Constitutive Modeling of Corneal Tissue: Influence of Three-Dimensional Collagen Fiber Microstructure

Biomechanics of soft biological tissues and organs, mechanobiology, homeostasis and modelling

The human body consists of many different soft biological tissues that exhibit diverse microstructures and functions and experience diverse loading conditions. Yet, under many conditions, the mechanical behaviour of these tissues can be described well with similar nonlinearly elastic or inelastic constitutive relations, both in health and some diseases. Such constitutive relations are essential for performing nonlinear stress analyses, which in turn are critical for understanding physiology, pathophysiology and even clinical interventions, including surgery. Indeed, most cells within load-bearing soft tissues are highly sensitive to their local mechanical environment, which can typically be quantified using methods of continuum mechanics only after the constitutive relations are determined from appropriate data, often multi-axial. In this review, we discuss some of the many experimental findings of the structure and the mechanical response, as well as constitutive formulations for 10 representative soft tissues or organs, and present basic concepts of mechanobiology to support continuum biomechanical studies. We conclude by encouraging similar research along these lines, but also the need for models that can describe and predict evolving tissue properties under many conditions, ranging from normal development to disease progression and wound healing. An important foundation for biomechanics and mechanobiology now exists and methods for collecting detailed multi-scale data continue to progress. There is, thus, considerable opportunity for continued advancement of mechanobiology and biomechanics.

Influence of Contact Lens Parameters on Cornea: Biomechanical Analysis

This study presents a finite element analysis to model ocular biomechanics and the interactions between the human eye and contact lenses in the closed-eye condition. The closed-eye state, where the eyelids are fully shut, presents challenges for experimental measurements due to the invasive nature of accessing and analysing the contact lens and corneal interface, making simulation tools valuable for accurate characterisation. The primary objective of this study was to examine how CLs fold and twist and their impact on the cornea when the eye is closed. The secondary aim of this study was to assess how crucial contact lens parameters (Young's modulus, base curve, and diameter) influence corneal stress distribution and the overall fit of the lens on the eye. The findings show that increasing Young's modulus significantly reduces corneal stress and promotes uniform stress distribution, making it the most influential factor for wearer comfort and safety. While base curve and diameter variations primarily affect contact area, their impact on stress distribution is minimal. This research provides insights for improving contact lens design and enhancing safety for contact lens wearers.

Contact lens fitting and changes in the tear film dynamics: mathematical and computational models review

PURPOSE

This review explores mathematical models, blinking characterization, and non-invasive techniques to enhance understanding and refine clinical interventions for ocular conditions, particularly for contact lens wear.

METHODS

The review evaluates mathematical models in tear film dynamics and their limitations, discusses contact lens wear models, and highlights computational mechanical models. It also explores computational techniques, customization of models based on individual blinking dynamics, and non-invasive diagnostic tools like high-speed cameras and advanced imaging technologies.

RESULTS

Mathematical models provide insights into tear film dynamics but face challenges due to simplifications. Contact lens wear models reveal complex ocular physiology and design aspects, aiding in lens development. Computational mechanical models explore eye biomechanics, often integrating tear film dynamics into a Multiphysics framework. While different computational techniques have their advantages and disadvantages, non-invasive tools like OCT and thermal imaging play a crucial role in customizing these Multiphysics models, particularly for contact lens wearers.

CONCLUSION

Recent advancements in mathematical modeling and non-invasive tools have revolutionized ocular health research, enabling personalized approaches. The review underscores the importance of interdisciplinary exploration in the Multiphysics approach involving tear film dynamics and biomechanics for contact lens wearers, promoting advancements in eye care and broader ocular health research.

Second harmonic generation microscopy, biaxial mechanical tests and fiber dispersion models in human skin biomechanics

Advanced numerical simulations of the mechanical behavior of human skin require thorough calibration of the material's constitutive models based on experimental ex vivo mechanical tests along with images of tissue microstructure for a variety of biomedical applications. In this work, a total of 14 human healthy skin samples and 4 additional scarred skin samples were experimentally analyzed to gain deep insights into the biomechanics of human skin. In particular, second harmonic generation (SHG) microscopy was used to extract detailed images of the distribution of collagen fibers, which were subsequently processed using a three-dimensional Fourier transform-based method recently proposed by the authors to quantify the distribution of fiber orientations. Mechanical tests under both biaxial and uniaxial loading were performed to calibrate the relevant mechanical parameters of two widely used constitutive models of soft fiber-reinforced biological tissues that account for non-symmetrical fiber dispersion. The calibration of the models allowed us to identify correlations between the mechanical parameters of the constitutive models considered. STATEMENT OF SIGNIFICANCE: Constitutive models for soft collagenous tissues can accurately reproduce the complex nonlinear and anisotropic mechanical behavior of skin. However, a comprehensive analysis of both microstructural and mechanical parameters is still missing for human skin. In this study, these parameters are determined by combining biaxial mechanical tests and SHG stacks of collagen fibers on ex vivo healthy human skin samples. The constitutive parameters are provided for two widely used hyperelastic models and enable accurate characterization of skin mechanical behavior for advanced numerical simulations.

Modeling biological growth of human keratoconus: On the effect of tissue degradation, location and size

Keratoconus is a non-inflammatory bilateral disease, that usually occurs in the inferior-temporal region, where the cornea bulges out and becomes thinner, due to the gradual loss of structural organization in corneal tissue. Degenerated extracellular matrix and fibers breakage have been observed in keratoconic corneas, that may promote the progression of the pathology. While keratoconus histopathology has been widely described in literature, its etiology is still not clear. Being able to fully understand keratoconus growing process could be crucial to detect its development and improve prevention strategies. This work proposes a novel continuum-based keratoconus growth model. The proposed framework accounts for the structural changes occurring in the underlying tissue during the progression of the disease, as indicated in experiments. The developed formulation is able to replicate the typical bulging and thinning of keratoconic corneas, as well as different forms in terms of shape, as they are commonly classified in clinics (nipple, oval and globus cones). The cone that is obtained constitutes a permanent deformed state, not pressure dependent. The resulting model may help to better understand the etiology of the behavior of this disease with the aim of improving the diagnosis and the treatment of the pathology.

Establishing Standardization Guidelines For Finite-Element Optomechanical Simulations of Refractive Laser Surgeries: An Application to Photorefractive Keratectomy

Purpose

Computational models can help clinicians plan surgeries by accounting for factors such as mechanical imbalances or testing different surgical techniques beforehand. Different levels of modeling complexity are found in the literature, and it is still not clear what aspects should be included to obtain accurate results in finite-element (FE) corneal models. This work presents a methodology to narrow down minimal requirements of modeling features to report clinical data for a refractive intervention such as PRK.

Methods

A pipeline to create FE models of a refractive surgery is presented: It tests different geometries, boundary conditions, loading, and mesh size on the optomechanical simulation output. The mechanical model for the corneal tissue accounts for the collagen fiber distribution in human corneas. Both mechanical and optical outcome are analyzed for the different models. Finally, the methodology is applied to five patient-specific models to ensure accuracy.

Results

To simulate the postsurgical corneal optomechanics, our results suggest that the most precise outcome is obtained with patient-specific models with a 100 µm mesh size, sliding boundary condition at the limbus, and intraocular pressure enforced as a distributed load.

Conclusions

A methodology for laser surgery simulation has been developed that is able to reproduce the optical target of the laser intervention while also analyzing the mechanical outcome.

Translational Relevance

The lack of standardization in modeling refractive interventions leads to different simulation strategies, making difficult to compare them against other publications. This work establishes the standardization guidelines to be followed when performing optomechanical simulations of refractive interventions.

Direct measurements of collagen fiber recruitment in the posterior pole of the eye

Collagen is the main load-bearing component of the peripapillary sclera (PPS) and lamina cribrosa (LC) in the eye. Whilst it has been shown that uncrimping and recruitment of the PPS and LC collagen fibers underlies the macro-scale nonlinear stiffening of both tissues with increased intraocular pressure (IOP), the uncrimping and recruitment as a function of local stretch have not been directly measured. This knowledge is crucial to understanding their functions in bearing loads and maintaining tissue integrity. In this project we measured local stretch-induced collagen fiber bundle uncrimping and recruitment curves of the PPS and LC. Thin coronal samples of PPS and LC of sheep eyes were mounted and stretched biaxially quasi-statically using a custom system. At each step, we imaged the PPS and LC with instant polarized light microscopy and quantified pixel-level (1.5 μm/pixel) collagen fiber orientations. We used digital image correlation to measure the local stretch and quantified collagen crimp by the circular standard deviation of fiber orientations, or waviness. Local stretch-recruitment curves of PPS and LC approximated sigmoid functions. PPS recruited more fibers than the LC at the low levels of stretch. At 10% stretch the curves crossed with 75% bundles recruited. The PPS had higher uncrimping rate and waviness remaining after recruitment than the LC: 0.9º vs. 0.6º and 3.1º vs. 2.7º. Altogether our findings support describing fiber recruitment of both PPS and LC with sigmoid curves, with the PPS recruiting faster and at lower stretch than the LC, consistent with a stiffer tissue. STATEMENT OF SIGNIFICANCE: Peripapillary sclera (PPS) and lamina cribrosa (LC) collagen recruitment behaviors are central to the nonlinear mechanical behavior of the posterior pole of the eye. How PPS and LC collagen fibers recruit under stretch is crucial to develop constitutive models of the tissues but remains unclear. We used image-based stretch testing to characterize PPS and LC collagen fiber bundle recruitment under local stretch. We found that fiber-level stretch-recruitment curves of PPS and LC approximated sigmoid functions. PPS recruited more fibers at a low stretch, but at 10% bundle stretch the two curves crossed with 75% bundles recruited. We also found that PPS and LC fibers had different uncrimping rates and non-zero waviness's when recruited.

Patient-specific finite element analysis of human corneal lenticules: An experimental and numerical study

The number of elective refractive surgeries is constantly increasing due to the drastic increase in myopia prevalence. Since corneal biomechanics are critical to human vision, accurate modeling is essential to improve surgical planning and optimize the results of laser vision correction. In this study, we present a numerical model of the anterior cornea of young patients who are candidates for laser vision correction. Model parameters were determined from uniaxial tests performed on lenticules of patients undergoing refractive surgery by means of lenticule extraction, using patient-specific models of the lenticules. The models also took into account the known orientation of collagen fibers in the tissue, which have an isotropic distribution in the corneal plane, while they are aligned along the corneal curvature and have a low dispersion outside the corneal plane. The model was able to reproduce the experimental data well with only three parameters. These parameters, determined using a realistic fiber distribution, yielded lower values than those reported in the literature. Accurate characterization and modeling of the cornea of young patients is essential to study better refractive surgery for the population undergoing these treatments, to develop in silico models that take corneal biomechanics into account when planning refractive surgery, and to provide a basis for improving visual outcomes in the rapidly growing population undergoing these treatments.

Damage and Fracture Mechanics of Porcine Subcutaneous Tissue Under Tensile Loading

Subcutaneous injection, which is a preferred delivery method for many drugs, causes deformation, damage, and fracture of the subcutaneous tissue. Yet, experimental data and constitutive modeling of these dissipation mechanisms in subcutaneous tissue remain limited. Here we show that subcutaneous tissue from the belly and breast anatomical regions in the swine show nonlinear stress-strain response with the characteristic J-shaped behavior of collagenous tissue. Additionally, subcutaneous tissue experiences damage, defined as a decrease in the strain energy capacity, as a function of the previously experienced maximum deformation. The elastic and damage response of the tissue are accurately described by a microstructure-driven constitutive model that relies on the convolution of a neo-Hookean material of individual fibers with a fiber orientation distribution and a fiber recruitment distribution. The model fit revealed that subcutaneous tissue can be treated as initially isotropic, and that changes in the fiber recruitment distribution with loading are enough to explain the dissipation of energy due to damage. When tested until failure, subcutaneous tissue that has undergone damage fails at the same peak stress as virgin samples, but at a much larger stretch, overall increasing the tissue toughness. Together with a finite element implementation, these data and constitutive model may enable improved drug delivery strategies and other applications for which subcutaneous tissue biomechanics are relevant.

The role of KS GAGs in the microstructure of CXL-treated corneal stroma; a transmission electron microscopy study

The mechanical and physical properties of the cornea originate from the microstructure and composition of its extracellular matrix. It is known that collagen fibrils, with a relatively uniform diameter, are organized in a pseudo-hexagonal array. It has been suggested that proteoglycans and the interaction of their glycosaminoglycan (GAG) side chains with themselves and collagen fibrils are important for collagen fibril organization inside the cornea. There are several diseases such as keratoconus in which the regular collagen fibrillar packing becomes distorted causing corneal optical and mechanical properties to be compromised. The primary purpose of the present work was to investigate the role of GAGs on the microstructure of corneal extracellular matrix before and after corneal crosslinking (CXL) treatment. For this purpose, keratan sulphates (KS) were removed from corneal samples using the keratanase enzyme and the CXL procedure was used to crosslink the specimens. The transmission electron microscopy was then used to characterize the diameter of collagen fibrils and their interfibrillar spacing. It was found that KS GAG depletion increased the collagen interfibrillar spacing while the CXL treatment significantly decreased the interfibrillar spacing. The enzyme and CXL treatments had an insignificant effect on the diameter of collagen fibrils. The underlying mechanisms responsible for these observations were discussed in terms of the assumption that GAG chains form duplexes that behave as tiny ropes holding collagen fibrils in place.

Direct measurements of collagen fiber recruitment in the posterior pole of the eye

Collagen is the main load-bearing component of the peripapillary sclera (PPS) and lamina cribrosa (LC) in the eye. Whilst it has been shown that uncrimping and recruitment of the PPS and LC collagen fibers underlies the macro-scale nonlinear stiffening of both tissues with increased intraocular pressure (IOP), the uncrimping and recruitment as a function of local stretch have not been directly measured. This knowledge is crucial for the development of constitutive models associating micro and macro scales. In this project we measured local stretch-induced collagen fiber bundle uncrimping and recruitment curves of the PPS and LC. Thin coronal samples of PPS and LC of sheep eyes were mounted and stretched biaxially quasi-statically using a custom system. At each step, we imaged the PPS and LC with instant polarized light microscopy and quantified pixel-level (1.5 μm/pixel) collagen fiber orientations. We used digital image correlation to measure the local stretch and quantified collagen crimp by the circular standard deviation of fiber orientations, or waviness. Local stretch-recruitment curves of PPS and LC approximated sigmoid functions. PPS recruited more fibers than the LC at the low levels of stretch. At 10% stretch the curves crossed with 75% bundles recruited. The PPS had higher uncrimping rate and waviness remaining after recruitment than the LC: 0.9° vs. 0.6° and 3.1° vs. 2.7°. Altogether our findings support describing fiber recruitment of both PPS and LC with sigmoid curves, with the PPS recruiting faster and at lower stretch than the LC, consistent with a stiffer tissue.

Orientation and depth dependent mechanical properties of the porcine cornea: Experiments and parameter identification

The porcine cornea is a standard animal model in ophthalmic research, making its biomechanical characterization and modeling important to develop novel treatments such as crosslinking and refractive surgeries. In this study, we present a numerical model of the porcine cornea based on experimental measurements that captures both the depth dependence and orientation dependence of the mechanical response. The mechanical parameters of the established anisotropic hyperelastic material models of Gasser, Holzapfel and Ogden (HGO) and Markert were determined using tensile tests. Corneas were cut with a femtosecond laser in the anterior (100 μm), central (350 μm), and posterior (600 μm) regions into nasal-temporal, superior-inferior, and diagonal strips of 150 μm thickness. These uniformly thick strips were tested at a low speed using a single-axis testing machine. The results showed that the corneal mechanical properties remained constant in the anterior half of the cornea regardless of orientation, but that the material softened in the posterior layer. These results are consistent with the circular orientation of collagen observed in porcine corneas using X-ray scattering. In addition, the parameters obtained for the HGO model were able to reproduce the published inflation tests, indicating that it is suitable for simulating the mechanical response of the entire cornea. Such a model constitutes the basis for in silico platforms to develop new ophthalmic treatments. In this way, researchers can match their experimental surrogate porcine model with a numerical counterpart and validate the prediction of their algorithms in a complete and accessible environment.

Anisotropic damage model for collagenous tissues and its application to model fracture and needle insertion mechanics

The analysis of tissue mechanics in biomedical applications demands nonlinear constitutive models able to capture the energy dissipation mechanisms, such as damage, that occur during tissue deformation. Furthermore, implementation of sophisticated material models in finite element models is essential to improve medical devices and diagnostic tools. Building on previous work toward microstructure-driven models of collagenous tissue, here we show a constitutive model based on fiber orientation and waviness distributions for skin that captures not only the anisotropic strain-stiffening response of this and other collagen-based tissues, but, additionally, accounts for tissue damage directly as a function of changes in the microstructure, in particular changes in the fiber waviness distribution. The implementation of this nonlinear constitutive model as a user subroutine in the popular finite element package Abaqus enables large-scale finite element simulations for biomedical applications. We showcase the performance of the model in fracture simulations during pure shear tests, as well as simulations of needle insertion into skin relevant to auto-injector design.

Modeling and experimental investigation of electromechanical properties of scleral tissue; a CEM model using an anisotropic hyperelastic constitutive relation

The sclera is a soft tissue primarily consisting of collagen fibers, elastin, and proteoglycans. The proteoglycans are composed of a core protein and negatively charged glycosaminoglycan side chains. The fixed electric charges inside the scleral extracellular matrix play a key role in its swelling and are expected to cause the tissue to deform in response to an electric field. However, the electroactive response of the sclera has not yet been investigated. The present work experimentally demonstrates that sclera behaves similar to an anionic electrosensitive hydrogel and develops a chemo-electro-mechanical (CEM) mathematical framework for its electromechanical response. In the numerical model, a hyperelastic constitutive law with distributed collagen fibers is used to capture the nonlinear mechanical properties of the sclera, and the coupled Poisson-Nernst-Planck equations represent the distribution of mobile ions throughout the domain. After calibrating the proposed numerical CEM model against the experimental measurements, we employ it to investigate the effects of different parameters on the scleral electromechanical response including the voltage and fixed charge density. The experimental and numerical findings of the present study confirm that sclera behaves as an electroactive hydrogel and provide new insight into the mechanical response of this ocular tissue.

Delineating Corneal Elastic Anisotropy in a Porcine Model Using Noncontact OCT Elastography and Ex Vivo Mechanical Tests

Purpose

To compare noncontact acoustic microtapping (AμT) OCT elastography (OCE) with destructive mechanical tests to confirm corneal elastic anisotropy.

Design

Ex vivo laboratory study with noncontact AμT-OCE followed by mechanical rheometry and extensometry.

Participants

Inflated cornea of whole-globe porcine eyes (n = 9).

Methods

A noncontact AμT transducer was used to launch propagating mechanical waves in the cornea that were imaged with phase-sensitive OCT at physiologically relevant controlled pressures. Reconstruction of both Young's modulus (E) and out-of-plane shear modulus (G) in the cornea from experimental data was performed using a nearly incompressible transversely isotropic (NITI) medium material model assuming spatial isotropy of corneal tensile properties. Corneal samples were excised and parallel plate rheometry was performed to measure shear modulus, G. Corneal samples were then subjected to strip extensometry to measure the Young's modulus, E.

Main Outcome Measures

Strong corneal anisotropy was confirmed with both AμT-OCE and mechanical tests, with the Young's (E) and shear (G) moduli differing by more than an order of magnitude. These results show that AμT-OCE can quantify both moduli simultaneously with a noncontact, noninvasive, clinically translatable technique.

Results

Mean of the OCE measured moduli were E = 12 ± 5 MPa and G = 31 ± 11 kPa at 5 mmHg and E = 20 ± 9 MPa and G = 61 ± 29 kPa at 20 mmHg. Tensile testing yielded a mean Young's modulus of 1 MPa - 20 MPa over a strain range of 1% to 7%. Shear storage and loss modulus (G'/G'') measured with rheometry was approximately 82/13 ± 12/4 kPa at 0.2 Hz and 133/29 ± 16/3 kPa at 16 Hz (0.1% strain).

Conclusions

The cornea is confirmed to be a strongly anisotropic elastic material that cannot be characterized with a single elastic modulus. The NITI model is the simplest one that accounts for the cornea's incompressibility and in-plane distribution of lamellae. AμT-OCE has been shown to be the only reported noncontact, noninvasive method to measure both elastic moduli. Submillimeter spatial resolution and near real-time operation can be achieved. Quantifying corneal elasticity in vivo will enable significant innovation in ophthalmology, helping to develop personalized biomechanical models of the eye that can predict response to ophthalmic interventions.

Multi-physics modeling and finite element formulation of corneal UV cross-linking

The UV cross-linking technique applied to the cornea is a popular and effective therapy for eye diseases such as keratoconus and ectatic disorders. The treatment strengthens the cornea by forming new cross-links via photochemical reactions and, in turn, prevents the disease from further developing. To better understand and capture the underlying mechanisms, we develop a multi-physics model that considers the migration of the riboflavin (i.e., the photo-initializer), UV light absorption, the photochemical reaction that forms the cross-links, and biomechanical changes caused by changes to the microstructure. Our model is calibrated to a set of nanoindentation tests on UV cross-linked corneas from the literature. Additionally, we implement our multi-physics model numerically into a commercial finite element software. We also compare our simulation against a set of inflation tests from the literature. The simulation capability allows us to make quantitative predictions of a therapy's outcomes in full 3-D, based on the actual corneal geometry; it also helps medical practitioners with surgical planning.

Numerical Investigation on the Role of Mechanical Factors Contributing to Globe Flattening in States of Elevated Intracranial Pressure

Flattening of the posterior eye globe in the magnetic resonance (MR) images is a sign associated with elevated intracranial pressure (ICP), often seen in people with idiopathic intracranial hypertension (IIH). The exact underlying mechanisms of globe flattening (GF) are not fully known but mechanical factors are believed to play a role. In the present study, we investigated the effects of material properties and pressure loads on GF. For this purpose, we used a generic finite element model to investigate the deformation of the posterior eyeball. The degree of GF in numerical models and the significance of different mechanical factors on GF were characterized using an automated angle-slope technique and a statistical measure. From the numerical models, we found that ICP had the most important role in GF. We also showed that the angle-slope graphs pertaining to MR images from five people with high ICP can be represented numerically by manipulating the parameters of the finite element model. This numerical study suggests that GF observed in IIH patients can be accounted for by the forces caused by elevation of ICP from its normal level, while material properties of ocular tissues, such as sclera (SC), peripapillary sclera (PSC), and optic nerve (ON), would impact its severity.