It is estimated that over 200 million people worldwide have osteoporosis. The prevalence of osteoporosis is continuing to escalate with the increasingly elderly population. The major complication of osteoporosis is an increase in fragility fractures leading to morbidity, mortality, and decreased quality of life. In the European Union, in 2000, the number of osteoporotic fractures was estimated at 3.79 million. A baseline fracture is a very strong predictor of further fractures with 20% of patients experiencing a second fracture within the first year. The costs to health care services are already considerable and, on current trends, are predicted to double by 2050. The direct costs of osteoporotic fractures to the health services in the European Union in the year 2000 were estimated at 32 billion Euros. Guidelines for the diagnosis and treatment of osteoporosis are available in many countries; however, implementation is generally poor despite the availability of treatments with proven efficacy. Programs to increase awareness of osteoporosis and its outcomes are necessary for healthcare specialists and the general public. Earlier diagnosis and intervention prior to the first fracture are highly desirable.

Osteoporosis: a still increasing prevalence

Recent bone quantitative ultrasound approaches exploit the multimode waveguide response of long bones for assessing properties such as cortical thickness and stiffness. Clinical applications remain, however, challenging, as the impact of soft tissue on guided waves characteristics is not fully understood yet. In particular, it must be clarified whether soft tissue must be incorporated in waveguide models needed to infer reliable cortical bone properties. We hypothesize that an inverse procedure using a free plate model can be applied to retrieve the thickness and stiffness of cortical bone from experimental data. This approach is first validated on a series of laboratory-controlled measurements performed on assemblies of bone- and soft tissue mimicking phantoms and then on in vivo measurements. The accuracy of the estimates is evaluated by comparison with reference values. To further support our hypothesis, these estimates are subsequently inserted into a bilayer model to test its accuracy. Our results show that the free plate model allows retrieving reliable waveguide properties, despite the presence of soft tissue. They also suggest that the more sophisticated bilayer model, although it is more precise to predict experimental data in the forward problem, could turn out to be hardly manageable for solving the inverse problem.

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Predicting bone strength with ultrasonic guided waves

The 2-D Fourier transform analysis of multichannel signals is a straightforward method to extract the dispersion curves of guided modes. Basically, the time signals recorded at several positions along the waveguide are converted to the wavenumber–frequency space, so that the dispersion curves (i.e., the frequency-dependent wavenumbers) of the guided modes can be extracted by detecting peaks of energy trajectories. In order to improve the dispersion curve extraction of low-amplitude modes propagating in a cortical bone, a multiemitter and multireceiver transducer array has been developed together with an effective singular vector decomposition (SVD)-based signal processing method. However, in practice, the limited number of positions where these signals are recorded results in a much lower resolution in the wavenumber axis than in the frequency axis. This prevents a clear identification of overlapping dispersion curves. In this paper, a sparse SVD (S-SVD) method, which combines the signal-to-noise ratio improvement of the SVD-based approach with the high wavenumber resolution advantage of the sparse optimization, is presented to overcome the above-mentioned limitation. Different penalty constraints, i.e., $l_{1}$ -norm, Frobenius norm, and revised Cauchy norm, are compared with the sparse characteristics. The regularization parameters are investigated with respect to the convergence property and wavenumber resolution. The proposed S-SVD method is investigated using synthetic wideband signals and experimental data obtained from a bone-mimicking phantom and from an ex-vivo human radius. The analysis of the results suggests that the S-SVD method has the potential to significantly enhance the wavenumber resolution and to improve the extraction of the dispersion curves.

/pdf/sparse-svd-method-for-high-resolution-extraction-of-the-1ni6q3dg1g.pdf

Sparse SVD Method for High-Resolution Extraction of the Dispersion Curves of Ultrasonic Guided Waves

Cortical bone loss is not fully assessed by the current X-ray methods, and there is an unmet need in identifying women at risk of osteoporotic fracture, who should receive a treatment. The last decade has seen the emergence of the ultrasound (US) axial transmission (AT) techniques to assess a cortical bone. Recent AT techniques exploit the multimode waveguide response of the long bones such as the radius. A recent ex vivo study by our group evidenced that a multimode AT approach can yield simultaneous estimates of cortical thickness (Ct.Th) and stiffness. The aim of this paper is to move one step forward to evaluate the feasibility of measuring multimode guided waves (GW) in vivo and to infer from it cortical thickness. Measurements were taken on the forearm of 14 healthy subjects with the goal to test the accuracy of the estimated thickness using the bidirectional AT method implemented on a dedicated 1-MHz linear US array. This setup allows determining in vivo the dispersion curves of GW transmitted in the cortical layer of the radius. An inverse procedure based on the comparison between the measured and modeled dispersion curves predicted by a 2-D transverse isotropic free plate waveguide model allowed an estimation of cortical thickness, despite the presence of soft tissue. The Ct.Th values were validated by comparison with the site-matched estimates derived from X-ray high-resolution peripheral quantitative computed tomography. Results showed a significant correlation between both measurements ( $r^{2} = 0.7$ , $p , and $\text {RMSE} = 0.21$ mm). This pilot study demonstrates the potential of bidirectional AT for the in vivo assessment of cortical thickness, a bone strength-related factor.

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In Vivo Characterization of Cortical Bone Using Guided Waves Measured by Axial Transmission

Porous Nb-Ti-Ta alloy scaffolds for bone tissue engineering: Fabrication, mechanical properties and in vitro/vivo biocompatibility.

Multichannel analysis of dispersive ultrasonic energy requires a reliable mapping of the data from the time-distance (t-x) domain to the frequency-wavenumber (f-k) or frequency-phase velocity (f-c) domain. The mapping is usually performed with the classic 2-D Fourier transform (FT) with a subsequent substitution and interpolation via c = 2πf/k. The extracted dispersion trajectories of the guided modes lack the resolution in the transformed plane to discriminate wave modes. The resolving power associated with the FT is closely linked to the aperture of the recorded data. Here, we present a linear Radon transform (RT) to image the dispersive energies of the recorded ultrasound wave fields. The RT is posed as an inverse problem, which allows implementation of the regularization strategy to enhance the focusing power. We choose a Cauchy regularization for the high-resolution RT. Three forms of Radon transform: adjoint, damped least-squares, and high-resolution are described, and are compared with respect to robustness using simulated and cervine bone data. The RT also depends on the data aperture, but not as severely as does the FT. With the RT, the resolution of the dispersion panel could be improved up to around 300% over that of the FT. Among the Radon solutions, the high-resolution RT delineated the guided wave energy with much better imaging resolution (at least 110%) than the other two forms. The Radon operator can also accommodate unevenly spaced records. The results of the study suggest that the high-resolution RT is a valuable imaging tool to extract dispersive guided wave energies under limited aperture.

Imaging Ultrasonic Dispersive Guided Wave Energy in Long Bones Using Linear Radon Transform

Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom and in vitro bone studies

Recent in vitro studies have shown that guided waves can characterize bone properties. However, for clinical applications to be viable, the soft-tissue layer should be considered. This study examined the effect of soft tissue on guided waves using a bovine bone plate over a water half-space and overlaid by a 4-mm gelatin- based soft-tissue mimic. The data (with and without soft tissue) clearly show a high-frequency, fast-propagating wave packet and a low-frequency, delayed phase group. The presence of soft tissue attenuates the signals signifi- cantly and increases mode density and number as predicted by theory. The data retain higher frequency content than the bone-plate data at large offsets. Using theoretical dispersion curves, the guided modes can be identified with mode 1 (similar to the A0 Lamb mode) minimally affected by the addition of soft tissue. There is infiltration of high-frequency, late-arriving energy within the low-velocity guided-wave regime. Results of travel-time calcu- lation suggest that P-wave and PP-reflections/multiples within the soft tissue may be responsible for the high- frequencyoscillations. (E-mail: lawrence.le@ualberta.ca) 2013 World Federation for Ultrasound in Medicine & Biology.

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Analysis of ultrasonic waves propagating in a bone plate over a water half-space with and without overlying soft tissue.

Computing dispersion curves of elastic/viscoelastic transversely-isotropic bone plates coupled with soft tissue and marrow using semi-analytical finite element (SAFE) method

Multichannel ultrasonic axial-transmission data are multimodal by nature. As guided waves are commonly used in nondestructive material testing, wave field filtering becomes important because the analysis is usually limited to a few lower-order modes and requires their extraction. An application of the Radon transform to enhance signal-to-noise ratio and separate wave fields in ultrasonic records is presented. The method considers guided wave fields as superpositions of plane waves defined by ray parameters (p) and time intercepts (τ) and stacks the amplitudes along linear trajectories, mapping time-offset (t−x) data to a τ−p or Radon panel. The transform is implemented using a least-squares strategy with Cauchy-norm regularization that serves to enhance the focusing power. The method was verified using simulated data and applied to an uneven spatially sampled bovine-bone-plate data set. The results demonstrate the Radon panels show isolated amplitude clusters and the Cauchy-norm constraint provides a more...

Tho N.H.T. Tran

Papers

Imaging Ultrasonic Dispersive Guided Wave Energy in Long Bones Using Linear Radon Transform

Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom and in vitro bone studies

Analysis of ultrasonic waves propagating in a bone plate over a water half-space with and without overlying soft tissue.

Computing dispersion curves of elastic/viscoelastic transversely-isotropic bone plates coupled with soft tissue and marrow using semi-analytical finite element (SAFE) method

Multichannel filtering and reconstruction of ultrasonic guided wave fields using time intercept-slowness transform.