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.     

Elastic anisotropy of bone lamellae as a function of fibril orientation pattern

In this study, the homogenized anisotropic elastic properties of single bone lamellae are computed using a finite element unit cell method. The resulting stiffness tensor is utilized to calculate indentation moduli for multiple indentation directions in the lamella plane which are then related to nanoindentation experiments. The model accounts for different fibril orientation patterns in the lamellae—the twisted and orthogonal plywood pattern, a 5-sublayer pattern and an X-ray diffraction-based pattern. Three-dimensional sectional views of each pattern facilitate the comparison to transmission electron (TEM) images of real lamella cuts. The model results indicate, that the 5-sublayer- and the X-ray diffraction-based patterns cause the lamellae to have a stiffness maximum between 0° and 45° to the osteon axis. Their in-plane stiffness characteristics are qualitatively matching the experimental findings that report a higher stiffness in the osteon axis than in the circumferential direction. In contrast, lamellae owning the orthogonal or twisted plywood fibril orientation patterns have no preferred stiffness alignment. This work shows that the variety of fibril orientation patterns leads to qualitative and quantitative differences in the lamella elastic mechanical behavior. The study is a step toward a deeper understanding of the structure—mechanical function relationship of bone lamellae.     

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.     

Spatial mapping of collagen fibril organisation in primate cornea-an X-ray diffraction investigation

New insights are presented into the collagenous structure of the primate cornea. Wide-angle X-ray diffraction was used to map the fibrillar arrangement and distribution of collagen over three common marmoset corneas. The maps provide a point of reference to help interpret data from pathological corneas or primate models of refractive surgery. The results herein disclose a circum-corneal annulus of highly aligned collagen, 0.5-1.5 mm wide, where the cornea and sclera fuse at the limbus; a feature similar to that observed in human tissue. As in humans, the annulus is not uniform, varying in width, fibril angular spread, and collagen density around its circumference. However, more centrally the marmoset cornea exhibits a preferred lamella orientation in which proportionally more fibrils are oriented along the superior-inferior corneal meridian. This observation is in striking contrast with the situation in human cornea, where there is an orthogonal arrangement of preferentially aligned fibrils. Investigation of a further 16 corneas confirmed that approximately 33% (+/-1%) (n = 76) of fibrils in the central marmoset cornea lie within a 45 degrees sector of the superior-inferior meridian. Implications for the mechanical and optical properties of the cornea are discussed.     

Hyperglycemia-Induced Abnormalities in Rat and Human Corneas: The Potential of Second Harmonic Generation Microscopy

Background

Second Harmonic Generation (SHG) microscopy recently appeared as an efficient optical imaging technique to probe unstained collagen-rich tissues like cornea. Moreover, corneal remodeling occurs in many diseases and precise characterization requires overcoming the limitations of conventional techniques. In this work, we focus on diabetes, which affects hundreds of million people worldwide and most often leads to diabetic retinopathy, with no early diagnostic tool. This study then aims to establish the potential of SHG microscopy for in situ detection and characterization of hyperglycemia-induced abnormalities in the Descemet’s membrane, in the posterior cornea.

Methodology/Principal Findings

We studied corneas from age-matched control and Goto-Kakizaki rats, a spontaneous model of type 2 diabetes, and corneas from human donors with type 2 diabetes and without any diabetes. SHG imaging was compared to confocal microscopy, to histology characterization using conventional staining and transmitted light microscopy and to transmission electron microscopy. SHG imaging revealed collagen deposits in the Descemet’s membrane of unstained corneas in a unique way compared to these gold standard techniques in ophthalmology. It provided background-free images of the three-dimensional interwoven distribution of the collagen deposits, with improved contrast compared to confocal microscopy. It also provided structural capability in intact corneas because of its high specificity to fibrillar collagen, with substantially larger field of view than transmission electron microscopy. Moreover, in vivo SHG imaging was demonstrated in Goto-Kakizaki rats.

Conclusions/Significance

Our study shows unambiguously the high potential of SHG microscopy for three-dimensional characterization of structural abnormalities in unstained corneas. Furthermore, our demonstration of in vivo SHG imaging opens the way to long-term dynamical studies. This method should be easily generalized to other structural remodeling of the cornea and SHG microscopy should prove to be invaluable for in vivo corneal pathological studies.     

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 viscoelastic anisotropic hyperelastic constitutive model of the human cornea

A constitutive model based on the continuum mechanics theory has been developed which represents interlamellar cohesion, regional variation of collagen fibril density, 3D anisotropy and both age-related viscoelastic and hyperelastic stiffening behaviour of the human cornea. Experimental data gathered from a number of previous studies on 48 ex vivo human cornea (inflation and shear tests) enabled calibration of the constitutive model by numerical analysis. Wide-angle X-ray scattering and electron microscopy provided measured data which quantify microstructural arrangements associated with stiffness. The present study measures stiffness parallel to the lamellae of the cornea which approximately doubles with an increase in strain rate from 0.5 to 5%/min, while the underlying stromal matrix provides a stiffness 2–3 orders of magnitude lower than the lamellae. The model has been simultaneously calibrated to within 3% error across three age groups ranging from 50 to 95 years and three strain rates across the two loading scenarios. Age and strain-rate-dependent material coefficients allow numerical simulation under varying loading scenarios for an individual patient with material stiffness approximated by their age. This present study addresses a significant gap in numerical representation of the cornea and has great potential in daily clinical practice for the planning and optimisation of corrective procedures and in preclinical optimisation of diagnostic procedures.     

Structural Characterization of Edematous Corneas by Forward and Backward Second Harmonic Generation Imaging

The purpose of this study was to image and quantify the structural changes of corneal edema by second harmonic generation (SHG) microscopy. Bovine cornea was used as an experimental model to characterize structural alterations in edematous corneas. Forward SHG and backward SHG signals were simultaneously collected from normal and edematous bovine corneas to reveal the morphological differences between them. In edematous cornea, both an uneven expansion in the lamellar interspacing and an increased lamellar thickness in the posterior stroma (depth > 200 μm) were identified, whereas the anterior stroma, composed of interwoven collagen architecture, remained unaffected. Our findings of heterogeneous structural alteration at the microscopic scale in edematous corneas suggest that the strength of collagen cross-linking is heterogeneous in the corneal stroma. In addition, we found that qualitative backward SHG collagen fiber imaging and depth-dependent signal decay can be used to detect and diagnose corneal edema. Our work demonstrates that SHG imaging can provide morphological information for the investigation of corneal edema biophysics, and may be applied in the evaluation of advancing corneal edema in vivo.     

A helicoidal cell wall texture in root hairs ofLimnobium stoloniferum

Summary The cell wall of root hairs ofLimnobium stoloniferum is composed of two fibrillar layers: an outer layer with a dispersed texture and an inner layer with a helicoidal texture. In stained oblique sections the helicoidal layer appears as a series of bow-shaped structures. In sections which were shadow-casted after the embedding medium was removed, the following properties of the helicoidal layer can be directly observed. (1) It is build up of superimposed lamellae. (2) Each lamella consists of parallel oriented microfibrils. (3) Going into the helicoidal layer, there is a counter-clockwise discontinuous rotation of the microfibril orientation in successive lamellae. (4) Between adjacent lamellae the average angular displacement of the microfibril orientation is about 23 degrees. The dispersed outer layer is also polylamellated, but with randomly arranged microfibrils in each lamella. Both layers are present in the lateral wall as well as in the apical wall of the root hairs. Observations indicate that in the cell wall of the tip the parallel oriented microfibrils of the outermost helicoidal lamellae become distorted towards a dispersed arrangement. The suggestion is made that the dispersed outer layer is derived from the helicoidal layer.     

Optical clearing for improved contrast in second harmonic generation imaging of skeletal muscle

Using second harmonic generation (SHG) imaging microscopy, we have examined the effect of optical clearing with glycerol to achieve greater penetration into specimens of skeletal muscle tissue. We find that treatment with 50% glycerol results in a 2.5-fold increase in achievable SHG imaging depth. Signal processing analyses using fast Fourier transform and continuous wavelet transforms show quantitatively that the periodicity of the sarcomere structure is unaltered by the clearing process and that image quality deep in the tissue is improved with clearing. Comparison of the SHG angular polarization dependence also shows no change in the supramolecular organization of acto-myosin complexes. By contrast, identical treatment of mouse tendon (collagen based) resulted in a strong decrease in SHG response. We suggest that the primary mechanism of optical clearing in muscle with glycerol treatment results from the reduction of cytoplasmic protein concentration and concomitant decrease in the secondary inner filter effect on the SHG signal. The lack of glycerol concentration dependence on the imaging depth indicates that refractive index matching plays only a minor role in the optical clearing of muscle. SHG and optical clearing may provide an ideal mechanism to study physiology in highly scattering skeletal or cardiac muscle tissue with significantly improved depth of penetration and achievable imaging depth.     

Microfibrillar structure of growing cell wall in a coenocytic green alga,Boergesenia forbesii

A fine structure of cell wall lamellae in a coenocytic green algaBoergesenia forbesii was examined by electron microscopy. The wall has a polylamellate structure containing cellulose microfibrils 25 to 30 nm in diameter. The outer surface of the cell was covered by a thin structureless lamella, underneath which existed a lamella containing randomly-oriented microfibrils. The major part of the wall consisted of two types of lamellae, multifibrillar lamella and a transitional, matrix-rich one. In the former, microfibrils were densely arranged more or less parallel with each other. In the transitional lamella, existing between the multifibrillar ones, the microfibril orientation shifted about 30° within the layer. The fibril orientation also shifted 30° between adjacent transitional and multifibrillar layers, and consequently the microfibril orientation in the neighboring multifibrillar layers shifted 90°. It was concluded that the orientation rotated counterclockwise when observed from inside the cell. Each lamella in the thallus wall become thinner with cell expansion, but no reorientation of microfibrils in the outer old layers was observed. In the rhizoid, the outer lamellae sloughed off with the tip growth.     

Structural characterization of wheat starch granules differing in amylose content and functional characteristics

Small-angle X-ray scattering (SAXS) together with several complementary techniques, such as differential scanning calorimetry and X-ray diffraction, have been employed to investigate the structural features that give diverse functional properties to wheat starches (Triticum aestivum L.) within a narrow range of enriched amylose content (36–43%). For these starches, which come from a heterogeneous genetic background, SAXS analysis of duplicate samples enabled structural information to be obtained about their lamellar architecture where differences in lamellar spacing among samples were only several tenths of nanometer. The SAXS analysis of these wheat starches with increased amylose content has shown that amylose accumulates in both crystalline and amorphous parts of the lamella. Using waxy starch as a distinctive comparison with the other samples confirmed a general trend of increasing amylose content being linked with the accumulation of defects within crystalline lamellae. We conclude that amylose content directly influences the architecture of semi-crystalline lamellae, whereas thermodynamic and functional properties are brought about by the interplay of amylose content and amylopectin architecture.     

Orientation of collagen fibers at the boundary between two successive osteonic lamellae and its mechanical interpretation

By applying an original technique, an investigation has been carried out to determine the orientation of collagen fibrils at the boundary between two successive lamellae in alternate osteons. Evidence is reported that the predominant fiber direction does not change abruptly from one lamella to the next; there is an intermediate system of criss-crossed fibers whose main orientation makes an angle of nearly 45 degrees with the direction of the fibers in the two adjacent lamellae. Taking a composite orthogonally reinforced laminate as a model, a mechanical interpretation of this intermediate system of collagen fibers is given.     

Quantification of biomechanical properties of human corneal scar using acoustic radiation force optical coherence elastography

Biomechanical properties of corneal scar are strongly correlated with many corneal diseases and some types of corneal surgery, however, there is no elasticity information available about corneal scar to date. Here, we proposed an acoustic radiation force optical coherence elastography system to evaluate corneal scar elasticity. Elasticity quantification was first conducted on ex vivo rabbit corneas, and the results validate the efficacy of our system. Then, experiments were performed on an ex vivo human scarred cornea, where the structural features, the elastic wave propagations, and the corresponding Young’s modulus of both the scarred region and the normal region were achieved and based on this, 2D spatial distribution of Young’s modulus of the scarred cornea was depicted. Up to our knowledge, we realized the first elasticity quantification of corneal scar, which may provide a potent tool to promote clinical research on the disorders and surgery of the cornea.     

The correlation of single-particle diffraction patterns as a continuous function of particle orientation

A statistical model for X-ray scattering of a non-periodic sample to high angles is introduced. It is used to calculate analytically the correlation of distinct diffraction measurements of a particle as a continuous function of particle orientation. Diffraction measurements with shot-noise are also considered. This theory provides a general framework for a deeper understanding of single particle imaging techniques used at X-ray free-electron lasers. Many of these techniques use correlations as a measure of diffraction-pattern similarity in order to determine properties of the sample, such as particle orientation.     

用二次谐波成像技术研究经飞秒激光切削后角膜变化

本文用二次谐波成像技术（second harmonic generation SHG）来研究飞秒激光切削后角膜结构的变化.在生物学研究,材料科学等方面都有很广泛应用的SHG成像技术能在不破坏的角膜情况下获得高对比度的角膜层析图像,分辨率为500 nm,实验装置是利用现有的双光子显微镜.本文还根据成像结果评价了飞秒激光在角膜切削中的质量,为飞秒激光微米级的精确切削和临床应用提供了实验支持.     

Collagen and mature elastic fibre organisation as a function of depth in the human cornea and limbus

A network of circumferentially oriented collagen fibrils exists in the periphery of the human cornea, and is thought to be pivotal in maintaining corneal biomechanical stability and curvature. However, it is unknown whether or not this key structural arrangement predominates throughout the entire corneal thickness or exists as a discrete feature at a particular tissue depth; or if it incorporates any elastic fibres and how, with respect to tissue depth, the circumcorneal annulus integrates with the orthogonally arranged collagen of the central cornea. To address these issues we performed a three-dimensional investigation of fibrous collagen and elastin architecture in the peripheral and central human cornea using synchrotron X-ray scattering and non-linear microscopy. This showed that the network of collagen fibrils circumscribing the human cornea is located in the posterior one-third of the tissue and is interlaced with significant numbers of mature elastic fibres which mirror the alignment of the collagen. The orthogonal arrangement of collagen in the central cornea is also mainly restricted to the posterior stromal layers. This information will aid the development of corneal biomechanical models aimed at explaining how normal corneal curvature is sustained and further predicting the outcome of surgical procedures.     

Three‐dimensional mapping of the orientation of collagen corneal lamellae in healthy and keratoconic human corneas using SHG microscopy

Keratoconus is an eye disorder that causes the cornea to take an abnormal conical shape, thus impairing its refractive functions and causing blindness. The late diagnosis of keratoconus is among the principal reasons for corneal surgical transplantation. This pathology is characterized by a reduced corneal stiffness in the region immediately below Bowman's membrane, probably due to a different lamellar organization, as suggested by previous studies. Here, the lamellar organization in this corneal region is characterized in three dimensions by means of second‐harmonic generation (SHG) microscopy. In particular, a method based on a three‐dimensional correlation analysis allows to probe the orientation of sutural lamellae close to the Bowman's membrane, finding statistical differences between healthy and keratoconic samples. This method is demonstrated also in combination with an epi‐detection scheme, paving the way for a potential clinical ophthalmic application of SHG microscopy for the early diagnosis of keratoconus.

SHG image acquired with sagittal optical sectioning ( A ) of a healthy cornea and ( B ) of a keratoconic cornea. Scale bars: 30 μm.     

Biomechanical model of human cornea based on stromal microstructure

The optical characteristics of the human cornea depends on the mechanical balance between the intra-ocular pressure and intrinsic tissue stiffness. A wide range of ophthalmic surgical procedures alter corneal biomechanics to induce local or global curvature changes for the correction of visual acuity. Due to the large number of surgical interventions performed every day, a deeper understanding of corneal biomechanics is needed to improve the safety of these procedures and medical devices. The aim of this study is to propose a biomechanical model of the human cornea, based on stromal microstructure. The constitutive mechanical law includes collagen fiber distribution based on X-ray scattering analysis, collagen cross-linking, and fiber uncrimping. Our results showed that the proposed model reproduced inflation and extensiometry experimental data [Elsheikh et al., Curr. Eye Res., 2007; Elsheikh et al., Exp. Eye Res., 2008] successfully. The mechanical properties obtained for different age groups demonstrated an increase in collagen cross-linking for older specimens. In future work such a model could be used to simulate non-symmetric interventions, and provide better surgical planning.     

Microfibril Orientation Dominates the Microelastic Properties of Human Bone Tissue at the Lamellar Length Scale

The elastic properties of bone tissue determine the biomechanical behavior of bone at the organ level. It is now widely accepted that the nanoscale structure of bone plays an important role to determine the elastic properties at the tissue level. Hence, in addition to the mineral density, the structure and organization of the mineral nanoparticles and of the collagen microfibrils appear as potential key factors governing the elasticity. Many studies exist on the role of the organization of collagen microfibril and mineral nanocrystals in strongly remodeled bone. However, there is no direct experimental proof to support the theoretical calculations. Here, we provide such evidence through a novel approach combining several high resolution imaging techniques: scanning acoustic microscopy, quantitative scanning small-Angle X-ray scattering imaging and synchrotron radiation computed microtomography. We find that the periodic modulations of elasticity across osteonal bone are essentially determined by the orientation of the mineral nanoparticles and to a lesser extent only by the particle size and density. Based on the strong correlation between the orientation of the mineral nanoparticles and the collagen molecules, we conclude that the microfibril orientation is the main determinant of the observed undulations of microelastic properties in regions of constant mineralization in osteonal lamellar bone. This multimodal approach could be applied to a much broader range of fibrous biological materials for the purpose of biomimetic technologies.