3718
Ind. Eng. Chem. Res. 2006, 45, 37183734
Modeling Ophthalmic Drug Delivery by Soaked Contact Lenses ChiChung Li and Anuj Chauhan* UniVersity of Florida, Chemical Engineering Department, Room 237 CHE, P.O. Box 116005, GainesVille, Florida 326116005
Approximately 90% of all ophthalmic drug formulations are now applied as eye drops. While eye drops are convenient and wellaccepted 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 integrodifferential 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 drugcontaining 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, betablockers such as timolol that treat wideangle 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) 3922592. Fax: (352) 3929513. Email:
[email protected]
gels, colloidal particles, and collagen shields.1 Addition of polymeric gels such as hyaluronic acid (HA),5 poly(acrylic acid) (PAA),68 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 drugdelivery vehicles.1316 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 drugdelivery vehicles to deliver a single drug.1820 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 PLTFlensPOLTF 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, presentday 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 crosslinked 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 drugladen 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.2737 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.3840 Also, recently Karlgard et al. measured the uptake and release of a number of ophthalmic drugs by both HEMAbased 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,2934 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 wellcharacterized both mathematically and experimentally.22,23,4244 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 drugrelease 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, twodimensional film bounded by an undeformable cornea and an undeformable but moving contact lens. The lens is treated as a twodimensional 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 twodimensional 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 lensPOLTFcornea 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 wellknown 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 blinkinterblink frequency, i.e., ω ) 2π/Tb, where Tb is the time for a blinkinterblink 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 semiinfinite. The flow is assumed to be periodicsteady because hf ≈ 10 µm and the tear kinematic viscosity ν ) 1.5 × 102 cm2/s,45 giving rise to a characteristic time for fully developed flow tchar ) hf2/ν ≈ 7 × 105 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
Cgy2)hg ) Cg,eq(Cfy1 ) hf) ) KCfy1)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 lenspostlens 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 timedependent POLTF thickness with a mean

∂Cf ∂x1
)0
(7a)
Cfx1)L ) 0
(7b)
x1)0
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3721
Df

∂Cf ∂y1
y1)0
) kcCfy1)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
Cft)0 ) 0
(8a)
Cgt)0 ) Ci
(8b)
The above set of equations can be solved by finitedifference or finiteelement 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 integrodifferential 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 drugrelease 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 )](tt*) 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 )](tt*) 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 )](tt*) 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 )](tt*) 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 integrodifferential 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 integrodifferential 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)
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Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006
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](ττ*) dτ*  Cf0(τ)
×
)]
(15)
∞
˜j0 ) [
∑4 e(2n+1) π τ  C˜ f0(τ) + n)0 2 2
∞
2
(∫0 C ˜ f0(τ*)n2π2 en π (ττ*) 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 ∼103 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 POLTFgel 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 POLTFgel 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 blinkinterblink 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 A158 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 blinkinterblink 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 A146 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
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3723
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 Integrodifferential 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](ττ*) dτ*  Cf0(τ)
]
(18)
and thus,
dIn ) λnCf0(τ)  λn2 dτ
∫0τCf0(τ*) eλ (ττ*) 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 en π (ττ*) 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.
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Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006
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 lensPLTF interface till the mass transfer boundary layers from both the POLTFlens and the PLTFlens 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* dx1fx)L hf dt ) dC
P1 ∞ ∫0∞D* dηfη)1 dτ ≈  P2 ∫0 D*
P1 P2
2 M0
(23)
dC
∫0∞ ∫0L Dg dy2gy )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.
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3725
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 drugcarrying 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. However, for a very small K, the flux from the lens is so large that, in the time that it takes for the concentration to reach the maximum, the loss to the tear lake is negligible and, thus, the dimensionless concentration reaches a value very close to 1. Since the loss to the PLTF does not affect the short time profiles, the reasoning for the effect of P2 on the maximum concentration and the time to attain it is the same for both case I (no loss to PLTF) and case II (maximum loss to PLTF). Effect of P3. Figure 12 plots the C ˜ f0(τ) at the center of the lens for both cases and for two different values of P3, and Figure 13 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 P3
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Figure 12. Effect of P3 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 P2 are 0.5 and 100, respectively, and the values of P3 are noted on the curves.
Figure 13. Effect of P3 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 and case II. The values of P1 and P2 are 0.5 and 100, respectively.
values. The parameter P3 is the ratio of the time scale for the diffusion in the lens and the drug uptake by the cornea. Thus, an increase in P3 can be interpreted as an increase in the permeability of the cornea to the solute. Accordingly, an increase in P3 leads to a larger flux into the cornea and also a reduction in the POLTF concentrations, which results in an increase in the fractional uptake by the cornea and a reduction in the fraction lost from the side to the tear lake. The trends are the same for the case of no flux to PLTF and zero concentration in the PLTF. As P3 becomes very large, the concentration in the POLTF is expected to be negligibly small (see Figure 12), and thus, the boundary condition at the lensPOLTF interface becomes almost identical to that at the lensPLTF interface for case II. Accordingly, the diffusion from the gel become symmetric, and thus, ∼50% of the drug goes to the PLTF and the other 50% goes to the POLTF. Since the loss from the edges due to dispersion is small, as P3 becomes very large, the fraction of drug that goes to the cornea is expected to be ∼100% for case I and 50% for case II. Results for Timolol. The values of Df and Dg for timolol, which is a commonly used glaucoma drug, are 5 × 1010 m2/s and 5 × 1012 m2/s,46,47 respectively, and the value of corneal permeability, kc, for timolol is 1.5 × 107 m/s.48 The partition coefficient for timolol in pHEMA gels depends on concentration,
and the average value is ∼5.47 Also the values of hf and hg are 10 and 100 µm, respectively. Thus, the parameters P1, P2, and P3 are 0.04, 50, and 30, respectively. The fractional flux to the cornea for these parameters is 0.1658 for the case of zero concentration in the PLTF and 0.7993 for the case of zero flux to the PLTF. The fractional losses to the lake are 0.0567 and 0.1923 for the two cases. For the case of a wellmixed PLTF, the loss to the PLTF is 0.8008. The durations of drug release are about 16 000 and 4 000 s for the cases of rapid PLTF breakup and the wellmixed PLTF, respectively. Since the real situation is expected to be somewhere between the two extreme cases, the fraction of timolol that enters the cornea may be around 50%, and the release time is perhaps ∼10 000 s. At a saturation loading of 2 mM,47 a contact lens can contain ∼0.01 mg of drug. Thus, for the mean release time of 10 000 s and 50% uptake by cornea, the delivery rates to cornea are 0.04 mg/day. The usual starting dose of timolol is one drop of 0.25% timolol maleate in the affected eye(s) twice a day. Assuming a volume of 25 µL for each drop, the daily dosage of timolol is 0.125 mg each day. Only ∼15% of this amount actually reaches the cornea. Thus, the dosage that needs to be delivered to cornea is at the maximum ∼0.0063 mg each day, which is much smaller than the rate that can be delivered by a saturated contact lens. These results show that, if a contact lens saturated with timolol is worn for a period of ∼23 h, it may provide adequate dosage of timolol and minimize side effects. Comparison of Model Predictions with Clinical Data. Most of the clinical studies on drug delivery by contact lenses focused on delivery of a glaucoma drug pilocarpine by soft contact lenses.2932 Pilocarpine is a hydrophilic drug that simply absorbs in the pores present in the hydrogel matrix of soft contact lenses and does not bind to the polymer matrix. Thus, the diffusion of pilocarpine through a swollen contact lens can be modeled as a purely Fickian process. Since pilocarpine simply absorbs in the soft contact lenses, the partition coefficient for this drug is equal to the fractional water content in the fully swollen lens, and this varies from ∼0.70.86 for the Sauflon lenses, which were used in most studies.31 The contact lenses used in most studies were afocal, 0.2 mm thick, with a diameter of 13.5 or 14 mm and back central radii (radius of curvature) between 7.8 and 8.6 mm.29,32 Thus, these contact lenses occupied a volume of ∼6575 µL. If these lenses are soaked in 1% pilocarpine solution, then, based on a saturation water content of 80%, the lenses should absorb ∼560 µg of drug. Hillman et al. reported a slightly larger value of 700 µg for the drug content in the lenses soaked in 1% solution.32 Hillman et al. compared clinical response of Sauflon lenses soaked in 1% pilocarpine solution with that of intensive pilocarpine 4% therapy, which comprises instilling 1 or 2 drops per min for 5 min, every 5 min for half an hour and then every 15 min for 90 min.32 The pilocarpine therapy amounts to instilling of ∼17 drops that contain ∼3040 µL of 4% pilocarpine, which deliver ∼23 mg of drug to the eyes, which is ∼33 times the amount of drug present in the contact lens. Hillman et al. reported that the clinical response to the contact lens soaked in 1% solution was better than that for pilocarpine therapy.32 In another study with the same type of lenses and the same treatment methodologies, Hillman observed a 54.8% drop in intraocular pressure with the contact lenses and a 49.7% reduction with the 4% pilocarpine regimen.29 Thus, both of these studies suggest that drug delivery by contact lenses is ∼33 times more efficient than that by drops. As stated above, the value of the partition coefficient of pilocarpine in hydrophilic soft contact lenses is expected to be ∼0.8. Also,
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3727
the in vitro data for release of pilocarpine from a soaked lens into saline gives a value of ∼1012 m2/s for the diffusivity in gel, Dg.32 Since the molecular weight of pilocarpine is ∼2/3 of that of timolol, its diffusivity in tears, Df, can be approximately taken as 1.5 times that of timolol, which as stated previously is 5 × 1010 m2/s. The corneal permeability of pilocarpine is 1.7 × 107 m/s.49 On the basis of these parameters and values of 10 and 200 µm for hf and hg, respectively, the values of P1, P2, and P3 are 0.003, 16, and 680, respectively. The fractional flux to the cornea for these parameters is 0.48 for the case of zero concentration in the PLTF and 0.98 for the case of zero flux to the PLTF. Since the real situation is expected to be somewhere between the two extreme cases, the fraction of pilocarpine that enters the cornea may be ∼70%. The fractional uptake for pilocarpine has been reported to be ∼2%.29 Thus, the model predicts that the drug uptake by contact lens is ∼35 times more than that by drops, and this is in reasonable agreement with the clinical results. Conclusions The model developed in this paper represents a first attempt to model drug release from a contact lens and the subsequent uptake by the cornea. The model assumes Fickian diffusion in the contact lens and takes into account the convective enhancement in mass transfer in the postlens tear film due to the flow driven by the oscillation of the contact lens during the blink. The fraction of drug that enters the cornea has been determined for two extreme cases. The first case corresponds to a rapid breakup of the prelens tear film, which prevents drug loss from the anterior lens surface. 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. The fraction of drug that enters the cornea varies from ∼70 to 95% for the first case and from 20 to 35% for the second case. It is thus reasonable to assume that, in normal physiological circumstances where the prelens tear film breaks in a few seconds, the fraction of drug that is absorbed by the corneal tissue will be ∼50%. The tear breakup on the anterior lens surface also causes partial dehydration of the lens near the surface, and the dehydration results in a thin glassy layer near the surface, which might suggest that case 1, which corresponds to no flux to the PLTF, may be more applicable under normal conditions of lens wear. The model predicts that the fraction of pilocarpine that enters the cornea may be ∼70%, which is ∼35 times larger than the reported value of ∼2% for delivery by drops.29 Clinical results show that contact lenses deliver ∼33 times more drug than eye drops, and thus, the value of 35 obtained by model is in reasonable agreement with the clinical results. The model presented in this paper takes into account the essential mechanisms but simplifies each one of those. For instance, the diffusion through the contact lens has been assumed to be Fickian. The diffusion of solutes through hydrogels is complex and depends on a number of factors such as the degree of swelling and the interaction of the solute with the gel. The contact lens is swollen when it is inserted in the eye, and although the water content of the lens changes after insertion, particularly near the anterior surface, this change is not expected to significantly alter the drug diffusion. However, many of the ophthalmic drugs, including timolol, have been shown to adsorb
on the lens matrix, and thus, the assumption of Fickian diffusion is not appropriate for these drugs. In fact, a partition coefficient value of >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 drugrelease 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 HEMAbased contact lenses. Second, even for drugs that can absorb in the lens matrix, the drugrelease 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.
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(
)
∂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 + ....
(A11) where
u˜ 1 )
(A12)
(
)
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
∫
(A13)
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
(A14)
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)
(A15)
Ω
∂C ˆf ∂C ˆf ∂C ˆf ∂C ˆf ∂C ˆf + ΩV˜ ) + Kc + 2 + Ωu˜ ∂ts ∂tm ∂t1 ∂η ∂ζ
(A17)
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 A16 and A17) and then using Taylor expansions, the dimensionless velocities can be expanded as
∂η2
+
∂2C ˆf ∂ζ2
(A111)
∂C ˆf ˆ ) k cC ∂y
(A112)
{
∂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 (A16) 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 A110) to yield a series of differential equations to different orders in . The boundary condition at the POLTFcornea 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 A12 and A13 by Lω and h0ω, respectively. The dimensionless velocities are
u˜ ≡
(A110)
By using the dedimensionalization described above and the multiple time scale form for concentration given in eq A14, the convectiondiffusion equation becomes
Df Cf
(A19)
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
(A18)
∂C ˆ f1 (ζ)0) ) KcCf0 ∂ζ
(A113)
∂C ˆ f2 (ζ)0) ) KcCf1 ∂ζ
The boundary condition at the lenscornea interface, i.e., at y ) hf, is
Df
∂Cf DgCi ˆj )j≡ ∂y hg
(A114)
which in dimensionless form becomes
[
]
hf(ts) DgKh0 ∂C ˆf ζ) ) ˆj ≡ Λjˆ ∂ζ h0 D f hg
(A115)
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3729
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 (A116) Thus the lensPOLTF boundary is located at
∫
N∆′ f(ts) dts ) h0 ∆′L ∆′L 1  N 2 f(ts) dts ≡ 1  N 2 g(ts) (A117) h0 h0
ζ)1
( )
( )
∫
∂C ˆ f1 (ζ)1) ) Λjˆ ∂ζ
[ ( ) ]
ˆf ∂C (ζ)1) ∆′L 2 ∂ζ3 2 N 2 g2(ts) + .... ) Λjˆ (A118) 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
(A119)
(A120)
(
)
where overbar denotes average over ts and ζ. Combining eqs A127 and A125 gives the following to leading order in
∂C ˆ f0 (ζ)0) ) 0 ∂ζ ˆ f0(tm,t1,η) wC ˆ f0 ) C
(A121)
(A122)
(A128)
To leading order, the above equation in dimensional form becomes
(A129)
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:
(A126)
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 (A127)
∂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 A110 into the governing equation (eq A111) 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 A113 and A119. Order: O(E0).
∂C ˆ f0 (ζ)0) ) 0 ∂ζ
(A125)
Now substituting the regular expansion for C ˆ f into eq A126 and integrating in ts from 02π and in ζ from 01 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 multipletimescale form of the concentration (eq A14) with time gives
∂2C ˆf (ζ)1) ∂C ˆf ∂ζ2 ∆′L (ζ)1) + N 2 g(ts) + ∂ζ 1! h0
2
(A124)
Integrating eq A123 in ts over a period 02π and in ζ over the domain 01 and applying the boundary conditions (eq A124) gives
where (∆′L/h02) ≈ O(1) and g(ts) ≡ ∫f(s) dts. Utilizing eq A117 in eq A115 and then expanding ∂C ˆ f/∂ζ[ζ)1  N(∆′L/h02)g(ts)] around ζ ) 1 gives
3
(A123)
(A130)
In this case, the dimensionless convectiondiffusion 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 (A131)
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Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006
{
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)∞
(A141)
n*0
(A132) 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
( ) ( )
(A133)
( )
∂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 A138 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) (A143)
(
idn n
(A134)
∂C ˆ f1 ∂C ˆ f0 ∂C ˆ f0 ∂2C ˆ f1 Ω + ΩV˜ 1 ) + Ωu˜ 1 (A135) ∂ts ∂η ∂ζ ∂ζ2
(
nΩ sinh (1 + i) ζ idn 2 ζG1,n ) n nΩ (1 + i) 2
First Order.
∞
G2 )
2
Ω
(A142)
n*0
ˆ f0 N∆′L ∂C ˆ f1 ∂ C ) g(ts) ∂ζ ∂ζ2 h02 2
s
∑ G2,n(ζ) eint
s
(A144)
n)∞ n*0
(A136)
and solving eq A139 gives the following expression for G2,n: On the basis of the form of the differential equation and u˜ 1 (eq A19), 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) (A137) ∂η 1 s L h0 On substituting in the chosen form of C ˆ f1 (eq A136), 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
(A138)
(A139)
(
(
)
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 (A145) 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
(A140)
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 ) + (A146) 2 ∂ζ ∂η ∂ζ2
We use continuity to write the above equation in the form
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3731
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 + (A147) 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
(A148)
∂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
}
(A149)
( )( ) ∫( )( )
∂2G1 ∂2G2 + G1 2 dζ ) 2 ∂ζ ∂ζ 2
D* ) 1 
(A156)
∂G1 ∂G2 dz ) ∂ζ ∂ζ
∂G1 ∂G2 ∂ζh
∂ζh
dζh ) 0 (A157)
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 
(A150)
{ ( )∫ ( )∫
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
(A158)
R(η) )
L2kc L2j C h0Df h0Df f0
(A151)
Multiplying eq A138 by G2 and eq A139 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 (A152) ∂ts 1 2 s
∫
dn ζG1,n dζ +
( )
]
∆ h ′N 2 η dn ζ(1  ζ)G2,n dζ (A160) h0
Ω6
∫(ζG2 + ζ(1  ζ)G1)f(ts) dts dζ )
(A159)
∫
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 (A153) ∂ζ ∂ζ
To evaluate the above expression, it is useful to rewrite eqs A138 and A139 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 A138 and A139 become
( ) ( )
( ) ( )
∂2 ∂G1 ∂ ∂G1 + Ωf(ts) ) 2 Ω ∂ts ∂ζh ∂ζh ∂ζh Ω
∂2 ∂G2 ∂ ∂G2  2Ωζhf(ts) ) 2 ∂ts ∂ζh ∂ζh ∂ζh
∑
(A161)
where
( )( )
∫
D* ) 1 +
(A154)
(A155)
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) (A162) where κ ≡ xΩ/2. The average equation in this case is
∂Cf0 L2kc ∂Cf0 L2j ∂ D* + ) C ∂t1 ∂η ∂η h0Df h0Df f0
(A163)
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Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006
which in dimensional form becomes
( )
The above solution can also be expressed as
∂Cf0 ∂Cf0 j  kcCf0 ∂ ) Df D* + ∂t ∂x ∂x h0
(A164)
Thus, it can be seen that the average equation for the case of O(), i.e., eq A128, is simply a special case of eq A164 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
(A27)
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 (A21)
Cg(t)0) ) Ci
(A23)
Taking Laplace transforms of the governing equation and the boundary conditions, we get
d2C ˆg
(A24)
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
[
(A25)
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(tτ) 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
(A22)
cosh
(2n + 1)π
e[((2n+1) π )/(4hg )]Dg(tτ) dτ (A28)
∂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(tτ) (A29)
(A26)
Thus, the concentration profile becomes
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3733
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(tτ) 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(tτ) dτ  Cf(t) 0 2 4hg (A211)
∫
t
Case (ii): WellMixed 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
(A212)
Cg(y)0) ) 0 Cg(y)hg) ) KCf(t)
(A213)
Cg(t)0) ) Ci
(A214)
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(tτ) dτ  Cf(t)
]
(A216)
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(tτ) dτ (A210)
j ) Dg
4DgCi hg
×
∂Cg  ∂y y)hg
sin nπy hg
]
nπy hg

2 2
2
e[(n π )/(hg )]Dg(tτ) dτ (A215)
Thus, the flux j can be expressed as
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ReceiVed for reView July 5, 2005 ReVised manuscript receiVed January 3, 2006 Accepted January 24, 2006 IE0507934