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

A signal-processing tool adapted to the periodic biphasic phenomena: the Dynalet transform

The linear functional analysis, historically founded by Fourier and Legendre (Fourier's supervisor), has provided an original vision of the mathematical transformations between functional vector spaces. Fourier, and later Laplace and Wavelet transforms, respectively, defined using the simple and damped pendulum have been successfully applied in numerous applications in Physics and engineering problems. However, the classical pendulum basis may not be the most appropriate in several problems, such as biological ones, where the modelling approach is not linked to the pendulum. Efficient functional transforms can be proposed by analyzing the links between the physical or biological problem and the orthogonal (or not) basis used to express a linear combination of elementary functions approximating the observed signals. In this study, an extension of the Fourier point of view called Dynalet transform is described. The approach provides robust approximated results in the case of relaxation signals of periodic biphasic organs in human physiology.

Solution of nonlinear Lamb waves in plates with discontinuous thickness

Nonlinear Lamb waves can propagate over long distances in plate and shell structures and are sensitive to the early fatigue damage of materials. Therefore, they offer unique advantages in the fields of nondestructive testing and material health monitoring. Plate and shell structures with discontinuous thicknesses (e.g., ribs, stiffeners, or joints) will cause nonlinear Lamb wave scattering, and it is necessary to study the scattering processes of nonlinear Lamb waves at discontinuities and how these processes impact the resulting signal characteristics. Thus, nonlinear Lamb waves can be used to identify the structural characteristics and defect characteristics of signals in practical applications. In this paper, the propagating and scattering processes of the second harmonic of a Lamb wave in a discontinuous plate are studied, including the contributions of the evanescent Lamb modes near the discontinuity and the nonlinear boundary effect at the discontinuity. The scattering characteristics of the second harmonics with respect to the frequency and geometry of the plate are analyzed. In addition, the integral formula is adjusted to improve the computational stability under different numbers of Lamb wave modes. Transient finite element simulation is used to validate the proposed method.

Numerical and experimental investigation of guided ultrasonic wave propagation in non-uniform plates with structural phase variations

The article presents the results of numerical and experimental investigations of guided wave propagation in aluminum plates with variable thickness. The shapes of plate surfaces have been specially designed and manufactured using a CNC milling machine. The shapes of the plates were defined by sinusoidal functions varying in phase shift, which forced the changes in thickness variability alongside the propagation path. The main aim of the study is to analyze the wave propagation characteristics caused by non-uniform thickness. In the first step, the influence of thickness variability on the time course of propagating waves has been analyzed theoretically. The study proves that the wave propagation signals can be determined based on knowledge about the statistical description of the specimen geometry. The histograms of thickness distribution together with the a priori knowledge of the dispersion curves were used to develop an iterative procedure assuming that the signal from the previous step becomes the excitation in the next step. Such an approach allowed for taking into account the complex geometry of the plate and rejecting the assumption about the constant average thickness alongside the propagation path. In consequence, it was possible to predict correctly the signal time course, as well as the time of flight and number of propagating wave modes in specimens with variable thickness. It is demonstrated that theoretical signals predicted in this way coincide well with numerical and experimental results. Moreover, the novel procedure allowed for the correct prediction of the occurrence of higher-order modes.

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.

Improvement of Patient Classification Using Feature Selection Applied to Bidirectional Axial Transmission

Osteoporosis is still a worldwide problem, particularly due to associated fragility fractures. Patients at risk of fracture are currently detected using the X-Ray gold standard dual-energy X-ray absorptiometry (DXA), based on a calibrated 2-D image. Different alternatives, such as 3-D X-rays, magnetic resonance imaging (MRI) or ultrasound, have been proposed, the latter having advantages of being portable and sensitive to mechanical and geometrical properties. Bidirectional axial transmission (BDAT) has been used to classify between patients with or without nontraumatic fractures using "classical" ultrasonic parameters, such as velocities, as well as cortical thickness and porosity, obtained from an inverse problems. Recently, complementary parameters acquired with structural and textural analysis of guided wave spectrum images (GWSIs) have been introduced. These parameters are not limited by solution ambiguities, as for inverse problem. The aim of the study is to improve the patient classification using a feature selection strategy for all available ultrasound features completed by clinical parameters. To this end, three classical feature ranking methods were considered: analysis of variance (ANOVA), recursive feature elimination (RFE), and extreme gradient boosting importance feature (XGBI). In order to evaluate the performance of the feature selection techniques, three classical classification methods were used: logistic regression (LR), support vector machine (SVM), and extreme gradient boosting (XGB). The database was obtained from a previous clinical study [Minonzio et al., 2019]. Results indicate that the best accuracy of 71 [66-76]% was achieved by using RFE and SVM with 22 (out of 43) ultrasonic and clinical features. This value outperformed the accuracy of 68 [64-73]% reached with 2 (out of 6) DXA and clinical features. These values open promising perspectives toward improved and generalizable classification of patients at risk of fracture.

Ex Vivo Assessment of Cortical Bone Properties Using Low-Frequency Ultrasonic Guided Waves

The early diagnosis of osteoporosis through bone quality assessment is a major public health challenge. Research in axial transmission using ultrasonic guided waves has shown the method to be sensitive to the geometrical and mechanical properties of the cortical layer in long bones. However, because of the asymmetric nature of cortical bone, the introduction of a more elaborate numerical model than the analytical plate and cylinder models, as well as its inversion, continues to be challenging. The aim of this article is, therefore, to implement a bone-like geometry using semianalytical finite-element (SAFE) modeling to perform the inverse characterization of ex vivo radii at low frequencies (< 60 kHz). Five cadaveric radiuses were taken from donors aged between 53 and 88 and tested using a typical axial transmission configuration at the middle of the diaphysis. The dispersion curves of the propagating modes were measured experimentally and then systematically compared with the solutions obtained with the SAFE method. For each sample, four parameters were estimated using a parameter identification procedure: 1) the bulk density; 2) the thickness; 3) the outer diameter; and 4) a shape factor (SF). The results showed a moderate agreement between the predicted bulk density and the average voxel value calculated from X-ray computed tomography images. Furthermore, a good agreement was observed between the geometrical parameters (thickness, outer diameter, and SF) and the reference images.

In Vivo Measurements of Cortical Thickness and Porosity at the Proximal Third of the Tibia Using Guided Waves: Comparison with Site-Matched Peripheral Quantitative Computed Tomography and Distal High-Resolution Peripheral Quantitative Computed Tomography

The aim of this study was to estimate cortical porosity (Ct.Po) and cortical thickness (Ct.Th) using 500-kHz bi-directional axial transmission (AT). Ct.Th_AT and Ct.Po_AT were obtained at the tibia in 15 patients from a 2-D transverse isotropic free plate model fitted to measured guided wave dispersion curves. The velocities of the first arriving signal (υ_FAS) and A₀ mode (υ_A0) were also determined. Site-matched peripheral quantitative computed tomography (pQCT) provided volumetric cortical bone mineral density (Ct.vBMD_pQCT) and Ct.Th_pQCT. Good agreement was found between Ct.Th_AT and Ct.Th_pQCT (R² = 0.62, root mean square error [RMSE] = 0.39 mm). Ct.vBMD_pQCT correlated with Ct.Po_AT (R² = 0.57), υ_FAS (R² = 0.43) and υ_A0 (R² = 0.28). Furthermore, a significant correlation was found between AT and distal high-resolution pQCT. The measurement ofcortical parameters at the tibia using guided waves might improve the prediction of bone fractures in a cost-effective and radiation-free manner.

The nonlinear S0 Lamb mode in a plate with a linearly-varying thickness

The aim of this paper is to investigate propagation characteristics and the generation mechanism of the nonlinear lowest-order symmetric Lamb mode (S₀) which propagates downslope in free elastic plates with slowly linearly varying-thickness. From theoretical analyses, in a low frequency-thickness product (fd) range, the S₀ mode is slightly dispersive, it is easy to generate, and it approximately satisfies the principle of the phase velocity matching. Therefore, if a S₀ mode is excited at a proper frequency in the low fd range, the amplitude of the second harmonic wave is linearly increasing in a certain propagating-distance, which is valuable for the practical NDE application of the second harmonic wave. Moreover, numerical simulations and experiments have been carried out to validate theoretical results. Our investigation of properties of the second harmonic wave can be applied to characterize and evaluate micro-structural damages in varying-thickness waveguides.

Effect of intracortical bone properties on the phase velocity and cut-off frequency of low-frequency guided wave modes (20-85 kHz)

The assessment of intracortical bone properties is of interest since early-stage osteoporosis is associated with resorption in the endosteal region. However, understanding the interaction between ultrasonic guided waves and the cortical bone structure remains challenging. The purpose of this work is to investigate the effect of intracortical bone properties on the ultrasonic response obtained at low-frequency (<100 kHz) using an axial transmission configuration. The semi-analytical finite element method was used to simulate the propagation of guided waves in a waveguide with realistic geometry and material properties. An array of 20 receivers was used to calculate the phase velocity and cut-off frequency of the excited modes using the two-dimensional Fourier transform. The results show that the position of the emitter around the circumference of the bone is an important parameter to control since it can lead to variations of up to 10 dB in the amplitude of the transmitted modes. The cut-off frequency of the high order modes was, however, only slightly affected by the circumferential position of the emitter, and was sensitive mainly to the axial shear modulus. The phase velocity and cut-off frequency in the 20-85 kHz range are promising parameters for the assessment of intracortical properties.

Ex vivo cortical porosity and thickness predictions at the tibia using full-spectrum ultrasonic guided-wave analysis

The estimation of cortical thickness (Ct.Th) and porosity (Ct.Po) at the tibia using axial transmission ultrasound was successfully validated ex vivo against site-matched micro-computed tomography. The assessment of cortical parameters based on full-spectrum guided-wave analysis might improve the prediction of bone fractures in a cost-effective and radiation-free manner.

PURPOSE

Cortical thickness (Ct.Th) and porosity (Ct.Po) are key parameters for the identification of patients with fragile bones. The main objective of this ex vivo study was to validate the measurement of Ct.Po and Ct.Th at the tibia using a non-ionizing, low-cost, and portable 500-kHz ultrasound axial transmission system. Additional ultrasonic velocities and site-matched reference parameters were included in the study to broaden the analysis.

METHODS

Guided waves were successfully measured ex vivo in 17 human tibiae using a novel 500-kHz bi-directional axial transmission probe. Theoretical dispersion curves of a transverse isotropic free plate model with invariant matrix stiffness were fitted to the experimental dispersion curves in order to estimate Ct.Th and Ct.Po. In addition, the velocities of the first arriving signal (υ_FAS) and A₀ mode (υ_A0) were measured. Reference Ct.Po, Ct.Th, and vBMD were obtained from site-matched micro-computed tomography. Scanning acoustic microscopy (SAM) provided the acoustic impedance of the axial cortical bone matrix.

RESULTS

The best predictions of Ct.Po (R² = 0.83, RMSE = 2.2%) and Ct.Th (R² = 0.92, RMSE = 0.2 mm, one outlier excluded) were obtained from the plate model. The second best predictors of Ct.Po and Ct.Th were vBMD (R² = 0.77, RMSE = 2.6%) and υ_A0 (R² = 0.28, RMSE = 0.67 mm), respectively.

CONCLUSIONS

Ct.Th and Ct.Po were accurately predicted at the human tibia ex vivo using a transverse isotropic free plate model with invariant matrix stiffness. The model-based predictions were not further enhanced when we accounted for variations in axial tissue stiffness as reflected by the acoustic impedance from SAM.

Bone cortical thickness and porosity assessment using ultrasound guided waves: An ex vivo validation study

Several studies showed the ability of the cortex of long bones such as the radius and tibia to guide mechanical waves. Such experimental evidence has given rise to the emergence of a category of quantitative ultrasound techniques, referred to as the axial transmission, specifically developed to measure the propagation of ultrasound guided waves in the cortical shell along the axis of long bones. An ultrasound axial transmission technique, with an automated approach to quantify cortical thickness and porosity is described. The guided modes propagating in the cortex are recorded with a 1-MHz custom made linear transducer array. Measurement of the dispersion curves is achieved using a two-dimensional spatio-temporal Fourier transform combined with singular value decomposition. Automatic parameters identification is obtained through the solution of an inverse problem in which the dispersion curves are predicted with a two-dimensional transverse isotropic free plate model. Thirty-one radii and fifteen tibiae harvested from human cadavers underwent axial transmission measurements. Estimates of cortical thickness and porosity were obtained on 40 samples out of 46. The reproducibility, given by the root mean square error of the standard deviation of estimates, was 0.11 mm for thickness and 1.9% for porosity. To assess accuracy, site-matched micro-computed tomography images of the bone specimens imaged at 9 μm voxel size served as the gold standard. Agreement between micro-computed tomography and axial transmission for quantification of thickness and porosity at the radius and tibia ranged from R²=0.63 for porosity (root mean square error RMSE=1.8%) to 0.89 for thickness (RMSE=0.3 mm). Despite an overall good agreement for porosity, the method performs less well for porosities lower than 10%. The heterogeneity and general complexity of cortical bone structure, which are not fully accounted for by our model, are suspected to weaken the model approximation. This study presents the first validation study for assessing cortical thickness and porosity using the axial transmission technique. The automatic signal processing minimizes operator-dependent errors for parameters determination. Recovering the waveguide characteristics, that is to say cortical thickness and porosity, could provide reliable information about skeletal status and future fracture risk.

Monitoring ice thickness and elastic properties from the measurement of leaky guided waves: A laboratory experiment

The decline of Arctic sea ice extent is one of the most spectacular signatures of global warming, and studies converge to show that this decline has been accelerating over the last four decades, with a rate that is not reproduced by climate models. To improve these models, relying on comprehensive and accurate field data is essential. While sea ice extent and concentration are accurately monitored from microwave imagery, an accurate measure of its thickness is still lacking. Moreover, measuring observables related to the mechanical behavior of the ice (such as Young's modulus, Poisson's ratio, etc.) could provide better insights in the understanding of sea ice decline, by completing current knowledge so far acquired mostly from radar and sonar data. This paper aims at demonstrating on the laboratory scale that these can all be estimated simultaneously by measuring seismic waves guided in the ice layer. The experiment consisted of leaving a water tank in a cold room in order to grow an ice layer at its surface. While its thickness was increasing, ultrasonic guided waves were generated with a piezoelectric source, and measurements were subsequently inverted to infer the thickness and mechanical properties of the ice with very good accuracy.

Numerical evaluation of the backward propagating acoustic field in healing long bones

The propagation of ultrasound in healing long bones induces complex scattering phenomena due to the interaction of an ultrasonic wave with the composite nature of callus and osseous tissues. This work presents numerical simulations of ultrasonic propagation in healing long bones using the boundary element method aiming to provide insight into the complex scattering mechanisms and better comprehend the state of bone regeneration. Numerical models of healing long bones are established based on scanning acoustic microscopy images from successive postoperative weeks considering the effect of the nonhomogeneous callus structure. More specifically, the scattering amplitude and the acoustic pressure variation are calculated in the backward direction to investigate their potential to serve as quantitative and qualitative indicators for the monitoring of the bone healing process. The role of the excitation frequency is also examined considering frequencies in the range 0.2-1 MHz. The results indicate that the scattering amplitude decreases at later stages of healing compared to earlier stages of healing. Also, the acoustic pressure could provide supplementary qualitative information on the interaction of the scattered energy with bone and callus.

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.

Multichannel processing for dispersion curves extraction of ultrasonic axial-transmission signals: Comparisons and case studies

Some pioneering studies have shown the clinical feasibility of long bones evaluation using ultrasonic guided waves. Such a strategy is typically designed to determine the dispersion information of the guided modes to infer the elastic and structural characteristics of cortical bone. However, there are still some challenges to extract multimode dispersion curves due to many practical limitations, e.g., high spectral density of modes, limited spectral resolution and poor signal-to-noise ratio. Recently, two representative signal processing methods have been proposed to improve the dispersion curves extraction. The first method is based on singular value decomposition (SVD) with advantages of multi-emitter and multi-receiver configuration for enhanced mode extraction; the second one uses linear Radon transform (LRT) with high-resolution imaging of the dispersion curves. To clarify the pros and cons, a face to face comparison was performed between the two methods. The results suggest that the LRT method is suitable to separate the guided modes at low frequency-thickness-product ( fh) range; for multimode signals in broadband fh range, the SVD-based method shows more robust performances for weak mode enhancement and noise filtering. Different methods are valuable to cover the entire fh range for processing ultrasonic axial transmission signals measured in long cortical bones.

A method for the measurement of dispersion curves of circumferential guided waves radiating from curved shells: experimental validation and application to a femoral neck mimicking phantom

Our long-term goal is to develop an ultrasonic method to characterize the thickness, stiffness and porosity of the cortical shell of the femoral neck, which could enhance hip fracture risk prediction. To this purpose, we proposed to adapt a technique based on the measurement of guided waves. We previously evidenced the feasibility of measuring circumferential guided waves in a bone-mimicking phantom of a circular cross-section of even thickness. The goal of this study is to investigate the impact of the complex geometry of the femoral neck on the measurement of guided waves. Two phantoms of an elliptical cross-section and one phantom of a realistic cross-section were investigated. A 128-element array was used to record the inter-element response matrix of these waveguides. This experiment was simulated using a custom-made hybrid code. The response matrices were analyzed using a technique based on the physics of wave propagation. This method yields portions of dispersion curves of the waveguides which were compared to reference dispersion curves. For the elliptical phantoms, three portions of dispersion curves were determined with a good agreement between experiment, simulation and theory. The method was thus validated. The characteristic dimensions of the shell were found to influence the identification of the circumferential wave signals. The method was then applied to the signals backscattered by the superior half of constant thickness of the realistic phantom. A cut-off frequency and some portions of modes were measured, with a good agreement with the theoretical curves of a plate waveguide. We also observed that the method cannot be applied directly to the signals backscattered by the lower half of varying thicknesses of the phantom. The proposed approach could then be considered to evaluate the properties of the superior part of the femoral neck, which is known to be a clinically relevant site.

Computational Study of the Effect of Cortical Porosity on Ultrasound Wave Propagation in Healthy and Osteoporotic Long Bones

Computational studies on the evaluation of bone status in cases of pathologies have gained significant interest in recent years. This work presents a parametric and systematic numerical study on ultrasound propagation in cortical bone models to investigate the effect of changes in cortical porosity and the occurrence of large basic multicellular units, simply called non-refilled resorption lacunae (RL), on the velocity of the first arriving signal (FAS). Two-dimensional geometries of cortical bone are established for various microstructural models mimicking normal and pathological tissue states. Emphasis is given on the detection of RL formation which may provoke the thinning of the cortical cortex and the increase of porosity at a later stage of the disease. The central excitation frequencies 0.5 and 1 MHz are examined. The proposed configuration consists of one point source and multiple successive receivers in order to calculate the FAS velocity in small propagation paths (local velocity) and derive a variation profile along the cortical surface. It was shown that: (a) the local FAS velocity can capture porosity changes including the occurrence of RL with different number, size and depth of formation; and (b) the excitation frequency 0.5 MHz is more sensitive for the assessment of cortical microstructure.

Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes

Theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous material on flexural wave in the plates of continuously varying thickness is developed. For the wedges with thickness increasing as a power law of distance from its edge strong modifications of the wave dynamics with propagation distance are predicted. It is found that nonlinear absorption progressively disappearing with diminishing wave amplitude leads to complete attenuation of acoustic waves in most of the wedges exhibiting black hole phenomenon. It is also demonstrated that black holes exist beyond the geometrical acoustic approximation. Applications include nondestructive evaluation of micro-inhomogeneous materials and vibrations damping.

Experimental analysis and prediction of antisymmetric wave motion in a tapered anisotropic waveguide

This paper presents experimental results for wave propagation in an anisotropic multilayered structure with linearly varying cross section. Knowing the dispersion and wave propagation properties in such a structure is of great importance for non-destructive material testing and structural health monitoring applications for accurate damage detection and localization. In the proposed study, the wavefield is generated by a circular piezoelectric wafer active sensor and measured by a scanning laser-Doppler-vibrometer. The measurements are compared with a theoretical group delay estimation and a signal prediction for the antisymmetric wave motion along the non-uniform propagation path. The required dispersion curves are derived from the well-known global matrix method for segments of constant thickness. A multidimensional frequency-wavenumber analysis of linescan data and the full wavefield provides further insight of the adiabatic wave motion because the wavenumber changes along the tapered geometry of the waveguide. In addition, it is demonstrated that a terahertz time-domain system can be used in glass-fiber reinforced plastic structures as a tool to estimate the thickness profile of thin structures by means of time-of-flight measurements. This information is particularly important for guided wave-based diagnostics of structures with unknown thickness.

Distribution of mesoscale elastic properties and mass density in the human femoral shaft

Cortical bone properties are determined by tissue composition and structure at several hierarchical length scales. In this study, the spatial distribution of micro- and mesoscale elastic properties within a human femoral shaft has been investigated. Microscale tissue degree of mineralization (DMB), cortical vascular porosity Ct.Po and the average transverse isotropic stiffness tensor C(Micro) of cylindrical-shaped samples (diameter: 4.4 mm, N = 56) were obtained from cortical regions between 20 and 85% of the total femur length and around the periphery (anterior, medial, posterior and lateral quadrants) by means of synchrotron radiation µCT (SRµCT) and 50-MHz scanning acoustic microscopy (SAM). Within each cylinder, the volumetric bone mineral density (vBMD) and the mesoscale stiffness tensor C(Meso) were derived using a numerical homogenization approach. Moreover, microelastic maps of the axial elastic coefficient c33 measured by SAM at distinct cross-sectional locations along the femur were used to construct a 3-D multiscale elastic model of the femoral shaft. Variations of vBMD (6.1%) were much lower than the variations of mesoscale elastic coefficients (11.1-21.3%). The variation of DMB was only a minor predictor for variations of the mesoscale elastic properties (0.05 ≤ R(2) ≤ 0.34). Instead, variations of the mesoscale elastic properties could be explained by variations of cortical porosity and microscale elastic properties. These data were suitable inputs for numerical evaluations and may help to unravel the relations between structure and composition on the elastic function in cortical bone.