A quasi-brittle continuum damage finite element model of the human proximal femur based on element deletion

Estimation of the in-plane ultimate stress of lamellar tissue as a function of bone mineral density and osteocyte lacunae porosity

BACKGROUND AND OBJECTIVE

Detailed finite element models based on medical images (μ-CT) are commonly used to analyze the mechanical behavior of bone at microscale. In order to simulate the tissue failure onset, isotropic failure criteria of lamellar tissue are often used, despite its non-isotropic and heterogeneous nature. The main goal of the present work is to estimate the in-plane ultimate stress of lamellar bone, considering the influence of mineral content and the porosity due to the osteocyte lacunae concentration.

METHODS

To this aim, a representative volume cell of lamellar tissue is modeled numerically, including: (1) non-isotropic elastic properties of tissue as a function of the bone mineral density and (2) explicit modeling of the osteocyte lacunae, considering the range of porosity content, size and orientation of ellipsoid-shaped lacunae. Firstly, the element size for the finite element models have been defined from a preliminary convergence analysis. Bounds on the ultimate stress of non-porous lamellar tissue are estimated for two values of bone mineral density, considering the results of tensile and compressive tests of wet osteons from the literature. Subsequently, the ultimate stress of lamellar tissue considering several values of micro-porosity is addressed.

RESULTS

Results obtained in this work show that the strength of lamellar bone decreases exponentially with the increase of lacunae porosity concentration. Ultimate stress of non-porous tissue (p=0%) increases with high mineral content, reaching a value of S¯transc=355.40±39.80 MPa for compression in the transversal direction of the fiber bundles, being BMD=1.246g/cm3. The mean value for the longitudinal to transverse strength ratio evaluated for porosity p=0%,1% and 5% and a mineral content BMD=1.2g/cm3, is 2.47:1 for tension and 1.55:1 for compression. These values are in agreement with literature.

CONCLUSIONS

Osteocyte lacunae act as stress concentrators, acting as potential stimulus for the bone regeneration process. A novel micromechanical model for the in-plane ultimate stress of lamellar tissue as a function of mineral content and lacunae concentration is presented. Additional considerations about the intralamellar shear stress evolution are also given.

Prediction of the critical energy release rate for rat femoral cortical bone structure under different failure conditions

BACKGROUND AND OBJECTIVE

Critical energy release rate is a global fracture parameter that could be measured during the failing process, and its value may change under different failure conditions even in the same bone structure. The aim of this study was to propose an approach that combined the experimental test and finite element analysis to predict the critical energy release rates in the femoral cortical bone structures under compression and three-point bending loads.

METHODS

Three-point bending and compression experiments and the corresponding fracture simulations were performed on the rat femoral cortical bone structures. Different values of energy release rate were repeatedly assigned to the finite element models to perform fracture simulations, and then the load-displacement curves predicted in each simulation were compared with the experimental data to back-calculate the critical energy release rate.

RESULTS

The predicted data were similar to the experimental results when the calibrated energy release rate was suitable. The results showed that the cortical bone structure occurred shear open failure under compression load, and the predicted critical energy release rate was 0.12 N/mm. The same cortical bone structure occurred tensile open failure under three-point bending load, and the predicted critical energy release rate was 0.16 N/mm.

CONCLUSIONS

The critical energy release rates were different under various failure conditions in one cortical bone structure. A comprehensive analysis from the perspectives of material mechanical properties, failure mode, and damage fracture mechanism was conducted to reveal the reasons for the differences in the critical energy release rate in the cortical bone structure, which provided a theoretical basis for the measurement of the critical energy release rate and the accurate fracture simulation.

Investigation on the Differences in the Failure Processes of the Cortical Bone under Different Loading Conditions

Cortical bone is a transversely isotropic material, and the mechanical properties may be related to the loading direction on the osteon. Therefore, analyzing the differences in the failure processes of cortical bone under different loading conditions is necessary to explore the measures for reducing the incidence of fracture. In this study, to investigate the effects of different loading directions on the fracture performance in the cortical bone, a numerical method that could simultaneously simulate the failure processes in the cortical bone structure under compression and bending loads was established based on continuum damage mechanics theory. The prediction accuracy and feasibility of the numerical method were first verified by comparing with the corresponding experimental results. Then, the differences in the failure process and fracture performance of the same cortical bone structure under compression and bending loads were investigated. The simulation results indicated that for the same structure, the slip-open failure mode appeared under compression load, and the crack propagated along a certain angle to the loading direction; the tension-open failure mode appeared under bending load, and the crack propagated along the direction perpendicular to the loading direction. Meanwhile, the fracture load was greater and the fracture time was later in the compression than in the bending condition. These phenomena stated that discrepant failure processes and fracture patterns occurred in the same cortical bone structure under different loading conditions. The main reason may be related to the tension–compression asymmetry and transversely isotropic characteristics in the cortical bone material. The fracture simulations in the cortical bone under different loading conditions could improve the prediction accuracy in bone biomechanics and provide the prevention method for cortical bone damage and fracture.

Numerical modelling of cancellous bone damage using an orthotropic failure criterion and tissue elastic properties as a function of the mineral content and microporosity

BACKGROUND AND OBJECTIVE

Elastic and strength properties of lamellar tissue are essential to analyze the mechanical behaviour of bone at the meso- or macro-scale. Although many efforts have been made to model the architecture of cancellous bone, in general, isotropic elastic constants are assumed for tissue modelling, neglecting its non-isotropic behaviour. Therefore, isotropic damage laws are often used to estimate the bone failure. The main goals of this work are: (1) to present a new model for the estimation of the elastic properties of lamellar tissue which includes the bone mineral density (BMD) and the microporosity, (2) to address the numerical modelling of cancellous bone damage using an orthotropic failure criterion and a discrete damage mechanics analysis, including the novel approach for the tissue elastic properties aforementioned.

METHODS

Numerical homogenization has been used to estimate the elastic properties of lamellar bone considering BMD and microporosity. Microcomputed Tomography (μ-CT) scans have been performed to obtain the micro-finite element (μ-FE) model of cancellous bone from a vertebra of swine. In this model, lamellar tissue is orientated by considering a unidirectional layer pattern being the mineralized collagen fibrils aligned with the most representative geometrical feature of the trabeculae network. We have considered the Hashin's failure criterion and the Material Property Degradation (MPDG) method for simulating the onset and evolution of bone damage.

RESULTS

The terms of the stiffness matrix for lamellar tissue are derived as functions of the BMD and microporosity at tissue scale. Results obtained for the apparent yield strain values agree with experimental values found in the literature. The influence of the damage parameters on the bone mechanics behaviour is also presented.

CONCLUSIONS

Stiffness matrix of lamellar tissue depends on both BMD and microporosity. The new approach presented in this work enables to analyze the influence of the BMD and porosity on the mechanical response of bone. Lamellar tissue orientation has to be considered in the mechanical analysis of the cancellous bone. An orthotropic failure criterion can be used to analyze the bone failure onset instead of isotropic criteria. The elastic property degradation method is an efficient procedure to analyze the failure propagation in a 3D numerical model.

Effect of coronal fracture angle on the stability of screw fixation in medial malleolar fractures: A finite element analysis

Malleolar screw fixation is the most widely used treatment method for medial malleolar (MM) fractures. Here, although buttress plate fixation is advocated for vertical MM fractures, the angular discrimination between oblique and vertical MM fractures is still not fully understood. The purpose of this study is to test the adequacy of screw fixation in MM fractures with different angles and determination of a ‘critical fracture angle’ to guide surgeons in the decision-making for screw fixation for MM fractures by utilizing an advanced engineering simulation approach. In addition to loading of the healthy tibia structure, various cases of the MM fracture double screw fixation (14 simulation scenarios in total with fracture angles between 30° and 90°, in 5° increments) were considered in this research and their static loading conditions just after fixation operation were simulated through nonlinear (geometric and contact nonlinearity) finite element analysis (FEA). Patient-specific computed tomography scan data, parametric three-dimensional solid modelling and finite element method (FEM) based engineering codes were employed in order to simulate the fixation scenarios. Visual and numerical outputs for the deformation and stress distributions, separation and sliding behaviours of the MM fracture fragments of various screw fixations were clearly exhibited through FEA results. Minimum and maximum separation distances (gap) of 3.75 and 150.34 µm between fracture fragments at fracture angles of 30° and 90° were calculated respectively against minimum and maximum sliding distances of 25.87 and 41.37 µm between fracture fragments at fracture angles of 90° and 35°, respectively. The FEA results revealed that while the separation distance was increasing, the sliding distance was decreasing and there were no distinct differences in sliding distances in the scenarios from fracture angles of 30°–90°. The limitations and errors in a FEA study are inevitable, however, it was interpreted that the FEA scenarios were setup in this study by utilizing acceptable assumptions providing logical outputs under pre-defined boundary conditions. Finally, the fracture healing threshold for separation and/or sliding distance between fracture fragments was assigned as 100 µm by referring to previous literature and it was concluded that the screws fixed perpendicular to the fracture in a MM fracture with more than 70° angle with the tibial plafond results in a significant articular separation (>100 µm) during single-leg stand. Below this critical angle of 70°, two screws provide sufficient fixation.

Stochastic failure analysis of proximal femur using an isogeometric analysis based nonlocal gradient-enhanced damage model

BACKGROUND AND OBJECTIVE

Medical imaging-based finite element methods are more accurate tools for fracture risk prediction than the traditional aBMD based methods. However, these methods have drawbacks like geometric errors, high computational cost, mesh-dependent results, etc. In this article, the authors have proposed an isogeometric analysis-based nonlocal gradient-enhanced damage model to overcome some of these issues. Moreover, there are uncertainties in the values of input parameters for such analysis due to various measurement errors. Hence, stochastic analysis is performed to quantify the effect of these parametric uncertainties on the fracture behavior of the proximal femur.

METHODS

Computed Tomography images of a patient are used to create a 2D proximal femur model with a heterogeneous description of material properties. A numerical model based on gradient-enhanced nonlocal continuum damage mechanics is used for fracture analysis of proximal femur to overcome the issues related to mesh dependency in traditional continuum damage mechanics models. Further, a multipatch isogeometric solver is developed to solve the governing equations. Monte Carlo simulations are used to understand the effect of parametric uncertainties on the fracture behavior of the proximal femur.

RESULTS

The developed numerical framework is used to solve the fracture problem of proximal femur under single leg stance loading conditions. The obtained results are validated by comparing the load-displacement response and the crack path with that given in the literature. Stochastic analysis is performed by considering a ±5% variation in the elastic modulus, damage initiation strain, and the neck-shaft angle values.

CONCLUSION

The proposed numerical framework can correctly predict the damage initiation and propagation in a proximal femur. The results reveal that the heterogeneous nature of material properties of bone plays a significant role in determining the fracture characteristics of the proximal femur. Further, the results of the stochastic analysis reveal that the parametric uncertainties in the neck-shaft angle have a much more significant influence on the results of the analysis than the parametric uncertainties in the elastic modulus and damage initiation strain.

Damage Function of a Quasi-Brittle Material, Damage Rate, Acceleration and Jerk during Uniaxial Compression: Model and Application to Analysis of Trabecular Bone Tissue Destruction

A diversity of quasi-brittle materials can be observed in various engineering structures and natural objects (rocks, frozen soil, concrete, ceramics, bones, etc.). In order to predict the condition and safety of these objects, a large number of studies aimed at analyzing the strength of quasi-brittle materials has been conducted and presented in publications. However, at the modeling level, the problem of estimating the rate and acceleration of destruction of a quasi-brittle material under loading remains relevant. The purpose of the study was to substantiate the function of damage to a quasi-brittle material under uniaxial compression, determine the rate, acceleration and jerk of the damage process, and also to apply the results obtained to predicting the destruction of trabecular bone tissue. In accordance with the purpose of the study, the basic concepts of fracture mechanics and standard methods of mathematical modeling were used. The proposed model is based on the application of the previously obtained differentiable damage function without parameters. The results of the study are presented in the form of plots and analytical relations for computing the rate, acceleration and jerk of the damage process. Examples are given. The predicted peak of the combined effect of rate, acceleration and jerk of the damage process are found to be of practical interest as an additional criterion for destruction. The simulation results agree with the experimental data known from the available literature.

Loss of longitudinal superiority marks the microarchitecture deterioration of osteoporotic cancellous bones

Osteoporosis (OP), a skeletal disease making bone mechanically deteriorate and easily fracture, is a global public health issue due to its high prevalence. It has been well recognized that besides bone loss, microarchitecture degradation plays a crucial role in the mechanical deterioration of OP bones, but the specific role of microarchitecture in OP has not been well clarified and quantified from mechanics perspective. Here, we successfully decoupled and identified the specific roles of microarchitecture, bone mass and tissue property in the failure properties of cancellous bones, through μCT-based digital modeling and finite element method simulations on bone samples from healthy and ovariectomy-induced osteoporotic mice. The results show that the microarchitecture of healthy bones exhibits longitudinal superiority in mechanical properties such as the effective stiffness, strength and toughness, which fits them well to bearing loads along their longitudinal direction. OP does not only reduce bone mass but also impair the microarchitecture topology. The former is mainly responsible for the mechanical degradation of bones in magnitude, wherever the latter accounts for the breakdown of their function-favorable anisotropy, the longitudinal superiority. Hence, we identified the microarchitecture-deterioration-induced directional mismatch between material and loading as a hazardous feature of OP and defined a longitudinal superiority index as measurement of the health status of bone microarchitecture. These findings provide useful insights and guidelines for OP diagnosis and treat assessment.

Combined Effects of Exercise and Denosumab Treatment on Local Failure in Post-menopausal Osteoporosis-Insights from Bone Remodelling Simulations Accounting for Mineralisation and Damage

Denosumab has been shown to increase bone mineral density (BMD) and reduce the fracture risk in patients with post-menopausal osteoporosis (PMO). Increase in BMD is linked with an increase in bone matrix mineralisation due to suppression of bone remodelling. However, denosumab anti-resorptive action also leads to an increase in fatigue microdamage, which may ultimately lead to an increased fracture risk. A novel mechanobiological model of bone remodelling was developed to investigate how these counter-acting mechanisms are affected both by exercise and long-term denosumab treatment. This model incorporates Frost's mechanostat feedback, a bone mineralisation algorithm and an evolution law for microdamage accumulation. Mechanical disuse and microdamage were assumed to stimulate RANKL production, which modulates activation frequency of basic multicellular units in bone remodelling. This mechanical feedback mechanism controls removal of excess bone mass and microdamage. Furthermore, a novel measure of bone local failure due to instantaneous overloading was developed. Numerical simulations indicate that trabecular bone volume fraction and bone matrix damage are determined by the respective bone turnover and homeostatic loading conditions. PMO patients treated with the currently WHO-approved dose of denosumab (60 mg administrated every 6 months) exhibit increased BMD, increased bone ash fraction and damage. In untreated patients, BMD will significantly decrease, as will ash fraction; while damage will increase. The model predicted that, depending on the time elapsed between the onset of PMO and the beginning of treatment, BMD slowly converges to the same steady-state value, while damage is low in patients treated soon after the onset of the disease and high in patients having PMO for a longer period. The simulations show that late treatment PMO patients have a significantly higher risk of local failure compared to patients that are treated soon after the onset of the disease. Furthermore, overloading resulted in an increase of BMD, but also in a faster increase of damage, which may consequently promote the risk of fracture, specially in late treatment scenarios. In case of mechanical disuse, the model predicted reduced BMD gains due to denosumab, while no significant change in damage occurred, thus leading to an increased risk of local failure compared to habitual loading.

Mesh-independent damage model for trabecular bone fracture simulation and experimental validation

We propose in this study a two-dimensional constitutive model for trabecular bone combining continuum damage with embedded strong discontinuity. The model is capable of describing the three failure phases of trabecular bone tissue which is considered herein as a quasi-brittle material with strains and rotations assumed to be small and without viscous, thermal or other non-mechanical effects. The finite element implementation of the present model uses constant strain triangle (CST) elements. The displacement jump vector is implicitly solved through a return mapping algorithm at the local (finite element) level, while the global equilibrium equations are dealt with by Newton-Raphson method. The performance, accuracy and applicability of the proposed model for trabecular bone fracture are evaluated and validated against experimental measurements. These comparisons include both global and local aspects through numerical simulations of three-point bending tests performed on 10 single bovine trabeculae in the quasi-static regime.

Prediction of mechanical properties of trabecular bone in patients with type 2 diabetes using damage based finite element method

Type-2 diabetic (T2D) and osteoporosis (OP) suffered patients are more prone to fragile fracture though the nature of alteration in areal bone mineral density (aBMD) in these two cases are completely different. Therefore, it becomes crucial to compare the effect of T2D and OP on alteration in mechanical and structural properties of femoral trabecular bone. This study investigated the effect of T2D, OP, and osteopenia on bone structural and mechanical properties using micro-CT, nanoindentation and compression test. Further, a nanoscale finite element model (FEM) was developed to predict the cause of alteration in mechanical properties. Finally, a damage-based FEM was proposed to predict the pathological related alteration of bone's mechanical response. The obtained results demonstrated that the T2D group had lower volume fraction (-18.25%, p = 0.023), young's modulus (-23.47%, p = 0.124), apparent modulus (-37.15%, p = 0.02), and toughness (-40%, p = 0.001) than the osteoporosis group. The damage-based FE results were found in good agreement with the compression experiment results for all three pathological conditions. Also, nanoscale FEM results demonstrated that the elastic and failure properties of mineralised collagen fibril decreases with increase in crystal size. This study reveals that T2D patients are more prone to fragile fracture in comparison to OP and osteopenia patients. Also, the proposed damage-based FEM can help to predict the risk of fragility fracture for different pathological conditions.

A Review on Multiscale Bone Damage: From the Clinical to the Research Perspective

The investigation of bone damage processes is a crucial point to understand the mechanisms of age-related bone fractures. In order to reduce their impact, early diagnosis is key. The intricate architecture of bone and the complexity of multiscale damage processes make fracture prediction an ambitious goal. This review, supported by a detailed analysis of bone damage physical principles, aims at presenting a critical overview of how multiscale imaging techniques could be used to implement reliable and validated numerical tools for the study and prediction of bone fractures. While macro- and meso-scale imaging find applications in clinical practice, micro- and nano-scale imaging are commonly used only for research purposes, with the objective to extract fragility indexes. Those images are used as a source for multiscale computational damage models. As an example, micro-computed tomography (micro-CT) images in combination with micro-finite element models could shed some light on the comprehension of the interaction between micro-cracks and micro-scale bone features. As future insights, the actual state of technology suggests that these models could be a potential substitute for invasive clinical practice for the prediction of age-related bone fractures. However, the translation to clinical practice requires experimental validation, which is still in progress.

Experimental and numerical investigation of secondary screw perforation in the human proximal humerus

Surgical treatment of proximal humerus fractures remains challenging, with a reported failure rate ranging from 15% to 35%. The dominant failure mode is secondary, i.e. post-operative screw perforation through the glenohumeral joint. A better understanding and the ability to predict this complication could lead to improved fracture fixation and decreased failure rate. The aims of this study were (1) to develop an experimental model for single screw perforation in the human humeral head and (2) to evaluate the ability of densitometric measures and micro finite element (microFE) analyses to predict the experimental failure event. Screw perforation was investigated experimentally in twenty cuboidal specimens cut from four pairs of fresh-frozen human cadaveric proximal humeral heads. A centrally inserted 3.5 mm screw was pushed quasi-statically at a constant displacement rate until perforation of the articular cartilage in each specimen. Force and displacement were recorded and evaluated at both initial screw loosening and perforation events. Bone volume was calculated around and in front of the screw and tip-to-joint distance was measured on the combined pre- and post-instrumentation micro computed tomography (microCT) scans. Implicit linear and explicit non-linear microFE models were created based on the microCT scans. The strength of these densitometric, geometrical and microFE methods to predict the experimental results was evaluated via correlation analysis. The bone volume measures were optimized in a parametric analysis to maximize correlation coefficients. The strongest and quantitatively correct predictions of perforation force (R² = 0.93) and displacement (R² = 0.77) were achieved using the explicit, non-linear microFE models. Linear microFE simulations provided the strongest predictions of loosening force (R² = 0.90). Correlation strengths reached by optimized bone volume measures for predicting experimental force and by tip-to-joint distance for predicting displacement were only slightly inferior compared to the results of microFE models. The strong correlations achieved with densitometric and geometric measures indicate that monotonic perforation of single screws through the articular surface of the humeral head can be well predicted with these easily accessible measures. However, non-linear microFE models delivered even stronger correlations and quantitatively correct predictions of perforation force and displacement. This indicates that if computational resources are available, non-linear simulations may have a high potential to investigate more complex fixations and loading scenarios.

Numerical analysis of hip fracture due to a sideways fall

The primary purpose of this paper is to outline a methodology for evaluating the likelihood of cortical bone fracture in the proximal femur in the event of a sideways fall. The approach includes conducting finite element (FE) analysis in which the cortical bone is treated as an anisotropic material, and the admissibility of the stress field is validated both in tension and compression regime. In assessing the onset of fracture, two methodologies are used, namely the Critical Plane approach and the Microstructure Tensor approach. The former is employed in the tension regime, while the latter governs the conditions at failure in compression. The propagation of localized damage is modeled using a constitutive law with embedded discontinuity (CLED). In this approach, the localized deformation is described by a homogenization procedure in which the average properties of cortical tissue intercepted by a macrocrack are established. The key material properties governing the conditions at failure are specified from a series of independent material tests conducted on cortical bone samples tested at different orientations relative to the loading direction. The numerical analysis deals with simulations of experiments involving the sideways fall, and the results are compared with the experimental data. This includes both the evolution of fracture pattern and the local load-displacement characteristics. The proposed approach is numerically efficient, and the results do not display a pathological mesh-dependency. Also, in contrast to the XFEM approach, the analysis does not require any extra degrees of freedom.

Subject-specific FE models of the human femur predict fracture path and bone strength under single-leg-stance loading

Hip fractures are a major health problem with high socio-economic costs. Subject-specific finite element (FE) models have been suggested to improve the fracture risk assessment, as compared to clinical tools based on areal bone mineral density, by adding an estimate of bone strength. Typically, such FE models are limited to estimate bone strength and possibly the fracture onset, but do not model the fracture process itself. The aim of this study was to use a discrete damage approach to simulate the full fracture process in subject-specific femur models under stance loading conditions. A framework based on the partition of unity finite element method (PUFEM), also known as XFEM, was used. An existing PUFEM framework previously used on a homogeneous generic femur model was extended to include a heterogeneous material description together with a strain-based criterion for crack initiation. The model was tested on two femurs, previously mechanically tested in vitro. Our results illustrate the importance of implementing a subject-specific material distribution to capture the experimental fracture pattern under stance loading. Our models accurately predicted the fracture pattern and bone strength (1% and 5% error) in both investigated femurs. This is the first study to simulate complete fracture paths in subject-specific FE femur models and it demonstrated how discrete damage models can provide a more complete picture of fracture risk by considering both bone strength and fracture toughness in a subject-specific fashion.

Comparison of femur strain under different loading scenarios: Experimental testing

Bone fracture, formation and adaptation are related to mechanical strains in bone. Assessing bone stiffness and strain distribution under different loading conditions may help predict diseases and improve surgical results by determining the best conditions for long-term functioning of bone-implant systems. In this study, an experimentally wide range of loading conditions (56) was used to cover the directional range spanned by the hip joint force. Loads for different stance configurations were applied to composite femurs and assessed in a material testing machine. The experimental analysis provides a better understanding of the influence of the bone inclination angle in the frontal and sagittal planes on strain distribution and stiffness. The results show that the surface strain magnitude and stiffness vary significantly under different loading conditions. For the axial compression, maximal bending is observed at the mid-shaft, and bone stiffness is also maximal. The increased inclination leads to decreased stiffness and increased magnitude of maximum strain at the distal end of the femur. For comparative analysis of results, a three-dimensional, finite element model of the femur was used. To validate the finite element model, strain gauges and digital image correlation system were employed. During validation of the model, regression analysis indicated robust agreement between the measured and predicted strains, with high correlation coefficient and low root-mean-square error of the estimate. The results of stiffnesses obtained from multi-loading conditions experiments were qualitatively compared with results obtained from a finite element analysis of the validated model of femur with the same multi-loading conditions. When the obtained numerical results are qualitatively compared with experimental ones, similarities can be noted. The developed finite element model of femur may be used as a promising tool to estimate proximal femur strength and identify the best conditions for long-term functioning of the bone-implant system in future study.

Image-based finite-element modeling of the human femur

Fracture is considered a critical clinical endpoint in skeletal pathologies including osteoporosis and bone metastases. However, current clinical guidelines are limited with respect to identifying cases at high risk of fracture, as they do not account for many mechanical determinants that contribute to bone fracture. Improving fracture risk assessment is an important area of research with clear clinical relevance. Patient-specific numerical musculoskeletal models generated from diagnostic images are widely used in biomechanics research and may provide the foundation for clinical tools used to quantify fracture risk. However, prior to clinical translation, in vitro validation of predictions generated from such numerical models is necessary. Despite adopting radically different models, in vitro validation of image-based finite element (FE) models of the proximal femur (predicting strains and failure loads) have shown very similar, encouraging levels of accuracy. The accuracy of such in vitro models has motivated their application to clinical studies of osteoporotic and metastatic fractures. Such models have demonstrated promising but heterogeneous results, which may be explained by the lack of a uniform strategy with respect to FE modeling of the human femur. This review aims to critically discuss the state of the art of image-based femoral FE modeling strategies, highlighting principal features and differences among current approaches. Quantitative results are also reported with respect to the level of accuracy achieved from in vitro evaluations and clinical applications and are used to motivate the adoption of a standardized approach/workflow for image-based FE modeling of the femur.

Patient-Specific Bone Multiscale Modelling, Fracture Simulation and Risk Analysis-A Survey

This paper provides a starting point for researchers and practitioners from biology, medicine, physics and engineering who can benefit from an up-to-date literature survey on patient-specific bone fracture modelling, simulation and risk analysis. This survey hints at a framework for devising realistic patient-specific bone fracture simulations. This paper has 18 sections: Section 1 presents the main interested parties; Section 2 explains the organzation of the text; Section 3 motivates further work on patient-specific bone fracture simulation; Section 4 motivates this survey; Section 5 concerns the collection of bibliographical references; Section 6 motivates the physico-mathematical approach to bone fracture; Section 7 presents the modelling of bone as a continuum; Section 8 categorizes the surveyed literature into a continuum mechanics framework; Section 9 concerns the computational modelling of bone geometry; Section 10 concerns the estimation of bone mechanical properties; Section 11 concerns the selection of boundary conditions representative of bone trauma; Section 12 concerns bone fracture simulation; Section 13 presents the multiscale structure of bone; Section 14 concerns the multiscale mathematical modelling of bone; Section 15 concerns the experimental validation of bone fracture simulations; Section 16 concerns bone fracture risk assessment. Lastly, glossaries for symbols, acronyms, and physico-mathematical terms are provided.

Previous Damage Accumulation Can Influence Femoral Fracture Strength: A Finite Element Study

To manage osteoporotic hip fracture risk, it is necessary to understand failure mechanisms of bone at both the material and organ level. The structural response of bone is dependent on load history. Repeated loading causes progressive microstructural cracking, resulting in reduced apparent-level stiffness and, if damage is significant, reductions to peak load bearing capability. However, the effect of previous damage accumulation has not been well explored at the organ level. It was hypothesized that femoral fracture load and fracture pattern may be sensitive to damage accumulation from previous loading events. Six cadaveric specimens were used to develop patient specific finite element (FE) models from quantitative tomographic (qCT) scans. Material properties were assigned from qCT intensity at each element location, and damage evolution was predicted using a previously validated quasi-brittle FE model. Three scenarios were investigated: stumble followed by another stumble (S-S), fall followed by another fall (F-F), and stumble followed by a fall (S-F). Fracture load and pattern were compared to FE predictions for a single stumble (S) or single fall (F) loading event. Most specimens were resilient to accumulated damage, showing little (<5%) change in fracture load from the multiple-load scenarios (S-S, F-F, and S-F) compared to an equivalent single load scenario (S or F). Only one specimen demonstrated moderate (5-15%) reductions in strength from all three multiple-load scenarios. However, two specimens experienced moderate (20-30%) increase in fracture load in some load cases. In these cases, initial damage caused the load to be more evenly distributed upon subsequent loading events. © 2019 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 37:2197-2203, 2019.

A comparative study of tapped and untapped pilot holes for bicortical orthopedic screws – 3D finite element analysis with an experimental test

The aim of this study was to compare the screw-to-bone fixation strength of two insertion techniques: self-tapping screw (STS) and non-self-tapping screw (NSTS). Finite element analysis (FEA) was used for the comparison by featuring three tests (insertion, pull-out and shear) in a human tibia bone model. A non-linear material behavior with ductile damage properties was chosen for the modeling. To validate the numerical models, experimental insertion and pull-out tests were carried out using a synthetic bone. The experimental and numerical results of pull-out tests correlated well. Thread forming was successfully simulated during the insertion process of STS and NSTS. It is demonstrated that the STS generates higher insertion torque, induces a higher amount of stress after the insertion process and relatively more strength under the pull-out and shear tests than the NSTS. However, the NSTS induces more stiffness under the two tests (pull-out and shear) and less damage to the screw-bone interface compared to the STS. It is concluded that the use of STS ensures tighter bony contact and enables higher pull-out strength; however, the use of NSTS improves the stiffness of the fixation and induces less damage to the cortical bone-screw fixation and thus minimum risk is obtained in terms of bone necrosis.

A novel phase field method for modeling the fracture of long bones

A proximal humerus fracture is an injury to the shoulder joint that necessitates medical attention. While it is one of the most common fracture injuries impacting the elder community and those who suffer from traumatic falls or forceful collisions, there are almost no validated computational methods that can accurately model these fractures. This could be due to the complex, inhomogeneous bone microstructure, complex geometries, and the limitations of current fracture mechanics methods. In this paper, we develop a novel phase field method to investigate the proximal humerus fracture. To model the fracture in the inhomogeneous domain, we propose a power-law relationship between bone mineral density and critical energy release rate. The method is validated by an in vitro experiment, in which a human humerus is constrained on both ends while subjected to compressive loads on its head, in the longitudinal direction, that lead to fracture at the anatomical neck. CT scans are employed to acquire the bone geometry and material parameters, from which detailed finite element meshes with inhomogeneous Young modulus distributions are generated. The numerical method, implemented in a high performance computing environment, is used to quantitatively predict the complex 3D brittle fracture of the bone and is shown to be in good agreement with experimental observations. Furthermore, our findings show that the damage is initiated in the trabecular bone-head and propagates outward towards the bone cortex. We conclude that the proposed phase field method is a promising approach to model bone fracture.

An explicit micro-FE approach to investigate the post-yield behaviour of trabecular bone under large deformations

Homogenised finite element (FE) analyses are able to predict osteoporosis-related bone fractures and become useful for clinical applications. The predictions of FE analyses depend on the apparent, heterogeneous, anisotropic, elastic, and yield material properties, which are typically determined by implicit micro-FE (μFE) analyses of trabecular bone. The objective of this study is to explore an explicit μFE approach to determine the apparent post-yield behaviour of trabecular bone, beyond the elastic and yield properties. The material behaviour of bone tissue was described by elasto-plasticity with a von Mises yield criterion closed by a planar cap for positive hydrostatic stresses to distinguish the post-yield behaviour in tension and compression. Two ultimate strains for tension and compression were calibrated to trigger element deletion and reproduce damage of trabecular bone. A convergence analysis was undertaken to assess the role of the mesh. Thirteen load cases using periodicity-compatible mixed uniform boundary conditions were applied to three human trabecular bone samples of increasing volume fractions. The effect of densification in large strains was explored. The convergence study revealed a strong dependence of the apparent ultimate stresses and strains on element size. An apparent quadric strength surface for trabecular bone was successfully fitted in a normalised stress space. The effect of densification was reproduced and correlated well with former experimental results. This study demonstrates the potential of the explicit FE formulation and the element deletion technique to reproduce damage in trabecular bone using μFE analyses. The proper account of the mesh sensitivity remains challenging for practical computing times.

Numerical modelling of hip fracture patterns in human femur

BACKGROUND AND OBJECTIVE

Hip fracture morphology is an important factor determining the ulterior surgical repair and treatment, because of the dependence of the treatment on fracture morphology. Although numerical modelling can be a valuable tool for fracture prediction, the simulation of femur fracture is not simple due to the complexity of bone architecture and the numerical techniques required for simulation of crack propagation. Numerical models assuming homogeneous fracture mechanical properties commonly fail in the prediction of fracture patterns. This paper focuses on the prediction of femur fracture based on the development of a finite element model able to simulate the generation of long crack paths.

METHODS

The finite element model developed in this work demonstrates the capability of predicting fracture patterns under stance loading configuration, allowing the distinction between the main fracture paths: intracapsular and extracapsular fractures. It is worth noting the prediction of different fracture patterns for the same loading conditions, as observed during experimental tests.

RESULTS AND CONCLUSIONS

The internal distribution of bone mineral density and femur geometry strongly influences the femur fracture morphology and fracture load. Experimental fracture paths have been analysed by means of micro-computed tomography allowing the comparison of predicted and experimental crack surfaces, confirming the good accuracy of the numerical model.

How a pilot hole size affects osteosynthesis at the screw-bone interface under immediate loading

An inappropriate pilot hole size (PHS) is one of several factors that affects the stiffness of the screw-bone fixation. The present study uses finite element models to investigate the effect of varying the PHS on the biomechanical environment of the screw-bone interface of the fractured bone, after the screw insertion and under the immediate body weight pressure (BWP). Four PHS from 71% up to 85% of the screw external diameter (SED) were considered for analysis. A non linear material behaviour of the bone with ductile damage properties was used in the study. To validate the numerical models, an experimental pull-out test was carried out using a synthetic bone. The results of the insertion process demonstrated that the relatively smaller holes (71% and 75.5% of SED) increased the insertion torque value within the recommended level, caused more bone radial extension deformation and maximized the contact area between the bone threads and the screw, in comparison to the PHS higher than 80% of SED. Under the immediate BWP after osteosynthesis, the stress level exceeds the elastic limit and becomes high enough to initiate the ductile damage of the bone. Also, enlarging PHS from 71% to 75.5% of SED increased the bone microdisplacement at the screw-bone interface from 75 up to 100 μm, and that reduced the stiffness of the fixation.

Using Non-linear Homogenization to Improve the Performance of Macroscopic Damage Models of Trabecular Bone

Realistic macro-level finite element simulations of the mechanical behavior of trabecular bone, a cellular anisotropic material, require a suitable constitutive model; a model that incorporates the mechanical response of bone for complex loading scenarios and includes post-elastic phenomena, such as plasticity (permanent deformations) and damage (permanent stiffness reduction), which bone is likely to experience. Some such models have been developed by conducting homogenization-based multiscale finite element simulations on bone micro-structure. While homogenization has been fairly successful in the elastic regime and, to some extent, in modeling the macroscopic plastic response, it has remained a challenge with respect to modeling damage. This study uses a homogenization scheme to upscale the damage behavior from the tissue level (microscale) to the organ level (macroscale) and assesses the suitability of different damage constitutive laws. Ten cubic specimens were each subjected to 21 strain-controlled load cases for a small range of macroscopic post-elastic strains. Isotropic and anisotropic criteria were considered, density and fabric relationships were used in the formulation of the damage law, and a combined isotropic/anisotropic law with tension/compression asymmetry was formulated, based on the homogenized results, as a possible alternative to the currently used single scalar damage criterion. This computational study enhances the current knowledge on the macroscopic damage behavior of trabecular bone. By developing relationships of damage progression with bone's micro-architectural indices (density and fabric) the study also provides an aid for the creation of more precise macroscale continuum models, which are likely to improve clinical predictions.

Assessment of mechanical properties of human head tissues for trauma modelling

Many discrepancies are found in the literature regarding the damage and constitutive models for head tissues as well as the values of the constants involved in the constitutive equations. Their proper definition is required for consistent numerical model performance when predicting human head behaviour, and hence skull fracture and brain damage. The objective of this research is to perform a critical review of constitutive models and damage indicators describing human head tissue response under impact loading. A 3D finite element human head model has been generated by using computed tomography images, which has been validated through the comparison to experimental data in the literature. The threshold values of the skull and the scalp that lead to fracture have been analysed. We conclude that (1) compact bone properties are critical in skull fracture, (2) the elastic constants of the cerebrospinal fluid affect the intracranial pressure distribution, and (3) the consideration of brain tissue as a nearly incompressible solid with a high (but not complete) water content offers pressure responses consistent with the experimental data.

Young's Modulus and Load Complexity: Modeling Their Effects on Proximal Femur Strain

Finite element analysis (FEA) is a powerful tool for evaluating questions of functional morphology, but the application of FEA to extant or extinct creatures is a non-trivial task. Three categories of input data are needed to appropriately implement FEA: geometry, material properties, and boundary conditions. Geometric data are relatively easily obtained from imaging techniques, but often material properties and boundary conditions must be estimated. Here we conduct sensitivity analyses of the effect of the choice of Young's Modulus for elements representing trabecular bone and muscle loading complexity on the proximal femur using a finite element mesh of a modern human femur. We found that finite element meshes that used a Young's Modulus between 500 and 1,500 MPa best matched experimental strains. Loading scenarios that approximated the insertion sites of hip musculature produced strain patterns in the region of the greater trochanter that were different from scenarios that grouped muscle forces to the superior greater trochanter, with changes in strain values of 40% or more for 20% of elements. The femoral head, neck, and proximal shaft were less affected (e.g. approximately 50% of elements changed by 10% or less) by changes in the location of application of muscle forces. From our sensitivity analysis, we recommend the use of a Young's Modulus for the trabecular elements of 1,000 MPa for the proximal femur (range 500-1,500 MPa) and that the muscular loading complexity be dependent on whether or not strains in the greater trochanter are the focus of the analytical question. Anat Rec, 301:1189-1202, 2018. © 2018 Wiley Periodicals, Inc.

Virtual Experimental Validation of Dynamic Osteosynthesis in the Treatment of Fractures of the Proximal Femur

Volume model of transtrochanteric fracture of the femoral bone fixed by Targon PF dynamic fixator was studied by the finite element analysis. The following parameters were measured: magnitude and direction of displacement of system elements, pressure between fragments, and von Mises stress distribution in the fixator depending on support screw plunging into the sleeve. The results indicate that stability of the bone-nail system increases during fracture union and 10-mm shortening of the femoral neck axis, which is seen from a decrease in system deformation under load by 16.8%, stress in the implant by 20.2%, and pressure in the zone of fragments contact by 19.8%.

Assessment of compressive failure process of cortical bone materials using damage-based model

The main failure factors of cortical bone are aging or osteoporosis, accident and high energy trauma or physiological activities. However, the mechanism of damage evolution coupled with yield criterion is considered as one of the unclear subjects in failure analysis of cortical bone materials. Therefore, this study attempts to assess the structural response and progressive failure process of cortical bone using a brittle damaged plasticity model. For this reason, several compressive tests are performed on cortical bone specimens made of bovine femur, in order to obtain the structural response and mechanical properties of the material. Complementary finite element (FE) model of the sample and test is prepared to simulate the elastic-to-damage behavior of the cortical bone using the brittle damaged plasticity model. The FE model is validated in a comparative method using the predicted and measured structural response as load-compressive displacement through simulation and experiment. FE results indicated that the compressive damage initiated and propagated at central region where maximum equivalent plastic strain is computed, which coincided with the degradation of structural compressive stiffness followed by a vast amount of strain energy dissipation. The parameter of compressive damage rate, which is a function dependent on damage parameter and the plastic strain is examined for different rates. Results show that considering a similar rate to the initial slope of the damage parameter in the experiment would give a better sense for prediction of compressive failure.

Fracture characterization of human cortical bone under mode II loading using the end-notched flexure test

Fracture characterization of human cortical bone under mode II loading was analyzed using a miniaturized version of the end-notched flexure test. A data reduction scheme based on crack equivalent concept was employed to overcome uncertainties on crack length monitoring during the test. The crack tip shear displacement was experimentally measured using digital image correlation technique to determine the cohesive law that mimics bone fracture behavior under mode II loading. The developed procedure was validated by finite element analysis using cohesive zone modeling considering a trapezoidal with bilinear softening relationship. Experimental load-displacement curves, resistance curves and crack tip shear displacement versus applied displacement were used to validate the numerical procedure. The excellent agreement observed between the numerical and experimental results reveals the appropriateness of the proposed test and procedure to characterize human cortical bone fracture under mode II loading. The proposed methodology can be viewed as a novel valuable tool to be used in parametric and methodical clinical studies regarding features (e.g., age, diseases, drugs) influencing bone shear fracture under mode II loading.

Modelling of bone fracture and strength at different length scales: a review

In this paper, we review analytical and computational models of bone fracture and strength. Bone fracture is a complex phenomenon due to the composite, inhomogeneous and hierarchical structure of bone. First, we briefly summarize the hierarchical structure of bone, spanning from the nanoscale, sub-microscale, microscale, mesoscale to the macroscale, and discuss experimental observations on failure mechanisms in bone at these scales. Then, we highlight representative analytical and computational models of bone fracture and strength at different length scales and discuss the main findings in the context of experiments. We conclude by summarizing the challenges in modelling of bone fracture and strength and list open topics for scientific exploration. Modelling of bone, accounting for different scales, provides new and needed insights into the fracture and strength of bone, which, in turn, can lead to improved diagnostic tools and treatments of bone diseases such as osteoporosis.

Linear Elastic Analysis of Bovine Cortical Bone under Compression Loading

Linear elastic response of the bovine cortical bone has been examined under compression load. Experimental and computational methods were used to observe and predict the response of cortical bone. In computational method, two mechanical behaviors of isotropic and orthotropic were considered to simulate the cortical bone deformation. In experimental process, the specimens were designed to show maximum stiffness and strength by specifying osteon direction along loading axis during tests. The tests were controlled by displacement rate of 0.5 mm/minute and the overall stiffness responses of the structures were recorded to extract mechanical properties and also for validation aims. Finite Element Method (FEM) was used to model the linear response of the structure by using ABAQUS6.9EF. The FE results using orthotropic definition shows a good correlation with experimental data. A discussion was given based on overall stiffness and effective stress variation for both mechanical behaviors. In order to design the optimal implant structure, the presented study was proposed for prediction of bone structure deformation that attached to the orthopedic implants.

Micro-CT finite element model and experimental validation of trabecular bone damage and fracture

Most micro-CT finite element modeling of human trabecular bone has focused on linear and non-linear analysis to evaluate bone failure properties. However, prediction of the apparent failure properties of trabecular bone specimens under compressive load, including the damage initiation and its progressive propagation until complete bone failure into consideration, is still lacking. In the present work, an isotropic micro-CT FE model at bone tissue level coupled to a damage law was developed in order to simulate the failure of human trabecular bone specimens under quasi-static compressive load and predict the apparent stress and strain. The element deletion technique was applied in order to simulate the progressive fracturing process of bone tissue. To prevent mesh-dependence that generally affects the damage propagation rate, regularization technique was applied in the current work. The model was validated with experimental results performed on twenty-three human trabecular specimens. In addition, a sensitivity analysis was performed to investigate the impact of the model factors' sensitivities on the predicted ultimate stress and strain of the trabecular specimens. It was found that the predicted failure properties agreed very well with the experimental ones.

Finite element prediction with experimental validation of damage distribution in single trabeculae during three-point bending tests

There is growing evidence that information on trabecular microarchitecture can improve the assessment of fracture risk. One current strategy is to exploit finite element (FE) analysis applied to experimental data of mechanically loaded single trabecular bone tissue obtained from non-invasive imaging techniques for the investigation of the damage initiation and growth of bone tissue. FE analysis of this type of bone has mainly focused on linear and non-linear analysis to evaluate the bone's failure properties. However, there is a lack of experimentally validated FE damage models at trabecular bone tissue level allowing for the simulation of the progressive damage process (initiation and growth) till complete fracture. Such models are needed to perform enhanced prediction of the apparent failure mechanical properties needed to assess the fracture risk of bone organs. In the current study, we develop a FE model based on a continuum damage mechanics (CDM) approach to simulate the damage initiation and propagation of a single trabecula till complete facture in quasi-static regime. Three-point bending experiments were performed on single bovine trabeculae and compared to FE results. In order to validate the proposed FE mode, (i) the force displacement curve was compared to the experimental one and (ii) the damage distribution was correlated to the measured one obtained by digital image correlation based on stress whitening in bone, reported to be correlated to microdamage. A very good agreement was obtained between the FE and experimental results, indicating that the proposed damage investigation protocol based on FE analysis and testing is reliable to assess the damage behavior of bone tissue and that the current damage model is able to accurately simulate the damaging and fracturing process of single trabeculae under quasi static load.