A model for the human cornea: constitutive formulation and numerical analysis

The human cornea (the external lens of the eye) has the macroscopic structure of a thin shell, originated by the organization of collagen lamellae parallel to the middle surface of the shell. The lamellae, composed of bundles of collagen fibrils, are responsible for the experimentally observed anisotropy of the cornea. Anomalies in the fibril structure may explain the changes in the mechanical behavior of the tissue observed in pathologies such as keratoconus. We employ a fiber-matrix constitutive model and propose a numerical model for the human cornea that is able to account for its mechanical behavior in healthy conditions or in the presence of keratoconus under increasing values of the intraocular pressure. The ability of our model to reproduce the behavior of the human cornea opens a promising perspective for the numerical simulation of refractive surgery.     

An analytically solvable model for biomechanical response of the cornea to refractive surgery

An analttically solvable model that considers the elasticity of the cornea is developed for use in the current and novel corneal refractive surgery procedures. The model assumes that the cornea is a thin spheroid shell with an elastic response to intraocular pressure. The value of the Young's modulus of the post-operative cornea and its dependence on the geometric parameters of the ablation zone are estimated employing "best-fit" approach to nomograms currently used in corneal refractive surgery. These elasticity parameters are applied for quantitative modeling of different types of refractive surgery for myopia.     

Biomechanical modeling of refractive corneal surgery

The aim of refractive corneal surgery is to modify the curvature of the cornea to improve its dioptric properties. With that goal, the surgeon has to define the appropriate values of the surgical parameters in order to get the best clinical results, i.e., laser and geometric parameters such as depth and location of the incision, for each specific patient. A biomechanical study before surgery is therefore very convenient to assess quantitatively the effect of each parameter on the optical outcome. A mechanical model of the human cornea is here proposed and implemented under a finite element context to simulate the effects of some usual surgical procedures, such as photorefractive keratectomy (PRK), and limbal relaxing incisions (LRI). This model considers a nonlinear anisotropic hyperelastic behavior of the cornea that strongly depends on the physiological collagen fibril distribution. We evaluate the effect of the incision variables on the change of curvature of the cornea to correct myopia and astigmatism. The obtained results provided reasonable and useful information in the procedures analyzed. We can conclude from those results that this model reasonably approximates the corneal response to increasing pressure. We also show that tonometry measures of the IOP underpredicts its actual value after PRK or LASIK surgery.     

The anisotropic material constitutive models for the human cornea

This paper presents an anisotropic analysis model for the human cornea. The model is based on the assumption that the fibrils in the cornea are organised into lamellae, which may have preferential orientation along the superior-inferior and nasal-temporal directions, while the alignment of lamellae with different orientations is assumed to be random. Hence, the cornea can be regarded as a laminated composite shell. The constitutive equation describing the relationships between membrane forces, bending moments, and membrane strains, bending curvatures are derived. The influences of lamella orientations and the random alignment of lamellae on the stiffness coefficients of the constitutive equation are discussed.     

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.     

A closed shell structured eyeball model with application to radial keratotomy

A closed shell structured eyeball model was developed for predicting the displacements and curvatures in an eyeball due to radial keratotomy. Both the cornea and sclera are modeled as an ellipsoidal cap, and the two caps are connected at the limbus to form a closed shell. The analysis of the number of corneal collagen laminae required for the tissue to be theoretically transversely isotropic was presented. The cornea, as well as the limbus and sclera, is considered as macroscopically homogeneous and isotropic in this study. A procedure to obtain the principal curvature at a point on the exterior surface was established. In the basic formulation, large displacements are contemplated. However, the FORTRAN computer program that was prepared to implement the procedure considers small displacements, and the resulting equations are linear. Although the results from this shell structured eyeball model are fairly good quantitatively, they do show vividly the following qualitative corneal behavior after the operation is performed: The opening of an incision has a V-shape, the radial displacements through the corneal thickness are nearly the same, and the largest in-plane displacement is only one-tenth of the largest radial displacement.     

Molecular level probing of preferential hydration and its modulation by osmolytes through the use of pyranine complexed to hemoglobin

Two spectroscopic probes are used to expose molecular level changes in hydration shell water interactions that directly relate to such issues as preferential hydration and protein stability. The major focus of the present study is on the use of pyranine (HPT) fluorescence to probe as a function of added osmolytes (PEG, urea, trehalose, and magnesium), the extent to which glycerol is preferentially excluded from the hydration shell of free HPT and HPT localized in the diphosphoglycerate (DPG) binding site of hemoglobin in both solution and in Sol-Gel matrices. The pyranine study is complemented by the use of vibronic side band luminescence from the gadolinium cation that directly exposes the changes in hydrogen bonding between first and second shell waters as a function of added osmolytes. Together the results form the basis for a water partitioning model that can account for both preferential hydration and water/osmolyte-mediated conformational changes in protein structure.     

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.     

Determination of the true intraocular pressure and modulus of elasticity of the human cornea in vivo

The purpose of this study was to determine the true intraocular pressure and modulus of elasticity of the human cornea in vivo. The cornea was modeled as a shell, and the equations for the deformations of a shell due to applanating and intraocular pressures were combined to model the behavior of the cornea during applanation tonometry. At certain corneal dimensions called the calibration dimensions, the applanating and intraocular pressures are considered to be equal. This relationship was used to determine the modulus of elasticity of the cornea and the relationship between the applanating and intraocular pressures. The true intraocular pressure (IOPT) was found to be related to Goldmann’s applanating pressure (IOPG) as (IOPT = IOPG/K, where K is a correction factor. For the calibration corneal thickness of 0.52 mm, the modulus of elasticity E in MPa of the human cornea was found to be related to the true intraocular pressure IOPT in mmHg as E = 0.0229IOPT. The generalization of the Imbert—Fick law that takes into account the effect of corneal dimensions and stiffness was found to be given by IOPT = 73.5W/(K A), where W is the applanating weight in gf (gram force) and A is the applanated area in mm². The calculated true intraocular pressure and modulus of elasticity were found to agree with published experimental results. The mathematical model developed may therefore be used to improve results from applanation tonometry and to estimate the mechanical property of the cornea in vivo.     

Enhancement of RNA labeling in iris and lens by wounding cornea

Lentectomy of the newt eye leads to formation of the lens from the iris. The initial event which occurs in the iris after lentectomy is enhancement of uridine incorporation into RNA. The present data demonstrate that surgery on the cornea without lentectomy enhances uridine incorporation into iris RNA. However, the profile of incorporation after cornea surgery is different from that after lentectomy. Furthermore, cornea surgery fails to cause the high level of incorporation of thymidine into iris which occurs after lentectomy. Cornea surgery also causes enhancement of uridine incorporation into lens RNA with a profile different from that in iris RNA.     

A quantitative analysis of astigmatism induced by pterygium

An analytical model has been developed for the localized corneal deformation produced in the region of the head of a pterygia. Astigmatism results when this localized deformation enters the central region or optical zone of the cornea. The amount and direction of the pterygia-induced astigmatism may be predicted from the values of the corneal curvature within the optical zone. The analytical solution of Lur'e based on the Papkovich-Neuber theory was applied to the anatomical and mechanical conditions affecting the cornea and conjunctiva. The force exerted by the head of a pterygia was measured experimentally for the first time. This force is of the order of that produced by the extraocular muscles in primary gaze. Using this model it is possible to predict that 2.35 diopters of the pterygia-induced astigmatism would result from a pterygia exerting 5 g of force along a meridian passing through the center of the cornea, and whose head is located 2.38 millimeters from the optical center of the cornea of an eye having a 4 mm pupillary diameter.     

Influence of needling conditions on the corneal insertion force

Abstract

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.     

CAUSE OF BIMODAL DISTRIBUTION IN THE SHAPE OF A TERRESTRIAL GASTROPOD

The distribution of a phenotypic state is often discontinuous and dispersed. An example of such a distribution can be found in the shell shapes of terrestrial gastropods, which exhibit a bimodal distribution whereby species possess either a tall shell or a flat shell. Here we propose a simple model to test the hypothesis that the bimodal distribution relates to the optimum shape for shell balance on the substrates. This model calculates the theoretical shell balance by moment and obtains empirical distribution of shell shape by compiling published data and performing a new analysis. The solution of the model supports one part of the hypothesis, showing that a low-spired shell is the best balanced and is better suited for locomotion on horizontal surface. Additionally, the model shows that both high- and low-spired shells are well balanced and suited on vertical surfaces. The shell with a spire index (shell height divided by diameter) of 1.4 is the least well balanced as a whole. Thus, spire index is expected to show a bimodal distribution with a valley at 1.4. This expectation was supported by empirical distribution of a spire index, suggesting that the bimodality of shell shape in terrestrial gastropods is related to shell balance.     

A theoretical and experimental study of the mechanical behavior of the cornea with application to the measurement of intraocular pressure

A theoretical and experimental study was made of the mechanical behavior of the cornea. The theoretical analysis included an analytical solution for the symmetrical constraint of a thin, shallow, spherical shell by a rigid indenter. The experimental study investigated the rheology of the cornea with particular emphasis on its compliance with the requirements of the Boltzmann Superposition Principle. Representative results of tests on twenty enucleated hog eyes and two human eyes have been reported. The corneas of the human and hog eyes behaved as linear viscoelastic solids; the human eyes differed from the hog eyes in having a long term creep component. Several eyes were tested at the site of procurement, six to seven minutes after the animal's death, and it was established that creep is not an artifact due to aging or enucleation. The analytical and experimental results were combined to study some instruments used to detect the level of pressure in the eye. The theoretical analysis predicted that a type of elastic instability occurs during the process of flattening a small portion of the cornea; this is discussed with reference to the Goldmann and Mackay-Marg tonometers. The role of corneal creep was considered with reference to the response of the Schiøtz indentation tonometer during the time dependent process known as tonography.     

Ultrastructural changes in the developing chicken cornea following caffeine administration

Caffeine is one of the most frequently consumed psychoactive substances. It has been known for many years that caffeine at high concentrations exerts harmful effects on both women's and laboratory animals' fertility, moreover it may impair normal development of many organs in the prenatal period. So far there have been few studies performed that demonstrate teratogenic effects of caffeine on structures of the developing eye, particularly the cornea. The aim of the study was to show ultrastructural changes in the developing cornea, as the effect of caffeine administration to chicken embryos. The experimental materials were 26 chicken embryos from incubated breeding eggs. Eggs were divided into two groups: control (n=30) in which Ringer liquid was administrated, and experimental (n=30) in which teratogenic dose of caffeine 3.5mg/egg was given. In 36th hour of incubation solutions were given with cannula through hole in an egg shell directly onto amniotic membrane. After closing the hole with a glass plate and paraffine, eggs were put back to incubator. In 10th and 19th day of incubation corneas were taken for morphological analysis with a use of electron microscopy. Administration of caffeine during chicken development causes changes of collagen fibers of Bowman's membrane patterns and of the corneal stroma but it also changes proportion of amount of collagen fibers and of the stromal cells.     

Particle-bubble attachment in yeast flotation by colloidal gas aphrons

Flotation of microorganisms by colloidal gas aphrons (CGAs) is recognised as an inexpensive and effective method of separation. CGAs are micro-sized gas bubbles of 25 7m in average diameter and each is encapsulated in an aqueous shell of surfactant solution. High flotation efficiency can be achieved when colloidal gas aphrons are employed in a dilute suspension containing yeast cells. Under certain experimental conditions, the adsorption of yeast on the aphrons follows the Langmuir model. With changes in the pH and feed concentration, the mechanism of bubble attachment and detachment changes from a monolayer to a multilayer adsorption.     

Hydration and Hydrodynamic Interactions of Lysozyme: Effects of Chaotropic versus Kosmotropic Ions

Using static and dynamic light scattering we have investigated the effects of either strongly chaotropic, nearly neutral or strongly kosmotropic salt ions on the hydration shell and the mutual hydrodynamic interactions of the protein lysozyme under conditions supportive of protein crystallization. After accounting for the effects of protein interaction and for changes in solution viscosity on protein diffusivity, protein hydrodynamic radii were determined with ±0.25 Å resolution. No changes to the extent of lysozyme hydration were discernible for all salt-types, at any salt concentration and for temperatures between 15-40°C. Combining static with dynamic light scattering, we also investigated salt-induced changes to the hydrodynamic protein interactions. With increased salt concentration, hydrodynamic interactions changed from attractive to repulsive, i.e., in exact opposition to salt-induced changes in direct protein interactions. This anti-correlation was independent of solution temperature or salt identity. Although salt-specific effects on direct protein interactions were prominent, neither protein hydration nor solvent-mediated hydrodynamic interactions displayed any obvious salt-specific effects. We infer that the protein hydration shell is more resistant than bulk water to changes in its local structure by either chaotropic or kosmotropic ions.     

Involvement of pigment epithelium-derived factor,docosahexaenoic acid and neuroprotectin D1 in corneal inflammation and nerve integrity after refractive surgery

Alterations in corneal innervations result in impaired corneal sensation, severe dry eye and damage to the epithelium that may in turn lead to corneal ulcers, melting and perforation. These alterations can occur after refractive surgery. We have discovered that pigment epithelium-derived factor (PEDF) plus docosahexaenoic acid (DHA or the docosanoid bioactive neuroprotectin D1 (NPD1)) induces nerve regeneration after corneal surgery that damages the stromal nerves. We found that PEDF is released from corneal epithelial cells after injury, and when DHA is provided to the cells it stimulates the biosynthesis of NPD1 by an autocrine mechanism. The combination of PEDF plus DHA also decreased the production of leukotriene B4 (LTB4), a neutrophil chemotactic factor, thereby decreasing the inflammation induced after corneal damage. These studies suggest that PEDF plus DHA and its derivative NPD1 hold promise as a future treatment to restore a healthy cornea after nerve damage.     

人眼波前像差表示、测量及矫正研究

目的:波前像差引导的准分子激光角膜消融是屈光手术的新方法,研究人眼波前像差的测量原理、方法、表示、人眼波前像差准分子激光矫正的原理,以此理论用于准分子激光人眼像差矫正系统。方法:采用理论研究、计算机模拟、实验室实验等手段。分析人眼像差的概念和产生的原因,用数学的Zern ike多项式来表示像差,理论上定量分析Zern ike多项式表示的波前像差与角膜切削深度的关系,研究准分子激光切削角膜的机理,研究准分子激光进行矫正人眼像差的原理框图。结果:通过计算机模拟和实验室实验,用准分子激光矫正低阶和高阶像差是可行的。结论:用波前像差来引导屈光手术,使人眼的视力能够达到20/10上,并能避免当前PRK、LASIK屈光手术前后像差增大而引起的对视觉质量的影响。