A Linear Theory of Transdermal Transport Phenomena

A Linear Theory of Transdermal Transport Phenomena DAVIDA. ED WARDS^^ AND ROBERTLANGER Received January 20, 1994, from the Deparfmenf of Chemical Engineering, Bldg €25-342, Massachusetts lnsfifufe of Technology, Accepted for publication June 3, 1994@. Cambridge, MA 02139. t Present address: Department of Chemical Engineering, The Pennsylvania State University, 158 Fenske Laboratory, University Park, PA 16802-4400. Abstract 0 A theory of charge, fluid-mass, and solute (including macromolecular) transport through porous media is applied to describe transport phenomena across the external layer of mammalian skin. Linear relationships are derived between transport fluxes and applied fields. These relationships introduce six effective transdermal transport coefficients. Formulas for each of these coefficients are provided. The practical relevance of these parameters is emphasized in the specific context of transdermal drug delivery. By employing typical physiological values for the various geometrical and physicochemical parameters that appear in the formulas for the transdermal transport coefficients, predictions are made for transport rates of charge, fluid mass, and solute species across a uniform-thickness skin sample contained within a diffusion-cell apparatus. These results are used to explore transdermal phenomena involving forced convection, current flow, electroosmosis, iontophoresis, and molecular diffusion (including convective dispersion). Comparisons with existing transdermal drug delivery data are made. On the basis of these comparisons, the theory suggests that transdermal transport in the presence of an electrical field may occur through corneocytes of the stratum corneum. The theory confirms the importance of a shunt route for small ion transport, as well as an intercellular route of transport for passive diffusion of noncharged substances. These latter conclusions, also based on comparisons with experimental data, are consistent with previous statements in the literature. A new form of solute transport enhancement, termed transdermal convective dispersion, is included in the theory, and methods for its measurement are described. Generalizations and future applications of the theory are discussed.

Introduction The relative impermeability of the outermost layer of mammalian skin to water, charge, and solute transport crucially assists the body’s regulation of water and electrolyte loss, serving as well its ability to ward off the transdermal invasion of foreign substances. A theoretical understanding of the origins of epidermal impermeability, while of general scientific value, is of particular practical importance in transdermal drug delivery’ applications. The delivery of drugs via a transdermal route, as contrasted to oral delivery, is advantageous2since drug loss due to liver metabolism is minimized and a sustained delivery of the drug over relatively long durations (up to 1week) can be obtained. However, clinical use of transdermal drug delivery has found only limited application. This owes to the fact that very few drugs are able, at least by passive diffusion alone, to penetrate the skin a t a rate sufficient to be viable for drug delivery purposes. A major avenue of drug delivery research therefore concerns the development of technologies for enhancing the transport of small as well as of large molecules across the skin. Emerging drug delivery technologies that capitalize upon the enhancement effects of electric field^,^-^ ultrasonic pressure waves,I and chemical modification of the transporting substances each exhibit a need for a predictive understanding of @

Abstract published in Advance ACS Abstracts, July 15, 1994.

0 1994, American Chemical Society and American Pharmaceutical Association

underlying transport mechanisms for their optimization and implementation. Numerous quantitative approaches to transdermal transport phenomena have been advanced in the past, many of which may be classified according to the following categories: (i) “circuit rn~dels”~-ll characterize the transdermal transport process by analogy with series and parallel arrangements of resistors and capacitors; (ii) “macroscale models”12-16 assume the validity of standard continuum transport laws at a homogeneous length scale of the epidermis, deducing from these laws various general features of transdermal transport phenomena; and (iii) “parametric models”17employ physical arguments to advance correlations between transdermal transport phenomena and underlying physicaUgeometrica1 parameters. The present approach may be distinguished from the former efforts insofar as it employs standard spatial-averaging methods of analysis1s to derive so-called “macrotransport” descriptions of the various transport phenomena on the basis of their underlying microscale (pore-level) descriptions. Our approach (the closest analog of which in the transdermal transport literature appears to be the work of Michaels and co-workerslg)possesses considerable precedent in the general field of transport phenomena in porous media.20,21Spatialaveraging theories of transport phenomena in porous media include the theories of macrotransport volume averaging,23and homogenizati~n.~~ Such theories have been used to deduce macroscale transport phenomena behavior on the basis of a detailed microscale transport description. (See the work of Sangani and AcrivosZ5for effective heat transfer calculations in heterogeneous media, Koch and Brady26 for analysis of convective dispersion in porous media, and Edwards2I for calculation of effective transport characteristics of gases and aerosols in human lungs.) Our application of these methods to transdermal transport phenomena is an attempt to build upon existing theoretical understanding by bringing the considerable scientific progress made in recent years in the general field of transport phenomena in porous media to bear upon the emerging field of transdermal transport. We begin with a brief description of the physical background t o linear, macroscale transdermal transport phenomena, focusing specifically upon the application to transdermal drug delivery. This is followed by a description of the theoretical approach and then by a summary of effective transport property formulas for two idealized transdermal pathways. (The explicit derivation of these formulas is discussed in the Appendix.) A parametric study of model results is presented next, along with a comparison of theoretical predictions with limited experimental findings. Finally, a discussioli of future extensions and applications is provided.

Background Macroscale Laws-Consider the scenario depicted in Figure la. This scenario may be regarded as a “macroscale” characterization of transdermal transport. (See Figures 3 and 4 and the captions thereof for a description of the “macroscale”

0022-3549/94/1200-1315$04.50/0

Journal of Pharmaceutical Sciences / 1315 Vol. 83, No. 9, September 1994

orthogonal directions in the skin. Since the internal structure of the skin is extremely anisotropic, the transport properties K and KEcannot, in general, be replaced by scalar equivalents; that is, these transport properties possess different magnitudes depending upon whether the flow of water is parallel or perpendicular t o the skin surface normal, n. However, principle attention in this article shall be directed to the special case of transport normal to a uniform-thickness skin sample contained within the diffusion cell apparatus depicted in Figure l b in this scenario, eq 1 adopts the simplified form

/Hsir) follicle

?I

L --

n 0 --

(b)

Figure 1-(a) A highly idealized, cross-sectional schematic of the epidermis of mammalian skin The epidermis is characterized by three roughly parallel layers the very thin stratum corneum, the viable epidermis, and the papillary layer of the epidermis The latter layer is rich in blood capillaries and represents, at least in the context of the present article, the ultimate destination for solute species that are able to traverse the epidermis from an origin at the surface of the skin (whose surface normal is n). Hair follicles are idealized as circular cylindrical (whereas in reality they are knownB to taper inwardly toward the papillaty layer), containing circular hair shafts The complex internal structure of the hair follicles, indeed of the sweat ducts as well, is not represented in this figure (b) A two-dimensional view of a diffusion cell A skin sample of thickness L separates fluidic donor and receptor compartments, the latter being distinguished from each other by virtue of unique electrical voltage potentials, solute concentrations, and fluid pressures

point of view in the context of our multiscale descriptios-of slun.) Pressure-gradient (Vp), concentration-gradient (VC), and electric (E) fields are assumed to act at arbitrary angles relative to the skin surface normal, n. These applied fields produces-fluxes of fluid mass (i.e., fluid velocity, V), solute species (J), and charge (J,) across the epidermal layer into the papillary layer of the dermis. In this paper, attention is constrained almost exclusively to &eaj relationships between the _applied external fields (Vp, VC, E)and resultant fluxes (V,J, J”). [Exceptions to this linear behavior arising from “convective dispersion” (i.e., convectively enhanced diffusion) phenomena are addressed by the present theory; nonlinearities arising ma field-dependent structural changes within the slun may easily be incorporated into the theory, as later discussed.] These linear, macroscale relationships are briefly summarized in a general vector form below, as well as in a simplified form for the special scenario of the “diffusion cell” depicted in Figure lb. Transdermal transport of fluid mass may be described by the generalized Darcy’s law equation v = --K.vp I - - - - -Kc.E IE

Pf

Pf

(3) where J, is the effective current flux density vector, a the effective electrical conductivity, and the effective streaming potential conductivity. This law simplifies for the case of the diffusion cell (Figure lb) to the relationship (4) Finally, transdermal solute transport obeys the constitutive form of Fick’s law J

=

u*c - D*.VC

(5)

where J is the effective solute flux vector density, U*the mean solute velocity vector (not necessarily the mean fluid vector 01, D* the effectiv- “dispersivity”(not necessarily the effective diffusivity), and C the effective solute concentration field. In general, C varies with time and from point to point within the skin. That is, eq 5 applies under unsteady as well as steady-state circumstances, requiring22,28 for its validity only that a sufficient elapse of time occurs for a certain “local equilibration”. For the case of the diffusion cell of Figure lb, we have (6)

(1)

where ii is the volume-average(or so-called“seepage”)velocity vector, ,uf (=pw in the notation-of the following sections) the viscosity of the flowing fluid, K the hydraulic permeability, and KE the electroosmostic permeability. Equation 1, like eqs 3 and 5 below, is a general vector equation. It can be applied to describe the flow of water in each of three mutually 1316 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

with L being the thickness of the skin sample, the voltage potential, P the pressure, and the A operation signifying AA = A1 - A2, with A a generic field quantity. Our convention is to regard the entire skin sample in the diffusion-cell experiment as the macroscale medium. This explains the appearance of the thickness L in eq 2 (as well as in eqs 4 and 6). However, under all circumstances subsequently addressed in this paper, the stratum corneum layer (assumed to be of thickness h ) provides the principal barrier to transport, as reflected in the appearance of the multiplicative factor (Llh) in many of the formulas derived (see, e.g., eqs 10 and 12) for the effectve transport properties. See also the discussion preceding eq 29. Transdermal transport of current satisfies the generalized Ohmic law

where C is found by solving the solute conservation equation

ac

aJ

at

an

- -- - - ( L r n r O )

(7

with appropriate initial condition, as well as boundary condi tions

where Kl, Kz are (possibly concentration-dependent) partition coefficients describing solute partitioning into the skin from the donor and receptor sides of the cell, respectively. The mean species velocity across the skin layer may frequently be placed in the form

with H being an effective hydrodynamic hindrance factor characterizing steric hindrance to molecular transport, z the charge of the transporting substance, and M E the effective ionic mobility of the substance across the skin. It should be noted that the boundary conditions (eq 8) will differ if transport of solute from the donor compartment into the skin and/or from the skin into the receptor compartment is relatively slow compared to the time of transport across the skin itself. In this case (which may perhaps have some application for very large and hydrophilic substances that are not able to easily partition into the skin), the boundary conditions at the skin surfaces will be of the form

with &, & being a pair of mass-transfer coefficients (units, c d s ) serving to characterize the mass-transfer permeability of the bounding surfaces of the skin sample (see, e.g., refs 68 and 71). Application of the Macroscale Laws: Transdermal Drug Delivery-Transdermal drug delivery offers a compelling physical example of the above macrotransport processes. In the simplest case of drug delivery by passive diffusion, a concentration-gradient field (VC) is applied across the skin. This field results in a net macroscale flux (J)of solute (i.e., drug) across the epidermis. Thus, the drug enters the papillary dermal layer, where it may be taken up by the bloodstream. Frequently, however, the magnitude of solute flux created solely by the concentration-gradient field (VC) is far too small for practical drug delivery purposes. Thus, additional driving forces may be applied. Enhanced d e g delivery schemes involving application of an electric field-(E) (typically in addition to a-concentration-gradient field, VC), give rise-to a solute flux, J, that is accompanied by a flow of current J,, across the skin, as well as a finite water velocity, t,the latter owing to the indigenous negative charge bocne by skin at physiological pH.65 Pressure-gradient fields (Vjj) may also be applied in the interests of enhancing drug delivery, as in sonoph~resis.~ Use of the above formulas for predicting transdermal transport fluxes, as in the case of drug delivery applications, Lequires knowledge of the six coefficients (K,PE, b, $,U*, D*). [In addition, we require knowledge _of (one or both of) the equilibrium partition coefficients (Kl,Kz). In this paper, where necessary, attention is limited to the ideal partitioning circumstances of eq 38.3 Explicit expressions for limiting pE, (5, ii“p, U*, D*), correspondforms of these coefficients (K, ing to the diffusion cell of Figure l b , are provided in a later section of this paper in terms of various structural characteristics of the epidermis. We have created Table 1to provide a_ way for estimating, on the basis of the K,pE, ii, 0,and D* results, transdermal fluxes in a variety of diffusion-cell transport scer*arios. Application of this table is discussed a t the beginning of the section entitled Comparison with Experimental Data. A few structural features of mammalian skin may serve to embed the subsequent results and discussion within a physiological context [see, for example, the work of J a 1 ~ e t ,Elia~,~O 2~

and Bodd&et aL31for more information]. The epidermis (see Figure l a ) is comprised of a stratum corneum layer (ca. 10 pm) and a viable epidermis layer (ca. 100 pm), each of whose thicknesses vary considerably over the surface of the body. The epidermis is bounded below by the papillary (bloodcapillary-rich) layer of the dermis (ca. 100-200 pm). Piercing these roughly parallel layers are various hair follicles and sweat ducts, whose densities also vary considerably over the surface of the body. The stratum corneum layer of the epidermis represents the principal barrier to transdermal transport phenomena. This owes largely to the fact that the stratum corneum is composed (see Figure 2) of densely packed hexagonal, disclike corneocytes (or keratinocytes), separated by thin, multilamellar lipid bilayers. The corneocytes are composed30of a-keratin filaments (ca. 6-8 nm in diameter) and, a t least under hydrated circumstances, are approximately 50% water-filled (by volume). Corneocytes possess a lateral dimension of approximately 30 pm and a longitudinal dimension of approximately 1 pm.2g The lamellar bilayer zones separating the corneocytes are approximately 0.05 pm (50 nm) in thickness and possess a small amount of bound water. In combination, the densely packed corneocytes and bilayer lipid regions function to minimize the transdermal transport of both lipophilic and hydrophilic substances.

Theoretical Determination of Effective Transport Coefficients A recent methodologyls permits the derivation of the transport laws 1, 3, and 5 for any spatially periodic medium characterized a t the microscale (most notably) by thin electrical double layers bounding charged pore surfaces, highly viscous fi.e., Stokes) flow in the interstices of the medium, and Fickian diffusion of the transporting (possibly charged) solute. In the course of these derivations, explicit schemes are @, U*,and Eutlined for calculating the coefficients K,PE, D*; these schemes are discussed in the Appendix. Our application of the paradigms described in the Appendix PE, b, $,U*, for determining the material parameters (K, D*) specific to transdermal transport phenomena involves the following principal assumptions: (i) skin microstructure (in particular, the stratum corneum of the epidermis) may be defined at two or more widely separated length scales; (ii) the geometry of the skin at each length scale may be characterized as spatially periodic (i.e., possessing a spatially repetitive geometrical structure), and the external boundaries of the tissue relative to each length scale may be very widely separated; (iii) the fmed charged surfaces at each length scale are bordered by electrical double layers that are extremely thin relative to pore dimensions; (iv) Stokes flow, Ohmic conduction, and Fickian diffusion prevail at all scales; and (v) transport processes possess a sufficiently short relaxation time scale so as to allow steady-state charge and fluid-mass transport, as well as to permit the requisite local diffusional sampling required for the validity of macrotransport theory.18,22 Condition i is easily met for typical transport pathways through the epidermis, as established in the specific transport scenarios considered below. The spatially peiiodic assumption ii is frequently made in effective medium calculations, and its restrictiveness can be relaxed c ~ n s i d e r a b l y Fortuitously, .~~ the substructure of the skin possesses a higher degree of order, hence the spatial periodicity assumption may perhaps be viewed as equally “realistic” as the assumption of infinitely extended boundaries. The assumption of thin equilibrium double layers is perhaps most easily faulted, seemingly representing the assumption

q,

Journal of Pharmaceutical Sciences / 1317 Vol. 83, No. 9, September 1994

Table 1-Transdermal Transport Phenomena Paradigm’ Phenomenon

Transport Law

Transport Coefficient

Route 1 ,b interlshunt

Route 2,b transcorneoc

Forced convection

Hydr perm., K

10 or 29

17 or 30

Current flow

Electr cond. ir

11 o r 3 l a

18 or 31b

Electroosmosis

Electr perm., @

12 or 33

19 or 34

Solute velo, u* Partit coeff, Ki

13or35 38

20 or 36

Solute diff, o*

15 or 42a

25 or 42a

Partit coeff, Ki

38

lontophoresis

Passive diffusion

-

_ - -

J % K, V C,

KP’K(P) ’ h

aThe formulas presented in this table pertain to transdermal transport phenomena in the diffusion cell apparatus shown in Figure Ib. bThe numbers listed below refer to equation numbers in the text.

the stratum corneum

bilayers

7

11 66 [ $ $ P I 11 66 666

i Figure 2-A two-dimensional idealization of the substructure of the stratum corneum. Thin corneocyte cells (hexagonally shaped in the plane perpendicular to the page) are separated by narrow layers of lipid material. These layers are comprised of multilamellar lipid bilayers separated by extremely thin bound-water layers.

most needful of generalization in future contributions (see, however, below). Polar lipid head groups within the lamellar regions of the stratum corneum, as well as charged keratin fibers within the corneocytes, are bordered by extremely thin aqueous pores. Thus, a nonuniform charge distribution may be presumed to extend, in general, throughout aqueous pore spaces bounding charged subsurfaces within the skin. Removal of this assumption constitutes a primary extension of the present theory, as discussed a t the article’s conclusion. It bears noting that the thin ionic double layer assumption underlies the linear quality of the laws 1,3, and 5 with respect to the applied electric field, E. Whereas a nonlinear current/ voltage-drop relationship is typically observed in excised human tissue experiments,1° this nonlinearity need not be a reflection of nonlinear double-layer phenomena; rather, it may point to voltage-drop-dependentstructural changes within the tissue. (In this context, see the discussion following eq 31a regarding electroosmosis.) Such structural changes may easily be introduced within the context of the present theory. Evidence for electric-field-inducedstructural alterations may exist, as demonstrated by the fact that iontophoretic solute fluxes typically vary linearly with current fluxes (at least for suitably low frequency ac curents14);thus the ratio of effective electrical conductivity to effective ionic mobility does not, under these circumstances, depend upon the strength of the 1318 /Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

electric field. This appears t o suggest that internal structural changes (expected to impact similarly upon small and large ion fluxes) may be responsible for the nonlinear current/ voltage-drop behavior and not, necessarily, nonlinear electrical double-layer effects. Finally, the existence of Ohmic conduction and Fickian diffusion (assumption iv) within the microinterstices of the skin may be viewed as realistic given the relative dilution with which most substances traverse the skin, and Stokes flow is assured owing to the smallness of the length scales encountered. It is presumed, particularly given the general impermeability of the skin, that “sufficiently slow” transport circumstances may always be identified for which the time constraints underlying validity of the theory can be met.

Idealized Transport Pathway Formulas The Appendix describes derivations of the effective trans@, ii“,, U*, and D* for the highly port coefficients K, PE, idealized transdermal transport scenarios of Figures 3 and 4. Table 1illustrates how these expressions may be used to predict transdermal transport fluxes in in uitro, diffusion-cell studies involving several types of transport phenomena. Intercellular/Shunt Route: Physical DescriptionFigure 3 provides a model idealization of the most commonly identified transdermal route for transport of charge, fluid mass, and solute species. Its principal features are discussed below in the context of current flow. Consider the transport of charge across the skin. The intercellular/shunt pathway attributes the macroscale (i.e., -1 mm) electrical conductivity (a) of the skin to the occurrence of ion flow through aqueous pathways within the stratum corneum, as well as t o parallel flow through the sweat ducts and annular spaces surrounding hair shafts. These three routes are revealed at the mesoscale (i.e., -50 pm) level of description. At this scale, the stratum corneum is seen to be a homogeneous medium possessing an anisotropic electrical conductivity, @SC. The sweat ducts are idealized as circular cylinders penetrating the stratum corneum. The actual continuation of the sweat ducts far into the dermis of the skin is accounted for in the following by associating the sweat ductswith the epidermal length scale ( L ) , rather than the length scale ( h )of the stratum corneum. (This same comment applies t o our idealization of hair follicles as well.) The hair

n

n

n.

n

I

Stratum comeu

\Stratum corneunl

t A ‘L

Corne phase

Lipid (‘oil’)layer

rG&z-

4 Lamall phase

I N

Water layer

’

g-@-[J pores

Lou ;< *-

jSubmlcroacale] 7)Lipid -\)’l io‘( layer

-

Figure 3-The intercellular/shunt route of transdermal transport. The macroscale

Aqueous pores through lipid bilayers

2RL

Lipid region

level of description of the skin corresponds to the view of the skin sample in the diffusion cell of Figure l b At the mesoscale level of description, the stratum corneum is revealed to be perforated by circular cylindrical sweat ducts and annular channels between hair shafts and follicles. At the microscale level, the stratum corneum consists of rectangular, nonconducting corneocyte cells (presumed to possess a relatively infinite length in the direction perpendicular to the page) separated by rectilinear lamellar layers. The lamellar layers are found, at the microscale level of description, to consist of parallel lipid and water layers Transport occurs either through the shunt (hair follicle and sweat duct) pathways or between the corneocyte cells via either the lipid or water layers

Figure 4-The transcorneocyte route of transdermal transport The macroscale level of description is identical to that depicted in Figure 3. The mesoscale level of description characterizes the stratum corneum as alternating layers of lamellar and corneocyte phases. Each of these phases possesses a substructure at the microscale level. The corneocyte cells are depicted as perforated by circular cylindrical aqueous pores through a nonconducting keratin environment. The lamellar regions consist of alternating lipid and water layers The lipid layers, at the submicroscale level of description, are shown to be perforated by circular cylindrical aqueous pores. Transport, according to this pathway, occurs through aqueous pores existing both in the corneocyte and lamellar layers.

shafts are also idealized as circular cylinders and centered within cylindrical follicles, with the annular spaces between the stratum corneum and the hair shafts being filled with an aqueous solution that, a t least insofar as its transport properties are concerned, is of identical composition to that which ~ ) . occupies the sweat-duct pathways (see Verde et ~ 1 . ~ The shunt pathways may be regarded as possessing an electrical conductivity, &H. Ion transport through the viable and papillary epidermal layers is presumed to be instantaneous relative to transport throughout the stratum corneum layer. At a finer, microscale (i.e., -1 pm) level, the origin of the anisotropic electrical conductivity within the stratum corneum is shown to be ion transport through the lamellar regions separating the corneocyte cells; that is, these regions possess an anisotropic electrical conductivity. The arrangement of the cells is idealized in the ordered, “brick-and-mortar’’ fashion shown in Figure 3. Transport through the lamellar regions is highly anisotropic since the electrical conductivity of the layers possesses a finite component only in the direction parallel to the corneocyte surfaces. At the submicroscale (i.e., -10 nm) level, the lamellar regions are observed to be composed of narrow (bound) water zones separating lipid bilayers. Although of molecular dimensions, the water and oil layers are idealized as plane-parallel, with the lipid headlwater interface characterized by a surface charge bordered by (according to the idealization) very thin double layers of ions. The layers are assumed to be arranged parallel to the corneocyte surfaces. (It is this arrangement which gives rise to the perpendicular anisotropic nature of the electrical conductivity of the lamellar layers at the microscale level of description.)

Similar comments as applied above to the case of charge transport may be applied to the cases of fluid mass and solute transport. Intercellular/Shunt Route: Summary of Results-The transport coefficients K,PE, a, pp, U*, and D* corresponding to this route of transport are summarized below. Further discussion of these formulas is provided in the next section, where comparisons are made with experimental data. EffectiveHydraulic Permeability-A finite effective hydraulic permeability of the skin may be attributed to the existence of three hydraulic pathways, as characterized by the formulas (Appendix)

The first term in the large brackets corresponds to water flow through sweat duct channels (radius, Rs; net, cross-sectional area fraction, &), the second term to water flow through the annular spaces surrounding the hair shafts (follicle radius, Journal of Pharmaceutical Sciences / 1319 Vol. 83, No. 9, September 1994

RF;shaft radius, RH;net area fraction, @H), and the last term

coefficients of the skin, namely (Appendix)

t o water flow through the bound-water channels of the , ~ S = H 4s stratum corneum itself (area fraction, 1 - ~ S H with &). Reference may be made to Figure 3 for a physical interpretation of the remaining geometrical factors. In general, the effective hydraulic permeability may be expressed in the form

+

r a r e a fractiod of transport hydraulic pathway relative to total no. of of pathway pathways total skin Lsurface a r e 4 This intuitive formulation, while valid for many of the following effective transport properties according to the present intercellular/shunt route, is not, however, valid in general (see the discussion following eq 17). Effective Electrical Conductivity-The effective electrical conductivity of the skin (Appendix),

c

arise on account of water and/or ion flow through the shunt spaces and intercellular water channels of the stratum corneum. Here, 5 denotes the "zeta potential", which is related to the surface charge of lipid bilayer head groups and shunt (i.e., protein) surfaces, and tw is the aqueous permittivity (i.e., dielectric) constant. Similar to the preceding coefficients, the above effective electrokinetic coefficients are of the form

Tiarea fractiod

PE=P+

c

total no. of pathways

of transport pathway relative to total skin burface a r e a j

electroosmotic

of pathway

Mean Solute Velocity-Attention is restricted t o cases of either wholly hydrophilic or entirely lipophilic solute transport. Hydrophilic substances exhibit a mean solute velocity characterized by the formula (Appendix) derives from ion transport through shunt (i.e., sweat-duct and hair-follicle) pathways and through intercellular pathways. Here, OW refers to the electrical conductivity of the aqueous solution presumed to occupy the sweat duct, hair follicle, and lamellar spaces. The function @ expresses the departure of the electrical conductivity of the intercellular (bound-water) pathways from the unbounded-volume value, UW, owing to hydrodynamic hindrance as well as steridelectrostatic exclusion effects. It is important to note that the double-layer current contribution t o eq l l a (explicitly included in eq A24) has been omitted as it is of an extremely small magnitude relative t o the terms retained. In the ideal situation of spherical co-ions of identical hydrodynamic radius, a , a n approximate expression for this function may be offered by assuming the ions to be constrained to move parallel to the midplane between the lipid-head surfaces (through the bound-water channels, of thickness lw) and neglecting the hindrance/exclusion effects caused by electrostatic interactions; thus, employing the partitioning result of Glandt52for spheres into pores confined by parallel plane boundaries and the hydrodynamic resistance result of F a ~ e nit, is ~~ found that @

= (1 -

?)[

1 - 1.004(?)

+ 0.418(?)3 + O(?r] (llb)

Equation l l a is of the generic form

BE

c

total no. of pathways

r a r e a fractiod of transport pathway relative to total skin b u r f a c e area_]

of pathway

Effective Electroosmotic Permeability and Streaming Potential Conductivity-The effective electrokinetic coupling 1320 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

Here, pw is the viscosity of water, KSHthe hydraulic permeability of the shunt routes (cf. eq A50), z the charge of the convecting/diffusing molecular species, F the Faraday constant, R the gas constant, T the absolute temperature, DW the (unhindered) diffusivity coefficient of the diffusing species in the aqueous regions, and How,H& a pair of "hindrance" coefficients (see the discussions following eqs A31 and A48), respectively characterizing the hindered convective and diffusive transport of the (relatively finite size, nonsphericalshaped) hydrophilic solute species through the bound-water regions between the lipid bilayers. Some insight into these latter factors may be gained by experimental lipid-bilayer transport studies involving hydrophilic substance^.^^ In general, our approach is to employ continuum theoretical analyses, of the type made by F a ~ e or n ~Saffman,36 ~ as has been done in arriving at eq l l b . Recent efforts37to estimate hydrodynamic hindrance effects during solute transport through human skin by use of the (cylindrical-pore)R e n k i r ~ equation ~~ constitute an alternative, less-detailed (though complementary) approach to characterizing hindered transdermal transI port. In the case of a lipophilic solute, we have

This result reflects an assumption that the lipid bilayers are immobile, or are a t least impermeable to convective flow.

I-’ ,

The mean solute velocity may be regarded as being of the generic form

C

U* =

total no. of pathways

[ i t y oftransport pathway

mean solute velocity through pathway

1

specifically having identified the factor

as representing the tortuosity of the intercellular pathways, and with the tortuosity of the (“linear”)shunt pathways given by a factor of unity. Effective DispersivitylDiffusivity-The effective diffusive transport of solute through the skin may either be of a convective-dispersive nature (see Discussion section),reflecting a nonlinear enhancement of diffusion by convection, or a purely diffusive nature. This depends both upon the nature of the applied external fields and the type (i.e., hydrophilic or lipophilic) of diffusing substance. Thus,

characterizes the effective dispersion coefficient of a hydrophilic solute species, whereas

D* = (&@o

describes the effective diffusion coefficient of a lipophilic substance. Here, DOis the (unhindered) diffusivity coefficient of the diffusing species in the oil lamellae, I is a convective dispersion parameter (whose explicit functionality may be found elsewherez2),and HL is the hindrance coefficient characterizing hindered diffusive transport of the lipophilic species through the lipid bilayers parallel to the bilayer surfaces. As with the aqueous analogs discussed following eq 13, the hindrance factor Hb may be estimated on the basis of continuum t h e ~ r i e s . ~Insight ~ , ~ ~ may also be gained by experimental lipid-bilayer transport studies involving lipophilic s~bstances.3~

transport necessary to permit transport via this route to any significant degree (together with the corneocyte-membrane transport necessary for penetration into the corneocytes)has often been implicitly regarded as extremely small relative to lateral bilayer and shunt transport. However, this may not necessarily be the case. As discussed in the following section, our results suggest that small ions in the presence of an electric field may be able to permeate the bilayers a t an appreciable rate. Moreover, in a subsequent article,4O evidence is offered to suggest that the transcorneocyte pathway may be the pathway for even large molecular transport, particularly under conditions of relatively “high-strength, pulsed electrical fields. In general, the transcorneocyte route may be viewed as representing a viable route for relatively large fluxes of charge and hydrophilic solute species, at least to the degree that the route is accessible (i.e., naturally, owing t o the smallness and hydrophilicity of the transporting substance, or unnaturally, say, by the “electroporation” of lipid bilayers, as described in a subsequent studPo). Consider the four-scale pathway depicted in Figure 4. As in the preceding case, it is useful to momentarily focus attention on current flow in order to clarify various physical features of this pathway. The effective electrical conductivity a t the macroscale (i.e., -1 mm), is seen to be a manifestation of ion transport through a “resistance-in-series” arrangement of corneocyte cells (thickness Z C ) and lamellar zones (thickness 1 ~ ) The . origin of the corneocytes’ conductivity, as is evident a t the microscale (i.e., -0.01 pm) level of description, lies in the existence of aqueous pores (idealized as circular cylindrical, of radius Rc) between nonconducting keratin fibers. Also at this scale, the lamellar zones are comprised of oil and water layers, of respective thicknesses lo and 1 ~ The . oil layers possess a finite electrical conductivity on account of the existence, a t the submicroscale (i.e., -1 nm) level, of small aqueous pores (radius, RL)piercing the lipid bilayers. It is important to point out that the present scenario neglects the parallel transport occurring in the lamellar spaces between the corneocytes as well as through the sweat ducts and hair follicles. This is justified to the extent that transport through the transcorneocyte route is of a sufficiently different rate than transport through the intercellularlshunt route. The results of the next section generally verify this postulate.

Transcorneocyte Route: Summary of Results- Expressions for the effective transport coefficients E, &$, a, pp,U*, and D*) appropriate to this case are summarized below. Effective Hydraulic Permeability-The effective hydraulic permeability of the skin (Appendix)

The effective diffusioddispersion coefficient is of the form

D

][

= total no. of pathways

solute diffusivity r s i t y oftransport x of dispersivity pathway in pathway

Transcorneocyte Route: Physical Description-This route (see Figure 4) of transport has not been considered seriously in the literature as offering a viable transdermal pathway for charge, fluid-mass, and/or solute transport. This would appear to owe in part to the fact that the translamellar

may be regarded as rate-limited by the permeability of the oil layers to water transport normal to their surfaces. This holds true so long as the area fraction (&) of aqueous pores in the corneocytes is significantly larger than the area fraction ( 4 ~of ) pores through the lipid bilayers. Journal of Pharmaceutical Sciences / 1321 Vol. 83, No. 9, September 1994

1

to permeate the skin according to this idealized route of transport. The mean solute velocity of hydrophilic substances (Appendix),

fractional thickness of corneocyte layers relative to stratum ,corneum thickness

fractional thickness ' of lamallar layers relative to stratum .corneum thickness

. derives from convective flow resulting from a pressure gradient (first term in the above) across the skin and iontophoresis (second term). A third, electroosmotic term has been omitted in the above owing t o its extremely small relative magnitude (see eq 13, where this term is retained). The hydrodynamic hindrance factors @, H:, with i = C and 0, respectively characterize hindered convective and diffusive transport in the circular-cylindrical water pores of the corneocytes (C) and lamellae. Theoretical estimathrough the pores of the oil (0) tion of these molecular-size-and shape-dependent coefficients is illustrated by the example of a transporting molecule possessing hydrodynamic radius a. In this limiting circumstance, the departure of the hindrance parameters @, Hi' from unity may be entirely expressed in terms of the parameters

The results of Brenner and Gaydos41and Mavrovouniotis and Brennel.42may be respectively used to show that, for relatively small hydrodynamic radii,

@ x 1+ 2 4 - 4.922;

+ O(ni3)(Ai<< 1)

(22)

whereas for large hydrodynamic radii,

1-1-

fractional thickness of corneocyte layers relative to stratum 'corneum thickness

1

(24) The discussions following eqs A31 and A48 provide further information regarding the calculation of the parameters @,

H,'. Effective Dispersitiuity-The effective dispersivity of hydrophilic solute through the tissue may, under the conditions of relatively small aqueous-pore convection, be approximated by

D*

1

electroosmotic coefficient of lamallae parallel t o surface normal n Mean Solute Velocity-Only hydrophilic substances are able

[

1322 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

E

H&D,

H& R&AF 2 +48DA hpw )

ZFR,AV

~

+

(

RTh

2

) DwH'

Table 2-Geometrical and Physicochemical Characteristics of Human Skine

Geometry Sweat ducts

Source

Physical Chemistry

Source

Electrical conductivities

Rs=5prn no./cm2= 100

#s = 7.9 x 10-5

Hair follicles RH= 50 pm

b

OW = 1.2

ref 74 (upper arm)

00 = 0.0 (Q m)-’

b

ref 74 ref 74 (abdomen)

RF = 51 pm no./cm2 = 60 +H=1.9~10-4

Shunt pathways 4 S H = 4s + 4H = 2.7 x I 0-4 Stratum corneum A/ (=k) = 0.05 pm h=23pm IN (=k) = 1 pm

L=7A /o = 58 8,

0.1 M NaCl

(Q m)-I

insulator

Aqueous permittivity GW = 8.9 x C2(N m2)-i Aqueous viscosity pw = 0.01 g (cm s)-i Zeta potential

ref 67 b

(=50mV

Molecular diffusivities &, = = 10-6 cm2 s-1

ref 73 ref 45 ref 73 ref 72 ref 72

ref 67

b

b

R~=l08, h=15pm L = 350 pm +c = 0.5 Rc=l88,

ref 72 ref 29 ref 45 ref 40

a The numerical estimates listed in this table should be viewed as merely representative. Large variations may occur owing to variations of anatomical location, ion composition, skin hydration, etc. Authors’own estimate.

with the first term characterizing pure molecular diffusion through the corneocytes, the second term representing convective dispersion driven by a pressure gradient, and the third term indicating convective dispersion driven by iontophoretic convection. The factors H& and H& (to be distinguished from @ and H:) reflect convective dispersive “hindrance” effects arising in the aqueous pores of the corneocytes (see discussion following eq A48). Employing the idealization of the transporting molecule as possessing a hydrodynamic ~ ~ ~again 4 ~ be used radius a,the former theoretical r e s ~ l t s may to show that, for small molecules,

H;

=

&0.158 ln2’ ;1

- 0.901 In AG1

(1- Awl6

+ 1.3851

(a,

<< 1)

(27) whereas for large molecules

Comparison with Experimental Data The purpose of this section is to compare predictions of the theory deriving from the “intercellular/shunt”and “transcorneocyte” pathway scenarios with experimental data appearing in the literature. These comparisons are used to probe the nature of underlying mechanisms for several types of transdermal transport phenomena. Table 1 identifies the five forms of transdermal transport considered below; these are (i) forced mass convection, (ii) current flow, (iii) electroosmosis, (iv) iontophoresis, and (v) passive diffusion. Each of these phenomena are distinguished by a single type of flux (charge, fluid-mass, or solute) and a single type applied field (electrical, pressure-gradient, or concentration-gradient). Thus, forced convection refers to the flow of fluid mass (water velocity, B) in response to a pressure drop (AP)across the skin, current flow to the flow of current

(2”)in response to a voltage drop ( A n , electroosmosis to the flow of fluid mass (water velocity, B) in response to a voltage drop ( A n , etc. Obviously, certain of these phenomena will occur simultaneously (e.g., current flow will occur simultaneously with electroosmosis). Moreover, each type of phenomenon is characterized by at least one transport coefficient (note that for iontophoretic and passive diffusicn phenomena, a second coefficient, the partition coefficient Kl, arises). In diffusion-cell experiments, fluxes of charge, fluid mass, and/or solute are measured as functions of various applied fields. By considering the forms of the transport laws shown in Table 1, it is then possible to experimentally deduce the following transport coefficients: (i)the hydraulic permeability, (ii)the electrical conductivity, (iii) the electroosmotic permeability, (iv) the product of solute velocity and partition coefficient, and (v) the product of solute diffusivity and partition coefficient. These same transport coefficients may be predicted on the basis of the present theory in terms of structural features of the stratum corneum, the nature of the transporting substance, the transdermal pathway, etc., by use of the formulas shown in Table 1. These formulas are listed for each type of phenomenon according to whether a n intercellular or shunt route is followed (route 1)or a transcorneocyte route is followed (route 2). Two equation numbers are listed under each classification. Thus, forced convection occurring by a transcorneocyte route is characterized by a hydraulic permeability that is given by either eq 17 or 30. The first of each pair of numbers listed in the columns labeled routes 1 and 2, designates a general formula, whereas the second employs the data shown in Table 2. Table 2 summarizes the estimates employed for the various geometrical and physicochemical parameters appearing in eqs 10-25. The actual magnitudes of these parameters may vary considerably from one skin sample to the next, depending on such factors as (i) anatomical location of the skin sample, (ii) subject-to-subject variability, (iii)internal structural integrity of the skin (e.g., which may be altered by passage of a sustained electrical current), and (iv) type of mammalian skin (i.e., some of the data sets used for comparison have been obtained using mouse skin samples). It should be pointed out that the multiplicative factor (Lh) appearing in many of these formulas is uniformly omitted in Journal of Pharmaceufical Sciences / 1323 Vol. 83, No. 9, September 1994

these calculations owing, a t least in part, to the fact that the data selected for comparison acknowledge (either explicitly or implicitly) the stratum corneum to be the chief barrier to transport; in other words, formulas 2, 4, and 6 underlying these studies (as often occurs in the transdermal drug delivery literature) are taken to apply over the thickness (h)of the stratum corneum and not over the thickness ( L )of the entire skin sample. Forced Convection-Forced convective mass transfer refers to the transport of matter following the application of a pressure gradient. As shown in Table 1, this mode _of transport is characterized by the hydraulic permeability, K. By employing the data of Table 2, the expressions 10 and 17 for the effective hydraulic skin permeabilities appropriate to intercellularhhunt and transcorneocyte pathways may respectively be written as

K = 1.33 1 0 ~ +~3.57 4 ~ 10-ll4~+ 9.74 x 1.2

10-l~

4.27 x 1.5 x 10-l'

+ 6.5 x

39.0 x 10-64, u--

~

2.5 x 10-'(58.0

+ 7.04,)

lop6&

1- 65.0 x

,.,

39.04, (31b) 1.45 65.04,

+

The units of 6 are (Q m1-l. The transcorneocyte pathway (cf. eq 31b) is observed to be far more conductive than the intercellularhhunt pathway, at least for lipid bilayer water fractions, on the order of approximately (PL 2 An estimate of the intercellular ion-resistance function Q, can be made on the basis of eq l l b . Assuming the ions to be of hydrated hydrodynamic radius a x 0.3 nm, then Q, x 0.08, and eq 31a gives

o = 3.7

(324

(29) with approximately 40%of the current passing through the shunt routes. A value for the specific skin resistance Q = h h may be calculated from eq 32a on the basis of the value of h shown in Table 1;this gives

and

K=

and

x 2.8 x 10-'44L

(30)

Q x 4000 (Q cm2)

(32b)

with both effective permeability coefficients possessing the This result is consistent with the experimental skin resistance units of cm2. values reported by Scott et aL7 in the limit of long current Equation 29 expresses the hydraulic permeability as a exposures. Moreover, their conclusion that a large amount function of the area fraction of sweat duct (4s) and hair follicle of current passes through the shunt routes of the skin is also ( 4 ~ pathways. ) It reveals that the shunt pathways are far borne out by the above result. It should be noted, however, more effective as hydraulic pathways than are the intercelthat a large degree of scatter exists in the reported experilular regions (represented by the third term in eq 29). This mental skin-resistance data,44 perhaps reflecting varying owes to the fact that water flow between the lipid bilayers is levels of structural damage incurred prior to or in the process accompanied by an extremely large viscous stress since the of measurement. Another aspect of skin resistance measurelength scale (ZW) is extremely small. Transcorneocyte water ments that is not addressed in the current model is the time flow, as characterized by eq 30, is seen t o be significantly dependence of skin resistance, a phenomenon frequently smaller than shunt transport (cf. eq 29). observed in the presence of a prolonged constant (dc) or Forced convective flow across the epidermis appears to be alternating (ac) c u r r e n t . l l ~ ~ ~ of lesser practical significance than the other forms of The possibility that ions traverse the corneocytes also exists. transport (considered below). Thus, we are unaware of Experimental evidence for the possibility of transcorneocyte published experimental data to which the results of eqs 29 ion transport is provided by the data of Campbell et Their and 30 may be directly compared. One example, however, results show a large variation of skin resistance with respect where this phenomenon may arise and possibly exhibit some to hydration of the stratum corneum. This directly implicates importance is in the application of sonophoresis4J for drug the transcorneocyte route of ion transport, inasmuch as water delivery enhancement across the skin. Although numerous in the stratum corneum is largely contained within the factors may ultimately be responsible for the enhancement corneocytes. Further, theoretical evidence for a transcorneoof drug transport in the presence of ultrasound, the propagacyte route of ion transfer is offered in the next section. tion of longitudinal pressure waves through the skin will Electroosmosis-Electroosmosis refers to the flow of fluid produce a rapidly oscillatory (as well as a steady) forcedaccompanying an applied electric field, typically arising as a convective motion. This will give-rise t o a rapidly oscillatory consequence of ionic motion within electrical double layers effective hydraulic permeability, K, of the skin, whose instangathered near fured charged surfaces within the medium. The taneous magnitude may be considered to be described by eqs occurrence of electroosmosis within human skin has fre29 or 30. The study of acoustic propagation in porous media, quently been observed, a t least since the study of and including the theoretical determination of the instantaneous may be explained by the fact that skin, under typical physi~ hydraulic permeability, is an ongoing research e f f ~ r t . ? ~ , ?ological conditions, bears a negative fixed charge (above a pH Since K serves as a measure of instantaneous convection of ca. 3).49 caused by a propagating acoustic wave, the effective hydraulic Some of the most direct observations of electroosmosis permeability, K, of the skin may play a role in the ultimate (though, see also refs 16, 46, and 48) are those reported by understanding of the still-poorly understood phenomenon of Pika1 and Shah49for hairless mouse skin. Their measureskin sonophoresis. ments (see particularly their Figure 8) of water flow acC u r r e n t Flow-Table 1 indicates that current flow is companying a steady applied current may be used to directly characterized by the effective electrical conductivity of the deduce a value of the effective electroosmotic coefficient pE skin, a, as described by eqs l l a and 18. The data of Table 2 (pE/p& = L,, in their notation). This value is shown in may be used to express these formulas in the respective forms Figure 5. Table 1 indicates that the coefficient pEis theoretically characterized by eqs 12 and 19, corresponding to b = 1.2(1.20 x 2.34 x lou4@,) (31a) intercelldadshunt and transcorneocyte routes. These for-

+

1324 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

We begin by noting, upon use of the data of Table 2, with eqs 13 and 20 (see Table 11, that convection of hydrophilic, macromolecular solute across the skin is governed by

U* x 0.02594(4,,

+ O.O42H&)zAV

(35)

in the case of the intercellularlshunt route, and

n*

= 1.0376Hk4,zAV (36) in the case of the tran_scorneocyte route. Here, H i = H,1 (tr_anscorneocyte),and U* is expressed in units of cuds and AV in volts. Dividing the latter by the former, gives 40.0H&,

x cntercellular 2.7 + O.O42H&

i7ranscorneocyte

x

x

Figure 5-Comparison of theoretical predictions and experimental measurements for the effective electroosmotic coefficient of mammalian skin.

mulas, with thedata of Table 2, give

1 --K",

= 2.8 x

(33)

PW

and

These effective electrokinetic parameters are dimensionless. Equation 33, based upon the intercellular route of transport, offers a prediction that is 4 orders of magnitude smaller than that reported by the It also significantly underpredicts the human skin electroosmotic data of Sims et U Z . ~ The ~ transcorneocyte formula (34), with 4~ 1 (i.e., 4~ 2 O . l ) , shows extremely good agreement (bearing in mind the fact that the data pertain to mouse, rather than human skin: see the experimental results of Phipps et aZ.,13where iontophoretic fluxes are as much as 1 order of magnitude smaller for human skin than for pig and/or mouse skin). This lends evidence for the transcorneocyte route as a possible mode of small ion transport. The possible predominance of a transcorneocyte route for electroosmotic flow may reflect the fact that the majority of water in the stratum corneum is contained within the corneocyte~.~~ Iontophoresis-Whereas the above comparisons suggest that small ions may potentially follow a transcorneocyte route (at least in the presence of an applied electrical field), abundant evidence in the literature has been offered50 to support the fact that larger molecules are most commonly forced to follow the intercellular/shunt route. This is confirmed by our calculations (as reported both in this and in the following subsection) as well. Iontophoresis refers t o the movement of ions owing to the application of an electric field. Large molecules undergo a similar phenomenon known as electrophoresis, which refers to the motion of a charged macromolecule or colloidal particle driven by convection of smaller ions gathered around the charge groups (or charged surfaces) of the larger entity. Both iontophoresis and electrophoresis occur in human skin, though typically both are described in the literature as iontophoretic phenomena. Below, our theory is employed to offer a prediction of the iontophoretic flux of a charged solute across the skin in a diffusion-cell experiment, resulting from application of an electric field.

-

where the limit H& 1 for the hindrance factor (in the aqueous zones separating the lipid bilayers) corresponds t o the case of an extremely small molecule and H& I to the case of a large macromolecule (as further discussed below). This relation reveals that iontophoretic velocities resulting from a transcorneocyte pathway may be several orders of magnitude larger than those resulting from an intercellular pathway (typically, H i x 1). As established by the data comparison made below, iontophoretic velocities of such a magnitude as those suggested by the transcorneocyte pathway far exceed typically observed iontophoretic velocity magnitudes. Hence, this pathway is not considered further. The flux, J , of solute across a skin membrane in a diffisioncell experiment is represented by formula 6. Equations 6-8 may be solved undergteady-state conditions, with Cz = 0, to obtain the result

J%:*cl

(37)

with the approximation symbol in 37 reflecting the fact that the diffusion contribution to eq 37 has been neglected owing to its relatively negligible magnitude [cf. eq 15 (with AP = 0) and the dgta of Table 21. To obtain an explicit numerical result for J , a value of the partition coefficient K1 is required, as well as the value of the solute concentration C1in the donor compartment of the diffusion cell. In addition, eq 35 requires specification of both the skin voltage drop and the hydrodynamic hindrance coefficient for solute diffusion in the aqueous regions separating the lipid bilayers. Comparison is to be made with published data for hydrophilic substances that are assumed to be equally soluble in the aqueous pore spaces as in the aqueous donor solution. In addition, steric-exclusion (and molecular-shape-dependent) effect^,^^^^^ as well as electrostatic repulsive effects, are neglected; thus, the partition coefficient (employing the data of Table 2) is given by

K '-

1,

+ A1

1,

+I,

= 2.34 x

lop4

(38)

(neglecting the shunt route contribution, for reasons discussed below), expressing the area fraction of the skin surface accessible to the hydrophilic substance. Combination of eqs 35, 37, and 38, with the data of Table 2, furnishes

J = 21.85(0.26 + 42.0H&)zAVC1 (nmol/cm2 h) with C1 in units of m o m and AV in units of volts. Observe that for very large, hydrophilic molecules that exhibit a Journal of Pharmaceutical Sciences / 1325 Vol. 83, No. 9, September 1994

sufficiently small hindrance coefficient, the shunt pathway contribution to the above flux expression may predominate, in which case the degree of hindrance that occurs within the aqueous pores of the lamellar zones of the stratum corneum will be of relatively small importance. This conclusion, however, must be modified by the fact that hindrance effects arising on passage of a hydrophilic molecule from the shunt pathways to the viable epidermis are neglected in the current pathway idealization. This comment also applies to the case of passive diffusion (cf. eq 43 and the discussion thereofl. In addition to the problem of accurately modeling partitioning phenomena, one of the principle difficulties that arises in offering predictions of macromolecular transport across the skin is the estimation of the various hindrance factors that enter into the theoretical description. In the present case of iontophoretic transport through the intercellular aqueous pathways of the stratum corneum, a value is required for the factor H& reflecting hydrodynamic hindrance effects accompanying solute transport through the thin bound-water layers separating the lipid bilayers. For very small ions that do not bind with the polar lipid head groups bordering the aqueous layers, this factor may be estimated from eq l l b . Estimation of H& for hydrophilic macromolecules which are sufficiently large (hydrodynamic radius exceeding a = 0.5 nm) to extend into the lipid regions is a challenging problem. Whereas a good deal of theoretical and experimental understanding has been gained in recent years regarding lateral hindered diffusion of macromolecules within single lipid b i l a y e r ~considerably ,~~ less is known of lateral macromolecular diffusion in multiple bilayers. The multiple bilayer study . ~a decrease ~ of approximately 3 orders of of Vaz et ~ 1reports magnitude in the translational diffusion coefficient for a lipid molecule of approximately 0.5-nm radius and with a rod length equivalent to the lipid bilayer thickness. In their study, the multibilayers were separated by highly viscous glycerol layers, rather than by water layers. For the case of water layers, this reports a decrease in translational diffusion of around two orders of magnitude. These results for large may suggest an estimate on the order of H& x macromolecules diffusing through the aqueous zones of the lipid multibilayers, particularly when it is noted that the “liquid-crystalline’’lipid regions bordering the aqueous spaces provide a highly viscous medium relative t o the aqueous medium through which the molecules travel. In any case, the following comparisons with experimental data are restricted to relatively small molecules (molecular weights ranging between 100 and 365 Da). It is assumed that H& z 1.0, whence the preceding equation yields

J

= 917.OzAVC1(nmol/crn2h )

In particular, we consider the data reported in the studies . former ~ ~ of Burnette and M a r r e r ~and ~ ~Green et ~ 1 The provides solute flux data across nude mouse skin for thyrotropin releasing hormone (MW 362.5, z = 1, C1 = 8.8 mM, C2 = 0 mM) at pH 4. The latter provide data for the average steady-state flux across hairless mouse skin of six different anionic permeants (molecular weights ranging between 100 and 200 Da, charge, z = -1, C1 = 10 mM, and Cz = 0 mM). A constant current of 0.31 d c m 2was applied in the first study and 0.36 d c m 2in the second. Since a skin resistance value was not reported in either study, we have used the voltagecurrent correlation provided by Kasting and Bowman44 in order to arrive at an estimated voltage drop across the skin of approximately 4 V for all the data sets considered. Figure 6 offers a comparison of our theoretical prediction (with z = -1, C1 = 10 mM, AV = 4 V) with the two sets of experimental data. The fair agreement between theory and 1326 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

8‘ O 0 0I

r

(nmoll cm ’/h)

Theory Experiments Experiments (intercellular) (Burnette & (Green M a ~ ~ e r o ~ ef ~ )dS4)

Figure 6-Comparison of theoretical predictions and experimental data for the effective transdermal iontophoretic flux of solute (presumed to equipartition into the aqueous pores, of sufficiently small molecular weight to be characterized by negligable hydrodynamic hindrance). 0 I

I

I

2

5 Kpo

I

0

0 ‘ 18

I

I

I

I

I

20

22

24

26

28

Lateral corneocyte dimension 4 ( p m )

Figure 7-Comparison of theoretical predictions and experimental measurements for the normalized solute permeability of water across human skin as a function of corneocyte surface area (represented by the lateral corneocyte dimension h).

experiment, acknowledging our uncertainty regarding the estimation of partitioning and hindrance effects, is especially meaningful when contrasted with the severalfold overestimate offered by the transcorneocyte-route prediction (eq 36 ). It indicates that the intercellular/shunt route may be the pathway followed by charged macromolecules in iontophoretic drug transport. Additional support for this conclusion will be offered by further iontophoretic comparisons between theory and experiment in a subsequent paper.40 Molecular Diffusion-Both molecular diffusion and convective-dispersion phenomena arise in the transport of small and large molecules across human skin. Whereas circumstances may arise in which a convective-dispersion phenomenon predominate^,^^ the more prevalent mode of solute spread through the skin appears to be that of molecular diffusion. Evidence supporting an intercellular route for small as well as large solute diffusion through the stratum corneum has They provide experimental been offered by Rougier et data, obtained by measurements performed on living human subjects at several different anatomic sites, relating corneocyte cell surface area to rates of transdermal water loss and benzoic acid permeation. Their data for transdermal water loss are provided in Figure 7 in terms of a relative water permeability value (see below),normalized by the permeability for the

largest corneocyte surface area. Note that we have recast the . ~ of~ corneocyte surface area values of Rougier et ~ 1in terms a lateral corneocyte dimension IT (see Figure 3) by equating the reported corneocyte surface areas with the expression 1/2?tlT2. Direct comparison with the present theory can be made upon defining a solute permeability coefficient -1

AC v

Kp =

(39)

Potts and provide an explicit expression for their membranejwater partition coefficient K,, as well as for the (“finite molecular volume”) hindrance factor H,! (with i = W or 0) underlying our coefficient D* (i.e., upon making the interpretation H,’ = D,/Do, in the terminology of ref 17). On the basis of these expressions, they offer correlations of a wide range of published experimental data for passive molecular diffusion across the epidermis. In the course of these correlations, they arrive a t the following pair of conclusions:

Dm = constant (i) -

Under the conditions of steady-state passive diffusion, eqs 6 and 8 may be substituted into the above (note the definition of the A-operator following eq 2) to yield (40) which assumes the stratum corneum (of thickness h ) to be the rate-limiting barrier for diffusive transport and where we have chosen as a reference C2 = 0. Employing eqs 38, 40, and 15 gives

corresponding to the transport of hydrophilic substances via an intercellular route (neglectingthe contribution of the shunt pathways). Relative to the solute permeability (Kpo)at a given lateral corneocyte dimension, IT (e.g., IT(,), holding all other parameters constant, the above formula gives

cm/s)

H,’a

(ii)a x 500 p m These results are compared with our theoretical predictions below. Focusing upon the intercellular/shunt route of-transport, we have from eqs 15 and 16 (with AV = 0 and A€’= 0)

The diffusion pathlength, a, may be defined as (cf. the discussion following eq 14)

= 360 pm

The advantage of examing relative permeabilities, as is made evident by this formula, is that hydrodynamic hindrance and nonideal partitioning effects cancel, and one is left with a purely geometrical relation between corneocyte size and relative permeability. This formula may be compared with ~ is done in Figure the experimental data of Rougier et U Z . , ~ as 7, with ZTO = 27 pm representing the largest lateral corneocyte dimension examined in their study; moreover, as in Table 2, IN = 1pm and A1 = 0.05 pm. The approximate agreement between theory and experiment lends further support to the validity of an intercellular route for passive diffusion of solute species through the skin, [Note that the observation of Rougier et al.55 that skin permeability appears to “double with a doubling of corneocyte surface area” (i.e., l / 2 ? t ~ ~is) confirmed by the above formula.] Potts and Guy17 have analyzed a large amount of transdermal data appearing in the literature (notably, the data summarized by Flynn56) for passive diffusion of compounds ranging in molecular weight from 18 to 750 Da. Their data are reported in terms of the solute permeability, Kp, which they further express in terms of various membrane transport, partitioning, and geometrical factors. In particular, their formula 1may be compared with eqs 39 and 40 above to give

(42b)

The latter value has been obtained by use of the estimates of Table 1and approximately confirms the estimate (ii)of Potts and Guy17(particularly when it is noted that the precise value of a depends upon lateral corneocyte dimension I T , which may vary considerably over the surface of the body, as in Figure 7). It should, however, be noted that this diffusion pathlength possesses a physical meaning only insofar as the intercellular pathway predominates over the shunt pathway. According to the relation preceding eq 42b, this will generally occur for lipophilic solutes and will also occur for hydrophilic solutes to the degree that the first term in the same relation above is much larger than the second (cf. eq 43 below and the discussion thereof). Combining the above results yields

h

(lipophilic)

Divide by Hi (with i = W or 0 )t o obtain (hydrophilic)

W+-

-_

(lipophilic) Next, employ the data of Table 2 to obtain (in units of c d s )

where (in the notation of ref 17) K, is a membrane partition coefficient (to be distinguished from K1, a t least insofar as K, does not include the area-fraction-exclusion effects inherrent in Kl,cf. eq 38), D, is the “permeant diffusivity within the membrane” and a is the “diffusion pathlength”.

D* - 12.9 x --

H:h

2.9

lop5+ 10-~

Hh

(hydrophilic)

(43)

(lipophilic)

Note that for sufficiently small values of the hindrance Journal of Pharmaceutical Sciences / 1327 Vol. 83, No. 9, September 1994

coefficient, H& (i.e., for large macromolecules), the shunt pathway (represented by the second term in eq 43, for hydrophilic solute) may predominate as a route of solute transport through the skin. To proceed with our comparison via eqs 41 and 43 with the result i,17 an explicit expression for K, in terms of K1 is required. This comparison will be postponed until a future study, which will be devoted, in-part, to the theoretical characterization of the coefficient K1.

Discussion The theory outlined in this article provides a mathematically rigorous, self-consistent framework for investigating transdermal transport phenomena on the basis of explicit mechanistic hypotheses, such as those underlying the proposed intercellularhhunt and transcorneocyte transport pathways. Two new features of transdermal transport that have arisen in the theory, and that deserve special experimental attention in the future, concern (i) the hydraulic permeability of the skin (cf. eqs 10 and 17) and (ii) the convective-dispersioncontribution to the effective diffusivity of solute (cf. eqs 15 and 25). As far as we are aware, no experimental measurements have yet been reported for the skin’s hydraulic permeability, K. Such measurements might entail (cf. eq 4) measurement of water flow across the stratum corneum under the conditions of a prescribed transdermal pressure drop. These results may then be compared with the theoretical predictions 10 and 17 (see also eqs 29 and 30) to investigate the pathway(s) of forcedconvection water transport. At the same time, as described following eq 30, these results may have indirect bearing on the nature of ultrasonic wave propagation through the skin. The convective dispersion contributions to solute diffusion characterized in eqs 15 and 25 are of two types: a forced convective type, pertinent to either shunt-route transport (cf. eq 15) or transcorneocyte transport (cf. eq 25), or a n iontophoretic type (cf. eq 25), corresponding to transcorneocyte transport. In general, convective dispersion refers to the enhancement of molecular diffusion caused by convectivevelocity gradients. Its physical origin may be identified with the ability of (spatial and/or temporal) convection inhomogeneities to magnify the random walk nature of the diffusing substance. It has been observed in numerous physical applications, as described thoroughly in a recent textzz on the subject. One possibility for measuring convective dispersion as described in eq 25, involves application of an oscillating electric field across the skin. For example, under the conditions of a sinusoidally varying electric field (possessing zero time-average), the charged solute (or even small ion) transport enhancement relative to the passive transport case may be identified with the transcorneocyte convective dispersion term appearing in eq 25. That is, there is no net iontophoretic convection under these alternating-field conditions since the solute velocities defined in eqs 13 and 20 depend linearly on the applied electric field. However, the convective dispersion term in eq 25 increases with the square of the potential drop; this means that the magnitude of the convective-dispersion term will not change sign with an alternating electric field. This potentially new form of charged solute transport enhancement should be examined experimentally. Other new features of the analysis include the possibility for explicitly accounting for the various kinds of hindrance effects that accompany solute convection and difision through the stratum corneum. The potential exists t o employ formulas of the type I l b and 26-28 to investigate the role of solute size, shape, charge, etc., on effective transport rates. 1328 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

While the comparisons made in the previous section between theoretical predictions and experimental data are encouraging, many refinements are required in order to encompass within the theory the wide range of experimental observations already reported in the literature. In the remainder of this section, we consider the significant physical limitations of the current theory (as well as of the transport mechanisms proposed) and discuss possibilities for removing these limitations in future studies. A distinction is made between transport phenomena involving small ions in the presence of an electrical field and transport phenomena involving macromolecules (as well as passive transport of small molecules). The transport of small ions across the skin in the presence of an applied electrical field is quantified in the present theory by the effective skin electrical conductivity (a)and streamingpotential conductivity (or electroosmotic permeability PE). These coefficients establish a linear relation between an applied electrical field and the net charge and mass fluxes. Frequently, however,lo such linear relationships are not observed. This may be due, in part, to the polarizability of the skin,ll a phenomenon related to the diffuse nature of electrical double-layers within the skin relative t o aqueous pore dimensions. Electrokinetic phenomena involving diffuse double layers require solution, a t the microscale level of aqueous pores within the skin, of the complete Maxwell equations. However, the nonlinearity of these equations is such that the ‘effectivemedium’ methods of analysis employed by EdwardsL8are not directly applicable, these methods being limited to the “linearized” circumstances of very thin double layers. New methods of analysis are therefore required (see ref 57) in order to develop direct relations, such as those summarized in the Appendix, between microscale physics and macroscale transport properties. A second reason for nonlinear voltage-droplcurrentbehavior is the occurrenceof voltage-drop-dependentstructural changes in the multilamellar regions of the stratum corneum. Ion transport across the lamellar regions separating the corneocytes may be anticipated to exhibit a nonlinear dependence upon electrical field strength owing to a dependence of lamellar permeability to ion flow upon the strength of the electric field. This effect, which may be examined by experimental studies of current flow across multilamellar systems, may be easily incorporated into the current theory since the fundamentally linear nature of the microscale transport laws is unchanged. Use of the present theory t o predict rates of passive solute diffusion, or macromolecular transport in general, requires knowledge of partition coefficient(s), as well as of various hydrodynamic hindrance coefficients. Considerable progress toward calculation of these parameters (in terms of solute size and shape and pore geometry) for relatively simple porous media has been reported in the literature. Partitioning of solutes of various sizes and (possibly flexible) shapes into porous membranes perforated by long circular cylindrical pores has been considered by several authors.51*52~59~60 Calculations of the various hindrance factors that enter into the present theory for spherical particles transporting through circular pores have existed in the l i t e r a t ~ r e . 4 ~Determina9~~ tion of the relevant partitioning and hindrance coefficients under the far more complex circumstances that arise in transdermal scenarios will require, as in previous analyses, application of standard statistical as well as continuum mechanical theories. Especially, we note the importance of a low Reynolds number61and macrotransport processesz2methods of analysis. Use of these theories for estimating partitioning and hindrance factors based upon molecular size and shape and pore dimensions is obviously necessary. Such

29. theoretical calculations will ideally be performed in conjunction with experiments, as has been done i n the p a s t (see, for 30. example, Colton et a1.62 and Malcone and A n d e r s o d 3 for 31. hindered diffusion in porous media and Hughes et for hindered diffusion in lipid bilayers). Owing t o the very small length scale of molecular interactions in lipid b i l a y e r ~ , ~ ~ 32. molecular dynamics70 calculations m a y also b e necessary for 33. estimating molecular hindrance effects. In general, f u t u r e developments of the c u r r e n t theory 34. should aim at employing more realistic microstructural 35. characterizations of the s t r a t u m corneum than those depicted 36. in Figures 3 a n d 4. O n e of the principal advantages of the 37. current theory is that it m a y be applied t o extremely complex geometrical models. F o r example, application of spatially 38. periodic paradigms such as those described in the Appendix 39. t o disordered a n d fractal type cellular geometries has been 40. performed in recent years, as discussed i n detail b y Alder.20 Chemical binding of solutes within the skin is recognized15 41. t o play an important role in t r a n s d e r m a l drug delivery. 42. Extension of the present theory to include these effects is promising, since the underlying theory of macrotransport 43. processes has often been applied to chemically reactive transport scenarios.22 44.

References and Notes 1. Langer, R. Science, 1990,249, 1527-1533. 2. Bronaugh, R. L.; Maibach, H. I. (eds) Percutaneous Absorption: Mechanisms, Methodology, Drug Delivery; Marcel Dekker: New York, 1989. 3. Masada, T.; Higuchi, W. I.; Srinivasan, V.; Rohr, U.; Fox, J.; Behl, C.; Pons, S. Int. J . Pharm. 1989, 49, 57-62. 4. Singh, J.; Roberts, M. S. Drug Design Delivery 1989,4, 1-12. 5. Prausnitz, M. R., Bose, V. G., Langer, R. & Weaver, J. C. Proc. Natl. Acad. Sci. 90, 10504-10508. 6. Potts, R. 0. In Electricity and Magnetism in Biology and Medicine; Martin Blank, M., Ed., San Francisco Press, Inc.: San Francisco. 1993: KID 1-4. 7. Levy, D.; Kost, J:;Meshulam, Y.; Langer, R. J . Clin. Invest. 1989, 83.2074-2078. 8. Jacob, J . N.; Hesse, G. W.; Shashoua, V. E. J . Med. Chem. 1990, 33, 733. 9. Stephens, W. G. S. Med. Electron. Biol. Eng. 1963,1,389-399. 10. Candia, 0 . A. Biophys. J . 1970, 10, 323-344. 11. Burnette, R. R.; Bagniefski, T. M. J . Pharm. Sci. 1988,77,492497. 12. Kasting, G. B.; Keister, J . C. J . Controlled Release 1989,8, 195210. 13. Phipps, J . B.; Padmanabhan, R. V.; Lattin, G. A. J . Pharm. Sci. 1989. 78. 365-369. 14. Wearley,'L. L.; Tojo, K.; Chien, Y. W. J . Pharm. Sci. 1990, 79, 992-998. 15. Srinivasan, V.; Higuchi, W. I.; Su, M. J . Controlled Release 1989, 10, 157-165. 16. Kubota, K.;Yamada, T. J . Pharm. Sci. 1990, 79, 1015-1019. 17. Potts, R., 0.;Guy, R. H. Pharm. Res. 1992, 9, 663-669. 18. Edwards, D. A. Proc. R . Soc. London A , 1994, in press. 19. Michaels, A. S.; Chandrasekaran, S. K.; Shaw, J . E. AlChE J . 1975,21,985-996. 20. Adler, P. M. Porous Media: Geometry and Transports. Butterworth-Heinemann, Boston, 1992. 21. Bear, J . Dynamics of Fluids in Porous Media; American Elsevier: New York, 1972. 22. Brenner, H.; Edwards, D. A. Macrotransport Processes; Butterworth-Heineman, Boston, 1993. 23. Quintard, M.; Whitaker, S. Transport in Porous Media 1990,5, 341-379. 24. Bensoussan, A.; Lions, J. L.; Papanicolau, G.; Asymptotic AnaZysis for Periodic Structures; North-Holland: Amsterdam, 1978. 25. Sangani, A. S.; Acrivos, A. Proc. R . Soc. London 1983, A386, 263-275. 26. Koch, D. L.; Brady, J . F. Fluid Mech. 1985,154, 399-427. 27. Edwards, D. A. J . Aerosol Sci. 1994,25, 533-565. 28. Taylor, G. I. Proc. R . Soc. 1953, A219, 186-203.

Jarrett, A. (Ed.) The Physiology and Pathology of the Skin; Academic Press: London, 1978; Vol 5. Elias, P. M. Drug Dev. Res. 1988,13, 97-105. BoddB, H. E.; Homan, B.; Spies, F.; Weerheim, A.; Kempenaar, J.; Mommaas, M.; Ponec, M. J . Invest. Dermatol. 1990,95,108116.

Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: New York, 1976. Verde, T.; Shephard, R. J.; Corey, P.; Moore, R. J . Appl. Physiol. 1982.53. 1540-1545. Chidichimo, G.;De Fazio, D.; Ranieri, G. A,; Terenzi, M. Chem. Phys. Lett. 1985, 117, 514-517. Faxen, H. Ann. Phys.' 1922, 68, 89-95. Saffman, P. G. J . Fluid Mech. 1976, 73, 593-602.

Peck, K. D.; Ghanem, A. H.; Higuchi, W. I. Presented at the 1993 AAPS Conference, Orlando, FL. Renkin, E. M. J . Gen. Phvsiol. 1954,38, 225-243. Tocanne, J. F.; Dupou-Cezanne, L.;'Lopez, A.; Tournier, J. F. Eur. J . Biochem. 1989,257, 10-16. Edwards, D. A,; Prausnitz, M. R.; Langer, R.; Weaver, J . J . Controlled Release. submitted. Brenner, H. & Gaydos, L. J. J . Colloid Interface Sci. 1977, 58, 312-356.

Mavrovouniotis, G. M.; Brenner, H. J . Colloid Interface Sci. 1988,124, 269-283.

Wells, P. N. T. Biomedical Ultrasonics. Academic Press: New York. 1977. Kasting, G. B.; Bowman, L. A. Pharm. Res. 1990, 7, 134-143. 45. Camubell. S. D.: Kranino. K. K.: Schibli.' E. G.: Momii. S. T. J . Inveit. Dermatd. 1977, 89, 290-296. 46. Burnette, R. R.; Marrero, D. J . Pharm. Sci. 1986, 75, 738-743. 47. Bagniefski, T.; Burnette, R. R. J . Controlled Release 1990, 11, 113-122. 48. Sims, S. M.; Higuchi, W. I.; Srinivasan, V. Int. J . Pharm. 1991, 69, 109-121. 49. Pikal, M. J.; Shah, S. Pharm. Res. 1990, 7, 213-221. 50. Bodde, H. E.; van den Brink, I.; Koerten, H. K.; de Haan, F. H. N. J . Controlled Release 1991, 15, 227-236. 51. Brochard, F.; de Gennes, P. G. J . Phys. 1979,40, L399-410. 52. Glandt, E. D. AlChE J . 1981,27, 51-59. 53. Vaz, W. L. C.; Stumpel, J.; Hallmann, D.; Gambocorta, A,; De Rosa, M. Eur. Biophys. J . 1987,15, 111-115. 54. Green, P. G.; Hinz, R. S.; Cullander, C.; Yamane, G.; Guy, R. in Vitro. Pharm. Res. 1991, 8, 1113-1120. 55. Rougier, A.; Lotte, C.; Corcuff, P.; Maibach, H. I. J . SOC.Cosmet. Chem. 1988,39, 15-26. 56. Flynn, G. L. In Principles of Route-to-Route Extrapolation for Risk Assessment; Gerrity, T. R., Henry, C. J., Eds.; Elsevier: New York. 1990. PP 93-97. 57. Carr, J. Applicaiions of Center Manifold Theory; SpringerVerlag: Berlin, 1981. N. A,; Meadows, D. L. J . Memb. Sci. 1983, 16, 36158. Peppas, ,,"" 31 I .

59. Casassa, E. F. J . Polym. Sci. 1967, B5, 773. 60. Casassa, E. F. Macromolecules 1976, 9, 182. 61. Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Nijhof: Dordrecht, The Netherlands, 1983. 62. Colton, C. K.; Satterfield, C. N.; Lai, C. J. ATChE J . 1975, 21, 289. 63. Malone, D. M.; Anderson, J. L. Chem. Eng. Sci. 1978,33, 14291440. 64. Hughes, B. D.; Pailthorpe, B. A,; White, L. R. J . Fluid Mech. 1981,110, 349-372. 65. Rein,.H. 2. Biol. 1924, 81, 125-140. 66. Balakotaiah, V.; Chang, H. C. Submitted to J . Colloid Interface Sci. 67. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. 68. Xiang, T.; Chen, X.; Anderson, B. D. Biophys. J . 1992, 63, 7888. 69. Israelachvili, J. N.; McGuiggan, P. M. Science 1988,241, 795800. 70. Bitsanis, I.; Somers, S. A,; Davis, H. T., Tirrell, M. J . Chem. Phys. 1990,93, 3427-3431. 71. Johnson, M.; Langer, R.; Blankshtein, D. Presented at the 2nd US-Japan Symposium on Drug Delivery Systems, 1993, Maui,

Hawaii.

72. Bouwstra, J . A,; de Vries, M. A,; Gooris, G. S.; Bras, W.; Brussee, J.; Ponec, M. J . Controlled Release 1991, 15, 209-220. 73. Scheuplein, R. In The Physiology and Pathophysiology of the Skin; Academic Press: New York, 1978; Vol. 5. 74. Montagna, W.; Parakkal, P. F. The Structure and Function of the Skin, 3rd ed.; Academic Press: New York, 1974.

Journal of Pharmaceutical Sciences / 1329 Vol. 83, No. 9, September 1994

75. Chapman, A. M.; Higdon, J. J. L. Phys. FluidsA 1992,4,20992116. 76. Santos, J. E.; Douglas, J.; Corbera, J.; Lovera, 0. M. J. Accoust. SOC.Am. 1990,87, 1439-1448. 77. Scott, E. R.; Laplaza, A. I.; White, H. S.;Phillips, J. Pharm. Res. 1993,10, 1699-1709. 78. Crank, J. The Mathematics of Diffusion; Oxford University Press: London, 1983.

APPENDIX-Effective Transport Properties of a Laminated Bilayer Medium Brief Summary of the Effective Transport P r o p e r t y Scheme-In this appendix, a general scheme18 is described for deriving formulas for the six, overall transport coefficients (K,KL,i?, U*, D*) of a spatially periodic, charged, porous tissue. Owing to the mathematical complexity of this scheme, many of its details have been omitted in what follows, attention being given to principal highlights of the scheme, rather than to exhaustive mathematical detail. Next, the calculation of the transport properties of a porous medium comprising parallel layers (labeled c and d) of unequal thicknesses I, and Zd (Figure A l ) is described. Finally, application of the layered-medium results to the transdermal scenarios of Figures 3 and 4 is discussed. The effective transport properties are to be obtained by performing the following integrations over a spatially periodic, unit cell (of net volume T, = t, t d where re is the ‘continuousphase’ volume and t d the ‘discontinuous-phase’volume):

q,

+

effective electrical conductivity

effective streaming-potential conductivity

effective hydraulic permeability - def 1 K = --JVdV

To

To

effective electroosmotic permeability

(A4)

mean solute velocity vector def

U* = J ( U q - D*VP;)dV to

1330 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

I

‘Unit cell’

Discontinue

phase

I

--

-i 1

Continuoub

ohase

-4-

I

Y

%/

G u

X Figure A1-A laminated bilayer medium comprised of “discontinuous” and “continuous” phases.

effective dispersivity dyadic

In the above, the following notation has been introduced: u is the (generally anisotropic) electrical conductivity dyadic (possessing the value a, in the continuous phase t cand a d in the discontinuous phase td), E is the dielectric permittivity constant (with the respective values E , and Ed in t, and Td), pf is the viscosity of the fluid that permeates both the continuous and discontinuous phases, 5 is the zeta potential on the continuous-phase side of the particulate surfaces sp,& is the mobile (double-layer)surface charge density (assumed to vary over the particulate surfaces sp)on the continuous-phase side of the surface sp,satisfying

with ( 1 / ~ )a measure of the mobile, Debye double-layer thickness, U is the velocity of solute species through the continuous phase of the porous medium, and D the solute diffusivity dyadic (possessing the respective values D, and Dd in t, and Zd). In many applications of the theory, solute particles may possess finite dimensions relative to the pore size. In this case, the nonelectrophoretic contribution t o U will not, in general, be equivalent to the solvent velocity v but will be somewhat less on account of hydrodynamic interactions between the pore boundaries and the finite-size solute particle. [The particle’s ionic (or electrophoretic) mobility, ME,will also generally differ from its unbound-fluid value.] The exact relation between U and v will thus depend upon the precise location of the (possibly nonspherical, charged, and/or flexible) particle within the pore space and may be determined by low Reynolds number calculations as described in detail by Happel and Brenner.61 See related discussions following eqs A31 and A48. In addition to the above parameters, six spatially varying scalar, vector, and dyadic fields appear in the expressions AlA6; these include the scalar PO, vectors g, h, and B, and dyadics V (with associated field V,)and VE.Each of these is t o be obtained by solution of an associated boundary-value problem (specific t o the particular porous medium being considered), problems which have been described in detail in ref 1. Two of these boundary-value problems, however, differ from the comparable equation sets derived previously, since in the present case fluid flow at the microscale is governed by Darcy’s law (in both t, and Td; with pf designating the viscosity of the Newtonian fluid presumed to permeate the

two phases), rather than the Stokes' equations, only in tc. These two cellular boundary value problems (for V and VE) are therefore given below. The dyadic V field (as well as its associated vector field Il) is to be obtained by solution of the following unit-cell boundary-value problem:

V = -K.(VII

+ I)

V V =0 (r E

(r E ro)

(A81

to)

(A91

bilayers, and

Choose the local, cellular position vector r as originating at the edge of a unit cell, as shown in the figure. Then,

r = ip

+ ig (0 5 x < L,

vE= - K . V H ~(r E z,) n.v; = mv;,

VE, IIE are spatially periodic

spl (A141 (A151

Here, IIE is a vector field defined to within an arbitrary constant vector by the above problem, and I, is the unit surface tensor (or surface idemfactor). Similar unit-cell, boundary-value problems existla for determination of the P,, g, h,and B fields. Explicit Calculations f o r a Laminated Bilayer Medium-Anticipating the transdermal geometries of Figures 3 and 4,we consider the laminated bilayer example of Figure A l . Each layer of the medium is assumed to possess an anisotropic electrical conductivity (i and hydraulic permeability K, moreover, a (possibly charged) solute is assumed to transport through the layers, with anisotropic ionic mobility ME and anisotropic diffusivity D. These coefficients may be expressed in the component forms (with i = c, d)

+

= ixixdiyIydi

(i

= ixixG,

+ iyiya,,

(A231

where

denotes a "resistances-in-series" type formula for the effective conductivity perpendicular to the bilayer surfaces, and

(A251

(A121

1;v; = 0, 1.y; = E,C(V,%- I,) ( r E

(A22)

Effective Electrical Conductivity-Calculation of the g field leads via eq A1 to the following expression for the effective electrical conductivity, (i, of the medium: (T

The dyadic V is uniquely defined by the above boundary-value problem (the vector n being defined to within an arbitrary constant vector). Here, K denotes the hydraulic permeability at the microscale (possessing values & and & in tcand Zd), n is the outward-drawn unit normal to the particulate phase surfaces sp, and r is a position vector defined wholly within the unit cell. I is the unit tensor (or so-called "idemfactor"). Similar to the above, the dyadic field VEuniquely satisfies the following unit-cell boundary-value problem:

0 5 y < h)

(A161

a "resistances-in-parallel" type formula for the effective conductivity parallel to the bilayer surfaces. Observe that the term &cc
where

(A271 is the "resistances-in-parallel" permeability component perpendicular to the interface normal, and in the parallel direction we have the "resistances-in-series'' permeability component

In arriving a t the above, note that the spatially periodic dyadic field V is easily deduced from eqs Ag-All, using eq A l , that is,

+ V, = iXi+ xiyiye e Vc = ixixV, iyiyq

D = i,iJlY

+ iyiyD;,

(A191

where,

Determination of the effective transport properties of this medium by use of the scheme previously described requires subdivision of the periodic medium into an infinite array of identical unit cells of total area, for example, to= hL

(A201

where L is an arbitrary length measured parallel to the

Substituting these results into eq A3 gives eqs, A!26-A28. Journal of Pharmaceufical Sciences / 1331 Vol. 83, No. 9, September 1994

Effective Electrokinetic Cross-Coupling CoefficientsThe boundary-value problem posed by eqs A12-Al5 is foundto furnish the solution

: V

= 0, V? = -cccixix

(-439) Here,

ipll Vp = ipL + ipll E = iJL +

which reflects the fact that electrokinetic phenomena occur only parallel to the charged surfaces of the medium. Substitution of this result into eq A4 gives = iXi$)ccc

(A29)

iA40) (A41)

With the above results, it is possible to show by solution of the Pr problem that

for the effective electroosmotic permeability. Moreover, solution of the associated h-problem leads to the equivalence @

- pE

(A301

P-

This equivalence is indeed anticipated on the grounds of irreversible thermodynamics. Solute velocity-From eq A5 it may be easily shown that

u*= lpo",(@Vc + ME:Ec)

r!dPyd(evd

+ MEd*E,)

(A31)

(i = c and d) where the hydrodynamic hindrance factors, are to be evaluated4I via knowledge of the precise relationship between the solute convective velocity U and the solvent velocity v as described in the footnote following eq A5. Moreover, implicit in the ionic mobilities, ME^, are additional hindrance factors (i.e., H:); that is ME^ = zH,'Di/kT, with z being the valence, k the Boltzmann's constant and T the temperature. The hindrance factors H,' are also to be dztermined42(observe that, in the notation of the latter,42dM = H!). See also the discussion following eq A48. To interpret the above relation (A31) it is necessary to note that the microscale electric E and velocity v fields are related to the g, h, V, and VE fields used above for calculating the effective electromechanical properties of the medium by1

E =VgE

1 + -Vh-Vjj 0

L

a result obtained only in the case of a charged solute (i.e., possessing finite ionic mobility, \ME\* 0). Otherwise, for either a noncharged solute o r in the absence of an applied electric field, we have, simply,

revealing that the solute is in this case equipartitioned. Solute Dispersivity-Finally, one may evaluate eq A6 to ultimately obtain

D* = i X Q L + i,,iJJl

(A441

for the effective dispersivity dyadic of the solute through the layered medium, where

(A321 is the effective dispersivity component perpendicular to the interface normal, and the parallel component is given by

and

1 + Pf-vE*E

1 -_

v = -v.vp Pf

(A331

This may be shown to lead to

+ v = ixuL + ipll

E = ip1 i&,

(-434) (-435)

where

Here,

and

1332 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

and

In a similar manner, the effective transport properties (eqs 17-28) according t o the transcorneocyte pathway (Figure 4) may be derived. It is necessary to explicitly note two parameters that arise in the derivation of the intercellular/shunt formulas (10- 16). These are

c;“= (A481 Observe that the diffusioddispersion coefficients ( D I , DlJ may contain, inter alia, hydrodynamic hindrance factors (HO, H2, H3) reflecting the finite size of the solute species; thus, HO is to be associated with hindered molecular diffusion (see the discussion following eq A31), H2 is associated with hindered convective dispersion (i.e., I-P is equivalent to d, in the notation of ref 41), and H3 identifies hindered convective dispersion resulting from application of an expternal (e.g., E field (i.e., the dc-parameter of ref 42). General properties of H2,S’) may be stated as the four hindrance factors (HO,H1, follows:

the cross-sectional area fraction of the sweat duct and hair follicle routes (with Lt envisioned as a unit-cell dimension serving to definite the side length of a paralellepipedal unit cell in two orthogonal directions perpendicular to n at the mesoscale level of description), and

for an infinitesimal-size solute, whereas

{@,HI, H2,H3}

-0

for the case of a solute whose size is comparable to the size of the pore through which it travels. Derivation of Effective Transdermal Transport Properties-The above results may be applied to the transdermal scenarios depicted in Figures 3 and 4 to derive the formulas presented in the main body of the paper. These derivations, as described below, are very similar to those outlined above for the layered medium. The derivations of eqs 10-25 are to be performed in four separate steps. Periodic media are identified at the submicro-, micro- and mesoscales of Figures 3 and 4. The geometries a t each of these scales are found to constitute some manner of layered medium (Figure All, hence it proves possible to use results similar (or identical to) the results provided in eqs A23-A48 to arrive a t effective transport properties, first at the microscale, then at the mesoscale, and finally at the macroscale level of the skin. In the case of the intercellular/shunt route, the initial task is to calculate the effective transport coefficients of the lamellar zones at the “microscale”level of description. Next, upon noting, at the microscale level, that the corneocytes are nonconducting (with the corneocyte surfaces assumed noncharged, the corneocytesnonconducting and impermeable, and with perfect hydrodynamic slip a t the corneocyte surfaces), the effective transport characteristics of the stratum corneum at the mesoscale level of description may be derived. This derivation, which substantially departs from the derivation of the laminated medium characteristics outlined above, is reminiscent of the effective medium calculations described in the text of Crank.78 With the effective transport characteristics of the stratum corneum known, and the transport characteristics pertinent to charge, fluid-mass, and species transport via the “shunt” pathways (i.e., that occurring through the hair follicles and sweat ducts) easily established on the basis of standard transport texts (see Bird et aL6I and Brenner and Edwards22),the layered medium results given in the preceding section of this appendix may be used to derive the macroscale properties describing transport parallel to the skin surface normal n. These results are given in eqs 1016.

the effective hydraulic permeability of the shunt routes of transport (cf. eq 13).

GLOSSARY a

c, c1, cz

effective hydraulic radius of molecule effective solute concentration (1 and 2 respectively refer to donor and receptor compartments of a diffusion cell apparatus) effective mean diffusivity or dispersivity (dyadic and scalar) molecular diffusivity in aqueous and lipid regions of skin effective electric field (vector and scalar) Faraday constant thickness of stratum corneum mean hindrance coefficient for solute transporting through skin hindrance coefficients describing the resistance to transport arising on account of the finite size of the solute molecule relative to the dimensions of the pore through which the molecule passes for forced convection, diffusion, convective dispersion caused by forced convection, and convective dispersion caused by the application of an electric field effective electrical current flux (vector and scalar) effective solute flux (vector and scalar) hydraulic permeability of skin (tensor and scalar) passive skin permeability electroosmotic permeability of skin (tensor and scalar) streaming potential conductivity of skin (tensor and scalar) Journal of Pharmaceutical Sciences / 1333 Vol. 83, No. 9, September 1994

L

T U", u*

v, u 2

AF

AV

mass-transfer coefficients relevant for very slow partitioning of solute into skin (1and 2 respectively refer to donor and receptor compartments of a diffusion cell apparatus) skin partition coefficients (1 and 2 respectively refer to donor and receptor compartments of a diffusion cell apparatus) thickness of the skin sample in the diffusion cell experiment characteristic length of bound-water zones separating head groups of lipid bilayers between corneocytes, thickness of lipid bilayers, lateral corneocyte dimension, and corneocyte thickness mean electrophoretic mobility of a charged molecule in skin unit normal vector and normal coordinate of skin surface gas constant characteristic radii of hair shaft and follicle characteristic radii of aqueous pathways through corneocyte and intercellular lamellae absolute temperature effective mean solute velocity (vector and scalar) mean fluid velocity (vector and scalar) valence of molecule pressure drop across the skin voltage potential across the skin

1334 / Journal of Pharmaceutical Sciences Vol. 83, No. 9, September 1994

4L @SH

electrical permittivity of aqueous pathways in skin ion-transport resistance function arising owing to hindered transport of ions through narrow bound-water regions between lipid head groups characteristic area fraction of hair follicle and sweat duct routes characteristic area fraction of ion pathways through lipid bilayers of skin characteristic area fraction of shunt routes (=@H

@C

Pf7 P W

i2 u, a

5 Qp

+ 4F)

characteristic aqueous area fraction of corneocytes viscosity of fluid (water) effective specific skin resistance effective electrical conductivity (dyadic and scalar) electrical conductivity of aqueous pathways in skin zeta potential of charged surfaces in skin effective pressure-gradient field

Acknowledgments The authors gratefully acknowledge numerous constructive comments made by Dr. Steven Schwendeman, Dr. Nicholas Peppas, Dr. Russell Potts, Dr. Ronald Siegel, and Dr. James Weaver during the preparation of this work. Mark Prausnitz and Mark Johnson, in addition to offering significant constructive input, were of invaluable aid in identifying many pertinent literature references. Support for this work was provided by NIH Grant GM 44884.