Effect of cornea material stiffness on measured intraocular pressure

Intraocular pressure (IOP) in the human eye as measured by a Goldmann applanation tonometer (GAT) is known to be affected by individual differences in central corneal thickness (CCT). However, data from clinical studies also show considerable scatter in the correlation between measured IOP and CCT. One possible implication of the large observed scatter is that the true IOP (IOPT) also depends significantly on individual variations in the material stiffness properties of the cornea. This hypothesis is explored and evaluated herein using computational simulation of applanation tonometry. A simplified 2D finite element model of the eye, which employs a calibrated nonlinear transversely isotropic material model for the cornea, is developed, and a series of GAT simulations is carried out to study the effect of geometry and material properties of the cornea on the IOP readings obtained via GAT. The results of this parametric study provide a simple correction equation, which quantifies the effect on measured IOP of variations in CCT and corneal material stiffness. In addition, several previously proposed IOP correction equations are compared with the one proposed here.     

The distribution of pressure behind a soft contact lens

We determine the pressure distribution behind a soft contact lens that is necessary to keep the lens in conformity with an axisymmetric substrate. The substrate consists of two regions: a central portion, the cornea, supposed to be an ellipsoid; and a peripheral region, the sclera, taken to be a sphere. The pressure is obtained as part of a numerical solution of the axisymmetric equilibrium equations for an initially curved, linearly elastic membrane. The relaxed shape of the lens is assumed to be an axisymmetric ellipsoid with a central curvature and a shape factor different from those of the cornea. The variation in the thickness of the lens from its center to edge is approximated by a polynomial. Pressure distributions are obtained for several typical soft contact lens fittings.     

Hydroxyproline concentrations in ocular tissues of Arabian camel (Camelus dromedarius Linn.)

Hydroxyproline (Hyp) concentrations (total, free, peptide-bound and protein-bound) in camel eye tissues were determined. Total Hyp concentration was highest in iris, followed by ciliary body, sclera, cornea, lens and retina; the difference between total Hyp concentration of iris and sclera (P < 0.05) and cornea, lens and retina (P < 0.001) was statistically significant. Cornea had the highest concentration of free Hyp, followed by ciliary body, retina, iris, sclera and lens (P < 0.001). Peptide-bound Hyp concentration was highest in iris, followed by lens, cornea, ciliary body, retina and sclera (P < 0.001). Iris also had the highest concentration of protein-bound Hyp, followed by ciliary body, sclera, cornea, retina and lens; the difference in the protein-bound Hyp concentration between iris and sclera (P < 0.05) and cornea, retina and lens (P < 0.001) was statistically significant. Iris was also found to have the highest concentration of collagen, followed by ciliary body, sclera, cornea, lens and retina; the difference between the collagen concentration of iris and sclera (P < 0.05) and cornea, lens and retina (P < 0.001) was statistically significant. These variations may result from differences in the collagen structure and/or composition in these tissues.     

Determination of optical parameters of the porcine eye and development of a simulated model

The eye is a very sophisticated system of optical elements for the preeminent sense of vision. In recent years, the number of laser surgery to correct the optical aberration such as myopia or astigmatism has significantly increased. Consequently, improving the knowledge related to the interactions of light with the eye is very important in order to enhance the efficiency of the surgery. For this reason, a complete optical characterization of the porcine eye is presented in this study. Kubelka‐Munk and Inverse Adding‐Doubling methods were applied to spectroscopy measurement to determine the absorption and scattering coefficients. Furthermore, the refractive index has been measured by ellipsometry. The different parameters were obtained for the cornea, lens, vitreous humor, sclera, iris, choroids and eyelid in the visible and infrared region. Thereafter, the results are implemented in a COMSOL Multiphysics® software to create an eye model. This model gives a better understanding of the propagation of light in the eye by adding optical parts such as the iris, the sclera or the ciliary bodies. Two simulations show the propagation of light from the cornea to the retina but also from the sclera to the retina. This last possibility provides a better understanding of light propagation during eye laser surgery such as, for example, transscleral cyclophotocoagulation. Figure: Eye simulation models allow the development of new laser treatments in a simple and safe way for patients. To this purpose, the creation of an eye simulated model based on optical parameters obtained from experimental data is presented in this study. This model will facilitate the understanding of the light propagation inside the porcine eye.     

Dynamic material properties of the human sclera

As a result of trauma, approximately 30,000 people become blind in one eye every year in the United States. A common injury prediction tool is computational modeling, which requires accurate material properties to produce reliable results. Therefore, the purpose of this study was to determine the dynamic material properties of the human sclera. A high-rate pressurization system was used to create dynamic pressure to the point of rupture in 12 human eyes. Measurements were obtained for the internal pressure, the diameter of the globe, the thickness of the sclera, and the changing coordinates of the optical markers using high-rate video. A relationship between true stress and true strain was determined for the sclera tissue in two directions. It was found that the average maximum true stress was 13.89±4.81 MPa for both the equatorial and meridional directions, the average maximum true strain along the equator was 0.041±0.014, and the average maximum true strain along the meridian was 0.058±0.018. Results show a significant difference in the maximum strain in the equatorial and meridional directions (p=0.02). In comparing these data with previous studies, it is concluded that the human sclera is both anisotropic and viscoelastic. The dynamic material properties presented in this study can be used for advanced models of the human eye to help prevent eye injuries in the future.     

An inverse finite element method for determining the anisotropic properties of the cornea

An inverse finite element method was developed to determine the anisotropic properties of bovine cornea from an in vitro inflation experiment. The experiment used digital image correlation (DIC) to measure the three-dimensional surface geometry and displacement field of the cornea at multiple pressures. A finite element model of a bovine cornea was developed using the DIC measured surface geometry of the undeformed specimen. The model was applied to determine five parameters of an anisotropic hyperelastic model that minimized the error between the measured and computed surface displacement field and to investigate the sensitivity of the measured bovine inflation response to variations in the anisotropic properties of the cornea. The results of the parameter optimization revealed that the collagen structure of bovine cornea exhibited a high degree of anisotropy in the limbus region, which agreed with recent histological findings, and a transversely isotropic central region. The parameter study showed that the bovine corneal response to the inflation experiment was sensitive to the shear modulus of the matrix at pressures below the intraocular pressure, the properties of the collagen lamella at higher pressures, and the degree of anisotropy in the limbus region. It was not sensitive to a weak collagen anisotropy in the central region.     

A lattice model for computing the transmissivity of the cornea and sclera.

The method of photonic band structure is used to calculate the frequencies of light that propagate in lattice models of the cornea and sclera of the mammalian eye, providing an explanation for transparency in the cornea that first properly accounts for multiple scattering of light. Each eye tissue is modeled as an ordered array of collagen rods, and photonic band structure methods are used to solve Maxwell's equations exactly for these models, a procedure that automatically effectively includes all orders of multiple scattering. These calculations show that the dispersion relation for the cornea is linear in the visible range, implying that the cornea is transparent. We show that the transmissivity is approximately 97% by using an effective medium approximation derived from the photonic band structure results and applicable in the visible region. In contrast, the dispersion relation for the model in the sclera is not linear in the visible region, and there are band gaps in this region that could play an important role in the transmission of light in the sclera.     

X-ray scattering used to map the preferred collagen orientation in the human cornea and limbus

Many properties of connective tissues are governed by the organization of the constituent collagen. For example, the organization of collagen in the cornea and the limbus, where the cornea and sclera meet, is an important determinant of corneal curvature and hence of the eye's focusing power. We have used synchrotron X-ray scattering to map the orientation of the collagen fibrils throughout the human cornea, limbus, and adjacent sclera. We demonstrate a preferred orientation of collagen in the vertical and horizontal directions that is maintained to within about 1 mm from the limbus, where a circular or tangential disposition of fibrils occurs. The data are also used to map the relative distribution of both the total and the preferentially aligned collagen in different parts of the tissue, revealing considerable anisotropy. The detailed structural information provided is an important step toward understanding the shape and the mechanical properties of the tissue.     

Mechanical characterization of porcine corneas

An experimental program has been carried out in order to investigate the mechanical behavior of porcine corneas. We report the results of inflation tests on the whole cornea and uniaxial tests on excised corneal strips, performed on 51 fresh porcine eyes. Uniaxial tests have been performed on specimens cut from previously inflated corneas. The cornea behavior is characterized by means of elastic stiffness, measured on both average pressure-apex displacement and average uniaxial stress-strain curves; and by means of transversal contraction coefficient, peak stress, and failure stress measured on uniaxial stress-strain curves. Uniaxial tests performed on excised strips allowed to measure the anisotropy in the corneal stiffness and to compare the stiffness of the cornea with the one of the sclera. Viscous properties of the cornea have been obtained through uniaxial relaxation curves on excised corneal strips. The relevant geometrical parameters have been measured and, with the aid of the elastic thin shell theory, a stress-strain curve has been derived from the average inflation test data and compared with similar data available in the literature. The experimental system has been developed in view of future applications to the mechanical testing of both porcine and human corneas.     

Increased aggrecan (cartilage proteoglycan) production in the sclera of myopic chicks

A previously characterized chick model of myopia was used to evaluate biochemical changes in the sclera which are associated with ocular enlargement and myopia. Chicks were monocularly occluded for 10 days and the DNA, hydroxyproline, and glycosaminoglycan contents of the sclera were compared between the normal and the myopic eyes. No significant differences could be detected in total DNA or hydroxyproline content. There was, however, a 34% increase in glycosaminoglycans and a 20.7% decrease in cell density within the posterior sclera of myopic eyes. The biosynthesis of scleral proteoglycans was determined by measuring 35SO4 incorporation in the sclera of chicks visually occluded for 5, 10, and 15 days. No differences could be detected in 35SO4 incorporation into the cornea or the anterior sclera. However, 35SO4 incorporation was significantly increased in the posterior sclera of myopic eyes by 64% at Day 5, 39% at Day 10, and 49% at Day 15. When fractionated on Sepharose CL-4B, scleral proteoglycans were resolved into two peaks which were identified by Western blot analysis as aggrecan (cartilage proteoglycan) and decorin. Furthermore, Western blot and dot blot analyses indicated that significantly more aggrecan core protein was present in the sclera of myopic eyes compared with equivalent amounts of sclera from control eyes. These results indicate that increased synthesis and accumulation of aggrecan, which increases the volume of extracellular matrix in the posterior sclera, are responsible for the ocular enlargement observed in this model of myopia.     

Quantifying nonlinear anisotropic elastic material properties of biological tissue by use of membrane inflation

Determination of material parameters for soft tissue frequently involves regression of material parameters for nonlinear, anisotropic constitutive models against experimental data from heterogeneous tests. Here, parameter estimation based on membrane inflation is considered. A four parameter nonlinear, anisotropic hyperelastic strain energy function was used to model the material, in which the parameters are cast in terms of key response features. The experiment was simulated using finite element (FE) analysis in order to predict the experimental measurements of pressure versus profile strain. Material parameter regression was automated using inverse FE analysis; parameter values were updated by use of both local and global techniques, and the ability of these techniques to efficiently converge to a best case was examined. This approach provides a framework in which additional experimental data, including surface strain measurements or local structural information, may be incorporated in order to quantify heterogeneous nonlinear material properties.     

The Effect of Spatial Inhomogeneity of the Cornea on the Deformation Properties of the Eyeball and the Results of Maklakoff Applanation Tonometry

A two-component model of the eyeball that represents the cornea as a momentless, linearly elastic deformable surface and the scleral region, as an elastic element that responds to intraocular pressure changes by volume changes, has been used to analyze the effect of spatial inhomogeneity in the distribution of effective corneal stiffness on the mechanical properties of the eye. The effective stiffness of the cornea characterized both the elastic properties and the thickness of the cornea within the framework of the model. Various axisymmetric forms of the effective stiffness distribution characterized by monotonic increase along the arc between a point on the corneal surface and the apex of the cornea were studied. The considered distributions simulated both natural inhomogeneity and apical region weakening due to surgical interventions. Numerical simulation yielded the dependences of deformation parameters on intraocular pressure changes. These parameters characterized the deformation properties of both the cornea (apex displacement) and the eyeball as a whole (intraocular volume change). In the case of moderate inhomogeneity, the dependences were only slightly different from those for a homogeneous cornea with an effective stiffness equal to the mean value for the corresponding inhomogeneous distribution. A noticeable increase in the integral response of the cornea and the eyeball as a whole to changes in pressure was observed if the effective stiffness amplitude was very high (two or more times higher than the mean value). The effect of inhomogeneity on the results of tonometric measurements with a Maklakoff tonometer (flat stamp) was studied. The tonometric difference, that is, the difference between the tonometric pressure (in the loaded eye) and the true pressure (before loading), mainly depended on the average stiffness of the cornea in this case as well, with a substantial increase observed at very high stiffness amplitudes only. Apical weakening of the cornea led to an increase (although not very pronounced) of the tonometric difference.     

The integration of the corneal and limbal fibrils in the human eye.

The precise orientation of the collagen fibrils in human cornea and sclera and the method by which these two areas fuse together at the limbus have never been determined, despite the importance of this information. From a consideration of the mechanics of the system, fibril orientation in the tissue has the potential to affect the curvature of the cornea so, by inference, refractive problems such as astigmatism involving an incorrect curvature of the cornea may be related to fibril orientation. The high intensity and small beam size of a synchrotron x-ray source has enabled us to study fibril orientation in post-mortem human cornea and sclera. Previously we have revealed two preferred directions of orientation in the cornea (Meek, K. M., T. Blamires, G. F. Elliot, T. Y. Gyi, and C. J. Nave. 1987. Curr. Eye Res. 6:841-846) and a circumcorneal annulus in the limbus (Newton, R. H., and K. M. Meek. 1998. Invest. Ophthalmol. & Visual Sci. 39: 1125-1134). Here we present the results of our investigation into the relationship between these two features. Our measurements indicate that the corneal fibrils oriented in the two preferred directions bend at the limbus to run circumferentially. On the basis of these results we propose a model as to how the human cornea and sclera fuse together.     

Swelling studies on the cornea and sclera: the effects of pH and ionic strength.

The biophysical properties of the cornea and sclera depend on the precise maintenance of tissue hydration. We have studied the swelling of the tissues as a function of pH and ionic strength of the bathing medium, using an equilibration technique that prevents the loss of proteoglycans during swelling. Synchrotron x-ray diffraction was used to measure the average intermolecular and interfibrillar spacings, the fibril diameters, and the collagen D-periodicity. We found that both tissues swelled least near pH 4, that higher hydrations were achieved at lower ionic strengths, and that sclera swelled about one-third as much as cornea under most conditions. In the corneal stroma, the interfibrillar spacing increased most with hydration at pH values near 7. Fibril diameters and D-periodicity were independent of tissue hydration and pH at hydrations above 1. Intermolecular spacings in both tissues decreased as the ionic strength was increased, and there was a significant difference between cornea and sclera. Finally, we observed that corneas swollen near pH 7 transmitted significantly more light than those swollen at lower pH levels. The results indicate that the isoelectric points of both tissues are close to pH 4. The effects of ionic strength can be explained in terms of chloride binding within the tissues. The higher light transmission achieved in corneas swollen at neutral pH may be related to the fact that the interfibrillar fluid is more evenly distributed under these conditions.     

A microstructurally-based finite element model of the incised human cornea.

A mechanical model of the human cornea is proposed and employed in a finite element formulation for simulating the effects of surgical procedures, such as radial keratotomy, on the cornea. The model assumes that the structural behavior of the cornea is governed by the properties of the stroma. Arguments based on the microstructural organization and properties of the stroma lead to the conclusion that the human cornea exhibits flexural and shear rigidities which are negligible compared to its membrane rigidity. Accordingly, it is proposed that to a first approximation, the structural behavior of the cornea is that of a thick membrane shell. The tensile forces in the cornea are resisted by very fine collagen fibrils embedded in the ground substance of the stromal lamellae. When the collagen fibrils are cut, as in radial keratotomy, it is argued that they become relaxed since there is negligible transfer of load between adjacent fibrils due to the low shear modulus of the ground substance. The forces in the cornea are then resisted only by the remaining uncut fibrils. The cutting of fibrils induces an anisotropy and inhomogeneity in the membrane rigidity. By assuming a uniform angular distribution of stromal lamellae through the corneal thickness, geometric arguments lead to a quantitative representation for the anisotropy and inhomogeneity. All material behavior is assumed to be in the linear elastic regime and with no time-dependency. The resulting constitutive model for the incised cornea has been employed in a geometrically non-linear finite element membrane shell formulation for small strains with moderate rotations. A number of numerical examples are presented to illustrate the effectiveness of the proposed constitutive model and finite element formulation. The dependence of the outcome of radial keratotomy, measured in terms of the immediate postoperative shift in corneal power, on a number of important factors is investigated. These factors include the value of the elastic moduli of the stromal lamellae (dependent on the patient's age), the incision depth, the optic zone size, the number of incisions and their positions, and the intraocular pressure. Results have also been compared with expected surgical corrections predicted by three expert surgeons and show an excellent correspondence.     

The dermatan sulfate proteoglycans of bovine sclera and their relationship to those of articular cartilage. An immunological and biochemical study

Dermatan sulfate proteoglycans were isolated from adult bovine sclera and adult bovine articular cartilage. Their immunological relationships were studied by enzyme-linked immunosorbent assays using polyclonal antibodies raised against the large and small dermatan sulfate proteoglycans from sclera and a polyclonal and monoclonal antibody directed against the small dermatan sulfate proteoglycans from cartilage. The small dermatan sulfate proteoglycans from sclera and cartilage displayed immunological cross-reactivity while there was no convincing evidence of shared epitope(s) with the larger dermatan sulfate proteoglycans, nor did these larger proteoglycans share any common epitopes with each other. A hyaluronic acid binding region was detected immunologically on the larger scleral dermatan sulfate proteoglycan but was absent from the larger dermatan sulfate proteoglycan of cartilage and both the small dermatan sulfate proteoglycans. These antibodies were used in immunofluorescence microscopy to localize the scleral proteoglycans and molecules containing these epitopes in the eye. The large scleral dermatan sulfate proteoglycan was restricted to sclera while molecules related to the small scleral and cartilage proteoglycans were found in the sclera, anterior uveal tract, iris, and cornea. Amino acid sequencing of the amino-terminal regions of the core proteins of the small dermatan sulfate proteoglycans from sclera and articular cartilage showed that all the first 14 amino acids analyzed were identical and the same as reported earlier for the small bovine skin and tendon dermatan sulfate proteoglycans. These studies demonstrate that the larger dermatan sulfate proteoglycans of sclera and cartilage are chemically unrelated to each other and to the smaller dermatan sulfate proteoglycans isolated from these tissues. The latter have closely related core proteins and probably represent a molecule with a widespread distribution in which the degree of epimerization of glucuronic acid and iduronic acid varies between tissues.     

The Expression of Tenascin-X in Developing and Adult Rat and Human Eye

Tenascin-X has been studied in developing and adult rat eye and in foetal and adult human eyes, using immunohistochemistry and frozen sections. The data were compared with the distribution of tenascin-C. The immunoreactivity for tenascin-X was seen in a basement membrane-like feature in different structures of embryonic (E) day 16–17 rat eyes. Postnatal (P) day 2 and older rat eyes showed immunoreactivity for tenascin-X in different connective tissues. In the epithelial basement membrane zone of the cornea, immunostaining was positive in P5 eyes, negative in P10 and P15 eyes and again positive in P30 and adult eyes. In the 20-week-old human foetus, immunoreactivity for the tenascin was seen in the posterior parts of the conjunctival stroma adjacent to the sclera and in a basement membrane-like fashion in anterior conjunctiva. In the adult human eye, immunoreactivity for tenascin-X was seen in the anterior one-third stroma of cornea as thin fibrils, in the stroma of the limbus and conjunctiva, and in blood vessels. Immunostaining for tenascin-C was seen in the posterior aspect of the further cornea, and in mesenchyme adjacent to cornea in E16–17 rat eyes. Corneal keratocytes and Descemet's membrane showed immunoreactivity for tenascin-C in P2–P15 rat eyes. Sclera and the junction of the cornea, and sclera expressed tenascin-C in P2 and older rat eyes. In human foetal eyes, immunostaining for tenascin-C was seen in the anterior parts of the corneal stroma, in the basement membrane zone and Bowman's membrane of the corneal epithelium, in the posterior one-fifth of the corneal stroma and the sclera starting from the junction of the cornea and sclera. In normal human adult eyes, immunostaining for tenascin-X was seen in the anterior one-third stroma of cornea, in the stroma of limbus and conjunctiva, and in blood vessels. The association of tenascin-X and basement membranes in early development evokes a question of its potential function in the development of the basement membrane. The results also suggest the association of tenascin-X with connective tissue development as well as the association of tenascin-C with the migration of keratocytes during the development of the corneal stroma.     

Finite element simulation of arcuates for astigmatism correction

In order to simulate the corneal incisions used to correct astigmatism, a three-dimensional finite element model was generated from a simplified geometry of the anterior half of the ocular globe. A hyperelastic constitutive behavior was assumed for cornea, limbus and sclera, which are collagenous materials with a fiber structure. Due to the preferred orientations of the collagen fibrils, corneal and limbal tissues were considered anisotropic, whereas the sclera was simplified to an isotropic one assuming that fibrils are randomly disposed. The reference configuration, which includes the initial strain distribution that balances the intraocular pressure, is obtained by an iterative process. Then the incisions are simulated. The final positions of the nodes belonging to the incised meridian and to the perpendicular one are fitted by both radii of curvature, which are used to calculate the optical power. The simulated incisions were those specified by Lindstrom's nomogram [Chu, Y., Hardten, D., Lindquist, T., Lindstrom, R., 2005. Astigmatic keratotomy. Duane's Ophthalmology. Lippincott Williams and Wilkins, Philadelphia] to achieve 1.5, 2.25, 3.0, 4.5 and 6.0D of astigmatic change, using the next values for the parameters: length of 45 degrees , 60 degrees and 90 degrees , an optical zone of 6mm, single or paired incisions. The model gives results similar to those in Lindstrom's nomogram [Chu et al., 2005] and can be considered a useful tool to plan and simulate refractive surgery by predicting the outcomes of different sorts of incisions and to optimize the values for the parameters involved: depth, length, position.     

Coupled Biomechanical Response of the Cornea Assessed by Non-Contact Tonometry. A Simulation Study

The mechanical response of the cornea subjected to a non-contact air-jet tonometry diagnostic test represents an interplay between its geometry, the corneal material behavior and the loading. The objective is to study this interplay to better understand and interpret the results obtained with a non-contact tonometry test. A patient-specific finite element model of a healthy eye, accounting for the load free configuration, was used. The corneal tissue was modeled as an anisotropic hyperelastic material with two preferential directions. Three different sets of parameters within the human experimental range obtained from inflation tests were considered. The influence of the IOP was studied by considering four pressure levels (10–28 mmHg) whereas the influence of corneal thickness was studied by inducing a uniform variation (300–600 microns). A Computer Fluid Dynamics (CFD) air-jet simulation determined pressure loading exerted on the anterior corneal surface. The maximum apex displacement showed a linear variation with IOP for all materials examined. On the contrary, the maximum apex displacement followed a cubic relation with corneal thickness. In addition, a significant sensitivity of the apical displacement to the corneal stiffness was also obtained. Explanation to this behavior was found in the fact that the cornea experiences bending when subjected to an air-puff loading, causing the anterior surface to work in compression whereas the posterior surface works in tension. Hence, collagen fibers located at the anterior surface do not contribute to load bearing. Non-contact tonometry devices give useful information that could be misleading since the corneal deformation is the result of the interaction between the mechanical properties, IOP, and geometry. Therefore, a non-contact tonometry test is not sufficient to evaluate their individual contribution and a complete in-vivo characterization would require more than one test to independently determine the membrane and bending corneal behavior.