Modeling of bone at a single lamella level

This paper focuses on the ultrastructure of bone at a single lamella level. At this scale, collagen fibrils reinforced with apatite crystals are aligned preferentially to form a lamella. At the next structural level, such lamella are stacked in different orientations to form either osteons in cortical bone or trabecular pockets in trabecular bone. We use a finite element model, which treats small strain elasticity of a spatially random network of collagen fibrils, and compute anisotropic effective stiffness tensors and deformations of such a single lamella as a function of fibril volume fractions (or porosities), prescribed microgeometries, and fibril geometric and elastic properties.     

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.     

A first estimation of prestress in so-called circularly fibered osteonic lamellae.

The existence and role of prestress in the various hierarchical structures of long bone are long standing questions. In this study, the prestress and associated strain that exist in a component of human bone microstructure, circularly fibered osteonic lamella, are estimated. Such estimates allow the formulation of hypotheses on prestress formation and lamellar stiffness. Dimensional measurements were obtained for eight fully calcified lamellae. These dimensions, before isolation from the surrounding alternate osteon and after strain relief by isolation and axial sectioning, furnish data upon which a geometric lamellar model is constructed. A material model is based on the most likely hypothesis as to lamellar structure. This geometric-material model allows estimation of the preexisting strain. The largest strains occur in shear circumferential-axial and normal axial strain directions, averaging 0.08 and 0.05, respectively. The geometric-material model expresses prestress in terms of as yet unknown elastic moduli. The average prestress magnitude is the largest in shear circumferential-axial direction, compensating for alternate osteon weakness in this direction. The estimated axial prestress confirms long hypothesized alternate osteon precompression, which impedes fractures of areas of collagen bundles transverse to the osteon axis at low stresses. The results of the model support the formulation of the following biological hypotheses: (a) lamellar prestress occurs at a supra-molecular level, namely through collagen bundles which are themselves likely to be prestressed; (b) collagen bundles oblique to the lamellar axis are responsible for shear prestress; (c) prestress ranges up to 0.11 GPa; and (d) the lamella is less stiff than alternate osteon.     

Specimen-specific multi-scale model for the anisotropic elastic constants of human cortical bone

The anisotropic elastic constants of human cortical bone were predicted using a specimen-specific micromechanical model that accounted for structural parameters across multiple length scales. At the nano-scale, the elastic constants of the mineralized collagen fibril were estimated from measured volume fractions of the constituent phases, namely apatite crystals and Type I collagen. The elastic constants of the extracellular matrix (ECM) were predicted using the measured orientation distribution function (ODF) for the apatite crystals to average the contribution of misoriented mineralized collagen fibrils. Finally, the elastic constants of cortical bone tissue were determined by accounting for the measured volume fraction of Haversian porosity within the ECM. Model predictions using the measured apatite crystal ODF were not statistically different from experimental measurements for both the magnitude and anisotropy of elastic constants. In contrast, model predictions using common idealized assumptions of perfectly aligned or randomly oriented apatite crystals were significantly different from the experimental measurements. A sensitivity analysis indicated that the apatite crystal volume fraction and ODF were the most influential structural parameters affecting model predictions of the magnitude and anisotropy, respectively, of elastic constants.     

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

Mineralized collagen fibrils have been usually analyzed like a two-phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that when Halpin–Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations, and a new finite element model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.     

Pore and matrix distribution in the fiber wall revealed by atomic force microscopy and image analysis

A method for the ultrastructural investigation of fiber cross-sections based on atomic force microscopy in combination with image analysis is presented. A uniform distribution of pores across the matrix material within the fiber wall was revealed by impregnation of pulp fibers with poly(ethylene glycol). The effects of chemical and mechanical processing on the pore and matrix structure and on the arrangement of the cellulose fibril aggregates were investigated. During chemical processing, changes in the fiber ultrastructure occur: a broadening of the pore and matrix lamella widths in combination with a reduction in their number and an enlargement of the cellulose fibril aggregates. It was found that pores formed during pulping are evenly distributed across the fiber wall in the transverse direction. In contrast, refining increases the pore and matrix lamella width in the fiber wall closest to the middle lamella an effect which gradually decrease in size toward the lumen side.     

'Universal' microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity

Bone materials are characterized by an astonishing variability and diversity. Still, because of 'architectural constraints' due to once chosen material constituents and their physical interaction, the fundamental hierarchical organization or basic building plans of bone materials remain largely unchanged during biological evolution. Such universal patterns of microstructural organization govern the mechanical interaction of the elementary components of bone (hydroxyapatite, collagen, water; with directly measurable tissue-independent elastic properties), which are here quantified through a multiscale homogenization scheme delivering effective elastic properties of bone materials: at a scale of 10nm, long cylindrical collagen molecules, attached to each other at their ends by approximately 1.5nm long crosslinks and hosting intermolecular water inbetween, form a contiguous matrix called wet collagen. At a scale of several hundred nanometers, wet collagen and mineral crystal agglomerations interpenetrate each other, forming the mineralized fibril. At a scale of 5-10microm, the extracellular solid bone matrix is represented as collagen fibril inclusions embedded in a foam of largely disordered (extrafibrillar) mineral crystals. At a scale above the ultrastructure, where lacunae are embedded in extracellular bone matrix, the extravascular bone material is observed. Model estimates predicted from tissue-specific composition data gained from a multitude of chemical and physical tests agree remarkably well with corresponding acoustic stiffness experiments across a variety of cortical and trabecular, extracellular and extravascular materials. Besides from reconciling the well-documented, seemingly opposed concepts of 'mineral-reinforced collagen matrix' and 'collagen-reinforced mineral matrix' for bone ultrastructure, this approach opens new possibilities in the exploitation of computer tomographic data for nano-to-macro mechanics of bone organs.     

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.     

A quantitative collagen fibers orientation assessment using birefringence measurements: calibration and application to human osteons

Even though mechanical properties depend strongly on the arrangement of collagen fibers in mineralized tissues, it is not yet well resolved. Only a few semi-quantitative evaluations of the fiber arrangement in bone, like spectroscopic techniques or circularly polarized light microscopy methods are available. In this study the out-of-plane collagen arrangement angle was calibrated to the linear birefringence of a longitudinally fibered mineralized turkey leg tendon cut at variety of angles to the main axis. The calibration curve was applied to human cortical bone osteons to quantify the out-of-plane collagen fibers arrangement. The proposed calibration curve is normalized to sample thickness and wavelength of the probing light to enable a universally applicable quantitative assessment. This approach may improve our understanding of the fibrillar structure of bone and its implications on mechanical properties.     

The problem of bone lamellation: An attempt to explain different proposed models

Collagen texture and osteocyte distribution were analyzed in human woven‐ and lamellar‐bone using scanning and transmission electron microscopy. We provide data substantiating the concept that lamellar bone is made up of an alternation of dense‐acellular lamellae and loose‐cellular lamellae, all exhibiting an interwoven texture of collagen fibers. An attempt is also made to explain how the present findings might conform to those of authors whose models propose orderly, geometric arrangements of collagen fibers inside bony lamellae. Such a comparison is possible because the present investigation analyzes split loose lamellae and tangentially‐sectioned dense lamellae. It emerged that only loose lamellae can be dissected, revealing a loose interwoven collagen texture and halved osteocyte lacunae. Dense lamellae cannot be split because of their compactness. The analysis of tangentially sectioned dense lamellae demonstrates that they consist of a network of interwoven collagen fiber bundles. Inside each bundle, collagen fibers run parallel to each other but change direction where they enter adjacent bundles, at angles as described by other authors whose TEM investigations were performed at a much higher magnification than those of the present study. Consequently, what these authors consider to be a lamella are, instead, bundles of collagen fibers inside a lamella. There is discussion of the role played by the manner of osteocyte‐recruitment in the deposition of lamellar‐ and woven‐bone and how the presence of these cells is crucial for collagen spatial arrangement in bone tissues. J. Morphol., 2013. © 2013 Wiley Periodicals, Inc.     

Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods

The key parameters determining the elastic properties of an unidirectional mineralized bone fibril-array decomposed in two further hierarchical levels are investigated using mean field methods. Modeling of the elastic properties of mineralized micro- and nanostructures requires accurate information about the underlying topology and the constituents’ material properties. These input data are still afflicted by great uncertainties and their influence on computed elastic constants of a bone fibril-array remains unclear. In this work, mean field methods are applied to model mineralized fibrils, the extra-fibrillar matrix and the resulting fibril-array. The isotropic or transverse isotropic elastic constants of these constituents are computed as a function of degree of mineralization, mineral distribution between fibrils and extra-fibrillar matrix, collagen stiffness and fibril volume fraction. The linear sensitivity of the elastic constants was assessed at a default set of the above parameters. The strain ratios between the constituents as well as the axial and transverse indentation moduli of the fibril-array were calculated for comparison with experiments. Results indicate that the degree of mineralization and the collagen stiffness dominate fibril-array elasticity. Interestingly, the stiffness of the extra-fibrillar matrix has a strong influence on transverse and shear moduli of the fibril-array. The axial strain of the intra-fibrillar mineral platelets is 30–90% of the applied fibril strain, depending on mineralization and collagen stiffness. The fibril-to-fibril-array strain ratio is essentially ~1. This study provides an improved insight in the parameters, which govern the fibril-array stiffness of mineralized tissues such as bone.     

In Vitro Orientation of Fibroblasts and Myoblasts on Aligned Collagen Film

A collagen film in which the collagen fibers were aligned was prepared and characterized by scanning electron microscopy. Cell orientation on this film was studied in vitro using human fibroblasts and chick embryo myoblasts. Ninety-four percent of innoculated fibroblasts were aligned along the direction of the collagen fiber. The cell orientation was disturbed when cytochalasin B or colchicine was added to the culture medium. The myoblasts showed a similar alignment along the direction of collagen fiber. The scanning electron microscopic observation revealed that none of the cytoplasmic extensions had consistent relationships to the direction of collagen fiber. Myoblast fusion was accelerated on the aligned membrane as compared to a randomly oriented film, suggesting some role of contact guidance in muscle cell differentiation.     

Ex-vivo observation of calcification process in chick tibia slice: Augmented calcification along collagen fiber orientation in specimens subjected to static stretch

Bone formation through matrix synthesis and calcification in response to mechanical loading is an essential process of the maturation in immature animals, although how mechanical loading applied to the tissue increases the calcification and improves mechanical properties, and which directions the calcification progresses within the tissue are largely unknown. To address these issues, we investigated the calcification of immature chick bone under static tensile stretch using a newly developed real-time observation bioreactor system. Bone slices perpendicular to the longitudinal axis obtained from the tibia in 2- to 4-day-old chick legs were cultured in the system mounted on a microscope, and their calcification was observed up to 24 h while they were stretched in the direction parallel to the slice. Increase in the calcified area, traveling distance and the direction of the calcification and collagen fiber orientation in the newly calcified region were analyzed. There was a significant increase in calcified area in the bone explant subjected to tensile strain over ∼3%, which corresponds to the threshold strain for collagen fibers showing alignment in the direction of stretch, indicating that the fiber alignment may enhance tissue calcification. The calcification progressed to a greater distance to the stretching direction in the presence of the loading. Moreover, collagen fiber orientation in the calcified area in the loaded samples was coincided with the progression angle of the calcification. These results clearly show that the application of static tensile strain enhanced tissue calcification, which progresses along collagen fibers aligned to the loading direction.     

Lamellar bone: structure-function relations.

The term "bone" refers to a family of materials that have complex hierarchically organized structures. These structures are primarily adapted to the variety of mechanical functions that bone fulfills. Here we review the structure-mechanical relations of one bone structural type, lamellar bone. This is the most abundant type in many mammals, including humans. A lamellar unit is composed of five sublayers. Each sublayer is an array of aligned mineralized collagen fibrils. The orientations of these arrays differ in each sublayer with respect to both collagen fibril axes and crystal layers, such that a complex rotated plywood-like structure is formed. Specific functions for lamellar bone, as opposed to the other bone types, could not be identified. It is therefore proposed that the lamellar structure is multifunctional-the "concrete" of the bone family of materials. Experimentally measured mechanical properties of lamellar bone demonstrate a clear-cut anisotropy with respect to the axis direction of long bones. A comparison of the elastic and ultimate properties of parallel arrays of lamellar units formed in primary bone with cylindrically shaped osteonal structures in secondary formed bone shows that most of the intrinsic mechanical properties are built into the lamellar structure. The major advantages of osteonal bone are its fracture properties. Mathematical modeling of the elastic properties based on the lamellar structure and using a rule-of-mixtures approach can closely simulate the measured mechanical properties, providing greater insight into the structure-mechanical relations of lamellar bone.     

Scanning microscopy of collagen in the basement lamella of normal and regenerating frog tadpoles

The arrangement of collagen fibers over the body surface in the basement lamella of Pseudaeris and Xenopus tadpoles is described. It can be viewed by scanning microscopy after removal of epidermis and basal lamina by trypsin treatment of alcohol fixed tissue. The orthogonal array is modified in regions where fiber direction changes extensively such as the base of the ventral fin or the posterior part of the head. In these regions “exceptional points” in the orthogonal pattern occur, as described by Rosin (1946). The pattern is bilaterally symmetrical. In the region of the nasal opening the orthogonal pattern is replaced by a mat of randomly oriented fibers. In tail regeneration the wound area is marked by aberrant disposition of collagen anteriorly then a mat of randomly disposed fibers followed posteriorly with a sharp transition to the orthogonal pattern of the regenerate. No fiber terminations could be seen in normal or regenerating regions of the lamella.     

Probing the role of water in lamellar bone by dehydration in the environmental scanning electron microscope

Water, collagen and mineral are the three major components of bone. The structural organization of water and its functions within the bone were investigated using the environmental scanning electron microscope and by analyzing dimensional changes that occur when fresh equine osteonal bone is dehydrated and then rehydrated. These changes are attributed mainly to loss of bulk and weakly bound water. In longitudinal sections a contraction of 1.2% was observed perpendicular to the lamellae, whereas no contraction occurred parallel to the lamellae. In transverse sections a contraction of 1.4% was observed both parallel and perpendicular to the lamellae. SEM back scattered electron images showed that about half of an individual lamella is less mineralized, and thus has more water than the other half. We therefore propose that contractions perpendicular to lamellae are due to the presence of more water-filled rather than mineral-filled channels within the mineralized collagen fibril arrays. As these channels are also aligned with the crystal planes, the crystal arrays, oriented as depicted in the rotated plywood model for lamellar bone, facilitate or hinder contraction in different directions.     

Mechanical properties of collagen fibrils

The formation of collagen fibers from staggered subfibrils still lacks a universally accepted model. Determining the mechanical properties of single collagen fibrils (diameter 50-200 nm) provides new insights into collagen structure. In this work, the reduced modulus of collagen was measured by nanoindentation using atomic force microscopy. For individual type 1 collagen fibrils from rat tail, the modulus was found to be in the range from 5 GPa to 11.5 GPa (in air and at room temperature). The hypothesis that collagen anisotropy is due to the subfibrils being aligned along the fibril axis is supported by nonuniform surface imprints performed by high load nanoindentation.     

A new model to simulate the elastic properties of mineralized collagen fibril

Bone, because of its hierarchical composite structure, exhibits an excellent combination of stiffness and toughness, which is due substantially to the structural order and deformation at the smaller length scales. Here, we focus on the mineralized collagen fibril, consisting of hydroxyapatite plates with nanometric dimensions aligned within a protein matrix, and emphasize the relationship between the structure and elastic properties of a mineralized collagen fibril. We create two- and three-dimensional representative volume elements to represent the structure of the fibril and evaluate the importance of the parameters defining its structure and properties of the constituent mineral and collagen phase. Elastic stiffnesses are calculated by the finite element method and compared with experimental data obtained by synchrotron X-ray diffraction. The computational results match the experimental data well, and provide insight into the role of the phases and morphology on the elastic deformation characteristics. Also, the effects of water, imperfections in the mineral phase and mineral content outside the mineralized collagen fibril upon its elastic properties are discussed.     

Elastic properties of woven bone: effect of mineral content and collagen fibrils orientation

Woven bone is a type of tissue that forms mainly during fracture healing or fetal bone development. Its microstructure can be modeled as a composite with a matrix of mineral (hydroxyapatite) and inclusions of collagen fibrils with a more or less random orientation. In the present study, its elastic properties were estimated as a function of composition (degree of mineralization) and fibril orientation. A self-consistent homogenization scheme considering randomness of inclusions’ orientation was used for this purpose. Lacuno-canalicular porosity in the form of periodically distributed void inclusions was also considered. Assuming collagen fibrils to be uniformly oriented in all directions led to an isotropic tissue with a Young’s modulus \(E = 1.90\) GPa, which is of the same order of magnitude as that of woven bone in fracture calluses. By contrast, assuming fibrils to have a preferential orientation resulted in a Young’s modulus in the preferential direction of 9–16 GPa depending on the mineral content of the tissue. These results are consistent with experimental evidence for woven bone in foetuses, where collagen fibrils are aligned to a certain extent.     

Two-Dimensional Nanoscale Structural and Functional Imaging in Individual Collagen Type I Fibrils

The piezoelectric properties of single collagen type I fibrils in fascia were imaged with sub-20 nm spatial resolution using piezoresponse force microscopy. A detailed analysis of the piezoresponse force microscopy signal in controlled tip-fibril geometry revealed shear piezoelectricity parallel to the fibril axis. The direction of the displacement is preserved along the whole fiber length and is independent of the fiber conformation. It is shown that individual fibrils within bundles in skeletal muscle fascia can have opposite polar orientations and are organized into domains, i.e., groups of several fibers having the same polar orientation. We were also able to detect piezoelectric activity of collagen fibrils in the high-frequency range up to 200 kHz, suggesting that the mechanical response time of biomolecules to electrical stimuli can be ∼5 μs.