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 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.     

The role of 3-D collagen organization in stromal elasticity: a model based on X-ray diffraction data and second harmonic-generated images

Examining the cross-section of the human cornea with second harmonic-generated (SHG) imaging shows that many lamellae do not lie parallel to the cornea’s anterior surface but have inclined trajectories that take them through the corneal thickness with a depth-dependent distribution. A continuum mechanics-based model of stromal elasticity is developed based on orientation information extracted and synthesized from both X-ray scattering studies and SHG imaging. The model describes the effects of inclined lamella orientation by introducing a probability function that varies with depth through the stroma, which characterizes the range and distribution of lamellae at inclined angles. When combined with the preferred lamellar orientations found from X-ray scattering experiments, a fully 3-D representation of lamella orientation is achieved. Stromal elasticity is calculated by a weighted average of individual lamella properties based on the spatially varying 3-D orientation distribution. The model is calibrated with in vitro torsional shear experiments and in vivo indentation data and then validated with an in vitro inflation study. A quantitative explanation of the experimentally measured depth dependence of mechanical properties emerges from the model. The significance of the 3-D lamella orientation in the mechanics of the human cornea is demonstrated by investigating and contrasting the effects of previous modeling assumptions made on lamella orientation.     

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.     

Collagen organization in the chicken cornea and structural alterations in the retinopathy, globe enlarged (rge) phenotype--an X-ray diffraction study

An investigation into the collagenous structure of the mature avian cornea is presented. Wide-angle X-ray diffraction is employed to assess collagen organization in 9-month-old chicken corneas. The central 2-4mm corneal region features a preponderance of fibrils directed along the superior-inferior and nasal-temporal orthogonal meridians. More peripherally the orientation of fibrils alters in favor of a predominantly tangential arrangement. The chicken cornea appears to be circumscribed by an annulus of fibrils that extends into the limbus. The natural arrangement of collagen in the chicken cornea is discussed in relation to corneal shape and the mechanical requirements of avian corneal accommodation. Equivalent data are also presented from age-matched blind chickens affected with the retinopathy, globe enlarged (rge) mutation, characterized by an abnormally thick and flat cornea. The data indicate considerable realignment and redistribution of collagen lamellae in the peripheral rge cornea. In contrast to normal chickens, no obvious tangential collagen alignment was evident in the periphery of rge corneas. In mammals, the presence of a limbal fibril annulus is believed to be important in corneal shape preservation. We postulate that corneal flattening in rge chickens may be related to biomechanical changes brought about by an alteration in collagen arrangement at the corneal periphery.     

On the relationship between the microstructure of bone and its mechanical stiffness.

A recent study of bone structure shows that the plate-shaped carbonate apatite crystals in individual lamellae are arranged in layers across the lamellae, and that the orientation of these layers are different in alternate lamellae. Based on these findings, a new micromechanical model for the Young's modulus of bone is proposed, which accounts for the anisotropy and geometrical characteristics of the material. The model incorporates the platelet-like geometry of the basic reinforcing unit, the presence of alternating thin and thick lamellae, and the orientations of the crystal platelets in the lamellae. The thin and thick lamellae are modeled as orthotropic composite layers made up of thin rectangular apatite platelets within a collagen matrix, and classical orthotropic elasticity theory is used to calculate the Young's modulus of the lamellae. Bone is viewed as an assembly of such orthotropic lamellae bent into cylindrical structures, and having a constant, alternating angle between successive lamellae. The micromechanical model employs a modified rule-of-mixtures to account for the two types of lamellae. The model provides a curve similar to the published experimental data on the angular dependence of Young's modulus, including a local maximum at an angle between 0 and 90 degrees. A rigorous testing of the model awaits additional experimental data.     

Quantitative changes and ultrastructural alterations of the cornea in response to ultraviolet light II. Effects on amphibia; elucidation of desmosomal structure and basement membrane synthesis

The cornea of the urodele amphibian Triturus c. cristatus was studied ultrastructurally in order to provide the basis for a comparison among corneas throughout the vertebrate phylum. The cornea of this salamander consists of relatively thick epithelium and basement membrane and thin Descemet's membrane, unlike the mammalian corneas. The outermost epithelial cells contain Ruthenium Red stainable extracellular filaments and intracellular vesicles which are thought to play a role in the process of lubricating the corneal surface. Occluding junctions have been observed in the apical region of the superficial epithelial cells and are considered as barriers to the intercellular passage of material. A thin substantia propria (stroma) consists of about 40 collagenous highly organized lamellae. The thicknesses of the basement membrane, Descemet's membrane and the epithelium are believed to represent the primitive situation in the process of corneal evolution.     

Aligning cells in arbitrary directions on a membrane sheet using locally formed microwrinkles

Sheets of cells can be used for tissue regenerative medicine. Cell alignment within the sheet is now a key factor in the next generation of this technology. Anisotropic cell sheets without random cell orientations have been conventionally produced with photolithographically, microfabricated substrates using special facilities and equipment. Here we demonstrate a more accessible approach to the fabrication of anisotropic substrates. We locally deformed part of an elastic membrane and simultaneously oxidized the surface to create microwrinkles as well as to enable adhesion to the extracellular matrix. The approach with the local loading made it possible to orient cells in controlled directions within a single membrane sheet depending on the strains determined by the controllable deformation. This technique potentially enables a versatile design of microwrinkles for target-compatible cell alignments.     

花背蟾蜍蝌蚪变态期角膜发育的研究

作者用光镜和电镜研究了花背蟾蜍蝌蚪变态期角膜的发育。在后肢发育晚期,内、外角膜在中央部位首先愈台,在完全变态期角膜完全愈合,此时角膜上皮细胞增殖,上皮基质变为Bowman’s膜,内、外角膜之间的成纤维细胞和由它分泌的胶原纤维形成角膜基质,内角膜细胞形成单层的角膜内皮,它与角膜基质间的Descemet’s膜最晚形成。     

Chlorophyll,glycerolipid and protein ratios in spinach chloroplast grana and stroma lamellae

The relative molar amounts of glycerolipids are similar in grana and stroma lamellae, as are the ratios of total glycerolipid to weight of membrane protein. However the chlorophyll content relative to protein of grana lamellae is about 40% higher than that of stroma lamellae from the same preparation. Previous reports of chemical composition or enzyme activity based on chlorophyll alone can be highly misleading. The large functional and conformational differences between these two membranes may be related to these differences in pigment content, but are likely to result primarily from qualitative protein differences. The data are in accord with a membrane model in which nonpolar regions of membrane protein bind lipid in fairly constant amounts.     

A mechanical model for the human intervertebral disc

Stress relaxation experiments were performed on specimens from a human intervertebral disc. Specimens were made from the nucleus pulposus and from the external lamellae of the anulus fibrosus in two different orientations. Tests were run with varying moisture content so as to develop a relaxation master curve. A model was developed based on the experimental data. It was found that the short term master curve for the lamellae of the anulus and nucleus are similar, whereas the long term rubbery plateau is different between the lamellae and the nucleus. It was also established that the master curves for different lamellae and the nucleus were shifted relative to each other in the time domain due to changes in water content. The average relaxation modulus of the whole disc was obtained by averaging the properties between the anulus and nucleus. This model was then used for studies of Schmorl's nodes, of degenerated discs and for circumstances in which hydration is considered to be important.     

A computational remodeling approach to predict the physiological architecture of the collagen fibril network in corneo-scleral shells

Organized collagen fibrils form complex networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of human eye tissues. Physiological adaptation of the collagen network and the mechanical condition within biological tissues are complex and mutually dependent phenomena. In this contribution, a computational model is presented to investigate the interaction between the collagen fibril architecture and mechanical loading conditions in the corneo-scleral shell. The biomechanical properties of eye tissues are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. Biomechanically induced remodeling of the collagen network is captured on the meso-scale by allowing for a continuous re-orientation of preferred fibril orientations and a continuous adaptation of the fibril dispersion. The presented approach is applied to a numerical human eye model considering the cornea and sclera. The predicted fibril morphology correlates well with experimental observations from X-ray scattering data.     

Native architecture of the photosynthetic membrane from Rhodobacter veldkampii

The photosynthetic membrane in purple bacteria contains several pigment–protein complexes that assure light capture and establishment of the chemiosmotic gradient. The bioenergetic tasks of the photosynthetic membrane require the strong interaction between these various complexes. In the present work, we acquired the first images of the native outer membrane architecture and the supramolecular organization of the photosynthetic apparatus in vesicular chromatophores of Rhodobacter (Rb.) veldkampii. Mixed with LH2 (light-harvesting complex 2) rings, the PufX-containing LH1–RC (light-harvesting complex 1 – reaction center) core complexes appear as C-shaped monomers, with random orientations in the photosynthetic membrane. Within the LH1 fence surrounding the RC, a remarkable gap that is probably occupied (or partially occupied) by PufX is visualized. Sequence alignment revealed that one specific region in PufX may be essential for PufX-induced core dimerization. In this region of ten amino acids in length all Rhodobacter species had five conserved amino acids, with the exception of Rb. veldkampii. Our findings provide direct evidence that the presence of PufX in Rb. veldkampii does not directly govern the dimerization of LH1–RC core complexes in the native membrane. It is indicated, furthermore, that the high membrane curvature of Rb. veldkampii chromatophores (Rb. veldkampii features equally small vesicular chromatophores alike Rb. sphaeroides) is not due to membrane bending induced by dimeric RC–LH1–PufX cores, as it has been proposed in Rb. sphaeroides.     

On the alignment of cellulose microfibrils by cortical microtubules: A review and a model

Summary The hypothesis that microtubules align microfibrils, termed the alignment hypothesis, states that there is a causal link between the orientation of cortical microtubules and the orientation of nascent microfibrils. I have assessed the generality of this hypothesis by reviewing what is known about the relation between microtubules and microfibrils in a wide group of examples: in algae of the family Characeae,Closterium acerosum, Oocystis solitaria, and certain genera of green coenocytes and in land plant tip-growing cells, xylem, diffusely growing cells, and protoplasts. The salient features about microfibril alignment to emerge are as follows. Cellulose microfibrils can be aligned by cortical microtubules, thus supporting the alignment hypothesis. Alignment of microfibrils can occur independently of microtubules, showing that an alternative to the alignment hypothesis must exist. Microfibril organization is often random, suggesting that self-assembly is insufficient. Microfibril organization differs on different faces of the same cell, suggesting that microfibrils are aligned locally, not with respect to the entire cell. Nascent microfibrils appear to associate tightly with the plasma membrane. To account for these observations, I present a model that posits alignment to be mediated through binding the nascent microfibril. The model, termed templated incorporation, postulates that the nascent microfibril is incorporated into the cell wall by binding to a scaffold that is oriented; further, the scaffold is built and oriented around either already incorporated microfibrils or plasma membrane proteins, or both. The role of cortical microtubules is to bind and orient components of the scaffold at the plasma membrane. In this way, spatial information to align the microfibrils may come from either the cell wall or the cell interior, and microfibril alignment with and without microtubules are subsets of a single mechanism.Dedicated to Professor Brian E. S. Gunning on the occasion of his 65th birthday     

De novo determination of internuclear vector orientations from residual dipolar couplings measured in three independent alignment media

The straightforward interpretation of solution state residual dipolar couplings (RDCs) in terms of internuclear vector orientations generally requires prior knowledge of the alignment tensor, which in turn is normally estimated using a structural model. We have developed a protocol which allows the requirement for prior structural knowledge to be dispensed with as long as RDC measurements can be made in three independent alignment media. This approach, called Rigid Structure from Dipolar Couplings (RSDC), allows vector orientations and alignment tensors to be determined de novo from just three independent sets of RDCs. It is shown that complications arising from the existence of multiple solutions can be overcome by careful consideration of alignment tensor magnitudes in addition to the agreement between measured and calculated RDCs. Extensive simulations as well applications to the proteins ubiquitin and Staphylococcal protein GB1 demonstrate that this method can provide robust determinations of alignment tensors and amide N-H bond orientations often with better than 10 degrees accuracy, even in the presence of modest levels of internal dynamics.     

Collagen orientation patterns in human secondary osteons, quantified in the radial direction by confocal microscopy

The composite structure of secondary osteon lamellae, key micro-mechanical components of human bone, has intrigued researchers for the last 300 years. Scanning confocal microscopy here for the first time systematically quantifies collagen orientations by location within the lamellar thickness. Fully calcified lamellar specimens, extinct or bright in cross-section under circularly polarized light, were gently flattened, and then examined along their thickness direction, the radial direction in the previously embedding osteon. Collagen orientation was measured from confocal image stacks. So-called extinct lamellae and so-called bright lamellae are found to display distinct, characteristic patterns of collagen orientation distribution. Orientations longitudinal to the osteon axis in extinct lamellae, transverse to the osteon axis in bright lamellae, and oblique to the osteon axis in both lamellar types, show parabolic distribution through specimen thickness. Longitudinal collagen in extinct lamellae, and transverse collagen in bright lamellae, peaks at middle third of lamellar thickness, while oblique collagen peaks at outer thirds of both types. Throughout the thickness, longitudinal collagen orientations characterize extinct lamellar specimens, while orientations oblique to the original osteon axis characterize bright lamellar specimens. Measured patterns complement previous indirect results by different methods and reinforce previously hypothesized differences in lamellar mechanical functions.     

由吸附的缩氨酸诱导的细胞膜的形变

研究了由一系列相互平行的吸附在细胞膜上的缩氨酸引起的膜的弹性形变,以及膜对缩氨酸的包裹行为,得到膜的平衡方程,用它可以来处理大尺度的形变,弯曲能量、吸附能量和弹性形变的相互竞争导致膜对缩氨酸发生从不吸附到部分吸附乃至完全包裹的结构转变．在膜的形变很小的时候,可以得到系统能量的解析解。     

Constitutive Model of Erythrocyte Membranes with Distributions of Spectrin Orientations and Lengths

We present an analytical hyperelastic constitutive model of the red blood cell (erythrocyte) membrane based on recently improved characterizations of density and microscopic structure of its spectrin network from proteomics and cryo-electron tomography. The model includes distributions of both orientations and natural lengths of spectrin and updated copy numbers of proteins. By applying finite deformation to the spectrin network, we obtain the total free energy and stresses in terms of invariants of shear and area deformation. We generalize an expression of the initial shear modulus, which is independent of the number of molecular orientations within the network and also derive a simplified version of the model. We apply the model and its simplified version to analyze micropipette aspiration computationally and analytically and explore the effect of local cytoskeletal density change. We also explore the discrepancies among shear modulus values measured using different experimental techniques reported in the literature. We find that the model exhibits hardening behavior and can explain many of these discrepancies. Moreover, we find that the distribution of natural lengths plays a crucial role in the hardening behavior when the correct copy numbers of proteins are used. The initial shear modulus values we obtain using our current model (5.9–15.6 pN/μm) are close to the early estimates (6–9 pN/μm). This new, to our knowledge, constitutive model establishes a direct connection between the molecular structure of spectrin networks and constitutive laws and also defines a new picture of a much denser spectrin network than assumed in prior studies.     

Corneal morphogenesis in the Indian garden lizard,Calotes versicolor (agamidae)

The present study traces corneal morphogenesis in a reptile, the lizard Calotes versicolor, from the lens placode stage (stage 24) until hatching (stage 42), and in the adult. The corneal epithelium separates from the lens placode as a double layer of peridermal and basal cells and remains bilayered throughout development and in the adult. Between stages 32– and 33+, the corneal epithelium is apposed to the lens, and limbic mesodermal cells migrate between the basement membrane of the epithelium and the lens capsule to form a monolayered corneal endothelium. Soon thereafter a matrix of amorphous ground substance and fine collagen fibrils, the presumptive stroma, is seen between the epithelium and the endothelium. Just before stage 34 a new set of limbic mesodermal cells, the keratocytes, migrate into the presumptive stroma. Migrating limbic mesodermal cells, both endothelial cells and keratocytes, use the basement membrane of the epithelium as substratum. Keratocytes may form up to six cell layers at stage 37, but in the adult stroma they form only one or two cell layers. The keratocytes sysnthesize collagen, which aggregates as fibrils and fibers organized in lamellae. The lamellae become condensed as dense collagen layers subepithelially or become compactly organized into a feltwork structure in the rest of the stroma. The basement membrane of the endothelium is always thin. Thickness of the entire cornea increases up to stage 38 and decreases thereafter until stage 41. In the adult the cornea is again nearly as thick as at stage 38.