M12.2 Pharmacokinetic Models

Pharmacokinetics is the study of the dynamic behavior of drug concentrations in blood and tissues. Compartmental models are widely used to conceptually describe this behavior. One common 3-compartment model is shown in Figure M12-1.

Compartment 1 is often used to represent blood plasma. The drug is assumed to be directly infused to compartment 1. The drug can then diffuse between compartment 1 and compartments 2 and 3, based on the transfer parameters, k_ij. For example, k₁₂ represents the kinetics for transfer of the drug from compartment 1 to compartment 2. Compartment 2 can be thought of as well-perfused tissues, such as muscle and brain, while compartment 3 represents residual tissues and bone.

Modeling Equations

If we let I represent the mass infusion rate (mass/unit time) of the drug, the overall material balance can be written as

Equation M12.1

This has the form of a state space model

Equation M12.2

One is normally concerned more about drug concentrations rather than the total amount of drug. The blood plasma concentration can be represented by C₁ = M₁/V₁. The state space model for the measured output is then

Equation M12.3

Alternatively, the states can be formulated as concentrations, resulting in the following model:

Equation M12.4