Pharmaceutical Research, Vol. 18, No. 11, November 2001 (© 2001)
`
`Research Paper
`
`Predicted Permeability of the Cornea
`to Topical Drugs
`
`Aure´lie Edwards1,3 and Mark R. Prausnitz2
`
`Received July 25, 2001; accepted August 3, 2001
`
`Purpose. To develop a theoretical model to predict the passive,
`steady-state permeability of cornea and its component layers (epithe-
`lium, stroma, and endothelium) as a function of drug size and distri-
`bution coefficient (F). The parameters of the model should represent
`physical properties that can be independently estimated and have
`physically interpretable meaning.
`Methods. A model was developed to predict corneal permeability
`using 1) a newly developed composite porous-medium approach to
`model transport through the transcellular and paracellular pathways
`across the epithelium and endothelium and 2) previous work on mod-
`eling corneal stroma using a fiber-matrix approach.
`Results. The model, which predicts corneal permeability for mol-
`ecules having a broad range of size and lipophilicity, was validated by
`comparison with over 150 different experimental data points and
`showed agreement with a mean absolute fractional error of 2.43,
`which is within the confidence interval of the data. In addition to
`overall corneal permeability, the model permitted independent
`analysis of transcellular and paracellular pathways in epithelium,
`stroma and endothelium. This yielded strategies to enhance corneal
`permeability by targeting epithelial paracellular pathways for hydro-
`philic compounds (F < 0.1 − 1), epithelial transcellular pathways for
`intermediate compounds, and stromal pathways for hydrophobic
`compounds (F > 10 − 100). The effects of changing corneal physical
`properties (e.g., to mimic disease states or animals models) were also
`examined.
`Conclusions. A model based on physicochemical properties of the
`cornea and drug molecules can be broadly applied to predict corneal
`permeability and suggest strategies to enhance that permeability.
`
`KEY WORDS: ophthalmic drug delivery; theoretical model; eye;
`ophthalmology; ocular transport prediction.
`
`INTRODUCTION
`
`Topical drug delivery to the eye is the most common
`treatment of ophthalmic diseases, and the cornea provides the
`dominant barrier to drug transport (1). For this reason, a
`large body of experimental work has characterized corneal
`permeability (2), and some models have been developed as a
`result to describe transcorneal transport (3–7). However,
`most existing models rely on parameters that are fitted to a
`small number of experimental measurements and are not ap-
`plicable to larger data sets, and many models do not account
`for all existing transport processes and routes. A model that
`can predict the corneal permeability of any drug based on its
`
`1 Department of Chemical and Biological Engineering, Tufts Univer-
`sity, 4 Colby Street, Medford, Massachusetts 02155.
`2 Schools of Chemical and Biomedical Engineering, Georgia Institute
`of Technology, Atlanta, Georgia 30332–0100.
`3 To whom correspondence should be addressed. (e-mail:
`aurelie.edwards@tufts.edu)
`
`physical properties and those of the cornea would be more
`broadly useful.
`If all the variables of a predictive model correspond to
`physical properties that are independently measured (such as
`molecular radius, width of intercellular spaces, etc.), the
`model then should not only describe the data used in its de-
`velopment but also predict corneal permeability to classes of
`compounds not considered during model development. With
`such a tool, the ability to deliver newly synthesized or even
`computer-generated drugs can be assessed without stepping
`into the laboratory. In addition, transport processes can be
`understood in physical terms (e.g., epithelial tight junctions
`are the rate-limiting barrier for a given drug), which helps
`develop appropriate ways to enhance or target delivery and
`facilitates predicting or analyzing delivery problems. Finally,
`because model variables have physical meaning, they can be
`easily changed to reflect the properties of diseased or injured
`cornea and to account for differences between humans and
`animals.
`Given the potential power of an approach based on phys-
`ics rather than statistics, we developed a model that builds off
`of previous work describing the permeability of corneal
`stroma (and sclera) by modeling it as a fiber matrix (8) and
`combines it with a new analysis presented for the corneal
`epithelium and endothelium. Both transcellular and paracel-
`lular transport pathways were considered and, whenever pos-
`sible, parameters were derived from independent experimen-
`tal observations.
`
`MODEL DEVELOPMENT
`
`Following the general approach taken by a number of
`previous studies (4,6,9), the steady-state permeability of cor-
`nea can be determined by considering the individual perme-
`abilities of the three primary tissues that make up cornea:
`endothelium, stroma, and epithelium (10). Because Bow-
`mann’s membrane and Descemet’s membranes are so thin
`and permeable, they do not contribute significantly to overall
`corneal permeability and are therefore not considered in this
`analysis. We have developed previously a model that predicts
`the permeability of stroma and that of the paracellular path-
`way across endothelium (8). In this section, we summarize
`these findings and develop new expressions for transport
`across the epithelium and the transcellular pathway across
`endothelium.
`
`Endothelium
`
`The corneal endothelium is a monolayer of hexagonal
`cells, each about 20 mm wide and 5 mm thick, found at the
`internal base of the cornea (10). Two pathways are available
`for solutes diffusing across the endothelium (Fig. 1): a para-
`cellular route (i.e., between cells), which is a water-filled path-
`way impeded by gap and tight junctions and is favored by
`hydrophilic molecules and ions; and a transcellular route (i.e.,
`within or across cell membranes), which involves partitioning
`into and diffusing within cell membranes and is the pathway
`of choice for hydrophobic molecules. The fraction of a given
`solute that goes through each route is determined primarily
`by its membrane-to-water distribution coefficient (F). The
`larger F (i.e., for hydrophobic molecules), the greater the
`
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`Edwards and Prausnitz
`
`the overall permeability of the transcellular route can be ex-
`pressed as:
`
`kt = klat + ktr
`
`(2)
`
`where klat and ktr are the permeabilities of the lateral and
`transverse routes, respectively.
`Lateral Diffusion. The permeability klat of the lateral
`transcellular pathway to a given solute is given by:
`
`klat =
`
`FDlat
`Llat
`
`(3)
`
`where Dlat is the lateral diffusivity of the solute in the cell
`membranes and Llat is the mean diffusion pathway length.
`Our estimate of the lateral diffusivity of a small solute
`(~ 5 Å in radius) in the cell membranes of the cornea is 2 ×
`10−8 cm2/s, based upon a compilation of experimental mea-
`surements reported by Johnson et al. (11), which were con-
`ducted using model cholesterol-containing lipid bilayers. This
`value is consistent with membrane diffusion studies in epithe-
`lial tissues (12). The mean diffusion pathway length, Llat, was
`calculated as described below (Eq. 12). The average length of
`the paracellular opening between adjacent endothelial cells is
`L 4 12.2 mm (13), and the average radius of an endothelial
`4 12.2 + 10/3 4 15.5 mm.
`cell is 10 mm, yielding Llat
`An empirical relationship between the membrane-to-
`water distribution coefficient and that between octanol and
`water (Kow) is given by Johnson (Mark Johnson, personal
`communication):
`
`F = Kow
`
`0.87
`
`(4)
`
`The octanol-to-water distribution coefficient was determined
`using experimental values for the octanol-to-water partition
`coefficient and calculated values of the degree of solute
`ionization (assuming ionized molecules do not partition
`into octanol), as described and tabulated in Prausnitz and
`Noonan (2).
`Transverse Diffusion. The permeability of the transverse
`route, ktr, was calculated by considering three steps in series:
`transport across the anterior endothelial cell membrane, dif-
`fusion through cell cytoplasm, and transport across the pos-
`terior cell membrane.
`
`(5)
`
`= 1
`kmem
`
`+ 1
`kcyt
`
`+ 1
`kmem
`
`= 2
`kmem
`
`+ 1
`kcyt
`
`1 k
`
`tr
`
`where kmem is cell membrane permeability and kcyt is trans-
`cytosol permeability.
`Endothelial cell membrane permeability was difficult to
`calculate because very few independent data are available.
`Most membrane permeability studies have used artificial lipid
`bilayers, which are more permeable than cell membranes.
`Lacking data on endothelial cell membrane permeability, we
`chose red blood cell (RBC) membranes as a model, which is
`the only cell membrane for which we could find useful data on
`transmembrane transport by passive mechanisms (i.e., diffu-
`sion).
`
`Fig. 1. An idealized representation of the cornea showing transport
`pathways across the epithelium, stroma, and endothelium (drawing
`not to scale). (A) In the epithelium, paracellular pathways follow the
`hydrophilic spaces between epithelial cells (1p), and transcellular
`pathways are either solely within the hydrophobic cell membranes
`(1tlat) or alternate crossing of cell membranes and cell cytosol (1ttr).
`The largely cell-free stroma offers only hydrophilic pathways among
`and between collagen fibers and proteoglycan matrix (2). The endo-
`thelium contains both hydrophilic paracellular (3p) and hydrophobic
`transcellular (3tlat and 3ttr) routes. (B) The paracellular spaces in
`epithelium and endothelium are modeled as slits with constrictions
`that represent tight and gap junctions.
`
`relative amount of solute that diffuses through the cells. Be-
`cause the two pathways are in parallel, the overall permeabil-
`ity of the cell layer (klayer) is the sum of the permeabilities of
`each pathway:
`
`klayer = kt + kp
`where kt and kp are the permeabilities of transcellular and
`paracellular routes, respectively. How to calculate the perme-
`ability of each of these pathways is described below.
`
`(1)
`
`Transcellular Pathway
`
`The transcellular pathway across endothelium involves
`two possible routes. The first route consists of 1) partitioning
`from the water-rich stroma into the lipid-rich plasma mem-
`branes of endothelial cells; 2) diffusing within the cell mem-
`branes across the endothelium; and 3) partitioning out of the
`membranes and into the aqueous humor bathing the internal
`surface of the cornea. This pathway, referred to as the lateral
`route of the transcellular pathway (pathway 3tlat in Fig. 1A),
`does not include transport within the cytosol of endothelial
`cells. The second route involves the following: 1) partitioning
`into, diffusing transversely across, and partitioning out of the
`anterior cell membrane; 2) diffusing through the cytosol; and
`3) partitioning into, diffusing across, and partitioning out of
`the posterior cell membrane. This second pathway is referred
`to as the transverse route (pathway 3ttr in Fig. 1A).
`For simplicity, we assume those two routes can be treated
`separately. In reality, they are not independent: a molecule
`may follow a path involving a combination of lateral transport
`along the membrane and diffusion in the cytosol. Given this,
`
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`
`Predicted Corneal Permeability
`
`1499
`
`prediction reduces our confidence in extrapolated predictions
`using Equation 6, it supports our overall conclusion that the
`transverse route is almost always negligible, because these
`measured permeabilities are even smaller than predicted.
`
`Paracellular Pathway
`
`0
`
`Reliable measurements of RBC basal permeability have
`been compiled by Lieb and Stein (14). The octanol-to-water
`partition coefficient of the solutes considered in that study
`varies over a broad range between 1.2 × 10−3 and 1.1 × 102,
`but their molecular volume does not exceed 73 cm3/mol (i.e.,
`~ 3 Å radius), meaning that data extrapolation was needed to
`include molecules typically delivered to the eye (i.e., radius of
`3.5 to 5.5 Å). Recognizing this limitation, we used the model
`of Lieb and Stein for non-Stokesian diffusion (14), whereby
`the permeability of a cell membrane can be written as
`(6)
`kmem = kmem
`10- mvV
`where k0
`mem is the membrane permeability for a theoretical
`molecule of infinitely small size and the term 10−mvV accounts
`for the effects of molecular size; mv is a measure of the size
`selectivity for diffusion within the membrane, which is 0.0516
`mol/cm3 for human RBCs (14), and V is the van der Waals
`molecular volume determined from molecular structure (2).
`The term K0
`mem has been empirically shown to depend on
`partition coefficient according to the following relationship.
`! = A log~Kow! + B
`(7)
`0
`log~kmem
`where k0
`mem has units of cm/s. Using octanol as a model for
`partitioning and assuming that the values of the distribution
`and partition coefficients are similar for the solutes being
`considered, the best-fit values of A and B were found to be
`1.323 and −0.834, respectively, by linear regression of the
`RBC permeability data presented by Lieb and Stein (14).
`Once the solute has crossed the cell membrane, it will
`diffuse within the cell cytoplasm. Cytosol-to-water diffusivity
`ratios have been found to be on the order of 1/4 (15), and we
`assumed that the average distance traveled within the cytosol
`4 5 mm. The
`from one side of the cell to the other is about lcyt
`transcytosol permeability can thus be estimated as follows:
`kcyt = D‘
`4lcyt
`where D‘ is the solute diffusivity in dilute bulk solution. Im-
`plicit in the equation given above is the assumption that solu-
`bility within the cytoplasm is equal to that within water.
`Equation 6 yields transverse permeability values that are
`very low and almost always negligible compared to lateral
`permeability values. Because the data of Lieb and Stein (14)
`were obtained for molecular radii smaller than about 3 Å, it
`is possible that the expression accounting for the size depen-
`dence of kmem does not apply to the larger solutes examined
`in this study. If Equation 6 overpredicts permeability (i.e., the
`actual transverse permeability is lower), overall model pre-
`dictions would remain unchanged, because the transverse
`route would remain negligible. In contrast, underprediction
`could be problematic because transverse permeabilities could
`become significant and thereby increase corneal permeability
`predictions.
`To assess the likelihood of over- or underprediction for
`larger molecules, we found literature values for transverse
`permeability across synthetic lipid bilayers for tryptophan
`4 1.9 × 10−2), both of
`4 9.1 × 10−2) and citric acid (Kow
`(Kow
`which have a radius of about 3.5 Å (16,17). These permeabili-
`ties have been measured as 4.1 × 10−10 and 3.1 × 10−11 cm/s,
`respectively, whereas Equation 6 predicts values of 1.6 × 10−8
`and 2.1 × 10−9 cm/s, respectively. Although this large over-
`
`(8)
`
`Solutes that do not partition extensively into endothelial
`cell membranes and thus cannot access the transcellular path-
`way follow the paracellular route between the cells. As de-
`scribed by Fischbarg (13), the intercellular space between two
`endothelial cells can be idealized as a slit channel consisting of
`a wide section and a narrow one, the latter corresponding to
`the gap junction (Fig. 1B). The half-width of the wide part
`(W1) and that of the narrow part (W2) have been measured as
`15 nm and 1.5 nm, respectively. The length of the wide part
`(L1) has been determined to be 12 mm and that of the gap
`junction (L2) is 0.24 mm (13).
`Based on the theoretical results of Panwar and Anderson
`(18), the permeability ki of a parallel-wall channel of half-
`width Wi and length Li is given by:
`DlnS rS
`S rs
`F1 + 9
`ki = fiD‘
`D3G
`+ 0.159317S rs
`Li
`
`D - 1.19358S rs
`
`Wi
`
`D
`
`(9)
`
`16
`
`Wi
`
`Wi
`
`Wi
`
`where fi is the fractional area of intercellular openings on the
`surface (i.e., porosity), and rs is the solute radius. The total
`4 1200 cm/cm2 of
`cell perimeter length per unit area is lc
`? lc. The
`endothelial surface (13), and fi is estimated as Wi
`term within brackets in Equation 9 represents the channel-
`to-free solution diffusivity ratio (including the effects of par-
`titioning). D‘ is determined using the Wilke-Chang equation
`(19) (for the molecules examined in this study, D‘ is between
`0.5 × 10−5 and 2 × 10−5 cm2/s). Electrostatic effects are ne-
`glected in this approach, as suggested by previous studies
`(20). Also, note that because of tortuosity, the total length of
`4 12.2 mm, is significantly
`the intercellular channel, L1 + L2
`greater than the cell thickness, which is equal to 5 mm.
`The overall permeability of the paracellular pathway,
`consisting of the wide and narrow slit channels in series, is
`then given by:
`
`kp =
`
`1
`1/k1 + 1/k2
`
`(10)
`
`Stroma
`
`The stroma is a fibrous tissue that forms the bulk of the
`cornea and is made up primarily of large collagen fibers em-
`bedded in a proteoglycan matrix. We previously developed a
`model for the permeability of the stroma to small solutes and
`macromolecules (8). The analysis was performed on three
`length scales; for each, we assumed a given arrangement of
`fibers of defined geometry and orientation, around and
`through which solutes diffuse. Corresponding calculations
`were based on a fiber matrix approach. At the macroscale,
`stromal collagen lamellae are arranged in parallel sheets. At
`the mesoscale, the lamellae contain collagen fibrils forming
`hexagonal arrays of parallel cylinders. At the microscale,
`ground substance that surrounds the fibrils and lamellae was
`modeled as a randomly oriented collection of fibers, repre-
`
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`
`

`
`1500
`
`Edwards and Prausnitz
`
`senting the proteoglycans. The corresponding equations are
`detailed in Edwards and Prausnitz (8) and are briefly sum-
`marized in the Appendix. It should be noted that in the pre-
`vious study (8), a stromal hydration of 86% was used because
`we considered transport across isolated stroma, which has an
`increased water content. For this study, stromal hydration was
`taken as 78%, which is the physiologic value for intact cornea
`(8,10).
`
`Epithelium
`
`The epithelium is a multi-layer of cells found at the ex-
`ternal surface of the cornea. The basal layer, separated from
`the stroma by a thin basement membrane, consists of a single
`sheet of columnar cells, about 20 mm high and 10 mm wide
`(10). Two or three layers of wing cells cover the basal cells,
`from which they are derived. As wing cells migrate towards
`the corneal surface, they flatten and give rise to two or three
`sheets of squamous cells that are about 4 mm thick and 20–45
`mm wide. The total thickness of corneal epithelium is approxi-
`mately 50–60 mm in humans (10). Similar to endothelium, the
`epithelium has two parallel pathways, a transcellular and a
`paracellular one (Fig. 1), meaning that Eq. 1 can be used.
`
`Transcellular Pathway
`
`epithelium. In addition, we assumed an idealized epithelial
`geometry having three layers of squamous cells with a mean
`surface radius of 20 mm, three layers of wing cells of mean
`radius 10 mm, and one layer of columnar cells, 5 mm in radius
`4 120 + 95/3 4 151.7 mm.
`(10). This yields Llat
`Transverse Diffusion. To determine the permeability of
`the transverse route across epithelium, the permeability of a
`single cell membrane and that of the cytosol in a single cell
`were taken to be the same as in endothelium. Because epi-
`thelial cell layers can be represented as resistances in series,
`lcyt was set equal to epithelium thickness, 50 mm, and Eq. 8
`was used to calculate cytosol permeability. The permeability
`of a single cell membrane was determined using Eq. 6, and
`Eq. 5 was used to calculate permeability of the transverse
`route with the term 1/kmem multiplied by 14, rather than 2,
`based on our above idealization of seven cell layers in the
`epithelium.
`
`Paracellular Pathway
`
`Freeze-fracture observations of the epithelium show that
`tight junctions are localized almost exclusively in the super-
`ficial layer, whereas larger gap junctions are found in deeper
`layers (21). In the absence of specific experimental data re-
`garding the dimensions of the different epithelial junctions,
`we chose to replace those multiple junctions by one narrow
`junction, such that the latter would account for the size se-
`lectivity imparted by all tight and gap junctions combined. We
`therefore modeled each intercellular opening as a parallel-
`wall slit with a narrow section, corresponding to the equiva-
`lent junction, followed by a wider one (Fig. 1B). The dimen-
`sions of the epithelial narrow junction were chosen so that its
`effects are equivalent to the combined effects of all epithelial
`junctions in series, as described below.
`Lacking independent data, the large half-width (W1) was
`taken to be 15 nm throughout the epithelium, which was
`based on measurements in endothelium. We estimated the
`narrow half-width (W2) based upon the data of Ha¨ma¨la¨inen et
`al. (22). In their study, the authors measured the permeability
`of cornea to small hydrophilic solutes, which diffuse predomi-
`nantly through the paracellular pathway and for which the
`epithelium should be by far the tightest barrier. We therefore
`fitted Eq. 10 to their data; the value of W2 that yielded the
`best agreement with experimental observations was 0.8 nm.
`We further assumed that the length of the equivalent narrow
`4 0.24 mm, that is, equal to
`junction in the epithelium was L2
`that of endothelial gap junctions. The combined length of the
`wide channel parts (L1) was assumed to be equal to the over-
`all distance covered by molecules diffusing alongside the edge
`4
`of cells minus the length of the tight junction, i.e., Llat − L2
`151.4 mm.
`2), the total cell perimeter
`Given a cell density of 1/(pRi
`length per unit area of epithelium, lc, was calculated as 2/Ri
`(i.e., 1000 cm/cm2 based on a 20 mm radius for squamous cells;
`? lc. With those param-
`Ref. 15), and fi was calculated as Wi
`eters, the permeability of the paracellular pathway in epithe-
`lium was determined using Eqs. 9 and 10.
`
`Whole Cornea
`
`In summary, solute permeability of epithelium and en-
`dothelium is each described by the sum of the transcellular
`
`The permeability of the transcellular pathway in the epi-
`thelium was calculated using Eq. 2; the values for klat and ktr
`were determined as follows.
`Lateral Diffusion. The permeability of the lateral route
`in the epithelium was calculated using Eq. 3, assuming that
`the only significant difference between lateral diffusion across
`corneal epithelium and endothelium is the pathway length,
`Llat. That is, F and Dlat have the same values as in endothe-
`lium. We assumed that epithelial cells are packed very tightly
`so that there is almost no discontinuity as a solute diffuses
`from one cell to the other, i.e., membrane-to-membrane par-
`titioning is not a rate-limiting step.
`The mean diffusion pathway length, Llat, was calculated
`assuming the cells are shaped like cylinders. Once a molecule
`partitions into a cell membrane somewhere on its upper sur-
`face, it must first diffuse across to the edge of the upper
`surface and then down along the side of the cell. The average
`distance <ri> a molecule must diffuse across the upper circu-
`lar surface of a cell i is equal to:
`Ri *
`
`2p
`u=0
`
`r2drdu = Ri
`3
`
`r=0
`
`(11)
`
`^ri& = Ri -
`
`2 *
`1
`pRi
`where Ri is the cell radius.
`Assuming that the cells do not form columns but are
`randomly aligned, the mean pathway length is then given by:
`Llat = (
`
`SLi + Ri
`
`3
`
`i
`
`(12)
`
`i
`
`D = L + 1
`3(
`Ri
`where Li is the distance that a molecule must diffuse down
`along the side of a cell and L is the total diffusion length along
`the side of the cells, which is greater than the thickness of the
`epithelium because of tortuosity. In the 5-mm-thick endothe-
`lium, L has been measured as approximately 12 mm, i.e., a
`tortuosity of 2.4 (13). In the absence of data for the epithe-
`lium, we assumed that the tortuosity factor was the same in
`both barriers, and L was taken as 120 mm for the 50-mm-thick
`
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`

`
`Predicted Corneal Permeability
`
`1501
`
`(Eq. 2) and paracellular (Eq. 10) contributions, where differ-
`ent geometric constants (e.g., Li, Llat, Wi) are used in each
`tissue, as summarized in Table I. The permeability of stroma
`is calculated using a fiber-matrix model developed previously
`(8) and briefly recapitulated in the Appendix. Finally, the
`overall permeability of the cornea (kcornea) is determined by
`the series combination of the resistance to transport of the
`three tissues:
`
`1
`+ 1
`1
`= 1
`kendo
`kstroma
`kepi
`kcornea
`where the subscripts “epi” and “endo” refer to the epithelium
`and endothelium, respectively.
`
`(13)
`
`+
`
`Calculations for Model Validation and Comparison with
`Other Theoretical Models
`
`Comparisons between model predictions and experimen-
`tal data are needed to validate the above model, which we
`performed using a compilation of experimentally determined
`permeabilities, as well as distribution coefficients, molecular
`radii, and other data collected by Prausnitz and Noonan (2).
`Whole cornea and isolated stroma permeabilities were deter-
`mined directly through experimentation. However, few direct
`experimental measurements of corneal endothelial or epithe-
`lial solute permeability were made. Instead, literature reports
`present permeability values for combined layers, such as
`stroma-plus-endothelium, i.e., de-epithelialized cornea. Indi-
`rect experimental values of the permeability of endothelium
`and epithelium were therefore obtained by considering resis-
`tances in series. For example, endothelial permeabilities were
`determined by subtracting the resistance (i.e., the inverse of
`the permeability) of stroma from that of stroma-plus-
`endothelium:
`
`=
`
`(14)
`
`1
`1
`1
`kstroma
`kstroma+endo
`kendo
`Epithelial permeabilities were calculated in a similar manner,
`based on reported values for stroma and stroma-plus-
`epithelium, or for full cornea and de-epithelialized cornea.
`These calculations require that the measured values that
`are combined to yield endothelial or epithelial permeabilities
`correspond to identical experimental conditions, i.e., same
`species, temperature, and hydration. We therefore only sub-
`tracted resistances when they were reported in the same
`study. Even in this case, it is, for example, likely that the
`hydration of stroma was higher when the cell layers were
`removed, but we did not account for this effect in the absence
`of specific hydration data. Results for endothelium and epi-
`thelium are given in Tables II and III, respectively. In some
`studies, the measured permeability of several layers com-
`
`bined was higher than that of one layer; such inconsistent data
`were not included in this analysis.
`Uncertainties in these permeability values can be very
`large, as illustrated by the following example. The reported
`permeability of stroma to corynanthine is 3.2 ± 0.6 × 10−5 cm/s
`and that of stroma-plus-endothelium is 3.1 ± 0.2 × 10−5 cm/s
`(23), yielding a calculated endothelial permeability of 9.9 ± 78
`× 10−4 cm/s. Even though experimental standard deviations
`were small, uncertainty in kendo is much larger than the per-
`meability itself because the two measured values are very
`close. This constrains our ability to validate model predictions
`given the large error bars on much of the test data calculated
`using Eq. 14.
`To compare the predictive ability of our model to that of
`others, we calculated the following sum of squared errors,
`which is related to a log-scale chi squared (24):
`
`n Fln~kcornea
`SSE = (
`
`- 1G2
`
`(15)
`
`calc
`
`!
`
`i=1
`
`meas !
`ln~kcornea
`
`where n is the number of solutes considered (e.g., 117 for the
`
`
`cornea) and k calccornea and k meascornea are the calculated and mea-
`sured permeabilities of cornea to solute i, respectively. As an
`additional characterization, we determined the mean absolute
`fractional error (MAFE) associated with differences between
`predicted and measured values, defined as the average abso-
`lute value of the residual divided by the actual (i.e., experi-
`mental) value (24).
`
`
`n |kcornea
`- kcorneameas |
`MAFE = 1
`n(
`meas
`kcornea
`i=1
`
`calc
`
`(16)
`
`RESULTS AND DISCUSSION
`
`Permeability Predictions
`
`To predict the passive, steady-state permeability of cor-
`nea to a broad range of compounds, we developed a theoret-
`ical model based on the physicochemical properties of the
`cornea and diffusing solutes, which were estimated, whenever
`possible, from independent literature data. Two types of path-
`ways were considered (Fig. 1). Paracellular pathways are wa-
`ter-filled routes between cells in epithelium and endothelium
`and between the fibers of the largely aqueous stroma. Per-
`meability of these routes is mainly a function of the size and
`geometry of the pathways and solutes. Transcellular pathways
`consist of lipid-filled routes within cell membranes. Although
`the permeability of those pathways also depends upon size
`and geometry, another important factor is the ability of a
`solute to partition into cell membranes, as determined by the
`solute’s membrane-to-water distribution coefficient.
`
`Table I. Parameters for Epithelial and Endothelial Permeability Calculations
`
`Transcellular mean diffusion pathway length (Llat)
`Lateral diffusivity in cell membranes (Dlat)
`Wide channel half-width (W1)
`Narrow channel half-width (W2)
`Wide channel length (L1)
`Narrow channel length (L2)
`
`Endothelium
`
`15.5 mm
`2 × 10−8 cm2/s
`15 nm
`1.5 nm
`12 mm
`0.24 mm
`
`Epithelium
`
`151.7 mm
`2 × 10−8/cm2/s
`15 nm
`0.8 nm
`151.4 mm
`0.24 mm
`
`Page 5
`
`-
`

`
`1502
`
`Acebutolol
`Atenolol
`Clonidine
`Corynanthine
`Fluorescein
`Lactate ion
`Mannitol
`Metoprolol
`Oxprenolol
`Phenylephrine
`Phosphate ion
`Propranolol
`Rauwolfine
`Rubidium ion
`SKF 72 223b
`SKF 86 466b
`Sucrose
`Thiocyanate ion
`Urea
`
`Table II. Measured and Calculated Permeabilities of Endotheliuma
`
`Measured
`kstroma (cm/s)
`
`Measured
`kstroma+endo (cm/s)
`
`Indirectly measured
`kendo (cm/s)
`
`Predicted
`kendo (cm/s)
`
`Reference(s)
`
`Edwards and Prausnitz
`
`3.0E-5
`3.3E-5
`4.9E-5
`3.2E-5
`
`3.4E-5
`3.7E-5
`5.8E-5
`
`3.5E-5
`3.6E-5
`
`4.2E-5
`5.7E-5
`
`9.3E-6
`1.6E-5
`4.7E-5
`3.1E-5
`
`2.8E-5
`3.1E-5
`2.1E-5
`
`3.1E-5
`2.3E-5
`
`3.9E-5
`5.3E-5
`
`1.4E-5
`3.1E-5
`1.2E-3
`9.9E-4
`5.0E-6c
`2.8E-5c
`9.2E-6c
`1.6E-4
`1.9E-4
`3.3E-5
`4.4E-6c
`2.7E-4
`6.4E-5
`3.4E-5c
`4.9E-4
`8.4E-4
`5.9E-6c
`2.5E-5c
`2.0E-5c
`
`1.8E-5
`7.9E-6
`1.0E-5
`1.2E-3
`6.4E-6
`1.8E-5
`1.1E-5
`1.0E-5
`2.0E-5
`1.1E-5
`2.5E-5
`6.6E-5
`5.9E-5
`7.2E-5
`1.9E-5
`2.6E-4
`7.3E-6
`2.5E-5
`2.1E-5
`
`6
`6
`6
`23
`31
`31
`32
`6
`6
`23
`33
`6
`23
`34
`23
`23
`32,35,36
`34
`35,36
`
`a As discussed in the “Model Development” section, experimental permeability measurements were obtained from the references listed and
`treated mathematically using Eq. 14.
`b SKF 72 223: 5,8 Dimethoxy-1,2,3,4-tetrahydroisoquinoline; SKF 86 466; 6-Chloro-3-methyl-2,3,4,5-tetrahydro-1H-3-benzazepine.
`c Measured directly.
`
`Figure 2A shows the predicted permeability as a function
`of solute radius and distribution coefficient in each of the
`cornea’s three primary layers: endothelium, stroma, and epi-
`thelium. Results are given for solutes ranging from 3.5 to 5.5
`Å in radius (i.e., about 100 to 500 Da molecular weight);
`macromolecules are too large to diffuse across cornea and are
`therefore not included. Figure 2A suggests that the perme-
`ability of stroma, a water-filled fibrous medium composed
`mostly of collagen and glycosaminoglycans, depends only on
`solute radius (rs) and not on distribution coefficient (F) be-
`cause there is no favored pathway for hydrophobic com-
`pounds (because F is a measure of partitioning between wa-
`ter and lipid, it does not influence partitioning between water
`and stroma). In the endothelium and epithelium, the perme-
`ability increases with both decreasing solute radius and in-
`creasing solute distribution coefficient.
`When distribution coefficients are small, solutes diffuse
`predominantly through the paracellular routes around cells so
`that endothelial and epithelial permeabilities are a function of
`size only. As F increases from approximately 0.01 to 10, the
`contribution of the transcelullar pathways becomes progres-
`sively more important, and permeabilities rise with the distri-
`bution coefficient. This increase is all the more pronounced
`for the larger solutes because they have a lower paracellular
`permeability as a result of hindered transport through tight
`and gap junctions. For values of F greater than 10, the trans-
`cellular route is dominant, and the permeability of the cell
`layers becomes mainly a function of distribution coefficient.
`Permeability predictions for the entire cornea are shown
`in Fig. 2B. Using the equations developed above, model pre-
`dictions can be made for any compound of known radius and
`distribution coefficient. For small F values, the permeability
`is essentially a function of solute size, since only hydrophilic
`pathways are accessible. Comparison with Fig. 2A shows that
`
`for such