Non-Linear Mixed Effects Modeling – From Methodology and Software Development to Driving Implementation in Drug Development Science

  • Goonaseelan (Colin) Pillai
  • France Mentré
  • Jean-Louis Steimer

Few scientific contributions have made significant impact unless there was a champion who had the vision to see the potential for its use in seemingly disparate areas—and who then drove active implementation. In this paper, we present a historical summary of the development of non-linear mixed effects (NLME) modeling up to the more recent extensions of this statistical methodology. The paper places strong emphasis on the pivotal role played by Lewis B. Sheiner (1940–2004), who used this statistical methodology to elucidate solutions to real problems identified in clinical practice and in medical research and on how he drove implementation of the proposed solutions. A succinct overview of the evolution of the NLME modeling methodology is presented as well as ideas on how its expansion helped to provide guidance for a more scientific view of (model-based) drug development that reduces empiricism in favor of critical quantitative thinking and decision making


algorithms NONMEM pharmacokinetics pharmacodynamics PK PKPD NLME model-based 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beal, S.L., Sheiner, L.B. 1980The NONMEM systemAmer. Statist34118119Google Scholar
  2. 2.
    Sheiner, L.B. 1969Computer-aided long-term anticoagulation therapyComput. Biomed. Res2507513CrossRefPubMedGoogle Scholar
  3. 3.
    Peck, C.C., Sheiner, L.B., Combs, D.T., Marton, C.M., Melmon, K.L. 1973Computer assisted digoxin therapyN. Engl. J. Med289441446PubMedCrossRefGoogle Scholar
  4. 4.
    Sheiner, L.B., Halkin, H., Peck, C., Rosenberg, B., Melmon, K.L. 1975Improved computer assisted digoxin therapyAnn. Intern. Med82619627PubMedGoogle Scholar
  5. 5.
    Sheiner, L.B., Rosenberg, B., Melmon, K.L. 1972Modeling of individual pharmacokinetics for computer-aided drug dosageComput. Biomed. Res5441459CrossRefGoogle Scholar
  6. 6.
    Sheiner, L.B., Rosenberg, B., Marathe, V.V. 1977Estimation of population characteristics of pharmacokinetic parameters from routine clinical dataJ. Pharmacokinet. Biopharm5445479CrossRefPubMedGoogle Scholar
  7. 7.
    Berman, M., Weiss, M.F. 1978SAAM ManualUSDHEW Publication No 78–180Washington, D.CGoogle Scholar
  8. 8.
    Berman, M., Beltz, W.F., Grief, P.C., Chabay, R., Boston, R.C. 1983CONSAM User’s GuideUSDHEW, NIH Lab for Math BiologyWashington, DCGoogle Scholar
  9. 9.
    Metzler, C.M. 1982Statistical properties of estimates of kinetic parametersBozler, G.Rossum, J.M eds. Pharmacokinetics During Drug Development: Data Analysis and Evaluation TechniquesFischer VerlagStuttgart138143Google Scholar
  10. 10.
    Bard, Y. 1974Nonlinear Parameter EstimationAcademic PressNew YorkGoogle Scholar
  11. 11.
    Beal, S.L., Sheiner, L.B. 1988Heteroscedastic nonlinear regressionTechnometrics30327338Google Scholar
  12. 12.
    Boxenbaum, H.G., Riegelman, S., Elashoff, R.M. 1974Statistical estimations in pharmacokineticsJ. Pharmacokinet. Biopharm.2123148CrossRefPubMedGoogle Scholar
  13. 13.
    Sheiner, L.B., Beal, S.L. 1981Estimation of pooled pharmacokinetic parameters describing populationsEndrenyi, L eds. Kinetic Data AnalysisPlenum PressNew York271284Google Scholar
  14. 14.
    L. B. Sheiner. Lewis Sheiner curriculum vitae., 2002
  15. 15.
    Beal, S.L., Sheiner, L.B. 1982Estimating population kineticsCRC Crit. Rev. Bioeng.8195222Google Scholar
  16. 16.
    Sheiner, L.B., Beal, S.L. 1980Evaluation of methods for estimating population pharmacokinetic parameters. I. Michaelis–Menten model: routine clinical pharmacokinetic dataJ. Pharmacokinet. Biopharm.8553571CrossRefPubMedGoogle Scholar
  17. 17.
    Sheiner, L.B., Beal, S.L. 1981Evaluation of methods for estimating population pharmacokinetic parameters. II. Biexponential model and experimental pharmacokinetic dataJ. Pharmacokinet. Biopharm.9635651CrossRefPubMedGoogle Scholar
  18. 18.
    Sheiner, L.B., Beal, S.L. 1983Evaluation of methods for estimating population pharmacokinetic parameters. III. Monoexponential model. Routine clinical pharmacokinetic dataJ. Pharmacokinet. Biopharm.11303319CrossRefPubMedGoogle Scholar
  19. 19.
    Pillai, G., Miller, R. 1988Clinical experience with theophylline. A case for monitoring serum concentrationsS. Afr. Med. J.73640642PubMedGoogle Scholar
  20. 20.
    Sheiner, L.B. 1997Learning versus confirming in clinical drug developmentClin. Pharmacol. Ther.61275291CrossRefPubMedGoogle Scholar
  21. 21.
    Harville, D.A. 1977Maximum likelihood approaches to variance component estimation and to related problemsJ. Am. Stat. Assoc.72320388Google Scholar
  22. 22.
    Laird, N.M., Ware, J.H. 1982Random effects models for longitudinal dataBiometrics38964973Google Scholar
  23. 23.
    Liang, K., Zeger, S.L. 1986Longitudinal data analysis using generalized linear modelsBiometrika731322Google Scholar
  24. 24.
    S. R. Searl, G. Grasela, and L. E. McCullogh. Variance Components. Wiley, 1992.Google Scholar
  25. 25.
    Dempster, A.P., Laird, N.M., Rubin, D.B. 1977Maximum likelihood from incomplete data via the EM algorithmJ. R. Stat. Soc. Ser. B. Stat. Methodol.1138Google Scholar
  26. 26.
    Lindstrom, M.J., Bates, D.M. 1988Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures dataJ. Am. Stat. Assoc.8310141022Google Scholar
  27. 27.
    Davidian, M., Giltinan, D.M. 1995Nonlinear Models for Repeated Measurement DataChapman and HallLondonGoogle Scholar
  28. 28.
    Steimer, J.-L., Mallet, A., Golmard, J.L., Boisvieux, J.F. 1994Alternative approaches to estimation of population pharmacokinetic parameters: comparison with the nonlinear mixed-effect modelDrug Metab. Rev.15265292Google Scholar
  29. 29.
    Davidian, M., Giltinan, D.M. 2003Non linear models for repeated measurement data: an overview and updateJ. Agric. Biol. Environ. Stat.8387419CrossRefGoogle Scholar
  30. 30.
    Lindstrom, M.J., Bates, D.M. 1990Nonlinear mixed effects models for repeated measures dataBiometrics46673687PubMedGoogle Scholar
  31. 31.
    Urien, S. 1995MicroPharm-K, a microcomputer interactive program for the analysis and simulation of pharmacokinetic processesPharm. Res.1212251230CrossRefPubMedGoogle Scholar
  32. 32.
    Mentre, F., Mallet, A., Steimer, J.-L. 1988Hyperparameter estimation using stochastic approximation with application to population pharmacokineticsBiometrics44673683PubMedGoogle Scholar
  33. 33.
    Vonesh, E.F., Carter, R.L. 1992Mixed-effects nonlinear regression for unbalanced repeated measuresBiometrics48117PubMedGoogle Scholar
  34. 34.
    Wolfinger, R. 1993Laplace’s approximation for nonlinear mixed modelsBiometrika80791795Google Scholar
  35. 35.
    Pinheiro, J.C., Bates, D.M. 1995Approximations to the log-likelihood function in the nonlinear mixed-effects modelJ. Comput. Graph. Statist.11235Google Scholar
  36. 36.
    Mentre, F., Gomeni, R. 1995A two-step algorithm for estimation on non-linear mixed-effects with an evaluation in population pharmacokineticsJ. Biopharm. Stat.5141158PubMedGoogle Scholar
  37. 37.
    Aarons, L. 1993The estimation of population pharmacokinetic parameters using an EM algorithmComput. Methods. Programs. Biomed.41916CrossRefPubMedGoogle Scholar
  38. 38.
    Walker, S.G. 1996An EM algorithm for non-linear random effects modelsBiometrics52934944Google Scholar
  39. 39.
    Mallet, A. 1986A maximum likelihood estimation method for random coefficient regression modelsBiometrika4110151023Google Scholar
  40. 40.
    Schumitzky, A. 1991Nonparametric EM algorithm for estimating prior distributionsAppl. Math. Comput.45141157CrossRefGoogle Scholar
  41. 41.
    Davidian, M., Gallant, A.R. 1993The nonlinear mixed effects model with a smooth random effects densityBiometrika80475488Google Scholar
  42. 42.
    Park, K., Verotta, D., Gupta, S.K., Sheiner, L.B. 1998Use of a pharmacokinetic/pharmacodynamic model to design an optimal dose input profileJ. Pharmacokinet. Biopharm.26471492CrossRefPubMedGoogle Scholar
  43. 43.
    Racine-Poon, A. 1985A Bayesian approach to nonlinear random effects modelsBiometrics4110151023PubMedGoogle Scholar
  44. 44.
    Gelfand, A.E., Smith, A.F.M. 1990Sampling-based approaches to calculating marginal densitiesJ. Am. Stat. Assoc.85398409Google Scholar
  45. 45.
    Gelfand, A.E., Hills, S.E., Racine-Poon, A., Smith, A.F.M. 1990Illustration of bayesian inference in normal data models using Gibbs samplingJ. Am. Stat. Assoc.85972975Google Scholar
  46. 46.
    Smith, A.F.M., Wakefield, J. 1994The hierarchical Bayesian approach to population pharmacokinetic modellingInt. J. Biomed. Comput.363542CrossRefPubMedGoogle Scholar
  47. 47.
    Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R. 1996BUGS 0.5: Bayesian Inference Using Gibbs Sampling - ManualMRC Biostatistics UnitCambridgeGoogle Scholar
  48. 48.
    Lunn, D., Wakefield, J., Thomas, A., Best, N.G., Spiegelhalter, D.J. 1999PKBugs User GuideDept. Epidemiology & Public Health, Imperial College School of MedicineLondonGoogle Scholar
  49. 49.
    Best, N.G., Tan, K.K.C., Gilks, W.R., Spiegelhalter, D.J. 1995Estimation of population pharmacokinetics using the Gibbs samplerJ. Pharmacokinet. Biopharm.23407424PubMedGoogle Scholar
  50. 50.
    Lunn, D., Aarons, L. 1997Markov chain Monte Carlo techniques for studying interoccasion and intersubject variability: application to pharmacokinetic dataAppl. Stat.467391Google Scholar
  51. 51.
    Wakefield, J., Walker, S.G. 1997Bayesian nonparametric population models: formulation and comparison with likelihood approachesJ. Pharmacokinet. Biopharm.25235253CrossRefPubMedGoogle Scholar
  52. 52.
    Rosner, G.L., Muller, P. 1997Bayesian population pharmacokinetic and pharmacodynamic analyses using mixture modelsJ. Pharmacokinet. Biopharm.25209233CrossRefPubMedGoogle Scholar
  53. 53.
    Gu, M., Kong, F. 1998A stochastic approximation algorithm with MCMC for incomplete data estimation problemsProc. Natl. Acad. Sci. U S A9572707274PubMedGoogle Scholar
  54. 54.
    Delyon, B., Laveille, M., Moulines, E. 1999Convergence of a stochastic approximation version of the EM procedureAnn. Stat.2794128Google Scholar
  55. 55.
    Chen, J., Zhang, D., Davidian, M. 2002A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distributionBiostatistics3347360CrossRefPubMedGoogle Scholar
  56. 56.
    Concordet, D., Nunez, O. 2002A simulated pseudo-maximum likelihood estimator for nonlinear mixed modelsComput. Statist. Data. Anal.39187201CrossRefGoogle Scholar
  57. 57.
    Bauer, R.J., Guzy, S. 2004Monte Carlo parametric expectation maximization method for analyzing population pharmacokinetic/pharmacodynamic (PK/PD) dataD’Argenio, D.Z eds. Advanced Methods of PK and PD Systems AnalysisKluwer Acacemic PublisherBoston135163Google Scholar
  58. 58.
    Kuhn, E., Laveille, M. 2005Maximum likelihood estimation in nonlinear mixed effects modelsComput. Statist. Data Anal.4910201038CrossRefGoogle Scholar
  59. 59.
    Schumitzky, A. 1995EM algorithms and two stage methods in pharmacokinetic population analysisD’Argenio, D.Z eds. Advanced Methods of Pharmacokinetic and Pharmacodynamic Systems AnalysisPlenum PressNew York145160Google Scholar
  60. 60.
    R. H. Leary, R. Jelliffe, A. Schumitzky, and R. E. Port. Accurate Maximum Likelihood Estimation for Parametric Population Analysis., 2004.
  61. 61.
    Roe, D. 1997Comparison of population pharmacokinetic modelling methods using simulated data: results from the population modelling workgroupStat. Med.1612411262CrossRefPubMedGoogle Scholar
  62. 62.
    Vonesh, E.F., Chinchilli, V.M. 1997Linear and Nonlinear Models for the Analysis of Repeated MeasurementsMarcel DekkerNew YorkGoogle Scholar
  63. 63.
    Pinheiro, J.C., Bates, D.M. 2000Mixed-Effect Models in S and SplusSpringer VerlagNew YorkGoogle Scholar
  64. 64.
    Whiting, B., Kelman, A.W., Grevel, J. 1986Population pharmacokinetics: theory and clinical applicationClin. Pharmacokinet.11387401PubMedGoogle Scholar
  65. 65.
    Whiting, B. 1992The population approach: applications to date An overviewRowland, M.Aarons, L eds. New Strategies in Drug Development and Clinical Evaluation: The population approachCommission of the European CommunitiesLuxembourg4158Google Scholar
  66. 66.
    Swiss Society for Pharmacology and Toxicology. Symposium on frontiers in pharmacokinetic data analysis. Drug Metab. Rev. 15(1&2) (1984).Google Scholar
  67. 67.
    M. Rowland L. B. Sheiner, and J.-L. Steimer. Variability in Drug Therapy: Description, Estimation and Control – A SANDOZ Workshop. Raven Press, 1985.Google Scholar
  68. 68.
    Rowland, M., Aarons, L. 1992New Strategies in Drug Development and Clinical Evaluation: The Population ApproachCommission of the European CommunitiesLuxembourgGoogle Scholar
  69. 69.
    Aarons, L., Balant, L., Danhof, M.,  et al. 1997The Population Approach: Measuring And Managing Variability in Response, Concentration and DoseOffice for Official Publications of the European CommunitiesLuxembourgGoogle Scholar
  70. 70.
    Sheiner, L.B. 1991The intellectual health of clinical drug evaluationClin. Pharmacol. Ther.5049PubMedCrossRefGoogle Scholar
  71. 71.
    G. E. P. Box, W. G. Hunter, and J. S. Hunter. Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. Wiley-Interscience, 1978.Google Scholar
  72. 72.
    G. E. P. Box. The Collected Works of George E.P. Box, Volume II. Chapman & Hall/CRC, 1984.Google Scholar
  73. 73.
    Food and Drug Administration. The Critical Path to New Medical Products., 2005.Google Scholar
  74. 74.
    Holford, N.H., Sheiner, L.B. 1981Understanding the dose–effect relationship: clinical application of pharmacokinetic-pharmacodynamic modelsClin. Pharmacokinet.6429453PubMedGoogle Scholar
  75. 75.
    Holford, N.H., Sheiner, L.B. 1981Pharmacokinetic and pharmacodynamic modeling in vivoCRC. Crit. Rev. Bioeng.5273322Google Scholar
  76. 76.
    Sheiner, L.B., Stanski, D.R., Vozeh, S., Miller, R.D., Ham, J. 1979Simultaneous modeling of pharmacokinetics and pharmacodynamics: application to d-tubocurarineClin. Pharmacol. Ther.25358371PubMedGoogle Scholar
  77. 77.
    Sheiner, L.B., Steimer, J.L. 2000Pharmacokinetic/pharmacodynamic modeling in drug developmentAnnu. Rev. Pharmacol. Toxicol.406795CrossRefPubMedGoogle Scholar
  78. 78.
    Sambol, N.C., Sheiner, L.B. 1991Population dose versus response of betaxolol and atenolol: a comparison of potency and variabilityClin. Pharmacol. Ther.492431PubMedCrossRefGoogle Scholar
  79. 79.
    Sheiner, L.B., Beal, S.L., Sambol, N.C. 1989Study designs for dose-rangingClin. Pharmacol. Ther.466377PubMedCrossRefGoogle Scholar
  80. 80.
    Sheiner, L.B., Benet, L.Z. 1989Premarketing observational studies of population pharmacokinetics of new drugsClin. Pharmacol. Ther.38481487Google Scholar
  81. 81.
    Steimer, J.-L., Vozeh, S., Racine-Poon, A., Holford, N., O’Neill, R. 1994The “population approach”: rationale, methods, applications in clinical pharmacology and drug developmentWelling, P.E.Balant, L.P eds. Handbook of Experimental PharmacologySpringer-VerlagHeidelberg405451Google Scholar
  82. 82.
    J.-L. Steimer, F. Mentre, and A. Mallet. Population studies for evaluation of pharmacokinetic variability: Why? How? When? In 2nd European Congress on Biopharmaceutics and Pharmacokinetics, J. M. Aiache and J. Hirtz (eds), Vol. 2. Paris, 1984, pp. 40–49.Google Scholar
  83. 83.
    Temple, R. 1983Discussion Paper on the Testing of Drugs in the Elderly. Memorandum of the Food and Drug AdministrationFood and Drug Administration DHHSWashington DCGoogle Scholar
  84. 84.
    Temple, R. 1987The clinical investigation of drug use by the elderly: food and drug guidelinesClin. Pharmacol. Ther.42681685PubMedCrossRefGoogle Scholar
  85. 85.
    Food and Drug Administration. Guidance for Industry: Population Pharmacokinetics., 1999.
  86. 86.
    Retout, S., Mentre, F. 2003Further developments of the Fisher information matrix in nonlinear mixed-effects models with evaluation in population pharmacokineticsJ. Biopharm. Stat.13209226CrossRefPubMedGoogle Scholar
  87. 87.
    Retout, S., Duffull, S.B., Mentre, F. 2001Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designsComput. Methods Programs Biomed.65141151CrossRefPubMedGoogle Scholar
  88. 88.
    Aarons, L., Karlsson, M.O., Mentre, F., Rombout, F., Steimer, J.-L., Peer, A. 2001Role of modelling and simulation in phase I drug developmentEur. J. Pharm. Sci.13115122PubMedCrossRefGoogle Scholar
  89. 89.
    Gries, J.M., Troconiz, I.F., Verotta, D., Jacobson, M., Sheiner, L.B. 1997A pooled analysis of CD4 response to zidovudine and zalcitabine treatment in patients with AIDS and AIDS-related complexClin. Pharmacol. Ther.617082CrossRefPubMedGoogle Scholar
  90. 90.
    Vanhove, G.F., Gries, J.M., Verotta, D.,  et al. 1997Exposure–response relationships for saquinavir, zidovudine, and zalcitabine in combination therapyAntimicrob. Agents. Chemother.4124332438PubMedGoogle Scholar
  91. 91.
    Holford, N.H., Peace, K. 1994The effect of tacrine and lecithin in Alzheimer′s disease A population Pharmacodynamic analysis of five clinical trials.Eur. J. Clin. Pharmacol.471724PubMedGoogle Scholar
  92. 92.
    Sheiner, L.B. 1994A new approach to the analysis of analgesic drug trials, illustrated with bromfenac dataClin. Pharmacol. Ther.5630983322CrossRefGoogle Scholar
  93. 93.
    Karlsson, M.O., Sheiner, L.B. 1993The importance of modeling interoccasion variability in population pharmacokinetic analysesJ. Pharmacokinet. Biopharm.21735750CrossRefPubMedGoogle Scholar
  94. 94.
    Girard, P., Blaschke, T.F., Kastrissios, H., Sheiner, L.B. 1998A Markov mixed effect regression model for drug complianceStat. Med.1723132333CrossRefPubMedGoogle Scholar
  95. 95.
    Girard, P., Sheiner, L.B., Kastrissios, H., Blaschke, T.F. 1996Do we need full compliance data for population pharmacokinetic analysis?J. Pharmacokinet. Biopharm.24265282PubMedGoogle Scholar
  96. 96.
    Kastrissios, H., Suarez, J.R., Katzenstein, D., Girard, P., Sheiner, L.B., Blaschke, T.F. 1998Characterizing patterns of drug-taking behavior with a multiple drug regimen in an AIDS clinical trialAIDS1222952303PubMedGoogle Scholar
  97. 97.
    Sheiner, L.B., Beal, S.L., Dunne, A. 1997Analysis of non-randomly censored ordered categorical longitudinal data from analgesic trials (with comment & rejoinder)J. Am. Stat. Assoc.9212351255Google Scholar
  98. 98.
    Cox, E.H., Veyrat-Follet, C., Beal, S.L., Fuseau, E., Kenkare, S., Sheiner, L.B. 1999A population pharmacokinetic-pharmacodynamic analysis of repeated measures time-to-event pharmacodynamic responses: the antiemetic effect of ondansetronJ. Pharmacokinet. Biopharm.27625644CrossRefPubMedGoogle Scholar
  99. 99.
    Hashimoto, Y., Sheiner, L.B. 1991Designs for population pharmacodynamics: value of pharmacokinetic data and population analysisJ. Pharmacokinet. Biopharm.19333353PubMedCrossRefGoogle Scholar
  100. 100.
    Sheiner, L.B. 1990Implications of an alternative approach to dose-response trialsJ. Acquir. Immune Defic. Syndr.3S20S26PubMedGoogle Scholar
  101. 101.
    Sheiner, L.B. 1989Clinical pharmacology and the choice between theory and empiricismClin. Pharmacol. Ther.46605615PubMedCrossRefGoogle Scholar
  102. 102.
    Jonsson, E.N., Sheiner, L.B. 2002More efficient clinical trials through use of scientific model-based statistical testsClin. Pharmacol. Ther.72603614CrossRefPubMedGoogle Scholar
  103. 103.
    T. Goggin, R. Gieschke, G. Pillai, P. Jordan, P. Jordan, and J.-L. Steimer. Modeling and simulation of clinical trials: an industry perspective. In Simulation for Designing Clinical Trials: A Pharmacokinetic-Pharmacodynamic Modeling Perspective, H. Kimko and S. B. Duffull (eds), Dekker, 2002.Google Scholar
  104. 104.
    The European Medicines Agency. The European Medicines Agency Road Map to 2010: Preparing the Ground for the Future. EMEA/H/34163/03/Final, 2005Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Goonaseelan (Colin) Pillai
    • 1
  • France Mentré
    • 2
  • Jean-Louis Steimer
    • 1
  1. 1.Modeling and SimulationNovartis Pharma AGBaselSwitzerland
  2. 2.INSERM U738, Department of Epidemiology, Biostatistics and Clinical ResearchBichat University HospitalParisFrance

Personalised recommendations