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Identifiability - Numerical Approaches

There are a number of approaches to detecting or confirming identifiability problems with a model or sample selection. These methods can be grouped as numerical and analytical. The numerical group includes an empirical approach and use of programs such as IDENT by Jacquez and Perry, 1990.

Empirical Method

The idea behind this method is that you should be able to fit simulated data and determine the original parameter values well if the parameters are identifiable. Repeated fitting of these data should result in the same parameter values if the parameters are identifiable. The method includes:

Example 1 - Drug and Metabolite Model

Consider the model from Figure 20.3.2.

Figure 20.4.1 Diagram representing and IV Administration, One Compartment Model with Metabolism

Data were simulated for drug and metabolite concentrations are shown in Table 20.4.1

Table 20.4.1 Data Simulated for Cp and Cm

The data were then fitted to the model in Figure 20.4.1 using Boomer.

Figure 20.4.1 Output from Boomer

Figure 20.4.1 shows just one result from fitting the data in Table 20.4.1. Note that the CV% values look good for most of the parameters. This generally indicates a good fit and well defined parameters. However, the value of zero for V2 (= Vm) is not a good sign. The real problem becomes apparent when the model is fit multiple times with the same data. Now we see a variety of values for ke, km and Vm. Curiously if the best-fit ke and km values are plotted it can be seen that they fall on a line indicating a common value of kel (= ke + km).

Figure 20.4.2 Plot of ke versus km

The blue squares represent the initial values used for each fit and the red circle represents the final, best-fit, value. Note that all the red circles fall on a straight line representing the value of kel (= ke + km). This indicates the neither ke nor km are identifiable but does suggest that kel is identifiable.

Example 2 - Michaelis-Menten Model

We can repeat the exercise above with the Michaelis-Menten model described on the previous page one dose at time. The results from high and low dose data are shown below.

Figure 20.4.3 Output from Boomer - 500 mg Dose

Figure 20.4.4 Output from Boomer - 5 mg Dose

Notice that the CV% for the low dose are quite large (720%) indicating that Vm (=Vmax) and Km are not identifiable with these low dose data. Note also that the parameters have hit reasonable upper limits. Increasing the limits resulted in the values increasing to the new limits. The value for the parameter CV% using the high dose data are much better. The final parameter values for all three estimated parameters are close to the starting values. This indicates that if data as good as the simulated data were available then these parameters would be identifiable.

Example 3 - Oral Administration Model

The final example demonstrates the effect of missing early data points after oral administration.

Figure 20.4.5 Output from Boomer - Oral Administration with Missing Early Data Points

Notice the high value for CV% for the ka parameter value compared with the CV% for kel and V. Clearly ka is not well defined.

Using the IDENT Program

The IDENT program is another numerical method of investigating identifiability problems. The steps include: