Half-life of elimination

Elimination of drug from the plasma is usually a first-order process (i.e. concentration-dependent); the larger the concentration of drug in the plasma, the faster the rate of elimination. As such, the process has a first order elimination rate constant (k_el for one-compartment drugs, β for two-compartment drugs) and, therefore, a half-life for elimination (0.693/k_el or 0.693/β). The half-life for elimination is the time it takes for the concentration in the plasma to drop to 50% of the concentration present when distribution is complete (C_p0 on a one-compartment graph, or the B-intercept on a two-compartment graph).

When movement of drug is very rapid between the central and tissue compartments (i.e when the drug shows one-compartment kinetics), the first order elimination rate constant for loss of drug from the central compartment in isolation is directly related to the first order elimination rate constant for loss of drug from the entire volume of distribution. For example, in the figure below on the right, 20 mg drug are present in a beaker containing 1 litre of water (representing the plasma) and 1 litre of oil (representing the tissues), and the AV_D is 5 litres (due to the concentration in the water layer being 4 mg/l at equilibrium, and thus giving an AV_D:plasma volume ratio of 5). The first-order rate constants for drug movement between the two compartments, k₁₂ (for movement from compartment 1, or water, to compartment 2, or oil) and k₂₁ (for movement in the opposite direction), are large. The tap is then turned on at a flow rate of 0.5 l/h, thereby achieving a clearance value of 0.5 l/h, with fresh water entering at the same flow rate to replace that drained via the tap. As a result, the rate constant (which is a proportionality constant) for loss of drug from the plasma only (with a volume of 1 litre), shown as k₁₀ in this example (indicating movement from compartment 1 to compartment 0, which is extracorporeal waste), is 5× larger than the k_el, the rate constant for elimination of drug from the entire 5 litres of the AV_D. This is because drug draining via the tap represents a 5× larger proportion of the 1 litre of water than of the 5 l AV_D, and k_el and k₁₀ are proportionality constants. Accordingly, the half-life of elimination from the entire volume of distribution (0.693/k_el) is 5× longer than the half-life for elimination of drug from the plasma (water) in isolation (0.693/k₁₀).

In contrast, if k₁₂ and k₂₁ were much smaller such that distribution was significantly slower, the drug would show two-compartment kinetics. As such, the elimination rate constant during the terminal elimination phase (which is β for a two-compartment or multi-compartment drug) would be smaller than k_el, i.e smaller than would be predicted by considering the AV_D:plasma volume ratio and the value for k₁₀, as was done for the one-compartment drug. Thus, the terminal half-life for elimination calculated from β is longer than the half-life for elimination calculated from k_el (vide supra; from the limited data provided here, you can’t calculate how much longer). This is due entirely to the slow redistribution of drug from the tissues back to the plasma, putting a “brake” on the rate of elimination.

Based on this discrepancy between values for k_el and β, it may have occurred to you that the mathematical relationship that exists between AV_D, k_el and Cl can not also describe the relationship between AV_D, β and Cl. In fact, the relationship does hold, but there are approaches for determining AV_D for a two-compartment drug that differ subtly from the approaches used for a one-compartment drug, and as such, there are also subtle differences in both the meaning and in the calculated values for AV_D, depending upon the approach chosen. This is discussed briefly at the end of the entry for apparent volume of distribution.