Probabilistic failure analysis of bone using a finite element model of mineral-collagen composites

Effect of geometrical structure variations on strength and damage onset of cortical bone using multi-scale cohesive zone based finite element method

Three-dimensional multi-scale finite element models were designed to examine the effects of geometrical structure variations on the damage onset in cortical bone at multiple structural scales. A cohesive zone finite element approach, together with anisotropic damage initiation criteria, is used to predict the onset of damage. The finite element models are developed to account for the onset of microdamage from the microscopic length scales consisting of collagen fibres, to the macroscopic level consisting of osteons and the Haversian canals. Numerical results indicated that the yield strain at the initiation of microcracks is independent of variations in the local mineral volume fraction at each structural scale. Further, the yield strain and strength properties of cortical bone are dependent on its structural anisotropy and hierarchical structure. A positive correlation is observed between bone strength and mineral content at each length scale.

Finite element analysis of EBM manufactured bespoke implants: A probabilistic study

This study analyses three additively manufactured canine implants designed for angular limb deformity correction procedure through probabilistic numerical analysis. These implants have produced excellent results in-vivo and are operational to-date. Therefore, this study uses finite element analysis in conjunction with statistical analysis in order to further validate these implants from a numerical perspective. Due to uncertainties associated with boundary conditions for a bespoke implant geometry, the analyses in this study were conducted on a range of input values. An interrogation of these parameters through sensitivity analysis enabled in identifying the vital inputs. These inputs were then employed to conduct robustness analysis in order to determine the mean value of stress on which these implants ideally operate. These mean values were then compared with the associated safety and failure limit to obtain the probability of reaching these limits through different reliability techniques. A low probability of failure computed from numerical analysis in combination with the continued performance of these implants, suggests a successful integration of the methodology in the design phase of bespoke implants.

An embedded element based 2D finite element model for the strength prediction of mineralized collagen fibril using Monte-Carlo type of simulations

A computationally efficient statistical model for the prediction of the strength of mineralized collagen fibril (a basic building block of bone) is presented by taking into account the uncertainties associated with the geometrical and material parameters of collagen and mineral phases. The mineral plates have been considered as one-dimensional bar elements embedded in the two-dimensional plane stress collagen matrix. The mineral phase is considered as linear elastic and a hyperelastic material model is adopted for the collagen phase. Further, the crack initiation and propagation in the collagen phase have been modeled using a damage plasticity approach. Different realizations of the arrangement of mineral plates have been generated to account for the associated geometrical uncertainties using an in-house MATLAB® code. Monte-Carlo type simulations have been performed on the different realizations of mineralized collagen fibril to predict its characteristic stress-strain response under tensile load. The characteristic strength of 3.64 GPa is obtained for mineralized collagen fibril using Weibull's analysis which is found to be in agreement with the molecular dynamics simulation data and numerical studies reported in the past. A parameter sensitivity analysis concluded that mineral modulus has a significant effect on the overall tangent modulus of mineralized collagen fibril in large strain regime.

The importance of intrinsic damage properties to bone fragility: a finite element study

As the average age of the population has increased, the incidence of age-related bone fracture has also increased. While some of the increase of fracture incidence with age is related to loss of bone mass, a significant part of the risk is unexplained and may be caused by changes in intrinsic material properties of the hard tissue. This investigation focused on understanding how changes to the intrinsic damage properties affect bone fragility. We hypothesized that the intrinsic (μm) damage properties of bone tissue strongly and nonlinearly affect mechanical behavior at the apparent (whole tissue, cm) level. The importance of intrinsic properties on the apparent level behavior of trabecular bone tissue was investigated using voxel based finite element analysis. Trabecular bone cores from human T12 vertebrae were scanned using microcomputed tomography (μCT) and the images used to build nonlinear finite element models. Isotropic and initially homogenous material properties were used for all elements. The elastic modulus (E(i)) of individual elements was reduced with a secant damage rule relating only principal tensile tissue strain to modulus damage. Apparent level resistance to fracture as a function of changes in the intrinsic damage properties was measured using the mechanical energy to failure per unit volume (apparent toughness modulus, W(a)) and the apparent yield strength (σ(ay), calculated using the 0.2% offset). Intrinsic damage properties had a profound nonlinear effect on the apparent tissue level mechanical response. Intrinsic level failure occurs prior to apparent yield strength (σ(ay)). Apparent yield strength (σ(ay)) and toughness vary strongly (1200% and 400%, respectively) with relatively small changes in the intrinsic damage behavior. The range of apparent maximum stresses predicted by the models was consistent with those measured experimentally for these trabecular bone cores from the experimental axial compressive loading (experimental: σ(max) = 3.0-4.3 MPa; modeling: σ(max) = 2-16 MPa). This finding differs significantly from previous studies based on nondamaging intrinsic material models. Further observations were that this intrinsic damage model reproduced important experimental apparent level behaviors including softening after peak load, microdamage accumulation before apparent yield (0.2% offset), unload softening, and sensitivity of the apparent level mechanical properties to variability of the intrinsic properties.

Increased susceptibility to microdamage in Brtl/+ mouse model for osteogenesis imperfecta

Osteogenesis imperfecta (OI) is a genetic disease of collagen or collagen-related proteins that adversely impacts bone mass and fracture resistance. Little is known regarding the role that microdamage plays in OI and whether or not OI bone is more prone to damage accumulation than bone with unaffected collagen. The Brtl/+ mouse is a heterozygous model for OI which contains a Gly349Cys substitution in one COL1A1 allele, and demonstrates a low ductility phenotype. At 8 weeks of age, Brtl/+ demonstrates an increase in osteoclast number, which mimics the upregulated bone turnover often found in OI patients. We hypothesize that upregulated osteoclast activity in Brtl/+ is due, in part, to increased remodeling associated with microdamage repair. In the present study, we used Brtl/+ to investigate the susceptibility of OI bone to microdamage. The mouse ulnar loading model was used to induce microdamage and to test the hypothesis that Brtl/+ is more susceptible to damage accumulation than age-matched wild type (WT) counterparts. Linear elastic fracture mechanics (LEFM) was used to investigate the fracture toughness properties of both Brtl/+ and WT bones to determine if there is any correlation with toughness and the degree of microdamage. Results show that Brtl/+ ulnae subject to normal cage activity demonstrate an inherently larger amount of microdamage than WT controls. Following axial compressive loading, Brtl/+ ulnae are more prone to damage than WT counterparts despite demonstrating a greater resistance to whole-bone deformation. Fracture toughness results demonstrate that Brtl/+ specimens, despite not exhibiting a significant difference, display a trend toward lower fracture toughness values than their WT counterparts. Correlations show that microdamage levels tend to increase as fracture toughness decreases. Together, these findings may have strong clinical implications for explaining increased fragility and remodeling activity in OI patients.

FINDING OF MICRODAMAGE MORPHOLOGY DIFFERENCES IN MOUSE FEMORAL BONES WITH DISTINCT MINERALIZATION LEVELS

Microdamage progression in bone is dependent on the ultrastructure of the tissue. Thus, any pathological changes in bone ultrastructure may be reflected in the pattern and capacity of microdamage accumulation. A previous numerical study of microdamage progression in bone using a probabilistic failure analysis approach predicts that the microdamage morphology (either linear microcracks or diffuse damage) is very sensitive to the level of mineralization in bone, which is also implicated in some experimental observations. To examine the prediction, femurs from two strains of mice (C57BL/6J, N = 10 and C3H/HeJ, N = 11) that have distinct mineralization levels were fatigued under four-point bending to create damage in the bone tissues. After testing, the microdamage morphology of the bone samples was examined using bulk-staining technique with basic fuchsin. The results demonstrate that more linear microcracks are observed in femurs of C3H/HeJ (higher mineralization), whereas more diffuse-like damage is found in C57BL/6J femurs (less mineralized). Compared with linear microcracks, the formation of diffuse damage tends to dissipate more energy and help bone to avoid catastrophic failures. Therefore, results from this study may help explain why highly mineralized bone tends to be more brittle. Observations from this study are consistent with the numerical prediction from the previous study, suggesting that mineralization has a significant effect on the microdamage morphology of bone.

Effect of mineral-collagen interfacial behavior on the microdamage progression in bone using a probabilistic cohesive finite element model

The interactions between mineral and collagen phases in the ultrastructural level play an important role in determining the mechanical properties of bone tissue. Three types of mineral-collagen interaction (i.e., ionic interactions, hydrogen/van der Waals bonds, and van der Waals/viscous shear in opening/sliding mode, respectively) have been simulated in this study, using cohesive zone-modeling techniques. Considering the inhomogeneity of bone, a probabilistic failure analysis approach has been also employed to account for the effect of mineral-collagen interfacial behavior on microdamage accumulation in lamellar bone tissues. The results of this study suggested that different interfacial behaviors cause different types of microdamage accumulation. The ionic interactions between the mineral and collagen phases lead to the formation of linear microcracks, while the van der Waals/viscous shear interactions may facilitate the formation of diffuse damage. In the case of hydrogen/van der Waals bonds, a transitional behavior of microdamage accumulation in bone was observed. The findings of this study may help in understanding the mechanisms of mineral-collagen interactions and its effects on the failure mechanism of bone.

Random field assessment of nanoscopic inhomogeneity of bone

Bone quality is significantly correlated with the inhomogeneous distribution of material and ultrastructural properties (e.g., modulus and mineralization) of the tissue. Current techniques for quantifying inhomogeneity consist of descriptive statistics such as mean, standard deviation and coefficient of variation. However, these parameters do not describe the spatial variations of bone properties. The objective of this study was to develop a novel statistical method to characterize and quantitatively describe the spatial variation of bone properties at ultrastructural levels. To do so, a random field defined by an exponential covariance function was used to represent the spatial uncertainty of elastic modulus by delineating the correlation of the modulus at different locations in bone lamellae. The correlation length, a characteristic parameter of the covariance function, was employed to estimate the fluctuation of the elastic modulus in the random field. Using this approach, two distribution maps of the elastic modulus within bone lamellae were generated using simulation and compared with those obtained experimentally by a combination of atomic force microscopy and nanoindentation techniques. The simulation-generated maps of elastic modulus were in close agreement with the experimental ones, thus validating the random field approach in defining the inhomogeneity of elastic modulus in lamellae of bone. Indeed, generation of such random fields will facilitate multi-scale modeling of bone in more pragmatic details.

Collagen mutation causes changes of the microdamage morphology in bone of an OI mouse model

Previous studies have postulated that ultrastructural changes may alter the pattern and capacity of microdamage accumulation in bone. Using an osteogenesis imperfecta (OI) mouse model, this study was performed to investigate the correlation of collagen mutation with the microdamage morphology and the associated brittleness of bone. In this study, femurs from mild OI and wild type mice were fatigued under four-point bending to create microdamage in the specimens. Then, the microdamage morphology of these specimens was examined using the bulk-staining technique with basic fuchsin. Similar with the results of previous studies, it was observed that linear microcracks were formed more easily in compression, whereas diffuse damage was induced more readily in tension for both wild-type and mild-type mice. However, less diffuse damage was found in the tensile side of mild OI mouse femurs (collagen mutation) compared with those of wild type mice, showing that the microdamage morphology is correlated to the brittleness of bone. The results of this study provide direct evidence that supports the prediction made by the previous numerical simulation studies, suggesting that microdamage morphology in bone is significantly correlated with the integrity of the collagen phase.

Activation of bone remodeling after fatigue: differential response to linear microcracks and diffuse damage

Recent experiments point to two predominant forms of fatigue microdamage in bone: linear microcracks (tens to a few hundred microns in length) and "diffuse damage" (patches of diffuse stain uptake in fatigued bone comprised of clusters of sublamellar-sized cracks). The physiological relevance of diffuse damage in activating bone remodeling is not known. In this study microdamage amount and type were varied to assess whether linear or diffuse microdamage has similar effects on the activation of intracortical resorption. Activation of resorption was correlated to the number of linear microcracks (Cr.Dn) in the bone (R(2)=0.60, p<0.01). In contrast, there was no activation of resorption in response to diffuse microdamage alone. Furthermore, there was no significant change in osteocyte viability in response to diffuse microdamage, suggesting that osteocyte apoptosis, which is known to activate remodeling at typical linear microcracks in bone, does not result from sublamellar damage. These findings indicate that inability of diffuse microdamage to activate resorption may be due to lack of a focal injury response. Finally, we found that duration of loading does not affect the remodeling response. In conclusion, our data indicate that osteocytes activate resorption in response to linear microcracks but not diffuse microdamage, perhaps due to lack of a focal injury-induced apoptotic response.