Two general methods for population pharmacokinetic modeling: non-parametric adaptive grid and non-parametric Bayesian

  • Tatiana Tatarinova
  • Michael Neely
  • Jay Bartroff
  • Michael van Guilder
  • Walter Yamada
  • David Bayard
  • Roger Jelliffe
  • Robert Leary
  • Alyona Chubatiuk
  • Alan Schumitzky
Original Paper


Population pharmacokinetic (PK) modeling methods can be statistically classified as either parametric or nonparametric (NP). Each classification can be divided into maximum likelihood (ML) or Bayesian (B) approaches. In this paper we discuss the nonparametric case using both maximum likelihood and Bayesian approaches. We present two nonparametric methods for estimating the unknown joint population distribution of model parameter values in a pharmacokinetic/pharmacodynamic (PK/PD) dataset. The first method is the NP Adaptive Grid (NPAG). The second is the NP Bayesian (NPB) algorithm with a stick-breaking process to construct a Dirichlet prior. Our objective is to compare the performance of these two methods using a simulated PK/PD dataset. Our results showed excellent performance of NPAG and NPB in a realistically simulated PK study. This simulation allowed us to have benchmarks in the form of the true population parameters to compare with the estimates produced by the two methods, while incorporating challenges like unbalanced sample times and sample numbers as well as the ability to include the covariate of patient weight. We conclude that both NPML and NPB can be used in realistic PK/PD population analysis problems. The advantages of one versus the other are discussed in the paper. NPAG and NPB are implemented in R and freely available for download within the Pmetrics package from


Population pharmacokinetic modeling Non-parametric Maximum likelihood Bayesian Stick-breaking Pmetrics RJags 



Support from NIH: GM068968, EB005803, EB001978, NIH-NICHD: HD070996 and Royal Society: TG103083 is gratefully acknowledged.


  1. 1.
    FDA (1999) FDA guidance for industry: population pharmacokineticsGoogle Scholar
  2. 2.
    Beal S, Sheiner L (1982) Estimating population kinetics. Crit Rev Biomed Eng 8(3):95–222Google Scholar
  3. 3.
    Beal S, Sheiner L (1992) NONMEM User’s Guide. In: Nonlinear mixed effects models for repeated measures. University of California, San FranciscoGoogle Scholar
  4. 4.
    Lavielle M, Mentré F (2007) Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the MONOLIX software. J Pharmacokinet Pharmacodyn 34(2):229–249PubMedCrossRefGoogle Scholar
  5. 5.
    D’Argenio D, Schumitzky A, Wang X (2009) ADAPT 5 User’s guide:pharmacokinetic/pharmacodynamic systems analysis Software. Biomedical Simulations Resource, Los AngelesGoogle Scholar
  6. 6.
    Wang A, Schumitzky A, DArgenio D (2007) Nonlinear random effects mixture models: maximum likelihood estimation via the EM algorithm. Comput Stat Data Anal 51:6614–6623PubMedCrossRefGoogle Scholar
  7. 7.
    Wang A, Schumitzky A, DArgenio D (2009) Population pharmacokinet-ic/pharmacodyanamic mixture models via maximum a posteriori estimation. Comput Stat Data Anal 53:3907PubMedCrossRefGoogle Scholar
  8. 8.
    Lindsay B (1983) The geometry of mixture likelihoods: a general theory. Ann Stat 11:86–94CrossRefGoogle Scholar
  9. 9.
    Mallet A (1986) A maximum likelihood estimation method for random coefficient regression models. Biometrika 73:645–656CrossRefGoogle Scholar
  10. 10.
    Kiefer J, Wolfowitz J (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann Math Stat 27(4):887–906CrossRefGoogle Scholar
  11. 11.
    Baverel P, Savic R, Karlsson M (2011) Two bootstrapping routines for obtaining imprecision estimates for nonparametric parameter distributions in nonlinear mixed effects models. J Pharmacokinet Pharmacodyn 38(1):63–82PubMedCrossRefGoogle Scholar
  12. 12.
    Spiegelhalter DJ, Thomas A, Best NG (2004) WinBUGS Version 1.4 User Manual, MRC Biostatistics UnitGoogle Scholar
  13. 13.
    Plummer M (2003) JAGS: A program for analysis of Bayesian Graphical Models Using Gibbs Sampling. Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), ViennaGoogle Scholar
  14. 14.
    Wakefield J, Walker S (1997) Bayesian nonparametric population models: formulation and comparison with likelihood approaches. J Pharmacokinet Biopharm 25:235–253PubMedCrossRefGoogle Scholar
  15. 15.
    Wakefield J, Walker S (1998) Population models with a nonparametric random coefficient distribution. Sankhya Ser B 60:196–212Google Scholar
  16. 16.
    Mueller P, Rosner G (1997) A Bayesian population model with hierarchical mixture priors applied to blood count data. J Am Stat Assoc 92:1279–1292Google Scholar
  17. 17.
    Rosner G, Mueller P (1997) Bayesian population pharmacokinetic and pharmacodynamic analyses using mixture models. J Pharmacokinet Biopharm 25:209–233PubMedCrossRefGoogle Scholar
  18. 18.
    Wang J (2010) Dirichlet processes in nonlinear mixed effects models. Commun Stat Simul Comput 39:539–556CrossRefGoogle Scholar
  19. 19.
    Yamada Y, Bartroff J, Bayard D, Burke J, Van Guilder M, RW J, et al. (2012) The nonparametric adaptive grid algorithm for population pharmacokinetic modeling. Technical Report, LAPK, USC, Laboratory of Applied PharmacokineticsGoogle Scholar
  20. 20.
    Neely M, Tatarinova T, Bartroff J, Van Guilder M, Yamada W, Bayard D, et al (2012) Non-parametric Bayesian fitting: a novel approach to population pharmacokinetic modeling. Poster presented at: Population Analysis Group in Europe, VeniceGoogle Scholar
  21. 21.
    Neely M, van Guilder M, Yamada W, Schumitzky A, Jelliffe R (2012) Accurate detection of outliers and subpopulations with Pmetrics, a nonparametric and parametric pharmacometric modeling and simulation package for R. Ther Drug Monit 34(4):467–476PubMedCrossRefGoogle Scholar
  22. 22.
    Bustad A, Terziivanov D, Leary R, Port R, Schumitzky A, Jelliffe R (2006) Parametric and nonparametric population methods: their comparative performance in analysing a clinical dataset and two Monte Carlo simulation studies. Clin Pharmacokinet 45(4):365–383PubMedCrossRefGoogle Scholar
  23. 23.
    Ishwaran H, James L (2001) Gibbs sampling methods for stick-breaking priors. J Am Stat Assoc 96:161–173CrossRefGoogle Scholar
  24. 24.
    Jelliffe R, Bayard D, Milman M, Van Guilder M, Schumitzky A (2000) Achieving target goals most precisely using nonparametric compartmental models and “multiple model” design of dosage regimens. Ther Drug Monit 22:346–353PubMedCrossRefGoogle Scholar
  25. 25.
    Schumitzky A (1991) Nonparametric EM algorithms for estimating prior distributions. Appl Math Comput 45:141–157CrossRefGoogle Scholar
  26. 26.
    Leary R, Jelliffe R, Schumitzky A, Van Guilder M (2001) An adaptive grid non-parametric approach to population pharmacokinetic/dynamic (PK/PD) population models. Proceedings, 14th IEEE symposium on computer based medical systems, pp 389–394Google Scholar
  27. 27.
    Baek Y (2006) An interior point approach to constrained nonparametric mixture models. PhD Dissertation, Thesis supervisor: Prof. James Burke, University of Washington, Department of MathematicsGoogle Scholar
  28. 28.
    Fox BL (1986) Algorithm 647: implementation and relative efficiency of Quasirandom sequence generators. Trans Math Softw 12(4):362–376CrossRefGoogle Scholar
  29. 29.
    Karush W (1939) Minima of functions of several variables with inequalities as side constraints. MSc disseertation, University of Chicago, Department of Mathematics, ChicagoGoogle Scholar
  30. 30.
    Kuhn H, Tucker A (1951) Nonlinear programming. Proceedings of the 2nd Berkeley Symposium, pp 481–492Google Scholar
  31. 31.
    Papaspiliopoulos O, Roberts GO (2008) Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika 95(1):169–186CrossRefGoogle Scholar
  32. 32.
    Sethuraman J (1994) A constructive definition of Dirichlet priors. Statistica Sinica 4:639–650Google Scholar
  33. 33.
    Tatarinova T (2006) Bayesian analysis of linear and nonlinear mixture models. USC, Los AngelesGoogle Scholar
  34. 34.
    Tatarinova T, Bouck J, Schumitzky A (2008) Kullback-Leibler Markov chain Monte Carlo—a new algorithm for finite mixture analysis and its application to gene expression data. J Bioinform Comput Biol 6(4):727–746PubMedCrossRefGoogle Scholar
  35. 35.
    Frühwirth-Schnatter S (2010) Finite mixture and markov switching models, 1st edn. Springer, New YorkGoogle Scholar
  36. 36.
    Ghosh P, Rosner G (2007) A semiparametric Bayesian approach to average bioequivalence. Stat Med 26:1224–1236PubMedCrossRefGoogle Scholar
  37. 37.
    Ohlssen D, Sharples L, Spiegelhalter D (2007) Flexible random-effects models using Bayesian semi-parametric models: applications to institutional comparisons. Stat Med 26:2088–2112PubMedCrossRefGoogle Scholar
  38. 38.
    Walker S (2007) Sampling the Dirichlet mixture model with slices. Commun Stat Simul Comput 36:45–54CrossRefGoogle Scholar
  39. 39.
    Kalli M, Griffen J, Walker S (2011) Slice sampling mixture models. Stat Comput 21:93–105CrossRefGoogle Scholar
  40. 40.
    Plummer M (2011) rjags: Bayesian graphical models using MCMCGoogle Scholar
  41. 41.
    Robert C (2007) The Bayesian choice, 2nd edn. Springer, New YorkGoogle Scholar
  42. 42.
    Robert C, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  43. 43.
    Dunson D (2010) Nonparametric Bayes applications to biostatistics. In: Hjort N, Holmes C, Muller P, Walker S (eds) Bayesian nonparametrics. Cambridge University Press, Cambridge, pp 223–268Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Tatiana Tatarinova
    • 1
  • Michael Neely
    • 1
  • Jay Bartroff
    • 1
    • 2
  • Michael van Guilder
    • 1
  • Walter Yamada
    • 1
    • 4
  • David Bayard
    • 1
    • 3
  • Roger Jelliffe
    • 1
  • Robert Leary
    • 1
    • 5
  • Alyona Chubatiuk
    • 2
  • Alan Schumitzky
    • 1
    • 2
  1. 1.Laboratory of Applied Pharmacokinetics, Keck School of MedicineUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsDornsife College of Letters and Science, University of Southern CaliforniaLos AngelesUSA
  3. 3.Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaUSA
  4. 4.Department of PsychologyAzusa Pacific UniversityAzusaUSA
  5. 5.Pharsight CorporationCaryUSA

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