## Calculation of ka

### Wagner-Nelson method

Another method of calculating ka is the Wagner-Nelson method[1]
**Advantages:** i) The absorption and elimination processes can be quite similar and still accurate determinations of ka can be made.

ii) The absorption process doesn't have to be first order. This method can be used to investigate the absorption process. I have used this type of method to investigate data obtained after IM administration and found that two absorption steps maybe appropriate. Possibly a fast step from drug in solution and a slower step from drug precipitated at the injection site.

**Disadvantages:** i) The major disadvantage of this method is that you need to know the elimination rate constant, from data collected following intravenous administration.

Theory: The working equations can be derived from the mass balance equation:-

Amount Absorbed (A) = Amount in Body (X) + Amount Eliminated (U)

or A = X + U

Differentiating each term with respect to time gives:-

or

or

dA = V * dCp + kel * V * Cp * dt

and integrating gives:-

or

Taking this to infinity, Cp = 0 and

Finally (Amax - A) is the amount remaining to be absorbed, Xg thus:-

We can use this equation to look at the absorption process. If absorption is first order

or

Thus a plot of ln (Amax - A) versus time will give a straight line for first order absorption with a slope = -ka

**Table IX-2 Example Data for the Method of Wagner-Nelson**

**kel (from IV data) = 0.2 hr**^{-1}

**Time**
(hr) | **Plasma**
Concentration
(mg/L) | **Column**
3
AUC | **Column**
4
AUC | **Column 5**
kel * AUC | **A/V**
[Col2 + Col5] | **(A**_{max } - A)/V |

**0.0** | **0.0** | **0.0** | **0.0** | **0.0** | **0.0** | **4.9** |

**1.0** | **1.2** | **0.6** | **0.6** | **0.12** | **1.32** | **3.58** |

**2.0** | **1.8** | **1.5** | **2.1** | **0.42** | **2.22** | **2.68** |

**3.0** | **2.1** | **1.95** | **4.05** | **0.81** | **2.91** | **1.99** |

**4.0** | **2.2** | **2.15** | **6.2** | **1.24** | **3.44** | **1.46** |

**5.0** | **2.2** | **2.2** | **8.4** | **1.68** | **3.88** | **1.02** |

**6.0** | **2.0** | **2.1** | **10.5** | **2.1** | **4.1** | **0.8** |

**8.0** | **1.7** | **3.7** | **14.2** | **2.84** | **4.54** | **0.36** |

**10.0** | **1.3** | **3.0** | **17.2** | **3.44** | **4.74** | **0.16** |

**12.0** | **1.0** | **2.3** | **19.5** | **3.9** | **4.9** | **-** |

| **0.0** | **5.0** | **24.5** | **4.9** | **4.9** | **-** |

Thus, ka = 0.3056 hr^{-1} from the straight line on semi-log graph paper, thus first order. Only points from 0 to 8 hours plotted.

**Figure IX-5, Semi-Log Plot of (Amax-A)/V versus Time**

**Figure IX-6, Linear Plot of (Amax - A)/V versus Time**

**For practice** try calculating the absorption rate constant, ka, using the Wagner-Nelson Method. Compare your answers with the computer!

This page was last modified: 9 Jul 2003
Copyright 2001 David W.A. Bourne