Computational approaches to parameter estimation and model selection in immunology. (English) Zbl 1072.92020

Summary: One of the significant challenges in biomathematics (and other areas of science) is to formulate meaningful mathematical models. Our problem is to decide on a parametrized model which is, in some sense, most likely to represent the information in a set of observed data. We illustrate the computational implementation of an information-theoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.
The approach is illustrated by modelling lymphocytic choriomeningitis virus (LCMV) infection using a family of models based on systems of ordinary differential and delay differential equations. The models (which use parameters that have a scientific interpretation) are chosen to fit data arising from experimental studies of virus-cytotoxic T lymphocyte kinetics; the parametrized models that result are arranged in a hierarchy by the computation of Akaike indices. The practical illustration is used to convey more general insight. Because the mathematical equations that comprise the models are solved numerically, the accuracy in the computation has a bearing on the outcome, and we address this and other practical details in our discussion.


92C50 Medical applications (general)
62P10 Applications of statistics to biology and medical sciences; meta analysis
65C20 Probabilistic models, generic numerical methods in probability and statistics
92C30 Physiology (general)


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[1] Akaike, H., A new look at the statistical model identification, IEEE trans. automat. control, 19, 716-723, (1974) · Zbl 0314.62039
[2] Altman, J.D.; Moss, P.A.H.; Goulder, P.J.R.; Barouch, D.H.; McHeyzer-Williams, M.G.; Bell, J.I.; McMichael, A.J.; Davis, M.M., Phenotypic analysis of antigen-specific T lymphocytes, Science, 274, 94-96, (1996)
[3] Armitage, P.; Beŕry, G.; Matthews, J.N.S., Statistical methods in medical research, (2001), Blackwell Science Oxford
[4] Audoly, S.; Belllu, G.; D’Angio, L.; Saccomni, M.; Cobelli, C., Global identifiablity of nonlinear biological systems, IEEE trans. biomed. eng., 48, 55-65, (2001)
[5] Baker, C.T.H., Retarded differential equations, J. comput. appl. math., 125, 309-335, (2000) · Zbl 0970.65079
[6] Baker, C.T.H.; Bocharov, G.A.; Paul, C.A.H., Mathematical modelling of the interleukin-2 T-cell system: a comparative study of approaches based on ordinary and delay differential equations, J. theoret. med., 2, 117-128, (1997) · Zbl 0904.92022
[7] Baker, C.T.H.; Bocharov, G.A.; Paul, C.A.H.; Rihan, F.A., Modelling and analysis of time-lags in some basic patterns of cell proliferation, J. math. biol., 37, 341-371, (1998) · Zbl 0908.92026
[8] C.T.H. Baker, G.A. Bocharov, C.A.H. Paul, F.A. Rihan, Computational modelling with functional differential equations: identification, selection, and sensitivity, Appl. Numer. Math., in press. Available online 19th October 2004. · Zbl 1069.65082
[9] C.T.H. Baker, G.A. Bocharov, F.A. Rihan, A report on the use of delay differential equations in numerical modelling in the biosciences, MCCM Technical Report 343, University of Manchester, 1999, 45pp, ISSN 1360-1725.
[10] C.T.H. Baker, G.A. Bocharov, F.A. Rihan, A report on the models with delays for cell population dynamics: identification, selection and analysis—Part I. MCCM Technical Report 425, University of Manchester, 2003, 28pp, ISSN 1360-1725.
[11] C.T.H. Baker, E.I. Parmuzin, Identification of the initial function for delay differential equations: Parts I, II, III, MCCM Technical Report 431, 443 & 444, University of Manchester, 2004, ISSN 1360-1725.
[12] Baker, C.T.H.; Parmuzin, E.I., Analysis via integral equations of an identification problem for delay differential equations, J. integral equations appl., 16, 111-135, (2004) · Zbl 1080.65121
[13] C.T.H. Baker, E.I. Parmuzin, Identification of the initial function for nonlinear delay differential equations, Russian J. Numer. Anal. Math. Modelling, 20 (2005) to appear. · Zbl 1088.34067
[14] Baker, C.T.H.; Paul, C.A.H., Pitfalls in parameter estimation for delay differential equations, SIAM J. sci. comput., 18, 305-314, (1997) · Zbl 0867.65032
[15] C.T.H. Baker, C.A.H. Paul, Discontinuous solutions of neutral delay differential equations, Appl. Numer. Math., to appear. · Zbl 1105.34055
[16] H.T. Banks, Delay systems in biological models: approximation techniques, Nonlinear systems and applications, in: V. Lakshmikantham (Ed.), Proceedings of the International Conference, University of Texas, Arlington, TX, 1976, Academic Press, New York, 1977, pp. 21-38.
[17] H.T. Banks, Approximation of delay systems with applications to control and identification. Functional differential equations and approximation of fixed points, in: H.-O. Peitgen, H.-O. Walther (Eds.), Proceedings of the Summer School and Conference, University of Bonn, Bonn, 1978, Lecture Notes in Mathematics, vol. 730, Springer, Berlin, 1979, pp. 65-76.
[18] Banks, H.T.; Bihari, K.L., Modelling and estimating uncertainty in parameter estimation, Inverse problems, 17, 95-111, (2001) · Zbl 1054.35121
[19] Banks, H.T.; Burns, J.A.; Cliff, E.M., Parameter estimation and identification for systems with delay, SIAM J. control optim., 19, 6, 791-828, (1981) · Zbl 0504.93019
[20] Banks, H.T.; Lamm, P.K.D., Estimation of delays and other parameters in nonlinear functional differential equations, SIAM J. control optim., 21, 895-915, (1983) · Zbl 0526.93015
[21] Bard, Y., Nonlinear parameter estimation, (1974), Academic Press New York · Zbl 0345.62045
[22] Battegay, M.; Cooper, S.; Althage, A.; Banziger, H.; Hengartner, H.; Zinkernagel, R.M., Quantification of lymphocytic choriomeningitis virus with an immunological focus assay in 24- or 96-well plates, J. virol. methods, 33, 191-198, (1991)
[23] A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Oxford University Press, New York, 2003. · Zbl 1038.65058
[24] Bertuzzi, A.; Gandolfi, A.; Vitelli, R., A regularization procedure for estimating cell kinetic parameters from flow-cytometry data, Math. biosci., 82, 63-85, (1986) · Zbl 0599.92010
[25] Bocharov, G.A., Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses, J. theoret. biol., 192, 283-308, (1998)
[26] Bocharov, G.A.; Ludewig, B.; Bertoletti, A.; Klenerman, P.; Junt, T.; Krebs, P.; Luzyanina, T.; Fraser, C.; Anderson, R.M., Underwhelming the immune response: effect of slow virus growth on CD\(8^+\)-T-lymphocyte responses, J. virol., 78, 2247-2254, (2004)
[27] Bocharov, G.A.; Rihan, F.A., Numerical modelling in biosciences using delay differential equations, J. comput. appl. math., 125, 183-199, (2000) · Zbl 0969.65124
[28] Borghans, J.A.; Taams, L.S.; Wauben, M.H.M.; De Boer, R.J., Competition for antigenic sites during T cell proliferation: a mathematical interpretation of in vitro data, Proc. natl. acad. sci. USA, 96, 10782-10787, (1999)
[29] Burnet, F.M., The clonal selection theory of acquired immunity, (1959), Cambridge University Press Cambridge
[30] Burnham, K.P.; Anderson, D.R., Model selection and inference—a practical information-theoretic approach, (1998), Springer New York · Zbl 0920.62006
[31] Chakraborty, A.K.; Dustin, M.L.; Shaw, A.S., In silico models for cellular and molecular immunology: successes, promises and challenges, Nat. immunol., 4, 933-936, (2003)
[32] Conn, A.R.; Gould, N.; Toint, P.L., LANCELOT: a Fortran package for large-scale nonlinear optimization, (1992), Springer New York · Zbl 0761.90087
[33] De Boer, R.J.; Oprea, M.; Antia, R.; Murali-Krishna, K.; Ahmed, R.; Perelson, A.S., Recruitment times, proliferation, and apoptosis rates during the CD\(8^+\) T-cell response to lymphocytic choriomeningitis virus, J. virol., 75, 10663-10669, (2001)
[34] Efron, B.; Tibshirani, R., Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statist. sci., 1, 54-77, (1986) · Zbl 0587.62082
[35] Ehl, S.; Klenerman, P.; Zinkernagel, R.M.; Bocharov, G., The impact of variation in the number of CD\(8^+\) T-cell precursors on the outcome of virus infection, Cell. immunol., 189, 67-73, (1998)
[36] Gershenfeld, N.A., The nature of mathematical modelling, (2000), Cambridge University Press Cambridge
[37] Gingerich, P.D., Arithmetic or geometric normality of biological variation: an empirical test of theory, J. theoret. biol., 204, 201-221, (2000)
[38] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations, I, nonstiff problems, (1993), Springer Berlin · Zbl 0789.65048
[39] Horbelt, W.; Timmer, J.; Voss, H.U., Parameter estimation in nonlinear delayed feedback systems from noisy data, Phys. lett. A, 299, 513-521, (2002) · Zbl 0996.37077
[40] Kesmir, C.; De Boer, R., Clonal exhaustion as a result of immune deviation, Bull. math. biol., 65, 359-374, (2003) · Zbl 1334.92234
[41] Kullback, S.; Leibler, R.A., On information and sufficiency, Ann. math. statist., 22, 79-86, (1951) · Zbl 0042.38403
[42] Marchuk, G.I., Mathematical modelling of immune response in infectious diseases, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0876.92015
[43] McLean, A.R.; Rosado, M.M.; Agenes, F.; Vascocellos, R.; Freitas, A.A., Resource competition as a mechanism for B cell homeostasis, Proc. natl. acad. sci. USA, 94, 5792-5797, (1997)
[44] Myung, I.J., Tutorial on maximum likelihood estimation, J. math. physiol., 47, 90-100, (2003) · Zbl 1023.62112
[45] U. Nowak, A. Grah, M. Schreier, Parameter estimation and accuracy matching strategies for 2-D reactor models, ZIB-Report 03-52, 2003, 13pp. · Zbl 1081.80008
[46] Numerical Algorithms Group The NAg {\scFORTRAN} Library _.
[47] Pascual, M.A.; Kareiva, P., Predicting the outcome of competition using experimental data: maximum likelihood and Bayesian approaches, Ecology, 77, 337-349, (1996)
[48] C.A.H. Paul, A User Guide to Archi, MCCM Report 283, University of Manchester. \(\sim\).
[49] Rabitz, H., Chemical sensitivity analysis theory with applications to molecular dynamics and kinetics, Comput. chem., 5, 167-180, (1981)
[50] Renshaw, E., Modelling biological populations in space and time, (1993), Cambridge University Press Cambridge · Zbl 0779.92016
[51] Schwarz, G., Estimating the dimension of a model, Ann. statist., 6, 461-464, (1978) · Zbl 0379.62005
[52] Venzon, D.J.; Mooogavkor, S.H., A method for computing profile-likelihood-based confidence intervals, Appl. statist., 37, 87-94, (1988)
[53] Verotta, D.; Schaedeli, F., Non-linear dynamics models characterizing long-term virological data from AIDS clinical trials, Math. biosci., 176, 163-183, (2002) · Zbl 1015.92022
[54] Willé, D.R.; Baker, C.T.H., The tracking of derivative discontinuities in systems of delay-differential equations, Appl. numer. math., 9, 209-222, (1992) · Zbl 0747.65054
[55] Wodarz, D.; Klenerman, P.; Nowak, M.A., Dynamics of cytotoxic T-lymphocyte exhaustion, Proc. R. soc. (London) ser. B, 265, 191-203, (1998)
[56] Wolpert, D.H., The bootstrap is inconsistent with probability theory, () · Zbl 0886.62005
[57] Wolters, L.M.M.; Hansen, B.E.; Niesters, H.G.M.; Levi-Drummer, R.S.L.; Neumann, A.U.; Schalm, S.W.; de Man, R.A., The influence of baseline characteristics on viral dynamic parameters in chronic hepatitis B patients treated with lamivudine, J. hepatol., 37, 253-258, (2002)
[58] Zinkernagel, R.M., Lymphocytic choriomeningitis virus and immunology, Curr. top. microbiol. immunol., 263, 1-5, (2002)
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