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Figure 7.18 Relative concentrations of accetor and donor compartments as a function of time for the gradient-pH ketoprofen model.

iso-pH case, 16 h shows only near meeting. Also, the gradient pH curves cross slightly below the 0.5 concentration ratio, since membrane retention is only 9%.

### 7.5.3.4 Gradient pH Summary

The benefits of an assay designed under gradient pH conditions are (1) less retention and thus more analytical sensitivity, (2) shorter permeation times and thus higher throughput possible, and (3) more realistic modeling of the in vivo pH gradients found in the intestinal tract and thus better modeling. Time savings with increased sensitivity are important additions to an assay designed for high-throughput applications. A double-sink condition created by the combination of a pH gradient and serum protein (or an appropriate surfactant) in the acceptor compartment is an important component of the biophysical GIT transport model. In contrast, a no-sink condition may be more suitable for a BBB transport model. This is discussed in greater detail later.

7.6 PERMEABILITY-LIPOPHILICITY RELATIONS 7.6.1 Nonlinearity

In the introductory discussion in Chapter 2, it was indicated that the effective permeability Pe linearly depends on the apparent membrane-water partition

log Kd log Kd

log Kd log Kd

Figure 7.19 Permeability-lipophilicity relations: (a) linear; (b) hyperbolic; (c) sigmoidal; (d) bilinear.

coefficient, Kd [Eq. (2.3)]. The simple model system considered there assumed the membrane barrier to be a structureless homogeneous oil, free of aqueous pores, and also assumed the aqueous solutions on both sides of the barrier to be well mixed by convection, free of the UWL (Section 7.7.6) effect. A log Pe/log Kd plot would be a straight line. Real membrane barriers are, of course, much more complicated. Studies of permeabilities of various artificial membranes and culture-cell monolayers indicate a variety of permeability-lipophilicity relations (Fig. 7.19). These relationships have been the subject of two reviews [49,54]. Figure 7.19 shows linear [579], hyperbolic [580-582], sigmoidal [552,583,584], and bilinear [23,581,585,586] permeability-lipophilicity relations.

Early efforts to explain the nonlinearity were based on drug distribution (equilibrium) or transport (kinetic) in multicompartment systems [21,22]. In this regard, the 1979 review by Kubinyi is highly recommended reading [23]. He analyzed the transport problem using both kinetic and equilibrium models. Let us consider the simple three-compartment equilibrium model first. Imagine an organism reduced to just three phases: water (compartment 1), lipid (compartment 2), and receptor (compartment 3). The corresponding volumes are v1, v2, and v3, respectively, and vj ^ v2 ^ v3. If all of the drug is added to the aqueous phase at time 0, concentration C1(0), then at equilibrium, the mass balance (see Section 7.5) would be v1C1(0) = v1C1(i) + v2C2(i) + v3C3(i). Two partition coefficients need to be defined: Kp2 = C2(i)/C1(i) and Kp3 = C3(i)/C1(i). With these, the mass balance may be rewritten as v1 C1(0) = v1C1 (1) + v2Kp2C1(i) + v3Kp3C1 (1) = C1 (i)(v1 + v2Kp2 + v3Kp3). If the organic : aqueous volume ratios are r2 and r3, then the equilibrium concentrations in the three phases can be stated as

Further reduction is possible. To a good approximation, partition coefficients from different organic solvents may be interrelated by the so-called Collander equation [364,587]: logKp3 = a log Kp2 + c, or Kp3 = 10cKp2, where a and c are constants. Equations (7.38)-(7.40) can be expressed in log forms as a function of just one partition coefficient (i.e., Kp = Kp2):

Lipid: log = log Kp - log(i + T2Kp + T3i0cK;) (7.42)

Receptor: log ^^ = a log Kp - log(i + T2Kp + T3i0cKap )+c (7.43)

Figure 7.20 is a sample plot of relative equilibrium concentrations, Eqs. (7.41)-(7.43). In the example, the three phases were picked to be water, octanol, and phos-phatidylcholine-based liposomes (vesicles consisting of a phospholipid bilayer), with the volumes vi = 1 mL (water), v2 = 50 p.L (octanol), and v3 = 10 p.L (liposomes). The Collander equation was deduced from Fig. 5.6: log Kp,liposome = 0.41 log Kp oct + 2.04. Figure 7.20 suggests that when very hydrophilic molecules (with log Kp, oct < 6) are placed into this three-phase mixture, most of them distribute into the water phase (solid curve), with only minor liposome phase occupation (dashed-dotted curve), but virtually no octanol phase occupation (dashed curve). In the example, molecules with log Kp oct of —4 to +3, mostly reside in the liposome fraction, schematically modeling the lipophilic property of a hypothetical receptor site, reaching maximum occupancy for compounds with log Kp,oct at about + 1.5. Very lipophilic molecules, with log Kp oct > 5 preferentially concentrate in the (more lipophilic) octanol compartment, becoming unavailable to the receptor region.

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