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Bernard S, Cai X, Grimal Q. Measurement of Cortical Bone Elasticity Tensor with Resonant Ultrasound Spectroscopy. ADVANCES IN EXPERIMENTAL MEDICINE AND BIOLOGY 2022; 1364:253-277. [DOI: 10.1007/978-3-030-91979-5_12] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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2
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Cai X, Bernard S, Grimal Q. Documenting the Anisotropic Stiffness of Hard Tissues with Resonant Ultrasound Spectroscopy. ADVANCES IN EXPERIMENTAL MEDICINE AND BIOLOGY 2022; 1364:279-295. [DOI: 10.1007/978-3-030-91979-5_13] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Wang R, Fan F, Shen F, Wang Y, Laugier P, Niu H. Application of differential evolution on elasticity measurement of low quality factor materials using FEM-based resonant ultrasound spectroscopy. J Mech Behav Biomed Mater 2021; 124:104848. [PMID: 34600428 DOI: 10.1016/j.jmbbm.2021.104848] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2021] [Revised: 08/31/2021] [Accepted: 09/17/2021] [Indexed: 11/25/2022]
Abstract
Finite element method based resonant ultrasound spectroscopy (FEM-based RUS) allows elasticity measurement for a material with high quality factor (Q) and arbitrary geometry by minimizing the differences between its theoretically calculated resonant frequencies and the corresponding experimentally measured ones. As Q decreases, some experimental frequencies remain undetermined, which makes it difficult to pair the calculated and experimental frequencies and to correctly identify the elastic constants. Additional difficulty need be tackled for irregularly-shaped low-Q materials due to the adoption of time-consuming FEM, thus efficiency of the identification method needs to be focused on. To apply FEM-based RUS to low-Q materials, a new elastic constant identification method is proposed based on a differential evolution algorithm in this paper. This method can perform a global search combining with local optimizations in the elastic constant space, and improve the overall efficiency by limiting the number of the frequency calculations. By using numerical experiments, the effectiveness of the proposed method under different frequency missing situations was verified and its efficiency was measured from the required frequency calculation numbers, showing an approximate two third reduction compared with an existing method. Finally, the elastic constants of an actual irregular cortical bone-mimicking material (Q ≈ 25) were measured using the two methods, yielding consistent Young's moduli (calculated from the identified constants) with the data provided by the manufacturer and a similar improvement in computational efficiency of the proposed method.
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Affiliation(s)
- Rui Wang
- School of Biological Science and Medical Engineering, Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing, 100083, China
| | - Fan Fan
- School of Biological Science and Medical Engineering, Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing, 100083, China
| | - Fei Shen
- School of Biological Science and Medical Engineering, Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing, 100083, China
| | - Yue Wang
- School of Biological Science and Medical Engineering, Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing, 100083, China
| | - Pascal Laugier
- Laboratoire d'Imagerie Biomédicale (LIB), Centre National de la Recherche Scientifique (CNRS), Institut National de la Santé et de la Recherche Médicale (INSERM), Sorbonne Université, 75006, Paris, France
| | - Haijun Niu
- School of Biological Science and Medical Engineering, Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing, 100083, China.
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Brémaud L, Cai X, Brenner R, Grimal Q. Maximum effect of the heterogeneity of tissue mineralization on the effective cortical bone elastic properties. Biomech Model Mechanobiol 2021; 20:1509-1518. [PMID: 33884512 DOI: 10.1007/s10237-021-01459-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 04/07/2021] [Indexed: 11/29/2022]
Abstract
The mineralization level is heterogeneous in cortical bone extracellular matrix as a consequence of remodeling. Models of the effective elastic properties at the millimeter scale have been developed based on idealizations of the vascular pore network and matrix properties. Some popular models do not take into account the heterogeneity of the matrix. However, the errors on the predicted elasticity when the difference in elastic properties between osteonal and interstitial tissues is not modeled have not been quantified. This work provides an estimation of the maximum error. We compare the effective elasticity of a representative volume element (RVE) assuming (1) different elastic properties in osteonal and interstitial tissues vs. (2) average matrix properties. In order to account for the variability of bone microstructure, we use a collection of high resolution images of the pore network to build RVEs. In each RVE we assumed a constant osteonal wall thickness and we artificially varied this thickness between 35 and 140 [Formula: see text]m to create RVEs with different amounts of osteonal tissue. The homogenization problem was solved with a fast Fourier transform (FFT)-based numerical scheme. We found that the error depends on pore volume fraction and varies on average from 1 to [Formula: see text] depending on the assumed diameter of the osteons. The results suggest that matrix heterogeneity may be disregarded in cortical bone models in most practical cases.
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Affiliation(s)
- Luc Brémaud
- Sorbonne Université, INSERM, CNRS, Laboratoire d'Imagerie Biomédicale, 75006, Paris, France.,Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, 75005, Paris, France
| | - Xiran Cai
- Sorbonne Université, INSERM, CNRS, Laboratoire d'Imagerie Biomédicale, 75006, Paris, France.,School of Information Science and Technology, ShanghaiTech University, Pudong District, 201210, Shanghai, China
| | - Renald Brenner
- Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, 75005, Paris, France
| | - Quentin Grimal
- Sorbonne Université, INSERM, CNRS, Laboratoire d'Imagerie Biomédicale, 75006, Paris, France.
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Peralta L, Maeztu Redin JD, Fan F, Cai X, Laugier P, Schneider J, Raum K, Grimal Q. Bulk Wave Velocities in Cortical Bone Reflect Porosity and Compression Strength. ULTRASOUND IN MEDICINE & BIOLOGY 2021; 47:799-808. [PMID: 33341302 DOI: 10.1016/j.ultrasmedbio.2020.11.012] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2020] [Revised: 11/16/2020] [Accepted: 11/16/2020] [Indexed: 06/12/2023]
Abstract
The goal of this study was to evaluate whether ultrasonic velocities in cortical bone can be considered as a proxy for mechanical quality of cortical bone tissue reflected by porosity and compression strength. Micro-computed tomography, compression mechanical testing and resonant ultrasound spectroscopy were used to assess, respectively, porosity, strength and velocity of bulk waves of both shear and longitudinal polarisations propagating along and perpendicular to osteons, in 92 cortical bone specimens from tibia and femur of elderly human donors. All velocities were significantly associated with strength (r = 0.65-0.83) and porosity (r = -0.64 to -0.77). Roughly, according to linear regression models, a decrease in velocity of 100 m/s corresponded to a loss of 20 MPa in strength (which is approximately 10% of the largest strength value) and to an increase in porosity of 5%. These results provide a rationale for the in vivo measurement of one or several velocities for the diagnosis of bone fragility.
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Affiliation(s)
- Laura Peralta
- Sorbonne Universite, INSERM, CNRS, Laboratoire d'lmagerie Biomedicale, LIB, F-75006 Paris, France; Department of Biomedical Engineering, School of Biomedical Engineering & Imaging Sciences, Kings College London, London, United Kingdom.
| | - Juan Deyo Maeztu Redin
- Sorbonne Universite, INSERM, CNRS, Laboratoire d'lmagerie Biomedicale, LIB, F-75006 Paris, France
| | - Fan Fan
- Sorbonne Universite, INSERM, CNRS, Laboratoire d'lmagerie Biomedicale, LIB, F-75006 Paris, France; Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing, China
| | - Xiran Cai
- Sorbonne Universite, INSERM, CNRS, Laboratoire d'lmagerie Biomedicale, LIB, F-75006 Paris, France
| | - Pascal Laugier
- Sorbonne Universite, INSERM, CNRS, Laboratoire d'lmagerie Biomedicale, LIB, F-75006 Paris, France
| | - Johannes Schneider
- Berlin-Brandenburg School for Regenerative Therapies, Charit-Universittsmedizin Berlin, Berlin, Germany
| | - Kay Raum
- Berlin-Brandenburg School for Regenerative Therapies, Charit-Universittsmedizin Berlin, Berlin, Germany
| | - Quentin Grimal
- Sorbonne Universite, INSERM, CNRS, Laboratoire d'lmagerie Biomedicale, LIB, F-75006 Paris, France
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Yousefian O, Karbalaeisadegh Y, Muller M. Frequency-dependent analysis of ultrasound apparent absorption coefficient in multiple scattering porous media: application to cortical bone. Phys Med Biol 2021; 66:035026. [PMID: 32937603 PMCID: PMC10851310 DOI: 10.1088/1361-6560/abb934] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
The effect of matrix viscoelastic absorption on frequency-dependent attenuation in porous structures mimicking simplified cortical bone is addressed in this numerical study. An apparent absorption is defined to quantify the difference between total attenuation (resulting from both absorption and scattering) and attenuation exclusively due to scattering. A power-law model is then used to describe the frequency-dependent apparent absorption as a function of pore diameter and density. The frequency response of the porous structures to a Gaussian pulse is studied to determine the frequency range over which the system can be considered linear. The results show that for low scattering regimes (normalized frequency [Formula: see text]0.80), the system and its apparent absorption can be considered linear. Hence, the total attenuation coefficient results from the summation of scattering and absorption coefficients. However, for highly scattering regimes, the system can no longer be considered linear, as the apparent absorption vs. frequency deviates from a linear trend. As the pore density increases, the apparent absorption coefficient increases as well.
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Affiliation(s)
- Omid Yousefian
- Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27606, United States of America
| | - Yasamin Karbalaeisadegh
- Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27606, United States of America
| | - Marie Muller
- Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27606, United States of America
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Cai X, Brenner R, Peralta L, Olivier C, Gouttenoire PJ, Chappard C, Peyrin F, Cassereau D, Laugier P, Grimal Q. Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity. J R Soc Interface 2020; 16:20180911. [PMID: 30958180 DOI: 10.1098/rsif.2018.0911] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
With ageing and various diseases, the vascular pore volume fraction (porosity) in cortical bone increases, and the morphology of the pore network is altered. Cortical bone elasticity is known to decrease with increasing porosity, but the effect of the microstructure is largely unknown, while it has been thoroughly studied for trabecular bone. Also, popular micromechanical models have disregarded several micro-architectural features, idealizing pores as cylinders aligned with the axis of the diaphysis. The aim of this paper is to quantify the relative effects on cortical bone anisotropic elasticity of porosity and other descriptors of the pore network micro-architecture associated with pore number, size and shape. The five stiffness constants of bone assumed to be a transversely isotropic material were measured with resonant ultrasound spectroscopy in 55 specimens from the femoral diaphysis of 29 donors. The pore network, imaged with synchrotron radiation X-ray micro-computed tomography, was used to derive the pore descriptors and to build a homogenization model using the fast Fourier transform (FFT) method. The model was calibrated using experimental elasticity. A detailed analysis of the computed effective elasticity revealed in particular that porosity explains most of the variations of the five stiffness constants and that the effects of other micro-architectural features are small compared to usual experimental errors. We also have evidence that modelling the pore network as an ensemble of cylinders yields biased elasticity values compared to predictions based on the real micro-architecture. The FFT homogenization method is shown to be particularly efficient to model cortical bone.
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Affiliation(s)
- Xiran Cai
- 1 Laboratoire d'Imagerie Biomédicale, Sorbonne Université , INSERM UMR S 1146, CNRS UMR 7371, 75006 Paris , France
| | - Renald Brenner
- 2 Institut Jean le Rond ∂'Alembert, Sorbonne Université , CNRS UMR 7190, 75005 Paris , France
| | - Laura Peralta
- 1 Laboratoire d'Imagerie Biomédicale, Sorbonne Université , INSERM UMR S 1146, CNRS UMR 7371, 75006 Paris , France
| | - Cécile Olivier
- 3 CREATIS, Université de Lyon , INSERM U1206, CNRS UMR 5220 , INSA-Lyon, UCBL, 69621 Villeurbanne , France.,4 ESRF , 38043 Grenoble , France
| | | | | | - Françoise Peyrin
- 3 CREATIS, Université de Lyon , INSERM U1206, CNRS UMR 5220 , INSA-Lyon, UCBL, 69621 Villeurbanne , France.,4 ESRF , 38043 Grenoble , France
| | - Didier Cassereau
- 1 Laboratoire d'Imagerie Biomédicale, Sorbonne Université , INSERM UMR S 1146, CNRS UMR 7371, 75006 Paris , France
| | - Pascal Laugier
- 1 Laboratoire d'Imagerie Biomédicale, Sorbonne Université , INSERM UMR S 1146, CNRS UMR 7371, 75006 Paris , France
| | - Quentin Grimal
- 1 Laboratoire d'Imagerie Biomédicale, Sorbonne Université , INSERM UMR S 1146, CNRS UMR 7371, 75006 Paris , France
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Mohanty K, Yousefian O, Karbalaeisadegh Y, Ulrich M, Grimal Q, Muller M. Artificial neural network to estimate micro-architectural properties of cortical bone using ultrasonic attenuation: A 2-D numerical study. Comput Biol Med 2019; 114:103457. [PMID: 31600691 PMCID: PMC6817400 DOI: 10.1016/j.compbiomed.2019.103457] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2019] [Revised: 09/18/2019] [Accepted: 09/19/2019] [Indexed: 01/10/2023]
Abstract
The goal of this study is to estimate micro-architectural parameters of cortical porosity such as pore diameter (φ), pore density (ρ) and porosity (ν) of cortical bone from ultrasound frequency dependent attenuation using an artificial neural network (ANN). First, heterogeneous structures with controlled pore diameters and pore densities (mono-disperse) were generated, to mimic simplified structure of cortical bone. Then, more realistic structures were obtained from high resolution CT scans of human cortical bone. 2-D finite-difference time-domain simulations were conducted to calculate the frequency-dependent attenuation in the 1-8 MHz range. An ANN was then trained with the ultrasonic attenuation at different frequencies as the input feature vectors while the output was set as the micro-architectural parameters (pore diameter, pore density and porosity). The ANN is composed of three fully connected dense layers with 24, 12 and 6 neurons, connected to the output layer. The dataset was trained over 6000 epochs with a batch size of 16. The trained ANN exhibits the ability to predict the micro-architectural parameters with high accuracy and low losses. ANN approaches could potentially be used as a tool to help inform physics-based modelling of ultrasound propagation in complex media such as cortical bone. This will lead to the solution of inverse-problems to retrieve bone micro-architectural parameters from ultrasound measurements for the non-invasive diagnosis and monitoring osteoporosis.
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Affiliation(s)
- Kaustav Mohanty
- Department of Mechanical and Aerospace Engineering, NC State University, Raleigh, NC, 27695, USA.
| | - Omid Yousefian
- Department of Mechanical and Aerospace Engineering, NC State University, Raleigh, NC, 27695, USA.
| | - Yasamin Karbalaeisadegh
- Department of Mechanical and Aerospace Engineering, NC State University, Raleigh, NC, 27695, USA.
| | - Micah Ulrich
- Department of Mechanical and Aerospace Engineering, NC State University, Raleigh, NC, 27695, USA.
| | - Quentin Grimal
- Sorbonne Université, INSERM UMR S 1146, CNRS UMR 7371, Laboratoire d'Imagerie Biomédicale, 75006, Paris, France.
| | - Marie Muller
- Department of Mechanical and Aerospace Engineering, NC State University, Raleigh, NC, 27695, USA.
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Cai X, Follet H, Peralta L, Gardegaront M, Farlay D, Gauthier R, Yu B, Gineyts E, Olivier C, Langer M, Gourrier A, Mitton D, Peyrin F, Grimal Q, Laugier P. Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly. Acta Biomater 2019; 90:254-266. [PMID: 30922952 DOI: 10.1016/j.actbio.2019.03.043] [Citation(s) in RCA: 21] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2018] [Revised: 03/19/2019] [Accepted: 03/20/2019] [Indexed: 02/08/2023]
Abstract
The strong dependence between cortical bone elasticity at the millimetre-scale (mesoscale) and cortical porosity has been evidenced by previous studies. However, bone is an anisotropic composite material made by mineral, proteins and water assembled in a hierarchical structure. Whether the variations of structural and compositional properties of bone affect the different elastic coefficients at the mesoscale is not clear. Aiming to understand the relationships between bone elastic properties and compositions and microstructure, we applied state-of-the-art experimental modalities to assess these aspects of bone characteristics. All elastic coefficients (stiffness tensor of the transverse isotropic bone material), structure of the vascular pore network, collagen and mineral properties were measured in 52 specimens from the femoral diaphysis of 26 elderly donors. Statistical analyses and micromechanical modeling showed that vascular pore volume fraction and the degree of mineralization of bone are the most important determinants of cortical bone anisotropic mesoscopic elasticity. Though significant correlations were observed between collagen properties and elasticity, their effects in bone mesoscopic elasticity were minor in our data. This work also provides a unique set of data exhibiting a range of variations of compositional and microstructural cortical bone properties in the elderly and gives strong experimental evidence and basis for further development of biomechanical models for human cortical bone. STATEMENT OF SIGNIFICANCE: This study reports the relationships between microstructure, composition and the mesoscale anisotropic elastic properties of human femoral cortical bone in elderly. For the first time, we provide data covering the complete anisotropic elastic tensor, the microstructure of cortical vascular porosity, mineral and collagen characteristics obtained from the same or adjacent samples in each donor. The results revealed that cortical vascular porosity and degree of mineralization of bone are the most important determinants of bone anisotropic stiffness at the mesoscale. The presented data gives strong experimental evidence and basis for further development of biomechanical models for human cortical bone.
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Semaan M, Mora P, Bernard S, Launay F, Payan C, Lasaygues P, Pithioux M, Baron C. Assessment of elastic coefficients of child cortical bone using resonant ultrasound spectroscopy. J Mech Behav Biomed Mater 2018; 90:40-44. [PMID: 30343169 DOI: 10.1016/j.jmbbm.2018.09.044] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2018] [Revised: 08/30/2018] [Accepted: 09/26/2018] [Indexed: 10/28/2022]
Abstract
The assessment of the anisotropic elastic properties of non-pathological child cortical bone remains a challenge for the biomechanical engineering community and an important clinical issue. Resonant ultrasound spectroscopy (RUS) can be used to determine bone stiffness coefficients from the mechanical resonances of bone specimens. Here, a RUS protocol was used on 7 fibula specimens from children (mean age 14 ± 3 years) to estimate the whole elastic stiffness tensor of non-pathological child cortical bone considered as orthotropic. Despite a small number of sample, results are consistent with this hypothesis, even if a trend towards transverse isotropy is discussed. Indeed, the average values of the 9 independent stiffness coefficients obtained in this study for child bone are: C11 = 16.73 ± 0.19 GPa, C22 = 16.19 ± 0.12 GPa, C33 = 24.47 ± 0.30 GPa, C44 = 4.14 ± 0.08 GPa, C55 = 4.16 ± 0.07 GPa, C66 = 3.13 ± 0.05 GPa, C12 = 10.14 ± 0.20 GPa, C13 = 10.67 ± 0.27 GPa, C23 = 10.25 ± 0.14 GPa.
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Affiliation(s)
- Marie Semaan
- Aix-Marseille Univ, CNRS, ISM, Marseille, France; Aix-Marseille Univ, APHM, CNRS, ISM, Sainte-Marguerite Hospital, Institute for Locomotion, Department of Orthopaedics and Traumatology, Marseille, France
| | - Pierric Mora
- Aix-Marseille Univ., CNRS, IUSTI, Marseille, France
| | - Simon Bernard
- Aix-Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France
| | - Franck Launay
- Aix-Marseille Univ, CNRS, ISM, Marseille, France; Department of Pediatric Orthopaedic Surgery APHM Timone Hospital, Marseille, France
| | - Cédric Payan
- Aix-Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France
| | | | - Martine Pithioux
- Aix-Marseille Univ, CNRS, ISM, Marseille, France; Aix-Marseille Univ, APHM, CNRS, ISM, Sainte-Marguerite Hospital, Institute for Locomotion, Department of Orthopaedics and Traumatology, Marseille, France.
| | - Cécile Baron
- Aix-Marseille Univ, CNRS, ISM, Marseille, France; Aix-Marseille Univ, APHM, CNRS, ISM, Sainte-Marguerite Hospital, Institute for Locomotion, Department of Orthopaedics and Traumatology, Marseille, France
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Niu H, Fan F, Wang R, Zhang Q, Shen F, Ren P, Liu T, Fan Y, Laugier P. Elastic properties measurement of human enamel based on resonant ultrasound spectroscopy. J Mech Behav Biomed Mater 2018; 89:48-53. [PMID: 30261480 DOI: 10.1016/j.jmbbm.2018.09.014] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2018] [Revised: 08/30/2018] [Accepted: 09/11/2018] [Indexed: 10/28/2022]
Abstract
OBJECTIVES To investigate the elastic properties of human enamel using resonant ultrasound spectroscopy (RUS). METHODS Six rectangular parallelepiped specimens were prepared from six human third molars. For all specimens, the theoretical resonant frequencies were calculated using the Rayleigh-Ritz method, knowing the specimen mass density and dimensions, and using a priori stiffness constants. The experimental resonant frequencies were measured and extracted by RUS. Then, the optimal stiffness constants were retrieved by adjustment of the theoretical resonant frequencies to the measured ones based on the Levenberg-Marquardt method. The engineering elastic moduli, including Young's moduli, shear moduli, and Poisson's ratios, were also calculated based on the optimal stiffness constants. RESULTS The five independent stiffness constants C11, C12, C13, C33, and C44 were 90.2 ± 6.65 GPa, 34.7 ± 6.90 GPa, 29.5 ± 4.82 GPa, 83.5 ± 8.93 GPa, and 37.0 ± 10.9 GPa, respectively. Young's moduli E11 and E33, shear moduli G13 and G12, and Poisson's ratios υ12 and υ13 were 71.7 ± 7.34 GPa, 69.2 ± 7.32 GPa, 37.0 ± 10.9 GPa, 28.1 ± 4.35 GPa, 0.303 ± 0.098, and 0.248 ± 0.060, respectively. SIGNIFICANCE Elastic properties are critical for developing dental materials and designing dental prostheses. The RUS method may provide more precise measurement of elastic properties of dental materials.
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Affiliation(s)
- Haijun Niu
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China.
| | - Fan Fan
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Rui Wang
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Qiang Zhang
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Fei Shen
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Pengling Ren
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Tao Liu
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Yubo Fan
- Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China
| | - Pascal Laugier
- Sorbonne Université, INSERM, CNRS, Laboratoire d'Imagerie Biomédicale (LIB), Paris 75006, France
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