Measurement and Prediction of Lateral Diffusion within Human Sclera

Abstract

purpose. Drug delivery via the sclera is a promising approach to retinal disorder treatments that require access to the posterior segment of the eye. To complement existing studies of transverse diffusion across the sclera, this study examined lateral diffusion within the sclera parallel to the scleral surface.

methods. Using sulforhodamine as a model hydrophilic drug, rates of diffusion were measured in strips of human cadaveric sclera for up to 1 week. Data were analyzed with a mathematical model based on theoretical expressions for one-dimensional diffusion.

results. Measurable amounts of sulforhodamine were detected at distances of 5 and 10 mm from the sulforhodamine donor reservoir at 4 hours and 3 days, respectively. The effective lateral diffusivity of sulforhodamine was determined to be 3.82 × 10⁻⁶ cm²/s, which is similar in magnitude to the transverse diffusivity. The theoretical model agreed with experimental values with an average error of 39%.

conclusions. This study shows that the lateral diffusion of sulforhodamine in human sclera is slow and localizes to the site of administration.

Targeted administration of drugs to the posterior segment of the eye remains a significant challenge in ocular drug delivery. Current treatment strategies include systemic delivery, by oral or parenteral routes, and local delivery using topical drops, subconjunctival and peribulbar injections, intravitreal injections, and implants. ^{1 2} However, none of these approaches provides fully satisfactory ocular delivery to the posterior part of the eye. 

Systemic delivery is often accompanied by side effects because of the high drug doses needed to reach the target tissues within the eye. Topical drops through the cornea generally cannot achieve adequate drug concentrations in the posterior segment due to slow diffusion across the cornea to the back of the eye and counterproductive convection of tear fluid and aqueous humor. ^{1 3} Although intraocular injection and implants can provide delivery targeted to the posterior segment, they can lead to complications such as retinal detachment, hemorrhage, endophthalmitis, and cataract, especially when repeated injections are required. ^{3 4}  

Because of these limitations, there is growing interest in drug delivery across the sclera, which avoids the complications associated with penetrating the globe and the diffusional barrier of the cornea. ² The sclera’s large surface area, which averages 17 cm² on the human eye, ⁵ is approximately 20 times larger than the cornea. Moreover, sclera is much more permeable, especially to large and hydrophilic drugs. ^{6 7} Conventional subconjunctival and peribulbar injections provide access to the transscleral route. Novel delivery systems, involving implants, gels, and patches applied to the scleral surface, and intrascleral injections are being developed to enable extended-release and better-targeted drug delivery via the sclera. ²  

Motivated by these opportunities, a number of studies have examined rates of diffusion across the sclera as a function of molecular size and other parameters. ^{8 9 10 11 12} However, little attention has been given to diffusion within the sclera in the lateral direction parallel to the scleral surface, which could affect drug distribution caused by lateral spread of the released drug—for example, from an extraocular implant or intrascleral injection. The nonisotropic architecture of collagen lamellae and other features of scleral microanatomy ¹³ suggest that lateral diffusion may behave differently from transscleral diffusion. 

This study presents the first experimental measurements of lateral diffusion within the sclera, with sulforhodamine used as the model drug, and provides a theoretical model that predicts the diffusion profile as a function of both time and distance along the sclera. Lateral diffusivity is also compared to transverse diffusivity across the sclera to identify possible differences. 

Experimental and Theoretical Methods

Lateral Diffusion Measurements

The lateral diffusion profile of a model drug, sulforhodamine (558 Da; Invitrogen, Eugene, OR), was measured through human cadaveric sclera using spectrofluorimetry. Human sclera was obtained from the Georgia Eye Bank (Atlanta, GA) and stored in a moist container for 2 to 5 days at 4°C. Adherent tissues associated with the retina, choroid, and episclera were gently removed with cotton swabs. Strips of full-thickness sclera measuring 10 to 15 mm in length and 3 to 5 mm in width were cut from the globes using surgical scissors and razor blades. 

In this study, we used human cadaveric sclera, which has the advantage of being human tissue and enabling better control over the experimental system through the in vitro environment. Although an in vivo animal study would provide an improved physiological environment, this study focused on the process of diffusion within the sclera, which is governed largely by the nonliving collagen and extracellular matrix structures and does not address the effects of, for example, blood flow or other active processes. 

A glass vial was filled with 1 mL of donor solution containing 9.0 × 10⁻⁵ M sulforhodamine in physiological saline (Balanced Salt Solution [BSS]; Alcon Laboratories, Fort Worth, TX). A scleral strip was suspended vertically in the glass vial such that the lower 3 mm of the tissue dipped into the donor solution (Fig. 1) . The vial was then capped and sealed with parafilm to maintain scleral hydration and placed in a 37°C water bath, although some tissue dehydration probably still occurred. After a designated experimental period (4, 24, 48, 72, or 168 hours), the scleral strip was removed from the vial, rinsed with BSS, placed in a sample block containing freezing agent (OCT; Sakura Finetechnical, Tokyo, Japan), and snap frozen with liquid nitrogen. 

The scleral tissue was sectioned into 50-μm-thick pieces with a cryostat microtome (Richard Allan Scientific, Kalamazoo, MI). Every 15 consecutive pieces (750 μm) were collected as one sample. To account for the difference in scleral thickness, the first piece from each sample was examined by bright-field microscopy to determine its cross-sectional area, which was assumed to be representative of the entire sample. The remaining 14 pieces from each sample were placed into a test tube containing 2 mL saline solution and allowed to incubate in the dark at 4°C for 12 hours, to extract the sulforhodamine from the tissue sections. 

The solution in each tube was then placed into a cuvette (V _cuvette = 1.8 mL) to determine its sulforhodamine concentration by calibrated spectrofluorimetry (Photon Technology International, Lawrenceville, NJ) at an excitation wavelength of 565 nm and emission spectra collected at 580 to 620 nm. The sulforhodamine concentration within the tissue, C _tissue, was calculated using the following equation:  

\[C_{\mathrm{tissue}}{=}\frac{C_{\mathrm{cuvette}}\ {\cdot}\ V_{\mathrm{cuvette}}}{V_{\mathrm{tissue}}},\]

where C _cuvette is the sulforhodamine concentration in the cuvette, determined by spectrofluorimetry, and V _tissue is the volume of each tissue sample, which is equal to the measured cross-sectional area multiplied by the total thickness of 14 tissue pieces analyzed per section (i.e., 700 μm). In this way, drug concentration within the tissue was determined as a function of distance along the sclera at each time point. 

Transscleral Diffusion Measurements

Sulforhodamine diffusion was also measured across the sclera with a flow-through permeation chamber. According to the procedure described previously, ¹⁴ scleral discs 10 to 15 mm in diameter were excised from human globes and mounted in two-compartment perfusion chambers. A 300-μL depot of 9.0 × 10⁻⁵ M sulforhodamine donor solution was added to the episcleral surface while BSS was perfused across the choroidal side. Every 1 hour, a fraction containing 2 mL of the perfusate was collected over a 24-hour period and its fluorescence concentration was measured by spectrofluorimetry. From these measurements, the effective transverse diffusivity (D _trans) was calculated as  

\[D_{\mathrm{trans}}{=}\frac{C_{\mathrm{cuvette}}\ {\cdot}\ V_{\mathrm{cuvette}}\ {\cdot}\ d}{C_{\mathrm{donor}}\ {\cdot}\ A\ {\cdot}\ {\Delta}t}\ ,\]

where d is scleral thickness (0.6 mm), ⁷ A is the scleral surface area exposed to donor solution (0.37 cm²), Δt is sampling time (1 hour) and C _donor is the sulforhodamine concentration in the donor solution, which was initially at 9.0 × 10⁻⁵ M and decreased over time. This effect was accounted for by correcting C _donor in the calculation by using equation 2 . The transscleral permeability coefficient can be obtained by dividing the transverse diffusivity by scleral thickness. ¹⁵  

Sclera-to-Saline Distribution Coefficient

An additional experiment was performed to determine the sclera-to-saline distribution coefficient to examine possible binding between sulforhodamine and scleral tissue. A full-thickness scleral strip was submerged in a 9.0 × 10⁻⁵ M sulforhodamine donor solution for 24 hours. After the tissue was removed and rinsed with the saline solution, it was sectioned into 50-μm-thick pieces on a cryostat. All tissue pieces were collected and incubated in 20 mL of BSS. The sulforhodamine concentration in the solution was measured over time by spectrofluorimetry until it reached a constant value (after ∼1 hour). Assuming 100% sulforhodamine extraction efficiency, the sclera-to-saline distribution coefficient (K _D) was obtained as  

\[K_{\mathrm{D}}{=}\frac{C_{\mathrm{sclera}}}{C_{\mathrm{bath}}},\]

where C _sclera is the concentration in the sclera determined by extraction, and C _bath is the original concentration of the donor solution (9.0 × 10⁻⁵ M). 

Theoretical Model

A theoretical model was developed to predict the concentration of sulforhodamine as a function of both time and distance during lateral diffusion within human sclera. A one-dimensional model was justified by the geometry of the experimental setup, which was symmetric in both of the horizontal dimensions of the scleral strip and provided a concentration-gradient driving force for diffusion only in the vertical direction (Fig. 1) . We further recognized that the sulforhodamine could be present within the sclera in two forms: free sulforhodamine that can diffuse and bound sulforhodamine immobilized at binding sites in the scleral tissue. Previous studies have suggested that compounds structurally similar to sulforhodamine bind within sclera. ⁸  

Transient, one-dimensional diffusion with binding in a semi-infinite slab can be modeled mathematically as ¹⁶  

\[\frac{dC_{\mathrm{free}}}{dt}{=}D_{\mathrm{lat}}\ \frac{d^{2}C_{\mathrm{free}}}{dx^{2}}{-}k_{1}(C_{\mathrm{free}}{-}K_{\mathrm{eq}}C_{\mathrm{bound}})\]

and  

\[\frac{dC_{\mathrm{bound}}}{dt}{=}k_{1}(C_{\mathrm{free}}{-}K_{\mathrm{eq}}C_{\mathrm{bound}}),\]

where C _free is the free sulforhodamine concentration in the sclera, C _bound is the bound sulforhodamine concentration in the sclera, k ₁ is the binding rate constant, D _lat is the effective lateral diffusivity of free sulforhodamine in the sclera, x is the lateral position in the sclera, and K _eq is the ratio of free-to-bound sulforhodamine at equilibrium in the sclera,  

\[K_{\mathrm{eq}}{=}\frac{C_{\mathrm{free}}}{C_{\mathrm{bound}}}.\]

 

For the evaluation of bound versus free sulforhodamine in the sclera, the sclera-to-saline distribution coefficient (equation 3)can be re-expressed as  

\[K_{\mathrm{D}}{=}\frac{C_{\mathrm{sclera}}}{C_{\mathrm{bath}}}{=}\frac{C_{\mathrm{free}}{+}C_{\mathrm{bound}}}{C_{\mathrm{bath}}}.\]

Because the liquid portion of the sclera is composed of saline of similar composition to that of the surrounding bath, we can assume C _free is equal to C _bath. Thus,  

\[K_{\mathrm{D}}{=}1{+}\frac{C_{\mathrm{bound}}}{C_{\mathrm{free}}}{=}1{+}\frac{1}{K_{\mathrm{eq}}}.\]

Using equation 8 , as well as the assumption that sulforhodamine binding in the sclera is at equilibrium (C _free = K _eq C _bound), Equations 4 and 5can be rewritten as  

\[\frac{d\left(C_{\mathrm{free}}{+}\frac{C_{\mathrm{free}}}{K_{\mathrm{eq}}}\right)}{dt}{=}D_{\mathrm{lat}}\frac{d^{2}C_{\mathrm{free}}}{dx^{2}}.\]

Rearranging equation 9yields the following expression  

\[\frac{dC_{\mathrm{free}}}{dt}{=}D_{\mathrm{lat}}\frac{K_{\mathrm{eq}}}{1{+}K_{\mathrm{eq}}}\frac{d^{2}C_{\mathrm{free}}}{dx^{2}}.\]

In solving equation 10 , the initial and boundary conditions are  

\[C_{\mathrm{free}}(x,0){=}0,\]

 

\[C_{\mathrm{free}}(0,t){=}C_{\mathrm{donor}}\ {\cdot}\ K_{D},\mathrm{and}\]

 

\[C_{\mathrm{free}}(z,t){=}0,\]

where z is the length of the scleral strip above the donor solution. Solving equation 10subject to the conditions in equation 11yields the final expression for sulforhodamine concentration in the sclera as a function of time and position ¹⁶ :  

\[C_{\mathrm{free}}(x,t){=}C_{\mathrm{donor}}\ {\cdot}\ K_{D}\ {\cdot}\ \left(1{-}erf\ \frac{x}{\sqrt{4\ {\cdot}\ D_{\mathrm{lat}}\ {\cdot}\ t\ \frac{K_{\mathrm{eq}}}{1{+}K_{\mathrm{eq}}}}}\right),\]

where erf is the error function. 

Mean Absolute Percent Error

Mean absolute percent error (MAPE) was used for statistical analysis of the difference between theoretical predictions and experimental data. MAPE is calculated by averaging the percentage difference between predicted values and experimental data:  

\[\mathrm{MAPE}{=}{\Sigma}\left|\frac{C_{\mathrm{exp}}{-}C_{\mathrm{theor}}}{C_{\mathrm{exp}}}\right|\ {\cdot}\ \frac{1}{n}\ {\cdot}\ 100\%,\]

where C _exp is the experimentally measured concentration, C _theor is the theoretically predicted concentration, and n is the number of measurements. 

Results

In this study, we sought to image and quantify lateral diffusion within the sclera and to compare lateral and transverse diffusivity. A series of experiments were conducted to measure the sulforhodamine concentration profile within human cadaveric sclera at several time points between 4 hours and 1 week. A mathematical model of one-dimensional, semi-infinite diffusion was also developed to predict the diffusion profiles and compare with the experimental data. 

Imaging Lateral Diffusion within the Sclera

An initial experiment was performed to provide visual images showing the progression of sulforhodamine diffusion along the sclera as a function of time and position. Figure 2shows representative cross-sectional views of sclera frozen after 24 hours of sulforhodamine diffusion and then sectioned for viewing by fluorescence microscopy. Within each scleral slice, the sulforhodamine concentration appears uniform, which indicates that vertical diffusion occurred at the same rate, independent of position in the horizontal direction. This observation is consistent with modeling sulforhodamine diffusion as a one-dimensional process. Scleral sections collected further from the sulforhodamine donor solution show progressively lower sulforhodamine concentrations over the ∼1-cm scleral strip. 

Quantifying Lateral Diffusion within the Sclera

To quantify the lateral diffusion profile within the sclera, sulforhodamine concentration was measured in the scleral sections as a function of both time and distance along the sclera. Figure 3shows the resultant concentration profiles over a distance of 11 mm along the sclera at time points between 4 hours and 1 week (i.e., 168 hours). After 4 hours, sulforhodamine diffusion was detected at a distance up to 5 mm along the sclera. After 1 week, sulforhodamine diffused farther than 1 cm along the sclera. At each time point, concentration decreased with increasing distance (analysis of variance (ANOVA), P < 0.0001). Over time, the sulforhodamine concentration at each position increased with time (ANOVA, P < 0.0001). 

Determining Lateral Diffusivity

To determine the effective lateral scleral diffusivity of sulforhodamine from the data in Figure 3 , we used a theoretical model of one-dimensional diffusion (equation 12 ). As parameters for this model, we measured the sclera-to-saline distribution coefficient (K _D) experimentally to be 13.6, which indicates a strong binding between sulforhodamine molecules and the sclera tissue. The free-to-bound sulforhodamine ratio (K _eq) was then calculated with equation 8to be 0.08. This left sulforhodamine diffusivity (D _lat) as the only unknown variable. The diffusion model was then fitted to the experimental data, as shown in Figure 4 , which yielded an effective sulforhodamine diffusivity of D _lat = 3.82 × 10⁻⁶ cm²/s. 

Visually, the predicted curves in Figure 4capture the trend of the data, but show some disagreement. The quality of this fit can be gauged quantitatively by its MAPE of 39%, which indicated that predicted values were on average within 39% of experimental values. This uncertainty can be compared to the average standard error associated with experimental measurement (i.e., the average of error bars in Fig. 3 ), which was calculated to be 60%. Thus, the error associated with the experimental measurements is greater than the disagreement between the theoretical model and the experimental data, which means that the theoretical model predictions are as good as possible, given the uncertainty in the data. 

Further examination shows that at early times (e.g., 4 hours), the model generally underpredicted the data, whereas at later times, it generally overpredicted the data. This finding can be explained by a changing diffusivity, which was initially larger than the overall fitted value and later was smaller. Diffusivity may have changed over time because of changes in tissue hydration. Although the sclera was maintained in a humid environment, some tissue dehydration could have occurred over the course of the 1-week experiment. Decreasing tissue hydration could progressively decrease diffusivity in the sclera as the aqueous diffusion pathways decrease in number and size. In addition, decreased water content of the sclera could also decrease average tissue sulforhodamine concentrations by decreasing the aqueous regions containing sulforhodamine relative to the collagen, GAG, and other insoluble regions. 

Comparing Lateral and Transverse Diffusivities

To compare lateral and transverse diffusion in the sclera, we measured the rate of transverse diffusion of sulforhodamine across the sclera, which provided an effective diffusivity of D _trans = 1.28 ± 0.22 × 10⁻⁶ cm²/s, which corresponds to a permeability of 2.15 ± 0.37 × 10⁻⁵ cm/s. These values compare well with previously reported experimental data for scleral permeability of other molecules of similar molecular weight ¹⁷ and to a predicted diffusivity of 2.5 × 10⁻⁶ cm²/s for sulforhodamine generated using an independent theoretical model described previously (calculation not shown). ¹⁵  

Comparing the lateral (D _lat = 3.82 × 10⁻⁶ cm²/s) and transverse (D _trans = 1.28 × 10⁻⁶ cm²/s) diffusivity values generated in this study indicates that diffusing in the lateral direction occurs approximately three times faster than in the transverse direction. Although a rigorous analysis of statistical significance is difficult due to the way that lateral diffusivity was determined, this threefold difference may be insignificant due to experimental variability. If there is a significant difference, then the larger diffusivity of lateral diffusion might be explained by the lateral orientation of collagen fibers in the sclera. Diffusion parallel to these fibers might encounter less hindrance than diffusion across the fibers, which might be more torturous. However, the importance of the collagen fibers is not clear, since diffusion in the sclera is expected to be governed by the extracellular glycosaminoglycan matrix, ¹⁵ which is randomly oriented ¹³ and thus should not favor diffusion in any particular direction. 

Discussion

This study provides the first measurements of lateral diffusion in the human sclera. As new modalities for the treatment of age-related macular degeneration and other retinal diseases become available, drug diffusion across and within the sclera to target the posterior segment will become increasingly important. For example, after peribulbar or other periocular injection, drug may diffuse across the sclera to reach targets in the choroid or retina. Because retinal diseases are often disseminated, it may be desirable for drug to diffuse laterally to cover a larger area of retina beyond the site of injection. This may be especially important in the case of localized, slow-release drug delivery devices placed on the sclera, such as a scleral buckle or other implanted devices. 

Data from this study showed that after just 4 hours, measurable concentrations of sulforhodamine were present at a distance of 5 mm from the donor solution, but measurable concentrations at 10 mm required 3 days. This is consistent with a calculation using the value of lateral diffusivity determined in this study: At a distance of 1 cm, the characteristic diffusion time is 1 cm² × (3.82 × 10⁻⁶ cm²/s)⁻¹ = 2.6 × 10⁵ s = 3.0 days. Based on a similar calculation, it should take at least 6 weeks for sulforhodamine to diffuse from a localized source throughout all the sclera in a human eye (using a characteristic distance of 3.75 cm, which is half the circumference of the human eye ¹³ ). 

Although validated using only one compound, the model developed in this study for scleral diffusion is general. It should be valid for both hydrophilic and lipophilic compounds, as well as small drugs and macromolecules, given knowledge of their effective diffusivity, distribution coefficient, and binding constant in the sclera, as shown in equation 12 . Effective diffusivity should be strongly reduced by increases in molecular size, but only weakly affected by lipophilicity. Binding constant and distribution coefficient (which is strongly influenced by the binding constant) should be strongly influenced by molecular properties, such as lipophilicity. 

Measurements and calculations of lateral diffusion in this study have assumed that diffusion is one-dimensional and no drug exits the sclera along its choroidal or episcleral surfaces. If this were to occur—and it naturally would in vivo—then lateral spread of drug in the sclera would be even slower, due to “leakage” of drug into the choroid and periocular space. Moreover, because diffusion in the sclera is known to be a strong inverse function of molecular size, ⁸ lateral diffusion of proteins and other macromolecules is expected to be one or more orders of magnitude slower than that of sulforhodamine. 

This slow lateral distribution indicates that if delivery localized on the millimeter scale is desirable, then drug administration to a particular site in or on sclera will remain highly localized on a timescale of hours to days. Conversely, if lateral distribution of drug over a larger area with faster kinetics is required, then a less localized injection or implant that covers a larger area of scleral surface may be needed. Moreover, drug distribution by the vasculature in the choroid has not been considered in this analysis and may provide a means for additional drug distribution over larger areas. 

Conclusions

Lateral diffusion of sulforhodamine, a hydrophilic model drug, was studied in human cadaveric sclera. Measurable amounts of drug were detected at distances of 5 and 10 mm from the drug donor reservoir at 4 hours and 3 days, respectively. Experimental data were used to calculate an effective lateral diffusivity of 3.82 × 10⁻⁶ cm²/s. This calculation enabled the prediction that a point source of sulforhodamine would require 6 weeks to diffuse throughout all the sclera in a human eye. Comparison of experimental measurements of lateral diffusion within the sclera to transverse diffusion across the sclera indicated similar effective diffusivities, although lateral diffusion was approximately three times faster. A theoretical model for one-dimensional diffusion in the sclera was developed and shown to match experimental data with an MAPE of 39%. This model can be used to predict rates of lateral diffusion in the sclera for various drug delivery scenarios. Altogether, this study shows that lateral diffusion in the sclera is a slow process that localizes drug distribution on the millimeter scale for hours to days. Lateral diffusion over larger surface areas could occur over longer times—for example, during extended release drug delivery from an implant. 

 Supported in part by National Eye Institute Grants T32-EY07092 and P30-EY06360. JJ and MRP are members of the Institute for Bioengineering and Bioscience and the Center for Drug Design, Development and Delivery at the Georgia Institute of Technology.

 Submitted for publication November 15, 2005; revised February 20, 2006; accepted May 3, 2006.

 Disclosure: J. Jiang, None; D.H. Geroski, None; H.F. Edelhauser, None; M.R. Prausnitz, None

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Experimental apparatus for the measurement of lateral diffusion profiles within human cadaveric sclera. A scleral strip was suspended vertically in a glass vial with the lower end of the tissue submerged in a donor solution of sulforhodamine. At different time points, the tissue was removed, rinsed, snap-frozen, sectioned, and analyzed by calibrated spectrofluorimetry to determine sulforhodamine concentration in the sclera as a function of time and position.

Figure 2.

 

Representative cross-sectional views of human cadaveric sclera containing sulforhodamine imaged by fluorescence microscopy. One end of the sclera (A), which had been submerged in a sulforhodamine donor solution for 24 hours, contained a large concentration of the model drug. Progressively less sulforhodamine was present in scleral sections located farther away at distances of (B) 3.25, (C) 6.50, and (D) 9.75 mm from the donor solution.

Figure 3.

 

Lateral diffusion profiles of sulforhodamine in human cadaveric sclera as a function of time and position. At each time point, the spatial distribution of sulforhodamine is shown, where the bar on the left of each set corresponds to sclera bathed in the donor solution and each consecutive bar to the right corresponds to 750-μm increments positioned away from the donor solution. Average results with SE bars are shown for n = 3 replicates.

Figure 4.

 

Experimental measurements and theoretical predictions of sulforhodamine concentration in human cadaveric sclera as a function of time and position. Experimental data points show good agreement with theoretically predicted curves (equation 12)using experimentally determined values for K _D = 13.6 and K _eq = 0.08 and a fitted value for diffusivity, D = 3.82 × 10⁻⁶ cm²/s at 4 (▪), 24 (□), 48 (▴), 72 (▵), and 168 (♦) hours. The experimental data are the same as shown in Figure 3 .

The authors thank Jake Gilbert for helpful advice and the Georgia Eye Bank for providing scleral tissue.