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

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

The effective elastic properties of human trabecular bone may be approximated using micro-finite element analyses of embedded volume elements

Boundary conditions (BCs) and sample size affect the measured elastic properties of cancellous bone. Samples too small to be representative appear stiffer under kinematic uniform BCs (KUBCs) than under periodicity-compatible mixed uniform BCs (PMUBCs). To avoid those effects, we propose to determine the effective properties of trabecular bone using an embedded configuration. Cubic samples of various sizes (2.63, 5.29, 7.96, 10.58 and 15.87 mm) were cropped from \(\mu \hbox {CT}\) scans of femoral heads and vertebral bodies. They were converted into \(\mu \hbox {FE}\) models and their stiffness tensor was established via six uniaxial and shear load cases. PMUBCs- and KUBCs-based tensors were determined for each sample. “In situ” stiffness tensors were also evaluated for the embedded configuration, i.e. when the loads were transmitted to the samples via a layer of trabecular bone. The Zysset–Curnier model accounting for bone volume fraction and fabric anisotropy was fitted to those stiffness tensors, and model parameters \(\nu _{0}\) (Poisson’s ratio) \(E_{0}\) and \(\mu _{0}\) (elastic and shear moduli) were compared between sizes. BCs and sample size had little impact on \(\nu _{0}\). However, KUBCs- and PMUBCs-based \(E_{0}\) and \(\mu _{0}\), respectively, decreased and increased with growing size, though convergence was not reached even for our largest samples. Both BCs produced upper and lower bounds for the in situ values that were almost constant across samples dimensions, thus appearing as an approximation of the effective properties. PMUBCs seem also appropriate for mimicking the trabecular core, but they still underestimate its elastic properties (especially in shear) even for nearly orthotropic samples.     

Effects of damage on the orthotropic material symmetry of bovine tibial trabecular bone

The macroscopic mechanical properties of trabecular bone can be predicted by its architecture using theoretical relationships between the elastic and architectural properties. Microdamage caused by overloading or fatigue decreases the apparent elastic moduli of trabecular bone requiring these relationships to be modified to predict the damaged elastic properties. In the case of isotropic damage, the apparent level elastic properties could be determined by multiplying all of the elastic constants by a single scalar factor. If the damage is anisotropic, the elastic constants may change by differing factors and the material coordinate system could become misaligned with the fabric coordinate system. High-resolution finite element models were used to simulate damage overloading on seven trabecular bone specimens subjected to pure shear strain in two planes. Comparison of the apparent elastic moduli of the specimens before and after damage showed that the reduction of the elastic moduli was anisotropic. This suggests that the microdamage within the specimens was inhomogeneous. However, after damage the specimens exhibited nearly orthotropic material symmetry as they did before damage. Changes in the orientation of the orthotropic material coordinate system were also small and occurred primarily in the transverse plane. Thus, while damage in trabecular bone is anisotropic, the material coordinate system remains aligned with the fabric tensor.     

Effect of boundary conditions on yield properties of human femoral trabecular bone

Trabecular bone plays an important mechanical role in bone fractures and implant stability. Homogenized nonlinear finite element (FE) analysis of whole bones can deliver improved fracture risk and implant loosening assessment. Such simulations require the knowledge of mechanical properties such as an appropriate yield behavior and criterion for trabecular bone. Identification of a complete yield surface is extremely difficult experimentally but can be achieved in silico by using micro-FE analysis on cubical trabecular volume elements. Nevertheless, the influence of the boundary conditions (BCs), which are applied to such volume elements, on the obtained yield properties remains unknown. Therefore, this study compared homogenized yield properties along 17 load cases of 126 human femoral trabecular cubic specimens computed with classical kinematic uniform BCs (KUBCs) and a new set of mixed uniform BCs, namely periodicity-compatible mixed uniform BCs (PMUBCs). In stress space, PMUBCs lead to 7–72 % lower yield stresses compared to KUBCs. The yield surfaces obtained with both KUBCs and PMUBCs demonstrate a pressure-sensitive ellipsoidal shape. A volume fraction and fabric-based quadric yield function successfully fitted the yield surfaces of both BCs with a correlation coefficient \(R^{2} \ge 0.93\). As expected, yield strains show only a weak dependency on bone volume fraction and fabric. The role of the two BCs in homogenized FE analysis of whole bones will need to be investigated and validated with experimental results at the whole bone level in future studies.     

Microarchitecture and Nanomechanical Properties of Trabecular Bone After Strontium Administration in Osteoporotic Goats

Strontium (Sr) ralenate is a new agent used for the prevention and treatment of osteoporosis. As a bone-seeking element, 98% of Sr is deposited in the bone and teeth after oral ingestion. However, the effect of Sr treatment on bone microarchitecture and bone nanomechanical properties remains unclear. In this study, 18 osteoporotic goats were divided into four groups according to the treatment regimen: control, calcium alone (Ca), calcium and Sr at 24 mg/kg (Ca + 24Sr), and calcium and Sr at 40 mg/kg (Ca + 40Sr). The effects of Sr administration on bone microarchitecture and nanomechanical properties of trabecular bones were analyzed with micro-CT and nanoindentation test, respectively. Serum Sr levels increased six- and tenfold in the Ca + 24Sr and Ca + 40Sr groups, respectively. Similarly, Sr in the bone increased four- and sixfold in these two groups. Sr administration significantly increased trabecular bone volume fraction, trabecular thickness, and double-labeled new bone area. Sr administration, however, did not significantly change the nanomechanical properties of trabecular bone (elastic modulus and hardness). The data suggested that Sr administration increased trabecular bone volume and improved the microarchitecture while maintaining the intrinsic tissue properties in the osteoporotic goat model.     

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.     

Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network

Trabecular bone is modeled as a cellular material with an idealized periodic structure made of open cubic cells, which is effectively orthotropic. We evaluate apparent couple-stress moduli of such a periodic material; apparent moduli refer to the moduli obtained using a domain smaller than a Representative Volume Element and they depend on boundary conditions. We conduct this analysis computationally (using ANSYS) by subjecting a unit cell of this periodic cellular material to either displacement or traction boundary conditions. Cell walls, representing bone tissue, and void space, representing bone marrow, are both modeled and they are assumed to be linear elastic. The applied loadings include a uniaxial extension (or uniaxial stress), a hydrostatic deformation (or hydrostatic stress) and a shear deformation (or shear stress) to evaluate the first stiffness (or compliance) tensor, and an applied curvature (or bending moment), a uniaxial twist (or torsion), and a triaxial twist (or triaxial torsion) to evaluate the second couple-stress stiffness (or compliance) tensor. Apparent couple-stress moduli are computed by equating the total strain energy stored in the unit cell with the energy of an equivalent homogeneous orthotropic couple-stress material for each applied loading. The moduli computed using displacement boundary conditions give upper bound, while those obtained using traction boundary conditions give lower bound on effective couple-stress moduli. These bounds are very wide due to a large mismatch in elastic moduli of bone tissue and bone marrow. These results are in agreement with our studies on composite materials with very stiff or very compliant inclusions.     

Pregnancy outcome of women in the vicinity of nuclear power plants in Taiwan

The purpose of the study was to investigate whether proximity to nuclear power plants may increase the risk of abnormal pregnant outcomes among the resident women. In this ecological study, data were used from the Health Services Birth Reports Database established by the Bureau of Health Promotion, National Department of Health, Taiwan, in 2001–2004. Chi-square-tests were carried out to investigate the “Plant-vicinity” and “Non plant-vicinity” group in terms of pregnancy outcome. Additionally, logistic regression was performed to investigate whether residence in the vicinity of a nuclear power plant was related to any abnormal pregnancy results. Based on data from 5,679 included subjects, no difference was observed between pregnancy outcomes of the “Plant-vicinity” and “Non plant-vicinity” groups. After accounting for possible confounders, the adjusted odds ratios were 1.20 (95% CI = 0.56–2.56) for stillbirth, 1.21 (95% CI = 0.95–1.53) for premature birth, 1.04 (95% CI = 0.79–1.37) for low birth weight, and 1.58 (95% CI = 0.85–2.93) for congenital deficiencies, respectively, when comparing the “Plant-vicinity” with the “Non plant-vicinity” group. The results of the study indicate that residence in the vicinity of a nuclear power plant is not a significant factor which will cause abnormal health situations during pregnancy.     

Accounting for spatial variation of trabecular anisotropy with subject-specific finite element modeling moderately improves predictions of local subchondral bone stiffness at the proximal tibia

IntroductionPreviously, a finite element (FE) model of the proximal tibia was developed and validated against experimentally measured local subchondral stiffness. This model indicated modest predictions of stiffness (R² = 0.77, normalized root mean squared error (RMSE%) = 16.6%). Trabecular bone though was modeled with isotropic material properties despite its orthotropic anisotropy. The objective of this study was to identify the anisotropic FE modeling approach which best predicted (with largest explained variance and least amount of error) local subchondral bone stiffness at the proximal tibia.MethodsLocal stiffness was measured at the subchondral surface of 13 medial/lateral tibial compartments using in situ macro indentation testing. An FE model of each specimen was generated assuming uniform anisotropy with 14 different combinations of cortical- and tibial-specific density-modulus relationships taken from the literature. Two FE models of each specimen were also generated which accounted for the spatial variation of trabecular bone anisotropy directly from clinical CT images using grey-level structure tensor and Cowin’s fabric-elasticity equations. Stiffness was calculated using FE and compared to measured stiffness in terms of R² and RMSE%.ResultsThe uniform anisotropic FE model explained 53–74% of the measured stiffness variance, with RMSE% ranging from 12.4 to 245.3%. The models which accounted for spatial variation of trabecular bone anisotropy predicted 76–79% of the variance in stiffness with RMSE% being 11.2–11.5%.ConclusionsOf the 16 evaluated finite element models in this study, the combination of Synder and Schneider (for cortical bone) and Cowin’s fabric-elasticity equations (for trabecular bone) best predicted local subchondral bone stiffness.     

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.     

Combination of topological parameters and bone volume fraction better predicts the mechanical properties of trabecular bone

Trabecular bone structure may complement bone volume/total volume fraction (BV/TV) in the prediction of the mechanical properties. Nonetheless, the direct in vivo use of information pertaining to trabecular bone structure necessitates some predictive analytical model linking structure measures to mechanical properties. In this context, the purpose of this study was to combine BV/TV and topological parameters so as to better estimate the mechanical properties of trabecular bone. Thirteen trabecular bone mid-sagittal sections were imaged by magnetic resonance (MR) imaging at the resolution of 117 x 117x 300 microm(3). Topological parameters were evaluated in applying the 3D-line skeleton graph analysis (LSGA) technique to the binary MR images. The same images were used to estimate the elastic moduli by finite element analysis (FEA). In addition to the mid-sagittal section, two cylindrical samples were cored from each vertebra along vertical and horizontal directions. Monotonic compression tests were applied to these samples to measure both vertical and horizontal ultimate stresses. BV/TV was found as a strong predictor of the mechanical properties, accounting for 89-94% of the variability of the elastic moduli and for 69-86% of the variability of the ultimate stresses. Topological parameters and BV/TV were combined following two analytical formulations, based on: (1) the normalization of the topological parameters; and on (2) an exponential fit-model. The normalized parameters accounted for 96-98% of the variability of the elastic moduli, and the exponential model accounted for 80-95% of the variability of the ultimate stresses. Such formulations could potentially be used to increase the prediction of the mechanical properties of trabecular bone.     

A global relationship between trabecular bone morphology and homogenized elastic properties.

An alternative concept of the relationship between morphological and elastic properties of trabecular bone is presented and applied to human tissue from several anatomical locations using a digital approach. The three-dimensional morphology of trabecular bone was assessed with a microcomputed tomography system and the method of directed secants as well as the star volume procedure were used to compute mean intercept length (MIL) and average bone length (ABL) of 4 mm cubic specimens. Assuming isotropic elastic properties for the trabecular tissue, the general elastic tensors of the bone specimens were determined using the homogenization method and the closest orthotropic tensors were calculated with an optimization algorithm. The assumption of orthotropy for trabecular bone was found to improve with specimen size and hold within 6.1 percent for a 4 mm cube size. A strong global relationship (r2 = 0.95) was obtained between fabric and the orthotropic elastic tensor with a minimal set of five constants. Mean intercept length and average bone length provided an equivalent power of prediction. These results support the hypothesis that the elastic properties of human trabecular bone from an arbitrary anatomical location can be estimated from an approximation of the anisotropic morphology and a prior knowledge of tissue properties.     

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.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

Terahertz vibrational properties of water nanoclusters relevant to biology

Water nanoclusters are shown from first-principles calculations to possess unique terahertz-frequency vibrational modes in the 1–6 THz range, corresponding to O–O–O “bending,” “squashing,” and “twisting” “surface” distortions of the clusters. The cluster molecular-orbital LUMOs are huge Rydberg-like “S,” “P,” “D,” and “F” orbitals that accept an extra electron via optical excitation, ionization, or electron donation from interacting biomolecules. Dynamic Jahn–Teller coupling of these “hydrated-electron” orbitals to the THz vibrations promotes such water clusters as vibronically active “structured water” essential to biomolecular function such as protein folding. In biological microtubules, confined water-cluster THz vibrations may induce their “quantum coherence” communicated by Jahn–Teller phonons via coupling of the THz electromagnetic field to the water clusters’ large electric dipole moments.     

Recruitment patterns in Antarctic Peninsula shelf sediments: evidence of decoupling from seasonal phytodetritus pulses

Summer phytoplankton blooms on the West Antarctic Peninsula (WAP) shelf result in episodic deposition of labile food material for benthic detritivores. This summer deposition is thought to enhance benthic recruitment of macro- and megafauna. To explore seasonality in benthic recruitment, juvenile invertebrates (>100 μm) were collected in a seasonal time series at three stations on the WAP continental shelf. 4,098 juveniles were collected (average densities 2,000–7,000 m⁻²), with polychaetes dominant (2,581 individuals). The majority of polychaetes showed evidence of enhanced recruitment prior to the summer bloom, although patterns varied across stations. Additional taxa showed recruitment peaks in summer, but again, patterns varied among stations. Based on observed patterns, polychaete taxa are classified as “seasonal”, “marginally seasonal”, and “non-seasonal” recruiters, with the latter two patterns predominating. The year-round presence of small juveniles in most taxa suggests that recruitment occurs continuously, with periodic enhancement. Year-round recruitment is consistent with the presence of a persistent “food bank” of labile organic material in WAP shelf sediments, allowing recruitment to be largely decoupled from seasonal bloom dynamics.     

Erratum to: Mechanical properties of femoral trabecular bone in dogs

After the publication of this work [1], we became aware of the fact that the frequency of the ultrasound transmitter that we used for determining the elastic moduli of the trabecular bone specimens was not correctly specified. The oscillation frequency of the ultrasound transmitter was 2 MHz (and not 100 MHz as stated in our work) while we used a sampling rate of 100 MHz. In our publication, the oscillation frequency and sampling rate were confounded. Therefore also the statement in the discussion that we might have determined elastic moduli of trabecular bone tissue rather than the elastic properties of whole specimens because we used an ultrasound frequency > 2 MHz is wrong and has to be omitted.     

Sp1 polymorphism in collagen I α1 gene is associated with osteoporosis in lumbar spine of Mexican women

The Sp1 binding site polymorphism in collagen type I alpha 1 gene (COLIA1) has been associated with osteoporosis (OP) and bone mineral density (BMD). The aim of this study was to explore the association of this polymorphism with OP and BMD in the Mexican population by polymerase chain reaction and restriction fragment length polymorphism (PCR–RFLP) procedure. Allelic and genotypic frequencies from the Sp1 polymorphism were determined in 100 women with OP, 100 women without OP and 500 subjects from general Mexican population (GMP). Distribution of Sp1 polymorphism was in Hardy–Weinberg equilibrium. In spite of population structure due to racial mix in Mexican population, associations with OP were demonstrated. The frequency of “s” allele was significantly higher in women with OP (35%) than in women without OP (11%; P < 0.00001). Interestingly, “ss” genotype, was exclusive of women with OP and was associated with low BMD (0.588 ± 0.077 g/cm²) in contrast to “SS” genotype (0.733 ± 0.039 g/cm²; P = 0.0001). This work confirms the association of Sp1 polymorphism with low BMD and OP in Mexican population and make sure to use Sp1 as a genetic marker for OP in our population.     

Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

We describe and analyze a model for a stochastic pulse-coupled neuronal network with many sources of randomness: random external input, potential synaptic failure, and random connectivity topologies. We show that different classes of network topologies give rise to qualitatively different types of synchrony: uniform (Erdős–Rényi) and “small-world” networks give rise to synchronization phenomena similar to that in “all-to-all” networks (in which there is a sharp onset of synchrony as coupling is increased); in contrast, in “scale-free” networks the dependence of synchrony on coupling strength is smoother. Moreover, we show that in the uniform and small-world cases, the fine details of the network are not important in determining the synchronization properties; this depends only on the mean connectivity. In contrast, for scale-free networks, the dynamics are significantly affected by the fine details of the network; in particular, they are significantly affected by the local neighborhoods of the “hubs” in the network.