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

Robust guided wave inversion for estimating bone thickness and elasticity

Accurately characterizing bone properties using quantitative ultrasound remains a significant challenge due to the dispersive nature of guided waves, limited observations, irregularity of bone structure, and heterogeneity of bone tissues. In this paper, an inversion technique is proposed that combines weighted mean absolute criteria and the simulated annealing algorithm to extract the thicknesses and elastic properties of a bilayer bone model. By utilizing the L1 norm with an appropriate weighting parameter, this method effectively reduces the influence of outliers and noises commonly encountered in ultrasonic data, leading to more accurate estimation. This paper also introduces an asymptotic scheme to significantly reduce the search domain, improving the speed and precision of the inversion process. This approach employs a spectral collocation method as a forward modeling technique to simulate guided waves in a bone plate coated by a soft tissue layer. This paper validates the inversion using simulated and ex vivo data and demonstrates its ability to estimate features of cortical bone and soft tissue with high accuracy. Results are presented for the isotropic model. These findings hold great promise for the accurate characterization of bone properties using quantitative ultrasound, with potential applications in clinical diagnosis and treatment of bone-related diseases and injuries.

Cortical bone plate properties assessment using inversion of axially transmitted low frequency ultrasonic guided waves

Over the past few decades, early osteoporosis detection using ultrasonic bone quality evaluation has gained prominence. Specifically, various studies focused on axial transmission using ultrasonic guided waves and have highlighted this technique's sensitivity to intrinsic properties of long cortical bones. This work aims to demonstrate the potential of low-frequency ultrasonic guided waves to infer the properties of the bone inside which they are propagating. A proprietary ultrasonic transducer, tailored to transmit ultrasonic guided waves under 500 kHz, was used for the data collection. The gathered data underwent two-dimensional fast Fourier transform processing to extract experimental dispersion curves. The proposed inversion scheme compares experimental dispersion curves with simulated dispersion curves calculated through the semi-analytical iso-geometric analysis (SAIGA) method. The numerical model integrates a bone phantom plate coupled with a soft tissue layer on its top surface, mimicking the experimental bone phantom plates. Subsequently, the mechanical properties of the bone phantom plates were estimated by reducing the misfit between the experimental and simulated dispersion curves. This inversion leaned heavily on the dispersive trajectories and amplitudes of ultrasonic guided wave modes. Results indicate a marginal discrepancy under 5% between the mechanical properties ascertained using the SAIGA-based inversion and those measured using bulk wave pulse-echo measurements.

Influence of optical transmissivity on signal characteristics of photoacoustic guided waves in long cortical bone

Long cortical bone allows axial transmission of ultrasonic guided waves, which has been utilized for osteoporosis evaluation. Benefiting structural and molecular sensitivity, photoacoustic has been used for tissue composition characterization. However, photoacoustic guided waves (PAGWs) in long cortical bone as well as the influence of optical transmissivity on PAGWs have not been thoroughly investigated. In the study, the influence of optical transmissivity on the signal characteristics of PAGWs was experimentally studied with a 1064 nm pulsed laser ultrasonic system and a tunable laser system (wavelength range: 650-2600 nm). Results show that dispersion curves of PAGWs are not significantly affected by the optical transmissivity; while photoacoustic guided modes and signal spectrum are sensitive to the optical transmissivity in cortical bone. In experiments, the lasers with high transmissivity can emit pure A₀ mode PAGWs at the low frequency, around 22 kHz, in the relatively thick 6.2 mm bone plate; on the contrary, both A₀ and S₀ modes are generated. The slope of power spectrum density (PSD) of PAGWs decreases with the increase of transmissivity, and the decline rate is around -0.229. The study proves the correlation between the signal characteristics of PAGWs and the optical transmissivity, it is helpful for the development of PAGWs in long cortical bone towards the osteoporosis evaluation.

Ultrasonic Guided Waves in Bone: A Decade of Advancement in Review

The use of guided wave ultrasonography as a means to assess cortical bone quality has been a significant practice in bone quantitative ultrasound for more than 20 years. In this article, the key developments within the technology of ultrasonic guided waves (UGW) in long bones during the past decade are documented. The covered topics include data acquisition configurations available for measuring bone guided waveforms, signal processing techniques applied to bone UGW, numerical modeling of ultrasonic wave propagation in cortical long bones, formulation of inverse approaches to extract bone properties from observed ultrasonic signals, and clinical studies to establish the technology's application and efficacy. The review concludes by highlighting specific challenging problems and future research directions. In general, the primary purpose of this work is to provide a comprehensive overview of bone guided-wave ultrasound, especially for newcomers to this scientific field.

Study on ultrasonic wave propagation in equine leg bone for screening bucked shin

For simple, safe, portable, and inexpensive evaluation suitable for leg bone diseases of racehorses in the field, an ultrasonic measurement technique was applied to evaluate wave velocities. A digital model of the third metacarpal bone with the bucked shin was fabricated using high-resolution peripheral quantitative computerized tomography data of a racehorse. This model was anisotropic and heterogeneous, and was constructed using the measured ultrasonic wave velocities in the bone. With this model, ultrasonic wave propagation along the bone axis was simulated using the elastic finite-difference time-domain method. We found two main waves with different propagation velocities. The fast-waves showed a wave velocity close to the longitudinal wave in the axial direction. However, the apparent velocities changed dramatically owing to bone surface irregularities (changes of the shape) in the area of bucked shin. The slow-waves showed a wave velocity close to the shear wave, which was unaffected by the bone surface irregularities. The simple comparison of different wave behaviors may be a suitable parameter for the initial in vivo screening of bucked shin in the legs of racehorses, which can be performed in the field.

Meta-Learning Analysis of Ultrasonic Guided Waves for Coated Cortical Bone Characterization

Due to its sensitivity to geometrical and mechanical properties of waveguides, ultrasonic guided waves (UGWs) propagating in cortical bones play an important role in the early diagnosis of osteoporosis. However, as impacts of overlaid soft tissues are complex, it remains challenging to retrieve bone properties accurately. Meta-learning, i.e., learning to learn, is capable of extracting transferable features from a few data and, thus, suitable to capture potential characteristics, leading to accurate bone assessment. In this study, we investigate the feasibility to apply the multichannel identification neural network (MCINN) to estimate the thickness and bulk velocities of coated cortical bone. It minimizes the effects of soft tissue by extracting specific features of UGW, which shares the same cortical properties, while the overlaid soft tissue varies. Distinguished from most reported methods, this work moves from the hand-design inversion scheme to data-driven assessment by automatically mapping features of UGW to the space of bone properties. The MCINN was trained and validated using simulated datasets produced by the finite-difference time-domain (FDTD) method and then applied to experimental data obtained from cortical bovine bone plates overlaid with soft tissue mimics. A good match was found between experimental trajectories and theoretical dispersion curves. The results demonstrated that the proposed method was feasible to assess the thickness of coated cortical bone plates.

Analysis of Ultrasonic Guided Wave Propagation in Multilayered Bone Structure With Varying Soft-Tissue Thickness in View of Cortical Bone Characterization

Noninvasive characterization of cortical long bones using axial transmission ultrasound is a promising diagnostic technology for osteoporotic cortical thinning assessment. However, the soft tissue-bone coupling effect remains to be a challenge and an ambiguity especially in vivo. The influence of the overlying soft tissue layer with a varying thickness on the propagation of ultrasonic guided waves in cortical bone is studied experimentally and theoretically in this article. The wave propagation is characterized based on waveform comparison, spectral density and decomposition, dispersion energy imaging, and particle displacement analysis. Good agreement between experimental observations with theoretical predictions by semi-analytical finite element simulations is observed. The sensitivity of propagation characteristics in response to the coupled tissue thickness is elucidated. As the thickness of the loading soft tissue grows, the guided wave signals exhibit greater attenuated amplitude and delayed arrival time; more complex dispersive wave patterns emerge; and the modal number and density increase. The research findings advance the fundamental comprehension of ultrasonic-guided-wave excitation and interaction in long bones and facilitate further technical development and clinical utility of quantitative guided-wave ultrasonography in routine healthcare services as a nondestructive imaging modality for cortical bone examination.

Experimental identification of high order Lamb waves and estimation of the mechanical properties of a dry human skull

We experimentally investigate and characterize high order Lamb wave modes in a dry human skull. Specifically, we show that the diploë supports distinct wave modes in the sub-1.0 MHz frequency regime, and we employ these modes for the estimation of equivalent mechanical properties of cortical and trabecular bones. These modes are efficiently generated in a parietal region by direct contact excitation with a wedge beam transducer, and are recorded via infrared laser vibrometry. Frequency/wavenumber data are estimated using a matrix pencil method applied to wavefield measurements recorded on the outer cortical surface. The semi-analytical finite element model of an equivalent three-layered plate provides the platform for the identification of wave modes based on their through-the-thickness profiles, and supports the estimation of equivalent mechanical properties in conjunction with an optimization algorithm developed for this purpose. The results presented herein illustrate how high order Lamb waves can be used to gain understanding of the wave properties of a human skull and to estimate the orthotropic and equivalent isotropic mechanical properties of cortical and trabecular bones.

Single Versus Multi-channel Dispersion Analysis of Ultrasonic Guided Waves Propagating in Long Bones

Ultrasonic guided wave techniques have been applied to characterize cortical bone for osteoporosis assessment. Compared with the current gold-standard X-ray-based diagnostic methods, ultrasound-based techniques pose some advantages such as compactness, low cost, lack of ionizing radiation, and their ability to detect the mechanical properties of the cortex. Axial transmission technique with a source-receiver offset is employed to acquire the ultrasound data. The dispersion characteristics of the guided waves in bones are normally analyzed in the transformed domains using the dispersion curves. The transformed domain can be time-frequency map using a single channel or wavenumber-frequency (or phase velocity-frequency) map with multi-channels. In terms of acquisition effort, the first method is more cost- and time-effective than the latter. However, it remains unclear whether single-channel dispersion analysis can provide as much quantitative guided-wave information as the multi-channel analysis. The objective of this study is to compare the two methods using numerically simulated and ex vivo data of a simple bovine bone plate and explore their advantages and disadvantages. Both single- and multi-channel signal processing approaches are implemented using sparsity-constrained optimization algorithms to reinforce the focusing power. While the single-channel data acquisition and processing are much faster than those of the multi-channel, modal identification and analysis of the multi-channel data are straightforward and more convincing.

Deep Learning Analysis of Ultrasonic Guided Waves for Cortical Bone Characterization

Ultrasonic guided waves (UGWs) propagating in the long cortical bone can be measured via the axial transmission method. The characterization of long cortical bone using UGW is a multiparameter inverse problem. The optimal solution of the inverse problem often includes a complex solving process. Deep neural networks (DNNs) are essentially powerful multiparameter predictors based on universal approximation theorem, which are suitable for solving parameter predictions in the inverse problem by constructing the mapping relationship between UGW and cortical bone material parameters. In this study, we investigate the feasibility of applying the multichannel crossed convolutional neural network (MCC-CNN) to simultaneously estimate cortical thickness and bulk velocities (longitudinal and transverse). Unlike the multiparameter estimation in most previous studies, the technique mentioned in this work avoids solving a multiparameter optimization problem directly. The finite-difference time-domain (FDTD) method is performed to obtain the simulated UGW array signals for training the MCC-CNN. The network that is exclusively trained on simulated data sets can predict cortical parameters from the experimental UGW data. The proposed method is confirmed by using FDTD simulation signals and experimental data obtained from four bone-mimicking plates and from ten ex vivo bovine cortical bones. The estimated root-mean-squared error (RMSE) in the simulated test data for the longitudinal bulk velocity ( V_L ), transverse bulk velocity ( V_T ), and cortical thickness (Th) is 97 m/s, 53 m/s, and 0.089 mm, respectively. The predicted RMSE in the bone-mimicking phantom experiments for V_L|| , V_T|| , and Th is 120 m/s, 80 m/s, and 0.14 mm, respectively. The experimental dispersion trajectories are matched with the theoretical dispersion curves calculated by the predicted parameters in ex vivo bovine cortical bone experiments. Our proposed method demonstrates a feasible approach for the accurate evaluation of long cortical bones based on UGW.

Nonlinear Inversion of Ultrasonic Dispersion Curves for Cortical Bone Thickness and Elastic Velocities

In this study, a nonlinear grid-search inversion has been developed to estimate the thickness and elastic velocities of long cortical bones, which are important determinants of bone strength, from axially-transmitted ultrasonic data. The inversion scheme is formulated in the dispersive frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve overlying soft tissue thickness, cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires systematic examination of a large set of trial dispersion-curve solutions within a pre-defined model space to match the data with minimum cost in a least-squares sense. The theoretical dispersion curves required to solve the inverse problem are computed for bilayered bone models using a semi-analytical finite-element method. The feasibility of the proposed approach was demonstrated by the numerically simulated data for a 1 mm soft tissue-5 mm bone bilayer and ex-vivo data from a bovine femur plate with an overlying 2 mm-thick soft-tissue mimic. The bootstrap method was employed to evaluate the inversion uncertainty and stability. Our results have shown that the cortical thickness and wave speeds could be recovered with fair accuracy.

Influence of guided waves in bone on pulse-inversion contrast-enhanced ultrasound

PURPOSE

Guided waves generated from bone cortex inevitably act on microbubbles flowing through skeletal muscle capillaries in contrast-enhanced ultrasound (CEUS) and might influence the image quality. However, the action mechanism underlying the guided waves influence is still unknown, especially under contrast pulse-inversion transmission mode. This study aimed to clarify the influence of guided waves on pulse-inversion CEUS, which was investigated via in vitro infusion experiments.

METHOD

Tibia guided waves were detected at pulse-inversion transmission and then characterized by using a short-time Fourier transform energy distribution. Using results at normal incidence as a baseline, the influence of guided wave dispersion on the contrast and resolution of pulse-inversion CEUS was investigated at an oblique incidence through continuous microbubbles infusion experiments in a vessel-tibia flow phantom.

RESULTS

Frequency-dispersive property of tibia guided waves was observed at phases 0° and 180°, which improved the contrast of CEUS and reduced its resolution. Pulse-inversion CEUS balanced the contrast enhancement and resolution degeneration induced by guided waves. By contrast, contrast-to-tissue ratio of pulse-inversion CEUS increased by up to 109.1 ± 13.2% (P < 0.05) due to guided waves and its resolution was up to 0.9 ± 0.1 times that of baseline.

CONCLUSIONS

Alterations of contrast and resolution in pulse-inversion CEUS induced by guided waves might provide an additional assessment for the capillary perfusion in the skeletal muscle near the bone cortex.

Fatigue evaluation of long cortical bone using ultrasonic guided waves

Bone fatigue fracture is a progressive disease due to stress concentration. This study aims to evaluate the long bone fatigue damage using the ultrasonic guided waves. Two-dimensional finite-difference time-domain method was employed to simulate the ultrasonic guided wave propagation in the long bone under different elastic modulus. The experiment was conducted on a 3.8 mm-thick bovine bone plate. The phase velocities of two fundamental guided modes, A1 and S1, were measured by using the axial transmission technique. Simulation shows that the phase velocities of guided modes A1 and S1 decrease with the increasing of the fatigue damage. After 20,000 cycles of fatigue loading on the bone plate, the average phase velocities of A1 and S1 modes were 6.6% and 5.3% respectively, lower than those of the intact bone. The study suggests that ultrasonic guided waves can be potentially used to evaluate the fatigue damage in long bones.

Relationships of the group velocity of the time-reversed Lamb wave with bone properties in cortical bone in vitro

The present study aims to investigate the feasibility of using the time-reversed Lamb wave as a new method for noninvasive characterization of long cortical bones. The group velocity of the time-reversed Lamb wave launched by using the modified time reversal method was measured in 15 bovine tibiae, and their correlations with the bone properties of the tibia were examined. The group velocity of the time-reversed Lamb wave showed significant positive correlations with the bone properties (r=0.55-0.81). The best univariate predictor of the group velocity of the time-reversed Lamb wave was the cortical thickness, yielding an adjusted squared correlation coefficient (r²) of 0.64. These results imply that the group velocity of the time-reversed Lamb wave, in addition to the velocities of the first arriving signal and the slow guided wave, could potentially be used as a discriminator for osteoporosis.

Predicting bone strength with ultrasonic guided waves

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.

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

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 ( r² = 0.7 , , and [Formula: see text] mm). This pilot study demonstrates the potential of bidirectional AT for the in vivo assessment of cortical thickness, a bone strength-related factor.

Propagation of time-reversed Lamb waves in bovine cortical bone in vitro

The present study aims to investigate the propagation of time-reversed Lamb waves in bovine cortical bone in vitro. The time-reversed Lamb waves were successfully launched at 200 kHz in 18 bovine tibiae through a time reversal process of Lamb waves. The group velocities of the time-reversed Lamb waves in the bovine tibiae were measured using the axial transmission technique. They showed a significant correlation with the cortical thickness and tended to follow the theoretical group velocity of the lowest order antisymmetrical Lamb wave fairly well, consistent with the behavior of the slow guided wave in long cortical bones.

A free plate model can predict guided modes propagating in tubular bone-mimicking phantoms

The goal of this work was to show that a non-absorbing free plate model can predict with a reasonable accuracy guided modes measured in bone-mimicking phantoms that have circular cross-section. Experiments were carried out on uncoated and coated phantoms using a clinical axial transmission setup. Adjustment of the plate model to the experimental data yielded estimates for the waveguide characteristics (thickness, bulk wave velocities). Fair agreement was achieved over a frequency range of 0.4 to 1.6 MHz. A lower accuracy observed for the thinnest bone-mimicking phantoms was caused by limitations in the wave number measurements rather than by the model itself.

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.

Wave dispersion and attenuation on human femur tissue

Cortical bone is a highly heterogeneous material at the microscale and has one of the most complex structures among materials. Application of elastic wave techniques to this material is thus very challenging. In such media the initial excitation energy goes into the formation of elastic waves of different modes. Due to “dispersion”, these modes tend to separate according to the velocities of the frequency components. This work demonstrates elastic wave measurements on human femur specimens. The aim of the study is to measure parameters like wave velocity, dispersion and attenuation by using broadband acoustic emission sensors. First, four sensors were placed at small intervals on the surface of the bone to record the response after pencil lead break excitations. Next, the results were compared to measurements on a bulk steel block which does not exhibit heterogeneity at the same wave lengths. It can be concluded that the microstructure of the tissue imposes a dispersive behavior for frequencies below 1 MHz and care should be taken for interpretation of the signals. Of particular interest are waveform parameters like the duration, rise time and average frequency, since in the next stage of research the bone specimens will be fractured with concurrent monitoring of acoustic emission.

Measuring the wavenumber of guided modes in waveguides with linearly varying thickness

Measuring guided waves in cortical bone arouses a growing interest to assess skeletal status. In most studies, a model of waveguide is proposed to assist in the interpretation of the dispersion curves. In all the reported investigations, the bone is mimicked as a waveguide with a constant thickness, which only approximates the irregular geometry of cortical bone. In this study, guided mode propagation in cortical bone-mimicking wedged plates is investigated with the aim to document the influence on measured dispersion curves of a waveguide of varying thickness and to propose a method to overcome the measurement limitations induced by such thickness variations. The singular value decomposition-based signal processing method, previously introduced for the detection of guided modes in plates of constant thickness, is adapted to the case of waveguides of slowly linearly variable thickness. The modification consists in the compensation at each frequency of the wavenumber variations induced by the local variation in thickness. The modified method, tested on bone-mimicking wedged plates, allows an enhanced and more accurate detection of the wavenumbers. Moreover, the propagation in the directions of increasing and decreasing thickness along the waveguide is investigated.