title: ADVAN-style analytical solutions for common pharmacokinetic models.
abstract: BACKGROUND
  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.
  
  
  CONCLUSIONS
  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.
authors.affiliation: Australian Centre for Pharmacometrics and Sansom Institute, School of Pharmacy and Medical Sciences, University of South Australia, SA 5000, Australia.
authors.forename: Ahmad Y
authors.initials: AY
authors.lastname: Abuhelwa
authors.affiliation: Australian Centre for Pharmacometrics and Sansom Institute, School of Pharmacy and Medical Sciences, University of South Australia, SA 5000, Australia.
authors.forename: David J R
authors.initials: DJ
authors.lastname: Foster
authors.affiliation: Australian Centre for Pharmacometrics and Sansom Institute, School of Pharmacy and Medical Sciences, University of South Australia, SA 5000, Australia. Electronic address: richard.upton@unisa.edu.au.
authors.forename: Richard N
authors.initials: RN
authors.lastname: Upton
mesh_terms: D007262:Infusions, Intravenous; D008954:Models, Biological; D008962:Models, Theoretical; D004364:Pharmaceutical Preparations / Q000008:administration & dosage / Q000378:metabolism*
chemical_list: D004364:Pharmaceutical Preparations
keywords: ADVAN; Analytical solutions; Intravenous; Methods; Modelling; NONMEM; Oral; Pharmacokinetics; R-language; Simulations