Identifiability of mathematical models in medical biology

Abstract

The analysis of biological data is a key topic in bioinformatics, computational genomics, molecular modeling, and systems biology. The methods covered in this article can reduce the cost of experiments aimed at obtaining biological data. The problem of the identifiability of mathematical models in physiology, pharmacokinetics, and epidemiology is considered. The processes considered are modeled using nonlinear systems of ordinary differential equations. The mathematical modeling of dynamic processes is based on the use of the mass conservation law. The problem of estimating the parameters characterizing the process under study raises the question of nonuniqueness. When the input and output data are known, it is useful to perform an a priori analysis of the relevance of these data. The definition of the identifiability of mathematical models is considered. Methods for the analysis of the identifiability of dynamic models are reviewed. In this review article, the following approaches are considered: the transfer function method applied to linear models (useful for the analysis of pharmacokinetic data, since a large class of drugs is characterized by linear kinetics); the Taylor series expansion method applied to nonlinear models; the differential algebra methods (the structure of this algorithm allows it to be run on a computer); and a method based on graph theory (this method allows for the analysis of the identifiability of the model and finding a proper reparametrization reducing the initial model to an identifiable one). The need to perform a priori identifiability analysis before estimating the parameters characterizing any process is demonstrated with several examples. The examples of identifiability analysis of mathematical models in medical biology are presented.

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

References

  1. Audoly, S. and D’Angio, L., On the identifiability of linear compartmental system: A revisited transfer function approach based on topological properties, Math. Biosci., 1983, vol. 66, pp. 201–228.

    Article  Google Scholar 

  2. Bellman, R. and Astrom, K., On structural identifiability, Math. Biosci., 1970, vol. 7, no. 3, pp. 329–339.

    Article  Google Scholar 

  3. Bellu, G., Saccomani, P., Audoly, S., and D’Angio, L., Daisy: A new software tool to test global identifiablity of biological and physiological system, Comput. Methods Programs Biomed., 2007, vol. 88, no. 1, pp. 52–61. doi 10.1016/j.cmpb.2007.07.002

    Article  PubMed  PubMed Central  Google Scholar 

  4. Ben-Zvi, A., McLellan, P.J., and McAuley, B.K., Identifability of linear time-invariant differential-algebraic systems, Ind. Eng. Chem. Res., 2004, vol. 43, no. 8, pp. 1251–1259.

    CAS  Article  Google Scholar 

  5. Brown, R., Compartmental system analysis: State of the art, IEEE Trans. Biomed. Eng., 1980, vol. 27, no. 1, pp. 1–38.

    CAS  Article  PubMed  Google Scholar 

  6. Brown, R.F., Identifiability: Role in design of pharmacokinetic experiments, IEEE Trans. Biomed. Eng., 1982, vol. 29, pp. 49–54.

    CAS  Article  PubMed  Google Scholar 

  7. Carson, E. and Cobelli, C., Modelling Methodology for Physiology and Medicine, San Diego: Academic Press, 2001.

    Google Scholar 

  8. Carson, E. and Cobelli, C., Introduction to Modelling in Physiology and Medicine, San Diego: Academic Press, 2008.

    Google Scholar 

  9. Cobelli, C., Lepschy, A., and Jacur, G.R., Identifiability of compartmental systems and related structural properties, Math. Biosci., 1976, vol. 48, pp. 1–18.

    Google Scholar 

  10. Cobelli, C. and DiStefano, J., Parameter and structural identifiability concepts and ambiguities: A critical review and analysis, Am. J. Physiol.-Regul. Integr. Comp. Physiol., 1980, vol. 23, no. 9, pp. 7–24.

    Google Scholar 

  11. Goodwin, G.C. and Payne, R.L., Dynamic System Identification: Experiment Design and Data Analysis, New York: Academic Press, 1977.

    Google Scholar 

  12. Jacquez, J.A. and Greif, P., Numerical parameter identifiability and estimability: Integrating identifiability, estimability and optimal design, Math. Biosci., 1985, vol. 77, pp. 201–227.

    Article  Google Scholar 

  13. Meshkat, N., Anderson, C., and DiStefano, J., Alternative to Ritt’s Pseudodivision for finding the input-output equations of multi-output models, Math. Biosci., 2012, vol. 239, no. 1, pp. 117–123. doi 10.1016/j.mbs.2012.04.008

    Article  PubMed  Google Scholar 

  14. Meshkat, N., Eisenberg, M., and DiStefano, J., On finding and using identifiable parameter combinations in nonlinear dynamic systems biology models and COMBOS: A novel web implementation, Plos One, 2014, vol. 9, no. 10. doi 10.1371/journal.pone.0110261

  15. Meshkat, N., Identifiable reparametrizations of linear compartment models, J. Symb. Comput., 2014, vol. 63, pp. 46–67. doi 2013.11.002.10.1016/j.jsc

    Article  Google Scholar 

  16. Tunali, T. and Tarn, T.J., New results for identifiability of nonlinear systems, IEEE Trans. Autom. Control, 1987, vol. 32, no. 2, pp. 146–154.

    Article  Google Scholar 

  17. Walter, E., Lecourtier, Y., and Happel, J., On the structural output distinguishability of parametric models, and its relation with structural identifiability, IEEE Trans. Autom. Control, 1984, vol. 29, pp. 56–57.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to D. A. Voronov.

Additional information

Original Russian Text © S.I. Kabanikhin, D.A. Voronov, A.A. Grodz, O.I. Krivorotko, 2015, published in Vavilovskii Zhurnal Genetiki i Selektsii, 2015, Vol. 19, No. 6, pp. 738–744.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kabanikhin, S.I., Voronov, D.A., Grodz, A.A. et al. Identifiability of mathematical models in medical biology. Russ J Genet Appl Res 6, 838–844 (2016). https://doi.org/10.1134/S2079059716070054

Download citation

Keywords

  • identifiability
  • mathematical models in medical biology
  • system of ordinary differential equations
  • pharmacokinetics
  • epidemiology
  • physiology
  • differential algebra