Application of new measures of nonlinearity to parameter estimation and simulations in individual pharmacokinetic analyses

  • Leonid KhinkisEmail author
  • Milburn Crotzer
Original Paper


Since simulation studies are widely used in pharmacokinetics, it is necessary to ascertain their validity. An important, well-documented, concern that may negatively impact the validity of estimated parameters in pharmacokinetic models is existence of multiple minima of the criterion used in estimation. A presence of multiple minima causes instability of the estimates, dependence of the parameter estimates on the initial values, and bimodal (or, generally, multimodal) distributions of the estimated parameters. This paper offers a method to identify when these issues may occur by applying two new measures of nonlinearity.


Pharmacokinetic models Nonlinear parameter estimation Simulation Compartmental models Michaelis–Menten model 


Supplementary material

10928_2018_9616_MOESM1_ESM.pdf (26 kb)
Electronic supplementary material 1 (PDF 26 kb)
10928_2018_9616_MOESM2_ESM.pdf (27 kb)
Electronic supplementary material 2 (PDF 27 kb)


  1. 1.
    Demidenko E (2000) Is this the least squares estimate? Biometrika 87:437–452CrossRefGoogle Scholar
  2. 2.
    Toulias T, Kitsos C (2016) Fitting the Michaelis–Menten model. J. Comput. Appl. Math. 296:303–319CrossRefGoogle Scholar
  3. 3.
    Cobelli C, Salvan A (1997) Parameter estimation in a biological two compartment model—a computer experimental study of the influence of the initial estimate in the parameter space and of the model representation. Math Biosci 33:51–62CrossRefGoogle Scholar
  4. 4.
    Ritz C, Streibig JC (2008) Nonlinear regression with R. SpringerGoogle Scholar
  5. 5.
    Chavent G (1990) A new sufficient condition for the wellposedness of non-linear least-square problems arising in identification and control. In: Analysis and optimization of systems, lecture notes in control and inform sci, vol 144. Springer, pp 452–463Google Scholar
  6. 6.
    Demidenko E (2006) Criteria for global minimum of sum of squares in nonlinear regression. Comput Stat Data Anal 51:1739–1753CrossRefGoogle Scholar
  7. 7.
    Pronzato L, Pazman A (2013) Design of experiments in nonlinear models. Springer, New YorkCrossRefGoogle Scholar
  8. 8.
    Pronzato L, Walter E (2001) Eliminating suboptimal local minimizers in nonlinear parameter estimation. Technometrics 43(4):434–442CrossRefGoogle Scholar
  9. 9.
    Toulias T, Kitsos C (2016) Estimation aspects of the Michaelis–Menten model. REVSTAT 14(2):101–118Google Scholar
  10. 10.
    Ratkowsky DA (1983) Nonlinear regression modeling, a uniform practical approach. Marcel Dekker, New YorkGoogle Scholar
  11. 11.
    Ratkowsky DA (1990) Handbook of nonlinear regression models. Marcel Dekker, New YorkGoogle Scholar
  12. 12.
    SAS/STAT(R) (2015) 14.1 User’s Guide, the NLIN procedure. SAS Institute Inc., Cary, NCGoogle Scholar
  13. 13.
    Box M (1971) Bias in nonlinear estimation. J R Stat Soc 33:171–201Google Scholar
  14. 14.
    Hougaard P (1985) The appropriateness of the asymptotic distribution in a nonlinear regression model in relation to curvature. J R Stat Soc B 47:103–114Google Scholar
  15. 15.
    Beale EML (1960) Confidence regions in nonlinear estimation. J R Stat Soc 22:41–76Google Scholar
  16. 16.
    Bates DM, Watts DG (1980) Relative curvature measures of nonlinearity (with discussion). J R Stat Soc B 42:1–25Google Scholar
  17. 17.
    Bates DM, Watts DG (1988) Nonlinear regression analysis and its applications. Wiley, New YorkCrossRefGoogle Scholar
  18. 18.
    Hamilton DC, Watts DG, Bates DM (1982) Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions. Ann Stat 10:386–393CrossRefGoogle Scholar
  19. 19.
    Hamilton D (1986) Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions. Biometrika 73:57–64CrossRefGoogle Scholar
  20. 20.
    Liu Y, Li XR (2015) Measure of nonlinearity for estimation. IEEE Trans Signal Process 63(9):2377–2388CrossRefGoogle Scholar
  21. 21.
    Khinkis L, Crotzer M, Oprisan A (2018) Sizing up the regions of unique minima in the least squares nonlinear regression. Math Appl 7:41–52CrossRefGoogle Scholar
  22. 22.
    Khinkis L, Crotzer M (2008) A new approach for finding global minima in nonlinear least squares regression. In: Proceedings of the American Statistical Association Biopharmaceutical Section—JSM, pp 2264–2271Google Scholar
  23. 23.
    Metzler CM (1981) Estimation of pharmacokinetic parameters: statistical considerations. Pharmacol Ther 13:543–556CrossRefGoogle Scholar
  24. 24.
    Laskarzewski DL, Weiner DL, Ott L (1982) A simulation study of parameter estimation in the one and two compartment models. J Pharmacokinet Biopharm 10(3):317–334CrossRefGoogle Scholar
  25. 25.
    Endrenyi L (1981) Design of experiments for estimating enzyme and pharmacokinetic parameters. In: Endrenyi L (ed) Kinetic data analysis: design and analysis of enzyme and pharmacokinetic experiments. Plenum Press, New York, NY, pp 137–167CrossRefGoogle Scholar
  26. 26.
    Watts DG (1981) An introduction to nonlinear least squares. In: Endrenyi L (ed) Kinetic data analysis: design and analysis of enzyme and pharmacokinetic experiments. Kluwer Academic Pub, Dordrecht, pp 1–24Google Scholar
  27. 27.
    Westlake WJ (1973) Use of statistical methods in evaluation of in vivo performance of dosage forms. Pharm Sci 62:1579–1589CrossRefGoogle Scholar
  28. 28.
    Pazman A (1993) Nonlinear statistical models. Kluwer, DordrechtCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Canisius CollegeBuffaloUSA

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