Modelling drug flux through microporated skin

A simple mathematical equation has been developed to predict drug flux through microporated skin. The theoretical model is based on an approach applied previously to water evaporation through leaf stomata. Pore density, pore radius and drug molecular weight are key model parameters. The predictions of the model were compared with results derived from a simple, intuitive method using porated area alone to estimate the flux enhancement. It is shown that the new approach predicts significantly higher fluxes than the intuitive analysis, with transport being proportional to the total pore perimeter rather than area as intuitively anticipated. Predicted fluxes were in good general agreement with experimental data on drug delivery from the literature, and were quantitatively closer to the measured values than those derived from the intuitive, area-based approach.


Graphical Abstract Introduction
The skin and its outer layer, the stratum corneum (SC), in particular, is a formidable barrier to the passive delivery of topical and transdermal drugs.The ideal physicochemical characteristics for 'good' skin penetrants are well-known: small molecular weight (MW), reasonable, but not excessive lipophilicity (log P = 2-3), adequate solubility in both oil and water, low melting point [1].
However, these criteria are seldom satisfied and, in essence, this has meant that the most successful drugs that are delivered into or through the skin for therapeutic effect are those with high pharmacological potency [2,3].
The aim to deliver drugs requiring higher doses, or those with MW > 500 (e.g., peptides and small proteins), has led to a substantial effort to enhance percutaneous absorption, and to reversibly undermine skin barrier function, using a variety of different approaches [4].Most recently, attention has focused on several so-called microporation techniques, based on the use of microneedles, laser poration and heat ablation to create entirely new pathways across the SC and circumvent thereby the major resistance to passive diffusion into the viable skin tissue [5,6].
The underlying principle here is that (for example) large molecules will never transport through intact skin at a rate necessary to be useful therapeutically and it is therefore necessary to by-pass the SC.The microporation approaches do so by creating pores of 20-200 μm in diameter and of a depth that they are minimally invasive (i.e., as sensation-free as possible) [7,8].The strategy is reversible as the skin very effectively 'seals' itself quite quickly following such a disruption [9].
A key question when contemplating the application of a microporation technique to enhance topical/transdermal drug delivery is "how many pores, and of what dimension, are required to achieve the desired flux"?The simple, intuitive approach to make such a prediction is to assume that the flux achieved will equal that through an aqueous domain, the surface area of which equals that of the total pore area created (and to assume that any contribution via intact SC is negligible).
However, this logical assumption has not undergone rigorous testing and the possibility that the enhancement required may be achievable with fewer pores than intuitively expected (and hence rendering the approach even less invasive to the patient) has not been explored.In this paper, therefore, a model, based on water evaporation via leaf stomata [10,11], for drug flux across the skin via micropores is developed and is compared both to the predictions from an intuitive, areabased model and to experimental measurements in the literature.

Model development
Following a poration event, it is assumed that round pores (of radius r p ), which completely penetrate the SC, are formed.It is further assumed that these pores are spaced at regular intervals as shown in Fig 1A, so that the pore density n p (dimension: 1/cm 2 or pore number per cm 2 ) results in the spacing between the neighbouring pores of 1 . The applied drug concentration in the vehicle is maintained constant (C v ) over a distance h v above the SC, while that at a depth h s below the SC is assumed to be zero (i.e., sink conditions) as a result, for example, of clearance by the dermal microcirculation (see Fig 1B).
The steady state flux in the case that all of the SC barrier is removed may be expressed as: where D v and D s are the drug's diffusivities in the vehicle and viable skin, respectively, and K s is the partition coefficient of the drug between the viable skin and the vehicle.In the absence of SC, therefore, the diffusional resistance (R no SC ) to drug absorption is given by: When the SC is only partially removed, an analytical solution to the diffusion equation cannot be obtained even for the steady state and an approach similar to that developed by Patlak [11] is therefore used to approximate the resistance in Eq. 1 by the integral: Here ( ) K x and ( ) D x are step functions describing drug partitioning and diffusion in the vehicle ( ) and skin ( ( ) , ( ) ).Assuming that the SC pores are filled with the vehicle, then the partition and diffusion coefficients therein are the same as those in the vehicle ( ).Further, in Eq 3, ( ) a x is the fraction of the total area available for diffusion in the direction x; ( ) a x is estimated by assuming that diffusion in the layers adjacent to the pores is limited to the region formed by the shaded "funnels" (see Fig 1B).
In terms of the geometry of these funnels, Patlak [11] has shown that tan 4 A B for the skin side and ( ) 4 for the vehicle side.Thus, in summary, the functions ( ) K x , ( ) D x and ( ) a x are given by: Substituting Eq 4 for , and into the integral Eq 3 yields: With the assumption that both h v and h s are greater than or equal to h f , integration of Eq 5 produces: where ( ) u x is the unit step function ( ( ) 1 , the transport resistance R depends on parameters related to the drug, its vehicle and the skin (D v , D s and K s ), the pore density and radius (n p and r p ), and the different layer thicknesses (h v , h s and h sc ).
While Eq 6 can be directly used to predict flux, for many practical cases of SC poration, further simplification is possible.First, it is reasonable to expect that the pore radius is much less than the distance between the pores ( 2 p p r l  ).Second, if the pore density is not large, such that ( ) , it follows that: Hence, if the pore radius is comparable to the distance between the pores ( p p r l  ), or the pore density is equal to or greater than the right-hand side of Eq 7, then the first two terms in Eq 6 dominate the resistance which then approaches that in Eq 2, i.e., no SC R .If, however, 2 p p r l  and Eq 7 is satisfied, then the first two terms in Eq 6 can be neglected and: Eq 8 is a valid approximation as long as R is significantly greater than no SC R (defined in Eq 2).The reciprocal of the resistance in Eq 8 is, of course, the drug's permeability coefficient across the porated skin: where D aq is the drug's aqueous diffusion coefficient.It is interesting to note that k p and therefore the flux through compromised skin is proportional to r p (providing that p sc r h ≥ ) and not 2 p r , as would have been expected for the intuitive model in which transport is proportional to the total area of pores.In the latter case, the permeability through porated SC would equal that across skin with no functional barrier multiplied by the fractional pore area (A pore /A total ): Eq 10 is derived from Eq 2 for noSC R and . Note that Eq 10 (but with v h replaced with v sc h h + ) can also be obtained by using 2 ( ) p p a x n r π = in the integral Eq 3 for all values of x .
The flux proportionality to r p rather than 2 p r for thin porated membranes is well-known and explains why water evaporation from a leaf is so high (about half of that expected from a fully exposed surface), even though the open surface area of the stomata of plant leaves is only about 1% of the total leaf area [11].
Eq 9 can be further simplified by assuming that D v ~ D aq , which can be deduced from the Wilke-Chang relationship [12]: where MW w is the molecular weight of water (18 Da), T is the absolute temperature (305K or 32 o C on the skin surface), η w is the water viscosity (0.77 cP at 305K) and V is the molar volume of the drug at its boiling point (in cm 3 mole -1 ).In the calculations below, V is replaced by the drug's MW (i.e., a density of 1 g cm -3 is assumed) and Eq 11 reduces to: The viable skin has been likened to an aqueous gel and it has been suggested that drug diffusivity here is about one-sixth of that in water [13]; it has therefore been assumed that ( ) 9 can now be simplified accordingly: if (as is the case experimentally) a typical pore diameter is between 10 and 200 µm, and assuming an SC thickness of 15 µm, then the term in parenthesis in Eq 9 has a value somewhere between 7 and 10.Using 8 as a reasonable compromise given the number of estimates already made, and calculating D aq with Eq 12 (having converted to units of cm 2 h -1 ), the model predicts the following expressions for the permeability coefficient (in cm h -1 ) and the flux of the drug through porated skin: 0.6 0.36 Using the same approximations, the corresponding expressions for the intuitive model are: Ultimately, these permeability coefficients can be compared with the approximation of that through skin which has no SC whatsoever, i.e.: In Eqs 13-16, the units for p r , s h and v h are cm and, for p n , cm -2 .

Results and discussion
By way of illustration, Figure 2 compares the predicted skin permeability coefficients for an example small drug (MW = 300 and log P = 2) using the new and intuitive (area-based) models of the dermal microcirculation [15].The thickness of the dermis is on the order of 1 mm and, while the capillary loops in the dermal papillae (right beneath the epidermal-dermal junction) may not be expected to fully clear all drug arriving from the epidermis, the rich plexus of blood vessels situated a few hundred microns further down [15] is a reasonable location where 'sink' conditions may be expected.In vitro, the value of h s depends on the type of skin used (dermatomed versus full-thickness) and is typically 500-1000 µm as there is no functional microcirculation in this case.In the final approximation of the model (Eqs 12 and 13), the principal contribution to the diffusional resistance is from inside the funnels, and any increase in resistance due to a larger h s is assumed negligible.
It is immediately apparent from Figure 2 that the new approach predicts a permeability coefficient that is approximately an order of magnitude greater than the intuitive estimate.Also, for the same model drug, the new model predicts that the porated skin presents no functional barrier (i.e., ) once about 8% of the SC is ablated (equivalent to about 4000 pores per cm 2 , in this case).Of course, the situation where k p > k no SC is physically impossible, as the permeability of porated skin can only approach that of the skin with all the SC removed.This violation is due to the pore density exceeding that required by Eq 7 (the right hand side is about 2000 pores per cm 2 ), and indicates the limit of applicability of the new model.
To assess the validity of the new model, it was used to predict published experimental results [16][17][18][19][20][21] for a variety of drugs permeating across mammalian skin that had been fully porated either using microneedles or by laser ablation; the measured values were also compared to estimates from the intuitive model and the results are presented in Tables 1 and 2. For laser-porated skin, the pore diameter was reported; in the case of microneedle poration, the pore radius (r p ) was calculated from geometric considerations: where d n is the diameter of the microneedle base, l in is the length of that part of the microneedle inserted into the skin, and l n is the total microneedle length (Figure 3).It has been reported that the depth of the conduits created by microneedles is 40-80 μm [17] and it was therefore assumed, in the predictions summarized in Tables 1 and 2, that l in could be reasonably approximated to a value of 50 µm.The experimental results in Tables 1 and 2 are, in general, better predicted by the new model than the intuitive, area-based approach.This is illustrated graphically in Figure 4 which (for both microneedle and laser poration) shows the ratio of predicted to experimental fluxes for a wide range of drugs.In the case of the new model (red bars), the ratio mostly falls within a factor of 5 of unity (i.e., 0.2 < ratio < 5) whereas, for the intuitive estimation (blue bars), the ratio is nearly always less than 0.2.It follows that the novel model described in this research provides a closer description of the measured fluxes of drugs of diverse physicochemical properties across porated skin.The approach may be valuable, therefore, in drug selection and in the design and optimisation of formulation strategies for transdermal delivery through selectively compromised skin.
Figure 4 shows that the new model accurately predicts the experimentally observed fluxes for many compounds across microneedle-porated skin based on logical and readily available parameters, such as the permeant's molecular weight, its concentration in the applied formulation, and the density and radius of the pores.Earlier publications have indicated that a chemical's flux through microneedle-porated skin depends on the length [16], tip radius (sharpness) [22], base width [16], duration of application [23] and insertion force of microneedles [24].Given the model derived here, these parameters are those which most directly influence the average radius of the created pores.
Flux predictions based on Equation 14 are particularly good for the peptides and proteins (carnitine, hexapeptide, Ac-hexapeptide, oxytocin, cytochrome C, and BSA) that have been investigated experimentally.Equally, good agreement is seen for two conventional, small molecular weight drugs, namely acyclovir and lidocaine.In contrast, the predictions for insulin, diclofenac, rhGH and FSH deviated by a factor of 5 or more from the measured values; in fact, for the latter three compounds, the intuitive model was a better predictor.While the lack of agreement for insulin may be, at least in part, due to the animal model used (rat abdominal skin), the exact reason(s) behind these examples of poor prediction remain, for now, unclear.and 2.

3 Fig 1 .
Fig 1.Schematic representation of pores in SC plane (A) and side view (B) with relevant dimensions defined in the text.

(
Eqs 13 and 15, respectively) as a function of both the number of pores (n p ) and of the fractional pore area (A pore /A total )h s = h v = 300 µm.The limiting permeability coefficient values of fully intact SC, determined using the Potts & Guy equation [14], and of skin with no SC whatsoever ( noSC k , Eq 16) are also included in the Figure.The value of h s was chosen on the basis of the situation in vivo, and the known ultrastructure and organisation

Fig 2 .
Fig 2. Predicted permeability coefficients of a model small drug as a function of the number of pores (n p ) based on both the new model described in this work (solid line), and an estimate based on an intuitive, areabased calculation (dashed line).Horizontal lines show the limiting cases of the permeability coefficients for fully intact (k SC ) and fully compromised SC (k noSC ).

Fig 3 .
Fig 3. Geometric parameters associated with insertion of a single microneedle into the skin.

Fig. 4 .
Fig. 4. Ratio of predicted to experimental fluxes using the new model described here (red bars) and the intuitive, area-based approach (blue bars).Data are provided in Tables 1

a
Pore radius calculated using Eq 18. b FITC-labelled insulin; experimental flux determined from amount penetrated in 3 hours.

Table 1 :
Experimental and predicted fluxes of various drugs through microneedle-porated skin

Table 2 :
Experimental and predicted fluxes of various drugs through laser-porated skin.
a Equine heart cytochrome C. b Recombinant human growth hormone.c Urinary follicle stimulating hormone.d FITC-labelled bovine serum albumin.