The mechanical behavior of PMMA/bone specimens extracted from augmented vertebrae: a numerical study of interface properties, PMMA shrinkage and trabecular bone damage

Recently published compression tests on PMMA/bone specimens extracted after vertebral bone augmentation indicated that PMMA/bone composites were not reinforced by the trabecular bone at all. In this study, the reasons for this unexpected behavior should be investigated by using non-linear micro-FE models. Six human vertebral bodies were augmented with either standard or low-modulus PMMA cement and scanned with a HR-pQCT system before and after augmentation. Six cylindrical PMMA/bone specimens were extracted from the augmented region, scanned with a micro-CT system and tested in compression. Four different micro-FE models were generated from these images which showed different bone tissue material behavior (with/without damage), interface behavior (perfect bonding, frictionless contact) and PMMA shrinkage due to polymerization. The non-linear stress-strain curves were compared between the different micro-FE models as well as to the compression tests of the PMMA/bone specimens. Micro-FE models with contact between bone and cement were 20% more compliant compared to those with perfect bonding. PMMA shrinkage damaged the trabecular bone already before mechanical loading, which further reduced the initial stiffness by 24%. Progressing bone damage during compression dominated the non-linear part of the stress-strain curves. The micro-FE models including bone damage and PMMA shrinkage were in good agreement with the compression tests. The results were similar with both cements. In conclusion, the PMMA/bone interface properties as well as the initial bone damage due to PMMA polymerization shrinkage clearly affected the stress-strain behavior of the composite and explained why trabecular bone did not contribute to the stiffness and strength of augmented bone.     

The effect of specimen geometry on the mechanical behaviour of trabecular bone specimens.

The effect of specimen geometry on the mechanical behaviour of trabecular bone specimens was studied by non-destructive uniaxial compression to 0.4% strain using cylindrical specimens with different sizes and length-to-diameter ratios, and by comparing cubic and cylindrical specimens with the same cross-sectional area. Both the length and the cross-sectional area of the specimen had a highly significant influence on the mechanical behaviour (p less than 0.0001). Within the actual range of length (2.75-11.0 mm) the normalized stiffness (Young's modulus) was related nearly linearly to the specimen length. This dependency on specimen length is suggested to be caused mainly by structural disintegrity of the trabecular specimens near the surface. The normalized stiffness (Young's modulus) was also positively correlated to the cross-sectional area. This dependency on cross-sectional area is probably due to friction-induced stress inhomogeneity at the platen-specimen interface. A cube with side length 6.5 mm or a cylindrical specimen with 7.5 mm diameter and 6.5 mm length are suggested as standard specimens for comparative studies on trabecular bone mechanics.     

Tensile strength of bovine trabecular bone

Data on the tensile and compressive properties of trabecular bone are needed to define input parameters and failure criteria for modeling total joint replacements. To help resolve differences in reports comparing tensile and compressive properties of trabecular bone, we have developed new methods, based on porous foam technology, for tensile testing of fresh/frozen trabecular bone specimens. Using bovine trabecular bone from an isotropic region from the proximal humerus as a model material, we measured ultimate strengths in tension and compression for two groups of 24 specimens each. The average ultimate strength in tension was 7.6 +/- 2.2 (95% C.I.) MPa and in compression was 12.4 +/- 3.2 MPa. This difference was statistically significant (p = 0.013) and was not related to density differences between the test groups (p = 0.28). Strength was related by a power-law function of the local apparent density, but, even accounting for density influences, isotropic bovine trabecular bone exhibits significantly lower strengths in tension than in compression.     

Compressive axial strain distributions in cancellous bone specimens

The compressive axial strain distribution in cylindrical trabecular bone specimens was studied using digitized images of the specimen surface. Specimens were tested with strain rate 0.00015 s-1. Images were taken at 0, 1, 2, 3, 4, 6, 8 and 10% strain. Using an optical illusion of movement by rapidly changing succeeding images, failures were classified as transverse (33%) or oblique collapses (67%). The location of failure was not determined by the specimen density gradient. Local axial strain in the distal, intermediate and proximal third was measured throughout the compression in the transversely failing specimens, whereas local strain in the obliquely failing specimens was measured in the pre-failure phase only. Axial strain inhomogeneity was observed in the pre-failure as well as in the post-failure phase. In the pre-failure phase the intermediate third was strained significantly less than the thirds near the ends. In the post-failure phase specimen strain occurred solely in the collapsed part. Ultimate strain of the transversely failing specimens was 2.5% and ultimate strain of the failing third was 3.7%. At failure less than 1% strain was observed in the intermediate third and at 10% specimen strain 1.5% local strain was found in the intermediate third. The results indicate unreliability of conventional stiffness and strain measurements in trabecular bone specimens probably due to lack of trabecular constraint at the end surfaces. Conventional measurements tend to underestimate stiffness and, by giving an average value of strain in spite of considerable strain inhomogeneity, to underestimate failure strain.     

Age-related differences in the morphology of microdamage propagation in trabecular bone

Microdamage density has been shown to increase with age in trabecular bone and is associated with decreased fracture toughness. Numerous studies of crack propagation in cortical bone have been conducted, but data in trabecular bone is lacking. In this study, propagation of severe, linear, and diffuse damage was examined in trabecular bone cores from the femoral head of younger (61.3±3.1 years) and older (75.0±3.9 years) men and women. Using a two-step mechanical testing protocol, damage was first initiated with static uniaxial compression to 0.8% strain then propagated at a normalized stress level of 0.005 to a strain endpoint of 0.8%. Coupling mechanical testing with a dual-fluorescent staining technique, the number and length/area of propagating cracks were quantified. It was found that the number of cycles to the test endpoint was substantially decreased in older compared to younger samples (younger: 77,372±15,984 cycles; older: 34,944±11,964 cycles, p=0.06). This corresponded with a greater number of severely damaged trabeculae expanding in area during the fatigue test in the older group. In the younger group, diffusely damaged trabeculae had a greater damage area, which illustrates an efficient energy dissipation mechanism. These results suggest that age-related differences in fatigue life of human trabecular bone may be due to differences in propagated microdamage morphology.     

Application of homogenization theory to the study of trabecular bone mechanics.

It is generally accepted that the strength and stiffness of trabecular bone is strongly affected by trabecular microstructure. It has also been hypothesized that stress induced adaptation of trabecular bone is affected by trabecular tissue level stress and/or strain. At this time, however, there is no generally accepted (or easily accomplished) technique for predicting the effect of microstructure on trabecular bone apparent stiffness and strength or estimating tissue level stress or strain. In this paper, a recently developed mechanics theory specifically designed to analyze microstructured materials, called the homogenization theory, is presented and applied to analyze trabecular bone mechanics. Using the homogenization theory it is possible to perform microstructural and continuum analyses separately and then combine them in a systematic manner. Stiffness predictions from two different microstructural models of trabecular bone show reasonable agreement with experimental results, depending on metaphyseal region, (R2 greater than 0.5 for proximal humerus specimens, R2 less than 0.5 for distal femur and proximal tibia specimens). Estimates of both microstructural strain energy density (SED) and apparent SED show that there are large differences (up to 30 times) between apparent SED (as calculated by standard continuum finite element analyses) and the maximum microstructural or tissue SED. Furthermore, a strut and spherical void microstructure gave very different estimates of maximum tissue SED for the same bone volume fraction (BV/TV). The estimates from the spherical void microstructure are between 2 and 20 times greater than the strut microstructure at 10-20% BV/TV.     

Stiffness behaviour of trabecular bone specimens

Trabecular bone specimens were tested by non-destructive technique with the purpose of investigating stiffness behaviour and optimizing stiffness determination. Cylindrical specimens (n = 25) were loaded repetitively (0.1 Hz, 30 cycles) by axial compression to 50% of predicted ultimate strength and finally compressed to failure. Analyses of single compression curves showed increasing stiffness (E') until a stress level about 50% of ultimate stress followed by decreasing stiffness. Curve fit analysis of the elastic part of the compression curve showed the best fit, when a second order polynomial was used (r = 0.94, p less than 0.001). The stiffness determined non-destructively at the 25% level of ultimate strength increased significantly to the tenth loading cycle followed by a steady state. The precision of stiffness determination as an average of five consecutive measurements at steady state was E' +/- less than 5% (95% confidence limits). A reproducibility test by repetition of the test sequence after 3 h rest showed qualitatively the same stiffness behaviour. The variation of stiffness determination between the two test sequences was +/- 27% at the first loading cycle falling to +/- 12% at steady state.     

Three-axial strain controlled testing applied to bone specimens from the proximal tibial epiphysis

Reproducibility of the determination of Young's modulus and energy absorption along the three axes of trabecular bone cubes was analysed by non-destructive compression to 0.5% strain using different testing protocols. These protocols included testing with and without pre-conditioning to a viscoelastic steady state, and different orders of test directions. Reproducibility of conditioned tests was generally better than that of non-conditioned tests. No major effect of changing the order of the test direction was found. Three-axial conditioned testing of cubes from the proximal tibial epiphysis of five humans revealed a global transverse isotrophy while most cubes showed orthotropy. The ratio between stiffness along the long axis of the tibia and the stiffness in the transverse plane was 3.7 +/- 0.4 (mean +/- 2 SE). The corresponding ratios for elastic energy storage and viscoelastic energy dissipation were 2.5 +/- 0.2. There was no difference between the relative energy loss during a testing cycle (loss tangent) in the three axes.     

The biomechanical effect of torsion on humeral shaft repair techniques for completed pathological fractures

In the presence of a tumor defect, completed humeral shaft fractures continue to be a major surgical challenge since there is no "gold standard" treatment. This is due, in part, to the fact that only one prior biomechanical study exists on the matter, but which only compared 2 repair methods. The current authors measured the humeral torsional performance of 5 fixation constructs for completed pathological fractures. In 40 artificial humeri, a 2-cm hemi-cylindrical cortical defect with a transverse fracture was created in the lateral cortex. Specimens were divided into 5 different constructs and tested in torsion. Construct A was a broad 10-hole 4.5-mm dynamic compression plate (DCP). Construct B was the same as A except that the screw holes and the tumor defect were filled with bone cement and the screws were inserted into soft cement. Construct C was the same as A except that the canal and tumor defect were filled with bone cement and the screws were inserted into dry cement. Construct D was a locked intramedullary nail inserted in the antegrade direction. Construct E was the same as D except that bone cement filled the defect. For torsional stiffness, construct C (4.45 ± 0.20 Nm/deg) was not different than B or E (p > 0.16), but was higher than A and D (p < 0.001). For failure torque, construct C achieved a higher failure torque (69.65 ± 5.35 Nm) than other groups (p < 0.001). For the failure angle, there were no differences between plating constructs A to C (p ≥ 0.11), except for B versus C (p < 0.05), or between nailing groups D versus E (p = 0.97), however, all plating groups had smaller failure angles than both nailing groups (p < 0.05). For failure energy, construct C (17.97 ± 3.59 J) had a higher value than other groups (p < 0.005), except for A (p = 0.057). Torsional failure always occurred in the bone in the classic "spiral" pattern. Construct C provided the highest torsional stability for a completed pathological humeral shaft fracture.     

Improvement of mechanical properties of acrylic bone cement by fiber reinforcement

Acrylic bone cement is significantly weaker and less stiff than compact bone. Bone cement is also weaker in tension than in compression. This limits its use in orthopaedics to areas where tensile stresses are minimum. We have attempted to improve the mechanical properties of PMMA by reinforcing it with metal wires, and graphite and aramid fibers. Normal, carbon fiber reinforced and aramid fiber reinforced bone cement specimens were tested in compression. Addition of a small percentage (1-2% by weight for carbon and up to 6% for aramid) of these fibers improved the mechanical properties significantly. Due to the improved mechanical properties of fiber reinforced bone cement, its clinical use may reduce the incidence of cement fracture and thus loosening of the prosthesis.     

Micro-mechanical damage of trabecular bone–cement interface under selected loading conditions: a finite element study

In this study, two micro finite element models of trabecular bone–cement interface developed from high resolution computed tomography (CT) images were loaded under compression and validated using the in situ experimental data. The models were then used under tension and shear to examine the load transfer between the bone and cement and the micro damage development at the bone–cement interface. In addition, one models was further modified to investigate the effect of cement penetration on the bone–cement interfacial behaviour. The simulated results show that the load transfer at the bone–cement interface occurred mainly in the bone cement partially interdigitated region, while the fully interdigitated region seemed to contribute little to the mechanical response. Consequently, cement penetration beyond a certain value would seem to be ineffective in improving the mechanical strength of trabecular bone–cement interface. Under tension and shear loading conditions, more cement failures were found in denser bones, while the cement damage is generally low under compression.     

Trabecular bone microdamage and microstructural stresses under uniaxial compression

The balance between local remodeling and accumulation of trabecular bone microdamage is believed to play an important role in the maintenance of skeletal integrity. However, the local mechanical parameters associated with microdamage initiation are not well understood. Using histological damage labeling, micro-CT imaging, and image-based finite element analysis, regions of trabecular bone microdamage were detected and registered to estimated microstructural von Mises effective stresses and strains, maximum principal stresses and strains, and strain energy density (SED). Bovine tibial trabecular bone cores underwent a stepwise uniaxial compression routine in which specimens were micro-CT imaged following each compression step. The results indicate that the mode of trabecular failure observed by micro-CT imaging agreed well with the polarity and distribution of stresses within an individual trabecula. Analysis of on-axis subsections within specimens provided significant positive relationships between microdamage and each estimated tissue stress, strain and SED parameter. In a more localized analysis, individual microdamaged and undamaged trabeculae were extracted from specimens loaded within the elastic region and to the apparent yield point. As expected, damaged trabeculae in both groups possessed significantly higher local stresses and strains than undamaged trabeculae. The results also indicated that microdamage initiation occurred prior to apparent yield at local principal stresses in the range of 88-121 MPa for compression and 35-43 MPa for tension and local principal strains of 0.46-0.63% in compression and 0.18-0.24% in tension. These data provide an important step towards understanding factors contributing to microdamage initiation and establishing local failure criteria for normal and diseased trabecular bone.     

Effect of trabecular curvature on the stiffness of trabecular bone

Simplified structural models of trabecular bone have been used to model various forms of trabecular variability. The structural effects of variability of direction, length and thickness of the trabeculae have been studied using 'lattice-type' finite element models. However, many of the trabeculae are not perfectly straight, and have a small degree of curvature. The objective of this study is to quantify the influence of small curvatures of the trabeculae on the effective modulus of trabecular bone, in the principal material direction. An analytical analysis of the effect of curvature on a single trabecula is performed, utilizing the concept of cellular-solid models. Closed-form expressions are derived for the effect of curvature on the flexibility in the principal material direction. For comparison, expressions are derived for the flexibility of a straight oblique element, representing angular variability. A quantitative comparison is presented, which is dependent on the thickness of the trabeculae. It was found that small curvatures have a large effect on the stiffness of the trabecular structure. This effect is largest for thin trabeculae, and decreases for thick trabeculae. The stiffness of the trabecular structure can be reduced by a factor of up to four for thin trabeculae and up to two for thick trabeculae, even for small curvatures. The flexibility of curved elements is found to be larger than the flexibility of oblique elements with similar eccentricities. Thus it seems that curvature might play a role in determining the effective modulus of trabecular bone.     

Validation of composite finite elements efficiently simulating elasticity of trabecular bone

Patient-specific analyses of the mechanical properties of bones become increasingly important for the management of patients with osteoporosis. The potential of composite finite elements (CFEs), a novel FE technique, to assess the apparent stiffness of vertebral trabecular bone is investigated in this study. Segmented volumes of cylindrical specimens of trabecular bone are compared to measured volumes. Elasticity under uniaxial loading conditions is simulated; apparent stiffnesses are compared to experimentally determined values. Computational efficiency is assessed and recommendations for simulation parameters are given. Validating apparent uniaxial stiffnesses results in concordance correlation coefficients 0.69 ≤ r_𝒸 ≤ 0.92 for resolutions finer than 168 μm, and an average error of 5.8% between experimental and numerical results at 24 μm resolution. As an application, the code was used to compute local, macroscopic stiffness tensors for the trabecular structure of a lumbar vertebra. The presented technique allows for computing stiffness using smooth FE meshes at resolutions that are well achievable in peripheral high resolution quantitative CT. Therefore, CFEs could be a valuable tool for the patient-specific assessment of bone stiffness.     

The effects of bone and pore volume fraction on the mechanical properties of PMMA/bone biopsies extracted from augmented vertebrae

Vertebroplasty forms a porous PMMA/bone composite which was shown to be weaker and less stiff than pure PMMA. It is not known what determines the mechanical properties of such composites in detail. This study investigated the effects of bone volume fraction (BV/TV), cement porosity (PV/(TV-BV), PV…pore volume) and cement stiffness. Nine human vertebral bodies were augmented with either standard or low-modulus PMMA cement and scanned with a HR-pQCT system before and after augmentation. Fourteen cylindrical PMMA/bone biopsies were extracted from the augmented region, scanned with a micro-CT system and tested in compression until failure. Micro-finite element (FE) models of the complete biopsies, of the trabecular bone alone as well as of the porous cement alone were generated from CT images to gain more insight into the role of bone and pores. PV/(TV-BV) and experimental moduli of standard/low-modulus cement (R(2)=0.91/0.98) as well as PV/(TV-BV) and yield stresses (R(2)=0.92/0.83) were highly correlated. No correlation between BV/TV (ranging from 0.057 to 0.138) and elastic moduli was observed (R(2)< 0.05). Interestingly, the micro-FE models of the porous cement alone reproduced the experimental elastic moduli of the standard/low-modulus cement biopsies (R(2)=0.75/0.76) more accurately than the models with bone (R(2)=0.58/0.31). In conclusion, the mechanical properties of the biopsies were mainly determined by the cement porosity and the cement material properties. The study showed that bone tissue inside the biopsies was mechanically "switched off" such that load was carried essentially by the porous PMMA.     

Modeling modulus reduction in bovine trabecular bone damaged in compression.

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

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

High-frequency oscillatory motions enhance the simulated mechanical properties of non-weight bearing trabecular bone

Extremely low-level oscillatory accelerations, applied without constraint, can increase bone formation. Here, we tested the hypothesis that high-frequency oscillations, applied in the absence of functional weight bearing, can be sensed by trabecular bone to produce a structure that is more efficient in sustaining applied loads. The left leg of anesthetized adult female mice (n=18) was subjected to high-frequency oscillations at 45 Hz, 0.6g for 20 min/day, 5 days/week for 3 weeks, while the contralateral leg served as an internal control. To remove the potential interference of the habitual strain environment with the imposed physical signal, the hindlimbs of these mice were chronically unloaded. In vivo microCT scans of the proximal metaphyseal region of the tibia were transformed into finite element meshes to evaluate trabecular and cortical mechanical properties. Simulated longitudinal compression tests showed that the short applications of high-frequency oscillations were sensed primarily by trabecular bone. At the end of the experimental period, apparent trabecular stiffness of the oscillated bones was 38% (p<0.001) greater than that of non-weight bearing controls. Simulated uniaxial loads applied to trabecular bone induced 21%, 52%, and 131% greater (p<0.05) median, peak compressive, and peak tensile longitudinal stresses in control than in stimulated bones. Non-weight bearing control bones were also characterized by greater transverse normal and shear stresses (77% and 54%, respectively, p<0.001) as well as 35% greater (p=0.03) longitudinal shear stresses. Compared to normal age-matched controls (n=18), oscillations were able to attenuate, but not fully prevent, the decline in trabecular mechanical properties associated with the removal of weight bearing. These data indicate not only that bone cells can sense low-level, high-frequency oscillatory accelerations, but also that they can orchestrate a structural response that produces a stiffer trabecular structure that may be less prone to fracture.     

Damage in trabecular bone at small strains

Evidence that damage decreases bone quality, increases fracture susceptibility, and serves as a remodeling stimulus motivates further study of what loading magnitudes induce damage in trabecular bone. In particular, whether damage occurs at the smaller strains characteristic of habitual, as opposed to traumatic, loading is not known. The overall goal of this study was to characterize damage accumulation in trabecular bone at small strains (0.20 - 0.45% strain). A continuum damage mechanics approach was taken whereby damage was quantified by changes in modulus and residual strain. Human vertebral specimens (n = 7) were tested in compression using a multi-cycle load - unload protocol in which the maximum applied strain for each cycle, epsilonmax, was increased incrementally from epsilonmax = 0.20% on the first loading cycle to epsilonmax = 0.45% on the last cycle. Modulus and residual strain were measured for each cycle. Both changes in modulus and residual strains commenced at small strains, beginning as early as 0.24 and 0.20% strain, respectively. Strong correlations between changes in modulus and residual strains were observed (r = 0.51 - 0.98). Fully nonlinear, high-resolution finite element analyses indicated that even at small apparent strains, tissue-level strains were sufficiently high to cause local yielding. These results demonstrate that damage in trabecular bone occurs at apparent strains less than half the apparent compressive yield strain reported previously for human vertebral trabecular bone. Further, these findings imply that, as a consequence of the highly porous trabecular structure, tissue yielding can initiate at very low apparent strains and that this local failure has detectable and negative consequences on the apparent mechanical properties of trabecular bone.     

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

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