Compartmental modeling techniques

Element losses together with element bioavailability determine how much of an element has to be provided through the diet to remain in element balance. For assessment of losses, a label needs to be administered once either orally or intravenously. Compartmental modeling techniques permit to calculate when the label has equilibrated with the natural element in all body compartments. When isotopic labeling has been achieved, continuous replacement of lost isotopic label with the natural element from the diet results in a continuous decline of the body s isotopic enrichment. The change in tracer to tracee ratio in blood corresponds directly to the fraction of the body s element inventory that has been lost and replaced. 

Extensive literature is available on general mathematical treatments of compartmental models [2], The compartmental system based on a set of differential equations may be solved by Laplace transform or integral calculus techniques. By far... 

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. 

The development of a successful pharmacokinetic model allows one to summarize large amounts of data into a few values that describe the whole data set. The general procedure used to develop a pharmacokinetic model is outlined in Table 10.1. Certain aspects of this procedure have been described previously in Chapters 3 and 8. For example, the technique of curve peeling" frequently is used to indicate the number of compartments that are included in a compartmental model. In any event, the eventual outcome should be a model that can be used to interpolate or extrapolate to other conditions. 

Other approaches have been used for more complex models. These include curve stripping or the method of residuals,either manually or using a computer program such as CSTRIP and ESTRIF. These techniques can separate a multiexponential curve into its component parts for initial estimates. Other techniques include deconvolution methods specific to the one and two compartment pharmacokinetic models. The objective of the deconvolution method is to mathematically subtract the results obtained after IV administration from the oral or extravascular data. This results in information about the input or absorption process alone. More general methods have been presented by various researchers that do not rely on a particular compartmental model. ... 

The two most commonly used methods for characterizing pharmacokinetic data are noncompartmental analysis and the fitting of compartmental models. The latter technique can range from simple one to three well-stirred compartments to physiologically-based pharmacokinetic (PBPK) models, which are covered in the next section. The choice of which method to utilize will be largely dictated by the goals and objectives of the analysis. For example, descriptions of major pharmacokinetic parameters for linear systems (i.e., net systemic exposure is dose-proportional) can be easily calculated from a noncompartmental... 

Model selection, application and validation are issues of major concern in mathematical soil and groundwater quality modeling. For the model selection, issues of importance are the features (physics, chemistry) of the model its temporal (steady state, dynamic) and spatial (e.g., compartmental approach resolution) the model input data requirements the mathematical techniques employed (finite difference, analytic) monitoring data availability and cost (professional time, computer time). For the model application, issues of importance are the availability of realistic input data (e.g., field hydraulic conductivity, adsorption coefficient) and the existence of monitoring data to verify model predictions. Some of these issues are briefly discussed below. 

Although, during the early applications of therapeutic mAbs, pharmacokinetic modeling was rarely applied, a variety of analytical techniques has been used over the years to characterize the pharmacokinetics of this class of compounds. The application and information derived from three different methods of noncompart-mental analysis, individual compartmental analysis, and population analysis will be discussed in the following sections. 

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. 

The Kona Conference and many other events that I could cite show that it is now time to end this compartmentalization of knowledge, get our act together, and understand that there is a common body of concepts and techniques that apply to a large domain of very important processes and situations. Professor Doraiswami Ramkrishna has made a major contribution to the needed unification of theory and computational techniques of population balances with the preparation of his book Population Balances Theory and Applications to Particulate Systems in Engineering. It should be, and I hope it will be, the source that workers from many diverse fields turn to when they seek to learn the concepts and techniques of population balance modeling of particulate systems. 

Noncompartmental pharmacokinetics has been developed as an alternative to data-intensive compartmental and physiologic models. While the latter techniques are useful in pharmacokinetic predictions if sufficient data are available, drugs with complex distribution and elimination may be difficult to properly model without additional experimental data. The noncompartmental techniques do not rely on specific distribution characteristics of a drug and therefore become useful when data are limited. 

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