Accurate and Efficient Plate and Rod Microfinite Element Models for Whole Bone Segments Based on High-Resolution Peripheral Computed Tomography.

The high-resolution peripheral quantitative computed tomography (HR-pQCT) provides unprecedented visualization of bone microstructure and the basis for constructing patient-specific microfinite element (µFE) models. Based on HR-pQCT images, we have developed a plate-and-rod µFE (PR µFE) method for whole bone segments using individual trabecula segmentation (ITS) and an adaptive cortical meshing technique. In contrast to the conventional voxel approach, the complex microarchitecture of the trabecular compartment is simplified into shell and beam elements based on the trabecular plate-and-rod configuration. In comparison to voxel-based µFE models of µCT and measurements from mechanical testing, the computational and experimental gold standards, nonlinear analyses of stiffness and yield strength using the HR-pQCT-based PR µFE models demonstrated high correlation and accuracy. These results indicated that the combination of segmented trabecular plate-rod morphology and adjusted cortical mesh adequately captures mechanics of the whole bone segment. Meanwhile, the PR µFE modeling approach reduced model size by nearly 300-fold and shortened computation time for nonlinear analysis from days to within hours, permitting broader clinical application of HR-pQCT-based nonlinear µFE modeling. Furthermore, the presented approach was tested using a subset of radius and tibia HR-pQCT scans of patients with prior vertebral fracture in a previously published study. Results indicated that yield strength for radius and tibia whole bone segments predicted by the PR µFE model was effective in discriminating vertebral fracture subjects from nonfractured controls. In conclusion, the PR µFE model of HR-pQCT images accurately predicted mechanics for whole bone segments and can serve as a valuable clinical tool to evaluate musculoskeletal diseases.

Regional Variations of HR-pQCT Morphological and Biomechanical Measurements of Bone Segments and Their Associations With Whole Distal Radius and Tibia Mechanical Properties.

High-resolution peripheral quantitative computed tomography (HR-pQCT) is a promising imaging modality that provides an in vivo three-dimensional (3D) assessment of bone microstructure by scanning fixed regions of the distal radius and tibia. However, how microstructural parameters and mechanical analysis based on these segment scans correlate to whole distal radius and tibia mechanics are not well-characterized. On 26 sets of cadaveric radius and tibia, HR-pQCT scans were performed on the standard scan segment, a segment distal to the standard segment, and a segment proximal to the standard segment. Whole distal radius and tibia stiffness were determined through mechanical testing. Segment bone stiffness was estimated using linear finite element (FE) analysis based on segment scans. Standard morphological and individual trabecula segmentation (ITS) analyses were used to estimate microstructural properties. Significant variations in microstructural parameters were observed among segments at both sites. Correlation to whole distal radius and tibia stiffness was moderate for microstructural parameters at the standard segment, but correlation was significantly increased for FE-predicted segment bone stiffness based on standard segment scans. Similar correlation strengths were found between FE-predicted segment bone stiffness and whole distal radius and tibia stiffness. Additionally, microstructural parameters at the distal segment had higher correlation to whole distal radius and tibia stiffness than at standard or proximal segments. Our results suggest that FE-predicted segment stiffness is a better predictor of whole distal radius and tibia stiffness for clinical HR-pQCT analysis and that microstructural parameters at the distal segment are more highly correlated with whole distal radius and tibia stiffness than at the standard or proximal segments.

Effect of variations in tissue-level ductility on human vertebral strength.

Although the ductility of bone tissue is a unique element of bone quality and varies with age and across the population, the extent to which and mechanisms by which typical population-variations in tissue-level ductility can alter whole-bone strength remains unclear. To provide insight, we conducted a finite element analysis parameter study of whole-vertebral (monotonic) compressive strength on six human L1 vertebrae. Each model was generated from micro-CT scans, capturing the trabecular micro-architecture in detail, and included a non-linear constitutive model for the bone tissue that allowed for plastic yielding, different strengths in tension and compression, large deformations, and, uniquely, localized damage once a specified limit in tissue-level ultimate strain was exceeded. Those strain limits were based on reported (mean ± SD) values from cadaver experiments (8.8 ± 3.7% strain for trabecular tissue and 2.2 ± 0.9% for cortical tissue). In the parameter study, the strain limits were varied by ±1 SD from their mean values, for a combination of nine analyses per specimen; bounding values of zero and unlimited post-yield strain were also modeled. The main outcomes from the finite element analysis were the vertebral compressive strength and the amount of failed (yielded or damaged) tissue at the overall structure-level failure. Compared to a reference case of using the mean values of ultimate strain, we found that varying both trabecular and cortical tissue ultimate strains by ±1 SD changed the computed vertebral strength by (mean ± SD) ±6.9 ± 1.1% on average. Mechanistically, that modest effect arose because the proportion of yielded tissue (without damage) was 0.9 ± 0.3% of all the bone tissue across the nine cases and the proportion of damaged tissue (i.e. tissue exceeding the prescribed tissue-level ultimate strain) was 0.2 ± 0.1%. If the types of variations in tissue-level ductility investigated here accurately represent real typical variations in the population, the consistency of our results across specimens and the modest effect size together suggest that typical variations in tissue-level ductility only have a modest impact on vertebral compressive strength, in large part because so few trabeculae are damaged at the load capacity of the bone.

High-resolution peripheral quantitative computed tomography (HR-pQCT) can assess microstructural and biomechanical properties of both human distal radius and tibia: Ex vivo computational and experimental validations.

High-resolution peripheral quantitative computed tomography (HR-pQCT) provides in vivo three-dimensional (3D) imaging at the distal radius and tibia and has been increasingly used to characterize cortical and trabecular bone morphology in clinical studies. In this study, we comprehensively examined the accuracy of HR-pQCT and HR-pQCT based micro finite element (µFE) analysis predicted bone elastic stiffness and strength through comparisons with gold-standard micro computed tomography (µCT) based morphological/µFE measures and direct mechanical testing results. Twenty-six sets of human cadaveric distal radius and tibia segments were imaged by HR-pQCT and µCT. Microstructural analyses were performed for the registered HR-pQCT and µCT images. Bone stiffness and yield strength were determined by both HR-pQCT and µCT based linear and nonlinear µFE predictions and mechanical testing. Our results suggested that strong and significant correlations existed between the HR-pQCT standard, model-independent and corresponding µCT measurements. HR-pQCT based nonlinear µFE overestimated stiffness and yield strength while the linear µFE underestimated yield strength, but both were strongly correlated with those predicted by µCT µFE and measured by mechanical testing at both radius and tibia (R(2)>0.9). The microstructural differences between HR-pQCT and µCT were also examined by the Bland-Altman plots. Our results showed HR-pQCT morphological measurements of BV/TV(d), Tb.Th, and Tb.Sp, can be adjusted by correction values to approach true values measured by gold-standard µCT. In addition, we observed moderate correlations of HR-pQCT biomechanical and microstructural parameters between the distal radius and tibia. We concluded that morphological and mechanical properties of human radius and tibia bone can be assessed by HR-pQCT based measures.

Cortical and trabecular load sharing in the human femoral neck.

The relative role of the cortical vs trabecular bone in the load-carrying capacity of the proximal femur-a fundamental issue in both basic-science and clinical biomechanics-remains unclear. To gain insight into this issue, we performed micro-CT-based, linear elastic finite element analysis (61.5-micron-sized elements; ~280 million elements per model) on 18 proximal femurs (5M, 13F, ages 61-93 years) to quantify the fraction of frontal-plane bending moment shared by the cortical vs trabecular bone in the femoral neck, as well as the associated spatial distributions of stress. Analyses were performed separately for a sideways fall and stance loading. For both loading modes and across all 18 bones, we found consistent patterns of load-sharing in the neck: most proximally, the trabecular bone took most of the load; moving distally, the cortical bone took increasingly more of the load; and more distally, there was a region of uniform load-sharing, the cortical bone taking the majority of the load. This distal region of uniform load-sharing extended more for fall than stance loading (77 ± 8% vs 51 ± 6% of the neck length for fall vs. stance; mean ± SD) but the fraction of total load taken by the cortical bone in that region was greater for stance loading (88 ± 5% vs. 64 ± 9% for stance vs. fall). Locally, maximum stress levels occurred in the cortical bone distally, but in the trabecular bone proximally. Although the distal cortex showed qualitative stress distributions consistent with the behavior of an Euler-type beam, quantitatively beam theory did not apply. We conclude that consistent and well-delineated regions of uniform load-sharing and load-transfer between the cortical and trabecular bone exist within the femoral neck, the details of which depend on the external loading conditions.

Theoretical effects of fully ductile versus fully brittle behaviors of bone tissue on the strength of the human proximal femur and vertebral body.

The influence of the ductility of bone tissue on whole-bone strength represents a fundamental issue of multi-scale biomechanics. To gain insight, we performed a computational study of 16 human proximal femurs and 12 T9 vertebral bodies, comparing the whole-bone strength for the two hypothetical bounding cases of fully brittle versus fully ductile tissue-level failure behaviors, all other factors, including tissue-level elastic modulus and yield stress, held fixed. For each bone, a finite element model was generated (60-82 µm element size; up to 120 million elements) and was virtually loaded in habitual (stance for femur, compression for vertebra) and non-habitual (sideways fall, only for femur) loading modes. Using a geometrically and materially non-linear model, the tissue was assumed to be either fully brittle or fully ductile. We found that, under habitual loading, changing the tissue behavior from fully ductile to fully brittle reduced whole-bone strength by 38.3±2.4% (mean±SD) and 39.4±1.9% for the femur and vertebra, respectively (p=0.39 for site difference). These reductions were remarkably uniform across bones, but (for the femur) were greater for non-habitual (57.1±4.7%) than habitual loading (p<0.001). At overall structural failure, there was 5-10-fold less failed tissue for the fully brittle than fully ductile cases. These theoretical results suggest that the whole-bone strength of the proximal femur and vertebra can vary substantially between fully brittle and fully ductile tissue-level behaviors, an effect that is relatively insensitive to bone morphology but greater for non-habitual loading.

Microstructural failure mechanisms in the human proximal femur for sideways fall loading.

The etiology of hip fractures remains unclear but might be elucidated by an improved understanding of the microstructural failure mechanisms of the human proximal femur during a sideways fall impact. In this context, we biomechanically tested 12 cadaver proximal femurs (aged 76 ± 10 years; 8 female, 4 male) to directly measure strength for a sideways fall and also performed micro-computed tomography (CT)-based, nonlinear finite element analysis of the same bones (82-micron-sized elements, ∼120 million elements per model) to estimate the amount and location of internal tissue-level failure (by ductile yielding) at initial structural failure of the femur. We found that the correlation between the directly measured yield strength of the femur and the finite element prediction was high (R(2) = 0.94, p < 0.0001), supporting the validity of the finite element simulations of failure. In these simulations, the failure of just a tiny proportion of the bone tissue (1.5% to 6.4% across all bones) led to initial structural failure of the femur. The proportion of failed tissue, estimated by the finite element models, decreased with decreasing measured femoral strength (R(2) = 0.88, p < 0.0001) and was more highly correlated with measured strength than any measure of bone volume, mass, or density. Volume-wise, trabecular failure occurred earlier and was more prominent than cortical failure in all femurs and dominated in the very weakest femurs. Femurs with low measured strength relative to their areal bone mineral density (BMD) (by dual-energy X-ray absorptiometry [DXA]) had a low proportion of trabecular bone compared with cortical bone in the femoral neck (p < 0.001), less failed tissue (p < 0.05), and low structural redundancy (p < 0.005). We conclude that initial failure of the femur during a sideways fall is associated with failure of just a tiny proportion of the bone tissue, failure of the trabecular tissue dominating in the very weakest femurs owing in part to a lack of structural redundancy.

Theoretical bounds for the influence of tissue-level ductility on the apparent-level strength of human trabecular bone.

The role of tissue-level post-yield behavior on the apparent-level strength of trabecular bone is a potentially important aspect of bone quality. To gain insight into this issue, we compared the apparent-level strength of trabecular bone for the hypothetical cases of fully brittle versus fully ductile failure behavior of the trabecular tissue. Twenty human cadaver trabecular bone specimens (5mm cube; BV/TV=6-36%) were scanned with micro-CT to create 3D finite element models (22-micron element size). For each model, apparent-level strength was computed assuming either fully brittle (fracture with no tissue ductility) or fully ductile (yield with no tissue fracture) tissue-level behaviors. We found that the apparent-level ultimate strength for the brittle behavior was only about half the value of the apparent-level 0.2%-offset yield strength for the ductile behavior, and the ratio of these brittle to ductile strengths was almost constant (mean±SD=0.56±0.02; n=20; R(2)=0.99 between the two measures). As a result of this small variation, although the ratio of brittle to ductile strengths was positively correlated with the bone volume fraction (R(2)=0.44, p=0.01) and structure model index (SMI, R(2)=0.58, p<0.01), these effects were small. Mechanistically, the fully ductile behavior resulted in a much higher apparent-level strength because in this case about 16-fold more tissue was required to fail than for the fully brittle behavior; also, there was more tensile- than compressive-mode of failure at the tissue level for the fully brittle behavior. We conclude that, in theory, the apparent-level strength behavior of human trabecular bone can vary appreciably depending on whether the tissue fails in a fully ductile versus fully brittle manner, and this effect is largely constant despite appreciable variations in bone volume fraction and microarchitecture.

Micromechanics of the human vertebral body for forward flexion.

To provide mechanistic insight into the etiology of osteoporotic wedge fractures, we investigated the spatial distribution of tissue at the highest risk of initial failure within the human vertebral body for both forward flexion and uniform compression loading conditions. Micro-CT-based linear elastic finite element analysis was used to virtually load 22 human T9 vertebral bodies in either 5° of forward flexion or uniform compression; we also ran analyses replacing the simulated compliant disc (E=8 MPa) with stiff polymethylmethacrylate (PMMA, E=2500 MPa). As expected, we found that, compared to uniform compression, forward flexion increased the overall endplate axial load on the anterior half of the vertebra and shifted the spatial distribution of high-risk tissue within the vertebra towards the anterior aspect of the vertebral body. However, despite that shift, the high-risk tissue remained primarily within the central regions of the trabecular bone and endplates, and forward flexion only slightly altered the ratio of cortical-to-trabecular load sharing at the mid-vertebral level (mean±SD for n=22: 41.3±7.4% compression; 44.1±8.2% forward flexion). When the compliant disc was replaced with PMMA, the anterior shift of high-risk tissue was much more severe. We conclude that, for a compliant disc, a moderate degree of forward flexion does not appreciably alter the spatial distribution of stress within the vertebral body.

Vertebral fragility and structural redundancy.

The mechanisms of age-related vertebral fragility remain unclear, but may be related to the degree of "structural redundancy" of the vertebra; ie, its ability to safely redistribute stress internally after local trabecular failure from an isolated mechanical overload. To better understand this issue, we performed biomechanical testing and nonlinear micro-CT-based finite element analysis on 12 elderly human thoracic ninth vertebral bodies (age 76.9 ± 10.8 years). After experimentally overloading the vertebrae to measure strength, we used nonlinear finite element analysis to estimate the amount of failed tissue and understand the failure mechanisms. We found that the amount of failed tissue per unit bone mass decreased with decreasing bone volume fraction (r(2) = 0.66, p < 0.01). Thus, for the weak vertebrae with low bone volume fraction, overall failure of the vertebra occurred after failure of just a tiny proportion of the bone tissue (<5%). This small proportion of failed tissue had two sources: the existence of fewer vertically oriented load paths to which load could be redistributed from failed trabeculae; and the vulnerability of the trabeculae in these few load paths to undergo bending-type failure mechanisms, which further weaken the bone. Taken together, these characteristics suggest that diminished structural redundancy may be an important aspect of age-related vertebral fragility: vertebrae with low bone volume fraction are highly susceptible to collapse because so few trabeculae are available for load redistribution if the external loads cause any trabeculae to fail.