Micro CT-based multiscale elasticity of double-porous (pre-cracked) hydroxyapatite granules for regenerative medicine

Hundred micrometers-sized porous hydroxyapatite globules have proved as a successful tissue engineering strategy for bone defects in vivo, as was shown in studies on human mandibles. These granules need to provide enough porous space for bone ingrowth, while maintaining sufficient mechanical competence (stiffness and strength) in this highly load-bearing organ. This double challenge motivates us to scrutinize more deeply the micro- and nanomechanical characteristics of such globules, as to identify possible optimization routes. Therefore, we imaged such a (pre-cracked) granule in a microCT scanner, transformed the attenuation coefficients into voxel-specific nanoporosities, from which we determined, via polycrystal micromechanics, voxel-specific (heterogeneous) elastic properties. The importance of the latter and of the presence of one to several hundred micrometers-sized cracks for realistically estimating the load-carrying behavior of the globule under a typical two-point compressive loading (as in a "splitting" test) is shown through results of large-scale Finite Element analyses, in comparison to analytical results for a sphere loaded at its poles: Use of homogeneous instead of heterogeneous elastic properties would overestimate the structure's stiffness by 5% (when employing a micromechanics-based process as to attain homogeneous properties)-the cracks, in comparison, weaken the structure by one to two orders of magnitudes.     

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.     

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.     

Mineral heterogeneity has a minor influence on the apparent elastic properties of human cancellous bone: a SRμCT-based finite element study

At the tissue level, the local material properties of human cancellous bone are heterogeneous due to constant remodelling. Since standard high-resolution computed tomography scanning methods are unable to capture this heterogeneity in detail, local differences in mineralisation are normally not incorporated in computational models. To investigate the effects of heterogeneous mineral distribution on the apparent elastic properties, 40 cancellous bone samples from the human femoral neck were scanned by means of synchrotron radiation microcomputed tomography (SRμCT). SRμCT-based micromechanical finite element models that accounted for mineral heterogeneity were compared with homogeneous models. Evaluation of the apparent stiffness tensor of both model types revealed that homogeneous models led to a minor but significant (p < 0.05) overestimation of the elastic properties of heterogeneous models by 2.18 ± 1.89%. Variation of modelling parameters did not affect the overestimation to a great extent. It was concluded that the heterogeneous mineralisation has only a minor influence on the apparent elastic properties of human cancellous bone.     

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.     

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.     

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.     

Multiscale,Converging Defects of Macro-Porosity,Microstructure and Matrix Mineralization Impact Long Bone Fragility in NF1

Bone fragility due to osteopenia, osteoporosis or debilitating focal skeletal dysplasias is a frequent observation in the Mendelian disease Neurofibromatosis type 1 (NF1). To determine the mechanisms underlying bone fragility in NF1 we analyzed two conditional mouse models, Nf1Prx1 (limb knock-out) and Nf1Col1 (osteoblast specific knock-out), as well as cortical bone samples from individuals with NF1. We examined mouse bone tissue with micro-computed tomography, qualitative and quantitative histology, mechanical tensile analysis, small-angle X-ray scattering (SAXS), energy dispersive X-ray spectroscopy (EDX), and scanning acoustic microscopy (SAM). In cortical bone of Nf1Prx1 mice we detected ectopic blood vessels that were associated with diaphyseal mineralization defects. Defective mineral binding in the proximity of blood vessels was most likely due to impaired bone collagen formation, as these areas were completely devoid of acidic matrix proteins and contained thin collagen fibers. Additionally, we found significantly reduced mechanical strength of the bone material, which was partially caused by increased osteocyte volume. Consistent with these observations, bone samples from individuals with NF1 and tibial dysplasia showed increased osteocyte lacuna volume. Reduced mechanical properties were associated with diminished matrix stiffness, as determined by SAM. In line with these observations, bone tissue from individuals with NF1 and tibial dysplasia showed heterogeneous mineralization and reduced collagen fiber thickness and packaging. Collectively, the data indicate that bone fragility in NF1 tibial dysplasia is partly due to an increased osteocyte-related micro-porosity, hypomineralization, a generalized defect of organic matrix formation, exacerbated in the regions of tensional and bending force integration, and finally persistence of ectopic blood vessels associated with localized macro-porotic bone lesions.     

Incorporating tissue anisotropy and heterogeneity in finite element models of trabecular bone altered predicted local stress distributions

Trabecular bone is composed of organized mineralized collagen fibrils, which results in heterogeneous and anisotropic mechanical properties at the tissue level. Recently, biomechanical models computing stresses and strains in trabecular bone have indicated a significant effect of tissue heterogeneity on predicted stresses and strains. However, the effect of the tissue-level mechanical anisotropy on the trabecular bone biomechanical response is unknown. Here, a computational method was established to automatically impose physiologically relevant orientation inherent in trabecular bone tissue on a trabecular bone microscale finite element model. Spatially varying tissue-level anisotropic elastic properties were then applied according to the bone mineral density and the local tissue orientation. The model was used to test the hypothesis that anisotropy in both homogeneous and heterogeneous models alters the predicted distribution of stress invariants. Linear elastic finite element computations were performed on a 3 mm cube model isolated from a microcomputed tomography scan of human trabecular bone from the distal femur. Hydrostatic stress and von Mises equivalent stress were recorded at every element, and the distributions of these values were analyzed. Anisotropy reduced the range of hydrostatic stress in both tension and compression more strongly than the associated increase in von Mises equivalent stress. The effect of anisotropy was independent of the spatial redistribution high compressive stresses due to tissue elastic heterogeneity. Tissue anisotropy and heterogeneity are likely important mechanisms to protect bone from failure and should be included for stress analyses in trabecular bone.     

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

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

'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.     

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.     

Femur EPA and DHA are correlated with femur biomechanical strength in young fat-1 mice

Evidence suggests that n-3 polyunsaturated fatty acids (PUFA) are beneficial for maintenance of bone health and possibly bone development. This study used the fat-1 mouse, a transgenic model that synthesizes n-3 PUFA from n-6 PUFA, to determine if outcomes of bone health were correlated with n-3 PUFA in femurs. Control and fat-1 mice were fed an AIN-93G diet containing 10% safflower oil from weaning through 12 weeks of age. Femur bone mineral content (BMC) and density were determined by dual-energy X-ray absorptiometry, and biomechanical strength properties, surrogate measures of fracture risk, were measured by a materials testing system. Femur fatty acid composition was determined by gas chromatography. At 12 weeks of age, femur n-3 PUFA were higher among fat-1 mice compared to control mice. The n-6/n-3 PUFA ratio in the femur was negatively correlated with BMC (r=-.57, P=.01) and peak load at femur midpoint (r=-.53, P=.02) and femur neck (r=-.52, P=.02). Moreover, long-chain n-3 PUFA, eicosapentaenoic acid, and docosahexaenoic acid were significantly and positively correlated or displayed a trend suggesting positive correlations, with BMC and peak load. In conclusion, the results of the present study suggest that n-3 PUFA have a favorable effect on mineral accumulation and functional measures of bone in fat-1 mice at young adulthood.     

Do mineral crystals stiffen bone by straitjacketing its collagen?

Theories that assume that the collagen and the mineral of bone have the same properties in bone as they do when they are by themselves have difficulty in accounting for bone's tensile stiffness. The collagen fibers are long but stretchy: the apatite crystals are stiff enough, but too short to be effective tension carriers. In bone the mineral is so finely divided that each collagen molecule may have a mineral crystal next to it. It is proposed that collagen is the prime tension carrier in bone, and that via short molecular struts the mineral crystals prevent the naturally kinky collagen molecules from straightening under tensile stress, which greatly increases the collagen's tensile stiffness.     

The dependence of the elastic properties of osteoporotic cancellous bone on volume fraction and fabric

Osteoporosis is a progressive systemic skeletal condition characterized by low bone mass and microarchitectural deterioration, with a consequent increase in susceptibility to fracture. Hence, osteoporosis would be best diagnosed by in vivo measurements of bone strength. As this is not clinically feasible, our goal is to estimate bone strength through the assessment of elastic properties, which are highly correlated to strength. Previously established relations between morphological parameters (volume fraction and fabric) and elastic constants could be applied to estimate cancellous bone stiffness in vivo. However, these relations were determined for normal cancellous bone. Cancellous bone from osteoporotic patients may require different relations. In this study we set out to answer two questions. First, can the elastic properties of osteoporotic cancellous bone be estimated from morphological parameters? Second, do the relations between morphological parameters and elastic constants, determined for normal bone, apply to osteoporotic bone as well? To answer these questions we used cancellous bone cubes from femoral heads of patients with (n=26) and without (n=32) hip fractures. The elastic properties of the cubes were determined using micro-finite element analysis, assuming equal tissue moduli for all specimens. The morphological parameters were determined using microcomputed tomography. Our results showed that, for equal tissue properties, the elastic properties of cancellous bone from fracture patients could indeed be estimated from morphological parameters. The morphology-based relations used to estimate the elastic properties of cancellous bone are not different for women with or without fractures.     

Bone mass, bone strength, and their relationship in developing CD-1 mice

Optimizing nutrition during development may provide effective prevention strategies to protect against osteoporosis during later life. Because the mouse model is commonly used to test nutritional interventions on bone health, the overall objective of this study was to determine how bone develops during the first 4 months of life by assessing bone mass (bone mineral content (BMC) and bone mineral density (BMD)) and biomechanical strength properties such as peak load in male and female CD-1 mice. Bone outcomes were assessed at 1 month intervals from 1 to 4 months of age. Femur and spine BMC and BMD at 3 months were similar to 4 months, indicating that the accumulation of bone mass occurs primarily during the first 3 months of life. In contrast, the timing of changes in peak load, a measure of bone strength, varied by skeletal site. Regression analyses demonstrated that femur BMC is a significant predictor of femur peak load at the femur midpoint and neck. The study findings suggest that nutritional interventions aimed at optimizing peak bone mass to prevent osteoporosis may be most effective during pubertal growth.     

Femur morphometry,densitometry, geometry and mechanical properties in young pigs fed a diet free of inorganic phosphorus and supplemented with phytase

The study investigated in piglets the effect of replacing dietary inorganic P by addition of microbial phytase and its impact on performance, nutrient digestibility and on the geometrical characteristics and mineralisation of the femur. Sixteen pigs on day 58 of age were divided into two groups and fed either a diet free of additional inorganic phosphorus (P) and supplemented with phytase (Diet LP, 4.23 g total P/kg diet) or a diet with a mineral source of P and not supplemented with phytase (Diet SP, 5.38 g total P/kg diet). Performance data and the apparent total tract digestibility of nutrients were estimated between days 58 and 114, and 72 and 86 of age, respectively. On day 114 of age, the pigs were slaughtered, the femur was dissected and the mineral content and mineral density, maximum strength and maximum elastic strength, cortical wall thickness, cross-sectional area and cortical index were analysed. The growth performance and digestibility of nutrient fractions (with exception of P) did not differ between treatment groups. The P-digestibility was significantly higher in Group LP. The femur of pigs in Group LP had significantly greater cortical wall thickness, cortical index, bone mineral content, bone mineral density, maximum strength and maximum elastic strength than Group SP. Femur maximum strength and maximum elastic strength were correlated with cortical wall thickness and cortical index. Resulting from the different supply of digestible P, the femur geometrical, densitometric and mechanical properties of Group LP were better than those of Group SP. The mechanical properties of the femur of pigs depended more on its geometrical characteristics than on the degree of its mineralisation.     

Adaptation of bone and tendon to prolonged hindlimb suspension in rats

The rat hindlimb suspension model was used to ascertain the importance of ground reaction forces in maintaining bone and tendon homeostasis. Young female Sprague-Dawley rats were randomly assigned to either a suspended or a nonsuspended group. After 28 days, femur bones and patellar tendons were obtained for morphological and biochemical analyses. Prolonged suspension induced a significant change in the geometric configuration of the femur middiaphysis by increasing the minimum diameter (12%) without any significant alterations in cortical area, density, mineral, and collagen concentrations. Femur wet weight, length, DNA, and uronic acid concentrations of suspended animals were not significantly different from bones of nonsuspended rats. However, the collagen and proteoglycan concentrations in patellar tendons of suspended rats were 28% lower than the concentrations of matrix proteins in tissues obtained from nonsuspended animals. These data suggest that elimination of ground reaction forces induces alterations in tendon composition and femur diaphyseal shape by changing regional rates in bone remodeling and localized tendon strain. Therefore it appears that ground reaction forces are an important factor in the maintenance of cortical bone and patellar tendon homeostasis during weight-bearing conditions.     

Composite bounds on the elastic modulus of bone

Advances in diagnosis and treatment of some bone disorders can be made by understanding the linkage between mineral content and mechanical function. Bone is approximately half by volume a hydrated protein network, and the remainder is a biomineral analogue of hydroxyapatite. In the current work, paired measurements of mechanical properties, using nanoindentation, and of bone mineral volume fraction, computed from quantitative back-scattered electron imaging, were made on six different types of normal and outlier bone samples. Local elastic modulus was plotted against mineral fraction and compared with predictions of engineering bounds for a two-phase composite material. Experimental data spanning the composite bounds showed no one-to-one relationship between mechanical stiffness and bone composition, excluding the possibility of any single, simple composites model for bone at nanometer length-scales.