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Using a model previously explored, the excretion and metabolism model, we can work through the Laplace transform method.

**Figure 20.5.1 Diagram Representing Excretion and Metabolism Model**

**Equation 20.5.1 Laplace of Drug Concentration in Plasma or Blood**

Note that this equation includes what could be called an intensity term (Dose/V_{1}) and one 's' term (s + ke + km). The intensity term can be evaluated from an intercept and the 's' term from a slope. Since we know 'Dose' the parameter V_{1} can be identified from the intercept. The slope provides information about ke + km (= kel) but not ke or km on their own. Thus, if we measure C_{1} two parameters are identifiable, V_{1} and kel. The parameters ke, km, kmu and V_{3} are either non-identifiable or non-observable.

If we collect another sample, metabolite concentration in blood or plasma (C_{3}), what additional parameters can be identified?

**Equation 20.5.2 Laplace of Metabolite Concentration in Plasma or Blood**

The intensity term is (Dose • km)/V_{3} but because we don't know km or V_{3} separately we can't identify either parameter. There are two 's' terms, (ke + km) and kmu. We still can't separate ke or km, however kmu can now be identified.

If we collect one more sample, metabolite amount in urine (X_{4}), more information may be available.

**Equation 20.5.3 Laplace of Metabolite Amount in Urine**

The intensity factor (km • kmu • Dose) provides information about km since kmu and Dose are known. There are three 's' terms, s, (s + ke + km) and (s+ kmu). Since we now know km we can identify ke from the second 's' term. Finally we can go back to the metabolite in blood or plasma data and use the intensity factor to estimate V_{3} since we now know km. With good drug in plasma and metabolite in plasma and urine data we should be able estimate all the parameters of the model shown in Figure 20.5.1. A similar approach could be followed with other models to determine the samples that must be collected to identify or estimate the model parameters.

- Godfrey, K.R. and Fitch, W.R. 1984
**The deterministic identifiability of nonlinear pharmacokinetic models**,*J. Pharmacokin. Biopharm.*,**12**, 177-191 - Wang, Y.M.C. and Reuning, R.H. 1992
**An experimental design strategy for quantitating complex pharmacokinetic models - enterohepatic circulation with time varying gallbladder emptying as an example**,*Pharm. Res.*,**9**, 169-177

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