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21 PERMEABILITY-SOLUBILITY-CHARGE STATE AND THE pH PARTITION HYPOTHESIS Fick s rst law applied to a membrane [18,40 42] shows that passive diffusion of a solute is the product of the diffusivity and the concentration gradient of the solute inside the membrane The membrane/water apparent partition coef cient relates the latter internal gradient to the external bulk water concentration difference between the two solutions separated by the membrane For an ionizable molecule to permeate by passive diffusion most ef ciently, the molecule needs to be in its uncharged form at the membrane surface This is the essence of the pH partition hypothesis [44] The amount of the uncharged form present at a given pH, which directly contributes to the ux, depends on several important factors, such as pH, binding to indigenous carriers (proteins and bile acids), self-binding (aggregate or micelle formation), and solubility (a solid-state form of self-binding) Low solubility enters the transport consideration as a thermodynamic speed attenuator, as a condition that lowers the opportunity for transport In this way, permeability and solubility are the linked kinetic and thermodynamic parts of transport across a membrane Consider a vessel divided into two chambers, separated by a homogeneous lipid membrane Figure 21 is a cartoon of such an arrangement The left side is the donor compartment, where the sample molecules are rst introduced; the right side is the acceptor compartment, which at the start has no sample molecules
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Absorption and Drug Development: Solubility, Permeability, and Charge State By Alex Avdeef ISBN 0-471-423653 Copyright # 2003 John Wiley & Sons, Inc
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Figure 21 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 coef cient, Kd 2 [Avdeef, A, Curr Topics Med Chem, 1, 277 351 (2001) Reproduced with permission from Bentham Science Publishers, Ltd]
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PERMEABILITY-SOLUBILITY-CHARGE STATE
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Fick s rst law applied to homogeneous membranes at steady state is a transport equation J
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0 h Dm dCm Dm Cm Cm h dx
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0 h where J is the ux, in units of mol cm 2 s 1, where Cm and Cm are the concentra3 tions, in mol/cm 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 21, 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 (21) 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 (21) 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 coef cients between bulk water and the membrane, log Kd (the pH-dependent apparent partition coef cient), we can convert Eq (21) into a more accessible form
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Dm Kd CD CA h
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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 (22) is still not suf ciently 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 : Pm Dm Kd h 2:3
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The relevance of Eq (22) (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 (22) reduces to the following ux equation J Pm C D 2:4
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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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