Advertisement

Identifiability and indistinguishability of nonlinear pharmacokinetic models

  • Keith R. Godfrey
  • Michael J. Chapman
  • Sandor Vajda
Article

Abstract

Three nonlinear model structures of interest in pharmacokinetics are analyzed to determine whether the unknown, independent, model parameters can be deduced if perfect input-output data were available. This is the problem of identifiability. The method used is based on the local state isomorphism theorem. In certain circumstances, the modeler may be undecided between several model structures and it is then of interest to determine whether different model structures can be distinguished from perfect input-output data. This is the problem of model indistinguishability. The technique used, again based on the local state isomorphism theorem, parallels the similarity transformation approach for linear systems described previously in this journal. The analysis is performed on three two-compartment examples having one linear and one nonlinear (Michaelis-Menten) elimination pathway. In each model there is, on physiological and other grounds, some uncertainty over the precise location (central compartment or peripheral compartment) of one of the elimination pathways.

Key words

compartmental models identification indistinguishability Michaelis-Menten kinetics model discrimination nonlinear systems pharmacokinetics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Bellman and K. J. Åström. On structural identifiability.Math. Biosci. 7:329–339 (1970).CrossRefGoogle Scholar
  2. 2.
    E. Walter.Identifiability of State Space Models, Lecture Notes in Biomathematics, Vol. 46, Springer, New York, 1982.Google Scholar
  3. 3.
    H. Pohjanpalo. System identifiability based on the power series expansion of the solution.Math. Biosci. 41:21–33 (1978).CrossRefGoogle Scholar
  4. 4.
    K. R. Godfrey and W. R. Fitch. The deterministic identifiability of nonlinear pharmacokinetic models.J. Pharmacokin. Biopharm. 12:177–191 (1984).CrossRefGoogle Scholar
  5. 5.
    A. Raksanyi, Y. Lecourtier, E. Walter, and A. Venot. Identifiability and distinguishability testing via computer algebra.Math. Biosci. 77:245–266 (1985).CrossRefGoogle Scholar
  6. 6.
    E. Walter, Y. Lecourtier, A. Raksanyi, and J. Happel. On the distinguishability of parametric models with different structures. In J. Eisenfeld and C. DeLisi (eds),Mathematics and Computers in Biomedical Applications, Elsevier North-Holland, Amsterdam, 1985.Google Scholar
  7. 7.
    A. Isidori.Nonlinear Control Systems: an Introduction, Springer, New York, 1985.CrossRefGoogle Scholar
  8. 8.
    H. J. Sussmann. Existence and uniqueness of minimal realizations of nonlinear systems.Math. Syst. Theory 10:263–284 (1977).CrossRefGoogle Scholar
  9. 9.
    S. Vajda, K. R. Godfrey, and H. Rabitz. Similarity transformation approach to identifiability analysis of nonlinear, compartmental models.Math. Biosci. 93:217–248 (1989).PubMedCrossRefGoogle Scholar
  10. 10.
    K. R. Godfrey and M. J. Chapman. The problem of model indistinguishability in pharmacokinetics.J. Pharmacokin. Biopharm. 17:229–267 (1989).CrossRefGoogle Scholar
  11. 11.
    H. G. Boxenbaum and S. Riegelman. Pharmacokinetics of isoniazid and some metabolites in man.J. Pharmacokin. Biopharm. 4:287–325 (1976).CrossRefGoogle Scholar
  12. 12.
    J. G. Wagner. Do you need a pharmacokinetic model, and if so, which one?J. Pharmacokin. Biopharm. 3:457–478 (1975).CrossRefGoogle Scholar
  13. 13.
    K. R. Godfrey and M. J. Chapman. Identifiability and indistinguishability of linear compartment models.Math. Comput. Simul. 32:273–295 (1990).CrossRefGoogle Scholar
  14. 14.
    J. G. Wagner, G. J. Szunpar, and J. J. Ferry. A nonlinear physiologic pharmacokinetic model: 1-Steady-state.J. Pharmacokin. Biopharm. 13:73–92 (1985).CrossRefGoogle Scholar
  15. 15.
    K. R. Godfrey,Compartmental Models and Their Application, Academic Press, London, and New York, 1983.Google Scholar
  16. 16.
    E. Walter, Y. Lecourtier, and J. Happel. On the structural output distinguishability of parametric models, and its relation with structural identifiability.IEEE Trans. Automat. Contr. 29:56–57 (1984).CrossRefGoogle Scholar
  17. 17.
    K. R. Godfrey and J. J. DiStefano III. Identifiability of model parameters. In E. Walter (ed.),Identifiability of Parametric Models, Pergamon, Oxford, 1987, pp. 1–20.Google Scholar
  18. 18.
    M. J. Chappell, K. R. Godfrey, and S. Vajda. Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods.Math. Biosci. 102:41–73 (1990).PubMedCrossRefGoogle Scholar
  19. 19.
    R. Hermann and A. J. Krener. Nonlinear controllability and observability.IEEE Trans. Automat. Contr. 22:728–740 (1977).CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Keith R. Godfrey
    • 1
  • Michael J. Chapman
    • 2
  • Sandor Vajda
    • 3
  1. 1.Department of EngineeringUniversity of WarwickCoventryEngland
  2. 2.Control Theory and Applications CentreCoventry UniversityCoventryEngland
  3. 3.Department of Biomedical EngineeringBoston UniversityBoston

Personalised recommendations