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Journal of Controlled Release 99 (2004) 241 – 258 www.elsevier.com/locate/jconrel Model of transient drug diffusion across cornea Wensheng Zhanga,*, Mark R. Prausnitzb, Aurélie Edwardsa,* a Department of Chemical and Biological Engineering, Tufts University, 4 Colby Street, Medford, MA 02155, USA b School of Chemical and Biomolecular Engineering and Center for Drug Design, Development, and Delivery, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA Received 26 February 2004; accepted 1 July 2004 Available online 14 August 2004 Abstract A mathematical model of solute transient diffusion across the cornea to the anterior chamber of the eye was developed for topical drug delivery. Solute bioavailability was predicted given solute molecular radius and octanol-to-water distribution coefficient (U), ocular membrane ultrastructural parameters, tear fluid hydrodynamics, as well as solute distribution volume (V d) and clearance rate (Cla) in the anterior chamber. The results suggest that drug bioavailability is primarily determined by solute lipophilicity. In human eyes, bioavailability is predicted to range between 1% and 5% for lipophilic molecules (UN1), and to be less than 0.5% for hydrophilic molecules (Ub0.01). The simulations indicate that the distribution coefficient that maximizes bioavailability is on the order of 10. It was also found that the maximum solute concentration in the anterior chamber (C max) and the time needed to reach C max significantly depend on U, V d, and Cla. Consistent with experimental findings, model predictions suggest that drug bioavailability can be increased by lowering the conjunctival-to-corneal permeability ratio and reducing precorneal solute drainage. Because of its mechanistic basis, this model will be useful to predict drug transport kinetics and bioavailability for new compounds and in diseased eyes. D 2004 Elsevier B.V. All rights reserved. Keywords: Eye; Ophthalmic drug delivery; Topical; Anterior chamber; Transport 1. Introduction Topical drug delivery is the most common treatment for diseases of the anterior segment of the eye, such as glaucoma, but is limited by the low perme* Corresponding authors. Wensheng Zhang is to be contacted at Tel.: +1-617-627-3528; fax: +1-617-627-3991. Aurélie Edwards, Tel.: +1-617-627-3731; fax: +1-617-627-3991. E-mail addresses: [email protected] (W. Zhang)8 [email protected] (A. Edwards). 0168-3659/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jconrel.2004.07.001 ability of the multilayered cornea, rapid clearance by tear drainage, and absorption into the conjunctiva. Hence, the bioavailability of topically administered drugs is very low. Drug transport processes and methods to improve bioavailability have been studied extensively for the past decades, as reviewed, for example, by Schoenwald [1], Lee and Robinson [2], and Jarvinen et al. [3]. Pharmacokinetic models have been widely used to predict the efficiency of new topical agents. Early physiological models of the eye were developed by 242 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 Himmelstein et al. [4] and Miller et al. [5]. Compartment models, in which each tissue is represented as a separate compartment and is linked sequentially to the next one as the drug passes through, are commonly employed today. Their underlying assumptions as well as their applications have been previously reviewed [1,2]. In these pharmacokinetic models, absorption into the cornea and conjunctiva is usually described as a first-order process, and corresponding rate constants are determined by fitting the model to a given set of experimental data. The models are therefore often valid only for a specific group of molecules, and their use for the design of new agents is limited. Taking a more detailed look at drug transport after topical instillation, drug is absorbed and diffuses in the cornea and conjunctiva/sclera. The cornea consists of three main barriers in series: the epithelium, the stroma, and the endothelium. The corneal stroma and sclera are highly hydrated fibrous acellular tissues. The conjunctiva, corneal epithelium, and endothelium are cellular layers that contain both transcellular and paracellular pathways; lipophilic molecules preferentially diffuse within cells, whereas hydrophilic molecules permeate mostly through the openings between cells. The authors previously developed theoretical approaches to predict solute diffusivity in the cornea and sclera based upon the ultrastructure of these membranes and solute physicochemical properties [6,7]. Building upon these predictions, in this study, a mathematical model was developed for topical drug delivery across the cornea that accounts for the ultrastructure of ocular membranes, solute molecular size and lipophilicity, tear fluid hydrodynamics, as well as drug distribution and clearance in the anterior chamber. This transient model is built upon the partial differential equation for diffusion across ocular membranes (i.e., Fick’s law) rather than the empirical first-order absorption equation commonly used. All the parameters of this model have a physical meaning and can be independently measured or estimated, so that the model can be applied to a variety of compounds. Since the actual mechanisms of drug transport and elimination are explicitly taken into consideration, this approach can yield insights into the main determinants of drug delivery and may therefore guide strategies to improve drug bioavailability. 2. Mathematical model The model developed in this study addresses the fate of drug molecules applied topically to the eye. As a drop of solution is instilled into the eyelid sac, it is assumed to mix instantly with the tear fluid due to reflex blinks [8]. As shown in Fig. 1, drug solution thus delivered to the precorneal area (i.e., the conjunctival sac and the tear film, considered as a single, homogeneous compartment) is diluted by lacrimal secretion and cleared by four different mechanisms: drainage with the tear fluid towards Fig. 1. Schematic representation of topical drug delivery model. W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 the nasal cavity, absorption into the palpebral conjunctiva, absorption into the bulbar conjunctiva/ sclera, and absorption into the cornea. Since solute absorbed into the conjunctiva/sclera generally does not subsequently diffuse into the anterior chamber [9,10], the contribution of that route to solute deposition into the anterior chamber is neglected, and the concentration at the interface between the sclera and the anterior chamber is assumed to be zero. Molecules that permeate into the cornea diffuse through the epithelium, stroma, and endothelium before reaching the aqueous humor in the anterior chamber. The anterior chamber is considered as a homogeneous compartment. In the anterior chamber, the solute partitions between the aqueous humor and other tissues, such as iris and ciliary body, and it is cleared either by aqueous humor drainage or by systemic absorption [1]. In the approach of this study, solute molecules are characterized by their size and lipophilicity. The latter is usually quantified using two parameters: the partition coefficient, which represents the nonionized molecule concentration ratio between the organic and aqueous phases, and the distribution coefficient, which represents the overall concentration ratio (i.e., including the ionized solute concentration in the aqueous phase). Here, the lipid bilayer-to-water distribution coefficient (U) is used to express concentration equilibrium at the interface between the tear film and the adjacent epithelial membranes. There are two pathways available for solutes diffusing across the epithelia and endothelia: transcellular routes, favored by lipophilic molecules that can easily penetrate the lipid bilayer of cells; and paracellular routes, favored by hydrophilic molecules. Accounting for simultaneous transport through both routes requires a full two-dimensional representation of ocular membranes, which lies beyond the scope of this study. Therefore, it is assumed that solutes permeate through only one or the other of these two pathways, which permits a simpler, one-dimensional approach. 243 portion in excess of the normal lacrimal volume), following the approach of Chrai et al. [11], so that: dVt ¼ kd ðVt VL Þ dt ð1Þ where V t is the tear volume at time t, V L is the normal lacrimal volume (so that V tV L is the excess tear volume), and k d is the solution drainage constant. Integration of Eq. (1) yields the tear volume as: Vt ¼ Vi expð kd t Þ þ VL ð2Þ where Vi is the instilled volume (i.e., the initial excess tear volume). Alternatively, conservation of tear fluid can be expressed as: dVt ¼SQ dt ð3Þ where S is the (fixed) lacrimal secretion rate and Q is the solution discharge rate. Combining Eqs. (1)–(3) yields the tear drainage rate as: Q ¼ S þ kd Vi expð kd t Þ ð4Þ The solute concentration in the tear film C t is assumed to remain homogeneous. Conservation of solute in the tear film can be written as: dðVt Ct Þ dCt dVt ¼ Vt þ Ct dt dt dt ¼ Fep fep Aep Fpj fpj Apj Fbj fbj Abj QCt ð5Þ where F i is the Fickian diffusional flux of solute across membrane i whose surface area is A i . The fraction of A i that is occupied by the diffusional route being considered (i.e., transcellular or paracellular) is denoted by f i . The subscripts bep,Q bpj,Q and bbjQ represent corneal epithelium, palpebral conjunctiva, and bulbar conjunctiva, respectively. Substituting Eqs. (2) and (3) into Eq. (5) yields: 2.1. Precorneal area After a drop is instilled, it is assumed that the change in precorneal tear volume over time is proportional to the excess tear volume (i.e., the Fep fep Aep Fpj fpj Apj Fbj fbj Abj SCt dCt ¼ dt VL þ Vi expðkd t Þ ð6Þ 244 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 where W is the tissue-to-water distribution coefficient. Fickian diffusional flux F i is expressed as: 2.2. Anterior chamber To determine solute concentration in the aqueous humor as a function of time, conservation of solute in the anterior chamber is expressed as: Fi ¼ Di dðVd Ca Þ ¼ Fen fen Aen Cla Ca dt 2.4. Anatomical and physiological parameters of the eye ð7Þ where V d is the distribution volume of solute in the anterior chamber (which accounts for partitioning between the aqueous humor and the iris/ciliary body), C a is the solute concentration in the anterior chamber (which is a weighted average that accounts for solute bound to tissue and free in solution), and Cla is the drug clearance rate in the anterior chamber. As described by Schoenwald [1], the clearance rate of solute can be several times higher than that of aqueous humor, as metabolism and systemic uptake by the vascular tissues of the anterior uvea constitute alternate routes of elimination. The subscript benQ denotes the endothelium at the endothelium–anterior chamber interface. Rearranging Eq. (7) yields: dCa Fen fen Aen Cla Ca ¼ dt Vd ð8Þ @Ci @x ð12Þ Anatomical and physiological parameters were determined based on experimental measurements found in the literature, or, when data were not available, using theoretical estimates as described in previous publications [6,7]. Baseline values for these parameters are summarized in Table 1. Note that the distribution volume (V d) and clearance (Cla) in the anterior chamber vary with each solute. For the baseline case, midrange values of 600 Al and 13.5 Al/min, respectively, were therefore assumed. The baseline instillation volume was taken as 30 Al. It was also assumed that the diffusional length, diffusivity of the solute, and its lipid bilayer-water distribution coefficient (U) were identical in the conjunctival epithelium and in the corneal epithelium and that the fractional surface area occupied by the paracellular route (i.e., the porosity f ) was 15 times greater in the 2.3. General approach to diffusion in the tissues Table 1 Anatomical and physiological parameters in human eyes The general solute conservation equation in tissue i is expressed as: Parameters Value Reference Normal tear volume (V L) [Al] Tear secretion rate (S) [Al/min] Solution drainage rate constant (k d) [min1] Corneal surface area (A c) [cm2] Conjunctival surface area (A j ) [cm2] Palpebral conjunctiva surface area (A pj) [cm2] Bulbar conjunctiva surface area (A bj) [cm2] Corneal thickness [mm] Volume of aqueous humor in anterior chamber (Va) [Al] Aqueous humor secretion rate, as a percentage of Va [%/min] Distribution volume in anterior chamber (V d) [Al] Clearance in anterior chamber (Cla) [Al/min] 7.0 1.2 1.45 [2] [2] [11] 1.04 17.65 8.82 [38] [38] See text 7.78 See Text 0.52 261–310 [38] [2,39,40] 1–2%/min [2] 150–3000 [1] 1–30 [1] @Ci @ 2 Ci ¼ Di @t @x2 ð9Þ where D i is the solute diffusivity across the corresponding layer and x is the axial coordinate as shown in Fig. 1. At the interface between two adjacent layers, interfacial boundary conditions can be expressed as: Cj Ci ¼ Wi Wj ð10Þ Fi fi Ai ¼ Fj fj Aj ð11Þ W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 conjunctival epithelium than in the corneal epithelium. These assumptions are discussed later. 2.5. Diffusion across cornea Diffusion across the cornea was modeled following the approach of Edwards and Prausnitz [6,7]. The detailed model can be found in those papers; the predicted parameters needed for this study are reported in Table 2. Briefly, transport across the three primary layers that make up the cornea was considered. The cellular tissues of corneal epithelium and endothelium were modeled based on two transport pathways: one following a tortuous route through tight junctions and around cells, utilized primarily by hydrophilic compounds, and the other, utilized primarily by lipophilic compounds, involving partitioning into, moving along, and partitioning out of cell membranes. The largely acellular tissue of stroma was modeled as a fiber matrix, where molecules diffuse around the lamellar collagen fibrils embedded in a fibrous extracellular matrix. The model was developed using independently determined parameters that represent physical properties of drugs and tissues. In this analysis, the corneal epithelium constitutes the main barrier to the transport of hydrophilic molecules across the cornea; for hydrophilic solutes with a radius between 0.35 and 0.55 nm (i.e., approximately 150–450 Da), the relative resistances to diffusion through the corneal epithelium, stroma, and endothelium are predicted to be approximately 92%, 1%, and 7%, respectively (Table 2), in reasonable agreement with experimentally measured relative resistances of 80:1:2 for mannitol [12]. For lipophilic molecules, the resistance of the cellular layers increases as U decreases. For highly lipophilic compounds (e.g., U=100), the relative resistance of the corneal epithelium is found to decrease from 41% to 20% as solute radius increases from 0.35 to 0.55 nm. For less lipophilic compounds (U=10), the relative resistance is 81– 67%. These predictions agree with the results of Huang et al. [13], who experimentally found that the resistance of the epithelium layer to h-blockers is 1– 20% that of cornea for highly lipophilic molecules (UN40), and 45–70% for less lipophilic compounds (Uc10). 245 Table 2 Diffusional parameters across cornea and scleraa Lipophilicity Hydrophilic Solute radius (nm) Epithelium L b (Am) f c (%) D (106 cm2/s) Wd P e (106 cm/s) k f (%) Stroma L (Am) f (%) D (106 cm2/s) W P (106 cm/s) k (%) Endothelium L (Am) f (%) D (106 cm2/s) W P (106 cm/s) k (%) Sclera D (106 cm2/s) W Corneal P (106 cm/s) 0.35 151.7 0.18 10.9 a b c 0.45 151.7 0.18 6.54 Lipophilic 0.55 151.7 0.18 4.33 0.35 151.7 1 0.02 0.45 151.7 1 0.02 0.55 151.7 1 0.02 0.86 0.83 0.79 10 1.1 0.64 0.40 13.2 10 10 13.2 13.2 92 450 1 6.1 75 450 1 3.5 92 450 1 3.5 92 450 1 2.2 81 450 1 6.1 67 450 1 2.2 0.73 0.71 0.69 0.73 98.9 55.5 33.9 98.9 0.71 0.69 55.5 33.9 1 12.2 0.18 11.1 11 15.5 1 0.02 18 15.5 1 0.02 26 15.5 1 0.02 0.88 0.84 0.81 10 14.4 8.37 5.33 129 10 129 10 129 7 7 7 8 6.96 4.16 2.72 6.96 7 7 4.16 2.72 1 12.2 0.18 6.71 1 12.2 0.18 4.46 0.762 0.752 0.741 0.762 0.752 0.741 1.03 0.59 0.37 10.7 9.84 8.84 Parameter evaluation is described in Refs. [6,7]. L=diffusional length (i.e., actual thickness times tortuosity). f=Fraction of the surface area occupied by the paracellular route. d W=Tissue-to-water distribution coefficient. This coefficient was taken as the octanol-to-water distribution coefficient (i.e., U) for partitioning of lipophilic molecules into epithelium or endothelium, which is determined by solubility. In contrast, W is governed by steric exclusion for lipophilic and hydrophilic molecules in the aqueous stroma, and for hydrophilic molecules in the water-filled paracellular spaces of cell layers. In these latter cases, W was calculated as described in Edwards and Prausnitz [6]. e P=permeability, calculated as DfW/L. For cornea, P c=1/(1/ P ep+1/P st+1/P en). For lipophilic compounds, P was based on W=10. f k=Ratio of the membrane diffusional resistance to that of the cornea. Resistance is the inverse of permeability. 2.6. Diffusion across sclera and conjunctiva Solute diffusion in the sclera was modeled as in stroma, following the approach of Edwards and Prausnitz [6]; the diffusional parameters are shown in Table 2. 246 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 The conjunctiva consists of ultrastructurally heterogeneous epithelia, with 10–15 cell layers towards the cornea and five to six layers at the eyelids [14]. Lacking more specific ultrastructural data, it was assumed that the surface area of the palpebral conjunctiva, which covers the inner surface of the eyelid, is the sum of the surface area of the bulbar conjunctiva (which covers the visible part of the sclera) and that of the cornea. The effective diffusional length across conjunctiva was also assumed to be the same as that across corneal epithelium, based on the transmission electron microscopy (TEM) pictures published by Breitbach and Spitznas [15], which suggest that the thickness of bulbar conjunctiva is close to 60 Am, comparable to that of the corneal epithelium. The porosity of the conjunctiva was estimated as 2.7%, based on experimental measurements by Hamalainen et al. [16] showing that the porosity of the conjunctiva is 15 times greater than those of the cornea. The porosity estimate in this study is several orders of magnitude larger than that of those investigators since theirs characterizes the superficial tight junction layer of the conjunctival (as well as corneal) epithelium, whereas the value in this study is an average for the entire epithelial layer. These porosity calculations are supported by the experimental measurements of pore size distribution by Doughty [17], which yield a calculated porosity between 0.25% and 44% (7.6% on average) for palpebral conjunctiva, and at least 0.8% for bulbar conjunctiva; these ranges comprise the estimate of 2.7% in this study. This porosity value corresponds to a conjunctival-to-corneal permeability ratio (c) for hydrophilic molecules of 17, based on the assumption that the diffusivities in both conjunctival and corneal epithelia are the same and the observation that the corneal epithelium accounts for 90% of the resistance of the entire cornea. This prediction is in good agreement with experimental data by Wang et al. [18], who measured the conjunctival-to-corneal permeability ratio of hydrophilic h-blockers as 20, and by other investigators who reported values in the range of 14–25 [12,16,19]. It also appears reasonable to assume that lipophilic molecules diffuse within the cell membranes of corneal and conjunctival epithelia at the same rate; thus, the permeabilities of these epithelia were taken to be equal, which yields predictions of c between 2 and 5 for solutes with a molecular radius between 0.35 and 0.55 nm and a distribution coefficient between 10 and 100. These estimates are comparable to the experimental data of Sasaki et al. [20] and Wang et al. [18], who reported c values between 2 and 10 for lipophilic molecules. It is possible that the uncertainty in conjunctival parameter values might influence the accuracy of predictions of corneal absorption and bioavailability. Indeed, this model predicts that the amount of solute absorbed by the conjunctiva is much greater than that diffusing across the cornea, due to the large value of c. In the baseline case, the fractions absorbed into the anterior chamber vs. the bulbar and palpebral conjunctiva were found to be equal to 0.13% vs. 16% and 22%, respectively, for hydrophilic molecules, and 3.2% vs. 31% and 43%, respectively, for lipophilic molecules. However, a sensitivity analysis showed that even a twofold increase in epithelial permeability of either conjunctiva alone, or both conjunctiva and cornea yielded changes in these fractional absorption values of less than 40%. 2.7. Overall fractions of solute clearance The overall amount of solute that has penetrated into a given ocular membrane (i.e., cornea, palpebral conjunctiva, or bulbar conjunctiva) at time t can be obtained from an interfacial balance at the boundary between the tear film and the epithelium (i.e., at x=0): Z t dCi M i ðt Þ ¼ A i f i D i ð13Þ dt: 0 dx x¼0 Mass conservation in the tear film at time t also yields: Mc ðt Þ þ Mpj ðt Þ þ Mbj ðt Þ þ R t 0 QCt dt þ Ct ðt ÞVt ðt Þ ðCt Vt Þjt¼0 ¼1 ð14Þ This equation can be used to verify the accuracy of the numerical results, as described later. Similarly, the overall amount of drug that has been absorbed by the aqueous humor at time t, M a(t), can be obtained either from an interfacial balance at the boundary between the corneal endothelium and the W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 aqueous humor, or from mass conservation in the aqueous humor: Z t dCen Ma ðt Þ ¼ Aen fen Den 0 dx Z t þ Cla Ca dt j dt ¼ Vd Ca eneq ð15Þ 247 (the time needed to reach C max), and M a (the fractional amount of solute diffusing into the chamber) are important to clinical applications and therefore are often measured in experimental studies, the effects of various parameters on these three measurements were predicted. 0 3.1. Effect of molecular size on drug transport where "en–aq" denotes the endothelium–aqueous humor interface. By comparing the right-hand side of Eq. (15) with the center term, the accuracy of the numerical method can be estimated. As time goes to infinity, the value of M a(t) should approach that of M c(t). 2.8. Numerical methods Eqs. (6), (8), and (9) constitute a system of differential equations that must be integrated to yield solute concentration in the precorneal area, the ocular membranes considered here, and the anterior chamber. At t=0, C t=C 0, and all other concentrations are equal to zero; interface (boundary) conditions are expressed by Eqs. (10) and (11). These differential equations were solved using second-order implicit finite difference approximations. The overall magnitude of solute clearance or delivery was calculated based upon Eqs. (13)–(15). The program was written using MatLab 6.1 (The Mathworks, Natick, MA). Simulations were run on a personal computer with an AMD Athlon 1800+ processor (Advanced Micro Devices, Sunnyvale, CA) and lasted typically between 10 and 20 min. The difference between the right-hand side and the center term of Eq. (15) was always less than 103 at each time step, and so was the difference between the righthand and left-hand side of Eq. (14), which shows good numerical accuracy. 3. Results To address the bioavailability of topical eye drops, the objective of this study was to model the transient absorption of a solute from the precorneal area into the anterior chamber, based upon the morphology and physiology of the eye, and upon the solute properties. Because C max (the maximum solute concentration in the anterior chamber), t max Increasing molecular size is known to decrease drug bioavailability and limit topical delivery applications [1–3]. Shown in Table 3 are predicted t max, C max, and M a values based on the diffusional parameters in Table 2. Increasing molecular radius of hydrophilic molecules from 0.35 to 0.95 nm decreases absorption into the ocular tissues. This has the effect of reducing drug transport into the anterior chamber (M a), palpebral conjunctiva (M pj), and bulbar conjunctiva (M bj), as well as C max. Likewise, it increases the time needed to diffuse into the anterior chamber (t max and t 90) and the fraction of solute cleared by tear fluid drainage (M t). For lipophilic compounds, the same trends are generally seen, except that more drugs are absorbed into the ocular tissues over longer periods of time and less are Table 3 Effect of molecular size on solute deliverya Molecular U radius (nm) b b c C max M ac M pj M bjc M tc t 90 t max 5 (min) (10 ) (%) (%) (%) (%) (min) 0.35 0.45 0.55 0.95 0.35 0.45 0.55 0.95 42 55 68 119 99 105 112 145 a Hydrophilic Hydrophilic Hydrophilic Hydrophilic 10 10 10 10 4.3 2.7 1.8 0.37 50 45 40 18 0.17 0.13 0.11 0.06 3.3 3.2 3.0 1.9 26 22 18 9 41 43 44 53 19 16 13 7 33 31 29 20 55 62 69 84 23 23 24 25 100 160 240 759 275 305 340 590 Parameter values are that of the baseline case (see text), unless otherwise specified. b C max is the maximum concentration in the anterior chamber, normalized by the initial concentration in the tear fluid, and t max is the time needed to reach this peak concentration. c M a, M pj, M bj, and M t represent the fraction of solute that has reached the anterior chamber through the corneal route, been absorbed into the palpebral conjunctiva, been absorbed into the bulbar conjunctiva, and has been eliminated by tear fluid drainage, respectively, at t 90 (i.e., when 90% of molecules that have been absorbed into the cornea have diffused into the anterior chamber). 248 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 drained away by tear fluid. As shown in Fig. 2, M a decreases by 65% and 43% for hydrophilic and lipophilic solutes, respectively, as molecular radius increases from 0.35 to 0.95 nm (i.e., from 125 to 2500 Da); for a given solute size, M a is about 20 times larger for lipophilic molecules relative to hydrophilic molecules. In the remainder of this study, results are given by assuming a molecular radius of 0.45 nm, an average representative of current therapeutic compounds, unless otherwise specified. 3.2. Effect of lipophilic vs. hydrophilic transport pathways on drug transport Notwithstanding the variations due to molecular size, bioavailability is predicted to be small for lipophilic molecules that follow transcellular pathways (on the order of 3%) and still smaller for hydrophilic compounds that follow paracellular pathways (on the order of 0.1%; see Table 3 and Fig. 2). This difference can be explained by the very small fraction of the epithelial surface area that is occupied by intercellular openings ( f ) available for hydrophilic solute transport, and the fact that the conjunctival-tocorneal permeability ratio (c) is significantly higher for hydrophilic molecules. The peak time t max is predicted to be close to 1 h for hydrophilic molecules and 2 h for lipophilic compounds in the range of solute radii examined (Table 3). These predictions agree with the exper- imental data of Fukuda and Sasaki [21], who measured t max as 2.25 h for the lipophilic molecule erythromycin lactobionate, and as ~1 h for the hydrophilic compounds chloramphenicol and levofloxacin. It is interesting to note that the corneal layer that controls steady-state permeability values is not necessarily the same as that which controls the kinetics of transient diffusion. Permeability scales with diffusivity (D) and the fractional cross-sectional area available for transport ( f ), as well as other parameters. Because D for lipophilic molecules within epithelial layers is low and f for hydrophilic molecules is also small, the epithelium controls corneal permeability for both lipophilic and hydrophilic molecules. In contrast, transport time lag (and therefore t max) depends on diffusivity (D) and tissue thickness (L), where the characteristic time is on the order of L 2/D. Given their low diffusivity in epithelial layers, t max for lipophilic molecules is also controlled by the epithelium. However, the epithelial diffusivity of hydrophilic molecules is much larger and it is the larger tissue thickness contributed by stroma that predominantly determines t max. The overall result is that the steadystate corneal permeability is greater for lipophilic molecules, whereas transient transport lag time is shorter for hydrophilic molecules. As expected, M a significantly depends on permeability. The results indicate that a twofold increase in the permeability of epithelial layers significantly increased bioavailability in the anterior chamber: for hydrophilic molecules, M a increased by 85%, and for lipophilic molecules, M a increased by 50%. In contrast, when the permeability of the layer that controls t max (i.e., epithelium for lipophilic molecules and stroma for hydrophilic molecules) was held constant, simultaneous twofold changes in diffusivity and membrane thickness (i.e., the parameters that affect transient transport lag time) yielded 30–40% changes in t max, but no significant variations in M a were observed. 3.3. Effect of distribution coefficient on drug transport Fig. 2. Effect of solute molecular radius on bioavailabiliy in the anterior chamber (M a). To more closely examine the effects of lipophilicity, the effects of distribution coefficient on solute transport are illustrated in Fig. 3, assuming that drug is transported exclusively through the transcellular path- W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 Fig. 3. Effect of lipid bilayer-to-water distribution coefficient M a and t max, assuming that solutes are exclusively transported across cellular membranes through transcellular pathways (i.e., neglecting delivery through paracellular pathways). ways across cellular layers even for U less than 1 (i.e., neglecting the possible delivery through paracellular pathways). As U rises from 0.01 to 1000, bioavailability initially increases until it reaches a maximum at U=15, and then it decreases. This functionality is due in part to the observation that increases in U raise the permeability of the conjunctiva to a greater extent than that of cornea, since the corneal permeability includes the acellular stroma (the intrinsic permeability of which is not affected by U). As U increases up to 15, the conjunctiva-to-cornea permeability ratio (c) remains sufficiently small so that corneal absorption experiences a net increase. Thereafter, the increases in permeability of cornea are outweighed by still greater increases in permeability of palpebral conjunctiva (which is composed of epithelium only), such that absorption into palpebral conjunctiva dominates at the expense of absorption into the cornea, as well as bulbar conjunctival absorption and precorneal drainage. Distribution coefficient also affects transport lag time. In these simulations in which U was varied, the clearance rate (Cla) and the distribution volume (V d) in the anterior chamber were fixed, whereas these two parameters are in fact dependent upon solute lipophilicity, among other factors. As described later, although Cla and V d have a small effect on solute bioavailability, they are significant determinants of C max and t max. The effect of U on C max and t max 249 should therefore be examined in conjunction with that of U on Cla and V d. However, it is not possible to predict the effect of U on clearance rate and distribution volume. Assuming that the latter two parameters do not vary with the distribution coefficient, the simulations indicate that as U increases from 0.01 to 1000, t max decreases initially, reaches a minimum for Uc5, and increases thereafter. Indeed, for molecules with UV1, increasing U raises the epithelial concentration at the epithelium–tear interface, thereby reducing the residence time in the precorneal area and decreasing t max. As U increases beyond 5, partitioning between the cell layers and the stroma is significantly reduced, so that diffusion across the stroma is considerably slower and t max increases. In agreement with these predictions, Huang et al. [22] found that timolol (U=2.2) diffuses more rapidly across the cornea than both acebutolol, which is hydrophilic (U=1.6), and bufuralol, which is highly lipophilic (U=200). The review of Lee and Robinson [2] also suggests that the optimum partition coefficient for corneal drug absorption is between 10 and 100. To facilitate analysis, as discussed in the previous section, this model assumes that any given solute permeates through either the transcellular or paracellular route, but not both. The model, therefore, does not apply to the entire range of distribution coefficients and is only valid when transport via one of these two types of routes is predominant. To identify the appropriate range of applicability, the predicted bioavailablity by each route at different distribution coefficients was compared. If U is varied from 1 to 0.1 to 0.01, M a is calculated as 1.2%, 0.14%, and 0.014%, respectively, assuming that all molecules follow the transcellular routes across the epithelium and endothelium. Assuming that solutes all follow the paracellular pathways, M a=0.13%, independently of U. Thus, at Uc0.1, the predicted M a values are comparable for both routes and the model is not rigorously correct. However, for much more lipophilic compounds (Uz1) or much more hydrophilic compounds (UV0.01), the model’s assumptions are valid. 3.4. Effect of anterior chamber distribution volume and clearance rate on drug transport Unlike solute properties, such as molecular size and distribution coefficient, which can be easily 250 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 controlled and varied experimentally, study of ocular physiological properties, such as anterior chamber distribution volume and clearance rate, is experimentally difficult and therefore especially benefits from theoretical analysis. Understanding the effects of anterior chamber fluid mechanics is especially important in cases of glaucoma, ocular inflammation, and pediatric patients. The anterior chamber distribution volume accounts for solute dissolved in the aqueous humor as well as partitioned into the iris/ciliary body, and the clearance rate takes into consideration not only aqueous humor turnover but also metabolic degradation and systemic uptake by vascular tissues. In the simulation results presented previously, effective distribution volume (V d) and clearance rate (Cla) were taken as 600 Al and 13.5 Al/min, respectively, which are the most frequently reported values; however, measurements of V d and Cla vary from 150–3000 Al and 1–30 Al/min, respectively [1]. The effects of V d and Cla were assessed by varying one of the two parameters, while keeping the other fixed at its baseline value. As shown in Fig. 4A and B, variations in Cla or V d have almost no effect on solute bioavailability in the anterior chamber; for lipophilic molecules, M a is slightly reduced as Cla decreases to ~1 Al/min. Indeed, at lower clearance, the accumulation of solute produces a higher concentration in the anterior chamber, which slows diffusion due to a reduced concentration gradient across the cornea; the concentration gradient is even reversed after a long period of time, so that backdiffusion of solute from the aqueous humor to the corneal endothelium occurs. However, as described previously, Cla is generally larger than the aqueous humor secretion rate, which is at least 2.6 Al/min, as shown in Table 1. The effects of Cla and V d on M a are therefore negligible in most cases. Not surprisingly, increases in Cla or V d significantly lower C max, as shown in Fig. 5A and B. Of greater impact on drug delivery, the model also predicts that t max decreases as the clearance rate increases, as illustrated in Fig. 6A. As indicated by Eq. (8), solute concentration in the anterior chamber (C a) is determined by the balance between the diffusional influx from the cornea ( F enA en f en) and the efflux by clearance mechanisms (ClaC a). Initially, the influx is larger than the efflux; however, the clearance rate continues to increase over time while the influx Fig. 4. Effect of (A) clearance rate (Cla) and (B) distribution volume (V d) in the anterior chamber on M a. reaches steady state and then decreases. When the clearance rate equals the diffusional influx, the concentration reaches its maximum value. Therefore, increasing clearance rate reduces the time needed for influx and clearance rate to become equal (i.e., t max). In contrast, t max increases with increasing distribution volume (Fig. 6B) because this slows the rate at which the anterior chamber concentration increases (i.e., dC a/dt; see Eq. (8)) and thereby increases the time needed for the clearance rate to catch up to the diffusional influx. Altogether, if Cla and V d are simultaneously varied over the entire interval (1–30 Al/min and 150–3000 Al, respectively), t max is calculated to range between 17 and 155 min for W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 251 rapidly partition into the cellular epithelia, t 1/2 is calculated to be as short as 5 s. Another consequence of partitioning is that the amount of solute cleared by precorneal drainage (M t) is much greater for hydrophilic than lipophilic molecules (62% vs. 23%). Precorneal drainage rate ( Q) can be altered by changes in instilled volume (Vi), drainage rate constant (k d), and tear turnover rate (S), as indicated in Eq. (4). As shown in Table 4, order of magnitude increases in k d or S affect hydrophilic and lipophilic molecules by reducing M a by ~40% and 10%, respectively, and increasing M t by ~20% and 140%, respectively. The different magnitudes of these effects depending on lipophilicity stem from the fact that lipophilic solutes are much more rapidly absorbed into Fig. 5. Effect of (A) clearance rate and (B) distribution volume in the anterior chamber on C max. hydrophilic molecules and between 63 and 340 min for lipophilic molecules. 3.5. Effect of precorneal solution drainage on drug transport Bioavailability can be dramatically affected by precorneal drainage, which can be altered by dry eye, tearing due to irritation by an eye drop, and drug formulation [2]. For hydrophilic molecules delivered in a conventional aqueous formulation, the half-life of solute concentration in the precorneal area t 1/2 is predicted as 70 s, which is close to that measured by Lee and Robinson [23] for pilocarpine in the rabbit eye (1–2 min). For lipophilic molecules, which Fig. 6. Effect of (A) clearance rate and (B) distribution volume in the anterior chamber on t max. 252 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 Table 4 Effect of precorneal drainage U Drainage rate Ma Tear turnover t max constant, k d (min) (%) rate, S (1.45 min1) (1.2 Al/min) Mt (%) Hydrophilic Hydrophilica Hydrophilic Hydrophilic Hydrophilic 10 10a 10 10 10 0 1 1 10 10 0 1 1 10 10 0 62 76 77 87 0 23 53 51 70 a 0 1 10 1 10 0 1 10 1 10 58 55 55 55 54 108 105 93 105 93 0.35 0.13 0.08 0.08 0.05 4.1 3.2 1.9 2.0 1.2 Baseline case. the corneal and conjunctival epithelium, which reduces their drainage by the tear fluid and makes the corneal epithelium a solute reservoir. As concentration in the precorneal area decreases over time, solute absorbed in this reservoir diffuses back to the precorneal area and is cleared by tear drainage. In contrast, changing precorneal drainage parameters has very little effect on t max, since the precorneal concentration does not affect the time needed to diffuse across the cornea. If solution drainage is totally eliminated due to occlusion of the puncta, which are the two nasal ducts through which fluid leaves the precorneal area, M a increases by 170% and ~30% for hydrophilic and lipophilic molecules, respectively, and M t is, of course, zero (Table 4). This is consistent with the observation of Patton and Robinson [24] that occlusion of the puncta should increase the amount of a mildly hydrophilic molecule, pilocarpine, delivered to the aqueous humor by onefold to threefold. It also agrees with Linden and Alm [25], who suggested that occlusion may increase fluorescein concentration in aqueous humor by a factor of four. Administration of eye drops, which increases precorneal fluid volume, causes increases in lacrimal drainage as well as tear turnover, which further dilute solutes in the tear film [11]. As shown in Fig. 7, an order-of-magnitude decrease in Vi from 50 to 5 Al increases M a by 150% and ~30% for hydrophilic and lipophilic molecules, respectively, because Q is proportional to Vi. However, since the overall amount of instilled drug is concomitantly reduced by a factor of 10, the actual amount of drug delivered to the anterior chamber (calculated as M aC i Vi, where C i Vi is the amount of solute being instilled) decreases by 75% and 90% for hydrophilic and lipophilic molecules, respectively. Therefore, a large instilled volume is preferred (at the maximum possible concentration). 3.6. Additional comparison of predictions with experimental measurements To assess the validity of the developed model, predictions were compared with experimental measurements. Predicting bioavailability is of paramount interest to applications and can be estimated by determining M a. However, opportunities for direct comparison with experimental data are limited, since experimental measurements of M a are difficult or costly, and therefore other parameters, such as C max and t max, are more often reported. Therefore, the predicted values of M a, C max, and t max were compared with reported measurements, despite intrinsic limitations in predicting C max and t max when the clearance rate and the distribution volume in the anterior chamber have not been measured. Since both calculations and measurements of AUC (i.e., the area under the concentration–time curve in the anterior chamber) vary with the duration of simulations and experiments, this parameter was not included in the later analysis. Results are shown in Table 5. As described later, the overall difference between the predicted estimates of M a, C max, and t max and experimental measurements for compounds whose V d and Cla have been measured is always less than a Fig. 7. Effect of instilled volume on M a. W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 253 Table 5 Comparison between model predictions and experimental observationsa Compound logU V d (Al) Cla (Al /min) t max (min) Clonidine Phenylephrine Thiamphenicol Timololc Pilocarpined Flubiprofen Fluorescein Chloramphenicole 2.2 1.7b 0.27b 0.34 0.460.22 0.75 4.3 2.5 Levofloxacine 1.3b 530 423 2200 446 580 620 600 600 200 600 150 600 300 600 150 Erythromycin lactobionatee 1.57 Bufuralolc 2.31 14.9 14.6 28 25.4 13.0 14.4 13.5 13.5 7 13.5 4.5 13.5 27 13.5 30 C max (105) M a (%) Reference O P O P O P 15–30 40–60 45 25–60 20–30 60–120 120 61 40 35 57 80 [9–111] 100 62 75 63 72 65 155 110 158 160 200 5.0 2.8 352 9.3 7.7 2.9 141 1.63 0.36 190 0.1 50 110 1.9 18 40 18 58 170 110 78 50 65 135 10 58 9.9 260 [27] [41] [28] [22] 0.62– 0.8 [0.41– 4.1] [24] 7.0 5.5 [42] [25] [21] [21] [21] [22] a Experimental measurements of V d and Cla are from Ref. [1] except for timolol [43]. For compounds with unknown V d and Cla values, either baseline values (600 Al and 13.5 Al/min, respectively) or fitted values were used in order to match experimental measurements of t max and C max. LogU values are from the listed references or from Ref. [26]; O=experimental observation; P=model prediction. b Although the model can only be rigorously applied to logUN0 or logUb2, predictions are given here for intermediate U values, assuming that all molecules were carried through paracellular routes. c Instilled three times at 2-min intervals. d Predictions for pilocarpine are given as a range (see text); the lower and upper limits are obtained by considering only the hydrophilic pathways across cell layers, and by adding the contributions of the hydrophilic and lipophilic pathways, respectively. e Instilled three times at 15-min intervals. factor of two, which represents good agreement for a mechanistic model that does not rely heavily on empirical fitting. The errors in M a were less than 50% in two of three cases; those in C max were less than 50% in four of five cases examined. Similarly, the errors in t max were less than 25% in three of six cases, and less than 50% in five cases. Since V d and Cla have small effects on M a, this model should give reasonable estimates of M a even for drugs with unknown V d and Cla values. Comparisons between predicted and measured M a values were possible for only three compounds: clonidine, pilocarpine and flubiprofen. There is good agreement in the case of flubiprofen, with a relative difference between the two values of M a equal to ~ 20% (t max and C max values are also in good agreement). The logU of pilocarpine has been reported to lie between 0.22 and to 0.46 [26]; therefore, the lower and upper limits of M a were calculated, based upon the diffusion across hydrophilic pathways only, or the contributions of both lipophilic and hydrophilic pathways, respectively. The value of M a reported by Patton and Robinson [24] falls within the predicted range, and is close to the lower limit, thereby suggesting that pilocarpine may be weakly hydrophilic. The measured M a of clonidine is 1.6% [27] (i.e., more than four times higher than the prediction). This large discrepancy may be due to the fact that clonidine is highly ionized [26], which is not accounted for in this model. The predicted value of t max for clonidine is, however, much closer to the experimental one. The values of V d and Cla were available for only 6 of 11 compounds examined, including the three solutes examined previously. Among the three other solutes phenylephrine, thiamphenicol, and timolol, the agreements between predicted and measured t max and C max values are good for the first two. Since the distribution coefficient of thiamphenicol is 0.54, the model does not strictly apply in that case, and predictions were based upon the assumption that all thiamphenicol molecules follow paracellular pathways within cellular layers. Aldana et al. [28] observed that the anterior chamber concentration of 254 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 thiamphenicol remained elevated several hours after t max (which they measured as 45 min), most likely because of the non-negligible fraction of molecules following transcellular pathways. The logU of timolol has been reported as ranging from 0.39 to 0.34 [22,26]. Assuming that timolol is lipophilic (i.e., assuming the upper limit value for U), the predicted t max (80 min) is close to that reported by Ellis et al. [29], but significantly greater than that measured by Huang et al. [22]. For the other five solutes in Table 5 for which V d and Cla values have not been reported, either the baseline values (600 Al and 13.5 Al/min, respectively) or fitted values of V d and Cla were used in order to match experimental measurements of t max and C max. Since V d and Cla are strong functions of solute lipophilicity, and since these two parameters significantly affect C max and t max values, the predictions should be viewed with caution. Using baseline values, the agreement between predictions and experimental data was generally poor. With fitted V d and Cla values, good agreement was obtained for chloramphenicol and levofloxacin. For fluorescein, however, t max is significantly underestimated and C max is greatly overestimated independently of V d and Cla values. A previous study suggested that the solute may adhere to the sclera [30]; since the composition of stroma and sclera is similar, it is possible that fluorescein binds to the corneal stroma as well, which would explain the discrepancy between calculated and measured values. For erythromycin lactobionate, the predicted t max is close to experimental data, but there is a factor of 10 difference in calculated and measured C max values; it is possible that erythromycin lactobionate is degraded within the cornea, as observed for levobunolol by Tang-Liu et al. [31]. Lastly, the discrepancy between calculated and measured t max values in the case of bufuralol (the latter is as low as 10 min) may stem from the fact that for such a lipophilic molecule (logUN 2.3), the amount of solute delivered to the anterior chamber through the conjunctiva/sclera may not be negligible. 4. Discussion The theoretical model developed in this study should be considered in the context of previous work. Drug delivery through ocular membranes has been modeled extensively [1–3]. A semiempirical approach involving compartment models has often been used due to the complicated morphology of the eye, the lack of specific ultrastructural data related to membrane composition, and the difficulty in accounting for certain transport phenomena such as binding, blinking, and tear drainage [4,5,23,32,33]. In those approaches, clearance and transport processes for solutes are described by first-order rate equations, and several rate constants are typically obtained by fitting the model to experimental data. Although such compartment models yield adequate predictions, they are generally limited to the specific type of molecules studied by the investigators; and because they do not explicitly account for the ultrastructure and the hydrodynamics of the eye, they must be validated for every case being considered. 4.1. Model strengths and limitations In this study, a mechanism-based transient diffusion model was developed in order to examine the effects of ultrastructure and hydrodynamics on topical drug delivery to the anterior chamber through the corneal route, and to provide a broadly predictive tool for estimating drug bioavailability. In contrast to compartment models, transport rates across the ocular tissues were calculated based upon the physicochemical properties of the solutes as well as the ultrastructural parameters of the cornea, sclera, and conjunctiva, without relying heavily on fitted empirical parameters. Its modeling ability should extend to any drugs within the broad range of physicochemical properties considered and requires knowledge of molecular radius, octanol–water distribution coefficient, as well as anterior chamber distribution volume and clearance. This model may also help predict the effect of transiently modifying the integrity of the corneal epithelium in order to enhance corneal drug permeability. For example, Grass et al. [34] have proposed using calcium chelators to dissolve extracellular matrix material to widen intercellular spaces and other compounds to perturb lipid bilayers. Chung et al. [35] have proposed the use of peptides to open hydrophilic pathways across cornea. Such changes in the ultrastructure of the cornea could be directly reflected in the parameters of the model. W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 Despite this model’s general ability to capture the important physical processes that govern topical drug delivery to the eye, it has limitations. Although it was assumed that solutes enter the anterior chamber through the corneal route exclusively, drugs can also be absorbed into the eye through conjunctiva and sclera. However, the assumption appears to be generally valid because studies indicated that the noncorneal route bypasses the anterior chamber and results primarily in delivery to the iris, ciliary body, lens, and possibly vitreous humor by mechanisms that may involve uptake and redistribution via the bloodstream [10,14,36]. To avoid the complexity of a full two-dimensional representation of ocular membranes, it was assumed that solutes permeate through either the paracellular or transcellular route, but not both. This assumption is valid for strongly hydrophilic molecules (Ub0.01) and strongly lipophilic molecules (UN1) (i.e., cases when one of the two routes carries at least 10 times more solute than the other one). Especially at intermediate levels of lipophilicity (0.01bUb1), model predictions are therefore underestimates. In this study, the octanol-to-water distribution coefficient was used, instead of the partition coefficient employed in many other studies, to characterize solute lipophilicity, which is viewed as a strength because the distribution coefficient accounts for the ratio between the organic and aqueous phase concentrations of both ionized and nonionized molecules. However, this model did not account for effects of ionization state on diffusion rates, which may be especially important for diffusion through the corneal epithelium, which is negatively charged at normal pH [37]. This model did not consider drug metabolism or binding within the cornea in the absence of specific data for each solute of interest. The corneal epithelium possesses drug-metabolizing enzymes [2], and possible binding of solute to the stroma was also reported [30]. Drug metabolism should lower M a and C max, whereas binding is expected to elongate t max and decrease C max. The effects of the mucin layer on solute penetration, not considered in this study, are likely to be significant for charged molecules as well as macromolecules. As charged molecules are sequestered by the mucin glycocalyx membrane, which forms the 255 exterior surface of the tear film, t max is expected to increase. Other factors likely to affect transport rates that are not considered in this analysis include molecular shape, viscosity of the instilled solution, as well as the effects of reflex blinks. 4.2. Improving the efficiency of drug delivery Topical drug delivery remains generally inefficient, as tear drainage and absorption into the conjunctiva are the dominant clearance mechanisms, in good agreement with experimental data. This model predicted that only a small fraction of solute present in the instilled drop reaches the anterior chamber: less than 5% for lipophilic molecules and less than 0.5% for hydrophilic molecules. Lacrimal drainage continuously depletes solute in the tear film, and the solute that remains is preferentially absorbed by the conjunctiva. Indeed, the latter membrane has a surface area that is 17 times greater than that of cornea [38], and the permeability of the conjunctiva is also manyfold greater than that of the thicker, three-layer cornea. Strategies to optimize ocular drug delivery usually involve one of the following approaches: increasing drug residence time in the tear film by using suspensions, gels, or inserts for instance; reducing lacrimal secretion and tear fluid discharge by eye closure or punctum occlusion; or enhancing corneal permeability. These strategies are consistent with the model predictions. Increasing the instilled volume may not necessarily improve corneal absorption because of spilling and more frequent blinking; moreover, the resulting higher systemic drug load may be accompanied by undesirable side effects. Of note, this study suggests that since simultaneous enhancement of the permeability of cornea and conjunctiva does not necessarily increase drug bioavailability, strategies should focus on reducing the conjunctival-to-corneal permeability ratio (c), such as designing drugs, and their formulations, to have lipidto-water distribution coefficients between 10 and 20. As U becomes very large, conjunctival absorption is increasingly favored to the detriment of corneal absorption, since increases in U reduce the diffusional resistance of epithelia but not that of stroma, thereby raising c. Hence, maximum corneal permeability does 256 W. Zhang et al. / Journal of Controlled Release 99 (2004) 241–258 not necessarily mean enhanced absorption by the cornea relative to conjunctiva, and there may not always be a proportional correlation between corneal permeability and bioavailability. Experimental observations by Huang et al. [22] support the findings: although the lipophilicity and corneal permeability of acebutolol, timolol, and bufuralol increase in that order, the mildly lipophilic timolol has the greatest C max of all three compounds. Experimental determination of the quantitative effects of tear secretion, lacrimal discharge, blinking frequency, and punctum occlusion on topical drug administration remains uncertain. This model indicated that a factor of 10 change in the rate of lacrimal secretion and solution drainage only yields a ~40% variation in the amount of solute delivered to the anterior chamber. That is because an increase or decrease in solute removal by tear drainage is primarily compensated for by changes in solute absorption into the conjunctiva, and vice versa. Punctum occlusion is predicted to increase the amount of solute that reaches the anterior chamber by a factor of two for hydrophilic molecules, but only by ~30% for lipophilic molecules. 5. Conclusion The developed model of transient transport across the cornea explicitly considers the ultrastructure and hydrodynamics of the eye as well as the physicochemical properties of diffusing solutes. Predictions in this study are in good agreement with experimental data on topical drug delivery. The model suggests promising strategies to increase the bioavailability of drugs applied topically to the eye, involving (1) increasing corneal permeability without a corresponding increase in conjunctival permeability, and (2) increasing drug–cornea contact time by reducing drug loss via tear fluid drainage and/or providing drug for an extended period of time (e.g., using controlledrelease approaches). The model’s greatest strength, however, is its expected ability to predict bioavailability and drug delivery kinetics for new drugs and abnormal ocular anatomical and physiological states due to the model’s mechanistic basis, use of parameters with true physical meaning, and avoidance of fitted empirical parameters. Nomenclature Ai Surface area of ocular membrane i Ci Solute concentration in compartment i C0 Initial solute concentration in tear fluid C max Maximum solute concentration in aqueous humor normalized by C 0 Cla Solute clearance rate in the anterior chamber Di Diffusivity in membrane i Fi Diffusional flux across membrane i f Fraction of surface area occupied by diffusional pathway kd Lacrimal discharge rate constant L Length of diffusional pathway across membrane Mi Fraction of solute clearance or absorption by compartment i P Permeability Q Lacrimal discharge rate rs Solute molecular radius S Lacrimal secretion rate V Volume Vd Distribution volume in the anterior chamber Vi Instilled drug solution volume VL Volume of tear film under normal physiological conditions Vt Tear volume or volume in precorneal area t Time t 90 Time when 90% of solute absorbed into the cornea has reached the anterior chamber t max Time when solute concentration in anterior chamber reaches its maximum Greek U W c k symbols Octanol-to-water distribution coefficient Tissue-to-water distribution coefficient Conjunctival-to-corneal permeability ratio Membrane-to-cornea diffusional resistance ratio Subscripts a Aqueous humor or anterior chamber c Cornea en Corneal endothelium en–aq Endothelium–aqueous humor interface ep Corneal epithelium bj Bulbar conjunctiva pj Palpebral conjunctiva st Corneal stroma t Tear film or tear flow W. 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