Ultrasound Computed Tomography.

This chapter presents theoretical, numerical, and experimental frameworks for the use of Ultrasound Computed Tomography (USCT) for cortical bone tissue imaging. Most of the research conducted on this topic concerns adult bone, although some work presented in this chapter is specific to the study of child bone. USCT is recognized as a powerful method for soft tissue imaging. In bone imaging, the difficulties arise from the very high impedance contrast between tissues which alters the propagation of the ultrasonic waves and limits the linear inversion algorithms used. Solutions consist in optimally assessing non-linear effects in an iterative approach aiming at local linearization. When the problem can be reduced to the study of a fluid-like cavity buried in an elastic cylinder surrounded by water, the signal processing and/or compound algorithms can be added as an extension to the linear algorithms. The main limitation of these methods is the heavy experimental costs involved. We have then suggested the introduction of purely numerical non-linear full-waveform inversion algorithms. The performances and the limitations of these linear and non-linear methods applied to cortical bone tissue imaging problems are overviewed and discussed.

Assessing the Elasticity of Child Cortical Bone.

A better understanding of the mechanical behaviour of child bone is essential to improve the diagnosis of pediatric bone disorders that may influence bone development. Even though the process of bone growth is well described, there are still lacks of knowledge on intrinsic material properties of child bone and particularly on child bone considered as "non-pathological". Geometry, material properties, microstructure and biochemical components are associated with child bone fragility and remain difficult to assess for two main reasons: the scarcity of the bone samples and their small dimensions. In this context, ultrasonic methods offer interesting possibilities by exploiting in particular their non-destructive character. In this chapter, the elasticity properties of Non Pathological Child Cortical Bone (NPCCB) obtained by ultrasonic methods are presented. The objective was to contribute to the construction of a reference database on NPCCB that would serve as a point of comparison for analyzing the effect of a pathology or treatment. After the presentation of the hypotheses on the elasticity and anisotropy of NPCCB, ultrasonic transmission-mode and resonance spectroscopy methods are described. Results are presented and discussed with respect to microstructural and biochemical properties.

Assessment of qualitative and quantitative S0 guided wave tomography of sharp thickness loss defects in the isotropic membrane regime.

Progress in instrumentation, computer hardware, and inversion methods is encouraging the development of more advanced guided wave tomography techniques, especially for nondestructive testing of plate structures to characterize corrosion. An experimental S0 tomography performance assessment in the membrane regime is reported. One of the main interests of the fundamental membrane regime is that in this regime, waves are propagated over long distances. A 2 mm thick steel disk containing calibrated sharp artificial defects (flat bottom holes) is tested in both reflection and extinction modes. A reconstruction algorithm derived from the membrane approximation is presented. We expose a complete reflection mode inversion approach that includes beam inversion, waveform deconvolution, and thickness loss calibration. Non-linear correction factors are introduced and discussed for quantitative imaging. A width-regularity-depth description of defects is introduced to put the results into perspective with other defect geometries. The results show the relevance of the inversion method to enhance the imaging performance with regard to defect localization and sizing. Crucial points concerning instrumentation such as coupling, signal-to-noise ratio, excitation mode, coupling, selection of frequency, are also discussed.

Ultrasonic Imaging of High-contrasted Objects Based on Full-waveform Inversion: Limits under Fluid Modeling.

Quantitative ultrasound techniques have been previously used to evaluate biological hard tissues, characterized by a large acoustic impedance contrast. Here, we are interested in the imaging of experimental data from different test-targets with high acoustic impedance contrast, using the Full Waveform Inversion (FWI) method to solve the inverse problem. This method is based on high-resolution numerical modeling of the forward problem of interaction between waves and medium, considering the full time series. To reduce the complexity of the numerical implementation, the model considers a fluid medium. Therefore, the aim is to evaluate the precision of the reconstruction under this assumption for materials with a different level of attenuation of shear waves, to study the limits of this hypothesis. Images of the sound speed obtained using the experimental data are presented, and the precision of the reconstruction is evaluated. Future work should include viscoelastic materials.

The benefits of compression methods in acoustic coherence tomography.

Pulse compression methods improve the quality of ultrasonic medical images. In comparison with standard broadband pulse techniques, these methods enhance the contrast-to-noise ratio (CNR) and increase the probing depth without any perceptible loss of spatial resolution. The Golay compression technique is analyzed here in the context of ultrasonic computed tomography, first on a one-dimensional target and second on a very low-contrast phantom probed using a half-ring array tomograph. The imaging performances were assessed based on the image CNR. The improvement obtained (up to 40%) depends, however, on the number of coherently associated diffraction projections. Beyond a certain number, few advantages were observed. Advances in ultrasound computed tomography suggest that pulse compression methods should provide a useful means of optimizing the trade-off between the image quality and the probing sampling density.

Computational model to address lens-based acoustic field aperture in the in vitro ultrasonic cell stimulation.

Low-Intensity Pulsed Ultrasound Stimulation (LIPUS) is a therapeutic modality used for bone tissue regeneration and healing. Its clinical efficacy is still debated, as the underlying physical phenomena remain poorly understood. The interaction between ultrasonic waves and cells, likely to trigger mechanotransduction inducing bone regeneration, is at the center of scientific concerns on the subject. In order to get new insights into these phenomena, the development of in vitro experiments is a key step but special attentions should be paid concerning to the actual acoustic area covered that has to be sufficiently large and homogeneous. To address this issue, an acoustic lens can be placed on the transducer to improve the homogeneity of the acoustic field over the entire cell culture area. A computational model is developed to test several shapes and heights of acoustic lenses and compare their effectiveness in order to find a compromise between the surface covered, the homogeneity of the intensity distribution and the acoustic pressure loss. All the lenses studied improve the enlargement of the field and its homogeneity but they all generate pressure acoustic loss. The best performing lens in terms of field homogeneity is the one that minimizes pressure acoustic loss but covers only 22% of the target surface. The best enlargement (68% of the surface covered) is obtained for a lens that produces a field that is 4 times less homogeneous and 3 times less efficient in terms of pressure acoustic loss. As no one lens is ideal, the choice of the lens should be the result of a compromise taking into account the prioritization of criteria.

Quantitative non-linear ultrasonic imaging of targets with significant acoustic impedance contrast--an experimental study.

This study deals with the reconstruction, from ultrasonic measured data, of the sound speed profile of a penetrable two-dimensional target of arbitrary cross-section embedded in an infinite medium. Green's theorem is used to obtain a domain integral representation of the acoustical scattered field, and a discrete formulation of the inverse problem is obtained using a moment method. An iterative non-linear algorithm minimizing the discrepancy between the measured and computed scattered fields is used to reconstruct the sound speed profile in the region of interest. The minimization process is performed using a conjugated-gradient method. An experimental study with significant acoustical impedance contrast targets immersed in water was performed. Images of the sound speed profile obtained by inversion of experimental data are presented.

Quantitative parametric imaging by ultrasound computed tomography of trees under anisotropic conditions: Numerical case study.

A method for the reconstruction of 2D tomographic images adapted to wood was presented, aiming to perform a nondestructive evaluation of standing trees. The proposed method takes into account the orthotropy property of wood material, performing an iterative process that approximated the curved rays. A slowness function was defined for every cell and a nonlinear regression allowed the mapping of the inner elastic constants. Four numerical configurations were tested representing real cases usually found in standing tree monitoring. These specific configurations allowed this work to focus on the analysis of the effect of anisotropy on image reconstruction. The reconstructed images using the proposed method were compared with a straight-ray reconstruction method (filtered back projection algorithm), highlighting a more detailed identification and quantification of the inner state of the anisotropic structure of the trunk.

Ultrasounds could be considered as a future tool for probing growing bone properties.

Juvenile bone growth is well described (physiological and anatomical) but there are still lacks of knowledge on intrinsic material properties. Our group has already published, on different samples, several studies on the assessment of intrinsic material properties of juvenile bone compared to material properties of adult bone. The purpose of this study was finally to combine different experimental modalities available (ultrasonic measurement, micro-Computed Tomography analysis, mechanical compression tests and biochemical measurements) applied on small cubic bone samples in order to gain insight into the multiparametric evaluation of bone quality. Differences were found between juvenile and adult groups in term of architectural parameters (Porosity Separation), Tissue Mineral Density (TMD), diagonal stiffness coefficients (C33, C44, C55, C66) and ratio between immature and mature cross-links (CX). Diagonal stiffness coefficients are more representative of the microstructural and biochemical parameters of child bone than of adult bone. We also found that compression modulus E was highly correlated with several microstructure parameters and CX in children group while it was not at all correlated in the adult group. Similar results were found for the CX which was linked to several microstructure parameters (TMD and E) only in the juvenile group. To our knowledge, this is the first time that, on a same sample, ultrasonic measurements have been combined with the assessment of mechanical and biochemical properties. It appears that ultrasonic measurements can provide relevant indicators of child bone quality (microstructural and biochemical parameters) which is promising for clinical application since, B-mode ultrasound is the preferred first-line modality over other more constraining imaging modalities (radiation, parent-child accessibility and access to the patient's bed) for pediatric patients.

Effect of wood anisotropy in ultrasonic wave propagation: A ray-tracing approach.

Ultrasonic nondestructive imaging methods allow analyzing the inner structures of trees, without altering their condition. In this study, we are interested in evaluating the influence of anisotropy condition in the wood on the ultrasonic waves time-of-flight (TOF) estimation, by means of a raytracing approach. This technique is used particularly in the field of exploration seismography to simulate wavefronts in elastic media. Wood sections from two species were tested. Defects in the wood were simulated by drilling holes. Defects were tested in two positions, centric and eccentric, and three different defect diameters were used for each position. First, experiments with healthy wood showed that the orthotropic behavior resulted in curved rays from the transmitter to every receiver, compared to the straight-line paths for the isotropic case, considering that the radial direction presents a higher wave velocity. Defects inside the wood resulted in low velocity propagation areas, that modified the trajectories compared to the healthy case. Centric defects resulted in larger TOF variations than eccentric defects. A combination of centric position and bigger size corresponded to a higher probability of decay detection using a tomographic image. To increase the tomographic image quality, curved rays should be considered when performing the image reconstruction.

Assessment of elastic coefficients of child cortical bone using resonant ultrasound spectroscopy.

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 ±â€¯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 ±â€¯0.19 GPa, C22 = 16.19 ±â€¯0.12 GPa, C33 = 24.47 ±â€¯0.30 GPa, C44 = 4.14 ±â€¯0.08 GPa, C55 = 4.16 ±â€¯0.07 GPa, C66 = 3.13 ±â€¯0.05 GPa, C12 = 10.14 ±â€¯0.20 GPa, C13 = 10.67 ±â€¯0.27 GPa, C23 = 10.25 ±â€¯0.14 GPa.

Accuracy of coded excitation methods for measuring the time of flight: Application to ultrasonic characterization of wood samples.

Ultrasound computed tomography (USCT) using the transmission mode is a way to detect and assess the extent of decay in wood structures. The resolution of the ultrasonic image is closely related to the different anatomical features of wood. The complexity of the wave propagation process generates complex signals consisting of several wave packets with different signatures. Wave paths, depth dependencies, wave velocities or attenuations are often difficult to interpret. For this kind of assessment, the focus is generally on signal pre-processing. Several approaches have been used so far including filtering, spectrum analysis and a method involving deconvolution using a characteristic transfer function of the experimental device. However, all these approaches may be too sophisticated and/or unstable. The alternative methods proposed in this work are based on coded excitation, which makes it possible to process both local and general information available such as frequency and time parameters. Coded excitation is based on the filtering of the transmitted signal using a suitable electric input signal. The aim of the present study was to compare two coded-excitation methods, a chirp- and a wavelet-coded excitation method, to determine the time of flight of the ultrasonic wave, and to investigate the feasibility, the robustness and the precision of the measurement of geometrical and acoustical properties in laboratory conditions. To obtain control experimental data, the two methods were compared with the conventional ultrasonic pulse method. Experiments were conducted on a polyurethane resin sample and two samples of different wood species using two 500 kHz-transducers. The relative errors in the measurement of thickness compared with the results of caliper measurements ranged from 0.13% minimum for the wavelet-coded excitation method to 2.3% maximum for the chirp-coded excitation method. For the relative errors in the measurement of ultrasonic wave velocity, the coded excitation methods showed differences ranging from 0.24% minimum for the wavelet-coded excitation method to 2.62% maximum for the chirp-coded excitation method. Methods based on coded excitation algorithms thus enable accurate measurements of thickness and ultrasonic wave velocity in samples of wood species.

An optimization method for quantitative impedance tomography.

A near-field ultrasonic tomography method providing high resolution imaging for soft tissue in the reflection mode is reported. When the Born approximation is valid, the main limitation of this method is that it requires an incident pulse with infinite bandwidth, whereas the incident pulses used in practice have a limited bandwidth, which makes quantitative reconstruction impossible. The reconstructed image is qualitative in the sense that it is a band-pass filtered reconstruction of the impedance distribution. An optimization method based on the use of the geometrical information provided by the tomographic reconstruction is developed to obtain the quantitative information required. The object was approximated locally by an equivalent canonical body, on the basis of the previous global estimation. The inversion procedure is then carried out using the minimization of a cost function, which is the average over frequency of the difference between the measured field scattered by the object and the estimated field scattered by the equivalent canonical body. Assuming the object to be homogeneous by regions, the last step consists of assigning the estimated local impedance value to the region of interest. When the geometry of the real body is almost canonical, the optimization method yields accurate impedance assessments.

Ultrasonic computed tomography based on full-waveform inversion for bone quantitative imaging.

We introduce an ultrasonic quantitative imaging method for long bones based on full-waveform inversion. The cost function is defined as the difference in the L 2-norm sense between observed data and synthetic results at a given iteration of the iterative inversion process. For simplicity, and in order to reduce the computational cost, we use a two-dimensional acoustic approximation. The inverse problem is solved iteratively based on a quasi-Newton technique called the Limited-memory Broyden-Fletcher-Goldfarb-Shanno method. We show how the technique can be made to work fine for benchmark models consisting of a single cylinder, and then five cylinders, the latter case including significant multiple diffraction effects. We then show pictures obtained for a tibia-fibula bone pair model. Convergence is fast, typically in 15 to 30 iterations in practice in each frequency band used. We discuss the so-called 'cycle skipping' effect that can occur in such full waveform inversion techniques and make them remain trapped in a local minimum of the cost function. We illustrate strategies that can be used in practice to avoid this. Future work should include viscoelastic materials rather than acoustic, and real data instead of synthetic data.

Assessing the cortical thickness of long bone shafts in children, using two-dimensional ultrasonic diffraction tomography.

Echography is one of the first-line techniques used in clinical practice to diagnose osteoarticular diseases in children. However, this technique involves the use of standard equipment, which is not adapted to the morphology or the acoustical properties of children's bones. In this study, we developed an ultrasonic tomography method for measuring the cortical thickness of children's long bones. Ultrasonic tomography gives cross-sectional images showing the spatial distribution of some of the physical components of an object, based on scattered ultrasound measurements. These measurements are carried out using variably dense sets of transmitter and receiver positions and various the wave frequencies. We solved this inverse scattering problem using a Born approximation, which yields an attractively simple linear relation between the object function and the scattered field, particularly in the far field. Experiments with a 2D-ring antenna show the applicability of the method and its various improvements to bone thickness imaging.

Measuring mass density and ultrasonic wave velocity: A wavelet-based method applied in ultrasonic reflection mode.

When assessing ultrasonic measurements of material parameters, the signal processing is an important part of the inverse problem. Measurements of thickness, ultrasonic wave velocity and mass density are required for such assessments. This study investigates the feasibility and the robustness of a wavelet-based processing (WBP) method based on a Jaffard-Meyer algorithm for calculating these parameters simultaneously and independently, using one single ultrasonic signal in the reflection mode. The appropriate transmitted incident wave, correlated with the mathematical properties of the wavelet decomposition, was determined using a adapted identification procedure to build a mathematically equivalent model for the electro-acoustic system. The method was tested on three groups of samples (polyurethane resin, bone and wood) using one 1-MHz transducer. For thickness and velocity measurements, the WBP method gave a relative error lower than 1.5%. The relative errors in the mass density measurements ranged between 0.70% and 2.59%. Despite discrepancies between manufactured and biological samples, the results obtained on the three groups of samples using the WBP method in the reflection mode were remarkably consistent, indicating that it is a reliable and efficient means of simultaneously assessing the thickness and the velocity of the ultrasonic wave propagating in the medium, and the apparent mass density of material.

Pore network microarchitecture influences human cortical bone elasticity during growth and aging.

Cortical porosity is a major determinant of bone strength. Haversian and Volkmann׳s canals are׳seen' as pores in 2D cross-section but fashion a dynamic network of interconnected channels in 3D, a quantifiable footprint of intracortical remodeling. Given the changes in bone remodeling across life, we hypothesized that the 3D microarchitecture of the cortical pore network influences its stiffness during growth and ageing. Cubes of cortical bone of 2 mm side-length were harvested in the distal 1/3 of the fibula in 13 growing children (mean age±SD: 13±4 yrs) and 16 adults (age: 75±13 yrs). The cubes were imaged using desktop micro-CT (8.14µm isotropic voxel size). Pores were segmented as a solid to assess pore volume fraction, number, diameter, separation, connectivity and structure model index. Elastic coefficients were derived from measurements of ultrasonic bulk compression and shear wave velocities and apparent mass density. The pore volume fraction did not significantly differ between children and adults but originates from different microarchitectural patterns. Compared to children, adults had 42% (p=0.033) higher pore number that were more connected (Connective Density: +205%, p=0.001) with a 18% (p=0.007) lower pore separation. After accounting for the contribution of pore volume fraction, axial elasticity in traction-compression mode was significantly correlated with better connectivity in growing children and with pore separation among adults. The changes in intracortical remodeling across life alter the distribution, size and connectedness of the channels from which cortical void fraction originates. These alterations in pore network microarchitecture participate in changes in compressive and shear mechanical behavior, partly in a porosity-independent manner. The assessment of pore volume fraction (i.e., porosity) provides only a limited understanding of the role of cortical void volume fraction in its mechanical properties.

Progress towards in vitro quantitative imaging of human femur using compound quantitative ultrasonic tomography.

The objective of this study is to make cross-sectional ultrasonic quantitative tomography of the diaphysis of long bones. Ultrasonic propagation in bones is affected by the severe mismatch between the acoustic properties of this biological solid and those of the surrounding soft medium, namely, the soft tissues in vivo or water in vitro. Bone imaging is then a nonlinear inverse-scattering problem. In this paper, we showed that in vitro quantitative images of sound velocities in a human femur cross section could be reconstructed by combining ultrasonic reflection tomography (URT), which provides images of the macroscopic structure of the bone, and ultrasonic transmission tomography (UTT), which provides quantitative images of the sound velocity. For the shape, we developed an image-processing tool to extract the external and internal boundaries and cortical thickness measurements. For velocity mapping, we used a wavelet analysis tool adapted to ultrasound, which allowed us to detect precisely the time of flight from the transmitted signals. A brief review of the ultrasonic tomography that we developed using correction algorithms of the wavepaths and compensation procedures are presented. Also shown are the first results of our analyses on models and specimens of long bone using our new iterative quantitative protocol.

Analyzing the anisotropic Hooke's law for children's cortical bone.

Child cortical bone tissue is rarely studied because of the difficulty of obtaining samples. Yet the preparation and ultrasonic characterization of the small samples available, while challenging, is one of the most promising ways of obtaining information on the mechanical behavior of non-pathological children׳s bone. We investigated children׳s cortical bone obtained from chirurgical waste. 22 fibula or femur samples from 21 children (1-18 years old, mean age: 9.7±5.8 years old) were compared to 16 fibula samples from 16 elderly patients (50-95 years old, mean age: 76.2±13.5 years old). Stiffness coefficients were evaluated via an ultrasonic method and anisotropy ratios were calculated as the ratio of C33/C11, C33/C22 and C11/C22. Stiffness coefficients were highly correlated with age in children (R>0.56, p<0.01). No significant difference was found between C11 and C22 for either adult or child bone (p>0.5), nor between C44 and C55 (p>0.5). We observe a transverse isotropy with C33>C22=C11>C44C55>C66. For both groups, we found no correlation between age and anisotropy ratios. This study offers the first complete analysis of stiffness coefficients in the three orthogonal bone axes in children, giving some indication of how bone anisotropy is related to age. Future perspectives include studying the effect of the structure and composition of bone on its mechanical behavior.

Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula.

As a determinant of skeletal fragility, the organic matrix is responsible for the post-yield and creep behavior of bone and for its toughness, while the mineral apatite acts on stiffness. Specific to the fibula and ulna in children, greenstick fractures show a plastic in vivo mechanical behavior before bone fracture. During growth, the immature form of collagen enzymatic cross-links gradually decreases, to be replaced by the mature form until adolescence, subsequently remaining constant throughout adult life. However, the link between the cortical bone organic matrix and greenstick fractures in children remains to be explored. Here, we sought to determine: 1) whether plastic bending fractures can occur in vitro, by testing cortical bone samples from children's fibula and 2) whether the post-yield behavior (ωp plastic energy) of cortical bone before fracture is related to total quantity of the collagen matrix, or to the quantity of mature and immature enzymatic cross-links and the quantity of non-enzymatic cross-links. We used a two-step approach; first, a 3-point microbending device tested 22 fibula machined bone samples from 7 children and 3 elderly adults until fracture. Second, biochemical analysis by HPLC was performed on the sample fragments. When pooling two groups of donors, children and elderly adults, results show a rank correlation between total energy dissipated before fracture and age and a linear correlation between plastic energy dissipated before fracture and ratio of immature/mature cross-links. A collagen matrix with more immature cross-links (i.e. a higher immature/mature cross-link ratio) is more likely to plastically deform before fracture. We conclude that this ratio in the sub-nanostructure of the organic matrix in cortical bone from the fibula may go some way towards explaining the variance in post-yield behavior. From a clinical point of view, therefore, our results provide a potential explanation of the presence of greenstick fractures in children.