# Chapter 9

# Calculation of Bioavailability Parameters

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## 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 Cp^{late}, Slope, and Intercept

then

**Equation 9.2.4 Cp**^{late} *versus* Time

and plotting Cp^{late} *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 Cp^{late}

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 | Cp^{late} = 5.5 * e^{
-0.2 * t} | |

7.0 | 1.37 | | |

**Figure 9.2.4 Figure Illustrating the Method of Residuals**

Homework 2

Homework 2

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