The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques

Acoustic microscopy (30-60 microm resolution) and nanoindentation (1-5 microm resolution) are techniques that can be used to evaluate the elastic properties of human bone at a microstructural level. The goals of the current study were (1) to measure and compare the Young's moduli of trabecular and cortical bone tissues from a common human donor, and (2) to compare the Young's moduli of bone tissue measured using acoustic microscopy to those measured using nanoindentation. The Young's modulus of cortical bone in the longitudinal direction was about 40% greater than (p<0.01) the Young's modulus in the transverse direction. The Young's modulus of trabecular bone tissue was slightly higher than the transverse Young's modulus of cortical bone, but substantially lower than the longitudinal Young's modulus of cortical bone. These findings were consistent for both measurement methods and suggest that elasticity of trabecular tissue is within the range of that of cortical bone tissue. The calculation of Young's modulus using nanoindentation assumes that the material is elastically isotropic. The current results, i.e., the average anisotropy ratio (E(L)/E(T)) for cortical bone determined by nanoindentation was similar to that determined by the acoustic microscope, suggest that this assumption does not limit nanoindentation as a technique for measurement of Young's modulus in anisotropic bone.     

Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur.

The mechanical properties of bone tissue are determined by composition as well as structural, microstructural and nanostructural organization. The aim of this study was to quantify the elastic properties of bone at the lamellar level and compare these properties among osteonal, interstitial and trabecular microstructures from the diaphysis and the neck of the human femur. A nanoindentation technique with a custom irrigation system was used for simultaneously measuring force and displacement of a diamond tip pressed 500 nm into the moist bone tissue. An isotropic elastic modulus was calculated from the unloading curve with an assumed Poisson ratio of 0.3, while hardness was defined as the maximal force divided by the corresponding contact area. The elastic moduli ranged from 6.9 +/- 4.3 GPa in trabecular tissue from the femoral neck of a 74 yr old female up to 25.0 +/- 4.3 GPa in interstitial tissue from the diaphyseal cortex of a 69 yr old female. The mean elastic modulus was found to be significantly influenced by the type of lamella (p < 10(-6)) and by donor (p < 10(-6)). The interaction between the type of lamella and the donor was also highly significant (p < 10(-6)). Hardness followed a similar distribution as elastic modulus among types of lamellae and donor, but with lower statistical contrast. It is concluded that the nanostructure of bone tissue must differ substantially among lamellar types, anatomical sites and individuals and suggests that tissue heterogeneity is of potential importance in bone fragility and adaptation.     

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.     

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.     

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 nu, 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.     

Multi-scale characterization of swine femoral cortical bone

Multi-scale experimental work was carried out to characterize cortical bone as a heterogeneous material with hierarchical structure, which spans from nanoscale (mineralized collagen fibril), sub-microscale (single lamella), microscale (lamellar structures), to mesoscale (cortical bone) levels. Sections from femoral cortical bone from 6, 12, and 42 months old swine were studied to quantify the age-related changes in bone structure, chemical composition, and mechanical properties. The structural changes with age from sub-microscale to mesoscale levels were investigated with scanning electron microscopy and micro-computed tomography. The chemical compositions at mesoscale were studied by ash content method and dual energy X-ray absorptiometry, and at microscale by Fourier transform infrared microspectroscopy. The mechanical properties at mesoscale were measured by tensile testing, and elastic modulus and hardness at sub-microscale were obtained using nanoindentation. The experimental results showed age-related changes in the structure and chemical composition of cortical bone. Lamellar bone was a prevalent structure in 6 months and 12 months old animals, resorption sites were most pronounced in 6 months old animals, while secondary osteons were the dominant features in 42 months old animals. Mineral content and mineral-to-organic ratio increased with age. The structural and chemical changes with age corresponded to an increase in local elastic modulus, and overall elastic modulus and ultimate tensile strength as bone matured.     

Elastic modulus variation in mandibular bone: a microindentation study of Macaca fascicularis

We characterized the heterogeneous anisotropic elastic properties of mandibular bone in an adult female specimen of Macaca fascicularis using the technique of microindentation. This approach used an indenter of known mass and geometry to sample bone hardness at a spatial resolution in the order of 100 mum. Hardness values were converted to elastic modulus using empirically derived regression. We determined properties in alveolar, midcorpus, and basal regions of coronal and transverse sections taken from multiple locations along the corpus and ramus. Within sections, we determined properties from endosteal, midcortical, and periosteal regions. We found regional variations in bone structure, including bands of orthotropic circumferential lamellar bone at the endosteal and periosteal corpus base, angular region, and ramus. Transversely isotropic osteonal bone characterizes the midcortices of alveolar and basal regions, with many resorption spaces in alveolar regions restricting sampling opportunities. Regional variations in elasticity include relatively compliant bone in the anterior corpus and ramus. Basal cortical bone is stiffer longitudinally than transversely or superoinferiorly, while the evidence for directional dependence in alveolar bone is equivocal. Alveolar bone appears to be relatively compliant with respect to bone found in midcorpus or basal regions. Considerable variation exists in structure and material properties on a highly localized scale, more so than is discernible through conventional approaches for determining material property variation.     

Transversely isotropic elasticity imaging of cancellous bone

To measure spatial variations in mechanical properties of biological materials, prior studies have typically performed mechanical tests on excised specimens of tissue. Less invasive measurements, however, are preferable in many applications, such as patient-specific modeling, disease diagnosis, and tracking of age- or damage-related degradation of mechanical properties. Elasticity imaging (elastography) is a nondestructive imaging method in which the distribution of elastic properties throughout a specimen can be reconstructed from measured strain or displacement fields. To date, most work in elasticity imaging has concerned incompressible, isotropic materials. This study presents an extension of elasticity imaging to three-dimensional, compressible, transversely isotropic materials. The formulation and solution of an inverse problem for an anisotropic tissue subjected to a combination of quasi-static loads is described, and an optimization and regularization strategy that indirectly obtains the solution to the inverse problem is presented. Several applications of transversely isotropic elasticity imaging to cancellous bone from the human vertebra are then considered. The feasibility of using isotropic elasticity imaging to obtain meaningful reconstructions of the distribution of material properties for vertebral cancellous bone from experiment is established. However, using simulation, it is shown that an isotropic reconstruction is not appropriate for anisotropic materials. It is further shown that the transversely isotropic method identifies a solution that predicts the measured displacements, reveals regions of low stiffness, and recovers all five elastic parameters with approximately 10% error. The recovery of a given elastic parameter is found to require the presence of its corresponding strain (e.g., a deformation that generates ??? is necessary to reconstruct C????), and the application of regularization is shown to improve accuracy. Finally, the effects of noise on reconstruction quality is demonstrated and a signal-to-noise ratio (SNR) of 40 dB is identified as a reasonable threshold for obtaining accurate reconstructions from experimental data. This study demonstrates that given an appropriate set of displacement fields, level of regularization, and signal strength, the transversely isotropic method can recover the relative magnitudes of all five elastic parameters without an independent measurement of stress. The quality of the reconstructions improves with increasing contrast, magnitude of deformation, and asymmetry in the distributions of material properties, indicating that elasticity imaging of cancellous bone could be a useful tool in laboratory studies to monitor the progression of damage and disease in this tissue.     

The bone tissue of the canine mandible is elastically isotropic

This paper reports experimental measurements which show that canine mandibular bone tissue is elastically isotropic. Earlier work has established that human, canine and bovine cortical bone tissue of the femur, tibia and skull are elastically anisotropic and therefore the reported isotropy of mandibular tissue was unexpected. The isotropic elastic moduli of the canine mandible are represented by a Young's modulus of 7.5 GPa and a Poisson's ratio of 0.4. Earlier work gave the three orthotropic Young's moduli of the cortical one of the canine femur as 12.8 GPa, 15.6 GPa and 20.1 GPa. The experimental technique employed is elastic wave propagation at ultrasonic frequencies.     

Mechanics of linear microcracking in trabecular bone

Microcracking in trabecular bone is responsible both for the mechanical degradation and remodeling of the trabecular bone tissue. Recent results on trabecular bone mechanics have demonstrated that bone tissue microarchitecture, tissue elastic heterogeneity and tissue-level mechanical anisotropy all should be considered to obtain detailed information on the mechanical stress state. The present study investigated the influence of tissue microarchitecture, tissue heterogeneity in elasticity and material separation properties and tissue-level anisotropy on the microcrack formation process. Microscale bone models were executed with the extended finite element method. It was demonstrated that anisotropy and heterogeneity of the bone tissue contribute significantly to bone tissue toughness and the resistance of trabecular bone to microcrack formation. The compressive strain to microcrack initiation was computed to increase by a factor of four from an assumed homogeneous isotropic tissue to an assumed anisotropic heterogenous tissue.     

Mapping anisotropy of the proximal femur for enhanced image based finite element analysis

Finite element (FE) models of bone derived from quantitative computed tomography (QCT) rely on realistic material properties to accurately predict bone strength. QCT cannot resolve bone microarchitecture, therefore QCT-based FE models lack the anisotropy apparent within the underlying bone tissue. This study proposes a method for mapping femoral anisotropy using high-resolution peripheral quantitative computed tomography (HR-pQCT) scans of human cadaver specimens. Femur HR-pQCT images were sub-divided into numerous overlapping cubic sub-volumes and the local anisotropy was quantified using a ‘direct-mechanics’ method. The resulting directionality reflected all the major stress lines visible within the trabecular lattice, and provided a realistic estimate of the alignment of Harvesian systems within the cortical compartment. QCT-based FE models of the proximal femur were constructed with isotropic and anisotropic material properties, with directionality interpolated from the map of anisotropy. Models were loaded in a sideways fall configuration and the resulting whole bone stiffness was compared to experimental stiffness and ultimate strength. Anisotropic models were consistently less stiff, but no statistically significant differences in correlation were observed between material models against experimental data. The mean difference in whole bone stiffness between model types was approximately 26%, suggesting that anisotropy can still effect considerable change in the mechanics of proximal femur models. The under prediction of whole bone stiffness in anisotropic models suggests that the orthotropic elastic constants require further investigation. The ability to map mechanical anisotropy from high-resolution images and interpolate information into clinical-resolution models will allow testing of new anisotropic material mapping strategies.     

Muscle sound frequencies of the frog are modulated by skeletal muscle tension.

The purpose of this study was to determine the relationships among muscle sound frequencies, muscle tension, and stiffness. Time-frequency transformations of nonstationary acoustic signals provided measures of resonant frequency during isometric contractions of frog (Rana pipiens) semitendinosus and gastrocnemius muscles. A mathematical expression for muscle transverse resonant frequency, elastic modulus and tension, based on elastic beam theory, was formulated by the Rayleigh method adapted for muscles. For thin muscles, the elastic modulus was found to have negligible influence on transverse muscle resonant frequency. Changes in muscle tension were the major determinants of changes in transverse resonant frequency. Consequently, for thin muscles, the time course of muscle tension, but not elastic modulus, can be monitored acoustically during the early phase of contraction when muscles give rise to sounds. Muscles were found to be anisotropic with a modulus of elasticity, EL, measured via length perturbations near 0.1% muscle length peak-to-peak, that was much larger than the modulus of elasticity, Eb, that resists the lateral bending that causes sound production. The elastic and resonant behavior of a thin muscle is similar to a tensioned fibrous cable with distributed mass.     

Mechano-acoustic determination of Young's modulus of articular cartilage

The compressive stiffness of an elastic material is traditionally characterized by its Young's modulus. Young's modulus of articular cartilage can be directly measured using unconfined compression geometry by assuming the cartilage to be homogeneous and isotropic. In isotropic materials, Young's modulus can also be determined acoustically by the measurement of sound speed and density of the material. In the present study, acoustic and mechanical techniques, feasible for in vivo measurements, were investigated to quantify the static and dynamic compressive stiffness of bovine articular cartilage in situ. Ultrasound reflection from the cartilage surface, as well as the dynamic modulus were determined with the recently developed ultrasound indentation instrument and compared with the reference mechanical and ultrasound speed measurements in unconfined compression (n=72). In addition, the applicability of manual creep measurements with the ultrasound indentation instrument was evaluated both experimentally and numerically. Our experimental results indicated that the sound speed could predict 47% and 53% of the variation in the Young's modulus and dynamic modulus of cartilage, respectively. The dynamic modulus, as determined manually with the ultrasound indentation instrument, showed significant linear correlations with the reference Young's modulus (r(2)=0.445, p<0.01, n=70) and dynamic modulus (r(2)=0.779, p<0.01, n=70) of the cartilage. Numerical analyses indicated that the creep measurements, conducted manually with the ultrasound indentation instrument, were sensitive to changes in Young's modulus and permeability of the tissue, and were significantly influenced by the tissue thickness. We conclude that acoustic parameters, i.e. ultrasound speed and reflection, are indicative to the intrinsic mechanical properties of the articular cartilage. Ultrasound indentation instrument, when further developed, provides an applicable tool for the in vivo detection of cartilage mechano-acoustic properties. These techniques could promote the diagnostics of osteoarthrosis.     

The impact of the parameters of the constitutive model on the distribution of strain in the femoral head

The rapid spread of the finite element method has caused that it has become, among other methods, the standard tool for pre-clinical estimates of bone properties. This paper presents an application of this method for the calculation and prediction of strain and stress fields in the femoral head. The aim of the work is to study the influence of the considered anisotropy and heterogeneity of the modeled bone on the mechanical fields during a typical gait cycle. Three material models were tested with different properties of porous bone carried out in literature: a homogeneous isotropic model, a heterogeneous isotropic model, and a heterogeneous anisotropic model. In three cases studied, the elastic properties of the bone were determined basing on the Zysset-Curnier approach. The tensor of elastic constants defining the local properties of porous bone is correlated with a local porosity and a second order fabric tensor describing the bone microstructure. In the calculations, a model of the femoral head generated from high-resolution tomographic scans was used. Experimental data were drawn from publicly available database “Osteoporotic Virtual Physiological Human Project.” To realistically reflect the load on the femoral head, main muscles were considered, and their contraction forces were determined based on inverse kinematics. For this purpose, the results from OpenSim packet were used. The simulations demonstrated that differences between the results predicted by these material models are significant. Only the anisotropic model allowed for the plausible distribution of stresses along the main trabecular groups. The outcomes also showed that the precise evaluation of the mechanical fields is critical in the context of bone tissue remodeling under mechanical stimulations.

     

The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone.

Conceptually, the elastic characteristics of cancellous bone could be predicted directly from the trabecular morphology--or architecture--and by the elastic properties of the tissue itself. Although hardly any experimental evidence exists, it is often implicitly assumed that tissue anisotropy has a negligible effect on the apparent elastic properties of cancellous bone. The question addressed in this paper is whether this is actually true. If it is, then micromechanical finite element analysis (micro-FEA) models, representing trabecular architecture, using an 'effective isotropic tissue modulus' should be able to predict apparent elastic properties of cancellous bone. To test this, accurate multi-axial compressive mechanical tests of 29 whale bone specimens were simulated with specimen-specific micro-FEA computer models built from true three-dimensional reconstructions. By scaling the micro-FEA predictions by a constant tissue modulus, 92% of the variation of Young's moduli determined experimentally could be explained. The correlation even increased to 95% when the micro-FEA moduli were scaled to the isotropic tissue moduli of individual specimens. Excellent agreement was also found in the elastic symmetry axes and anisotropy ratios. The prediction of Poisson's ratios was somewhat less precise at 85% correlation. The results support the hypothesis; for practical purposes, the concept of an 'effective isotropic tissue modulus' concept is a viable one. They also suggest that the value of such a modulus for individual cases might be inferred from the average tissue density, hence the degree of mineralization. Future studies must clarify how specific the tissue modulus should be for different types of bone if adequate predictions of elastic behavior are to be made in this way.     

Tensile experiments and SEM fractography on bovine subchondral bone

Subchondral bone undecalcified samples, extracted from bovine femoral heads, are subjected to a direct tensile load. The Young's modulus of each sample is determined from repeated tests within the elastic limit. In a last test, the tensile load is increased up to the specimen failure, determining the ultimate tensile strength. The investigation is performed on both dry and wet specimens. The measured Young's modulus for dry samples is 10.3+/-2.5GPa, while that of wet samples is 3.5+/-1.2GPa. The ultimate tensile strengths are 36+/-10 and 30+/-7.5MPa for dry and wet specimens, respectively. SEM micrographs of failure surfaces show characteristic lamellar bone structures, with lamellae composed of calcified collagen fibers. Rudimentary osteon-like structures are also observed. Failure surfaces of wet samples show a marked fiber pull-out, while delamination predominates in dry samples. The obtained results are interpreted on the basis of the deformation mechanisms typical of fiber-reinforced laminated composite materials.     

Consideration of anisotropic elasticity minimizes volumetric rather than shear deformation in human mandible

This article is focused on the role of anisotropic elasticity in the simulation of the load distribution in a human mandible, due to a lateral bite on the leftmost premolar. Based on experimental evidence, orthotropy of the elastic properties of the bone tissue has been adopted. The trajectories of anisotropic elasticity are reconstructed from (i) the organ's geometry and (ii) from coherent structures which can be recognized from the spatial distribution of the grey values coming from computer tomography (CT). A sensitivity analysis comprising various three-dimensional (3D) finite element (FE) simulations reveals the relevance of elastic anisotropy for the load carrying behavior of a human mandible: comparison of the load distributions in isotropic and anisotropic simulations indicates that anisotropy seems to "spare" the mandible from loading. Moreover, a maximum degree of anisotropy leads to kind of load minimization of the mandible, expressed by a minimum of different norms of local volumetric strain, evaluated throughout the organ. The observed optimization with respect to volumetric rather than shear strain seems to confirm the frequently emphazised role of volumetric-strain-induced fluid flow for the stimulation of cellular activity.     

Isotropy and anisotropy of the arterial wall

The passive biomechanical response of intact cylindrical rat carotid arteries is studied in vitro and compared with the mechanical response of rubber tubes. Using true stress and natural strain in the definition of the incremental modulus of elasticity, the tissue wall properties are analyzed over wide ranges of simultaneous circumferential and longitudinal deformations. The type of loading chosen is 'physiological' i.e. symmetric: the cylindrical segments are subjected to internal pressure and axial prestretch without torsion or shear. Several aspects pertaining to the choice of parameters characterizing the material are discussed and the analysis pertaining to the deformational behavior of a hypothetical compliant tube with Hookean wall material is presented. The experimental results show that while rubber response can be adequately represented as linearly elastic and isotropic, the overall response of vascular tissue is highly non-linear and anisotropic. However, for states of deformation that occur in vivo, the elasticity of arteries is quite similar to that of rubber tubes and as such the arterial wall may be viewed as incrementally isotropic for the range of deformations that occur in vivo.     

Estimation of the effective transversely isotropic elastic constants of a material from known values of the material's orthotropic elastic constants

A method is illustrated for determining the effective transversely isotropic (or isotropic) elastic constants from measured orthotropic elastic constants. This method consists of constructing upper and lower bounds on the effective transversely isotropic (or isotropic) elastic constants using the known orthotropic values. This method is illustrated using three sets of elastic constants for bone. Fortunately, the upper and lower bounds are very close. Thus very good approximations for the effective transversely isotropic (or isotropic) elastic constants for cortical and cancellous bone are obtained from previously published data on the orthotropic elastic constants for those tissue types. This work is undertaken to build a greater database for the transversely isotropic elastic constants of bone with the intention of employing them in a transversely isotropic model of bone poroelasticity. An interesting aspect of the present result is that the Voigt and Reuss bounds are very tight for these anisotropic materials. This is not always the case for these bounds. Received: 14 November 2001 / Accepted: 25 February 2002