Compartmental modeling parameters

The EDBD variant of BP was used in several chemical kinetic studies. 2-154 This method gave much better estimates of kinetic analytical parameters than either nonlinear regression or principal components regression. EDBD has also been applied to multicomponent kinetic determinations and to the estimation of kinetic compartmental model parameters.i The EDBD method was found to offer increased modeling power for nonlinear multivariate data compared to partial least squares and principal components regression, provided the training set is extensive enough to adequately sample the nonlinear features of the data.i55 Finally, EDBD has been successfully applied to the prediction of retention indices, i ... 

At this point one has a compartmental model structure, a description of the measurement error, and a numerical value of the parameters together with the precision with which they can be estimated. It is now appropriate to address the optimal experiment design issue. The rationale of optimal experiment design is to act on design variables such as number of test input and outputs, form of test inputs, number of samples and sampling schedule, and measurement errors so as to maximize, according to some criterion, the precision with which the compartmental model parameters can be estimated [DiStefano, 1981 Carson et al, 1983 Landaw and DiStefano, 1984 Walter and Pronzato, 1990]. 

We now turn our attention to the graphical determination of the various parameters of our two-compartmental model, i.e. the plasma volume of distribution Vp,... 

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 cumulative curve obtained from the transit time distribution in Figure 9 was fitted by Eq. (48) to determine the number of compartments. An additional compartment was added until the reduction in residual (error) sum of squares (SSE) with an additional compartment becomes small. An F test was not used, because the compartmental model with a fixed number of compartments contains no parameters. SSE then became the only criterion to select the best compartmental model. The number of compartments generating the smallest SSE was seven. The seven-compartment model was thereafter referred to as the compartmental transit model. 

Absorbed lead is distributed in various tissue compartments. Several models of lead pharmacokinetics have been proposed to characterize such parameters as intercompartmental lead exchange rates, retention of lead in various pools, and relative rates of distribution among the tissue groups. See Section 2.3.5 for a discussion of the classical compartmental models and physiologically based pharmacokinetic models (PBPK) developed for lead risk assessments. 

PBPK and classical pharmacokinetic models both have valid applications in lead risk assessment. Both approaches can incorporate capacity-limited or nonlinear kinetic behavior in parameter estimates. An advantage of classical pharmacokinetic models is that, because the kinetic characteristics of the compartments of which they are composed are not constrained, a best possible fit to empirical data can be arrived at by varying the values of the parameters (O Flaherty 1987). However, such models are not readily extrapolated to other species because the parameters do not have precise physiological correlates. Compartmental models developed to date also do not simulate changes in bone metabolism, tissue volumes, blood flow rates, and enzyme activities associated with pregnancy, adverse nutritional states, aging, or osteoporotic diseases. Therefore, extrapolation of classical compartmental model simulations... 

Analysis of data using simple mammillary, compartmental models allows the estimation of all of the basic parameters mentioned here, if data for individual tissues are analyzed with one or two compartment models, and combined with results from... 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

Various PK parameters such as CL, Vd, F%, MRT, and T /2 can be determined using noncompartmental methods. These methods are based on the empirical determination of AUC and AUMC described above. Unlike compartmental models (see below), these calculation methods can be applied to any other models provided that the drug follows linear PK. However, a limitation of the noncompartmental method is that it cannot be used for the simulation of different plasma concentration-time profiles when there are alterations in dosing regimen or multiple dosing regimens are used. 

As previously discussed, compartmental models can be effectively used to project plasma concentrations that would be achieved following different dosage regimens and/or multiple dosing. However, for these projections to be accurate, the drug PK profile should follow first-order kinetics where various PK parameters such as CL, V,h T /2, and F% do not change with dose. 

In pharmacokinetics empirical models are for example compartment models where the body is sub-divided into one or more compartments and the drug is assumed to distribute and be eliminated with first-order rate constants. Typical model parameters are the rate constants and the volumes of the compartments. The compartmental models reflect the physiological reality only to a very limited degree. Despite this limitation compartment models are essential in drug development and have received considerable attention and showed huge utility and impact on the labeling of drugs on the market [4]. 

Figure 3. This kinetic model for zinc in humans was based on averaged data obtained following oral and i.v. administration of Zn to 17 patients with abnormalities of taste and smell. The compartmental model used all kinetic data from Zn activity in plasma, red blood cells, urine, liver, and thigh as well as stable zinc parameters, including dietary intake, serum, and urinary concentration. The SAAM27 computer program was used to obtain the simplest set of mathematical relationships that would satisfy the data characteristics for each measurement time in the study and remain consistent with accepted concepts of zinc metabolism. Although the short physical half-life of Zn limited the data collection period, this model allowed for analysis of the rapid phases of zinc metabolism (about 10% of total body zinc) and derivation of a number of fundamental steady state...

From previous chapters it is clear that the evaluation. of pharmacokinetic parameters is an essential part of understanding how drugs function in the body. To estimate these parameters studies are undertaken in which transient data are collected. These studies can be conducted in animals at the preclinical level, through all stages of clinical trials, and can be data rich or sparse. No matter what the situation, there must be some common means by which to communicate the results of the experiments. Pharmacokinetic parameters serve this purpose. Thus, in the field of pharmacokinetics, the definitions and formulas for the parameters must be agreed upon, and the methods used to calculate them understood. This understanding includes assumptions and domains of validity, for the utility of the parameter values depends upon them. This chapter focuses on the assumptions and domains of validity for the two commonly used methods — noncompartmental and compartmental analysis. Compartmental models have been presented in earlier chapters. This chapter expands upon this, and presents a comparison of the two methods. 

Most of the theoretical details of the material covered in this chapter can be found in Coveil et al. (4), Jacquez and Simon (5), and Jacquez (6). Of particular importance to this chapter is the material covered in Coveil et al. (4) in which the relationships between the calculation of kinetic parameters from statistical moments and the same parameters calculated from the rate constants of a linear, constant-coefficient compartmental model are derived. Jacquez and Simon (5) discuss in detail the mathematical properties of systems that depend upon local mass balance this forms the basis for understanding compartmental models and the simplifications that result from certain assumptions about a system under study. Berman (7) gives examples using metabolic turnover data, while the examples provided in Gibaldi and Perrier (8) and Rowland and Tozer (9) are more familiar to clinical phar maco logists. 

The pharmacokinetic parameters descriptive of the system are as follows (although these definitions apply to both noncompartmental and compartmental models, some modification will be needed for two accessible pool models as well as compartmental models) ... 

The following discussion will describe how AUC and AUMC are estimated, how they are used to estimate specific pharmacokinetic parameters (including the assumptions), and what their relationship is to specific pharmacokinetic parameters estimated from compartmental models. Both moments, however, are used for other purposes. For example, AUC acts as a surrogate for exposure, and values of AUC from different dose levels of a drug have been used to justify assumptions of pharmacokinetic linearity. These uses will not be reviewed. 

Calculating Model Parameters from a Compartmental Model... 

Realizing the full generality of the compartmental model, consider now only the limited situation of linear, constant-coefficient models. What parameters can be calculated from a model The answer to this question can be addressed in the context of Figure 8.5. 

This discussion will rely heavily on the following sources. First, the publications of DiStefano and Landaw (22, 23) deal with issues related to compartmental versus single accessible pool noncompartmental models. Second, Cobelli and Toffolo (3) discuss the two accessible pool noncompartmental model. Finally, Coveil et al. (4) provide the theory to demonstrate the link between noncompartmental and compartmental models in estimating the pharmacokinetic parameters. 

When are the parameter estimates from the noncompartmental model equal to those from a linear, constant-coefficient compartmental model As DiStefano and Landaw (22) explain, they are equal when the equivalent sink and source constraints are valid. The equivalent source constraint means that all... 

The equivalent sink constraint is illustrated in Figure 8.8. In Figure 8.8A, the constraint holds and hence the parameters estimated from either the noncompartmental model (left) or the multicompart-mental model (right) will be equal. If the multi-compartmental model is a model of the system, then, of course, the information about the drug s disposition will be much richer, since many more specific parameters can be estimated to describe each compartment. 

Recovering Pharmacokinetic Parameters from Compartmental Models... 

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