(6.1)

where

P

is the partition coefficient of the permeant between the membrane and the donor vehicle;

C

is the concentration (in this case, the solubility) of the permeant in the donor solution.

Validation of this model produced excellent fits between experimental and predicted permeabilities (r

^{2}> 0.94):(6.2)

They also applied their approach to previously published data (by Diez-Sales et al. 1996) and found a significant fit to a linear model, which was similar to Eq. 6.2, for the skin penetration enhancer Azone

^{®}, which is known to enhance permeation based on the lipophilicity of the permeant:(6.4)

They commented that, despite significant chemical differences between SLS and Azone

^{®}, particularly in their hydrophilic domains which might imply different interactions with the stratum corneum, in a qualitative sense—and in the context of the nature and size of the data set used for this study—their effects on skin permeability were very similar and could be predicted very well by the models proposed (Eqs. 6.1–6.4).In such a context of model range and/or limitation, particularly in the context of vehicles more complex than those normally associated with Flynn-based permeability models, Gregoire et al. (2009) addressed a significant issue in the development of predictive algorithms of skin permeation—the lack of applicability to a wider range of vehicles. They developed a predictive model which estimated the cumulative mass of a chemical absorbed into and across the skin from topical formulations (i.e. cosmetic or dermatological preparations). In doing so, they assumed that a steady state was achieved despite the application of a finite dose, that vehicle effects were small relative to the precision (or otherwise) of the prediction and that each formulation could be treated as an oil-in-water emulsion in which only the aqueous fraction of the chemical was available for permeation into the stratum corneum. In analysing a data set of 101 ex vivo human skin experiments for 36 chemicals they found that, in most cases, the difference between experimental and predicted permeability was less than fivefold and that the model was able to accurately estimate permeation for two chemicals not in the data set. Nevertheless, their model highlights the complex issues associated with predicting the permeability of exogenous chemicals from a range of formulations and, in doing so, addresses the limitations of current models—which focus mostly on saturated aqueous solutions and highlights the challenges ahead in this field. This is an issue that has, more broadly, been discussed by others (Selzer et al. 2013) and which will be examined in more detail in subsequent chapters.

Another theme touched on by Flynn (1990) was the nature of models which were not “global” in the sense that they were represented by a single algorithm. An elegant example of how this approach has been taken forward is Mitragotri’s (2003) discussion, in the context of a porous pathway approach, of multiple permeation pathways based on permeant physicochemical properties. This approach attempts to discuss the permeation of hydrophilic molecules as, in general, permeation of hydrophobic molecules is reasonably well described by lipid-based models. Several models have described this approach as a “porous pathway” model which properly accounts for the permeation of hydrophilic permeants. For example, Peck et al. (1994) introduced the concept of “hindered diffusion” of polar molecules through the skin by examining a small group of model, hydrophilic, compounds (urea, mannitol, sucrose and raffinose) and describing their permeation. This approach has been developed in other studies (e.g. Hatanaka et al. 1990; Kim et al. 1992; Morimoto et al. 1992; Lai and Roberts 1998, 1999). Thus, the general expression for permeability via the porous pathway is given by:

where

(6.5)

ε, τ and Δx

are the porosity, tortuosity and thickness of the membrane, respectively;

is the diffusion coefficient of the permeant in the liquid-filled pores of the membrane.

The “hindered diffusion” model considers that is a function both of the membrane and the permeant, being dependent upon pore size (radii) and the diffusion coefficient of the permeant at infinite dilution. Despite being able to clearly characterise porosity and its influence on the permeation of highly hydrophilic permeants, such models have found limited application due mainly to the lack of defined links between pore radii and aspects of skin morphology, such as pore density.

Thus, Mitragotri (2003) approached this problem by examining solute permeation through four possible routes in the stratum corneum: free volume diffusion through lipid bilayers (using scaled particle theory), lateral diffusion along lipid bilayers (determined from literature data), diffusion through pores (from the “hindered transport” theory) and diffusion through shunts (via the application of a simple diffusion model). Mitragotri’s analysis resulted in a series of models which described each pathway. Solute permeability across the stratum corneum for hydrophobic solutes was described by the expression:

where r is the radius (units Å), which can be calculated as described by van der Bondi (1964) or approximated from the molecular weight (MW) of the penetrant, based on the relationship (Mitragotri et al. 1999).

(6.6)

They also proposed a method to estimate the lateral diffusion of lipids, proposing that the diffusion of large solutes that are incorporated into the bilayer is related to the lateral diffusion coefficients of lipid molecules:

(6.7)

Equation 6.7 can, in the context of several assumptions (i.e. consideration of r = 4.3 Å, where which is comparable to lateral lipid diffusion in other systems), be rewritten as:

(6.8)

Mitragotri highlighted that lateral lipid diffusion is not dependent on permeant size and that the transition from free volume diffusion to lateral diffusion occurs at a radius of approximately 4.3 Å, which corresponds to a value of r

^{2}= 18.5, or an approximate molecule weight of 380 Da.Transport through pores was assumed to play a major role in the permeation of hydrophilic permeants. Based on the assumption that polar or aqueous pathways—often considered as “pore” pathways—exist and will favour the permeation of hydrophilic molecules (Cornwell and Barry 1993; Edwards and Langer 1994; Menon and Elias 1997), Mitragotri placed such a pathway in the context of lipid bilayer imperfections which may be observed as grain boundaries, lattice vacancies, defects in lattice structures or any combination of such features, and which may provide a “polar” pathway for the permeation of hydrophilic molecules. Equation 6.5 shows the general expression for permeability of a solute through a porous membrane, from which the hindrance factor may be determined, following the method of Deen (1987):

where λ is the ratio of the hydrodynamic radius of the permeant and the effective pore radius of the membrane.

(6.9)

And

where a

(6.10)

_{1}= −1.217, a_{2}= 1.534, a_{3}= −22.51, a_{4}= −5.612, a_{5}= −0.3363, a_{6}= −1.216 and a_{7}= 1.647Thus, to estimate permeability via this pathway ε, τ, and Δx, the porosity, tortuosity and thickness of the membrane, respectively, need to be known. Finally, the last part of the four-compartment model, which considers transport through skin appendages, may ultimately be represented by the expression:

(6.11)

Mitragotri commented that this route is only of significance for the permeation of large (MW > 100,000 Da) hydrophilic molecules.

Thus, this specific and comprehensive model compares well with experimental results in the data sets used in this study. In addition, different permeants have different weightings for each pathway based on their physicochemical properties. Using the Johnson-modified Flynn data set (Flynn 1990; Johnson et al. 1995; n = 83), they found excellent correlations between measured and estimated permeability, with a mean error of approximately 6 %. They found that the contribution of free volume diffusion decreases exponentially and is dominant for small permeants (less than 4 Å), which is related to the radii of the pores and the solutes. Skin permeability to hydrophobic solutes exhibits significant size selectivity, whereas the contribution of lateral lipid diffusion was considered to be significant for larger solutes but to not be characterised by size dependence. The transport of hydrophilic drugs is hypothesised to occur through pores in the stratum corneum lipid bilayers, which may be the result of structural imperfections within the bilayer. Transport through such “pores” is characterised by porosity, tortuosity and pore size distribution. Pore size and porosity are characteristics associated entirely with the skin morphology, whereas tortuosity depends on the stratum corneum structure as well as the solute size.

Different solutes were shown to differ in the relative contributions each pathway makes to their overall permeability, and the contribution of each pathway to skin permeability is a function of size and lipophilicity. Thus, permeation of small, hydrophobic solutes is mostly via free volume diffusion. As solute size increases, the free volume pathway diminishes to insignificance and permeability is defined predominately by lateral lipid diffusion. For highly hydrophilic solutes where, for example, K

_{o/w }< 0.01, skin permeability is a function of pore and shunt permeability. Finally, for permeants of moderate hydrophilicity (K_{o/w }~ 0.01–1), permeability is related mostly to free volume diffusion through lipid bilayers.For further details of this excellent study, the reader is referred to Mitragotri’s excellent paper (Mitragotri 2003).

One of the most original, and important, models that sit outside the context of Potts- and Guy-type algorithms based on the permeability coefficient was reported by Magnusson et al. (2004). They commented that the delivery rate at which the solute is absorbed into and across the skin is highly significant in terms of systemic and local therapeutic or toxicological endpoints. More so than the permeability coefficient (k

_{p }) as, in practice, the maximum flux (J_{max}, with units of amount/time/surface area), usually at steady-state, is of most interest in determining the maximum absorption.