Spatial distribution of acoustic and elastic properties of human femoral cortical bone

The purpose of this study is to quantify the spatial distribution of acoustic velocities and elastic properties (elastic constants) on Human femoral cortical bone. Four cross sections (average thickness of 2.09+/-0.27 mm) have been cut transversally between 40% and 70% of the total length and between them parallelepiped samples in each quadrant have been cut. Ultrasonic technique in transmission with immersion focused transducers at 5 MHz and contact transducers 2.25 MHz were used on the cross sections and parallelepiped samples, respectively. The first technique allows relative spatial distribution of velocities to be obtained, meanwhile the second technique allows the direct assessment of elastic constants. For both techniques, bulk velocities were found to be lower at the posterior side with an increase along the length (from 40% to 70% total length) (p < 0.05). Densities and elastic constants show equivalent pattern of variation. These variations are mainly due the cortical porosity related to vascularisation environment. The spatial distribution of velocities exhibits significant radial variation from the endosteal to the periosteal region. This is in agreement with variation of the porosity at that location. Same range of velocities was obtained with both techniques. The range of longitudinal velocities values varies from 3548 to 3967 m/s and between 18.5 and 33.1 GPa for the bulk velocities and axial elastic constants, respectively. Our results are within the range with those found in the literature. However, it must be noted that the range of acoustic and elastic properties variation is concerning the same bone. So, our new results show the ability of the technique to quantify accurately local variation of acoustic and elastic properties (within the section and along the length) of human cortical bone. Furthermore, our immersion technique could be used to assess the spatial distribution of the elastic constants with the knowledge of spatial distribution of densities.     

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.     

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.     

Determination of mechanical properties of human femoral cortical bone by the Hopkinson bar stress technique

The Hopkinson bar stress technique and a universal testing machine (Instron 1125) have been used to investigate the dynamic and static mechanical properties of cortical bone taken from a human femur respectively. We found that the average dynamic Young's modulus value (Ed = 19.9 GPa) to be 23% higher than the average static Young's modulus value (Ed = 16.2 GPa). Furthermore, the Poisson's ratio did not exhibit any significant variation for the two different types of loading. No difference was observed between the values of the dynamic Young's modulus in tension and those found in compression. A comparison was made of the results of this study with those found by other researchers using different techniques, such as ultrasonics, and it was found that they agree well with most of the results of previous studies. Finally, the viscosity for cortical bone found in this study correlates with viscosity reported by Tennyson et al. [Expl Mech. 12, 502-507 (1972)] for ten days post mortem age specimens.     

Mechanical properties of cancellous bone in the human mandibular condyle are anisotropic

The objective of the present study was (1) to test the hypothesis that the elastic and failure properties of the cancellous bone of the mandibular condyle depend on the loading direction, and (2) to relate these properties to bone density parameters. Uniaxial compression tests were performed on cylindrical specimens (n=47) obtained from the condyles of 24 embalmed cadavers. Two loading directions were examined, i.e., a direction coinciding with the predominant orientation of the plate-like trabeculae (axial loading) and a direction perpendicular to the plate-like trabeculae (transverse loading). Archimedes' principle was applied to determine bone density parameters. The cancellous bone was in axial loading 3.4 times stiffer and 2.8 times stronger upon failure than in transverse loading. High coefficients of correlation were found among the various mechanical properties and between them and the apparent density and volume fraction. The anisotropic mechanical properties can possibly be considered as a mechanical adaptation to the loading of the condyle in vivo.     

Determination of dynamically adapting anisotropic material properties of bone under cyclic loading

Because bone tissue adapts to loading conditions, finite element simulations of remodelling bone require a precise prediction of dynamically changing anisotropic elastic parameters. We present a phenomenological theory that refers to the tissue in terms of the tendency of the structure to align with principal stress directions. We describe the material parameters of remodelling bone. This work follows findings by the same research group and independently by Danilov (1971) in the field of plasticity, where the dependencies of the components of the stiffness tensor in terms of time are based on Hill's anisotropy. We modify such an approach in this novel theory that addresses bone tissue that can regenerate. The computational assumption of the theory is that bone trabeculae have the tendency to orient along one of the principal stress directions but during remodelling the principal stresses change continuously and the resulting orientation of the trabeculae can differ from the principal stress direction at any given time. The novelty of this work consists in the limited number of parameters needed to compute the twenty-one anisotropic material parameters at any given location in the bone tissue. In addition to the theory, we present here two cases of simplified geometry, loading and boundary conditions to show the effect of (1) time on the material properties; and (2) change of loading conditions on the anisotropic parameters. The long term goal is to experimentally verify that the predictions generated by theory provide a reliable simulation of cancellous bone properties.     

Erratum to: Mechanical properties of femoral trabecular bone in dogs

After the publication of this work [1], we became aware of the fact that the frequency of the ultrasound transmitter that we used for determining the elastic moduli of the trabecular bone specimens was not correctly specified. The oscillation frequency of the ultrasound transmitter was 2 MHz (and not 100 MHz as stated in our work) while we used a sampling rate of 100 MHz. In our publication, the oscillation frequency and sampling rate were confounded. Therefore also the statement in the discussion that we might have determined elastic moduli of trabecular bone tissue rather than the elastic properties of whole specimens because we used an ultrasound frequency > 2 MHz is wrong and has to be omitted.     

Identifying heterogeneous anisotropic properties in cerebral aneurysms: a pointwise approach

The traditional approaches of estimating heterogeneous properties in a soft tissue structure using optimization-based inverse methods often face difficulties because of the large number of unknowns to be simultaneously determined. This article proposes a new method for identifying the heterogeneous anisotropic nonlinear elastic properties in cerebral aneurysms. In this method, the local properties are determined directly from the pointwise stress–strain data, thus avoiding the need for simultaneously optimizing for the property values at all points/regions in the aneurysm. The stress distributions needed for a pointwise identification are computed using an inverse elastostatic method without invoking the material properties in question. This paradigm is tested numerically through simulated inflation tests on an image-based cerebral aneurysm sac. The wall tissue is modeled as an eight-ply laminate whose constitutive behavior is described by an anisotropic hyperelastic strain energy function containing four parameters. The parameters are assumed to vary continuously in the sac. Deformed configurations generated from forward finite element analysis are taken as input to inversely establish the parameter distributions. The delineated and the assigned distributions are in excellent agreement. A forward verification is conducted by comparing the displacement solutions obtained from the delineated and the assigned material parameters at a different pressure. The deviations in nodal displacements are found to be within 0.2% in most part of the sac. The study highlights some distinct features of the proposed method, and demonstrates the feasibility of organ level identification of the distributive anisotropic nonlinear properties in cerebral aneurysms.     

Effect of boundary conditions on yield properties of human femoral trabecular bone

Trabecular bone plays an important mechanical role in bone fractures and implant stability. Homogenized nonlinear finite element (FE) analysis of whole bones can deliver improved fracture risk and implant loosening assessment. Such simulations require the knowledge of mechanical properties such as an appropriate yield behavior and criterion for trabecular bone. Identification of a complete yield surface is extremely difficult experimentally but can be achieved in silico by using micro-FE analysis on cubical trabecular volume elements. Nevertheless, the influence of the boundary conditions (BCs), which are applied to such volume elements, on the obtained yield properties remains unknown. Therefore, this study compared homogenized yield properties along 17 load cases of 126 human femoral trabecular cubic specimens computed with classical kinematic uniform BCs (KUBCs) and a new set of mixed uniform BCs, namely periodicity-compatible mixed uniform BCs (PMUBCs). In stress space, PMUBCs lead to 7–72 % lower yield stresses compared to KUBCs. The yield surfaces obtained with both KUBCs and PMUBCs demonstrate a pressure-sensitive ellipsoidal shape. A volume fraction and fabric-based quadric yield function successfully fitted the yield surfaces of both BCs with a correlation coefficient \(R^{2} \ge 0.93\). As expected, yield strains show only a weak dependency on bone volume fraction and fabric. The role of the two BCs in homogenized FE analysis of whole bones will need to be investigated and validated with experimental results at the whole bone level in future studies.     

Heterogeneous linear elastic trabecular bone modelling using micro-CT attenuation data and experimentally measured heterogeneous tissue properties

High-resolution voxel-based finite element software, such as FEEBE developed at the NCBES, is widely used for studying trabecular bone at the micro-scale. A new approach to determine heterogeneous bone tissue material properties for computational models was proposed in this study. The specimen-specific range of tissue moduli across strut width was determined from nanoindentation testing. This range was mapped directly using linear interpolation to that specimen's micro-computed tomography (microCT) grey value range as input material properties for finite element analysis. The method was applied to cuboid trabecular bone samples taken from eight, 4-year-old (skeletally mature) ovine L5 vertebrae. Before undergoing experimental uniaxial compression tests, the samples were microCT scanned and 30 microm resolution finite element models were generated. The linear elastic finite element models were compressed to 1% strain. This material property assignment method for computational models accurately reproduced the experimentally determined apparent modulus and concentrations of stress at locations of failure.     

Spatial distribution of tissue level properties in a human femoral cortical bone

The mechanical properties of cortical bone are determined by a combination bone tissue composition, and structure at several hierarchical length scales. In this study the spatial distribution of tissue level properties within a human femoral shaft has been investigated. Cylindrically shaped samples (diameter: 4.4mm, N=56) were prepared from cortical regions along the entire length (20-85% of the total femur length), and around the periphery (anterior, medial, posterior and lateral quadrants). The samples were analyzed using scanning acoustic microscopy (SAM) at 50MHz and synchrotron radiation micro computed tomography (SRμCT). For all samples the average cortical porosity (Ct.Po), tissue elastic coefficients (c(ij)) and the average tissue degree of mineralization (DMB) were determined. The smallest coefficient of variation was observed for DMB (1.8%), followed by BV/TV (5.4%), c(ij) (8.2-45.5%), and Ct.Po (47.5%). Different variations with respect to the anatomical position were found for DMB, Ct.Po and c(ij). These data address the anatomical variations in anisotropic elastic properties and link them to tissue mineralization and porosity, which are important input parameters for numerical multi-scale bone models.     

Tissue mineral density measured at the sub-millimetre scale can provide reliable statistics of elastic properties of bone matrix

Reliability of multiscale models of bone is related to the accuracy of the experimental information available on bone microstructure. X-ray-based imaging techniques allow to inspect bone structure and mineralization in vitro at the micrometre scale. However, spatial resolution achievable in vivo is much coarser and can produce blurry, uncertain information on bone microstructure. Working with uncertain data calls for new modelling paradigms able to propagate uncertainty through the scales. In this paper we investigate the effects of uncertain bone mineralization on the elastic coefficients of the bone matrix. To this aim, some stochastic concepts were developed and compared with one another in order to identify the best way to account for uncertain input data. These concepts step from a deterministic micromechanical model of bone matrix which was extended in order to account for uncertain bone composition. Uncertainty was introduced by assuming to know only mean value and dispersion of the parameters describing bone composition. Thus, these parameters were modelled as random variables and their distribution functions were obtained using the maximum entropy principle. Either the tissue mineral density (TMD) or the ensuing volume fractions of collagen and mineral were used to describe uncertain bone composition. Moreover, mean value and dispersion were estimated at the scales of either 10 or a few 100 \(\upmu \)m, representative of standard in vitro and in vivo spatial resolutions, respectively. Analysis of these modelling concepts suggests that TMD measured at the sub-millimetre scale can be used to obtain reliable statistical information about the elastic coefficients of bone matrix.     

Assessment of composition and anisotropic elastic properties of secondary osteon lamellae

Measurement of the elastic properties of single osteon lamellae is still one of the most demanding tasks in bone mechanics to be solved. By means of site-matched Raman microspectroscopy, acoustic microscopy and nanoindentation the structure, chemical composition and anisotropic elasticity of individual lamellae in secondary osteons were investigated. Acoustic impedance images (911-MHz) and two-dimensional Raman spectra were acquired in sections of human femoral bone. The samples were prepared with orientations at various observation angles theta relative to the femoral long axis. Nanoindentations provided local estimations of the elastic modulus and landmarks necessary for spatial fusion of the acoustic and spectral Raman images. Phosphate nu(1) (961 cm(-1)) and amide I (1665 cm(-1)) band images representing spatial distributions of mineral and collagen were fused with the acoustic images. Acoustic impedance was correlated with the indentation elastic modulus E(IT) (R(2)=0.61). Both parameters are sensitive to elastic tissue anisotropy. The lowest values were obtained in the direction perpendicular to the femoral long axis. Acoustic images exhibit a characteristic bimodal lamellar pattern of alternating high and low impedance values. Since this undulation was not associated with a variation of the phosphate nu(1)-band intensity in the Raman images, it was attributed to variations of the lamellar orientation. After threshold segmentation and conversion to elastic modulus the orientation and transverse isotropic elastic constants were derived for individual ensembles of apparent thin and thick lamellae. Our results suggest that this model represents the effective anisotropic properties of an asymmetric twisted plywood structure made of transverse isotropic fibrils. This is the first report that proves experimentally the ability of acoustic microscopy to map tissue elasticity in two dimensions with micrometer resolution. The combination with Raman microspectroscopy provides a unique way to study bone and mineral metabolism and the relation with mechanical function at the ultrastructural tissue level.     

Anatomic variation in the elastic inhomogeneity and anisotropy of human femoral cortical bone tissue is consistent across multiple donors

Numerical models commonly account for elastic inhomogeneity in cortical bone using power-law scaling relationships with various measures of tissue density, but limited experimental data exists for anatomic variation in elastic anisotropy. A recent study revealed anatomic variation in the magnitude and anisotropy of elastic constants along the entire femoral diaphysis of a single human femur (Espinoza Orías et al., 2009). The objective of this study was to confirm these trends across multiple donors while also considering possible confounding effects of the anatomic quadrant, apparent tissue density, donor age, and gender. Cortical bone specimens were sampled from the whole femora of 9 human donors at 20%, 50%, and 80% of the total femur length. Elastic constants from the main diagonal of the reduced fourth-order tensor were measured on hydrated specimens using ultrasonic wave propagation. The tissue exhibited orthotropy overall and at each location along the length of the diaphysis (p < 0.0001). Elastic anisotropy increased from the mid-diaphysis toward the epiphyses (p < 0.05). The increased elastic anisotropy was primarily caused by a decreased radial elastic constant (C(11)) from the mid-diaphysis toward the epiphyses (p < 0.05), since differences in the circumferential (C(22)) and longitudinal (C(33)) elastic constants were not statistically significant (p > 0.29). Anatomic variation in intracortical porosity may account for these trends, but requires further investigation. The apparent tissue density was positively correlated with the magnitude of each elastic constant (p < 0.0001, R(2) > 0.46), as expected, but was only weakly correlated with C(33)/C(11) (p < 0.05, R(2) = 0.04) and not significantly correlated with C(33)/C(22) and C(11)/C(22).     

Critical evaluation of known bone material properties to realize anisotropic FE-simulation of the proximal femur

PURPOSE: In a meta-analysis of the literature we evaluated the present knowledge of the material properties of cortical and cancellous bone to answer the question whether the available data are sufficient to realize anisotropic finite element (FE)-models of the proximal femur. MATERIAL AND METHOD: All studies that met the following criteria were analyzed: Young's modulus, tensile, compressive and torsional strengths, Poisson's ratio, the shear modulus and the viscoelastic properties had to be determined experimentally. The experiments had to be carried out in a moist environment and at room temperature with freshly removed and untreated human cadaverous femurs. All material properties had to be determined in defined load directions (axial, transverse) and should have been correlated to apparent density (g/cm(3)), reflecting the individually variable and age-dependent changes of bone material properties. RESULTS: Differences in Young's modulus of cortical [cancellous] bone at a rate of between 33% (58%) (at low apparent density) and 62% (80%) (at high apparent density), are higher in the axial than in the transverse load direction. Similar results have been seen for the compressive strength of femoral bone. For the tensile and torsional strengths, Poisson's ratio and the shear modulus, only ultimate values have been found without a correlation to apparent density. For the viscoelastic behaviour of bone only data of cortical bone and in axial load direction have been described up to now. CONCLUSIONS: Anisotropic FE-models of the femur could be realized for most part with the summarized material properties of bone if characterized by apparent density and load directions. Because several mechanical properties have not been correlated to these main criteria, further experimental investigations will be necessary in future.     

Identification of heterogeneous elastic properties in stenosed arteries: a numerical plane strain study

Assessing the vulnerability of atherosclerotic plaques requires an accurate knowledge of the mechanical properties of the plaque constituents. It is possible to measure displacements in vivo inside a plaque using magnetic resonance imaging. An important issue is to solve the inverse problem that consists in estimating the elastic properties inside the plaque from measured displacements. This study focuses on the identifiability of elastic parameters, e.g. on the compromise between identification time and identification accuracy. An idealised plane strain finite element (FE) model is used. The effects of the FE mesh of the a priori assumptions about the constituents, of the measurement resolution and of the data noise are numerically investigated.