Personalized loading conditions for homogenized finite element analysis of the distal sections of the radius.

The microstructure of trabecular bone is known to adapt its morphology in response to mechanical loads for achieving a biomechanical homeostasis. Based on this form-function relationship, previous investigators either simulated the remodeling of bone to predict the resulting density and architecture for a specific loading or retraced physiological loading conditions from local density and architecture. The latter inverse approach includes quantifying bone morphology using computed tomography and calculating the relative importance of selected load cases by minimizing the fluctuation of a tissue loading level metric. Along this concept, the present study aims at identifying an optimal, personalized, multiaxial load case at the distal section of the human radius using in vivo HR-pQCT-based isotropic, homogenized finite element (hFE) analysis. The dataset consisted of HR-pQCT reconstructions of the 20 mm most distal section of 21 human fresh-frozen radii. We simulated six different unit canonical load cases (FX palmar-dorsal force, FY ulnar-radial force, FZ distal-proximal force, MX moment about palmar-dorsal, MY moment about ulnar-radial, MZ moment about distal-proximal) using a simplified and efficient hFE method based on a single isotropic bone phase. Once we used a homogeneous mean density (shape model) and once the original heterogeneous density distribution (shape + density model). Using an analytical formulation, we minimized the deviation of the resulting strain tensors ε(x) to a hydrostatic compressive reference strain ε0, once for the 6 degrees of freedom (DOF) optimal (OPT) load case and for all individual 1 DOF load cases (FX, FY, FZ, MX, MY, MZ). All seven load cases were then extended in the nonlinear regime using the scaled displacements of the linear load cases as loading boundary conditions (MAX). We then compared the load cases and models for their objective function (OF) values, the stored energies and their ultimate strength using a specific torsor norm. Both shape and shape + density linear-optimized OPT models were dominated by a positive force in the z-direction (FZ). Transversal force DOFs were close to zero and mean moment DOFs were different depending on the model type. The inclusion of density distribution increased the influence and changed direction of MX and MY, while MZ was small in both models. The OPT load case had 12-15% lower objective function (OF) values than the FZ load case, depending on the model. Stored energies at the optimum were consistently 142-178% higher for the OPT load case than for the FZ load case. Differences in the nonlinear response maximum torsor norm ‖t‖ were heterogeneous, but consistently higher for OPT_MAX than FZ_MAX. We presented the proof of concept of an optimization procedure to estimate patient-specific loading conditions for hFE methods. In contrast to similar models, we included canonical load cases in all six DOFs and used a strain metric that favors hydrostatic compression. Based on a biomechanical analysis of the distal joint surfaces at the radius, the estimated load directions are plausible. For our dataset, the resulting OPT load case is close to the standard axial compression boundary conditions, usually used in HR-pQCT-based FE analysis today. But even using the present simplified hFE model, the optimized linear six DOF load case achieves a more homogeneous tissue loading and can absorb more than twice the energy than the standard uniaxial load case. The ultimate strength calculated with a torsor norm was consistently higher for the 6-DOF nonlinear model (OPT_MAX) than for the 1-DOF nonlinear uniaxial model (FZ_MAX). Defining patient-specific boundary conditions may decrease angulation errors during CT measurements and improve repeatability as well as reproducibility of bone stiffness and strength estimated by HR-pQCT-based hFE analysis. These results encourage the extension of the present method to anisotropic hFE models and their application to repeatability data sets to test the hypothesis of reduced angulation errors during measurement

A constitutive law for trabecular bone

Motivated by applications in orthopaedic surgery, new constitutive laws for trabecular (or spongious) bone are developed in the framework of continuum mechanics, implemented in a mechanical analysis computer program, validated by a number of in vitro experiments and illustrated by the simulation of a femoral total hip component. Current knowledge about the morphological and mechanical properties of trabecular bone is reviewed for setting the background and clarifying the contributions of the thesis. Comprehensive 1D and 3D theoretical models based on the approach of standard generalized materials are developed with a specific attention towards irreversible phenomena. The 1D model includes linear elasticity and rate-independent as well as rate-dependent plastic strain flow with damage. Based on a second order fabric tensor, the 3D model includes inhomogeneous, orthotropic linear elasticity and rate-independent plasticity with damage. In order to solve boundary value problems involving complex bone or bone-implant structures, implicit projection algorithms are developed for integrating the plastic flow rules with damage and implemented in the computer program TACT combining the finite element method, the linear iteration method and the finite difference method. The resulting numerical models are illustrated by the means of traction, bending and torsion benchmark tests. A number of pilot in vitro experiments are undertaken on human and bovine trabecular bone specimens in order to validate the theoretical models and identify the material constants. Quasistatic uniaxial and torsion experiments are performed with a method avoiding artefacts due to the inhomogeneous boundary conditions associated with porosity. Anisotropic elasticity, plasticity and damage of trabecular bone prove to be successfully described by the models in terms of structural density and morphology. Finally, the 3D constitutive law is applied to the biomechanical problem of primary stability of a cementless femoral total hip component in order to illustrate its potential

Investigating the post-yield behavior of mineralized bone fibril arrays using a 3D non-linear finite element unit-cell model.

In this study, we propose a 3D non-linear finite element (FE) unit-cell model to investigate the post-yield behavior of mineralized collagen fibril arrays (FAY). We then compare the predictions of the model with recent micro-tensile and micropillar compression tests in both axial and transverse directions. The unit cell consists of mineralized collagen fibrils (MCFs) embedded in an extrafibrillar matrix (EFM), and the FE mesh is equipped with cohesive interactions and a custom plasticity model. The simulation results confirm that MCF plays a dominant role in load bearing prior to yielding under axial tensile loading. Damage was initiated via debonding in shear and progressive sliding at the MCF/EFM interface, and resulted in MCF pull-out until brittle failure. In transverse tensile loading, EFM carried most of the load in pre-yield deformation, and then mixed normal/shear debonding between MCF and EFM began to form, which eventually produced brittle delamination of the two phases. The loading/unloading FE analysis in compression along both axial and transverse directions demonstrated perfect plasticity without any reduction in elastic modulus, i.e., damage due to the interfaces as seen in micropillar compression. Beyond the brittle and ductile nature of the stress-strain curves, in tensile and compressive loading, the simulated post-yield behavior and failure mechanism are in good quantitative agreement with the experimental observations. Our rather simple but efficient unit-cell FE model can reproduce qualitatively and quantitatively the mechanical behavior of bone ECM under tensile and compressive loading along the two main orientations. The model's integration into higher length scales may be useful in describing the macroscopic post-yield and failure behavior of trabecular and cortical bone in greater detail

Microtensile properties and failure mechanisms of cortical bone at the lamellar level.

Bone features a remarkable combination of toughness and strength which originates from its complex hierarchical structure and motivates its investigation on multiple length scales. Here, in situ microtensile experiments were performed on dry ovine osteonal bone for the first time at the length scale of a single lamella. The micromechanical response was brittle and revealed larger ultimate tensile strength compared to the macroscale (factor of 2.3). Ultimate tensile strength for axial and transverse specimens was 0.35 ± 0.05 GPa and 0.13 ± 0.02 GPa, respectively. A significantly greater strength anisotropy relative to compression was observed (axial to transverse strength ratio of 2.7:1 for tension, 1.3:1 for compression). Fracture surface and transmission electron microscopic analysis suggested that this may be rationalized by a change in failure mode from fibril-matrix interfacial shearing for axial specimens to fibril-matrix debonding in the transverse direction. An improved version of the classic Hashin's composite failure model was applied to describe lamellar bone strength as a function of fibril orientation. Together with our experimental observations, the model suggests that cortical bone strength at the lamellar level is remarkably tolerant to variations of fibrils orientation of about ±30°. This study highlights the importance of investigating bone's hierarchical organization at several length scales for gaining a deeper understanding of its macroscopic fracture behavior. STATEMENT OF SIGNIFICANCE: Understanding bone deformation and failure behavior at different length scales of its hierarchical structure is fundamental for the improvement of bone fracture prevention, as well as for the development of multifunctional bio-inspired materials combining toughness and strength. The experiments reported in this study shed light on the microtensile properties of dry primary osteonal bone and establish a baseline from which to start further investigations in more physiological conditions. Microtensile specimens were stronger than their macroscopic counterparts by a factor of 2.3. Lamellar bone strength seems remarkably tolerant to variations of the sub-lamellar fibril orientation with respect to the loading direction (±30°). This study underlines the importance of studying bone on all length scales for improving our understanding of bone's macroscopic mechanical response

Tension-compression strength asymmetry of bone extracellular matrix

Bone features a hierarchical architecture, as a result of which antagonistic properties like toughness and strength are achieved. On the macroscale, bone exhibits a distinct anisotropy and loading mode dependence, with a considerably lower strength in tension compared to compression. To better understand the mechanisms leading to this behavior, anisotropic tensile yield and failure properties of ovine bone were characterized on the length scale of a single lamella (3-7 μm) and then compared to compression data for the same scale [1]. In situ microtensile experiments were carried out using an improved testing methodology, developed to overcome typical issues encountered during small scale testing related to sample fabrication, sample handling and misalignment [2]. The methodology is based on self-aligning silicon grippers prepared by means of reactive ion etching and an optimized microtensile sample geometry that can be fabricated via focused ion beam (FIB) milling. The measured elastic modulus, strength, yield stress and strain at maximum stress are summarized in table 1. Please click Additional Files below to see the full abstract

Influence of aging on mechanical properties of the femoral neck using an inverse method.

Today, we are facing rapid aging of the world population, which increases the incidence of hip fractures. The gold standard of bone strength assessment in the laboratory is micro-computed finite element analysis (μFEA) based on micro-computed tomography (μCT) images. In clinics, the standard method to assess bone fracture risk is based on areal bone mineral density (aBMD), measured by dual-energy X-ray absorptiometry (DXA). In addition, homogenized finite element analysis (hFEA) constructed from quantitative computed tomography reconstructions (QCT) predicts clinical bone strength more accurately than DXA. Despite considerable evidence of degradation of bone material properties with age, in the past fifty years of finite element analysis to predict bone strength, bone material parameters remained independent of age. This study aims to assess the influence of age on apparent modulus, yield stress, and strength predictions of the human femoral neck made by laboratory-available bone volume fraction (BV/TV) and μFEA; and by clinically available DXA and hFEA. Using an inverse method, we test the hypothesis that FEA material parameters are independent of age. Eighty-six human femora were scanned with DXA (aBMD) and with QCT. The femoral necks were extracted and scanned at 16 μm resolution with μCT. The grayscale images were downscaled to 32 μm and 65 μm for linear and non-linear analyses, respectively, and segmented. The μFE solver ParOSolNL (non-linear) and a standard hFEA method were applied to the neck sections with the same material properties for all samples to compute apparent modulus, yield stress, and strength. Laboratory-available BV/TV was a good predictor of apparent modulus (R2 = 0.76), almost as good as μFEA (R2 = 0.79). However, yield stress and strength were better predicted by μFEA (R2 = 0.92, R2 = 0.86, resp.) than BV/TV (R2 = 0.76, R2 = 0.76, resp.). For clinically available variables, prediction of apparent modulus was better with hFEA than aBMD (R2 = 0.67, R2 = 0.58, resp.). hFEA outperformed aBMD for predictions of yield stress (R2 = 0.63 vs R2 = 0.34 for female and R2 = 0.55 for male) and strength (R2 = 0.48 vs R2 = 0.33 for female and R2 = 0.15 for male). The inclusion of age did not improve the multiple linear models for apparent modulus, yield stress, and strength. The resolution of the μFE meshes seems to account for most morphological changes induced by aging. The errors between the simulation and the experiment for apparent modulus, yield stress, and strength were age-independent, suggesting no rationale for correcting tissue material parameters in the current FE analysis of the aging femoral neck

Nanoscale compressive deformation mechanisms and yield properties of hydrated bone extracellular matrix

Bone features a hierarchical architecture combining antagonistic properties like toughness and strength. In order to better understand the mechanisms leading to this advantageous combination of properties, its postyield and failure behavior was analyzed on the length scale of a single lamella. Micropillars were compressed to large strains in saline solution to measure their anisotropic yield and post-yield behavior under near-physiological conditions. An increase in strength compared to the macroscale by a factor of 1.55 was observed in line with theory for size effects in quasi-brittle materials. Furthermore, a clear influence of hydration with a reduction by 60% compared to in vacuo results was found. This is well in line with literature nanoindentation data and the known change in properties of the organic phase upon hydration. Please click Additional Files below to see the full abstract

Development of a personalized fall rate prediction model in community-dwelling older adults: a negative binomial regression modelling approach.

BACKGROUND Around a third of adults aged 65 and older fall every year, resulting in unintentional injuries in 30% of the cases. Fractures are a frequent consequence of falls, primarily caused in individuals with decreased bone strength who are unable to cushion their falls. Accordingly, an individual's number of experienced falls has a direct influence on fracture risk. The aim of this study was the development of a statistical model to predict future fall rates using personalized risk predictors. METHODS In the prospective cohort GERICO, several fall risk factor variables were collected in community-dwelling older adults at two time-points four years apart (T1 and T2). Participants were asked how many falls they experienced during 12 months prior to the examinations. Rate ratios for the number of reported falls at T2 were computed for age, sex, reported fall number at T1, physical performance tests, physical activity level, comorbidity and medication number with negative binomial regression models. RESULTS The analysis included 604 participants (male: 122, female: 482) with a median age of 67.90 years at T1. The mean number of falls per person was 1.04 and 0.70 at T1 and T2. The number of reported falls at T1 as a factor variable was the strongest risk factor with an unadjusted rate ratio [RR] of 2.60 for 3 falls (95% confidence interval [CI] 1.54 to 4.37), RR of 2.63 (95% CI 1.06 to 6.54) for 4 falls, and RR of 10.19 (95% CI 6.25 to 16.60) for 5 and more falls, when compared to 0 falls. The cross-validated prediction error was comparable for the global model including all candidate variables and the univariable model including prior fall numbers at T1 as the only predictor. CONCLUSION In the GERICO cohort, the prior fall number as single predictor information for a personalized fall rate is as good as when including further available fall risk factors. Specifically, individuals who have experienced three and more falls are expected to fall multiple times again. TRIAL REGISTRATION ISRCTN11865958, 13/07/2016, retrospectively registered

2D-3D reconstruction of the proximal femur from DXA scans: Evaluation of the 3D-Shaper software.

Introduction: Osteoporosis is currently diagnosed based on areal bone mineral density (aBMD) computed from 2D DXA scans. However, aBMD is a limited surrogate for femoral strength since it does not account for 3D bone geometry and density distribution. QCT scans combined with finite element (FE) analysis can deliver improved femoral strength predictions. However, non-negligible radiation dose and high costs prevent a systematic usage of this technique for screening purposes. As an alternative, the 3D-Shaper software (3D-Shaper Medical, Spain) reconstructs the 3D shape and density distribution of the femur from 2D DXA scans. This approach could deliver a more accurate estimation of femoral strength than aBMD by using FE analysis on the reconstructed 3D DXA. Methods: Here we present the first independent evaluation of the software, using a dataset of 77 ex vivo femora. We extend a prior evaluation by including the density distribution differences, the spatial correlation of density values and an FE analysis. Yet, cortical thickness is left out of this evaluation, since the cortex is not resolved in our FE models. Results: We found an average surface distance of 1.16 mm between 3D DXA and QCT images, which shows a good reconstruction of the bone geometry. Although BMD values obtained from 3D DXA and QCT correlated well (r 2 = 0.92), the 3D DXA BMD were systematically lower. The average BMD difference amounted to 64 mg/cm3, more than one-third of the 3D DXA BMD. Furthermore, the low correlation (r 2 = 0.48) between density values of both images indicates a limited reconstruction of the 3D density distribution. FE results were in good agreement between QCT and 3D DXA images, with a high coefficient of determination (r 2 = 0.88). However, this correlation was not statistically different from a direct prediction by aBMD. Moreover, we found differences in the fracture patterns between the two image types. QCT-based FE analysis resulted mostly in femoral neck fractures and 3D DXA-based FE in subcapital or pertrochanteric fractures. Discussion: In conclusion, 3D-Shaper generates an altered BMD distribution compared to QCT but, after careful density calibration, shows an interesting potential for deriving a standardized femoral strength from a DXA scan