Ultrasonic Velocities Measurement

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Table of contents

1 Introduction 
1.1 Context and motivation
1.2 Hierarchical structure of bone and anisotropy
1.3 Assessment of cortical bone elasticity at the millimeter-scale
1.3.1 Mechanical testing
1.3.2 Ultrasonic waves methods
1.4 Resonant Ultrasound Spectroscopy
1.4.1 Basic principles
1.4.2 Historical review
1.4.3 RUS and attenuative materials such as bone
1.5 Outline of the thesis
2 RUS measurement of cortical bone elasticity: a feasibility study 
2.1 Introduction
2.2 Materials and Methods
2.2.1 Specimen
2.2.2 Resonant Frequencies Calculation
2.2.3 Resonant Frequencies Measurement
2.2.4 Signal Processing
2.2.5 Elastic Constants Estimation
2.2.6 Uncertainty on the elastic constants
2.2.7 Engineering moduli
2.2.8 Ultrasonic Velocities Measurement
2.3 Results
2.4 Discussion
2.5 Conclusion
3 RUS for viscoelastic characterization of anisotropic attenuative solid materials 
3.1 Introduction
3.2 Resonant frequencies computation
3.2.1 Rectangular Parallelepiped
3.2.2 Cylinder
3.3 Samples and measurement setup
3.3.1 Samples
3.3.2 Setup for the measurement of the frequency responses
3.3.3 Additional elasticity measurements
3.4 Processing of the measured spectra
3.4.1 Estimation of the resonant frequencies in time domain
3.4.2 Non-linear fitting in frequency domain
3.5 Elastic constants estimation
3.5.1 Bayesian formulation of the RUS inverse problem
3.5.2 Automated pairing of the resonant modes
3.5.3 Implementation
3.6 Damping factors estimation
3.7 Results
3.8 Discussion
3.9 Conclusion
4 Bayesian mode identification and stiffness estimation using Markov Chain Monte Carlo 
4.1 Introduction
4.2 Method
4.2.1 Forward problem
4.2.2 Bayesian formulation of the inverse problem
4.2.3 Gibbs sampling
4.3 Application 1 – Data from Ogi et al., 2002
4.3.1 Prior distributions
4.3.2 Results and discussion
4.3.3 Additional results
4.4 Application 2 – Data from Bernard al., 2013
4.4.1 Prior distributions
4.4.2 Results and discussion
4.5 Conclusion
5 Application to a collection of human tibial cortical bone specimens 
5.1 Introduction
5.2 Materials and methods
5.2.1 Specimens
5.2.2 Measurement setup and signal processing
5.2.3 Estimation of the elastic properties
5.2.4 Viscoelasticity
5.3 Results
5.4 Statistical analysis
5.5 Discussion
5.6 Conclusion
Summary and conclusion
A Appendix: Transformed elastic parameters 
A.1 Isotropic symmetry
A.2 Cubic symmetry
A.3 Transversely Isotropic symmetry
A.4 Orthotropic symmetry
B Appendix: Cutting protocol for human tibia specimens 
B.1 Material
B.2 Specimen preparation
B.3 Transversal cuts
B.4 Radial cuts
B.5 Axial cuts
B.6 C+2/3 specimen
Bibliography