Model mammillary

Two-compartment mammillary model for intravenous administration using Laplace transform... 

Fig. 39.12. (a) Two-compartment mammillary model for single intravenous injection of a doseD. The buffer compartment exchanges with plasma with transfer constants and k p. (b) Time courses of the... 

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.)... 

By way of illustration, we apply the y-method to the two-compartment mammillary model for intravenous administration which we have already seen in Section 39.1.6. The matrix K of transfer constants for this case is defined by means of ... 

LZ Benet. General treatment of linear mammillary models with elimination from any compartment as used in pharmacokinetics. J Pharm Sci 6 536-541, 1972. 

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). 

Traditionally, linear pharmacokinetic analysis has used the n-compartment mammillary model to define drug disposition as a sum of exponentials, with the number of compartments being elucidated by the number of exponential terms. More recently, noncompartmental analysis has eliminated the need for defining the rate constants for these exponential terms (except for the terminal rate constant, Xz, in instances when extrapolation is necessary), allowing the determination of clearance (CL) and volume of distribution at steady-state (Vss) based on geometrically estimated Area Under the Curves (AUCs) and Area Under the Moment Curves (AUMCs). Numerous papers and texts have discussed the values and limitations of each method of analysis, with most concluding the choice of method resides in the richness of the data set. 

Wagner [1] has shown, with IV bolus dosing, that the Vss for a n-compartment open mammillary model with first-order elimination from the central compartment is ... 

A good review of the master equation approach to chemical kinetics has been given by McQuarrie [383]. Jacquez [335] presents the master equation for the general ra-compartment, the catenary, and the mammillary models. That author further develops the equation for the one- and two-compartment models to obtain the expectation and variance of the number of particles in the model. Many others consider the m-compartment case [342,345,384], and Matis [385] gives a complete methodological rule to solve the Kolmogorov equations. 

Figure 1.15 Identifiable and unidentifiable 3-compartment mammillary models.

Figure 3.2-3. Schematic of a two-compartment mammillary model commonly used to characterize macromolecule pharmacokinetics. Drug input and elimination (CL) occur to and from a central compartment (Ap) and may distribute (CLo) to a peripheral site (A V,). Plasma drug concentrations reflect the amount in the central compartment relative the volume of distribution of that site (K)-...

Although there are three possible types of two-compartment model based on the site(s) of elimination (as seen in Fig. 13.6), the most useful and common two-compartment model (called the mammillary model) has drug elimination occurring from the central compartment (Fig. 13.7). 

Pharmacokinetics emerged as a discipline in the 1960s with its foundation in compartmental modeling, although earlier origins of pharmacokinetics can be traced [1], Mammillary compartmental models provided the framework for phar-... 

It has been shown that linear mammillary compartment models can readily be represented by products of input and disposition functions in the Laplace domain [20], Solutions for the drug concentration or amount in any compartment are obtained by taking the inverse of the Laplace function. This approach avoids the use of differential equations and their potentially tedious solution. The La-... 

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... 

In most pharmacokinetic applications, one can assume that the system is open and at least weakly connected. This is the case of mammillary compartmental models, where the compartment n° 1 is referred to as the central compartment and the other compartments are referred to as the distribution compartments, characterized by kio = 0 and kij = 0 for i,j = 2,..., to. For open mammillary compartmental configurations, the eigenvalues of K are distinct, real, and negative, implying that... 

Identifiability is not a common problem in pharmacokinetics since the type of models that are usually dealt with are mammillary compartment models, in which all the peripheral compartments are directly connected to the central compartment and drug elimination occurs only from the central compartment (Model A in Fig. 

Under this single input-single output scheme, all mammillary compartment models are identifiable. If, however, transfer to any peripheral compartment is unidirectional or irreversible, the model will become unidentifiable (Model B in Fig. 1.15). The model will become identifiable if a sample is collected from the compartment having irreversible gain from the central compartment, e.g., Compartment 2 in Model B. If the transfer between compartments is reversible, i.e., material can flow to the peripheral compartment from the central compartment and then back to the central compartment, and if only one peripheral compartment has irreversible loss from the compartment the model is identifiable (Model C in Fig. 1.15). But if two or more peripheral compartments have irreversible loss from them, the model will become unidentifiable (Model D in Fig. 

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