Micromechanics modeling of Haversian cortical bone properties.

A finite-element micromechanics model for Haversian cortical bone tissue has been developed and studied. The model is an extension of two-dimensional micromechanics techniques for fiber-reinforced composite materials. Haversian systems, or secondary osteons, are considered to be the fiber component, and interstitial lamellar bone the matrix material. The cement line is included as an 'interphase' component along the fiber/matrix interface. The model assumes a regular repeatable spacing of the longitudinally aligned continuous fibers and is, therefore, restricted to approximating Haversian cortical bone in its present form. Haversian porosity is modeled explicitly by incorporating a hollow fiber to represent the Haversian canal. Solutions have been obtained by applying uniform macroscopic stresses to the boundaries of the repeating unit cell model. Macroscopic mechanical property predictions correspond reasonably well with the experimental data for cortical bone, but are necessarily dependent on the input properties for each constituent, which are not well established. The predicted variation in the elastic modulus with porosity is not as sensitive as that observed experimentally. Stresses within the constituents can also be modeled with this method and are demonstrated to deviate from the macroscopic applied stress levels.     

Impact of the porous microstructure on the overall elastic properties of the osteonal cortical bone

Mechanical properties of bones are largely determined by their microstructure. The latter comprises a large number of diverse pores. The present paper analyzes a connection between structure of the porous space of the osteonal cortical bone and bone's overall anisotropic elastic moduli. The analysis is based on recent developments in the theory of porous materials that predict the anisotropic effective moduli of porous solids in terms of pores' shapes, orientations and densities. Bone's microstructure is modeled using available micrographs. The calculated anisotropic elastic constants for porous cortical bone are, mostly, in agreement with available experimental data. The influence of each of the pore types on the overall moduli is examined. The results of the analysis can also be used to estimate the extent of mineralization (hydroxyapatite content) if the overall porosity and the effective moduli are known and, vice versa, to estimate porosity from the measured moduli and the extent of mineralization.     

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.     

Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale

This paper addresses the relationships between the microscopic properties of bone and its elasticity at the millimetre scale, or mesoscale. A method is proposed to estimate the mesoscale properties of cortical bone based on a spatial distribution of acoustic properties at the microscopic scale obtained with scanning acoustic microscopy. The procedure to compute the mesoscopic stiffness tensor involves (i) the segmentation of the pores to obtain a realistic model of the porosity; (ii) the construction of a field of anisotropic elastic coefficients at the microscopic scale which reflects the heterogeneity of the bone matrix; (iii) finite element computations of mesoscopic homogenized properties. The computed mesoscopic properties compare well with available experimental data. It appears that the tissue anisotropy at the microscopic level has a major effect on the mesoscopic anisotropy and that assuming the pores filled with an incompressible fluid or, alternatively, empty, leads to significantly different mesoscopic properties.     

Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method

A two-level micromechanical model of cortical bone based on a generalized self-consistent method was developed to take into consideration the transversely isotropic elasticity of many microstructural features in cortical bone, including Haversian canals, resorption cavities, and osteonal and interstitial lamellae. In the first level, a single osteon was modeled as a two-phase composite such that Haversian canals were represented by elongated pores while the surrounding osteonal lamellae were considered as matrix. In the second level, osteons and resorption cavities were modeled as multiple inclusions while interstitial lamellae were regarded as matrix. The predictions of cortical bone elasticity from this two-level micromechanical model were mostly in agreement with experimental data for the dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. However, variation in cortical bone elastic constants was greater in experimental data than in model predictions. This could be attributed to variations in the elastic properties of microstructural features in cortical bone. The present micromechanical model of cortical bone will be useful in understanding the contribution of cortical bone porosity to femoral neck fractures.     

How is the indentation modulus of bone tissue related to its macroscopic elastic response? A validation study

This work consists of the validation of a novel approach to estimate the local anisotropic elastic constants of the bone extracellular matrix using nanoindentation. For this purpose, nanoindentation on two planes of material symmetry were analyzed and the resulting longitudinal elastic moduli compared to the moduli measured with a macroscopic tensile test. A combined lathe and tensile system was designed that allows machining and testing of cylindrical microspecimens of approximately 4mm in length and 300 microm in diameter. Three bovine specimens were tested in tension and their outer geometry and porosity assessed by synchrotron radiation microtomography. Based on the results of the traction test and the precise outer geometry, an apparent longitudinal Young's modulus was calculated. Results between 20.3 and 27.6 GPa were found that match with previously reported values for bovine compact bone. The same specimens were then characterized by nanoindentation on a transverse and longitudinal plane. A longitudinal Young's modulus for the bone matrix was then derived using the numerical scheme proposed by Swadener and Pharr and the fabric-elasticity relationship by Zysset and Curnier. Based on the matrix modulus and a power law effective volume fraction, an apparent longitudinal Young's modulus was predicted for each microspecimen. This alternative approach provided values between 19.9 and 30.0 GPa, demonstrating differences between 2% and 13% to the values provided by the initial tensile test. This study therefore raises confidence in our nanoindentation protocol of the bone extracellular matrix and supports the underlying hypotheses used to extract the anisotropic elastic constants.     

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.     

Micromechanical modelling of cortical bone

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.     

The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity

The objective of this study was to examine the dependence of the elastic properties of cortical bone as a transversely isotropic material on its porosity. The longitudinal Young's modulus, transverse Young's modulus, longitudinal shear modulus, transverse shear modulus, and longitudinal Poisson's ratio of cortical bone were determined from eighteen groups of longitudinal and transverse specimens using tensile and torsional tests on a servo-hydraulic material testing system. These cylindrical waisted specimens of cortical bone were harvested from the middle diaphysis of three pairs of human femora. The porosity of these specimens was assessed by means of histology. Our study demonstrated that the longitudinal Young's and shear moduli of human femoral cortical bone were significantly (p<0.01) negatively correlated with the porosity of cortical bone. Conversely, the elastic properties in the transverse direction did not have statistically significant correlations with the porosity of cortical bone. As a result, the transverse elastic properties of cortical bone were less sensitive to changes in porosity than those in the longitudinal direction. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly (p<0.01) negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone.     

A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

At its highest level of microstructural organization—the mesoscale or millimeter scale—cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.     

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.     

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.     

Regional,ontogenetic, and sex‐related variations in elastic properties of cortical bone in baboon mandibles

Understanding the mechanical features of cortical bone and their changes with growth and adaptation to function plays an important role in our ability to interpret the morphology and evolution of craniofacial skeletons. We assessed the elastic properties of cortical bone of juvenile and adult baboon mandibles using ultrasonic techniques. Results showed that, overall, cortical bone from baboon mandibles could be modeled as an orthotropic elastic solid. There were significant differences in the directions of maximum stiffness, thickness, density, and elastic stiffness among different functional areas, indicating regional adaptations. After maturity, the cortical bone becomes thicker, denser, and stiffer, but less anisotropic. There were differences in elastic properties of the corpus and ramus between male and female mandibles which are not observed in human mandibles. There were correlations between cortical thicknesses and densities, between bone elastic properties and microstructural configuration, and between the directions of maximum stiffness and bone anatomical axes in some areas. The relationships between bone extrinsic and intrinsic properties bring us insights into the integration of form and function in craniofacial skeletons and suggest that we need to consider both macroscopic form, microstructural variation, and the material properties of bone matrix when studying the functional properties and adaptive nature of the craniofacial skeleton in primates. The differences between baboon and human mandibles is at variance to the pattern of differences in crania, suggesting differences in bone adaption to varying skeletal geometries and loading regimes at both phylogenetic and ontogenetic levels. Am J Phys Anthropol, 2010. © 2009 Wiley‐Liss, Inc.     

Nanoindentation testing and finite element simulations of cortical bone allowing for anisotropic elastic and inelastic mechanical response

Anisotropy is one of the most peculiar aspects of cortical bone mechanical behaviour, and the numerical approach can be successfully used to investigate aspects of bone tissue mechanics that analytical methods solve in approximate way or do not cover. In this work, nanoindentation experimental tests and finite element simulations were employed to investigate the elastic-inelastic anisotropic mechanical properties of cortical bone. The model allows for anisotropic elastic and post-yield behaviour of the tissue. A tension-compression mismatch and direction-dependent yield stresses are allowed for. Indentation experiments along the axial and transverse directions were simulated with the purpose to predict the indentation moduli and hardnesses along multiple orientations. Results showed that the experimental transverse-to-axial ratio of indentation moduli, equal to 0.74, is predicted with a ～3% discrepancy regardless the post-yield material behaviour; whereas, the transverse-to-axial hardness ratio, equal to 0.86, can be correctly simulated (discrepancy ～6% w.r.t. the experimental results) only employing an anisotropic post-elastic constitutive model. Further, direct comparison between the experimental and simulated indentation tests evidenced a good agreement in the loading branch of the indentation curves and in the peak loads for a transverse-to-axial yield stress ratio comparable to the experimentally obtained transverse-to-axial hardness ratio. In perspective, the present work results strongly support the coupling between indentation experiments and FEM simulations to get a deeper knowledge of bone tissue mechanical behaviour at the microstructural level. The present model could be used to assess the effect of variations of constitutive parameters due to age, injury, and/or disease on bone mechanical performance in the context of indentation testing.     

Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue

The ability to determine trabecular bone tissue elastic and failure properties has biological and clinical importance. To date, trabecular tissue yield strains remain unknown due to experimental difficulties, and elastic moduli studies have reported controversial results. We hypothesized that the elastic and tensile and compressive yield properties of trabecular tissue are similar to those of cortical tissue. Effective tissue modulus and yield strains were calibrated for cadaveric human femoral neck specimens taken from 11 donors, using a combination of apparent-level mechanical testing and specimen-specific, high-resolution, nonlinear finite element modeling. The trabecular tissue properties were then compared to measured elastic modulus and tensile yield strain of human femoral diaphyseal cortical bone specimens obtained from a similar cohort of 34 donors. Cortical tissue properties were obtained by statistically eliminating the effects of vascular porosity. Results indicated that mean elastic modulus was 10% lower (p<0.05) for the trabecular tissue (18.0+/-2.8 GPa) than for the cortical tissue (19.9+/-1.8 GPa), and the 0.2% offset tensile yield strain was 15% lower for the trabecular tissue (0.62+/-0.04% vs. 0.73+/-0.05%, p<0.001). The tensile-compressive yield strength asymmetry for the trabecular tissue, 0.62 on average, was similar to values reported in the literature for cortical bone. We conclude that while the elastic modulus and yield strains for trabecular tissue are just slightly lower than those of cortical tissue, because of the cumulative effect of these differences, tissue strength is about 25% greater for cortical bone.     

A computational and cellular solids approach to the stiffness-based design of bone scaffolds

We derive a cellular solids approach to the design of bone scaffolds for stiffness and pore size. Specifically, we focus on scaffolds made of stacked, alternating, orthogonal layers of hydroxyapatite rods, such as those obtained via micro-robotic deposition, and aim to determine the rod diameter, spacing and overlap required to obtain specified elastic moduli and pore size. To validate and calibrate the cellular solids model, we employ a finite element model and determine the effective scaffold moduli via numerical homogenization. In order to perform an efficient, automated execution of the numerical studies, we employ a geometry projection method so that analyses corresponding to different scaffold dimensions can be performed on a fixed, non-conforming mesh. Based on the developed model, we provide design charts to aid in the selection of rod diameter, spacing and overlap to be used in the robotic deposition to attain desired elastic moduli and pore size.     

A comparison of cortical elastic properties in the craniofacial skeletons of three primate species and its relevance to the study of human evolution

When a force is applied to an object, the resulting pattern of strain is a function of both the object's geometry and its elastic properties. Thus, knowledge of elastic properties in craniofacial cortical bone is indispensable for exploring the biomechanics and adaptation of primate skulls. However, elastic properties, such as density and stiffness, cannot be measured in all species, particularly extinct species known only from fossils. In order for advanced engineering techniques such as finite element analysis (FEA) to be applied to questions of primate and hominid craniofacial functional morphology, it is important to understand interspecific patterns of variation in elastic properties. We hypothesized that closely related species would have similar patterns of bone elastic properties, and that similarities with extant species should allow reasonable predictions of elastic properties in the skeletons of extinct primate species. In this study, we tested this hypothesis by measuring elastic properties in five areas of the external cortex of the baboon craniofacial skeleton using an ultrasonic technique, and by comparing the results to existing data from macaque and human crania. Results showed that cortical density, thickness, elastic and shear moduli, and anisotropy varied among areas in the baboon cranium. Similar variation had previously been found in rhesus and human crania, suggesting area-specific elastic patterns in the skulls of each species. Comparison among species showed differences, suggesting species-specific patterns. These patterns were more similar between macaques and baboons for density, maximum elastic and shear stiffness, and anisotropy than between either of these and humans. This finding demonstrates that patterns of cortical elastic properties are generally similar in closely related primate species with similar craniofacial morphology. Thus, reasonable estimates of cortical bone elastic properties should be possible for extinct species through the study of phylogenetically related and functionally similar modern forms. For example, reasonable elastic property estimates of cortical bone from fossil hominid skulls should be possible once adequate information about such properties in extant great apes is added to our current data from humans, macaques, and baboons. Such data should eventually allow FEA of craniofacial function in fossil hominids.     

Elastic properties of cancellous bone: measurement by an ultrasonic technique

The high degree of porosity of cancellous bone makes elastic property measurement difficult by traditional mechanical testing methods. An ultrasonic technique is described with which mechanical properties of anisotropic, rigid, porous materials, such as cancellous bone, can be measured. The technique utilizes unique piezoelectric transducers operated in a continuous wave mode at a frequency of approximately 50 kHz. Both longitudinal and shear waves can be propagated and received with the transducers allowing both Young's moduli and shear moduli to be determined with the technique. A comparison between moduli measured with the ultrasonic technique and moduli measured with traditional mechanical testing shows the new method to be quite accurate in elastic property determination, (r2 = 0.935, Emech = 1.00E1dt + 23.3 MPa) (r2 = 0.656, Gmech = 1.08 Gult--3.3MPa).     

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.