Transient ultrasound propagation in porous media using Biot theory and fractional calculus: application to human cancellous bone

Frequency dependence of the ultrasonic power reflected from the water-tissue interface of human cancellous bone in vitro

Numerous studies have performed in vitro ultrasonic measurements of cancellous bone in water to develop techniques for ultrasonic bone assessment. Because cancellous bone is a highly porous medium, ultrasonic reflections at the water-bone interface may be frequency dependent. The goal of this study was to investigate the effect of porosity on the frequency dependence of the reflected power. Ultrasonic measurements were performed in a water tank at room temperature on 15 specimens of cancellous bone prepared from the proximal end of 9 human femurs using single element, broadband transducers with center frequencies of 3.5, 5, 7.5, and 10 MHz. Power spectra of pulses reflected from the water-specimen interface were corrected for the frequency response of the measurement system to obtain the reflected power in decibels R^dB(f). To suppress random phase cancellation effects, R^dB(f) was averaged over multiple sites on multiple specimens. A frequency dependence of R^dB(f) was observed in the 2.6-10 MHz range. The frequency dependence was moderate, with a maximum change of less than 6 dB over the entire frequency range. R^dB(f) was greatest for low porosity specimens. The frequency averaged intensity reflection coefficient ranged from 7.4 × 10^-4 to 7.8 × 10^-3 for high and low porosity specimen groups, respectively.

Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not.

Wave equations for porous media described by the Biot model

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatial loss terms, whereas in the high frequency regime, it is represented with fractional order temporal and spatial loss terms. In the current work, a time domain representation in terms of a partial differential equation is proposed for all three wave solutions of the Biot model across all frequencies, and it is derived from the material response function of the Biot poroelastic medium with suitable approximations for the compressional modes and the dynamic permeability. The dynamic permeability in the time domain is represented by a fractional pseudo-differential operator. Optimal correction factors are introduced into the wave equation to compensate for the discrepancies in the compressional wave dispersion and attenuation. Additionally, the method for incorporating the squirt flow mechanism into the wave equation via the Extended Biot poroviscoelastic model is described. The proposed wave equation has a physical basis and satisfies the passivity criterion.

Establishing density-dependent longitudinal sound speed in the vertebral lamina

Focused ultrasound treatments of the spinal cord may be facilitated using a phased array transducer and beamforming to correct spine-induced focal aberrations. Simulations can non-invasively calculate aberration corrections using x-ray computed tomography (CT) data that are correlated to density (ρ) and longitudinal sound speed (c_L). We aimed to optimize vertebral lamina-specific c_L(ρ) functions at a physiological temperature (37 °C) to maximize time domain simulation accuracy. Odd-numbered ex vivo human thoracic vertebrae were imaged with a clinical CT-scanner (0.511 × 0.511 × 0.5 mm), then sonicated with a transducer (514 kHz) focused on the canal via the vertebral lamina. Vertebra-induced signal time shifts were extracted from pressure waveforms recorded within the canals. Measurements were repeated 5× per vertebra, with 2.5 mm vertical vertebra shifts between measurements. Linear functions relating c_L with CT-derived density were optimized. The optimized function was c_L(ρ)=0.35(ρ-ρ_w)+ c_L,w m/s, where w denotes water, giving the tested laminae a mean bulk density of 1600 ± 30 kg/m³ and a mean bulk c_L of 1670 ± 60 m/s. The optimized lamina c_L(ρ) function was accurate to λ/16 when implemented in a multi-layered ray acoustics model. This modelling accuracy will improve trans-spine ultrasound beamforming.

Compressional wave propagation in saturated porous media and its numerical analysis using a space-time conservation element and solution element method

Compressional waves in saturated porous media are relevant to many fields from oil exploration to diagnostic of human cancellous bone and can be used to interpret physical behaviors of materials. In this work, based on Biot's theory in the low frequency range, a key finding is that there exists a critical frequency of Biot's theory in the low frequency range, which determines the coincidence of the properties of Biot waves of the first and second kinds. Furthermore, we have investigated the dispersion and attenuation of the coalescence of the first and second compressional waves in the low frequency range. The coalescence of the first and second waves is strongly attenuated with a moderate phase velocity and shows the in-phase feature. In addition, acoustic wave propagation has been calculated numerically using the space-time conservation element and solution element (CESE) method. The CESE-simulated results are compared to the experimental data and to those of the classical transfer function approach. We show that the CESE scheme preserves the local and global flux conservations in the solution procedure of Biot's theory. It is found that the CESE method provides more accurate predictions of high dispersion and strong attenuation of compressional waves in the low frequency range and is well suitable for predicting compressional wave fields in saturated porous media.

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.

Bayesian inference of a human bone and biomaterials using ultrasonic transmitted signals

Ultrasonic techniques could be good candidates to aid the assessment of osteoporosis detection, due to their non-intrusiveness and low cost. While earlier studies made use of the measured ultrasonic phase velocity and attenuation inside the bone, very few have considered an inverse identification of both the intrinsic pore microstructure and the mechanical properties of the bone, based on Biot's model. The main purpose of this work is to present an in vitro methodology for bone identification, adopting a statistical Bayesian inference technique using ultrasonic transmitted signals, which allows the retrieval of the identified parameters and their uncertainty. In addition to the bone density, Young's modulus and Poisson's ratio, the bone pore microstructure parameters (porosity, tortuosity, and viscous length) are identified. These additional microstructural terms could improve the knowledge on the correlations between bone microstructure and bone diseases, since they provide more information on the trabecular structure. In general, the exact properties of the saturating fluid are unknown (bone marrow and blood in the case of bone study) so in this work, the fluid properties (water) are identified during the inference as a proof of concept.

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.

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

The characteristic time-dependent viscosity of the intergranular pore-fluid in Buckingham's grain-shearing (GS) model [Buckingham, J. Acoust. Soc. Am. 108, 2796-2815 (2000)] is identified as the property of rheopecty. The property corresponds to a rare type of a non-Newtonian fluid in rheology which has largely remained unexplored. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and the shear wave equation derived from the GS model are shown to take the form of the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation, respectively. Therefore, an analogy is drawn between the dispersion relations obtained from the fractional framework and those from the GS model to establish the equivalence of the respective wave equations. Further, a physical interpretation of the characteristic fractional order present in the wave equations is inferred from the GS model. The overall goal is to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials. Rather, it can also be derived from real physical processes as illustrated in this work by the example of GS.

Measuring mass density and ultrasonic wave velocity: A wavelet-based method applied in ultrasonic reflection mode

When assessing ultrasonic measurements of material parameters, the signal processing is an important part of the inverse problem. Measurements of thickness, ultrasonic wave velocity and mass density are required for such assessments. This study investigates the feasibility and the robustness of a wavelet-based processing (WBP) method based on a Jaffard-Meyer algorithm for calculating these parameters simultaneously and independently, using one single ultrasonic signal in the reflection mode. The appropriate transmitted incident wave, correlated with the mathematical properties of the wavelet decomposition, was determined using a adapted identification procedure to build a mathematically equivalent model for the electro-acoustic system. The method was tested on three groups of samples (polyurethane resin, bone and wood) using one 1-MHz transducer. For thickness and velocity measurements, the WBP method gave a relative error lower than 1.5%. The relative errors in the mass density measurements ranged between 0.70% and 2.59%. Despite discrepancies between manufactured and biological samples, the results obtained on the three groups of samples using the WBP method in the reflection mode were remarkably consistent, indicating that it is a reliable and efficient means of simultaneously assessing the thickness and the velocity of the ultrasonic wave propagating in the medium, and the apparent mass density of material.

A generalized recursive convolution method for time-domain propagation in porous media

An efficient numerical method, referred to as the auxiliary differential equation (ADE) method, is proposed to compute convolutions between relaxation functions and acoustic variables arising in sound propagation equations in porous media. For this purpose, the relaxation functions are approximated in the frequency domain by rational functions. The time variation of the convolution is thus governed by first-order differential equations which can be straightforwardly solved. The accuracy of the method is first investigated and compared to that of recursive convolution methods. It is shown that, while recursive convolution methods are first or second-order accurate in time, the ADE method does not introduce any additional error. The ADE method is then applied for outdoor sound propagation using the equations proposed by Wilson et al. in the ground [(2007). Appl. Acoust. 68, 173-200]. A first one-dimensional case is performed showing that only five poles are necessary to accurately approximate the relaxation functions for typical applications. Finally, the ADE method is used to compute sound propagation in a three-dimensional geometry over an absorbing ground. Results obtained with Wilson's equations are compared to those obtained with Zwikker and Kosten's equations and with an impedance surface for different flow resistivities.

Dependences of ultrasonic properties on frequency and trabecular spacing in trabecular-bone-mimicking phantoms

The dependences of ultrasonic properties on the frequency and the trabecular spacing were investigated in 20 trabecular-bone-mimicking phantoms consisting of cellular copper foams. The strong slow waves were consistently observed in the signals transmitted through all of the phantoms. The frequency-dependent phase velocity and attenuation coefficient of the slow wave were measured at frequencies from 0.7 to 1.3 MHz. The phase velocity decreased approximately linearly with increasing frequency while the attenuation coefficients increased with increasing frequency. The phase velocity increased monotonically with increasing trabecular spacing from 1337 to 2931 μm while the attenuation coefficient decreased with increasing spacing.

Reflection and transmission coefficients of a single layer in poroelastic media

Wave propagation in poroelastic media is a subject that finds applications in many fields of research, from geophysics of the solid Earth to material science. In geophysics, seismic methods are based on the reflection and transmission of waves at interfaces or layers. It is a relevant canonical problem, which has not been solved in explicit form, i.e., the wave response of a single layer, involving three dissimilar media, where the properties of the media are described by Biot's theory. The displacement fields are recast in terms of potentials and the boundary conditions at the two interfaces impose continuity of the solid and fluid displacements, normal and shear stresses, and fluid pressure. The existence of critical angles is discussed. The results are verified by taking proper limits-zero and 100% porosity-by comparison to the canonical solutions corresponding to single-phase solid (elastic) media and fluid media, respectively, and the case where the layer thickness is zero, representing an interface separating two poroelastic half-spaces. As examples, it was calculated the reflection and transmission coefficients for plane wave incident at a highly permeable and compliant fluid-saturated porous layer, and the case where the media are saturated with the same fluid.

Numerical investigation of ultrasonic attenuation through 2D trabecular bone structures reconstructed from CT scans and random realizations

In this paper, we compare ultrasound interrogations of actual CT-scanned images of trabecular bone with artificial randomly constructed bone. Even though it is known that actual bone does not have randomly distributed trabeculae, we find that the ultrasound attenuations are close enough to cast doubt on any microstructural information, such as trabeculae width and distance between trabeculae, being gleaned from such experiments. More precisely, we perform numerical simulations of ultrasound interrogation on cancellous bone to investigate the phenomenon of ultrasound attenuation as a function of excitation frequency and bone porosity. The theoretical model is based on acoustic propagation equations for a composite fluid-solid material and is solved by a staggered-grid finite-difference scheme in the time domain. Numerical experiments are performed on two-dimensional bone samples reconstructed from CT-scanned images of real human calcaneus and from random distributions of fluid-solid particles generated via the turning bands method. A detailed comparison is performed on various parameters such as the attenuation rate and speed of sound through the bone samples as well as the normalized broadband ultrasound attenuation coefficient. Comparing results from these two types of bone samples allows us to assess the role of bone microstructure in ultrasound attenuation. It is found that the random model provides suitable bone samples for ultrasound interrogation in the transverse direction of the trabecular network.