Journal of Pharmacokinetics and Pharmacodynamics

, Volume 44, Issue 5, pp 415–423 | Cite as

Deterministic identifiability of population pharmacokinetic and pharmacokinetic–pharmacodynamic models

  • Vijay K. SiripuramEmail author
  • Daniel F. B. Wright
  • Murray L. Barclay
  • Stephen B. Duffull
Original Paper


Identifiability is an important component of pharmacokinetic–pharmacodynamic (PKPD) model development. Structural identifiability is concerned with the uniqueness of the model parameters for a set of perfect input–output data and deterministic identifiability with the precision of parameter estimation given imperfect input–output data. We introduce two subcategories of deterministic identifiability, external and internal, and consider factors that distinguish between these forms. We define external deterministic identifiability as a function of externally controllable variables, i.e., the design, and internal deterministic identifiability as a function of the model and its parameter values. The concepts are explored using three common PK and PKPD models, and verified for their precision for the selected set of parameter values under optimal design.


Identifiability Parameter precision Population analysis PKPD models Optimal study design NONMEM 



Vijay K Siripuram is supported by University of Otago postgraduate scholarship.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

10928_2017_9530_MOESM1_ESM.docx (293 kb)
Supplementary material 1 (DOCX 293 kb)


  1. 1.
    Cobelli C (1981) A priori identifiability analysis in pharmacokinetic experiment design. In: Endrenyi L (ed) Kinetic data analysis: design and analysis of enzyme and pharmacokinetic experiments. Springer, Boston, pp 181–208. doi: 10.1007/978-1-4613-3255-8_12 CrossRefGoogle Scholar
  2. 2.
    Bellman R, Åström KJ (1970) On structural identifiability. Math Biosci 7(3–4):329–339. doi: 10.1016/0025-5564(70)90132-X CrossRefGoogle Scholar
  3. 3.
    Evans N, Godfrey K, Chapman M, Chappell M, Aarons L, Duffull SB (2001) An identifiability analysis of a parent-metabolite pharmacokinetic model for ivabradine. J Pharmacokinet Pharmacodyn 28(1):93–105. doi: 10.1023/A:1011521819898 CrossRefPubMedGoogle Scholar
  4. 4.
    Godfrey KR, Chapman MJ (1990) Identifiability and indistinguishability of linear compartmental models. Math Comput Simul 32(3):273–295. doi: 10.1016/0378-4754(90)90185-L CrossRefGoogle Scholar
  5. 5.
    Godfrey KR, Fitch WR (1984) The deterministic identifiability of nonlinear pharmacokinetic models. J Pharmacokinet Biopharm 12(2):177–191. doi: 10.1007/BF01059277 CrossRefPubMedGoogle Scholar
  6. 6.
    Jacquez JA (1987) Identifiability: the first step in parameter estimation. Fed Proc 46(8):2477–2480PubMedGoogle Scholar
  7. 7.
    Merino JA, De Biasi J, Plusquellec Y, Houin G (1998) Local identifiability for two and three-compartment pharmacokinetic models with time-lags. Med Eng Phys 20(4):261–268. doi: 10.1016/S1350-4533(98)00015-0 CrossRefPubMedGoogle Scholar
  8. 8.
    Yates JWT (2006) Structural identifiability of physiologically based pharmacokinetic models. J Pharmacokinet Pharmacodyn 33(4):421–439. doi: 10.1007/s10928-006-9011-7 CrossRefPubMedGoogle Scholar
  9. 9.
    Yates JWT, Jones RDO, Walker M, Cheung SYA (2009) Structural identifiability and indistinguishability of compartmental models. Expert Opin Drug Metab Toxicol 5(3):295–302. doi: 10.1517/17425250902773426 CrossRefPubMedGoogle Scholar
  10. 10.
    Jacquez JA (1991) Identifiability and parameter estimation. J Parenter Enter Nutr 15(3):55S–59S. doi: 10.1177/014860719101500355s CrossRefGoogle Scholar
  11. 11.
    Cheung SY, Majid O, Yates JW, Aarons L (2012) Structural identifiability analysis and reparameterisation (parameter reduction) of a cardiovascular feedback model. Eur J Pharm Sci 46(4):259–271. doi: 10.1016/j.ejps.2011.12.017 CrossRefPubMedGoogle Scholar
  12. 12.
    Cheung SY, Yates JW, Aarons L (2013) The design and analysis of parallel experiments to produce structurally identifiable models. J Pharmacokinet Pharmacodyn 40(1):93–100. doi: 10.1007/s10928-012-9291-z CrossRefPubMedGoogle Scholar
  13. 13.
    Janzen DL, Jirstrand M, Chappell MJ, Evans ND (2016) Three novel approaches to structural identifiability analysis in mixed-effects models. Comput Methods Programs Biomed. doi: 10.1016/j.cmpb.2016.04.024 PubMedGoogle Scholar
  14. 14.
    Shivva V, Korell J, Tucker IG, Duffull SB (2013) An approach for identifiability of population pharmacokinetic–pharmacodynamic models. CPT Pharmacomet Syst Pharmacol 2(6):1–9. doi: 10.1038/psp.2013.25 CrossRefGoogle Scholar
  15. 15.
    Shivva V, Korell J, Tucker IG, Duffull SB (2014) Parameterisation affects identifiability of population models. J Pharmacokinet Pharmacodyn 41(1):81–86. doi: 10.1007/s10928-013-9347-8 CrossRefPubMedGoogle Scholar
  16. 16.
    Lavielle M, Aarons L (2016) What do we mean by identifiability in mixed effects models? J Pharmacokinet Pharmacodyn 43(1):111–122. doi: 10.1007/s10928-015-9459-4 CrossRefPubMedGoogle Scholar
  17. 17.
    Cobelli C, Lepschy A, Jacur GR (1979) Identifiability results on some constrained compartmental systems. Math Biosci 47(3):173–195. doi: 10.1016/0025-5564(79)90036-1 CrossRefGoogle Scholar
  18. 18.
    Chappell MJ, Godfrey KR, Vajda S (1990) Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods. Math Biosci 102(1):41–73. doi: 10.1016/0025-5564(90)90055-4 CrossRefPubMedGoogle Scholar
  19. 19.
    Janzén D, Jirstrand M, Evans ND, Chappell M (2015) Structural identifiability in mixed-effects models: two different approaches. IFAC-Papers OnLine 48(20):563–568. doi: 10.1016/j.ifacol.2015.10.201 CrossRefGoogle Scholar
  20. 20.
    Bellu G, Saccomani MP, Audoly S, D’Angio L (2007) DAISY: a new software tool to test global identifiability of biological and physiological systems. Comput Methods Programs Biomed 88(1):52–61. doi: 10.1016/j.cmpb.2007.07.002 CrossRefPubMedPubMedCentralGoogle Scholar
  21. 21.
    Jacquez JA, Greif P (1985) Numerical parameter identifiability and estimability: integrating identifiability, estimability, and optimal sampling design. Math Biosci 77(1):201–227. doi: 10.1016/0025-5564(85)90098-7 CrossRefGoogle Scholar
  22. 22.
    Karlsson J, Anguelova M, Jirstrand M (2012) An efficient method for structural identifiability analysis of large dynamic systems. IFAC Proc Vol 45(16):941–946. doi: 10.3182/20120711-3-BE-2027.00381 CrossRefGoogle Scholar
  23. 23.
    Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmuller U, Timmer J (2009) Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25(15):1923–1929. doi: 10.1093/bioinformatics/btp358 CrossRefPubMedGoogle Scholar
  24. 24.
    Mentré F, Mallet A, Baccar D (1997) Optimal design in random-effects regression models. Biometrika 84(2):429–442CrossRefGoogle Scholar
  25. 25.
    Duffull SB, Waterhouse T, Eccleston J (2005) Some considerations on the design of population pharmacokinetic studies. J Pharmacokinet Pharmacodyn 32(3):441–457. doi: 10.1007/s10928-005-0034-2 CrossRefPubMedGoogle Scholar
  26. 26.
    Sharma A, Jusko WJ (1998) Characteristics of indirect pharmacodynamic models and applications to clinical drug responses. Br J Clin Pharmacol 45(3):229–239CrossRefPubMedPubMedCentralGoogle Scholar
  27. 27.
    Dayneka NL, Garg V, Jusko WJ (1993) Comparison of four basic models of indirect pharmacodynamic responses. J Pharmacokinet Biopharm 21(4):457–478. doi: 10.1007/bf01061691 CrossRefPubMedPubMedCentralGoogle Scholar
  28. 28.
    Sharma A, Jusko WJ (1996) Characterization of four basic models of indirect pharmacodynamic responses. J Pharmacokinet Biopharm 24(6):611–635. doi: 10.1007/bf02353483 CrossRefPubMedGoogle Scholar
  29. 29.
    Dost FH (1968) Grundlagen der pharmakokinetik. G Thieme, Stuttgart: 2 Gufl 2:38–47Google Scholar
  30. 30.
    Nyberg J, Bazzoli C, Ogungbenro K, Aliev A, Leonov S, Duffull S, Hooker AC, Mentre F (2015) Methods and software tools for design evaluation in population pharmacokinetics–pharmacodynamics studies. Br J Clin Pharmacol 79(1):6–17. doi: 10.1111/bcp.12352 CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Otago Pharmacometrics Group, School of PharmacyUniversity of OtagoDunedinNew Zealand
  2. 2.Departments of Gastroenterology & Clinical PharmacologyChristchurch HospitalChristchurchNew Zealand

Personalised recommendations