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     

Effects of damage on the orthotropic material symmetry of bovine tibial trabecular bone

The macroscopic mechanical properties of trabecular bone can be predicted by its architecture using theoretical relationships between the elastic and architectural properties. Microdamage caused by overloading or fatigue decreases the apparent elastic moduli of trabecular bone requiring these relationships to be modified to predict the damaged elastic properties. In the case of isotropic damage, the apparent level elastic properties could be determined by multiplying all of the elastic constants by a single scalar factor. If the damage is anisotropic, the elastic constants may change by differing factors and the material coordinate system could become misaligned with the fabric coordinate system. High-resolution finite element models were used to simulate damage overloading on seven trabecular bone specimens subjected to pure shear strain in two planes. Comparison of the apparent elastic moduli of the specimens before and after damage showed that the reduction of the elastic moduli was anisotropic. This suggests that the microdamage within the specimens was inhomogeneous. However, after damage the specimens exhibited nearly orthotropic material symmetry as they did before damage. Changes in the orientation of the orthotropic material coordinate system were also small and occurred primarily in the transverse plane. Thus, while damage in trabecular bone is anisotropic, the material coordinate system remains aligned with the fabric tensor.     

The fabric dependence of the orthotropic elastic constants of cancellous bone

It has been proposed that the orthotropic elastic constants of cancellous bone depend upon a tensorial measure of anisotropy called fabric as well as the tissue's structural density. Cowin (1985, Mechanics Mater, 4, 137-147; 1986, J. biomech. Engng 108, 83-88) developed explicit relationships for the elastic constant, structural density and fabric relationship. In this study the orthotropic elastic moduli, structural density, and fabric components were measured for 11 cancellous bone specimens from five bovine femora and for 75 specimens from three human proximal tibiae and fitted to these relationships using a least squares analysis. The relationships explained between 72 and 94% of the variance in the elastic constants. The relationships between the elastic constants and squared or cubed power functions of structural density had better predictive value over the entire distribution of the data than did expressions with linear functions of structural density.     

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.     

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 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.     

Errors in the orientation of the principal stress axes if bone tissue is modeled as isotropic

The error in the prediction of the orientation of the principal axes of stress in bone tissue is determined in the case when the tissue is modeled as elastically isotropic rather than as orthotropic, the probable symmetry of bone tissue. Results are two-dimensional and assume the same underlying strain state for both the orthotropic and isotropic cases. The maximum error is 45 degrees, and the typical error is generally significant.     

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.     

Mechanical testing of cancellous bone from the femoral head: experimental errors due to off-axis measurements

The aim of this study was to verify whether a misalignment between the testing direction and the trabecular main direction has a significant effect on the compressive behaviour of cancellous bone. Ten cylindrical specimens were extracted from femoral heads with a misalignment to the trabecular main direction of approximately 20 degrees. Each specimen was paired with a specimen extracted aligned with the main direction of the trabeculae on the basis of the closest bone volume fraction, obtaining two groups, one 'aligned' and one 'misaligned'. The average off-axis angle was 6.1 degrees and 21.6 degrees for the 'aligned' and 'misaligned' group, respectively. The specimens underwent micro-tomographic analysis, compressive testing, micro-indentation testing and ashing. No significant differences were found in histomorphometric parameters, hardness and ash density between the two groups, whereas significant differences were found in Young's modulus and ultimate stress: both parameters, measured for the 'misaligned' group, were about 40% lower than those measured for the 'aligned' group. These results demonstrate a great effect of the angle between the testing direction and the main direction of the trabecular structure on the compressive behaviour of cancellous bone. This angle should be reduced as much as possible (in the present work the average value was 6.6+/-3.3 degrees), in any case measured, and always reported together with the mechanical parameters of cancellous bone.     

Elastic properties of external cortical bone in the craniofacial skeleton of the rhesus monkey

Knowledge of elastic properties and of their variation in the cortical bone of the craniofacial skeleton is indispensable for creating accurate finite-element models to explore the biomechanics and adaptation of the skull in primates. In this study, we measured elastic properties of the external cortex of the rhesus monkey craniofacial skeleton, using an ultrasonic technique. Twenty-eight cylindrical cortical specimens were removed from each of six craniofacial skeletons of adult Macaca mulatta. Thickness, density, and a set of longitudinal and transverse ultrasonic velocities were measured on each specimen to allow calculation of the elastic properties in three dimensions, according to equations derived from Newton's second law and Hooke's law. The axes of maximum stiffness were determined by fitting longitudinal velocities measured along the perimeter of each cortical specimen to a sinusoidal function. Results showed significant differences in elastic properties between different functional areas of the rhesus cranium, and that many sites have a consistent orientation of maximum stiffness among specimens. Overall, the cortical bones of the rhesus monkey skull can be modeled as orthotropic in many regions, and as transversely isotropic in some regions, e.g., the supraorbital region. There are differences from human crania, suggesting that structural differences in skeletal form relate to differences in cortical material properties across species. These differences also suggest that we require more comparative data on elastic properties in primate craniofacial skeletons to explore effectively the functional significance of these differences, especially when these differences are elucidated through modeling approaches, such as finite-element modeling.     

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.     

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.     

Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods

The key parameters determining the elastic properties of an unidirectional mineralized bone fibril-array decomposed in two further hierarchical levels are investigated using mean field methods. Modeling of the elastic properties of mineralized micro- and nanostructures requires accurate information about the underlying topology and the constituents’ material properties. These input data are still afflicted by great uncertainties and their influence on computed elastic constants of a bone fibril-array remains unclear. In this work, mean field methods are applied to model mineralized fibrils, the extra-fibrillar matrix and the resulting fibril-array. The isotropic or transverse isotropic elastic constants of these constituents are computed as a function of degree of mineralization, mineral distribution between fibrils and extra-fibrillar matrix, collagen stiffness and fibril volume fraction. The linear sensitivity of the elastic constants was assessed at a default set of the above parameters. The strain ratios between the constituents as well as the axial and transverse indentation moduli of the fibril-array were calculated for comparison with experiments. Results indicate that the degree of mineralization and the collagen stiffness dominate fibril-array elasticity. Interestingly, the stiffness of the extra-fibrillar matrix has a strong influence on transverse and shear moduli of the fibril-array. The axial strain of the intra-fibrillar mineral platelets is 30–90% of the applied fibril strain, depending on mineralization and collagen stiffness. The fibril-to-fibril-array strain ratio is essentially ~1. This study provides an improved insight in the parameters, which govern the fibril-array stiffness of mineralized tissues such as bone.     

A three-dimensional finite element analysis of the upper tibia

A three-dimensional finite element model of the proximal tibia has been developed to provide a base line for further modeling of prosthetic resurfaced tibiae. The geometry for the model was developed by digitizing coronal and transverse sections made with the milling machine, from one fresh tibia of average size. The load is equally distributed between the medial and lateral compartments over contact areas that were reported in the literature. An indentation test has been used to measure the stiffness and the ultimate strength of cancellous bone in four cadaver tibiae. These values provided the statistical basis for characterising the inhomogeneous distribution of the cancellous bone properties in the proximal tibia. All materials in the model were assumed to be linearly elastic and isotropic. Mechanical properties for the cortical bone and cartilage have been taken from the literature. Results have been compared with strain gage tests and with a two-dimensional axisymmetric finite element model both from the literature. Qualitative comparison between trabecular alignment, and the direction of the principal compressive stresses in the cancellous bone, showed a good relationship. Maximum stresses in the cancellous bone and cortical bone, under a load which occurs near stance phase during normal gait, show safety factors of approximately eight and twelve, respectively. The load sharing between the cancellous bone and the cortical bone has been plotted for the first 40 mm distally from the tibial eminence.     

Ultrasound propagation in anisotropic soft tissues: the application of linear elastic theory

Linear elastic theory has served well in modeling the mechanical properties of numerous materials. In modeling ultrasonic wave propagation in biological soft tissues, an isotropic model has usually been employed. Many tissues, however, possess a lower order of symmetry, and the speed of sound in muscle is known to vary with the direction of propagation. In this study, by applying linear regression to acoustic microscopic data from seven frog sartorius specimens, four observable elastic constants associated with a transversely isotropic model were obtained. The average values of these constants were c11 = 2.64, c13 = 3.39 and c33 = 4.40 Nm-2 for resting muscles and c11 = 2.65, c13 = 3.43 and c33 = 4.57 Nm-2 for muscles undergoing tetanic contraction, where '1' and '3' represent the transverse and longitudinal axes, respectively. In all cases, c44 was 0, indicating a minimal contribution from longitudinal shear. For all seven specimens, the model of transverse isotropy provided a better fit of the data than that of isotropy.     

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.     

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.     

Correction formulae for the misalignment of axes in the measurement of the orthotropic elastic constants

In the experimental determination of the orthotropic elastic constants, one often encounters the situation in which the symmetry axes of the material are not coincident with specimen axes along which the material testing is accomplished. The problem of calculating the compliance coefficients in the symmetry coordinate system from measurements of the compliance coefficients made in an arbitrary, specimen fixed, coordinate system is considered here.     

The dependence of the elastic properties of osteoporotic cancellous bone on volume fraction and fabric

Osteoporosis is a progressive systemic skeletal condition characterized by low bone mass and microarchitectural deterioration, with a consequent increase in susceptibility to fracture. Hence, osteoporosis would be best diagnosed by in vivo measurements of bone strength. As this is not clinically feasible, our goal is to estimate bone strength through the assessment of elastic properties, which are highly correlated to strength. Previously established relations between morphological parameters (volume fraction and fabric) and elastic constants could be applied to estimate cancellous bone stiffness in vivo. However, these relations were determined for normal cancellous bone. Cancellous bone from osteoporotic patients may require different relations. In this study we set out to answer two questions. First, can the elastic properties of osteoporotic cancellous bone be estimated from morphological parameters? Second, do the relations between morphological parameters and elastic constants, determined for normal bone, apply to osteoporotic bone as well? To answer these questions we used cancellous bone cubes from femoral heads of patients with (n=26) and without (n=32) hip fractures. The elastic properties of the cubes were determined using micro-finite element analysis, assuming equal tissue moduli for all specimens. The morphological parameters were determined using microcomputed tomography. Our results showed that, for equal tissue properties, the elastic properties of cancellous bone from fracture patients could indeed be estimated from morphological parameters. The morphology-based relations used to estimate the elastic properties of cancellous bone are not different for women with or without fractures.     

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.