Three-Dimensional Modeling of Metabolic Species Transport in the Cornea With a Hydrogel Intrastromal Inlay

Abstract

Purpose.: Intrastromal inlays for refractive correction of presbyopia are being adopted into clinical practice. An important concern is the effect of the inlay on the long-term health of the cornea due to disturbances in the concentration profiles of metabolic species. A three-dimensional metabolic model for the cornea is employed to investigate oxygen, glucose, and lactate ion transport in the cornea and to estimate changes in species concentrations induced by the introduction of a hydrogel inlay.

Methods.: A reaction-diffusion metabolic model, appropriate for highly oxygen-permeable hydrogel inlays, is used to describe cellular consumption of oxygen and glucose and production of lactic acid. A three-layer corneal geometry (epithelium, stroma, endothelium) is employed with a hydrogel inlay placed under a lamellar flap. The model is solved numerically by the finite element method.

Results.: For a commercially available hydrogel material with a relative inlay diffusivity of 43.5%, maximum glucose depletion and lactate ion accumulation occur anterior to the inlay and both are less than 3%. Below 20% relative diffusivity, glucose depletion and lactate ion accumulation increase exponentially. Glucose depletion increases slightly with increasing depth of inlay placement.

Conclusions.: The flux of metabolic species is modified by an inlay, depending on the inlay relative diffusivity. For commercially available hydrogel materials and a typical inlay design, predicted changes in species concentrations are small when compared to the variation of concentrations across the normal cornea. In general, glucose depletion and lactate ion accumulation are highly sensitive to inlay diffusivity and somewhat insensitive to inlay depth.

Introduction

Various designs of intrastromal inlays are currently being adopted into clinical practice for treatment of presbyopia and other refractive conditions. ^1–3 Compared with alternative refractive surgical options, the use of intrastromal inlays for the improvement of near and intermediate vision in presbyopes has the advantage that the removal of tissue is not required. Experience has shown that inlays can be safely removed at any time, should that become necessary. ⁴ They also appear to preserve the patient's ability to undergo future ophthalmic procedures, including cataract surgery and retinal imaging. The current generation of intrastromal inlays includes three mechanisms of action: corneal reshaping inlays provide an additive volume that reshapes the central anterior curvature, producing a cornea with a gradual focal gradient; refractive optic inlays employ a bifocal optic with a refractive index that differs from the stroma; and small aperture opaque inlays, which have a small central aperture that increases depth of field through the pinhole optical principle. All three inlay designs are typically implanted in the nondominant eye, concentric with the visual axis or pupil center, and under a lamellar flap or in a pocket created by a mechanical or femtosecond laser platform. 

In this study, we consider corneal reshaping inlays. A design that illustrates current technology for these inlays is provided by the Raindrop Near Vision Inlay ⁵ (ReVision Optics, Lake Forest, CA, USA). This hydrogel polymer-based inlay is permeable to metabolic species (including oxygen and glucose) and has high light transmissibility with a refractive index close to that of the cornea. The inlay has a meniscus lens-like shape that after implantation alters the corneal anterior surface curvature above the inlay. Current approximate dimensions for this implant are a diameter of 2 mm with a thickness that varies from 14 μm at the periphery to 34 μm at the center. These dimensions may be compared with the diameter of the corneal limbus, which is approximately 11.5 mm, and to the corneal central thickness of approximately 0.5 mm. 

Because the cornea is avascular, the metabolic requirements of corneal cells must be met by the diffusive transport of glucose from the aqueous humor and oxygen primarily from the air, or from the palpebral conjunctiva in the closed eye condition. ⁶ Anaerobic breakdown of glucose in glycolysis produces lactic acid that must also diffuse out of the cornea through the endothelium and into the aqueous chamber. Early experience with impermeable or low-permeable inlays showed that anterior stromal thinning and keratolysis can result, ¹ suggesting the importance of maintaining the transport of glucose, oxygen, and metabolic products such as lactic acid. It is also necessary that the mobility of stromal salt ions be maintained since these ions, in concert with the fixed charges of the glycosaminoglycans, produce the electrolyte character of the stroma and are crucial to the creation of osmotic pressure and optimal tissue hydration. ⁷ Thus, a significant barrier to flow can produce both cell stress and hydration imbalance. 

The lateral dimension of the cornea is much greater than its thickness and therefore oxygen and glucose will transport directly across the cornea in opposite directions, driven by concentration gradients in an essentially one-dimensional (1D) manner. Introduction of an intrastromal inlay will modify metabolic species concentrations in the surrounding tissue by three principal mechanisms. Flux directions of metabolic species in the stroma will change in the vicinity of the inlay due to diffusivity mismatch; for example, if the inlay has relatively lower diffusivity than the stroma, species flux vectors in the stroma will tend to develop tangential components as species flow around the obstacle. Further, cellular metabolic reactions will be suspended within the implant volume (assuming cells do not migrate into the implant), and the length of the diffusion path across the cornea-inlay system will have been changed by the introduction of the implant. Although current inlay diameters are small compared with the limbal diameter, they lie directly in the path of metabolic species transporting across the central cornea. For the reasons noted above, modeling of metabolic species transport in the cornea with an intrastromal inlay must be treated as a three-dimensional (3D) problem. 

Because it is difficult to measure metabolic processes experimentally, a numerical modeling approach can be useful. The oxygen distribution across the thickness of the cornea has been modeled as a reaction-diffusion problem using experimental data primarily derived from in vitro rabbit corneas. ^8–11 Corneal lactate production, ¹² glucose consumption, ^13,14 and change in corneal pH ^15–17 in response to hypoxia were also investigated for rabbit corneas. A coupled reaction-diffusion model for corneal metabolism has more recently been proposed by Chhabra et al. ¹⁸ that describes the metabolic consumption of glucose and oxygen to produce lactic acid, bicarbonate ion, hydrogen ion, and carbon dioxide through Monod kinetics-based reactions for both aerobic respiration and anaerobic fermentation. The theory makes it possible to model corneal acidosis induced by hypoxia and was applied by the authors to analyze 1D metabolic transport during contact lens wear. Leung et al. ¹⁹ adopted their framework to model the effect of hypoxia on corneal edema. In the present work, we adapt the Chhabra et al. ¹⁸ model in a restricted form that is suitable for application to hydrogel inlays while generalizing the model to 3D and solving the model numerically using the finite element method. 

Larrea et al. ²⁰ introduced the first numerical model for assessing the 3D steady-state distribution of oxygen and glucose in the cornea with an intrastromal hydrogel inlay. The two-species model assumes oxygen and glucose transport to be governed by the reaction-diffusion equation in steady-state with both reaction (consumption) rate expressions stated in terms of a single reactant. As a result, there is no reaction coupling of species in this model. The formulation is based on an axisymmetric model including the three principal corneal layers (epithelium, stroma, and endothelium), and a hydrogel inlay domain. The oxygen consumption rate was modeled using Michaelis-Menten kinetics following Lin ²¹ and specialized for each corneal layer. The glucose consumption rate was assumed to be a linear function of glucose concentration in each corneal layer, with rate constants derived by analysis of the experimental rabbit data in Zurawski et al. ¹³ The finite element method was used to solve the model equations and obtain oxygen and glucose concentration profiles and to study the influence of implant diffusivity and depth of placement on those profiles. 

In the model proposed by Chhabra et al. ¹⁸ (and employed in this study), oxygen and lactate reaction rates are modeled directly using Monod kinetics and the glucose reaction rate is derived indirectly according to the balance demanded by the aerobic and anaerobic metabolic reactions. In order to provide an assessment of the proposed metabolic model, we compare predicted glucose consumption rates to estimates obtained from in vitro measurements of rabbit cornea cellular glucose consumption rates reported by Zurawski et al. ¹³ Following that analysis, we extend the model to the case of a cornea with an implanted hydrogel inlay and analyze changes in oxygen tension and glucose and lactate ion concentrations induced by the presence of the inlay. 

Materials and Methods

Corneal Metabolism Model

Metabolic consumption of glucose by cells in the corneal layers occurs through aerobic respiration and anaerobic fermentation. Aerobic respiration comprises the three metabolic processes of glycolysis, the Krebs cycle (also called the tricarboxylic acid cycle or citric acid cycle), and oxidative phosphorylation. Glycolysis is a purely anaerobic reaction and while it can occur in the presence of oxygen, oxygen is not involved in the reaction. In glycolysis, one mole of 6-carbon glucose is converted to two moles of 3-carbon pyruvate (pyruvic acid). The Embden-Meyerhof-Parnas pathway is the primary glycolytic pathway in the normal cornea. The pyruvate resulting from glycolysis is converted to acetyl-CoA, enters the Krebs cycle, and is oxidized to carbon dioxide. Finally, oxidative phosphorylation via the electron transport chain produces water by electron acceptance by oxygen atoms. Aerobic respiration may be summarized by the following simplified reaction formula,  

 

 

Equation 1 implies that 1 mole of glucose is consumed with 6 moles of oxygen to produce 6 moles of carbon dioxide, 6 moles of water and 36 net moles (theoretical upper limit) of adenosine triphosphate (ATP). 

In the absence of oxygen, respiration occurs by anaerobic fermentation consisting of the two metabolic pathways of glycolysis and fermentation. In this case, the pyruvate produced by glycolysis is converted by fermentation to lactate (lactic acid) and hydrogen ions. This is described by the following reaction formula,  

 

 

In this case, the anaerobic conversion of 1 mole of glucose produces 2 moles of lactate ions, 2 moles of hydrogen ions, and 2 moles of ATP. 

Metabolic Species Transport

A general 3D reaction-diffusion model for coupled oxygen, glucose, and lactate ion transport in the cornea-inlay system is provided by the following system of reaction-diffusion equations,  

 

 

 

 

 

where x = {x₁, x₂, x₃} is the position of a typical point in the cornea-implant domain, p_O is the oxygen tension (equilibrium partial pressure), and related to the oxygen concentration c_O by Henry's constant k_O such that c_O = k_Op_O. Glucose and lactate ion concentrations are denoted c_G, and c_L, respectively. In Equations 3 through 5, the diffusivity of all metabolic species is assumed to be isotropic in all layers. The oxygen permeability D_Ok_O and glucose and lactate diffusivity D_G and D_L, respectively, vary by layer as indicated by the dependence on position x. Likewise, the oxygen and glucose reaction (consumption) rates, Q_O and Q_G, respectively, and lactate reaction (production) rate, Q_L, must be positive functions that vary, in general, by corneal layer and depend on the concentrations of all three metabolic species. To reduce the complexity of the model, Descemet's and Bowman's membranes are modeled as having the same transport (reaction and diffusion) properties as the stroma.⁸ Thus the cornea is represented by the epithelial, stromal, and endothelial layers alone.  

Observing the metabolic consumption of glucose described in Equations 1 and 2, and assuming that all glucose is consumed during the Krebs cycle or in reduction to lactate, it follows that the oxygen, glucose and lactate reaction terms in Equations 3 through 5 must be related as follows:  

 

 

Because the reaction terms are not independent by virtue of this constraint, any two can be modeled, and the third may then be determined from Equation 6. In the Chhabra et al.¹⁸ model, reaction functions are written for oxygen and lactate, Q_O and Q_L, respectively, and the glucose consumption rate is then found by use of Equation 6. If Q_O and Q_L are assumed to depend on p_O and c_G only, Equations 3 through 6—specialized for steady-state conditions—reduce to  

 

 

 

 

 

 

At the interfaces between the corneal layers and between stroma and implant, the three metabolic species must satisfy continuity conditions on concentration and flux. In the absence of data to the contrary, it is assumed that the partition coefficient governing concentration jumps across all interfaces—including the stroma–inlay interface—is unity. This assumption implies simple continuity of all concentrations across all interfaces. Conservation of mass requires continuity of normal fluxes. Letting the interface between any two regions be arbitrarily designated with + and − faces, the following conditions must be satisfied across that interface:  

 

 

 

 

 

 

All terms contained in a parenthesis are taken to be evaluated at the parenthesis subscript face; (n_i )₊ and (n_i )₋ are the i-th components of the unit vectors n ₊ and n , normal to the interface and pointing out from the + and − faces, respectively, such that n ₊ = − n ₋. 

Boundary conditions on the epithelial–air interface Γ_air and endothelial–aqueous chamber interface Γ_AC are specified as follows:  

 

 

 

 

 

where Display Formula and Display Formula correspond to constant values of oxygen tension at the cornea–air and cornea–aqueous humor interfaces, respectively. For glucose, a zero normal flux at the epithelium–air interface is assumed and at the endothelium–aqueous interface glucose concentration is assumed to be constant with a value denoted by Display Formula . For lactate ion, a zero normal flux at the epithelium–air interface is also assumed and Display Formula corresponds to an assumed constant value at the endothelium–aqueous humor interface. The values used for the above-noted physical constants are given in Table 1.  

Table 1

 

Metabolic Model Parameters

Parameter	Value	Units
	24.0	mm Hg
	155.0	mm Hg
	6.9	mM
	7.7	mM
	2.2	mm Hg
	2.2	mm Hg
	0.4	mM
	0.4	mM

Reaction Model

Because both aerobic and anaerobic reactions are catalyzed by enzymes, the oxygen reaction term Q_O in Equation 7 and the lactate reaction term Q_L in Equation 9 can be expressed using nonlinear Monod kinetics in the form,¹⁸ 

 

 

 

 

In Equation 16, Display Formula is the saturation oxygen consumption rate and the first and second bracketed terms on the right-hand side of Equation 16 represents the Monod dependence on oxygen tension and glucose concentration, respectively. The oxygen consumption rate is zero when there is no oxygen tension (p_O = 0). At high levels of oxygen tension, the reaction saturates and tends toward being independent of oxygen tension. The oxygen Monod dissociation equilibrium constant Display Formula in Equation 16 is set from the condition that the oxygen consumption rate attains 90% of the saturation rate at a transition oxygen tension of 20 mm Hg,^18,22 giving a value of Display Formula = 2.2 mm Hg. Similarly, the oxygen consumption rate is zero when glucose concentration is zero and tends toward being independent of glucose concentration at high values of glucose concentration. The glucose Monod dissociation equilibrium constant in Equation 16 is denoted Display Formula and its value is taken to be 0.4 mM which corresponds to an oxygen consumption rate of 93.75% of the saturation rate at a glucose concentration of 6 mM.^18,23 The rate constants Display Formula and Display Formula are taken to be the same in all corneal layers (due to lack of experimental data), whereas Display Formula varies by layer; see Tables 1 and 2.  

Table 2

 

Values of Physical Constants

Parameter	Units	Endothelium	Stroma	Epithelium
DOkO	Fatt Dk (Barrer)	5.3	29.5	18.8
DG	10−6 cm2/s	3.0	3.0	3.0
DL	10−6 cm2/s	4.4	4.4	4.4
	10−5 mLO2/mL/s	47.78	2.29	25.9
	10−6 mol/mL/h	4.7	23.4	28.1

It is noted that Equation 16 is a specialized form of a more general reaction function proposed by Chhabra et al., ¹⁸ which features a third factor for describing the oxygen consumption rate dependence on pH. This factor, not included in the current work, reflects the observation that the oxygen consumption rate increases with reduction in pH below a value of 7.6 and makes it possible to model corneal acidosis induced by hypoxia (and important for contact lens applications). pH is determined from the hydrogen ion molar concentration c_H as pH = −log₁₀(c_H [mol/L]). In the present study, we limit consideration to hydrogel materials that have oxygen permeability properties that match corneal permeability closely and assume that the cornea is maintained at a normal pH of 7.6. 

Equation 17 represents the kinetic rate of lactate ion production, where Display Formula represents the minimum lactate ion production rate. The first and second terms on the right-hand side of Equation 17 describe dependence on oxygen tension and glucose concentration, respectively. The “shift” constant Display Formula reflects how glycolysis shifts from aerobic respiration to anaerobic fermentation as oxygen tension falls; at low oxygen tension, the lactate ion production rate is assumed to be double the minimum value¹²Display Formula , whereas for high oxygen tension, the production rate tends to Display Formula . The shift constant is evaluated to Display Formula = 2.2 mm Hg by assuming that at an oxygen tension of p_O = 20 mm Hg, the lactate ion production rate is 10% above the minimum value¹⁸Display Formula . Finally, Display Formula is the Monod dissociation equilibrium constant for glucose, which is assumed to have the same value as Display Formula . As for the case of oxygen consumption, the lactate ion production rate constants Display Formula and Display Formula are assumed to be the same in all layers, whereas Display Formula varies by layer; see Tables 1 and 2.  

Because of lack of experimental data, we have assumed values for the four reaction constants such that Display Formula = Display Formula : = K_G and Display Formula = Display Formula : = K_O, reducing the number of reaction constants to two. In this case (but not generally), the glucose consumption rate may be found from Equations 6, 16, and 17 as  

 

 

This rate expression describes the consumption of glucose by aerobic and anaerobic respiration. The glucose consumption rate is zero when glucose concentration is zero. In the absence of oxygen, respiration occurs by glycolysis and fermentation and glucose is then consumed at a rate given by  

 

 

At high levels of oxygen tension and glucose concentration, the glucose consumption rate tends to Display Formula /6 + Display Formula /2, as required.  

Geometric Model and Numerical Solution

The assumed spherical geometry of the three-layer normal cornea comprises a 5-μm thick endothelium, a 500-μm thick stroma, and a 50-μm thick epithelium; total corneal thickness is 0.555 mm. The anterior radius of the cornea is taken to be 7.53 mm. Solution of the metabolic model for the normal cornea provides the baseline for the study of the effects of the inlay. The metabolic model comprises solution of Equations 7 through 9, implemented with axisymmetry and subject to boundary and interface conditions (Equations 10–15). The nonlinear and coupled system of equations was solved by the finite element method using COMSOL Multiphysics 4.3a (2012), version 4.3.1.161. The mesh employed approximately 80,000 triangular elements that fully resolve the solution in all layers. 

The cornea model was extended to include a centrally located hydrogel inlay placed under a lamellar flap. The hydrogel polymer-based inlay ⁵ (ReVision Optics) was selected as a representative inlay device and the dimensions and transport parameters of that inlay (Table 3) have been employed as reference values for the study. In order to facilitate a direct comparison of metabolic species concentrations in the normal cornea and cornea with inlay, it was necessary to introduce a modified normal cornea model. This model is designed to correspond to the normal cornea solution but displayed on an “expanded” cornea that matches the geometry of the corneal tissue surrounding the inlay. This can be accomplished by solving the cornea-inlay model with the inlay domain modeled with: (1) stromal oxygen permeability and stromal glucose and lactate diffusivities scaled by a factor of 10⁶; and (2) suspending metabolic reactions, so that Q_O = Q_L = Q_G = 0. This has the effect of shunting the concentrations across the implant domain at constant values and exactly reproduces the normal cornea concentration profiles on the cornea-implant geometry, thereby facilitating a direct comparison on the two solutions. 

Table 3

 

Material Parameters for Reference Hydrogel Inlay Material

Parameter	Value	Units
Oxygen permeability	50.8	Fatt Dk (or Barrer)
Glucose diffusivity	1.304	10−6cm2/s
Lactate ion diffusivity	1.913	10−6cm2/s

Results

Concentration Profiles for Normal Cornea

The solution for the normal cornea is 1D in character, with species gradients in the thickness direction only. Figure 1 illustrates oxygen tension, glucose concentration, and lactate ion concentration profiles across the central corneal thickness (CCT). Note that the thickness coordinate (mm) has its origin at the endothelium–aqueous humor interface and ends at 0.555 mm on the anterior surface of the epithelium. Predicted glucose and lactate ion concentrations at this interface are 4.93 mM and 10.08 mM, respectively. Steady-state consumption rates for oxygen and glucose and production rate for lactate ion were computed from Equations 16, 18, and 17, respectively, and their profiles across the CCT are shown in Figure 2. 

Figure 1

 

Predicted metabolic species concentration profiles across central corneal thickness for the normal cornea. (A) Oxygen tension. (B) Glucose concentration. (C) Lactate ion concentration. The endothelium, stroma, and epithelium are 0.005 mm, 0.5 mm, and 0.05 mm thick, respectively. The thickness coordinate (mm) has its origin at the endothelial–aqueous humor interface and ends at the anterior surface of the epithelium with a value of 0.555 mm.

Figure 2

 

Predicted metabolic species reaction rate profiles across the central stromal thickness for the normal cornea. (A) Oxygen consumption rate. (B) Glucose consumption rate. (C) Lactate ion production rate. The thickness scale starts at 0.005 mm (endothelium–stroma interface) and ends at 0.505 mm (stroma–epithelium interface).

Assessment of Metabolic Model for Normal Cornea

In this section, we provide a measure of the accuracy of the metabolic model by comparing predicted glucose consumption rates to direct measurements for rabbit in vitro. Zurawski et al. ¹³ collected and cultured rabbit stromal keratocyte, epithelial, and endothelial cells that were grown to confluence in a gas flow humidified incubator with Krebs-Ringers solution at a pH of 7.2 and with varying initial glucose concentrations. For each of the three cell types, glucose concentration versus time is reported for three initial glucose concentrations of 0.9, 0.6, and 0.3 μg/mm³, giving a total of nine trials. Each of the trials comprises five consecutive measurements taken with an interval of 20 minutes. Zurawski et al. ¹³ report the average glucose consumption rate for each cell monolayer of each cell line. If it is assumed, following Zurawski et al., ¹³ that the average number of cell layers found in the epithelium, stroma, and endothelium are 5, 7.5, and 1, respectively, the glucose consumption rate per unit volume in each corneal layer may be estimated by direct calculation. Zurawski et al. ¹³ labeled the cell monolayer consumption rates with the approximate initial glucose concentration targeted at the start of each trial. We examined the data presented in Zurawski et al. ¹³ and estimated the average glucose concentration during each of the nine individual trials. The average glucose concentrations, converted to mM units by assuming a glucose molar mass M_G = 0.18016 kg/mol are given in Table 4 along with the corresponding glucose consumption rates for each of the corneal layers. As might be anticipated by cellular density, glucose consumption rates in the epithelial and endothelial layers exceed that of the stroma. For example, at a glucose concentration of 5 mM, the epithelial and endothelial glucose consumption rates exceed the stromal rate by factors of 5.0 and 3.7, respectively. 

Table 4

 

Glucose Consumption Rate Q_G (mM/s) Versus Glucose Concentration c_G (mM) Based on In Vitro Rabbit Data in Zurawski et al. ¹³

Epithelium		Stroma		Endothelium
cG , mM	QG , mM/s	cG , mM	QG , mM/s	cG , mM	QG , mM/s
1.49867	0.00262	1.27664	0.00116	1.33215	0.00288
3.16385	0.00756	2.66430	0.00160	2.88632	0.00555
4.44050	0.01033	4.16297	0.00208	4.44050	0.00760

The data in Table 4 can be modeled with reasonable accuracy using the following (dimensionally-dependent) expressions:  

 

 

 

 

 

 

Use of these expressions allows mild extrapolation of the data to the range of glucose concentration found in the normal human cornea. Figure 3 shows a comparison of the experimental glucose consumption data summarized in Table 4 with modeling provided by Equations 20 through 22. The agreement of the two sets of curves and the required degree of extrapolation is apparent from the plot. 

Figure 3

 

Glucose consumption rates versus glucose concentration are plotted for the epithelium, stroma, and endothelium. The solid curves are based on the experimentally measured values for in vitro rabbit in Zurawski et al. ¹³ Note that the concentration values correspond to the mean concentration during the tests and not the initial (label) concentrations reported in Zurawski et al. ¹³ The experimental data is modeled by Equations 20 through 22, which are plotted as dashed curves. These functions are used to extrapolate the experimental data to the range of glucose concentration found in the normal cornea (6.9 mM at the endothelium and 4.93 mM at the epithelium); the extent of the dashed curves indicates the range of extrapolation required.

Using the metabolic model, oxygen tension and glucose and lactate ion concentration profiles were computed for all layers of the cornea as described above. Reaction Equations 16 and 17 were then evaluated, giving predicted oxygen consumption and lactate ion production rate profiles across all corneal layers. These were converted to a predicted glucose consumption rate by use of Equation 6 or 18. Having obtained both the predicted glucose concentration and the glucose consumption rate profiles across the corneal thickness, these two quantities can be directly related at each point through the CCT. The result for the stroma is shown in Figure 4. 

Figure 4

 

Predicted glucose consumption rate versus glucose concentration in the normal corneal stroma.

Glucose consumption rates estimated from the Zurawski et al. ¹³ data, summarized in Table 4, are given for the experimental range of glucose concentration of 1.7 through 5.0 mM. The range of glucose concentration across the normal cornea predicted by the model is 4.93 to 6.9 mM. In order to compare the estimated and predicted rates, the results in Table 3 were mildly extrapolated by employing Equations 20 through 22. A comparison of estimated and predicted glucose consumption rates versus glucose concentration at the center of the three principal corneal layers is given in Table 5. 

Table 5

 

Comparison of Model Prediction and Experimental Measurement for Glucose Consumption Rate [mM/s] in the Corneal Layers

Layer and Glucose Concentration	Current Model	Zurawski et al.13
Mid-epithelium cG = 4.9 mM	0.00564	0.0113
Mid-stroma cG = 5.6 mM	0.00332	0.0022
Mid-endothelium cG = 6.9 mM	0.00381	0.0095

Concentration Profiles With Hydrogel Intrastromal Implant

The geometric model of the cornea with the hydrogel polymer-based inlay ⁵ (ReVision Optics) is shown in Figure 5. The posterior face of the inlay is curved to conform with the stromal bed. The inlay has thickness which varies from 14 μm at the periphery to 34 μm at the center. The 150-μm thick corneal flap above the inlay lies directly on the inlay and is assumed to conform to the inlay shape, which slightly alters the corneal anterior surface above the inlay. The stroma–inlay interfaces are assumed to be tightly contacting boundaries for species transport and continuity conditions described by Equations 10 through 12 are assumed to hold. The oxygen permeability and glucose diffusivity of the hydrogel inlay were measured by the manufacturer but lactate diffusivity for the hydrogel is unknown. This value was chosen by requiring the inlay glucose-lactate ion diffusivity ratio to match the value for the stroma; from Table 2, the ratio is D_G /D_L = 0.682. Values of the inlay transport parameters used in the study are summarized in Table 3. 

Figure 5

 

Geometry of the cornea and reference hydrogel inlay used in calculations. All dimensions are in mm. (A) Shows the inlay placed within the cornea extended to the limbus. (B) A detailed view of the cornea and inlay with dimensions used in the model. Note that the stromal thicknesses is preserved, which requires the shape of the stroma and epithelium anterior to the inlay to be slightly modified to accommodate the volume of the inlay, and which produces changes in the anterior surface curvature.

Oxygen tension and glucose and lactate ion concentration profiles across the CCT for the normal cornea and cornea with inlay are shown in Figure 6. Percentage changes in concentration for each species due to the introduction of the inlay was computed by evaluating (c_inlay − c_normal ) * 100/c_normal pointwise over the entire corneal longitudinal cross-section. Fringe plots showing areas of depletion and accumulation for oxygen, glucose and lactate ion are shown in Figure 7. Changes in concentrations remote from the inlay are zero. Maximum glucose depletion was 2.5% in the tissue anterior to the inlay and maximum glucose accumulation was 0.6% in the tissue posterior to the inlay (Fig. 7B). Maximum lactate accumulation was 1.4% anterior to the inlay (Fig. 7C). Maximum oxygen depletion was 3.5% at mid-stroma and maximum oxygen accumulation was 3.3% and occurs immediately anterior to the inlay (Fig. 7A). 

Figure 6

 

Metabolic species concentration profiles across the central corneal thickness with and without inlay illustrating regions of depletion (concentration reduction) and accumulation (concentration increase). (A) Oxygen tension. (B) Glucose. (C) Lactate ion. The profile for the cornea without inlay (blue) has a horizontal segment in which the concentrations are constant across the inlay thickness (see “Methods”). The profile for the cornea with inlay is shown in red.

Figure 7

 

Species concentration percent change were determined from (c_inlay − c_normal ) * 100/c_normal (accumulation is positive and depletion is negative). The percent changes are plotted as fringes in the corneal region containing the inlay. (A) Oxygen tension. (B) Glucose. (C) Lactate ion.

Sensitivity to Inlay Glucose and Lactate Ion Diffusivity

In the long-term use of hydrogel inlays it is possible that the transport properties of the hydrogel could change with time as pore structure and organization become modified by chemical or other processes. For this reason, it is useful to analyze the sensitivity of metabolic species transport to variations in the properties of the inlay. Since the molecular size of oxygen is less than that of glucose by an order, it is probable that glucose and lactate ion transport modification would occur well before oxygen transport modification. To model this condition, we introduce the inlay relative diffusivity defined as Γ _G = (D_G ) _inlay /(D_G ) _stroma . From Tables 2 and 3, the hydrogel inlay material used in the present study corresponds to Γ _G = 0.435. Lactate ion and glucose inlay relative diffusivities will have the same value by virtue of our assumption on inlay lactate ion diffusivity. We now consider the situation in which glucose (and lactate ion) transport properties of the implant are varied over the range Γ _G ∈ [0.01, 1.0] while oxygen permeability is maintained at the value of D_Ok_O = 50.8 Fatt Dk units, where 1 Fatt Dk unit = [10⁻¹¹(cm²/s)(mL/O₂ at STP/mL·mm Hg)]. At Γ _G = 1.0, the inlay glucose diffusivity exactly matches the corneal diffusivity (for all layers), Γ _G = 0.435 corresponds to the reference hydrogel material, and at Γ _G = 0.01 the inlay glucose diffusivity is reduced to 1% of the corneal diffusivity. 

Figure 8 depicts maximum glucose and lactate ion percentage change versus inlay relative diffusivity Γ _G . For each value of the inlay relative diffusivity, the glucose depletion and lactate ion accumulation values plotted in Figure 8 are taken from those positions in the cornea where they achieve their maximum values; in general, both these positions change with the relative diffusivity. Note that at Γ _G = 1.0, the changes are not zero, reflecting the increase in the diffusion path due to the inlay. It is seen that maximum glucose depletion and lactate ion accumulation remains below 5.5% for Γ _G > 0.2. For Γ _G < 0.2, the maximum glucose depletion and maximum lactate ion accumulation grow exponentially. This analysis can provide a check for inlay design. If, for example, 5% depletion in glucose concentration is deemed safe, then the implant must have a glucose and lactate ion inlay relative diffusivity no lower than Γ _G = 0.2. To illustrate the effect of low inlay diffusivity, predicted glucose and lactate ion concentration profiles across the cornea computed with Γ _G = 0.05 are shown in Figure 9. This solution gives approximately 18.6% depletion in glucose concentration and 10.7% accumulation in lactate ion concentration, with both maximums located anterior of the inlay. Glucose concentration fringe plots for Γ _G = 0.2 and Γ _G = 0.05 are presented in Figure 10. 

Figure 8

 

Maximum glucose and lactate ion percent concentration depletion and accumulation versus inlay relative diffusivity. For each value of the inlay relative diffusivity, the depletion and accumulation values reported are taken from those positions in the cornea where they achieve maximum values; in general, these positions change with the relative diffusivity. Thus, for any relative diffusivity, the plot shows the maximum glucose depletion that occurs anywhere over the entire cornea; similarly with the lactate ion accumulation.

Figure 9

 

Comparison of concentration profiles across the central corneal thickness for the normal cornea and cornea with an inlay having a relative diffusivity of 5%. (A) Glucose concentration profile; the maximum percent depletion reaches 18.6% in the stroma and epithelium anterior to the inlay. (B) Lactate ion concentration profile; the maximum percent accumulation reaches 10.7% anterior to the inlay.

Figure 10

 

Glucose concentration fringes for the cornea with inlay when the inlay has relative diffusivity of 20% (A) and 5% (B). Notice that the concentration contours remain approximately parallel to the anterior surface at 20% (A), whereas they turn almost perpendicular to the anterior surface at 5% (B).

Sensitivity to Depth of Placement, Corneal Thickness, and Inlay Dimensions

Depth of Placement.

In order to characterize sensitivity to depth of placement, three geometric models for the cornea-inlay system corresponding to lamellar flap thicknesses of 100 μm, 150 μm, and 250 μm were analyzed. The dimensions of the inlay were maintained at their reference values (Fig. 5B) and the inlay oxygen permeability was taken to be constant at D_Ok_O = 50.8 Fatt Dk units. For each flap thickness, maximum species concentration percentage change was determined for inlay relative diffusivities Γ _G of 43.5% (hydrogel reference value), 20%, and 5%. The results for glucose, lactate ion, and oxygen tension are summarized in Figures 11A through 11C. Glucose depletion and lactate ion accumulation both increase with: (1) an increase in inlay depth of placement, and (2) a reduction in inlay relative diffusivity. Further, sensitivity to depth of placement increases for lower values of the inlay relative diffusivity. 

Figure 11

 

Maximum metabolic species percent concentration change when depth of inlay placement (flap thickness) and inlay relative diffusivity (RD) is varied. (A) Glucose depletion. (B) Lactate ion accumulation. (C) Oxygen depletion. The inlay glucose (and lactate ion) relative diffusivity were varied while the inlay oxygen permeability is maintained constant.

Central Corneal Thickness.

Three modified models of the cornea-inlay system were analyzed corresponding to CCT of 450 μm, 555 μm, and 650 μm, in order to characterize sensitivity to CCT. The inlay was modeled under a 150-μm-thick lamellar flap in all cases and the changes in CCT were modeled as changes in the thickness of the stroma only. The metabolic model was solved using the inlay oxygen permeability of D_Ok_O = 50.8 Fatt Dk units and two values of inlay relative diffusivity Γ _G of 43.5% (hydrogel reference value), and 20%. The results for maximum glucose depletion and maximum lactate ion accumulation are summarized in Tables 6 and indicate that a small increase in glucose depletion occurs with increasing CCT. 

Table 6

 

Sensitivity to CCT

Thickness, μm	Maximum Glucose Depletion, %		Maximum Lactate Ion Accumulation, %
	ΓG = 0.435	ΓG = 0.2	ΓG = 0.435	ΓG = 0.2
450	2.29	4.98	1.60	3.5
555	2.50	5.43	1.41	3.09
650	2.84	6.15	1.23	2.74

Inlay Dimensions.

The inlay reference dimensions (Fig. 5B) were systematically varied to characterize sensitivity to inlay dimensions. Four new inlay shapes were created: half-thickness, half-diameter; half-thickness, double-diameter; double-thickness, half-diameter; and double-thickness, double-diameter. The metabolic model was solved for these geometric cases with all inlays placed under a 150-μm lamellar flap. The inlay oxygen permeability of D_Ok_O = 50.8 Fatt Dk units was used with two values of inlay relative diffusivity Γ _G of 43.5% (hydrogel reference value), and 20%. The results for maximum glucose depletion and maximum lactate ion accumulation are summarized in Table 7. Not surprisingly, the double-thickness, double-diameter condition produced the greatest glucose depletion of around 6% and 13% for Γ _G of 43.5% and 20%, respectively. 

Table 7

 

Sensitivity to Inlay Dimensions

Dimensions		Maximum Glucose Depletion, %		Maximum Lactate Ion Accumulation, %
Diameter, mm	Max Thickness, mm	ΓG = 0.435	ΓG = 0.2	ΓG = 0.435	ΓG = 0.2
1.0	0.017	1.07	2.25	0.61	1.29
4.0	0.017	1.49	3.24	0.84	1.83
1.0	0.068	4.22	7.80	2.35	4.36
4.0	0.068	5.94	12.69	3.34	7.13

Discussion

A contact lens placed on the eye impedes the flow of oxygen from the atmosphere into the cornea. An intrastromal hydrogel inlay typically provides oxygen permeability that is equal to or greater than that of the stroma. In the normal cornea, glucose is not significantly transported out of the epithelium into the tear film (as modeled by condition Equation 14A) and a contact lens will therefore not directly modify the glucose state at the cornea–lens interface. In contrast, a hydrogel inlay placed within the stroma could impede the flow of glucose and lactic acid as they move across the cornea following their concentration gradients. In the analysis of the safety and metabolic impact of contact lenses, oxygen is of prime interest, whereas for intrastromal inlays, glucose is of prime interest. 

Reduction in the flow of metabolic species resulting from the inlay may cause changes in the tissue and cells. An extreme example is provided by the early intrastromal inlays which used materials that were impermeable to water and metabolic species. These implants caused anterior stromal necrosis followed by implant extrusion. ^24,25 Modern hydrogel materials, such as that analyzed in this study, have oxygen permeability that is very close to that of the stroma, or even exceeding it. However, the glucose diffusivity of these materials is inferior to the stroma and small changes in glucose concentration in adjacent stroma will certainly be generated, generally as depletion anterior to the inlay and accumulation posterior to the inlay. In this study we have analyzed these changes as a function of the inlay relative diffusivity and, as discussed below, for currently available inlay materials, the changes in glucose concentration are small. However, there exists the theoretical possibility that the pores of the hydrogel material may become obstructed in time ²⁴ with a concomitant reduction in diffusivity. As far as we know, no explanted hydrogel inlay has been tested for changes in diffusivity. As shown in this study, if the inlay material is reduced to relative diffusivities below 20%, there is an exponential increase in glucose depletion. An earlier theoretical study by Larrea et al. ²⁰ drew a similar conclusion when the relative diffusivity approached 10%. 

Complications associated with hydrogel inlays for presbyopic correction have included intracorneal deposits, corneal haze, epithelial perilenticular opacity, and vision-related effects such as irregular astigmatism. ²⁴ Some of the complications have been identified with the surgical implant procedure ²⁵ and some with an immunological or inflammatory ²⁶ response to the hydrogel material. It is probable that the latter, mediated by macrophages or activated keratocytes, does not result from low levels of oxygen or glucose. It is currently not clear to what extent small changes in metabolic species concentrations, such as seen in the current study, contribute to the development of complications. However, if an inlay exhibits progressive reduction in diffusivity, the resulting glucose depletion will likely become an important factor in the long-term health of the tissue. 

In this work, we have employed the metabolic model proposed by Chhabra et al.,¹⁸ but restricted by the assumption that the corneal pH is maintained at a normal and constant value of 7.6. To provide an assessment of the model, we have computed the glucose consumption rate versus glucose concentration in the normal cornea and compared the results to estimates based on rabbit data presented by Zurawski et al.¹³ The agreement is considered fair for the stroma (Table 5). While the predicted glucose consumption rates in the epithelium and endothelium exceed the predicted glucose consumption rate in the stroma, agreeing with the trend in the experimental data, the predicted consumption rates are nevertheless underestimated by approximately 50% and 60%, respectively. (It may be noted that because the bounding cellular layers are thin compared to the stroma, the total glucose consumed in those layers will be less than the total glucose consumed in the stroma.) These results are influenced by the values assigned to the epithelial and endothelial maximum oxygen consumption rate Display Formula and minimum lactate production rate Display Formula . The data summarized in Table 5 could be employed to revise these values, although we have not pursued such a study in the present work. Because glucose is the only reactant in glycolysis, it is also suggested that that the glucose consumption rate data provided by Zurawski et al.¹³ could be directly modeled using simple Monod or Michaelis-Menten kinetics. The oxygen consumption rate would retain the form of Equation 16 and the lactate ion production rate would be derived from Equation 6.  

The current model has assumed isotropic diffusion in all corneal layers. The corneal stroma has a lamella structure that suggests that anisotropic diffusion of metabolic species might be important. No experimental measurement of anisotropy in stromal tissue is available. Stromal lamellae contain parallel arrays of collagen fibrils in a regular lattice arrangement with an average fibril diameter of approximately 25 nm and center-to-center spacing of approximately 53 nm. The question for isotropy is whether diffusive transport across the fibril direction is significantly hindered by the fibril “obstacles” compared with transport along the fibril direction. However, the clear distance between adjacent fibril faces of approximately 28 nm relative to the molecule diameter (which is on the order of 1 nm for glucose), seems to suggest that isotropy will hold to first order and that transport is governed primarily by the properties of the interstitial fluid of the extracellular matrix. 

A further assumption in the model is that the transport and reaction properties of Descemet's and Bowman's layers are the same as the stroma. Descemet's and Bowman's have collagen fibril arrangement and density that is comparable with the stroma and both exhibit elasticity. In the absence of experimental measurements, we have taken their diffusive transport and metabolic reaction properties to be similar to the stroma and the model simply combines their volumes with that of the stroma. 

With regard to accuracy of the model for application to the hydrogel inlay, it is noted that the principal results are given as the difference of two solutions; the cornea with inlay and the normal cornea are subtracted. In this case, predictive accuracy is increased because the subtraction of the two solutions removes most of the leading order error term. In other words, while the normal cornea/inlay solutions have modeling inaccuracies, the accuracy of solution differences will be better. This tells us that the accuracy of estimates for concentration changes (depletion or accumulation) will be more reliable that the underlying concentrations. 

For the reference hydrogel material, the calculation of glucose concentration change over the corneal cross-section reveals a maximum depletion of approximately 3% in the stroma and epithelium anterior to the inlay. The reference inlay material is reasonably well matched to the cornea (relative inlay diffusivity of 43.5%) and the results show minimal disturbance of metabolic species. For example, in the normal cornea, glucose concentration is predicted to vary from 4.9 to 6.9 mM. Expressing this natural variation as a percentage of the mean value gives 33%, well above the 3% depletion value associated with the inlay. 

For the reference inlay hydrogel material with a relative inlay diffusivity of 43.5%, it may be seen from Figure 7B that there is no significant lateral transport of glucose through the stroma anterior to the inlay as indicated by the strong depletion gradient above the inlay. Thus, achievement of low glucose depletion depends on the inlay having sufficiently high diffusivity; lateral diffusion around the inlay is not an effective mechanism for transporting glucose to the central anterior stroma. To illustrate, glucose concentration fringes are shown in Figure 10 for inlay relative diffusivities of Γ _G of 20% and 5%. As noted above, the value of 20% appears to represent a threshold value (see Fig. 8) below which depletion grows exponentially. At 20% diffusivity, the concentration contours remain level (Fig. 10A) and glucose flux follows the normal posterior-to-anterior path, while at 5% diffusivity lateral gradients in glucose concentration anterior to the inlay are set up (Fig. 10B), but these are not adequate to generate sufficient lateral mass transport. The latter point is further illustrated in Figure 9, which depicts glucose and lactate ion concentration profiles when the inlay has a relative diffusivity of 5%. 

Finally, it is remarked that our analysis indicates that glucose depletion always increases with increasing depth of placement and that this sensitivity increases as the inlay relative diffusivity is reduced (in our study from 43.5% to 5%; see Fig. 11). Larrea at al. ²⁰ have performed an analogous sensitivity study in which the relative diffusivities are varied over a lower range from 10% to 0%. Results for 5% inlay relative diffusivity are common to both studies and our findings agree, namely that glucose depletion increases with depth of inlay placement. That is, shallow placement is always favored with respect to glucose concentration. However, at inlay diffusivities lower than 0.1%, Larrea et al. ²⁰ show a reversal in the trend whereby glucose depletion reduces with depth of placement. Although we have not reported results for such low values of inlay diffusivity here—our analyses at these low values do not support that finding—we find that glucose depletion always increases slightly with depth of placement for any value of inlay diffusivity. From a practical point of view, and for commercially available hydrogel materials, Figure 11A shows that glucose depletion is essentially insensitive to depth of inlay placement. 

Acknowledgments

Disclosure: P.M. Pinsky, ReVision Optics (C)