DERIVATION OF MEMBRANE-RETENTION PERMEABILITY EQUATIONS in Visual Studio .NET

Encoder Data Matrix in Visual Studio .NET DERIVATION OF MEMBRANE-RETENTION PERMEABILITY EQUATIONS
DERIVATION OF MEMBRANE-RETENTION PERMEABILITY EQUATIONS
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that passes across the membrane then binds strongly to the serum protein Consider phenazopyridine (pKa 515; see Table 31) in a pH 74 PAMPA assay, where the acceptor solution contains 3% wt/vol BSA (bovine serum albumin) As soon as the free base reaches the acceptor compartment, it binds to the BSA The unbound fraction becomes very low, even though the total concentration of the base in the acceptor compartment may be relatively high This may be called a bindingmaintained sink In this chapter we use the term sink to mean any process that can signi cantly lower the concentration of the neutral form of the sample molecule in the acceptor compartment Under the right conditions, the ionization and the binding sinks serve the same purpose as the physically maintained sink often used in Caco-2 measurements We will develop several transport models to cover these chemical sink conditions When both of the chemical sink conditions (ionization and binding) are imposed, we will use the term double sink in this chapter The chemical sink may be thought of as a method used to increase the volume of distribution of species in the acceptor solution beyond the geometric volume of the receiving compartment As such, this extension of terminology should be clear to traditional Caco-2 users The use of the chemical sinks in PAMPA is well suited to automation, and allows the assay to be conducted at high-throughput speeds As mentioned above, the one-point-in-time (single-timepoint) sampling can lead to errors if not properly executed We will show that when multitimepoint PAMPA is done (see Fig 715), the equations developed in this chapter for high-speed single-timepoint applications are acceptably good approximations 751 Thin-Membrane Model (without Retention)
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Perhaps the simplest Fick s law permeation model consists of two aqueous compartments, separated by a very thin, pore-free, oily membrane, where the unstirred water layer may be disregarded and the solute is assumed to be negligibly retained in the membrane At the start (t 0 s), the sample of concentration CD (0), in mol/cm3 units, is placed into the donor compartment, containing a volume (VD , in cm3 units) of a buffer solution The membrane (area A, in cm2 units) separates the donor compartment from the acceptor compartment The acceptor compartment also contains a volume of buffer (VA , in cm3 units) After a permeation time, t (in seconds), the experiment is stopped The concentrations in the acceptor and donor compartments, CA t and CD t , respectively, are determined Two equivalent ux expressions de ne such a steady-state transport model [41] J t P CD t CA t and J t VD dCD t dt A 7:2 7:1
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PERMEABILITY
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where P denotes either the effective or the apparent permeability, Pe or Pa, depending on the context (see later), in units of cm/s These expressions may be equated to get the differential equation   dCD t A P CD t CA t dt VD 7:3
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It is useful to factor out CA t and solve the differential equation in terms of just CD t This can be done by taking into account the mass balance, which requires that the total amount of sample be preserved, and be distributed between the donor and the acceptor compartments (disregarding the membrane for now) At t 0, all the solute is in the donor compartment, which amounts to VD CD (0) moles At time t, the sample distributes between two compartments: VD CD 0 VD CD t VA CA t 7:4
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This equation may be used to replace CA t in Eq (73) with donor-based terms, to get the simpli ed differential equation dCD t aCD t b 0 dt 7:5
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where a AP= VA VD = VA VD t 1 , teq is the time constant, and b eq APCD 0 =VA Sometimes, t 1 is called the rst-order rate constant, k [in s 1 units eq (reciprocal seconds)] The ordinary differential equation may be solved by standard techniques, using integration limits from 0 to t, to obtain an exponential solution, describing the disappearance of solute from the donor compartment as a function of time CD t mD t VA VD exp t=teq CD 0 mD 0 VA VD VA ! 7:6
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where mD t refers to the moles of solute remaining in the donor compartment at time t Note that when VA ) VD , Eq (76) approximately equals exp( t=teq ) Furthermore, exp( t=teq ) % 1 t=teq when t is near zero Using the mole balance relation [Eq (74)], the exponential expression above [Eq (76)] may be converted into another one, describing the appearance of solute in the acceptor compartment   CA t VD t 1 exp CD 0 VA VD teq 7:7
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