The finite element method for micro-scale modeling of ultrasound propagation in cancellous bone

Quantitative ultrasound for bone assessment is based on the correlations between ultrasonic parameters and the properties (mechanical and physical) of cancellous bone. To elucidate the correlations, understanding the physics of ultrasound in cancellous bone is demanded. Micro-scale modeling of ultrasound propagation in cancellous bone using the finite-difference time-domain (FDTD) method has been so far utilized as one of the approaches in this regard. However, the FDTD method accompanies two disadvantages: staircase sampling of cancellous bone by finite difference grids leads to generation of wave artifacts at the solid–fluid interface inside the bone; additionally, this method cannot explicitly satisfy the needed perfect-slip conditions at the interface. To overcome these disadvantages, the finite element method (FEM) is proposed in this study. Three-dimensional finite element models of six water-saturated cancellous bone samples with different bone volume were created. The values of speed of sound (SOS) and broadband ultrasound attenuation (BUA) were calculated through the finite element simulations of ultrasound propagation in each sample. Comparing the results with other experimental and simulation studies demonstrated the capabilities of the FEM for micro-scale modeling of ultrasound in water-saturated cancellous bone.     

The effect of blood acceleration on the ultrasound power Doppler spectrum

The purpose of the present work was to study the influence of blood acceleration and time window length on the power Doppler spectrum for Gaussian ultrasound beams. The work has been carried out on the basis of continuum model of the ultrasound scattering from inhomogeneities in fluid flow. Correlation function of fluctuations has been considered for uniformly accelerated scatterers, and the resulting power Doppler spectra have been calculated. It is shown that within the initial phase of systole uniformly accelerated slow blood flow in pulmonary artery and aorta tends to make the correlation function about 4.89 and 7.83 times wider, respectively, than the sensitivity function of typical probing system. Given peak flow velocities, the sensitivity function becomes, vice versa, about 4.34 and 3.84 times wider, respectively, then the correlation function. In these limiting cases, the resulting spectra can be considered as Gaussian. The optimal time window duration decreases with increasing acceleration of blood flow and equals to 11.62 and 7.54 ms for pulmonary artery and aorta, respectively. The width of the resulting power Doppler spectrum is shown to be defined mostly by the wave vector of the incident field, the duration of signal and the acceleration of scatterers in the case of low flow velocities. In the opposite case geometrical properties of probing field and the average velocity itself are more essential. In the sense of signal–noise ratio, the optimal duration of time window can be found. Abovementioned results may contribute to the improved techniques of Doppler ultrasound diagnostics of cardiovascular system.     

Time reversal of speckle noise

Focusing a wave in an unknown inhomogeneous medium is an open problem in wave physics. This work presents an iterative method able to focus in pulse-echo mode in an inhomogeneous medium containing a random distribution of scatterers. By performing a coherent summation of the random echoes backscattered from a set of points surrounding the desired focus, a virtual bright pointlike reflector is generated. A time-reversal method enables an iterative convergence towards the optimal wave field focusing at the location of this virtual scatterer. Thanks to this iterative time-reversal process, it is possible to focus at any arbitrary point in the heterogeneous medium even in the absence of pointlike source. An experimental demonstration is given for the correction of strongly distorted images in the field of medical ultrasound imaging. This concept enables envisioning many other applications in wave physics.     

Wigner distribution of a transducer beam pattern within a multiple scattering formalism for heterogeneous solids

Diffuse ultrasonic backscatter measurements have been especially useful for extracting microstructural information and for detecting flaws in materials. Accurate interpretation of experimental data requires robust scattering models. Quantitative ultrasonic scattering models include components of transducer beam patterns as well as microstructural scattering information. Here, the Wigner distribution is used in conjunction with the stochastic wave equation to model this scattering problem. The Wigner distribution represents a distribution in space and time of spectral energy density as a function of wave vector and frequency. The scattered response is derived within the context of the Wigner distribution of the beam pattern of a Gaussian transducer. The source and receiver distributions are included in the analysis in a rigorous fashion. The resulting scattered response is then simplified in the single-scattering limit typical of many diffuse backscatter experiments. Such experiments, usually done using a modified pulse-echo technique, utilize the variance of the signals in space as the primary measure of microstructure. The derivation presented forms a rigorous foundation for the multiple scattering process associated with ultrasonic experiments in heterogeneous media. These results are anticipated to be relevant to ultrasonic nondestructive evaluation of polycrystalline and other heterogeneous solids.     

Determination of the Disperse Composition of a PbO Suspension Containing Aggregates of Particles of Lamellar Shape by the Laser-Polarimetry Method

The results of measurements of the scattering matrix at a wavelength of 0.63 μm in the range of scattering angles of 10°–155° are presented for an aqueous suspension of lead oxide containing particles of plate form and their aggregates of monomers with dimensions of ~5 nm. The results of the measurements are compared with the results of calculations for axially symmetric scatterers (ellipsoids of rotation, cylinders). It is shown that the presence of aggregates affects the scattering properties of such a medium. The results of reconstructing the distribution of particles of a disperse medium in sizes from the measurements data of the scattering matrix are presented. The reconstruction of the distributions was carried out by solving the problem of optimizing the sum of the squared deviations of the experimental and calculated values of matrix elements in the framework of the model of axially symmetric scatterers. It is shown that the distribution of particles by sizes is more accurately reconstructed by minimizing the sum of the squares of the deviations for the sum of the diagonal elements. The obtained distribution is compared with the distribution measured by the method of dynamic light scattering.     

Optical theorem detectors for active scatterers

We develop a new theory of the optical theorem for scalar fields in nonhomogeneous media which can be bounded or unbounded. It applies to arbitrary lossless backgrounds and quite general probing fields. The derived formulation holds for arbitrary passive scatterers, which can be dissipative, as well as for the more general class of active scatterers which are composed of a (passive) scatterer component and an active, radiating (antenna) component. The generalization of the optical theorem to active scatterers is relevant to many applications such as surveillance of active targets including certain cloaks and invisible scatterers and wireless communications. The derived theoretical framework includes the familiar real power optical theorem describing power extinction due to both dissipation and scattering as well as a novel reactive optical theorem related to the reactive power changes. The developed approach naturally leads to three optical theorem indicators or statistics which can be used to detect changes or targets in unknown complex media. The paper includes numerical simulation results that illustrate the application of the derived optical theorem results to change detection in complex and random media.     

Estimation of ultrasonic attenuation in a bone using coded excitation

This paper describes a novel approach to estimate broadband ultrasound attenuation (BUA) in a bone structure in human in vivo using coded excitation. BUA is an accepted indicator for assessment of osteoporosis. In the tested approach a coded acoustic signal is emitted and then the received echoes are compressed into brief, high amplitude pulses making use of matched filters and correlation receivers. In this way the acoustic peak pressure amplitude probing the tissue can be markedly decreased whereas the average transmitted intensity increases proportionally to the length of the code. This paper examines the properties of three different transmission schemes, based on Barker code, chirp and Golay code. The system designed is capable of generating 16 bits complementary Golay code (CGC), linear frequency modulated (LFM) chirp and 13-bit Barker code (BC) at 0.5 and 1 MHz center frequencies. Both in vivo data acquired from healthy heel bones and in vitro data obtained from human calcaneus were examined and the comparison between the results using coded excitation and two cycles sine burst is presented. It is shown that CGC system allows the effective range of frequencies employed in the measurement of broadband acoustic energy attenuation in the trabecular bone to be doubled in comparison to the standard 0.5 MHz pulse transmission. The algorithm used to calculate the pairs of Golay sequences of the different length, which provide the temporal side-lobe cancellation is also presented. Current efforts are focused on adapting the system developed for operation in pulse-echo mode; this would allow examination and diagnosis of bones with limited access such as hip bone.     

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

This article quantitatively investigates ultrasound propagation in numerical anisotropic porous media with finite-difference simulations in 3D. The propagation media consist of clusters of ellipsoidal scatterers randomly distributed in water, mimicking the anisotropic structure of cancellous bone. Velocities and attenuation coefficients of the ensemble-averaged transmitted wave (also known as the coherent wave) are measured in various configurations. As in real cancellous bone, one or two longitudinal modes emerge, depending on the micro-structure. The results are confronted with two standard theoretical approaches: Biot’s theory, usually invoked in porous media, and the Independent Scattering Approximation (ISA), a classical first-order approach of multiple scattering theory. On the one hand, when only one longitudinal wave is observed, it is found that at porosities higher than 90% the ISA successfully predicts the attenuation coefficient (unlike Biot’s theory), as well as the existence of negative dispersion. On the other hand, the ISA is not well suited to study two-wave propagation, unlike Biot’s model, at least as far as wave speeds are concerned. No free fitting parameters were used for the application of Biot’s theory. Finally we investigate the phase-shift between waves in the fluid and the solid structure, and compare them to Biot’s predictions of in-phase and out-of-phase motions.     

The reflection of ultrasound from partially contacting rough surfaces

Ultrasound is commonly used to detect and size cracks in a range of engineering components. Modeling techniques are well established for smooth and open cracks. However, real cracks are often rough (relative to the ultrasonic wavelength) and closed due to compressive stress. This paper describes an investigation into the combined effects of crack face roughness and closure on ultrasonic detectability. A contact model has been used to estimate the size and shape of scatterers (voids) at the interface of these rough surfaces when loaded. The response of such interfaces to excitation with a longitudinal ultrasonic pulse over a wide range of frequencies has been investigated. The interaction of ultrasound with this scattering interface is predicted using a finite-element model and good agreement with experiments on rough surfaces is shown. Results are shown for arrays of equi-sized scatterers and a distribution of scatterer sizes. It is shown that the response at high frequencies is dependent on the size, shape, and distribution of the scatterers. It is also shown that the finite-element results depart from the mass-spring model predictions when the product of wave number and scatterer half-width is greater than 0.4.     

Ultrasonic properties of random media under uniaxial loading.

Acoustic properties of two types of soft tissue-like media were measured as a function of compressive strain. Samples were subjected to uniaxial strains up to 40% along the axis of the transducer beam. Measurements were analyzed to test a common assumption made when using pulse-echo waveforms to track motion in soft tissues--that local properties of wave propagation and scattering are invariant under deformation. Violations of this assumption have implications for elasticity imaging procedures and could provide new opportunities for identifying the sources of backscatter in biological media such as breast parenchyma. We measured speeds of sound, attenuation coefficients, and echo spectra in compressed phantoms containing randomly positioned scatterers either stiffer or softer than the surrounding gelatin. Only the echo spectra of gel media with soft scatterers varied significantly during compression. Centroids of the echo spectra were found to be shifted to higher frequencies in proportion to the applied strain up to 10%, and increased monotonically up to 40% at a rate depending on the scatterer size. Centroid measurements were accurately modeled by assuming incoherent scattering from oblate spheroids with an eccentricity that increases with strain. While spectral shifts can be accurately modeled, recovery of lost echo coherence does not seem possible. Consequently, spectral variance during compression may ultimately limit the amount of strain that can be applied between two data fields in heterogeneous media such as lipid-filled tissues. It also appears to partially explain why strain images often produce greater echo decorrelation in tissues than in commonly used graphite-gelatin test phantoms.     

基于递归T矩阵的离散随机散射体散射特性研究

本文根据电磁场矢量球波函数多极点展开原理及矢量叠加定理提出了递归T矩阵算法的矢量形式,并且基于矢量递归T矩阵算法建立了多散射球模拟离散随机散射体散射的三维电磁散射模型.通过计算不同尺寸、随机分布散射球的散射以及分析散射球间的高阶散射效应,结果表明:矢量递归T矩阵算法具有很高的计算精度,算法中包含多散射体间的高阶散射效应,因此能够精确计算多散射体总的散射效应.本文所建模型可应用于土壤湿度探测工程中评估地表下掩埋离散随机散射体散射对雷达回波信号产生的影响.     

The scanning acoustic microprobe: I. Analysis and synthesis of a spherically symmetric point spread function

The scanning acoustic microprobe is a novel system which probes and characterizes, from a limited acoustic window, the fine-scale structural features of an object, point-by-point, using a multiplicity of acoustic pulses all aimed and focused at that point. Spherically symmetric, three-dimensional Gaussian pulses are synthesized to measure the backscatter diffraction pattern of the least-resolvable volume of scatterers centered at the point in question. The size and distribution of the scattering volume is forced to be constant, independent of frequency and angle. This method is analytically simple, compared with other pulse-echo techniques, and is applicable to scatterers ranging continuously in size from Rayleigh scatterers to specular reflectors. This is the first of a number of papers describing the development and application of systems based on these concepts. The analytical principles will be described herein for examination of one point at a time. In a companion paper appearing in this issue [F. E. Barber, J. Acoust. Soc. Am. 90, 11-19 (1991)], application to measurement and characterization of a discrete, flat, circular "piston" will be presented. Application to human tissue imaging and tissue characterization will be described in a subsequent third paper. The primary features detected experimentally are the strength of nondirective patterns, and the strength, orientation, and directivity of angle-dependent echo functions associated with planar or layered structures. Fine-scale structural features of a scattering center are obtained either by pattern recognition in k(data) space or inverse Fourier transformation. It is shown that when the bandwidth criteria are met to produce a spherically symmetric point spread function, scattering phenomena are completely described by only two parameters, namely the center frequency of the pulse-echo system and the characteristic diameter of the Gaussian point spread function.     

Ultrasonically determined thickness of long cortical bones: Three-dimensional simulations of in vitro experiments

It was reported in a previous study that simulated guided wave axial transmission velocities on two-dimensional (2D) numerically reproduced geometry of long bones predicted moderately real in vitro ultrasound data on the same bone samples. It was also shown that fitting of ultrasound velocity with simple analytical model yielded a precise estimate (UTh) for true cortical bone thickness. This current study expands the 2D bone model into three dimensions (3D). To this end, wave velocities and UTh were determined from experiments and from time-domain finite-difference simulations of wave propagation, both performed on a collection of 10 human radii (29 measurement sites). A 3D numerical bone model was developed with tuneable fixed material properties and individualized geometry based on X-ray computed tomography reconstructions of real bones. Simulated UTh data were in good accordance (root-mean-square error was 0.40 mm; r(2)=0.79, p<0.001) with true cortical thickness, and hence the measured phase velocity can be well estimated by using a simple analytical inversion model also in 3D. Prediction of in vitro data was improved significantly (by 10% units) and the upgraded bone model thus explained most of the variability (up to 95% when sites were carefully matched) observed in in vitro ultrasound data.     

Measurement of dependence of backscatter coefficient from cylinders on frequency and diameter using focused transducers--with applications in trabecular bone

A theory for the elastic scattering response from a cylinder insonified by a plane wave was previously derived by Faran. In the present paper, the empirical relationship between Faran's theory and measurements of backscatter coefficient from cylindrical targets using focused transducers is investigated. Experimental measurements of dependence of backscatter coefficient on frequency and diameter for nylon wires are reported. It is found that, under certain conditions (including weak, incoherent scattering), backscatter coefficient measurements from collections of cylindrical scatterers may be meaningfully compared with Faran's model predictions. At low frequencies, the theory and experimental measurements exhibit similar dependences on frequency and diameter, provided that the scatterers are not too densely packed. At higher frequencies, the fine structure of Faran's predictions becomes difficult to reproduce experimentally with a focused transducer. Implications regarding applications to characterization of trabecular bone are discussed.     

微纳粒子光学散射分析

为实现利用光学方式对微纳尺度粒子性质的研究,探讨了亚微米线及亚微米球对光电磁波的散射效应.微纳米尺度粒子的光学散射,散射粒子尺寸与入射光波长尺寸可满足米氏(Mie)散射条件.利用Matlab数值模拟的方式,将分析结果以模拟图的形式清晰地展现出来.满足尺寸条件的层状粒子以及任意多个散射粒子存在时对电磁波的散射都可采用Mie散射分析方法,并且针对多粒子散射,分析了散射体位于不同位置时对散射造成的影响.通过分析光学散射光场相关的微分散射截面及近场散射电磁场分布,可得出散射光场随散射角度的变化趋势,以及散射光场受各类因素的影响,包括入射光偏振态、散射粒子尺寸、散射粒子结构及粒子构成层数、散射粒子数量等的影响,也包括一些隐含因素对散射光场的影响,如散射粒子与周围介质的相对折射率.本文的科学意义体现在:与入射光波长尺寸可比的亚微米尺度的粒子,可用作传感器,对于其位移的探测可通过光学方式来实现,而由于粒子本身特性对散射光的影响具有一定的参考价值,从而使通过光学方式对机械位移的读出具有更高准确度.研究结果对于光学方式探测亚微米线机械振动具有指导意义.     

Scattering of longitudinal waves (sound) by defects in fluids. Rough surface

The classical theory of scattering of longitudinal waves (sound) by small inhomogeneities (scatterers) in an ideal fluid is generalized to a distribution of scatterers and such as to include the effect of the inhomogeneities on the elastic properties of the fluid. The results are obtained by a new method of solving the wave equation with spatial restrictions (caused by the presence of the scatterers), which can also be applied to other types of inhomogeneities (like surface roughness, for instance). A coherent forward scattering is identified for a uniform distribution of scatterers (practically equivalent with a mean-field approach), which is due to the fact that our treatment does not include multiple scattering. The reflected wave is obtained for a half-space (semi-infinite fluid) of uniformly distributed scatterers, as well as the field diffracted by a perfect lattice of scatterers. The same method is applied to a (inhomogeneous) rough surface of a semi-infinite ideal fluid. A perturbation-theoretical scheme is devised, with the roughness function as a perturbation parameter, for computing the waves scattered by the surface roughness. The waves scattered by the rough surface are both waves localized (and propagating only) on the surface (two-dimensional waves) and waves reflected back in the fluid. They exhibit directional effects, slowness, attenuation or resonance phenomena, depending on the spatial characteristics of the roughness function. The reflection coefficients and the energy carried on by these waves are calculated both for fixed and free surfaces. In some cases, the surface roughness may generate waves confined to the surface (damped, rough-surface waves).     

Relationships between the anisotropy of longitudinal wave velocity and hydroxyapatite crystallite orientation in bovine cortical bone

Quantitative ultrasound (QUS) is now widely used for evaluating bone in vivo, because obtained ultrasonic wave properties directly reflect the visco-elasticity. Bone tissue is composed of minerals like hydroxyapatite (HAp) and a collagen matrix. HAp crystallites orientation is thus one parameter of bone elasticity. In this study, we experimentally investigated the anisotropy of ultrasonic wave velocity and the HAp crystallites orientation in the axial-radial and axial-tangential planes in detail, using cylindrical specimens obtained from the cortical bone of three bovine femurs. Longitudinal bulk wave propagation was investigated by using a conventional ultrasonic pulse system. We used the one cycle of sinusoidal pulse which was emitted from wide band transmitter. The nominal frequency of the pulse was 1 MHz. First, we investigated the anisotropy of longitudinal wave velocity, measuring the anisotropy of velocity in two planes using cylindrical specimens obtained from identical bone areas. The wave velocity changed due to the rotation angle, showing the maximum value in the direction a little off the bone axis. Moreover, X-ray pole figure measurements also indicated that there were small tilts in the HAp crystallites orientation from the bone axis. The tilt angles were similar to those of the highest velocity direction. There were good correlations between velocity and HAp crystallites orientation obtained in different directions. However, a comparatively low correlation was found in posterior bone areas, which shows the stronger effects of bone microstructure. In the radial-tangential plane, where the HAp crystallites hardly ever align, weak anisotropy of velocity was found which seemed to depend on the bone microstructure.     

随机介质中电磁波空间分布特性研究

采用随机介质统计模型和FDTD方法模拟了随机介质电场空间分布根据光子局域化理论,讨论了电场的空间分布和介电常数涨落的关系,结果表明,在一定的介质密度下,电磁场是局域化的,其空间分布和介质介电常数的涨落密切相关.