Measurements of ultrasonic phase velocities and attenuation of slow waves in cellular aluminum foams as cancellous bone-mimicking phantoms

The water-saturated aluminum foams with an open network of interconnected ligaments were investigated by ultrasonic transmission technique for the suitability as cancellous bone-mimicking phantoms. The phase velocities and attenuation of nine samples covering three pores per inch (5, 10, and 20 PPI) and three aluminum volume fractions (5, 8, and 12% AVF) were measured over a frequency range of 0.7-1.3 MHz. The ligament thickness and pore sizes of the phantoms and low-density human cancellous bones are similar. A strong slow wave and a weak fast wave are observed for all samples while the latter is not visible without significant amplification (100x). This study reports the characteristics of slow wave, whose speeds are less than the sound speed of the saturating water and decrease mildly with AVF and PPI with an average 1469 m/s. Seven out of nine samples show positive dispersion and the rest show minor negative dispersion. Attenuation increases with AVF, PPI, and frequency except for the 20 PPI samples, which exhibit non-increasing attenuation level with fluctuations due to scattering. The phase velocities agree with Biot's porous medium theory. The RMSE is 16.0 m/s (1%) at n = 1.5. Below and above this value, the RMSE decreases mildly and rises sharply, respectively.     

Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone

Previous studies have shown that interference between fast waves and slow waves can lead to observed negative dispersion in cancellous bone. In this study, the effects of overlapping fast and slow waves on measurements of the apparent attenuation as a function of propagation distance are investigated along with methods of analysis used to determine the attenuation properties. Two methods are applied to simulated data that were generated based on experimentally acquired signals taken from a bovine specimen. The first method uses a time-domain approach that was dictated by constraints imposed by the partial overlap of fast and slow waves. The second method uses a frequency-domain log-spectral subtraction technique on the separated fast and slow waves. Applying the time-domain analysis to the broadband data yields apparent attenuation behavior that is larger in the early stages of propagation and decreases as the wave travels deeper. In contrast, performing frequency-domain analysis on the separated fast waves and slow waves results in attenuation coefficients that are independent of propagation distance. Results suggest that features arising from the analysis of overlapping two-mode data may represent an alternate explanation for the previously reported apparent dependence on propagation distance of the attenuation coefficient of cancellous bone.     

Inverse problems in cancellous bone: estimation of the ultrasonic properties of fast and slow waves using Bayesian probability theory

Quantitative ultrasonic characterization of cancellous bone can be complicated by artifacts introduced by analyzing acquired data consisting of two propagating waves (a fast wave and a slow wave) as if only one wave were present. Recovering the ultrasonic properties of overlapping fast and slow waves could therefore lead to enhancement of bone quality assessment. The current study uses Bayesian probability theory to estimate phase velocity and normalized broadband ultrasonic attenuation (nBUA) parameters in a model of fast and slow wave propagation. Calculations are carried out using Markov chain Monte Carlo with simulated annealing to approximate the marginal posterior probability densities for parameters in the model. The technique is applied to simulated data, to data acquired on two phantoms capable of generating two waves in acquired signals, and to data acquired on a human femur condyle specimen. The models are in good agreement with both the simulated and experimental data, and the values of the estimated ultrasonic parameters fall within expected ranges.     

A multiscale poromicromechanical approach to wave propagation and attenuation in bone

Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density–elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcy’s law for fluid flow and micro–poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.     

Numerical and experimental study on the wave attenuation in bone--FDTD simulation of ultrasound propagation in cancellous bone

In cancellous bone, longitudinal waves often separate into fast and slow waves depending on the alignment of bone trabeculae in the propagation path. This interesting phenomenon becomes an effective tool for the diagnosis of osteoporosis because wave propagation behavior depends on the bone structure. Since the fast wave mainly propagates in trabeculae, this wave is considered to reflect the structure of trabeculae. For a new diagnosis method using the information of this fast wave, therefore, it is necessary to understand the generation mechanism and propagation behavior precisely. In this study, the generation process of fast wave was examined by numerical simulations using elastic finite-difference time-domain (FDTD) method and experimental measurements. As simulation models, three-dimensional X-ray computer tomography (CT) data of actual bone samples were used. Simulation and experimental results showed that the attenuation of fast wave was always higher in the early state of propagation, and they gradually decreased as the wave propagated in bone. This phenomenon is supposed to come from the complicated propagating paths of fast waves in cancellous bone.     

Influence of cancellous bone microstructure on two ultrasonic wave propagations in bovine femur: an in vitro study

The influence of cancellous bone microstructure on the ultrasonic wave propagation of fast and slow waves was experimentally investigated. Four spherical cancellous bone specimens extracted from two bovine femora were prepared for the estimation of acoustical and structural anisotropies of cancellous bone. In vitro measurements were performed using a PVDF transducer (excited by a single sinusoidal wave at 1 MHz) by rotating the spherical specimens. In addition, the mean intercept length (MIL) and bone volume fraction (BV/TV) were estimated by X-ray micro-computed tomography. Separation of the fast and slow waves was clearly observed in two specimens. The fast wave speed was strongly dependent on the wave propagation direction, with the maximum speed along the main trabecular direction. The fast wave speed increased with the MIL. The slow wave speed, however, was almost constant. The fast wave speeds were statistically higher, and their amplitudes were statistically lower in the case of wave separation than in that of wave overlap.     

Negative dispersion in bone: the role of interference in measurements of the apparent phase velocity of two temporally overlapping signals

In this study the attenuation coefficient and dispersion (frequency dependence of phase velocity) are measured using a phase sensitive (piezoelectric) receiver in a phantom in which two temporally overlapping signals are detected, analogous to the fast and slow waves typically found in measurements of cancellous bone. The phantom consisted of a flat and parallel Plexiglas plate into which a step discontinuity was milled. The phase velocity and attenuation coefficient of the plate were measured using both broadband and narrowband data and were calculated using standard magnitude and phase spectroscopy techniques. The observed frequency dependence of the phase velocity and attenuation coefficient exhibit significant changes in their frequency dependences as the interrogating ultrasonic field is translated across the step discontinuity of the plate. Negative dispersion is observed at specific spatial locations of the plate at which the attenuation coefficient rises linearly with frequency, a behavior analogous to that of bone measurements reported in the literature. For all sites investigated, broadband and narrowband data (3-7 MHz) demonstrate excellent consistency. Evidence suggests that the interference between the two signals simultaneously reaching the phase sensitive piezoelectric receiver is responsible for this negative dispersion.     

In vitro estimation of fast and slow wave parameters of thin trabecular bone using space-alternating generalized expectation-maximization algorithm

In testing cancellous bone using ultrasound, two types of longitudinal Biot’s waves are observed in the received signal. These are known as fast and slow waves and their appearance depend on the alignment of bone trabeculae in the propagation path and the thickness of the specimen under test (SUT). They can be used as an effective tool for the diagnosis of osteoporosis because wave propagation behavior depends on the bone structure. However, the identification of these waves in the received signal can be difficult to achieve.In this study, ultrasonic wave propagation in a 4 mm thick bovine cancellous bone in the direction parallel to the trabecular alignment is considered. The observed Biot’s fast and slow longitudinal waves are superimposed; which makes it difficult to extract any information from the received signal. These two waves can be separated using the space alternating generalized expectation maximization (SAGE) algorithm. The latter has been used mainly in speech processing.In this new approach, parameters such as, arrival time, center frequency, bandwidth, amplitude, phase and velocity of each wave are estimated. The B-Scan images and its associated A-scans obtained through simulations using Biot’s finite-difference time-domain (FDTD) method are validated experimentally using a thin bone sample obtained from the femoral-head of a 30 months old bovine.     

Predictions of the modified Biot-Attenborough model for the dependence of phase velocity on porosity in cancellous bone

The modified Biot–Attenborough (MBA) model for acoustic wave propagation in porous media has been found useful to predict wave properties in cancellous bone. The present study is aimed at applying the MBA model to predict the dependence of phase velocity on porosity in cancellous bone. The MBA model predicts a phase velocity that decreases nonlinearly with porosity. The optimum values for input parameters of the MBA model, such as compressional speed c_m of solid bone and phase velocity parameter s₂, were determined by comparing the predictions with previously published measurements in human calcaneus and bovine cancellous bone. The value of the phase velocity parameter s₂ = 1.23 was obtained by curve fitting to the experimental data for 53 human calcaneus samples only, assuming a compressional speed c_m = 2500 m/s of solid bone. The root-mean-square error (RMSE) of the curve fit was 15.3 m/s. The optimized value of s₂ for all 75 cancellous bone samples including 22 bovine samples was 1.42 with a value of 55 m/s for the RMSE of the curve fit. The latter fit was obtained by using of a value of c_m = 3200 m/s. Although the MBA model relies on the empirical parameters determined from experimental data, it is expected that the model can be usefully employed as a practical tool in the field of clinical ultrasonic bone assessment.     

Ultrasonic slow waves in air-saturated cancellous bone

This paper describes preliminary observations of ultrasonic wave propagation in air-saturated defatted cancellous bone from the human vertebra. Using a broadband pulse transmission system, attenuation and phase velocity were measured over a wide frequency range (100 kHz-1 MHz). The observed behaviour was consistent with that expected for the decoupled slow wave predicted by Biot's theory. Velocity was lower than that of free air, and there was marked frequency-dependent attenuation and velocity dispersion. The tortuosity (alpha) of the trabecular microstructure was estimated from the high frequency limit of the dispersion curve, with a mean value of alpha = 1.040 +/- 0.004 obtained in five specimens. Ultrasonic measurements in air represent a valuable new approach, capable of yielding parameters that directly characterise bone structure. Furthermore, they may give useful insights into wave propagation in bone in vivo, where the trabecular framework is saturated with marrow fat rather than air.     

Ultrasonic wave propagation in human cancellous bone: application of Biot theory

Ultrasonic wave propagation in human cancellous bone is considered. Reflection and transmission coefficients are derived for a slab of cancellous bone having an elastic frame using Biot's theory modified by the model of Johnson et al. [J. Fluid Mech. 176, 379-402 (1987)] for viscous exchange between fluid and structure. Numerical simulations of transmitted waves in the time domain are worked out by varying the modified Biot parameters. The variation is applied to the governing parameters and is about 20%. From this study, we can gain an insight into the sensitivity of each physical parameter used in this theory. Some parameters play an important role in slow-wave wave form, such as the viscous characteristic length lambda and pore fluid bulk modulus Kf. However, other parameters play an important role in the fast-wave wave form, such as solid density rhos and shear modulus N. We also note from these simulations that some parameters such as porosity phi, tortuosity alpha(infinty), thickness, solid bulk modulus Ks, and skeletal compressibility frame Kb, play an important role simultaneously in both fast and slow wave forms compared to other parameters which act on the wave form of just one of the two waves. The sensitivity of the modified Biot parameters with respect to the transmitted wave depends strongly on the coupling between the solid and fluid phases of the cancellous bone. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.     

Propagation of two longitudinal waves in a cancellous bone with the closed pore boundary

Ultrasound propagation in cancellous bone (porous media) under the condition of closed pore boundaries was investigated. A cancellous bone and two plate-like cortical bones obtained from a racehorse were prepared. A water-immersion ultrasound technique in the MHz range and a three-dimensional elastic finite-difference time-domain (FDTD) method were used to investigate the waves. The experiments and simulations showed a clear separation of the incident longitudinal wave into fast and slow waves. The findings advance the evaluation of bones based on the two-wave phenomenon for in vivo assessment.     

Elastic waves in non-Newtonian (Maxwell) fluid-saturated porous media

This paper studies the elastic waves in non-Newtonian (Maxwell) fluid-saturated porous media with the nonzero boundary slip velocity for pore size distribution. The coefficient bF_m(ω) that measures the deviation from Poiseuille flow friction in such media is presented. Based on this coefficient, we investigate the properties of elastic waves by calculating their phase velocities and attenuation coefficients as functions of frequency and the behaviour of the dynamic permeability. The study shows that the pore size distribution removes oscillations in all physical quantities in the non-Newtonian regime. Consideration of the nonzero boundary slip effect in non-Newtonian (Maxwell) fluid-saturated porous media results in (a) an overall increase of the dynamic permeability, (b) an increase of phase velocities of fast Biot waves and shear waves except in the low frequency domain and an overall increase of phase velocity of slow Biot waves and (c) an overall increase of the attenuation of three Biot waves in the intermediate frequency domain except in the deeply non-Newtonian regime. The study also shows that the attenuation coefficient of slow Biot waves is small in the deeply non-Newtonian regime at higher frequency, which encourages us to detect slow Biot waves in oil-saturated porous rock.     

Estimation of critical and viscous frequencies for Biot theory in cancellous bone

The use of Biot theory for modelling ultrasonic wave propagation in porous media involves the definition of a "critical frequency" above which both fast and slow compressional waves will, in principle, propagate. Critical frequencies have been evaluated for healthy and osteoporotic cancellous bone filled with water or marrow, using data from the literature. The range of pore sizes in bone gives rise to a critical frequency band rather than a single critical frequency, the mean of which is lower for osteoporotic bone than normal bone. However, the critical frequency is a theoretical concept and previous researchers considered a more realistic "viscous frequency" above which both fast and slow waves may be experimentally observed. Viscous frequencies in bone are found to be several orders of magnitude greater than calculated critical frequencies. Whereas two waves may well be observed at all ultrasonic frequencies for water-filled cancellous bone at 20 degrees C, it is probable megahertz frequencies would be needed for observation of two waves in vivo.     

Investigation of an anisotropic tortuosity in a biot model of ultrasonic propagation in cancellous bone

The modeling of ultrasonic propagation in cancellous bone is relevant to the study of clinical bone assessment. Historical experiments revealed the importance of both the viscous effects of bone marrow and the anisotropy of the porous microstructure. Of those propagation models previously applied to cancellous bone, Biot's theory incorporates viscosity, but has only been applied in isotropic form, while Schoenberg's anisotropic model does not include viscosity. In this paper we present an approach that incorporates the merits of both models, by utilizing the tortuosity, a key parameter describing pore architecture. An angle-dependent tortuosity for a layered structure is used in Biot's theory to generate the "Stratified Biot Model" for cancellous bone, which is compared with published bone data. While the Stratified Biot model was inferior to Schoenberg's model for slow wave velocity prediction, the proposed model improved agreement fast wave velocity at high propagation angles, particularly when sorted for porosity. An attempt was made to improve the fast wave agreement at low angles by introducing an angle-dependent Young's Modulus, which, while improving the agreement of predicted fast wave velocity at low angles, degraded agreement at high angles. In this paper the utility of the tortuosity in characterizing the architecture of cancellous bone is highlighted.     

Phase and group velocities of fast and slow compressional waves in trabecular bone

This Letter is an extension to a multilayer model of porous bone first proposed by Hughes et al. [Ultrasound Med. Biol. 25, 811-821 (1999)]. Both slow and fast compressional waves propagate when the acoustic wave propagation is parallel to the trabecular alignment. However, a slow wave disappears at high refraction angles. To explain this phenomenon, the multilayer model is extended to compute group velocity surface and arrival times with an angle. Two major effects are highlighted as the refraction angle increases. First, the energy of the slow wave is refracted from the phase propagation direction. Second, the signals of fast and slow waves overlap. As a consequence, the slow wave may not be observed for a refraction angle greater than 40 degrees, which is in agreement with previous experimental data published by Hughes et al. and others.     

Dispersion of interface waves in sediments with power-law shear speed profiles. I. Exact and approximate analytical results.

In the upper tens of meters of ocean bottom, unconsolidated marine sediments consisting of clay, silt, or fine sand with high porosity are "almost incompressible" in the sense that the shear wave velocity is much smaller than the compressional wave velocity. The shear velocity has very large gradients close to the ocean floor leading to strong coupling of compressional and shear waves in such "soft" sediments. The weak compressibility opens an avenue for developing a theory of elastic wave propagation in continuously stratified soft sediments that fully accounts for the coupling. Elastic waves in soft sediments consist of "fast" waves propagating with velocities close to the compressional velocity and "slow" waves propagating with velocities on the order of the shear velocity. For the slow waves, the theory predicts the existence of surface waves at the ocean-sediment boundary. In the important special case of the power-law depth-dependence of shear rigidity, phase and group velocities of the interface waves are shown to scale as a certain power of frequency. An explicit, exact solution was obtained for the surface waves in sediments characterized by constant density and a linear increase of shear rigidity with depth, that is, for the case of shear speed proportional to the square root of the depth below the sediment-water interface. Asymptotic and perturbation techniques were used to extend the result to more general environments. Theoretical dispersion relations agreed well with numerical simulations and available experimental data and, as demonstrated in a companion paper [D. M. F. Chapman and O. A. Godin, J. Acoust. Soc. Am 110, 1908 (2001)] led to a simple and robust inversion of interface wave travel times for shear velocity profiles in the sediment.     

Ultrasonic pulse waves in cancellous bone analyzed by finite-difference time-domain methods

The trabecular frame of cancellous bone has a high degree of porosity, anisotropy and inhomogeneity. The propagation of ultrasonic waves in cancellous bone is significantly affected by the trabecular structure. In this paper, two two-dimensional finite-difference time-domain (FDTD) methods, which were the popular viscoelastic FDTD method for a viscoelastic medium and Biot's FDTD method for a fluid-saturated porous medium, have been applied to numerically analyze the ultrasonic pulse waves propagating through bovine cancellous bone in the directions parallel and perpendicular to the trabecular alignment. The Biot's fast and slow longitudinal waves, which were identified in previous experiments for the propagation parallel to the trabecular orientation, could be analyzed using Biot's FDTD method rather than the viscoelastic FDTD method. For the single wave propagation in the perpendicular direction, on the other hand, the viscoelastic FDTD result was found to be in more good agreement with the experimental result.     

Fabric dependence of quasi-waves in anisotropic porous media

Assessment of bone loss and osteoporosis by ultrasound systems is based on the speed of sound and broadband ultrasound attenuation of a single wave. However, the existence of a second wave in cancellous bone has been reported and its existence is an unequivocal signature of poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as bone mineral density (BMD), a fabric-dependent anisotropic poroelastic wave propagation theory was recently developed for pure wave modes propagating along a plane of symmetry in an anisotropic medium. Key to this development was the inclusion of the fabric tensor--a quantitative stereological measure of the degree of structural anisotropy of bone--into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of mixed wave modes along an arbitrary direction in anisotropic porous media called quasi-waves. It was found that differences between phase and group velocities are due to the anisotropy of the bone microarchitecture, and that the experimental wave velocities are more accurately predicted by the poroelastic model when the fabric tensor variable is taken into account. This poroelastic wave propagation theory represents an alternative for bone quality assessment beyond BMD.     

Flexural wave motion in beam-type disordered periodic systems: Coincidence phenomenon and sound radiation

This paper deals with flexural wave motion in uniform beam-type periodic systems whose repeating units are identical finite beams with multiple beam-length disorders. A general expression derived for the propagation constants has been employed to study its variation with frequency for a beam system having 4-span disordered repeating units. This is helpful in understanding flexural wave motion in disordered periodic beams. Free flexural waves have been studied as wave groups consisting of a large number of harmonic components of different wavelengths, phase velocities and directions. Phase velocities have been computed and plotted for different frequencies in the propagation zones in which the free waves progress without attenuation. This has been found to be useful in understanding and predicting the coincidence phenomenon in disordered periodic beams under convected pressure field loading. The excitation of wave groups in disordered periodic beam-type systems by a slow (subsonic) convecting pressure field can include fast (supersonic) moving flexural wave components which can radiate sound. It has been pointed out that sound radiation from a disordered periodic beam (or plate) can be quite different as compared to that from a periodic beam under similar convected pressure field loading.