The sensitivity of nonlinear computational models of trabecular bone to tissue level constitutive model

Microarchitectural finite element models have become a key tool in the analysis of trabecular bone. Robust, accurate, and validated constitutive models would enhance confidence in predictive applications of these models and in their usefulness as accurate assays of tissue properties. Human trabecular bone specimens from the femoral neck (n = 3), greater trochanter (n = 6), and lumbar vertebra (n = 1) of eight different donors were scanned by μ-CT and converted to voxel-based finite element models. Unconfined uniaxial compression and shear loading were simulated for each of three different constitutive models: a principal strain-based model, Drucker–Lode, and Drucker–Prager. The latter was applied with both infinitesimal and finite kinematics. Apparent yield strains exhibited minimal dependence on the constitutive model, differing by at most 16.1%, with the kinematic formulation being influential in compression loading. At the tissue level, the quantities and locations of yielded tissue were insensitive to the constitutive model, with the exception of the Drucker–Lode model, suggesting that correlation of microdamage with computational models does not improve the ability to discriminate between constitutive laws. Taken together, it is unlikely that a tissue constitutive model can be fully validated from apparent-level experiments alone, as the calculations are too insensitive to identify differences in the outcomes. Rather, any asymmetric criterion with a valid yield surface will likely be suitable for most trabecular bone models.     

Setup of a bone aging experimental model in the rabbit comparing changes in cortical and trabecular bone: Morphological and morphometric study in the femur

Bone aging was studied in an experimental model (rabbit femur) in three populations aged 0.5, 1.5, and 7.5 years. Cortical bone histology was compared with a data set from a 1.5‐month‐old population of an earlier published paper. From 0.5‐year‐old onward, the mean femur length did not increase further. Thereafter, the mean marrow area increased and the cortical area decreased significantly with aging. This was associated with a structural pattern transformation from plexiform to laminar and then Haversian‐like type. The distal meta‐epiphysis bone trabecular density of the oldest populations also was significantly lower in specific regions of interest (ROI). Percentage sealed primary vascular canals in laminar bone significantly increased with aging without variation of percentage sealed secondary osteons. Remodeling rate reflected by the density of cutting cones did not significantly change among the age populations. These data suggest that laminar bone vascular pattern is more functional in the fast diaphyseal expansion but not much streamlined with the renewal of blood flow during secondary remodeling. Bone aging was characterized by: 1) secondary remodeling subendosteally; 2) increment of sealed primary vascular canals number; 3) increased calcium content of the cortex; 4) cortical and trabecular bone mass loss in specific ROIs. Taken together, the present data may give a morphological and morphometric basis to perform comparative studies on experimental models of osteoporosis in the rabbit. J. Morphol. 276:733–747, 2015. © 2015 Wiley Periodicals, Inc.     

Medieval trabecular bone architecture: the influence of age, sex, and lifestyle

Osteoporosis has become a growing health concern in developed countries and an extensive area of research in skeletal biology. Despite numerous paleopathological studies of bone mass, few studies have measured bone quality in past populations. In order to examine age- and sex-related changes in one aspect of bone quality in the past, a study was made of trabecular bone architecture in a British medieval skeletal sample. X-ray images of 5-mm-thick coronal lumbar vertebral bone sections were taken from a total of 54 adult individuals divided into three age categories (18-29, 30-49, and 50+ years), and examined using image analysis to evaluate parameters related to trabecular bone structure and connectivity. Significant age-related changes in trabecular bone structure (trabecular bone volume (BV/TV), trabecular number (Tb.N), trabecular separation (Tb.Sp), and anisotropic ratio (Tb.An)) were observed to occur primarily by middle age with significant differences between the youngest and two older age groups. Neither sex showed continuing change in trabecular structure between the middle and old age groups. Age-related changes in bone connectivity (number of nodes (N.Nd) and node-to-node strut length (Nd.Nd)) similarly indicated a change in bone connectivity only between the youngest and two older age groups. However, females showed no statistical differences among the age groups in bone connectivity. These patterns of trabecular bone loss and fragility contrast with those generally found in modern populations that typically report continuing loss of bone structure and connectivity between middle and old age, and suggest greater loss in females. The patterns of bone loss in the archaeological samples must be interpreted cautiously. We speculate that while nutritional factors may have initiated some bone loss in both sexes, physical activity could have conserved bone architecture in old age in both sexes, and reproductive factors such as high parity and extended periods of lactation could have played a key role in female bone maintenance in this historic population. The study of qualitative elements (such as trabecular architecture) is vital if we are to understand bone maintenance and fragility in the past.     

Mechanical significance of femoral head trabecular bone structure in Loris and Galago evaluated using micromechanical finite element models

Work on the interspecific and intraspecific variation of trabecular bone in the proximal femur of primates demonstrates important architectural variation between animals with different locomotor behaviors. This variation is thought to be related to the processes of bone adaptation whereby bone structure is optimized to the mechanical environment. Micromechanical finite element models were created for the proximal femur of the leaping Galago senegalensis and the climbing and quadrupedal Loris tardigradus by converting bone voxels from high-resolution X-ray computed tomography scans of the femoral head to eight-noded brick elements. The resulting models had approximately 1.8 million elements each. Loading conditions representing takeoff phase of a leap and more generalized load orientations were applied to the models, and the models were solved using the iterative "row-by-row" matrix-vector multiplication algorithm. The principal strain and Von Mises stress results for the leaping model were similar for both species at each load orientation. Similar hip joint reaction forces in the range of 4.9 x to 12 x body weight were calculated for both species under each loading condition, but the hip reaction values estimated for Loris were higher than predicted based on locomotor behavior. These results suggest that functional adaptation to hip joint loading may not fully explain the differences in femoral head trabecular bone structure in Galago and Loris. The finite element method represents a unique and useful tool for analyzing the functional adaptation of trabecular bone in a diversity of animals and for reconstructing locomotor behavior in extinct taxa.     

On the interrelationship of permeability and structural parameters of vertebral trabecular bone: a parametric computational study

To characterise the flow of a fluid through a uniform porous medium, the medium may be completely described by its permeability, a measure of flow resistance. Fluid flow in the intertrabecular spaces of cancellous bone has been recognised as an important factor in a number of physical phenomena. In order to investigate the interdependence of permeability, porosity and the structural parameters, we adapted a morphological model and systematically varied its structural parameters. By simulating a viscous Stokes flow regime, we were able to estimate the anisotropic permeability tensor and performed an extensive, stepwise multivariate regression analysis to establish empirical relationships between the morphological parameters and the permeability for the anatomical directions individually. The regression analysis indicated high values of determination coefficients [0.88 < R² < 0.89 (transversal directions) and R² = 0.60 (longitudinal direction), porosity-based prediction and R² = 0.98 for all directions and information presented to the regression model]. We conclude that a pooled set of structural parameters may explain up to 98% of the permeability variability, the regression model predicts permeability values that match experimental data, and a good prediction performance could be achieved by only incorporating the porosity and either the degree of anisotropy (0.89 < R² < 0.91) or the trabecular spacing predictor (0.96 < R² < 0.97). These conclusions imply that trabecular thickness and shape parameters only play a minor role in the determination of vertebral trabecular bone permeability. However, a major limitation of the model is that it reflects an idealisation of the real, regionally varying structure of trabecular bone. Therefore, the goodness-of-fit estimates we are presenting should be considered as an upper bound limitation regarding the prediction performance.     

Angular orientation of trabecular bone in the femoral head and its relationship to hip joint loads in leaping primates

The elastic properties and mechanical behavior of trabecular bone are largely determined by its three-dimensional (3D) fabric structure. Recent work demonstrating a correlation between the primary mechanical and material axes in trabecular bone specimens suggests that fabric orientation may be used to infer directional components of the material strength and, by extension, the hypothetical loading regime. Here we quantify the principal orientation of trabecular bone in the femoral head and relate these principal fabric directions to loading patterns during various locomotor behaviors. The proximal femora of a diverse sample of prosimians were scanned using a high-resolution X-ray computed tomography scanner with resolution of better than 50 mum. Spherical volumes of interest were defined within the femoral heads and the 3D fabric anisotropy was calculated using the mean intercept length and star volume distribution methods. In addition to differences in bone volume and anisotropy, significant differences were found in the spatial orientation of the principal trabecular axes depending on locomotor behavior. The principal orientations for leapers (Galago, Tarsius, Avahi) are relatively tightly clustered (alpha(95) confidence limit: 8.2; angular variance s: 18.2 degrees ) and oriented in a superoanterior direction, while those of nonleapers are more variable across a range of directions (alpha(95): 16.8; s: 42.0 degrees ). The mean principal directions are significantly different for leaping vs. nonleaping taxa. These results further suggest a relationship between bone microstructure in the hip joint and locomotor behavior and indicate a similarity of loading across leapers despite differences in kinematics and phylogeny.     

The effect of restriction of dietary calcium on trabecular and cortical bone mineral density in the rats

This study aimed to investigate effects of restricted calcium intake on cortical and trabecular bone density in white rats. Low Ca diet was fed for six weeks, and bone density and bone metabolism parameters were assessed in blood. This study was carried out on 12 male white rats aged 12 weeks (Sprague-Dawley; SD). These rats were bred for 1 week and randomly assigned to the standard calcium diet group (SCa group, n = 6) and the low calcium diet group (LCa group; n = 6). The SCa group was given a modified AIN-93M mineral mix (with 0.5% Ca), which was made by adding calcium to a standard AIN93 diet, and the LCa Group was fed a modified AIN-93 Mineral mix (with 0.1% Ca). Femoral BMD and BMC were measured by DEXA in each rat. After trabecular bone was separated from cortical bone, volumetric bone mineral density (vBMD) was measured using pQCT. Serum Ca and P levels were measured as parameters of bone metabolism, and S-ALP, S-TrACP and-Dpd levels were also measured. The results revealed no significant differences in weight, growth rate, feed consumption and feed efficiency between the two groups before and after calcium-restricted diet (p > .05). No significant differences were also observed in bone length and bone mass between the two groups (p > .05). Although bilateral femoral BMDs were not significantly different between the two groups, bilateral femoral BMCs significantly decreased in the LCa group, compared with the SCa group (p = .023, p = .047). Bilateral cortical MDs were not significantly different between the two groups, either. However, trabecular BMD significantly decreased in the LCa group, compared with the SCa group (p = .041). U-Dpd and S-TrACP levels significantly declined in the LCa group, compared to the SCa group (p = .039, p = .010). There were no significant differences in serum Ca and P levels between the two groups (p > .05). However, a significant decrease in urinary Ca level (p = .001) and a significant increase in urinary P (p = .001) were observed in the LCa group, compared to the Sca group. These findings described that six-week low calcium diet led to decreased trabecular bone density, reduced urinary excretion of Ca and increased urinary excretion of P. As a result, Ca hemeostasis can be maintained.     

Stochastic multi-scale prediction on the apparent elastic moduli of trabecular bone considering uncertainties of biological apatite (BAp) crystallite orientation and image-based modelling

An assessment of the mechanical properties of trabecular bone is important in determining the fracture risk of human bones. Many uncertainty factors contribute to the dispersion of the estimated mechanical properties of trabecular bone. This study was undertaken in order to propose a computational scheme that will be able to predict the effective apparent elastic moduli of trabecular bone considering the uncertainties that are primarily caused by image-based modelling and trabecular stiffness orientation. The effect of image-based modelling which focused on the connectivity was also investigated. A stochastic multi-scale method using a first-order perturbation-based and asymptotic homogenisation theory was applied to formulate the stochastically apparent elastic properties of trabecular bone. The effective apparent elastic modulus was predicted with the introduction of a coefficient factor to represent the variation of bone characteristics due to inter-individual differences. The mean value of the predicted effective apparent Young's modulus in principal axis was found at approximately 460 MPa for respective 15.24% of bone volume fraction, and this is in good agreement with other experimental results. The proposed method may provide a reference for the reliable evaluation of the prediction of the apparent elastic properties of trabecular bone.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

Mechanical analysis of a rodent segmental bone defect model: The effects of internal fixation and implant stiffness on load transfer

Segmental bone defect animal models are often used for evaluating the bone regeneration performance of bone substituting biomaterials. Since bone regeneration is dependent on mechanical loading, it is important to determine mechanical load transfer after stabilization of the defect and to study the effects of biomaterial stiffness on the transmitted load. In this study, we assess the mechanical load transmitted over a 6 mm femur defect that is stabilized with an internal PEEK fixation plate. Subsequently, three types of selective laser melted porous titanium implants with different stiffness values were used to graft the defect (five specimens per group). In one additional group, the defect was left empty. Micro strain gauges were used to measure strain values at four different locations of the fixation plate during external loading on the femoral head. The load sharing between the fixation plate and titanium implant was highly variable with standard deviations of measured strain values between 31 and 93% of the mean values. As a consequence, no significant differences were measured between the forces transmitted through the titanium implants with different elastic moduli. Only some non-significant trends were observed in the mean strain values that, consistent with the results of a previous finite element study, implied the force transmitted through the implant increases with the implant stiffness. The applied internal fixation method does not standardize mechanical loading over the defect to enable detecting small differences in bone regeneration performances of bone substituting biomaterials. In conclusion, the fixation method requires further optimization to reduce the effects of the operative procedure and make the mechanical loading more consistent and improve the overall sensitivity of this rat femur defect model.     

Simulating the effects of localized red:far-red ratio on tillering in spring wheat (Triticum aestivum) using a three-dimensional virtual plant model

The outgrowth of tiller buds in Poaceae is influenced by the ratio of red to far-red light irradiance (R:FR). At each point in the plant canopy, R:FR is affected by light scattered by surrounding plant tissues. This paper presents a three-dimensional virtual plant modelling approach to simulate local effects of R:FR on tillering in spring wheat (Triticum aestivum). R:FR dependence of bud outgrowth was implemented in a wheat model, using three hypothetical responses of bud extension to R:FR (unit step, curvilinear and linear response). Bud break occurred when a threshold bud length was reached. Simulations were performed for three plant population densities. In accordance with experimental observations, fewer tillers per plant were simulated for higher plant population densities. The linear and curvilinear responses caused a delay in the increase in tiller number compared with experimental data. The unit step response approached experimental results best. It is suggested that a model based on relatively simple relations can be used to simulate degree of tillering. This study has shown that the virtual plant approach is a promising tool with which to address crop morphological and ecological research questions where the determining factors act at the level of the individual plant organ.     

A revised secondary structure model for the internal transcribed spacer 2 of the green algae Scenedesmus and Desmodesmus and its implication for the phylogeny of these algae

Secondary structure analysis of 34 internal transcribed spacer 2 (ITS-2) sequences showed that the current model for the green algae Scenedesmus and Desmodesmus is not accurate. In particular, helix I of the currently used model showed considerable deviations from our new model. The newly proposed model is supported by many two-sided compensated base pair changes and fully compensated insertions in all four helices. Phylogenetic analysis by maximum parsimony based on the new alignment confirmed the recent division of the old genus Scenedesmus into the new genera Scenedesmus and Desmodesmus. However, the analysis was not able to show phylogenetic relationships within these two genera. Hence, the ITS-2 region alone is not suitable for clarifying the phylogeny of Scenedesmus and Desmodesmus and new regions have to be found for future sequence analyses.     

Preliminary study of Lead (Pb) immobilization by meat and bone meal combustion residues (MBMCR) in soil: Assessment of Pb toxicity (phytotoxicity and genotoxicity) using the tobacco model (Nicotiana tabacum var. xanthi Dulieu)

Lead (Pb) is a major chemical pollutant in the environment. The present investigation evaluates the possible use of Meat and Bone Meal Combustion Residues (MBMCR), to sequester Pb from the soil compartment using the heterozygous tobacco model (Nicotiana tabacum var. xanthi Dulieu) characterized by the a1+ /a1 a2+ /a2 system. The toxic potential of Pb-contaminations (50, 100, 1,000, 2,000 and 10,000 mg Pb kg(-1)) as Pb(NO3) in standard soil was investigated in lab conditions according to three endpoints: (i) acute toxicity of plants (mortality, height and surface area parameters), (ii) Pb-accumulation in roots, stems and leaves, and (iii) genetic effects as the expression of reversion in the leaf of plants. Moreover, chemical investigations of Pb interactions with soil were realized to complete the toxicity evaluation. The results demonstrated that: (i) MBMCR were not acutely toxic or genotoxic to tobacco plants, (ii) Pb is acutely toxic to tobacco plants at 10,000 mg Pb kg(-1) of soil, (ii) but is not genotoxic, and (iii) Pb-bioaccumulation is significant in leaves, stems and roots (from 1,000, 2,000, and 50 mg Pb kg(-1) of soil, respectively). In contrast, in the presence of MBMCR, the toxic impacts of Pb were inhibited and Pb-accumulation in tobacco plants was reduced. In complement, chemical analyses highlighted the high capacity of the standard soil to immobilize Pb. The results suggest that even if Pb is bioavailable from soils to plants, complex mechanisms could occur in plants protecting them from the toxic impact of Pb.