NACOB presentation to ASB Young Scientist Award: Postdoctoral. The impact of boundary conditions and mesh size on the accuracy of cancellous bone tissue modulus determination using large-scale finite-element modeling. North American Congress on Biomechanics

A combined experimental and numerical method to estimate the elastic modulus of single trabeculae

The elastic modulus at the single trabecular level is an important parameter for the understanding of the mechanical behavior of trabecular bone. Current methods are commonly limited by the irregular trabecular shape and the accuracy of displacement measurement. The aim of this study was to propose a method to estimate the trabecular modulus overcoming some of these limitations. For high-precision displacement measurements, in-situ compression within a synchrotron radiation based X-ray tomograph was used. Trabecular displacements were subsequently estimated by a global digital volume correlation algorithm, followed by high-resolution finite element analyses to account for the irregular geometry. The trabecular elastic moduli were then estimated by comparing the loads from the finite element analyses with those of the experiments. With this strategy, the average elastic modulus was estimated to 3.83 ± 0.54 GPa for three human trabeculae samples. Though limited by the sample size, the demonstrated method shows a potential to estimate the mechanical properties at the single trabecular level.

Effect of CT imaging on the accuracy of the finite element modelling in bone

The finite element (FE) analysis is a highly promising tool to simulate the behaviour of bone. Skeletal FE models in clinical routine rely on the information about the geometry and bone mineral density distribution from quantitative computed tomography (CT) imaging systems. Several parameters in CT imaging have been reported to affect the accuracy of FE models. FE models of bone are exclusively developed in vitro under scanning conditions deviating from the clinical setting, resulting in variability of FE results (< 10%). Slice thickness and field of view had little effect on FE predicted bone behaviour (≤ 4%), while the reconstruction kernels showed to have a larger effect (≤ 20%). Due to large interscanner variations (≤ 20%), the translation from an experimental model into clinical reality is a critical step. Those variations are assumed to be mostly caused by different “black box” reconstruction kernels and the varying frequency of higher density voxels, representing cortical bone. Considering the low number of studies together with the significant effect of CT imaging on the finite element model outcome leading to high variability in the predicted behaviour, we propose further systematic research and validation studies, ideally preceding multicentre and longitudinal studies.

Micro-finite-element method to assess elastic properties of trabecular bone at micro- and macroscopic level

OBJECTIVE OF THE STUDY

The objective of the present study is to assess the mechanical behavior of trabecular bone based on microCT imaging and micro-finite-element analysis. In this way two methods are detailed: (i) direct determination of macroscopic elastic property of trabecular bone; (ii) inverse approach to assess mechanical properties of trabecular bone tissue.

PATIENTS

Thirty-five females and seven males (forty-two subjects) mean aged (±SD) 80±11.7 years from hospitals of Assistance publique-Hôpitaux de Paris (AP-HP) diagnosed with osteoporosis following a femoral neck fracture due to a fall from standing were included in this study.

MATERIALS AND METHODS

Fractured heads were collected during hip replacement surgery. Standardized bone cores were removed from the femoral head's equator by a trephine in a water bath. MicroCT images acquisition and analysis were performed with CTan^® software and bone volume fraction was then determined. Micro-finite-element simulations were per-formed using Abaqus 6.9-2^® software in order to determine the macroscopic mechanical behaviour of the trabecular bone. After microCT acquisition, a longitudinal compression test was performed and the experimental macroscopic Young's Modulus was extracted. An inverse approach based on the whole trabecular bone's mechanical response and micro-finite-element analysis was performed to determine microscopic mechanical properties of trabecular bone.

RESULTS

In the present study, elasticity of the tissue was shown to be similar to that of healthy tissue but with a lower yield stress.

CONCLUSION

Classical histomorphometric analysis form microCT imaging associated with an inverse micro-finite-element method allowed to assess microscopic mechanical trabecular bone parameters.

Relationship between sample volumes and modulus of human vertebral trabecular bone in micro-finite element analysis

Micro-finite element (μFE) models have been widely used to assess the biomechanical properties of trabecular bone. How to choose a proper sample volume of trabecular bone, which could predict the real bone biomechanical properties and reduce the calculation time, was an interesting problem. Therefore, the purpose of this study was to investigate the relationship between different sample volumes and apparent elastic modulus (E) calculated from μFE model. 5 Human lumbar vertebral bodies (L1-L5) were scanned by micro-CT. Cubic concentric samples of different lengths were constructed as the experimental groups and the largest possible volumes of interest (VOI) were constructed as the control group. A direct voxel-to-element approach was used to generate μFE models and steel layers were added to the superior and inferior surface to mimic axial compression tests. A 1% axial strain was prescribed to the top surface of the model to obtain the E values. ANOVA tests were performed to compare the E values from the different VOIs against that of the control group. Nonlinear function curve fitting was performed to study the relationship between volumes and E values. The larger cubic VOI included more nodes and elements, and more CPU times were needed for calculations. E values showed a descending tendency as the length of cubic VOI decreased. When the volume of VOI was smaller than (7.34mm(3)), E values were significantly different from the control group. The fit function showed that E values approached an asymptotic values with increasing length of VOI. Our study demonstrated that apparent elastic modulus calculated from μFE models were affected by the sample volumes. There was a descending tendency of E values as the length of cubic VOI decreased. Sample volume which was not smaller than (7.34mm(3)) was efficient enough and timesaving for the calculation of E.

Elastic modulus of the femoral trochanteric region measured by scanning acoustic microscopy in elderly women

PURPOSE

We have devised a method that can obtain continuous detailed distributions of the elastic modulus along the measurement line in a non-decalcified specimen of human bone tissue. The aim of this study was to determine whether local variations exist in the distribution of mechanical properties within the trochanteric region of the femur of elderly females using a newly developed form of scanning acoustic microscopy (SAM) technology.

METHODS

Human proximal femurs were harvested from seven female cadavers aged between 67 and 88 years at death. Using data collected with SAM, the elastic modulus of cortical and trabecular bone tissue of the lateral and medial trochanter was calculated and statistically analyzed.

RESULTS

The longitudinal and transverse elastic moduli in cortical bone tissue of the lateral trochanter were found to be significantly lower than those of the medial trochanter in all specimens from individuals over age 70 (p < 0.05). Compared to that of the distal region, the longitudinal and transverse elastic moduli of trabecular bone tissue of the proximal region of the lateral trochanter were significantly lower in all specimens from individuals over age 80 (p < 0.05).

CONCLUSION

Our new method allows obtaining detailed distributions of the elastic modulus of bone tissue.

Biomechanics and mechanobiology of trabecular bone: a review

Trabecular bone is a highly porous, heterogeneous, and anisotropic material which can be found at the epiphyses of long bones and in the vertebral bodies. Studying the mechanical properties of trabecular bone is important, since trabecular bone is the main load bearing bone in vertebral bodies and also transfers the load from joints to the compact bone of the cortex of long bones. This review article highlights the high dependency of the mechanical properties of trabecular bone on species, age, anatomic site, loading direction, and size of the sample under consideration. In recent years, high resolution micro finite element methods have been extensively used to specifically address the mechanical properties of the trabecular bone and provide unique tools to interpret and model the mechanical testing experiments. The aims of the current work are to first review the mechanobiology of trabecular bone and then present classical and new approaches for modeling and analyzing the trabecular bone microstructure and macrostructure and corresponding mechanical properties such as elastic properties and strength.

Computer-aided tissue engineering: benefiting from the control over scaffold micro-architecture

Minimization schema in nature affects the material arrangements of most objects, independent of scale. The field of cellular solids has focused on the generalization of these natural architectures (bone, wood, coral, cork, honeycombs) for material improvement and elucidation into natural growth mechanisms. We applied this approach for the comparison of a set of complex three-dimensional (3D) architectures containing the same material volume but dissimilar architectural arrangements. Ball and stick representations of these architectures at varied material volumes were characterized according to geometric properties, such as beam length, beam diameter, surface area, space filling efficiency, and pore volume. Modulus, deformation properties, and stress distributions as contributed solely by architectural arrangements was revealed through finite element simulations. We demonstrated that while density is the greatest factor in controlling modulus, optimal material arrangement could result in equal modulus values even with volumetric discrepancies of up to 10%. We showed that at low porosities, loss of architectural complexity allows these architectures to be modeled as closed celled solids. At these lower porosities, the smaller pores do not greatly contribute to the overall modulus of the architectures and that a stress backbone is responsible for the modulus. Our results further indicated that when considering a deposition-based growth pattern, such as occurs in nature, surface area plays a large role in the resulting strength of these architectures, specifically for systems like bone. This completed study represents the first step towards the development of mathematical algorithms to describe the mechanical properties of regular and symmetric architectures used for tissue regenerative applications. The eventual goal is to create logical set of rules that can explain the structural properties of an architecture based solely upon its geometry. The information could then be used in an automatic fashion to generate patient-specific scaffolds for the treatment of tissue defects.

The relationship between trabecular bone structure modeling methods and the elastic modulus as calculated by FEM

Trabecular bone cores were collected from the femoral head at the time of surgery (hip arthroplasty). Investigated were 42 specimens, from patients with osteoporosis and coxarthrosis. The cores were scanned used computer microtomography (microCT) system at an isotropic spatial resolution of 36 microns. Image stacks were converted to finite element models via a bone voxel-to-element algorithm. The apparent modulus was calculated based on the assumptions that for the elastic properties, E = 10 MPa and ν = 0.3. The compressive deformation as calculated by finite elements (FE) analysis was 0.8%. The models were coarsened to effectively change the resolution or voxel size (from 72 microns to 288 microns or from 72 microns to 1080 microns). The aim of our study is to determine how an increase in the distance between scans changes the elastic properties as calculated by FE models. We tried to find a border value voxel size at which the module values were possible to calculate. As the voxel size increased, the mean voxel volume increased and the FEA-derived apparent modulus decreased. The slope of voxel size versus modulus relationship correlated with several architectural indices of trabecular bone.

Mechanical and microarchitectural analyses of cancellous bone through experiment and computer simulation

The relationship between microarchitecture to the failure mechanism and mechanical properties can be assessed through experimental and computational methods. In this study, both methods were utilised using bovine cadavers. Twenty four samples of cancellous bone were extracted from fresh bovine and the samples were cleaned from excessive marrow. Uniaxial compression testing was performed with displacement control. After mechanical testing, each specimen was ashed in a furnace. Four of the samples were exemplarily scanned using micro-computed tomography (μCT) and three dimensional models of the cancellous bones were reconstructed for finite element simulation. The mechanical properties and the failure modes obtained from numerical simulations were then compared to the experiments. Correlations between microarchitectural parameters to the mechanical properties and failure modes were then made. The Young's modulus correlates well with the bone volume fraction with R² = 0.615 and P value 0.013. Three different types of failure modes of cancellous bone were observed: oblique fracture (21.7%), perpendicular global fracture (47.8%), and scattered localised fracture (30.4%). However, no correlations were found between the failure modes to the morphological parameters. The percentage of error between computer predictions and the actual experimental test was from 6 to 12%. These mechanical properties and information on failure modes can be used for the development of synthetic cancellous bone.

Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid

We study apparent elastic moduli of trabecular bone, which is represented, for simplicity, by a two- or three-dimensional periodic cellular network. The term "apparent" refers to the case when the region used in calculations (or specimen size) is smaller than a representative volume element and the moduli depend on the size of that region and boundary conditions. Both the bone tissue forming the network and the pores (represented by a very soft material) are assumed, for simplicity, as homogeneous, linear elastic, and isotropic. In order to investigate the effects of scale and boundary conditions on the moduli of these networks we vary the specimen size and apply four different boundary conditions: displacement, traction, mixed, and periodic. The analysis using periodic boundary conditions gives the effective moduli, while the displacement, traction, and mixed boundary conditions give apparent moduli. The apparent moduli calculated using displacement and traction boundary conditions bound the effective moduli from above and below, respectively. The larger is the size of the region used in our calculations, the closer are the bounds. Our choice of mixed boundary conditions gives results that are very close to those obtained using periodic boundary conditions. We conduct this analysis computationally using a finite element method. We also investigate the effect of mismatch in elastic moduli of bone tissue and soft fill, trabecular bone structure geometry, and bone tissue volume fraction on the apparent elastic moduli of idealized periodic models of trabecular bone. This study gives guidance on how the size of the specimen and boundary conditions (used in experiments or simulations) influence elastic moduli of cellular materials. This approach is applicable to heterogeneous materials in general.

The influence of boundary conditions and loading mode on high-resolution finite element-computed trabecular tissue properties

A widely used technique for determining the material properties of trabecular tissue is to perform combined experimental and computational testing of trabecular structures in order to calibrate effective tissue properties. To better understand the nature of such properties, we tested n=25 cores of human vertebral trabecular bone under two different boundary conditions (endcap and PMMA embedding) and loading modes (compression and torsion). High-resolution (20 microm) finite element models that explicitly modeled the different experimental conditions were constructed and sensitivity studies were performed to quantify errors arising from uncertainties between model and experiment. Mean (+/-S.D.) effective tissue modulus for the four groups ranged from 9.6+/-1.9 to 11.5+/-3.5 GPa, and the overall mean was 10.3+/-2.4 GPa. For the endcap tests, mean values were the same regardless of loading mode, suggesting that the effective tissue modulus is representative of true material behavior. However, on a specimen-specific basis, the various repeated measures of effective tissue modulus were only moderately correlated with each other (R2=27% to 81%), indicating that the individual measures can be subject to appreciable random errors. The sensitivity studies on the endcap tests indicated that models using lower resolution (40 microm element size) and roller-type platens boundary conditions overestimated effective tissue modulus by 42% on average, although preliminary tests with higher-density femoral neck bone indicated less sensitivity to modeling issues. We conclude that effective tissue properties derived from micro-finite element models do have biomechanical significance if measured correctly, although individual measures of tissue properties may have poor precision.

Tissue modulus calculated from beam theory is biased by bone size and geometry: implications for the use of three-point bending tests to determine bone tissue modulus

Current practice to determine bone tissue modulus of murine cortical bone is to estimate it from three-point bending tests, using Euler-Bernoulli beam theory. However, murine femora are not perfect beams; hence, results can be inaccurate. Our aim was to assess the accuracy of beam theory, which we tested for two commonly used inbred strains of mice, C57BL/6 (B6) and C3H/He (C3H). We measured the three-dimensional structure of male and female B6 and C3H femora (N=20/group) by means of micro-computed tomography. For each femur five micro-finite element (micro-FE) models were created that simulated three-point bending tests with varying distances between the supports. Tissue modulus was calculated from beam theory using micro-FE results. The accuracy of beam theory was assessed by comparing the beam theory-derived moduli with the modulus as used in the micro-FE analyses. An additional set of fresh-frozen femora (10 B6 and 12 C3H) was biomechanically tested and subjected to the same micro-FE analyses. These combined experimental-computational analyses enabled an unbiased assessment of specimen-specific tissue modulus. We found that by using beam theory, tissue modulus was underestimated for all femora. Femoral geometry and size had strong effects on beam theory-derived tissue moduli. Owing to their relatively thin cortex, underestimation was markedly higher for B6 than for C3H. Underestimation was dependent on support width in a strain-specific manner. From our combined experimental-computational approach we calculated tissue moduli of 12.0+/-1.3 GPa and 13.4+/-2.1 GPa for B6 and C3H, respectively. We conclude that tissue moduli in murine femora are strongly underestimated when calculated from beam theory. Using image-based micro-FE analyses we could precisely quantify this underestimation. We showed that previously reported murine inbred strain-specific differences in tissue modulus are largely an effect of geometric differences, not accounted for by beam theory. We suggest a re-evaluation of the tissue properties obtained from three-point bending tests, especially in mouse genetics.

Effects of intracortical porosity on fracture toughness in aging human bone: a microCT-based cohesive finite element study

The extent to which increased intracortical porosity affects the fracture properties of aging and osteoporotic bone is unknown. Here, we report the development and application of a microcomputed tomography based finite element approach that allows determining the effects of intracortical porosity on bone fracture by blocking all other age-related changes in bone. Previously tested compact tension specimens from human tibiae were scanned using microcomputed tomography and converted to finite element meshes containing three-dimensional cohesive finite elements in the direction of the crack growth. Simulations were run incorporating age-related increase in intracortical porosity but keeping cohesive parameters representing other age-related effects constant. Additional simulations were performed with reduced cohesive parameters. The results showed a 6% decrease in initiation toughness and a 62% decrease in propagation toughness with a 4% increase in porosity. The reduction in toughnesses became even more pronounced when other age-related effects in addition to porosity were introduced. The initiation and propagation toughness decreased by 51% and 83%, respectively, with the combined effect of 4% increase in porosity and decrease in the cohesive properties reflecting other age-related changes in bone. These results show that intracortical porosity is a significant contributor to the fracture toughness of the cortical bone and that the combination of computational modeling with advanced imaging improves the prediction of the fracture properties of the aged and the osteoporotic cortical bone.

The Effect of Regional Variations of the Trabecular Bone Properties on the Compressive Strength of Human Vertebral Bodies

Cancellous centrum is a major component of the vertebral body and significantly contributes to its structural strength and fracture risk. We hypothesized that the variability of cancellous bone properties in the centrum is associated with vertebral strength. Microcomputed tomography (micro-CT)-based gray level density (GLD), bone volume fraction (BV/TV), and finite element modulus (E) were examined for different regions of the trabecular centrum and correlated with vertebral body strength determined experimentally. Two sets of images in the cancellous centrum were digitally prepared from micro-CT images of eight human vertebral bodies (T10-L5). One set included a cubic volume (1 per vertebral centrum, n = 8) in which the largest amount of cancellous material from the centrum was included but all the shell materials were excluded. The other set included cylindrical volumes (6 per vertebral centrum, n = 48) from the anterior (4 regions: front, center, left, and right of the midline of vertebra) and the posterior (2 regions: left and right) regions of the centrum. Significant positive correlations of vertebral strength with GLD (r (2) = 0.57, p = 0.03) and E (r (2) = 0.63, p = 0.02) of the whole centrum and with GLD (r (2) = 0.65, p = 0.02), BV/TV (r (2) = 0.72, p = 0.01) and E (r (2) = 0.85, p = 0.001) of the central region of the vertebral centrum were found. Vertebral strength decreased with increasing coefficient of variation of GLD, BV/TV, and E calculated from subregions of the vertebral centrum. The values of GLD, BV/TV, and E in centrum were significantly smaller for the anterior region than for the posterior region. Overall, these findings supported the significant role of regional variability of centrum properties in determining the whole vertebral strength.

Subject-specific finite element models of long bones: An in vitro evaluation of the overall accuracy

The determination of the mechanical stresses induced in human bones is of great importance in both research and clinical practice. Since the stresses in bones cannot be measured non-invasively in vivo, the only way to estimate them is through subject-specific finite element modelling. Several methods exist for the automatic generation of these models from CT data, but before bringing them in the clinical practice it is necessary to assess their accuracy in the predictions of the bone stresses. Particular attention should be paid to those regions, like the epiphyseal and metaphyseal parts of long bones, where the automatic methods are typically less accurate. Aim of the present study was to implement a general procedure to automatically generate subject-specific finite element models of bones from CT data and estimate the accuracy of this general procedure by applying it to one real femur. This femur was tested in vitro under five different loading scenarios and the results of these tests were used to verify how the adoption of a simplified two-material homogeneous model would change the accuracy with respect to the density-based inhomogeneous one, with special attention paid to the epiphyseal and metaphyseal proximal regions of the bone. The results showed that the density-based inhomogeneous model predicts with a very good accuracy the measured stresses (R(2)=0.91, RMSE=8.6%, peak error=27%), while the two-material model was less accurate (R(2)=0.89, RMSE=9.6%, peak error=35%). The results showed that it is possible to automatically generate accurate finite element models of bones from CT data and that the strategy of material properties mapping has a significant influence on its accuracy.

Vertebral osteoporosis and trabecular bone quality

Vertebral fractures due to osteoporosis commonly occur under non-traumatic loading conditions. This problem affects more than 1 in 3 women and 1 in 10 men over a lifetime. Measurement of bone mineral density (BMD) has traditionally been used as a method for diagnosis of vertebral osteoporosis. However, this method does not fully account for the influence of changes in the trabecular bone quality, such as micro-architecture, tissue properties and levels of microdamage, on the strength of the vertebra. Studies have shown that deterioration of the vertebral trabecular architecture results in a more anisotropic structure which has a greater susceptibility to fracture. Transverse trabeculae are preferentially thinned and perforated while the remaining vertical trabeculae maintain their thickness. Such a structure is likely to be more susceptible to buckling under normal compression loads and has a decreased ability to withstand unusual or off-axis loads. Changes in tissue material mechanical properties and levels of microdamage due to osteoporosis may also compromise the fracture resistance of vertebral trabecular bone. New diagnostic techniques are required which will account for the influence of these changes in bone quality. This paper reviews the influence of the trabecular architecture, tissue properties and microdamage on fracture risk for vertebral osteoporosis. The morphological characteristics of normal and osteoporotic architectures are compared and their potential influence on the strength of the vertebra is examined. The limitations of current diagnostic methods for osteoporosis are identified and areas for future research are outlined.

The effects of side-artifacts on the elastic modulus of trabecular bone

Determining accurate density-mechanical property relationships for trabecular bone is critical for correct characterization of this important structure-function relation. When testing any excised specimen of trabecular bone, an unavoidable experimental artifact originates from the sides of the specimen where peripheral trabeculae lose their vertical load-bearing capacity due to interruption of connectivity, a phenomenon denoted here as the 'side-artifact'. We sought in this study to quantify the magnitude of such side-artifact errors in modulus measurement and to do so as a function of the trabecular architecture and specimen size. Using parametric computational analysis of high-resolution micro-CT-based finite-element models of cores of elderly human vertebral trabecular bone, a specimen-specific correction factor for the side-artifact was quantified as the ratio of the side-artifact-free apparent modulus (Etrue) to the apparent modulus that would be measured in a typical experiment (Emeasured). We found that the width over which the peripheral trabeculae were mostly unloaded was between 0.19 and 0.58 mm. The side-artifact led to an underestimation error in Etrue of over 50% in some specimens, having a mean (+/-SD) of 27+/-11%. There was a trend for the correction factor to linearly increase as volume fraction decreased (p=0.001) and as mean trabecular separation increased (p<0.001). Further analysis indicated that the error increased substantially as specimen size decreased. Two methods used for correcting for the side-artifact were both successful in bringing Emeasured into statistical agreement with Etrue. These findings have important implications for the interpretation of almost all literature data on trabecular bone mechanical properties since they indicate that such properties need to be adjusted to eliminate the substantial effects of side-artifacts in order to provide more accurate estimates of in situ behavior.

Evaluation of filler materials used for uniform load distribution at boundaries during structural biomechanical testing of whole vertebrae

This study was designed to compare the compressive mechanical properties of filler materials, Wood's metal, dental stone, and polymethylmethacrylate (PMMA), which are widely used for performing structural testing of whole vertebrae. The effect of strain rate and specimen size on the mechanical properties of the filler materials was examined using standardized specimens and mechanical testing. Because Wood's metal can be reused after remelting, the effect of remelting on the mechanical properties was tested by comparing them before and after remelting. Finite element (FE) models were built to simulate the effect of filler material size and properties on the stiffness of vertebral body construct in compression. Modulus, yield strain, and yield strength were not different between batches (melt-remelt) of Wood's metal. Strain rate had no effect on the modulus of Wood's metal, however, Young's modulus decreased with increasing strain rate in dental stone whereas increased in PMMA. Both Wood's metal and dental stone were significantly stiffer than PMMA (12.7 +/- 1.8 GPa, 10.4 +/- 3.4 GPa, and 2.9 +/- 0.4 GPa, respectively). PMMA had greater yield strength than Wood's metal (62.9 +/- 8.7 MPa and 26.2 +/- 2.6 MPa). All materials exhibited size-dependent modulus values. The FE results indicated that filler materials, if not accounted for, could cause more than 9% variation in vertebral body stiffness. We conclude that Wood's metal is a superior moldable bonding material for biomechanical testing of whole bones, especially whole vertebrae, compared to the other candidate materials.

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.

Computer-Aided Tissue Engineering of a Human Vertebral Body

Tissue engineering is developing into a less speculative science involving the careful interplay of numerous design parameters and multidisciplinary professionals. Problem solving abilities and state of the art research tools are required to develop solutions for a wide variety of clinical issues. One area of particular interest is orthopedic biomechanics, a field that is responsible for the treatment of over 700,000 vertebral fractures in the United States alone last year. Engineers are currently lacking the technology and knowledge required to govern the subsistence of cells in vivo, let alone the knowledge to create a functional tissue replacement for a whole organ. Despite this, advances in computer-aided tissue engineering are continually growing. Using a combinatory approach to scaffold design, patient-specific implants may be constructed. Computer-aided design, optimization of geometry using voxel finite element models or other optimization routines, creation of a library of architectures with specific material properties, rapid prototyping, and determination of a defect site using imaging modalities highlight the current availability of design resources. This study proposes a novel methodology from start to finish which could, in the future, be used to design a tissue-engineered construct for the replacement of an entire vertebral body.

Mechanical properties of femoral trabecular bone in dogs

Background

Studying mechanical properties of canine trabecular bone is important for a better understanding of fracture mechanics or bone disorders and is also needed for numerical simulation of canine femora. No detailed data about elastic moduli and degrees of anisotropy of canine femoral trabecular bone has been published so far, hence the purpose of this study was to measure the elastic modulus of trabecular bone in canine femoral heads by ultrasound testing and to assess whether assuming isotropy of the cancellous bone in femoral heads in dogs is a valid simplification.

Methods

From 8 euthanized dogs, both femora were obtained and cubic specimens were cut from the centre of the femoral head which were oriented along the main pressure and tension trajectories. The specimens were tested using a 100 MHz ultrasound transducer in all three orthogonal directions. The directional elastic moduli of trabecular bone tissue and degrees of anisotropy were calculated.

Results

The elastic modulus along principal bone trajectories was found to be 11.2 GPa ± 0.4, 10.5 ± 2.1 GPa and 10.5 ± 1.8 GPa, respectively. The mean density of the specimens was 1.40 ± 0.09 g/cm³. The degrees of anisotropy revealed a significant inverse relationship with specimen densities. No significant differences were found between the elastic moduli in x, y and z directions, suggesting an effective isotropy of trabecular bone tissue in canine femoral heads.

Discussion

This study presents detailed data about elastic moduli of trabecular bone tissue obtained from canine femoral heads. Limitations of the study are the relatively small number of animals investigated and the measurement of whole specimen densities instead of trabecular bone densities which might lead to an underestimation of Young's moduli. Publications on elastic moduli of trabecular bone tissue present results that are similar to our data.

Conclusion

This study provides data about directional elastic moduli and degrees of anisotropy of canine femoral head trabecular bone and might be useful for biomechanical modeling of proximal canine femora.

Effect of Microcomputed Tomography Voxel Size on the Finite Element Model Accuracy for Human Cancellous Bone

The level of structural detail that can be acquired and incorporated in a finite element (FE) analysis might greatly influence the results of microcomputed tomography (μCT)-based FE simulations, especially when relatively large bones, such as whole vertebrae, are of concern. We evaluated the effect of scanning and reconstruction voxel size on the μCT-based FE analyses of human cancellous tissue samples for fixed- and free-end boundary conditions using different combinations of scan/reconstruction voxel size. We found that the bone volume fraction (BV/TV) did not differ considerably between images scanned at 21 and 50 μm and reconstructed at 21, 50, or 110 μm (−0.5% to 7.8% change from the 21/21 μm case). For the images scanned and reconstructed at 110 μm, however, there was a large increase in BV/TV compared to the 21/21 μm case (58.7%). Fixed-end boundary conditions resulted in 1.8% [coefficient of variation (COV)] to 14.6% (E) difference from the free-end case. Dependence of model output parameters on scanning and reconstruction voxel size was similar between free- and fixed-end simulations. Up to 26%, 30%, 17.8%, and 32.3% difference in modulus (E), and average (VMExp), standard deviation (VMSD) and coefficient of variation (COV) of von Mises stresses, respectively, was observed between the 21/21 μm case and other scan/reconstruction combinations within the same (free or fixed) simulation group. Observed differences were largely attributable to scanning resolution, although reconstruction resolution also contributed significantly at the largest voxel sizes. All 21/21 μm results (taken as the gold standard) could be predicted from the 21/50 radj2=0.91-0.99;p<0.001, 21/110 radj2=0.58-0.99;p<0.02 and 50/50 results radj2=0.61-0.97;p<0.02. While BV/TV, VMSD, and VMExp/σz from the 21/21 could be predicted by those from the 50/110 radj2=0.63-0.93;p<0.02 and 110/110 radj2=0.41-0.77;p<0.05 simulations as well, prediction of E, VMExp, and COV became marginally significant 0.04<p<0.13 at 50/110 and nonsignificant at 110/110 0.21<p<0.70. In conclusion, calculation of cancellous bone modulus, mean trabecular stress, and other parameters are subject to large errors at 110/110 μm voxel size. However, enough microstructural details for studying bone volume fraction, trabecular shear stress scatter, and trabecular shear stress amplification VMExp/σz can be resolved using a 21/110 μm, 50/110 μm, and 110/110 μm voxels for both free- and fixed-end constraints.

Bone tissue stiffness in the mandibular condyle is dependent on the direction and density of the cancellous structure

Variation in the apparent stiffness of cancellous bone is generally ascribed to variation in cancellous structure and density, while the bone tissue stiffness is assumed to be constant. The purpose of the present study was to examine whether the bone tissue stiffness is dependent on the direction and density of the cancellous structure. Bone tissue stiffness was estimated by combining mechanical testing and micro-finite element (micro-FE) modeling on cylindrical bone specimens obtained from the human mandibular condyle. One set of specimens was tested in the vertical direction of the condyle (n = 39) and another set in the transverse direction (n = 30). The cancellous structure of the specimens was characterized by micro-CT. The apparent bone stiffnesses predicted by the FE model correlated strongly (r2 = 0.91) with the measured apparent bone stiffnesses. Apparent bone stiffness in the transverse direction was considerably smaller than that in the vertical direction. In contrast, the predicted bone tissue stiffness was significantly larger in the transverse direction (E = 13.70 GPa) than in the vertical direction (E = 11.87 GPa). In addition, bone tissue stiffness correlated negatively with the bone volume fraction and directional sensitivity of the bone tissue stiffness increased with a decrease of bone volume fraction. The results suggest that the transversely oriented trabeculae in the mandibular condyle are stiffer and more mineralized than the vertically oriented trabeculae and that bone loss is compensated by an increase in the degree of mineralization.

Bisphosphonate treatment affects trabecular bone apparent modulus through micro-architecture rather than matrix properties

Bisphosphonates are emerging as an important treatment for osteoporosis. But whether the reduced fracture risk associated with bisphosphonate treatment is due to increased bone mass, improved trabecular architecture and/or increased secondary mineralization of the calcified matrix remains unclear. We examined the effects of bisphosphonates on both the trabecular architecture and matrix properties of canine trabecular bone. Thirty-six beagles were divided into a control group and two treatment groups, one receiving risedronate and the other alendronate at 5-6 times the clinical dose for osteoporosis treatment. After one year, the dogs were killed, and samples from the first lumbar vertebrae were examined using a combination of micro-computed tomography, finite element modeling, and mechanical testing. By combining these methods, we examined the treatment effects on the calcified matrix and trabecular architecture independently. Conventional histomorphometry and microdamage data were obtained from the second and third lumbar vertebrae of the same dogs [Bone 28 (2001) 524]. Bisphosphonate treatment resulted in an increased apparent Young's modulus, decreased bone turnover, increased calcified matrix density, and increased microdamage. We could not detect any change in the effective Young's modulus of the calcified matrix in the bisphosphonate treated groups. The observed increase in apparent Young's modulus was due to increased bone mass and altered trabecular architecture rather than changes in the calcified matrix modulus. We hypothesize that the expected increase in the Young's modulus of the calcified matrix due to the increased calcified matrix density was counteracted by the accumulation of microdamage.

Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue

The ability to determine trabecular bone tissue elastic and failure properties has biological and clinical importance. To date, trabecular tissue yield strains remain unknown due to experimental difficulties, and elastic moduli studies have reported controversial results. We hypothesized that the elastic and tensile and compressive yield properties of trabecular tissue are similar to those of cortical tissue. Effective tissue modulus and yield strains were calibrated for cadaveric human femoral neck specimens taken from 11 donors, using a combination of apparent-level mechanical testing and specimen-specific, high-resolution, nonlinear finite element modeling. The trabecular tissue properties were then compared to measured elastic modulus and tensile yield strain of human femoral diaphyseal cortical bone specimens obtained from a similar cohort of 34 donors. Cortical tissue properties were obtained by statistically eliminating the effects of vascular porosity. Results indicated that mean elastic modulus was 10% lower (p<0.05) for the trabecular tissue (18.0+/-2.8 GPa) than for the cortical tissue (19.9+/-1.8 GPa), and the 0.2% offset tensile yield strain was 15% lower for the trabecular tissue (0.62+/-0.04% vs. 0.73+/-0.05%, p<0.001). The tensile-compressive yield strength asymmetry for the trabecular tissue, 0.62 on average, was similar to values reported in the literature for cortical bone. We conclude that while the elastic modulus and yield strains for trabecular tissue are just slightly lower than those of cortical tissue, because of the cumulative effect of these differences, tissue strength is about 25% greater for cortical bone.

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.

Finite element modeling of trabecular bone damage

This paper presents a finite element-based, computational model for analysis of structural damage to trabecular bone tissues. A modulus reduction method was formulated from elasto-plasticity theory, and was used to account for site-specific trabecular bone tissue damage. Trabecular bone tissue damage is illustrated using a large-scale, anatomically accurate, two-dimensional, microstructural finite element model of a human thoracic vertebral body. Four models with varying specifications for damage accumulation were subjected to compressive loading and unloading cycles. The numerical results and experimental validation demonstrated that the modulus reduction method reproduced the non-linear mechanical behaviour of vertebal trabecular bone. The iterative computational approach presented provides a methodology to study trabecular bone damage, and should provide researchers with a computational approach to study bone fracture and repair and to predict vertebral fragility.

Micro-FE analyses of bone: state of the art

The ability to provide a complete characterization of elastic properties of bone has vastly improved our understanding of trabecular bone mechanical properties. Based on this information, it was possible to validate several mechanical concepts related to the elastic behavior of trabecular bone that could not be validated earlier. With recently developed micro-CT scanners and the availability of large parallel computer systems, this technique has also enabled the determination of physiological bone tissue loading conditions from very large microFE models that can represent whole human bones in detail. Such analyses can provide the data needed for a better understanding of bone failure processes or cell mediated load adaptive remodeling processes. Computational demands for whole bone analyses, however, are still excessive. Unlike linear stress and strain analyses, the application of PFE to study non-linear processes, in particular bone failure mechanisms, is still in an early phase Results of recent studies, however, are promising and indicate that an accurate prediction of bone failure with these techniques is possible. Compelling features of such analyses are that they enable multi-axial failure criteria at the apparent level to be developed using primarily computational methods as well as that they can provide a basis for detailed analysis of micro-mechanics associated with trabecular failure at the apparent level. The application of microFE techniques to analyze bone in vivo is in an early stage as well. First results have indicated that, although the resolution of presently available in vivo imaging techniques (i.e. pQCT and MR) is much less than that of images used so far for uFE analyses, the technique can provide meaningful elastic properties of trabecular bone in vivo in most cases. It is expected that the remaining uncertainties in the microFE results can be eliminated as soon as the resolution of in vivo images is improved. With the fast developments in pQCT(47) and MR scanning, this will probably be in the near future.

Estimation of distal radius failure load with micro-finite element analysis models based on three-dimensional peripheral quantitative computed tomography images

There is increasing evidence that, in addition to bone mass, bone microarchitecture and its mechanical load distribution are important factors for the determination of bone strength. Recently, it has been shown that new high-resolution imaging techniques in combination with new modeling algorithms based on the finite element (FE) method can account for these additional factors. Such models thus could provide more relevant information for the estimation of bone failure load. The purpose of the present study was to determine whether results of whole-bone micro-FE (microFE) analyses with models based on three-dimensional peripheral quantitative computer tomography (3D-pQCT) images (isotropic voxel resolution of 165 microm) could predict the failure load of the human radius more accurately than results with dual-energy X-ray absorptiometry (DXA) or bone morphology measurements. For this purpose, microFE models were created using 54 embalmed cadaver arms. It was assumed that bone failure would be initiated if a certain percentage of the bone tissue (varied from 1% to 7%) would be strained beyond the tissue yield strain. The external force that produced this tissue strain was calculated from the FE analyses. These predictions were correlated with results of real compression testing on the same cadaver arms. The results of these compression tests were also correlated with results of DXA and structural measurements of these arms. The compression tests produced Colles-type fractures in the distal 4 cm of the radius. The predicted failure loads calculated from the FE analysis agreed well with those measured in the experiments (R(2) = 0.75 p < 0.001). Lower correlations were found with bone mass (R(2) = 0.48, p < 0.001) and bone structural parameters (R(2) = 0.57 p < 0.001). We conclude that application of the techniques investigated here can lead to a better prediction of the bone failure load for bone in vivo than is possible from DXA measurements, structural parameters, or a combination thereof.

Cancellous bone mechanical properties from normals and patients with hip fractures differ on the structure level, not on the bone hard tissue level

Osteoporosis is currently defined in terms of low bone mass. However, the source of fragility leading to fracture has not been adequately described. In particular, the contributions of bone tissue properties and architecture to the risk or incidence of fracture are poorly understood. In an earlier experimental study, it was found that the architectural anisotropy of cancellous bone from the femoral heads of fracture patients was significantly increased compared with age- and density-matched control material (Ciarelli et al., J Bone Miner Res 15:32-40; 2000). Using a combination of compression testing and micro-finite element analysis on a subset of cancellous bone specimens from that study, we calculated the hard tissue mechanical properties and the apparent (macroscopic) mechanical properties. The tissue modulus was 10.0 GPa (SD 2.2) for the control group and 10.8 GPa (SD 3.3) for the fracture group (not significant). There were no differences in either the apparent yield strains, percentages of highly strained tissue, or the relationship between apparent yield stress and apparent elastic modulus. Hence, a difference in the tissue yield properties is unlikely. At the apparent level, the fracture group had a significantly decreased transverse stiffness, resulting in increased mechanical anisotropy. These changes suggest that bone in the fracture group was "overadapted" to the primary load axis, at the cost of fragility in the transverse direction. We conclude that individuals with a history of osteoporotic fractures do not have weaker bone tissue. Architectural and mechanical anisotropy alone renders their bone weaker in the nonprimary loading direction.

Biomechanics of trabecular bone

Trabecular bone is a complex material with substantial heterogeneity. Its elastic and strength properties vary widely across anatomic sites, and with aging and disease. Although these properties depend very much on density, the role of architecture and tissue material properties remain uncertain. It is interesting that the strains at which the bone fails are almost independent of density. Current work addresses the underlying structure-function relations for such behavior, as well as more complex mechanical behavior, such as multiaxial loading, time-dependent failure, and damage accumulation. A unique tool for studying such behavior is the microstructural class of finite element models, particularly the "high-resolution" models. It is expected that with continued progress in this field, substantial insight will be gained into such important problems as osteoporosis, bone fracture, bone remodeling, and design/analysis of bone-implant systems. This article reviews the state of the art in trabecular bone biomechanics, focusing on the mechanical aspects, and attempts to identify important areas of current and future research.

A voxel-based formulation for contact finite element analysis

To date, voxel-based finite element models have not been feasible for contact problems, owing to the inherent stair-step boundary discontinuities. New preprocessing techniques are reported herein to mesh these boundaries smoothly, for purposes of contact stress analysis. Further, new techniques are reported to concentrate the mesh resolution automatically near the articular surface, thus reducing the problem size to levels compatible with executing nonlinear problems on contemporary engineering workstations. Close approximations to Hertzian analytical solutions were obtained for spherical and cylindrical geometries meshed in this manner, and an illustrative anatomical contact problem of the human hip joint is presented.

Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions

Strong correspondence between the uniaxial apparent strength and stiffness of cancellous bone allows the use of stiffness as a predictor of bone strength. Measured values of mechanical properties in cancellous bone can be different between experiments due to different experimental conditions. In the current study, bone volume fraction, experimentally determined and finite element (FE) predicted stiffness were examined as predictors of cancellous bone ultimate strength in two different groups each of which was tested using a different end constraint. It is demonstrated that, although always significant, the relationships of strength with bone volume fraction and experimentally determined stiffness are different between test groups. Apparent stiffness, estimated by FE modeling, predicts the ultimate strength of human cancellous bone consistently for all examined experimental protocols.

Trabecular shear stress in human vertebral cancellous bone: intra- and inter-individual variations

Correlation of the mean and standard deviation of trabecular stresses has been proposed as a mechanism by which a strong relationship between the apparent strength and stiffness of cancellous bone can be achieved. The current study examined whether the relationship between the mean and standard deviation of trabecular von Mises stresses can be generalized for any group of cancellous bone. Cylindrical human vertebral cancellous bone specimens were cut in the infero-superior direction from T12 of 23 individuals (inter-individual group). Thirty nine additional specimens were prepared similarly from the T4-T12 and L2-L5 vertebrae of a 63 year old male (intra-individual group). The specimens were scanned by micro-computed tomography (microCT) and trabecular von Mises stresses were calculated using finite element modeling. The expected value, standard deviation and coefficient of variation of the von Mises stress were calculated form a three-parameter Weibull function fitted to von Mises stress data from each specimen. It was found that the average and standard deviation of trabecular von Mises shear stress were: (i) correlated with each other, supporting the idea that high correlation between the apparent strength and stiffness of cancellous bone can be achieved through controlling the trabecular level shear stress variations, (ii) dependent on anatomical site and sample group, suggesting that the variation of stresses are correlated to the mean stress to different degrees between vertebrae and individuals, and (iii) dependent on bone volume fraction, consistent with the idea that shear stress is less well controlled in bones with low BV/TV. The conversion of infero-superior loading into trabecular von Mises stresses was maximum for the tissue at the junction of the thoracic and lumbar spine (T12-L1) consistent with this junction being a common site of vertebral fracture.

A decreased subchondral trabecular bone tissue elastic modulus is associated with pre-arthritic cartilage damage

In osteoarthritis, one postulate is that changes in the mechanical properties of the subchondral bone layer result in cartilage damage. The goal of this study was to examine changes in subchondral trabecular bone properties at the calcified tissue level in the early stages of cartilage damage. Finite element models were constructed from microCT scans of trabectilar bone from the proximal tibia of donors with mild cartilage damage and from normal donors. In the donors with cartilage damage, macroscopic damage was present only in the medial compartment. The effective tissue elastic moduli were determined using a combination of finite element models and mechanical testing. The bone tissue modulus was reduced by 60% in the medial condyle of the cases with cartilage damage compared to the control specimens. Neither the presence of cartilage damage nor the anatomic site (medial vs. lateral) affected the elastic modulus at the apparent level. The volume fraction of trabecular bone was higher in the medial compartment compared to the lateral compartment of tibiae with cartilage damage (but not the controls), suggesting that mechanical properties were preserved in part at the apparent level by an increase in the bone volume fraction. It seems likely that the normal equilibrium between cartilage properties, bone tissue properties and bone volume fraction is disrupted early in the development of osteoarthritis.

Increase in bone volume fraction precedes architectural adaptation in growing bone

In mature trabecular bone, both density and trabecular orientation are adapted to external mechanical loads. Few quantitative data are available on the development of architecture and mechanical adaptation in juvenile trabecular bone. We studied the hypothesis that a time lag occurs between the adaptation of trabecular density and the adaptation of trabecular architecture during development. To investigate this hypothesis we used ten female pigs at 6, 23, 56, 104, and 230 weeks of age. Three-dimensional morphological and mechanical parameters of trabecular bone samples from the vertebra and proximal tibia were studied using microcomputed tomography and micro-finite element analysis. Both bone volume fraction and stiffness increased rapidly in the initial growth phase (from 6 weeks on), whereas the morphological anisotropy started increasing only after 23 weeks of age. In addition, the anisotropy reached its highest value much later in the development than did bone volume fraction. Hence, the alignment of trabeculae was still progressing at the time of peak bone mass. Therefore, our hypothesis was supported by the time lag between the increase in trabecular density and the adaptation of the trabecular architecture. The rapid increase of bone volume fraction in the initial growth phase can be explained by the enormous weight increase of the pigs. The trabeculae aligned at later stages when the increase in weight, and thus the loading, was slowed considerably compared with the early growth stage. Hence, the trabecular architecture was more efficient in later years. We conclude that density is adapted to external load from the early phase of growth, whereas the trabecular architecture is adapted later in the development.

Convergence behavior of high-resolution finite element models of trabecular bone

The convergence behavior of finite element models depends on the size of elements used, the element polynomial order, and on the complexity of the applied loads. For high-resolution models of trabecular bone, changes in architecture and density may also be important. The goal of this study was to investigate the influence of these factors on the convergence behavior of high-resolution models of trabecular bone. Two human vertebral and two bovine tibial trabecular bone specimens were modeled at four resolutions ranging from 20 to 80 microns and subjected to both compressive and shear loading. Results indicated that convergence behavior depended on both loading mode (axial versus shear) and volume fraction of the specimen. Compared to the 20 microns resolution, the differences in apparent Young's modulus at 40 microns resolution were less than 5 percent for all specimens, and for apparent shear modulus were less than 7 percent. By contrast, differences at 80 microns resolution in apparent modulus were up to 41 percent, depending on the specimen tested and loading mode. Overall, differences in apparent properties were always less than 10 percent when the ratio of mean trabecular thickness to element size was greater than four. Use of higher order elements did not improve the results. Tissue level parameters such as maximum principal strain did not converge. Tissue level strains converged when considered relative to a threshold value, but only if the strains were evaluated at Gauss points rather than element centroids. These findings indicate that good convergence can be obtained with this modeling technique, although element size should be chosen based on factors such as loading mode, mean trabecular thickness, and the particular output parameter of interest.