ADVAN-style analytical solutions for common pharmacokinetic models

Ahmad Y. Abuhelwa, David J.R. Foster, Richard N. Upton

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Introduction: The analytical solutions to compartmental pharmacokinetic models are well known, but have not been presented in a form that easily allows for complex dosing regimen and changes in covariate/parameter values that may occur at discrete times within and/or between dosing intervals. Methods: Laplace transforms were used to derive ADVAN-style analytical solutions for 1, 2, and 3 compartment pharmacokinetic linear models of intravenous and first-order absorption drug administration. The equations calculate the change in drug amounts in each compartment of the model over a time interval (t; t=t2-t1) accounting for any dose or covariate events acting in the time interval. The equations were coded in the R language and used to simulate the time-course of drug amounts in each compartment of the systems. The equations were validated against commercial software [NONMEM (Beal, Sheiner, Boeckmann, & Bauer, 2009)] output to assess their capability to handle both complex dosage regimens and the effect of changes in covariate/parameter values that may occur at discrete times within or between dosing intervals. Results: For all tested pharmacokinetic models, the time-course of drug amounts using the ADVAN-style analytical solutions were identical to NONMEM outputs to at least four significant figures, confirming the validity of the presented equations. Discussion: To our knowledge, this paper presents the ADVAN-style equations for common pharmacokinetic models in the literature for the first time. The presented ADVAN-style equations overcome obstacles to implementing the classical analytical solutions in software, and have speed advantages over solutions using differential equation solvers. The equations presented in this paper fill a gap in the pharmacokinetic literature, and it is expected that these equations will facilitate the investigation of useful open-source software for modelling pharmacokinetic data.

Original languageEnglish
Pages (from-to)42-48
Number of pages7
Early online date2 Apr 2015
Publication statusPublished - May 2015
Externally publishedYes


  • Analytical solutions
  • Intravenous
  • Methods
  • Modelling
  • Oral
  • Pharmacokinetics
  • R-language
  • Simulations


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