An experimental and theoretical analysis of unconfined compression of corneal stroma

Ex vivo, in vivo and in silico studies of corneal biomechanics: a systematic review

Healthy cornea guarantees the refractive power of the eye and the protection of the inner components, but injury, trauma or pathology may impair the tissue shape and/or structural organization and therefore its material properties, compromising its functionality in the ocular visual process. It turns out that biomechanical research assumes an essential role in analysing the morphology and biomechanical response of the cornea, preventing pathology occurrence, and improving/optimising treatments. In this review, ex vivo, in vivo and in silico methods for the corneal mechanical characterization are reported. Experimental techniques are distinct in testing mode (e.g., tensile, inflation tests), samples' species (human or animal), shape and condition (e.g., healthy, treated), preservation methods, setup and test protocol (e.g., preconditioning, strain rate). The meaningful results reported in the pertinent literature are discussed, analysing differences, key features and weaknesses of the methodologies adopted. In addition, numerical techniques based on the finite element method are reported, incorporating the essential steps for the development of corneal models, such as geometry, material characterization and boundary conditions, and their application in the research field to extend the experimental results by including further relevant aspects and in the clinical field for diagnostic procedure, treatment and planning surgery. This review aims to analyse the state-of-art of the bioengineering techniques developed over the years to study the corneal biomechanics, highlighting their potentiality to improve diagnosis, treatment and healing process of the corneal tissue, and, at the same, pointing out the current limits in the experimental equipment and numerical tools that are not able to fully characterize in vivo corneal tissues non-invasively and discourage the use of finite element models in daily clinical practice for surgical planning.

Design and Experiment of an Ultrasound-Assisted Corneal Trephination System

According to the advantages of ultrasonic vibration cutting, an ultrasound-assisted corneal trepanation robotic system is developed to improve the accuracy of corneal trephination depth and corneal incision quality in corneal trephination operations. Firstly, we analyzed the reasons for the difficulty in controlling the depth of trephination in corneal transplantations from the perspective of the biomechanical properties of the cornea. Based on the advantages of ultrasonic vibration cutting, we introduced an ultrasonic-vibration-assisted cutting method for corneal trephination and analyzed the cutting mechanism. Secondly, we described the surgical demands of corneal trephination and listed the design requirements of a robotic system. Thirdly, we introduced the design details of said system, including the system's overall structure, the ultrasound-assisted end effector, the key mechanisms of the robotic system, and the human-machine interaction interface. We designed the end effector based on ultrasonic vibration cutting and its eccentric adjustment system in an innovative way. Additionally, we then presented a procedure for robot-assisted corneal trephination. Finally, we performed several cutting experiments on grapes and porcine eyeballs in vitro. The results show that, compared with manual trephine, ultrasound-assisted corneal trephination has a better operation effect on the accuracy of corneal trephination depth and corneal incision quality.

The short-term effects of wearing swimming goggles on corneal biomechanics

PURPOSE

This study aimed to assess the impact of wearing swimming goggles (SG) on corneal biomechanics.

METHODS

Corneal deformation response, central corneal thickness (CCT), intraocular pressure (IOP) and biomechanically corrected intraocular pressure (bIOP) were measured with the Corvis system (Oculus Optikgeräte GmbH, Wetzlar, Germany) in thirty-one healthy young adults while wearing a drilled SG. All measurements were obtained before, at 30 s, 2 min, 3.5 min and 5 min of wearing SG, just after SG removal and after 2 min of SG removal.

RESULTS

The corneal biomechanics is sensitive to SG wear, observing lower corneal deformability during SG use. Specifically, wearing SG caused an increase in the time and length of the first applanation and radius curvature at the highest concavity, as well as a decrease and in the velocity of the first applanation and time and deformation amplitude of the second applanation (p < 0.001 in all cases). After SG removal, corneal biomechanical parameters showed a rebound-effect, obtaining a higher corneal deformability in comparison with baseline reading (p-corrected < 0.05 in all cases). Additionally, IOP and bIOP significantly increased while wearing SG (p < 0.001 in both cases), whereas CCT remained stable (p = 0.850).

CONCLUSIONS

Wearing SG modifies the biomechanical properties of the cornea, with reduced corneal deformability during SG wear. The outcomes of this study should be taken into consideration when making clinical decisions in subjects at high risk of developing corneal ectasias or glaucoma, as well as in the post-surgical management of these ocular conditions.

Indentation of the cornea: A Bi-layer contact problem

Histological observations of the cornea have identified the presence of multiple layers with differing thickness and function. The composition of the cornea consists primarily of collagen fibrils held together with proteoglycans but with an aqueous interstitial component being dominant. Indentation provides a means to quantify the spatial variation of the mechanical properties of the cornea, however the role of the different layers on the indentation response has barely been addressed. In addition, the response of the fluid content and its displacement during indentation has not been adequately considered. In this study indentation of the cornea with a relatively large spherical tipped indenter (R = 500 μm) is considered. It was observed that the initial phase of loading did not fit a classic Hertz elastic response but showed an initial steeper slope that gradually declines with increasing force and displacement. A relatively simple approach is developed that initially considers the cornea as a poro-elastic bi-layer contact problem, that is the presence of an outer thin stiffer Bowman's layer overlaying the thicker less stiff stroma.

A biphasic approach for characterizing tensile, compressive and hydraulic properties of the sclera

Measuring the biomechanical properties of the mouse sclera is of great interest: altered scleral properties are features of many common ocular pathologies, and the mouse is a powerful tool for studying genetic factors in disease, yet the small size of the mouse eye and its thin sclera make experimental measurements in the mouse difficult. Here, a poroelastic material model is used to analyse data from unconfined compression testing of both pig and mouse sclera, and the tensile modulus, compressive modulus and permeability of the sclera are obtained at three levels of compressive strain. Values for all three properties were comparable to previously reported values measured by tests specific for each property. The repeatability of the approach was evaluated using a test-retest experimental paradigm on pig sclera, and tensile stiffness and permeability measurements were found to be reasonably repeatable. The intrinsic material properties of the mouse sclera were measured for the first time. Tensile stiffness and permeability of the sclera in both species were seen to be dependent on the state of compressive strain. We conclude that unconfined compression testing of sclera, when analysed with poroelastic theory, is a powerful tool to phenotype mouse scleral changes in future genotype-phenotype association studies.

Detecting Mechanical Anisotropy of the Cornea Using Brillouin Microscopy

Purpose

The purpose of this study was to detect the mechanical anisotropy of the cornea using Brillouin microscopy along different perturbation directions.

Methods

Brillouin frequency shift of both whole globes (n = 10) and cornea punches (n = 10) were measured at different angles to the incident laser, thereby probing corneal longitudinal modulus of elasticity along different directions. Frequency shift of virgin (n = 26) versus cross-linked corneas (n = 15) over a large range of hydration conditions were compared in order to differentiate the contributions to Brillouin shift due to hydration from those due to stromal tissue.

Results

We detected mechanical anisotropy of corneas, with an average frequency shift increase of 53 MHz and 96 MHz when the instrument probed from 0° to 15° and 30° along the direction of the stromal fibers. Brillouin microscopy did not lose sensitivity to mechanical anisotropy up to 96% water content. We experimentally measured and theoretically modeled how mechanical changes independent of hydration affect frequency shift as a result of corneal cross-linking by isolating an approximately 100 MHz increase in frequency shift following a cross-linking procedure purely due to changes of stromal tissue mechanics.

Conclusions

Brillouin microscopy is sensitive to mechanical anisotropy of the stroma even in highly hydrated corneas. The agreement between model and experimental data suggested a quantitative relationship between Brillouin frequency shift, hydration state of the cornea, and stromal tissue stiffness.

Translational Relevance

The protocol and model validated throughout this study offer a path for comprehensive measurements of corneal mechanics within the clinic; allowing for improved evaluation of the long-term mechanical efficacy of cross-linking procedures.

Optical Coherence Elastography-Based Corneal Strain Imaging During Low-Amplitude Intraocular Pressure Modulation

Purpose: Optical coherence elastography (OCE) is a promising technique for high-resolution strain imaging in ocular tissues. A major strain-inducing factor in the eye is intraocular pressure (IOP), with diurnal physiological fluctuations reaching up to 5 mmHg. We study herein low-amplitude IOP modulation to assess local corneal strain patterns.

Methods:Ex vivo porcine eye globes were adjusted to an initial IOP of 15 mmHg and subsequently 25 mmHg. Corneal strain was induced by two subsequent pressure cycles, in which IOP was first increased and then decreased, each by a total of 5 mmHg. Two-dimensional optical coherence tomography (2D-OCT) B-scans were recorded after each loading step. Axial strain maps were obtained from magnitude and phase changes and supra-pixel displacements from cross-correlation. The strain detection sensitivity was evaluated in an isotropic material.

Results: Deformations arising from a single 1-mmHg step could be resolved. The largest strain amplitudes (5.11·10⁻³) were observed in the posterior stroma at a low initial IOP. Strain amplitude was 1.34 times higher at 15 mmHg than at 25 mmHg (p = 0.003). Upon IOP increase, the anterior cornea was compressed, whereas the posterior cornea showed axial expansion. Both morphological images and strain maps were sensitive to postmortem time. Strains that are larger than 2.44·10⁻⁵ could be reliably measured.

Conclusions: Low-amplitude IOP modulation, similar to diurnal physiological changes, induced measurable deformations in corneal tissue. Axial strain maps permit a localized comparison of the corneal biomechanical response. Small-strain OCE can likely be extended to other domains.

Influence of needling conditions on the corneal insertion force

Needle insertion plays an important part in the process of corneal graft surgery. In this paper, a three-dimensional symmetry model of the human cornea is constructed using the finite element method. Simplification of specific optic physiology is defined for the model: The cornea constrained by the sclera is presented as two layers consisting of epithelium and stroma. A failure criterion based on the distortion energy theory has been proposed to predict the insertion process of the needle. The simulation results show a good agreement with the experimental data reported in the literature. The influence of needling conditions (e.g. insertion velocity, rotation parameters and vibration parameters) on the insertion force are then discussed. In addition, a multi-objective optimization based on particle swarm optimization (PSO) is applied to reduce the insertion force. The numerical results provide guidelines for selecting the motion parameters of the needle and a potential basis for further developments in robot-assisted surgery.

Estimating three-dimensional outflow and pressure gradients within the human eye

In this paper we set the previously reported pressure-dependent, ordinary differential equation outflow model by Smith and Gardiner for the human eye, into a new three-dimensional (3D) porous media outflow model of the eye, and calibrate model parameters using data reported in the literature. Assuming normal outflow through anterior pathways, we test the ability of 3D flow model to predict the pressure elevation with a silicone oil tamponade. Then assuming outflow across the retinal pigment epithelium is normal, we test the ability of the 3D model to predict the pressure elevation in Schwartz-Matsuo syndrome. For the first time we find the flow model can successfully model both conditions, which helps to build confidence in the validity and accuracy of the 3D pressure-dependent outflow model proposed here. We employ this flow model to estimate the translaminar pressure gradient within the optic nerve head of a normal eye in both the upright and supine postures, and during the day and at night. Based on a ratio of estimated and measured pressure gradients, we define a factor of safety against acute interruption of axonal transport at the laminar cribrosa. Using a completely independent method, based on the behaviour of dynein molecular motors, we compute the factor of safety against stalling the dynein molecule motors, and so compromising retrograde axonal transport. We show these two independent methods for estimating factors of safety agree reasonably well and appear to be consistent. Taken together, the new 3D pressure-dependent outflow model proves itself to capable of providing a useful modeling platform for analyzing eye behaviour in a variety of physiological and clinically useful contexts, including IOP elevation in Schwartz-Matsuo syndrome and with silicone oil tamponade, and potentially for risk assessment for optic glaucomatous neuropathy.

Transient viscous response of the human cornea probed with the Surface Force Apparatus

Knowledge of the biomechanical properties of the human cornea is crucial for understanding the development of corneal diseases and impact of surgical treatments (e.g., corneal laser surgery, corneal cross-linking). Using a Surface Force Apparatus we investigated the transient viscous response of the anterior cornea from donor human eyes compressed between macroscopic crossed cylinders. Corneal biomechanics was analyzed using linear viscoelastic theory and interpreted in the framework of a biphasic model of soft hydrated porous tissues, including a significant contribution from the pressurization and viscous flow of fluid within the corneal tissue. Time-resolved measurements of tissue deformation and careful determination of the relaxation time provided an elastic modulus in the range between 0.17 and 1.43 MPa, and fluid permeability of the order of 10⁻¹³ m⁴/(N∙s). The permeability decreased as the deformation was increased above a strain level of about 10%, indicating that the interstitial space between fibrils of the corneal stromal matrix was reduced under the effect of strong compression. This effect may play a major role in determining the observed rate-dependent non-linear stress-strain response of the anterior cornea, which underlies the shape and optical properties of the tissue.

Corneal biomechanics after laser refractive surgery: Unmasking differences between techniques

The hypothesis that small-incision lenticule extraction provides better preservation of corneal biomechanics than previous laser refractive techniques has led to a growth in the interest in clinical and experimental research in this field. This hypothesis is based on the fact that corneal layers with greater stiffness are preserved with this new technique. However, this hypothesis is controversial because clinical research has shown a great disparity in the outcomes. In this review, we performed an in-depth analysis of the factors that might affect corneal biomechanics in laser refractive surgery procedures from a macrostructural to a microstructural viewpoint. New advances in algorithms with current devices or the introduction of new devices might help unmask the possible advantages of small-incision lenticule extraction in corneal biomechanics.

Depth-Dependent Out-of-Plane Young's Modulus of the Human Cornea

Purpose/Aim: Despite their importance in accurate mechanical modeling of the cornea, the depth-dependent material properties of the cornea have only been partially elucidated. In this work, we characterized the depth-dependent out-of-plane Young's modulus of the central and peripheral human cornea with high spatial resolution.

MATERIALS AND METHODS

Central and peripheral corneal buttons from human donors were subjected to unconfined axial compression followed by stress relaxation for 30 min. Sequences of fluorescent micrographs of full-thickness corneal buttons were acquired throughout the experiment to enable tracking of fluorescently labeled stromal keratocyte nuclei and measurements of depth-dependent infinitesimal strains. The nominal (gross) out-of-plane Young's modulus and drained Poisson's ratio for each whole specimen was computed from the equilibrium stress and overall tissue deformation. The depth-dependent (local) out-of-plane Young's modulus was computed from the equilibrium stress and local tissue strain based on an anisotropic model (transverse isotropy).

RESULTS

The out-of-plane Young's modulus of the cornea exhibited a strong dependence on in-plane location (peripheral versus central cornea), but not depth. The depth-dependent out-of-plane Young's modulus of central and peripheral specimens ranged between 72.4-102.4 kPa and 38.3-58.9 kPa. The nominal out-of-plane Young's modulus was 87 ± 41.51 kPa and 39.9 ± 15.28 kPa in the central and peripheral cornea, while the drained Poisson's ratio was 0.05 ± 0.02 and 0.07 ± 0.04.

CONCLUSIONS

The out-of-plane Young's modulus of the cornea is mostly independent of depth, but not in-plane location (i.e. central vs. peripheral). These results may help inform more accurate finite element computer models of the cornea.

A simple basis for determination of the modulus and hydraulic conductivity of human ocular surface using nano-indentation

This paper presents a simple analysis based upon Darcy's Law and indentation contact mechanics to determine the effective hydraulic conductivity and elastic modulus of fluid filled tissues. The approach is illustrated with the mechanical response of the human ocular surface using a 500μm radius spherical tipped indenter. Indentations of various regions of the ocular surface including the corneal stroma, limbal region and sclera have been conducted. Force-control indentations were made to a maximum force, which was maintained before unloading. Measurements of the indentation response of cornea at three different loading rates were also made. Elastic like response was observed during loading, which was followed by extensive creep prior to unloading.

STATEMENT OF SIGNIFICANCE

This manuscript attempts to provide a relatively simply model for the contact loading of fluid containing tissues and materials. It shows that the response of such materials provides a basis for determining the effective modulus and effective hydraulic conductivity (permeability) in much the same manner that hardness and modulus do for the indentation of elastic-plastic materials. Eye tissue with its anisotropic elastic and permeability properties is used to illustrate the approach.

Characterizing white matter tissue in large strain via asymmetric indentation and inverse finite element modeling

Characterizing the mechanical properties of white matter is important to understand and model brain development and injury. With embedded aligned axonal fibers, white matter is typically modeled as a transversely isotropic material. However, most studies characterize the white matter tissue using models with a single anisotropic invariant or in a small-strain regime. In this study, we combined a single experimental procedure - asymmetric indentation - with inverse finite element (FE) modeling to estimate the nearly incompressible transversely isotropic material parameters of white matter. A minimal form comprising three parameters was employed to simulate indentation responses in the large-strain regime. The parameters were estimated using a global optimization procedure based on a genetic algorithm (GA). Experimental data from two indentation configurations of porcine white matter, parallel and perpendicular to the axonal fiber direction, were utilized to estimate model parameters. Results in this study confirmed a strong mechanical anisotropy of white matter in large strain. Further, our results suggested that both indentation configurations are needed to estimate the parameters with sufficient accuracy, and that the indenter-sample friction is important. Finally, we also showed that the estimated parameters were consistent with those previously obtained via a trial-and-error forward FE method in the small-strain regime. These findings are useful in modeling and parameterization of white matter, especially under large deformation, and demonstrate the potential of the proposed asymmetric indentation technique to characterize other soft biological tissues with transversely isotropic properties.

A Biphasic Transversely Isotropic Poroviscoelastic Model for the Unconfined Compression of Hydrated Soft Tissue

The unconfined compression experiments are commonly used for characterizing the mechanical behavior of hydrated soft tissues such as articular cartilage. Several analytical constitutive models have been proposed over the years to analyze the unconfined compression experimental data and subsequently estimate the material parameters. Nevertheless, new mathematical models are still required to obtain more accurate numerical estimates. The present study aims at developing a linear transversely isotropic poroviscoelastic theory by combining a viscoelastic material law with the transversely isotropic biphasic model. In particular, an integral type viscoelastic model is used to describe the intrinsic viscoelastic properties of a transversely isotropic solid matrix. The proposed constitutive theory incorporates viscoelastic contributions from both the fluid flow and the intrinsic viscoelasticity to the overall stress-relaxation behavior. Moreover, this new material model allows investigating the biomechanical properties of tissues whose extracellular matrix exhibits transverse isotropy. In the present work, a comprehensive parametric study was conducted to determine the influence of various material parameters on the stress–relaxation history. Furthermore, the efficacy of the proposed theory in representing the unconfined compression experiments was assessed by comparing its theoretical predictions with those obtained from other versions of the biphasic theory such as the isotropic, transversely isotropic, and viscoelastic models. The unconfined compression behavior of articular cartilage as well as corneal stroma was used for this purpose. It is concluded that while the proposed model is capable of accurately representing the viscoelastic behavior of any hydrated soft tissue in unconfined compression, it is particularly useful in modeling the behavior of those with a transversely isotropic skeleton.

Collagen cross-linking treatment effects on corneal dynamic biomechanical properties

Cornea is a soft tissue with the principal function of transmitting and refracting light rays. The objective of the current study was to characterize possible effects of the riboflavin/UVA collagen cross-linking on corneal dynamic properties. The original corneal cross-linking protocol was used to induce cross-links in the anterior portion of the bovine cornea. A DMA machine was used to conduct mechanical tensile experiments at different levels of tensile strains. The samples were divided into a control group (n = 5) and a treated group (n = 5). All specimens were first stretched to a strain of 5% and allowed to relax for twenty minutes. After completion of the stress-relaxation experiment, a frequency sweep test with oscillations ranging from 0.01 to 10 Hz was performed. The same procedure was repeated to obtain the stress-relaxation and dynamic properties at 10% strain. It was observed that the collagen cross-linking therapy significantly increased the immediate and equilibrium tensile behavior of the bovine cornea (P < 0.05). Furthermore, for all samples in control and treated groups and throughout the whole range of frequencies, a significantly larger tensile storage modulus was measured at an axial strain of 10% compared to what was obtained at a tensile strain of 5%. Finally, it was noted that although this treatment procedure resulted in a significant increase in the storage and loss modulus at any axial strain and frequency (P < 0.05), it significantly reduced the ratio of the dissipated and stored energy during a single cycle of deformation. Therefore, it was concluded that while the riboflavin/UVA collagen cross-linking increased significantly corneal stiffness, it decreased significantly its damping capability and deformability. This reduced damping ability might adversely interfere with corneal mechanical performance.

Effect of non-uniform thickness of samples in stress relaxation tests under unconfined compression of samples of articular discs

A precise information of the biomechanical properties of soft tissues is required to develop a suitable simulation model, with which the distribution of stress and strain in the complex structures can be estimated. Many soft tissues have been mechanically characterized by stress relaxation tests under unconfined or confined compression. In general, full-thickness samples are extracted to reduce the damage in the tissue as much as possible. However, it is not guaranteed that these samples have a uniform thickness or, in other words, planar parallel faces. In particular, in the articular disc of the temporomandibular joint, many studies can be found testing full-thickness samples for which that thickness is known to be non-uniform, while making the assumption of uniaxial stress state to extract the mechanical properties from those tests. That inaccuracy may have a strong influence in some cases and needs a profound revision. The main goal of this work is to quantify the error committed in that assumption and the influence of the variation of thickness on that error in a particular test: stress relaxation tests under unconfined compression. Based on this error and defining an allowable tolerance, a criterion is established to reject samples depending on their aspect ratio.

Hydration dependent viscoelastic tensile behavior of cornea

The cornea is a protective transparent connective tissue covering the front of the eye. The standard uniaxial tensile experiments are among the most popular techniques for investigating biomechanical properties of the cornea. This experimental method characterizes the stress-strain response of corneal strips immersed in a bathing solution. In the present study, the important roles of corneal hydration on tensile viscoelastic properties were investigated. The thickness was used as a surrogate for hydration and uniaxial tensile experiments were performed on bovine corneal samples with four different average thickness (hydration), i.e., 1100 μm (4.87 mg water/mg dry tissue), 900 μm (4.13 mg water/mg dry tissue), 700 μm (3.20 mg water/mg dry tissue), and 500 μm (1.95 mg water/mg dry tissue). The samples were immersed in mineral oil in order to prevent their swelling during the experiments. A quasilinear viscoelastic (QLV) model was used to analyze the experimental measurements and determine viscoelastic material constants. It was observed that both maximum and equilibrium (relaxed) stresses were exponentially increased with decreasing tissue thickness (hydration). Furthermore, the QLV model successfully captured the corneal viscoelastic response with an average R (2) value greater than 0.99. Additional experiments were conducted in OBSS in order to confirm that these significant changes in viscoelastic properties were because of corneal hydration and not the bathing solution. The findings of this study suggest that extra care must be taken in interpreting the results of earlier uniaxial tensile testings and their correspondence to the corneal biomechanical properties.

A new method to determine rate-dependent material parameters of corneal extracellular matrix

The cornea protects internal ocular contents against external insults while refracting and transmitting the incoming light onto the lens. The biomechanical properties of the cornea are largely governed by the composition and structure of the stromal layer which is an extracellular matrix composed of collagen fibrils embedded in a hydrated soft matrix. The mechanical behavior of the corneal stroma has commonly been characterized using uniaxial tensile tests and inflation experiments. In the present study, unconfined compression experiments were used to investigate the influence of loading rates on compressive behavior of nineteen porcine corneal specimens. The experiments were performed at ramp displacement rates 0.15 μm/s (eight samples), 0.5 μm/s (six samples), and 1.0 μm/s (five samples). For all tests, a maximum compressive strain of 50% (five strain increments of 4% followed by three strain increments of 10%) was selected. The experimental data was analyzed by a transversely isotropic biphasic model and material parameters, i.e., the in-plane Young's modulus, the out-of-plane Young's modulus, and the permeability coefficient were calculated. It was observed that while the permeability coefficient decreased exponentially with increasing compressive strain, the in-plane and out-of-plane Young's moduli increased exponentially with increasing strain. Furthermore, it was found that the equilibrium stress was almost rate independent.