Towards patient-specific material modeling of trabecular bone post-yield behavior

Semi-automatic algorithm to build finite element numerical models of the human hearing system from Micro-CT data

Finite Element modeling has been an extended methodology to build numerical model to simulate the behavior of the hearing system. Due to the complexity of the system and the difficulties to reduce the uncertainties of the geometric data, they result in computationally expensive models, sometimes generic, representative of average geometries. It makes it difficult to validate the model with direct experimental data from the same specimen or to establish a patient-oriented modeling strategy. In the present paper, a first attempt to automatize the process of model building is made. The source information is geometrical information obtained from CT of the different elements that compose the system. Importing that data, we have designed the complete procedure to build a model including tympanic membrane, ossicular chain and cavities. The methodology includes the proper coupling of all the elements and the generation of the corresponding finite element model. The whole automatic procedure is not complete, as we need to make some human-assisted decisions; however, the model development time is reduced from 4 weeks to approximately 3 days. The goal of the modeling algorithm is to build a Finite Element Model with a limited computational cost. Several tasks as contour identification or model decimation are designed and integrated in order to follow a semi-automated process that allows generating a patient-oriented model.

Mechanical loading of ex vivo bovine trabecular bone in 3D printed bioreactor chambers

Previous ex vivo bone culture methods have successfully implemented polycarbonate (PC) bioreactors to investigate bone adaptation to mechanical load; however, they are difficult to fabricate and have been limited to a 5 mm maximum specimen height. The objective of this study was to validate a custom-made 3D printed MED610TM bioreactor system that addresses the limitations of the PC bioreactor and assess its efficacy in ex vivo bone culture. Twenty-three viable trabecular bone cores (10 mm height by 10 mm diameter) from an 18-month-old bovine sternum were cultured in MED610TM bioreactors with culture medium at 37 °C and 5% CO2 for 21-days. Bone cores were ranked based on their day 0 apparent elastic modulus (Eapp) and evenly separated into a "Load" group (n = 12) and a control group (n = 11). The Load group was loaded five times per week with a sinusoidal strain waveform between -1000 and -5000 με for 120 cycles at 2 Hz. Eapp was assessed on day 0, 8, and 21 using quasi-static tests with a -4000 με applied strain. Over 21-days, the Eapp of Load group samples tended to increase by more than double the control group (53.4% versus 20.9%) and no visual culture contamination was observed. This study demonstrated that bone organ culture in 3D printed MED610TM bioreactors replicated Eapp trends found in previous studies with PC bioreactors. However, further studies are warranted with a larger sample size to increase statistical power and histology to assess cell viability and bone mineral apposition rate.

Efficient materially nonlinear [Formula: see text]FE solver for simulations of trabecular bone failure

An efficient solver for large-scale linear \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \hbox {FE}$$\end{document}μFE simulations was extended for nonlinear material behavior. The material model included damage-based tissue degradation and fracture. The new framework was applied to 20 trabecular biopsies with a mesh resolution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${36}\,{{\upmu }\hbox {m}}$$\end{document}36μm. Suitable material parameters were identified based on two biopsies by comparison with axial tension and compression experiments. The good parallel performance and low memory footprint of the solver were preserved. Excellent correlation of the maximum apparent stress was found between simulations and experiments (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2 > 0.97$$\end{document}R2>0.97). The development of local damage regions was observable due to the nonlinear nature of the simulations. A novel elasticity limit was proposed based on the local damage information. The elasticity limit was found to be lower than the 0.2% yield point. Systematic differences in the yield behavior of biopsies under apparent compression and tension loading were observed. This indicates that damage distributions could lead to more insight into the failure mechanisms of trabecular bone.

Young's modulus of trabecular bone at the tissue level: A review

The tissue-level Young's modulus of trabecular bone is important for detailed mechanical analysis of bone and bone-implant mechanical interactions. However, the heterogeneity and small size of the trabecular struts complicate an accurate determination. Methods such as micro-mechanical testing of single trabeculae, ultrasonic testing, and nanoindentation have been used to estimate the trabecular Young's modulus. This review summarizes and classifies the trabecular Young's moduli reported in the literature. Information on species, anatomic site, and test condition of the samples has also been gathered. Advantages and disadvantages of the different methods together with recent developments are discussed, followed by some suggestions for potential improvement for future work. In summary, this review provides a thorough introduction to the approaches used for determining trabecular Young's modulus, highlights important considerations when applying these methods and summarizes the reported Young's modulus for follow-up studies on trabecular properties.

STATEMENT OF SIGNIFICANCE

The spongy trabecular bone provides mechanical support while maintaining a low weight. A correct measure of its mechanical properties at the tissue level, i.e. at a single-trabecula level, is crucial for analysis of interactions between bone and implants, necessary for understanding e.g. bone healing mechanisms. In this study, we comprehensively summarize the Young's moduli of trabecular bone estimated by currently available methods, and report their dependency on different factors. The critical review of different methods with recent updates is intended to inspire improvements in estimating trabecular Young's modulus. It is strongly suggested to report detailed information on the tested bone to enable statistical analysis in the future.

Phase-field boundary conditions for the voxel finite cell method: Surface-free stress analysis of CT-based bone structures

The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field-based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

Calculation of cancellous bone elastic properties with the polarization-based FFT iterative scheme

The Fast Fourier Transform-based method, originally introduced by Moulinec and Suquet in 1994 has gained popularity for computing homogenized properties of composites. In this work, the method is used for the computational homogenization of the elastic properties of cancellous bone. To the authors' knowledge, this is the first study where the Fast Fourier Transform scheme is applied to bone mechanics. The performance of the method is analyzed for artificial and natural bone samples of 2 species: bovine femoral heads and implanted femurs of Hokkaido rats. Model geometries are constructed using data from X-ray tomographies, and the bone tissue elastic properties are measured using microindentation and nanoindentation tests. Computed results are in excellent agreement with those available in the literature. The study shows the suitability of the method to accurately estimate the fully anisotropic elastic response of cancellous bone. Guidelines are provided for the construction of the models and the setting of the algorithm.

Mimetization of the elastic properties of cancellous bone via a parameterized cellular material

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (Comput Methods Biomech Biomed Eng 9:135-147, 2006. doi: 10.1080/10255840600751473 ) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.

Contribution of fluid in bone extravascular matrix to strain-rate dependent stiffening of bone tissue - A poroelastic study

Osteoporotic fractures represent an increasing cost to society, and its diagnosis methods based on bone density still lack accuracy in identifying risk of fracture. This is why a better understanding of mechanical behavior of bone tissue is of importance, especially when it comes to relating experimental observations to realistic physiological fall loading conditions. This study aims at exploring the stiffening effect of pore fluid in bone extravascular matrix subject to high strain rate loading that is more realistic to simulate a physiological fall. A computational approach is used, where bone tissue microstructure extracted from micro-CT images is modeled using finite elements. The solid phase of bone tissue is modeled as a poroelastic material, a porous matrix filled with fluid. When the extravascular matrix experiences certain volumetric deformation, the fluid in pores presents load carrying capacity, which consequently varies the apparent stiffness of bone tissue. It is shown that effects of fluid stiffening in bone can be significant, depending on the chosen material properties, the amount of volumetric strain in tissue and the loading rate with respect to hydraulic conductivity and drainage conditions. It is also shown that such stiffening effect is influenced by bone microstructure, and is more significant in cortical bone than in trabecular bone.

Prediction of local ultimate strain and toughness of trabecular bone tissue by Raman material composition analysis

Clinical studies indicate that bone mineral density correlates with fracture risk at the population level but does not correlate with individual fracture risk well. Current research aims to better understand the failure mechanism of bone and to identify key determinants of bone quality, thus improving fracture risk prediction. To get a better understanding of bone strength, it is important to analyze tissue-level properties not influenced by macro- or microarchitectural factors. The aim of this pilot study was to identify whether and to what extent material properties are correlated with mechanical properties at the tissue level. The influence of macro- or microarchitectural factors was excluded by testing individual trabeculae. Previously reported data of mechanical parameters measured in single trabeculae under tension and bending and its compositional properties measured by Raman spectroscopy was evaluated. Linear and multivariate regressions show that bone matrix quality but not quantity was significantly and independently correlated with the tissue-level ultimate strain and postyield work (r = 0.65–0.94). Principal component analysis extracted three independent components explaining 86% of the total variance, representing elastic, yield, and ultimate components according to the included mechanical parameters. Some matrix parameters were both included in the ultimate component, indicating that the variation in ultimate strain and postyield work could be largely explained by Raman-derived compositional parameters.

Nanostructure and elastic modulus of single trabecula in bovine cancellous bone

We aimed to investigate the elastic modulus of trabeculae using tensile tests and assess the effects of nanostructure at the hydroxyapatite (HAp) crystal scale on the elastic modulus. In the experiments, 18 trabeculae that were at least 3mm in length in the proximal epiphysis of three adult bovine femurs were used. Tensile tests were conducted using a small tensile testing device coupled with microscopy under air-dried condition. The c-axis orientation of HAp crystals and the degree of orientation were measured by X-ray diffraction. To observe the deformation behavior of HAp crystals under tensile loading, the same tensile tests were conducted in X-ray diffraction measurements. The mineral content of specimens was evaluated using energy dispersive X-ray spectrometry. The elastic modulus of a single trabecula varied from 4.5 to 23.6 GPa, and the average was 11.5 ± 5.0 GPa. The c-axis of HAp crystals was aligned with the trabecular axis and the crystals were lineally deformed under tensile loading. The ratio of the HAp crystal strain to the tissue strain (strain ratio) had a significant correlation with the elastic modulus (r=0.79; P<0.001). However, the mineral content and the degree of orientation did not vary widely and did not correlate with the elastic modulus in this study. It suggests that the strain ratio may represent the nanostructure of a single trabecula and would determine the elastic modulus as well as mineral content and orientation.