The Annals of Applied Statistics

Modeling long-term longitudinal HIV dynamics with application to an AIDS clinical study

Yangxin Huang and Tao Lu

Full-text: Open access


A virologic marker, the number of HIV RNA copies or viral load, is currently used to evaluate antiretroviral (ARV) therapies in AIDS clinical trials. This marker can be used to assess the ARV potency of therapies, but is easily affected by drug exposures, drug resistance and other factors during the long-term treatment evaluation process. HIV dynamic studies have significantly contributed to the understanding of HIV pathogenesis and ARV treatment strategies. However, the models of these studies are used to quantify short-term HIV dynamics (< 1 month), and are not applicable to describe long-term virological response to ARV treatment due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. Long-term therapy with ARV agents in HIV-infected patients often results in failure to suppress the viral load. Pharmacokinetics (PK), drug resistance and imperfect adherence to prescribed antiviral drugs are important factors explaining the resurgence of virus. To better understand the factors responsible for the virological failure, this paper develops the mechanism-based nonlinear differential equation models for characterizing long-term viral dynamics with ARV therapy. The models directly incorporate drug concentration, adherence and drug susceptibility into a function of treatment efficacy and, hence, fully integrate virologic, PK, drug adherence and resistance from an AIDS clinical trial into the analysis. A Bayesian nonlinear mixed-effects modeling approach in conjunction with the rescaled version of dynamic differential equations is investigated to estimate dynamic parameters and make inference. In addition, the correlations of baseline factors with estimated dynamic parameters are explored and some biologically meaningful correlation results are presented. Further, the estimated dynamic parameters in patients with virologic success were compared to those in patients with virologic failure and significantly important findings were summarized. These results suggest that viral dynamic parameters may play an important role in understanding HIV pathogenesis, designing new treatment strategies for long-term care of AIDS patients.

Article information

Ann. Appl. Stat., Volume 2, Number 4 (2008), 1384-1408.

First available in Project Euclid: 8 January 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

AIDS antiretroviral drug therapy Bayesian nonlinear mixed-effects models time-varying drug efficacy long-term HIV dynamics longitudinal data


Huang, Yangxin; Lu, Tao. Modeling long-term longitudinal HIV dynamics with application to an AIDS clinical study. Ann. Appl. Stat. 2 (2008), no. 4, 1384--1408. doi:10.1214/08-AOAS192.

Export citation


  • Acosta, E. P., Wu, H., Walawander, A., Eron, J., Pettinelli, C., Yu, S., Neath, D., Ferguson, E., Saah, A. J., Kuritzkes, D. R. and Gerber, J. G., for the Adult ACTG 5055 Protocol Team. (2004). Comparison of two indinavir/ritonavir regimens in treatment-experienced HIV-infected individuals. J. Acquired Immune Deficiency Syndromes 37 1358–1366.
  • Bangsberg, D. R. et al. (2000). Adherence to protease inhibitors, HIV-1 viral load, and development of drug resistance in an indigent population. AIDS 14 357–366.
  • Besch, C. L. (1995). Compliance in clinical trials. AIDS 9 1–10.
  • Bonhoeffer, S., Lipsitch, M. and Levin, B. R. (1997). Evaluating treatment protocols to prevent antibiotic resistance. Proc. Natl. Acad. Sci. USA 94 12106–12111.
  • Callaway, D. S. and Perelson, A. S. (2002). HIV-1 infection and low steady state viral loads. Bull. Math. Biol. 64 29–64.
  • Carlin, B. P. and Louis, T. A. (1996). Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. Chapman and Hall, London.
  • Cobelli, C., Lepschy, A. and Jacur, G. R. (1979). Identifiability of compartmental systems and related structural properties. Math. Biosci. 44 1–18.
  • Davidian, M. and Giltinan, D. M. (1995). Nonlinear Models for Repeated Measurement Data. Chapman and Hall, London.
  • Ding, A. A. and Wu, H. (1999). Relationships between antiviral treatment effects and biphasic viral decay rates in modeling HIV dynamics. Math. Biosci. 160 63–82.
  • Ding, A. A. and Wu, H. (2000). A comparison study of models and fitting procedures for biphasic viral decay rates in viral dynamic models. Biometrics 56 16–23.
  • Ding, A. A. and Wu, H. (2001). Assessing antiviral potency of anti-HIV therapies in vivo by comparing viral decay rates in viral dynamic models. Biostatistics 2 13–29.
  • Gamerman, D. (1997). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Chapman and Hall, London.
  • Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. J. Amer. Statist. Assoc. 85 972–985.
  • Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398–409.
  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Recognition Machine Intelligence 6 721–741.
  • Han, C., Chaloner, K. and Perelson, A. S. (2002). Bayesian analysis of a population HIV dynamic model. In Case Studies in Bayesian Statistics 6 (C. Gatsonis, A. Carriquiry, A. Gelman, et al., eds.) 223–237. Springer, New York.
  • Heitjan, D. F. and Basu, S. (1996). Distinguishing “missing at random” and “missing completely at random.” Amer. Statist. 50 207–213.
  • Higgins, M., Davidian, M. and Giltinan, D. M. (1997). A two-step approach to measurement error in time-dependent covariates in nonlinear mixed-effects models, with application to IGF-I pharmacokinetics. J. Amer. Statist. Assoc. 92 436–448.
  • Ho, D. D., Neuman, 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., Rosenkranz, S. L. and Wu, H. (2003). Modeling HIV dynamics and antiviral responses with consideration of time-varying drug exposures, sensitivities and adherence. Math. Biosci. 184 165–186.
  • Huang, Y. and Wu, H. (2006). A Bayesian approach for estimating antiviral efficacy in HIV dynamic model. J. Appl. Statist. 33 155–174.
  • Hsu, A., Issacson, J., Kempf, D. J. et al. (2000). Trough concentrations-EC50 relationship as a predictor of viral response for ABT-378/ritonavir in treatment-experienced patients. In 40th Interscience Conference on Antimicrobial Agents and Chemotherapy, San Francisco, CA, Poster Session 171.
  • Ickovics, J. R. and Meisler, A. W. (1997). Adherence in AIDS clinical trial: A framework for clinical research and clinical care. J. Clinical Epidemiology 50 385–391.
  • IMSL MATH/LIBRARY (1994). FORTRAN Subroutines for Mathematical Applications. 2. Visual Numerics, Houston.
  • Jackson, R. C. (1997). A pharmacokinetic–pharmacodynamic model of chemotherapy of human immunodeficiency virus infection that relates development of drug resistance to treatment intensity. J. Pharmacokinetics Pharmacodynamics 25 713–730.
  • Klenernam, P., Phillips, R. E. et al. (1996). Cytotoxic T lymphocytes and viral turnover in HIV type 1 infrction. Proc. Natl. Acad. Sci. USA 93 15323–15328.
  • Labbé, L. and Verttoa, D. (2006). A nonlinear mixed effect dynamic model incorporating prior exposure and adherence to treatment to describe long-term therapy outcome in HIV-patients. J. Pharmacokinetics and Pharmacodynamics 33 519–542.
  • Markowitz, M., Louie, M., Hurley, A., Sun, E., Mascio, M. D., Perelson, A. S. and Ho, D. D. (2003). A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T-cell decay in vivo. J. Virology 77 5037–5038.
  • Molla, A. et al. (1996). Ordered accumulation of mutations in HIV protease confers resistance to ritonavir. Nature Medicine 2 760–766.
  • Notermans, D. W., Goudsmit, J., Danner, S. A. et al. (1998). Rate of HIV-1 decline following antiretroviral therapy is related to viral load at baseline and drug regimen. AIDS 12 1483–1490.
  • Nowak, M. A., Bonhoeffer, S. et al. (1995). HIV results in the frame. Nature 375 193.
  • Nowak, M. A. and May, R. M. (2000). Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford Univ. Press.
  • Perelson, A. S., Essunger, P. et al. (1997). Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387 188–191.
  • Perelson, A. S. and Nelson, P. W. (1999). Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41 3–44.
  • 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.
  • Pinheiro, J. and Bates, D. M. (2000). Mixed-Effects Models in S and S-plus. Springer, New York.
  • Roberts, G. O. (1996). Markov chain concepts related to sampling algorithms. In Markov Chain Monte Carlo in Practice (W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds.) 45–57. Chapman and Hall, London.
  • SAS institute inc. (2000). SAS/STAT User’s Guide, Version 8. SAS Publishing.
  • Sheiner, L. B. (1985). Modeling pharmacodynamics: Parametric and nonparametric approaches. In Variability in Drug Therapy: Description, Estimation, and Control (M. Rowland et al., eds.) 139–152. Raven Press, New York.
  • Stafford, M. A. et al. (2000). Modeling plasma virus concentration during primary HIV infection. J. Theoret. Biol. 203 285–301.
  • Venables, W. N. and Ripley, B. D. (1999). Modern Applied Statistics with S-Plus, 3rd ed. Springer, New York.
  • Verotta, D. (2005). Models and estimation methods for clinical HIV-1 data. J. Comput. Appl. Math. 184 275–300.
  • Wahl, L. M. and Nowak, M. A. (2000). Adherence and resistance: Predictions for therapy outcome. Proc. Roy. Soc. Biol. 267 835–843.
  • Wainberg, M. A. et al. (1996). Effectiveness of 3TC in HIV clinical trials may be due in part to the M184V substition in 3TC-resistant HIV-1 reverse transcriptase. AIDS 10(suppl) S3–S10.
  • Wakefield, J. C. (1996). The Bayesian analysis to population Pharmacokinetic models. J. Amer. Statist. Assoc. 91 62–75.
  • Wein, L. M., Damato, R. M. and Perelson, A. S. (1998). Mathematical analysis of antiretroviral therapy aimed at HIV-1 eradication or maintenance of low viral loads. J. Theoret. Biol. 192 81–98.
  • Wu, H., Ding, A. A. and De Gruttola, V. (1998). Estimation of HIV dynamic parameters. Statistics in Medicine 17 2463–2485.
  • Wu, H. and Ding, A. 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., Kuritzkes, D. R., Mcclernon, D. R. et al. (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. J. Infectious Diseases 179 799–807.
  • Wu, H., Mellors, J., Ruan, P. et al. (2003). Viral Dynamics and their relations to baseline factors and logn-term virologic responses in treatment-naive HIV-1 infected patients receiving abacavir in combination with HIV-1 protease inhibitors. JAIDS 32 557–564.
  • Wu, H., Lathey, J., Ruan, P. et al. (2004). Relationship of plasma HIV-1 RNA dynamics to baseline factors and virological responses to Highly Active Antiretroviral Therapy (HAART) in adolescents infected through risk behavior. J. Infectious Diseases 189 593–601.
  • Wu, H., Huang, Y., Acosta, E. P., Rosenkranz, S. L., Kuritzkes, D. R., Eron, J. J., Perelson, A. S. and Gerber, J. G. (2005). Modeling long-term HIV dynamics and antiretroviral response: effects of drug potency, pharmacokinetics, adherence and drug resistance. J. Acquired Immune Deficiency Syndromes 39 272–283.
  • Wu, L. (2002). A joint model for nonlinear mixed-effects models with censoring and covariates measured with error, with application to AIDS studies. J. Amer. Statist. Assoc. 97 955–964.