Skip to main content
SpringerLink
Log in

Parameter estimation in nonlinear mixed effect models based on ordinary differential equations: an optimal control approach

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

We present a method for parameter estimation for nonlinear mixed-effects models based on ordinary differential equations (NLME-ODEs). It aims to regularize the estimation problem in the presence of model misspecification and practical identifiability issues, while avoiding the need to know or estimate initial conditions as nuisance parameters. To this end, we define our estimator as a minimizer of a cost function that incorporates a possible gap between the assumed population-level model and the specific individual dynamics. The computation of the cost function leads to formulate and solve optimal control problems at the subject level. Compared to the maximum likelihood method, we show through simulation examples that our method improves the estimation accuracy in possibly partially observed systems with unknown initial conditions or poorly identifiable parameters with or without model error. We conclude this work with a real-world application in which we model the antibody concentration after Ebola virus vaccination.

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

Similar content being viewed by others

Availability of data and materials

Not applicable

Code availability

Our estimation method is implemented in R and a code reproducing the examples of Sect. 3 is available on a GitHub repository located here (https://github.com/QuentinClairon/NLME_ODE_estimation_via_optimal_control.git).

References

  • Aliyu M (2011) Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations, CRC Press

  • Balelli I, Pasin C, Prague M et al (2020) A model for establishment, maintenance and reactivation of the immune response after vaccination against Ebola virus. J Theor Biol 495:110254

    Article  MathSciNet  Google Scholar 

  • Brynjarsdottir J, O’Hagan A (2014) Learning about physical parameters: the importance of model discrepancy. Inverse Prob 30:24

    Article  MathSciNet  MATH  Google Scholar 

  • Campbell D (2007) Bayesian collocation tempering and generalized profiling for estimation of parameters from differential equation models. PhD thesis, McGill University Montreal, Quebec

  • Cimen T (2008) State−dependent Riccati equation (SDRE) control: a survey. IFAC Proc 41:3761–3775

    Google Scholar 

  • Cimen T, Banks S (2004) Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria. Syst Control Lett 53:327–346

    Article  MathSciNet  MATH  Google Scholar 

  • Clairon Q (2020) A regularization method for the parameter estimation problem in ordinary differential equations via discrete optimal control theory. J Stat Plan Inference 210:1–9

    Article  MathSciNet  MATH  Google Scholar 

  • Clarke F (2013) Functional analysis, calculus of variations and optimal control, Graduate Texts in Mathematics, Springer, London

  • Comets E, Lavenu A, Lavielle M (2017) Parameter estimation in nonlinear mixed effect models using saemix, an R implementation of the SAEM algorithm. J Stat Softw 80:1–42

    Article  Google Scholar 

  • De Gaetano A, Arino O (2000) Mathematical modelling of the intravenous glucose tolerance test. J Math Biol 40(2):136–168

    Article  MathSciNet  MATH  Google Scholar 

  • Donnet S, Samson A (2006) Estimation of parameters in incomplete data models defined by dynamical systems. J Stat Plan Inference 137(9):2815–2831

    Article  MathSciNet  MATH  Google Scholar 

  • Engelhardt B, Kschischo M, Fröhlich H (2017) A Bayesian approach to estimating hidden variables as well as missing and wrong molecular interactions in ordinary differential equation-based mathematical models. J R Soc Interface 14(131):20170332

    Article  Google Scholar 

  • Engl H, Flamm C, Kügler P et al (2009) Inverse problems in systems biology. Inverse Prob 25(12):123014

    Article  MathSciNet  MATH  Google Scholar 

  • Hooker G, Ellner SP, Roditi LD, Earn DJ (2011) Parameterizing state−space models for infectious disease dynamics by generalized profiling: measles in Ontario. J R Soc 8:961–974

    Google Scholar 

  • Guedj J, Thiebaut R, Commenges D (2007) Maximum likelihood estimation in dynamical models of HIV. Biometrics 63:1198–206

    Article  MathSciNet  MATH  Google Scholar 

  • Gutenkunst RN, Waterfall J, Casey F et al (2007) Universally sloppy parameter sensitivities in systems biology models. Public Libr Sci Comput Biol 3:e189

    MathSciNet  Google Scholar 

  • Hooker G, Ellner SP et al (2015) Goodness of fit in nonlinear dynamics: Misspecified rates or misspecified states? Ann Appl Stat 9(2):754–776

    Article  MathSciNet  MATH  Google Scholar 

  • Huang Y, Dagne G (2011) A Bayesian approach to joint mixed-effects models with a skew normal distribution and measurement errors in covariates. Biometrics 67:260–269

    Article  MathSciNet  MATH  Google Scholar 

  • Huang Y, Lu T (2008) Modeling long-term longitudinal HIV dynamics with application to an aids clinical study. Ann Appl Stat 2:1348–1408

    Article  MathSciNet  MATH  Google Scholar 

  • Kampen NV (1992) Stochastic process in physics and chemistry. Elsevier

    Google Scholar 

  • Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc Ser B (Stat Methodol) 63(3):425–464

    Article  MathSciNet  MATH  Google Scholar 

  • Kurtz T (1978) Strong approximation theorems for density dependent Markov chains. Stoch Process Appl 6:223–240

    Article  MathSciNet  MATH  Google Scholar 

  • Lavielle M, Aarons L (2015) What do we mean by identifiability in mixed effects models? J Pharmacokinet Pharmacodyn 43:111–122

    Article  Google Scholar 

  • Lavielle M, Mentré F (2007) Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the monolix software. J Pharmacokinet Pharmacodyn 34:229–249

    Article  Google Scholar 

  • Leary TO, Sutton A, Marder E (2015) Computational models in the age of large datasets. Curr Opin Neurobiol 32:87–94

    Article  Google Scholar 

  • Lunn D, Thomas A, Best N et al (2000) Winbugs—a Bayesian modelling framework: concepts, structure and extensibility. Stat Comput 10:325–337

    Article  Google Scholar 

  • Lavielle M, Samson A, Karina Fermin A, Mentre F (2011) Maximum likelihood estimation of long terms HIV dynamic models and antiviral response. Biometrics 67:250–259

    Article  MathSciNet  MATH  Google Scholar 

  • Murphy S, der Vaart AV (2000) On profile likelihood. J Am Stat Assoc 95:449–465

    Article  MathSciNet  MATH  Google Scholar 

  • Nash JC (2016) Using and extending the optimr package

  • Pasin C, Balelli I, Van Effelterre T et al (2019) Dynamics of the humoral immune response to a prime−boost Ebola vaccine: quantification and sources of variation. J Virol 93(18):e00579-19

    Article  Google Scholar 

  • Perelson A, Neumann A, Markowitz M et al (1996) Hiv-1 dynamics in vivo: virion clearance rate, infected cell life−span, and viral generation time. Science 271:1582–1586

    Article  Google Scholar 

  • Pinheiro J, Bates DM (1994) Approximations to the loglikelihood function in the nonlinear mixed effects model. J Comput Graph Stat 4:12–35

    Google Scholar 

  • Prague M, Commenges D, Drylewicz J et al (2012) Treatment monitoring of HIV-infected patients based on mechanistic models. Biometrics 68(3):902–911

    Article  MathSciNet  MATH  Google Scholar 

  • Prague M, Commengues D, Guedj J et al (2013) Nimrod: a program for inference via a normal approximation of the posterior in models with random effects based on ordinary differential equations. Comput Methods Programs Biomed 111:447–458

    Article  Google Scholar 

  • Raftery A, Bao L (2010) Estimating and projecting trends in HIV/aids generalized epidemics using incremental mixture importance sampling. Biometrics 66:1162–1173

    Article  MathSciNet  MATH  Google Scholar 

  • Ramsay J, Hooker G, Cao J et al (2007) Parameter estimation for differential equations: a generalized smoothing approach. J R Stat Soc 69:741–796

    Article  MathSciNet  MATH  Google Scholar 

  • Sartori N (2003) Modified profile likelihood in models with stratum nuisance parameters. Biometrika 90:533–549

    Article  MathSciNet  MATH  Google Scholar 

  • Sontag E (1998) Mathematical control theory: deterministic finite−dimensional systems. Springer, New York

    Book  MATH  Google Scholar 

  • Tornoe C, Agerso H, Jonsson EN et al (2004) Non-linear mixed-effects pharmacokinetic/pharmacodynamic modelling in NLME using differential equations. Comput Methods Programs Biomed 76:31–41

    Article  Google Scholar 

  • Tuo R, Wu C (2015) Efficient calibration for imperfect computer models. Ann Stat

  • van der Vaart A (1998) Asymptotic Statistics, Cambridge Series in Statistical and Probabilities Mathematics, Cambridge University Press

  • Varah JM (1982) A spline least squares method for numerical parameter estimation in differential equations. SIAM J Sci Stat Comput 3(1):28–46

    Article  MathSciNet  MATH  Google Scholar 

  • Villain L, Commenges D, Pasin C et al (2019) Adaptive protocols based on predictions from a mechanistic model of the effect of IL7 on CD4 counts. Stat Med 38(2):221–235

    Article  MathSciNet  Google Scholar 

  • Wang L, Cao J, Ramsay J et al (2014) Estimating mixed-effects differential equation models. Stat Comput 24:111–121

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d’ Aquitaine (see https://www.plafrim.fr/). This manuscript was developed under WP4 of EBOVAC3.

Funding

This work has received funding from the Innovative Medicines Initiative 2 Joint Undertaking under projects EBOVAC1 and EBOVAC3 (respectively grant agreement No 115854 and No 800176). The IMI2 Joint Undertaking receives support from the European Union’s Horizon 2020 research and innovation programme and the European Federation of Pharmaceutical Industries and Association.

Author information

Authors and Affiliations

  1. Inria Bordeaux Sud-Ouest, Inserm, Bordeaux Population Health Research Center, SISTM Team, UMR1219, University of Bordeaux, 33000, Bordeaux, France

    Quentin Clairon,  Rodolphe Thiébaut &  Mélanie Prague

  2. Vaccine Research Institute, 94000, Créteil, France

    Quentin Clairon,  Rodolphe Thiébaut &  Mélanie Prague

  3. Department of Infectious Diseases and Hospital Epidemiology, University Hospital, Collegium Helveticum, Institute of Medical Virology, University of Zurich, Zurich, Switzerland

    Chloé Pasin

  4. Centre Inria d’Université Côte d’Azur, EPIONE Research Project, Valbonne, France

    Irene Balelli

Authors
  1. Quentin Clairon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chloé Pasin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Irene Balelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rodolphe Thiébaut
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mélanie Prague
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quentin Clairon.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clairon, Q., Pasin, C., Balelli, I. et al. Parameter estimation in nonlinear mixed effect models based on ordinary differential equations: an optimal control approach. Comput Stat (2023). https://doi.org/10.1007/s00180-023-01420-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00180-023-01420-x

Keywords

Search

Navigation