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.     

Compressive axial mechanical properties of rat bone as functions of bone volume fraction,apparent density and micro-ct based mineral density

Mechanical testing has been regarded as the gold standard to investigate the effects of pathologies on the structure–function properties of the skeleton. With recent advances in computing power of personal computers, virtual alternatives to mechanical testing are gaining acceptance and use. We have previously introduced such a technique called structural rigidity analysis to assess mechanical strength of skeletal tissue with defects. The application of this technique is predicated upon the use of relationships defining the strength of bone as a function of its density for a given loading mode. We are to apply this technique in rat models to assess their compressive skeletal response subjected to a host of biological and pharmaceutical stimulations. Therefore, the aim of this study is to derive a relationship expressing axial compressive mechanical properties of rat cortical and cancellous bone as a function of equivalent bone mineral density, bone volume fraction or apparent density over a range of normal and pathologic bones.We used bones from normal, ovariectomized and partially nephrectomized animals. All specimens underwent micro-computed tomographic imaging to assess bone morphometric and densitometric indices and uniaxial compression to failure.We obtained univariate relationships describing 71–78% of the mechanical properties of rat cortical and cancellous bone based on equivalent mineral density, bone volume fraction or apparent density over a wide range of density and common skeletal pathologies. The relationships reported in this study can be used in the structural rigidity analysis introduced by the authors to provide a non-invasive method to assess the compressive strength of bones affected by pathology and/or treatment options.     

Shear strength and toughness of trabecular bone are more sensitive to density than damage

Microdamage occurs in trabecular bone under normal loading, which impairs the mechanical properties. Architectural degradation associated with osteoporosis increases damage susceptibility, resulting in a cumulative negative effect on the mechanical properties. Treatments for osteoporosis could be targeted toward increased bone mineral density, improved architecture, or repair and prevention of microdamage. Delineating the relative roles of damage and architectural degradation on trabecular bone strength will provide insight into the most beneficial targets. In this study, damage was induced in bovine trabecular bone samples by axial compression, and the effects on the mechanical properties in shear were assessed. The damaged shear modulus, shear yield stress, ultimate shear stress, and energy to failure all depended on induced damage and decreased as the architecture became more rod-like. The changes in ultimate shear strength and toughness were proportional to the decrease in shear modulus, consistent with an effective decrease in the cross-section of trabeculae based on cellular solid analysis. For typical ranges of bone volume fraction in human bone, the strength and toughness were much more sensitive to decreased volume fraction than to induced mechanical damage. While ultimately repairing or avoiding damage to the bone structure and increasing bone density both improve mechanical properties, increasing bone density is the more important contributor to bone strength.     

Effect of bone mineral content on the tensile properties of cortical bone: experiments and theory

The effect of mineral volume fraction on the tensile mechanical properties of cortical bone tissue is investigated by theoretical and experimental means. The mineral content of plexiform, bovine bone was lowered by 18% and 29% by immersion in fluoride solutions for 3 days and 12 days, respectively. The elastic modulus, yield strength and ultimate strength of bone tissue decreased, while the ultimate strain increased with a decrease in mineral content. The mechanical behavior of bone tissue was modeled by using a micromechanical shear lag theory consisting of overlapped mineral platelets reinforcing the organic matrix. The decrease in yield stress, by the 0.002 offset method, of the fluoride treated bones were matched in the theoretical curves by lowering the shear yield stress of the organic matrix. The failure criterion used was based on failure stresses determined from a failure envelope (Mohr's circle), which was constructed using experimental data. It was found that the model predictions of elastic modulus got worse with a decrease in mineral content (being 7.9%, 17.2% and 33.0% higher for the control, 3-day and 12-day fluoride-treated bones). As a result, the developed theory could not fully predict the yield strain of bones with lowered mineral content, being 12.9% and 21.7% lower than the experimental values. The predicted ultimate stresses of the bone tissues with lower mineral contents were within +/- 10% of the experimental values while the ultimate strains were 12.7% and 26.3% lower than the experimental values. Although the model developed in this study did not take into account the presence of hierarchical structures, voids, orientation of collagen molecules and micro cracks, it still indicated that the mechanical properties of the organic matrix depend on bone mineral content.     

A study of the interface strength between protein and mineral in biological materials

Bone, tooth, mineralized tendon and sea shells are nanocomposites of protein and mineral with superior mechanical properties. As the mineral is so small at nanoscale, the volume fraction of the protein-mineral interface in the bulk materials can be enormously large; therefore, the mechanics of the interface should be critically important for the integrity of these biomaterials. Currently, people do not have a good understanding of the interface between protein and mineral, a hybrid interface between organic and inorganic constituents in biological materials. In this paper, a tension-shear chain (TSC) model is introduced into the Dugdale model for estimating the fracture energy of biomaterials. The strength of the hybrid interface is then studied with a "soft-hard" bi-layer fracture model, by which we find for the first time that the interface strength depends on both the size and geometry of the mineral crystal, and has been highly optimized through the miniaturization of mineral at nanoscale. This study may provide important insights into the mechanics of bone and tooth at small scale for tissue engineering in biomedical applications.     

Tensile behavior of cortical bone: dependence of organic matrix material properties on bone mineral content

A porous composite model is developed to analyze the tensile mechanical properties of cortical bone. The effects of microporosity (volksman's canals, osteocyte lacunae) on the mechanical properties of bone tissue are taken into account. A simple shear lag theory, wherein tensile loads are transferred between overlapped mineral platelets by shearing of the organic matrix, is used to model the reinforcement provided by mineral platelets. It is assumed that the organic matrix is elastic in tension and elastic-perfectly plastic in shear until it fails. When organic matrix shear stresses at the ends of mineral platelets reach their yield values, the stress-strain curve of bone tissue starts to deviate from linear behavior. This is referred as the microscopic yield point. At the point where the stress-strain behavior of bone shows a sharp curvature, the organic phase reaches its shear yield stress value over the entire platelet. This is referred as the macroscopic yield point. It is assumed that after macroscopic yield, mineral platelets cannot contribute to the load bearing capacity of bone and that the mechanical behavior of cortical bone tissue is determined by the organic phase only. Bone fails when the principal stress of the organic matrix is reached. By assuming that mechanical properties of the organic matrix are dependent on bone mineral content below the macroscopic yield point, the model is used to predict the entire tensile mechanical behavior of cortical bone for different mineral contents. It is found that decreased shear yield stresses and organic matrix elastic moduli are required to explain the mechanical behavior of bones with lowered mineral contents. Under these conditions, the predicted values (elastic modulus, 0.002 yield stress and strain, and ultimate stress and strain) are within 15% of experimental data.     

Long-term dose response of trabecular bone in mice to proton radiation

Astronauts on exploratory missions will experience a complex environment, including microgravity and radiation. While the deleterious effects of unloading on bone are well established, fewer studies have focused on the effects of radiation. We previously demonstrated that 2 Gy of ionizing radiation has deleterious effects on trabecular bone in mice 4 months after exposure. The present study investigated the skeletal response after total doses of proton radiation that astronauts may be exposed to during a solar particle event. We exposed mice to 0.5, 1 or 2 Gy of whole-body proton radiation and killed them humanely 117 days later. Tibiae and femora were analyzed using microcomputed tomography, mechanical testing, mineral composition and quantitative histomorphometry. Relative to control mice, mice exposed to 2 Gy had significant differences in trabecular bone volume fraction (-20%), trabecular separation (+11%), and trabecular volumetric bone mineral density (-19%). Exposure to 1 Gy radiation induced a nonsignificant trend in trabecular bone volume fraction (-13%), while exposure to 0.5 Gy resulted in no differences. No response was detected in cortical bone. Further analysis of the 1-Gy mice using synchrotron microCT revealed a significantly lower trabecular bone volume fraction (-13%) than in control mice. Trabecular bone loss 4 months after exposure to 1 Gy highlights the importance of further examination of how space radiation affects bone.     

Evaluation of trabecular mechanical and microstructural properties in human calcaneal bone of advanced age using mechanical testing, microCT, and DXA

Early detection of fracture risk is important for initiating treatment and improving outcomes from both physiologic and pathologic causes of bone loss. While bone mineral density (a quantity measure) has traditionally been used for this purpose, alternative structural imaging parameters (quality measures) are proposed to better predict bone's true mechanical properties. To further elucidate this, trabecular bone from cadaveric human calcanei were used to evaluate the interrelationship of mechanical and structural parameters using mechanical testing, dual energy X-ray absorptiometry (DXA) scanning, and micro computed tomography (microCT) imaging. Directional specific structural properties were assessed in three-dimensional (3-D) and correlated to mechanical testing and DXA. The results demonstrated that microCT-derived indices of bone quality (i.e., volume fraction and structural model index) are better than DXA-derived bone mineral density for the prediction of the mechanical parameters of bone (i.e., elastic modulus, yield stress, and ultimate stress). Diagnostically, this implies that future work on the early prediction of fracture risk should focus as much on bone quality as on quantity. Furthermore, the results of this study show that a loss of bone primarily affects the connectedness and overall number of trabeculae. Ultimate stress, however, is better correlated with trabecular number than thickness. As such, primary prevention of osteoporosis may be more important than later countermeasures for bone loss.     

Six months of disuse during hibernation does not increase intracortical porosity or decrease cortical bone geometry,strength, or mineralization in black bear (Ursus americanus) femurs

Disuse typically uncouples bone formation from resorption, leading to bone loss which compromises bone mechanical properties and increases the risk of bone fracture. Previous studies suggest that bears can prevent bone loss during long periods of disuse (hibernation), but small sample sizes have limited the conclusions that can be drawn regarding the effects of hibernation on bone structure and strength in bears. Here we quantified the effects of hibernation on structural, mineral, and mechanical properties of black bear (Ursus americanus) cortical bone by studying femurs from large groups of male and female bears (with wide age ranges) killed during pre-hibernation (fall) and post-hibernation (spring) periods. Bone properties that are affected by body mass (e.g. bone geometrical properties) tended to be larger in male compared to female bears. There were no differences (p>0.226) in bone structure, mineral content, or mechanical properties between fall and spring bears. Bone geometrical properties differed by less than 5% and bone mechanical properties differed by less than 10% between fall and spring bears. Porosity (fall: 5.5±2.2%; spring: 4.8±1.6%) and ash fraction (fall: 0.694±0.011; spring: 0.696±0.010) also showed no change (p>0.304) between seasons. Statistical power was high (>72%) for these analyses. Furthermore, bone geometrical properties and ash fraction (a measure of mineral content) increased with age and porosity decreased with age. These results support the idea that bears possess a biological mechanism to prevent disuse and age-related osteoporoses.     

Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions

Strong correspondence between the uniaxial apparent strength and stiffness of cancellous bone allows the use of stiffness as a predictor of bone strength. Measured values of mechanical properties in cancellous bone can be different between experiments due to different experimental conditions. In the current study, bone volume fraction, experimentally determined and finite element (FE) predicted stiffness were examined as predictors of cancellous bone ultimate strength in two different groups each of which was tested using a different end constraint. It is demonstrated that, although always significant, the relationships of strength with bone volume fraction and experimentally determined stiffness are different between test groups. Apparent stiffness, estimated by FE modeling, predicts the ultimate strength of human cancellous bone consistently for all examined experimental protocols.     

Nonlinear hierarchical multiscale modeling of cortical bone considering its nanoscale microstructure

We have used a hierarchical multiscale modeling scheme for the analysis of cortical bone considering it as a nanocomposite. This scheme consists of definition of two boundary value problems, one for macroscale, and another for microscale. The coupling between these scales is done by using the homogenization technique. At every material point in which the constitutive model is needed, a microscale boundary value problem is defined using a macroscopic kinematical quantity and solved. Using the described scheme, we have studied elastic properties of cortical bone considering its nanoscale microstructural constituents with various mineral volume fractions. Since the microstructure of bone consists of mineral platelet with nanometer size embedded in a protein matrix, it is similar to the microstructure of soft matrix nanocomposites reinforced with hard nanostructures. Considering a representative volume element (RVE) of the microstructure of bone as the microscale problem in our hierarchical multiscale modeling scheme, the global behavior of bone is obtained under various macroscopic loading conditions. This scheme may be suitable for modeling arbitrary bone geometries subjected to a variety of loading conditions. Using the presented method, mechanical properties of cortical bone including elastic moduli and Poisson's ratios in two major directions and shear modulus is obtained for different mineral volume fractions.     

The long-term effect of ovariectomy on the quality and quantity of cancellous bone in young macaques

The effect of ovariectomy on the quality and quantity of cancellous bone using the young cynomolgus monkey was evaluated after a 2-year period. The bodies of the second lumbar vertebrae were analyzed for changes in bone mineral quality using density fractionation, chemical analysis, and X-ray diffraction techniques. Changes in bone tissue quality and quantity were evaluated using bone histomorphometry and image analysis. The experimental group (n=14) was made surgically menopausal (bilaterally ovariectomized), compared with intact controls (n=16), and then sacrificed after a 2-year period. There was a non-significant shift in the mineralization profile towards less dense bone in the ovariectomized (OVX) vertebrae compared with controls. Physical characteristics of the bone mineral in terms of crystal size or strain were unaffected by OVX. There was a parallel increase in mineral content with fractions of increasing density, however there was no difference in mineral content or the Ca/P ratio in each fraction between treatment groups. Histomorphometric analysis for structural parameters demonstrated no difference in bone volume between control and OVX groups. There was no significant change in trabecular width in the OVX vertebrae compared with controls. There was a significant increase in both osteoid volume and osteoid surface in the OVX vertebrae (P<0.001). Trabecular architecture as measured by image analysis was unchanged. There was a significant increase in eroded surface in the OVX vertebrae (P<0.03) compared with the controls. In conclusion, young, ovariectomized female cynomolgus macaques do not appear to be a useful animal model for the study of postmenopausal bone loss, however they may be a useful model to evaluate skeletal pathology which might be observed after surgical ovariectomy in young human females.     

Effects of differences in mineralization on the mechanical properties of bone

There is a considerable variation in the mineralization of bone; normal, non-pathological compact bone has ash masses ranging from 45 to 85% by mass. This range of mineralization results in an even greater range of mechanical properties. The Young modulus of elasticity can range from 4 to 32 GPa, bending strength from 50 to 300 MPa, and the work of fracture from 200 to 7000 Jm-2. It is not possible for any one type of bone to have high values for all three properties. Very high values of mineralization produce high values of Young modulus but low values of work of fracture (which is a measure of fracture toughness). Rather low values of mineralization are associated with high values of work of fracture but low values of Young modulus and intermediate values of bending strength. The reason for the high value for the Young modulus associated with high mineralization is intuitively obvious, but has not yet been rigorously modelled. The low fracture toughness associated with high mineralization may be caused by the failure of various crack-stopping mechanisms that can act when the mineral crystals in bone have not coalesced, but which become ineffective when the volume fraction of mineral becomes too high. The adoption of different degrees of mineralization by different bones, leading to different sets of mechanical properties, is shown to be adaptive in most cases studied, but some puzzles still remain.     

A potential role for the myeloid lineage in leptin-regulated bone metabolism

Leptin influences bone formation centrally through the hypothalamus and peripherally by acting on osteoblasts or their precursors. However, neither mechanism explains the divergent, gender-specific correlation between leptin and bone mineral density in humans. Although leptin is a potent regulator of pro-inflammatory immune responses, a potential role for leptin as an osteoimmunologic intermediate in bone metabolism has not been tested. Mice with myeloid-specific ablation of the long-form leptin receptor (ObRb) were generated using mice expressing cre-recombinase from the lysoszyme M promoter. At 12 weeks of age, the conditional knockout mice did not display any appreciable phenotype. However, at 52 weeks 2 changes were noted. First, there was a mild increase in liver inflammation. Second, a gender-specific, divergent bone phenotype was observed. Female mice displayed a consistent trend toward decreased trabecular bone parameters including reductions in bone volume fraction, trabecular number, and bone mineral content as well as a significant increase in marrow adipogenesis. Conversely, male mice lacked trabecular changes, but had statistically significant increases in cortical bone volume, thickness, and bone mineral density with equivalent total cortical volume. Since the year 2000, over 25 studies on more than 10,000 patients have sought to determine the correlation between leptin and bone mineral density. The results revealed a gender-specific correlation similar to that observed in our LysM transgenic animals. We hypothesize and show new evidence that regulation of myeloid lineage cells by leptin may facilitate their actions as an osteoimmunologic intermediate and contribute to leptin-regulated bone formation and metabolism in a gender-specific manner.     

Dietary pseudopurpurin improves bone geometry architecture and metabolism in red-bone Guishan goats

Red-colored bones were found initially in some Guishan goats in the 1980s, and they were designated red-boned goats. However, it is not understood what causes the red color in the bone, or whether the red material changes the bone geometry, architecture, and metabolism of red-boned goats. Pseudopurpurin was identified in the red-colored material of the bone in red-boned goats by high-performance liquid chromatography-electrospray ionization-mass spetrometry and nuclear magnetic resonance analysis. Pseudopurpurin is one of the main constituents of Rubia cordifolia L, which is eaten by the goats. The assessment of the mechanical properties and micro-computed tomography showed that the red-boned goats displayed an increase in the trabecular volume fraction, trabecular thickness, and the number of trabeculae in the distal femur. The mean thickness, inner perimeter, outer perimeter, and area of the femoral diaphysis were also increased. In addition, the trabecular separation and structure model index of the distal femur were decreased, but the bone mineral density of the whole femur and the mechanical properties of the femoral diaphysis were enhanced in the red-boned goats. Meanwhile, expression of alkaline phosphatase and osteocalcin mRNA was higher, and the ratio of the receptor activator of the nuclear factor kappa B ligand to osteoprotegerin was markedly lower in the bone marrow of the red-boned goats compared with common goats. To confirm further the effect of pseudopurpurin on bone geometry, architecture, and metabolism, Wistar rats were fed diets to which pseudopurpurin was added for 5 months. Similar changes were observed in the femurs of the treated rats. The above results demonstrate that pseudopurpurin has a close affinity with the mineral salts of bone, and consequently a high level of mineral salts in the bone cause an improvement in bone strength and an enhancement in the structure and metabolic functions of the bone.     

Interfacial mechanical behaviour of protein–mineral nanocomposites: A molecular dynamics investigation

Biological composite materials, such as bone, tooth and nacre, are comprised of a mixture of nano-sized hard components (e.g. mineral platelets) and soft components (e.g. protein molecules). Their mechanical behaviour greatly depends on the protein–mineral interfaces. This paper investigates the effects of mineral surface nanostructures on the interfacial interaction and mechanical behaviour of protein–mineral nanocomposites. Interfacial shear between osteopontin (OPN) and hydroxyapatite (HA) mineral layers with surface nanostructures is investigated using the atomistic molecular dynamics (MD) simulations. The results indicate that the OPN residues can be attached to HA surfaces but the surface nanostructures greatly affect the interfacial interaction and mechanical behaviour. The HA layers with a higher number of nano-sized grooves (defects) increase the surface roughness but reduce the pulling force and energy dissipation.     

Modern analysis of bone loss mechanisms in microgravity

A summary of results of investigations by the author and a brief review of some literature data on human bone tissue deprived of mechanical loading (spaceflight, hypokinesia) is given. The direction and markedness of changes in bone mass--the bone mineral density and the bone mineral content--in different skeletal segments depend on their position relative to the gravity vector. A theoretically expected bone mass reduction was revealed in the trabecular structures of the bones of the lower part of the skeleton (local osteopenia). In the upper part of the skeleton, an increase in the bone mineral content is observed, which is considered as a secondary response and is due to redistribution of body fluids cephalad. The main cause of osteopenia is mechanical unloading. Arguments are presented that osteocyte osteolysis, delayed osteoblast histogenesis, and osteoclast resorption provoked by rearrangement in the hierarchy of the systems of fluid volume and ion regulation, and the endocrine control of calcium homeostasis are the main mechanisms of osteopenia.     

Possible effect between the molecular packing of collagen and the composition of bony tissues

The weight fractions of the organic, mineral and water components of bone have been shown to be uniquely related to the wet bone density, except for a small variation possibly due to structure, for the range of bone densities from 1.7 g/cm³ for deer antler to 2.7 g/cm³ for porpoise petrosal. In this report the mathematical expression for the organic weight fraction is shown to depend on three factors, each a function of bone density. The first factor can be ralated to the mineral fraction, the second to the volume fraction of the organic component and the third to the density of the organic component. The influence of these factors is not obvious, since the change in the organic weight fraction could be due to an absolute loss of organic matter alone, or to a combination of increased mineral concentration together with some loss of organic matter. The mathematical development is based on the generalized packing model for collagen. It is demonstrated that the mineralization process requires a decrease of the organic component as well as a compaction of the collagen fibrils and these vary with the bone density.     

A two-parameter model of the effective elastic tensor for cortical bone

Multiscale models of cortical bone elasticity require a large number of parameters to describe the organization and composition of the tissue. We hypothesize that the macro-scale anisotropic elastic properties of different bones can be modeled retaining only two variable parameters, and setting the others to universal values identical for all bones. Cortical bone is regarded as a two-phase composite material: a dense mineralized matrix (ultrastructure) and a soft phase (pores). The ultrastructure is assumed to be a homogeneous and transversely isotropic tissue whose elastic properties in different directions are mutually dependent and can be scaled with a single parameter driving the overall rigidity. This parameter is taken to be the volume fraction of mineral f(ha). The pore network is modeled as an ensemble of water-filled cylinders and described only by the porosity p. The effective macroscopic elasticity tensor C(ij)(f(ha),p) is calculated with a multiscale micromechanics approach starting from existing models. The modeled stiffness coefficients compare favorably to four literature datasets which were chosen because they provide the full stiffness tensors of groups of human samples. Since the physical counterparts of f(ha) and p were unknown for the datasets, their values which allow the best fit of experimental tensors by the modeled ones were determined by optimization. Optimum values of f(ha) and p are found to be unique and realistic. These results suggest that a two-parameter model may be sufficient to model the elasticity of different samples of human femora and tibiae. Such a model would in particular be useful in large-scale parametric studies of bone mechanical response.