Gradient moduli lens models: how material properties and application of forces can affect deformation and distributions of stress

Characterization of mechanical stress in the occurrence of cortical opacification in age-related cataracts using three-dimensional finite element model of the human lens and RNA-seq

Cataract is the leading cause of blindness across the world. Age-related cataract (ARC) is the most common type of cataract, but its pathogenesis is not fully understood. Using three-dimensional finite element modeling combining experimental biotechnology, our study demonstrates that external forces during accommodation cause mechanical stress predominantly in lens cortex, basically matching the localization of opacities in cortical ARCs. We identified the cellular senescence and upregulation of PIEZO1 mRNA in HLECs under mechanical stretch. This mechano-induced senescence in HLECs might be mediated by PIEZO1-related pathways, portraying a potential biomechanical cause of cortical ARCs. Our study updates the fundamental insight towards cataractogenesis, paving the way for further exploration of ARCs pathogenesis and nonsurgical treatment.

The Effect of Lens Shape, Zonular Insertion and Finite Element Model on Simulated Shape Change of the Eye Lens

The process of lens shape change in the eye to alter focussing (accommodation) is still not fully understood. Modelling approaches have been used to complement experimental findings in order to determine how constituents in the accommodative process influence the shape change of the lens. An unexplored factor in modelling is the role of the modelling software on the results of simulated shape change. Finite element models were constructed in both Abaqus and Ansys software using biological parameters from measurements of shape and refractive index of two 35-year-old lenses. The effect of zonular insertion on simulated shape change was tested on both 35-year-old lens models and with both types of software. Comparative analysis of shape change, optical power, and stress distributions showed that lens shape and zonular insertion positions affect the results of simulated shape change and that Abaqus and Ansys show differences in their respective models. The effect of the software package used needs to be taken into account when constructing finite element models and deriving conclusions.

The major influence of anterior and equatorial zonular fibres on the far-to-near accommodation revealed by a 3D pre-stressed model of the anterior eye

PURPOSE

To explore the synergistic function of the ligaments in eye, the zonular fibres, that mediate change in eye lens shape to allow for focussing over different distances.

METHODS

A set of 3D Finite Element models of the anterior eye together with a custom developed pre-stress modelling approach was proposed to simulate vision for distant objects (the unaccommodated state) to vision for near objects (accommodation). One of the five zonular groups was cut off in sequence creating five models with different zonular arrangements, the contribution of each zonular group was analysed by comparing results of each specific zonular-cut model with those from the all-zonules model in terms of lens shape and zonular tensions.

RESULTS

In the all-zonular model, the anterior and equatorial zonules carry the highest tensions. In the anterior zonular-cut model, the equatorial zonular tension increases while the posterior zonular tension decreases, resulting in an increase in the change in Central Optical Power (COP). In the equatorial zonular-cut model, both the anterior and posterior zonular tensions increase, causing a decreasing change in COP. The change in COP decreases only slightly in the other models. For vitreous zonular-cut models, little change was seen in either the zonular tension or the change in COP.

CONCLUSIONS

The anterior and the equatorial zonular fibres have the major influence on the change in lens optical power, with the anterior zonules having a negative effect and the equatorial zonules contributing a positive effect. The contribution to variations in optical power by the equatorial zonules is much larger than by the posterior zonules.

Design of an Automatically Controlled Multi-Axis Stretching Device for Mechanical Evaluations of the Anterior Eye Segment

The young eye has an accommodative ability involving lens shape changes to focus over different distances. This function gradually decreases with age, resulting in presbyopia. Greater insights into the mechanical properties of anterior eye structures can improve understanding of the causes of presbyopia. The present study aims to develop a multi-axis stretching device for evaluating the mechanical properties of the intact eye lens. A stretching device integrating the mechanical stretcher, motor, torque sensor and data transmission mechanism was designed and developed by 3D printing. The mechanical stretcher can convert rotation into radial movement, both at constant speeds, according to the spiral of Archimedes. The loading unit equipped with eight jaws can hold the eye sample tightly. The developed device was validated with a spring of known constant and was further tested with anterior porcine eye segments. The validation experiment using the spring resulted in stiffness values close to the theoretical spring constant. Findings from measurements with porcine eye samples indicated that the measured forces are within the ranges reported in the literature. The developed multi-axis stretching device has good repeatability during experiments with similar settings and can be reliably used for mechanical evaluations of the intact eye lens.

Influence of anterior capsulorhexis shape, centration, size, and location on intraocular lens position: finite element model

PURPOSE

To evaluate the influence of anterior capsulorhexis shape, dimension, and eccentricity on intraocular lens (IOL) position.

SETTING

Laboratory investigation.

DESIGN

Computational model.

METHODS

A finite element model of the human crystalline lens capsule and zonule was created and the anterior capsule opened to simulate centered and decentered circular and elliptic rhexis. The model calculated capsular bag stress, IOL rotation, tilt, decentration, and vaulting, related to both capsular landmarks (absolute) and a reference IOL position defined as that obtained with a 5.0 mm circular and centered rhexis.

RESULTS

Mean von Mises stress along the IOL major z-axis was significantly higher than that along the perpendicular x-axis in all cases (P < .001), both at the equator and at the rhexis edge. Stress at the equator was always greater than that at the rhexis edge (P < .001) regardless of the rhexis shape and position. As rhexis eccentricity increased, the stress difference between the z- and x-axes increased. Absolute IOL tilt (range 10-1 to 10-7 degrees), decentration (10-3 to 10-7 mm), rotation (10-2 to 10-3 degrees), and vaulting (10-1 mm) were negligible from an optical standpoint, but all of them were significantly greater for decentered rhexis (both round and elliptic) compared with centered (P < .05).

CONCLUSIONS

Anterior capsulorhexis irregularity and/or eccentricity increase IOL tilt, decentration, rotation, and vaulting in a numerically significant but optically negligible way. Von Mises stress is much greater at the capsular bag equator compared with the rhexis edge and highly asymmetrically distributed in all cases. Stress asymmetry may influence postoperative biologic processes of capsular bag shrinking and further IOL tilting or decentration.

Characterisation and Modelling of an Artificial Lens Capsule Mimicking Accommodation of Human Eyes

A synthetic material of silicone rubber was used to construct an artificial lens capsule (ALC) in order to replicate the biomechanical behaviour of human lens capsule. The silicone rubber was characterised by monotonic and cyclic mechanical tests to reveal its hyper-elastic behaviour under uniaxial tension and simple shear as well as the rate independence. A hyper-elastic constitutive model was calibrated by the testing data and incorporated into finite element analysis (FEA). An experimental setup to simulate eye focusing (accommodation) of ALC was performed to validate the FEA model by evaluating the shape change and reaction force. The characterisation and modelling approach provided an insight into the intrinsic behaviour of materials, addressing the inflating pressure and effective stretch of ALC under the focusing process. The proposed methodology offers a virtual testing environment mimicking human capsules for the variability of dimension and stiffness, which will facilitate the verification of new ophthalmic prototype such as accommodating intraocular lenses (AIOLs).

A validated finite element model to reproduce Helmholtz's theory of accommodation: a powerful tool to investigate presbyopia

PURPOSE

To reproduce human in vivo accommodation numerically. For that purpose, a finite element model specific for a 29-year-old subject was designed. Once the proposed numerical model was validated, the decrease in accommodative amplitude with age was simulated according to data available in the literature.

METHODS

In contrast with previous studies, the non-accommodated eye condition was the reference configuration. Consequently, two aspects were specifically highlighted: contraction of the ciliary muscle, which was simulated by a continuum electro-mechanical model and incorporation of initial lens capsule stresses, which allowed the lens to become accommodated after releasing the resting zonular tension.

RESULTS

The morphological changes and contraction of the ciliary muscle were calibrated accurately according to the experimental data from the literature. All dynamic optical and biometric lens measurements validated the model. With the proposed numerical model, presbyopia was successfully simulated.

CONCLUSIONS

The most widespread theory of accommodation, proposed by Helmholtz, was simulated accurately. Assuming the same initial stresses in the lens capsule over time, stiffening of the lens nucleus is the main cause of presbyopia.

A numerical investigation of changes in lens shape during accommodation

The purpose of this study was to investigate how the mechanical properties and geometry of the lens influence the changes in lens shape during accommodation. To do so, ex vivo stretching tests of the isolated lens were simulated via finite element analysis. In these tests, the lens is stretched from the accommodated state to the non-accommodated state. Several key characteristics of the lens were studied: the stiffness gradient of the lens material, the distribution of the capsule thickness, the mechanical properties of the capsule and the material comprising the lens, nucleus and cortex, and the influence of two different age-related lens geometries (17 and 29 y/o subjects). To determine the effects on the changes in lens shape during accommodation, changes in the anterior and posterior radius, the lens and nucleus thicknesses and the equatorial lens diameter were analysed. The results suggest that multiple factors exert statistically significant influences on how the lens changes its shape, but two factors predominate over the rest: the stiffness ratio between the nucleus and cortex and the stiffness of the capsule, specifically the posterior surface.

Deformations and Ruptures in Human Lenses With Cortical Cataract Subjected to Ex Vivo Simulated Accommodation

Purpose

Human cortical opacities are most commonly accompanied by changes in lens fiber structure in the equatorial region at the lens nucleus–cortex interface. Cortex and nucleus have different elastic properties, which change with age. We therefore subjected ex vivo lenses to simulated accommodation and studied the internal deformations to better understand the mechanism of cortical cataract formation.

Methods

Nine human donor lenses (33–88 years old) were tested using a bespoke radial stretching device for anterior eye segments. Seven of the lenses exhibited cortical cataracts. The other two lenses, without cataract, were used as controls. Frontal and cross-sectional images of the lens obtained during stretching facilitated measurements on equatorial lens diameter and central lens thickness in the stretched and unstretched states.

Results

Stretching caused the lens equatorial diameter to increase in all cases. Conversely, the lens central thickness showed no systematic variation during stretching. For four of the lenses with cortical cataract, ruptures were observed during stretching at the nucleus–cortex boundary adjacent to the cortical cataracts. Ruptures were not observed in the control lenses or in the three other lenses with cortical cataract.

Conclusions

Internal ruptures can occur in aged ex vivo lenses subjected to simulated disaccommodation. These ruptures occur at the nucleus–cortex interface; at this location, a significant stiffness discontinuity is expected to develop with age. It is hypothesized that ruptures occur in in vivo lenses during accommodation—or attempted accommodation.

The Structure of the Lens and Its Associations with the Visual Quality

In humans, the lens is the organ with the ability to change morphology and refractive power, designated as accommodation, to focus light from various distances and obtain clear retinal image. The accommodative ability of the lens depends on its structure and biological parameters. The lens grows throughout the life, forming specific lens sutures and a unique gradient refractive index, and possesses regenerative ability under certain circumstances. Minimally invasive lens surgery that preserves endogenous lens epithelial stem/progenitor cells (LECs) can achieve functional lens regeneration in humans. The lens is the main source of intraocular aberration, especially intraocular higher-order aberrations (IHOAs) which is found to be binocularly symmetrical in phakic eyes. There is a compensation mechanism between corneal aberrations and lens aberrations. Therefore, the structure and the biological parameters of the lens, the binocular relationship of the lens and the correlation between the lens and cornea affect visual quality. This paper summarises the above findings and their current and potential applications in refractive surgeries, providing a comprehensive understanding of the lens as a strong determinant of visual quality in the optical system.

Image registration of the human accommodating eye demonstrates equivalent increases in lens equatorial radius and central thickness

AIM

To compare the results of in vivo human high resolution image registration studies of the eye during accommodation to the predictions of mathematical and finite element models of accommodation.

METHODS

Data from published high quality image registration studies of pilocarpine induced accommodative changes of equatorial lens radius (ELR) and central lens thickness (CLT) were statistically analyzed.

RESULTS

The mean changes in ELR and CLT were 6.76 µm/diopter and 6.51 µm/diopter, respectively. The linear regressions, reflecting the association between ELR and accommodative amplitude (AAELR) was: slope=6.58 µm/diopter, r2 =0.98, P<0.0001 and between CLT and AACLT was: slope=6.75 µm/diopter, r2 =0.83, P<0.001. On the basis of these relationships, the CLT slope and the AAELR were used to predict the measured change in ELR (ELRpredicted). There was no statistical difference between ELRpredicted and the measured ELR as demonstrated by a Student's paired t-test: P=0.96 and linear regression analysis: slope=0.97, r2 =0.98, P<0.00001.

CONCLUSION

Image registration with invariant positional references demonstrates that ELR and CLT equivalently minimally increase ∼7.0 µm/diopter during accommodation. The small equivalent increases in ELR and CLT are associated with a large accommodative amplitude. These findings are consistent with the predictions of mathematical and finite element models that specified the stiffness of the lens nucleus is the same or greater than the lens cortex and that accommodation involves a small force (<5 g).

A Modeling Approach for Investigating Opto-Mechanical Relationships in the Human Eye Lens

OBJECTIVE

The human visual system alters its focus by a shape change of the eye lens. The extent to which the lens can adjust ocular refractive power is dependent to a significant extent on its material properties. Yet, this fundamental link between the optics and mechanics of the lens has been relatively under-investigated. This study aims to investigate this opto-mechanical link within the eye lens to gain insight into the processes of shape alteration and their respective decline with age.

METHODS

Finite Element models based on biological lenses were developed for five ages: 16, 35, 40, 57, and 62 years by correlating in vivo measurements of the longitudinal modulus using Brillouin scattering with in vitro X-ray interferometric measurements of refractive index and taking into account various directions of zonular force.

RESULTS

A model with radial cortical Young's moduli provides the same amount of refractive power with less change in thickness than a model with uniform cortical Young's modulus with a uniform stress distribution and no discontinuities along the cortico-nuclear boundary. The direction of zonular angles can significantly influence curvature change regardless of the modulus distribution.

CONCLUSIONS

The present paper proposes a modelling approach for the human lens, coupling optical and mechanical properties, which shows the effect of parameter choice on model response.

SIGNIFICANCE

This advanced modelling approach, considering the important interplay between optical and mechanical properties, has potential for use in design of accommodating implant lenses and for investigating non-biological causes of pathological processes in the lens (e.g., cataract).

Refractive index redistribution with accommodation based on finite volume-constant age-dependent mechanical modeling

The human lens is considered to have a gradient refractive index (GRIN) distribution. The recently developed accommodating volume-constant age-dependent optical (AVOCADO) model can accurately describe the separate GRIN distributions in the axial and radial directions. Our study uses a finite element method to simulate the accommodation process and calculate the GRIN redistribution based on the AVOCADO model for 25-, 35- and 45-year-old lenses. The parameter p describes the steepness of the GRIN profile towards the lens periphery. The results show that axial p values increase with age. Under accommodation, the axial p value increases, while the radial p value decreases. We also use a ray tracing method to evaluate the optical performance of the lens. The aim of this paper is thus to provide an anatomically finite mechanical lens model with separate axial and radial refractive index profiles for a better understanding of accommodation at different ages.

Contributions of shape and stiffness to accommodative loss in the ageing human lens: a finite element model assessment

Ageing changes to the various components of the accommodative system of the eye lens contribute to the loss of focusing power. The relative contributions of each ageing component, however, are not well defined. This study investigates the contribution of geometric parameters and material properties on accommodation, simulated using models based on human lenses aged 16, 35, and 48 years. Each model was tested using two different sets of material properties and a range of zonular fiber angles and was compared to results from in vivo measurements. The geometries and material parameters of older and younger lens models were interchanged to investigate the role of shape and material on accommodative capacity. Results indicate that geometry has the greater role in accommodation.

The importance of parameter choice in modelling dynamics of the eye lens

The lens provides refractive power to the eye and is capable of altering ocular focus in response to visual demand. This capacity diminishes with age. Current biomedical technologies, which seek to design an implant lens capable of replicating the function of the biological lens, are unable as yet to provide such an implant with the requisite optical quality or ability to change the focussing power of the eye. This is because the mechanism of altering focus, termed accommodation, is not fully understood and seemingly conflicting theories require experimental support which is difficult to obtain from the living eye. This investigation presents finite element models of the eye lens based on data from human lenses aged 16 and 35 years that consider the influence of various modelling parameters, including material properties, a wide range of angles of force application and capsular thickness. Results from axisymmetric models show that the anterior and posterior zonules may have a greater impact on shape change than the equatorial zonule and that choice of capsular thickness values can influence the results from modelled simulations.