Modeling Ophthalmic Drug Delivery by Soaked Contact Lenses

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Ind. Eng. Chem. Res. 2006, 45, 3718-3734

Modeling Ophthalmic Drug Delivery by Soaked Contact Lenses Chi-Chung Li and Anuj Chauhan* UniVersity of Florida, Chemical Engineering Department, Room 237 CHE, P.O. Box 116005, GainesVille, Florida 32611-6005

Approximately 90% of all ophthalmic drug formulations are now applied as eye drops. While eye drops are convenient and well-accepted by patients, ∼95% of the drug contained in the drops is lost due to absorption through the conjunctiva or through the tear drainage. Ophthalmic drug delivery via contact lenses is more effective because it increases the residence time of the drug in the eye and leads to a larger fractional intake of drug by the cornea. In this paper, we model the drug release from the contact lens into the pre- and postlens tear films and the subsequent uptake by the cornea. The motion of the contact lens, which is driven by the eyelid motion during a blink, enhances the mass transfer in the postlens tear film (POLTF). We use regular perturbation methods to obtain the Taylor dispersion coefficient for mass transfer in the POLTF. The diffusion of drug in the gel is assumed to obey Fick’s law, and the diffusion in the gel and the mass transfer in the POLTF are combined to yield an integro-differential equation that is solved numerically by finite difference. Two extreme cases are considered in this paper. The first case corresponds to a rapid breakup of the prelens tear film (PLTF) that prevents drug loss from the anterior lens surface into the PLTF. The second case corresponds to a situation in which the prelens tear film exists at all times and, furthermore, the mixing and the tear drainage in the blink ensure that the concentration in this film is zero at all times. These two cases correspond to the minimum and the maximum loss to the prelens tear film and, thus, represent the highest and the lowest estimations for the fraction of the entrapped drug that diffuses into the cornea. Results show that the dispersion coefficient of the drug in the postlens tear film is unaffected by the release of the drug from the gel. Furthermore, simulation results show that drug delivery from a contact lens is more efficient than drug delivery by drops. The fraction of drug that enters the cornea varies from ∼70 to 95% for the first case (no flux to the PLTF) and from 20 to 35% for the second case (zero concentration in the PLTF). The model predicts that delivery of pilocarpine by soaked contact lenses is ∼35 times more efficient than delivery by drops, and this result matches clinical observations. Introduction Topical delivery via eye drops, which accounts for ∼90% of all ophthalmic formulations, is extremely inefficient and, in certain instances, leads to serious side effects.1 Only ∼5% of the drug applied as drops penetrates the corneal epithelium and reaches the ocular tissue, while the rest is lost due to tear drainage.2 Upon instillation, the drug mixes with the fluid present in the tear film and has a short residence time of ∼2 min in the film. During this time, ∼5% of the drug gets absorbed, and the remaining flows through the upper and the lower canaliculi into the lacrymal sac.2 The drug-containing tear fluid is carried from the lacrymal sac into the nasolacrimal duct where the drug gets absorbed and reaches the bloodstream. The transnasal absorption leads to drug wastage, and more importantly, the presence of certain drugs in the bloodstream leads to undesirable side effects. For example, beta-blockers such as timolol that treat wide-angle glaucoma have a deleterious effect on the heart.3 Furthermore, application of ophthalmic drugs through drops results in a rapid variation in drug delivery rates to the cornea that limits the efficacy of therapeutic systems.4 Thus, there is a need for new ophthalmic drug delivery systems that increase the residence time of the drug in the eye, thereby reducing wastage and eliminating side effects. The two most important requirements for effective ophthalmic drug delivery systems are comfort and a long residence time in the tear film. Most of the previous efforts at developing new ophthalmic drug delivery systems focused on using polymeric * Corresponding author. Tel.: (352) 392-2592. Fax: (352) 392-9513. E-mail: [email protected].

gels, colloidal particles, and collagen shields.1 Addition of polymeric gels such as hyaluronic acid (HA),5 poly(acrylic acid) (PAA),6-8 and cellulose9 to ophthalmic drug formulations increases the residence time of the drug in contact with the ocular tissue. Liposomes are spherical lipid bilayers with a hydrophobic annulus and a hydrophilic core.10 These colloidal particles can encapsulate a hydrophilic drug in the core and entrap a hydrophobic drug in the annulus. Liposomes are positively or negatively charged depending on the charge on the lipid molecules. Since the mucin covering the corneal surface is negatively charged liposomes with positive charge electrostatically bind to the corneal surface. This increases the residence time of these liposomes in the tear film and, hence, increases the bioavailability of the drug encapsulated in liposomes.11,12 Nanocapsules containing a central core of drug surrounded by a polymeric membrane and nanoparticles that exhibit bioadhesive properties have also been used to serve as drug-delivery vehicles.13-16 These systems are very promising, comfortable, and nontoxic, but they have been known to damage the corneal epithelium in some instances.17 Furthermore, these systems can increase the drug residence time to a few hours but cannot provide extended drug delivery. In a number of studies, soaked collagen shields have been successfully utilized as ophthalmic drug-delivery vehicles to deliver a single drug.18-20 But these are not recommended for multidrug applications, because in certain instances the drug mixtures form aggregates in the shield, which can potentially cause corneal toxicity. Also, these systems can only be used for certain drugs. Both biodegradable and nonerodible implants have also been tried for extended oph-

10.1021/ie0507934 CCC: $33.50 © 2006 American Chemical Society Published on Web 04/12/2006

Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3719

Figure 1. (a) The idealized geometry and (b) the real geometry utilized in the model for the PLTF-lens-POLTF system.

thalmic drug release.21 These are therapeutically very effective but are difficult to insert into the eye, leading to a lack of acceptance by patients. A hydrogel contact lens is an ideal vehicle for delivering drugs to the eye for a number of reasons. First, present-day soft contact lenses can be worn comfortably and safely for an extended period of time, varying from ∼1 day to 15 days. Second, once the contact lens is placed on the eye, the drug from the lens will diffuse into a thin fluid layer trapped between the lens and the cornea, namely, the postlens tear film (POLTF). There is limited mixing between the fluid in the POLTF and the outside tear fluid.22,23 Thus, the drug released from the lens will have a long residence time in the eye. Third, the contact lenses are made up of cross-linked gels, and thus, it is easy to entrap the drug in the gel matrix either by soaking the contact lens in a drug solution or by adding the drug during the polymerization process.24 An alternate method is to entrap the drug in nanoparticles and disperse the drug-laden nanoparticles in the gel during the polymerization.25,26 There have been a number of attempts in the past to use contact lenses for ophthalmic drug delivery; most of these focused on soaking the lens in drug solution followed by insertion into the eye.27-37 A number of researchers have studied the uptake of timolol, which is a common glaucoma medication, in pure hydroxyl ethyl methacrylate (HEMA) and copolymers of HEMA and methacrylic acid (MAA) or methyl methacrylate hydrogels.38-40 Also, recently Karlgard et al. measured the uptake and release of a number of ophthalmic drugs by both HEMA-based and silicone contact lenses in in vitro studies.41 While a number of in vitro studies have been done to study drug delivery by contact lenses, there are relatively few in vivo studies on ophthalmic drug delivery by contact lenses. There are a few clinical studies in which it was shown that the soaked contact lenses can achieve desired therapeutic results,29-34 but only a few of these studies quantified the actual amount of drug that was taken up by the cornea.29 The lack of quantitative studies is mainly due to the difficulty of conducting these experiments in humans. The dynamics of tear flow and the contact lens motion are well-characterized both mathematically and experimentally.22,23,42-44 Thus, drug delivery from a contact lens can be modeled by coupling the models developed in the studies listed above with a mass transfer model for the contact lens. The purpose of this study is to develop a mathematical model for the process of drug delivery by soaked contact lenses and use the model to determine the effectiveness of this method in

comparison to drug delivery by drops. To solve the drug-release problem, one has to model the mass transfer in the postlens tear film (POLTF). The dispersion in the POLTF is driven by the lid motion during the blink, and the mass transfer model developed in this paper is valid for any arbitrary lid velocity, which can be obtained from experiments. It thus represents an improvement over an earlier model for dispersion in the POLTF that was developed by Creech et al.,22 which was valid only for a specific form of blink velocity. Thus, the two main contributions of this paper are as follows. First, it represents the first attempt to model ophthalmic drug delivery by soaked contact lenses and predict the efficacy of this modality in comparison to drops. Second, this paper presents a rigorous approach based on multiple time scales to develop a model for mass transfer in the POLTF and develops an expression for the dispersion coefficient that is more accurate than those derived earlier. Also, our approach clearly points out the parameter regimes in which the results for the dispersion coefficient are valid. While this model cannot replace clinical studies, it can serve as a very useful tool to gauge the effectiveness of soaked contact lenses and also to design these to achieve the desired therapeutic dosages in clinical studies. We also hope that the encouraging results predicted by this model, which also match the limited clinical data available in the literature, will lead to more clinical studies in this area. Model We develop a model to predict the drug release from a presoaked contact lens into the postlens tear film and its subsequent uptake by the cornea. Figure 1 shows the real and the model geometry of the lens and the tear film. The postlens tear film (POLTF) is pictured as a flat, two-dimensional film bounded by an undeformable cornea and an undeformable but moving contact lens. The lens is treated as a two-dimensional body of length L and thickness hg and is assumed to extend infinitely in the third direction. The postlens tear film has a thickness hf, which may depend on x as the front surface of the eye has a complicated geometry, but for simplicity, it is taken to be independent of x in this paper. The curvatures of the cornea and the lens have been neglected because the thicknesses of the tear film (∼10 µm) and of the contact lens (∼100 µm) are much smaller than the corneal radius of curvature of ∼1.2 cm. The assumption of a two-dimensional geometry has been made to simplify the problem. The effect of gravity is negligible in the POLTF. Thus, for our purposes, the prelens tear film-

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Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006

contact lens-POLTF-cornea system is a flat, horizontally oriented channel. These assumptions have been utilized in the past to model mass transfer in the POLTF.22 The drug concentrations in the gel matrix of the contact lens and the tear film are Cg and Cf, respectively. To determine the drug flux to the cornea, we need to simultaneously solve the convectivediffusion equation in the gel matrix and in the postlens tear film. The governing equations for the mass transfer in the tear film and lens are

(

)

∂Cf ∂Cf ∂Cf ∂2Cf ∂2Cf +u +V ) Df + ∂t ∂x1 ∂y1 ∂x12 ∂y12

(1)

∂2Cg ∂2Cg ∂Cg ) Dg + ∂t ∂x22 ∂y22

(2)

(

)

where x1 and y1 are the lateral and transverse coordinates in the POLTF, respectively, and x2 and y2 are the lateral and transverse coordinates in the lens, respectively. In the above equations, Df and Dg are the drug diffusivities in the tear film and the contact lens hydrogel matrix, respectively, and u and V are velocity components in the x1 and y1 directions, respectively, in the postlens tear film. The approximate values of L, hf, and hg are 1 cm, 10 µm, and 100 µm, respectively. The diffusive time scale in the gel is hg2/Dg, which is typically a few hours. The dispersive time scale in the postlens tear film, i.e., the time required for a solute to leave the film by a combination of diffusion and convection is ∼30 min. Thus, a soaked contact lens is expected to supply drug to the tear film for a period of a couple of hours. In such a short period, the diffusion in the axial (x2) direction can be neglected in the lens, and thus, the diffusion equation in the lens simplifies to

( )

∂2Cg ∂Cg ) Dg ∂t ∂y22

(3)

Equation 2 assumes that the diffusion of the drug through the contact lens gel matrix can be modeled as a purely diffusive process. It is well-known that diffusion of solutes through hydrogels exhibits complex mechanisms and is governed by an interplay of swelling of the gel, adsorption and desorption of the solute molecules on the gel, surface diffusion along the polymer that comprises the gel, and bulk diffusion through the free water. Incorporation of each of these effects into our model is feasible; however, it significantly increases the complexity of the problem. Since our model is the first attempt to model the process of ophthalmic drug delivery by contact lenses, we keep the gel model simple and treat diffusion in the contact lens as purely Fickian. The fluid flow in the postlens tear film is driven by blinking. During a blink, the motion of the upper eyelid drives contact lens and POLTF tear fluid motion in both lateral (x1) and transverse (y1) directions. In the interblink period, both the lens and the POLTF tear fluid are stationary. The velocity profiles in the POLTF are a combination of squeeze flow (transverse motion) and shear flow (lateral motion) and are given by22

V′x1 (ζ - ζ2)f(t) + Vζf(t) hf

u)6

V ) V′(2ζ3 - 3ζ2)f(t)

value h0 and ζ ) y1/h0 (Figure 1). The POLTF thickness hf is a function of time due to the motion of the contact lens during blinking and is equal to h0 - V′ ∫f(t) dt. In the above expressions, V and V′ are the amplitude of the lens velocities in the lateral and the transverse directions, respectively. We now define ω as the blink-interblink frequency, i.e., ω ) 2π/Tb, where Tb is the time for a blink-interblink cycle. The function f(t) in the above equations characterizes the velocity of the contact lens driven by the blink; it is equal to zero during the interblink period, and it can be approximated as cos(ωbt) during the blink, where ωb ) Nω is defined as 2π/(blink time) and N is therefore the ratio of the time between two blinks and the actual blink time. The velocity amplitudes V and V′ can also be related to the amplitude of lens motion by V ) ∆Nω and V′ ) ∆′Nω, where ∆ and ∆′ are the lens displacements in the lateral (x1) and the transverse (y1) directions, respectively. As discussed above, the model contact lens is not circular but translationally invariant in the direction out of the paper in Figure 1, and thus, the lens and the tear film are effectively semi-infinite. The flow is assumed to be periodic-steady because hf ≈ 10 µm and the tear kinematic viscosity ν ) 1.5 × 10-2 cm2/s,45 giving rise to a characteristic time for fully developed flow tchar ) hf2/ν ≈ 7 × 10-5 s, much shorter than the blink period T ) 2π/ωb ≈ 0.1 s. The boundary conditions for Cg are

-Dg

|

∂Cg ∂y2

y2)hg

) Df

|

∂Cf ∂y1

(6a) y1)hf

Cg|y2)hg ) Cg,eq(Cf|y1 ) hf) ) KCf|y1)hf

|

∂Cg ∂y2

)0

(6b) (6c)

y2)0

where K is the partition coefficient, i.e., the ratio of the concentration of the drug in the gel and in the POLTF at equilibrium. The boundary conditions in eqs 6a and 6b ensure continuity of flux and equilibrium at the lens-postlens tear film interface, respectively. The boundary condition in eq 6c assumes that there is no loss of drug from the lens to the prelens tear film (PLTF) that lies between the lens and the air. This assumption may be reasonable because the PLTF breaks very rapidly, and the breakup of the PLTF prevents any further drug loss from the front surface. Additionally, the PLTF breakup causes partial dehydration of the lens in the region close to the front surface, and consequently, the front surface of the contact lens is expected to be glassy, which may further reduce drug flux from the front surface. This is clearly the scenario that will maximize the fraction of the drug trapped in the lens that will eventually be delivered to the cornea. To determine the fraction of trapped drug that will go to the cornea for the other extreme, we investigate the case in which we assume that the drug can diffuse into the PLTF and that rapid mixing and drainage from the PLTF keeps the drug concentration in PLTF ∼zero. Thus the boundary condition in eq 6c gets modified to

Cg(y2)0) ) 0

(6c′)

The boundary conditions for Cf are

(4) (5)

where hf is the time-dependent POLTF thickness with a mean

|

∂Cf ∂x1

)0

(7a)

Cf|x1)L ) 0

(7b)

x1)0


Df

|

∂Cf ∂y1

y1)0

) kcCf|y1)0

(7c)

The first boundary condition (eq 7a) arises due to symmetry, the condition in eq 7b assumes that the drug concentration in the tear meniscus is very small because of the large volume, and the last boundary condition (eq 7c) assumes that the drug flux to the cornea can be quantified as kcCf, where kc is the mass transfer coefficient for drug transport into the cornea. The mass transfer into the cornea is assumed to be irreversible, which is reasonable because the concentration of the drug in the corneal tissue is small and the drug is perhaps bound to the cellular or the extracellular components. The initial conditions for the drug concentrations are

Cf|t)0 ) 0

(8a)

Cg|t)0 ) Ci

(8b)

The above set of equations can be solved by finite-difference or finite-element methods to predict the drug flux to the cornea. However, because of the disparate length and time scales involved in the problem, it is possible to reduce the problem to a single integro-differential equation that can be solved numerically. By using a perturbation expansion in the aspect ratio and by using a multiple time scale analysis, the transport problem in the film can be simplified to a dispersion equation of the form

∂Cf0 j - kcCf0 ∂Cf0 ∂ ) D* + ∂t ∂x1 ∂x1 h0

(9)

where Cf0 is the leading order term in the regular expansion for Cf in and is independent of y1 (see Appendix 1), D* is the effective dispersion coefficient, and j is the flux of the drug entering the postlens tear film from the contact lens. The details of the derivation are shown in Appendix 1. Now, we solve separately the transport problem in the contact lens hydrogel matrix. The transport problem in the gel is

( )

∂2Cg ∂Cg ) Dg ∂t ∂ y22

(10)

with the following boundary conditions,

Cg(y2)hg) ) KCf(x1)

(11a)

∂Cg -D (y )h ) ) j ∂y2 2 g

(11b)

∂Cg (y )0) ) 0 ∂y2 2

(11c)

The boundary conditions in eq 11a assume equilibrium between the concentration in the contact lens and that in the tear fluid in the POLTF and in eq 11b impose flux continuity, thus coupling the mass transfer problems in the POLTF and in the contact lens. The boundary condition in eq 11c assumes that there is no loss of drug from the lens to the prelens tear film (PLTF) that lies between the lens and the air. As described previously, we also consider the other extreme case in which the boundary condition in eq 11c is replaced by

Cg(y2)0) ) 0

(11c′)

We solve the drug-release problem for both of the extreme cases. The above set of equations can be solved by using convolution theorem to yield an analytical expression for Cg in terms of Cf(x1). The derivation is shown in Appendix 2. As shown in Appendix 2, for case I, which corresponds to zero flux to the PLTF, the expression for the drug flux to the POLTF, j, becomes

[

Dg

j)

hg

∞

∑2Ci e-(2n+1) π D t/4h n)0 2 2

∞

2K

(

g

2

g

+

(2n + 1)2π2Dg

∑ ∫0 Cf(t*)

n)0

t

4hg2

×

)]

2

2 2

e[(-(2n+1) π Dg)/(4hg )](t-t*) dt* - Cf(t)

[

Dg

)

hg

∞

2Ci e-(2n+1) π D t/4h ∑ n)0 2 2

∞

2K

(

g

2

g

+

(2n + 1)2π2Dg

∑ ∫0 Cf0(t*)

n)0

t

4hg2

2

2 2

×

e[(-(2n+1) π Dg)/(4hg )](t-t*) dt* - Cf0(t)

)]

+ O() (12)

and for case II, which corresponds to zero concentration in the PLTF, the expression for j becomes

[

Dg

j)

hg

∞

4Ci e-(2n+1) π D t/h ∑ n)0 2 2

2DgK hg

[

Dg

)

(

hg

∫

t

∞

(

g

2

g

n2π2Dg

∑ ∫0 Cf(t*) n)1 t

Cf(t)

- D gK 2 2

∞

2 2

g

)]

2

e[(-n π Dg)/(hg )](t-t*) dt* - Cf(t)

hg2

∑4Ci e-(2n+1) π D t/h n)0

+

hg

2

g

Cf0(t)

- D gK

+

hg

2DgK hg

2 2

n π Dg

C (t*) 0 f0

2

2 2

2

)]

e[(-n π Dg)/(hg )](t-t*) dt* - Cf0(t)

hg

∞

∑× n)1 + O() (13)

where the O() errors in the flux are due to approximation of the film concentration Cf by the leading order concentration Cf0. These expression can be substituted into eq 9 to get a single integro-differential equation which can be solved numerically to determine the drug concentration in the tear film as a function of x and t. We now use the following scales for dedimensionalizing the integro-differential equation:

Cg ≈ Ci, Cf ≈

Ci hg2 DgCi , x1 ≈ L, D* ≈ Df, t ≈ , j ≈ K Dg hg

The dimensionless forms of the equations are

∂C ˜ f0 ∂C ˜ f0 ∂ ) P1 D ˜* + P2 ˜j0 - P3C ˜ f0 ∂τ ∂η ∂η

(14)

3722


where

Dfhg2

hg kchg2 P1 ) , P2 ) K , P3 ) hf hfDg DgL2 and ˜j0 is the dimensionless O(0) flux that can be determined from eqs 12 and 13. The ˜j0 for case I and case II are given below by eqs 15 and 16, respectively.

[∑ ∞

˜j0 ) 2

e

-(2n+1)2π2τ/4

∞

+2

n)0

(

(2n + 1)2π2

∑ ∫0 C˜ f0(τ*)

n)0

τ

2 2

4

e[(-(2n+1) π )/4](τ-τ*) dτ* - Cf0(τ)

×

)]

(15)

∞

˜j0 ) [

∑4 e-(2n+1) π τ - C˜ f0(τ) + n)0 2 2

∞

2

(∫0 C ˜ f0(τ*)n2π2 e-n π (τ-τ*) dτ* - C ˜ f0(τ))] ∑ n)1 2

2 2

(16)

Results and Discussion Scales for Various Parameters. The analysis presented in Appendix I required the following conditions to be satisfied:

≡

h0 ,1 L

∆ ) O() L

(17)

∆′ ) O() h0 The typical values of hf and L are 10 microns and 1 cm, respectively. Thus, is ∼10-3 and is thus ,1. Both ∆/L and ∆′/hf have a value of ∼0.05. Thus, these too are ,1 and can be treated as O(). Depending on the drug of interest, the parameters (Dg/Df)Khf/hg and kch0/Df could be either O() or O(2). When these parameters are O(2), the coupled POLTF-gel problem can be represented by eq 14. Also as shown in the Appendix, under the condition that kch0/Df and DgKh0/Dfhg are O(), the coupled POLTF-gel problem is simply a limiting case of eq 14 and can be obtained by setting P1 ) 0. This occurs because, if (Dg/Df)(hf/hg) is O(), then (Dg/Df)(hf/hg)(L/hf) ) O(1), which implies that L2/Df ≈ (hg2/Dg)(L/hg). Thus, in this limit, the time scale for lateral diffusion in the POLTF (L2/Df) is much larger than the time scale for the drug to diffuse out of the gel (hg2/ Dg), and accordingly, the dispersion of the drug in the POLTF can be neglected. Dispersion Coefficient. The expression for the dispersion coefficient that is obtained in Appendix I is valid for any arbitrary f(t). Creech et al. had analyzed the problem of dispersion in the POLTF and had obtained the dispersion coefficient for the case when f(t) ) cos(Nωt),22 where Nω is equivalent to ω defined by Creech et al. They had argued that, since dispersion in the interblink period is negligible, the overall dispersion coefficient can be obtained by simply calculating the dispersion for the case of repeated blinks without any interblink period in between and then multiplying the resulting dispersion coefficient by the ratio of blink time to the total time for a blink-interblink cycle. Since our approach can be used to

Figure 2. The function f(ts) that characterizes the time dependence of the velocities in the POLTF (thick line) and the Fourier representation of the function (thin line). The two functions match everywhere except at locations where f(ts) is discontinuous. At ts ) 0 and 2π, the Fourier series sums up to 0.5, which is the mean value at ts ) 0 and 2π. The value of N is taken to be 10 for the curves in this figure and everywhere else in this paper.

determine dispersion for an arbitrary f, we can test the validity of the above assumption. It is noted that, for a pure cosine function, the dispersion coefficient obtained in this paper reduces to that obtained by Creech et al. It is also noted that, in the analysis of Creech et al., there was no flux of solute from the lens or to the cornea, while in our problem, the drug diffuses from the lens and also enters the cornea. In fact, Creech et al. claimed that the dispersion coefficient of the solute in the POLTF would change if the solute can diffuse in/out of the lens or the cornea. However, it was not clear as to how the analysis would be altered to include the flux of the drug from the lens and into the cornea. More importantly, the analysis of Creech et al. is only valid for a certain range of parameters, but their analysis does not clearly reflect the region of applicability of the results. Since the analysis presented in the appendix is based on the method of multiple time scales and regular expansions, the region of validity of the results is clear. We now compute the dispersion coefficient for the case when f(ts) is given by cos(Nts) during a blink and is zero during the interblink. The function f(ts) and the Fourier series representation of the function are shown in Figure 2 for N ) 10. The coefficients of the Fourier series expansion can be used in eq A1-58 to determine the dispersion coefficient. The dispersion coefficient for f given in Figure 2 is calculated for a range of Ω values and is plotted in Figure 3 for the lateral velocity profile. Also, the dispersion coefficient predicted by Creech et al., which as explained previously can be obtained by setting f(ts) ) cos(Nts) with no interblinks and then multiplying the dispersion coefficient by the factor 1/N, is the ratio of blink time to total blink-interblink cycle time. The figure also contains results for the dispersion coefficient that were computed by numerically solving the O() equations in the POLTF and then computing the dispersion coefficient by computing the integral required in eq A1-46 numerically. As seen in Figure 3, the numerical computations agree with the analytical results, which is expected. Additionally, the predictions of Creech et al. are only ∼10% larger than the exact solution. The results of Creech et al. overestimate dispersion because they neglect the lateral diffusion during the interblink, and this diffusion equilibrates the concentration in the lateral direction and, thus, reduces the dispersion. Since the expression of Creech et al. is a good approximation to the exact solution, it is used to calculate dispersion


dCf0(τ) dIn + λnIn ) dτ dτ

Figure 3. Comparison of the exact solution for the dispersion coefficient (solid line) with the approximate expression assumed by Creech et al.22 (dashed line). The markers on each curve are the dispersion coefficients calculated by numerically solving the O() problem. In this figure, the results are shown only for the component of dispersion caused by the lateral flow.

(22)

where λn ) n2π2. Concentration Profiles. The concentration profiles (Cf0(x,t)) in the POLTF depend on the following parameters: D*(N, Ω ≡ ωh02/Df, ∆′/h0, ∆/L), P1, P2, and P3. In the results shown below, the values of N, Ω, ∆/L, and ∆′/h0 are kept constant at 10, 6.28, 0.02, and 0.028, respectively, which correspond to normal physiological values, and the values of P1, P2, and P3 are varied within a range of possible values. Figures 4 and 5 show typical concentration vs time plots at different positions for cases I and II, respectively. The insets in each figure show the magnified view of the plots near t ) 0. The concentration starts at zero and then very quickly increases to a value of ∼0.9. Since the concentrations are dedimensionalized by Ci/K, the maximum possible value of Cf0 in the POLTF is 1. During the period in which the concentration is increasing, the drug flux from the gel is larger than the sum of the drug loss from the sides (x ) (L) to the outer tear lake and the drug uptake by the cornea. The maximum value of the concentration is reached

for the remaining part of this paper. It is noted that Figure 3 shows the results for only the lateral flow, but the matching between the numerical and the analytical results and the comparison between the general expressions and the predictions of Creech et al. are similar for the case of squeeze flow. Solution of the Integro-differential Equation. The integrodifferential equations were solved numerically by finite difference. As shown below, the convolution integrals were evaluated by converting those to ordinary differential equations and by further solving those equations numerically. This was done to improve the convergence of the series sum of the convolution integrals. For case I, let ∞

I) ∞

∑0 In )

[

∑∫ n)0

τ

(2n + 1)2π2

C (τ*) 0 f0

4

2 2

e[(-(2n+1) π )/4](τ-τ*) dτ* - Cf0(τ)

]

(18)

and thus,

dIn ) λnCf0(τ) - λn2 dτ

∫0τCf0(τ*) e-λ (τ-τ*) dτ* n

Figure 4. Typical concentration transients in the POLTF at different axial locations for case 1, i.e., no flux to the PLTF. The inset shows a magnified view near t ) 0. The values of P1, P2, and P3 are 0.5, 100, and 100, respectively.

dCf0(τ) (19) dτ

where λn ) (2n + 1)2π2/4. We can then write an equation for solving In as follows:

-dCf0(τ) dIn + λnIn ) dτ dτ

(20)

This equation can be solved simultaneously with eq 14 to determine Cf(x,t). The same procedure can be followed for case II. Let ∞

I)

∞

In ) ∑[∫0 Cf0(τ*)n2π2 e-n π (τ-τ*) dτ* - Cf0(τ)] ∑ n)1 n)1 τ

2 2

(21) As in case I, we can write an equation for solving In for case II as follows:

Figure 5. Typical concentration transients in the POLTF at different axial locations for case 2, i.e., maximum flux to the PLTF. The inset shows a magnified view near t ) 0. The values of P1, P2, and P3 are 0.5, 100, and 100, respectively.

3724


Fc )

∫0∞ ∫0L kcCf dx1 dt )

2 M0

P3 P2 Fs )

2 M0

FP )

Figure 6. Concentration profiles in POLTF for both case I (solid) and case II (dashed) as a function of time. The values of P1, P2, and P3 are 0.5, 100, and 100, respectively.

in a very short period of time because the volume of the POLTF is small as reflected in the large value of P2. At initial times, the drug flux into the cornea and the drug loss from the sides are much less than the drug flux from the lens, causing drug concentration in the postlens tear film to increase. However, as the drug concentration in the POLTF builds up, the flux of the drug from the lens decreases and the drug loss from the sides due to dispersion and the drug flux into cornea increases. Consequently, very quickly, the drug concentrations in the POLTF begin to decrease. The concentrations finally approach zero in a dimensionless time of about 8 and 2 for case I and case II, which correspond to no flux and zero concentration in the PLTF, respectively. The total release time is larger for case I because there is no drug loss to the PLTF. The same data as shown in Figures 4 and 5 is shown as Cf0(x) plots at various times in Figure 6. The plots show the growth of the dispersive boundary layer from x ) L toward the lens center. The boundary layer thickness is about the same for both cases because the dispersion coefficient is independent of the flux from the lens. Also the concentration vs position plots are almost identical for both the cases for short times. This is expected because the flux from the lens to the POLTF will be unaffected by the boundary condition at the lens-PLTF interface till the mass transfer boundary layers from both the POLTF-lens and the PLTF-lens merge. We now investigate the effect of the variation of the three main parameters P1, P2, and P3 on the concentration profiles. The values of P1, P2, and P3 used in the figures are representative values for a wide variety of ophthalmic drugs. The variations in these parameters arise primarily due to variations in Dg, Df, K, and kc, and each of these can vary by more than an order of magnitude. Additionally, we calculate the fraction of the entrapped drug that enters the cornea and the fraction that is lost to the tear lake from the sides of the POLTF and, for case II, also the fraction that is lost to the PLTF. The amount of drug that diffuses into the cornea is simply equal to 2∫∞0 ∫L0 kcCf dx1 dt. The mass of drug that diffuses into the tear lake from the edges of the POLTF is given by 2∫∞0 -DfD*(dCf/dx1)|x1)L hf dt. Finally, the amount of the drug that is lost to the PLTF is given by 2∫∞0 ∫L0 Dg(dCg/dy2)|y2)0 dx2 dt. On dedimensionalizing each of these masses by the initial mass of drug in the gel ()2CiLhg), we get the following equations,

P3 ∞ 1 ∫0∞ ∫01Cf dη dτ ≈ P2 ∫0 ∫0 Cf0 dη dτ

∫0∞ - DfD* dx1f|x)L hf dt ) dC

P1 ∞ ∫0∞D* dηf|η)1 dτ ≈ - P2 ∫0 D*

P1 P2

2 M0

(23)

dC

∫0∞ ∫0L Dg dy2g|y )0 dx1 dt ) ∫0∞ ∫01 dC

2

|

dCf0 dη

η)1

|

dCg dζ

dτ (24)

ζ)0

dη dτ (25)

where Fc is the fraction of the total drug that enters cornea, Fs is the fraction of the total drug wasted from the side to the outer tear lake, and FP is the fraction of the total drug wasted to the PLTF. Effect of P1. Figure 7 plots the C ˜ f0(τ) at different locations at three different times for two different values of P1 for case II, and Figure 8 plots the fractions of the drug that are either delivered to the cornea or lost to the tear lake or to the PLTF

Figure 7. Effect of P1 on concentration profiles for case II. The concentration profiles are shown at three different times and for two different values of P1 (0.05, solid line; 0.5, dashed line). The values of P2 and P3 are 100 and 100, respectively.

Figure 8. Effect of P1 on the fractions of the drug that enter the cornea or are lost from the PLTF or from the POLTF. Results are shown for both case I (solid) and case II (dashed). The values of P2 and P3 are 100 and 100, respectively.


Figure 9. Effect of P2 on concentration profiles in the POLTF for case I. The profiles are shown at different times and for two different values of P2 (20, dashed line; 100, solid line). The values of P1 and P3 are 0.5 and 100, respectively.

Figure 10. Effect of P2 on the fractions of the drug that enter the cornea or are lost from the PLTF or from the POLTF. Results are shown for both case I (solid) and case II (dashed). The values of P1 and P3 are 0.5 and 100, respectively.

for a range of P1 values. The parameter P1 is the ratio of the diffusive time scales in the gel and the diffusive time scales in the POLTF in the lateral direction. Thus, an increase in P1 can be interpreted as an increase in the diffusion coefficient of the solute in tears. Since an increase in Df will lead to an enhancement of dispersion, the thickness of the edge region will increase. This effect is shown in Figure 7. Also, an increase in dispersion will increase the drug loss from the sides, and thus, the fraction of the drug that enters the cornea will be reduced. The trends are the same for both the cases of no flux to PLTF and zero concentration in the PLTF. Effect of P2. Figure 9 plots the C ˜ f0(η) at four different values of dimensionless time τ for two different values of P2, and Figure 10 plots the fractions of the drug that are either delivered to the cornea or lost to the tear lake or to the PLTF for a range of P2 values. The parameter P2 is the ratio of the amount of drug present in the contact lens and the amount present in the tear film at equilibrium. Thus, an increase in P2 can be interpreted as an increase in the partition coefficient of the solute, K. As shown in Figure 10, for case I, changes in P2 have a negligible effect on the fraction of drug delivered to the cornea. The reason for this behavior is that changes in P2 change the concentrations in the POLTF, but the change is very similar at

Figure 11. Effect of P2 on concentration transients in the POLTF at x ) 0 for case I (solid lines) and case II (dashed lines). The inset shows a magnified view near t ) 0. The values of P1 and P3 are 0.5 and 100, respectively, and the values of P2 are noted on the curves.

every axial location. Thus, the mass delivered to the cornea and the mass that gets out from the side are affected in an almost identical way, and accordingly, the fractions remain unchanged. For case 2, an increase in P2 leads to a reduction in the fluxes to the cornea and the fraction that is lost to the tear lake. The reason is that an increase in P2 is equivalent to an increase in K and, thereby, a reduction in the POLTF concentrations. Accordingly, for a given Ci, the duration of drug release into the POLTF increases with an increase in K. Thus, there is more time for the drug to diffuse into the PLTF, and so the flux that goes to the PLTF increases. The results for the temporal profiles (C ˜ f0(τ) at η ) 0) for two different values of P2 for both case I and case II are shown in Figure 11. Figure 11 again shows that, at short times (inset), the concentration profiles in the POLTF are the same for both cases. Also, the curves in this figure show that, as the value of P2 decreases, the time to achieve the maximum concentration in the POLTF increases slightly and the value of the maximum dimensionless concentration in the film decreases. To understand this issue, let us first consider the case when there is no loss of drug to the PLTF and also when the loss of the drug from the POLTF to the tear lake and also to the cornea is negligible. The parameter P2 is essentially the ratio of the drug-carrying capacity of the lens and the drugcarrying capacity of the POLTF, and as stated above, an increase in P2 can be interpreted as an increase in K. Thus, if loss to the tear lake, the cornea, and the PLTF is negligible, the concentration in the POLTF will reach equilibrium, i.e., C ˜ f0 will become 1, and the time to reach equilibrium will be shorter for a larger K, or a larger P2. In the results shown in Figure 11, the loss to the tear lake and to the cornea is not negligible, and this loss prevents the POLTF concentration from reaching equilibrium; thus, the value for the dimensionless concentration stays 1, which is required to achieve sufficient loading in the contact lens, shows that the drug must be adsorbing to the gel. The adsorption of the drug modifies the diffusion equation for the gel, and the modified Ficks law can be incorporated into the framework proposed in this paper. In addition to the simplification of the gel diffusion, a number of assumptions have been made for the mass transfer in the POLTF. First, the model assumes a simplified flat 2D geometry. This assumption has been used previously to model mass transfer in the POLTF with satisfactory results and, thus, is perhaps reasonable because of the thin POLTF. Additionally, the motion of the contact lens has been simplified to correspond to periodic squeeze flow and Couette flow. These assumptions have also been used previously with satisfactory results. While each of the assumptions listed above are expected to impact the drug-release profiles, none of them introduce new physics or mechanism, and furthermore, each of these can be incorporated into the framework developed in this paper. In future, we plan on incorporating some of the modifications discussed above. The results of this study show that a soaked contact lens can significantly reduce the drug wastage and the side effects associated with the entry of the drug into the systemic circulation and, thus, is a big improvement over ophthalmic drug delivery by eye drops. We note that, while soaked contact lenses are more efficient than drops, they still suffer from a number of drawbacks. First, when a lens is soaked in drug solution, the maximum drug concentration obtained in the lens matrix is limited to the equilibrium concentration. Thus, a soaked lens can supply only a limited amount of drug. This technique is especially inefficient in delivering hydrophobic drugs by HEMA-based contact lenses. Second, even for drugs that can absorb in the lens matrix, the drug-release time scale is only a few hours. Thus, a soaked contact lens cannot deliver drugs for extended period of time. However, despite these deficiencies, it is clear that a soaked contact lens is a much more efficient vehicle for ophthalmic drug delivery than the conventional eye drops. Acknowledgment This research is partly funded by support from the National Science Foundation (CTS 0426327). Appendix 1 The geometry of the POLTF is shown in Figure 14. For convenience, the coordinates x1 and y1 have been replaced by x and y in this Appendix. The governing equation for mass transfer in the POLTF is

Figure 14. Geometry of the postlens tear film.

3728


(

)

∂Cf ∂Cf ∂2Cf ∂2Cf ∂Cf +V ) Df + +u ∂t ∂x1 ∂y1 ∂x12 ∂y12

As noted in the main text, the fluid flow in the postlens tear film is driven by blinking and the velocity profiles are given by eqs 3 and 4,

N∆′ω x(ζ - ζ2)f(t) + N∆ωζf(t) u)6 t h0 - ∆′ω 0 f(t) dt

∫

V ) N∆′ω(2ζ3 - 3ζ2)f(t)

u˜ ) u˜ 1 + 2u˜ 2 + .... V˜ ) V˜ 1 + 2V˜ 2 + ....

(A1-1) where

u˜ 1 )

(A1-2)

(

)

N∆ h N∆ h′ 2 η(ζ - ζ) f(ts) ζ+6 L h0

u˜ 2 ) 6

N∆ h ′2 2 η(ζ - ζ)f(ts) f(ts) dts h02

∫

(A1-3)

There are four main time scales for mass transfer in the POLTF: the time scale for axial diffusion (L2/Df), the time scale for lateral diffusion (h02/Df), the time scale for the oscillatory blink (1/ω), and the time scale for drug uptake by the cornea (h0/kc). The time scale for the blink is comparable to the time scale for lateral diffusion, i.e., Ω ≡ ωh02/Df ) O(1), and both of these time scales are significantly smaller than the time scale for lateral diffusion. The ratio for the time scale for lateral equilibration and that for axial equilibration is (h02/Df)/(L2/Df) ) (h0/L)2 ≡ 2. Since ≡ h0/L , 1, it can be used as a parameter for regular expansion, and the mass transfer problem can be expanded in and then solved to different orders in . The time scale for drug uptake by the cornea varies, and for typical ophthalmic drugs, the ratio of the time scale for corneal uptake and that for axial equilibration, i.e., (h0/kc)(Df/L2), can be either O() or O(1). Below, we consider these two cases separately. First, we consider the case when (h0/kc)(Df/L2) is O(). In this case, since there are three different time scales, we use the method of multiple time scales and define the concentration to be of the form

ˆ f(ts,tm,tl,ζ,η) C ˆf ) C

(A1-4)

V˜ 1 )

The concentration can also be expanded as a regular expansion in , i.e.,

ˆ f0 + C ˆ f1 + 2C ˆ f2 + ... C ˆf ) C

C ˆf )

(Ci/K)

(A1-5)

Ω

∂C ˆf ∂C ˆf ∂C ˆf ∂C ˆf ∂C ˆf + ΩV˜ ) + Kc + 2 + Ωu˜ ∂ts ∂tm ∂t1 ∂η ∂ζ

(A1-7)

For normal blinking, both ∆/L and ∆′/h0 are ,1, and thus, these can be treated as O(). Accordingly, we define ∆ h t ∆/ and ∆ h′ t ∆′/, and thus, both ∆ h /L and ∆ h ′/h0 are O(1). After substituting ∆ ) ∆ h and ∆′ ) ∆ h ′ in the expressions for dimensionless velocity (eqss A1-6 and A1-7) and then using Taylor expansions, the dimensionless velocities can be expanded as

∂η2

+

∂2C ˆf ∂ζ2

(A1-11)

∂C ˆf ˆ ) k cC ∂y

(A1-12)

{

∂C ˆ f0 (ζ)0) ) 0 ∂ζ

∫

V ∆′ ) N (2ζ3 - 3ζ2)f(t) h0ω h0

∂2C ˆf

After dedimensionalizing the above equation and substituting the regular expansion for Cf, we get the following boundary conditions for different orders in :

N∆′ ∆ u )6 η(ζ - ζ2)f(t) + N ζf(t) t Lω L h0 - ∆′ω 0 f(t) dt (A1-6) V˜ ≡

2

where Ω ≡ ωh20/Df and Kc ≡ kcL/Df. The concentration Cf in the above equation can be expressed as a regular expansion in (eq A1-10) to yield a series of differential equations to different orders in . The boundary condition at the POLTF-cornea interface, i.e., at y ) 0 is

In the above expressions, Ci is the initial drug concentration in the contact lens and K is the partition coefficient, i.e., the ratio of concentration in the lens and in fluid at equilibrium. We dedimensionalize u and V given in eqs A1-2 and A1-3 by Lω and h0ω, respectively. The dimensionless velocities are

u˜ ≡

(A1-10)

By using the dedimensionalization described above and the multiple time scale form for concentration given in eq A1-4, the convection-diffusion equation becomes

Df Cf

(A1-9)

N∆ h′ 3 (2ζ - 3ζ2)f(ts) h0

where

k ct D ft y x ts ≡ ωt, tm ≡ , tl ≡ 2 , ζ ≡ , η ≡ , h0 h L L 0

(A1-8)

∂C ˆ f1 (ζ)0) ) KcCf0 ∂ζ

(A1-13)

∂C ˆ f2 (ζ)0) ) KcCf1 ∂ζ

The boundary condition at the lens-cornea interface, i.e., at y ) hf, is

Df

∂Cf DgCi ˆj )j≡ ∂y hg

(A1-14)

which in dimensionless form becomes

[

]

hf(ts) DgKh0 ∂C ˆf ζ) ) ˆj ≡ Λjˆ ∂ζ h0 D f hg

(A1-15)


where the dimensionless parameter DgKh0/Dfhg has been assumed to be O() and, thus, Λ ≡ DgKL/Dfhg is O(1). Since the boundary at y ) hf(ts) moves in time, we expand the boundary condition in a Taylor series to convert the problem to a stationary boundary. As noted in the main text

∫

Order: O(E1).

∂C ˆ f1 ∂C ˆ f0 ∂C ˆ f0 ∂2C ˆ f1 Ω + Kc + Ωu˜ 1 ) ∂ts ∂tm ∂η ∂ζ2 Boundary conditions: ∂C ˆ f1 ˆ f0 (ζ)0) ) KcC ∂ζ

∫

hf(ts) ) h0 - N∆′ω f(t) dt ) h0 - N∆′ f(ts) dts (A1-16) Thus the lens-POLTF boundary is located at

∫

N∆′ f(ts) dts ) h0 ∆′L ∆′L 1 - N 2 f(ts) dts ≡ 1 - N 2 g(ts) (A1-17) h0 h0

ζ)1-

( )

( )

∫

∂C ˆ f1 (ζ)1) ) Λjˆ ∂ζ

[ ( ) ]

ˆf ∂C (ζ)1) ∆′L 2 ∂ζ3 -2 N 2 g2(ts) + .... ) Λjˆ (A1-18) 2! h0

[ ( ) ]

By substituting the regular expansion of C ˆ f in the above equation and comparing various powers of in the above equation, we get the following boundary conditions at ζ ) 1 for different orders of the concentration in ,

{

∂C ˆ f0 )0 ∂ζ

( ) ( )

ˆ f0 N∆′L ∂C ˆ f1 ∂ C g(ts) + Λjˆ ) ∂ζ ∂ζ2 h02

∂C ˆ f0 ) KcC ˆ f0 - Λjˆ ∂tm

∂C ˆ f kc ∂C ˆ f D ∂C ˆf ∂C ˆf )ω + + 2 ∂t ∂ts h0 ∂tm L ∂tl

(A1-19)

(A1-20)

(

)

where overbar denotes average over ts and ζ. Combining eqs A1-27 and A1-25 gives the following to leading order in

∂C ˆ f0 (ζ)0) ) 0 ∂ζ ˆ f0(tm,t1,η) wC ˆ f0 ) C

(A1-21)

(A1-22)

(A1-28)

To leading order, the above equation in dimensional form becomes

(A1-29)

If the flux coming out of the lens ˆj is known, the above equation can be solved to determine the concentration in the POLTF as a function of time. It can be seen that when DgKh0/Dfhg and h0Df/kcL2 are O(), the concentration profile in the POLTF is independent of the axial direction. Now we consider the case when the time scale for drug uptake by the cornea and that for drug release by the lens are both comparable to the time scale for axial equilibration, i.e., DgKh0/ Dfhg and kcL2/h0Df are both O(1). In this case, tm ≈ t1, and accordingly, we define the concentration to be of the form

ˆ f(ts,t1,ζ,η) C ˆf ) C

Boundary conditions:

(A1-26)

hˆ f1 hˆ f kc ∂C ∂C kc ∂C ˆ f0 ˆ f0 ˆ f0 ∂C ∂C + + ) + O() ) ∂t h0 ∂tm ∂tm Kc ∂tl h0 ∂tm (A1-27)

∂Cf0 j - kcCf0 ) ∂t h0

where ˆj1 is the O() flux from the contact lens into the POLTF. Now we substitute the regular expansion for C ˆ f from eq A1-10 into the governing equation (eq A1-11) and then obtain differential equations to different orders in . We then solve these equations subject to the boundary conditions at the same order in as given by eqs A1-13 and A1-19. Order: O(E0).

∂C ˆ f0 (ζ)0) ) 0 ∂ζ

(A1-25)

Now substituting the regular expansion for C ˆ f into eq A1-26 and integrating in ts from 0-2π and in ζ from 0-1 gives

hˆ f Df ∂C ) (K C ˆ - Λjˆ) ∂t Lh0 c f0

ˆ f1 N∆′L ∂C ˆ f2 ∂2C ) g(ts) + Λjˆ1 ∂ζ ∂ζ2 h02

ˆ f0 ∂C ˆ f0 ∂2C ) Ω ∂ts ∂ζ2

Kc

Differentiating the multiple-time-scale form of the concentration (eq A1-4) with time gives

∂2C ˆf (ζ)1) ∂C ˆf ∂ζ2 ∆′L (ζ)1) + -N 2 g(ts) + ∂ζ 1! h0

2

(A1-24)

Integrating eq A1-23 in ts over a period 0-2π and in ζ over the domain 0-1 and applying the boundary conditions (eq A124) gives

where (∆′L/h02) ≈ O(1) and g(ts) ≡ ∫f(s) dts. Utilizing eq A1-17 in eq A1-15 and then expanding ∂C ˆ f/∂ζ[ζ)1 - N(∆′L/h02)g(ts)] around ζ ) 1 gives

3

(A1-23)

(A1-30)

In this case, the dimensionless convection-diffusion equation becomes

∂C ˆf ∂C ˆf ∂C ˆf ∂2C ˆ f ∂ 2C ˆf ∂C ˆf 2 2 + ΩV˜ ) + + Ωu˜ + 2 Ω 2 ∂ts ∂t1 ∂η ∂ζ ∂η ∂ζ (A1-31)

3730


{

The boundary conditions for this case are

expressed in the Fourier series as

∂C ˆ f0 (ζ)0) ) 0 ∂ζ

{

and at ζ ) 1

∞

f)

∂C ˆ f1 (ζ)0) ) 0 ∂ζ

∞

an sin(nts) + bn cos(nts) ) ∑ dn eint ∑ n)1 n)-∞

(A1-41)

n*0

(A1-32) Thus, G1 can also be expressed as

∂Cf2 (ζ)0) ) KcCf0 ∂ζ

∞

G1 )

∑G1s,n(ζ) sin(nts) + G1c,n(ζ) cos(nts) )

n)1

∞

G1,n(ζ) eint ∑ n)-∞

∂C ˆ f0 )0 ∂ζ

s

( ) ( )

(A1-33)

( )

∂3C ˆ f1 N∆′L ˆ f0 N∆′L ∂C ˆ f2 ∂2C g(ts) + Λjˆ ) 2 2 ∂ζ ∂ζ h0 ∂ζ3 h02

Substituting G1 from the above expression into eq A1-38 and then solving the resulting differential equation for G1,n gives

(

2

where, in this case, Λ ≡ DgKL2/Dfh0hg and Kc ≡ kcL2/h0Df. Now we solve the governing equations to different orders in . Zero Order. The zero order solution is unchanged, i.e.,

ˆ f0(t1,η) C ˆ f0 ) C

)

Similarly G2 can be expressed in a complex Fourier series as

Since C ˆ f0 is not a function of ζ, the above equation simplifies to

∂C ˆ f0 ∂ C ˆ f1 ∂C ˆ f1 + Ωu˜ 1 ) ∂ts ∂η ∂ζ2

)

x )x 1 - cosh((1 + i) xnΩ2 ) × (1 + i) xnΩ2 sinh((1 + i)xnΩ2 ) cosh((1 + i) xnΩ2 ζ) (A1-43)

(

idn n

(A1-34)

∂C ˆ f1 ∂C ˆ f0 ∂C ˆ f0 ∂2C ˆ f1 Ω + ΩV˜ 1 ) + Ωu˜ 1 (A1-35) ∂ts ∂η ∂ζ ∂ζ2

(

nΩ sinh (1 + i) ζ idn 2 ζG1,n ) n nΩ (1 + i) 2

First Order.

∞

G2 )

2

Ω

(A1-42)

n*0

ˆ f0 N∆′L ∂C ˆ f1 ∂ C ) g(ts) ∂ζ ∂ζ2 h02 2

s

∑ G2,n(ζ) eint

s

(A1-44)

n)-∞ n*0

(A1-36)

and solving eq A1-39 gives the following expression for G2,n: On the basis of the form of the differential equation and u˜ 1 (eq A1-9), we assume C ˆ f1 to be of the following form:

(

)

∂C ˆ f0 ∆ h ′N ∆ hN C ˆ f1 ) +6 G (ζ,t ) ηG2(ζ,ts) (A1-37) ∂η 1 s L h0 On substituting in the chosen form of C ˆ f1 (eq A1-36), we get the following equations for G1 and G2,

Ω

∂2G1 ∂G1 + Ωζf(ts) ) 2 ∂ts ∂ζ

∂2G2 ∂G2 + Ωζ(1 - ζ)f(ts) ) 2 Ω ∂ts ∂ζ

(A1-38)

(A1-39)

(

(

)

x ) x ( x ) x ( x ) ( x )

nΩ sinh (1 + i) ζ idn 2 2i ζ- ζ2 + G2,n ) n nΩ nΩ (1 + i) 2 nΩ -1 - cosh (1 + i) idn 2 × n nΩ nΩ sinh (1 + i) (1 + i) 2 2 nΩ cosh (1 + i) ζ (A1-45) 2

(

)

Second Order.

with the following boundary conditions:

∂G2 ∂G1 (ζ)0) ) (ζ)0) ) 0 ∂ζ ∂ζ ∂G1 ∂G2 (ζ)1) ) (ζ)1) ) 0 ∂ζ ∂ζ

Ω

ˆ f0 ∂C ˆ f2 ∂C ∂C ˆ f0 ∂Cf1 ∂C ˆ f1 + + Ωu˜ 2 + ΩV˜ 1 + ΩV˜ 1 + ∂ts ∂t1 ∂η ∂η ∂ζ

(A1-40)

Since f is a periodic function and ∫2π 0 f dts ) 0, it can be

ΩV˜ 2

∂C ˆ f0 ∂2C ˆ f0 ∂2C ˆ f2 ) + (A1-46) 2 ∂ζ ∂η ∂ζ2

We use continuity to write the above equation in the form


Ω

∂C ˆ f2 ∂C ˆ f0 ∂C ˆ f0 ∂ ∂ + + Ωu˜ 2 ˆ f1) + Ω (V˜ 1C ˆ f1) + + Ω (u˜ 1C ∂ts ∂tl ∂η ∂η ∂ζ

with the following boundary conditions:

∂G1

∂C ˆ f0 ∂2C ˆ f0 ∂2C ˆ f2 + (A1-47) ΩV˜ 2 ) 2 ∂ζ ∂η ∂ζ2 We now integrate the above equation over a period with respect to ts and over 0 to 1 with respect to ζ. After considerable algebraic manipulations, the above equation reduces to the form

∂Cf0 ∂Cf0 ∂Cf0 ∂ ) D* + R(η) + uj ∂t1 ∂η ∂η ∂η

(A1-48)

∂ζh

(ζ ) (1/2) )

∂G2 (ζ ) (1/2) ) 0 ∂ζh

From the above differential equations and the boundary conditions, it is clear that ∂G1/∂ζh and ∂G2/∂ζh are symmetric and antisymmetric, respectively, and thus,

(

∫ G2

)

∫

where

{ ( )∫ ( )∫ ( )( )∫

Ω ∆ hN 2 ζG1(ζ,ts;Ω)f(ts) dts dζ + 2π L h ′N 2 Ω ∆ 6 η ζ(1 - ζ)G2(ζ,ts;Ω)f(ts) dtsdζ + 2π h0 Ω ∆ h ′N hN ∆ 6 η (ζG2(ζ,ts;Ω) + 2π L h0 ζ(1 - ζ)G1(ζ,ts;Ω))f(ts) dts dζ uj ) 0

}

(A1-49)

( )( ) ∫( )( )

∂2G1 ∂2G2 + G1 2 dζ ) -2 ∂ζ ∂ζ -2

D* ) 1 -

(A1-56)

∂G1 ∂G2 dz ) ∂ζ ∂ζ

∂G1 ∂G2 ∂ζh

∂ζh

dζh ) 0 (A1-57)

The above derivation shows that the interaction of the lateral and the squeeze flows in the POLTF do not contribute to dispersion, and thus, the dispersion coefficient for mass transfer in the POLTF is simply a sum of the dispersion coefficients for the lateral and the squeeze flow. Thus, the expressions for D* and uj simplify to the following:

D* ) 1 -

(A1-50)

{ ( )∫ ( )∫

Ω ∆ hN 2 ζG1f(ts) dts dζ + 2π L h ′N 2 Ω ∆ 6 η ζ(1 - ζ)G2f(ts) dts dζ 2π h0

and

}

uj ) 0

(A1-58)

R(η) )

L2kc L2j C h0Df h0Df f0

(A1-51)

Multiplying eq A1-38 by G2 and eq A1-39 by G1 and then adding and integrating the equation gives

(

∫ G2

∂2G1 ∂ζ

2

+ G1

∂2G2 ∂ζ

2

-Ω

Substituting the Fourier expansions for G1, G2, and f in the above equations gives

D* ) 1 -

[( ) ∆ hN

∑Ω n*0

2

L

)

∂ G G dt dζ (A1-52) ∂ts 1 2 s

∫

d-n ζG1,n dζ +

( )

]

∆ h ′N 2 η d-n ζ(1 - ζ)G2,n dζ (A1-60) h0

Ω6

∫(ζG2 + ζ(1 - ζ)G1)f(ts) dts dζ )

(A1-59)

∫

After considerable algebra, the above expression reduces to

(an2 + bn2) D*(nΩ) n)1 n2 ∞

The above expression can be simplified by noting that ∫((∂/∂ts) G1G2) dts dζ ) 0 due to periodicity. Furthermore, integrating by parts and using the boundary conditions gives

(

∫ G2

2

∂ G1 ∂ζ

2

2

+ G1

)

∂ G2 ∂ζ

2

dζ ) -2

∂G1 ∂G2 dζ (A1-53) ∂ζ ∂ζ

To evaluate the above expression, it is useful to rewrite eqs A1-38 and A1-39 in a transformed coordinate ζh ) ζ - 1/2 and then to differentiate the equations with respect to ζh. After the coordinate transformation and differentiation, eqs A1-38 and A1-39 become

( ) ( )

( ) ( )

∂2 ∂G1 ∂ ∂G1 + Ωf(ts) ) 2 Ω ∂ts ∂ζh ∂ζh ∂ζh Ω

∂2 ∂G2 ∂ ∂G2 - 2Ωζhf(ts) ) 2 ∂ts ∂ζh ∂ζh ∂ζh

∑

(A1-61)

where

( )( )

∫

D* ) 1 +

(A1-54)

(A1-55)

D*(Ω) )

( )( ∆ hN L

2

1 1 + (sin κ cosh κ 2 x2Ω(cos κ - cosh κ)

) ( ) ∆ h ′N η h0

cos κ sinh κ + cosh κ sinh κ - cos κ sin κ) + 6

(

2

1 1 + (-sin κ cosh κ + cos κ sinh κ + 6 x2Ω(cos κ - cosh κ)

)

cosh κ sinh κ - cos κ sin κ) (A1-62) where κ ≡ xΩ/2. The average equation in this case is

∂Cf0 L2kc ∂Cf0 L2j ∂ D* + ) C ∂t1 ∂η ∂η h0Df h0Df f0

(A1-63)

3732


which in dimensional form becomes

( )

The above solution can also be expressed as

∂Cf0 ∂Cf0 j - kcCf0 ∂ ) Df D* + ∂t ∂x ∂x h0

(A1-64)

Thus, it can be seen that the average equation for the case of O(), i.e., eq A1-28, is simply a special case of eq A1-64 in the limit of negligible dispersion.

Ci C ˆg ) 1s

x x

cosh

s y Dg

cosh

s h Dg g

+

[

]

dC ˆf 1 K + Cf(0) dt s

Appendix 2 The geometry of the contact lens is shown in Figure 15. For convenience, the coordinates x2 and y2 have been replaced by x and y in this Appendix.

x x

cosh

s y Dg

cosh

s h Dg g

(A2-7)

The solution in the Laplace domain can now be inverted to give the solution in the time domain. The inverted solution is

C g ) Ci

(-1)n 4

∑(2n + 1)π cos

[

Case (i): Rapid PLTF Breakup. The governing equation and the boundary and initial conditions for diffusion in the gel are

{

K (A2-1)

Cg(t)0) ) Ci

(A2-3)

Taking Laplace transforms of the governing equation and the boundary conditions, we get

d2C ˆg

(A2-4)

dy2

∂C ˆg (y)0) ) 0 ∂y

K

∫0

t

(

( )

Ci ) 1s

cosh

s y Dg

cosh

s h Dg g

∫

(2n + 1)π

[

(-1)n 4

∑(2n + 1)π cos

(2n + 1)π

cos

2hg 2

(2n + 1)πy 2hg

]

2

2 2

[

(A2-5)

n

KCf(t) 1 -

s y Dg

cosh

s y Dg

cosh

s h Dg g

[

(-1) 4

∑(2n + 1)π cos

KCf(0) 1 e

s h Dg g

1 + sK C ˆf s

(-1)n 4

y×

]

×

e[(-(2n+1) π )/(4hg )]Dg(t-τ) dτ )

) x x x ( ) x x x cosh

[

dCf (τ) 1 dτ

dCf (τ) 1 dτ

where C ˆ g is the Laplace transform of Cg. The solution to the above set of differential equation and boundary conditions is

Ci Ci ˆf + KC s s

∫0

t

To simplify the above expression, we integrate the convolution integral by parts. This gives

C ˆ g(y)hg) ) KC ˆf

C ˆg )

]

2

y×

2hg

2 2

(A2-2)

cosh

(2n + 1)π

e[(-(2n+1) π )/(4hg )]Dg(t-τ) dτ (A2-8)

∂Cg (y)0) ) 0 ∂y Cg(y)hg) ) KCf(t)

sC ˆ g - Cg(t)0) ) Dg

(-1) 4

∑(2n + 1)π cos 2 2

2

e[(-(2n+1) π )/(4hg )]Dgt +

2

∂Cg ∂ Cg ) Dg 2 ∂t1 ∂y

2hg

2 2

y e[(-(2n+1) π )/(4hg )]Dgt +

n

KCf(0) 1 -

Figure 15. Geometry of the lens and the prelens tear film.

(2n + 1)π

∫

(2n + 1)π

]

2hg

cos

2hg

∫0 C(τ) ∑ t

×

(-1)n 4

(2n + 1)π

2 2 (2n + 1)πy (2n + 1) π Dg

(2n + 1)πy cos

+K

2hg

(2n + 1)πy

(-1)n 4

[(-(2n+1)2π2)/(4hg2)]Dgt

]

(2n + 1)πy

cos

2hg

4hg2 2 2

2

× ×

e[(-(2n+1) π )/(4hg )]Dg(t-τ) (A2-9)

(A2-6)

Thus, the concentration profile becomes


C g ) Ci

(2n + 1)πy

(-1)n 4

∑(2n + 1)π

[

cos

2hg n

KCf(t) 1 K

(-1) 4

∑(2n + 1)π cos

2 2

∫

C (τ) 0 f

2hg

∑(2n + 1)π

(2n + 1)πy (2n + 1) π Dg

]

(2n + 1)πy

(-1)n 4

t

j ) -Dg

2

e[(-(2n+1) π )/(4hg )]Dgt +

)

+

[

2 2

cos

2hg

4hg2

2 2

( ) hg

DgK

2 2 2 2 e[(-(2n+1) π )/(4hg )]Dgt - DgKCf(t) + hg (2n + 1)2π2Dg

∫0 Cf(τ) t

3

2 2

2

e[(-(2n+1) π )/(4hg )]Dg(t-τ) dτ

2hg

2DgK ∞ 2Dg ∞ [(-(2n+1)2π2)/(4hg2)]Dgt [ Ci e ]+ × ) hg n)0 hg n)0

[

∑

∑

]

(2n + 1)2π2Dg 2 2 2 C(t) e[(-(2n+1) π )/(4hg )]Dg(t-τ) dτ - Cf(t) 0 2 4hg (A2-11)

∫

t

Case (ii): Well-Mixed PLTF. The governing equation, the boundary conditions, and the initial condition of the mass transfer problem for case II are the following:

{

∂2Cg ∂Cg ) Dg 2 ∂t1 ∂y

(A2-12)

Cg(y)0) ) 0 Cg(y)hg) ) KCf(t)

(A2-13)

Cg(t)0) ) Ci

(A2-14)

By following the same procedure as outlined previously for case (i), one can solve the above set of equations to obtain the following expression for the concentration profile ∞

C g ) Ci

4

sin ∑ n)0 (2n + 1)π KCf(t) ∞

K

[

y

hg

(2n + 1)πy hg ∞

+

∑ n)1

n)1

nπ 2

sin

hg

2

2 2

e[(-(2n+1) π )/(hg )]Dgt +

2(-1)n

nπDg

∫0tCf(τ)∑2(-1)n

∞

∑ ∫0 Cf(τ)n)1 ∑ n)1 t

2)]D

g

gt

Cf(t)

- D gK

hg

+

2DgK hg

2 2

n π Dg hg

2

2 2

2

e[(-n π )/(hg )]Dg(t-τ) dτ - Cf(t)

]

(A2-16)

Literature Cited

∂Cg | ∂y y)hg

2DgCi

∑

∞

2 2

2

Thus, the flux j can be expressed as

)

∞

∑ e[(-(2n+1) π )/(4h n)0

e[(-(2n+1) π )/(4hg )]Dg(t-τ) dτ (A2-10)

j ) -Dg

4DgCi hg

×

∂Cg | ∂y y)hg

sin nπy hg

]

nπy hg

-

2 2

2

e[(-n π )/(hg )]Dg(t-τ) dτ (A2-15)

Thus, the flux j can be expressed as

(1) Bourlais, C. L.; Acar, L.; Zia, H.; Sado P. A.; Needham, T.; Leverge, R. Ophthalmic drug delivery systems. Prog. Retinal Eye Res. 1998, 17 (1), 33-58. (2) Urtti, A.; Salminen, L. Minimizing systemic absorption of topically administered ophthalmic drugs. SurV. Ophthalmol. 1993, 37 (6), 435-456. (3) TIMPOTIC prescribing information, supplied by Merck: Whitehouse Station, NJ. (4) Segal, M. Patches Pumps and timed release. FDA Consum. 1991, October. (5) Saettone, M. F.; Chetoni, P.; Torraca, M. T.; Burgalassi, S.; Giannaccini, B. Evaluation of mucoadhesive properties and in ViVo activity of ophthalmic vehicles based on hyaluronic acid. Int. J. Pharm. 1989, 51, 203-212. (6) Saettone, M. F.; Teneggi, A.; Savigni, P.; Tellini, N. Vehicle effects on ophthalmic bioavailability. The influence of different polymers. J. Pharm. Pharmacol. 1982, 34, 464-466. (7) Davies, N. M.; Farr, S. J.; Hadgraft, J.; Kellaway, I. W. Evaluation of mucoadhesive polymers in ocular drug delivery. I. Viscous solutions. Pharm. Res. 1982, 9 (9), 1137-1144. (8) Greaves, J. L.; Olejnik, O.; Wilson, C. G. Polymers and the precorneal tear film. STP Pharma Sci. 1992, 2 (1), 13-33. (9) Kyyronen, K.; Urtti, A. Improved ocular: Systemic absorption ratio of Timolol by viscous vehicle and phenylephrine. InVest. Ophthalmol. Visual Sci. 1990, 31 (9), 1827-1833. (10) Hiementz, P. C.; Rajagopalan R. Principles of colloid and surface chemistry, 3rd ed.; Marcel Dekker: New York, 1997. (11) Fitzerald, P.; Hadgraft, J.; Kreuter, J.; Wilcon, C. G. A gamma scintigraphic evaluation of microparticulate ophthalmic delivery systems: Liposomes and nanoparticles. Int. J. Pharm. 1987, 40, 81-84. (12) Lee, V. H. L. Applications of liposomes in ocular drug delivery. Liposomes in Ophthalmology and Dermatology; Pleyer, U., Schmidt, K., Thiel, H. J., Eds; 1993; pp 53-59. (13) Marchal-Heussler, L.; Sirbat, D.; Hoffman, M.; Maincent, P. Poly(-caprolactone) nanocapsules in carteolol ophthalmic drug delivery. Pharm. Res. 1993, 10 (3), 386-390. (14) Calvo, P.; Vila-Lato, J.; Alonso, M. J. Comparative in vitro evaluation of several colloidal systems, nanoparticles, nanocapsules, and nanoemulsiuons as ocular drug carriers. J. Pharm. Sci. 1996, 85 (5), 530536. (15) Calvo, P.; Sanchez, A.; Martinez, J.; Lopez, M. I.; Calonge, M.; Pastor, J. C.; Alonso, M. J. Polyster nanocapsules as new topical ocular delivery systems for cyclosporin A. Pharm. Res. 1996, 13 (2), 311-315. (16) Li, V. H. K.; Wood, R. W.; Kreuter, J.; Harmia, T.; Robinson, J. R. Ocular drug delivery of progesterone using nanoparticles. J. Microencapsulation 1986, 3 (3), 213-218. (17) Zimmer, A. K.; Kreuter, J.; Robinson, J. R. Studies on transport pathways of PBCA nanoparticles in ocular tissues. J. Microencapsulation 1991, 8 (4), 497-504. (18) Lee, V. H. L. Review: New directions in the optimization of ocular drug delivery. J. Ocul. Pharmacol. 1990, 6 (2), 157-164. (19) Sawusch, M. R.; O’Brien, T. P.; Dick, J. D.; Ottsch, J. D. Use of collagen corneal shields in the treatment of bacterial kearatitis. Am. J. Ophthalmol. 1988, 106, 279-281. (20) Vasantha, R.; Sehgal, P. K.; Panduranga Rao, K. Collagen ophthalmic inserts for pilocarpine drug delivery systems. Int. J. Pharm. 1988, 47, 95-102. (21) Gurtler, F.; Gurny, R. Patent literature review of ophthalmic inserts. Drug. DeV. Ind. Pharm. 1995, 21 (1), 1-18. (22) Creech, J. L.; Chauhan, A.; Radke, C. J. Dispersive mixing in the posterior tear film under a soft contact lens. Ind. Eng. Chem. Res. 2001, 40, 3015-3026.

3734


(23) McNamara, N. A.; Polse, K. A.; Brand, R. D.; Graham, A. D.; Chan, J. S.; McKenney, C. D. Tear mixing under a soft contact lens: Effects of lens diameter. Am. J. Ophthalmol. 1999, 127 (6), 659-665. (24) Peppas, N. A. Hydrogels in medicine and pharmacy, Vol. 1 & 2; CRC Press Inc.: Boca Raton, FL, 1986. (25) Gulsen, D.; Chauhan, A. Dispersion of microemulsion drops in HEMA hydrogel: a potential ophthalmic drug delivery vehicle. Int. J. Pharm. 2005, 292 (1-2), 95-117. (26) Gulsen, D.; Chauhan, A. Ophthalmic Drug Delivery through Contact Lenses. InVest. Ophthalmol. Visual Sci. 2004, 45 (7), 2342-2347. (27) Hehl, E. M.; Beck, R.; Luthard, K.; Guthoff, R.; et al. Improved penetration of aminoglycosides and fluoroquinolones into the aqueous humour of patients by means of Acuvue contact lenses. Eur. J. Clin. Pharmacol. 1999, 55 (4), 317-323. (28) Nakada, K.; Sugiyama, A. Process for producing controlled drugrelease contact lens, and controlled drug-release contact lens thereby produced. U.S. Patent 6,027,745, May 29, 1998. (29) Hillman, J. S. Management of acute glaucoma with Pilocarpinesoaked hydrophilic lens. Brit. J. Ophthalmol. 1974, 58, 674-679. (30) Ramer, R.; Gasset, A. Ocular Penetration of Pilocarpine. Ann. Ophthalmol. 1974, 6, 1325-1327. (31) Montague, R.; Wakins, R. Pilocarpine dispensation for the soft hydrophilic contact lens. Brit. J. Ophthalmol. 1975, 59, 455-458. (32) Hillman, J.; Masters, J.; Broad, A. Pilocarpine delivery by hydrophilic lens in the management of acute glaucoma. Trans. Ophthalmol. Soc. U.K. 1975, 79-84. (33) Giambattista, B.; Virno, M.; Pecori-Giraldi, J.; Pellegrino, N.; Motolese, E. Possibility of Isoproterenol Therapy with Soft Contact Lenses: Ocular Hypotension Without Systemic Effects. Ann. Ophthalmol. 1976, 8, 819-829. (34) Marmion, V. J.; Yardakul, S. Pilocarpine administration by contact lens Trans. Ophthalmol. Soc. U.K. 1977, 97, 162-163. (35) Schultz, C. L.; Mint, J. M. Drug delivery system for antiglaucomatous medication. U.S. Patent 6,410,045. (36) Rosenwald, P. L. Ocular device. U.S. Patent 4,484,922. (37) Schultz, C. L.; Nunez, I. M.; Silor, D. L.; Neil, M. L. Contact lens containing a leachable absorbed material. U.S. Patent 5,723,131. (38) Garcia, D. M.; Escobar, J. L.; Noa, Y.; Bada, N.; Hernaez, E.; Katime, I. Timolol maleate release from pH-sensible poly(2-hydroxyethyl methacrylate-co-methacrylic acid) hydrogels. Eur. Polym. J. 2004, 40 (8), 1683-1690.

(39) Hiratani, H.; Alvarez-Lorenzo, C. The nature of backbone monomers determines the performance of imprinted soft contact lenses as timolol drug delivery systems. Biomaterials 2004, 25 (6), 1105-1113. (40) Alverez-Lorenzo, C.; Hiratani, H.; Gomez-Amoza, J. L.; MartinezPacheco, R.; Souto, C.; Concheiro, A. Soft contact lenses capable of sustained delivery of timolol. J. Pharm. Sci. 2002, 91 (10), 2182-2192. (41) Karlgard, C.; Wong, N. S.; Jones, L.; Moresoli, C. In vitro uptake and release studies of ocular pharmaceutical agents by silicon-containing and p-HEMA hydrogel contact lens materials. Int. J. Pharm. 2003, 257, 141-151. (42) Chauhan, A.; Radke, C. J. Modeling the vertical motion of a soft contact lens. Curr. Eye Res. 2001, 22 (2), 102-108. (43) Chauhan, A.; Radke, C. J. Settling and deformation of a soft contact lens. InVest. Ophthalmol. Visual Sci. 2001, 42 (4), S589; J. Colloid Interface Sci. 2002, 245, 187-197. (44) Chauhan, A.; Radke, C. J. The Role of Fenestrations and Channels on the Transverse Motion of a Soft Contact Lens. Optometry Vision Sci. 2001, 78 (10), 732-743. (45) Tiffany, J. M. The Viscosity of Human Tears. Int. Ophthalmol. 1991, 15, 371. (46) Fantini, S.; Clohessy, J.; Gorgy, K.; Fusalba, F.; Johans, C.; Kontturi, K.; Cunnane, V. J. Influence of the presence of a gel in the water phase on the electrochemical transfer of ionic forms of b-blockers across a large water-1,2-dichloroethane interface. Eur. J. Pharm. Sci. 2003, 18, 251-257. (47) Hiratani, H.; Alvarez-Lorenzo, C. Timolol uptake and release by imprinted soft contact lenses made of N,N-diethylacrylamide and methacrylic acid. J. Controlled Release 2002, 83, 223-230. (48) Edwards, A.; Prausnitz, M. R. Predicted permeability of the cornea to topical drugs. Pharm. Res. 2001, 18 (11), 1497-1508. (49) Prausnitz, M. R.; Noonan, J. S. Permeability of cornea, sclera, and conjunctiva: a literature analysis for drug delivery to the eye. J. Pharm. Sci. 1998, 87 (12), 1479-1488.

ReceiVed for reView July 5, 2005 ReVised manuscript receiVed January 3, 2006 Accepted January 24, 2006 IE0507934

Modeling Ophthalmic Drug Delivery by Soaked Contact Lenses

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