Aspects of eye accommodation evaluated by finite elements

The potential influence of the ligament of Wieger on the crystalline lens shape

This research uses mathematical modelling to evaluate the influence of the ligament of Wieger on the crystalline lens shape at rest, and during accommodation. An axisymmetric model of the anterior segment, including the ligament of Wieger, was created using the finite element method. Different conditions including variations of stiffness and positions of the ligament, with and without the ligament, were tested to see how they affected lens curvature and optical power. Adding the ligament of Wieger to the simulation had a noticeable impact on the optical power of the lens, particularly on the posterior surface power and total power. Ligament stiffness and width significant influenced the accommodative range of the eye by - 0.95D and - 2.39D for ligaments with the same and 3× the stiffness of the capsular bag, respectively. Ligament width and inner diameter had negligible effects on lens thickness but did have significant effects on posterior surface power and accommodation. In this simulation, we found that the ligament of Wieger can significantly affect the lens shape, both at rest and during accommodation, and may need to be considered in lens models.

A Biomechanical Model for Evaluating the Performance of Accommodative Intraocular Lenses

Accommodation alters the shape of the eye lens to change focus from distant to near vision. This function declines with age in the development of presbyopia and most people experience a near total loss of accommodative ability by 55 years. Currently, there are no surgical procedures that correct presbyopia, but considerable work has been done in the development of accommodative intraocular lenses (AIOLs) implanted during cataract surgery. Despite these efforts, AIOLs only restore ∼ 20% of youthful accommodative amplitude and they suffer from high rates of visually-debilitating fibrosis. An important design tool that is lacking that could aid in improving AIOL designs is modeling. Herein, we addressed this need through the development of a fully 3-D finite element model that was used to predict the behavior of a dual-optic AIOL implanted within the post-surgical lens capsule. Models of the native human lens were developed to identify the stress-free configuration of the lens capsule needed to accurately predict the accommodated state of the lens and the configuration of the zonular traction needed for the disaccommodated state. The AIOL model demonstrated the functional importance of implant stiffness and predicted an approximately linear relationship between zonular traction magnitude and axial displacement of the optics. To our knowledge, this is the first model that can be used to gain insights into AIOL efficacy. It provides a foundation for continued development of a predictive tool that could ultimately improve AIOL designs that seek to restore youthful accommodative function.

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 study for lens capsule tearing during capsulotomy by finite element simulation

BACKGROUND AND OBJECTIVE

During capsulotomy, the force applied to the anterior capsule is a crucial parameter controlling capsule tears, that affects the clinical performance. This study aims to investigate the tear force in capsulotomy and analyze the effects of different tearing conditions on the tear force.

METHODS

A three-dimensional model of the human lens was constructed based on published clinical data using the finite element (FE) method. The lens model consisted of four layers: the anterior and posterior lens capsule, the cortex, and the nucleus. Distortion energy failure criterion combined with the bilinear interface law was used to express the crack propagation process at the edge of the anterior lens capsule. At the clamping position, a local coordinate system was established to parameterize the capsule tearing. The simulation results were then validated by conducting a capsulorhexis experiment using isolated porcine eyes with force-sensing forceps.

RESULTS

The simulation results showed a good agreement with the experimental data of two porcine specimens (No. 6 and 9) during a stable tearing process (p-values = 0.76 and 0.10). The mean force differences between the experimental data and the simulation were 3.10 ± 2.24 mN and 2.14 ± 1.73 mN, respectively. The tear direction with a minimum mean tear force was at θ₁ = 0° and θ₂ = 30°. The tear velocity was not significantly different to the variation in the tear force. However, an appropriate capsulorhexis diameter was found to contribute to the reduction of tear force.

CONCLUSIONS

The outcome of this paper demonstrates that our FE model could be used in modeling lens capsule tearing and the theoretical study of tear mechanism.

The action of ciliary muscle contraction on accommodation of the lens explored with a 3D model

The eye's accommodative mechanism changes optical power for near vision. In accommodation, ciliary muscle excursion relieves lens tension, allowing it to return to its more convex shape. Lens deformation alters its refractive properties, but the mechanics of ciliary muscle actions are difficult to intuit due to the complex architecture of the tissues involved. The muscle itself comprises three sections of dissimilarly oriented cells. These cells contract, transmitting forces through the zonule fibers and extralenticular structures. This study aims to create a finite element model (FEM) to predict how the action of the ciliary muscle sections leads to lens displacement. The FEM incorporates initialization of the disaccommodated lens state and ciliary muscle contraction, with three muscle sections capable of independent activation, to drive accommodative movement. Model inputs were calibrated to replicate experimentally measured disaccommodated lens and accommodated ciliary muscle shape changes. Additional imaging studies were used to validate model predictions of accommodative lens deformation. Models were analyzed to quantify mechanical actions of ciliary muscle sections in lens deformation and position modulation. Analyses revealed that ciliary muscle sections act synergistically: the circular section contributes most to increasing lens thickness, while longitudinal and radial sections can oppose this action. Conversely, longitudinal and radial sections act to translate the lens anteriorly with opposition from the circular section. This FEM demonstrates the complex interplay of the three sections of ciliary muscle in deforming and translating the lens during accommodation, providing a useful framework for future investigations of accommodative dysfunction that occurs with age in presbyopia.

Altered stress field of the human lens capsule after cataract surgery

The lens capsule of the eye is important in focusing light onto the retina during the process of accommodation and, in later life, housing a prosthetic lens implanted during cataract surgery. Though considerable modeling work has characterized the mechanics of accommodation, little has been done to understand the mechanics of the lens capsule after cataract surgery. As such, we present the first 3-D finite element model of the post-surgical human lens capsule with an implanted tension ring and, separately, an intraocular lens to characterize the altered stress field compared to that in a model of the native lens capsule. All finite element models employed a Holzapfel hyperelastic constitutive model with regional variations in anisotropy. The post-surgical lens capsule demonstrated a dramatic perturbation to the stress field with mostly large reductions in stresses (except at the equator where the implant contacts the capsule) compared to native, wherein maximal changes in Cauchy stress were -100% and -145% for the tension ring and intraocular lens, respectively. However, implantation of the tension ring produced a more uniform stress field compared to the IOL. The magnitudes and distribution of the perturbed stress field may be an important driver of the fibrotic response of inhabiting lens epithelial cells and associated lens capsule remodeling after cataract surgery. Thus, the mechanical effects of an implant on the lens capsule could be an essential consideration in the design of intraocular lenses, particularly those with an accommodative feature.

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.

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

The human lens provides one-third of the ocular focussing power and is responsible for altering focus over a range of distances. This ability, termed accommodation, defines the process by which the lens alters shape to increase or decrease ocular refractive power; this is mediated by the ciliary muscle through the zonule. This ability decreases with age such that around the sixth decade of life it is lost rendering the eye unable to focus on near objects. There are two opponent theories that provide an explanation for the mechanism of accommodation; definitive support for either of these requires investigation. This work aims to elucidate how material properties can affect accommodation using Finite Element models based on interferometric measurements of refractive index. Gradients of moduli are created in three models from representative lenses, aged 16, 35 and 48 years. Different forms of zonular attachments are studied to determine which may most closely mimic the physiological form by comparing stress and displacement fields with simulated shape changes to accommodation in living lenses. The results indicate that for models to mimic accommodation in living eyes, the anterior and posterior parts of the zonule need independent force directions. Choice of material properties affects which theory of accommodation is supported.

Stretch-dependent changes in surface profiles of the human crystalline lens during accommodation: a finite element study

BACKGROUND

A non-linear isotropic finite element (FE) model of a 29-year-old human crystalline lens was constructed to study the effects of various geometrical parameters on lens accommodation.

METHODS

The model simulates dis-accommodation by stretching of the lens and predicts the change in surface profiles of the lens capsule, cortex and nucleus at select states of stretching/accommodation. Multiple regression analysis (MRA) is used to develop a stretch-dependent mathematical model relating the lens sagittal height to the radial position of the lens surface as a function of dis-accommodative stretch. A load analysis is performed to compare the finite element results to empirical results from lens stretcher studies. Using the predicted geometrical changes, the optical response of the whole eye during accommodation was analysed by ray-tracing.

RESULTS

Aspects of lens shape change relative to stretch were evaluated, including change in diameter, central thickness and accommodation. Maximum accommodation achieved was 10.29 D. From the multiple regression analysis, the stretch-dependent mathematical model of the lens shape related lens curvatures as a function of lens ciliary stretch well (maximum mean-square residual error 2.5 × 10(-3 ) μm, p < 0.001). The results are compared with those from in vitro studies.

CONCLUSIONS

The finite element and ray-tracing predictions are consistent with Ex Vivo Accommodation Simulator (EVAS) studies in terms of load and power change versus change in thickness. The mathematical stretch-dependent model of accommodation presented may have utility in investigating lens behaviour at states other than the relaxed or fully accommodated states.

Corneal biomechanics: a review

Biomechanics is often defined as 'mechanics applied to biology'. Due to the variety and complexity of the behaviour of biological structures and materials, biomechanics is better defined as the development, extension and application of mechanics for a better understanding of physiology and physiopathology and consequently for a better diagnosis and treatment of disease and injury. Different methods for the characterisation of corneal biomechanics are reviewed in detail, including those that are currently commercially available (Ocular Response Analyzer and CorVis ST). The clinical applicability of the parameters provided by these devices are discussed, especially in the fields of glaucoma, detection of ectatic disorders and orthokeratology. Likewise, other methods are also reviewed, such as Brillouin microscopy or dynamic optical coherence tomography and others with potential application to clinical practice but not validated for in vivo measurements, such as ultrasonic elastography. Advantages and disadvantages of all these techniques are described. Finally, the concept of biomechanical modelling is revised as well as the requirements for developing biomechanical models, with special emphasis on finite element modelling.

Materials characterization and mechanobiology of the eye

The eye responds to a great deal of internal and external stimuli throughout its normal function. Due to this, a mechanical or chemical analysis alone is insufficient. A systematic materials characterization is needed. A mechanobiological approach is required for a full understanding of the unique properties and function of the eye. This review compiles the mechanical properties of select eye components, summarizes mechanical and chemical testing platforms, and overviews modeling approaches. Analysis is done across studies, experimental methods, and between species in order to summarize what is known about the mechanobiology of the eye. Several opportunities for future research are identified.

Hyperelastic modelling of the crystalline lens: Accommodation and presbyopia

Purpose

The modification of the mechanical properties of the human crystalline lens with age can be a major cause of presbyopia. Since these properties cannot be measured in vivo, numerical simulation can be used to estimate them. We propose an inverse method to determine age-dependent change in the material properties of the tissues composing the human crystalline lens.

Methods

A finite element model of a 30-year-old lens in the accommodated state was developed. The force necessary to achieve full accommodation in a 30-year-old lens of known external geometry was computed using this model. Two additional numerical models of the lens corresponding to the ages of 40 and 50 years were then built. Assuming that the accommodative force applied to the lens remains constant with age, the material properties of nucleus and cortex were estimated by inverse analysis.

Results

The zonular force necessary to reshape the model of a 30-year-old lens from the accommodated to the unaccommodated geometry was 0.078 newton (N). Both nucleus and cortex became stiffer with age. The stiffness of the nucleus increased with age at a higher rate than the cortex.

Conclusions

In agreement with the classical theory of Helmholtz, on which we based our model, our results indicate that a major cause of presbyopia is that both nucleus and cortex become stiffer with age; therefore, a constant value of the zonular forces with aging does not achieve full accommodation, that is, the accommodation capability decreases.