# One Compartment IV Bolus

## Elimination Rate Constant, kel

The elimination rate constant (abbreviated as kel, k10, and sometimes ke) is the first order rate constant describing drug elimination from the body. This is an overall elimination rate constant describing removal of the drug by all elimination processes including excretion and metabolism. Metabolites are different chemical entities and have their own elimination rate constant.

The elimination rate constant is the proportionality constant relating the rate of change drug concentration and concentration OR the rate of elimination of the drug and the amount of drug remaining to be eliminated.

## Defining Equations:

Table 4.9.1 Equations Defining kel  Equation 4.9.1 Rate of Change of Cp versus Cp Equation 4.9.2 Elimination Rate versus Amount Remaining, X

## Units:

By inspection of Equations 4.9.1. or 4.9.2 it can be seen that the units for kel are time-1, for example hr-1, min-1, or day-1. In both equations the rate expression is divided by Cp or X, respectively to provide units for kel. Thus:

Table 4.9.2 Units for kel  Units for kel from Equation 4.9.1 Units for kel from Equation 4.9.2

## Determining Values of kel:

From the integrated equation presented on the previous page: Equation 4.9.3 Ln(Cp) versus time

Plotting ln(Cp) values versus t, time should result in a straight line with a slope equal to -kel, thus kel can be calculated as: Equation 4.9.4 kel from -slope

Thus with two value for Cp and time data a value for kel can be determined. With more than two Cp - time data points it is possible to plot the data on semi-log graph paper and draw a 'best-fit' line through the data points. This plot is shown on the previous page. The best answer for kel can be calculated by taking points at either end of the 'best-fit' line. This approach has been covered in more detail earlier in Chapter 2.

Equation 4.6.6, presented earlier on the page describing apparent volume of distribution, can be used to calculate concentration at any time, t, after an IV dose with kel and V. Click on Figure 4.8.1 to try this out. Figure 4.8.1. Concentration versus time

Click on the figure to view the interactive graph

Note: It is important to distinguish between the elimination rate and the elimination rate constant. The rate (tangent or slope, dCp/dt) changes as the concentration changes, however, for a linear model the rate constant (kel) is constant, it does not change.
Table 4.9.3. Example Values for Elimination Rate Constant (Ritschel, 1992)

 Drug kel, hr-1 Acetaminophen 0.277 Diazepam 0.021 Digoxin 0.0161 Gentamicin 0.347 Lidocaine 0.39 Theophylline 0.126

References
• Ritschel, W.A. 1992 Handbook of Basic Pharmacokinetics, 4th ed., Drug Intelligence Publications, p 533-549
• Elimination rate constant has been discussed on the PharmPK listserv.