North Carolina State University, February 2008 ; Journal of Biological Dynamics, 3 (2009), 130–148. Zbl1342.92156
- H.T. Banks, J.L. Davis, S. Hu, A computational comparison of alternatives to including uncertainty in structured population models. CRSC-TR09-14, North Carolina State University June 2009 ; in Three Decades of Progress in Systems and Control, X. Hu, U. Jonsson, B. Wahlberg, B. Ghosh (Eds.), Springer, 2010, 19–33.
- H.T. Banks, B.F. Fitzpatrick. Estimation of growth rate distributions in size-structured population models. CAMS Tech. Rep. 90-2, Univ. of Southern California, January 1990 ; Quart. Appl. Math., 49 (1991), 215–235. Zbl0731.92021
- H.T. Banks, B.G. Fitzpatrick, L.K. Potter, Y. Zhang, Estimation of probability distributions for individual parameters using aggregate population observations. CRSC-TR98-06, North Carolina State University, January 1998 ; Stochastic Analysis, Control, Optimization and Applications (W.McEneaney, G. Yin, and Q. Zhang, eds.), Birkhauser, 1998, 353–371. Zbl0922.92016
- H.T. Banks, N.L. Gibson, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters. CRSC-TR04-01, North Carolina State University, January 2004 ; Applied Math. Letters, 18 (2005), 423–430. Zbl1069.78006
- H.T. Banks, N.L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters. CRSC-TR05-29, North Carolina State University, August2005 ; Quarterly of Applied Mathematics, 64 (2006), 749–795. Zbl1124.35090
- H.T. Banks, K. Holm, F. Kappel, Comparison of optimal design methods in inverse problems. CRSC-TR10-11, North Carolina State University, May 2011 ; Inverse Problems, 27 (2011), 075002. Zbl1271.62169
- H.T. Banks, S. Hu. Nonlinear stochastic Markov processes and modeling uncertainty in populations. CRSC-TR11-02, North Carolina State University, January 2011 ; Mathematical Bioscience and Engineering, 9 (2012), 1–25. Zbl1259.60090
- H.T. Banks, S. Hu. Uncertainty propagation in physiologically structured population models. CRSC-TR12-08, North Carolina State University, Raleigh, NC, March 2012 ; Journal on Mathematical Modelling of Natural Phenomena, submitted.
- H.T. Banks, K. Kunisch. Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 1989. Zbl0695.93020
- H.T. Banks and G.A. Pinter. A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue. CRSC-TR04-03, North Carolina State University, January 2004 ; IAM J. Multiscale Modeling and Simulation, 3 (2005), 395–412. Zbl1140.93313
- H.T. Banks, L.K. Potter. Probabilistic methods for addressing uncertainty and variability in biological models : Application to a toxicokinetic model. CRSC-TR02-27, North Carolina State University, September 2002 ; Math. Biosci., 192 (2004), 193–225. Zbl1073.92051
- H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, G. Bocharov, Marie Doumic, Tim Schenkel, Jordi Argilaguet, Sandra Giest, Cristina Peligero, Andreas Meyerhans. A New Model for the Estimation of Cell Proliferation Dynamics Using CFSE Data. CRSC-TR11-05, North Carolina State University, Revised July 2011 ; J. Immunological Methods, 373 (2011), 143–160 ; DOI :. URI10.1016/j.jim.2011.08.014
- H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, G. Bocharov, D. Roose, T. Schenkel, A. Meyerhans. Estimation of cell proliferation dynamics using CFSE data. CRSC-TR09-17, North Carolina State University, August 2009 ; Bull. Math. Biol., 70 (2011), 116–150. Zbl1209.92012
- H.T. 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. CRSC-TR12-03, North Carolina State University, January 2012 ; Math Biosci. Eng., to appear. Zbl1259.92017
- H.T. Banks, W. Clayton Thompson. A division-dependent compartmental model with cyton and intracellular label dynamics. CRSC-TR12-12, North Carolina State University, May 2012 ; Intl. J. Pure and Appl. Math77 (2012), 119–147. Zbl1247.92007
- H.T. Banks, H.T. Tran, D.E. Woodward. Estimation of variable coefficients in the Fokker-Planck equations using moving node finite elements. SIAM J. Numer. Anal., 30 (1993), 1574–1602. Zbl0791.65098
- B. Basse, B. Baguley, E. Marshall, G. Wake, D. Wall. Modelling the flow cytometric data obtained from unperturbed human tumour cell lines : Parameter fitting and comparison. Bull. Math. Biol., 67 (2005), 815–830. Zbl1334.92217
- F. Bekkal Brikci, J. Clairambault, B. Ribba, B. Perthame. An age-and-cyclin-structured cell population model for healthy and tumoral tissues. Math. Biol., 57 (2008), 91–110. Zbl1148.92014
- G. Bell, E. Anderson. Cell Growth and Division I. A Mathematical Model with Applications to Cell Volume Distributions in Mammalian Suspension Cultures, Biophysical Journal, 7 (1967), 329–351.
- S. Bernard, L. Pujo-Menjouet, M.C. Mackey. Analysis of cell kinetics using a cell division marker : Mathematical modeling of experimental data. Biophysical Journal, 84 (2003), 3414–3424.
- S. Bonhoeffer, H. Mohri, D. Ho, A.S. Perelson. Quantification of cell turnover kinetics using 5-Bromo-2’-deoxyuridine. Immunology, 64 (2000), 5049–5054.
- Jose A. M. Borghans, R.J. de Boer. Quantification of T-cell dynamics : from telomeres to DNA labeling. Immunological Reviews, 216 (2007), 35–47.
- K.P. Burnham, D.R. Anderson. Model Selection and Multimodel Inference : A Practical Information-Theoretic Approach (2nd Edition), Springer, New York, 2002. Zbl1005.62007
- R. Callard, P.D. Hodgkin. Modeling T- and B-cell growth and differentiation. Immunological Reviews, 216 (2007), 119–129.
- “Cyton Calculator”, Walter and Eliza Ball Institute of Medical Research. Available Online. Accessed 16 March 2012. URIhttp://www.wehi.edu.au/faculty_members/research_projects/cyton_calculator
- R.J. DeBoer, V.V. Ganusov, D. Milutinovic, P.D. Hodgkin, A.S. Perelson. Estimating lymphocyte division and death rates from CFSE data. Bull. Math. Biol., 68 (2006), 1011–1031. Zbl1334.92112
- R.J. DeBoer, A. S. Perelson. Estimating division and death rates from CFSE data. Comp. and Appl. Mathematics, 184 (2005), 140–164. Zbl1074.92016
- E.K. Deenick, A.V. Gett, P.D. Hodgkin. Stochastic model of T cell proliferation : a calculus revealing IL-2 regulation of precursor frequencies, cell cycle time, and survival. Immunology, 170 (2003), 4963–4972.
- K. Duffy, V. Subramanian. On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics. Math. Biol., 59 (2009), 255–285. Zbl1311.92061
- J.Z. Farkas. Stability conditions for the non-linear McKendrick equations. Appl. Math. and Comp., 156 (2004), 771–777. Zbl1068.45017
- J.Z. Farkas. Stability conditions for a non-linear size-structured modelNonlinear Analysis : Real World Applications, 6 (2005), 962–969. Zbl1082.35027
- V. V. Ganusov, D. Milutinovi, R. J. De Boer. IL-2 regulates expansion of CD4+ T cell populations by affecting cell death : insights from modeling CFSE data. Immunology, 179 (2007), 950–957.
- V.V. Ganusov, S.S. Pilyugin, R.J. De Boer, K. Murali-Krishna, R. Ahmed, R. Antia. Quantifying cell turnover using CFSE data. Immunological Methods, 298 (2005), 183–200.
- A.V. Gett, P.D. Hodgkin. A cellular calculus for signal integration by T cells. Nature Immunology, 1 (2000), 239–244.
- M. Gyllenberg, G.F. Webb. Age-size structure in populations with quiescence. Mathematical Biosciences, 86 (1987), 67–95. Zbl0632.92014
- M. Gyllenberg, G.F. Webb. A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol., 28 (1990), 671–694. Zbl0744.92026
- J. Hasenauer, D. Schittler, F. Allgöwer. A computational model for proliferation dynamics of division- and label-structured populations. arXive.org, arXiv :1202.4923v1,22Feb,2012. Zbl06134764
- E.D. Hawkins, Mirja Hommel, M.L. Turner, F. Battye, J. Markham, P.D. Hodgkin. Measuring lymphocyte proliferation, survival and differentiation using CFSE time-series data. Nature Protocols, 2 (2007), 2057–2067.
- E.D. Hawkins, M.L. Turner, M.R. Dowling, C. van Gend, P.D. Hodgkin. A model of immune regulation as a consequence of randomized lymphocyte division and death times. Proc. Natl. Acad. Sci., 104 (2007), 5032–5037.
- E.D. Hawkins, J.F. Markham, L.P. McGuinness, P.D. Hodgkin. A single-cell pedigree analysis of alternative stochastic lymphocyte fates. Proc. Natl. Acad. Sci., 106 (2009), 13457–13462.
- O. Hyrien, M.S. Zand. A mixture model with dependent observations for the analysis of CFSE-labeling experiments. American Statistical Association, 103 (2008), 222–239. Zbl05564482
- O. Hyrien, R. Chen, M.S. Zand. An age-dependent branching process model for the analysis of CFSE-labeling experiments. Biology Direct, 5 (2010), Published Online.
- H.Y. Lee, E.D. Hawkins, M.S. Zand, T. Mosmann, H. Wu, P.D. Hodgkin, A.S. Perelson. Interpreting CFSE obtained division histories of B cells in vitro with Smith-Martin and Cyton type models. Bull. Math. Biol., 71 (2009), 1649–1670. Zbl1173.92011
- H.Y. Lee, A.S. Perelson. Modeling T cell proliferation and death in vitro based on labeling data : generalizations of the Smith-Martin cell cycle model. Bull. Math. Biol., 70 (2008), 21–44. Zbl1281.92020
- K. Leon, J. Faro, J. Carneiro. A general mathematical framework to model generation structure in a population of asynchronously dividing cells. Theoretical Biology, 229 (2004), 455–476.
- T. Luzyanina, D. Roose, G. Bocharov. Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data. Math. Biol., 59 (2009), 581–603. Zbl1231.92027
- T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans, G. Bocharov. Numerical modelling of label-structured cell population growth using CFSE distribution data. Theoretical Biology and Medical Modelling, 4 (2007), Published Online.
- A.B. Lyons. Divided we stand : tracking cell proliferation with carboxyfluorescein diacetate succinimidyl ester. Immunology and Cell Biology, 77 (1999), 509–515.
- A.B. Lyons, J. Hasbold, P.D. Hodgkin. Flow cytometric analysis of cell division history using diluation of carboxyfluorescein diacetate succinimidyl ester, a stably integrated fluorescent probe. Methods in Cell Biology, 63 (2001), 375–398.
- A.B. Lyons, K.V. Doherty. Flow cytometric analysis of cell division by dye dilution. Current Protocols in Cytometry, (2004), 9.11.1-9.11.10.
- A.B. Lyons, C.R. Parish. Determination of lymphocyte division by flow cytometry. Immunol. Methods, 171 (1994), 131–137.
- G. Matera, M. Lupi, P. Ubezio. Heterogeneous cell response to topotecan in a CFSE-based proliferative test. Cytometry A, 62 (2004), 118–128.
- J.A. Metz, O. Diekmann. The Dynamics of Physiologically Structured Populations. Springer Lecture Notes in Biomathematics 68, Heidelberg, 1986. Zbl0614.92014
- H. Miao, X. Jin, A. Perelson, H. Wu. Evaluation of multitype mathemathematical modelsfor CFSE-labeling experimental data. Bull. Math. Biol., 74 (2012), 300–326 ; DOI 10.1007/s11538-011-9668-y. Zbl1317.92028
- K. Murphy, aneway’s Immunobiology, 8th[entity !#x20 !]Edition. Garland Science, London New York, 2012.
- R.E. Nordon, Kap-Hyoun Ko, R. Odell, T. Schroeder. Multi-type branching models to describe cell differentiation programs. Theoretical Biology, 277 (2011), 7–18.
- R.E. Nordon, M. Nakamura, C. Ramirez, R. Odell. Analysis of growth kinetics by division tracking. Immunology and Cell Biology, 77 (1999), 523–529.
- C. Parish. Fluorescent dyes for lymphocyte migration and proliferation studies. Immunology and Cell Biol., 77 (1999), 499–508.
- B. Perthame. Transport Equations in Biology. Birkhauser Frontiers in Mathematics, Basel, 2007. Zbl1185.92006
- S. S. Pilyugin, V. V. Ganusov, K. Murali-Krishnac, R. Ahmed, R. Antia. The rescaling method for quantifying the turnover of cell populations. Theoretical Biology, 225 (2003), 275–283.
- B.J.C. Quah, C.R. Parish. New and improved methods for measuring lymphocyte proliferation in vitro and in vivo using CFSE-like fluorescent dyes. Immunological Methods, (2012), to appear. Zbl06149134
- B. Quah, H. Warren, C. Parish. Monitoring lymphocyte proliferation in vitro and in vivo with the intracellular fluorescent dye carboxyfluorescein diacetate succinimidyl ester. Nature Protocols, 2 (2007), 2049–2056.
- P. Revy, M. Sospedra, B. Barbour, A. Trautmann. Functional antigen-independent synapses formed between T cells and dendritic cells. Nature Immunology, 2 (2001), 925–931.
- M. Roederer. Interpretation of cellular proliferation data : Avoid the panglossian, Cytometry A, 79 (2011), 95–101.
- D. Schittler, J. Hasenauer, F. Allgöwer. A generalized model for cell proliferation : Integrating division numbers and label dynamics. Proc. Eighth International Workshop on Computational Systems Biology (WCSB 2011), June 2001, Zurich, Switzerland, p. 165–168.
- J. Sinko, W. Streifer. A New Model for Age-Size Structure of a Population. Ecology, 48 (1967), 910–918.
- J.A. Smith, L. Martin. Do Cells Cycle ?Proc. Natl. Acad. Sci., 70 (1973), 1263–1267.
- V.G. Subramanian, K.R. Duffy, M.L. Turner, P.D. Hodgkin. Determining the expected variability of immune responses using the cyton model. Math. Biol., 56 (2008), 861–892. Zbl1206.92013
- H. Veiga-Fernandez, U. Walter, C. Bourgeois, A. McLean, B. Rocha. Response of naive and memory CD8+ T cells to antigen stimulation in vivo, Nature Immunology. 1 (2000), 47–53.
- W. C. Thompson. Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based Proliferation Assays. Ph.D. Dissertation, Dept. of Mathematics, North Carolina State University, Raleigh, December, 2011.
- P.K. Wallace, J.D. Tario, Jr., J.L. Fisher, S.S. Wallace, M.S. Ernstoff, K.A. Muirhead. Tracking antigen-driven responses by flow cytometry : monitoring proliferation by dye dilution. Cytometry A, 73 (2008), 1019–1034.
- H. S. Warren. Using carboxyfluorescein diacetate succinimidyl ester to monitor human NK cell division : Analysis of the effect of activating and inhibitory class I MHC receptors. Immunology and Cell Biology, 77 (1999), 544–551.
- C. Wellard, J. Markham, E.D. Hawkins, P.D. Hodgkin. The effect of correlations on the population dynamics of lymphocytes. Theoretical Biology, 264 (2010), 443–449.
- J.M. Witkowski. Advanced application of CFSE for cellular tracking. Current Protocols in Cytometry, 44 (2008), 9.25.1–9.25.8.
- A. Yates, C. Chan, J. Strid, S. Moon, R. Callard, A.J.T. George, J. Stark. Reconstruction of cell population dynamics using CFSE. BMC Bioinformatics, 8 (2007), Published Online.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.