Compartmental model, discussion

How does one resolve the difficulty associated with partial differential equations The most common way is to reduce the system into a finite number of components. This can be accomplished by lumping together processes based upon time or location, or a combination of the two. One thus moves from partial derivatives to ordinary derivatives, where space is not taken directly into account. This reduction in complexity results in the compartmental models discussed later in this chapter. The same lumping process also forms the basis for the noncompartmental models discussed in the next section, although the reduction is much simpler than for compartmental models. 

Stirred tank models have been widely used in pharmaceutical research. They form the basis of the compartmental models of traditional and physiological pharmacokinetics and have also been used to describe drug bioconversion in the liver [1,2], drug absorption from the gastrointestinal tract [3], and the production of recombinant proteins in continuous flow fermenters [4], In this book, a more detailed development of stirred tank models can be found in Chapter 3, in which pharmacokinetic models are discussed by Dr. James Gallo. The conceptual and mathematical simplicity of stirred tank models ensures their continued use in pharmacokinetics and in other systems of pharmaceutical interest in which spatially uniform concentrations exist or can be assumed. 

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. 

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. 

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. 

The simplest model is the one compartmental model, in which the compartment represents the circulatory system and all the tissues perfused by the drug (Figure 8.3). Other types of model are in use but unless stated otherwise all discussions in this text will be based on a one compartment model. 

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 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. 

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. 

The present drug of choice for this purpose is cis-platin [cis-diamminedichloroplatinum (II)], or its analog carboplatin. As will be discussed in Chapter 30, early compartmental models predicted a substantial pharmacokinetic advantage of intraperitoneal over intravenous delivery (8). A later Phase III trial (7) confirmed that a comparative survival advantage... 

The study of pharmacokinetics may be pursued at a number of levels. Considering the detail of the mathematical models involved one may use a noncompartmental, classical, or physiologically based approach. All three approaches require some measure of mathematical description or assumptions with the classical, compartmental approach intermediate in complexity. This chapter describes the development and use of compartmental models in pharmacokinetic research. Noncompartmental and physiologically based pharmacokinetic models are discussed in Chapters 13 and 14, respectively. 

We began modeling under the assumption that the introduction of the tracer (mass) into the system did not affect the mechanisms present for metabolism of the tracee. The compartmental model was compatible with the assumption that non-steady-state mechanisms for metabolism of /3-carotene were not induced by the tracer because the model prediction of the tracer state, the tracee state, and the steady state could be achieved using the same set of fractional transfer coefficients (FTCs). The appropriateness of this assumption is discussed again under Statistical Considerations. FTC is the fraction of analyte in a donor compartment that is transferred to a recipient compartment per unit of time, in this case per day. 

Questions regarding the extent of postabsorptive bioconversion of /3C to vitamin A persist. Animal data indicate the liver possesses this capability, but the relative importance of intestinal mucosa versus liver is unknown. Novotny and co-workers (1995) reported a compartmental model which predicted that both liver and intestinal mucosa were important sites for biotransformation of /SC in the human, with 43% of total conversion occurring in the liver and 56% in the intestinal mucosa. However, the model assumed a stoichiometry of 1 mol retinol per mole /SC, and the effect on the model assuming a 2 1 ratio was not discussed. 

An interesting application of linear compartmental models at the organ level is in describing the exchange of materials between blood, interstitial fluid, and cell of a tissue from multiple tracer indicator dilution data. Compartmental models provide a finite difference approximation in the space dimension of a system described by partial differential equations, which may be easier resolvable from the data. These models are discussed in facquez [1996] and an example of a model describing glucose transport and metabolism in the human skeletal muscle can be found in Saccomani et al. [1996]. 

As stated, a number of PBPK/PD models have been developed for individual nerve agents (sarin, VX, soman, and cyclosarin) in multiple species. Chapter 58 in the current volume discusses tiie development of such models. Standalone PBPK or compartmental models have also been developed that describe the pharmacokinetics of certain countermeasures, such as diazepam (Igari et al., 1983 Gueorguieva et al., 2004) and oximes (Stemler et al., 1990 Sterner et al., 2013). However, to date, few models for specific countermeasures have been harmonized and linked to NA PBPK/PD models to be able to quantitatively describe their pharmacokinetic and pharmacodynamic interactions. This is partly due to the fact that most PBPK/ PD models developed for NAs and other OPs focus on the inhibition of ChEs as the critical endpoint. The lack of a mathematical description of the disruption of other complex biochemical pathways presents a problem for linking these NA models to those of many countermeasures. For example, the conventional NA countermeasures, atropine and diazepam, as well as many novel countermeasures, do not directly impact ChE kinetics because they act at sites distinct from the active site of the esterases, such as muscarinic, GABA, or NMDARs (Figure 69.2). 

In what follows, both macromixing and micromixing models will be introduced and a compartmental mixing model, the segregated feed model (SFM), will be discussed in detail. It will be used in Chapter 8 to model the influence of the hydrodynamics on a meso- and microscale on continuous and semibatch precipitation where using CFD, diffusive and convective mixing parameters in the reactor are determined. 

Fig. 10. Mechanisms of steady-slqte kinetics of sugar phosphorylation catalyzed by E-IIs in a non-compartmentalized system. (A) The R. sphaeroides 11 model. The model is based on the kinetic data discussed in the text. Only one kinetic route leads to phosphorylation of fructose. (B) The E. coli ll " model. The model in Fig. 8 was translated into a kinetic scheme that would describe mannitol phosphorylation catalyzed by Il solubilized in detergent. Two kinetic routes lead to phosphorylation of mannitol. Mannitol can bind either to state EPcy, or EPpe,. E represents the complex of SF (soluble factor) and 11 and II in A and B, respectively. EP represents the phosphorylated states of the E-IIs. Subscripts cyt and per denote the orientation of the sugar binding site to the cytoplasm and periplasm, respectively. PEP, phosphoenolpyruvate.

Usually, the buffer compartment is not accessible and, consequently, the absolute amount of X cannot be determined experimentally. For this reason, we will only focus our discussion on the plasma concentration Cp. It is important to know, however, that the time course of the contents in the two compartments is the sum of two exponentials, which have the same positive hybrid transfer constants a and p. The coefficients A and B, however, depend on the particular compartment. This statement can be generalized to mammillary systems with a large number of compartments that exchange with a central compartment. The solutions for each of n compartments in a mammillary model are sums of n exponential functions, having the same n positive hybrid transfer constants, but with n different coefficients for each particular compartment. (We will return to this property of linear compartmental systems during the discussion of multi-compartment models in Section 39.1.7.)... 

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. 

An ecosystem can be thought of as a representative segment or model of the environment in which one is interested. Three such model ecosystems will be discussed (Figures 1 and 2). A terrestrial model, a model pond, and a model ecosystem, which combines the first two models, are described in terms of equilibrium schemes and compartmental parameters. The selection of a particular model will depend on the questions asked regarding the chemical. For example, if one is interested in the partitioning behavior of a soil-applied pesticide the terrestrial model would be employed. The model pond would be selected for aquatic partitioning questions and the model ecosystem would be employed if overall environmental distribution is considered. 

The discussion above provides a brief qualitative introduction to the transport and fate of chemicals in the environment. The goal of most fate chemists and engineers is to translate this qualitative picture into a conceptual model and ultimately into a quantitative description that can be used to predict or reconstruct the fate of a chemical in the environment (Figure 27.1). This quantitative description usually takes the form of a mass balance model. The idea is to compartmentalize the environment into defined units (control volumes) and to write a mathematical expression for the mass balance within the compartment. As with pharmacokinetic models, transfer between compartments can be included as the complexity of the model increases. There is a great deal of subjectivity to assembling a mass balance model. However, each decision to include or exclude a process or compartment is based on one or more assumptions—most of which can be tested at some level. Over time the applicability of various assumptions for particular chemicals and environmental conditions become known and model standardization becomes possible. 

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. 

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