A structural model for the in vivo human cornea including collagen-swelling interaction

Abstract

A structural model of the in vivo cornea, which accounts for tissue swelling behaviour, for the three-dimensional organization of stromal fibres and for collagen-swelling interaction, is proposed. Modelled as a binary electrolyte gel in thermodynamic equilibrium, the stromal electrostatic free energy is based on the mean-field approximation. To account for active endothelial ionic transport in the in vivo cornea, which modulates osmotic pressure and hydration, stromal mobile ions are shown to satisfy a modified Boltzmann distribution. The elasticity of the stromal collagen network is modelled based on three-dimensional collagen orientation probability distributions for every point in the stroma obtained by synthesizing X-ray diffraction data for azimuthal angle distributions and second harmonic-generated image processing for inclination angle distributions. The model is implemented in a finite-element framework and employed to predict free and confined swelling of stroma in an ionic bath. For the in vivo cornea, the model is used to predict corneal swelling due to increasing intraocular pressure (IOP) and is adapted to model swelling in Fuchs' corneal dystrophy. The biomechanical response of the in vivo cornea to a typical LASIK surgery for myopia is analysed, including tissue fluid pressure and swelling responses. The model provides a new interpretation of the corneal active hydration control (pump-leak) mechanism based on osmotic pressure modulation. The results also illustrate the structural necessity of fibre inclination in stabilizing the corneal refractive surface with respect to changes in tissue hydration and IOP.

Keywords: collagen-swelling interaction; cornea; hydration; osmotic pressure; proteoglycan; pump-leak.

Figure 1.

(a) The cornea is a fibre-reinforced electrolyte gel that resists the IOP. The collagen lamellae mostly follow the corneal curvature except in the anterior region where the lamellae are interweaving and inclined relative to the corneal surface, and are seen to insert into Bowman's layer. (b) An illustration of the organization of the corneal stroma showing several lamellae and a keratocyte cell. Collagen fibrils within lamellae are found in parallel arrays following the fibril direction; keratocyte cells are found interspersed between adjacent lamellae. (c) Cross-section of an idealized unit cell representing the hexagonal collagen fibril lattice. The coating region around the central cylinder (representing the collagen fibril) and the irregular lines in between are illustrative of the coating GAGs and interfibrillar GAGs, respectively.

Figure 2.

The in vivo osmotic pressure P_os at normal hydration (i.e. J = 1) as a function of the ionic transport parameter Q₊Q₋. The osmotic pressure P_os reduces monotonically with increasing active ionic flux values (decreasing Q₊Q₋). The calibrated value of Q₊Q₋ is found to be 0.965 for the normo-hydrated in vivo cornea. (Online version in colour.)

Figure 3.

(a) The X-ray scattering intensity can be additively decomposed into uniform and aligned parts. (b) The distribution interpolated from surrounding scan points is offset based on the depth within the stroma to increase alignment in the posterior.

Figure 4.

Data from [33] of alignment percentage for each third in the central cornea is fit with a linear curve (a). The depth-dependence of alignment is approximated from a full-thickness X-ray scan near the limbus by fitting with a line of the same slope (b). (Online version in colour.)

Figure 5.

SHG images were processed by Winkler et al. [37] to obtain quantitative depth-dependent inclination distributions. Images were scanned for lamella direction by first isolating a narrow section of the image and rotating it so the anterior surface was horizontal. Then the image processing algorithm searched for fibres by scanning through the depth. Data were processed to discard artefacts and then plotted in a histogram for each depth. A Gaussian distribution was fit to the data to define the FWHM value for each depth and radial position. The FWHM value was then used to define the inclination distribution (Online version in colour.)

Figure 6.

FWHM values were estimated in the posterior half by fitting a linear trend line that intersected 0° at the posterior surface. (Online version in colour.)

Figure 7.

(a) Illustrations of confined (i) and unconfined (ii) swelling pressure experiments for ex vivo cornea, respectively [20,52]. In both experiments, a mechanical pressure needs to be applied in the transverse direction to maintain a specified thickness. (b) The predicted confined and unconfined swelling pressure P_s versus corneal thickness t and their comparison with experimental measurements [20,52] over thickness range The swelling pressure data have been extracted from the original papers by free graphics software Plot Digitizer, and the hydration data H from [20] have been transformed to thickness t by the observed linear relation H = 7.09t − 0.44 [52,53]. (Online version in colour.)

Figure 8.

(a) The effect of lamella inclination on stromal swelling ratio during free swelling of the sample in a bath ionic concentration range of 10⁻³–10⁵ mM. At each value of C₀, the swelling ratio is calculated as the ratio of the swollen thickness t_m to the original thickness t₀ at which the swelling pressure is zero. (b) Predicted depth-dependent swelling at physiological bath concentration (C₀ = 150 mM) and deionized water (C₀ = 150 × 10⁻⁶ mM). (Online version in colour.)

Figure 9.

(a) The computed osmotic pressure P_os versus volume dilation J for three representative values of Q₊Q₋ and comparison with swelling pressure measurements [20,52]. (b) The predicted osmotic compressibility K_os versus volume dilation J and its comparison with experimental measurements [52]. The measured modulus is given by which is computed by the power law fit function for swelling pressure from [52]. (Online version in colour.)

Figure 10.

The predicted swollen CCT in Fuchs' dystrophy when (a) the endothelial ionic permeability is increased up to a factor of 4.5 () with normal active ionic flux () and (b) the active ionic flux rate is reduced up to a factor of 0.2 () with normal anionic permeability (). (Online version in colour.)

Figure 11.

Fringe plot of the vertical displacement field u_z at for a cornea with Fuchs' dystrophy. The anterior surface of the cornea deforms much less than that of the posterior surface. It is noted that the sclera elements are distorted at the limbus. This artefact is due to the abrupt transition from swelling to non-swelling tissue, and may be removed by defining a transition zone between the cornea and sclera.

Figure 12.

Illustration of the stress state in the normal living cornea. A nearly uniform fluid pressure exists in the stroma at the magnitude of the IOP. The in-plane lamellae are responsible for resisting such pressure in the lateral direction. In the transverse direction, the applied IOP at the posterior surface balances with the fluid pressure. At the anterior surface, the inclined lamellae insert into Bowman's layer and act as anchors resisting the stromal fluid pressure applied to the epithelium. (Online version in colour.)

Figure 13.

The predicted CCT response to increased IOP from 15 to 120 mmHg. The absence of inclined lamellae results in a much more swollen cornea. (Online version in colour.)

Figure 14.

Illustration of the flap geometry and ablation profile in the LASIK surgery designed for an intended correction of approximately −5 dioptres. The anterior flap diameter F = 9 mm, transition zone diameter T = 6.16 mm, sculpted surface diameter A = 6 mm and optic zone diameter (for least-squares curvature extraction) O = 5 mm. The CCT is 520 µm, and the maximum ablation depth of 87 µm combines with the 150 µm flap thickness to produce a residual central stromal thickness of d₂ = 283 µm.

Figure 15.

Fringe plots of (a) the fluid pressure and (b) volume dilation of a postsurgical cornea after LASIK designed for an intended correction of −5 dioptres.