Computer Modeling of Drug Delivery to the Posterior Eye: Effect of Active Transport and Loss to Choroidal Blood Flow

Abstract

Purpose

The direct penetration route following transscleral drug administration presents several barrier and clearance mechanisms—including loss to choroidal blood flow, active transport by the retinal pigment epithelium (RPE), and loss to the conjunctival lymphatics and episcleral blood vessels. The objective of this research was to quantify the role of choroidal and episcleral losses.

Materials and Methods

A finite element model was created for drug distribution in the posterior human eye. The volumetric choroidal loss constant, active transport component and mass transfer from the scleral surface were unknown parameters in the model. The model was used to simulate drug distribution from a systemic source, and the results were compared to existing experimental results to obtain values for the parameters.

Results

The volumetric choroidal loss constant, mass transfer coefficient from the scleral surface and active transport component were evaluated to be (2.0 ± 0.6) × 10⁻⁵ s⁻¹, (2.0 ± 0.35) × 10⁻⁵ cm/s and 8.54 × 10⁻⁶ cm/s respectively.

Conclusion

Loss to the choroidal circulation was small compared to loss from the scleral surface. Active transport was predicted to induce periscleral movement of the drug, resulting in more rapid distribution and elevated drug concentrations in the choroid and sclera.

Appendix

Sensitivity Computation for Gauss–Newton Scheme

Squared curve-fitting problem amounts to minimizing the error between the simulations and experimental results,

$$E = \left( {c_{\exp } - \frac{1}{V}\int\limits_v {c{\text{ }}dv} } \right)^2 $$

(11)

where c _exp is the average concentration observed experimentally. The Gauss–Newton scheme finds the minimum iteratively, requiring partial derivatives of the objective function E, with respect to the parameters, e.g.,

$$\frac{\partial }{{\partial \gamma }}\left( {c_{\exp } - \frac{1}{V}\int\limits_v {c{\text{ }}dv} } \right)^2 = - 2\left( {c_{\exp } - \frac{1}{V}\int\limits_v {c{\text{ }}dv} } \right)\left( {\frac{1}{V}\int\limits_v {\frac{{\partial {\text{c}}}}{{\partial \gamma }}{\text{ }}dv} } \right)$$

(12)

The sensitivities s _γ and s _ksc are defined as

$$s_\gamma \equiv \frac{{\partial c}}{{\partial \gamma }}{\text{ and }}s_{{\text{ksc}}} \equiv \frac{{\partial c}}{{\partial k_{{\text{sc}}} }}$$

(13)

The mass balance Eqs. 3, 4 and 5 along with boundary conditions 7, 8 and 9 are differentiated with respect to γ and k _sc to obtain the following relationships which would be solved for s _γ and s _ksc respectively.

Equations and Boundary Conditions for s _γ

• 1. Domain equations

  $${\text{Vitreous: }}\frac{{\partial s_\gamma }}{{\partial t}} + v \cdot \nabla s_\gamma - D_v \nabla ^2 s_\gamma = 0$$

  (14)
  $${\text{RPE: }}\frac{{\partial s_\gamma }}{{\partial t}} + \left( {v + k_{{\text{act}}} } \right) \cdot \nabla s_\gamma - D_{{\text{rpe}}} \nabla ^2 s_\gamma = - \frac{{\partial k_{{\text{act}}} }}{{\partial \gamma }} \cdot \nabla c$$

  (15)
  $${\text{CS: }}\frac{{\partial s_\gamma }}{{\partial t}} + v \cdot \nabla s_\gamma - D_{{\text{cs}}} \nabla ^2 s_\gamma + \gamma s_\gamma = \left( {c_{{\text{bl}}} - c} \right)$$

  (16)

• 2. Boundary conditions

  $${\text{Lens: }}n \cdot \left( { - D\nabla s_\gamma + vs_\gamma } \right) = 0$$

  (17)
  $${\text{Hyaloid: }}n \cdot \left( { - D\nabla s_\gamma + vs_\gamma } \right) = k_{{\text{hy}}} s_\gamma $$

  (18)
  $${\text{Sclera: }}n \cdot \left( { - D\nabla s_\gamma + vs_\gamma } \right) = k_{{\text{sc}}} s_\gamma $$

  (19)

where $\left\| {\frac{{\partial k_{{\text{act}}} }}{{\partial \gamma }}} \right\| = \frac{{{\text{Pe}}^* L_{{\text{rpe}}} }}{{\left( {\frac{{L_{{\text{rpe}}}^2 }}{{D_{{\text{rpe}}} }} + \frac{{L_{{\text{cs}}}^2 }}{{D_{{\text{cs}}} }} + \frac{1}{\gamma }} \right)^2 }}\left( {\frac{1}{{\gamma ^2 }}} \right)$ from Eq. 10, and the vector points in the same direction as k_act.

Equations and Boundary Conditions for s _ksc

• 1. Domain equations

  $${\text{Vitreous: }}\frac{{\partial s_{{\text{ksc}}} }}{{\partial t}} + v \cdot \nabla s_{{\text{ksc}}} - D_v \nabla ^2 s_{{\text{ksc}}} = 0$$

  (20)
  $${\text{RPE: }}\frac{{\partial s_{{\text{ksc}}} }}{{\partial t}} + \left( {v + k_{{\text{act}}} } \right) \cdot \nabla s_{{\text{ksc}}} - D_{{\text{rpe}}} \nabla ^2 s_{{\text{ksc}}} = 0$$

  (21)
  $${\text{CS: }}\frac{{\partial s_{ksc} }}{{\partial t}} + \left( {v + k_{{\text{act}}} } \right) \cdot \nabla s_{{\text{ksc}}} - D_{{\text{rpe}}} \nabla ^2 s_{{\text{ksc}}} = 0$$

  (22)

• 2. Boundary conditions

  $${\text{Lens: }}n \cdot \left( { - D\nabla s_{{\text{ksc}}} + vs_{{\text{ksc}}} } \right) = 0$$

  (23)
  $${\text{Hyaloid: }}n \cdot \left( { - D\nabla s_{{\text{ksc}}} + vs_{{\text{ksc}}} } \right) = k_{{\text{hy}}} s_{{\text{ksc}}} $$

  (24)
  $${\text{Sclera: }}n \cdot \left( { - D\nabla s_{{\text{ksc}}} + vs_{{\text{ksc}}} } \right) = k_{{\text{sc}}} s_{{\text{ksc}}} + c$$

  (25)

The average values of s _γ and s _ksc over the vitreous volume were calculated and used for the Gauss–Newton curve fitting algorithm.