Numerical simulation of wave propagation in cancellous bone

Numerical simulation of wave propagation is performed through 31 3D volumes of trabecular bone. These volumes were reconstructed from high synchrotron microtomography experiments and are used as the input geometry in a simulation software developed in our laboratory. The simulation algorithm accounts for propagation into both the saturating fluid and bone but absorption is not taken into account. We show that 3D simulation predicts phenomena observed experimentally in trabecular bones : linear frequency dependence of attenuation, increase of attenuation and speed of sound with the bone volume fraction, negative phase velocity dispersion in most of the specimens, propagation of fast and slow wave depending on the orientation of the trabecular network compared to the direction of propagation of the ultrasound. Moreover, the predicted attenuation is in very close agreement with the experimental one measured on the same specimens. Coupling numerical simulation with real bone architecture therefore provides a powerful tool to investigate the physics of ultrasound propagation in trabecular structures.     

基于弹性模量检测骨疲劳的超声导波方法研究

研究早期诊断骨疲劳的方法是当前骨质评价方面的研究热点之一. 本文对不同弹性模量下长骨中超声导波的传播特性进行了理论分析和仿真研究.首先, 通过数值计算得到导波在管状长骨中的理论解析解.然后对管状长骨进行了时域有限差分(FDTD) 仿真, 并验证了它与理论解析解的一致性, 同时得到长骨中不同模式导波群速度、 中心频率和衰减与弹性模量的关系.研究结果表明各个导波模式的群速度和中心频率均随弹性模量的增加而增加, 而衰减随弹性模量的增加而减小.说明超声导波的传播特性参量可以反映长骨弹性模量的变化, 从而为长骨的早期疲劳诊断提供理论依据.     

曲线坐标系下孔隙介质圆柱纵向表面波的传播特性

为了研究孔隙介质圆柱纵向表面波的传播规律,分析其频散和衰减特性,在正交曲线坐标系下建立了表面波的频散方程,通过数值计算得到频散曲线,将纵向导波最低模态与表面波进行对比,并分析了曲率半径及孔隙参数对表面波频散和衰减的影响。结果表明,当频率足够大时,导波最低模态的频散曲线与表面波近似;在同一频率下,表面波的相速度随曲率半径的增大而增大,随孔隙度的增大而减小;表面波的衰减随孔隙度的增大而增大。研究结果为开展孔隙介质圆柱结构的超声无损评价提供了一定的理论参考。     

Attenuation in trabecular bone: A comparison between numerical simulation and experimental results in human femur

Numerical simulations (finite-difference time domain) are compared to experimental results of ultrasound wave propagation through human trabecular bones. Three-dimensional high-resolution microcomputed tomography reconstructions served as input geometry for the simulation. The numerical simulation took into account scattering, but not absorption. Simulated and experimental values of the attenuation coefficients (alpha, dB/cm) and the normalized broadband ultrasound attenuation (nBUA, dB/cm/MHz) were measured and compared on a set of 28 samples. While experimental and simulated nBUA values were highly correlated (R(2)=0.83), and showed a similar dependence with bone volume fraction, the simulation correctly predicted experimental nBUA values only for low bone volume fraction (BV/TV). Attenuation coefficients were underestimated by the simulation. The absolute difference between experimental and simulated alpha values increased with both BV/TV and frequency. As a function of frequency, the relative difference between experimental and simulated alpha values decreased from 60% around 400 kHz to 30% around 1.2 MHz. Under the assumption that the observed discrepancy expresses the effect of the absorption, our results suggests that nBUA and its dependence on BV/TV can be mostly explained by scattering, and that the relative contribution of scattering to alpha increases with frequency, becoming predominant (>50 %) over absorption for frequencies above 600 kHz.     

利用井中偶极弯曲波反演渗透率的影响因素及灵敏度分析

本文以各向同性饱和流体孔隙介质充液井孔中声场传播理论为基础,系统考察了影响偶极弯曲波特征的主要因素,分析了快慢地层条件下弯曲波衰减和速度对不同参数的敏感度,对比结果表明,弯曲波的速度基本不受地层渗透率的影响,其衰减对渗透率的敏感性要远高于速度。弯曲波中心频率会随着波的衰减而发生偏移,说明利用弯曲波衰减和频移特征联合反演地层渗透率是可行的。通过对比快慢地层下的弯曲波衰减和速度灵敏度曲线,发现在慢速地层中反演渗透率的结果要好于快速地层。同时,本文也指出在低孔低渗储层中,还可以考虑利用弯曲波衰减识别气层。     

Generalized thermoelastic extensional and flexural wave motions in homogenous isotropic plate by using asymptotic method

In this paper the asymptotic method has been applied to investigate propagation of generalized thermoelastic waves in an infinite homogenous isotropic plate. The governing equations for the extensional, transversal and flexural motions are derived from the system of three-dimensional dynamical equations of linear theories of generalized thermoelasticity. The asymptotic operator plate model for extensional and flexural free vibrations in a homogenous thermoelastic plate leads to sixth and fifth degree polynomial secular equations, respectively. These secular equations govern frequency and phase velocity of various possible modes of wave propagation at all wavelengths. The velocity dispersion equations for extensional and flexural wave motion are deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate. The approximation for long and short waves along with expression for group velocity has also been obtained. The Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations and its equivalence established with that of asymptotic method. The numeric values for phase velocity, group velocity and attenuation coefficients has also been obtained using MATHCAD software and are shown graphically for extensional and flexural waves in generalized theories of thermoelastic plate for solid helium material.     

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.     

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Electromagnetic wave propagation in a circular waveguide with an axially-placed discharge tube is discussed. Fundamental properties of slow and fast waves are derived from the dispersion formula and a simple oscilloscopic method for the identification of these two waves is demonstrated. The electron density is obtained from the measured phase velocity by numerical calculation of dispersion equation (for real plasma permittivity). Collision frequency is then found from the attenuation factor.     

Dispersion characteristics of surface magnetostatic waves with dissipation

It is shown that unlike undamped waves, the dispersion characteristics of spin surface waves with dissipation have a maximum wave number at which there is a downward reversal in the dispersion curve of a wave number. This forms the upper branch of a dispersion curve with inverse dispersion and high attenuation, leading to an unclear frequency dependence of the wave vector. The lower primary dispersion branch corresponds to waves with forward dispersion, and attenuation is proportional to the small parameter of dissipation. However, the coefficient of wave attenuation grows sharply near the maximum wave number. Some angular and frequency limits of surface wave propagation change as well.     

Anisotropy of ultrasonic backscatter and attenuation from human calcaneus: implications for relative roles of absorption and scattering in determining attenuation

Although bone sonometry has been demonstrated to be useful in the diagnosis of osteoporosis, much remains to be learned about the processes governing the interactions between ultrasound and bone. In order to investigate these processes, ultrasonic attenuation and backscatter in two orientations were measured in 43 human calcaneal specimens in vitro at 500 kHz. In the mediolateral (ML) orientation, the ultrasound propagation direction is approximately perpendicular to the trabecular axes. In the anteroposterior (AP) orientation, a wide range of angles between the ultrasound propagation direction and trabecular axes is encountered. Average attenuation slope was 18% greater while average backscatter coefficient was 50% lower in the AP orientation compared with the ML orientation. Backscatter coefficient in both orientations approximately conformed to a cubic dependence on frequency, consistent with a previously reported model. These results support the idea that absorption is a greater component of attenuation than scattering in human calcaneal trabecular bone.     

On the velocity of propagation of a frequency-modulated wave packet in a dispersive absorbing medium

It is shown that the velocity of propagation of a frequency-modulated wave packet through a strongly dispersive absorbing medium can be significantly different from (either higher or lower than) that of a non-frequency-modulated wave packet. This difference is attributed to the absorption dispersion of the medium. The easiest way to take the absorption dispersion into account is to use the formalism of the complex group velocity of a wave packet. This paper considers the propagation of a linear frequency-modulated wave packet, whose carrier frequency is close to the frequency of a spectral absorption line of the medium.     

Scattering by a fluid cylinder in a porous medium: application to trabecular bone

In a trabecular bone, considered as a nondissipative porous medium, the scattering of an incident wave by cylindrical pores larger than the wavelength is studied. The goal is to know if scattering alone may cause such a high attenuation as that observed in calcaneus. The porous medium is modelized via Biot's theory and the scattering by a single pore is characterized from the definition of a scattering matrix. An approximation of weakly disordered medium is then discussed to estimate the effective attenuation and dispersion as a function of frequency. These effective properties are shown to be different of those measured on calcaneus, due to the neglect of wave conversions during the scattering process.     

Multiple scattering in a trabecular bone: influence of the marrow viscosity on the effective properties

The Foldy and the Waterman and Truell approximations are used to determine the effective properties of the coherent wave that emerges after multiple scattering of a plane longitudinal fast wave by the largest pores in a trabecular bone. The unit scattering cell considered is either a single pore or two close cylindrical pores (cluster), at a fixed overall bone porosity. In the cluster case, the effective attenuation is about twice that obtained with one single pore per scatterer. It is shown that taking into account the marrow viscosity leads only to minor differences on the effective dispersion and attenuation.     

Precise measurements of bulk-wave ultrasonic velocity dispersion and attenuation in solid materials in the VHF range

A general method was established for precisely measuring velocity dispersion and attenuation in solid specimens with acoustic losses in the very high frequency (VHF) range, using the complex-mode measurement method and the diffraction correction method. Experimental procedures were presented for implementing such a method and demonstrated this measurement method in the frequency range of 50-230 MHz, using borosilicate glass (C-7740) as a dispersive specimen and synthetic silica glass (C-7980) as a nondispersive standard specimen. C-7980 exhibited no velocity dispersion; velocity was constant at 5929.14 +/- 0.03 m/s. C-7740 exhibited velocity dispersion, from 5542.27 m/s at 50 MHz to 5544.47 m/s at 230 MHz with an increase of about 2 m/s in the measured frequency range. When frequency dependence of attenuation was expressed as alpha = alpha(0)f(beta), the results were as follows: alpha0 = 1.07 x 10(-16) s2/m and beta = 2 for C-7980 and alpha0 = 5.16 x 10(-9) s(1.25)/m and beta = 1.25 for C-7740.     

Causality and the velocity of acoustic signals in bubbly liquids

Several versions of the dispersion formula governing the acoustic propagation in bubbly liquids are shown to exhibit acausal behavior. The cause of this behavior is due to the inappropriate application of a low frequency approximation in the determination of the extinction of the signal from radiative scattering. Using a corrected causal formula, several principles of wave propagation in bubbly media consistent with the general theory of wave propagation in dispersive media are demonstrated: There exist two precursors to any finite signal. Both propagate without regard to the source characteristics at velocities, frequencies, and amplitudes dependent wholly upon the characteristics of the medium supporting the wave motion. The first travels at the infinite frequency phase velocity that is coincident with the infinite frequency limit of the group velocity. That part of a propagating wave oscillating at the source frequency arrives at a time determined by the signal velocity. Analogous to the well known signal velocity of electromagnetic wave propagation in conducting media, the value of the signal velocity depends on the detailed structure of the dispersion formula in the complex frequency plane.     

Phase velocities and attenuations of shear, Lamb, and Rayleigh waves in plate-like tissues submerged in a fluid (L)

In the past several decades, the fields of ultrasound and magnetic resonance elastography have shown promising results in noninvasive estimates of mechanical properties of soft tissues. These techniques often rely on measuring shear wave velocity due to an external or internal source of force and relating the velocity to viscoelasticity of the tissue. The mathematical relationship between the measured velocity and material properties of the myocardial wall, arteries, and other organs with non-negligible boundary conditions is often complicated and computationally expensive. A simple relationship between the Lamb-Rayleigh dispersion and the shear wave dispersion is derived for both the velocity and attenuation. The relationship shows that the shear wave velocity is around 20% higher than the Lamb-Rayleigh velocity and that the shear wave attenuation is about 20% lower than the Lamb-Rayleigh attenuation. Results of numerical simulations in the frequency range 0-500 Hz are presented.     

无限流体中孔隙介质圆柱周向导波的传播特性

为研究无限大流体约束的孔隙圆柱中周向导波的传播规律,分析孔隙参数对导波传播特性的影响,建立了无限流体中孔隙介质圆柱的理论模型,利用孔隙介质弹性波动理论,建立了周向导波频散方程,通过数值模拟计算得到无限流体中孔隙介质圆柱的频散曲线,探讨了圆柱半径和孔隙参数对导波传播特性的影响,并对导波的衰减特性进行了分析;通过数值计算,得到了周向导波的时域波形,讨论了孔隙参数对波形的影响.结果表明,孔隙介质圆柱半径的改变影响圆柱尺度,孔隙度的改变影响孔隙介质中体声波的波速,都对周向导波频散曲线产生一定的影响,所得到的频散曲线特征及衰减曲线与时域波形吻合.研究结果对开展无限流体中孔隙介质圆柱的超声无损评价提供了一定的理论参考.     

Attenuation characteristics of the fundamental modes that propagate in buried iron water pipes

The attenuation of the fundamental non-torsional modes that propagate down buried iron water pipes has been studied. The mode shapes, mode attenuation due to leakage into the surrounding medium and the scattering of the modes as they interact with pipe joints and fittings have been investigated. In the low frequency region the mode predicted to dominate over significant propagation distances approximates a plane wave in the water within the pipe. The established acoustic technique used to locate leaks in buried iron water pipes assumes that leak noise propagates as a single non-dispersive mode at a velocity related to the low frequency asymptote of this water borne mode. Experiments have been conducted on buried water mains at test sites in the UK to verify the attenuation and velocity dispersion predictions.     

The nonlinear propagation of acoustic waves in a viscoelastic medium containing cylindrical micropores

Based on an equivalent medium approach, this paper presents a model describing the nonlinear propagation of acoustic waves in a viscoelastic medium containing cylindrical micropores. The influences of pores' nonlinear oscillations on sound attenuation, sound dispersion and an equivalent acoustic nonlinearity parameter are discussed. The calculated results show that the attenuation increases with an increasing volume fraction of micropores. The peak of sound velocity and attenuation occurs at the resonant frequency of the micropores while the peak of the equivalent acoustic nonlinearity parameter occurs at the half of the resonant frequency of the micropores. Furthermore, multiple scattering has been taken into account, which leads to a modification to the effective wave number in the equivalent medium approach. We find that these linear and nonlinear acoustic parameters need to be corrected when the volume fraction of micropores is larger than 0.1%.