The Well-Stirred Model

The well-stirred model [eqn (13.5)] is a simplified model of hepatic drug clearance and relates in vivo intrinsic clearance, which is the clearance of the drug experiences at the enzyme level, to blood flow and liver distribution. The in vivo C/int can be estimated from an in vitro C/ nt measurement by scaling the in vitro measurement to the in vivo situation, using estimates of microsomal protein yield per gram of liver and liver weights (for microsomal estimates of C/int) or hepatocyte numbers per gram of liver for in vitro C/int values derived from hepatocytes. See eqn (13.5). 

The well-stirred model therefore allows prediction of in vivo clearance from intrinsic clearance estimates, generated from simple enzyme, sub-cellular or cellular assays. 

This model for predicting hepatic clearance has allowed simple enzyme (e.g. cytochrome P450 enzymes for phase 1 metabolism), sub-cellular (microsomes for phase 1 metabolism) or cellular assays (hepatocytes for phase 1 and 2 metabolism) to be established in drug discovery cascades to enable the identification of low clearance compounds, with the ability to predict the consequent expected in vivo clearance. Results from in vitro assays based on rat or dog microsomes or hepatocytes for a particular compound can be validated with 

These in vitro-in vivo methods have become more sophisticated, for example to allow for non-specific binding effects in the in vitro assay which otherwise may cause an underestimate of clearance.  

The well-stirred model, including corrections for non-specific binding is shown in eqn (13.6). 

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KS Pang, M Rowland. Hepatic clearance of drugs. II. Experimental evidence for acceptance of the well-stirred model over the parallel tube model using lido-caine in the perfused rat liver in situ preparation. J Pharmacokin Biopharm 5/6 655— 699, 1977. 

An alternative approach to relying simply upon allometric approaches for metaboli-cally-cleared compounds is to take into consideration their relative stability in vitro. Clearance by P450 enzymes observed in hepatic microsomes from different species provides a measure of the relative intrinsic clearance in different species. Using the equation for the well-stirred model ... 

Figure 7 Influence of changes in (a) organ blood flow on clearance, (b) fraction of the drug unbound in plasma (/u) on extraction ratio, and (c) intrinsic clearance on extraction ratio as predicted by the well-stirred model of hepatic clearance.

Figure 2 Comparison between the uptake clearance obtained in vivo and that extrapolated from the in vitro transport study of endothelin antagonists. In vivo uptake clearance of endothelin antagonists (BQ-123, BQ-518, BQ-485, compound A) was evaluated by integration plot analysis using the plasma concentration-time profile after intravenous administration (500 nmol/kg) and the amount of drug in the liver and that excreted in the bile. In vitro hepatic uptake clearance was measured using isolated rat hepatocytes and was extrapolated to the in vivo uptake clearance assuming the well-stirred model. Source From Ref. 5.

Mizuma T, Tsuji A, Hayashi M. Does the well-stirred model assess the intestinal first-pass effect well J Pharm Pharmacol 2004 56(12) 1597-1599. 

For a drug that is eliminated exclusively by the liver and that is completely absorbed following oral administration, the intrinsic clearance can be related to the area under the plasma concentration-time curve (AUCp0) if the well-stirred model of hepatic elimination is assumed (81,82) ... 

Panel 5.2 Summary of the predictions of the well-stirred model of hepatic ... 

The well-stirred model, shown in Figure 7.1, is the model of hepatic clearance that is used most commonly in pharmacokinetics. If we apply the Pick equation (see Chapter 6) to this model, hepatic clearance can be defined as follows (2) ... 

In addition to the well-stirred model that is the basis for Equation 7.6, several other kinetic models of hepatic clearance have been developed (4). However, the following discussion will be based on the relationships defined by Equation 7.6, and the limiting cases represented by Equations 7.7 and 7.8. 

A reasonable assumption is that the active secretion mechanism in the kidney can also be described by the well-stirred model. However, the kidneys have several mechanisms that may determine renal clearance of a drug, including passive filtration and reabsorption. 

Thus far, for compound X, we have obtained good results in this context with the simplest of these, the well-stirred model (see Table 8.1 for the equations, with and without significant plasma... 

Table 8.1 Equations for predicting hepatic clearance using the well-stirred model...

Predicted values were scaled from in vitro half-life data using liver microsomes and the well-stirred model of hepatic extraction. Hepatic Cl predictions were corrected for plasma and microsomal protein binding. Predicted total Cl was obtained by adding in renal Cl estimates which were, in turn, scaled allometrically (Y = aW015). 

Figure 9.2 Schematic diagram of three different perfusion methodologies for human use (a) open (b) semiopen and (c) double balloon. For the open and semiopen, the hydrodynamics is best described by the parallel-tube model (see the dotted line for the concentration profile over the intestinal length). The well-stirred model is the best hydrodynamic model for the doubleballoon perfusion technique.

To overcome the shortcomings of the intestinal well-stirred model, Yang et al. proposed the Qgut model [4]. This model maintains the basic equation of the well-stirred model, but expands the flow term into a hybrid of both permeability through the enterocyte membrane defined by CLperm and villous blood flow (Q m) according to Equation 12.8 ... 

CLperm can be derived from in vitro permeability experiments (Caco-2, MDCK cells, and PAMPA) or in silico from polar surface area and the number of hydrogen-bond donors [4]. Yang et al. found a clear superiority of the Qj, i model over the well-stirred model in terms of accuracy and precision, particularly when plasma protein binding was ignored [4]. This simplified the model to (12.9) ... 

The blood clearance can then be obtained through one of the several approaches, for example, the well-stirred model and fraction unbound (/[, obtained from plasma... 

Oxidative Biotransformation in Microsomes The rapid determination of pharmacokinetic parameters, solubility, permeability, and in vitro stability in plasma or liver tissue can often provide a reasonable explanation of the mechanisms limiting oral bioavailability. An approach that is often used is to extrapolate the in vitro rate of metabolism to estimate the hepatic clearance using in vitro-in vivo correlation methods.82-86 These methods use in vitro kinetic parameters, usually Vmllx/Km or in vitro t ji, to determine the intrinsic clearance, which is then scaled to hepatic clearance using the amount of tissue in the in vitro incubation, the weight of the liver, and the well-stirred model for hepatic clearance. 

From the material presented so far, one may intuitively imagine that hepatic drug clearance will be influenced by hepatic blood flow, fraction unbound, and intrinsic clearance that is, the intrinsic ability of the organ to clear unbound drug. The simplest model that describes hepatic clearance in terms of these physiological parameters is the well-stirred model (6). Assuming instantaneous and complete mixing, the well-stirred model states that hepatic clearance (with respect to blood concentration) is... 

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