Pharmacokinetics compartmental analysis

As we have shown above, pharmacokinetic compartmental analysis requires estimates of the transfer constants and the volume of distribution Vp from... 

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. 

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. 

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 contrast to noncompartmental analysis, in compartmental analysis a decision on the number of compartments must be made. For mAbs, the standard compartment model is illustrated in Fig. 3.11. It comprises two compartments, the central and peripheral compartment, with volumes VI and V2, respectively. Both compartments exchange antibody molecules with specific first-order rate constants. The input into (if IV infusion) and elimination from the central compartment are zero-order and first-order processes, respectively. Hence, this disposition model characterizes linear pharmacokinetics. For each compartment a differential equation describing the change in antibody amount per time can be established. For... 

Absorption of a drug into the theoretical central or main compartment may be followed by distribution into one or more peripheral compartments, or the drug may undergo excretion or metabolism from the central compartment. While compartmental analysis of drug distribution can be informative, it is beyond the scope of this book. For more details on the effect of multicompartmental distribution of a drug on pharmacokinetics, see references in the Bibliography. 

Let s look at pharmacokinetics and the work done with compartmental analysis. We should become involved with the biology of clinical trials in a quantitative way. Can we do more than just say the profile in compartments of the body follows a certain pattern Engineers have to start looking into this other side of regulatory activities. 

Where applicable, pharmacokinetic parameters (Cmax, Tmax, AUCo-24, hn) were calculated using a non-compartmental analysis employing a linear/log trapezoidal method. 

Muir KT, Gomeni RO. Non-compartmental analysis. In Pharmacokinetics in Drug Development. Volume 1 Clinical Study Design and Analysis. Bonate PL, Howard DR, eds. 2004. AAPS Press, Arlington, VA. pp. 235-265. 

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. 

The quantitative parameters require not only a mathematical formalism but also data from which to estimate them. As noted, the two most common methods used for pharmacokinetic estimation are noncompartmental and compartmental analysis. A comparison of the two methods has been given by Gillespie (1). Comparisons regarding the two methodologies as applied to metabolic studies have been provided by DiStefano III (2) and Cobelli and Toffolo (3). Coveil et al. (4) have made an extensive theoretical comparison of the two methods. 

Using the definition of pharmacokinetics given in terms of spatial and temporal distributions, one can easily progress to a description of the underlying assumptions and mathematics of noncompartmental and compartmental analysis, and, from there, proceed to the processes involved in estimating the pharmacokinetic parameters. This will permit a better understanding of the domain of validity of noncompartmental vs compartmental parameter estimation. 

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. 

Kiwada, H. Morita, K. Hayashi, M. Awazu, S. Hanano, M. A new numerical calculation method for deconvolution in linear compartmental analysis of pharmacokinetics. Chem. Pharm. Bull. 1977, 25, 1312-1318. 

The purpose of this article is to introduce the reader to simple basic concepts and principles of pharmacokinetic/toxicokinetic analysis using both types of models - compartmental and physiologically based. 

Gompartmental models have been the foundation of pharmacokinetic data analysis. The compartmental approach is presented here to facilitate definition of important pharmacokinetic parameters. Gompartmental models are determin-... 

Pharmacokinetics After Oral and Intravenous Administration. For proper characterization of an inhalation drug, information on the systemic pharmacokinetic properties needs to be provided. One of the major challenges for such studies is to provide a suitable formulation for injection, especially because new drug candidates are often very lipophilic. The resulting parameters of such studies (systemic clearance, volume of distribution, half-life, mean residence time) can then easily be extracted from concentration-time profiles after IV administration and subsequent standard pharmacokinetic analysis by noncompartmental approaches. In addition, a detailed compartmental analysis based on concentration-time profiles will be useful in evaluating the systemic distribution processes in sufficient detail. This will be especially important if deconvolution procedures (see later) are included for the assessment of the pulmonary absorption profiles. 

FIGURE 3.6 Compartmental analysis for different terms of volume of distribution. (Adapted from Kwon, Y., Handbook of Essential Pharmacokinetics, Pharmacodynamics and Drug Metabolism for Industrial Scientists, Kluwer Academic/Plenum Publishers, New York, 2001. With permission.) (a) Schematic diagram of two-compartment model for compound disposition. Compound is administrated and eliminated from central compartment (compartment 1) and distributes between central compartment and peripheral compartment (compartment 2). Vj and V2 are the apparent volumes of the central and peripheral compartments, respectively. kI0 is the elimination rate constant, and k12 and k21 are the intercompartmental distribution rate constants, (b) Concentration versus time profiles of plasma (—) and peripheral tissue (—) for two-compartmental disposition after IV bolus injection. C0 is the extrapolated concentration at time zero, used for estimation of V, The time of distributional equilibrium is fss. Ydss is a volume distribution value at fss only. Vj, is the volume of distribution value at and after postdistribution equilibrium, which is influenced by relative rates of distribution and elimination, (c) Time-dependent volume of distribution for the corresponding two-compart-mental disposition. Vt is the starting distribution space and has the smallest value. Volume of distribution increases to Vdss at t,s. Volume of distribution further increases with time to Vp at and after postdistribution equilibrium. Vp is influenced by relative rates of distribution and elimination and is not a pure term for volume of distribution. 

Liang, E. and Derendorf, H. Pitfalls in pharmacokinetic multi-compartmental analysis. Journal of Pharmacokinetics and Biopharmaceutics 1998 26 247-260. 

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. 

In recent years, non-compartmental or model-independent approaches to pharmacokinetic data analysis have been increasingly utilized since this approach permits the analysis of data without the use of a specific compartment model. Consequently, sophisticated, and often complex, computational methods are not required. The statistical or non-compartmental concept was first reported by Yamaoka in a general manner and by Cutler with specific application to mean absorption time. Riegelman and Collier reviewed and clarified these concepts and applied statistical moment theory to the evaluation of in vivo absorption time. This concept has many additional significant applications in pharmacokinetic calculations. 

In many cases pharmacokinetic data (i.e. plasma drug concentration versus time data) cannot be fitted to an explicit equation equivalent to a system containing a discrete number of compartments into which dmg distributes. This data analysis requires some form of non-compartmental analysis (also referred to as model-independent analysis.) This is achieved by the use of statistical moment theory. 

Deleu D, Sarre S, Michotte Y, Ebinger G. Simultaneous in vivo microdialysis in plasma and skeletal muscle a study of the pharmacokinetic properties of levodopa by non-compartmental analysis. J Pharm Sd 1994 83 25-8. 

Flooker, A.C., Foracchia, M., Dodds, M.G., and Vicini, P. 2003. An evaluation of population D-optimal designs via pharmacokinetic simulations. Ann. Biomed. Eng. 31 98-111. lacquez,l.A. 1996. Compartmental Analysis in Biology and Medicine. 3rded.,Biomedware,AnnArbor,MI. lacquez, l.A. and Simon, C.P. 1993. QuaUtative theory of compartmental systems. Siam. Rev., 35 43-79. Landaw, E.M. and DiStefano III, LI. 1984. Multiexponential, multicompartmental, and noncompartmental modeling. II. Data analysis and statistical considerations. Am. J. Physiol 246 R665-R677. 

A basic assumption related to both methods of analysis is that the elimination of drug from the body is exclusively from the sampling compartment (i. e., blood/ plasma), and that rate constants are first order. However, when some or all of the elimination occurs outside the sampling compartment - that is, in the peripheral or tissue compartment(s) - these types of analysis are prone to error in the estimation of Vss, but not CL. In compartmental modeling, the error is related to the fact that no longer do the exponents accurately reflect the inter-compartmental and elimination (micro) rate constants. This model mis specification will result in an error that is related to the relative magnitudes of the distribution rate constants and the peripheral elimination rate constant. However, less widely understood is the fact that this model mis specification will also result in errors in noncompartmental pharmacokinetic analysis. 

The primary analysis examined pharmacokinetic parameters calculated from plasma concentrations of CYS-conjugated XYZ1234 using non-compartmental techniques. The secondary analysis examined the pharmacokinetic parameters of unconjugated XYZ1234. 

Noncomp artmental versus Compartmental Approaches to Pharmacokinetic Analysis... 

The use of the mean residence time matrix can be a powerful tool in pharmacokinetic analysis with a compartmental model, especially if one is dealing with a model of the system in which physiological and/or anatomical correlates are being assigned to specific compartments (2). Modeling software tools such as SAAM II (21) automatically calculate the mean residence time matrix from the compartmental matrix, making the information easily available. 

When the drug is administered as an intravenous bolus dose, the total areas under the curves can be calculated from the coefficients and exponents of the equation describing the disposition curve and obtained by compartmental pharmacokinetic analysis of the plasma concentration-time data. 

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