A structural constitutive model for the human lens capsule

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.

Simulation of defects, flexibility and rupture in biopolymer networks

Networks of biopolymers occur often in nature, and are vulnerable to damage over time. In this work, a coarse grained model of collagen IV molecules is applied in a 2D hexagonal network and the mechanisms by which these networks can rupture are explored. The networks are stretched linearly in order to study their structural limits and mechanism of rupture over timescale of up to 100 μs. Metrics are developed to track the damage networks suffer over time, and qualitatively analyse ruptures that occur. Further simulations repeatedly stretch the networks sinusoidally to mimic the in vivo strains. Defects of increasing levels of complexity are introduced into an ordered network, and their effect on the rupturing behaviour of the biopolymer networks studied. The effect of introducing holes of varying size in the network, as well as strips of finite width to mimic surgical damage are studied. These demonstrate the importance of the flexibility of the networks to preventing damage.

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).

Self assembly of model polymers into biological random networks

The properties of biological networks, such as those found in the ocular lens capsule, are difficult to study without simplified models. Model polymers are developed, inspired by "worm-like" curve models, that are shown to spontaneously self assemble to form networks similar to those observed experimentally in biological systems. These highly simplified coarse-grained models allow the self assembly process to be studied on near-realistic time-scales. Metrics are developed (using a polygon-based framework) which are useful for describing simulated networks and can also be applied to images of real networks. These metrics are used to show the range of control that the computational polymer model has over the networks, including the polygon structure and short range order. The structure of the simulated networks are compared to previous simulation work and microscope images of real networks. The network structure is shown to be a function of the interaction strengths, cooling rates and external pressure. In addition, "pre-tangled" network structures are introduced and shown to significantly influence the subsequent network structure. The network structures obtained fit into a region of the network landscape effectively inaccessible to random (entropically-driven) networks but which are occupied by experimentally-derived configurations.

A numerical model of capsulorhexis to assess the relevance of size and position of the rhexis on the IOL decentering and tilt

Finite element models of human crystalline capsular bag and zonular fibers are used to estimate the mechanical response of the capsule to the presence of a C-loop Intra-Ocular Lens (IOL) after cataract surgery, to assess the influence of capsulorhexis size, shape and location on IOL decentration and tilt. The model includes, in the anterior capsule, a hole with variable size, shape and position, which represents the rhexis obtained in a manual or laser-assisted manner. The IOL is not explicitly modelled, but its action is reproduced by means of a set of elastic ties, connecting the opposite sides of the bag and exerting the force corresponding to an expanded IOL. Numerical simulations show that IOL decentration and tilt are not related to the size of the rhexis. A decentered rhexis induces IOL decentration <40 μm and a tilt <12°; the combination of non-circular shape and decentering of the rhexis induces IOL decentration <47 μm. While the introduction of a circular central rhexis causes an increment of the stresses in the capsule up to 100% relative to the physiological state, the combination of rhexis decentering and non-circular shape causes an additional 10% stress change. The values of IOL decentering and IOL tilt obtained from simulations are of scarce relevance in the clinical practice.

Simplified computational model for generating biological networks

A method to generate and simulate biological networks is discussed. An expanded Wooten–Winer–Weaire bond switching methods is proposed which allows for a distribution of node degrees in the network while conserving the mean average node degree. The networks are characterised in terms of their polygon structure and assortativities (a measure of local ordering). A wide range of experimental images are analysed and the underlying networks quantified in an analogous manner. Limitations in obtaining the network structure are discussed. A “network landscape” of the experimentally observed and simulated networks is constructed from the underlying metrics. The enhanced bond switching algorithm is able to generate networks spanning the full range of experimental observations.

We discuss a Monte Carlo method to simulate biological networks and compare to the underlying networks in experimental images.

Biomechanical Evaluations of Ocular Injury Risk for Blast Loading

Ocular trauma is one of the most common types of combat injuries resulting from the exposure of military personnel with improvised explosive devices. The injury mechanism associated with the primary blast wave is poorly understood. We employed a three-dimensional computational model, which included the main internal ocular structures of the eye, spatially varying thickness of the cornea-scleral shell, and nonlinear tissue properties, to calculate the intraocular pressure and stress state of the eye wall and internal ocular structure caused by the blast. The intraocular pressure and stress magnitudes were applied to estimate the injury risk using existing models for blunt impact and blast loading. The simulation results demonstrated that blast loading can induce significant stresses in the different components of the eyes that correlate with observed primary blast injuries in animal studies. Different injury models produced widely different injury risk predictions, which highlights the need for experimental studies evaluating mechanical and functional damage to the ocular structures caused by the blast loading.

Anterior lens capsule strains during simulated accommodation in porcine eyes

Experimental protocols have been developed to measure the spatial variation of the mechanical strains induced in the lens capsule during ex vivo lens stretching. The paper describes the application of these protocols to porcine lenses. The deformations and mechanical strains developed in the anterior capsule during each experiment were determined using full field digital image correlation techniques, by means of a speckle pattern applied to the lens surface. Several speckling techniques and illumination methods were assessed before a suitable combination was found. Additional data on the cross section shape of the anterior lens surface were obtained by Scheimpflug photography, to provide a means of correcting for lens curvature effects in the determination of the strains developed in the plane of the capsule. The capsule strains in porcine lenses exhibit non-linear behaviour, and hysteresis during loading and unloading. Peripheral regions experience higher magnitude strains than regions near the lens pole. The paper demonstrates the successful application of a procedure to make direct measurements of capsule strains simultaneously with ex vivo radial lens stretching. This experimental technique is applicable to future investigations on the mechanical characteristics of human lenses.

The lens growth process

The factors that regulate the size of organs to ensure that they fit within an organism are not well understood. A simple organ, the ocular lens serves as a useful model with which to tackle this problem. In many systems, considerable variance in the organ growth process is tolerable. This is almost certainly not the case in the lens, which in addition to fitting comfortably within the eyeball, must also be of the correct size and shape to focus light sharply onto the retina. Furthermore, the lens does not perform its optical function in isolation. Its growth, which continues throughout life, must therefore be coordinated with that of other tissues in the optical train. Here, we review the lens growth process in detail, from pioneering clinical investigations in the late nineteenth century to insights gleaned more recently in the course of cell and molecular studies. During embryonic development, the lens forms from an invagination of surface ectoderm. Consequently, the progenitor cell population is located at its surface and differentiated cells are confined to the interior. The interactions that regulate cell fate thus occur within the obligate ellipsoidal geometry of the lens. In this context, mathematical models are particularly appropriate tools with which to examine the growth process. In addition to identifying key growth determinants, such models constitute a framework for integrating cell biological and optical data, helping clarify the relationship between gene expression in the lens and image quality at the retinal plane.

Finite element implementation of a multiscale model of the human lens capsule

An axisymmetric finite element implementation of a previously described structural constitutive model for the human lens capsule (Burd in Biomech Model Mechanobiol 8(3):217-231, 2009) is presented. This constitutive model is based on a hyperelastic approach in which the network of collagen IV within the capsule is represented by an irregular hexagonal planar network of hyperelastic bars, embedded in a hyperelastic matrix. The paper gives a detailed specification of the model and the periodic boundary conditions adopted for the network component. Momentum balance equations for the network are derived in variational form. These balance equations are used to develop a nonlinear solution scheme to enable the equilibrium configuration of the network to be computed. The constitutive model is implemented within a macroscopic finite element framework to give a multiscale model of the lens capsule. The possibility of capsule wrinkling is included in the formulation. To achieve this implementation, values of the first and second derivatives of the strain energy density with respect to the in-plane stretch ratios need to be computed at the local, constitutive model, level. Procedures to determine these strain energy derivatives at equilibrium configurations of the network are described. The multiscale model is calibrated against previously published experimental data on isolated inflation and uniaxial stretching of ex vivo human capsule samples. Two independent example lens capsule inflation analyses are presented.

Imaging shear stress distribution and evaluating the stress concentration factor of the human eye

Healthy eyes are vital for a better quality of human life. Historically, for man-made materials, scientists and engineers use stress concentration factors to characterise the effects of structural non-homogeneities on their mechanical strength. However, such information is scarce for the human eye. Here we present the shear stress distribution profiles of a healthy human cornea surface in vivo using photo-stress analysis tomography, which is a non-intrusive and non-X-ray based method. The corneal birefringent retardation measured here is comparable to that of previous studies. Using this, we derive eye stress concentration factors and the directional alignment of major principal stress on the surface of the cornea. Similar to thermometers being used for monitoring the general health in humans, this report provides a foundation to characterise the shear stress carrying capacity of the cornea, and a potential bench mark for validating theoretical modelling of stresses in the human eye in future.

Geometric characterization and simulation of planar layered elastomeric fibrous biomaterials

Many important biomaterials are composed of multiple layers of networked fibers. While there is a growing interest in modeling and simulation of the mechanical response of these biomaterials, a theoretical foundation for such simulations has yet to be firmly established. Moreover, correctly identifying and matching key geometric features is a critically important first step for performing reliable mechanical simulations. The present work addresses these issues in two ways. First, using methods of geometric probability, we develop theoretical estimates for the mean linear and areal fiber intersection densities for 2-D fibrous networks. These densities are expressed in terms of the fiber density and the orientation distribution function, both of which are relatively easy-to-measure properties. Secondly, we develop a random walk algorithm for geometric simulation of 2-D fibrous networks which can accurately reproduce the prescribed fiber density and orientation distribution function. Furthermore, the linear and areal fiber intersection densities obtained with the algorithm are in agreement with the theoretical estimates. Both theoretical and computational results are compared with those obtained by post-processing of scanning electron microscope images of actual scaffolds. These comparisons reveal difficulties inherent to resolving fine details of multilayered fibrous networks. The methods provided herein can provide a rational means to define and generate key geometric features from experimentally measured or prescribed scaffold structural data.

Evaluation of the mechanical behaviour and estimation of the elastic properties of porcine zonular fibres

The mechanical behaviour of zonular fibres greatly affects the accommodation process in mammalian eyes. This paper introduces a detailed measurement procedure for the purpose of obtaining the force-displacement diagram necessary to evaluate the mechanical properties of porcine zonular fibres in situ. It is a complex technique, keeping the integrity of the zonular bundles between the crystalline lens and the ciliary muscle cells. We present a brief description of the measurement procedure both in theory and in practice, along with the force-displacement diagrams acquired from a porcine sample group. The strengths of this newly developed method are the unequivocal force transmission between the sample and the transducer, and the intact connection between the ciliary body and the crystalline lens via zonular fibres. With the aid of these measurements, we define an estimated material model for the zonular apparatus both analytically and using the finite element method. The two different evaluation methods show close agreement in the calculated Young's modulus for the zonular fibres. The range of the calculated elastic modulus is 200-250 kPa. This new measuring method is adaptable to human specimens. Despite its complexity, the entire procedure and the evaluation part are reproducible. The constitutive model aims to shed light on the mechanics of the accommodation process.

A quantitative geometric mechanics lens model: insights into the mechanisms of accommodation and presbyopia

This study expands on a geometric model of ocular accommodation (Reilly and Ravi, Vision Res. 50:330-336; 2010) by relaxing assumptions regarding lens symmetry about the equator. A method for predicting stretching force was derived. Two models were then developed: Model 1 held the equatorial geometry constant at all stages of accommodation, while Model 2 allowed localized deformation at the equator. Both models were compared to recent data for axial thickness, anterior and posterior radii of curvature, surface area, cross-sectional area, volume, and stretching force for the 29-year-old lens. Age-related changes in accommodation were also simulated. Model 1 gave predictions which agreed with the Helmholtz theory of accommodation, while Model 2's predictions agreed with the Schachar mechanism of accommodation. Trends predicted by Model 1 agreed with all available experimental data, while Model 2 disagreed with recent surface area measurements. Further analysis indicated that Model 1 was fundamentally more efficient in that it required less force per diopter change in optical power than Model 2. Model 1 more accurately predicted age-related changes in accommodation amplitude. This implies that the zero-force (fully accommodated) state geometry changes with age due to a shifting balance in residual stresses between the lens and capsule.

Mechanical response of wild-type and Alport murine lens capsules during osmotic swelling

The mechanical support of basement membranes, such as the lens capsule, is believed to arise from one of their main constituents - collagen IV. The basement membranes of the lens, kidney, and ear normally contain two different types of collagen IV networks, referred to as the major and minor chain networks. In Alport syndrome, a mutation in one of the minor chain COL4 genes leads to the absence of the minor chain network, causing life-threatening disturbances. We hypothesized that the absence of the minor chain network increases basement membrane distensibility, as measured in wild-type (n = 25) and Alport syndrome (n = 21) mice using the lens capsule as a model. Osmotic swelling experiments revealed direction-dependent changes. As a reflection of lens capsule properties, Alport lenses strained significantly more than wild-type lenses in the anterior-posterior direction, i.e. along their thickness, but not in the equatorial direction (p = 0.03 and p = 0.08, respectively). This is consistent with clinical data: Alport patients develop conical protrusions on the anterior and posterior lenticular poles. There was no evidence of significant change in total amount of collagen between Alport and wild-type lenses (p = 0.6). The observed differences in distensibility could indicate that the major chain network alone cannot fully compensate for the absence of the more highly cross-linked minor chain network, which is believed to be stronger, more stable, and resistant to deformation. The addition of mechanical information on Alport syndrome to the currently available biological data provides a fuller picture into the progression of the disease.

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.

Impact of heparan sulfate chains and sulfur-mediated bonds on the mechanical properties of bovine lens capsule

We assessed the importance of glycosaminoglycans and sulfur-mediated bonds for the mechanical properties of lens capsules by comparing the stress-strain responses from control and treated pairs of bovine source. No significant change in mechanical properties was observed upon reduction of disulfide bonds. However, removal of glycosaminoglycan chains resulted in a significantly stiffer lens capsule, whereas high concentrations of reducing agent, which is expected to reduce the recently reported sulfilimine bond of collagen IV, resulted in a significantly less stiff lens capsule. A comparison of the diffraction patterns of the control and strongly reduced lens capsules indicated structural rearrangements on a nanometer scale.

Computational model of evolving lens capsule biomechanics following cataract-like surgery

Cataract surgery is an invasive procedure whereby lens fibers are removed through a permanent central hole, or capsulorhexis, in the surrounding lens capsule and replaced with an artificial intraocular lens (IOL). Remnant lens epithelial cells subsequently transdifferentiate to a more contractile and synthetic wound-healing phenotype, which causes significant structural and mechanical adaptations of the residual lens capsule. The goal of this study is to present a computational model capable of capturing salient features of the biomechanical evolution of the lens capsule following cataract-like surgery. The model is shown to predict marked long-term increases in thickness and stiffness of the lens capsule nearest the edge of the capsulorhexis comparable to reported measurements. Such models represent a first step toward understanding better the long-term interactions between the residual lens capsule and implanted IOL, thus initiating a new paradigm for the design of improved IOLs, including those having an accommodative feature.

On the biomechanical function of scaffolds for engineering load-bearing soft tissues

Replacement or regeneration of load-bearing soft tissues has long been the impetus for the development of bioactive materials. While maturing, current efforts continue to be confounded by our lack of understanding of the intricate multi-scale hierarchical arrangements and interactions typically found in native tissues. The current state of the art in biomaterial processing enables a degree of controllable microstructure that can be used for the development of model systems to deduce fundamental biological implications of matrix morphologies on cell function. Furthermore, the development of computational frameworks which allow for the simulation of experimentally derived observations represents a positive departure from what has mostly been an empirically driven field, enabling a deeper understanding of the highly complex biological mechanisms we wish to ultimately emulate. Ongoing research is actively pursuing new materials and processing methods to control material structure down to the micro-scale to sustain or improve cell viability, guide tissue growth, and provide mechanical integrity, all while exhibiting the capacity to degrade in a controlled manner. The purpose of this review is not to focus solely on material processing but to assess the ability of these techniques to produce mechanically sound tissue surrogates, highlight the unique structural characteristics produced in these materials, and discuss how this translates to distinct macroscopic biomechanical behaviors.

Characterization of the complete fiber network topology of planar fibrous tissues and scaffolds

Understanding how engineered tissue scaffold architecture affects cell morphology, metabolism, phenotypic expression, as well as predicting material mechanical behavior has recently received increased attention. In the present study, an image-based analysis approach that provides an automated tool to characterize engineered tissue fiber network topology is presented. Micro-architectural features that fully defined fiber network topology were detected and quantified, which include fiber orientation, connectivity, intersection spatial density, and diameter. Algorithm performance was tested using scanning electron microscopy (SEM) images of electrospun poly(ester urethane)urea (ES-PEUU) scaffolds. SEM images of rabbit mesenchymal stem cell (MSC) seeded collagen gel scaffolds and decellularized rat carotid arteries were also analyzed to further evaluate the ability of the algorithm to capture fiber network morphology regardless of scaffold type and the evaluated size scale. The image analysis procedure was validated qualitatively and quantitatively, comparing fiber network topology manually detected by human operators (n = 5) with that automatically detected by the algorithm. Correlation values between manual detected and algorithm detected results for the fiber angle distribution and for the fiber connectivity distribution were 0.86 and 0.93 respectively. Algorithm detected fiber intersections and fiber diameter values were comparable (within the mean +/- standard deviation) with those detected by human operators. This automated approach identifies and quantifies fiber network morphology as demonstrated for three relevant scaffold types and provides a means to: (1) guarantee objectivity, (2) significantly reduce analysis time, and (3) potentiate broader analysis of scaffold architecture effects on cell behavior and tissue development both in vitro and in vivo.

The lens capsule

The lens capsule is a modified basement membrane that completely surrounds the ocular lens. It is known that this extracellular matrix is important for both the structure and biomechanics of the lens in addition to providing informational cues to maintain lens cell phenotype. This review covers the development and structure of the lens capsule, lens diseases associated with mutations in extracellular matrix genes and the role of the capsule in lens function including those proposed for visual accommodation, selective permeability to infectious agents, and cell signaling.