Elastic Properties Of Cortical Bone Research Articles

The effect of porosity on the elastic properties of cortical bone and ultrasound propagation

This paper focuses on the modeling of the effect of different porosity on the overall elastic properties of saturated cortical bone and ultrasound wave propagation in such bone. We first utilize micromechanical model of Zhou, Cui and Sevostianov (2020) to model anisotropic effective elastic stiffness of saturated cortical bone and evaluate the effect of partial porosities. It is shown that the moduli C33satand C44satof saturated bone are more sensitive to canalicular porosity, while the moduli C11satand C66satof saturated bone are more sensitive to Haversian porosity. This model has been validated by experimental data. We compare the extents of anisotropy of the saturated bones for different levels of partial porosity. Last but not the least, we apply this micromechanical model to the elastic field of saturated cortical bone to evaluate the effect of partial porosity on the velocity of different bulk waves and guided waves. Results show that the velocity of the fast longitudinal wave and the slower SH wave along the horizontal direction are more sensitive to Haversian porosity ϕHav, but the velocity of the fast longitudinal wave and SH wave along the vertical direction and the velocity of the SV wave along the horizontal direction are more sensitive to the canalicular porosity ϕCan. Further, it is found that, the plateau region of the axisymmetric longitudinal guided waves mode L(0, 1) and L(0, 2) phase velocity are more sensitive to canalicular porosity ϕCan, and the phase velocity of the lowest nonaxisymmetric flexural mode F(1, 1) in low frequencie is more sensitive to Haversian porosity ϕHav.

Biomechanical Evaluation of Stress Distribution in Equicrestal and Sub-crestally Placed, Platform-Switched Morse Taper Dental Implants in D3 Bone: Finite Element Analysis.

AimThe aim of the study was to assess the effect of implant placement depth on stress distribution in bone around a platform-switched and Morse taper dental implants placed at the equi-crestal and 1 mm and 2 mm sub-crestal levels in a D3 bone using the 3D finite element analysis.MethodologyA mechanical model of a partially edentulous maxilla was generated from a computerized tomography (CT) scan of an edentulous patient, as it can give exact bony contours of cortical bone. Also, from accurate geometric measurements obtained from the manufacturer, 3D models of Morse taper and platform-switched implants were manually drawn. The implant and bone models were then superimposed to simulate implant insertion in bone. Three implant positioning levels such as the equi-crestal, 1 mm sub-crestal, and 2 mm sub-crestal models were created, and meshing was done to create the number of elements for distribution of applying loads. The elastic properties of cortical bone and implant, such as Young's modulus and Poisson's ratio (µ), were determined. A load (axial and oblique) of 200N that simulated masticatory force was applied.ResultsOn comparing stresses within the bone around the equi-crestal and 1 mm and 2 mm sub-crestal implants, it was observed that the maximum stresses were seen within cortical bone around the equi-crestally placed implant (21.694), the least in the 2 mm sub-crestally placed implant (18.85), and intermediate stresses were seen within the 1 mm sub-crestally placed implant (18.876).ConclusionSub-crestal (1-2mm) placement of a Morse taper and a platform-switched implant is recommended for long-term success, as maximum von Mises stresses were found within cortical bone around the equi-crestal implant followed by the 1 mm sub-crestal implant and then the 2 mm sub-crestal implant.

Effect of saturation on the elastic properties and anisotropy of cortical bone

This paper focuses on the modeling of the effect of saturation on the overall elastic properties of cortical bone. We first use micromechanical model of Salguero, Saadat & Sevostianov (2014) to model anisotropic effective elastic stiffness of drained cortical bone and then apply replacement relations (see review of Sevostianov, 2020) to evaluate effect of the saturation. The model is verified by comparison with the experimental data of Granke et al. (2011). It is shown that accounting for the saturation makes the model consistent with the experiment data. We also compared the extents of anisotropy of the saturated and drained bones and show that presence of the biological fluid in pores reduces the overall anisotropy.

Estimation of the effective bone-elasticity tensor based on μ CT imaging by a stochastic model. A multi-method validation

Abstract In previous work, we proposed a stochastic model to describe the elasticity of bone matrix (so-called ultrastructure, US) based on basic statistical information on the tissue mineral density ( TMD ). This information was obtained by analyzing high-resolution images of a human femoral neck realized by means of synchrotron radiation micro-computed tomography ( SR − μ CT ). In this paper, we extend this study by focusing at the upper scale where cortical bone is described as a two-phase mixture made up of water-filled Haversian pores (HP) embedded in the surrounding solid US. The goal of this paper is to develop a stochastic model of cortical bone elasticity accounting for the effect of uncertainty affecting both phases, the US via the TMD and the HP. Experimental information was assumed to be given in terms of mean values and dispersions of the average TMD (denoted TMD ¯ ) and HP at the millimeter scale. To this aim, SR − μ CT images were used to extract several representative volume elements (RVEs) spanning the whole cortical tissue which, in turn, were analyzed to obtain the required statistical information on TMD ¯ and HP. This information has been used for constructing a stochastic multiscale model of cortical bone based on the Maximum Entropy (denoted MaxEnt) principle. This stochastic multiscale model is used in the estimation of the effective elastic properties of cortical bone (CB-S μ M) based on continuum micromechanics ( μM). In parallel, a deterministic nominal multiscale model of cortical bone (CB-NμM) was developed by using as input data the mean values of TMD ¯ and HP. The elastic moduli estimated with CB-NμM has been compared with the average value of the same counterpart obtained with the CB-SμM model. The two estimates differ for less than 1 % , proving the robustness of the CB-NμM model to uncertainties. Moreover, the accuracy of the SμM stochastic model has been tested. Estimations of the cortical bone elasticity of each RVE with the μ M (cylindrical HP and homogeneous US), has been compared with other estimation obtained through others homogenization techniques: finite elements (FEM) and fast Fourier transform (FFT), both able to account for the real morphology of Havesian porosity and the heterogeneity of the ultrastructure. Results show that (1) the transverse isotropic (TI) elastic μM model correctly approximates the FEM and FFT estimates (∼98% TI); (2) the μM model accurately estimates the axial modulus ( Y 3 ) in longitudinal direction and the lateral contraction ratio ν 31 .

Elasticity-density and viscoelasticity-density relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements.

Cortical bone tissue is an anisotropic material characterized by typically five independent elastic coefficients (for transverse isotropy) governing shear and longitudinal deformations in the different anatomical directions. It is well established that the Young's modulus in the direction of the bone axis of long bones has a strong relationship with mass density. It is not clear, however, whether relationships of similar strength exist for the other elastic coefficients, for they have seldom been investigated, and the results available in the literature are contradictory. The objectives of the present work were to document the anisotropic elastic properties of cortical bone at the tibia mid-diaphysis and to elucidate their relationships with mass density. Resonant ultrasound spectroscopy (RUS) was used to measure the transverse isotropic stiffness tensor of 55 specimens from 19 donors. Except for Poisson's ratios and the non-diagonal stiffness coefficient, strong linear correlations between the different elastic coefficients (0.7 < r(2) < 0.99) and between these coefficients and density (0.79 < r(2) < 0.89) were found. Comparison with previously published data from femur specimens suggested that the strong correlations evidenced in this study may not only be valid for the mid-tibia. RUS also measures the viscous part of the stiffness tensor. An anisotropy ratio close to two was found for damping coefficients. Damping increased as the mass density decreased. The data suggest that a relatively accurate estimation of all the mid-tibia elastic coefficients can be derived from mass density. This is of particular interest (1) to design organ-scale bone models in which elastic coefficients are mapped according to Hounsfield values from computed tomography scans as a surrogate for mass density and (2) to model ultrasound propagation at the mid-tibia, which is an important site for the in vivo assessment of bone status with axial transmission techniques.

Quantification of nonlinear elasticity for the evaluation of submillimeter crack length in cortical bone

The objective of this study was to investigate the sensitivity of the nonlinear elastic properties of cortical bone to the presence of a single submillimetric crack. Nonlinear elasticity was measured by nonlinear resonant ultrasound spectroscopy (NRUS) in 14 human cortical bone specimens. The specimens were parallelepiped beams (50×2×2mm3). A central notch of 500µm was made to control crack initiation and propagation during four-point bending. The nonlinear hysteretic elastic and dissipative parameters αf and αQ, and Young׳s modulus Eus were measured in dry condition for undamaged (control) specimens and in dry and wet conditions for damaged specimens. The length of the crack was assessed using synchrotron radiation micro-computed tomography (SR-μCT) with a voxel size of 1.4μm. The initial values of αf, measured on the intact specimens, were remarkably similar for all the specimens (αf =−5.5±1.5). After crack propagation, the nonlinear elastic coefficient αf increased significantly (p<0.006), with values ranging from −4.0 to –296.7. Conversely, no significant variation was observed for αQ and Eus. A more pronounced nonlinear elastic behavior was observed in hydrated specimens compared to dry specimens (p<0.001) after propagation of a single submillimetric crack. The nonlinear elastic parameter αf was found to be significantly correlated to the crack length both in dry (R=0.79, p<0.01) and wet (R=0.84, p<0.005) conditions. Altogether these results show that nonlinear elasticity assessed by NRUS is sensitive to a single submillimetric crack induced mechanically and suggest that the humidity must be strictly controlled during measurements.

In vitro ultrasonic and mechanic characterization of the modulus of elasticity of children cortical bone

The assessment of elastic properties in children’s cortical bone is a major challenge for biomechanical engineering community, more widely for health care professionals. Even with classical clinical modalities such as X-ray tomography, MRI, and/or echography, inappropriate diagnosis can result from the lack of reference values for children bone. This study provides values for elastic properties of cortical bone in children using ultrasonic and mechanical measurements, and compares them with adult values. 18 fibula samples from 8 children (5–16years old, mean age 10.6years old ±4.4) were compared to 16 fibula samples from 3 elderly adults (more than 65years old). First, the dynamic modulus of elasticity (Edyn) and Poisson’s ratio (ν) are evaluated via an ultrasonic method. Second, the static modulus of elasticity (Esta) is estimated from a 3-point microbending test. The mean values of longitudinal and transverse wave velocities measured at 10MHz for the children’s samples are respectively 3.2mm/μs (±0.5) and 1.8mm/μs (±0.1); for the elderly adults’ samples, velocities are respectively 3.5mm/μs (±0.2) and 1.9mm/μs (±0.09). The mean Edyn and the mean Esta for the children’s samples are respectively 15.5GPa (±3.4) and 9.1GPa (±3.5); for the elderly adults’ samples, they are respectively 16.7GPa (±1.9) and 5.8GPa (±2.1). Edyn, ν and Esta are in the same range for children’s and elderly adults’ bone without any parametric statistical difference; a ranking correlation between Edyn and Esta is shown for the first time.

Assessment of the Elastic Properties of Human Femoral Bone with Artificial Hip Joint by Ultrasound Transmission

Following artificial hip joint implantation, a stress inhibition, applied to bone in the surroundings of implants, causes a structural change in bone called bone loss. To evaluate the bone mechanical characteristics, it is essential to investigate the elastic properties of cortical bone. In this article a pair of donor femora was investigated, one with an implant and the other without. Differences in Speed of Sound (SOS), a parameter reflecting elastic properties, were measured in both femora by ultrasound transmission. As a result, in almost all areas, the femur that was implanted showed significantly lower cortical SOS. Our results indicated that the change in the mechanical function of bone, due to the introduction of femoral implants, could be evaluated by the measurement of SOS.

Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone

We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori-Tanaka scheme and the Hashin-Rosen bounds. The methods concur for specific engineering moduli such as the axial Young's modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.

Effect of porosity and mineral content on the elastic constants of cortical bone: a multiscale approach

A micromechanical multiscale model which estimates the elastic properties of cortical bone as a function of porosity and mineral content is presented. The steps of the model are divided into two main phases. In the first one, the elastic properties of the collagen fibril, collagen fiber and lamella are given. In the second phase, porosity is included in the lamella in the form of canaliculi, lacunae and Haversian canals, to provide the elastic properties of the osteonal tissue. Then, a symmetrization technique is used to estimate the transversely isotropic elasticity tensor of the osteon. Osteons are superimposed using a self-consistent scheme, and finally, the fluid filling the pores is included to estimate the elastic constants of the undrained cortical tissue. The main novelty of the model presented here is the possibility of varying the mineral content of bone, considering that mineralization begins from the inner levels, initially intrafibrillar and then interfibrillar. Correlations of the elastic properties of cortical bone obtained with this model on the one hand, and porosity and ash fraction on the other hand, are estimated.

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.

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.

Guided waves analysis using multiemitter and multireceiver arrays.

Structural and material properties of elastic waveguides can be characterized by fitting measured to theoretical guided wave phase velocities. Here emitters and receivers are placed in contact on the same side of the waveguide (i.e., axial transmission geometry). Multireceiver arrays allow the determination of phase velocities using two‐dimensional spatial‐temporal Fourier transform, which requires a large distance probed by the receivers. Practical constraints, as in clinical inspection of cortical bones, may reduce the inspected spatial length and therefore the efficiency of this technique. We propose a technique which takes benefit of using both multiple emitters and multiple receivers. This approach derived from the DORT method adapted to the transmission problem. The singular values decomposition at different frequencies of the transmission matrix between the arrays give the invariants of transmission, which are linked with the guided modes. First, experiments carried out on different metallic plates are shown. Experimental velocities are in good agreement with Lamb waves theoretical values. In particular, zero group velocity resonance and negative phase velocities are shown. These results allows a good evaluation of the thickness and the transverse and longitudinal bulk waves velocities. Further applications concerning evaluation of elastic properties of cortical bone are finally mentioned.

Microstructural basis of elastic properties of cortical bone in baboon mandibles

Successful bone regeneration in reconstructive surgery should go beyond reproducing shape and structure to restore biological and mechanical function as well as long‐term integration with surrounding native tissues. Recent work has demonstrated that the three‐dimensional material properties of cortical bone vary throughout craniofacial skeletons. However, how the bone microstructure is related to the mechanical properties of bone at the tissue level throughout the skull needs systematic investigation. Cortical samples from symphyseal, corporal and ramal areas of a baboon mandible were studied using micro‐CT and ultrasonic techniques. Results demonstrated that there were significant variations in elastic properties, porosity, and morphology of osteonal canals throughout the baboon mandible, and the principal orientation of osteonal canals agreed well with the axis of maximum stiffness, and further with the long axis of the bone in three functional areas. These findings on the relationships between bone material properties and microstructure, and their relationships to the skeletal form and function at the organ level will greatly enhance our ability to address questions derived from both evolutional and craniofacial medicine fields about the development and biomechanics of skulls. Supported by NSF HOMINID grants (BCS 0725141, 0725183), MEDCEN, and Mercer Seed.

90 Continuum Level Elastic Properties of Cortical Bone Derived from Acoustic Microscopy Images

90 Continuum Level Elastic Properties of Cortical Bone Derived from Acoustic Microscopy Images

A generalized self-consistent estimate for the effective elastic moduli of fiber-reinforced composite materials with multiple transversely isotropic inclusions

The effective elastic properties of a fiber-reinforced composite material with multiple transversely isotropic inclusions are estimated by the use of a generalized self-consistent method, which considers strong interactions between the inclusion and matrix as well as among inclusions. The accuracy of this method is established by comparing to the closed-form analytic solutions by Christensen when the matrix and inclusion are isotropic. Furthermore, current predictions from the generalized self-consistent method for a composite with multiple inclusions correspond well with the numerical results from finite element analysis. The generalized self-consistent method can be particularly useful in establishing micromechanics models of natural biological composite materials such as cortical bone to examine the dependence of the elastic properties of cortical bone on its porosity.

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.

Material properties of the human cranial vault and zygoma.

The material properties of cortical bone from the diaphyses of long bones (e.g., the femur and tibia) vary by direction, such that bone is stiffer and stronger along its long axis. This configuration improves the abilities of these structures to resist axial compressive loads coupled with bending. As in long bones, cortical bone from the cranial vault is subject to mechanical loads from various orofacial functions and the contraction of attached muscles. However, experimental studies suggest that the resulting bone strains are at least an order of magnitude smaller than those found in the midshafts of the femur or tibia. The characteristics of the three-dimensional elastic properties of cortical bone are largely unexplored in regions of low bone strain, including the cranial vault, in which little is known regarding cortical structure and function. In the present study we examined variations in the cortical microstructure and material properties of the bone of the human cranial vault, including the parietal, frontal, temporal, and occipital bones. A facial bone, the zygoma, was also included to contrast the properties of the cranial vault with another craniofacial intramembranous bone that experiences larger strains. Cortical specimens from the outer cortical plate of the cranial vault were removed from 15 frozen human crania. We measured cortical thicknesses and densities, and determined the primary direction of stiffness within the bone specimens prior to ultrasonic testing to determine their elastic properties. There were statistically significant differences in elastic properties between bones and, in some cases, sites within bones, which for most variables were clustered by bone or region. In striking contrast to this pattern, elastic moduli in the direction of primary stiffness were larger in cortical regions underlying muscle attachments than in regions without muscle attachments. Few sites in the cranial vault or zygoma showed a consistent orientation of the material axes among individuals, although specimens from many regions had directional differences similar to those in cortical bone from the mandible, femur, or tibia.

An alternative ultrasonic method for measuring the elastic properties of cortical bone

We studied the elastic properties of bone to analyze its mechanical behavior. The basic principles of ultrasonic methods are now well established for varying isotropic media, particularly in the field of biomedical engineering. However, little progress has been made in its application to anisotropic materials. This is largely due to the complex nature of wave propagation in these media. In the present study, the theory of elastic waves is essential because it relates the elastic moduli of a material to the velocity of propagation of these waves along arbitrary directions in a solid. Transducers are generally placed in contact with the samples which are often cubes with parallel faces that are difficult to prepare. The ultrasonic method used here is original, a rough preparation of the bone is sufficient and the sample is rotated. Moreover, to analyze heterogeneity of the structure we measure velocities in different points on the sample. The aim of the present study was to determine in vitro the anisotropic elastic properties of cortical bones. For this purpose, our method allowed measurement of longitudinal and transverse velocities ( C L and C T) in longitudinal (fiber direction) and the radial directions (orthogonal to the fiber direction) of compact bones. Young's modulus E and Poisson's ratio ν, were then deduced from the velocities measured considering the compact bone as transversely isotropic or orthotropic. The results are in line with those of other methods.

A continuous wave technique for the measurement of the elastic properties of cortical bone

A continuous wave technique is described for measuring the nine independent orthotropic elastic coefficients from a single small cubic specimen. The side dimensions of this cubic specimen are on the order of 5 mm. Because of the small size of the specimen, the spatial resolution of material inhomogeneity using this technique is quite good. Although it is possible to apply this technique to any elastic material such as woods or metals, the elastic properties of human and canine cortical femora are presented here. The orthotropic elastic coefficients and the variation of these coefficients are presented as a function of anatomical position.