Elastic properties of cancellous bone: measurement by an ultrasonic technique

The high degree of porosity of cancellous bone makes elastic property measurement difficult by traditional mechanical testing methods. An ultrasonic technique is described with which mechanical properties of anisotropic, rigid, porous materials, such as cancellous bone, can be measured. The technique utilizes unique piezoelectric transducers operated in a continuous wave mode at a frequency of approximately 50 kHz. Both longitudinal and shear waves can be propagated and received with the transducers allowing both Young's moduli and shear moduli to be determined with the technique. A comparison between moduli measured with the ultrasonic technique and moduli measured with traditional mechanical testing shows the new method to be quite accurate in elastic property determination, (r2 = 0.935, Emech = 1.00E1dt + 23.3 MPa) (r2 = 0.656, Gmech = 1.08 Gult--3.3MPa).     

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.     

Alterations in damage processes in dense cancellous bone following gamma-radiation sterilization

Structurally intact cancellous bone allograft is an attractive tissue form because its high porosity can provide space for delivery of osteogenic factors and also allows for more rapid and complete in-growth of host tissues. Gamma radiation sterilization is commonly used in cancellous bone allograft to prevent disease transmission. Commonly used doses of gamma radiation sterilization (25–35 kGy) have been shown to modify cortical bone post-yield properties and crack propagation but have not been associated with changes in cancellous bone material properties. The purpose of this study was to determine the effects of irradiation on the elastic and yield properties and microscopic tissue damage processes in dense cancellous bone. Cancellous bone specimens (13 control, 14 irradiated to 30 kGy) from bovine proximal tibiae were tested in compression to 1.3% apparent strain and examined for microscopic tissue damage. The yield strain in irradiated specimens (0.93±0.11%, mean±SD) did not differ from that in control specimens (0.90±0.11%, p=0.44). No differences in elastic modulus were observed between groups after accounting for differences in bone volume fraction. However, irradiated specimens showed greater residual strain (p=0.01), increased number of microfractures (p=0.02), and reduced amounts of cross-hatching type damage (p<0.01). Although gamma radiation sterilization at commonly used dosing (30 kGy) does not modify elastic or yield properties of dense cancellous bone, it does cause modifications in damage processes, resulting in increased permanent deformation following isolated overloading.     

Why fine-wire fixators work: an analysis of pressure distribution at the wire-bone interface

Tensioned fine-wire external fixator systems have been used successfully for the treatment of fractures, mal-unions and for limb lengthening for many years. When used in metaphyseal bone, this type of fixator has a lower loosening rate than half-pin fixators. The differing mechanical properties of these fixator systems have been investigated extensively; however, most studies have centered on the mechanical properties of the fixator as a whole. Our knowledge of the interactions occurring at the interface between implant and bone remains sparse. In order to investigate this interaction, we devised a simple experimental model to characterise the distribution of pressure in cancellous bone surrounding a tensioned wire under loading conditions. Comparison was then made to a similar model of a half-pin fixator. Pressure was measured using pressure sensitive film at various distances away from the implant-bone interface. Static, single cycle loading of the model was performed with a Universal Testing Machine. Pressure distribution in the fine-wire model was found to occur in three regions: polar, beam loading and uniform. Polar patterns were seen closest to the wire with pressure concentrated at the entry and exit points of the wire. Beam loading was seen at a distance of 1.5 mm from the wire and pressure reached a uniform distribution at 4.0 mm. Most of the pressure measured was less than 2 MPa, which is less than the yield strength of cancellous bone (2-7 MPa). Higher loads produced higher stresses but the distribution pattern was similar. In contrast, the half-pin model showed far higher pressures (20 MPa), which were present deeper in the bone specimen. These results further our understanding of the biomechanics of fine-wire fixators and may explain the lower loosening rate of this type of fixator when compared to half-pin fixators used in metaphyseal bone.     

Estimation of the effective yield properties of human trabecular bone using nonlinear micro-finite element analyses

Micro-finite element (\(\upmu \)FE) analyses are often used to determine the apparent mechanical properties of trabecular bone volumes. Yet, these apparent properties depend strongly on the applied boundary conditions (BCs) for the limited size of volumes that can be obtained from human bones. To attenuate the influence of the BCs, we computed the yield properties of samples loaded via a surrounding layer of trabecular bone (“embedded configuration”). Thirteen cubic volumes (10.6 mm side length) were collected from \(\upmu \)CT reconstructions of human vertebrae and femora and converted into \(\upmu \)FE models. An isotropic elasto-plastic material model was chosen for bone tissue, and nonlinear \(\upmu \)FE analyses of six uniaxial, shear, and multi-axial load cases were simulated to determine the yield properties of a subregion (5.3 mm side length) of each volume. Three BCs were tested. Kinematic uniform BCs (KUBCs: each boundary node is constrained with uniform displacements) and periodicity-compatible mixed uniform BCs (PMUBCs: each boundary node is constrained with a uniform combination of displacements and tractions mimicking the periodic BCs for an orthotropic material) were directly applied to the subregions, while the embedded configuration was achieved by applying PMUBCs on the larger volumes instead. Yield stresses and strains, and element damage at yield were finally compared across BCs. Our findings indicate that yield strains do not depend on the BCs. However, KUBCs significantly overestimate yield stresses obtained in the embedded configuration (+43.1 ± 27.9%). PMUBCs underestimate (?10.0 ± 11.2%), but not significantly, yield stresses in the embedded situation. Similarly, KUBCs lead to higher damage levels than PMUBCs (+51.0 ± 16.9%) and embedded configurations (+48.4 ± 15.0%). PMUBCs are better suited for reproducing the loading conditions in subregions of the trabecular bone and deliver a fair estimation of their effective (asymptotic) yield properties.     

Effect of thermodisinfection on mechanic parameters of cancellous bone

Revision surgery of joint replacements is increasing and raises the demand for allograft bone since restoration of bone stock is crucial for longevity of implants. Proceedings of bone grafts influence the biological and mechanic properties differently. This study examines the effect of thermodisinfection on mechanic properties of cancellous bone. Bone cylinders from both femoral heads with length 45 mm were taken from twenty-three 6–8 months-old piglets, thermodisinfected at 82.5 °C according to bone bank guidelines and control remained native. The specimens were stored at ?20 °C immediately and were put into 21 °C Ringer’s solution for 3 h before testing. Shear and pressure modulus were tested since three point bending force was examined until destruction. Statistical analysis was done with non-parametric Wilcoxon, t test and SPSS since p < 0.05 was significant. Shear modulus was significantly reduced by thermodisinfection to 1.02 ± 0.31 GPa from 1.28 ± 0.68 GPa for unprocessed cancellous bone (p = 0.029) since thermodisinfection reduced pressure modulus not significantly from 6.30 ± 4.72 GPa for native specimens to 4.97 ± 2.23 GPa and maximum bending force was 270.03 ± 116.68 N for native and 228.80 ± 70.49 N for thermodisinfected cancellous bone. Shear and pressure modulus were reduced by thermodisinfection around 20 % and maximum bending force was impaired by about 15 % compared with native cancellous bone since only the reduction of shear modulus reached significance. The results suggest that thermodisinfection similarly affects different mechanic properties of cancellous bone and the reduction of mechanic properties should not relevantly impair clinical use of thermodisinfected cancellous bone.     

On Wolff's law of trabecular architecture.

Several studies suggest that the yield strain in cancellous bone may be uniformly distributed and isotropic. Yield strain was reported to be independent of textural anisotropy in bovine cancellous bone [Turner, J. biomech. Engng 111, 1-5 (1989)] and it is plausible that yield strain is isotropic in human cancellous bone as well. In this paper, it is hypothesized that uniform, isotropic strain represents a goal of cancellous bone adaptation, i.e. cancellous bone alters its structure to maintain uniform, isotropic peak strains. Therefore, textural anisotropy must exactly cancel the anisotropy of the peak principal stresses imposed upon cancellous bone. When evaluating the relationships between mechanical properties of cancellous bone and trabecular architecture, it was found that over 90% of the variance of yield strength can be explained by one term--rho 2H3 (where rho is apparent density and H is the normalized anisotropy (fabric) constant). Furthermore, this single term explains 70-78% of the variance in Young's modulus of cancellous bone. Based upon these findings, it was postulated that fabric adaptation goes as Hi/Hj = [ sigma i/sigma j[, where Hi and Hj are fabric eigenvalues in the i- and the j-direction and sigma i and sigma j are peak principal stresses.     

Specimen size effect in the volumetric shrinkage of cancellous bone measured at two levels of dehydration

Water is commonly removed from bone to study its effect on mechanical behaviour; however, dehydration also alters the bone structure. To make matters worse, measuring structural changes in cancellous bone is complicated by a number of factors. Therefore, the goals of this study were to address these issues by (1) comparing Archimedes' method and a helium pycnometer as methods for measuring cancellous bone volume; (2) measuring the apparent dimensional and volumetric tissue shrinkage of cancellous bone at two levels of dehydration; and, (3) identifying whether a size effect exists in cancellous bone shrinkage. Cylindrical specimens (3, 5 and 8.3 mm diameters) of cancellous bone were taken from the distal bovine femur. The apparent dimensions of each cylindrical specimen were measured in a fully hydrated state (HYD), after drying at room temperature (AIR), and after oven drying at 105 degrees C (OVEN). Tissue volume measurements for those three hydration states were obtained using both a helium pycnometer and Archimedes' method. Aluminium foams, which mimic the cancellous structure, were used as controls. The results suggest that the helium pycnometer and Archimedes' method yield identical results in the HYD and AIR states, but that Archimedes' method under-predicts the nominal OVEN volume by incorporating the collagen-apatite porosity. A distinct size effect on volumetric shrinkage is observed (p<0.025) using the pycnometer in both AIR and OVEN states. Apparent dimensional shrinkage (2% and 7%) at the two dehydration levels is much smaller than the measured volumetric tissue shrinkage (16% and 29%), which results in a reduced dehydrated bone volume fraction.     

Differences in vertebral structure and strength of inbred female mouse strains

This study assessed mouse strain-related differences in vertebral biomechanics and histomorphometry in inbred mice strains shown to differ in bone mineral content (BMC) and areal density (BMD) (as measured by pDEXA). Lumbar vertebrae L3 to L5 were collected from three mice strains (C3H/HeJ[C3], C57BL/6J[B6], and DBA/2J[D2], n=12/strain, 4-month-old female, 22.2 +/- 0.3g). BMC and BMD were measured in L3 and L4 using peripheral dual energy x-ray absorptiometry. The L4 vertebral body was then mechanically tested in compression to determine structural properties (ultimate/yield load, stiffness) from load-displacement curves and derive apparent material properties (ultimate/yield stress, and modulus of elasticity). L5 was processed for histomorphometric evaluation. Vertebral BMC and BMD were greater in C3 than in B6 and D2 mice. Vertebral trabecular/cancellous bone volume was smaller in C3 than in D2 and B6 mice. Trabecular bone formation rates were greater in D2 than in B6 and C3 mice. Osteoid surface was smaller in C3 mice than in B6 and D2 mice. Differences in osteoclast and mineralizing surfaces were not detected among the three mouse strains. In addition, there were no significant differences in biomechanical properties between the three strains. Despite the greatest BMC and areal BMD in C3 mice, the lack of strain-related differences in vertebral body strength data suggests that the biomechanical properties may be affected by the bone distribution and/or complex combination of cortical and cancellous bone at this site.     

Concept and development of an orthotropic FE model of the proximal femur

PURPOSE: In contrast to many isotropic finite-element (FE) models of the femur in literature, it was the object of our study to develop an orthotropic FE "model femur" to realistically simulate three-dimensional bone remodelling. METHODS: The three-dimensional geometry of the proximal femur was reconstructed by CT scans of a pair of cadaveric femurs at equal distances of 2mm. These three-dimensional CT models were implemented into an FE simulation tool. Well-known "density-determined" bony material properties (Young's modulus; Poisson's ratio; ultimate strength in pressure, tension and torsion; shear modulus) were assigned to each FE of the same "CT-density-characterized" volumetric group.In order to fix the principal directions of stiffness in FE areas with the same "density characterization", the cadaveric femurs were cut in 2mm slices in frontal (left femur) and sagittal plane (right femur). Each femoral slice was scanned into a computer-based image processing system. On these images, the principal directions of stiffness of cancellous and cortical bone were determined manually using the orientation of the trabecular structures and the Haversian system. Finally, these geometric data were matched with the "CT-density characterized" three-dimensional femur model. In addition, the time and density-dependent adaptive behaviour of bone remodelling was taken into account by implementation of Carter's criterion. RESULTS: In the constructed "model femur", each FE is characterized by the principal directions of the stiffness and the "CT-density-determined" material properties of cortical and cancellous bone. Thus, on the basis of anatomic data a three-dimensional FE simulation reference model of the proximal femur was realized considering orthotropic conditions of bone behaviour. CONCLUSIONS: With the orthotropic "model femur", the fundamental basis has been formed to realize realistic simulations of the dynamical processes of bone remodelling under different loading conditions or operative procedures (osteotomies, total hip replacements, etc).     

Design and validation of a testing system to assess torsional cancellous bone failure in conjunction with time-lapsed micro-computed tomographic imaging

When compressed axially, cancellous bone often fails at an oblique angle along well-defined bands, highlighting the importance of cancellous bone shear properties. Torsion testing to determine shear properties of cancellous bone has often been conducted under conditions appropriate only for axis-symmetric specimens comprised of homogeneous and isotropic materials. However, most cancellous bone specimens do not meet these stringent test conditions. Therefore, the aim of this study was to design and validate a uniaxial, incremental torsional testing system for non-homogeneous orthotropic or non-axis-symmetric specimens.Precision and accuracy of the newly designed torsion system was validated by using Plexiglas rods and beams, where obtained material properties were compared to those supplied by the manufacturer. Additionally, the incremental step-wise application of angular displacement and simultaneous time-lapsed μCT imaging capability of the system was validated using whale cancellous bone specimens, with step-wise application of angular displacement yielding similar torsional mechanical properties to continuous application of angular displacement in a conventional torsion study.In conclusion, a novel torsion testing system for non-homogeneous, orthotropic materials using the incremental step-wise application of torsion and simultaneous time-lapsed μCT imaging was designed and validated.     

Adaptive changes in micromechanical environments of cancellous and cortical bone in response to in vivo loading and disuse

The skeleton accommodates changes in mechanical environments by increasing bone mass under increased loads and decreasing bone mass under disuse. However, little is known about the adaptive changes in micromechanical behavior of cancellous and cortical tissues resulting from loading or disuse. To address this issue, in vivo tibial loading and hindlimb unloading experiments were conducted on 16-week-old female C57BL/6J mice. Changes in bone mass and tissue-level strains in the metaphyseal cancellous and midshaft cortical bone of the tibiae, resulting from loading or unloading, were determined using microCT and finite element (FE) analysis, respectively. We found that loading- and unloading-induced changes in bone mass were more pronounced in the cancellous than cortical bone. Simulated FE-loading showed that a greater proportion of elements experienced relatively lower longitudinal strains following load-induced bone adaptation, while the opposite was true in the disuse model. While the magnitudes of maximum or minimum principal strains in the metaphyseal cancellous and midshaft cortical bone were not affected by loading, strains oriented with the long axis were reduced in the load-adapted tibia suggesting that loading-induced micromechanical benefits were aligned primarily in the loading direction. Regression analyses demonstrated that bone mass was a good predictor of bone tissue strains for the cortical bone but not for the cancellous bone, which has complex microarchitecture and spatially-variant strain environments. In summary, loading-induced micromechanical benefits for cancellous and cortical tissues are received primarily in the direction of force application and cancellous bone mass may not be related to the micromechanics of cancellous bone.     

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.     

High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone

The ability to predict trabecular failure using microstructure-based computational models would greatly facilitate study of trabecular structure–function relations, multiaxial strength, and tissue remodeling. We hypothesized that high-resolution finite element models of trabecular bone that include cortical-like strength asymmetry at the tissue level, could predict apparent level failure of trabecular bone for multiple loading modes. A bilinear constitutive model with asymmetric tissue yield strains in tension and compression was applied to simulate failure in high-resolution finite element models of seven bovine tibial specimens. Tissue modulus was reduced by 95% when tissue principal strains exceeded the tissue yield strains. Linear models were first calibrated for effective tissue modulus against specimen-specific experimental measures of apparent modulus, producing effective tissue moduli of (mean±S.D.) 18.7±3.4 GPa. Next, a parameter study was performed on a single specimen to estimate the tissue level tensile and compressive yield strains. These values, 0.60% strain in tension and 1.01% strain in compression, were then used in non-linear analyses of all seven specimens to predict failure for apparent tensile, compressive, and shear loading. When compared to apparent yield properties previously measured for the same type of bone, the model predictions of both the stresses and strains at failure were not statistically different for any loading case (p>0.15). Use of symmetric tissue strengths could not match the experimental data. These findings establish that, once effective tissue modulus is calibrated and uniform but asymmetric tissue failure strains are used, the resulting models can capture the apparent strength behavior to an outstanding level of accuracy. As such, these computational models have reached a level of fidelity that qualifies them as surrogates for destructive mechanical testing of real specimens.     

The mechanical behaviour of cancellous bone

Cancellous bone has a cellular structure: it is made up of a connected network of rods and plates. Because of this, its mechanical behaviour is similar to that of other cellular materials such as polymeric foams. A recent study on the mechanisms of deformation in such materials has led to an understanding of how their mechanical properties depend on their relative density, cell wall properties and cell geometry. In this paper, the results of this previous study are applied to cancellous bone in an attempt to further understand its mechanical behaviour. The results of the analysis agree reasonably well with experimental data available in the literature.     

Critical evaluation of known bone material properties to realize anisotropic FE-simulation of the proximal femur

PURPOSE: In a meta-analysis of the literature we evaluated the present knowledge of the material properties of cortical and cancellous bone to answer the question whether the available data are sufficient to realize anisotropic finite element (FE)-models of the proximal femur. MATERIAL AND METHOD: All studies that met the following criteria were analyzed: Young's modulus, tensile, compressive and torsional strengths, Poisson's ratio, the shear modulus and the viscoelastic properties had to be determined experimentally. The experiments had to be carried out in a moist environment and at room temperature with freshly removed and untreated human cadaverous femurs. All material properties had to be determined in defined load directions (axial, transverse) and should have been correlated to apparent density (g/cm(3)), reflecting the individually variable and age-dependent changes of bone material properties. RESULTS: Differences in Young's modulus of cortical [cancellous] bone at a rate of between 33% (58%) (at low apparent density) and 62% (80%) (at high apparent density), are higher in the axial than in the transverse load direction. Similar results have been seen for the compressive strength of femoral bone. For the tensile and torsional strengths, Poisson's ratio and the shear modulus, only ultimate values have been found without a correlation to apparent density. For the viscoelastic behaviour of bone only data of cortical bone and in axial load direction have been described up to now. CONCLUSIONS: Anisotropic FE-models of the femur could be realized for most part with the summarized material properties of bone if characterized by apparent density and load directions. Because several mechanical properties have not been correlated to these main criteria, further experimental investigations will be necessary in future.     

A physical model for simulating fusion of impaction-grafted morselized cancellous bone

This article describes a method to simulate fusion of morselized cancellous bone. The morselized cancellous bone is mixed with an epoxy adhesive, in empirically determined proportions. The mixture is then impacted into a construct. When the epoxy cures, the morselized material fuses into a cohesive, contiguous structure with a compressive modulus equivalent to that of intact cancellous bone. This model can be used to study biomechanical aspects of fused impaction grafts.     

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.     

Mechanical behavior of a new biphasic calcium phosphate bone graft

The aim of this study was to create a new porous calcium phosphate implant for use as a synthetic bone graft substitute. Porous bioceramic was fabricated using a foam-casting method. By using polyurethane foam and a slurry containing hydroxyapatite-dicalcium phosphate powder, water, and additives, a highly porous structure (66 ± 5%) was created. The porous specimens possess an elastic modulus of 330 ± 32 MPa and a compressive strength of 10.3 ± 1.7 MPa. The X-ray diffraction patterns show hydroxyapatite and beta-pyrophosphate phases after sintering. A rabbit model was developed to evaluate the compressive strength and elastic modulus of cancellous bone defects treated with these porous synthetic implants. The compressive mechanical properties became weaker until the second month post implantation. After the second month, these properties increased slightly and remained higher than control values. New bone formed on the outside surface and on the macropore walls of the specimens, as osteoids and osteoclasts were evident two months postoperatively. Considering these properties, these synthetic porous calcium phosphate implants could be applicable as cancellous bone substitutes.     

Multiaxial strength characteristics of trabecular bone

Bovine trabecular bone specimens were tested in multiaxial stress, including pure shear, in a special test device. Shear strength was proportional to apparent density to the exponent 1.65, in approximate agreement with theoretical and experimental studies on the shear strengths of porous foams. The mean shear strength was 6.60 +/- 1.66 MPa, after normalizing for apparent density. This compares well with normalized shear strengths from Saha and Gorman's (1981) study on human femora. A scanning electron microscope study indicated random trabecular architecture and a complex fracture mechanism at the level of the individual trabecular struts. Hoffmann's (1967) 3-D isotropic failure criterion was applied to the multiaxial test data, along with data from uniaxial compression tests, indicating a compressive strength approximately three times the tensile strength.