Hierarchical modeling of the elastic properties of bone at submicron scales: the role of extrafibrillar mineralization

We model the elastic properties of bone at the level of mineralized collagen fibrils via step-by-step homogenization from the staggered arrangement of collagen molecules up to an array of parallel mineralized fibrils. A new model for extrafibrillar mineralization is proposed, assuming that the extrafibrillar minerals are mechanically equivalent to reinforcing rings coating each individual fibril. Our modeling suggests that no more than 30% of the total mineral content is extrafibrillar and the fraction of extrafibrillar minerals grows linearly with the overall degree of mineralization. It is shown that the extrafibrillar mineralization considerably reinforces the fibrils’ mechanical properties in the transverse directions and the fibrils’ shear moduli. The model predictions for the elastic moduli and constants are found to be in a good agreement with the experimental data reported in the literature.     

Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model

Mineralized collagen fibrils are the basic building blocks of bone tissue at the supramolecular level. Several disease states, manipulation of the expression of specific proteins involved in biomineralization, and treatment with different agents alter the extent of mineralization as well as the morphology of mineral crystals which in turn affect the mechanical function of bone tissue. An experimental assessment of mineralized fibers' mechanical properties is challenged by their small size, leaving analytical and computational models as a viable alternative for investigation of the fibril-level mechanical properties. In the current study the variation of the elastic stiffness tensor of mineralized collagen fibrils with changing mineral volume fraction and mineral aspect ratios was predicted via a micromechanical model. The partitioning of applied stresses between mineral and collagen phases is also predicted for normal and shear loading of fibrils. Model predictions resulted in transversely isotropic collagen fibrils in which the modulus along the longer axis of the fibril was the greatest. All the elastic moduli increased with increasing mineral volume fraction whereas Poisson's ratios decreased with the exception of v12 (=v21). The partitioning of applied stresses were such that the stresses acting on mineral crystals were about 1.5, 15, and 3 times greater than collagen stresses when fibrils were loaded transversely, longitudinally, and in shear, respectively. In the overall the predictions were such that: (a) greatest modulus along longer axis; (b) the greatest mineral/collagen stress ratio along the longer axis of collagen fibers (i.e., greatest relief of stresses acting on collagen); and (c) minimal lateral contraction when fibers are loaded along the longer axis. Overall, the pattern of mineralization as put forth in this model predicts a superior mechanical function along the longer axis of collagen fibers, the direction which is more likely to experience greater stresses.     

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.     

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.     

'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.     

Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles

Both elastic modulus and fracture stress are known to increase with the amount of mineral deposited within collagen fibrils. Current mechanical models of mineralized fibrils, where mineral platelets are arranged in parallel arrays, reproduce the first effect but fail to predict an increase in fracture stress. Here, we propose a model with a staggered array of platelets that is in better agreement with results on molecular packing in collagen fibrils and that accounts for an increase of both elastic modulus and fracture stress with the amount of mineral in the fibril. Finally, we explore the dependence of the mechanical properties within the model, when the degree of mineralization and the thickness of the platelets as well as their distance varies.     

Mechanical properties of mineralized collagen fibrils as influenced by demineralization

Dentin and bone derive their mechanical properties from a complex arrangement of collagen type-I fibrils reinforced with nanocrystalline apatite mineral in extra- and intrafibrillar compartments. While mechanical properties have been determined for the bulk of the mineralized tissue, information on the mechanics of the individual fibril is limited. Here, atomic force microscopy was used on individual collagen fibrils to study structural and mechanical changes during acid etching. The characteristic 67 nm periodicity of gap zones was not observed on the mineralized fibril, but became apparent and increasingly pronounced with continuous demineralization. AFM-nanoindentation showed a decrease in modulus from 1.5 GPa to 50 MPa during acid etching of individual collagen fibrils and revealed that the modulus profile followed the axial periodicity. The nanomechanical data, Raman spectroscopy and SAXS support the hypothesis that intrafibrillar mineral etches at a substantially slower rate than the extrafibrillar mineral. These findings are relevant for understanding the biomechanics and design principles of calcified tissues derived from collagen matrices.     

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.     

Are mineralized tissues open crystal foams reinforced by crosslinked collagen? Some energy arguments

Which of the elementary components (hydroxyapatite (HA) crystals, collagen, non-collagenous organic matter, water) do significantly contribute to the ultrastructural elastic stiffness magnitude and anisotropy of mineralized tissues; and how, i.e. through which shapes and assemblages (which micromechanical morphology)? We suggest answers to these questions by analyzing stiffness-volume fraction relationships of wet and dry tissue specimens in the framework of strain energy considerations. Radial stiffness values of both isotropic and anisotropic tissues are found to depend linearly to quadratically on only the mineral volume fraction. This suggests the isotropic contribution of HA to the ultrastructural stiffness. An energy-based analysis of the difference between the axial and radial stiffness values of anisotropic, collagen-rich tissues allows us to assess the collagen elasticity contribution, which is found to depend linearly on the extra-collagenous mineral concentration. These results suggest that collagen and hydroxyapatite are the elementary components governing the ultrastructural elastic stiffness magnitude and anisotropy of bone and mineralized tendons. The elastic stiffness of water and non-collagenous organic matter does not play a significant role. As for the morphological issue, we suggest that mineralized tissues are isotropic open crystal foams; and that these foams are reinforced unidirectionally by collagen molecules which are mechanically activated through tight links between these molecules and HA-crystals. The HA crystals are mechanically activated through stretching and bending in long bone tissues, they are predominantly stretched in mineralized tendons, and bent in hyperpycnotic tissues.     

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.     

Dentin matrix protein 1 immobilized on type I collagen fibrils facilitates apatite deposition in vitro

During bone and dentin mineralization, the crystal nucleation and growth processes are considered to be matrix regulated. Osteoblasts and odontoblasts synthesize a polymeric collagenous matrix, which forms a template for apatite initiation and elongation. Coordinated and controlled reaction between type I collagen and bone/dentin-specific noncollagenous proteins are necessary for well defined biogenic crystal formation. However, the process by which collagen surfaces become mineralized is not understood. Dentin matrix protein 1 (DMP1) is an acidic noncollagenous protein expressed during the initial stages of mineralized matrix formation in bone and dentin. Here we show that DMP1 bound specifically to type I collagen, with the binding region located at the N-telopeptide region of type I collagen. Peptide mapping identified two acidic clusters in DMP1 responsible for interacting with type I collagen. The collagen binding property of these domains was further confirmed by site-directed mutagenesis. Transmission electron microscopy analyses have localized DMP1 in the gap region of the collagen fibrils. Fibrillogenesis assays further demonstrated that DMP1 accelerated the assembly of the collagen fibrils in vitro and also increased the diameter of the reconstituted collagen fibrils. In vitro mineralization studies in the presence of calcium and phosphate ions demonstrated apatite deposition only at the collagen-bound DMP1 sites. Thus specific binding of DMP1 and possibly other noncollagenous proteins on the collagen fibril might be a key step in collagen matrix organization and mineralization.     

Amelogenin-Collagen Interactions Regulate Calcium Phosphate Mineralization in Vitro

Collagen and amelogenin are two major extracellular organic matrix proteins of dentin and enamel, the mineralized tissues comprising a tooth crown. They both are present at the dentin-enamel boundary (DEB), a remarkably robust interface holding dentin and enamel together. It is believed that interactions of dentin and enamel protein assemblies regulate growth and structural organization of mineral crystals at the DEB, leading to a continuum at the molecular level between dentin and enamel organic and mineral phases. To gain insight into the mechanisms of the DEB formation and structural basis of its mechanical resiliency we have studied the interactions between collagen fibrils, amelogenin assemblies, and forming mineral in vitro, using electron microscopy. Our data indicate that collagen fibrils guide assembly of amelogenin into elongated chain or filament-like structures oriented along the long axes of the fibrils. We also show that the interactions between collagen fibrils and amelogenin-calcium phosphate mineral complexes lead to oriented deposition of elongated amorphous mineral particles along the fibril axes, triggering mineralization of the bulk of collagen fibril. The resulting structure was similar to the mineralized collagen fibrils found at the DEB, with arrays of smaller well organized crystals inside the collagen fibrils and bundles of larger crystals on the outside of the fibrils. These data suggest that interactions between collagen and amelogenin might play an important role in the formation of the DEB providing structural continuity between dentin and enamel.     

Strain-rate dependent stiffness of articular cartilage in unconfined compression

The stiffness of articular cartilage is a nonlinear function of the strain amplitude and strain rate as well as the loading history, as a consequence of the flow of interstitial water and the stiffening of the collagen fibril network. This paper presents a full investigation of the interplay between the fluid kinetics and fibril stiffening of unconfined cartilage disks by analyzing over 200 cases with diverse material properties. The lower and upper elastic limits of the stress (under a given strain) are uniquely established by the instantaneous and equilibrium stiffness (obtained numerically for finite deformations and analytically for small deformations). These limits could be used to determine safe loading protocols in order that the stress in each solid constituent remains within its own elastic limit. For a given compressive strain applied at a low rate, the loading is close to the lower limit and is mostly borne directly by the solid constituents (with little contribution from the fluid). In contrast, however in case of faster compression, the extra loading is predominantly transported to the fibrillar matrix via rising fluid pressure with little increase of stress in the nonfibrillar matrix. The fibrillar matrix absorbs the loading increment by self-stiffening: the quicker the loading the faster the fibril stiffening until the upper elastic loading limit is reached. This self-protective mechanism prevents cartilage from damage since the fibrils are strong in tension. The present work demonstrates the ability of the fibril reinfored poroelastic models to describe the strain rate dependent behavior of articular cartilage in unconfined compression using a mechanism of fibril stiffening mainly induced by the fluid flow.     

Microstructural Modeling of Collagen Network Mechanics and Interactions with the Proteoglycan Gel in Articular Cartilage

Cartilage matrix mechanical function is largely determined by interactions between the collagen fibrillar network and the proteoglycan gel. Although the molecular physics of these matrix constituents have been characterized and modern imaging methods are capable of localized measurement of molecular densities and orientation distributions, theoretical tools for using this information for prediction of cartilage mechanical behavior are lacking. We introduce a means to model collagen network contributions to cartilage mechanics based upon accessible microstructural information (fibril density and orientation distributions) and which self-consistently follows changes in microstructural geometry with matrix deformations. The interplay between the molecular physics of the collagen network and the proteoglycan gel is scaled up to determine matrix material properties, with features such as collagen fibril pre-stress in free-swelling cartilage emerging naturally and without introduction of ad hoc parameters. Methods are developed for theoretical treatment of the collagen network as a continuum-like distribution of fibrils, such that mechanical analysis of the network may be simplified by consideration of the spherical harmonic components of functions of the fibril orientation, strain, and stress distributions. Expressions for the collagen network contributions to matrix stress and stiffness tensors are derived, illustrating that only spherical harmonic components of orders 0 and 2 contribute to the stress, while orders 0, 2, and 4 contribute to the stiffness. Depth- and compression-dependent equilibrium mechanical properties of cartilage matrix are modeled, and advantages of the approach are illustrated by exploration of orientation and strain distributions of collagen fibrils in compressed cartilage. Results highlight collagen-proteoglycan interactions, especially for very small physiological strains where experimental data are relatively sparse. These methods for determining matrix mechanical properties from measurable quantities at the microscale (composition, structure, and molecular physics) may be useful for investigating cartilage structure-function relationships relevant to load-bearing, injury, and repair.     

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.     

Ultrastructure of mineralized nodules formed in rat bone marrow stromal cell culture in vitro.

Rat bone marrow stromal cells were cultured in vitro. At days 14-15 of culture, dense clusters of polygonal cells were formed, and they mineralized 2-3 days later. The cells resembling osteoblasts or young osteocytes were histologically observed to be embedded in mineralized or unmineralized extracellular matrices of the nodules. Next, these mineralized nodules were electron-microscopically examined. The osteoblastic cells associated with the nodules had a well-developed rough endoplasmic reticulum, an evident Golgi apparatus and some mitochondria as their intracellular organellae. Some lysosomes and microfilaments were also visible in the cytoplasms. Moreover, some cells protruded cell processes toward the neighboring cells through the extracellular matrix. The extracellular matrix consisted of numerous collagen fibrils which were striated with 60-70 nm axial periodicity and which was similar to bone tissue collagen. A large number of matrix vesicles were scattered among the collagen fibrils in the unmineralized area of the nodules. In contrast, in the mineralized area, numerous matrix vesicles at different stages of maturation and many calcified spherules were observed. That is the mineralization in this culture system was considered to be initiated in association with the matrix vesicles and to progress along the collagen fibrils. From these findings, it was confirmed by the present study that the mineralized nodules formed in this bone marrow stromal cell culture were ultrastructurally similar to bone and that the mineralization also proceeded by going through the normal calcification process. This culture system is considered to be available to study osteogenic differentiation and calcification mechanisms.     

In situ atomic force microscopy of partially demineralized human dentin collagen fibrils

Dentin collagen fibrils were studied in situ by atomic force microscopy (AFM). New data on size distribution and the axial repeat distance of hydrated and dehydrated collagen type I fibrils are presented. Polished dentin disks from third molars were partially demineralized with citric acid, leaving proteins and the collagen matrix. At this stage collagen fibrils were not resolved by AFM, but after exposure to NaOCl(aq) for 100-240 s, and presumably due to the removal of noncollagenous proteins, individual collagen fibrils and the fibril network of dentin connected to the mineralized substrate were revealed. High-aspect-ratio silicon tips in tapping mode were used to image the soft fibril network. Hydrated fibrils showed three distinct groups of diameters: 100, 91, and 83 nm and a narrow distribution of the axial repeat distance at 67 nm. Dehydration resulted in a broad distribution of the fibril diameters between 75 and 105 nm and a division of the axial repeat distance into three groups at 67, 62, and 57 nm. Subfibrillar features (4 nm) were observed on hydrated and dehydrated fibrils. The gap depth between the thick and thin repeating segments of the fibrils varied from 3 to 7 nm. Phase mode revealed mineral particles on the transition from the gap to the overlap zone of the fibrils. This method appears to be a powerful tool for the analysis of fibrillar collagen structures in calcified tissues and may aid in understanding the differences in collagen affected by chemical treatments or by diseases.     

Vitamin K2, a gamma-carboxylating factor of gla-proteins, normalizes the bone crystal nucleation impaired by Mg-insufficiency

It has been reported that the Mg-insufficient bone is fragile upon mechanical loading, despite its high bone mineral density, while vitamin K2 (MK-4: menatetrenone) improved the mechanical strength of Mg-insufficient bone. Therefore, we aimed to elucidate the ultrastructural properties of bone in rats with dietary Mg insufficiency with and without MK-4 supplementation. Morphological examinations including histochemistry, transmission electron microscopy, electron probe microanalysis (EPMA) and X-ray diffraction were conducted on the femora and tibiae of 4-week-old Wistar male rats fed with 1) a normal diet (control group, 0.09% Mg), 2) a Mg-insufficient diet (low Mg group, 0.006% Mg), or 3) a Mg-insufficient diet supplemented with MK-4 (MK-4 group, 0.006% Mg, 0.03% MK-4). MK-4 appeared to inhibit the osteoclastic bone resorption that is stimulated by Mg insufficiency. EPMA analysis, however, revealed an increased concentration of Ca paralleling Mg reduction in the low Mg group. Assessment by X-ray diffraction revealed an abundance of a particular synthetic form of hydroxyapatite in the low Mg group, while control bones featured a variety of mineralized crystals. In addition, Mg-deficient bones featured larger mineral crystals, i.e., crystal overgrowth. This crystalline aberration in Mg-insufficient bones induced collagen fibrils to mineralize easily, even in the absence of mineralized nodules, which therefore led to an early collapse of the fibrils. MK-4 prevented premature collagen mineralization by normalizing the association of collagen fibrils with mineralized nodules. Thus, MK-4 appears to rescue the impaired collagen mineralization caused by Mg insufficiency by promoting a re-association of the process of collagen mineralization with mineralized nodules.     

Effects of damage on the orthotropic material symmetry of bovine tibial trabecular bone

The macroscopic mechanical properties of trabecular bone can be predicted by its architecture using theoretical relationships between the elastic and architectural properties. Microdamage caused by overloading or fatigue decreases the apparent elastic moduli of trabecular bone requiring these relationships to be modified to predict the damaged elastic properties. In the case of isotropic damage, the apparent level elastic properties could be determined by multiplying all of the elastic constants by a single scalar factor. If the damage is anisotropic, the elastic constants may change by differing factors and the material coordinate system could become misaligned with the fabric coordinate system. High-resolution finite element models were used to simulate damage overloading on seven trabecular bone specimens subjected to pure shear strain in two planes. Comparison of the apparent elastic moduli of the specimens before and after damage showed that the reduction of the elastic moduli was anisotropic. This suggests that the microdamage within the specimens was inhomogeneous. However, after damage the specimens exhibited nearly orthotropic material symmetry as they did before damage. Changes in the orientation of the orthotropic material coordinate system were also small and occurred primarily in the transverse plane. Thus, while damage in trabecular bone is anisotropic, the material coordinate system remains aligned with the fabric tensor.