Computational Modeling of Drug Transport Across the In Vitro Cornea

Abstract

A novel quasi-3D (Q3D) modeling approach was developed to model networks of one dimensional structures like tubes and vessels common in human anatomy such as vascular and lymphatic systems, neural networks, and respiratory airways. Instead of a branching network of the same tissue type, this approach was extended to model an interconnected stack of different corneal tissue layers with membrane junction conditions assigned between the tissues. The multi-laminate structure of the cornea presents a unique barrier design and opportunity for investigation using Q3D modeling. A Q3D model of an in vitro rabbit cornea was created to simulate the drug transport across the cornea, accounting for transcellular and paracellular pathways of passive and convective drug transport as well as physicochemistry of lipophilic partitioning and protein binding. Lipophilic Rhodamine B and hydrophilic fluorescein were used as drug analogs. The model predictions for both hydrophilic and lipophilic tracers were able to match the experimental measurements along with the sharp discontinuities at the epithelium-stroma and stroma-endothelium interfaces. This new modeling approach was successfully applied towards pharmacokinetic modeling for use in topical ophthalmic drug design.

1. Introduction

The eye has many physical and biological barriers that limit a drug’s ophthalmic bioavailability, making targeted ophthalmic drug delivery design a major challenge. Topical instillation is the most convenient approach to treat common ophthalmic diseases, but it is also the most inefficient route as the majority of drug applied is lost via the nasolacrimal duct along with the tears facilitated by blinking or are absorbed into the conjunctiva where they are lost into the circulatory system. The small percentage of drug that manages to permeate into the eye has to pass through the multi-laminate structure of the cornea consisting of the epithelium, stroma and endothelium, with each having different barrier properties with respect to different drug chemistry [1–3]. As such, a better understanding of transcorneal permeation mechanisms would help in predicting drug ocular disposition for the development of improved drug delivery vehicles and dosage forms, enhancing bioavailability while minimizing side effects [4–8].

Conventional development and evaluation of drugs require in vitro and in vivo animal studies which can be expensive, time consuming and even ethically questionable. A more efficient approach is with computational modeling whereby drugs can be virtually designed and tested in large permutations. Computational Biology (CoBi) tools are a robust and adaptable computational framework which is developed to solve multiscale, multiphysics systems and allows for coupling of models of various granularity including compartmental, 1D, 2D, 3D and quasi-3D (Q3D) models [9–11]. In this paper, a CoBi Q3D model of an in vitro rabbit cornea was created to simulate transcorneal drug transport, taking into account physicochemical processes involved in drug delivery, distribution, binding and clearance. Isolated rabbit corneas are often mounted on Ussing chambers to evaluate the flux of drugs and the effects of formulations on the drug transport across the cornea [12–18]. The multi-laminate structure of the cornea presents a unique barrier design and opportunity for investigation using Q3D modeling.

Multiple attempts have been made to simulate the transport of drugs in the eye [19–23]. Traditional compartmental models used in PK typically employ ordinary differential equations (ODEs) with first order rate constants to model the mass transport between compartmentalized tissue layers. The rate constants are empirically derived from in vitro measurements (i.e. concentration vs time profiles) [24]. These models, however, cannot capture the spatial resolution across tissue layers. Reduced-order physiologically-based pharmacokinetic (PBPK) models, like the Q3D modeling approach, are improvements on classical compartmental models by taking into account physiological and drug physicochemical values (i.e. spatial variation in tissue geometry and drug diffusivity). Q3D modeling uses interconnecting cylindrical representative units to emulate the geometric profile of the tissue and simultaneously solves conservation equations for flow and species transport and other physical fields, e.g. temperature. The most geometrically accurate approach is with high-fidelity computational fluid dynamics (CFD) modeling defined with meaningful boundary conditions and tissue properties (e.g. hydraulic permeability) [25]. However, these models are computationally intensive, requiring longer time to solve, difficult for parametric simulations, and other complexities. To demonstrate the robustness of Q3D modeling in ophthalmic pharmacokinetics, two tracers, one lipophilic (Rhodamine B) and another hydrophilic (fluorescein), were used as drug analogs in the study of transcorneal permeation. The predicted concentrations permeated across the cornea were compared to experimentally measured results [22,26].

2. Quasi-3D Modeling Approach

CoBi tools provide a unique approach to model networks of 1D structures like tubes and vessels common in human anatomy such as vascular and lymphatic systems, neural networks and respiratory airways [9]. Traditional methods of modeling these structural networks employ either 1D modeling of the 3D structural network, which introduces inaccuracies due to its geometric oversimplification, or suffer from higher computation cost for accuracy in high-fidelity 3D computational simulations. The Q3D modeling approach in CoBi breaks down the structural network into a series of connecting 3D arbitrary cylinders, or Q3D segments, whose radii and height can be independently varied for a more accurate representation of the 3D geometry of interest. Figure 1 illustrates a simple example of a Q3D model where cylindrical Q3D segments represent different cell types of varying cross-sectional areas and lengths. One Q3D segment can be connected to one or more segments (e.g. cell type A to B and C) or merge from two or more segments into one (cell types B and C to D). Fluid and species transport are solved along each segment height and are exchanged between different segments at the boundary ends.

Figure 1.

An example of the CoBi Q3D meshing framework where each Q3D segment, representing different cell types in the form of a cylinder of arbitrary radius and height, can be branched off or merged together. Solved fluid and species fluxes along each segment are exchanged from the Q3D cylinder end boundary conditions (shown with a black circle in the center). Arrows indicate the direction of exchange from top to bottom.

CoBi solves for the complex multiphysics mass (continuity), momentum, energy, species and structural biomechanics conservation equations in 1D/2D/3D and Q3D topologies with heterogeneous properties, e.g. porous media or interfacial discontinuities. The transport equations account for convection, diffusion, fluid-structures interaction, electrostatic drift, interfacial friction and other effects. For this ocular drug delivery application, the governing flow and species conservation equations were derived from the generalized form of Navier-Stokes Equations and the convection-diffusion equation (Eqns (1), (2), (3)).

$$\text{Continuity}:\frac{\partial \rho }{\partial t}+\nabla (\rho \overrightarrow{\nu })=0$$ (1)

$$\text{Momentum}:\rho (\frac{\partial \overrightarrow{v}}{\partial t}+\overrightarrow{v}·\nabla v)=\nabla P+\mu {\nabla }^2\overrightarrow{v}+\overrightarrow{F}$$ (2)

$$\text{Species conservation}:\frac{\partial C}{\partial t}=\nabla ·(D\nabla C+\overrightarrow{v}C)+S$$ (3)

where P is the pressure, t is time, ρ is the fluid density, v⃗ is the bulk fluid velocity, μ is the fluid viscosity, F⃗ is the additional body force per unit mass, C is species concentration, D is the species diffusivity, and S is the source term. ρ and μ of porous ocular tissues were predefined, along with P assigned based on atmospheric (at tear film) and intraocular pressures (at aqueous chamber) and D of different solutes in each ocular tissue. v⃗ is solved as well as C according to the species source, S. CoBi also has built-in modules to assign hydrodynamics (e.g. pressure, volumetric flux, and porous medium) and diffusion (e.g. partition coefficients, permeability, and diffusivity) properties.

The Q3D approach has the advantages of ease of model setup, fast computational speed, simple visualization of results, and easy linking to compact models such as spring/mass/damper devices, valves, pumps, controllers, and compartmental reaction models. Previous CoBi Q3D model of transport in human airways showed three to four orders of magnitude faster computation time than a high-fidelity CFD model of the same airway, with minor loss of accuracy [9]. The CoBi solver and the Q3D modeling capability have been designed for computational physiology and pharmacology applications such as cardiovascular hemodynamics, respiratory physiology, neuronal electrophysiology, tissue biomechanics as well as in vitro and in vivo drug delivery, pharmacokinetics/pharmacodynamics and in vitro to in vivo extrapolation [27].

In this study, CoBi Q3D modeling approach was applied towards simulating drug transport across the cornea. Traditional compartments representing the different corneal tissue layers are broken down into Q3D segments with membrane junction conditions assigned at interfaces between various Q3D segments (Figure 2). The spatial components of the tissues, not captured in traditional compartmental modeling, are accounted for in Q3D modeling, thereby adding more realism to the model. The membrane junction condition is assigned with the following,

$$J=-D\frac{\partial C}{\partial x}$$ (4)

$$K_e=\frac{K_R}{K_L}$$ (5)

where J is the species flux, x is the diffusional distance, K_e is the equivalent partition coefficient at the interface, and K_L and K_R are the oil-water partition coefficients of the left and right compartments, respectively. If there is convective flux v⃗, the term v⃗C is added to the right of Eqn (4).

Figure 2.

CoBi Q3D modeling representation of two-compartment model with a membrane interface between compartments. C_L and C_R are the species concentrations in the left and right compartments, respectively.

The CoBi Q3D model solutions for the membrane junction have been validated against analytical solutions for different case scenarios (Figure 3) and for multiple membranes (Figure 4). For each CoBi Q3D model, boundary conditions were set such that the species concentration at the left domain boundary is set to 1 and to 0 at the end of the right boundary. The model in Figure 3a has a lower K_e of 0.1, which resulted in less species partitioning into the Q3D on the right. For the model in Figure 3b, which has K_e at a much higher value of 10, the partitioning causes the species concentration to be much higher on the right. The model in Figure 4 has two membranes and the left two Q3Ds have essentially a K_e of 10 between them, while the right 2 Q3Ds have K_e of 0.1. These differences in K_e have the similar behavior as seen in Figure 3, where the higher K_e results in higher permeation while the opposite is true for lower K_e. Note that in this analysis, D and P_s are kept constant.

Figure 3.

CoBi Q3D model steady-state solutions to 2 different test cases with one membrane (at X = 1.0) are in perfect agreement with analytical solution. C is the species concentration and X is the distance along the Q3D segment.

Figure 4.

CoBi Q3D model steady-state solution test case with two membranes (at X = 1.0 and 2.0) is in perfect agreement with standard finite element method (FEM) solution. C is the species concentration and X is the distance along the Q3D segment. K_c is the partition coefficient of the center Q3D segment.

In the following section, CoBi Q3D modeling is demonstrated in the study of transcorneal permeation of two sample solutes, one hydrophilic and the other lipophilic.

3. Modeling Drug Transport Across In Vitro Cornea

The cornea is an optically transparent tissue that allows transmission and refraction of light from the environment into the eye. It is a multilayered membrane comprised of epithelium, stroma and endothelium layers [28]. The layers of the cornea serve as principal barriers to topical drug delivery in the eye. Drug transport through the cornea involves molecules traveling via the transcellular (through cells) and/or paracellular (around cells) routes, and is influenced by the area available for absorption, thickness, porosity and tortuosity of each corneal layer as well as the drug lipophilicity [1].

The multicellular epithelium is made up of non-keratinized squamous epithelial cells [29] and is characterized as being very lipophilic, causing low permeation for hydrophilic drugs. The epithelium’s low porosity and high tortuosity, due to the many tight junctions between cells, minimizes paracellular transport, resulting in transcellular transport to dominate [22]. The corneal stroma is a largely acellular region representing approximately 90% of the corneal thickness [1]. The stroma is naturally hydrophilic due to its composition of mainly water, collagen, glycosaminoglycan and proteins. The hydrophilic stroma is a significant barrier to lipophilic drugs [22]. The corneal endothelium is the thinnest layer of the cornea, and is a relatively “leaky” basement layer due to its high porosity. Compared with the epithelium, the endothelium has a higher hydraulic conductivity [30] facilitating increased paracellular transport [31].

The physical dimensions of the corneal tissues, including the transcellular and paracellular pathways for passive and convective drug transport (Table 1) are included in the CoBi Q3D rabbit cornea in vitro model (Figure 5). The cornea geometric model displayed uses a scaled height for easier visualization and differentiation of the corneal tissue layers. Additional Q3Ds are appended to the front and back of the cornea to represent the tear film and aqueous humor, respectively.

Table 1.

Geometric parameters to CoBi Q3D rabbit eye model.

Parameter	Value	Units
Corneal transcellular surface area	1.537 [26]	cm2
Corneal epithelium thickness	38.5 [26]	µm
Corneal stroma thickness	372.5 [26]	µm
Corneal endothelium thickness	5.0 [26]	µm
Corneal paracellular surface area	0.0028*	cm2

^*Calculated based on 0.18% of transcellular membrane surface area [32].

Figure 5.

CoBi Q3D model of in vitro rabbit cornea (10 times scaled height vs. radius for easier viewing) with transcellular and paracellular compartments for corneal epithelium and endothelium, corneal stroma interfacing with both paracellular and transcellular routes, and the baths (i.e. apical and basolateral reservoirs).

3.1. Test Case 1: Transcorneal transport of hydrophilic solute, Fluorescein

Protein Binding

Drug binding (such as binding of drug to protein or sugars) can cause significant reduction in the fraction of free drug transported, requiring larger doses of the drug in order to achieve the desired physiological effect [33]. In the corneal stroma, Glycosaminoglycan (GAG) bound collagen fibers can bind to hydrophilic molecules as they move through due to the charge based attraction of polar functional groups between the molecule and GAG [22].

Protein binding occurs based on the affinity of the drug for the protein as determined by the physicochemical properties of the drug. At any given time, the total concentration of a drug in a region,C_total, can be determined as

$$C_{\mathit{\text{total}}}=C_{\mathit{\text{free}}}+C_{\mathit{\text{bound}}}$$ (6)

where C_free is the total concentration of the unbound drug molecule, and C_bound is the total concentration of the protein bound drug. C_bound is determined using a Michaelis-Menten kinetics equation with constants for maximum binding capacity, B_max, and the equilibrium dissociation constant, K_D (Eqn (7)) [34].

$$C_{\mathit{\text{bound}}}=\frac{B_{\mathit{\text{max}}}{C}_{\mathit{\text{free}}}}{K_D+C_{\mathit{\text{free}}}}$$ (7)

The bound drug concentration is then subtracted from the total drug concentration to give the free/unbound drug concentration. Drug binding is an important aspect of transport analysis and needs to be accounted for.

Fluorescein Transport Across the Rabbit Cornea

The CoBi Q3D model of the rabbit cornea model developed was used to simulate the transcorneal transport of a tracer dye, fluorescein (molecular weight of 376 Da and log K ranging from −2.4 to −1.57 [35,36]) and predictions were validated against experimental measurements from Gupta et al. [22] (Figure 8). In that experiment, rabbit corneas were extracted and mounted on a custom-built scanning microfluorometer (Figure 6a), which is used to measure the transcorneal penetration of the tracer across the rabbit cornea height. The corneas were continuously perfused with Ringers solutions on the endothelium side at a rate similar to aqueous humor clearance rate of 2 to 3 µL/min and held at a constant pressure of 20 mmHg to mimic intraocular pressure. Fluorescein was supplied on the endothelial side instead of the epithelial side due to the extremely low corneal epithelium permeability of fluorescein, which would make any fluorescein concentrations within the stroma or endothelium, if any, too low to accurately detect [22]. Fluorescein was constantly infused from the endothelial side to maintain a constant concentration and fluorescence measurements across the cornea were recorded over a period of 4 hours. However, after 200 minutes, the swelling of the cornea significantly skewed the depth-dependent measurements. As such, only measurements taken under 200 minutes were compared to our simulations. The CoBi Q3D model predicted fluorescein concentrations across the cornea are directly correlated to fluorescence measured from the experiment based on the assumption that the measured fluorescence is directly associated to the concentration [22]. The tear film and aqueous humor baths had the same volume with heights of 150 µm and cross-sectional areas of 1.53 cm².

Figure 8.

Spatial fluorescence of free fluorescein across the rabbit cornea predicted from CoBi Q3D model matched that obtained experimentally [22]. The corneal endothelium is around the 870 µm mark while the epithelium is near 550 µm. Tissue fluorescence displayed as arbitrary units (AU).

Figure 6.

Schematic of a custom confocal scanning microfluorometer used to experimentally measure the transcorneal penetration of tracer across the rabbit cornea mounted in vitro [22,26]. (a) In one experiment, a tracer, fluorescein was perfused with Ringers solution at the aqueous humor side [22]. (b) A different tracer, Rhodamine B (RhB), was held at a constant concentration by periodic replacement of the dye on the tear side [26]. Constant Perfusion was used to keep the cornea inflated under its normal pressure in both experiments.

The transport parameters used in the simulation are given in Table 2. The tissue-water distribution coefficient is represented by Ψ, which is the ratio of concentrations between lipid bilayers and water, and is more fitting than the partition coefficient when studying the transport of a drug through a tissue [37]. The hydraulic permeability values, P_f, in the model were calibrated based on Darcy’s law such that the fluorescein predicted matches the experimental data and the stroma velocity computed is close to literature value of about 1×10⁻⁸ m/s [22]. Since paracellular and transcellular pathways have been separated in the epithelium and endothelium, fluid flow in the transcellular route is very low. Therefore, P_f in the transcellular pathways of both corneal epithelium and endothelium were set to low values of 1×10⁻³⁰ m²,to simulate impermeability. The predicted fluorescein fluorescence across the cornea (Figure 7) was observed to be much more sensitive to P_fin the paracellular path at the epithelium than the endothelium since the former’s tight junctions limit fluid flow as it moves along the pressure gradient from the aqueous humor into the endothelium. Furthermore, the “leaky” endothelium, as expected, has a much higher permeability than at the epithelium (5×10⁻¹⁰ m² vs. 1.3×10⁻¹⁵ m²). The P_f of the stroma given in the experiment was 2.4×10⁻¹⁸ m², which is similar to the 4.0×10⁻¹⁸m² calibrated. The calibrated P_s of epithelium is also very close to the value optimized by Gupta et al. [22] at 9.2×10⁻¹⁰ m/s and within the range from other studies at 4.3×10⁻⁹ m/s [14] and 0.5×10⁻⁹ m/s [38]. At the endothelium, calibrated P_s is 18 times higher than at the epithelium, which is expected due to the “leaky” nature of the former. It is also within the range measured from another study at 8.55×10⁻⁸ m/s [14]. The calibrated D in the stroma is close to 7.76×10⁻¹² m²/s, the optimized value obtained by Gupta et al. [22].

Table 2.

Parameters used in the simulation of the transport of fluorescein through the cornea [22,32].

	Diffusivity, D (m2/s) [22] (Calibrated)	Hydraulic permeability, Pf (m2) (Calibrated)	Permeability, Ps (m/s) (Calibrated)	Tissue-water distribution coefficient, ψ [32]
Epithelium (transcellular)	3.68 × 10−14	1.0 × 10−30	6.34 × 10−10	0.79
Epithelium (paracellular)	4.25 × 10−10	1.3 × 10−15	-	-
Stroma	12.00 × 10−12	4.0 × 10−18	1.12 × 10−6	1.79
Endothelium (transcellular)	1.48 × 10−15	1.0 × 10−30	1.15 × 10−8	0.81
Endothelium (paracellular)	4.25 × 10−10	5.0 × 10−13	-	-

Figure 7.

Q3D Snapshots (10 times scaled heights) displaying the fluorescein concentration at 41 min (a), 81 min (b), 121 min (c) and 162 min (d). Fluorescein is continually fed along with the perfusion media in the basolateral (aqueous humor) bath. Fluorescein’s hydrophilic nature causes slower perfusion through the cells, but quickly travels through the paracellular route.

Protein binding plays a role in the transport of fluorescein, especially in the stroma where serum proteins and GAG binds to hydrophilic solutes. In the experiment, the excitation wavelength was 485 ± 10 nm and the emission wavelength recorded was 530 ± 10 nm, which is emitted by free fluorescein. Protein bound fluorescein, on the other hand, fluoresces at a different wavelength between 465 and 490 nm [39]. This is because a bound molecule will emit the same wavelength which it absorbs due to rotational restriction, while an unbound molecule, free to rotate, will emit a different wavelength than it absorbs [40]. The CoBi Q3D model results accounted for both total fluorescein concentrations in all layers of the cornea and bound fluorescein concentration in the stroma (Figure 7 and Figure 8). B_max of 0.1 and K_D of 0.005 were determined by calibrating the predicted fluorescein concentration to experimental data. Adjusting B_max changes the magnitude of the fluorescein concentration, while K_D influences the saturation concentration for fluorescein binding. The bound fluorescein concentration was subtracted from the total concentration to give the free (unbound) concentration of fluorescein, which was then compared to the experimental results. The predicted and experimentally determined spatial fluorescence of free fluorescein are compared in Figure 8. There is a good agreement between the predicted and measured fluorescein fluorescence. The concentration of hydrophilic fluorescein is lower in the hydrophobic regions (epithelium and endothelium) and higher in the hydrophilic regions (stroma).

3.2. Test Case 2: Transcorneal transport of lipophilic solute, Rhodamine B

Transport into Lipid Bilayer

The epithelium and endothelium cells contain lipid bilayer membranes, which separate the cytoplasm of the cell from the environment [41]. These cell membranes contain phospholipid bilayers made up of amphiphilic phospholipids and serves as a primary barrier to hydrophilic solutes. A drug with partition coefficient, K, of greater 1 will partition into the membrane more easily than one with K less than 1. Hydrophilic compounds generally show minimal partitioning and poor membrane permeability [42]. Solute accumulation within the phospholipid bilayers and hydrophobic domains is a factor that can play a significant role in drug transport studies [26] and depends on the log K of the solute molecule [43]. This occurs when a lipophilic drug remains inside the hydrophobic regions of a cell, such as inside organelle membranes.

CoBi Q3D modeling accounts for the accumulation of solute molecules in the lipids by solving for the concentration of solutes in the phospholipid bilayers, C^B. The flux of the solute in the cell membrane, J (Eqn (8)), and its intracellular lipophilic compartments, J^B (Eqn (9)), can be written as [26],

$$J=-D\frac{\partial C}{\partial x}-J^B$$ (8)

$$J^B=k^B(C-\frac{C^B}{R^B})$$ (9)

The variable k^B (s⁻¹) is the first-order rate constant describing the permeability of the cytoplasmic region separating the cell membrane bilayer from the internal hydrophobic domains. R^B is the ratio of the equilibrium concentration in the cell membrane bilayer to the concentration in the internal hydrophobic domains. These 2 terms, along with D, are solute specific.

Rhodamine B Transport Across the Rabbit Cornea

The CoBi Q3D model of the cornea developed includes both paracellular and transcellular pathways for any solute to travel through. The hydraulic permeabilities assigned to each pathway in both membranes were calibrated to experimental measurements of fluorescein conducted by Gupta et al. [22]. In this experiment, the epithelium sides of the corneas were bathed in Ringers solution, opened to the atmosphere, and dosed with 5.5 mol/m³ of well-mixed Rhodamine B (RhB, molecular weight of 479 Da, log K = 2.43) (Figure 6b). RhB’s high lipophilicity and the cornea’s opposing fluid flow along the pressure gradient from endothelium to epithelium in the small surface area (consisting of only 0.18% of cell membrane surface area [32]) paracellular pathway contribute to the transcellular dominance of RhB transport across the cornea. The same transport parameters used in the experiment were also used in the simulation (Table 3). Additionally, for the transport into the lipid bilayers, within the epithelium, k^B = 4 × 10⁻⁴ s⁻¹ and R^B = 1.6, while in the endothelium, k^B = 3×10⁻⁴ s⁻¹ and R^B = 1.2 [26].

Table 3.

Parameters used in the experiment [26] and simulation of the transport of RhB across the cornea. Hydraulic permeability values were previously calibrated (Table 2).

	Diffusivity, D (m2/s) [26]	Hydraulic permeability, Pf (m2) (Calibrated)	Permeability, Ps (m/s)	Tissue-water distribution coefficient, ψ [32]
Epithelium (transcellular)	7.9 × 10−12	1.0 × 10−30	1.0 × 10−3	9.8
Epithelium (paracellular)	4.5 × 10−12	1.3 × 10−15	-	-
Stroma	22.8 × 10−12	4.0 × 10−18	6.0 × 10−8 [26]	10.6
Endothelium (transcellular)	1.5 × 10−12	1.0 × 10−30	-	2.8
Endothelium (paracellular)	4.5 × 10−12	5.0 × 10−10	1.0 × 10−5	-

The CoBi Q3D model predicted RhB spatial concentration across the cornea, which is directly correlated to fluorescence measured experimentally (Figure 9). Since the RhB measured makes no distinction whether it is from intracellular lipids or cytosol, the RhB concentrations predicted in the tissue layer’s (epithelium and endothelium) cytosol and their associated intracellular lipids (C_B) were summed to give the total RhB in each volume. The predicted and experimentally measured spatial fluorescence of RhB across the rabbit cornea are compared in Figure 10. The multi-layer oil:water:oil structure of the cornea is highlighted by the fluorescence profile across the cornea, with distinct discontinuities at the cellular boundaries between epithelium and stroma as well as between stroma and endothelium [26]. As expected in a transient case with lipophilic tracer accumulation, the corneal epithelium and endothelium have a higher fluorescence compared to the stroma and baths. There is a small discrepancy observed in the spatial concentration profile measured at 6 mins after instillation of RhB, where the model under-predicts the experimentally measured RhB concentration. This under-prediction occurs for the first time-point, while the other predictions for the other time points (30, 60 and 140 mins) match the experimental results very well. This could be an experimental error possibly caused by mixing introduced in the apical chamber when the buffer and RhB-loaded buffer solutions were swapped after the cornea was stabilized. Both simulated and experimental results show RhB accumulates quickly into the epithelium from the tear reservoir, diffuses through the stroma with a linear concentration gradient after 45 minutes, and partitions into the endothelium, with a smaller peak value than the epithelium. A nonlinear stroma concentration gradient is expected at early time points as the stroma accumulates tracer, however, the gradient linearizes over time.

Figure 9.

Q3D Snapshots (10× scaled heights) displaying the RhB fluorescence at 6 minutes (a), 30 minutes (b), 60 minutes (c), and 140 minutes (d). Compared to fluorescein, RhB’s lipophilic nature caused faster transport through the corneal tissue layers as RhB favors the transcellular pathway compared to the paracellular. The Apical (tear) bath contains a fixed fluorescence of 5.5 AU, which shows up as a “low” concentration (i.e. blue) in the legend.

Figure 10.

Spatial fluorescence of RhB across the rabbit cornea predicted from CoBi Q3D model matched that obtained experimentally [26]. Smaller fluorescence is observed at the apical bath, due to the high fluorescence in the lipophilic corneal epithelium and endothelium, found between approximately 150–200 µm and 500–505 µm depth, respectively. Tissue fluorescence displayed as arbitrary units (AU).

4. Conclusion

The CoBi Q3D modeling framework is successfully applied towards modeling pharmacokinetics with the development of the in vitro rabbit cornea model to investigate transcorneal drug permeation that can be used for ophthalmic drug design. The geometrically accurate and detailed rabbit cornea model includes its multi-laminate structure consisting of epithelium, stroma and endothelium, along with transcellular and paracellular routes. Flow and species transport equations are solved to account for passive partition and diffusion as well as the advection influence in the paracellular pathways. Pressure boundary conditions are assigned to pressurize the posterior side of the corneal endothelium to replicate physiological intraocular pressure and to simulate the flow between tissue layers based on hydraulic permeability. The transcorneal spatial and temporal predictions of this in vitro rabbit cornea transport model were validated against experimentally measurements of lipophilic (RhB) and hydrophilic (fluorescein) tracers. Partitioning of lipophilic drugs into the hydrophobic domains of the corneal epithelium and endothelium were accounted for as well as fluorescein binding in the stroma.

The use of cylindrical representative (Q3D) units to emulate the tissue geometry helps to preserve the 3D spatial components of the tissue of interest while improving computational efficiency. However, the solution of the Q3D is not fully 3D, hence quasi-3D, as the solution to each computational cell of the Q3D is homogeneous radially (across the diameter). With multiple cells defined per Q3D, solutions can vary spatially lengthwise. This limitation of Q3D must be acknowledged in the design of engineering problem. Future research aims to expand the cornea model to include the rest of the eye.