Introduction of Ophthalmic Drug Delivery
Approximately 90% of all ophthalmic drug formulations are now applied as eye drops. While eye drops are convenient and well-accepted by patients, ∼95% of the drug contained in the drops is lost due to absorption through the conjunctiva or through the tear drainage. Ophthalmic drug delivery via contact lenses is more effective because it increases the residence time of the drug in the eye and leads to a larger fractional intake of drug by the cornea. We aim to model the drug release from the contact lens into the pre- and postlens tear films and the subsequent uptake by the cornea. The motion of the contact lens, which is driven by the eyelid motion during a blink, enhances the mass transfer in the postlens tear film (POLTF). We use regular perturbation methods to obtain the Taylor dispersion coefficient for mass transfer in the POLTF. The diffusion of drug in the gel is assumed to obey Fick’s law, and the diffusion in the gel and the mass transfer in the POLTF are combined to yield an integro-differential equation that is solved numerically by finite difference.
Fig.1 (a) The idealized geometry and (b) the real geometry utilized in the model for the PLTF-lens-POLTF system.
Drug Diffusion Model
We develop a model to predict the drug release from a presoaked contact lens into the postlens tear film and its subsequent uptake by the cornea. Figure 1 shows the real and the model geometry of the lens and the tear film. The postlens tear film (POLTF) is pictured as a flat, two-dimensional film bounded by an undeformable cornea and an undeformable but moving contact lens. The effect of gravity is negligible in the POLTF. Thus, for our purposes, the prelens tear film contact lens-POLTF-cornea system is a flat, horizontally oriented channel. 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
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. 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
It is well-known that diffusion of solutes through hydrogels exhibits complex mechanisms and is governed by an interplay of swelling of the gel, adsorption and desorption of the solute molecules on the gel, surface diffusion along the polymer that comprises the gel, and bulk diffusion through the free water. Incorporation of each of these effects into our model is feasible; however, it significantly increases the complexity of the problem. Thus, 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 by
where hf is the time-dependent POLTF thickness with a mean value h0 and y1/h0. The POLTF thickness hf is a function of time due to the motion of the contact lens during blinking. 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. The boundary conditions for Cg are
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 ensure continuity of flux and equilibrium at the lens-postlens tear film interface, respectively. 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 gets modified to
The boundary conditions for Cf are
The first boundary condition arises due to symmetry, the condition in the second assumes that the drug concentration in the tear meniscus is very small because of the large volume.
The initial conditions for the drug concentrations are
The above set of equations can be solved by finite-difference or finite-element methods to predict the drug flux to the cornea. However, because of the disparate length and time scales involved in the problem, it is possible to reduce the problem to a single integro-differential equation that can be solved numerically.
By using a perturbation expansion in the aspect ratio and by using a multiple time scale analysis, the transport problem in the film can be simplified to a dispersion equation of the form
Now, we solve separately the transport problem in the contact lens hydrogel matrix. The transport problem in the gel is
with the following boundary conditions,
The boundary conditions in eq1 assume equilibrium between the concentration in the contact lens and that in the tear fluid in the POLTF and in eq2 impose flux continuity, thus coupling the mass transfer problems in the POLTF and in the contact lens. The boundary condition in eq3 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.
For case I, which corresponds to zero flux to the PLTF, the expression for the drug flux to the POLTF, j, becomes
For case II, which corresponds to zero concentration in the PLTF, the expression for j becomes
Solution of the Integro-differential Equation
The integrodifferential equations were solved numerically by finite difference. As shown below, the convolution integrals were evaluated by converting those to ordinary differential equations and by further solving those equations numerically. This was done to improve the convergence of the series sum of the convolution integrals
For case I, let
and thus,
We can then write an equation for solving I n as follows:
The same procedure can be followed for case II. Let
As in case I, we can write an equation for solving In for case II as follows:
Protein modeling
We investigated the literature for salidroside transport proteins and found just one paper that speculated that salidroside might enter cells through human sodium dependent glucose transport proteins. Given that salidroside is synthesized in Saccharomyces cerevisiae by UDP-glucose and tyrosol via glycosyltransferase, we hypothesize that salidroside has a structure similar to glucose and can be transferred out of vesicles by glucose transporters. In order not to consume energy, we chose the Zymomonas mobilis glucose enhancing diffusion protein GLF - and used molecular docking to confirm the interaction of GLF and salidroside.
To analyze the binding affinities and modes of interaction between the salidroside and its target protein GLF, AutodockVina 1.2.2, a silico protein–ligand docking software was employed [1]. The molecular structures of salidroside was retrieved from PubChem Compound [2]. The 3D coordinates of GLF (Uniprot ID, P21906·GLF_ZYMMO) was downloaded from the Uniprot. For docking analysis, GLF and salidroside files were converted into PDBQT format with all water molecules excluded and polar hydrogen atoms were added. The grid box was centered to cover the domain of each protein and to accommodate free molecular movement. The grid box was set to 30 Å × 30 Å × 30 Å, and grid point distance was 0.05nm. Molecular docking studies were performed by Autodock Vina 1.2.2.
To evaluate the affinity of salidroside for GLF, we performed molecular docking analysis. The binding poses and interactions of salidroside with GLF were obtained with Autodock Vina v.1.2.2 and binding energy for interaction was generated. Results showed that salidroside bound to GLF through visible hydrogen bonds (Ser72, Ser75, Ser134, Thr135). For GLF, salidroside had low binding energy of -7.21 kcal/mol, indicating highly stable binding.
Fig.2 Binding mode of screened salidroside to GLF by molecular docking. (A) Binding mode of salidroside to GLF. (B) Three-dimensional structures of the binding pockets were showed by PyMOL software. Dashed lines represent hydrogen bonds.