Pharmacokinetic models, classical

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... 

Summary of Physiologically Based and Classical Pharmacokinetic Models. 

While these models simulate the transfer of lead between many of the same physiological compartments, they use different methodologies to quantify lead exposure as well as the kinetics of lead transfer among the compartments. As described earlier, in contrast to PBPK models, classical pharmacokinetic models are calibrated to experimental data using transfer coefficients that may not have any physiological correlates. Examples of lead models that use PBPK and classical pharmacokinetic approaches are discussed in the following section, with a focus on the basis for model parameters, including age-specific blood flow rates and volumes for multiple body compartments, kinetic rate constants, tissue dosimetry,... 

As with classic compartment pharmacokinetic models, PBPK models can be used to simulate drug plasma concentration versus time profiles. However, PBPK models differ from classic PK models in that they include separate compartments for tissues involved in absorption, distribution, metabolism and elimination connected by physiologically based descriptions of blood flow (Figure 10.1). 

Emond, C, J.E. Michalek, L.S. Birnbaum, and M.J. DeVito. 2005b. Comparison of the use of a physiologically based pharmacokinetic model and a classical pharmacokinetic model for dioxin exposure assessment. Environ. Health Perspect. 113(12) 1666-1668. 

Due to its compartmental nature, the CAT model can be easily coupled with the disposition of drug in the body using classical pharmacokinetic modeling. In this respect the CAT model has been used to interpret the saturable small-intestinal absorption of cefatrizine in humans [175]. 

The final area of computational toxicology to be discussed in this chapter applied to dermal absorption is the general area of pharmacokinetic models (see also Chapter 3). These are usually developed as extensions of whole animal-based models using classic compartmental [19,31] or physiological-based [32] approaches. These texts should be consulted for an overview of the... 

Attempts were made to fit the in vivo serum concentration results to pharmacokinetic models based on revised diffusion-model equations for the uncoated and coated valve-rim Inserts. To couple the in vitro diffusion equations to an in vivo simulation, a classical one-compartment model of drug distribution was employed. The mean apparent volume of distribution, = 5.63 liters, and the first order eliminating constant, K = 0.695/hr, for gentamicin were determined in the intravenous fnjection-clearance studies. 

There is at least one major area of activity pertaining directly to the environment for which the reader will seek in vain. The complexity of environmental problems and the availability of personal computers have led to extensive studies on models of varying sophistication. A discussion and evaluation of these lie well beyond the competence of an old-fashioned experimentalist this gap is left for others to fill but attention is drawn to a review that covers recent developments in the application of models to the risk assessment of xenobiotics (Barnthouse 1992), a book (Mackay 1991) that is devoted to the problem of partition in terms of fugacity — a useful term taken over from classical thermodynamics — and a chapter in the book by Schwarzenbach et al. (1993). Some superficial comments are, however, presented in Section 3.5.5 in an attempt to provide an overview of the dissemination of xenobiotics in natural ecosystems. It should also be noted that pharmacokinetic models have a valuable place in assessing the dynamics of uptake and elimination of xenobiotics in biota, and a single example (Clark et al. 1987) is noted parenthetically in another context in Section 3.1.1. In similar vein, statistical procedures for assessing community effects are only superficially noted in Section 7.4. Examples of the application of cluster analysis to analyze bacterial populations of interest in the bioremediation of contaminated sites are given in Section 8.2.6.2. 

Lumped parameter models for tissues are also referred to as compart-mental or pharmacokinetic models. These models can be divided into two categories—classical and physiological (or whole-body) compartmental analyses. In the classical approach, the body or a region of body is represented by one or more compartments, without relating the content or inputs or outputs associated with these compartments to available physiological and physicochemical information. These models are useful when it is sufficient to know and to be able to control the blood (plasma) concentration, and the details of any distribution in various tissues are not necessary (for details on classical pharmacokinetics, see Wagner, 1971). 

Classical pharmacokinetic models of systemicaUy administered drugs (see Chapter 1) do not fuUy apply to many ophthalmic drugs. Most ophthalmic medications are formulated to be apphed topically or may be injected by subconjunctival, sub-Tenon s, and retrobulbar routes (Figure 63-1 and Table 63-1). Although similar principles of absorption, distribution, metabolism, and excretion determine drug disposition in the eye, these alternative routes of drug administration introduce other variables in compartmental analysis. 

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. 

Classical pharmacokinetic modeling involves the representation of the subject s body as a collection of compartments. - Drug concentrations within each compartment are considered to be in rapid equilibrium. That is, they are not necessarily equal concentrations throughout the compartment but are in rapid equilibrium. Rapid is defined empirically. Compartments are defined empirically. Rapid means that the data collected do not support more compartments. The investigator starts with the minimum number of compartments for the entities studied and the route of administration. Additional compartments are included only if the data suggest that they are necessary. 

There are a number of assumptions with each approach. The assumptions, and their justifications, for classical pharmacokinetic modeling include ... 

FIGURE 13.1 Classical pharmacokinetic modeling of a two-compartment open model. The symbols C and P represent the central and peripheral compartments, whereas K represents the first-order transfer constants between compartments. The subscripts, a and e, represent absorption and elimination, whereas 12 and 21 represent transfer of drug between compartments 1 and 2 (central to peripheral) or 2 and 1 (peripheral to central). 

Whereas classical pharmacokinetic models utilize a relatively small number of compartments (see Figure. 6.10), PBPK models seek to mimic physiological pathways and processes controlling the time-course of plasma and tissue concentrations and represent the state-of-the-art in advanced pharmacokinetic systems analysis. Basic concepts of PBPK modeling and a hybrid application to composite nanodevice pharmacokinetics are described in this section. 

Pharmacokinetics is closely related to pharmacodynamics, which is a recent development of great importance to the design of medicines. The former attempts to model and predict the amount of substance that can be expected at the target site at a certain time after administration. The latter studies the relationship between the amount delivered and the observable effect that follows. In some cases the observable effect can be related directly to the amount of drug delivered at the target site [2]. In many cases, however, this relationship is highly complex and requires extensive modeling and calculation. In this text we will mainly focus on the subject of pharmacokinetics which can be approached from two sides. The first approach is the classical one and is based on so-called compartmental models. It requires certain assumptions which will be explained later on. The second one is non-compartmental and avoids the assumptions of compartmental analysis. 

Two classical methods used in the analysis of pharmacokinetic data are the fitting of sums of exponential functions (2- and 3-compartment mammillary models) to plasma and/or tissue data, and less frequently, the fitting of arbitrary polynomial functions to the data (noncompartmental analysis). 

As a possible alternative to in vitro metabolism studies, QSAR and molecular modelling may play an increasing role. Quantitative stracture-pharmacokinetic relationships (QSPR) have been studied for nearly three decades [42,45-52]. These are often based on classical QSAR approaches based on multiple linear regression. In its most simple form, the relationship between PK properties and lipophilicity has been discussed by various workers in the field [36, 49, 50]. 

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. 

The classical compartmental and more complex PBPK models require actual pharmacokinetic data to calibrate some parameters such as metabolic rate constants. However, PBPK models are more data-intensive and require greater numbers of chemical-specific and receptor-specific inputs. Although PBPK models have been used extensively in the last 20 years to address cross-species differences and other uncertainties, there are cases in which simpler one- or two-compartment models have been sufficient for risk assessment, for example for methyl mercury (EPA 2001). 

Fig. 1.3 Pharmacokinetic/pharmacodynamic (PK/PD) modeling as combination of the classic pharmacological disciplines pharmacokinetics and pharmacodynamics (from [5]).

The classical three-compartment model describes pharmacokinetics of 5-HT1A receptor agonists. By means of a sigmoidal function E (c), the 5-HT1A agonist concentration c (t) influences the set-point signal that dynamically interacts with the body temperature. By using x (t) and y (t) as dimensionless state variables for the set-point and temperature, respectively, the model is expressed by the set of two nonlinear differential equations ... 

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