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.     

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.     

Elastic constants of composites formed from PMMA bone cement and anisotropic bovine tibial cancellous bone

An ultrasonic pulse-transit time technique is used to determine the nine orthotropic engineering constants of 32 cement-cancellous bone composites as a function of volume fractions of bone ranging from 0.0 to 0.4. The composites are manufactured using well-aligned bovine cancellous bone from the proximal end of the tibia and low viscosity bone cement. Selected composites are also subjected to mechanical compression tests to compare with the ultrasonic results. There is excellent correlation between the dynamic or ultrasonically determined moduli and the static or mechanically determined moduli; the dynamic moduli are approximately twice the static moduli and this difference is thought to be due to the effect of strain rate. An orthotropic model is assumed requiring nine independent elastic constants to be determined. The dynamic Young's modulus in the direction of major trabecular alignment, E1, increases linearly from 4.9 to 10.4 GPa as bone volume fraction increases from 0 to 0.4; dynamic E2 and E3 values increase from 4.9 to 7 GPa as bone volume fractions increase from 0 to 0.4, with E2 being slightly higher than E3. The dynamic shear modulus, G12, increases from 1.8 to 3.0 GPa, and G31 and G23 increase slightly from 1.8 to 2.2 GPa as bone volume fractions increase from 0 to 0.4. The Poisson's ratios are more sensitive than the Young's moduli and shear moduli to experimental error in the velocity measurements. The mechanically tested modulus (static modulus) in the direction of major trabecular alignment, E1, increases with volume fraction of bone from 2.4 to 4.4 GPa as the bone volume fraction increases from 0 to 0.25; static E2 and E3 values are either equal to or lower than that of pure PMMA.     

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.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

Mechanical properties of porcine femoral cortical bone measured by nanoindentation

This study uses a nanoindentation technique to examine variations in the local mechanical properties of porcine femoral cortical bone under hydrated conditions. Bone specimens from three age groups (6, 12 and 42 months), representing developing bone, ranging from young to mature animals, were tested on the longitudinal and transverse cross-sectional surfaces. Elastic modulus and hardness of individual lamellae within bone's microstructure: laminar bone, interstitial bone, and osteons, were measured. Both the elastic modulus and hardness increased with age. However, the magnitudes of these increases were different for each microstructural component. The longitudinal moduli were higher than the transverse moduli. Dehydrated samples were also tested to allow a comparison with hydrated samples and these resulted in higher moduli and hardness than the hydrated samples. Again, the degree of variation was different for each microstructural component. These results indicate that the developmental changes in bone have different rates of mechanical change within each microstructural component.     

Intrinsic mechanical properties of trabecular calcaneus determined by finite-element models using 3D synchrotron microtomography

To determine intrinsic mechanical properties (elastic and failure) of trabecular calcaneus bone, chosen as a good predictor of hip fracture, we looked for the influence of image's size on a numerical simulation. One cubic sample of cancellous bone (9 x 9 x 9 mm(3)) was removed from the body of the calcaneus (6 females, 6 males, 79+/-9 yr). These samples were tested under compressive loading. Before compressive testing, these samples were imaged at 10.13 microm resolution using a 3D microcomputed tomography (muCT) (ESRF, France). The muCT images were converted to finite-element models. Depending on the bone density values (BV/TV), we compared two different finite element models: a linear hexahedral and a linear beam finite element models. Apparent experimental Young's modulus (E(app)(exp)) and maximum apparent experimental compressive stress (sigma(max)(exp)) were significantly correlated with bone density obtained by Archimedes's test (E(app)(exp)=236+/-231 MPa [19-742 MPa], sigma(max)(exp)=2.61+/-1.97 MPa [0.28-5.81 MPa], r>0.80, p<0.001). Under threshold at 40 microm, the size of the numerical samples (5.18(3) and 6.68(3)mm(3)) seems to be an important parameter on the accuracy of the results. The numerical trabecular Young's modulus was widely higher (E(trabecular)(num)=34,182+/-22,830 MPa [9700-87,211 MPa]) for the larger numerical samples and high BV/TV than those found classically by other techniques (4700-15,000 MPa). For rod-like bone samples (BV/TV<12%, n=7), Young's modulus, using linear beam element (E(trabecular)(num-skeleton): 10,305+/-5500 MPa), were closer to the Young's modulus found by other techniques. Those results show the limitation of hexahedral finite elements at 40 microm, mostly used, for thin trabecular structures.     

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.     

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.     

Elasticity of alveolar bone near dental implant-bone interfaces after one month's healing

Information is scarce about Young's modulus of healing bone surrounding an implant. The purpose of this preliminary study is to quantify elastic properties of pig alveolar bone that has healed for 1 month around titanium threaded dental implants, using the nanoindentation method. Two 2-year-old Sinclair miniswine were used for the study. Nanoindentation tests perpendicular to the bucco-lingual cross section were performed on harvested implant-bone blocks using the Hysitron TriboScope III. Nomarski differential interference contrast microscopy was used to identify pyramidal indentation measurements that were from bone. Reduced moduli, averaged for all anatomical regions, were found to start low (6.17 GPa) at the interface and gradually increase (slope=0.014) to a distance of 150 microm (7.89 GPa) from the implant surface, and then flatten to a slope of 0.001 from 150 to 1500 microm (10.13 GPa). Mean reduced modulus and its relationship to distance did not differ significantly by anatomic location (e.g., coronal, middle, and apical third; P>/=0.28 for all relevant tests) at 1 month after implantation.     

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 comparison of the fatigue behavior of human trabecular and cortical bone tissue

The fatigue properties of trabecular bone tissue (single trabeculae) and similarly sized cortical bone specimens from human tibia were experimentally determined on a microstructural level using four-point bending cyclic tests, and they were compared based on modulus, mineral density, and microstructural characteristics. The results showed that trabecular specimens had significantly lower moduli and lower fatigue strength than cortical specimens, despite their higher mineral density values. Fracture surface and microdamage analyses illustrated different fracture and damage patterns between trabecular and cortical bone tissue, depending upon their microstructural characteristics. Based on the results from mechanical tests and qualitative observations, a possible mechanical role of the cement lines in trabecular tissue microfracture was suggested.     

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.     

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.     

Errors induced by off-axis measurement of the elastic properties of bone

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.     

In vivo measurement of age-related stiffening in the crystalline lens by Brillouin optical microscopy

The biophysical and biomechanical properties of the crystalline lens (e.g., viscoelasticity) have long been implicated in accommodation and vision problems, such as presbyopia and cataracts. However, it has been difficult to measure such parameters noninvasively. Here, we used in vivo Brillouin optical microscopy to characterize material acoustic properties at GHz frequency and measure the longitudinal elastic moduli of lenses. We obtained three-dimensional elasticity maps of the lenses in live mice, which showed biomechanical heterogeneity in the cortex and nucleus of the lens with high spatial resolution. An in vivo longitudinal study of mice over a period of 2 months revealed a marked age-related stiffening of the lens nucleus. We found remarkably good correlation (log-log linear) between the Brillouin elastic modulus and the Young's modulus measured by conventional mechanical techniques at low frequencies (~1 Hz). Our results suggest that Brillouin microscopy is potentially useful for basic and animal research and clinical ophthalmology.     

The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus

The elastic moduli of human subchondral, trabecular, and cortical bone tissue from a proximal tibia were experimentally determined using three-point bending tests on a microstructural level. The mean modulus of subchondral specimens was 1.15 GPa, and those of trabecular and cortical specimens was 4.59 GPa and 5.44 GPa respectively. Significant differences were found in the modulus values between bone tissues, which may have mainly resulted from the differences in the microstructures of each bone tissue rather than in the mineral density. Furthermore, the size-dependency of the modulus was examined using eight different sizes of cortical specimens (heights h = 100-1000 microns). While the modulus values for relatively large specimens (h greater than 500 microns) remained fairly constant (approximately 15 GPa), the values decreased as the specimens became smaller. A significant correlation was found between the modulus and specimen size. The surface area to volume ratio proved to be a key variable to explain the size-dependency.     

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.     

An application of nanoindentation technique to measure bone tissue Lamellae properties

Measuring the microscopic mechanical properties of bone tissue is important in support of understanding the etiology and pathogenesis of many bone diseases. Knowledge about these properties provides a context for estimating the local mechanical environment of bone related cells thait coordinate the adaptation to loads experienced at the whole organ level. The objective of this study was to determine the effects of experimental testing parameters on nanoindentation measures of lamellar-level bone mechanical properties. Specifically, we examined the effect of specimen preparation condition, indentation depth, repetitive loading, time delay, and displacement rate. The nanoindentation experiments produced measures of lamellar elastic moduli for human cortical bone (average value of 17.7 +/- 4.0 GPa for osteons and 19.3 +/- 4.7 GPa for interstitial bone tissue). In addition, the hardness measurements produced results consistent with data in the literature (average 0.52 +/- 0.15 GPa for osteons and 0.59 +/- 0.20 GPa for interstitial bone tissue). Consistent modulus values can be obtained from a 500-nm-deep indent. The results also indicated that the moduli and hardnesses of the dry specimens are significantly greater (22.6% and 56.9%, respectively) than those of the wet and wet and embedded specimens. The latter two groups were not different. The moduli obtained at a 5-nm/s loading rate were significantly lower than the values at the 10- and 20-nm/s loading rates while the 10- and 20-nm/s rates were not significantly different. The hardness measurements showed similar rate-dependent results. The preliminary results indicated that interstitial bone tissue has significantly higher modulus and hardness than osteonal bone tissue. In addition, a significant correlation between hardness and elastic modulus was observed.     

A method for measuring linearly viscoelastic properties of human tympanic membrane using nanoindentation

A viscoelastic nanoindentation technique was developed to measure both in-plane and through-thickness viscoelastic properties of human tympanic membrane (TM). For measurement of in-plane Young's relaxation modulus, the TM sample was clamped on a circular hole and a nanoindenter tip was used to apply a concentrated force at the center of the TM sample. In this setup, the resistance to nanoindentation displacement can be considered due primarily to the in-plane stiffness. The load-displacement curve obtained was used along with finite element analysis to determine the in-plane viscoelastic properties of TM. For measurements of Young's relaxation modulus in the through-thickness (out-of-plane) direction, the TM sample was placed on a relatively rigid solid substrate and nanoindentation was made on the sample surface. In this latter setup, the resistance to nanoindentation displacement arises primarily due to out-of-plane stiffness. The load-displacement curve obtained in this manner was used to determine the out-of-plane relaxation modulus using the method appropriate for viscoelastic materials. From our sample tests, we obtained the steady-state values for in-plane moduli as approximately 17.4 MPa and approximately 19.0 MPa for posterior and anterior portions of TM samples, respectively, and the value for through-thickness modulus as approximately 6.0 MPa for both posterior and anterior TM samples. Using this technique, the local out-of-plane viscoelastic modulus can be determined for different locations over the entire TM, and the in-plane properties can be determined for different quadrants of the TM.