Diffusion aqueous

A series of studies have been reported modeling the diffusion process in water.49 Using the rules previously defined, we examined several characteristics of a system to determine their influence on diffusion. The first study revealed that solutes of high lipophilicity (low polarity) diffuse faster than those of low lipophilicity (high polarity). This result is not commonly considered or reported. Diffusion studies are numerous in the literature, but comparisons with solute polarity are very scarce. Two such studies, however, support the cellular automata model of this phenomenon.50,51 

A model of relative diffusion rates as a function of water temperature produced the expected result of greater diffusion with higher temperature. Additional studies49 were conducted in an attempt to model the influence of solution characteristics on solute diffusion. A series of dilute solutions were 

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Aqueous diffusion coefficients are usually on the order of 5 x 10 cm /s. A second is typically a long time to an electrochemist, so 6 = 30 fim. The definition of far is then 30 ]lni. Short is less than a second, perhaps a few milliseconds. Microseconds are not uncommon. Small, referring to the diameter of the electrode, is about a millimeter for microelectrodes, or perhaps only a few micrometers for ultramicroelectrodes (13). 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

Figure 7.37 Log aqueous diffusivities versus log molecular weights.

The in vitro measurements of permeability by the cultured-cell or PAMPA model underestimate true membrane permeability, because of the UWL, which ranges in thickness from 1500 to 2500 pm. The corresponding in vivo value is 30-100 pm in the GIT and nil in the BBB (Table 7.22). The consequence of this is that highly permeable molecules are (aqueous) diffusion limited in the in vitro assays, whereas the membrane-limited permeation is operative in the in vivo case. Correcting the in vitro data for the UWL effect is important for both GIT and BBB absorption modeling. 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

C. Steady Diffusion Across a Membrane with Aqueous Diffusion Layers... 

In the example above, the solutions are assumed to be well stirred and mixed the aqueous resistance is negligible, and the membrane is the only transport barrier. However, in any real case, the solutions on both sides of the membrane become less and less stirred as they approach the surface of the membrane. The aqueous diffusion resistance, therefore, very often needs to be considered. For example, for very highly permeable drugs, the resistance to absorption from the gastrointestinal tract is mainly aqueous diffusion. In the section, we give a general solution to steady diffusion across a membrane with aqueous diffusion resistance [5],... 

A stagnant diffusion layer is often assumed to approximate the effect of aqueous transport resistance. Figure 4 shows a membrane with a diffusion layer on each side. The bulk solutions are assumed to be well mixed and therefore of uniform concentrations cM and cb2. Adjacent to the membrane is a stagnant diffusion layer in which a concentration gradient of the solute may exist between the well-mixed bulk solution and the membrane surface the concentrations change from Cm to c,i for solution 1 and from cb2 to cs2 for solution 2. The membrane surface concentrations are cml and cm2. The membrane has thickness hm, and the aqueous diffusion layers have thickness ht and h2. 

Applying the concentration profile of Eq. (32) obtained from a thin film to both aqueous diffusion layers at steady state, we have... 

Figure 4 Diffusion across a membrane with aqueous diffusional layers. cbI and cb2 are the concentrations of bulk solutions 1 and 2, respectively. The thicknesses of the aqueous diffusion layers are /i, and h2. The membrane has a thickness of hm. Equilibrium is assumed at the interfaces of the membrane and the aqueous diffusion layers. At steady state, the concentrations remain constant at all points in the membrane and in the aqueous diffusion layers. The concentration profiles inside the membrane and aqueous diffusion layers are linear, and the flux is constant.

In Eqs. (40)-(42), cM and cb2 are experimentally measurable and the aqueous diffusion layer thickness can be estimated theoretically. Therefore, the only unknowns are the solute concentrations at the interfaces, csl and cs2. Their estimation is shown below. 

The steady flux across each aqueous diffusion layer is... 

We have discussed steady diffusion across a membrane with or without aqueous diffusion resistance. If the membrane is extremely thick or if solute diffusion in the membrane is extremely slow, the membrane may behave as if it is almost... 

It may be appropriate here to introduce film theory. As mentioned in reference to the steady diffusion across a thin film, we often hypothesize a film called an unstirred layer to account for the aqueous diffusion resistance to mass transfer. Film theory is valuable not only because of its simplicity but also because of its practical utility. However, the thickness of the film is often difficult to determine. In the following, we try to answer the question, What does the thickness of the film represent ... 

Therefore, the film thickness is the z-axis intercept of the tangent curve of the concentration profile. The calculated thickness of the aqueous diffusion layer is... 

Table 1 Calculated Thickness of the Aqueous Diffusion Layer Based on Eq. (82), Assuming a Diffusion Coefficient of 5 X I0 6 cm2/sec... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

In practice, estimation of Laq requires information on the rate of solute removal at the membrane since aqueous resistance is calculated from experimental data defining the solute concentration profile across this barrier [7], Mean /.aq values calculated from the product of aqueous diffusivity (at body temperature) and aqueous resistance obtained from human and animal intestinal perfusion experiments in situ are in the range of 100-900 pm, compared to lumenal radii of 0.2 cm (rat) and 1 cm (human). These estimates will necessarily be a function of perfusion flow rate and choice of solute. The lower Laq estimated in vivo is rationalized by better mixing within the lumen in the vicinity of the mucosal membrane [6],... 

JH Kou, D Fleisher, GL Amidon. Calculation of the aqueous diffusion layer resistance for absorption in a tube Application to intestinal membrane permeability determination. Pharm Res 8 298-305, 1991. 

D+, D = aqueous diffusion coefficients of the cation and anion, respectively, cm2/sec... 

The aqueous diffusivities of charged permeants are equivalent to those of uncharged species in a medium of sufficiently high ionic strength. The product DF(r/R) is the effective diffusion coefficient for the pore. It is implicit in k that adsorption of the cations does not occur, so that the fixed surface charges on the wall of the pore are not neutralized. Adsorption is more likely to occur with multivalent cations than with univalent ones. 

Aqueous diffusion coefficient calculated using Eqs. (41) and 42). Note PC = partition coefficient. 

Daq = aqueous diffusion coefficient, cm2/sec hAEL = effective thickness of the ABL, cm... 

DJ Lyman, SW Kim. Aqueous diffusion through partition membranes. J Polym Sci, Polym Symp 41 139-144, 1973. 

Levitt, M., C. Fine, J. Furne, and D. Levitt. Use of maltose hydrolysis measurements to characterize the interaction between the aqueous diffusion barrier and the epithelium in the rat jejunum., /. Clin. Invest. 1996, 97, 2308-2315... 

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