Advertisement

Journal of Pharmacokinetics and Pharmacodynamics

, Volume 45, Issue 1, pp 139–157 | Cite as

Understanding and reducing complex systems pharmacology models based on a novel input–response index

  • Jane Knöchel
  • Charlotte Kloft
  • Wilhelm HuisingaEmail author
Original Paper

Abstract

A growing understanding of complex processes in biology has led to large-scale mechanistic models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input–response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents as well as the size of the models. An approach that quantifies the importance of the different constituents for a given input–output relationship and allows to reduce the dynamics to its essential features is therefore highly desirable. In this article, we present a novel state- and time-dependent quantity called the input–response index that quantifies the importance of state variables for a given input–response relationship at a particular time. It is based on the concept of time-bounded controllability and observability, and defined with respect to a reference dynamics. In application to the brown snake venom–fibrinogen (Fg) network, the input–response indices give insight into the coordinated action of specific coagulation factors and about those factors that contribute only little to the response. We demonstrate how the indices can be used to reduce large-scale models in a two-step procedure: (i) elimination of states whose dynamics have only minor impact on the input–response relationship, and (ii) proper lumping of the remaining (lower order) model. In application to the brown snake venom–fibrinogen network, this resulted in a reduction from 62 to 8 state variables in the first step, and a further reduction to 5 state variables in the second step. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The input–response indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system. In summary, the novel measure of importance provides a powerful tool for analysing the complex dynamics of large-scale systems and a means for very efficient model order reduction of nonlinear systems.

Keywords

Control theory Model order reduction Blood coagulation network Nonlinear systems 

Notes

Acknowledgements

The authors would like to thank Niklas Hartung (Computational Physiology Group, Institute of Mathematics, University of Potsdam) for valuable discussions and comments on the manuscript.

References

  1. 1.
    Schoeberl B, Eichler-Jonsson C, Gilles ED, Muller G (2002) Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors. Nat Biotechnol 20:370–375CrossRefPubMedGoogle Scholar
  2. 2.
    Wajima T, Isbister G, Duffull SB (2009) A comprehensive model for the humoral coagulation network in humans. Clin. Pharmacol. Ther. 86:290–298CrossRefPubMedGoogle Scholar
  3. 3.
    Kloft C, Trame MN, Ritter C (2016) Systems pharmacology in drug development and therapeutic use—a forthcoming paradigm shift. Eur J Pharm Sci 94:1–3CrossRefPubMedGoogle Scholar
  4. 4.
    Antoulas A (2005) Approximation of large-scale dynamical systems. In: Advances in design and control. SIAM, PhiladelphiaGoogle Scholar
  5. 5.
    Okino MS, Mavrovouniotis ML (1998) Simplification of mathematical models of chemical reaction systems. Chem Rev 98(2):391–408CrossRefPubMedGoogle Scholar
  6. 6.
    Rabitz H, Kramer M, Dacol D (1983) Sensitivity analysis in chemical kinetics. Annu Rev Phys Chem 34:419–461CrossRefGoogle Scholar
  7. 7.
    Perumal TM, Gunawan R (2011) Understanding dynamics using sensitivity analysis: caveat and solution. BMC Syst Biol 5(1):41CrossRefPubMedPubMedCentralGoogle Scholar
  8. 8.
    Zi Z (2011) Sensitivity analysis approaches applied to systems biology models. IET Syst Biol 5(6):336–346CrossRefPubMedGoogle Scholar
  9. 9.
    Lam SH, Goussis DA (1994) The CSP method for simplifying kinetics. Int J Chem Kinet 26:461–486CrossRefGoogle Scholar
  10. 10.
    Zagaris A, Kaper HG, Kaper TJ (2004) Fast and slow dynamics for the computational singular perturbation method. Multiscale Model Simul 2(4):613–638CrossRefGoogle Scholar
  11. 11.
    Dokoumetzidis A, Aarons L (2009) Proper lumping in systems biology models. IET Syst Biol 3(1):40–51CrossRefPubMedGoogle Scholar
  12. 12.
    Brochot C, Tóth J, Bois FY (2005) Lumping in pharmacokinetics. J Pharmacokinet Pharmacodyn 32(5–6):719–736CrossRefPubMedGoogle Scholar
  13. 13.
    Li G, Tomlin AS, Rabitz H, Tóth J (1994) A general analysis of approximate nonlinear lumping in chemical kinetics. I. Unconstrained lumping. J Chem Phys 101(2):1172–1187CrossRefGoogle Scholar
  14. 14.
    Pilari S, Huisinga W (2010) Lumping of physiologically-based pharmacokinetic models and a mechanistic derivation of classical compartmental models. J Pharmacokinet Pharmacodyn 37(4):365–405CrossRefPubMedGoogle Scholar
  15. 15.
    Snowden TJ, van der Graaf PH, Tindall MJ (2017) A combined model reduction algorithm for controlled biochemical systems. BMC Syst Biol 11(1):17CrossRefPubMedPubMedCentralGoogle Scholar
  16. 16.
    Lall S, Marsden JE, Glavaški S (1999) Empirical model reduction of controlled nonlinear systems. In: Proceedings of the IFAC world congress, p 473–478Google Scholar
  17. 17.
    Gulati A, Isbister GK, Duffull SB (2013) Effect of Australian elapid venoms on blood coagulation: Australian Snakebite Project (ASP-17). Toxicon 61:94–104CrossRefPubMedGoogle Scholar
  18. 18.
    Gulati A, Isbister GK, Duffull SB (2014) Scale reduction of a systems coagulation model with an application to modeling pharmacokinetic–pharmacodynamic data. CPT Pharmacomet Syst Pharmacol 3(1):e90CrossRefGoogle Scholar
  19. 19.
    Tanos P, Isbister G, Lalloo D, Kirkpatrick C, Duffull S (2008) A model for venom-induced consumptive coagulopathy in snake bite. Toxicon 52:769–780CrossRefPubMedGoogle Scholar
  20. 20.
    Cawthern KM, van’t Veer C, Lock JB, DiLorenzo ME, Branda RF, Mann KG (1998) Blood coagulation in hemophilia A and hemophilia C. Blood 91(12):4581–4592PubMedGoogle Scholar
  21. 21.
    Dokoumetzidis A, Aarons L (2009) A method for robust model order reduction in pharmacokinetics. J Pharmacokinet Pharmacodyn 36(6):613–628CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Jane Knöchel
    • 1
    • 3
  • Charlotte Kloft
    • 2
  • Wilhelm Huisinga
    • 3
    Email author
  1. 1.Graduate Research Training Program PharMetrX: Pharmacometrics & Computational Disease ModelingFreie Universität Berlin and Universität PotsdamPotsdamGermany
  2. 2.Department of Clinical Pharmacy and Biochemistry, Institute of PharmacyFreie Universität BerlinBerlinGermany
  3. 3.Institute of MathematicsUniversität PotsdamPotsdamGermany

Personalised recommendations