Pharmacokinetic one-compartment model

Pharmacokinetics refers to the rate and extent of distribution of a drug to different tissues, and the rate of elimination of the drug. Pharmacokinetics can be reduced to mathematical equations, which describe the transit of the drug throughout the body, a net balance sheet from absorption and distribution to metabolism and excretion.

Widmark (1919) did experimental studies with bolus and constant administration of compounds, and analyzed the results based on model with a single compartment, which can be easily calculated with exponential function. A certain amount of drug (A₀) is administered at constant rate k₀₁. The amount of drug in the body (A₁) and the amount of cleared drug (A₂) are dependent also on the elimination rate constant k₁₂.

Pharmacokinetic one body compartment model — **Figure 1.** One body compartment model.

Differential equations for A₁ and A₂ are:

$\frac{dA_{1}(t)}{dt} = k_{01} A_0(t) - k_{12} A_{1}(t)$ $\frac{dA_{2}(t)}{dt} = k_{12} A_{1}(t)$

If drug is given intravenously as a fast bolus injection, it is often assumed that the drug is immediately distributed in the compartment A₁. Then the remaining amount of the drug at time t can be calculated as:

$A_1(t) = A_1(0) \times e^{-k_{12} \: t}$

Samples of plasma can be easily obtained, and if plasma concentrations, C_P(t), are measured after the bolus administration, the clearance and V_D of the drug can be estimated by fitting the model (Figure 2) to the data.

Pharmacokinetic one-compartment clearance model — **Figure 2.** One-compartment model with first-order elimination.

$C_P(t) = \frac{Dose}{V_D} \times e^{-\frac{Clearance}{V_D}t}$

In case of one-compartment model with oral first-order absorption the plasma concentrations can be modelled with two exponential functions

$C_P(t) = \frac{A}{k_a - k_e} \times \left( e^{-k_e t} - e^{-k_a t} \right)$

The differential equations can be used to simulate the concentration or amount of drug in the body at different time points with different drug administration regimes. If one compartment model is not sufficient to describe drug kinetics, then two or three compartment model can be used (Krüger-Thiemer, 1968).

References:

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Tags: Pharmacokinetics, Drug development, Plasma, Clearance

Updated at: 2023-05-05
Created at: 2019-01-08
Written by: Vesa Oikonen

Pharmacokinetic one-compartment model

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