Measurements of ultrasound velocity and attenuation in numerical anisotropic porous media compared to Biot's and multiple scattering models

Energy landscape interpretation of universal linearly increasing absorption with frequency

Absorption of elastic waves in complex media is commonly found to increase linearly with frequency, for both longitudinal and shear waves. This ubiquitous property is observed in media such as rocks, unconsolidated sediments, and human tissue. Absorption is due to relaxation processes at the level of atomic scales and up to the sub-micron scale of biological materials. The effect of these processes is usually expressed as an integral over relaxation frequencies or relaxation times. Here, this paper argues that these processes are thermally activated. Unusually for ultrasonics and seismics, the expression for absorption from the frequency or time domains can therefore be transformed to an integral over an activation energy landscape weighted by an energy distribution. The universal power-law property surprisingly corresponds to a flat activation energy landscape. This is the solution that maximizes entropy or randomness. Therefore, the linearly increasing absorption corresponds to the energy landscape with the fewest possible constraints.

The respective and dependent effects of scattering and bone matrix absorption on ultrasound attenuation in cortical bone

Cortical bone is characterized by a dense solid matrix permeated by fluid-filled pores. Ultrasound scattering has potential for the non-invasive evaluation of changes in bone porosity. However, there is an incomplete understanding of the impact of ultrasonic absorption in the solid matrix on ultrasound scattering. In this study, maps were derived from scanning acoustic microscopy images of human femur cross-sections. Finite-difference time domain ultrasound scatter simulations were conducted on these maps. Pore density, diameter distribution of the pores, and nominal absorption values in the solid and fluid matrices were controlled. Ultrasound pulses with a central frequency of 8.2 MHz were propagated, both in through-transmission and backscattering configurations. From these data, the scattering, bone matrix absorption, and attenuation extinction lengths were calculated. The results demonstrated that as absorption in the solid matrix was varied, the scattering, absorption, and attenuation extinction lengths were significantly impacted. It was shown that for lower values of absorption in the solid matrix (less than 2 dB mm-1), attenuation due to scattering dominates, whereas at higher values of absorption (more than 2 dB mm-1), attenuation due to absorption dominates. This will impact how ultrasound attenuation and scattering parameters can be used to extract quantitative information on bone microstructure.

Using ultrasonic attenuation in cortical bone to infer distributions on pore size

In this work we infer the underlying distribution on pore radius in human cortical bone samples using ultrasonic attenuation data. We first discuss how to formulate polydisperse attenuation models using a probabilistic approach and the Waterman Truell model for scattering attenuation. We then compare the Independent Scattering Approximation and the higher-order Waterman Truell models' forward predictions for total attenuation in polydisperse samples. Following this, we formulate an inverse problem under the Prohorov Metric Framework coupled with variational regularization to stabilize this inverse problem. We then use experimental attenuation data taken from human cadaver samples and solve inverse problems resulting in nonparametric estimates of the probability density function on pore radius. We compare these estimates to the "true" microstructure of the bone samples determined via microCT imaging. We find that our methodology allows us to reliably estimate the underlying microstructure of the bone from attenuation data.

Ultrasound Scattering in Cortical Bone

Recent advances in imaging of bone microstructure have led to a growing recognition of the role of cortical microstructure in osteoporosis. It is now accepted that the assessment of the microstructure of cortical porosity is essential to assess bone mechanical competence and predict fracture risk. Cortical porosity affects the propagation of ultrasound waves because pores act as ultrasound scatterers. Scattering by the porosity is an opportunity that should be leveraged to extract quantitative information about cortical microstructure. Scattering by the pores affects a number of ultrasound parameters that should be quantified, including attenuation, backscatter coefficient, ultrasound diffusivity, and their frequency dependence. Measuring these ultrasound parameters and developing models that describe their dependence upon parameters of cortical microstructure is the key to solve inverse problems that will allow the quantitative assessment of cortical porosity and ultimately will improve the non-invasive ultrasound-based evaluation of bone mechanical competence and fracture risk. In this chapter, we present recent advances in measuring and modeling those parameters in cortical bone.

Frequency-dependent analysis of ultrasound apparent absorption coefficient in multiple scattering porous media: application to cortical bone

The effect of matrix viscoelastic absorption on frequency-dependent attenuation in porous structures mimicking simplified cortical bone is addressed in this numerical study. An apparent absorption is defined to quantify the difference between total attenuation (resulting from both absorption and scattering) and attenuation exclusively due to scattering. A power-law model is then used to describe the frequency-dependent apparent absorption as a function of pore diameter and density. The frequency response of the porous structures to a Gaussian pulse is studied to determine the frequency range over which the system can be considered linear. The results show that for low scattering regimes (normalized frequency [Formula: see text]0.80), the system and its apparent absorption can be considered linear. Hence, the total attenuation coefficient results from the summation of scattering and absorption coefficients. However, for highly scattering regimes, the system can no longer be considered linear, as the apparent absorption vs. frequency deviates from a linear trend. As the pore density increases, the apparent absorption coefficient increases as well.

Inferring pore radius and density from ultrasonic attenuation using physics-based modeling

This work proposes the use of two physics-based models for wave attenuation to infer the microstructure of cortical bone-like structures. One model for ultrasound attenuation in porous media is based on the independent scattering approximation (ISA) and the other model is based on the Waterman Truell (WT) approximation. The microstructural parameters of interest are pore radius and pore density. Attenuation data are simulated for three-dimensional structures mimicking cortical bone using the finite-difference time domain package SimSonic. These simulated structures have fixed sized pores (monodisperse), allowing fine-tuned control of the microstructural parameters. Structures with pore radii ranging from 50 to 100  μm and densities ranging from 20 to 50 pores/mm3 are generated in which only the attenuation due to scattering is considered. From here, an inverse problem is formulated and solved, calibrating the models to the simulated data and producing estimates of pore radius and density. The estimated microstructural parameters closely match the values used to simulate the data, validating the use of both the ISA and WT approximations to model ultrasonic wave attenuation in heterogeneous structures mimicking cortical bone. Furthermore, this illustrates the effectiveness of both models in inferring pore radius and density solely from ultrasonic attenuation data.

Evaluation of Structural Anisotropy in a Porous Titanium Medium Mimicking Trabecular Bone Structure Using Mode-Converted Ultrasonic Scattering

The mode-converted (longitudinal to transverse, L-T) ultrasonic scattering method was utilized to characterize the structural anisotropy of a phantom mimicking the structural properties of trabecular bone. The sample was fabricated using metal additive manufacturing from high-resolution computed tomography (CT) images of a sample of trabecular horse bone with strong anisotropy. Two focused transducers were used to perform the L-T ultrasonic measurements. A normal incidence transducer was used to transmit longitudinal ultrasonic waves into the sample, while the scattered transverse signals were received by an oblique incidence transducer. At multiple locations on the sample, four L-T measurements were performed by collecting ultrasonic scattering from four directions. The amplitude of the root mean square (rms) of the collected ultrasonic scattering signals was calculated for each L-T measurement. The ratios of rms amplitudes for L-T measurements in different directions were calculated to characterize the anisotropy of sample. The results show that the amplitude of L-T converted scattering is highly dependent on the direction of microstructural anisotropy. A strong anisotropy of the microstructure was observed, which coincides with simulation results previously published on the same structure as well as with the anisotropy estimated from the CT images. These results suggest the potential of mode-converted ultrasonic scattering methods to assess the anisotropy of materials with porous, complex structures, including trabecular bone.

Mechanisms of Interaction of Ultrasound With Cancellous Bone: A Review

Ultrasound is now a clinically accepted modality in the management of osteoporosis. The most common commercial clinical devices assess fracture risk from measurements of attenuation and sound speed in cancellous bone. This review discusses fundamental mechanisms underlying the interaction between ultrasound and cancellous bone. Because of its two-phase structure (mineralized trabecular network embedded in soft tissue-marrow), its anisotropy, and its inhomogeneity, cancellous bone is more difficult to characterize than most soft tissues. Experimental data for the dependencies of attenuation, sound speed, dispersion, and scattering on ultrasound frequency, bone mineral density, composition, microstructure, and mechanical properties are presented. The relative roles of absorption, scattering, and phase cancellation in determining attenuation measurements in vitro and in vivo are delineated. Common speed of sound metrics, which entail measurements of transit times of pulse leading edges (to avoid multipath interference), are greatly influenced by attenuation, dispersion, and system properties, including center frequency and bandwidth. However, a theoretical model has been shown to be effective for correction for these confounding factors in vitro and in vivo. Theoretical and phantom models are presented to elucidate why cancellous bone exhibits negative dispersion, unlike soft tissue, which exhibits positive dispersion. Signal processing methods are presented for separating "fast" and "slow" waves (predicted by poroelasticity theory and supported in cancellous bone) even when the two waves overlap in time and frequency domains. Models to explain dependencies of scattering on frequency and mean trabecular thickness are presented and compared with measurements. Anisotropy, the effect of the fluid filler medium (marrow in vivo or water in vitro), phantoms, computational modeling of ultrasound propagation, acoustic microscopy, and nonlinear properties in cancellous bone are also discussed.

Artificial neural network to estimate micro-architectural properties of cortical bone using ultrasonic attenuation: A 2-D numerical study

The goal of this study is to estimate micro-architectural parameters of cortical porosity such as pore diameter (φ), pore density (ρ) and porosity (ν) of cortical bone from ultrasound frequency dependent attenuation using an artificial neural network (ANN). First, heterogeneous structures with controlled pore diameters and pore densities (mono-disperse) were generated, to mimic simplified structure of cortical bone. Then, more realistic structures were obtained from high resolution CT scans of human cortical bone. 2-D finite-difference time-domain simulations were conducted to calculate the frequency-dependent attenuation in the 1-8 MHz range. An ANN was then trained with the ultrasonic attenuation at different frequencies as the input feature vectors while the output was set as the micro-architectural parameters (pore diameter, pore density and porosity). The ANN is composed of three fully connected dense layers with 24, 12 and 6 neurons, connected to the output layer. The dataset was trained over 6000 epochs with a batch size of 16. The trained ANN exhibits the ability to predict the micro-architectural parameters with high accuracy and low losses. ANN approaches could potentially be used as a tool to help inform physics-based modelling of ultrasound propagation in complex media such as cortical bone. This will lead to the solution of inverse-problems to retrieve bone micro-architectural parameters from ultrasound measurements for the non-invasive diagnosis and monitoring osteoporosis.

Modeling ultrasound attenuation in porous structures with mono-disperse random pore distributions using the independent scattering approximation: a 2D simulation study

The validity of the independent scattering approximation (ISA) to predict the frequency dependent attenuation in 2D models of simplified structures of cortical bone is studied. Attenuation of plane waves at central frequencies ranging from 1 to 8 MHz propagating in structures with mono-disperse random pore distributions with pore diameter and pore density in the range of those of cortical bone are evaluated by finite difference time domain numerical simulations. An approach to assess the multiple scattering of waves in random media is discussed to determine the pore diameter ranges at which the ISA is applicable. A modified version of the ISA is proposed to more accurately predict the attenuation in porosity ranges where it would traditionally fail. The results show that the modified ISA can model the frequency-dependent attenuation of ultrasonic wave with pore diameter and density ranges comparable to those of cortical bone.

Predicting Structural Properties of Cortical Bone by Combining Ultrasonic Attenuation and an Artificial Neural Network (ANN): 2-D FDTD Study

The goal of this paper is to predict the micro-architectural parameters of cortical bone such as pore diameter ( ϕ ) and porosity ( v ) from ultrasound attenuation measurements using an artificial neural network (ANN). Slices from a 3-D CT scan of human femur are obtained. The micro-architectural parameters of porosity such as average pore size and porosity are calculated using image processing. When ultrasound waves propagate in porous structures, attenuation is observed due to scattering. Two-dimensional finite-difference time-domain simulations are carried out to obtain frequency dependent attenuation in those 2D structures. An artificial neural network (ANN) is then trained with the input feature vector as the frequency dependent attenuation and output as pore diameter ( ϕ ) and porosity ( v ) . The ANN is composed of one input layer, 3 hidden layers and one output layer, all of which are fully connected. 340 attenuation data sets were acquired and trained over 2000 epochs with a batch size of 32. Data was split into train, validation and test. It was observed that the ANN predicted the micro-architectural parameters of the cortical bone with high accuracies and low losses with a minimum R2 (goodness of fit) value of 0.95. ANN approaches could potentially help inform the solution of inverse-problems to retrieve bone porosity from ultrasound measurements. Ultimately, those inverse-problems could be used for the non-invasive diagnosis and monitoring of osteoporosis.

The effect of pore size and density on ultrasonic attenuation in porous structures with mono-disperse random pore distribution: A two-dimensional in-silico study

This work proposes a power law model to describe the attenuation of ultrasonic waves in non-absorbing heterogeneous media with randomly distributed scatterers, mimicking a simplified structure of cortical bone. This paper models the propagation in heterogeneous structures with controlled porosity using a two-dimensional finite-difference time domain numerical simulation in order to measure the frequency dependent attenuation. The paper then fits a phenomenological model to the simulated frequency dependent attenuation by optimizing parameters under an ordinary least squares framework. Local sensitivity analysis is then performed on the resulting parameter estimates in order to determine to which estimates the model is most sensitive. This paper finds that the sensitivity of the model to various parameter estimates depends on the micro-architectural parameters, pore diameter (ϕ) and pore density (ρ). In order to get a sense for how confidently model parameters are able to be estimated, 95% confidence intervals for these estimates are calculated. In doing so, the ability to estimate model-sensitive parameters with a high degree of confidence is established. In the future, being able to accurately estimate model parameters from which micro-architectural ones could be inferred will allow pore density and diameter to be estimated via an inverse problem given real or simulated ultrasonic data to be determined.

An iterative fullwave simulation approach to multiple scattering in media with randomly distributed microbubbles

Ultrasound contrast agents (UCA), such as microbubbles, enhance the scattering properties of blood, which is otherwise hypoechoic. The multiple scattering interactions of the acoustic field with UCA are poorly understood due to the complexity of the multiple scattering theories and the nonlinear microbubble response. The majority of bubble models describe the behavior of UCA as single, isolated microbubbles suspended in infinite medium. Multiple scattering models such as the independent scattering approximation can approximate phase velocity and attenuation for low scatterer volume fractions. However, all current models and simulation approaches only describe multiple scattering and nonlinear bubble dynamics separately. Here we present an approach that combines two existing models: (1) a full-wave model that describes nonlinear propagation and scattering interactions in a heterogeneous attenuating medium and (2) a Paul-Sarkar model that describes the nonlinear interactions between an acoustic field and microbubbles. These two models were solved numerically and combined with an iterative approach. The convergence of this combined model was explored in silico for 0.5 × 10⁶ microbubbles ml^-1, 1% and 2% bubble concentration by volume. The backscattering predicted by our modeling approach was verified experimentally with water tank measurements performed with a 128-element linear array transducer. An excellent agreement in terms of the fundamental and harmonic acoustic fields is shown. Additionally, our model correctly predicts the phase velocity and attenuation measured using through transmission and predicted by the independent scattering approximation.

Characterization of the Lung Parenchyma Using Ultrasound Multiple Scattering

The purpose of the study described here was to showcase the application of ultrasound to quantitative characterization of the micro-architecture of the lung parenchyma to predict the extent of pulmonary edema. The lung parenchyma is a highly complex and diffusive medium for which ultrasound techniques have remained qualitative. The approach presented here is based on ultrasound multiple scattering and exploits the complexity of ultrasound propagation in the lung structure. The experimental setup consisted of a linear transducer array with an 8-MHz central frequency placed in contact with the lung surface. The diffusion constant D and transport mean free path L* of the lung parenchyma were estimated by separating the incoherent and coherent intensities in the near field and measuring the growth of the incoherent diffusive halo over time. Significant differences were observed between the L* values obtained in healthy and edematous rat lungs in vivo. In the control rat lung, L* was found to be 332 μm (±48.8 μm), whereas in the edematous lung, it was 1040 μm (±90 μm). The reproducibility of the measurements of L* and D was tested in vivo and in phantoms made of melamine sponge with varying air volume fractions. Two-dimensional finite difference time domain numerical simulations were carried out on rabbit lung histology images with varying degrees of lung collapse. Significant correlations were observed between air volume fraction and L* in simulation (r = -0.9542, p < 0.0117) and sponge phantom (r = -0.9932, p < 0.0068) experiments. Ex vivo measurements of a rat lung in which edema was simulated by adding phosphate-buffered saline revealed a linear relationship between the fluid volume fraction and L*. These results illustrate the potential of methods based on ultrasound multiple scattering for the quantitative characterization of the lung parenchyma.

Numerical analysis of longitudinal ultrasonic attenuation in sintered materials using a simplified two-phase model

Techniques of quantitative nondestructive evaluation using attenuation of ultrasonic waves have been proposed as a potential tool for monitoring sintering processes because of the direct connection between the changes of wave propagation characteristics and microstructure properties. However, the influence of these changes during sintering on sound propagation remains unclear. In addition to theoretical investigations, numerical models can be utilized to provide key information for interpreting experimental data quantitatively. In this article, a simplified two-phase model using Voronoi polycrystals is applied to study wave propagation through sintered materials. Finite element simulations are developed with various material and geometric parameters of the two-phase model. Example longitudinal attenuation results are obtained and compared with the scattering theory for different input wave frequencies. The comparison of the numerical results with the theory shows the dependence of the attenuation on the parameters of the correlation function and the two-phase geometry. The results also validate the correlation function formula used in the theory. The influence of the input wave frequency and material properties on the correlation lengths is also discussed. Such numerical models can be used to verify theoretical models efficiently and to design further experimental methods for characterization of microstructures.

Multiple scattering in random dispersions of spherical scatterers: Effects of shear-acoustic interactions

The propagation of acoustic waves through a suspension of spherical particles in a viscous liquid is investigated, through application of a multiple scattering model. The model is based on the multiple scattering formulation of Luppé, Conoir, and Norris [J. Acoust. Soc. Am. 131, 1113-1120 (2012)] which incorporated the effects of thermal and shear wave modes on propagation of the acoustic wave mode. Here, the model is simplified for the case of solid particles in a liquid, in which shear waves make a significant contribution to the effective properties. The relevant scattering coefficients and effective wavenumber are derived in analytical form. The results of calculations are presented for a system of silica particles in water, illustrating the dependence of the scattering coefficients, effective wavenumber, speed, attenuation on particle size and frequency. The results demonstrate what has already been shown experimentally; that the shear-mediated processes have a very significant effect on the effective attenuation of acoustic waves, especially as the concentration of particles increases.

Ultrasound characterization of the mastoid for detecting middle ear effusion: A preliminary clinical validation

Ultrasound detection of middle ear effusion (MEE) is an emerging technique in otolaryngology. This study proposed using ultrasound characterization of the mastoid to noninvasively measure MEE-induced mastoid effusion (ME) as a new strategy for determining the presence of MEE. In total, 53 patients were enrolled (Group I: normal, n = 20; Group II: proven MEE through both otoscopy and tympanometry, n = 15; Group III: patients with MEE having effusions observed during grommet surgery, n = 18). A 2.25-MHz delay-line transducer was used to measure backscattered signals from the mastoid. The Nakagami parameter was estimated using the acquired signals to model the echo amplitude distribution for quantifying changes in the acoustic structures of mastoid air cells. The median Nakagami parameter and interquartile range were 0.35 (0.34-0.37) for Group I, 0.39 (0.37-0.41) for Group II, and 0.43 (0.39-0.51) for Group III. The echo amplitude distribution observed for patients with MEE was closer to Rayleigh distribution than that without MEE. Receiver operating characteristic (ROC) curve analysis further revealed that the area under the ROC was 0.88, sensitivity was 72.73%, specificity was 95%, and accuracy was 81.13%. The proposed method has considerable potential for noninvasive and comfortable evaluation of MEE.

Experimental observation of ultrasound fast and slow waves through three-dimensional printed trabecular bone phantoms

In this paper, ultrasound measurements of 1:1 scale three-dimensional (3D) printed trabecular bone phantoms are reported. The micro-structure of a trabecular horse bone sample was obtained via synchrotron x-ray microtomography, converted to a 3D binary data set, and successfully 3D-printed at scale 1:1. Ultrasound through-transmission experiments were also performed through a highly anisotropic version of this structure, obtained by elongating the digitized structure prior to 3D printing. As in real anisotropic trabecular bone, both the fast and slow waves were observed. This illustrates the potential of stereolithography and the relevance of such bone phantoms for the study of ultrasound propagation in bone.

New perspectives in echographic diagnosis of osteoporosis on hip and spine

Currently, the accepted "gold standard" method for bone mineral density (BMD) measurement and osteoporosis diagnosis is dual-energy X-ray absorptiometry (DXA). However, actual DXA effectiveness is limited by several factors, including intrinsic accuracy uncertainties and possible errors in patient positioning and/or post-acquisition data analysis. DXA employment is also restricted by the typical issues related to ionizing radiation employment (high costs, need of dedicated structures and certified operators, unsuitability for population screenings). The only commercially-available alternative to DXA is represented by "quantitative ultrasound" (QUS) approaches, which are radiation-free, cheaper and portable, but they cannot be applied on the reference anatomical sites (lumbar spine and proximal femur). Therefore, their documented clinical usefulness is restricted to calcaneal applications on elderly patients (aged over 65 y), in combination with clinical risk factors and only for the identification of healthy subjects at low fracture risk. Literature-reported studies performed some QUS measurements on proximal femur, but their clinical translation is mostly hindered by intrinsic factors (e.g., device bulkiness). An innovative ultrasound methodology has been recently introduced, which performs a combined analysis of B-mode images and corresponding "raw" radiofrequency signals acquired during an echographic scan of the target reference anatomical site, providing two novel parameters: Osteoporosis Score and Fragility Score, indicative of BMD level and bone strength, respectively. This article will provide a brief review of the available systems for osteoporosis diagnosis in clinical routine contexts, followed by a synthesis of the most promising research results on the latest ultrasound developments for early osteoporosis diagnosis and fracture prevention.

Fast and slow wave detection in bovine cancellous bone in vitro using bandlimited deconvolution and Prony's method

Fast and slow waves were detected in a bovine cancellous bone sample for thicknesses ranging from 7 to 12 mm using bandlimited deconvolution and the modified least-squares Prony's method with curve fitting (MLSP + CF). Bandlimited deconvolution consistently isolated two waves with linear-with-frequency attenuation coefficients as evidenced by high correlation coefficients between attenuation coefficient and frequency: 0.997 ± 0.002 (fast wave) and 0.986 ± 0.013 (slow wave) (mean ± standard deviation). Average root-mean-squared (RMS) differences between the two algorithms for phase velocities were 5 m/s (fast wave, 350 kHz) and 13 m/s (slow wave, 750 kHz). Average RMS differences for signal loss were 1.6 dB (fast wave, 350 kHz) and 0.4 dB (slow wave, 750 kHz). Phase velocities for thickness = 10 mm were 1726 m/s (fast wave, 350 kHz) and 1455 m/s (slow wave, 750 kHz). Results show support for the model of two waves with linear-with frequency attenuation, successful isolation of fast and slow waves, good agreement between bandlimited deconvolution and MLSP + CF as well as with a Bayesian algorithm, and potential variations of fast and/or slow wave properties with bone sample thickness.