Skip to main content

Advertisement

Log in

Pharmacokinetic–Pharmacodynamic Modelling: History and Perspectives

  • Published:
Journal of Pharmacokinetics and Pharmacodynamics Aims and scope Submit manuscript

Abstract

A major goal in clinical pharmacology is the quantitative prediction of drug effects. The field of pharmacokinetic–pharmacodynamic (PK/PD) modelling has made many advances from the basic concept of the dose–response relationship to extended mechanism-based models. The purpose of this article is to review, from a historical perspective, the progression of the modelling of the concentration–response relationship from the first classic models developed in the mid-1960s to some of the more sophisticated current approaches. The emphasis is on general models describing key PD relationships, such as: simple models relating drug dose or concentration in plasma to effect, biophase distribution models and in particular effect compartment models, models for indirect mechanism of action that involve primarily the modulation of endogenous factors, models for cell trafficking and transduction systems. We show the evolution of tolerance and time-variant models, non- and semi-parametric models, and briefly discuss population PK/PD modelling, together with some example of more recent and complex pharmacodynamic models for control system and nonlinear HIV-1 dynamics. We also discuss some future possible directions for PK/PD modelling, report equations for general classes of novel semi-parametric models, as well as describing two new classes, additive or set-point, of regulatory, additive feedback models in their direct and indirect action variants

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mager D., Wyska E., Jusko W. (2003). Diversity of mechanism-based pharmacodynamic models. Am. Soc. Pharmacol. Exp. Therapeut. 31:510–519

    CAS  Google Scholar 

  2. Tallarida R.J. (1984). Receptor theories and quantitative effect versus dose–concentration relationship. Drug Metab. Rev. 15:345–363

    Article  PubMed  Google Scholar 

  3. Verotta D., Sheiner L.B. (1995). A general conceptual model for non-steady state pharmacokinetic/pharmacodynamic data. J. Pharmacokin. Biopharm. 23:1–4

    Article  CAS  Google Scholar 

  4. Dayneka N., Garg V., Jusko W. (1993). Comparison of four basic models of indirect pharmacodynamic responses. J. Pharmacokinet. Biopharm. 21:457–478

    Article  PubMed  CAS  Google Scholar 

  5. Levy G. (1994). Mechanism-based pharmacodynamic modeling. Clin. Pharmacol. Therapeut. 56:356–358

    Article  CAS  Google Scholar 

  6. Verotta D. (1996). Concepts, properties, and applications of linear systems to describe the distribution, identify input, and control endogenous substances and drugs in biological systems. CRC Crit. Rev. Biomol. Eng. 24:73–139

    CAS  Google Scholar 

  7. D. Verotta. Semi-parametric direct and indirect action models for pharmacokinetics/pharmacodynamic data. In: Proceedings Society Computer Simulation Western Multiconference, Las Vegas, Nevada, USA, 1995.

  8. Segre G. (1968). Kinetics of interaction between drugs and biological system. Il Farmaco Ed. Sci. 23:907–918

    CAS  Google Scholar 

  9. Levy G. (1966). Kinetics of pharmacologic effects. Clin Pharm Therapeut. 7:362–372

    CAS  Google Scholar 

  10. Levy G. (1964). Relationship between elimination rate of drugs and rate of decline of their pharmacological effect. J. Pharm. Sci. 53:343–343

    Article  PubMed  Google Scholar 

  11. Levy G., Nelson E. (1965). Theoretical relationship between dose, elimination rate, and duration of pharmacologic effect of drugs. J. Pharm. Sci. 54:812

    CAS  Google Scholar 

  12. Levy G. (1965). Apparent potentiating effect of a second dose of drug. Nature 206:517–519

    CAS  Google Scholar 

  13. Wagner J. (1968). Kinetics of pharmacologic response. I. Proposed relationships between response and drug concentration in the intact animal and man. J. Theoret. Biol. 20:173–201

    Article  CAS  Google Scholar 

  14. Ariens E. (1964). The mode of action of biologically active compounds. In: De Stevens G (eds). Molecular Pharmacology. Academic Press, New York

    Google Scholar 

  15. Black J.W., Leff P. (1983). Operational models of pharmacological agonism. Proc. R. Soc. Lond. B. 220:141–162

    PubMed  CAS  Google Scholar 

  16. Tuk B., Van Oostenbruggen M., Herben V.M.M. et al. (1999). Characterization of the pharmacodynamic interaction between parent drug and active metabolite in vivo: Midazolam and ∝-OH-midazolam. J. Pharmacol. Exp. Therapeut. 289:1067–1074

    CAS  Google Scholar 

  17. Van der Graaf P., Van Schaick E., Mathot R. et al. (1997). Mechanism-based pharmacokinetic–pharmacodynamic modeling of the effects of N 6-cyclopentylasenosine analogs on heart rate in rat: Estimation of in vivo operational affinity and efficacy at adenosine A1 receptors. J. Pharmacol. Exp. Therapeut. 283:809–816

    Google Scholar 

  18. Furchgott R. (1955). The pharmacology of vascular smooth muscle. Pharm. Res. 7:183–265

    CAS  Google Scholar 

  19. Ferguson J. (1939). The use of chemical potentials as indices of toxicity. Proc. Roy. Soc. London, s.B. 127:387–404

    Article  CAS  Google Scholar 

  20. Berman M., M.F. Weiss. SAAM manual. U.S. Department of Health, Education and Welfare, Public Health Service Publication No. 1703. Washington, DC, U.S. Government Printing Office, 200 pp. 1967.

  21. Boston C., Greif P.C., Berman M. (1981). Conversational SAAM—An interactive program for kinetic analysis of biological systems. Comput. Programs Biomed. 1981: 111–119

    Article  Google Scholar 

  22. SAAM. In Series. Seattle, WA: SAAM Institute, Inc. http://www.saam.com/software/saam2/saam2software.htm.

  23. Dahlstrom B., Paalzow L., Segre G. et al. (1978). Relation between morphine pharmacokinetics and Analgesia. J. Pharmacokinet. Biopharm. 6:41–53

    Article  PubMed  CAS  Google Scholar 

  24. Levy G., Gibaldi M., Jusko W. (1969). Multicompartment pharmacokinetic models and pharmacologic effects. J. Pharm. Sci. 58:422–424

    Article  PubMed  CAS  Google Scholar 

  25. Galeazzi R., Benet L., Sheiner L. (1976). Relationship between the pharmacokinetics and pharmacodynamics of procainamide. Clin. Pharmacol. Ther. 20:278–289

    PubMed  CAS  Google Scholar 

  26. Kramer W., Kolibash A., Lewis R. et al. (1979). Pharmacokinetics of digoxin: Relationship between response intensity and predicted compartmental drug levels in man. J. Pharmacokinet. Biopharm. 7:47–61

    Article  PubMed  CAS  Google Scholar 

  27. Mandema J., Veng-Pedersen P., Danhof M. (1991). Estimation of amobarbital plasma-effect site equilibration kinetics. Relevance of polyexponential conductance functions. J. Pharmacokinet. Biopharm. 19:617–634

    Article  PubMed  CAS  Google Scholar 

  28. Sheiner L., Stanski D., Vozeh S. et al. (1979). Simultaneous modeling of pharmacokinetics and pharmacodynamics: Application to d-tubocurarine. Clin. Pharmacol. Ther. 25:358–371

    PubMed  CAS  Google Scholar 

  29. Jusko W. (1971). Pharmacodynamics of chemotherapeutic effects: Dose–time–response relationships for phase-non-specific agents. J. Pharm. Sci. 60:892–895

    CAS  Google Scholar 

  30. Jusko W. (1973). A pharmacodynamic model for cell-cycle-specific chemotherapeutic agents. J. Pharmacokinet. Biopharm. 1:175–200

    Article  CAS  Google Scholar 

  31. Nagashima R., O’Reilly R., Levy G. (1969). Kinetics of pharmacologic effects in man: The anticoagulant action of warfarin. Clin. Pharmacol. Ther. 10:22–35

    PubMed  CAS  Google Scholar 

  32. Sheiner L. (1969). Computer-aided long-term anticoagulation therapy. Comput. Biomed. Res. 2:507–518

    Article  PubMed  CAS  Google Scholar 

  33. Theophanus T., Barile R. (1973). Multiple-dose kinetics of oral anticoagulants: Methods of analysis and optimised dosing. J. Pharm. Sci. 62:261–266

    Article  PubMed  Google Scholar 

  34. Abbrecht P. O’Leary T., Behrendt D. (1982). Evaluation of a computer-assisted method for individualized anticoagulation: Retrospective an prospective studies with a pharmacodynamic model. Clin. Pharmacol. Ther. 32: 129–136

    Article  PubMed  Google Scholar 

  35. O’Leary T., Abbrecht P. (1981). Predicting oral anticoagulant response using a pharmacodynamic model. Ann. Biomed. Eng. 9: 199–216

    Article  PubMed  CAS  Google Scholar 

  36. Sharma A. (1998). Precursor-dependant indirect pharmacodynamic response model for tolerance and rebound phenomena. J. Pharm. Sci. 87:1577–1584

    Article  PubMed  CAS  Google Scholar 

  37. Earp J., Krzyzanski W., Chakraborty A. et al. (2004). Assessment of drug interactions relevant to pharmacodynamic indirect response models. J. Pharmacokine. Pharmacodynam. 31:345–380

    Article  CAS  Google Scholar 

  38. Wald J. (1991). Two-compartment basophil cell trafficking model for methylprednisolone pharmacodynamics. J. Pharmacokinet. Biopharm. 19:521–536

    Article  PubMed  CAS  Google Scholar 

  39. Kong A., Ludwig E., Slaughter R. et al. (1989). Pharmacokinetics and pharmacodynamic modeling of direct suppression effects of methylprednisolone on serum cortisol and blood histamine in human subjects. Clin. Pharmacol. Therpeut. 46:616–628

    CAS  Google Scholar 

  40. Dunn T., Ludwig E., Slaughter R. et al. (1991). Pharmacokinetics and pharmacodynamics of methylprednisolone in obese and on-obese men. Clin. Pharmacol. Therpeut. 50:536–549

    Google Scholar 

  41. Frey B., Walker C., Frey F. et al. (1984). Pharmacokinetics of three different prednisolone produrgs: effect on circulating lymphocyte subsets and function. J. Immunol. 133: 2479–2487

    PubMed  CAS  Google Scholar 

  42. Munck A., Leung K. (1977). Glucocorticoid receptors and mechanisms of action of steroid hormones Receptors and Mechanisms of Action of Steroid Hormones. Marcel Dekker, New York pp. 311–397

    Google Scholar 

  43. Evans R. (1988). The steroid and thyroid hormone receptor superfamily. Science 240: 889–895

    Article  PubMed  CAS  Google Scholar 

  44. Boudinot F., D’Ambrosio R., Jusko W. (1986). Receptor-mediated pharmacodynamics of prednisolone in the rat. J. Pharmacokinet. Biopharm. 14: 469–493

    Article  PubMed  CAS  Google Scholar 

  45. Nichols A. (1989). Second generation model for prednisolone pharmacodynamics in the rat. J. Pharmacokinet. Biopharm. 17:209–227

    Article  PubMed  CAS  Google Scholar 

  46. Sun Y., Jusko W. (1998). Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics. J. Pharm. Sci. 87:732–737

    Article  PubMed  CAS  Google Scholar 

  47. Oosterhuis B., Braat M., Roos C. et al. (1986). Pharmacokinetic–pharmacodynamic modeling of terbutaline bronchodilation in asthma. Clin. Pharmacol. Therpeut 40:469–475

    CAS  Google Scholar 

  48. Mager D., Jusko W. (2001). Pharmacodynamic modeling of time-dependent transduction systems. Clin. Pharmacol. Therpeut. 70:210–216

    Article  CAS  Google Scholar 

  49. Lobo E., Balthasar J. (2002). Pharmacodynamic modeling of chemotherapeutic effects: Application of a transit compartment model to characterize methotrexate effects in vitro. AAPS Pharmaceut. Sci. 4:1–11

    Article  Google Scholar 

  50. Zahler R., Wachter P., Jatlow P. et al. (1982). Kinetics of drug effect by distributed lags analysis: An application to cocaine. Clin. Pharmacol. Therpeut. 31:775–782

    CAS  Google Scholar 

  51. Ekbad E., Licko V. (1984). A model eliciting transient responses. Am. J. Physiol. 246:R114–R121

    PubMed  Google Scholar 

  52. Chow M., Ambre J., Ruo T. et al. (1985). Kinetics of cocaine distribution, elimination, and chronotropic effects. Clin. Pharmacol. Therpeutic. 38:318–324

    Article  CAS  Google Scholar 

  53. Hammarlund M., Odling B., Paalzow L. (1985). Acute tolerance to furosemide diuresis in humans. Pharmacokinetic–Pharmacodynamic modelling. J. Pharmacol. Exp. Therpeut. 233:447–453

    CAS  Google Scholar 

  54. Porchet H., Benowitz N., Sheiner L. (1988). Pharmacodynamic model of tolerance: Application to nicotine. J. Pharmacol. Exp. Therpeut. 244:231

    CAS  Google Scholar 

  55. Shi J., Benowitz N., Denaro C. et al. (1993). Pharmacokinetic–pharmacodynamic modeling of caffeine: Tolerance to pressor effects. Clin. Pharmacol. Therpeut. 53:6–14

    CAS  Google Scholar 

  56. Mandema J.W., Wada D. (1995). Pharmacodynamic model for acute tolerance development to the electroencephalographic effects of alfentanil in the rat. J. Pharmacol. Exp. Therpeut. 279:1035–1042

    Google Scholar 

  57. Shafer S.L., Siegel L.C., Cooke J.E. et al. (1988). Testing computer-controlled infusion pumps by simulation. Anesthesiology 68:261–266

    Article  PubMed  CAS  Google Scholar 

  58. Shafer S.L., Gregg K.M. (1992). Algorithms to rapidly achieve and maintain stable drug concentrations at the site of drug effect with a computer controlled infusion pump. J. Pharmacokinet Biopharm. 20:147–169

    Article  PubMed  CAS  Google Scholar 

  59. Verotta D. (1999). A general solution for non-parametric control of a linear system using computer controlled infusion pumps. IEEE Trans. Biomed. Eng. 46:44–50

    Article  PubMed  CAS  Google Scholar 

  60. Bauer J., Balthasar J., Fung H. (1997). Application of pharmacodynamic modeling for designing time variant dosing regimens to overcome nitroglycerin tolerance in experimental heart failure. Pharm. Res. 14:114–145

    Google Scholar 

  61. Francheteau P., Steimer J., Dubray C. et al. (1991). Mathematical model for in vivo pharmacodynamics integrating fluctuation of the response: Application to the prolactin suppressant effect of the dopaminomimetic drug DCN203–922. J. Pharmacokinet. Biopharm. 19:287–309

    Article  PubMed  CAS  Google Scholar 

  62. Lew K., Ludwig E., Milad M. et al. (1993). Gender-based effects on methylprednisolone pharmacokinetics and pharmacodynmamics. Clin. Pharmacol. Therpeut. 54: 402–414

    CAS  Google Scholar 

  63. Gries J., Benowitz N., Verotta D. (1996). Chronopharmacokinetics of nicotine. Clin. Pharmacol. Therpeut. 60:385–395

    Article  CAS  Google Scholar 

  64. Hull C., Van Beem H., and McLeod K. et al. (1978). A pharmacodynamic model for pancuronium. Brit. J. Anaesth 50:1113–1123

    Article  PubMed  CAS  Google Scholar 

  65. Fuseau E. and Sheiner L. (1984). Simultaneous modeling of pharmacokinetic and pharmacodynamics with a nonparametric model. Clin. Pharmacol. Therpeut. 35:733–741

    CAS  Google Scholar 

  66. Unadkat J., Bartha F., and Sheiner L. (1986). Simultaneous modeling of pharmacokinetics and pharmacodynamics with nonparametric and dynamic models. Clin. Pharmacol. Therpeut. 40:86–93

    CAS  Google Scholar 

  67. Verotta D. and Sheiner L. (1987). Simultaneous modeling of pharmacokinetics and pharmacodynamics: An improved algorithm. Comput. Appl. Biostat 3:345–349

    CAS  Google Scholar 

  68. Verotta D., Beal S., and Sheiner L. (1989). Semiparametric approach to pharmacokinetic-pharmacodynamic data. Am. J. Physiol. 256:R1005–R1010

    PubMed  CAS  Google Scholar 

  69. Verotta D. (1989). An inequality-constrained least-squares deconvolution method. J. Pharmacokinet. Biopharm. 17:269–289

    Article  PubMed  CAS  Google Scholar 

  70. Veng-Pedersen P., Mandema J., and Danhof M. (1991). A system approach to pharmacodynamics. III: An algorithm and computer program, COLAPS, for pharmacodynamic modeling. J. Pharm. Sci. 80:488–495

    Article  PubMed  CAS  Google Scholar 

  71. Tuk B., Danhof M., and Mandema J. (1997). The impact of arteriovenous concentration differences on pharmacodynamic parameter estimates. J. Pharmacokinet. Biopharm. 25:39–60

    Article  PubMed  CAS  Google Scholar 

  72. Tuk B., Herben V., and Mandema J. et al. (1998). Relevance of arteriovenous concentration differences in pharmacokinetic–pharmacodynamic modeling of midazolam. J.Pharmacol. Exp. Therpeut. 284:202–207

    CAS  Google Scholar 

  73. Verotta D. and Sheiner L.B. (1991). Semiparametric analysis of non-steady-state pharmacodynamic data. J. Pharmacokin. Biopharm 19:691–712

    Article  CAS  Google Scholar 

  74. Troconiz I.F., Sheiner L.B. and Verotta D. (1994). Semiparametric models for drug interactions. J. Appl. Physiol 76:2224–2233

    PubMed  CAS  Google Scholar 

  75. Holford N. (1990). Concepts and usefulness of pharmacokinetic–pharmacodynamic modeling. Fundament. Clin. Pharmacol. Therapeut 4:93–101

    Article  Google Scholar 

  76. Rowland M. (1985). Variability in Drug Therapy–Description, Estimation and Control. Raven Press Boks Ltd, New York

    Google Scholar 

  77. Sheiner L., Rosenberg B., and Melmon K. (1972). Modeling of individual pharmacokinetics for computer-aided drug dosage. Comput. Biomed. Res. 5:441–459

    Article  Google Scholar 

  78. Beal S. and Sheiner L.B. NONMEM I Users Guide. Technical Report, Division of Clinical Pharmacology, University of California at San Francisco, 1984.

  79. Yuh L., Beal S. and Davidian M. et al. (1994). Population pharmacokinetic/pharmacodynamic methodology and applications: A bibliography. Biometrics 50:566–575

    Article  PubMed  CAS  Google Scholar 

  80. Sheiner L. and Ludden T. (1992). Population pharmacokinetics/dynamics. Annu. Rev. Pharmacol. Toxicol. 32:185–209

    PubMed  CAS  Google Scholar 

  81. Sheiner L. and Beal S. (1982). Bayesian individualisation of pharmacokinetics: Simple implementation and comparison with non-Bayesian methods. J. Pharm. Sci. 71:1344–1348

    Article  PubMed  CAS  Google Scholar 

  82. Minto C. (1998). Expanding clinical applications of population pharmacodynamic modeling. Brit. J. Clin. Pharmaco 46:321–333

    Article  CAS  Google Scholar 

  83. Boeckmann A.J., Beal S.L., and Sheiner L.B. NONMEM V Users Guides. Technical Report, Division of Clinical Pharmacology, University of California at San Francisco, 1998.

  84. WinNonmix® Pharsight. http://www.pharsight.com/products/winnonmix/index.php.

  85. D. D’Argenio and A. Schumitzky. ADAPT II User’s Guide: Pharmacokinetic/Pharmacodynamic Systems Analysis Software. In Series ADAPT II User’s Guide: Pharmacokinetic/Pharmacodynamic Systems Analysis Software. Biomedical Simulations Resource, Los Angeles, 1997.

  86. PopKinetics. SI, INC. http://www.saam.com/software/popKinetics/popKineticsSoftware.htm.

  87. PKBUGS. MRC Biostatistics Unit, Cambridge, UK. http://www.mrc-bsu.cam.ac. uk/bugs/winbugs/pkbugs.shtml.

  88. Fattinger K., Verotta D., and Porchet H. et al. (1996). Modelling a bivariate control system: LH and testosterone response to the GnRH antagonist Antide. Am. J. Physiol. 271: E775–E787

    PubMed  CAS  Google Scholar 

  89. Zuideveld K., Maas H., and Treijtel N. et al. (2001). A set-point model with oscillatory behaviour predicts the time course of (8-)-OH-DPAT induced hypothermia. Am. J. Physiol. 281: R2059–R2071

    CAS  Google Scholar 

  90. Bridges N.A., Hindsmarsh P.C., and Pringle P.J. et al. (1993). The relationship between endogenous testosterone and gonadotropin secretion. Clin. Endocrinol 38:373–378

    Article  CAS  Google Scholar 

  91. Bonhoeffer S., May R.M. and Shaw G.M. et al. (1997). Virus dynamics and drug therapy. Proc. Natl. Acad. Sci. USA. 94:6971–6976

    Article  PubMed  CAS  Google Scholar 

  92. Nowak M.A. and May R.M. (1991). Mathematical biology of HIV infections: Antigenetic variation and diversity threshold. Math. Biosci. 106:1–21

    Article  PubMed  CAS  Google Scholar 

  93. Nowak M.A., Bonhoeffer S., and Shaw G.M. et al. (1997). Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell population. J. Theoret. Biol. 184:203–217

    Article  CAS  Google Scholar 

  94. Wein L.M., DAmato R.M., and Perelson A.S. (1998). Mathematical analysis of antiretroviral therapy aimed at HIV-1 eradication or maintenance of low viral loads. J. Theoret. Biol 192: 81–98

    Article  CAS  Google Scholar 

  95. Stafford M.A., Corey L., and Cao Y. et al. (2000). Modelling plasma virus concentration during primary HIV infection. J. Theoret. Biol. 203: 285–301

    Article  CAS  Google Scholar 

  96. Y. Huang and L. Wu. Mechanistic PK/PD modeling of antiretroviral therapies in AIDS clinical trials. In Advanced Methods of Pharmacokinetic and Pharmacodynamic Systems Analysis, Vol. III. Kluwer Academic Publishers, Boston, 2004, pp. 221–238.

  97. Nowak M.A. and May R.M. (1992). Coexistence and competition in HIV infections. J. Theoret. Biol. 159:329–342

    Article  CAS  Google Scholar 

  98. Perelson A.S. and Essunger P. (1997). Decay characteristics of HIV-1 infected compartments during combination therapy. Nature 387:188–191

    Article  PubMed  CAS  Google Scholar 

  99. Perelson A.S., Neumann A.U., and Markowitz M. et al. (1996). HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time. Science 271:1582–1587

    Article  PubMed  CAS  Google Scholar 

  100. Wu H. and Ding A.A. (1999). Population HIV-1 dynamics in vivo: Applicable models and inferential tools for virological data from AIDS clinical trials. Biometrics 55:410–418

    Article  PubMed  CAS  Google Scholar 

  101. Ding A.A. and Wu H. (2000). A comparison study of models and fitting procedures for biphasic viral dynamics in HIV-1 infected patients treated with antiviral therapies. Biometrics 56:293–300

    Article  PubMed  CAS  Google Scholar 

  102. Verotta D. and Schaedeli F. (2002). Non-linear dynamics models characterizing long-term virological data from AIDS clinical trials. Math. Biosci. 176: 163–183

    Article  PubMed  MathSciNet  Google Scholar 

  103. May R.M. (1987). Nonlinearity and complex behavior in simple ecological and epidemiological models. Perspect. Biol. Dynam. Theoret. Med 504:1–15

    CAS  Google Scholar 

  104. D. Verotta. Models and estimation methods for clinical HIV-1 data. J. Comput. Appl. Math., in press.

  105. Efron B. and Feldman D. (1991). Compliance as an explanatory variable in clinical trial. J. Am. Stat.Assoc 86:9–17

    Article  Google Scholar 

  106. Urquhart J. and De Klerk E. (1998). Contending paradigms for the interpretation of data on patient compliance with therapeutic drug regimens. Stat. Med. 17:251–267; discussion 387–259

    Article  PubMed  CAS  Google Scholar 

  107. Homer. Iliad, Book XIII; c. 850 BC.

  108. C. M. Metzler. A user’s manual for NONLIN. The Upjohn Co. Techn. Rep. 7292/69/7292/005. Kalama 700. Mich., 1969.

  109. Gelman A., Carlin J.B., and Stern H. et al. (1995). Bayesian Data Analysis. Chapman & Hall, London

    Google Scholar 

  110. Distefano-III J.J. and Landaw E.M. Multiexponential, multicompartmental, and noncompartmental modeling. I. Methodological limitations and physiological interpretations. Am. J. Physiol. Regulat. Integrat. Comp. Physiol. 15(1984).

  111. Verotta D. (2003). Volterra series in pharmacokinetics and pharmacodynamic. J. Pharmacokinet. Pharmacodynam 30:337–362

    Article  CAS  Google Scholar 

  112. Volterra V. (1959). Theory of Functionals and of integral and Integro-Differential Equations. Dover, New York

    Google Scholar 

  113. Marmarelis V.Z. (1993). Identification of nonlinear biological systems using laguerre expansions of kernels. Ann. Biomed. Eng. 21:573–589

    Article  PubMed  CAS  Google Scholar 

  114. C. DeBoor.A Practical Guide to Splines, 1978.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Davide Verotta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Csajka, C., Verotta, D. Pharmacokinetic–Pharmacodynamic Modelling: History and Perspectives. J Pharmacokinet Pharmacodyn 33, 227–279 (2006). https://doi.org/10.1007/s10928-005-9002-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10928-005-9002-0

Keywords

Navigation