Compartmental modeling identifiability

Respiratory Tract Clearance. This portion of the model identifies the principal clearance pathways within the respiratory tract. The model was developed to predict the retention of various radioactive materials. Figure 3-4 presents the compartmental model and is linked to the deposition model (see Figure 3-2) and to reference values presented in Table 3-5. This table provides clearance rates, expressed as a fraction per day and also as half-time (Part A), and deposition fractions (Part B) for each compartment for insoluble... 

Model development is intimately linked to correctly assigning model parameters to avoid problems of identifiability and model misspecification [27-29], A full understanding of the objectives of the modeling exercise, combined with carefully planned study protocols, will limit errors in model identification. Compartmental models, as much as any other modeling technique, have been associated with overzealous interpretation of the model and parameters. 

Figure 1.18 Indistinguishable parent-metabolite models for isoniazid. Parent drug is administered into Compartment 1 with parent and metabolite concentrations measured in Compartment 1 and 2, respectively. The models in the first column are pairwise symmetric to those in the second column by interchanging Compartments 2 and 4. The same is true for the third and fourth columns. All 18 of these models are kinetically indistinguishability and identifiable. Reprinted from Mathematical Biosciences, vol. 103, Zhang L-Q, Collins KC, and King PH Indistinguishability and identifiability analysis of linear compartmental models, pp. 77-95. Copyright (1971) with permission of Elsevier.

Audoly, S., D Angio, L., Saccomani, M.P., and Cobelli, C. Global identifiability of linear compartmental models— A computer algebra algorithm. IEEE Transactions on Biomedical Engineering 1998 45 36-47. 

Zhang, L.-Q., Collins, J.C., and King, P.H. Indistinguishabil-ity and identifiability analysis of linear compartmental models. Mathematical Biosciences 1991 103 77-95. 

To introduce the idea of identifiability, we start with two very simple problems, one from enz3rme kinetics and one from compartmental modeling. In the next section, we classify the parameters section IV then gives a general statement of the identifiability problem. 

Perry, T. (1991). IDENTII. Identifiability for Compartmental Models. RFKA Documentation for IDENT. University of Michigan, Ann Arbor. 

Vajda, S., Godfrey, K. R., and Rabitz, H. (1989). Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci 93,217-248. Walter, E. (1982). Identifiability of State Space Models, Lect. Notes Biomath. No. 46. Springer-Verlag, New York... 

Vajda, S. (1979). Comments on structural identifiability in linear time-invariant systems. IEEE Trans, on Automatic Control, AC-24, 495-Vajda, S. (1981). Structural equivalence of linear systems and compartmental models. 

Vajda, S. (1982). On parameter and structural identifiability—Nonunique observability, reconstructibility and equivalence of linear compartmental models. IEEE Trans. Autom. Control, 27, 1136. 

Miller, L.V., Krebs, N.F., and Hambidge, K.M. (2000) Development of a compartmental model of human zinc metabolism identifiability and multiple studies analyses. Am. 

Boens N, Novikov E, Ameloot M (2006) Compartmental modeling of the fluorescence anisotropy decay of a cylindrically symmetric Brownian rotor identifiability analysis. Chemphyschem 7(12) 2559-2566. doi 10.1002/cphc.200600309... 

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. 

Given an uniquely identifiable compartmental model, one can proceed by estimating the values of the unknown parameters from the experimental data. Assuming, for the sake of simpKcity, that the model is hnear and that only one output variable is observed, that is, m = 1, by integrating Equation 9.6 one can obtain the expHcit solution for the model output, that is, y t) = g t, 0), where 0 is a p-dimension vector which contains the p unknown parameters of the model. For instance, if the model of Figure 9.1 is used to describe the kinetics of a tracer after the administration of a pulse of known ampKtude D (i.e., u(r) = D8 t) and qi(0) = qjiO) = 0), the vector 0 is a 4-dimension vector containing the fractional... 

Figure 1 illustrates the simplified compartmental model for dissolution and absorption from the lung from the hitemational Commission on Radiological Protection (ICRP) publication 66 This identifies fractions of actinide dissolving rapidly and slowly ((f and fs), a bound state fr)), with associated rate constants (Sr, Ss and Sb). For simplicity, the fraction in the bound state (fb) is generally assumed to be 0. In vitro dissolution tests can be used to measure the rate of dissolution of actinides immobilised on a filter paper and equation 2 used to calculate the solubility parameters Sr, Ss, 6 and fs. ... 

In order to deal with these complex problems all data from the oral Zn studies obtained in patients with taste and smell dysfunction were organized and submitted to compartmental analysis (68,69) with the subsequent development of a model (Figure 1) which accounted for all the data obtained over the entire period of these studies, both prior to and after treatment with exogenous zinc (69). These results, compared in normal volunteers, demonstrated that not only was absorption of zinc significantly impaired in the patients compared with the normal volunteers (Table IV) but also that the rate at which zinc was absorbed was significantly lower in the patients than in the normals (3j5 6 and that their total body level of zinc was lower than in the normals (6 6. By the use of this model it was also possible to specify those conditions which were both necessary and sufficient to identify patients with zinc deficiency (60.69). With these techniques it was possible to identify, by objective criteria, laboratory tests by which patients with subacute zinc deficiency could be defined quantitatively. It was also possible to measure various tissue and total body zinc levels and to compare patients with normals so that patients with zinc deficiency could be identified. The major problems presented with these techniques are that they are time consuming, cumbersome, expensive and are presently unavailable in many areas of the U.S. 

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. 

Reaction rate parameters required for the distributed pharmacokinetic model generally come from independent experimental data. One source is the analysis of rates of metabolism of cells grown in culture. However, the parameters from this source are potentially subject to considerable artifact, since cofactors and cellular interactions may be absent in vitro that are present in vivo. Published enzyme activities are a second source, but these are even more subject to artifact. A third source is previous compartmental analysis of a tissue dosed uniformly by intravenous infusion. If a compartment in such a study can be closely identified with the organ or tissue later considered in distributed pharmacokinetic analysis, then its compartmental clearance constant can often be used to derive the required metabolic rate constant. 

---