Tho N.H.T. Tran

Imaging ultrasonic dispersive guided wave energy in long bones using linear radon transform.

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.

Analysis of Ultrasonic Waves Propagating in a Bone Plate over a Water Half-Space with and without Overlying Soft Tissue

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 significantly 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 calculation suggest that P-wave and PP-reflections/multiples within the soft tissue may be responsible for the high-frequency oscillations.

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

Long bones are good waveguides to support the propagation of ultrasonic guided waves. The low-order guided waves have been consistently observed in quantitative ultrasound bone studies. Selective excitation of these low-order guided modes requires oblique incidence of the ultrasound beam using a transducer-wedge system. It is generally assumed that an angle of incidence, θi, generates a specific phase velocity of interest, co, via Snells law, θi=sin(-1)(vw/co) where vw is the velocity of the coupling medium. In this study, we investigated the excitation of guided waves within a 6.3-mm thick brass plate and a 6.5-mm thick bovine bone plate using an ultrasound phased array system with two 0.75-mm-pitch array probes. Arranging five elements as a group, the first group of a 16-element probe was used as a transmitter and a 64-element probe was a receiver array. The beam was steered for six angles (0°, 20°, 30°, 40°, 50°, and 60°) with a 1.6-MHz source signal. An adjoint Radon transform algorithm mapped the time-offset matrix into the frequency-phase velocity dispersion panels. The imaged Lamb plate modes were identified by the theoretical dispersion curves. The results show that the 0° excitation generated many modes with no modal discrimination and the oblique beam excited a spectrum of phase velocities spread asymmetrically about co. The width of the excitation region decreased as the steering angle increased, rendering modal selectivity at large angles. The phenomena were well predicted by the excitation function of the source influence theory. The low-order modes were better imaged at steering angle ⩾30° for both plates. The study has also demonstrated the feasibility of using the two-probe phased array system for future in vivo study.

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

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 focused Radon image than the damped least-squares regularization. Wave field separation can be achieved by selectively windowing the τ - p signals and inverse transformation, which is illustrated by the successful extraction of the A0 mode in bone plate. In addition, the method effectively attenuates noise, enhances the coherency of the guided wave modes, and reconstructs the missing records. The proposed transform presents a powerful signal-enhancement tool to process guided waves for further analysis and inversion.

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

We present a semi-analytical finite element (SAFE) scheme for accurately computing the velocity dispersion and attenuation in a trilayered system consisting of a transversely-isotropic (TI) cortical bone plate sandwiched between the soft tissue and marrow layers. The soft tissue and marrow are mimicked by two fluid layers of finite thickness. A Kelvin-Voigt model accounts for the absorption of all three biological domains. The simulated dispersion curves are validated by the results from the commercial software DISPERSE and published literature. Finally, the algorithm is applied to a viscoelastic trilayered TI bone model to interpret the guided modes of an ex-vivo experimental data set from a bone phantom.

Micro-scale finite element modeling of ultrasound propagation in aluminum trabecular bone-mimicking phantoms: A comparison between numerical simulation and experimental results.

The present study investigated the accuracy of micro-scale finite element modeling for simulating broadband ultrasound propagation in water-saturated trabecular bone-mimicking phantoms. To this end, five commercially manufactured aluminum foam samples as trabecular bone-mimicking phantoms were utilized for ultrasonic immersion through-transmission experiments. Based on micro-computed tomography images of the same physical samples, three-dimensional high-resolution computational samples were generated to be implemented in the micro-scale finite element models. The finite element models employed the standard Galerkin finite element method (FEM) in time domain to simulate the ultrasonic experiments. The numerical simulations did not include energy dissipative mechanisms of ultrasonic attenuation; however, they expectedly simulated reflection, refraction, scattering, and wave mode conversion. The accuracy of the finite element simulations were evaluated by comparing the simulated ultrasonic attenuation and velocity with the experimental data. The maximum and the average relative errors between the experimental and simulated attenuation coefficients in the frequency range of 0.6-1.4 MHz were 17% and 6% respectively. Moreover, the simulations closely predicted the time-of-flight based velocities and the phase velocities of ultrasound with maximum relative errors of 20 m/s and 11 m/s respectively. The results of this study strongly suggest that micro-scale finite element modeling can effectively simulate broadband ultrasound propagation in water-saturated trabecular bone-mimicking structures.

Excitability of ultrasonic lamb waves in a cortical bone plate: A simulation study

Application of ultrasonic axial transmission techniques to assess human elongated cortical bones has gained considerable attention in the area of bone tissue imaging and characterization. The cortical shell of long bones acts as a natural waveguide and supports the propagation of multiple guided modes. In practice, not all guided modes exist. For the existing ones, their energies are absent at high frequencies. For low frequencies, their dispersion energies are segmented and discontinuous. The excitation depends on the mode and its position on the dispersion curve. This investigation is the first attempt to provide some formal explanations of the phenomena using the concept of attenuation and excitability. Numerical data of a simple bone plate, generated by a semi-analytical finite element algorithm, is used to illustrate the concept.

Sensitivity analysis of ultrasonic guided waves propagating in trilayered bone models: a numerical study

The fundamental ultrasonic guided modes are consistently observed in long bones ex vivo and in vivo. However, the responses of ultrasonic guided waves to the changes of cortical thickness, cortical elastic parameters, and thickness of the overlying soft tissues are not comprehensively understood. This paper systematically presents a sensitivity analysis of leaky Lamb modes to the geometry and material characteristics of layered bone model by means of semi-analytical finite element modeling. The stratified bone model is consisted of a transversely isotropic cortex with an overlying soft tissue and underlying marrow. The study is important as it offers guidance to the parameter inversion process about the optimal selection of guided modes and regions of sensitivity for better inversion results.

High-Resolution Imaging of Dispersive Ultrasonic Guided Waves in Human Long Bones Using Regularized Radon Transforms

Ultrasound is an indispensable imaging modality to monitor soft tissues in diagnostic radiology. Research into ultrasound has resulted in technology developments and extension of its use beyond soft tissue imaging. The use of ultrasound to probe hard tissues is not yet a common practice but in recent years modest interest has been generated to use quantitative ultrasound in bone evaluation. This is due to several advantages of ultrasound over ionizing techniques: lack of ionizing radiation, sensitivity to the mechanical elasticity of bone tissues, portability, and low cost. Recent studies using axial-transmission technique have shown that ultrasonic guided waves, which propagate within cortical bone, have great potential to characterize mechanical and structural properties of the cortical waveguide. Multi-channel dispersion analysis of ultrasonic guided waves requires a reliable mean to map the data from the time-distance domain to the frequency-phase velocity domain. In this work, linear Radon transform using various regularization strategies is considered to enhance the transform focusing power to image dispersive guide-wave energies. Four forms of linear Radon solution: adjoint, damped least-squares, Cauchy-regularized, and l1-regularized Radon transform were applied to the simulated and in-vivo experimental data and the results were compared. Among the regularization strategies, the l1-regularization renders a highly-sparse solution and images dispersion energies with the best focusing resolution. The high-resolution dispersion maps allow better wave-mode discrimination and separation. The results of this study suggest the l1-regularized Radon transform as a valuable tool to image dispersive ultrasonic guided-wave energies propagating in long bones.

A nonlinear grid-search inversion for cortical bone thickness and ultrasonic velocities

Cortical thickness and elasticity are important determinants of bone strength. This study proposes a nonlinear grid-search inversion to estimate the thickness and ultrasonic velocities of long cortical bones from axially-transmitted data. The inversion scheme has been developed in the frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires to systematically examine a set of trial dispersion curves within a pre-defined model space to match the data which minimizes the objective function in a least-squares sense. The theoretical dispersion curves required by the inversion are computed for bilayered bone models using a semi-analytical finite-element method. Its application demonstrates the feasibility of the proposed approach on synthetic data for a 5 mm-thick bone plate, and in-vitro dataset from a bovine femur plate with 2 mm-thick soft-tissue m...