Analysis of X-ray scattering microstructure data for implementation in numerical simulations of ocular biomechanical behaviour

Typical localised element-specific finite element anterior eye model

Purpose

The study presents an averaged anterior eye geometry model combined with a localised material model that is straightforward, appropriate and amenable for implementation in finite element (FE) modelling.

Methods

Both right and left eye profile data of 118 subjects (63 females and 55 males) aged 22-67 years (38.5 ± 7.6) were used to build an averaged geometry model. Parametric representation of the averaged geometry model was achieved through two polynomials dividing the eye into three smoothly connected volumes. This study utilised the collagen microstructure x-ray data of 6 ex-vivo healthy human eyes, 3 right eyes and 3 left eyes in pairs from 3 donors, 1 male and 2 females aged between 60 and 80 years, to build a localised element-specific material model for the eye.

Results

Fitting the cornea and the posterior sclera sections to a 5th-order Zernike polynomial resulted in 21 coefficients. The averaged anterior eye geometry model recorded a limbus tangent angle of 37° at a radius of 6.6 mm from the corneal apex. In terms of material models, the difference between the stresses generated in the inflation simulation up to 15 mmHg in the ring-segmented material model and localised element-specific material model were significantly different (p < 0.001) with the ring-segmented material model recording average Von-Mises stress 0.0168 ± 0.0046 MPa and the localised element-specific material model recording average Von-Mises stress 0.0144 ± 0.0025 MPa.

Conclusions

The study illustrates an averaged geometry model of the anterior human eye that is easy to generate through two parametric equations. This model is combined with a localised material model that can be used either parametrically through a Zernike fitted polynomial or non-parametrically as a function of the azimuth angle and the elevation angle of the eye globe. Both averaged geometry and localised material models were built in a way that makes them easy to implement in FE analysis without additional computation cost compared to the limbal discontinuity so-called idealised eye geometry model or ring-segmented material model.

In Vivo Corneal Stiffness Mapping by the Stress-Strain Index Maps and Brillouin Microscopy

The study of corneal stiffness in vivo has numerous clinical applications such as the measurement of intraocular pressure, the preoperative screening for iatrogenic ectasia after laser vision correction surgery and the diagnosis and treatment of corneal ectatic diseases such as keratoconus. The localised aspect of the microstructure deterioration in keratoconus leading to local biomechanical softening, corneal bulging, irregular astigmatism and ultimately loss of vision boosted the need to map the corneal stiffness to identify the regional biomechanical failure. Currently, two methods to map the corneal stiffness in vivo are integrated into devices that are either already commercially available or about to be commercialised: the stress-strain index (SSI) maps and the Brillouin Microscopy (BM). The former method produces 2D map of stiffness across the corneal surface, developed through numerical simulations using the corneal shape, its microstructure content, and the deformation behaviour under air-puff excitation. It estimates the whole stress-strain behaviour, making it possible to obtain the material tangent modulus under different intraocular pressure levels. On the other hand, BM produces a 3D map of the corneal longitudinal modulus across the corneal surface and thickness. It uses a low-power near-infrared laser beam and through a spectral analysis of the returned signal, it assesses the mechanical compressibility of the tissue as measured by the longitudinal modulus. In this paper, these two techniques are reviewed, and their advantages and limitations discussed.

In vivo Assessment of Localised Corneal Biomechanical Deterioration With Keratoconus Progression

Purpose: To evaluate the regional corneal biomechanical deterioration with keratoconus (KC) progression as measured by the Stress-Strain Index (SSI) maps.

Methods: The preoperative examinations of 29 progressive KC cases that were submitted to corneal cross-linking (CXL) were evaluated. The examinations included the tomography and the SSI measured by the Pentacam HR and the Corvis ST (Oculus, Wetzlar, Germany), respectively. The results were recorded twice, the latter of which was at the last visit before the CXL procedure. The patient-specific SSI maps were built, using data at each examination, based on finite element modelling and employing inverse analysis to represent the regional variation of biomechanical stiffness across the cornea.

Results: All cases presented significant shape progression (above the 95% CI of repeatability) in anterior and posterior curvatures and minimum thickness. The overall corneal stiffness as measured by the SSI within the central 8 mm-diameter area underwent slight but significant reductions from the first to the last examination (−0.02 ± 0.02, range: −0.09 to 0, p < 0.001). In all 29 cases, the reduction in stiffness was localised and concentred in the area inside the keratoconus cone. The SSI values inside the cone were significantly lower in the last examination (by 0.15 ± 0.09, range: −0.42 to −0.01, p < 0.001), while the SSI outside the cone presented minimal, non-significant variations (0 ± 0.01, range: −0.04 to 0.01, p = 0.999).

Conclusion: It has been observed through the SSI maps that the regional deterioration in stiffness was concerted inside the area of pathology, while only mild non-significant alterations were observed outside the area of pathology.

Stress-Strain Index Map: A New Way to Represent Corneal Material Stiffness

Purpose

To introduce a new method to map the mechanical stiffness of healthy and keratoconic corneas.

Methods

Numerical modeling based on the finite element method was used to carry out inverse analysis of simulated healthy and keratoconic corneas to determine the regional variation of mechanical stiffness across the corneal surface based on established trends in collagen fibril distribution. The Stress–Strain Index (SSI), developed and validated in an earlier study and presented as a parameter that can estimate the overall stress–strain behavior of corneal tissue, was adopted in this research as a measure of corneal stiffness. The regional variation of SSI across the corneal surface was estimated using inverse analysis while referring to the common features of collagen fibrils’ distribution obtained from earlier x-ray scattering studies. Additionally, for keratoconic corneas, a method relating keratoconic cone features and cornea’s refractive power to the reduction in collagen fibril density inside the cone was implemented in the development of SSI maps. In addition to the simulated cases, the study also included two keratoconus cases, for which SSI maps were developed.

Results

SSI values varied slightly across corneal surface in the simulated healthy eyes. In contrast, both simulated and clinical keratoconic corneas demonstrated substantial reductions in SSI values inside the cone. These SSI reductions depended on the extent of the disease and increased with more considerable simulated losses in fibril density in the cone area. SSI values and their regional variation showed little change with changes in IOP, corneal thickness, and curvature.

Conclusion

SSI maps provide an estimation of the regional variation of biomechanical stiffness across the corneal surface. The maps could be particularly useful in keratoconic corneas, demonstrating the dependence of corneal biomechanical behavior on the tissue’s microstructure and offering a tool to fundamentally understand the mechanics of keratoconus progression in individual patients.

Fibril density reduction in keratoconic corneas

This study aims to estimate the reduction in collagen fibril density within the central 6 mm radius of keratoconic corneas through the processing of microstructure and videokeratography data. Collagen fibril distribution maps and topography maps were obtained for seven keratoconic and six healthy corneas, and topographic features were assessed to detect and calculate the area of the cone in each keratoconic eye. The reduction in collagen fibril density within the cone area was estimated with reference to the same region in the characteristic collagen fibril maps of healthy corneas. Together with minimum thickness and mean central corneal refractive power, the cone area was correlated with the reduction in the cone collagen fibrils. For the corneas considered, the mean area of keratoconic cones was 3.30 ± 1.90 mm². Compared with healthy corneas, fibril density in the cones of keratoconic corneas was lower by as much as 35%, and the mean reduction was 17 ± 10%. A linear approximation was developed to relate the magnitude of reduction to the refractive power, minimum corneal thickness and cone area (R² = 0.95, p < 0.001). Outside the cone area, there was no significant difference between fibril arrangement in healthy and keratoconic corneas. The presented method can predict the mean fibril density in the keratoconic eye's cone area. The technique can be applied in microstructure-based finite-element models of the eye to regulate its stiffness level and the stiffness distribution within the areas affected by keratoconus.

Role of radially aligned scleral collagen fibers in optic nerve head biomechanics

Collagen fibers organized circumferentially around the canal in the peripapillary sclera are thought to provide biomechanical support to the sensitive tissues within the optic nerve head (ONH). Recent studies have demonstrated the existence of a family of fibers in the innermost sclera organized radially from the scleral canal. Our goal was to determine the role of these radial fibers in the sensitivity of scleral canal biomechanics to acute increases in intraocular pressure (IOP). Following the same general approach of previous parametric sensitivity studies, we created nonlinear generic finite element models of a posterior pole with various combinations of radial and circumferential fibers at an IOP of 0 mmHg. We then simulated the effects of normal and elevated IOP levels (15 and 30 mmHg). We monitored four IOP-induced geometric changes: peripapillary sclera stretch, scleral canal displacement, lamina cribrosa displacement, and scleral canal expansion. In addition, we examined the radial (maximum tension) and through-thickness (maximum compression) strains within the ONH tissues. Our models predicted that: 1) radial fibers reduced the posterior displacement of the lamina, especially at elevated IOP; 2) radial fibers reduced IOP-induced radial strain within the peripapillary sclera and retinal tissue; and 3) a combination of radial and circumferential fibers maintained strains within the ONH at a level similar to those conferred by circumferential fibers alone. In conclusion, radial fibers provide support for the posterior globe, additional to that provided by circumferential fibers. Most importantly, a combination of both fiber families can better protect ONH tissues from excessive IOP-induced deformation than either alone.

Numerical Simulation of Corneal Fibril Reorientation in Response to External Loading

Purpose: To simulate numerically the collagen fibril reorientation observed experimentally in the cornea. Methods: Fibril distribution in corneal strip specimens was monitored using X-ray scattering while under gradually increasing axial loading. The data were analysed at each strain level in order to quantify the changes in the angular distribution of fibrils with strain growth. The resulting relationship between stain and fibril reorientation was adopted in a constitutive model to control the mechanical anisotropy of the tissue material. The outcome of the model was validated against the experimental measurements before using the model in simplified representations of two surgical procedures. Results: The numerical model was able to reproduce the experimental measurements of specimen deformation and fibril reorientation under uniaxial loading with errors below 8.0%. With tissue removal simulated in a full eye numerical model, fibril reorientation could be predicted around the affected area, and this change both increased with larger tissue removal and reduced gradually away from that area. Conclusion: The presented method can successfully simulate fibril reorientation with changes in the strain regime affecting cornea tissue. Analyses based on this method showed that fibrils tend to align parallel to the tissue cut following keratoplasty operations. With the ability to simulate fibril reorientation, numerical modelling can have a greater potential in modelling the behaviour following surgery and injury to the cornea.

Scleral structure and biomechanics

As the eye's main load-bearing connective tissue, the sclera is centrally important to vision. In addition to cooperatively maintaining refractive status with the cornea, the sclera must also provide stable mechanical support to vulnerable internal ocular structures such as the retina and optic nerve head. Moreover, it must achieve this under complex, dynamic loading conditions imposed by eye movements and fluid pressures. Recent years have seen significant advances in our knowledge of scleral biomechanics, its modulation with ageing and disease, and their relationship to the hierarchical structure of the collagen-rich scleral extracellular matrix (ECM) and its resident cells. This review focuses on notable recent structural and biomechanical studies, setting their findings in the context of the wider scleral literature. It reviews recent progress in the development of scattering and bioimaging methods to resolve scleral ECM structure at multiple scales. In vivo and ex vivo experimental methods to characterise scleral biomechanics are explored, along with computational techniques that combine structural and biomechanical data to simulate ocular behaviour and extract tissue material properties. Studies into alterations of scleral structure and biomechanics in myopia and glaucoma are presented, and their results reconciled with associated findings on changes in the ageing eye. Finally, new developments in scleral surgery and emerging minimally invasive therapies are highlighted that could offer new hope in the fight against escalating scleral-related vision disorder worldwide.