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Figure 7.16 Relative concentrations of accetor and donor compartments as a function of time for the iso-pH ketoprofen model.

liposome-water partition studies: the surface ion pair (SIP) constant, log K®IP 0.70, corresponding to the partitioning of the anionic form of the drug in bilayers at high pH, and the neutral-species partition coefficient, log Kpf 2.14, evident at low pH [149]. For example, at pH 7.4, Kd is 5 and at pH 4.3, Kd is 58. Also used for the simulation calculation were the intrinsic permeability coefficient, P0 1.7 x 10—4 cm/s, corresponding to the transport property of the uncharged form of ketoprofen, and the unstirred water layer permeability coefficient, Pu 2.2 x 10—5 cm/s. (These two types of permeability are described later in this chapter.)

At pH 3, ketoprofen is mostly in an uncharged state in solution. The dashed curve in Fig. 7.16 corresponding to pH 3 shows a rapid decline of the sample in the donor well in the first half-hour; this corresponds to the membrane loading up with the drug, to the extent of 56%. The corresponding appearance of the sample in the acceptor well is shown by the solid line at pH 3. The solid curve remains at zero for t < tLAG. After the lag period, the acceptor curve starts to rise slowly, mirroring in shape the donor curve, which decreases slowly with time. The two curves nearly meet at 16 h, at a concentration ratio near 0.22, far below the value of 0.5, the expected value had the membrane retention not taken a portion of the material out of the aqueous solutions.

### 7.5.2.2 Sink Condition in Acceptor Wells

In Section 7.7.5.4, we discuss the effects of additives in the acceptor wells that create a sink condition, by strongly binding lipophilic molecules that permeate across the membrane. As a result of the binding in the acceptor compartment, the transported molecule has a reduced ''active'' (unbound) concentration in the acceptor compartment, cA (t), denoted by the lowercase letter c. The permeability equations in the preceding section, which describe the nonsink process, are inappropriate for this condition. In the present case, we assume that the reverse transport is effectively nil; that is, CA(t) in Eq. (7.1) may be taken as cA(t) « 0. As a result, the permeability equation is greatly simplified:

Note that we call this the ''apparent'' permeability, since there is a hidden assumption (unbound concentration is zero).

7.5.2.3 Precipitated Sample in the Donor Compartment

When very insoluble samples are used, sometimes precipitate forms in the donor wells, and the solutions remain saturated during the entire permeation assay. Equations (7.20) and (7.21) would not appropriately represent the kinetics. One needs to consider the following modified flux equations [see, Eqs. (7.1) and (7.2)]

The donor concentration becomes constant in time, represented by the solubility, S = CD (0) = CD (t). Reverse flux can still occur, but as soon as the sample reaches the donor compartment, it would be expected to precipitate. Furthermore, the concentration in the acceptor compartment would not be expected to exceed the solubility limit: CA(t) < S. After equating the two flux expressions, and solving the differential equation, we have the saturated-donor permeability equation

2.303 VA

Ordinarily it is not possible to determine the membrane retention of solute under the circumstances of a saturated solution, so no R terms appear in the special equation [Eq. (7.25)], nor is it important to do so, since the concentration gradient across the membrane is uniquely specified by S and CA(t). The permeability coefficient is ''effective'' in this case.

7.5.3 Gradient pH Equations with Membrane Retention: Single and Double Sinks

When the pH is different on the two sides of the membrane, the transport of ioniz-able molecules can be dramatically altered. In effect, sink conditions can be created by pH gradients. Assay improvements can be achieved using such gradients between the donor and acceptor compartments of the permeation cell. A three-compartment diffusion differential equation can be derived that takes into account gradient pH conditions and membrane retention of the drug molecule (which clearly still exists—albeit lessened—in spite of the sink condition created). As before, one begins with two flux equations

It is important to note that two different permeability coefficients need to be considered, one denoted by the superscript (D ! A), associated with donor (e.g., pHD

5.0, 6.5, or 7.4)-to-acceptor (pHA 7.4) transport, and the other denoted by the superscript (A ! D), corresponding to the reverse-direction transport. The two equivalent flux relationships can be reduced to an ordinary differential equation in CD(t), following a route similar to that in Section 7.5.2.1.

The gradient pH (2-Pe) model developed here implies that some backflux (A ! D) is possible. As far as we know, reported literature studies generally considered backflux to be nil under gradient pH conditions. That is, either Eq. (7.10) or (7.11) were used to interpret the membrane transport under a pH gradient conditions. If it can be assumed that CA(t) in Eq. (7.26) represents a fully charged (i.e., impermeable) form of the solute, then its contribution to backflux may be neglected, and an effective sink condition would prevail; that is, the concentration of the uncharged form of the solute, cA(t), is used in place of CA(t), where cA(t) k 0. Under such circumstances, the generic sink equation, Eq. (7.22), may be used to determine an apparent permeability coefficient, Pa—"apparent" so as to draw attention to hidden assumptions (i.e., no reverse flux). However, valid use of Eq. (7.22) is restricted to strictly maintained sink conditions and presumes the absence of membrane retention of solute. This is a rather impractical constraint in high-throughput applications, where molecules with potentially diverse transport properties may be assayed at the same time.

A more general analysis requires the use of two effective permeability coefficients, one for each pH, each of which would be valid in the respective iso-pH conditions. Since fewer limiting assumptions are made, the more general method may be more suitable for high-throughput applications. We continue to derive the appropriate new model.

### The donor-acceptor membrane mass balance is

Each side of the barrier has a different membrane-buffer apparent partition coefficient Kd, defined at t = 1 as molTOT = VdCd (0) = VACA(i) + VdCd(i) + VmCm (1) (7.28)

The moles lost to the membrane are derived from Eqs. (7.28)—(7.30):

The membrane retention fraction R may be defined as membrane-bound moles of sample, divided by the total moles of sample in the system:

The membrane saturates with solute early in the transport process. So, for t ^ 20 min, we may assume that (1) « (t) is reasonably accurate. With this assumption, the acceptor concentration may be expressed in terms of the donor concentration as

A differential equation as a function of CD (t) only, similar to Eq. (7.5), can be derived, where the specific constants a = A(p(A!D)/VA + PeD!A)/VD) and b = CD(0)(1 — R)APeA!D)/VA. The solution to the ordinary differential equation is

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