# American Institute of Mathematical Sciences

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:
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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. 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##### 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. 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