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

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.
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
References:
[1]

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.

[2]

H. T. Banks and Kathleen Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems, 17 (2001), 95-111. doi: 10.1088/0266-5611/17/1/308.

[3]

H. T.Banks, V. A. Bokil, S. Hu, F. C. T. Allnutt, R. Bullis, A. K. Dhar and 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.

[4]

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), 63-91. doi: 10.1016/S0025-5564(02)00218-3.

[5]

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, North Carolina State University, January 2003; Chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, (eds. H. T. Banks and C. Castillo-Chavez), Frontiers in Applied Math, FR28, SIAM, Philadelphia, PA, 2003, 129-154.

[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, Brown University, March 1987; Proc. 2nd Course on Math. Ecology, (Trieste, December 8-12, 1986) World Scientific Press, Singapore, 1988, 521-541.

[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), 1412-1415. doi: 10.1016/j.aml.2010.07.009.

[8]

H. T. Banks, M. Davidian, J. Samuels and K. L. Sutton, An inverse problem statistical methodology summary, CRSC-TR08-01, North Carolina State University, January 2008; Chapter 11 in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, et al.), Berlin Heidelberg New York, 2009, 249-302.

[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), 753-777. doi: 10.1016/j.apnum.2006.07.016.

[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, North Carolina State University, February 2008; Journal of Biological Dynamics, 3 (2009), 130-148.

[11]

H. T. Banks and B. G. Fitzpatrick, Inverse problems for distributed systems: statistical tests and ANOVA, LCDS/CSS Report 88-16, Brown University, July 1988; Proc. International Symposium on Math. Approaches to Envir. and Ecol. Problems, Springer Lecture Notes in Biomath., 81 (1989), 262-273.

[12]

H. T. Banks and 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.

[13]

H. T. Banks and 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.

[14]

H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters, CRSC-TR05-29, North Carolina State University, August 2005; Quarterly of Applied Mathematics, 64 (2006), 749-795.

[15]

H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems," Birkhauser, Boston, 1989.

[16]

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; SIAM J. Multiscale Modeling and Simulation, 3 (2005), 395-412.

[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, North Carolina State University, September 2002; Math. Biosci., 192 (2004), 193-225.

[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, North Carolina State University, Revised July 2011; J. Immunological Methods, 373 (2011), 143-160. doi: 10.1016/j.jim.2011.08.014.

[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, North Carolina State University, August 2009; Bull. Math. Biol., 70 (2011), 116-150. doi: 10.1007/s11538-010-9524-5.

[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, North Carolina State University, January 2012.

[21]

H. T. Banks and H. T. Tran, "Mathematical and Experimental Modeling of Physical and Biological Processes," CRC Press, Boca Raton London New York, 2009.

[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, North Carolina State University, January 1998; Stochastic Analysis, Control, Optimization and Applications, (eds. W. McEneaney, G. Yin and Q. Zhang), Birkhäuser, (1998), 353-371.

[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), 329-351.

[24]

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

[25]

Nigel J. Burroughs and P. Anton van der Merwe, Stochasticity and spatial heterogeneity in T-cell activation, Immunological Reviews, 216 (2007), 69-80.

[26]

R. Callard and P. D. Hodgkin, Modeling T- and B-cell growth and differentiation, Immunological Reviews, 216 (2007), 119-129.

[27]

Robin E. Callard, Jaroslav Stark and Andrew J. Yates, Fratricide: a mechanism for T memory-cell homeostasis, Trends in Immunology, 24 (2003), 370-375. doi: 10.1016/S1471-4906(03)00164-9.

[28]

R. J. Carroll and D. Ruppert, "Transformation and Weighting in Regression," Chapman Hall, London, 2000.

[29]

M. Davidian and D. M. Giltinan, "Nonlinear Models for Repeated Measurement Data," Chapman and Hall, London, 2000.

[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), 1011-1031. doi: 10.1007/s11538-006-9094-8.

[31]

R. J. DeBoer and Alan S. Perelson, Estimating division and death rates from CFSE data, J. Comp. and Appl. Mathematics, 184 (2005), 140-164. doi: 10.1016/j.cam.2004.08.020.

[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), 4963-4972.

[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), 517-526.

[34]

K. Duffy and V. Subramanian, On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics, J. Math. Biol., 59 (2009), 255-285. doi: 10.1007/s00285-008-0231-x.

[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), 559-564. doi: 10.1046/j.1440-1711.1999.00870.x.

[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), 950-957.

[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), 183-200. doi: 10.1016/j.jim.2005.01.011.

[38]

A. V. Gett and P. D. Hodgkin, A cellular calculus for signal integration by T cells, Nature Immunology, 1 (2000), 239-244.

[39]

M. Kot, "Elements of Mathematical Ecology," Cambridge University Press, Cambridge, UK, 2001.

[40]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. doi: 10.1007/BF00160231.

[41]

J. Hasenauer, D. Schittler, and F. Allgöwer, A computational model for proliferation dynamics of division- and label-structured populations, arXiv:1202.4923v1, 22 Feb, 2012.

[42]

E. D. Hawkins, Mirja Hommel, M. L Turner, Francis Battye, J. Markham and P. D Hodgkin, Measuring lymphocyte proliferation, survival and differentiation using CFSE time-series data, Nature Protocols, 2 (2007), 2057-2067.

[43]

E. D. Hawkins, J. F. Markham, L. P. McGuinness and P. D. Hodgkin, A single-cell pedigree analysis of alternative stochastic lymphocyte fates, Proc. Natl. Acad. Sci., 106 (2009), 13457-13462. doi: 10.1073/pnas.0905629106.

[44]

Mirja Hommel and Philip D. Hodgkin, TCR affinity promotes CD8+ T-cell expansion by regulating survival, J. Immunology, 179 (2007), 2250-2260.

[45]

O. Hyrien and M. S. Zand, A mixture model with dependent observations for the analysis of CFSE-labeling experiments, J. American Statistical Association, 103 (2008), 222-239. doi: 10.1198/016214507000000194.

[46]

O. Hyrien, R. Chen and M. S. Zand, An age-dependent branching process model for the analysis of CFSE-labeling experiments, Biology Direct, 5 (2010), Published Online.

[47]

D. E. Kirschner, S. T. Chang, T. W. Riggs, N. Perry and J. J. Linderman, Toward a multiscale model of antigen presentation in immunity, Immunological Reviews, 216 (2007), 93-118.

[48]

H. Y. Lee, E. D. Hawkins, M. S. Zand, T. Mosmann, H. Wu, P. D. Hodgkin and 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. doi: 10.1007/s11538-009-9418-6.

[49]

H. Y. Lee and 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. doi: 10.1007/s11538-007-9239-4.

[50]

K. Leon, J. Faro and J. Carneiro, A general mathematical framework to model generation structure in a population of asynchronously dividing cells, J. Theoretical Biology, 229 (2004), 455-476. doi: 10.1016/j.jtbi.2004.04.011.

[51]

Y. Louzoun, The evolution of mathematical immunology, Immunological Reviews, 216 (2007), 9-20.

[52]

T. Luzyanina, D. Roose and G. Bocharov, Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data, J. Math. Biol., 59 (2009), 581-603. doi: 10.1007/s00285-008-0244-5.

[53]

T. Luzyanina, M. Mrusek, J. T. Edwards, D. Roose, S. Ehl and G. Bocharov, Computational analysis of CFSE proliferation assay, J. Math. Biol., 54 (2007), 57-89. doi: 10.1007/s00285-006-0046-6.

[54]

T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans and G. Bocharov, Numerical modelling of label-structured cell population growth using CFSE distribution data, Theoretical Biology and Medical Modelling, 4 (2007), Published Online.

[55]

A. B. Lyons and C. R. Parish, Determination of lymphocyte division by flow cytometry, J. Immunol. Methods, 171 (1994), 131-137. doi: 10.1016/0022-1759(94)90236-4.

[56]

A. B. Lyons, J. Hasbold and 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. doi: 10.1016/S0091-679X(01)63021-8.

[57]

G. Matera, M. Lupi and P. Ubezio, Heterogeneous cell response to topotecan in a CFSE-based proliferative test, Cytometry A, 62 (2004), 118-128. doi: 10.1002/cyto.a.20097.

[58]

J. A. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations," Springer Lecture Notes in Biomathematics, 68, 1986.

[59]

H. Miao, X. Jin, A. Perelson and 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.

[60]

Robert E. Nordon, Kap-Hyoun Ko, Ross Odell and Timm Schroeder, Multi-type branching models to describe cell differentiation programs, J. Theoretical Biology, 277 (2011), 7-18. doi: 10.1016/j.jtbi.2011.02.006.

[61]

R. E. Nordon, M. Nakamura, C. Ramirez and R. Odell, Analysis of growth kinetics by division tracking, Immunology and Cell Biology, 77 (1999), 523-529. doi: 10.1046/j.1440-1711.1999.00869.x.

[62]

C. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies, Immunology and Cell Biol., 77 (1999), 499-508. doi: 10.1046/j.1440-1711.1999.00877.x.

[63]

Sergei S. Pilyugin, Vitaly V. Ganusov, Kaja Murali-Krishnac, Rafi Ahmed and Rustom Antia, The rescaling method for quantifying the turnover of cell populations, J. Theoretical Biology, 225 (2003), 275-283. doi: 10.1016/S0022-5193(03)00245-5.

[64]

B. Quah, H. Warren and 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.

[65]

P. Revy, M. Sospedra, B. Barbour and A. Trautmann, Functional antigen-independent synapses formed between T cells and dendritic cells, Nature Immunology, 2 (2001), 925-931.

[66]

G. A. Sever and C. J. Wild, "Nonlinear Regression," Wiley, Hoboken, NJ, 2003.

[67]

D. Schittler, J. Hasenauer and 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, 165-168

[68]

J. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.

[69]

V. G. Subramanian, K. R. Duffy, M. L. Turner and P. D. Hodgkin, Determining the expected variability of immune responses using the cyton model, J. Math. Biol., 56 (2008), 861-892. doi: 10.1007/s00285-007-0142-2.

[70]

David T. Terrano, Meenakshi Upreti and Timothy C. Chambers, Cyclin-dependent kinase 1-mediated $Bcl-x_L$/Bcl-2 phosphorylation acts as a functional link coupling mitotic arrest and apoptosis, Mol. Cell. Biol., 30 (2010), 640-656. doi: 10.1128/MCB.00882-09.

[71]

W. Clayton Thompson, "Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based Proliferation Assays," Ph.D. Dissertation, North Carolina State University, December, 2011.

[72]

B. Tummers, DataThief III. 2006. (http://datathief.org/)

[73]

M. L. Turner, E. D. Hawkins and P. D. Hodgkin, Quantitative regulation of B cell division destiny by signal strength, J. Immunology, 181 (2008), 374-382.

[74]

H. Veiga-Fernandez, U. Walter, C. Bourgeois, A. McLean and B. Rocha, Response of naive and memory CD8+ T cells to antigen stimulation in vivo, Nature Immunology, 1 (2000), 47-53.

[75]

P. K. Wallace, J. D. Tario, Jr., J. L. Fisher, S. S. Wallace, M. S. Ernstoff and K. A. Muirhead, Tracking antigen-driven responses by flow cytometry: monitoring proliferation by dye dilution, Cytometry A, 73 (2008), 1019-1034.

[76]

C. Wellard, J. Markham, E. D. Hawkins and P. D. Hodgkin, The effect of correlations on the population dynamics of lymphocytes, J. Theoretical Biology, 264 (2010), 443-449. doi: 10.1016/j.jtbi.2010.02.019.

[77]

J. M. Witkowski, Advanced application of CFSE for cellular tracking, Current Protocols in Cytometry, (2008), 9.25.1-9.25.8.

[78]

A. Yates, C. Chan, J. Strid, S. Moon, R. Callard, A. J. T. George and J. Stark, Reconstruction of cell population dynamics using CFSE, BMC Bioinformatics, 8 (2007), Published Online.

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, Boca Raton London New York, 2012.

[2]

H. T. Banks and Kathleen Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems, 17 (2001), 95-111. doi: 10.1088/0266-5611/17/1/308.

[3]

H. T.Banks, V. A. Bokil, S. Hu, F. C. T. Allnutt, R. Bullis, A. K. Dhar and 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.

[4]

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), 63-91. doi: 10.1016/S0025-5564(02)00218-3.

[5]

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, North Carolina State University, January 2003; Chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, (eds. H. T. Banks and C. Castillo-Chavez), Frontiers in Applied Math, FR28, SIAM, Philadelphia, PA, 2003, 129-154.

[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, Brown University, March 1987; Proc. 2nd Course on Math. Ecology, (Trieste, December 8-12, 1986) World Scientific Press, Singapore, 1988, 521-541.

[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), 1412-1415. doi: 10.1016/j.aml.2010.07.009.

[8]

H. T. Banks, M. Davidian, J. Samuels and K. L. Sutton, An inverse problem statistical methodology summary, CRSC-TR08-01, North Carolina State University, January 2008; Chapter 11 in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, et al.), Berlin Heidelberg New York, 2009, 249-302.

[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), 753-777. doi: 10.1016/j.apnum.2006.07.016.

[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, North Carolina State University, February 2008; Journal of Biological Dynamics, 3 (2009), 130-148.

[11]

H. T. Banks and B. G. Fitzpatrick, Inverse problems for distributed systems: statistical tests and ANOVA, LCDS/CSS Report 88-16, Brown University, July 1988; Proc. International Symposium on Math. Approaches to Envir. and Ecol. Problems, Springer Lecture Notes in Biomath., 81 (1989), 262-273.

[12]

H. T. Banks and 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.

[13]

H. T. Banks and 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.

[14]

H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters, CRSC-TR05-29, North Carolina State University, August 2005; Quarterly of Applied Mathematics, 64 (2006), 749-795.

[15]

H. T. Banks and K. Kunisch, "Estimation Techniques for Distributed Parameter Systems," Birkhauser, Boston, 1989.

[16]

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; SIAM J. Multiscale Modeling and Simulation, 3 (2005), 395-412.

[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, North Carolina State University, September 2002; Math. Biosci., 192 (2004), 193-225.

[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, North Carolina State University, Revised July 2011; J. Immunological Methods, 373 (2011), 143-160. doi: 10.1016/j.jim.2011.08.014.

[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, North Carolina State University, August 2009; Bull. Math. Biol., 70 (2011), 116-150. doi: 10.1007/s11538-010-9524-5.

[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, North Carolina State University, January 2012.

[21]

H. T. Banks and H. T. Tran, "Mathematical and Experimental Modeling of Physical and Biological Processes," CRC Press, Boca Raton London New York, 2009.

[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, North Carolina State University, January 1998; Stochastic Analysis, Control, Optimization and Applications, (eds. W. McEneaney, G. Yin and Q. Zhang), Birkhäuser, (1998), 353-371.

[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), 329-351.

[24]

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

[25]

Nigel J. Burroughs and P. Anton van der Merwe, Stochasticity and spatial heterogeneity in T-cell activation, Immunological Reviews, 216 (2007), 69-80.

[26]

R. Callard and P. D. Hodgkin, Modeling T- and B-cell growth and differentiation, Immunological Reviews, 216 (2007), 119-129.

[27]

Robin E. Callard, Jaroslav Stark and Andrew J. Yates, Fratricide: a mechanism for T memory-cell homeostasis, Trends in Immunology, 24 (2003), 370-375. doi: 10.1016/S1471-4906(03)00164-9.

[28]

R. J. Carroll and D. Ruppert, "Transformation and Weighting in Regression," Chapman Hall, London, 2000.

[29]

M. Davidian and D. M. Giltinan, "Nonlinear Models for Repeated Measurement Data," Chapman and Hall, London, 2000.

[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), 1011-1031. doi: 10.1007/s11538-006-9094-8.

[31]

R. J. DeBoer and Alan S. Perelson, Estimating division and death rates from CFSE data, J. Comp. and Appl. Mathematics, 184 (2005), 140-164. doi: 10.1016/j.cam.2004.08.020.

[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), 4963-4972.

[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), 517-526.

[34]

K. Duffy and V. Subramanian, On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics, J. Math. Biol., 59 (2009), 255-285. doi: 10.1007/s00285-008-0231-x.

[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), 559-564. doi: 10.1046/j.1440-1711.1999.00870.x.

[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), 950-957.

[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), 183-200. doi: 10.1016/j.jim.2005.01.011.

[38]

A. V. Gett and P. D. Hodgkin, A cellular calculus for signal integration by T cells, Nature Immunology, 1 (2000), 239-244.

[39]

M. Kot, "Elements of Mathematical Ecology," Cambridge University Press, Cambridge, UK, 2001.

[40]

M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. doi: 10.1007/BF00160231.

[41]

J. Hasenauer, D. Schittler, and F. Allgöwer, A computational model for proliferation dynamics of division- and label-structured populations, arXiv:1202.4923v1, 22 Feb, 2012.

[42]

E. D. Hawkins, Mirja Hommel, M. L Turner, Francis Battye, J. Markham and P. D Hodgkin, Measuring lymphocyte proliferation, survival and differentiation using CFSE time-series data, Nature Protocols, 2 (2007), 2057-2067.

[43]

E. D. Hawkins, J. F. Markham, L. P. McGuinness and P. D. Hodgkin, A single-cell pedigree analysis of alternative stochastic lymphocyte fates, Proc. Natl. Acad. Sci., 106 (2009), 13457-13462. doi: 10.1073/pnas.0905629106.

[44]

Mirja Hommel and Philip D. Hodgkin, TCR affinity promotes CD8+ T-cell expansion by regulating survival, J. Immunology, 179 (2007), 2250-2260.

[45]

O. Hyrien and M. S. Zand, A mixture model with dependent observations for the analysis of CFSE-labeling experiments, J. American Statistical Association, 103 (2008), 222-239. doi: 10.1198/016214507000000194.

[46]

O. Hyrien, R. Chen and M. S. Zand, An age-dependent branching process model for the analysis of CFSE-labeling experiments, Biology Direct, 5 (2010), Published Online.

[47]

D. E. Kirschner, S. T. Chang, T. W. Riggs, N. Perry and J. J. Linderman, Toward a multiscale model of antigen presentation in immunity, Immunological Reviews, 216 (2007), 93-118.

[48]

H. Y. Lee, E. D. Hawkins, M. S. Zand, T. Mosmann, H. Wu, P. D. Hodgkin and 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. doi: 10.1007/s11538-009-9418-6.

[49]

H. Y. Lee and 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. doi: 10.1007/s11538-007-9239-4.

[50]

K. Leon, J. Faro and J. Carneiro, A general mathematical framework to model generation structure in a population of asynchronously dividing cells, J. Theoretical Biology, 229 (2004), 455-476. doi: 10.1016/j.jtbi.2004.04.011.

[51]

Y. Louzoun, The evolution of mathematical immunology, Immunological Reviews, 216 (2007), 9-20.

[52]

T. Luzyanina, D. Roose and G. Bocharov, Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data, J. Math. Biol., 59 (2009), 581-603. doi: 10.1007/s00285-008-0244-5.

[53]

T. Luzyanina, M. Mrusek, J. T. Edwards, D. Roose, S. Ehl and G. Bocharov, Computational analysis of CFSE proliferation assay, J. Math. Biol., 54 (2007), 57-89. doi: 10.1007/s00285-006-0046-6.

[54]

T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans and G. Bocharov, Numerical modelling of label-structured cell population growth using CFSE distribution data, Theoretical Biology and Medical Modelling, 4 (2007), Published Online.

[55]

A. B. Lyons and C. R. Parish, Determination of lymphocyte division by flow cytometry, J. Immunol. Methods, 171 (1994), 131-137. doi: 10.1016/0022-1759(94)90236-4.

[56]

A. B. Lyons, J. Hasbold and 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. doi: 10.1016/S0091-679X(01)63021-8.

[57]

G. Matera, M. Lupi and P. Ubezio, Heterogeneous cell response to topotecan in a CFSE-based proliferative test, Cytometry A, 62 (2004), 118-128. doi: 10.1002/cyto.a.20097.

[58]

J. A. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations," Springer Lecture Notes in Biomathematics, 68, 1986.

[59]

H. Miao, X. Jin, A. Perelson and 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.

[60]

Robert E. Nordon, Kap-Hyoun Ko, Ross Odell and Timm Schroeder, Multi-type branching models to describe cell differentiation programs, J. Theoretical Biology, 277 (2011), 7-18. doi: 10.1016/j.jtbi.2011.02.006.

[61]

R. E. Nordon, M. Nakamura, C. Ramirez and R. Odell, Analysis of growth kinetics by division tracking, Immunology and Cell Biology, 77 (1999), 523-529. doi: 10.1046/j.1440-1711.1999.00869.x.

[62]

C. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies, Immunology and Cell Biol., 77 (1999), 499-508. doi: 10.1046/j.1440-1711.1999.00877.x.

[63]

Sergei S. Pilyugin, Vitaly V. Ganusov, Kaja Murali-Krishnac, Rafi Ahmed and Rustom Antia, The rescaling method for quantifying the turnover of cell populations, J. Theoretical Biology, 225 (2003), 275-283. doi: 10.1016/S0022-5193(03)00245-5.

[64]

B. Quah, H. Warren and 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.

[65]

P. Revy, M. Sospedra, B. Barbour and A. Trautmann, Functional antigen-independent synapses formed between T cells and dendritic cells, Nature Immunology, 2 (2001), 925-931.

[66]

G. A. Sever and C. J. Wild, "Nonlinear Regression," Wiley, Hoboken, NJ, 2003.

[67]

D. Schittler, J. Hasenauer and 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, 165-168

[68]

J. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.

[69]

V. G. Subramanian, K. R. Duffy, M. L. Turner and P. D. Hodgkin, Determining the expected variability of immune responses using the cyton model, J. Math. Biol., 56 (2008), 861-892. doi: 10.1007/s00285-007-0142-2.

[70]

David T. Terrano, Meenakshi Upreti and Timothy C. Chambers, Cyclin-dependent kinase 1-mediated $Bcl-x_L$/Bcl-2 phosphorylation acts as a functional link coupling mitotic arrest and apoptosis, Mol. Cell. Biol., 30 (2010), 640-656. doi: 10.1128/MCB.00882-09.

[71]

W. Clayton Thompson, "Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based Proliferation Assays," Ph.D. Dissertation, North Carolina State University, December, 2011.

[72]

B. Tummers, DataThief III. 2006. (http://datathief.org/)

[73]

M. L. Turner, E. D. Hawkins and P. D. Hodgkin, Quantitative regulation of B cell division destiny by signal strength, J. Immunology, 181 (2008), 374-382.

[74]

H. Veiga-Fernandez, U. Walter, C. Bourgeois, A. McLean and B. Rocha, Response of naive and memory CD8+ T cells to antigen stimulation in vivo, Nature Immunology, 1 (2000), 47-53.

[75]

P. K. Wallace, J. D. Tario, Jr., J. L. Fisher, S. S. Wallace, M. S. Ernstoff and K. A. Muirhead, Tracking antigen-driven responses by flow cytometry: monitoring proliferation by dye dilution, Cytometry A, 73 (2008), 1019-1034.

[76]

C. Wellard, J. Markham, E. D. Hawkins and P. D. Hodgkin, The effect of correlations on the population dynamics of lymphocytes, J. Theoretical Biology, 264 (2010), 443-449. doi: 10.1016/j.jtbi.2010.02.019.

[77]

J. M. Witkowski, Advanced application of CFSE for cellular tracking, Current Protocols in Cytometry, (2008), 9.25.1-9.25.8.

[78]

A. Yates, C. Chan, J. Strid, S. Moon, R. Callard, A. J. T. George and J. Stark, Reconstruction of cell population dynamics using CFSE, BMC Bioinformatics, 8 (2007), Published Online.

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