Bayesian mode identification and stiffness estimation using Markov Chain Monte Carlo 

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Hierarchical structure of bone and anisotropy

This section briefly reviews the hierarchical structure of bone across the scales, and its consequences on the elastic anisotropy, mainly based on Weiner and Wagner (1998) and Rho et al. (1998). The bone structure is illustrated on Fig. 1.2.
The nanoscale is the length scale of the basic constituents of bone: min-eral crystals (mainly hydroxyapatite), collagen molecules (type I), non-collagenous proteins, and water. The mineral crystals form plate-like structures of about 50 25 1:5 4 nm called platelets. Collagen molecules are made of three long amino acids chains organized in a triple helix and are approximately 1 1:5 nm in diameter and 300 nm long. To form the so-called mineralized collagen fibrils, col-lagen molecules align along a given direction, and mineral crystal grow in between the collagen molecules with their crystallographic c-axis – the stiffest direction – aligned with the collagen molecules. Mineralized collagen fibrils then assemble into arrays or bundles called fibers. The mineral part provides stiffness and strength to the more compliant collagen part.
At the microscale, fibers arrange parallel to each other in layers of about 3 7 m thick called lamellae. In cortical bone, the lamellae are arranged concentrically around the Haversian canals ( 50 m) to form cylindrical layered structures called osteons ( 100 – 300 m in diameter). The Harvesian canals are aligned with the long axis of bone and contains a blood vessel and nerve fibers. Inside a lamella, the main direction of the fibers is tilted. Several patterns of tilting angle along the radius of the osteons have been described in the literature, with different expected consequences on anisotropy (Reisinger et al., 2011a). However, Reisinger et al. (2011b) observed using nanoindentation that osteons are generally stiffer in the axial direction. The space between osteons is filed with the interstitial matrix, which is made of remnants of osteons.
At the mesoscale (millimeter scale), which is the intermediate scale between the highest level of microstructure (lamellae and osteons) and the whole organ, bone tissue can be seen as a two phase composite: long cylindrical soft fibers (Haversian canals) are aligned in a hard tissue matrix (osteons and interstitial tissue). Bone is stiffer in the axial direction at this length scale, due to a cumulative effect of the preferred orientation of the Harversian porosity and the preferred direction of alignment of the mineralized collagen fibrils inside the matrix.

Measurement of cortical bone elasticity at the millimeter-scale

This section reviews state-of-the-art methods for the measurement of cortical bone elasticity on millimeter-sized specimens and discusses their respective advantages and shortcomings. These methods can be classified in two categories: 1) mechanical testing methods, in which elasticity is obtained from the stress-strain relationship in low frequency or quasi-static regime and 2) ultrasonic methods, based on the fact that, under some particular conditions, the velocity of a wave propagating in an elastic medium can be directly related to a stiffness coefficient.
For both approaches, basic equations are recalled. An extensive discussion of the elasticity of anisotropic media, and particularly of wave propagation, can be found in textbooks, such as Auld (1990) and Royer and Dieulesaint (2000).

Resonant Ultrasound Spectroscopy

Summarizing the particularities of bone elasticity assessment and the limitations of the conventional methods described above, the three main requirements for an enhanced method are: The method should be able to completely characterize the anisotropic elasticity of a material from a single specimen, without requiring 45 -oriented faces. This is linked to the spatial variability of bone elasticity.
It should be fully applicable on small specimens (of the order of a few mm3), since the thickness of the cortical shell of bones limits the specimen size. It should be as accurate and reproducible as possible. Hence, it should not assume idealized state of stress and strain but rather take into account the complete stress-strain relationship. Moreover, delicate contact or bounding between the sample and the apparatus should be avoided.
Resonant Ultrasound spectroscopy (RUS) is a method that has been developed to satisfies these requirements, originally for the characterization of geological materials (see the historical review in section 1.4.2), and that could therefore be beneficial for the study of bone mechanics. The following section introduces the basic principles of RUS, followed by a brief historical review. Finally, the difficulties of the application of RUS to bone tissue samples are introduced.

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RUS and attenuative materials such as bone

In this section, we discuss the resonant behavior of materials that are highly attenu-ative for mechanical waves at ultrasonic frequencies, resulting in important damping of the resonant modes. The few studies in which RUS was applied to highly attenu-ative materials, including bone, were in purpose excluded from the above historical review and will be considered here. Many RUS applications, and all that have been cited in the historical review above, were done on materials in which the damping of the mechanical vibrations at ultrasonic frequencies is very low, such as crystals. The quality factor Q, usually introduced to characterize the level of damping, is defined as inverse of the relative width of a resonant peak at half amplitude Q = f = f (Lakes, 2009). For a metal Table 1.1: Q factor measured using RUS for some materials, from very weakly atten-uative to highly attenuative. Some results of this thesis are included.

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 

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