Priori identifiability

The number of quadruple, quintuple and sextuple excitations is however still a very large number for a zeroth-order wavefunction and even these configuration spaces still contain more dead than live wood. Our goal is therefore the development of a method for a priori identifying and selecting the latter. It will be achieved by... 

With regards to the analysis of the quality of the various parts of the model, one may use the same methods as are used for practical identifiability analysis. Since the same methods are used, albeit with different objectives, one sometimes refers to this model quality analysis as a posteriori identifiability (and the previous analysis as a priori identifiability). Now, however, one is also interested in how the parametric uncertainty translates to an uncertainty in the various model predictions. For instance, it might be so that even though two individual parameters have a high uncertainty, they are correlated in such a manner that their effect on a specific (non-measured) model output is always the same. Such a translation may be obtained by simulations of the model using parameters within the determined confidence ellipsoids. A global alternative to this is to consider the outputs for all parameters that correspond to a cost function that is below a certain threshold, for example 2% above the found minimum. 

To emphasize this point, once a model structure is postulated, the compartmentai matrix is known, since it depends only upon the transfers and losses. The input, the F q, comes from the experimental input and thus is determined by the investigator. In addition, the units of the differential equation (i.e., the units of the Xj) are determined by the units of the input. The point is that if the parameters of the model can be estimated from the data from a particular experimental design li.e., if the model is a priori identifiable see... 

The next question is whether the parameters characterizing a modei can be estimated from a set of pharmacokinetic data. The answer to this question has two parts. The first is caUed a priori identifiability. This answers the question given a particufar modei structure and experimentai design if the data are perfect/ can the modei parameters be estimated " The second part is a posteriori identifiability. This answers the ques-tioip "given a particuiar modei structure and a set of pharmacokinetic data can the modei parameters be estimated within a reasonabie degree of statisticaf precision "... 

Model A is a standard two-compartment model with input and sampling from a "plasma" compartment. There are three kjj and a volume term to be estimated. This model can be shown to be a priori identifiable. Model B has four ky and a volume term to be estimated. These parameters cannot be estimated from a single set of pharmacokinetic data, no matter how information rich they are. In fact, there are an infinite number of values for the ky and volume term that will produce the same fit of the data. If one insists on using this model structure, then some constraint will have to be placed on the parameters, such as fixing the volume or defining a relationship among the ky. Model C, while a priori identifiable, will have a different compart-mental matrix from that of model A, and hence, as discussed previously, some of the pharmacokinetic parameters will be different for the two models. 

Two commonly used three-compartment models are shown in Figures 8.6D and E. Of the two peripheral compartments, one exchanges rapidly and one changes slowly with the central compartment. Model D is (7 priori identifiable while model E is not. Model E will have two different compartmental matrices that will produce the same fit of the data. The reason is that the loss is from a peripheral compartment. Finally, model F, a model very commonly used to describe the pharmacokinetics of drug absorption, is not a priori identifiable. Again, there are two values for the compartmental K matrix that will produce the same fit to the data. 

Cobelli, C.. and Saccomani, M. P. (1990). Unappreciation of a priori identifiability in software packages causes ambiguities in numerical estimates. Am. J. Physiol. 2S8, E10S8-E10S9. Davidian, M., and Gallant, A. R. (1992). Smooth nonparametric maximum likelihood estimation for population pharmacokinetics, with application to quinidine. J. Pharmacokinet. Biopharm. 20, 529-556. 

A priori identifiability thus examines whether, given the ideal noise-free data y. Equation 9.13, and the error-free compartmental model structure. Equation 9.5 or Equation 9.6, it is possible to make unique estimates of aU the unknown model parameters. A model can be uniquely (globally) identifiable — that is all its parameters have one solution — or nonuniquely (locally) identifiable — that is, one or more of its parameters has more than one but a finite number of possible values — or nonidentifiable — that is, one or more of its parameters has an infinite number of solutions. For instance, the model of Figure 9.1 is uniquely identifiable, while that of Figure 9.3 is nonidentifiable. 

Similarly with the previous case, we can determine some conditions on the model parameters such that we can a priori identify the chattering behavior ... 

A Priori Identifiability Parameter Estimation Optimal Experiment Design Validation References... 

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