Extended three-dimensional impedance map methods for identifying ultrasonic scattering sites

The frequency-dependent ultrasound backscatter from tissues contains information about the microstructure that can be quantified. In many cases, the anatomic microstructure details responsible for ultrasonic scattering remain unidentified. However, their identification would lead to potentially improved methodologies for characterizing tissue and diagnosing disease from ultrasonic backscatter measurements. Recently, three-dimensional (3D) acoustic models of tissue microstructure, termed 3D impedance maps (3DZMs), were introduced to help to identify scattering sources [J. Mamou, M. L. Oelze, W. D. O'Brien, Jr., and J. F. Zachary, "Identifying ultrasonic scattering sites from 3D impedance maps," J. Acoust. Soc. Am. 117, 413-423 (2005)]. In the current study, new 3DZM methodologies are used to model and identify scattering structures. New processing procedures (e.g., registration, interpolations) are presented that allow more accurate 3DZMs to be constructed from histology. New strategies are proposed to construct scattering models [i.e., form factor (FF)] from 3DZMs. These new methods are tested on simulated 3DZMs, and then used to evaluate 3DZMs from three different rodent tumor models. Simulation results demonstrate the ability of the extended strategies to accurately predict FFs and estimate scatterer properties. Using the 3DZM methods, distinct FFs and scatterer properties were obtained for each tumor examined.     

Identifying ultrasonic scattering sites from three-dimensional impedance maps

Ultrasonic backscattered signals contain frequency-dependent information that is usually discarded to produce conventional B-mode images. It is hypothesized that parametrization of the quantitative ultrasound frequency-dependent information (i.e., estimating scatterer size and acoustic concentration) may be related to discrete scattering anatomic structures in tissues. Thus, an estimation technique is proposed to extract scatterer size and acoustic concentration from the power spectrum derived from a three-dimensional impedance map (3DZM) of a tissue volume. The 3DZM can be viewed as a computational phantom and is produced from a 3D histologic data set. The 3D histologic data set is constructed from tissue sections that have been appropriately stained to highlight specific tissue features. These tissue features are assigned acoustic impedance values to yield a 3DZM. From the power spectrum, scatterer size and acoustic concentration estimates were obtained by optimization. The 3DZM technique was validated by simulations that showed relative errors of less than 3% for all estimated parameters. Estimates using the 3DZM technique were obtained and compared against published ultrasonically derived estimates for two mammary tumors, a rat fibroadenoma and a 4T1 mouse mammary carcinoma. For both tumors, the relative difference between ultrasonic and 3DZM estimates was less than 10% for the average scatterer size.     

Temperature dependent ultrasonic characterization of biological media

Quantitative ultrasound (QUS) is an imaging technique that can be used to quantify tissue microstructure giving rise to scattered ultrasound. Other ultrasonic properties, e.g., sound speed and attenuation, of tissues have been estimated versus temperature elevation and found to have a dependence with temperature. Therefore, it is hypothesized that QUS parameters may be sensitive to changes in tissue microstructure due to temperature elevation. Ultrasonic backscatter experiments were performed on tissue-mimicking phantoms and freshly excised rabbit and beef liver samples. The phantoms were made of agar and contained either mouse mammary carcinoma cells (4T1) or chinese hamster ovary cells (CHO) as scatterers. All scatterers were uniformly distributed spatially at random throughout the phantoms. All the samples were scanned using a 20-MHz single-element f/3 transducer. Quantitative ultrasound parameters were estimated from the samples versus increases in temperature from 37 °C to 50 °C in 1 °C increments. Two QUS parameters were estimated from the backscatter coefficient [effective scatterer diameter (ESD) and effective acoustic concentration (EAC)] using a spherical Gaussian scattering model. Significant increases in ESD and decreases in EAC of 20%-40% were observed in the samples over the range of temperatures examined. The results of this study indicate that QUS parameters are sensitive to changes in temperature.     

Wavelet transforms in estimating scatterer spacing from ultrasound echoes

Ultrasound echoes from organs such as the liver display resolvable periodicity due to regular scattering centers within tissue. The spacing among such scattering centers has been proposed as a signature to characterize diffuse and focal diseases of the liver. Even though it is highly desirable to be able to estimate an inter-scatterer-spacing (ISS) distribution, current methods can estimate only the mean value of scatterer spacing (MSS) over a tissue length. In this paper, we propose a wavelet transform-based technique that is capable of estimating the location of each scattering center, making it possible to obtain the ISS distribution. We represent liver tissue with a point scatterer model, and show, via computer simulations, that the use of multi-scale information in the wavelet scale-space allows us to estimate the locations of regular scattering centers. We show that both the observation noise and random ultrasound returns from unresolvable tissue microstructure can be removed successfully in the wavelet domain via the properties of the modulus maxima sequence of observation across different scales.     

Inverse scattering for surface impedance from phase-less far field data

We present a numerical comparison of a regularized Newton-type method and a direct method for reconstructing the surface impedance function of a three dimensional acoustic scatterer with known shape from the full far field pattern for scattering of one incident time-harmonic plane wave. Furthermore, we propose a modification of the Newton-type algorithm to recover a real-valued surface impedance from phase-less far field data. Numerical reconstructions illustrate the feasibility of the methods.     

A new integral equation method for direct electromagnetic scattering in homogeneous media and its numerical confirmation

In this paper, we derive a new integral equation method for direct electromagnetic scattering in homogeneous media and present a numerical confirmation of the new method via a computer simulation. The new integral equation method is based on a paper written by DeSanto [1], originally for scattering from an infinite rough surface separating homogeneous dielectric half-spaces. Here, it is applied to a bounded scatterer, which can be an ohmic conductor or a dielectric, with some simplification of the continuity conditions for the fields. The new integral equation method is developed by choosing the electric field and its normal derivative as boundary unknowns, which are not the usual boundary unknowns. The new integral equation method may provide significant computational advantages over the standard Stratton-Chu method [2] because it leads to a 50% sparse, rather than 100% dense, impedance (collocation) matrix. Our theoretical development of the new integral equation method is exact.     

Ultrasonic backscatter coefficient quantitative estimates from Chinese hamster ovary cell pellet biophantoms

A cell pellet biophantom technique is introduced, and applied to the ultrasonic backscatter coefficient (BSC) estimate using Chinese hamster ovary (CHO) cells. Also introduced is a concentric sphere scattering model because of its geometrical similarities to cells with a nucleus. BSC comparisons were made between the concentric sphere model and other well-understood models for mathematical verification purposes. BSC estimates from CHO cell pellet biophantoms of known number density were performed with 40 and 80 MHz focused transducers (overall bandwidth: 26-105 MHz). These biophantoms were histologically processed and then evaluated for cell viability. Cell pellet BSC estimates were in agreement with the concentric sphere model. Fitting the model to the BSC data yielded quantitative values for the outer sphere and inner sphere. The radius of the cell model was 6.8 ± 0.7 μm; the impedance of the cytoplasm model was 1.63 ± 0.03 Mrayl and the impedance of the nuclear model was 1.55 ± 0.09 Mrayl. The concentric sphere model appears as a new tool for providing quantitative information on cell structures and will tend to have a fundamental role in the classification of biological tissues.     

Impedance theory of sound absorption: The best absorber and the black body

The impedance theory of scattering (which was proposed by the author—see Acoust. Phys. 52, No. 5 (2006)) is used as the basis for studying the extremum properties of the absorption and scattering sound powers of arbitrary elastic bodies in an arbitrary acoustic medium. Impedance-type conditions are obtained for the surfaces of a best absorber, a perfect scatterer, and a so-called Macdonald body. The absorbing and scattering properties of these objects are studied in comparison with the Kirchhoff black body. Illustrative examples are presented.     

Analysis of microstructural alterations of normal and pathological breast tissue in vivo using the AR cepstrum

The mean scatterer spacing is considered to be an important parameter for describing ultrasonic scattering and characterization of biological tissue. Autoregressive models are widely used in parametric techniques for spectral estimation. In this paper, we describe the results of a careful examination of the mean scatterer spacing parameter in normal and pathological breast tissues in vivo using the autoregressive cepstrum. Our experimental results carried out at 4.5 MHz using weakly focused pulse-echo single element transducer show that the mean scatterer spacing in normal breast tissues in vivo is 1.25+/-0.21 mm whereas in several pathological breast tissues, it is between 0.82+/-0.10 and 1.09+/-0.07 mm. These results indicate good correlation with microstructure of breast tissue characterization, and hence the AR cepstrum holds promise that it could be used as an effective method for signal analysis of ultrasonic scattering and characterization of breast tissues scatterers.     

Acoustic Pulse Diffraction by a Sphere in a Planar Layered Waveguide with a Gradient Layer

Consideration of the vertical sound velocity profile is highly important for solving problems of sound propagation in waveguides and scattering problems. A pulsed echo signal reflected from a spherical scatterer in a waveguide is modeled for the case of a waveguide characterized by sound velocity increasing with depth. The simplest model of the medium is considered in which the scatterer, the source, and the receiver are positioned in a layer with constant sound velocity. Below this layer, the sound velocity increases with depth so that the square of refractive index varies according to linear law. The scattering coefficients for the sphere are calculated using the normal wave method. The number of normal waves forming the echo signal is determined by the preset directionality of the source. Modeling is performed in a frequency band of 70?90 kHz for distances between the scatterer and the transmitter-receiver within 500?1000 m. The transmitted signal has the form of a pulse with cosine envelope and central frequency of 80 kHz.     

Method for solving the problems of multiple scattering by several bodies in a homogeneous unbounded medium

A method is developed for solving problems of multiple scattering by an aggregate of bodies in a homogeneous unbounded medium. For this purpose, the problem on the multiple scattering produced by two bodies in the field of a plane wave is first considered under the assumption that the initial unperturbed scattering amplitudes of both scatterers are known. The solution is constructed by considering plane waves multiply rescattered by the scatterers. Integral equations are obtained that allow one to calculate the resulting scattering amplitude of each scatterer and the combined scattering amplitude of the system of two scatterers. It is shown that knowledge of the solution to this problem is sufficient to solve the problem on the scattering field of a system consisting of an arbitrary number of scatterers. Expressions for the scattering amplitude in the case of an arbitrary primary field are presented. The relationship between the integral equations describing the multiple scattering in a homogeneous space and the multiple scattering by a single scatterer located near an interface is demonstrated. Approximate expressions are given for calculating the scattering amplitude in the case of multiple scattering.     

Peculiarities of elastic scattering of intermediate-energy electrons related to their capture at quasi-stationary levels of an atom in a continuous spectrum

A theoretical model of electron scattering on an atom is constructed to study elastic atomic scattering of intermediate-energy electrons. The proposed model is based upon the combined Mensing potential with two spheres of atomic electrons, which admits analytical solutions of the radial Schröbinger equation. A procedure for matching the parameters of this scatterer to an approximate electrostatic potential of an atom in the form of a screened Coulomb potential has been determined. The screening radius of the latter potential has been calculated proceeding from the properties corresponding to the Thomas-Fermi method. A model of a scatterer determined according to the aforementioned procedure can be used to calculate the energy dependence of the cross section of elastic electron scattering on some atoms with s, p, and d shells representing elements neighboring zirconium. The main result is the establishment of factors responsible for the appearance of maxima on the energy dependences of the cross section of elastic electron scattering. These maxima are related to the resonant trapping of impinging electrons by quasi-stationary levels in a continuous spectrum.     

Numerical study of the effects of scatterer sizes and distributions on multiple backscattered intensity patterns of polarized light

We investigate numerically the effects of scatterer sizes on backscattered polarization patterns using the third-order scattering model developed. The calculated results show that both parallel and cross polarization patterns from water suspensions of polystyrene spheres have four-lobe structures of the azimuth dependence of intensities. Particularly, the parallel polarization pattern is sensitive to scatterer sizes, exhibiting good agreement with prior experimental measurements. Furthermore, the polarization patterns from the dysplastic and normal cells with different size distribution widths are calculated and analyzed. The results show that the polarization patterns of dysplastic and normal cells have distinct differences, which might be used for identification of the morphological structure changes of cancer, dysplasia, and regeneration cells.     

Impedance theory of sound scattering: General relations

The problem of sound scattering by an elastic body of arbitrary geometry in an acoustic medium is solved by the impedance method. It is shown that, for a complete solution, three impedance matrices are necessary: one of them characterizes the scatterer and the other two, the medium. The scattering matrices and other characteristics of the solution are expressed through the incident field and these three impedance matrices. The necessary general relations are presented, and the most important particular cases are considered. Three new representations of the diffraction field are proposed in the form of a sum of two components obtained as solutions to two simpler boundary-value problems. Original Russian Text ￠ Yu.I. Bobrovnitskiĭ, 2006, published in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 5, pp. 601–606.     

An inverse scattering problem from an impedance obstacle

In this paper we consider an inverse scattering problem from an obstacle with impedance boundary condition. Our aim is to recover the unknown scatterer from the far field pattern iteratively assuming the impedance function. Our method, while remaining in the framework of Newton’s method, based on a system of two nonlinear integral equations which is equivalent to the original inverse problem, avoids the need of calculating a direct problem at each iteration. Because of the ill-posedness of this problem, regularization method for example, Tikhonov regularization, is incorporated in our solution scheme. Several numerical examples with only one incident wave are given at the end of the paper to show the feasibility of our method.     

A stochastic discrete model of tissue and its ultrasonic backscattering coefficient

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Effects of matrices on Mie scattering in the mid-infrared region

Various Mie scattering systems, each having a transparent matrix, are studied in the mid-infrared region. Our three theoretical scattering systems correspond to a lossless scatterer, an anomalous dispersive dielectric scatterer and a metal scatterer, each in a non-air usual matrix. The refractive-index effects of the matrix on scattering and extinction efficiencies in the mid-infrared region are found to be quite different in different cases. Although the non-air usual matrix reduces scattering and extinction efficiencies in the first kind of system, it may or may not help scatter and extinguish the mid-infrared radiation in the second, and it has little effect on them in the third.     

Computation of the far-field time domain solution of the scattering problem.

This paper presents a new technique for calculating the time domain (transient) far-field scattered pressure. The scattering problem is divided in two steps; the first step evaluates the field distribution inside the scatterer, and the second step generates the far-field scattered pressure by 3-D Radon transform of these data for each time step and summing over time. The algorithm results in considerable saving in CPU time and memory by simplifying the calculation along the path from scatterer to receiver. This technique can also be used in two dimensions.     

A monostatic inverse scattering model based on polarization utilisation

The solution of the inverse problem of electromagnetic scattering by smooth, convex shaped, perfectly conducting, 3-dimensional scatterers is analysed. Certain geometrical as well as physical-optics approximations were used to incorporate the concepts of the “Minkowski problem” of differential geometry into the space-time integral solution of electromagnetic scattering to yield the formal solution for the recovery of the surface profile of the scatterer from the scattered field data. Although various efficient solutions for target identification are available, still information contained in polarization-depolarization characteristics of the scatterer is not yet exploited to its full extent. Therefore the underlying assumption in this investigation was based on the fact that the “depolarization characteristics” of the scattered field do necessarily contain information regarding the surface profile of the scatterer.     

The fixed scatterer approximation and nucleon deuteron scattering

A method of solving the Schroedinger Equation for the scattering from two fixed local potentials is presented. The solutions are used within the framework of the fixed scatterer approximation to perform model calculations of N-D scattering using both effective range theory potentials and a “semi-realistic” potential with a strong repulsive core. For smooth potentials approximate solutions to the fixed scatterer problem are proposed and found to be quite accurate.Other procedures for calculating elastic scattering were compared with the exact fixed scatterer approximation. The results show that the neglect of longitudinal momentum transfer in the Glauber multiple diffraction theory is a severe effect except in the forward direction. For small angle scattering the Glauber prediction for the double scattering amplitude is quite accurate, and does not depend strongly upon either the extent of potential overlap, or the ratio $\text{V}\text{E}$. Comparisons with the Agassi and Gal results for nonoverlapping potentials indicate that the effects of potential overlap are important, at least for the lowest partial waves. Conclusions about the overall importance of off-energy shell effects in nucleon-deuteron scattering, and about the interference of these effects with the determination of the correlation structure of nuclei are not free from ambiguities.