How to Model Drug Distribution and Elimination Using a Two‑Compartment Pharmacokinetic Model
This article explains the assumptions, equations, and parameter‑estimation methods of a two‑compartment pharmacokinetic model for describing drug distribution, transfer between central and peripheral spaces, and elimination, and compares common dosing routes such as rapid IV bolus, constant‑rate infusion, and oral or intramuscular administration.
Problem
Describe drug distribution and elimination in the body.
Model Assumptions
Two compartments: central (1) and peripheral (2) with constant volumes.
Drug enters the central compartment from outside, transfers between compartments, and is eliminated from the central compartment.
Transfer and elimination rates are proportional to the drug concentration in the respective compartment.
Model Formulation
The model relates dose, concentration, and volume in each compartment, leading to a system of linear differential equations whose homogeneous solution can be expressed analytically.
Common Dosing Regimens
Rapid Intravenous Bolus
An instantaneous dose is injected into the central compartment, causing an immediate rise in blood concentration.
Constant‑Rate Intravenous Infusion
Drug is infused at a constant rate into the central compartment, and the concentration approaches a steady state before declining to zero after the infusion stops.
Oral or Intramuscular Injection
The administered dose first enters an absorption compartment and then transfers to the central compartment.
Parameter Estimation
Parameters such as transfer rates and elimination constants are estimated from concentration‑time data using the least‑squares method.
Reference
Source: “Mathematical Models (5th Edition)” by Jiang Qiyuan, Xie Jinxing, Ye Jun.
Model Perspective
Insights, knowledge, and enjoyment from a mathematical modeling researcher and educator. Hosted by Haihua Wang, a modeling instructor and author of "Clever Use of Chat for Mathematical Modeling", "Modeling: The Mathematics of Thinking", "Mathematical Modeling Practice: A Hands‑On Guide to Competitions", and co‑author of "Mathematical Modeling: Teaching Design and Cases".
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