An Analytical Approach to Corneal Mechanics for Determining Practical, Clinically-Meaningful Patient-Specific Tissue Mechanical Properties in the Rehabilitation of Vision

Abstract

Patient-specific biomechanical properties of the human cornea are rarely used with finite elements analysis. In order for that to be possible, a proper formulation for biomechanical properties that is based on patient-specific measurable values must be used. In this study, we propose a formula that simulates hyperelastic stress–strain curves based on non-invasive clinical measurements that can be acquired in vivo. These consist of, but are not limited to, center corneal thickness and center corneal curvature as well as corneal resistance factor and applanation diameter that are measured during non-contact tonometry. The presented formulation was demonstrated and validated through several computer simulations. First, mean values that were reported in literature were inputted into the formula to simulate a curve that represents a healthy case. This case was compared to two independent in vitro studies. Then, a sensitivity analysis was carried to identify inputs that have the most dominant effect. Finally, a finite element analysis simulating elevations in intraocular pressure was conducted; the corneal model comprised of patient-specific corneal geometry that was measured in vivo in our clinic as well as the current formulation for patient-specific corneal biomechanics. “Strong” and “weak” corneal tissue cases were simulated and deformations as well as instantaneous curvature optical maps were derived. Results for the simulated healthy curve showed good agreement with the in vitro studies. The sensitivity analysis found the corneal resistance factor and applanation diameter to have the most dominant influence. The finite element analysis of strong and weak biomechanical properties resulted in corneal deformations and instantaneous curvature optical maps that are common for healthy and pathological conditions respectively. In conclusion, the presented modeling technique can be used to assess corneal biomechanics in vivo and therefor may enhance follow-up on the effectiveness of clinical treatments, rehabilitation of vision and perhaps improve the diagnosis of pathologies that are related to corneal biomechanics.

Appendix

Expansion of Eq. (4) to Include IOP₀ (IOP at Time of Measurement)

Solving the differential Eq. (4) using non-homogenous boundary conditions results in:

$$S_{m} \left( {E_{m} } \right) = \frac{{E_{0} }}{a}\left[ {\exp \left( {a \cdot E_{m} } \right) - 1} \right] + S_{{{m}_{0} }}$$

(13)

Note that Eq. (13) becomes Eq. (5) by zeroing IOP₀

Use Eq. (2) to give the residual stress for measured IOP₀

$$S_{{m_{0} }} = 0.5 \cdot \left( {\frac{{{\text{IOP}}_{0} }}{7501}} \right) \cdot \frac{{\left( {R - \frac{t}{2}} \right) }}{t}$$

An Explicit Definition of α and β

$$\alpha = \frac{{2\pi \cdot A(\mu ) \cdot {\text{CCR}} \cdot \left( {{\text{CCR}} - \frac{\text{CCT}}{2}} \right)\sqrt {1 - \upsilon^{2} } }}{{{\text{CCT}}^{2} }}\,\,\,\, \beta = \frac{{\pi \cdot {\text{CCR}} \cdot \left( {{\text{CCR}} - \frac{\text{CCT}}{2}} \right)^{2} \left( {1 - \upsilon } \right)}}{{{\text{Area}} \cdot {\text{CCT}}}}$$

An Explicit Definition for A(μ)

A(μ) is a geometrical coefficient that was approximated using Taylor’s approximation as follows5,35:

(for 0 < μ < 1.4)

$$A\left( \mu \right) \cong 0.433047 - 0.001859\mu - 0.228169\mu^{2}$$

$$+ 0.237752\mu^{3} - 0.135992\mu^{4} + 0.032129\mu^{5}$$

where $\mu = \frac{AD}{2}\left[ {\frac{{12\left( {1 - \nu^{2} } \right)}}{{\left( {R - \frac{t}{2}} \right)^{2} \cdot t^{2} }}} \right]^{0.25}$ (AD is in our case the applanation diameter).

Limitations Imposed by A(μ)

Special attention was given to the Taylor’s approximation, in order to clarify the constraints it imposes on the model. For that matter, a simulation was carried in order to show the dependency of $A(\mu )$ on the range of possible inputs (CCT, CCR and AD). In each analysis (Figs. 9a, 9b and 9c) the horizontal dashed lines depict where the approximation limits are. Passing these margins will add a growing approximation error to the results. The CCT, CCR and AD were varied between 0.34–0.65 mm, 6–9 mm and 0.5–4 mm respectively. These ranges were chosen in order to cover a very wide of possible inputs, including very extreme pathological conditions.12–14,30,38

Figure 9

In each simulation the horizontal dashed lines depict where the approximation limits are. Passing these margins will add a growing error to the results. The CCT, CCR and applanation diameter were varied between 0.34–0.65 mm, 6–9 mm and 0.5–4 mm respectively. These ranges were chosen in order to cover a very wide of possible inputs, including very extreme pathological conditions12–14,30,38

The asterisk in Fig. 9 marks the values of the reference simulation (Fig. 2). Low CCT and CCR values move the asterisk towards lower A(μ) values. In contrast, low AD values move the asterisk upwards towards higher A(μ) values. The resulting lower limit for the CCT (Fig. 9a) is 0.4 mm and there is no upper limit. The CCR (Fig. 9b) has a lower limit of 6.1 mm and no upper limit as well. A(μ) is inside the boundaries for applanation diameters that are up to 3.5 mm and has no lower limit (Fig. 9c).