' log10

where ra =

is the sink asymmetry ratio (gradient-pH-induced). When the aqueous solution conditions are identical in the two chambers of the permeation cell (apart from the sample), rfl = rV, and Eq. (7.34) becomes equivalent to Eq. (7.20). This presumes that the system is free of serum proteins or surfactants in the acceptor well. We discuss such assay extensions later. Single Sink: Eq. (7.34) in the Absence of Serum Protein or Sink in Acceptor Wells

In general, Eq. (7.34) has two unknowns: P^A and P^D . In serum protein-free assays, the following method is used to solve Eq. (7.34). At least two assays are done: one as gradient pH (e.g., pH 5.0donor-7.4acceptor) and the other as iso-pH (e.g., pH 7.4donor-7.4acceptor), with one pH common to the two assays. For iso-pH, PeA!D)=p^d!a). This case can be solved directly using Eq. (7.20). Then, iteratively, Eq. (7.34) is solved for P^D!A). Initially ra is assumed to be rV, but with each iteration, the ra estimate is improved by using the calculated P^D!A) and the P^A!D) taken from the iso-pH case. The process continues until self-consistency is reached within the accuracy required.

In iso-pH serum protein- and surfactant-free solutions, the concentration of the sample in the acceptor wells cannot exceed that in the donor wells. With gradient-pH conditions, this limitation is lifted. At very long times, the concentrations in the donor and acceptor chambers reach equilibrium values, depending on the pH gradient or in terms of mole ratios

This limiting ratio can be predicted for any gradient-pH combination, provided the pKa values of the molecule, the unstirred water layer (UWL) Pu, and the intrinsic P0 permeabilities were known [25]. (The topic of the UWL are discussed in greater detail in Section 7.7.6.) In gradient pH assays, it is sometimes observed that nearly all the samples move to the acceptor side, due to the sink conditions created, sometimes limiting the determination of concentrations. Shorter permeation times solve the problem, a welcome prospect in a high-throughput application. A 3-4-h period suffices, which is a considerable reduction over the original 15 h permeation time [547,550]. Shorter times would lead to greater uncertainties in the calculated permeability, since the approximate lag time tLAG can be as long as one hour for the most lipophilic molecules. Double Sink: Eq. (7.34) in the Presence of Serum Protein or Sink in Acceptor Wells

If serum protein or surfactant is added to the acceptor wells, then, in general, pf!D) and P?!A) are not the same, even under iso-pH conditions. The acceptor-to-donor permeability needs to be solved by performing a separate iso-pH assay, where the serum protein or surfactant is added to the donor side, instead of the acceptor side. The value of Pe is determined, using Eq. (7.20), and used in gradient-pH cases in place of p'f^0, as described in the preceding section. The gradient-pH calculation procedure is iterative as well.

Figure 7.17 shows the asymmetry ratios of a series of compounds (acids, bases, and neutrals) determined at iso-pH 7.4, under the influence of sink conditions created not by pH, but by anionic surfactant added to the acceptor wells (discuss later in the chapter). The membrane barrier was constructed from 20% soy lecithin in dodecane. All molecules show an upward dependence on lipophilicity, as estimated by octanol-water apparent partition coefficients, log Kd(74). The bases are extensively cationic at pH 7.4, as well as being lipophilic, and so display the highest responses to the sink condition. They are driven to interact with the surfactant by both hydrophobic and electrostatic forces. The anionic acids are largely indifferent

Anionic drugs

Figure 7.17 Surfactant-induced sink asymmetry ratio versus octanol-water apparent partition coefficient at pH 7.4.

to the presence of the anionic surfactant in the acceptor wells, with a slight suggestion of repulsion in one case (Fig. 7.17).

For ionizable lipophilic molecules, the right pH gradients can drive the solute in the acceptor compartment to the charged (impermeable) form; the uncharged fraction is then further diminished in concentration by binding to the serum protein or surfactant, in the double-sink assay. Simulation Examples

Ketoprofen was selected to illustrate the properties of the gradient-pH permeability equation, Eq. (7.34), in a series of simulation calculations, as shown in Fig. 7.18. The membrane-buffer apparent partition coefficients, Kd(pH), were calculated at various pH values, using the approach described in Section The pH in the acceptor well was pHA 7.4 in all cases, while that in the donor wells was pHD 3-7.4. It is interesting to compare the transport properties of ketoprofen under iso-pH (Fig. 7.16) and gradient pH (Fig. 7.18) conditions. Under gradient pH conditions, at pHD 3, ketoprofen is mostly in an uncharged state in solution. The dashed curve in Fig. 7.18 corresponding to pHD 3 shows a rapid but not extensive decline of the sample in the donor well in the first few minutes; this corresponds to the membrane loading up with the drug, to the extent of only 9%, compared to 56% for iso-pH 3 conditions. The corresponding appearance of the sample in the acceptor well is shown by the solid line corresponding to pHD 3, pHA 7.4. After a short lag period, the acceptor curve starts to rise rapidly, mirroring in shape the donor curve, which decreases with time. The two curves cross at 7 h, whereas in the

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