# Calculation of Bioavailability Parameters

## Method of Residuals

Starting with the equation for Cp versus time Equation 9.2.1 Cp versus Time after Oral Administration

this can be written as Equation 9.2.2 Simplified Equation for Cp versus Time Equation 9.2.3 The Intercept, A Figure 9.2.1 Semi-log plot of Cp versus Time after Oral Administration

Click on the figure to view the interactive graph

If one of the rate constants (ka or kel) is much larger than the other, the method works best if the difference is at least five times, then the faster differential will approach zero more quickly, and at later times can be ignored. If we plot Cp versus time on semi-log graph paper we will see that the slope will approach a straight line.

The equation for this straight line portion can be obtained from the equation for Cp by setting the faster term (usually e-ka • t) to zero: Figure 9.2.2 Semi-log Plot of Cp versus Time Showing Cplate, Slope, and Intercept

then Equation 9.2.4 Cplate versus Time

and plotting Cplate versus time gives a straight line on semi-log graph paper, with a slope (ln) = -kel and intercept = A.

Now looking at the equation for Cp versus time again. Equation 9.2.5 Cp versus Time including Cplate

therefore Equation 9.2.6 Difference or Residual versus Time

Plotting the ln (Residual) versus time should give another straight line graph with a slope (ln) equal to - ka and the same intercept as before, i.e. A Equation 9.2.7 ln(Residual) versus Time Figure 9.2.3 Semi-log of Plot of Residual versus Time

This is the method of residuals or "feathering".

It can give quite accurate values of kel, ka, and V/F if :

i) one rate constant is at least five times larger than the other

and

ii) both absorption and elimination are first order processes.

### An Example Calculation Using the Method of Residuals

Table 9.2.1 Example Data for the Method of Residuals

 Time (hr) Plasma Concentration (mg/L) Cp(late) (mg/L) Residual [Col3 - Col2] (mg/L) 0.25 1.91 5.23 3.32 0.5 2.98 4.98 2.00 0.75 3.54 4.73 1.19 1.0 3.80 4.50 0.70 1.5 3.84 4.07 0.23 2.0 3.62 3.69 0.07 3.0 3.04 4.0 2.49 5.0 2.04 Residual = 5.5 * e -2.05 * t 6.0 1.67 Cplate = 5.5 * e -0.2 * t 7.0 1.37 Figure 9.2.4 Figure Illustrating the Method of Residuals Figure 9.2.5. Concentration versus time after oral administration. Method of residuals

Click on the figure to view the interactive graph