## Membrane

Figure 2.1 Transport model diagram, depicting two aqueous cells separated by a membrane barrier. The drug molecules are introduced in the donor cell. The concentration gradient in the membrane drives the molecules in the direction of the acceptor compartment. The apparent partition coefficient, Kd = 2. [Avdeef, A., Curr. Topics Med. Chem., 1, 277-351 (2001). Reproduced with permission from Bentham Science Publishers, Ltd.]

Figure 2.1 Transport model diagram, depicting two aqueous cells separated by a membrane barrier. The drug molecules are introduced in the donor cell. The concentration gradient in the membrane drives the molecules in the direction of the acceptor compartment. The apparent partition coefficient, Kd = 2. [Avdeef, A., Curr. Topics Med. Chem., 1, 277-351 (2001). Reproduced with permission from Bentham Science Publishers, Ltd.]

Fick's first law applied to homogeneous membranes at steady state is a transport equation j _ DmdCm _ Dm[Cm — Cm] (2 1)

dx h where J is the flux, in units of mol cm-2 s-1, where Cm and Cm are the concentrations, in mol/cm3 units, of the uncharged form of the solute within the membrane at the two water-membrane boundaries (at positions x _ 0 and x _ h in Fig. 2.1, where h is the thickness of the membrane in centimeters) and where Dm is the diffusivity of the solute within the membrane, in units of cm2/s. At steady state, the concentration gradient, dCm/dx, within the membrane is linear, so the difference may be used in the right side of Eq. (2.1). Steady state takes about 3 min to be established in a membrane of thickness 125 mm [19,20], assuming that the solution is very well stirred.

The limitation of Eq. (2.1) is that measurement of concentrations of solute within different parts of the membrane is very inconvenient. However, since we can estimate (or possibly measure) the distribution coefficients between bulk water and the membrane, log Kd (the pH-dependent apparent partition coefficient), we can convert Eq. (2.1) into a more accessible form

where the substitution of Kd allows us to use bulk water concentrations in the donor and acceptor compartments, CD and CA, respectively. (With ionizable molecules, CA and CD refer to the concentrations of the solute summed over all forms of charge state.) These concentrations may be readily measured by standard techniques. Equation (2.2) is still not sufficiently convenient, since we need to estimate Dm and Kd. It is common practice to lump these parameters and the thickness of the membrane into a composite parameter, called membrane permeability Pm:

The relevance of Eq. (2.2) (which predicts how quickly molecules pass through simple membranes) to solubility comes in the concentration terms. Consider ''sink'' conditions, where CA is essentially zero. Equation (2.2) reduces to the following flux equation

Flux depends on the product of membrane permeability of the solute times the concentration of the solute (summed over all charge state forms) at the water side of the donor surface of the membrane. This concentration ideally may be equal to the dose of the drug, unless the dose exceeds the solubility limit at the pH considered, in

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