A Dynamical Systems Analysis of the Indirect Response Model with Special Emphasis on Time to Peak Response

  • Lambertus A. Peletier
  • Johan Gabrielsson
  • Jacintha den Haag


In this paper we present a mathematical analysis of the four classical indirect response models. We focus on characteristics such as the evolution of the response R(t) with time t, the time of maximal/minimal response Tmax and the area between the response and the baseline AUC R , and the way these quantities depend on the drug dose, the dynamic parameters such as Emax and EC50 and the ratio of the fractional turnover rate kout to the elimination rate constant k of drug in plasma. We find that depending on the model and on the drug mechanism function, Tmax may increase, decrease, decrease and then increase, or stay the same, as the drug dose is increased. This has important implications for using the shift in Tmax as a diagnostic tool in the selection of an appropriate model


indirect response models turnover models time of maximal response peak shift differential equations pharmacodynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ackerman, E., Rosevear, J.W., McGuckin, W.F. 1964A mathematical model of the glucose-tolerance testPhys. Med. Biol.9203213CrossRefGoogle Scholar
  2. 2.
    Nagashima, R., O’Reilly, R. A., Levy, G. 1969Kinetics of pharmacological effects in man: The anticoagulant action of warfarinClin. Phamacol. Ther.1022Google Scholar
  3. 3.
    Ekblad, E.B.M., Licko, V. 1984A model eliciting transient responsesAm. J. Physiol.246R114R121(Regulatory Integrative Comp. Physiol. 15)PubMedGoogle Scholar
  4. 4.
    Dayneka, N.L., Garg, V., Jusko, W.J. 1993Comparison of four basic models of indirect pharmacodynamic responsesJ. Pharmacokin. Biopharm.21457478CrossRefGoogle Scholar
  5. 5.
    N. H. G. Holford Gabrielsson J.L., Sheiner L.B., Benowitz N., and Jones R.,. A physiological pharmacologicodynamic model for tolerance to cocaine effects on systolic blood pressure, heart rate and euphoria in human volunteers. Presented at Measurement and Kinetics of in vivo Drug Effects, 28–30 June, 1990, Noordwijk, The Netherlands.Google Scholar
  6. 6.
    Wakelkamp, M., Alvan, G., Gabrielsson, J., Paintaud, G. 1996Pharmacodynamic modeling of furosemide tolerance after multiple intravenous administrationClin. Pharmacol. Ther.607588CrossRefPubMedGoogle Scholar
  7. 7.
    Sun, Y.-N., DuBois, D.C., Almon, R.R., Pyszczynski, N.A., Jusko, W.J. 1998Dose dependence and repeated-dose studies for receptor/gene-mediated pharmacodynamics of methylprednisolone on glucocorticoid receptor down-regulation and tyrosine aminotransferase induction in rat liverJ. Pharmacokin. Biopharm.26619648CrossRefGoogle Scholar
  8. 8.
    Zuideveld, K.P., Maas, H.J., Treijtel, N., Hulshof, J., Graaf, P.H., Peletier, L.A., Danhof, M. 2001A set-point model with oscillatory behavior predicts the time-course of 8-OH-DPAT induced hypothermiaAm. J. Physiol. Regulatory Comp. Physiol.281R2059R2071Google Scholar
  9. 9.
    Gabrielsson J., Weiner D. Pharmacokinetic/Pharmacodynamic Data Analysis: Concepts and Applications, 2nd and 3rd edns. Swedish Pharmaceutical Press, Stockholm, 1997, 2000.Google Scholar
  10. 10.
    Mager, D.E., Wyska, E., Jusko, W.J. 2003Diversity of mechanism-based pharmacodynamic modelsDrug Metab. Dispos.31510519CrossRefPubMedGoogle Scholar
  11. 11.
    Sharma, A., Jusko, W.J. 1996Characterization of four basic models of indirect pharmacodynamic responsesJ. Pharmacokin. Biopharm.24611635Google Scholar
  12. 12.
    Krzyzanski, W., Jusko, W.J. 1997Mathematical formalism for the properties of four basic models of indirect pharmacodynamic responsesJ. Pharmacokin. Biopharm.25107123CrossRefGoogle Scholar
  13. 13.
    Krzyzanski, W., Jusko, W.J. 1998Mathematical formalism and characteristics of four basic models of indirect pharmacodynamic responses for drug infusionsJ. Pharmacokin. Biopharm.26385408Google Scholar
  14. 14.
    Krzyzanski, W., Jusko, W.J. 1998Integrated functions for four basic models of indirect pharmacodynamic responseJ. Pharmaceut. Sci.876772CrossRefGoogle Scholar
  15. 15.
    Krzyzanski, W. 2000Asymptotics of the total net pharmacological effect for large drug dosesJ. Math. Biol.41477492CrossRefPubMedGoogle Scholar
  16. 16.
    Wakelkamp, M., Alvan, G., Paintaud, G. 1998The time of maximum effect for model selection in pharmacokinetic–pharmacodynamic analysis applied to frusemide. [Clinical Trial Journal Article. Randomized Controlled Trial].British Journal of Clinical Pharmacology.456370CrossRefPubMedGoogle Scholar
  17. 17.
    Blanchard, P., Devaney, R.L., Hall, G.R. 1997Differential EquationsBrooks/Cole Publishing ompanyBostonGoogle Scholar
  18. 18.
    Ermentraut G.B., XPPAUT, Scholar
  19. 19.
    Maple 9, Waterloo Maple Inc.Google Scholar
  20. 20.
    Rescigno A., Segre G. Drug and Tracer Kinetics. Blaisdell Publishing Company, London, 1961, 1966.Google Scholar
  21. 21.
    Gibaldiand, M., Perrier, D. 1982Pharmacokinetics2Marcel Dekker Inc.New YorkRevised and Expanded.Google Scholar
  22. 22.
    Finkelstein, L., Carson, E.R. 1985Mathematical Modelling of Dynamic Biological SystemsResearch Studies Press Ltd.LetchworthGoogle Scholar
  23. 23.
    Majumdar, A. 1998Characterization of the dose-dependent time peak effect in Indirect response modelsJ. Pharmacokin. Biopharm.26183206CrossRefGoogle Scholar
  24. 24.
    Hansen, R., Balthasar, J.P. 2003Pharmacokinetic/pharmacodynamic modeling of the effects of intravenous immunoglobin on the disposition of antiplatelet antibodies in a rat model of immune thrombocytopeniaJ. Pharmaceut. Sci.9212061215CrossRefGoogle Scholar
  25. 25.
    Äbelö, A., Eriksson, U.G., Karlsson, M.O., Larsson, H., Gabrielsson, J. 2000A turnover model of irreversible inhibition of gastric acid secretion by Omeprazole in the dogJPET295662669Google Scholar
  26. 26.
    Bender, C.M., Orszag, S.A. 1978Advanced Mathematical Methods for Scientists and EngineersMcGraw-Hill International EditionsSingaporeGoogle Scholar
  27. 27.
    Rudin, W. 1964Principles of Mathematical AnalysisMcGraw-HillNew YorkGoogle Scholar
  28. 28.
    Simmons, G.F. 1963Introduction to Topology and Modern AnalysisMcGraw-HillNew YorkGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Lambertus A. Peletier
    • 1
    • 2
  • Johan Gabrielsson
    • 3
  • Jacintha den Haag
    • 4
  1. 1.Mathematical InstituteLeiden UniversityLeidenThe Netherlands
  2. 2.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  3. 3.DMPK & Bioanalytical ChemistryAstraZeneca R&D MölndahlMölndahlSweden
  4. 4.Wilde Wingerdlaan 22GoudaThe Netherlands

Personalised recommendations