Skip to main content

Advertisement

SpringerLink
Log in

Stochastic nonlinear mixed effects: a metformin case study

  • Original Paper
  • Published:
Journal of Pharmacokinetics and Pharmacodynamics Aims and scope Submit manuscript

Abstract

In nonlinear mixed effect (NLME) modeling, the intra-individual variability is a collection of errors due to assay sensitivity, dosing, sampling, as well as model misspecification. Utilizing stochastic differential equations (SDE) within the NLME framework allows the decoupling of the measurement errors from the model misspecification. This leads the SDE approach to be a novel tool for model refinement. Using Metformin clinical pharmacokinetic (PK) data, the process of model development through the use of SDEs in population PK modeling was done to study the dynamics of absorption rate. A base model was constructed and then refined by using the system noise terms of the SDEs to track model parameters and model misspecification. This provides the unique advantage of making no underlying assumptions about the structural model for the absorption process while quantifying insufficiencies in the current model. This article focuses on implementing the extended Kalman filter and unscented Kalman filter in an NLME framework for parameter estimation and model development, comparing the methodologies, and illustrating their challenges and utility. The Kalman filter algorithms were successfully implemented in NLME models using MATLAB with run time differences between the ODE and SDE methods comparable to the differences found by Kakhi [10] for their stochastic deconvolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Aarons L (1999) Pharmacokinetic and pharmacodynamic modelling in drug development. Stat Methods Med Res 8(3):181–182

    Article  PubMed  CAS  Google Scholar 

  2. Buse J, DeFronzo RA, Kim T, Skare S, Baron A, Fineman M (2013) Dissociation between metformin plasma exposure and its glucose-lowering effect: a novel gut-mediated mechanism for action. Presentation at the 49th Annual European Association for the Study of Diabetes Meeting, Barcelona, Spain

  3. Chi EM, Reinsel GC (1989) Models for longitudinal data with random effects and AR(1) errors. J Am Stat Assoc 84:452–459

    Article  Google Scholar 

  4. Davidian M, Giltinan DM (1993) Some general estimation methods for nonlinear mixed-effects model. J Biopharm Stat 3(1):23–55

    Article  PubMed  CAS  Google Scholar 

  5. Duong JK, Kumar SS, Kirkpatrick CM, Greenup LC, Arora M, Lee TC, Timmins P, Graham GG, Furlong TJ, Greenfield JR, Williams KM, Day RO (2013) Population pharmacokinetics of metformin in healthy subjects and patients with type 2 diabetes mellitus: simulation of doses according to renal function. Clin Pharmacokinet 53:373–384

    Article  Google Scholar 

  6. Grimmett G, Stirzaker D (2001) Probability and random processes. Oxford University Press, New York

    Google Scholar 

  7. Haykin S (2001) Kalman filtering and neural networks. Wiley, New York

    Book  Google Scholar 

  8. Jazwinski AH (1970) Stochastic processes and filtering theory. Academic Press, New York

    Google Scholar 

  9. Julier SJ, Uhlmann JK (1997) A new extension of the Kalman filter to nonlinear systems. In: Proceedings of Aerosense: The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Controls

  10. Kakhi M, Chittenden J (2013) Modeling of pharmacokinetic systems using stochastic deconvolution. J Pharm Sci 102:4433–4443

    Article  PubMed  CAS  Google Scholar 

  11. Karlsson MO, Beal SL, Sheiner LB (1995) Three new residual error models for population PK/PD analyses. J Pharmacokinet Biopharm 23(6):651–672

    Article  PubMed  CAS  Google Scholar 

  12. Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASME 82:35–45

    Article  Google Scholar 

  13. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. Trans ASME 83:95–108

    Article  Google Scholar 

  14. Klebaner FC (2005) Introduction to stochastic calculus with applications. Imperial College Press, London

    Book  Google Scholar 

  15. Klim S, Mortensen SB, Kristensen NR, Overgaard RV, Madsen H (2009) Population stochastic modelling (PSM)—an R package for mixed-effects models based on stochastic differential equations. Comput Methods Programs Biomed 94(3):279–289

    Article  PubMed  Google Scholar 

  16. Kristensen NR, Madsen H, Jorgensen SB (2004) Parameter estimation in stochastic grey-box models. Automatica 40:225–237

    Article  Google Scholar 

  17. Lindsey JK, Jones B, Jarvis P (2001) Some statistical issues in modeling pharmacokinetic data. Stat Med 20:2775–2783

    Article  PubMed  CAS  Google Scholar 

  18. Majda AJ, Harlim J (2012) Filtering turbulent complex systems. Cambridge University Press, Cambridge

    Book  Google Scholar 

  19. Matzuka B (2014) Nonlinear filtering methodologies for parameter estimation and uncertainty quantification in noisy, complex biological systems. PhD thesis, North Carolina State University

  20. Mortensen S, Klim S. Population stochastic modelling (PSM): model definition, description and examples. http://www2.imm.dtu.dk/projects/psm/doc/PSM. Published September 18, 2008, Accessed 21 Sept 2015

  21. Myung IJ (2003) Tutorial on maximum likelihood estimation. J Math Psychol 47:90–100

    Article  Google Scholar 

  22. Overgaard RV, Jonsson N, Tornoe CW, Madsen H (2005) Nonlinear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm. J Pharmacokinet Pharmacodyn 32(1):85–107

    Article  PubMed  Google Scholar 

  23. Pinhiero JC, Bates DM (1995) Approximations to the log-likelihood function in the nonlinear mixed effects model. J Comput Graph Stat 4(1):12–35

    Google Scholar 

  24. Racine-Poon A, Wakfield J (1998) Statistical methods for population pharmacokinetic modeling. Stat Methods Med Res 7(1):63–84

    Article  PubMed  CAS  Google Scholar 

  25. Sheiner L, Wakefield J (1999) Population modeling in drug development. Stat Methods Med Res 8(3):183–193

    Article  PubMed  CAS  Google Scholar 

  26. Sheiner LB, Steimer JL (2000) Pharmacokinetic/pharmacodynamic modeling in drug development. Ann Rev Pharmacol Toxicol 40:67–95

    Article  CAS  Google Scholar 

  27. Sloan IH, Wozniakowski H (1998) When are Quasi-Monte Carlo algorithms efficient for high dimensional integrals? J Complex 14(1):1–33

    Article  Google Scholar 

  28. Smith R (2014) Uncertainty quantification: theory, implementation, and applications. SIAM, Philadelphia

    Google Scholar 

  29. Taylor A, Chigutsa E, Monteleone J,Fineman M (2013) Population pharmacokinetic modeling of a novel delayed-release formulation of metformin (MetDR). Poster presented at the American Conference on Pharmacometrics, Fort Lauderdale, FL, USA

  30. Tornoe CW, Agerso H, Jonsson EN, Madson H, Nielsen HA (2004) Nonlinear mixed-effects pharmacokinetic/pharmacodynamic modelling in NLME using differential equations. Comput Methods Prog Biomed 76:31–40

    Article  Google Scholar 

  31. Tornoe CW, Jacobsen JL, Pedersen O, Hansen T, Madsen H (2004) Grey-box modelling of pharmacokinetic/pharmacodynamic systems. J Pharmacokinet Pharmacodyn 31:401–417

    Article  PubMed  CAS  Google Scholar 

  32. Tornoe CW, Overgaard RV, Agerso H, Nielsen HA, Madsen H, Jonsson EN (2005) Stochastic differential equations in NONMEM: implementation, application and comparison with ordinary differential equations. Pharm Res 22(8):1247–1258

    Article  PubMed  CAS  Google Scholar 

  33. Tucker GT, Casey C, Phillips PJ, Connor H, Ward JD, Woods HF (1981) Metformin kinetics in healthy subjects and in patients with diabetes mellitus. Br J Clin Pharm 12:235–246

    Article  CAS  Google Scholar 

  34. Wann E, Van Der Merwe R (2001) The unscented Kalman filter for nonlinear estimation. Adaptive Systems for Signal Processing, Communications, and Control Symposium, IEEE. 2001. p 153–158

Download references

Acknowledgments

Professor Hien Tran was supported in part by the National Science Foundation under Grant NSF-DMS 1022688 and by the National Institute of Allergy and Infectious Diseases under Grant NIAID 9R01AI071915. Metformin data is proprietary property of Elcelyx Therapeutics and used with permission from Dr. Terri Kim.

Author information

Author notes

  1. Brett Matzuka

    Present address: Department of Pharmacology and Therapeutic innovation, Children’s Mercy Hospital, Kansas City, MO, USA

Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Brett Matzuka & Hien Tran

  2. qPharmetra, LLC, Raleigh, NC, USA

    Jason Chittenden

  3. Alexion Pharmaceuticals Inc., New Haven, CT, USA

    Jonathan Monteleone

Authors
  1. Brett Matzuka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jason Chittenden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonathan Monteleone
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hien Tran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brett Matzuka.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matzuka, B., Chittenden, J., Monteleone, J. et al. Stochastic nonlinear mixed effects: a metformin case study. J Pharmacokinet Pharmacodyn 43, 85–98 (2016). https://doi.org/10.1007/s10928-015-9456-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10928-015-9456-7

Keywords

Search

Navigation