## Pp

where Pe refers to the measured effective permeability, Pu refers to the total UWL permeability, Pm is the permeability of the membrane (which would be measured if the UWL were made vanishingly thin). If it is possible to separate the donor and acceptor contributions to the UWL, then the total Pu can be allocated between its parts according to 1/Pu = 1/pUD + VP^. In Caco-2 literature, equations like Eq. (7.46) often have a fitter, Pf, component, to account for resistance of the water-filled pores of the fitter. In PAMPA, all pores are filled with lipid, and no consideration of filter contributions are needed.

The UWL permeability is nearly the same for drugs of comparable size, and is characterized by the water diffusivity (Daq) of the drug divided by twice the thickness of the layer (haq), Pu = Daq / (2 haq), in a symmetric permeation cell [40]. The unstirred water layer permeability can be determined experimentally in a number of ways: based on pH dependency of effective permeability [25,509,535-538], stirring rate dependence [511-514,552,578], and transport across lipid-free microfilters

7.7.6.2 Determination of UWL Permeability using pH Dependence (pf) Method

The membrane permeabilities Pm may be converted to intrinsic permeabilities P0, when the pKa is taken into consideration. An ionizable molecule exhibits its intrinsic permeability when it is in its uncharged form and there is no water layer resistance. The relationship between Pm and P0 is like that between the pH-dependent apparent partition coefficient (log Kd) and the true partition coefficient (log Kp), respectively. This relationship can be rationalized by the mass balance. Take, for example, the case of a monoprotic acid, HA. The total substance concentration is

Using the ionization quotient expression [Eq. (3.1)], [A ] may be expressed in terms of [HA]:

In the UWL, HA and A diffuse in parallel; the total UWL flux, Ju, is the sum of the two individual flux components. If it is assumed that the transport is under steady state and that the aqueous diffusivities of HA and A are the same, the UWL flux becomes

where ACha represents the drop in total concentration across the entire trilamellar barrier. If the pH partition hypothesis holds, then the flux in the membrane is related to the concentration gradient of the uncharged solute

where A [HA] represents the drop in concentration of the uncharged species in the membrane. Since the membrane and the UWL are in series, the total flux J may be expressed as

PeACHA ^ J

Multiplying this expression by the total sample concentration change, we obtain

Equating Eqs. (7.52) and (7.46) reveals the relationship between intrinsic and membrane permeabilities, Eq. (7.53), for the case of weak acids. Similar steps lead to expressions for weak bases and ampholytes, Eqs. (7.54) and (7.55):

P0 — Pm(1 + 10pKa1-pH + 10 pKa2+pH) (ampholyte) (7.55)

For ionizable molecules, the intrinsic P0 and the UWL Pu can be deduced from the pH dependence of Pe, as shown by Gutknecht and co-workers [535-537].

As can be seen from the second line of Eq. (7.52), a plot of 1/Pe versus 1/[H+] is expected to be linear (for a weak acid), with the intercept: 1/Pu + 1/P0 and the slope Ka/P0. When the pKa of the molecule is known, then both P0 and Pu can be determined. If Pu can be independently determined, then, in principle, the ionization constant may be determined from the pH dependence of the effective permeability.

Figure 7.34 shows the pH dependence of the effective permeability of ketopro-fen (measured using pION's PAMPA system with 2% DOPC in dodecane membrane lipid) [558], a weak acid with pKa 4.12 (0.01 M ionic strength, 25°C). Figure 7.34a shows that the log Pe curve has a flat region for pH < pKa and a region with a slope of —1 for pH > pKa. At pH 7.4, ketoprofen has a very low permeability, since it is almost entirely in a charged form. The molecule shows increasing permeabilities with decreasing pH, approaching 18 x 10 6 cm/s (thick curve, Fig. 7.34b inset). This is close to the value of the UWL permeability, 21 x 10 6 cm/s (log Pe — 4.68). The small difference vanishes for very lipophilic molecules, such as imipramine. For lipophilic acids, when pH < pKa, the transport is said to be ''diffusion-limited.'' For pH > pKa, the Pe curve coincides with the Pm curve, where transport is ''membrane-limited.'' In general, highly permeable molecules all show nearly the same maximum effective permeability when measured in the same apparatus. In order to deduce the uncharged molecule membrane permeability (top of the dashed curve in Fig. 7.34a), it is necessary to analyze the Pe-pH curve by the Gutknecht method [535-537]; thus, Eq. (7.52) is solved for Pu and P0, when pKa is known. Such analysis produces the dashed curve in Figs. 7.34a,b.

The Pm curve (dashed line) is not shifted to the right of the ''fraction neutral substance'' curve fu, (see inset in Fig. 7.34b). It just looks that way when unmatched scaling is used [554]. The two curves are exactly superimposed when the vertical coordinates of the Pm and fu are normalized to a common value. The Pe curve, in contrast, is shifted to the right for weak acids and to the left for weak bases. In the log-log plot (log Pe vs. pH), the pH value at the intersection of the slope 0 and slope —1 curve segments indicates an apparent pKa (Fig. 7.34a).

We have seen many instances of slope-(0, ±1) log-log plots (e.g., Figs. 2.2, 4.24.4, 4.6, 5.7, 5.11, 6.1-6.4, 6.12). Behind each tetrad equilibrium (e.g., Figs. 4.1, 5.1, 6.5) there is such a log-log plot, and associated with each such log-log plot is an apparent pKa. We have called these pK^, pKa°em, pKgibbs. In permeability, there is yet another one: pK^ (Fig. 7.34a). If we take the difference between pKa and pK^™, we can deduce the difference between log P0 and log Pu:

The shapes of permeability-pH profiles mirror those of solubility-pH (see, Figs. 6.1a, 6.2a, and 6.3a), with slopes of opposite signs. In solutions saturated with an insoluble compound, the product of solubility and permeability ("flux," as described in Chapter 2) is pH-independent! This is indicated in Fig. 2.2 as the maximum flux portions of the curves.

Figure 7.34 Permeability-pH profiles of ketoprofen: (a) log-log plot; solid curve represents effective permeability, and the dashed curve is the membrane permeability, calculated by Eq. (7.53). The latter curve levels off at the intrinsic permeability, P0. The effective curve levels off to approximately the unstirred water layer permeability, Pu. (b) Direct plot; the inset curve for the fraction neutral substance levels of at 100% (scale not shown). [Avdeef, A., Curr Topics Med. Chem., 1, 277-351 (2001). Reproduced with permission from Bentham Science Publishers, Ltd.]

Figure 7.34 Permeability-pH profiles of ketoprofen: (a) log-log plot; solid curve represents effective permeability, and the dashed curve is the membrane permeability, calculated by Eq. (7.53). The latter curve levels off at the intrinsic permeability, P0. The effective curve levels off to approximately the unstirred water layer permeability, Pu. (b) Direct plot; the inset curve for the fraction neutral substance levels of at 100% (scale not shown). [Avdeef, A., Curr Topics Med. Chem., 1, 277-351 (2001). Reproduced with permission from Bentham Science Publishers, Ltd.]

Figure 7.35 shows the characteristic log Pe-pH curve for a weak base, phenazo-pyridine (pKa 5.15). With bases, the maximum permeability is realized at high pH values. As in Fig. 7.34, the PAMPA assays were performed under iso-pH conditions (same pH in donor and acceptor wells), using the 2% DOPC in dodecane lipid system.

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