Compartmental models are mathematical representations of the body that simplify its complex physiology into one or more interconnected compartments for the purpose of describing drug concentration over time. These models allow pharmacokineticists to estimate important parameters such as clearance, volume of distribution, and half-life from concentration-time data. Despite their simplifying assumptions, compartmental models provide clinically useful predictions and form the foundation of pharmacokinetic analysis.
One-Compartment Model
The one-compartment model assumes that the body behaves as a single, well-mixed compartment. After drug administration, the drug distributes instantaneously throughout the compartment, and elimination occurs at a rate proportional to the drug concentration. This model is most appropriate for drugs that distribute rapidly relative to their elimination rate, so that the distribution phase is too brief to be distinguished from the elimination phase.
For an intravenous bolus dose in a one-compartment model, the plasma concentration declines monoexponentially according to the equation: C equals C0 multiplied by e to the power of negative k times t, where C0 is the initial concentration, k is the elimination rate constant, and t is time. The elimination rate constant relates to half-life as k equals 0.693 divided by t½. The volume of distribution is calculated as dose divided by C0, and clearance equals k multiplied by Vd.
For intravenous infusion, the one-compartment model predicts that concentration rises during the infusion and reaches a steady state when the infusion rate equals the elimination rate. After the infusion stops, concentration declines monoexponentially. For oral administration, the model incorporates an absorption phase with a first-order absorption rate constant, and the concentration-time profile shows a rising phase, a peak, and a declining phase.
Two-Compartment Model
The two-compartment model divides the body into a central compartment, representing plasma and highly perfused organs, and a peripheral compartment, representing less perfused tissues such as muscle, fat, and skin. Drug distributes between the central and peripheral compartments at rates governed by intercompartmental rate constants. This model is necessary for drugs that exhibit a distinct distribution phase before the terminal elimination phase.
After an intravenous bolus, the two-compartment model produces a biexponential concentration-time curve. The initial rapid decline represents drug distributing from the central to the peripheral compartment, while the later slower decline represents elimination from the central compartment with simultaneous redistribution from the peripheral compartment. The equation for the concentration-time curve has two exponential terms: C equals A multiplied by e to the power of negative alpha times t plus B multiplied by e to the power of negative beta times t, where alpha is the distribution rate constant and beta is the terminal elimination rate constant.
The two-compartment model provides additional parameters beyond the one-compartment model, including the volumes of the central and peripheral compartments, the intercompartmental clearance, and the microconstants that describe the transfer rates between compartments. These parameters provide a more complete picture of drug disposition, particularly for drugs such as digoxin, aminoglycosides, and many anesthetic agents.
Rate Constants: Microconstants and Macroconstants
In compartmental models, microconstants describe the elementary rate processes for drug transfer between compartments and elimination. In a two-compartment model, k12 represents the transfer rate from central to peripheral compartment, k21 represents the return rate, and k10 represents the elimination rate from the central compartment. These microconstants cannot be directly observed but are estimated by fitting the model to concentration-time data.
Macroconstants are the hybrid rate constants that describe the observable exponents of the concentration-time curve. In the two-compartment model, alpha and beta are macroconstants that are functions of the microconstants. The relationships between microconstants and macroconstants are mathematically defined, allowing estimation of the underlying rate processes from the observed data.
Volume Terms in Compartmental Models
Multiple volume terms arise from compartmental analysis. The central compartment volume (Vc) is the volume of the compartment from which samples are drawn, typically the plasma plus highly perfused tissues. The volume of distribution at steady state (Vss) is the total volume of distribution when distribution equilibrium has been achieved between central and peripheral compartments. The volume of distribution during the terminal phase (Vbeta or Vd area) is calculated as clearance divided by beta.
These different volume terms have distinct clinical meanings. Vss is the most physiologically relevant measure of distribution space, as it reflects the equilibrium partitioning between all tissues and plasma. Vbeta is commonly reported in the literature and is used for calculating half-life, but it can exceed Vss for drugs with very slow terminal elimination.
Application to Different Routes of Administration
Compartmental models can be adapted for any route of administration. For intravenous bolus dosing, the initial concentration is determined by the dose and the central compartment volume. For intravenous infusion, the model incorporates the infusion rate and duration. For extravascular routes, an absorption compartment with a first-order or zero-order absorption rate constant is added.
The model must be identifiable from the available data. With sparse sampling or data that does not adequately characterize the distribution phase, a two-compartment model may not be distinguishable from a one-compartment model. Modern pharmacokinetic software uses nonlinear regression to fit compartmental models to concentration-time data, estimating the parameters that best describe the observed profile and enabling predictions for alternative dosing regimens.
