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.     

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 moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur

Elastic moduli, yield stress and ultimate compressive stress were determined for cancellous bone from the femoral head and neck regions of the canine femur. Unconfined compression tests were performed on 5 mm cubic samples which were cut from two femurs. Elastic moduli were measured in three orthogonal directions, and the yield stress and ultimate stress were measured along the proximal-distal axis. The results from this investigation support previous assumptions that the mechanical behavior of canine cancellous bone is qualitatively similar to human cancellous bone. The canine cancellous bone was observed to be anisotropic in elastic modulus. For two thirds of the cubic specimens tested, the elastic modulus was largest in the load-bearing, proximal-distal direction. A linear relationship between yield stress and elastic modulus was observed for canine bone, as is typical of human bone. A similar linear relationship between ultimate stress and elastic modulus was observed. Thus, for canine bone as well as for human bone, failure appears to be governed by a strain level which is position independent. The yield strain of 0.0259 and ultimate strain of 0.0288 for canine bone were both less than the yield strain of 0.0395 reported for human bone.     

Modeling modulus reduction in bovine trabecular bone damaged in compression.

Loading bone beyond its yield point creates microdamage, leading to reduction in stiffness. Previously, we related microdamage accumulation to changes in mechanical properties. Here, we develop a model that predicts stiffness loss based on the presence of microdamage. Modeling is done at three levels: (1) a single trabecula, (2) a cellular solid consisting of intact, damaged, and fractured trabeculae, and (3) a specimen with a localized damage band. Predictions of a reduced modulus agree well with experimental measured modulus reductions of post-yield compression of bovine trabecular bone. The predicted reduced modulus is relatively insensitive to changes in the input parameters.     

Contribution, development and morphology of microcracking in cortical bone during crack propagation

A fracture mechanics study of cortical bone is presented to investigate the contribution, development morphology of microcracking in cortical bone during crack propagation. Post-hoc analyses of microcrack orientation, crack propagation velocity and fracture surface roughness were conducted on previously tested human and bovine bone compact tension specimens. It was found that, consistent with its higher toughness, bovine bone formed significantly more longitudinal, transverse and inclined microcracks than human bone. However, in human bone more of the microcracks that formed were longitudinal than transverse or inclined, a feature that would optimise bone's toughness. Crack propagation velocity in human and bovine bone displayed the same characteristic pattern with crack extension, where an increase in velocity is followed by a consequent decrease and vice versa. On the basis of this pattern, a model or crack propagation has been proposed. It provides a detailed account of mocrocrack formation and contribution towards the propagation of a fracture crack. Analyses of fracture surfaces indicated that, consistent with its higher toughness, bovine bone displays a rougher surface than human bone but they both have the same basic fractured element, i.e. a mineralised collagen fibril.     

Nonlinear behavior of trabecular bone at small strains

Study of the behavior of trabecular bone at strains below 0.40 percent is of clinical and biomechanical importance. The goal of this work was to characterize, with respect to anatomic site, loading mode, and apparent density, the subtle concave downward stress-strain nonlinearity, that has been observed recently for trabecular bone at these strains. Using protocols designed to minimize end-artifacts, 155 cylindrical cores from human vertebrae, proximal tibiae, proximal femora, and bovine proximal tibiae were mechanically tested to yield at 0.50 percent strain per second in tension or compression. The nonlinearity was quantified by the reduction in tangent modulus at 0.20 percent and 0.40 percent strain as compared to the initial modulus. For the pooled data, the mean +/- SD percentage reduction in tangent modulus at 0.20 percent strain was 9.07+/- 3.24 percent in compression and 13.8 +/- 4.79 percent in tension. At 0.40 percent strain, these values were 23.5 +/- 5.71 and 35.7+/- 7.10 percent, respectively. The magnitude of the nonlineari't depended on both anatomic site (p < 0.001) and loading mode (p < 0.001), and in tension was positively correlated with density. Calculated values of elastic modulus and yield properties depended on the strain range chosen to define modulus via a linear curve fit (p < 0.005). Mean percent differences in 0.20 percent offset yield strains were as large as 10.65 percent for some human sites. These results establish that trabecular bone exhibits nonlinearity at low strains, and that this behavior can confound intersite comparisons of mechanical properties. A nonlinear characterization of the small strain behavior of trabecular bone was introduced to characterize the initial stress-strain behavior more thoroughly.     

Virus inactivation in bone tissue transplants (femoral heads) by moist heat with the 'Marburg bone bank system'

Several virus inactivation procedures like heat treatment, gamma irradiation and chemical sterilization are used to increase the safety of bone tissue transplants. In this study we present data on the virus-inactivating effect of heat disinfection on human femoral heads, using the Marburg bone bank system 'Lobator sd-2'. Three enveloped viruses (human immunodeficiency virus type 2 [HIV-2], bovine viral diarrhoea virus as a model for Hepatitis C virus [HCV], and the herpesvirus pseudorabies virus), and three non-enveloped viruses (hepatitis A virus, poliomyelitis virus, and bovine parvovirus) were investigated.In a model system the central part of human femoral heads was contaminated with the respective cell-free virus suspension, establishing a direct contact between virus and native bone tissue. The core temperature in the femoral heads during the sterilization process was determined in additional model experiments. A temperature of 82.5 degrees C, given by the manufacturer as the effective temperature for virus inactivation, was maintained for at least 15 min in decartilaged femoral heads with a diameter of < or = 56 mm. Heat treatment using the Lobator sd-2 inactivated all viruses in human femoral heads below the detection limit (at least by a factor of > or =4 log(10)).By combining a well-focussed anamnesis of the donors and serological testing for relevant infection markers (anti-HIV-1/2, HBsAg, anti-HBcore, anti-HCV, TPHA) with heat treatment of femoral heads in the Lobator sd-2 system, a high safety level is achieved. To further increase virus safety of cadaveric bone transplants, it is recommended that multi-organ donors are tested by nucleic acid testing (i.e. polymerase chain reaction) for HIV, HBV and HCV genome.     

Anatomical variation of orthotropic elastic moduli of the proximal human tibia

The anatomical variation of orthotropic elastic moduli of the cancellous bone from three human proximal tibiae was investigated using an ultrasonic technique. With this technique, it was possible to measure three orthogonal elastic moduli and three shear moduli from cubic specimens of cancellous bone as small as 8 mm per side. Correlation with mechanical tensile testing has shown this technique to offer a precise measure of cancellous modulus (Eten = 0.94Eult + 144.6 MPa, r2 = 0.96, n = 34). The cancellous bone of the proximal tibia was found to be very inhomogeneous, with the axial modulus ranging between 340 and 3350 MPa. A course map is presented, showing measured Young's moduli as a function of anatomical position. The anisotropy of the cancellous bone, determined by the relative differences between the three orthogonal moduli, was shown to be relatively constant over the entire range of cancellous densities tested. The relationship between the axial elastic modulus and the apparent density was found to be approximately linear, as reported by others for proximal tibial cancellous bone.     

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.     

Static and dynamic finite element analyses of an idealized structural model of vertebral trabecular bone.

An idealized three-dimensional finite element model of a rodlike trabecular bone structure was developed to study its static and dynamic responses under compressive loading, considering the effects of bone marrow and apparent density. Static analysis of the model predicted hydraulic stiffening of trabecular bone due to the presence of bone marrow. The predicted power equation relating trabecular bone apparent elastic modulus to its apparent density was in good agreement with those of the reported experimental investigations. The ratio of the maximum stress in the trabecular bone tissue to its apparent stress had a high value, decreasing with increasing bone apparent density. Frequency analyses of the model predicted higher natural frequencies for the bone without marrow than those for the bone with marrow. Adding a mass relatively large compared to that of bone rendered a single-degree-of-freedom response. In this case, the resonant frequency was higher for the bone with marrow than that for the bone without marrow. The predicted vibrational measurement of apparent modulus was in good agreement with that of the static measurement, suggesting vibrational testing as a method for nondestructive measurement of trabecular bone elastic moduli.     

Similarity in the fatigue behavior of trabecular bone across site and species

Within the context of improving knowledge of the structure-function relations for trabecular bone for cyclic loading, we hypothesized that the S-N curve for cyclic compressive loading of trabecular bone, after accounting for differences in monotonic strength behavior, does not depend on either site or species. Thirty-five cores of fresh-frozen elderly human vertebral trabecular bone, harvested from nine donors (mean+/-S.D., age=74+/-17 years), were biomechanically tested in compression at sigma/E(0) values (ratio of applied stress to pre-fatigue elastic modulus) ranging from 0.0026 to 0.0070, and compared against literature data (J. Biomech. Eng. 120 (1998) 647-654) for young bovine tibial trabecular bone (n=37). As reported for the bovine bone, the number of cycles to failure for the human vertebral bone was related to sigma/E(0) by a power-law relation (r(2)=0.54, n=35). Quantitative comparison of these data against those reported for the bovine bone supported our hypothesis. Namely, when the differences in mean monotonic yield strain between the two types of bone were accounted for, a single S-N curve worked well for the pooled data (r(2)=0.75, n=72). Since elderly human vertebral and young bovine tibial trabecular bone represent two very different types of trabecular bone in terms of volume fraction and architecture, these findings suggest that the dominant failure mechanisms in trabecular bone for cyclic loading occur at the ultrastructural level.     

Sensitivity of multiple damage parameters to compressive overload in cortical bone

Damage accumulation plays a key role in weakening bones prior to complete fracture and in stimulating bone remodeling. The goal of this study was to characterize the degradation in the mechanical properties of cortical bone following a compressive overload. Longitudinally oriented, low-aspect ratio specimens (n=24) of bovine cortical bone were mechanically tested using an overload-hold-reload protocol. No modulus reductions greater than 5% were observed following overload magnitudes less than 0.73% strain. For each specimen, changes in strength and Poisson's ratio were greater (p=0.02) than that in modulus by 10.8- and 26.6-fold, respectively, indicating that, for the specimen configuration used in this study, longitudinal elastic modulus is one of the least sensitive properties to a compressive overload. Residual strains were also proportionately greater by 6.4-fold (p=0.01) in the transverse than axial direction. These results suggest that efforts to relate microcrack density and morphology to changes in compressive mechanical properties of cortical bone may benefit from considering alternative parameters to modulus reductions.     

The dynamic shear modulus of bone in dependence on the form.

Torsional testing of fresh cortical bovine bone showed highly divergent shear moduli at various sites in the middle part of the diaphysis of the right metacarpus. Two hundred samples taken from twenty-five bones were investigated with a free oscillating torsional pendulum to determine the dynamic shear modulus. The results clearly demonstrate the mutual dependence between the shape of the metacarpus and the modulus of the material. On the basis of the analysis of forty-eight samples, a positive correlation was found between the amount of Ca2+ and PO4(3)-ions in the bone and the shear modulus. The hypothesis is put forward that in bone, normal adaptation to increasing stress is obtained by a change in the modulus values, Wolff's law only being operative for extreme cases.     

Tensile and compressive behaviour of the bovine periodontal ligament

The mechanical response of the bovine periodontal ligament (PDL) subjected to uniaxial tension and compression is reported. Several sections normal to the longitudinal axis of bovine incisors and molars were extracted from different depths. Specimens with dimensions 10 x 5 x 2 mm including dentine, PDL and alveolar bone were obtained from these sections. Scanning electron microscopy suggested a strong similarity between the bovine PDL and the human PDL microstructure described in the literature. The prepared specimens were tested in a custom made uniaxial testing machine. They were clamped on their bone and dentine extremities and immersed in a saline solution at 37 degrees C. Stress-strain curves indicated that the PDL is characterized by a non-linear and time-dependent mechanical behaviour with the typical features of collagenous soft tissues. The curves exhibited hysteresis and preconditioning effects. The mechanical parameters evaluated in tension were maximum tangent modulus, strength, maximizer strain and strain energy density. For the molars, all these parameters increased with depth except for the apical region. For the incisors, all parameters increased with depth except ultimate strain which decreased. It was assumed that collagen fibre density and orientation were responsible for these findings.     

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.     

Stochastic multi-scale prediction on the apparent elastic moduli of trabecular bone considering uncertainties of biological apatite (BAp) crystallite orientation and image-based modelling

An assessment of the mechanical properties of trabecular bone is important in determining the fracture risk of human bones. Many uncertainty factors contribute to the dispersion of the estimated mechanical properties of trabecular bone. This study was undertaken in order to propose a computational scheme that will be able to predict the effective apparent elastic moduli of trabecular bone considering the uncertainties that are primarily caused by image-based modelling and trabecular stiffness orientation. The effect of image-based modelling which focused on the connectivity was also investigated. A stochastic multi-scale method using a first-order perturbation-based and asymptotic homogenisation theory was applied to formulate the stochastically apparent elastic properties of trabecular bone. The effective apparent elastic modulus was predicted with the introduction of a coefficient factor to represent the variation of bone characteristics due to inter-individual differences. The mean value of the predicted effective apparent Young's modulus in principal axis was found at approximately 460 MPa for respective 15.24% of bone volume fraction, and this is in good agreement with other experimental results. The proposed method may provide a reference for the reliable evaluation of the prediction of the apparent elastic properties of trabecular bone.     

Comparison of acoustic characteristics predicted by Biot's theory and the modified Biot-Attenborough model in cancellous bone

Biot's theory and the modified Biot-Attenborough (MBA) model are applied to predict the dependences of acoustic characteristics on frequency and on porosity in cancellous bone. The phase velocities and the attenuation coefficients predicted by both theories are compared with the experimental data of bovine cancellous bone specimens published in the literature. Biot's theory successfully predicts the dependences of the phase velocity on frequency and on porosity in cancellous bone, whereas a significant discrepancy is observed between the predicted and the measured attenuation coefficients. The MBA model agrees well with the frequency and the porosity dependences of the phase velocity and the attenuation coefficient experimentally measured in bovine bones. Although the MBA model relies on phenomenological parameters derived from the experimental data, its approach to cancellous bone can be usefully employed in the field of clinical ultrasonic bone assessment.     

A phenomenological model for predicting fatigue life in bovine trabecular bone

Cyclic loading of bone during daily activities can lead to fatigue degradation and increased risk of fracture in both the young and elderly population. Damage processes under cyclic loading in trabecular bone result in the reduction of the elastic modulus and accumulation of residual strain. These effects increase with increasing stress levels, leading to a progressive reduction in fatigue life. The present work analyzes the effect of stress and strain variation on the above damage processes in bovine trabecular bone, and develops a phenomenological model relating fatigue life to the imposed stress level. The elastic modulus reduction of the bone specimens was observed to depend on the maximum compressive strain, while the rate of residual strain accumulation was a function of the stress level. A model was developed for the upper and lower bounds of bone elastic modulus reduction with increasing number of cycles, at each stress range. The experimental observations were described well by the model. The model predicted the bounds of the fatigue life with change in fatigue stress. The decrease in the fatigue life with increasing stress was related to corresponding increases in the residual strain accumulation rates at the elevated stress levels. The model shows the validity of fatigue predictions from relatively few cyclic experiments, by combining trends observed in the monotonic and the cyclic tests. The model also presents a relatively simple procedure for predicting the endurance limit for bovine trabecular bone specimens.     

Fast and accurate specimen-specific simulation of trabecular bone elastic modulus using novel beam-shell finite element models

Elastic modulus and strength of trabecular bone are negatively affected by osteoporosis and other metabolic bone diseases. Micro-computed tomography-based beam models have been presented as a fast and accurate way to determine bone competence. However, these models are not accurate for trabecular bone specimens with a high number of plate-like trabeculae. Therefore, the aim of this study was to improve this promising methodology by representing plate-like trabeculae in a way that better reflects their mechanical behavior. Using an optimized skeletonization and meshing algorithm, voxel-based models of trabecular bone samples were simplified into a complex structure of rods and plates. Rod-like and plate-like trabeculae were modeled as beam and shell elements, respectively, using local histomorphometric characteristics. To validate our model, apparent elastic modulus was determined from simulated uniaxial elastic compression of 257 cubic samples of trabecular bone (4mm×4mm×4mm; 30μm voxel size; BIOMED I project) in three orthogonal directions using the beam-shell models and using large-scale voxel models that served as the gold standard. Excellent agreement (R(2)=0.97) was found between the two, with an average CPU-time reduction factor of 49 for the beam-shell models. In contrast to earlier skeleton-based beam models, the novel beam-shell models predicted elastic modulus values equally well for structures from different skeletal sites. It allows performing detailed parametric analyses that cover the entire spectrum of trabecular bone microstructures.     

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.