Exact solutions and equi-dosing regimen regions for multi-dose pharmacokinetics models with transit compartments
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ORIGINAL PAPER
Exact solutions and equi-dosing regimen regions for multi-dose pharmacokinetics models with transit compartments F. Hof1,2 • L. J. Bridge3 Received: 8 July 2020 / Accepted: 17 September 2020 Ó The Author(s) 2020
Abstract Compartmental models which yield linear ordinary differential equations (ODEs) provide common tools for pharmacokinetics (PK) analysis, with exact solutions for drug levels or concentrations readily obtainable for low-dimensional compartment models. Exact solutions enable valuable insights and further analysis of these systems. Transit compartment models are a popular semi-mechanistic approach for generalising simple PK models to allow for delayed kinetics, but computing exact solutions for multi-dosing inputs to transit compartment systems leading to different final compartments is nontrivial. Here, we find exact solutions for drug levels as functions of time throughout a linear transit compartment cascade followed by an absorption compartment and a central blood compartment, for the general case of n transit compartments and M equi-bolus doses to the first compartment. We further show the utility of exact solutions to PK ODE models in finding constraints on equi-dosing regimen parameters imposed by a prescribed therapeutic range. This leads to the construction of equi-dosing regimen regions (EDRRs), providing new, novel visualisations which summarise the safe and effective dosing parameter space. EDRRs are computed for classical and transit compartment models with two- and three-dimensional parameter spaces, and are proposed as useful graphical tools for informing drug dosing regimen design. Keywords Mathematical pharmacology Pharmacokinetics Compartment models Differential equations Transit compartments Regimen design
Introduction Mathematical models for the absorption, distribution and elimination of drugs are common in the pharmacokinetics (PK) literature. Typically, a drug’s route through the body to its pharmacological effect site is modelled as a number of compartments, with transfer between compartments being governed by pharmacokinetic rate laws. It is common to consider only one or two compartments, with linear pharmacokinetics, resulting in low-dimensional linear
& L. J. Bridge [email protected] F. Hof [email protected] 1
Swansea University, Swansea, UK
2
Medical University of Vienna, Vienna, Austria
3
Department of Engineering Design and Mathematics, University of the West of England, Bristol, UK
ordinary differential equation (ODE) systems. However, such models are not sufficient to capture delay-type effects, whereby some time passes before the drug appears at measurable levels in the systemic circulation [44]. If a significant ‘‘drug absorption delay’’ [44, 46] is observed, then a lag-time is sometimes introduced into solutions to the simple models to account for the delay, while avoiding any mechanistic considerations of the underlying delay roaaataaat esses. This simple approach may be used to paramaterise a system delay, b
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