The Annals of Applied Statistics

Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model

Hua Liang, Hongyu Miao, and Hulin Wu

Full-text: Open access

Abstract

Modeling viral dynamics in HIV/AIDS studies has resulted in a deep understanding of pathogenesis of HIV infection from which novel antiviral treatment guidance and strategies have been derived. Viral dynamics models based on nonlinear differential equations have been proposed and well developed over the past few decades. However, it is quite challenging to use experimental or clinical data to estimate the unknown parameters (both constant and time-varying parameters) in complex nonlinear differential equation models. Therefore, investigators usually fix some parameter values, from the literature or by experience, to obtain only parameter estimates of interest from clinical or experimental data. However, when such prior information is not available, it is desirable to determine all the parameter estimates from data. In this paper we intend to combine the newly developed approaches, a multi-stage smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares (SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear differential equation model. In particular, to the best of our knowledge, this is the first attempt to propose a comparatively thorough procedure, accounting for both efficiency and accuracy, to rigorously estimate all key kinetic parameters in a nonlinear differential equation model of HIV dynamics from clinical data. These parameters include the proliferation rate and death rate of uninfected HIV-targeted cells, the average number of virions produced by an infected cell, and the infection rate which is related to the antiviral treatment effect and is time-varying. To validate the estimation methods, we verified the identifiability of the HIV viral dynamic model and performed simulation studies. We applied the proposed techniques to estimate the key HIV viral dynamic parameters for two individual AIDS patients treated with antiretroviral therapies. We demonstrate that HIV viral dynamics can be well characterized and quantified for individual patients. As a result, personalized treatment decision based on viral dynamic models is possible.

Article information

Source
Ann. Appl. Stat. Volume 4, Number 1 (2010), 460-483.

Dates
First available in Project Euclid: 11 May 2010

Permanent link to this document
http://projecteuclid.org/euclid.aoas/1273584463

Digital Object Identifier
doi:10.1214/09-AOAS290

Zentralblatt MATH identifier
1189.62171

Mathematical Reviews number (MathSciNet)
MR2758180

Citation

Liang, Hua; Miao, Hongyu; Wu, Hulin. Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model. Ann. Appl. Stat. 4 (2010), no. 1, 460--483. doi:10.1214/09-AOAS290. http://projecteuclid.org/euclid.aoas/1273584463.


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References

  • Akaike, H. (1973). Information theory as an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (B. N. Petrov and F. Csaki, eds.) 267–281. Akademiai Kiado, Budapest.
  • Audoly, S., Bellu, G., D’Angio, L., Saccomani, M. P. and Cobelli, C. (2001). Global identifiability of nonlinear models of biological systems. IEEE Trans. Biomed. Eng. 48 55–65.
  • Bard, Y. (1974). Nonlinear Parameter Estimation. Academic, London.
  • Bellman, R. and Åström, K. J. (1970). On structural identifiability. Math. Biosci. 7 329–339.
  • Burnham, K. P. and Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociol. Methods Res. 33 261.
  • Chappel, M. J. and Godfrey, K. R. (1992). Structural identifiability of the parameters of a nonlinear batch reactor model. Math. Biosci. 108 245–251.
  • Chen, J. and Wu, H. (2008). Efficient local estimation for time-varying coefficients in deterministic dynamic models with applications to HIV-1 dynamics. J. Amer. Statist. Assoc. 103 369–384.
  • Chen, J. and Wu, H. (2009). Estimation of time-varying parameters in deterministic dynamic models. Statist. Sinica 18 987–1006.
  • Cobelli, C., Lepschy, A. and Jacur, R. (1979). Identifiability of compartmental systems and related structural properties. Math. Biosci. 44 1–18.
  • de Boor, C. (1978). A Practical Guide to Splines. Springer, New York.
  • Englezos, P. and Kalogerakis, N. (2001). Applied Parameter Estimation for Chemical Engineers. Dekker, New York.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications. Chapman & Hall, London.
  • Filter, R. A., Xia, X. and Gray, C. M. (2005). Dynamic HIV/AIDS parameter estimation with appliation to a vaccine readiness study in Southern Africa. IEEE Trans. Biomed. Eng. 52 284–291.
  • Glover, F. (1977). Heuristics for integer programming using surrogate constraints. Decision Sciences 8 156–166.
  • Gray, C. M., Williamson, C., Bredell, H., Puren, A., Xia, X., Filter, R., Zijenah, L., Cao, H., Morris, L., Vardas, E., Colvin, M., Gray, G., McIntyre, J., Musonda, R., Allen, S., Katzenstein, D., Mbizo, M., Kumwenda, N., Taha, T., Karim, S. A, Flores, J. and Sheppard, H. W. (2005). Viral dynamics and CD4+ T cell counts in subtype C human immunodeficiency virus type 1-infected individuals from Southern Africa. AIDS Research and Human Retroviruses 21 285–291.
  • Hemker, P. W. (1972). Numerical methods for differential equations in system simulation and in parameter estimation. In Analysis and Simulation of Biochemical Systems (H. C. Hemker and B. Hess, eds.) 59–80. North-Holland, Amsterdam.
  • Ho, D. D., Neumann, A. U., Perelson, A. S., Chen, W., Leonard, J. M. and Markowitz, M. (1995). Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373 123–126.
  • Huang, Y. and Wu, H. (2006). A Bayesian approach for estimating antiviral efficacy in HIV dynamic models. J. Appl. Statist. 33 155–174.
  • Huang, Y., Liu, D. and Wu, H. (2006). Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system. Biometrics 62 413–423.
  • Jeffrey, A. M. and Xia, X. (2005). Identifiability of HIV/AIDS model. In Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections With Intervention (W. Y. Tan and H. Wu, eds.). World Scientific, Singapore.
  • Joshi, M., Seidel-Morgenstern, A. and Kremling, A. (2006). Exploiting the bootstrap method for quantifying parameter confidence intervals in dynamic systems. Metabolic Engineering 8 447–455.
  • Kolchin, E. (1973). Differential Algebra and Algebraic Groups. Academic Press, New York.
  • Laguna, M. and Marti, R. (2003). Scatter Search: Methodology and Implementations in C. Kluwer Academic, Boston.
  • Laguna, M. and Marti, R. (2005). Experimental testing of advanced scatter search designs for global optimization of multimodal functions. J. Global Optim. 33 235–255.
  • Li, L., Brown, M. B., Lee, K. H. and Gupta, S. (2002). Estimation and inference for a spline-enhanced population pharmacokinetic model. Biometrics 58 601–611.
  • Li, Z., Osborne, M. and Prvan, T. (2005). Parameter estimation in ordinary differential equations. IMA J. Numer. Anal. 25 264–285.
  • Liang, H. and Wu, H. (2008). Parameter estimation for differential equation models using a framework of measurement error in regression models. J. Amer. Statist. Assoc. 103 1570–1583.
  • Ljung, L. and Glad, T. (1994). On global identifiability for arbitrary model parametrizations. Automatica 30 265–276.
  • Miao, H., Dykes, C., Demeter, L. M., Cavenaugh, J., Park, S. Y., Perelson, A. S. and Wu, H. (2008). Modeling and estimation of kinetic parameters and replicative fitness of HIV-1 from flow-cytometry-based growth competition experiments. Bull. Math. Biol. 70 1749–1771.
  • Miao, H., Dykes, C., Demeter, L. M. and Wu, H. (2009). Differential equation modeling of HIV viral fitness experiments: Model identification, model selection, and multi-model inference. Biometrics 65 292–300.
  • Moles, C. G., Banga, J. R. and Keller, K. (2004). Solving nonconvex climate control problems: Pitfalls and algorithm performances. Appl. Soft. Comput. 5 35–44.
  • Nocedal, J. and Wright, S. J. (1999). Numerical Optimization. Springer, New York.
  • Nowak, M. A. and May, R. M. (2000). Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford Univ. Press, Oxford.
  • Ogunnaike, B. A. and Ray, W. H. (1994). Process Dynamics, Modeling, and Control. Oxford Univ. Press, New York.
  • Ollivier, F. (1990). Le problème de l’identifiabilité globale: Étude thé orique, méthodes effectives et bornes de complexité. Ph.D. thesis, École Polytechnique, Paris, France.
  • Ouattara, D. A., Mhawej, M. J. and Moog, C. H. (2008). Clinical tests of therapeutical failures based on mathematical modeling of the HIV infection. Joint Special Issue of IEEE Trans. Circuits Syst. and IEEE Trans. Automat. Control (Special Issue on Systems Biology) 53 230–241.
  • Perelson, A. S. and Nelson, P. W. (1999). Mathematical analysis of HIV-1 dynamics in vivo. SIAM Review 41 3–44.
  • Perelson, A. S., Essunger, P., Cao, Y. Z., Vesanen, M., Hurley, A., Saksela, K., Markowitz, M. and Ho, D. D. (1997). Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387 188–191.
  • Perelson, A. S., Neumann, A. U., Markowitz, M., Leonard, J. M. and Ho, D. D. (1996). HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time. Science 271 1582–1586.
  • Pohjanpalo, H. (1978). System identifiability based on the power series expansion of the solution. Math. Biosci. 41 21–33.
  • Poyton, A. A., Varziri, M. S., McAuley, K. B., McLellan, P. J. and Ramsay, J. O. (2006). Parameter estimation in continuous-time dynamic models using principal differential analysis. Comput. Chem. Eng. 30 698–708.
  • Putter, H., Heisterkamp, S. H., Lange, J. M. A. and Wolf, F. (2002). A Bayesian approach to parameter estimation in HIV dynamic models. Statist. Med. 21 2199–2214.
  • Ramsay, J. O. (1996). Principal differential analysis: Data reduction by differential operators. J. Roy. Statist. Soc. Ser. B 58 495–508.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Ramsay, J. O., Hooker, G., Campbell, D. and Cao, J. (2007). Parameter estimation for differential equations: A generalized smoothing approach (with discussion). J. Roy. Statist. Soc. Ser. B 69 741–796.
  • Ritt, J. F. (1950). Differential Algebra. Amer. Math. Soc., Providence, RI.
  • Rodriguez-Fernandez, M., Egea, J. A. and Banga, J. R. (2006). Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems. BMC Bioinformatics 7 483.
  • Runge, C. (1901). Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. Math. Phys. 46 224–243.
  • Schwarz, G. (1978). Estimating the dimensions of a model. Ann. Statist. 6 461–464.
  • Storn, R. and Price, K. (1997). Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11 341–359.
  • Tan, W. Y. and Wu, H. (2005). Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections With Intervention. World Scientific, Singapore.
  • Vajda, S., Godfrey, K. and Rabitz, H. (1989). Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci. 93 217–248.
  • Varah, J. M. (1982). A spline least squares method for numerical parameter estimation in differential equations. SIAM J. Sci. Comput. 3 131–141.
  • Vrugt, J. A. and Robinson, B. A. (2007). Improved evolutionary optimization from genetically adaptive multimethod search. PNAS 104 708–711.
  • Walter, E. (1987). Identifiability of Parameteric Models. Pergamon Press, Oxford.
  • Wei, X., Ghosh, S. K., Taylor, M. E., Johnson, V. A., Emini, E. A., Deutsch, P., Lifson, J. D., Bonhoeffer, S., Nowak, M. A., Hahn, B. H., Saag, M. S. and Shaw, G. M. (1995). Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373 117–122.
  • Wu, H. (2005). Statistical methods for HIV dynamic studies in AIDS clinical trials. Statist. Methods Med. Res. 14 171–192.
  • Wu, H. and Ding, A. (1999). Population HIV-1 dynamics in vivo: Applicable models and inferential tools for virological data from AIDS clinical trials. Biometrics 55 410–418.
  • Wu, H. and Zhang, J. T. (2006). Nonparametric Regression Methods for Longitudinal Data Analysis. Wiley, Hoboken, NJ.
  • Wu, H., Kuritzkes, D. R., McClernon, D. R., Kessler, H., Connick, E., Landay, A., Spear, G., Heath-Chiozzi, M., Rousseau, F., Fox, L., Spritzler, J., Leonard, J. M. and Lederman, M. M. (1999). Characterization of viral dynamics in human immunodeficiency virus type 1-infected patients treated with combination antiretroviral therapy: Relationships to host factors, cellular restoration and virological endpoints. Journal of Infectious Diseases 179 799–807.
  • Wu, H., Zhu, H., Miao, H. and Perelson, A. S. (2008). Parameter identifiability and estimation of HIV/AIDS dynamic models. Bull. Math. Biol. 70 785–799.
  • Xia, X. (2003). Estimation of HIV/AIDS parameters. Automatica 39 1983–1988.
  • Ye, Y. (1987). Interior algorithms for linear, quadratic and linearly constrained non-linear programming. Ph.D. thesis, Dept. ESS, Stanford Univ.