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Identifiability - Analytical Approaches

As with the numerical methods there are a number of analytical approach to investigating identifiability problems. There is a Laplace transform method and a Taylor series approach. Although the Taylor series method is more general in application it is more cumbersome and won't be described and further at this time.

Laplace Transform Method

The Laplace transform method is less complex to apply but it doesn't work with time-varying or non-linear systems. Fortunately, most pharmacokinetics systems contain parameters that do not vary with time and linear models are more common.

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

First let us consider what we can determine (identify) if we measure just the drug concentration in plasma or blood. Starting with the differential equation for this model component and taking the Laplace provides:

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/V₁) 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 V₁ 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 C₁ two parameters are identifiable, V₁ and kel. The parameters ke, km, kmu and V₃ are either non-identifiable or non-observable.

If we collect another sample, metabolite concentration in blood or plasma (C₃), what additional parameters can be identified?

Equation 20.5.2 Laplace of Metabolite Concentration in Plasma or Blood

The intensity term is (Dose • km)/V₃ but because we don't know km or V₃ 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 (X₄), 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 V₃ 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.

References

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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