Retina-Choroid-Sclera Permeability for Ophthalmic Drugs in the Vitreous to Blood Direction: Quantitative Assessment

Abstract

Purpose

To determine the outward permeability of retina-choroid-sclera (RCS) layer for different ophthalmic drugs and to develop correlations between drug physicochemical properties and RCS permeability.

Methods

A finite volume model was developed to simulate pharmacokinetics in the eye following drug administration by intravitreal injection. The RCS permeability was determined for 32 compounds by best fitting the drug concentration-time profile obtained by simulation with previously reported experimental data. Multiple linear regression was then used to develop correlations between best fit RCS permeability and drugs physicochemical properties.

Results

The RCS drug permeabilities had values that ranged over 3 × 10⁻⁶ m/s. Regression analysis for hydrophilic compounds showed that more than 92% of the variation in permeability values can be explained by correlative models of drug properties that include logarithm of the octanol-water partition coefficient (LogP), protein binding (PB), number of hydrogen bond acceptors (HBA), hydrogen bond donors (HBD), polar surface area (PSA) and dissociation constant (pKa) as independent variables. Regression analysis for lipophilic compounds showed that no significant correlation can be found between just physicochemical properties and RCS permeability.

Conclusion

Using the RCS permeability obtained from this study for different drugs, one can predict pharmacokinetics of intravitreal drug delivery systems such as solid implants or colloidal systems. Furthermore, the developed correlations between RCS permeability and physicochemical properties of drugs are useful in early drug development by predicting RCS permeability and drug concentration in the vitreous without experimental data.

Appendices

APPENDIX I

CFD Model Construction

Governing Equations

Bulk flow through the vitreous body occurs due to the pressure gradient from the anterior part of the eye towards the posterior pole. This low level convective flow does not play any significant role for the distribution of low-molecular weight drugs. However, high-molecular weight drugs move through the vitreous as a result of bulk flow.

In order to evaluate the convective–diffusive drug transport within the eye, the fluid velocity was obtained by solving for creeping flow in a porous medium using Darcy’s Law. According to Darcy’s law, in laminar flows through porous media, the pressure drop is proportional to velocity as:

$$ \overrightarrow v = - {\left( {\frac{{{K_h}}}{\mu }} \right)_{{vit}}}\nabla P $$

(a1)

The aqueous humor is incompressible, hence:

$$ \nabla .\overrightarrow v = 0\quad \Rightarrow \quad {\left( {\frac{{{{\text{K}}_{\text{h}}}}}{\mu }} \right)_{{vit}}}{\nabla^2}P = 0 $$

(a2)

where, $\overrightarrow v$ is the velocity of the fluid, K _h is the hydraulic conductivity of the vitreous gel, μ is the viscosity of the fluid, P is the pressure, and ρ is the density. The flow is assumed to be steady and independent of the drug concentration.

To obtain the drug distribution, the species mass conservation equation was coupled with the flow field as:

$$ \frac{{\partial C}}{{\partial t}} + \overrightarrow v .\nabla C - D{\nabla^2}C + q = 0 $$

(a3)

where, C is the concentration of the drug, D is the diffusion coefficient of drug in the vitreous, and q is the generation/consumption rate of the drug. It is assumed that the drug is not metabolized or degraded within the vitreous, so q was set to zero.

Boundary Conditions

There are three main tissues that bound the vitreous humor: the hyaloid membrane, lens, and RCS layer. Boundary conditions are summarized in Table VIII.

Table VIII Summary of Boundary Conditions

The hyaloid membrane, which separates the vitreous humor from the anterior segment of the eye, is a suspensory ligament connecting lens and ciliary processes and is composed of loosely packed noncollageneous protein. Following Balachandran and Barocas (29), a flux boundary condition has been considered to represent the loss of the drug to the anterior segment. Assuming that the aqueous humor in the posterior chamber is well mixed, the flux of the drug at the hyaloid surface is expressed as:

$$ n.\left( { - D\frac{{\partial C}}{{\partial n}} + \overrightarrow v C} \right) = \frac{f}{A} \times C = {{\text{k}}_{\text{hy}}} \times {\text{C}} $$

(a4)

Where f is the flow rate of aqueous humor [3 μl/min], A is the hyaloid membrane area (1.78 cm²), and C is the concentration at the hyaloid surface.

Pressure at the hyaloid membrane is considered to be the same as that of aqueous humor, which is close to the intraocular pressure (IOP) of the eye. For a healthy eye, IOP is generally between 15 and 20 mmHg (2000–2666 Pa), whereas for a glaucomatous eye it can rise up to 40–80 mmHg (5333–10666 Pa) (30). In this study, pressure on hyaloid membrane is considered to be 15 mmHg (2000 Pa) for the normal eye.

The lens is assumed to be impermeable to both flow and the drug concentration. Therefore, at the surface of the lens a no-flux boundary condition for concentration and no slip boundary condition for flow have been applied.

The vitreous cavity is surrounded by several layers which give it mechanical support. Liquid leaving the vitreous chamber passes first through the retina, then the pigment epithelium, then through a loose capillary bed (the choroid and suprachoroid), then through the sclera, and finally through the loose episeleral tissue covering the eye. For the purpose of modelling, we have lumped the different tissues around the vitreous into one entity (RCS layer).

The boundary condition for momentum equation can be expressed by using Darcy’s law as below:

$$ n.\overrightarrow v = n.\left[ {{{\left( { - \frac{{{K_h}}}{\mu }} \right)}_{{RCS}}}\nabla P} \right] = n.\left[ {{{\left( {\frac{{{K_h}}}{\mu }} \right)}_{{RCS}}}\frac{{P - {P_v}}}{L}} \right] = {K_p}\left( {P - {P_v}} \right) $$

(a5)

where K _h is the hydraulic conductivity of RCS layer, μ is the aqueous humor viscosity, P is the pressure on the RCS layer adjacent to the vitreous, P_v is the pressure of the episcleral tissue (27) that is 9 mmHg (1200 Pa) and L is RCS thickness.

For the concentration boundary condition, both convective and diffusive transport of the drug from the vitreous to the retina has been considered. Various transport processes in RCS membrane control the movement of drug out of the vitreous chamber. The influence of all transport mechanisms out of the vitreous is incorporated in an unknown parameter which is referred to as RCS permeability, K_RCS, as:

$$ n.\left( { - D\frac{{\partial C}}{{\partial n}} + \overrightarrow v C} \right) = {K_{{RCS}}} \times C $$

(a6)

Initial Condition

To model an intravitreal bolus injection of a specific value of drug, it is assumed that the drug is injected at the center of the vitreous chamber. This assumption is based on the method of injection used in all the previous pharmacokinetic experiments. The injected drug initially has a homogenous distribution within a spherical region. These assumptions are based on the injection method which was mentioned in related papers. The initial size of the spherical region is according to the volume of injected drug. The density of the drug solution is considered to be same as that of water i.e., 1000 kg/m³. The initial normalized mass fraction of the drug is assumed to be 1 (normalized with respect to the concentration of the drug in the water base) within the domain of the drug injection site while it is 0 in rest of the vitreous. In mathematical words:

$$ \matrix{ {{{\text{C}}_{{{\text{t}} = 0}}} = 1} \hfill &{{\text{in}}\;{\text{the}}\;{\text{sphereical}}\;{\text{drug}}\;{\text{source}}} \hfill \\ {{{\text{C}}_{{{\text{t}} = 0}}} = 0} \hfill &{{\text{outside}}\;{\text{of}}\;{\text{the}}\;{\text{drug}}\;{\text{source}}} \hfill \\ }<!end array> $$

Model Parameters

The parameters required to solve the model are intraocular pressure (IOP), viscosity of the aqueous humor, hydraulic conductivity of the vitreous and RCS layer, drug diffusivity in the vitreous and RCS permeability. Of these parameters, two parameters depend on drug physicochemical properties and need to be calculated. These two parameters are drug vitreous diffusivity (D) and RCS layer permeability (K_RCS). The Stokes-Einstein equation was used to estimate drug vitreous diffusivity;

$$ D = \frac{{kT}}{{6\pi r\eta }} $$

(a7)

where k is the Boltzman Constant (1.38 × 10⁻²³ J/K), T is the temperature, η is the viscosity of the solvent, and r is the solute molecule radius which was calculated by:

$$ r = {\left[ {\frac{{3\left( {MW} \right)}}{{4\pi N\rho }}} \right]^{{\frac{1}{3}}}} $$

(a8)

where MW is the solute molecular weight, N is the Avogadro’s number and ρ is the density. Substituting Equation a8 into Stokes-Einstein equation gives:

$$ D = \frac{{kT}}{{6\pi \eta {{\left[ {\frac{{3\left( {MW} \right)}}{{4\pi N\rho }}} \right]}^{{\frac{1}{3}}}}}} $$

(a9)

Therefore, drug diffusivity is inversely proportional to the cube-root of the molecular weight. The diffusivity of fluorescein in the vitreous humor has been found experimentally by Araie et al.(5) to be 6 × 10⁻¹⁰ m²/s. This value was used as base and vitrous diffusivity of other drugs was calculated as below:

$$ D \cong 6 \times {10^{{ - 10}}} \times {\left( {\frac{{M{W_{{fluorescein}}}}}{{MW}}} \right)^{{\frac{1}{3}}}} $$

(a10)

For estimation of K_RCS, an initial guess for K_RCS was made based on the mean residence time of the drug in the vitreous. Later on the optimum value of K_RCS was obtained by minimizing the objective function y as described before.

The hydraulic conductivity of vitreous (K_h/μ)_vit and RCS membrane (K_p) were considered to be 8.4 × 10⁻¹¹ m²/Pa.s and 5 × 10⁻¹² m/Pa.s, respectively (31).

Computer Code

CFD calculations were conducted using FLUENT software version 12.1.4 (ANSYS, Inc., Canonsburg, PA). This code is based on a control volume approach where the computational domain is divided to a number of cells and the governing equations are discretized into algebraic equations in each cell. These equations satisfy the integral conservation of the mass and the momentum over each control volume. Geometrical models and meshing were constructed using GAMBIT version 2.2.30.

Grid Independence Study

To check whether the size of grid is sufficient, grid size independence study has been done. A good grid size is one that has no influence on the results. When the grid is too fine numerical round off errors appear and when it is too course one could get truncation errors. Grid independence means that the converged solution obtained from a CFD calculation is independent of the grid density. As it is shown in Fig. 5 simulation with 604 cells and 2322 cells, creates less than 1.5% variation in the magnitude of velocity in the centreline of vitreous. This good agreement between the results from the two grid levels justifies the use of 604 cells for the simulation.

Fig. 5

Velocity magnitude along the center of vitreous chamber for 604 and 2322 cells.

Velocity and Pressure Contour

Fig. 6 Shows the predicted fluid velocity contour within the vitreous chamber. The average velocity across the RCS surface was 4.0 × 10⁻⁹ m/s, and the average velocity in the vitreous was 7.12 × 10⁻⁹ m/s, which are consistent with earlier analysis (31).

Fig. 6

Contour plot of velocity magnitudes.

Fig. 7 Shows the pressure distribution within the vitreous. The higher pressure specified at the hyaloid membrane (15 mmHg = 2000 Pa) drives the aqueous from the hyaloid to the outer sclera which is maintained at a venous pressure of 9 mmHg (1200 Pa). It is observed that the pressure drop across the vitreous is about 0.5 Pa (0.004 mmHg). Therefore, most of the pressure drop occurs in the RCS membrane.

Fig. 7

Contour plot of pressure within vitreous (Pa) (pressure values are mentioned relative to the pressure at hyaloid membrane surface which is 2000 Pa).

APPENDIX II