Inverse computational analysis of in vivo corneal elastic modulus change after collagen crosslinking for keratoconus

Abstract

Corneal collagen crosslinking with riboflavin photosensitization and ultraviolet irradiation is a novel approach to limiting the progression of keratoconus in patients by increasing the elastic modulus of the degenerate cornea. Beneficial reductions in corneal steepness and aberrations after crosslinking also frequently occur. In a previous study, we described a computational modeling approach to simulating topographic progression in keratoconus and regression of disease with corneal collagen crosslinking. In the current study, this model has been expanded and applied to the inverse problem of estimating longitudinal time-dependent changes in the corneal elastic modulus after crosslinking using in vivo measurements from 16 human eyes. Topography measured before crosslinking was used to construct a patient-specific finite element model with assumed hyperelastic properties. Then the properties of the cornea were altered using an inverse optimization method to minimize the difference between the model-predicted and in vivo corneal shape after crosslinking. Effects of assumptions regarding sclera-to-cornea elastic modulus ratio and spatial attenuation of treatment effect due to ultraviolet beam characteristics on the predicted change in elastic modulus were also investigated. Corneal property changes computed by inverse finite element analysis provided excellent geometric agreement with clinical topography measurements in patient eyes post-crosslinking. Over all post-treatment time points, the estimated increase in corneal elastic modulus was 110.8 ± 48.1%, and slightly less stiffening was required to produce the same amount of corneal topographic regression of disease when the sclera-to-cornea modulus ratio was increased. Including the effect of beam attenuation resulted in greater estimates of stiffening in the anterior cornea. Corneal shape responses to crosslinking varied considerably and emphasize the importance of a patient-specific approach.

Keywords: biomechanics; collagen crosslinking; cornea; elastic modulus; inverse finite element analysis; keratoconus; modeling.

Figure 1

(a) and (b) Whole-eye FE model showing the cornea and sclera; (c) A schematic representation of the model showing the weaker keratoconic zone and the crosslinked zone in a standard 9mm-diameter simulated treatment.

Figure 2

The inverse FE optimization approach.

Figure 3

Comparison of in vivo and inverse FE (iFE) estimated optical parameters of the anterior surface of the analyzed eyes. Data from all 16 eyes is shown for different sclera-tocornea modulus ratio (S/C RMS for higher order aberrations): a) S/C = 3; b) S/C = 4; c) S/C = 5. Regressions suggest strong agreement between in vivo and iFE results.

Figure 4

Comparison of in vivo and inverse FE estimated wavefront aberration of the anterior surface of the analyzed eyes. Data from all 16 eyes is shown for different sclera-to-cornea modulus ratio (S/C): a) S/C = 3; b) S/C = 4; c) S/C = 5. Better correlation between in vivo and iFE RMS for higher order aberrations (HORMS) was seen than for RMS for lower order aberrations (LORMS).

Figure 5

Change in f_max [(f_max - 1) × 100 = % increase in modulus of the cornea at the geometric apex in the follow-up months after CXL estimated by the inverse FE. The effect of different sclera-to-cornea modulus ratios (S/C) were illustrated by the linear regression lines. There was considerable variability both intra-patient as well as inter-patient in the follow-up months. As S/C increased, iFE predicted that slightly less stiffening would occur to produce the same corneal curvature change with crosslinking.

Figure 6

A comparison of in vivo and inverse FE (iFE) estimate of tangential curvature (in Diopters) of one of the eyes showing progressive flattening (regression of keratoconic steepening) in the months following crosslinking. Vertical and horizontal axis units are in millimeters.

Figure 7

A comparison of in vivo and inverse FE estimate of tangential curvature (Diopters) of one of the three eyes showing progressive steepening (keratoconic progression) at least 6 months after crosslinking. Vertical and horizontal axis units are in millimeters.

Figure 8

A comparison of change in f_max [(f_max - 1) × 100 = % increase in modulus of the cornea at the geometric apex) and sensitivity to sclera-to-cornea elastic modulus ratio using the data from the two cases described in Figure 6 and 7. Note that the case that with an estimated loss of stiffening effect over time and possible progression of keratoconus showed no sensitivity to the assumed sclera-to-cornea elastic modulus ratio (overlapping data points). At 14 month, the eye with disease progression showed a stiffening of ~10% whereas the other eye had stiffened by over 200%.

Figure 9

Variation in f_max [(f_max - 1) × 100 = % increase in modulus of the cornea at the geometric apex] though the depth of the cornea using a depth-dependent inverse FE method. Data from the eye presented in Figure 6 were used in these analyses. Two different coefficients, β = 60 and 120 cm⁻¹ [see Eq. 6] were used to simulate depth-dependent stiffening effect for all sclera-to-cornea modulus ratios (S/C). With β = 120 cm⁻¹, the stiffening in the anterior stroma was significantly higher than with β = 60 cm⁻¹.