Exponential curve
An exponential curve is observed on a plot of concentration (on the Y-axis) versus time (on the X-axis) where drug concentrations are changing in a first order manner. Such a curve has a half life and this is related to the first order elimination rate constant. The exponential curve indicates that the rate of drug elimination is dependent on the concentration – as drug is eliminated, the concentration falls and so the rate of elimination falls. The rate of elimination at any point on the curve may be found from the slope of the tangent to the exponential curve; at any point on the curve, the rate of elimination is equal to the concentration of drug at that time multiplied by the first order rate constant for elimination (kel or β).
The reason the rate of elimination falls is that as drug is eliminated and concentration falls, the occupation of protein(s) responsible for elimination of that drug (e.g. a CYP enzyme) is reduced, and/or less drug is filtered at Bowman’s capsule in unit time as the concentration of drug in the blood passing through the glomerulus falls. With respect to occupation of proteins responsible for elimination, this is seen as a drop in b/Bmax, in the Hill-Langmuir equation. If less drug is bound to the CYP enzyme population (in other words, fewer CYP enzyme proteins have a drug molecule bound at the active site at any given moment in time), the drug is metabolised less quickly by the CYP enzyme population, and so the rate of elimination falls.
A mirror-image of the exponential curve for elimination can be seen when drug is administered at a constant rate (e.g. by IV infusion) whereby the drug concentration increases but the rate at which the concentration increases becomes lower as b/Bmax for the occupation of protein(s) responsible for elimination of that drug (e.g. a CYP enzyme) increases towards the plateau of a hyperbola. The infusion curve reaches a plateau when the rate of drug elimination increases to the point where it is equal to the rate of drug administration.
When a single exponential curve is plotted on a logarithmic Y-axis (i.e. concentrations are expressed as log values) then the exponential curve becomes a straight line. This allows for easy extrapolation of the line back to the Y-intercept to determine Cp0. If a biexponential curve (i.e. data from a drug showing two-compartment behaviour) is plotted on a logarithmic Y-axis, only the portion of the curve that corresponds to a single exponential process (elimination) will appear as a straight line. It is straightforward to determine from such a hockey stick curve at which point distribution had reached equilibrium and elimination became solely responsible for loss of drug from the plasma.
The slope of the straight line (when concentration data are plotted on a logarithmic Y-axis) allow calculation of the first order rate constant. When a log10 axis is used, kel (or β) = slope x 2.303, while plotting on a natural logarithm axis yields the rate constant directly from the slope, without the need for the 2.303 factor. An example slope calculation is shown under the entry for β.
