The speed of sound through trabecular bone predicted by Biot theory

Cancellous bone is a highly porous material filled with fluid. The mechanical properties of cancellous bone determine whether the bone is normal or osteoporotic. Wave propagation can be used to measure the elastic constants of cancellous bone. Recently, poroelasticity theory has been used to predict the elastic constants of cancellous bone from the wave velocities. In this study, it is shown that the fast wave, predicted by the Biot theory, corresponds to the wave penetrating the trabeculae, while the slow wave is determined by the interaction between the trabeculae and the fluid. The trabecular shape does not affect the wave velocity significantly when using the variable, which is determined by the microstructure, and the slow wave velocity decreases after the porosity reaches 80%.     

Effect of vertebral body stiffness before and after vertebroplasty on intradiscal pressure]

Fractures of osteoporotic vertebral bodies are increasingly stabilized with bone cement. The effects of vertebral-body stiffness before and after augmentation with bone cement and of wedge-shaped vertebral body fractures on intradiscal pressure are insufficiently known. In a finite element model of the lumbar spine the elastic modulus of cancellous bone as well as the amount and the elastic modulus of bone cement were varied and the dependency of intradiscal pressure on these parameters was calculated. In addition, a wedge-shaped vertebral-body fracture was simulated. The bulge of the vertebral-body endplate and thus the intradiscal pressure depends strongly on the grade of osteoporosis in the vertebral body. The influence of amount and elastic modulus of bone cement on intradiscal pressure is small. A wedge-shaped vertebral-body fracture causes an anterior shift of upper-body centre of gravity. If this shift is not compensated, it leads to an increased flexion moment that has to be balanced by muscle forces. In addition, this shift leads to a stronger increase of intradiscal pressure than the augmentation of the vertebral body with bone cement.     

Material changes in osteoporotic human cancellous bone following infiltration with acrylic bone cement for a vertebral cement augmentation

Bone cement infiltration can be effective at mechanically augmenting osteoporotic vertebrae. While most published literature describes the gain in mechanical strength of augmented vertebrae, we report the first measurements of viscoelastic material changes of cancellous bone due to cement infiltration. We infiltrated cancellous core specimen harvested from osteoporotic cadaveric spines with acrylic bone cement. Bone specimen before and after cement infiltration were subjected to identical quasi-static and relaxation loading in confined and free compression. Testing data were fitted to a linear viscoelastic model of compressible material and the model parameters for cement, native cancellous bone, and cancellous bone infiltrated (composite) with cement were identified. The fitting demonstrated that the linear viscoelastic model presented in this paper accurately describes the mechanical behaviour of cement and bone, before and after infiltration. Although the composite specimen did not completely adopt the properties of bulk bone cement, the stiffening of cancellous bone due to cement infiltration is considerable. The composite was, for example, 8.5 times stiffer than native bone. The local stiffening of cancellous bone in patients may alter the load transfer of the augmented motion segment and may be the cause of subsequent fractures in the vertebrae adjacent to the ones infiltrated with cement. The material model and parameters in this paper, together with an adequate finite-element model, can be helpful to investigate the load shift, the mechanism for subsequent fractures, and filling patterns for ideal cement infiltration.     

Material Changes in Osteoporotic Human Cancellous Bone Following Infiltration with Acrylic Bone Cement for a Vertebral Cement Augmentation

Bone cement infiltration can be effective at mechanically augmenting osteoporotic vertebrae. While most published literature describes the gain in mechanical strength of augmented vertebrae, we report the first measurements of viscoelastic material changes of cancellous bone due to cement infiltration. We infiltrated cancellous core specimen harvested from osteoporotic cadaveric spines with acrylic bone cement. Bone specimen before and after cement infiltration were subjected to identical quasi-static and relaxation loading in confined and free compression. Testing data were fitted to a linear viscoelastic model of compressible material and the model parameters for cement, native cancellous bone, and cancellous bone infiltrated (composite) with cement were identified. The fitting demonstrated that the linear viscoelastic model presented in this paper accurately describes the mechanical behaviour of cement and bone, before and after infiltration. Although the composite specimen did not completely adopt the properties of bulk bone cement, the stiffening of cancellous bone due to cement infiltration is considerable. The composite was, for example, 8.5 times stiffer than native bone. The local stiffening of cancellous bone in patients may alter the load transfer of the augmented motion segment and may be the cause of subsequent fractures in the vertebrae adjacent to the ones infiltrated with cement. The material model and parameters in this paper, together with an adequate finite-element model, can be helpful to investigate the load shift, the mechanism for subsequent fractures, and filling patterns for ideal cement infiltration.     

A surrogate long-bone model with osteoporotic material properties for biomechanical testing of fracture implants

In vitro comparative testing of fracture fixation implants is limited by the highly variable material properties of cadaveric bone. Bone surrogate specimens are often employed to avoid this confounding variable. Although validated surrogate models of normal bone (NB) exist, no validated bone model simulating weak, osteoporotic bone (OPB) is available. This study presents an osteoporotic long-bone model designed to match the lower cumulative range of mechanical properties found in large series of cadaveric femora reported in the literature. Five key structural properties were identified from the literature: torsional rigidity and strength, bending rigidity and strength, and screw pull-out strength. An OPB surrogate was designed to meet the low range for each of these parameters, and was mechanically tested. For comparison, the same parameters were determined for surrogates of NB. The OPB surrogate had a torsional rigidity and torsional strength within the lower 2% and 16%, respectively, of the literature based cumulative range reported for cadaveric femurs. Its bending rigidity and bending strength was within the lower 11% and 8% of the literature-based range, respectively. Its pull-out strength was within the lower 2% to 16% of the literature based range. With all five structural properties being within the lower 16% of the cumulative range reported for native femurs, the OPB surrogate reflected the diminished structural properties seen in osteoporotic femora. In comparison, surrogates of NB demonstrated structural properties within 23-118% of the literature-based range. These results support the need and utility of the OPB surrogate for comparative testing of implants for fixation of femoral shaft fractures in OPB.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of “equivalent” trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of "equivalent" trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

Atypical subtrochanteric femoral shaft fractures: role for mechanics and bone quality

Bisphosphonates are highly effective agents for reducing osteoporotic fractures in women and men, decreasing fracture incidence at the hip and spine up to 50%. In a small subset of patients, however, these agents have recently been associated with ''atypical femoral fractures'' (AFFs) in the subtrochanteric region or the diaphysis. These fractures have several atypical characteristics, including occurrence with minimal trauma; younger age than typical osteoporotic fractures; occurrence at cortical, rather than cancellous sites; early radiographic appearance similar to that of a stress fracture; transverse fracture pattern rather than the familiar spiral or transverse-oblique morphologies; initiation on the lateral cortex; and high risk of fracture on the contralateral side, at the same location as the initial fracture. Fracture is a mechanical phenomenon that occurs when the loads applied to a structure such as a long bone exceed its load-bearing capacity, either due to a single catastrophic overload (traumatic failure) or as a result of accumulated damage and crack propagation at sub-failure loads (fatigue failure). The association of AFFs with no or minimal trauma suggests a fatigue-based mechanism that depends on cortical cross-sectional geometry and tissue material properties. In the case of AFFs, bisphosphonate treatment may alter cortical tissue properties, as these agents are known to alter bone remodeling. This review discusses the use of bisphosphonates, their effects on bone remodeling, mechanics and tissue composition, their significance as an effective therapy for osteoporosis, and why these agents may increase fracture risk in a small population of patients.     

Enhanced bone screw fixation with biodegradable bone cement in osteoporotic bone model

Purpose: The purpose of this study was to study the potential of novel biodegradable PCL bone cement to improve bone screw fixation strength in osteoporotic bone. Methods: The biomechanical properties of bone cement (ε-polycaprolactone, PCL) and fixation strength were studied using biomechanical tests and bone screws fixed in an osteoporotic bone model. Removal torques and pullout strengths were assessed for cortical, self-tapping, and cancellous screws inserted in the osteoporotic bone model (polyurethane foam blocks with polycarbonate plate) with and without PCL bone cement. Open cell and cellular rigid foam blocks with a density of 0.12 g/cm3 were used in this model. Results: Removal torques were significantly (more than six-fold) improved with bone cement for cancellous screws. Furthermore, the bone cement improved pullout strengths three to 12 times over depending on the screw and model material.?Conclusions: Biodegradable bone cement turned out to be a very potential material to stabilize screw fixation in osteoporotic bone. The results warrant further research before safe clinical use, especially to clarify clinically relevant factors using real osteoporotic bone under human body conditions and dynamic fatigue testing for long-term performance.     

Predicting changes in mechanical properties of trabecular bone by adaptive remodeling

Because changes in the mechanical properties of bone are closely related to trabecular bone remodeling, methods that consider the temporal morphological changes induced by adaptive remodeling of trabecular bone are needed to estimate long-term fracture risk and bone quality in osteoporosis. We simulated bone remodeling using simplified and pig trabecular bone models and estimated the morphology of healthy and osteoporotic cases. We then displayed the fracture risk of the remodeled models based on a cumulative histogram from high stress. The histogram showed more elements had higher stresses in the osteoporosis model, indicating that the osteoporosis model had a greater risk.     

An approximate model for cancellous bone screw fixation

This paper presents a finite element (FE) model to identify parameters that affect the performance of an improved cancellous bone screw fixation technique, and hence potentially improve fracture treatment. In cancellous bone of low apparent density, it can be difficult to achieve adequate screw fixation and hence provide stable fracture fixation that enables bone healing. Data from predictive FE models indicate that cements can have a significant potential to improve screw holding power in cancellous bone. These FE models are used to demonstrate the key parameters that determine pull-out strength in a variety of screw, bone and cement set-ups, and to compare the effectiveness of different configurations. The paper concludes that significant advantages, up to an order of magnitude, in screw pull-out strength in cancellous bone might be gained by the appropriate use of a currently approved calcium phosphate cement.     

Simulated influence of osteoporosis and disc degeneration on the load transfer in a lumbar functional spinal unit

As life expectancy increases, age-related disorders and the search for related medical care will expand. Osteoporosis is the most frequent skeletal disease in this context with the highest fracture risk existing for vertebrae. The aging process is accompanied by systemic changes, with the earliest degeneration occurring in the intervertebral discs. The influence of various degrees of disc degeneration on the load transfer was examined using the finite element method. The effect of different possible alterations of the bone quality due to osteoporosis was simulated by adjusting the corresponding material properties and their distribution and several loadings were applied. An alteration of the load transfer, characterised by changed compression stiffness and strain distributions as well as magnitudes, due to osteoporotic bone and degenerated discs was found. When osteoporosis was simulated, the stiffness was substantially decreased, larger areas of the cancellous bone were subjected to higher strains and strain maxima were increased. Increasing ratios of transverse isotropy in the osteoporotic bone yielded smaller effects than reduced bone properties. Including a degenerated disc mainly altered the strain distribution. Combining osteoporosis and degenerated discs reduced the areas of cancellous bone subjected to substantial strain. Based on these results, it can be concluded that the definition of a healthy disc in osteoporotic spines might be considered as a worst-case scenario. One attempt to evaluate the progress of osteoporosis can be made by introducing increasing degrees of anisotropy. If several parameters in a model are changed to simulate degeneration, it should be pointed out how each individual definition influences the overall result.     

In-Vivo Assessment of Femoral Bone Strength Using Finite Element Analysis (FEA) Based on Routine MDCT Imaging: A Preliminary Study on Patients with Vertebral Fractures

PurposeTo experimentally validate a non-linear finite element analysis (FEA) modeling approach assessing in-vitro fracture risk at the proximal femur and to transfer the method to standard in-vivo multi-detector computed tomography (MDCT) data of the hip aiming to predict additional hip fracture risk in subjects with and without osteoporosis associated vertebral fractures using bone mineral density (BMD) measurements as gold standard.MethodsOne fresh-frozen human femur specimen was mechanically tested and fractured simulating stance and clinically relevant fall loading configurations to the hip. After experimental in-vitro validation, the FEA simulation protocol was transferred to standard contrast-enhanced in-vivo MDCT images to calculate individual hip fracture risk each for 4 subjects with and without a history of osteoporotic vertebral fractures matched by age and gender. In addition, FEA based risk factor calculations were compared to manual femoral BMD measurements of all subjects.ResultsIn-vitro simulations showed good correlation with the experimentally measured strains both in stance (R² = 0.963) and fall configuration (R² = 0.976). The simulated maximum stress overestimated the experimental failure load (4743 N) by 14.7% (5440 N) while the simulated maximum strain overestimated by 4.7% (4968 N). The simulated failed elements coincided precisely with the experimentally determined fracture locations. BMD measurements in subjects with a history of osteoporotic vertebral fractures did not differ significantly from subjects without fragility fractures (femoral head: p = 0.989; femoral neck: p = 0.366), but showed higher FEA based risk factors for additional incident hip fractures (p = 0.028).ConclusionFEA simulations were successfully validated by elastic and destructive in-vitro experiments. In the subsequent in-vivo analyses, MDCT based FEA based risk factor differences for additional hip fractures were not mirrored by according BMD measurements. Our data suggests, that MDCT derived FEA models may assess bone strength more accurately than BMD measurements alone, providing a valuable in-vivo fracture risk assessment tool.     

Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells

Evaluation of constitutive properties of cancellous bone and their relationships to microstructural parameters is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of 'equivalent' trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. Two cell types are analysed: cube- and prism-based. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with three shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Six static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Multi-point boundary conditions imposed on the models ensure mutual fitting of deformed neighbouring cells. Values of computed stresses allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement, though evidently better in the case of the prism-based cell model.     

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.     

Relevance of osteoporosis in women with fracture of the femoral neck.

A prospective study of fractures of the femoral neck was conducted over 12 months in order to ascertain the relevance of generalised osteoporosis as determined by metacarpal morphometry. A series of some 200 women sustaining a fracture of the femoral neck after minor trauma had bone mass measurements similar to those of a control population of normal women, and 16% were not osteoporotic. A history of previous fractures was documented in one third of the women, but this was unrelated to the presence or severity of osteoporosis, although over half of the fractures had occurred within the previous four years. Trochanteric fractures were seen more commonly in severely osteoporotic women (p less than 0.005), whereas cervical fractures predominated in those who were not osteoporotic. These findings support the hypothesis that postural instability is the major determinant for femoral neck fracture and that generalised osteoporosis, rather than being a prerequisite for fracture, merely determines the type of fracture sustained.     

Effects of bisphosphonates on matrix mineralization

Bone strength is determined not only by the volume of bone tissue and the microarchitectural organization of this bone, but also by the degree of mineralization of bone matrix. The mineralization process consists of a primary deposition of mineral substance on the calcification front, followed by a slow and progressive increase of the mineral deposition named secondary mineralization. In osteoporosis, there is a negative imbalance between bone resorption and bone formation, resulting in bone loss, and microarchitectural deterioration of the trabecular network. Therapeutic agents for osteoporosis could increase bone strength by three separate, but interrelated effects on bone tissue: 1) the prevention of bone loss and thus the preservation of bone microarchitecture, 2) an increase in the volume of bone matrix, and 3) an increase in the degree of mineralization to a level similar to that seen in healthy premenopausal women, through a prolongation of the duration of secondary mineralization. Therefore the use of antiresorptive agents that reduce bone turnover, as bisphosphonates, provide a rational approach to treatment of osteoporosis. Extensive phase III clinical trials have shown that osteoporotic women treated orally with alendronate (ALN) for 3 years or more had substantial increases in bone mineral density (BMD) of approximately 10% at the spine together with reductions of about 50% in the incidence of vertebral fractures. Since a marked reduction in activation frequency was evidenced in the transiliac biopsies taken after treatment with ALN compared to placebo (PLA), without detectable increase in cancellous bone volume, it was hypothesized that the increase in BMD and the reduction in the incidence of fragility fractures were due, in a substantial part, to an increase in the degree of mineralization of bone (DMB). The mean DMB was measured by quantitative microradiography on transiliac bone biopsies taken from 53 postmenopausal osteoporotic women who had been treated with ALN (10 mg/day) during 2 (9 patients) or 3 years (16 patients) or with PLA (15 and 13 patients, respectively). In the same patients, BMD values were obtained by dual-energy X-ray absorptiometry on lumbar spine at the beginning and end of treatment. Histomorphometric parameters and activation frequency of new remodeling units were also measured on the biopsies. After 2 years of ALN, mean DMB in compact bone was 9.3% (p=0.0035) and in cancellous bone was 7.3% (p=0.0009) higher, respectively, versus PLA. After 3 years of ALN, mean DMB in compact bone was 11.6% (p=0.0002) and in cancellous bone was 11.4% (p=0.0001) higher, respectively, versus PLA. After 2 and 3 years of ALN and compared to the corresponding PLA, the distribution of the DMB clearly showed a shift towards the highest mineralization values and a decrease of the number of bone structure units having low values of mineralization. The between group differences in mean DMB were similar to those of BMD at the lumbar spine level (+8.7% after 2 years +9.6% after 3 years, respectively), suggesting that mean DMB augmentation probably accounts for the major part of the increase in BMD seen with ALN. These results support our model that the reduction in the activation frequency caused by the antiresorptive effect of ALN is followed by a prolonged secondary mineralization which increases the percentage of bone structure units having reached a maximum degree of secondary mineralization and, through this mechanism, mean DMB. That these effects contribute to improved bone strength is demonstrated by the reduction in fracture incidence previously demonstrated in these patients. In conclusion, quantitative microradiography gives access to the mineral dimension of bone tissue which has been insufficiently taken into account until now as an important determinant of bone strength and quality of bone.     

The cancellous bone multiscale morphology-elasticity relationship

The cancellous bone effective properties relations are analysed on multiscale across two aspects; properties of representative volume element on micro scale and statistical measure of trabecular trajectory orientation on mesoscale. Anisotropy of the microstructure is described across fabric tensor measure with trajectory orientation tensor as bridging scale connection. The scatter measured data (elastic modulus, trajectory orientation, apparent density) from compression test are fitted by stochastic interpolation procedure. The engineering constants of the elasticity tensor are estimated by last square fitt procedure in multidimensional space by Nelder-Mead simplex. The multiaxial failure surface in strain space is constructed and interpolated by modified super-ellipsoid.     

How morphology predicts mechanical properties of trabecular structures depends on intra-specimen trabecular thickness variations

Two observations underlie this work. First, that the architecture of trabecular bone can accurately predict the mechanical stiffness characteristics of bone specimens when considering the combination of volume fraction and fabric, which is a measure of architectural anisotropy. Second, that the same morphological measures could not accurately predict the mechanical properties of porous structures in general. We hypothesize that this discrepancy can be explained by the special nature of trabecular bone as a structure in remodeling equilibrium relative to the external loads. We tested this hypothesis using a generic model of trabecular bone. Five series of 153 different architectures were created with this model. Each architecture was subjected to morphological analysis, and four different fabric measures were calculated to evaluate their effectiveness in characterizing the architecture. Relationships were determined relating morphology to the elastic constants. The quality of these relationships was tested by correlating the predicted elastic constants with those determined from finite element analysis. We found that the four fabric measures used could estimate the mechanical properties almost equally well. So the suggestion that fabric measures based on trabecular bone volume better represent the architecture than mean intercept length could not be affirmed. We conclude that for structures with equally sized elliptical voids the mechanical properties can be predicted well only if trabecular thickness variations within each structure are limited. These structures closely resemble previously developed models of trabecular bone. Furthermore, they are stiff in the principal fabric direction, hence, according to Cowin (J. Biomech. Eng. (108) (1986) 83), they are in remodeling equilibrium. These structures are also stiff over a large range of loading orientations, hence, are relatively insensitive to deviations in direction of loading.     

Determination of the heterogeneous anisotropic elastic properties of human femoral bone: From nanoscopic to organ scale

Cortical bone is a multiscale composite material. Its elastic properties are anisotropic and heterogeneous across its cross-section, due to endosteal bone resorption which might affect bone strength. The aim of this paper was to describe a homogenization method leading to the estimation of the variation of the elastic coefficients across the bone cross-section and along the bone longitudinal axis. The method uses the spatial variations of bone porosity and of the degree of mineralization of the bone matrix (DMB) obtained from the analysis of 3-D synchrotron micro-computed tomography images. For all three scales considered (the foam (100 nm), the ultrastructure (5 μm) and the mesoscale (500 μm)), the elastic coefficients were determined using the Eshelby’s inclusion problem. DMB values were used at the scale of the foam. Collagen was introduced at the scale of the ultrastructure and bone porosity was introduced at the mesoscale. The pores were considered as parallel cylinders oriented along the bone axis. Each elastic coefficient was computed for different regions of interest, allowing an estimation of its variations across the bone cross-section and along the bone longitudinal axis. The method was applied to a human femoral neck bone specimen, which is a site of osteoporotic fracture. The computed elastic coefficients for cortical bone were in good agreement with experimental results, but some discrepancies were obtained in the endosteal part (trabecular bone). These results highlight the importance of accounting for the heterogeneity of cortical bone properties across bone cross-section and along bone longitudinal axis.