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Experimental determination of Young modulus and Poisson ratio in cortical bone tissue using high resolution scanning acoustic microscopy and nanoindentation

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Abstract

Nanoindentation allows measurements of local mechanical properties of bone tissue. Scanning acoustic microscopy (SAM) provides images related to bone density and elasticity. In both techniques, the estimation of Young modulus (E) relies on the accuracy of Poisson's ratio value (sigma). In cortical bone, sigma varies between 0.15 and 0.45 but, is classically set to 0.3, resulting in an approximate value of E. This study describes a new method combining SAM and nanoindentation techniques to locally evaluate sigma in human femoral cortex. A 200 MHz SAM-based acoustic impedance (8 mum lateral resolution) was combined with synchrotron microtomography (to provide local bone mineral density) to map the distribution of near surface elastic modulus. Whereas, nanoindentation modulus was calculated on several osseous regions. Assuming the equalization rule, the intersection of both modulus curves versus sigma permits to accurately derive sigma. The method was tested on aluminium, PMMA and polycarbonate samples of known sigma and provided experimental sigma values with a precision better than 3%. In bone, sigma was 0.42+/-0.01 corresponding to E=20+/-1 GPa. Our preliminary results indicate that combination of high-resolution SAM and nanoindentation may be relevant to accurately determine both Poisson ratio and Young modulus of bone tissue.
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... In FE models, the one-level model is formed by a 3 Â 3 Â 3 onelevel unit cell, while the two-level model is a two-level unit cell composed of 3 Â 3 Â 3 one-level cells. The basic mechanical constants of the bovine cortical bone are E s ¼15 GPa [13], s s ¼225 MPa [14] and υ s ¼0.3, which is an intermediate value according to Rupin et al. [15]. ...
... (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) structure′s stiffness and strength on porosity (corresponding correlation coefficient are 0.994 and 0.988) for the one-level scaffold, which is confirmed by the micromechanical approaches6789101112131415161718, while this relationship is not so clear for the two-level scaffold. In particular, for the one-level structure, its normalized Young′s modulus ratio E ð1Þ =E s varies from 0.283 to 0.001 (the red solid line inFig. ...
... Therefore, the densities of the skin, subcutaneous tissue and phalanx in each finger region model were defined as 1090, 919.6, 1920 kg/m 3 , respectively; the average elastic moduli of the finger phalanx and skin were defined as 14.7 GPa and 18.0 MPa, respectively. Taking into consideration that the finger skin might have the maximum Poisson's ratio, followed by the subcutaneous tissue and the phalanx, and with the Poisson's ratios of some tissues obtained by Ren et al. (2018), Thieulin et al. (2020) and Rupin et al. (2008), the Poisson's ratios of the skin, subcutaneous tissue and phalanx in each finger region model were assumed in the ranges of 0.45-0.49, 0.30-0.45 ...
... Hence, only the thickness of the subcutaneous tissue was set as a possible independent variable to reflect the geometric size of different tissues in a finger region model. According to the studies by Oehman et al. (2011), Ren et al. (2018), Thieulin et al. (2020), Rupin et al. (2008) and Yeh et al. (2014), each potential independent variable was set as three levels. To reduce the number of simulations, a L 18 (3 7 ) orthogonal design was used. ...
Article
Soft grasping is a great challenge for picking robots and its bionic inspiration originates from human fingers. In this study, the hand was scanned to obtain the internal structure of fingers by a computerized tomography (CT) scanner, and the soft contact mechanical index a was defined for characterizing the degree of softness of a finger region during gentle grasping. The effects of mechanics and structure of finger tissues on the soft contact mechanical index were investigated by finite element analysis and multiple linear regression. The finite element models of the 14 finger regions were split into 6 different groups by a hierarchical cluster analysis. In each group, a mathematical model was established to link the soft contact mechanical index with the mechanics as well as the structure of finger tissues. In most finger regions, their soft contact mechanical index significantly depended on the elastic moduli of the skin and subcutaneous tissue (Eskin, Etissue), the Poisson’s ratio ʋtissue and the thickness Ttissue of the subcutaneous tissue (p < 0.05). The Etissue showed the most contribution on the soft contact mechanical index of a finger region, followed by ʋtissue, Ttissue, and Eskin. This study demonstrated how the mechanics and structure of the human finger quantitatively affect its soft contact mechanical behavior during gentle grasping and further provided a bionic basis for developing robotic fingers with varying degrees of softness, particularly for fruit picking.
... However, these data measured in the femur of a 72-year old female human donor may not be representative for other bone tissues, but rather demonstrates that Eq. 16.35 should be used with caution. Experimentally, the impact of the Poisson ratio has been investigated by sitematched analyses of the acoustic impedance and Young's modulus in human femoral cortical bones [44, 45]. Young's modulus E IT is usually derived from nanoindentation measurements. ...
... caused by surface roughness, viscous and contact effects, not perfectly matched interaction volumes, a considerable amount has been suggested to be caused by variations of the Poisson ratios. Indeed, Rupin et al. [45] have recently suggested that a site-matched analysis of Z and E IT may be used to assess the Poisson ratios experimentally. They reported isotropic Poisson ratios in the range between 0.18 and 0.46 (mean and standard deviation: 0.41 ± 0.04). ...
Chapter
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Several high frequency ultrasound techniques have been developed during the last decade with the intention to assess elastic properties of bone at the tissue level. In this chapter three major principles are described with exemplary measurements in the frequency range from 50MHz to 1.2GHz. The methods are compared and their application potentials and limitations are discussed with respect to the hierarchical structure of cortical bone. While highly focused transducers with frequencies between 50 and 200MHz are suitable for the assessment of microscale elastic properties, frequencies in the gigahertz range are dedicated to the investigation of the anisotropic lamellar bone structure. The relations between tissue mineralization, acoustic properties and anisotropic elastic coefficients at the micro- and nanoscales will be summarized. KeywordsAcoustic impedance-Acoustic lens-Analogue-to-digital (A/D)-Anisotropy-Aperture-Numerical aperture-N.A.-Attenuation-Bandpass-Beam axis-Beam width-Bone matrix-Collagen-Compressional wave-Confocal-Cortical bone-C-scan-Degree of mineralization of bone-Depth of focus-Elastic coefficient-Elasticity-Embedding-Fast Fourier transformation (FFT)-Femur-Fibril-Filter-Focal plane-Focal point-Group velocity-Haversian canal
... The material parameters, namely the Young's modulus is set to 20 GPa and the Poisson's ratio to 0.42 (see e.g. [228]). In addition, the load is given by f = (0, 0, 0), the Dirichlet data by u D = 0 on Γ D Γ represented as the left dark gray region of the boundary in fig. ...
Thesis
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This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refinement methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of– the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations.
... The remaining connective ligaments were also modeled using this average stress strain data with the same mechanical property parameters, sans damage. To model the bones, a Young's modulus of E = 20 GPa with an assumed nearly incompressible Poisson's ratio of v = 0.42 from Rupin et al., was used [27]. ...
Article
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The anterior cruciate ligament (ACL) plays a pivotal role in support of the knee under loading. When damaged, it is known that substantial changes in the mechanics of the neighboring ligaments can be observed. However, a localized damage approach to investigating how ACL deficiency influences the neighboring ligaments has not been carried out. To do this, a finite element model, incorporating a continuum damage material model of the ACL, was implemented. Localized ACL damage was induced using high quadriceps force loading. Once damaged, anterior shear forces or tibial torque loadings were applied to the knee joint. The relative changes in stress contour and average mid-substance stress were examined for each of the neighboring ligaments following localized ACL damage. It was observed that localized ACL damage could produce notable changes in the mechanics of the neighboring knee ligaments, with non-homogenous stress contour shape changes and average stress magnitude being observed to increase in most cases, with a notable exception occurring in the MCL for both loading modes. In addition, the ligament bearing the most loading also changed with ACL deficiency. These changes carry implications as to morphological effects that may be induced following localized ACL damage, indicating that early diagnosis of ACL injury may be helpful in mitigating other complications post injury.
... Elastic modulus of cortical bone has been reported roughly in the range 10 GPa to 32 GPa, when measured with different test methods such as destructive, ultrasound and non-indentation method (Choi et al., 1990;Hunt et al., 1998). The Poisson's ratio varies in the range 0.15 to 0.45, however it is classically set to 0.3 (Rupin et al., 2008). Since the elastic modulus of bone is considerably higher than the surrounding soft tissues, the bones are considered rigid more often (Beidokhti et al., 2017;Halloran, Petrella, and Rullkoetter, 2005;Pena et al., 2005). ...
Thesis
Musculoskeletal disorder of the lower limb is one of the most common health burdens that may lead to functional impairment in an individual. Although various operative management options are available, there seems no unanimity on a particular procedure that serves the best. To objectively assess disorders and effectively plan surgeries, it is essential to understand lower limb biomechanics under physiological loading conditions. With that motivation, this PhD aims to develop a comprehensive finite element based musculoskeletal modeling framework of the lower limb. The first phase of the PhD focuses on the development and evaluation of subject-specific finite element models under passive flexion. Novel approaches are proposed and evaluated for fast model development focusing on geometry and ligament properties. In the second phase, a novel finite element based approach for soft tissue artifact compensation is proposed and evaluated. This contribution allowed to effectively compensate for soft tissue artifact in motion analysis by taking subject specificity into account. The third phase of the PhD is dedicated to clinical application, where the utility of the biplanar X-ray system in evaluating Total Knee Arthroplasty implant alignment is briefly explored. Overall, this PhD may help to accurately estimate and understand lower limb biomechanics under clinically relevant loading conditions, and bring the model a step closer to clinical routine.
... The parameters of the linear elasticity problem for this test cases are the following: the Young's modulus is set to 20 GPa and the Poisson's ratio to 0.42 (see e.g. [64]). In addition, the load is given by f = (0, 0, 0), the Dirichlet data by u D = 0 on Γ D Γ represented as the left dark gray region of the boundary in fig. ...
Preprint
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In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material.
... Including this deformability improves the model because strong deformations of the softer layers would violate a general elastic model. Here, the shear modulus of the PC is taken as that of a slack muscle and its bulk modulus (K) is calculated using the shear modulus (G) and poisson ratio (v) according to the following equation: [35] 520.4 0.49 Bone [36] 20,000 0.42 Base (steel) [37] 210,000 0.3 Urethane [33] 0.025 0.0 After running simulations with a steel base to simulate a sturdy support, we also ran simulations using the properties of urethane foam, a material commonly used in seating and mattresses. When using Neo-Hookean hyperelastic properties, the simulations took 14e24 h to complete on a PC with Intel I7 4 GHz processor and 16 Gb of RAM. ...
... An experimental approach is to assess the Poisson ratio at this length scale is a site-matched analysis of the acoustic impedance and the indentation modulus E IT by SAM and nanoindentation, respectively [33,37,38]. Interestingly, in all studies consistent, rather moderate correlations between Z and the indentation modulus E IT (0.61 ≤ R 2 ≤ 0.67) have been observed. ...
Chapter
Scanning acoustic microscopy (SAM) with frequencies between 50 MHz and 2 GHz is adapted to the investigation of structure and elastic properties of the bone tissue matrix, that is, the linear elastic interaction of acoustic waves with the material under investigation can be used to visualize the microarchitecture, and also to measure sound velocities and acoustic impedances at various length scales. When combined with local density estimates (e.g., derived from quantitative X-ray micro-computed tomography data), acoustic impedance estimates can be used to derive tissue stiffness. By combining multiscale experimental data with numerical and continuum mechanical homogenization approaches, structural properties can be decoupled from material properties and their respective impacts on the elastic functional behavior of the compound can be studied. The underlying principles and application are described in this chapter.
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