## Info

ITrv

' log1C

log10

where the aqueous compartment volume ratio, rv = Vd/Va. Often, rv = 1. From analytical considerations, Eq. (7.13) is better to use than (7.12) when only a small amount of the compound reaches the acceptor wells; analytical errors in the calculated Pa, based on Eq. (7.13), tend to be lower.

Palm et al. [578] derived a two-way flux equation which is equivalent to Eq. (7.13), and applied it to the permeability assessment of alfentanil and cimeti-dine, two drugs that may be transported by passive diffusion, in part, as charged species. We will discuss this apparent violation of the pH partition hypothesis (Section 7.7.7.1).

### 7.5.2 Iso-pH Equations with Membrane Retention

The popular permeability equations [(7.10) and (7.11)] derived in the preceding section presume that the solute does not distribute into the membrane to any appreciable extent. This assumption may not be valid in drug discovery research, since most of the compounds synthesized by combinatorial methods are very lipophilic and can substantially accumulate in the membrane. Neglecting this leads to underestimates of permeability coefficients. This section expands the equations to include membrane retention.

7.5.2.1 Without Precipitate in Donor Wells and without Sink Condition in Acceptor Wells

When membrane retention of the solute needs to be considered, one can derive the appropriate permeability equations along the lines described in the preceding section: Eqs. (7.1)-(7.3) apply (with P designated as the effective permeability, Pe). However, the mass balance would need to include the membrane compartment, in addition to the donor and acceptor compartments. At time t, the sample distributes (mol amounts) between three compartments:

The partition coefficient is needed to determine the moles lost to the membrane, VM CM (t). If ionizable compounds are considered, then one must decide on the types of partition coefficient to use -Kp (true pH-independent partition coefficient) or Kd (pH-dependent apparent partition coefficient). If the permeability assay is based on the measurement of the total concentrations, CD(t) and CA(t), summed over all charge-state forms of the molecule, and only the uncharged molecules transport across the membrane to an appreciable extent, it is necessary to consider the apparent partition (distribution) coefficient, Kd, in order to explain the pH dependence of permeability.

The apparent membrane-buffer partition (distribution) coefficient Kd, defined at t = i, is

since at equilibrium, CD (1) = CA (1), in the absence of a pH gradient and other sink conditions. At equilibrium (t = 1), the mole balance equation [Eq. (7.14)] can be expanded to factor in the partition coefficient, Eq. (7.15):

VdCd(0) = VdCd(I) + VaCa (1) + Vm^Cd(I) = VdCd (1) + VaCd (1) + VMKdCD (1) = Cd (1)(Vd + Va + VMKd ) (7.16)

It is practical to make the approximation that CM (1) ~ CM (t). This is justified if the membrane is saturated with the sample in a short period of time. This lag (steady-state) time may be approximated from Fick's second law as tlag = h2/(p2Dm), where h is the membrane thickness in centimeters and Dm is the sample diffusivity inside the membrane, in cm2/s [40,41]. Mathematically, tLAG is the time at which Fick's second law has transformed into the limiting situation of Fick's first law. In the PAMPA approximation, the lag time is taken as the time when solute molecules first appear in the acceptor compartment. This is a tradeoff approximation to achieve high-throughput speed in PAMPA. With h = 125 mm and Dm « 10~7 cm2/s, it should take ^3 min to saturate the lipid membrane with sample. The observed times are of the order of 20 min (see below), short enough for our purposes. Cools and Janssen [545] reported 10-30-min lag times with octanol-impregnated filters. With thinner BLM membranes, the time to reach steady state under sink conditions was reported to be 3-6 min [537]. Times as short as 50 s have been reported in BLM membranes [84].

From Eq. (7.16), one can deduce CD(i), and apply it in the next step. Before equilibrium is reached, at time t > tLAG, the moles of solute in the membrane may be estimated from

At this point, we introduce the retention fraction R, which is defined as the mole fraction of solute ''lost'' to the membrane. Equation (7.16) is used in the steps leading to Eq. (7.18):

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