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.     

Impact of local attenuation approximations when estimating correlation length from backscattered ultrasound echoes

Estimating the characteristic correlation length of tissue microstructure from the backscattered power spectrum could improve the diagnostic capability of medical ultrasound. Previously, size estimates were obtained after compensating for source focusing, the frequency-dependent attenuation along the propagation path (total attenuation), and the frequency-dependent attenuation in the scattering region (local attenuation). In this study, the impact of approximations of the local attenuation on the scatterer size estimate was determined using computer simulations and theoretical analysis. The simulations used Gaussian impedance distributions with an effective radius of 25 microm randomly positioned in a homogeneous half-space sonified by a spherically focused source (f/1 to f/4). The approximations of the local attenuation that were assessed neglected local attenuation (i.e., assume 0 dB/cm-MHz) neglected frequency dependence of the local attenuation, and assumed a finite frequency dependence (i.e., 0.5 dB/cm-MHz) independent of the true attenuation of the medium. Errors in the scatterer size estimate due to the local attenuation approximations increased with increasing window length, increasing true local attenuation and increasing f number. The most robust estimates were obtained when the local attenuation was approximated by a tissue-independent attenuation value that was greater than 70% of the largest attenuation expected in the tissue region of interest.     

Method of improved scatterer size estimation and application to parametric imaging using ultrasound

The frequency dependence of RF signals backscattered from random media (tissues) has been used to describe the microstructure of the media. The frequency dependence of the backscattered RF signal is seen in the power spectrum. Estimates of scatterer properties (average scatterer size) from an interrogated medium are made by minimizing the average squared deviation (MASD) between the measured power spectrum and a theoretical power spectrum over an analysis bandwidth. Estimates of the scatterer properties become increasingly inaccurate as the average signal to noise ratio (SNR) over the analysis bandwidth becomes smaller. Some frequency components in the analysis bandwidth of the measured power spectrum will have smaller SNR than other frequency components. The accuracy of estimates can be improved by weighting the frequency components that have the smallest SNR less than the frequencies with the largest SNR in the MASD. A weighting function is devised that minimizes the noise effects on the estimates of the average scatterer sizes. Simulations and phantom experiments are conducted that show the weighting function gives improved estimates in an attenuating medium. The weighting function is applied to parametric images using scatterer size estimates of a rat that had developed a spontaneous mammary tumor.     

Three-dimensional impedance map analysis of rabbit liver

Three-dimensional impedance maps (3DZMs) are computational models of acoustic impedance of tissue constructed from histology images. 3DZMs can be analyzed to estimate model-based quantitative ultrasound parameters such as effective scatterer diameter (ESD). In this study, 3DZMs were constructed from normal and fatty rabbit livers. Estimates of ESD were made using the fluid-filled sphere scattering model. Weighting toward smaller scatterer sizes produced ESD estimates of 7.5 ± 1.3 and 7.0 ± 0.3 μm for normal and fatty liver, respectively, approximately the size of a liver cell nucleus. This suggests the nucleus could be a primary source of scattering in liver.     

Scatterer size estimation in pulse-echo ultrasound using focused sources: calibration measurements and phantom experiments

In a companion paper [T. A. Bigelow and W. D. O'Brien Jr., J. Acoust. Soc. Am. 116, 578 (2004)], theory, supported by simulations, showed that accurate scatterer size estimates could be obtained using highly focused sources provided that the derived generalized attenuation-compensation function was used and the velocity potential field near the focus could be approximated as a three-dimensional Gaussian. Herein, the theory is further evaluated via experimental studies. A calibration technique is developed to find the necessary equivalent Gaussian dimensions for a focused source using reflections obtained from a rigid plane scanned through the focus. Then, the theoretical analysis of focused sources is validated experimentally using three spherically focused ultrasound transducers to estimate the radius of glass beads imbedded in tissue mimicking phantoms. Both the impact of focusing (f/1, f/2, and f/4) and the effect of scatterer type (comparing glass bead results to simulation results that used scatterers with Gaussian impedance distributions) were tested. The simulated differences agree with the measured differences to within 2.5% provided that the comparison is made between the same scatterer type and sources with the same equivalent Gaussian dimensions. The improvement provided by the generalized attenuation-compensation function is greatly influenced by the type of scatterer whose size is being estimated and decreases as the wavelength dependence of the Gaussian depth of focus is reduced.     

Improved scatterer property estimates from ultrasound backscatter for small gate lengths using a gate-edge correction factor

Backscattered rf signals used to construct conventional ultrasound B-mode images contain frequency-dependent information that can be examined through the backscattered power spectrum. The backscattered power spectrum is found by taking the magnitude squared of the Fourier transform of a gated time segment corresponding to a region in the scattering volume. When a time segment is gated, the edges of the gated regions change the frequency content of the backscattered power spectrum due to truncating of the waveform. Tapered windows, like the Hanning window, and longer gate lengths reduce the relative contribution of the gate-edge effects. A new gate-edge correction factor was developed that partially accounted for the edge effects. The gate-edge correction factor gave more accurate estimates of scatterer properties at small gate lengths compared to conventional windowing functions. The gate-edge correction factor gave estimates of scatterer properties within 5% of actual values at very small gate lengths (less than 5 spatial pulse lengths) in both simulations and from measurements on glass-bead phantoms. While the gate-edge correction factor gave higher accuracy of estimates at smaller gate lengths, the precision of estimates was not improved at small gate lengths over conventional windowing functions.     

Defining optimal axial and lateral resolution for estimating scatterer properties from volumes using ultrasound backscatter

The rf signals used to construct conventional ultrasound B-mode images contain frequency-dependent information that can be examined through the backscattered power spectrum. Typically, the backscattered power spectrum is calculated from a region of interest (ROI) within some larger volume. The dimensions of the ROI are defined axially by the spatial length corresponding to the time gate and laterally by the number of scan lines included in the ROI. Averaging the backscattered power spectra from several independent scan lines can reduce the presence of noise caused by electronics and by the random scatterer spacings, but also decreases the lateral resolution of the interrogation region. Furthermore, larger axial gate lengths can be used to reduce the effects of noise and improve the precision and accuracy of scatterer property estimates but also decreases the axial resolution. A trade-off exists between the size of the ROI (the number of scan lines used, the separation distance between each scan line, the axial gate length) and the accuracy and precision of scatterer property estimates. A series of simulations and measurements from physical phantoms were employed to examine these trade-offs. The simulations and phantom measurements indicated the optimal lateral and axial sizes of the ROI, where estimate accuracy and precision were better than 10% and 5%, respectively, occurred at 4 to 5 beamwidths laterally and 15 to 20 spatial pulse lengths axially.     

Scatterer size estimation in pulse-echo ultrasound using focused sources: theoretical approximations and simulation analysis

The speckle in ultrasound images has long been thought to contain information related to the tissue microstructure. Many different investigators have analyzed the frequency characteristics of the backscattered signals to estimate the scatterer acoustic concentration and size. Previous work has been mostly restricted to unfocused or weakly focused ultrasound sources, thus limiting its implementation with diagnostically relevant fields. Herein, we derive equations capable of estimating the size of a scatterer for any reasonably focused source provided that the velocity potential field in the focal region can be approximated as a three-dimensional Gaussian beam, scatterers are a sufficient distance from the source, and the field is approximately constant across the scatterer. The calculations show that, when estimating the scatterer size, correcting for focusing requires a generalized attenuation-compensation function that includes both attenuation and focusing along the beam axis. The Gaussian approximation is validated by comparing the ideal velocity potential field for three spherically focused sources with f-numbers of 1, 2, and 4 to the Gaussian approximation for frequencies from 2 to 14 MHz. The theoretical derivations are evaluated by simulating the backscatter by using spherically focused sources (f-numbers of 1, 2, and 4) adjacent to attenuating media (0.05 to 1 dB/cm/MHz) that contain scatterers with Gaussian impedance distributions. The generalized attenuation-compensation function yielded results accurate to 7.2% while the traditional attenuation-compensation functions that neglected focusing had errors as high as 103%.     

On the estimation of backscatter coefficients using single-element focused transducers

The ultimate goal of quantitative ultrasound (QUS) imaging methods based on backscatter coefficient (BSC) estimates is to obtain system-independent structural information about samples. In the current study, three BSC estimation methods were compared and evaluated using the same backscattered pressure datasets in order to assess their consistency. BSC estimates were obtained from two phantoms with embedded glass spheres and compared to theoretical BSCs calculated using size distributions estimated using optical microscopy. Effective scatterer diameter and concentration estimates of the glass spheres were also obtained from the estimated BSCs. One estimation method needed to be compensated by more than an order of magnitude in amplitude in order to produce BSCs comparable to the other two methods. All calibration methods introduced different frequency-dependent effects, which could have noticeable effects on the bias of QUS estimates derived from experimental BSCs. Although in most cases the experimental QUS estimates obtained with all three methods were observed to differ by less than 10%, larger differences are expected depending on both the pressure focusing gain of the transducer (proportional to the ratio of the square of the aperture radius to the product of the wavelength and focal length) and ka range used in the estimation.     

The DORT solution in acoustic inverse scattering problem of a small elastic scatterer

The DORT (French acronym for Décomposition de l’Opérateur de Retournement Temporel) method is a novel approach for active detection and focusing of acoustic waves on the targets in the scattering medium. This technique involves the determination of the invariant of the time-reversal operator obtained by measurement of the scattering data in a pulse-echo mode. In this paper, a proposed approach based on the DORT method is developed to solve the acoustic inverse scattering problem of a small metallic scatterer. The proposed approach not only estimates the position of the scatterer, but also determines the physical properties of an unknown metallic scatterer such as the shape (cylinder or sphere), the material (density), and the size (radius) in an anisotropic scattering case. Theoretical and numerical simulation results are also studied and investigated to show that the proposed approach can simultaneously characterize all those properties of an unknown metallic scatterer. Moreover, the advantage of the proposed approach is to avoid the complex iterative scheme in solving the direct scattering problem and results in smaller computational load and faster implementation.     

A stochastic discrete model of tissue and its ultrasonic backscattering coefficient

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Correlation of ultrasonic scatterer size estimates for the statistical analysis and optimization of angular compounding

Ultrasonic scatterer size estimates generally have large variances due to the inherent noise of spectral estimates used to calculate size. Compounding partially correlated size estimates associated with the same tissue, but produced with data acquired from different angles of incidence, is an effective way to reduce the variance without making dramatic sacrifices in spatial resolution. This work derives theoretical approximations for the correlation between these size estimates, and the coherence between their associated spectral estimates, as functions of ultrasonic system parameters. A Gaussian spatial autocorrelation function is assumed to adequately model scatterer shape. Both approximations compare favorably with simulation results, which consider validation near the focus. Utilization of the correlation/coherence expressions for statistical analysis and optimization is discussed. Approximations, such as the invariance of phase and amplitude terms with angle, are made to obtain closed-form solutions to the derived spectral coherence near the focus and permit analytical optimization analysis. Results indicate that recommended parameter adjustments for performance improvement generally depend upon whether, for the system under consideration, the primary source of change in total coherence with rotation is phase term variation due to the change in the relative position of scattering sites, or field amplitude term variation due to beam movement.     

Acoustic impedance microscopy for biological tissue characterization

A new method for two-dimensional acoustic impedance imaging for biological tissue characterization with micro-scale resolution was proposed. A biological tissue was placed on a plastic substrate with a thickness of 0.5 mm. A focused acoustic pulse with a wide frequency band was irradiated from the “rear side” of the substrate. In order to generate the acoustic wave, an electric pulse with two nanoseconds in width was applied to a PVDF-TrFE type transducer. The component of echo intensity at an appropriate frequency was extracted from the signal received at the same transducer, by performing a time–frequency domain analysis. The spectrum intensity was interpreted into local acoustic impedance of the target tissue. The acoustic impedance of the substrate was carefully assessed prior to the measurement, since it strongly affects the echo intensity. In addition, a calibration was performed using a reference material of which acoustic impedance was known. The reference material was attached on the same substrate at different position in the field of view. An acoustic impedance microscopy with 200 × 200 pixels, its typical field of view being 2 × 2 mm, was obtained by scanning the transducer. The development of parallel fiber in cerebella cultures was clearly observed as the contrast in acoustic impedance, without staining the specimen. The technique is believed to be a powerful tool for biological tissue characterization, as no staining nor slicing is required.     

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.     

Resonances in the system “piezoelectric emitter-multilayer ultrasonic chamber with losses”

An approach to calculating resonance frequencies and acoustic characteristics of ultrasonic multilayer liquid chambers bordering an air or liquid half-space and loaded to a piezoelectric emitter is proposed. The assumption on the validity of plane motion is accepted. The approach allows for obtaining comprehensive data on physical processes in the chamber. The equation relating electrical admittance of a piezoplate with input impedance of the acoustic load is derived. An unloaded emitter and an emitter loaded to acoustic resistance with constant and frequency-dependent impedances are considered as examples.     

Statistics of ultrasonic scatterer size estimation with a reference phantom

A theoretical expression for the variance of scatterer size estimates is derived for a modified least squares size estimator used in conjunction with a reference phantom method for backscatter coefficient measurement. A Gaussian spatial autocorrelation function is assumed. Simulations and phantom experiments were performed to verify the results for backscatter and size variances. The dependence of size estimate errors upon free experimental parameters is explored. Implications of the findings for the optimization of scatterer size estimation are discussed. The utility of scatterer size parametric imaging is examined through the signal to noise ratio comparison with standard ultrasonic B-mode imaging.     

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.     

Dielectric relaxation of ZnO nanostructure synthesized by soft chemical method

Thioglycerol capped nanoparticles of ZnO have been prepared in methanol through chemical technique. Nanostructures of the prepared ZnO particles have been confirmed through X-ray diffraction measurement. The Debye–Scherrer formula is used to obtain the particle size. The average size of the prepared ZnO nanoparticles is found to be 50 nm. The frequency-dependent dielectric dispersion of the sample is investigated in the temperature range from 293 to 383 K and in a frequency range from 100 Hz to 1 MHz by impedance spectroscopy. An analysis of the complex permittivity (ε′ and ε′′) and loss tangent (tan δ) with frequency is performed assuming a distribution of relaxation times. The frequency-dependent maxima of the imaginary part of impedance are found to obey Arrhenius law with activation energy ∼1 eV. The scaling behavior of dielectric loss spectra suggests that the relaxation describes the same mechanism at various temperatures. The frequency-dependent electrical data are analyzed in the framework of conductivity and modulus formalisms. The frequency-dependent conductivity spectra obey the power law.     

Reduction of acoustic radiation by impedance control with a perforated absorber system

A model of sound radiation from an infinite plate with an absorptive facing is proposed and investigated theoretically from the viewpoint of acoustic power. Acoustic characteristics on the plate surface are represented by impedance derived from iso-absorption curves. A parametric study is carried out to clarify the effect of the impedance on the acoustic power. Results derived from this model show that acoustic radiation depends on change in impedance as well as the absorption coefficient, and there is a possibility of reducing the radiation from vibrating surface by introducing an appropriate impedance surface. In order to realize this effect, a model using a perforated board with a back cavity attached to the vibrating surface is proposed, in which the motion of the perforated board is made equal to that of the vibrating surface. To obtain fundamental data, a theoretical study is performed under a simplified condition, assuming an infinite plane piston. The calculated results are compared to experimental data measured by using an acoustic tube. The results, which are in good agreement in the reduction effect, show that this system can achieve the reduction of radiated sound power at arbitrary frequencies.     

Recent approaches to the measurement of acoustic impedance and materials characterization

Three methods are discussed: an automated pulse tube system; a direct, point measurement technique; and the application of a parametric array for oblique angle measurement.The first of these extends the capability of a proven impedance measurement technique using a waterborne acoustic waveguide (pulse tube). Data obtained in a frequency range 3 to 100 kHz, determined from complex reflection coefficients, are presented, via a transfer function analyser interfacing with a computer and plotter, to produce impedance diagrams.A direct, point impedance technique based on sensing particle velocity, or displacement of a surface and associated acoustic pressure is next discussed. Use is made of laser interferometry to measure the vector quantity, while scalar values are determined from a pressure sensor. This data affords a direct measurement of point impedance and can be applied in obtaining complex response information from heterogeneous materials or structures.The last method employs a non-linear acoustic device to obtain a requisite acoustic beam-width allowing characterization of materials at oblique angles, with samples of limited size, at low ultrasonic frequencies.