DERIVATION OF MEMBRANE-RETENTION PERMEABILITY EQUATIONS in VS .NET

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DERIVATION OF MEMBRANE-RETENTION PERMEABILITY EQUATIONS
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In mole fraction units, this is   mA t VA t 1 exp mD 0 VA VD teq 7:8
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Note that when VA ) VD , Eq (78) approximately equals 1 exp t=teq Furthermore, 1 exp t=teq % t=teq when t is near zero Figure 714 shows the forms of Eqs (76) and (78) under several conditions When less than $10% of the compound has been transported, the reverse ux due to CA t term in Eq (71) is nil This is effectively equivalent to a sink state, as though VA ) VD Under these conditions, Eq (78) can be simpli ed to mA t t APt % % mD 0 teq VD 7:9
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and the apparent permeability coef cient can be deduced from this one-way ux equation, Pa VD mA t At mD 0 7:10
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mD(t)/mD(0)
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-5 3 Pa = 10 cm/s, VD = VA = 02 cm
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Pa = 10-5 cm/s, VD = 02 cm3, VA = 20 cm3
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mA(t)/mD(0)
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-4 3 Pa = 10 cm/s, VD = VA = 02 cm
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Figure 714 Relative concentrations of accetor and donor compartments as a function of time for the thin-membrane model
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PERMEABILITY
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We de ne this permeability as apparent, to emphasize that there are important but hidden assumptions made in its derivation This equation is popularly (if not nearly exclusively) used in culture cell in vitro models, such as Caco-2 The sink condition is maintained by periodically moving a detachable donor well to successive acceptor wells over time At the end of the total permeation time t, the mass of solute is determined in each of the acceptor wells, and the mole sum mA t is used in Eq (710) Another variant of this analysis is based on evaluating the slope in the early part of the appearance curve (eg, solid curves in Fig 714): Pa VD mA t = t mD 0 A 7:11
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It is important to remember that Eqs (710) and (711) are both based on assumptions that (1) sink conditions are maintained, (2) data are taken early in the transport process (to further assure sink condition), and (3) there is no membrane retention of solute In discovery settings where Caco-2 assays are used, the validity of assumption 3 is often untested The more general solutions (but still neglecting membrane retention, hence still apparent ) are given by two-way ux in Eqs 712 (disappearance kinetics) and (713) (appearance kinetics)   ! 2:303 VD 1 CD t Pa log10 rV 1 rV 7:12 1 rV CD 0 At   ! 2:303 VD 1 CA t 1 log10 1 1 rV 7:13 1 rV CD 0 At where the aqueous compartment volume ratio, rV VD =VA Often, rV 1 From analytical considerations, Eq (713) is better to use than (712) when only a small amount of the compound reaches the acceptor wells; analytical errors in the calculated Pa, based on Eq (713), tend to be lower Palm et al [578] derived a two-way ux equation which is equivalent to Eq (713), and applied it to the permeability assessment of alfentanil and cimetidine, 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 7771) 752 Iso-pH Equations with Membrane Retention
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The popular permeability equations [(710) and (711)] 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 coef cients This section expands the equations to include membrane retention
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