Scatterer Number Density Research Articles

Scatterer Number Density Considerations in Reference Phantom-Based Attenuation Estimation

Attenuation estimation and imaging have the potential to be a valuable tool for tissue characterization, particularly for indicating the extent of thermal ablation therapy in the liver. Often the performance of attenuation estimation algorithms is characterized with numerical simulations or tissue-mimicking phantoms containing a high scatterer number density (SND). This ensures an ultrasound signal with a Rayleigh distributed envelope and a signal-to-noise ratio (SNR) approaching 1.91. However, biological tissue often fails to exhibit Rayleigh scattering statistics. For example, across 1647 regions of interest in five ex vivo bovine livers, we obtained an envelope SNR of 1.10 ± 0.12 when the tissue was imaged with the VFX 9L4 linear array transducer at a center frequency of 6.0 MHz on a Siemens S2000 scanner. In this article, we examine attenuation estimation in numerical phantoms, tissue-mimicking phantoms with variable SNDs and ex vivo bovine liver before and after thermal coagulation. We find that reference phantom-based attenuation estimation is robust to small deviations from Rayleigh statistics. However, in tissue with low SNDs, large deviations in envelope SNR from 1.91 lead to subsequently large increases in attenuation estimation variance. At the same time, low SND is not found to be a significant source of bias in the attenuation estimate. For example, we find that the standard deviation of attenuation slope estimates increases from 0.07 to 0.25 dB/cm-MHz as the envelope SNR decreases from 1.78 to 1.01 when estimating attenuation slope in tissue-mimicking phantoms with a large estimation kernel size (16 mm axially × 15 mm laterally). Meanwhile, the bias in the attenuation slope estimates is found to be negligible (<0.01 dB/cm-MHz). We also compare results obtained with reference phantom-based attenuation estimates in ex vivo bovine liver and thermally coagulated bovine liver.

Evaluation of the impact of backscatter intensity variations on ultrasound attenuation estimation

Quantitative ultrasound based approaches such as attenuation slope estimation can be used to determine underlying tissue properties and eventually used as a supplemental diagnostic technique to B-mode imaging. The authors investigate the impact of backscatter intensity and frequency dependence variations on the attenuation slope estimation accuracy. The authors compare three frequency domain based attenuation slope estimation algorithms, namely, a spectral difference method, the reference phantom method, and two spectral shift methods: a hybrid method and centroid downshift method. Both the reference phantom and hybrid method use a tissue-mimicking phantom with well-defined acoustic properties to reduce system dependencies and diffraction effects. The normalized power spectral ratio obtained is then filtered by a Gaussian filter centered at the transmit center frequency in the hybrid method. A spectral shift method is then used to estimate the attenuation coefficient from the normalized and filtered spectrum. The centroid downshift method utilizes the shift in power spectrum toward lower frequencies with depth. Numerical phantoms that incorporate variations in the backscatter intensity from -3 to 3 dB, by varying the scatterer number density and variations in the scatterer diameters ranging from 10 to 100 μm are simulated. Experimental tissue mimicking phantoms with three different scatterer diameter ranges (5-40, 75-90, and 125-150 μm) are also used to evaluate the accuracy of the estimation methods. The reference phantom method provided accurate results when the acoustical properties of the reference and the sample are well matched. Underestimation occurs when the reference phantom possessed a higher sound speed than the sample, and overestimation occurs when the reference phantom had a lower sound speed than the sample. The centroid downshift method depends significantly on the bandwidth of the power spectrum, which in turn depends on the frequency dependence of the backscattering. The hybrid method was the least susceptible to changes in the sample's acoustic properties and provided the lowest standard deviation in the numerical simulations and experimental evaluations. No significant variations in the estimation accuracy of the attenuation coefficient were observed with an increase in the scatterer number density in the simulated numerical phantoms for the three methods. Changes in the scatterer diameters, which result in different frequency dependence of backscatter, do not significantly affect attenuation slope estimation with the reference phantom and hybrid approaches. The centroid method is sensitive to variations in the scatterer diameter due to the frequency shift introduced in the power spectrum.

SU‐E‐I‐101: Multi Taper Coherence Measurements for Characterizing Scatterer Number Density

Purpose: Scatterer number density (SND) per resolution cell has been investigated by many authors as a useful tool for tissue classification in diagnostic ultrasound. SND has been shown to be capable of distinguishing between benign and malignant tumors in the breast, and of determining the degree of lens hardening in cataract formation. Low SND results in coherent scattering, which may be measured using the generalized spectrum (GS). The GS has been measured in ultrasound by correlating windowed FFT's of ultrasound radiofrequency (RF) data, where the data windows have been chosen ad hoc. We seek to improve GS calculations for distinguishing between tissue containing different number densities using Thomson's multi‐taper method. Methods: Three tissue‐mimicking (TM) phantoms consisting of glass beads and graphite powder suspended in agar were constructed. Scattering was mainly due to the glass beads and attenuation was mainly due to the graphite powder. The phantoms had number densities of 2, 10, and 400 glass beads per mm3. RF data was recorded using a 9L4 and 18L6 transducer from a Siemens S2000 system. In this study we compare two different methods of calculating spectral coherence. In one method we use a single taper and convolutionally smooth the GS. In the second method we compute the GS using Thomson's multi‐taper (MT) method. We show that the MT method outperforms diagonally smoothed single‐taper spectral estimates for distinguishing between the different phantoms. Results: We find that the MT method results in a greater separation between the mean coherence values with a lower standard deviation than single‐taper calculation methods, as the SND is varied. Conclusion: When SND is used to distinguish between different tissues, differences will be better captured if coherence is calculated with Thomson's multi‐taper method rather than a single taper.This work was supported by NIH‐NCI grants R01CA112192‐06 and T32 CA09206‐32.

Computer-Aided Diagnosis Based on Speckle Patterns in Ultrasound Images

For breast ultrasound, the scatterer number density from backscattered echo was demonstrated in previous research to be a useful feature for tumor characterization. To take advantage of the scatterer number density in B-mode images, spatial compound imaging was obtained, and the statistical properties of speckle patterns were analyzed in this study for use in distinguishing between benign and malignant lesions. A total of 137 breast masses (95 benign cases and 42 malignant cases) were used in the proposed computer-aided diagnosis (CAD) system. For each mass, the average number of speckle pixels in a region of interest (ROI) was calculated to use the concept of scatterer number density. In addition, the first-order and second-order statistics of the speckle pixels were quantified to obtain the distributions of the pixel values and the spatial relations among the pixels. The performance of the speckle features extracted from each ROI was compared with the performance of the segmentation features extracted from each segmented tumor. As a result, the proposed CAD system using the speckle features achieved an accuracy of 89.1% (122/137); a sensitivity of 81.0% (34/42); and a specificity of 92.6% (88/95). All of the differences between the speckle features and the segmentation features are not statistically significant (p > 0.05). In a receiver operating characteristic (ROC) curve analysis, the Az value, area under ROC curve, of the speckle features was significantly better than the Az value of the segmentation features (0.93 vs. 0.86, p = 0.0359). The performance of this approach supports the notion that the speckle patterns induced by the scatterers in tissues can provide information for classifying tumors. The proposed speckle features, which were extracted readily from drawing an ROI without any preprocessing, also provide a more efficient classification approach than tumor segmentation.

A theoretical comparison of attenuation measurement techniques from backscattered ultrasound echoes

Accurate characterization of tissue pathologies using ultrasonic attenuation is strongly dependent on the accuracy of the algorithm that is used to obtain the attenuation coefficient estimates. In this paper, computer simulations were used to compare the accuracy and the precision of the three methods that are commonly used to estimate the local ultrasonic attenuation within a region of interest (ROI) in tissue; namely, the spectral log difference method, the spectral difference method, and the hybrid method. The effects of the inhomgeneities within the ROI on the accuracy of the three algorithms were studied, and the optimal ROI size (the number of independent echoes laterally and the number of pulse lengths axially) was quantified for each method. The three algorithms were tested for when the ROI was homogeneous, the ROI had variations in scatterer number density, and the ROI had variations in effective scatterer size. The results showed that when the ROI was homogeneous, the spectral difference method had the highest accuracy and precision followed by the spectral log difference method and the hybrid method, respectively. Also, when the scatterer number density varied, the spectral difference method completely failed, while the log difference method and hybrid method still gave good results. Lastly, when the scatterer size varied, all of the methods failed.

Clustered scatterers: the effect on the mean acoustic field

Clustering, or spatially dependent correlation in the fluctuating number density of discrete scatterers of sound, is commonly found to occur in nature. In the ocean, clustering can occur as a result of scatterer entrainment in fluid flows, as in the case of particles and bubbles, or, in the case of fish, as a result of the behaviour of the scatterer. In either case, the presence of clustering can dramatically alter the effect of the scatterers on acoustic fields from that which would be predicted from the same scatterer population but without clustering. The effect of scatterers on the mean acoustic field is often treated using an effective medium wave number. In this paper, a modification to the effective medium number that accounts for clustering is presented and examined for the case of propagation through bubble clouds. Attenuation predictions over a broad range of frequencies (10‐350 kHz) using observations of bubble clustering under oceanic breaking waves will be discussed.

Local-field effects in magnetodielectric media: Negative refraction and absorption reduction

We give a microscopic derivation of the Clausius-Mossotti relations for a homogeneous and isotropic magnetodielectric medium consisting of radiatively broadened atomic oscillators. To this end the diagram series of electromagnetic propagators is calculated exactly for an infinite bicubic lattice of dielectric and magnetic dipoles for a small lattice constant compared to the resonance wavelength $\ensuremath{\lambda}$. Modifications of transition frequencies and linewidth of the elementary oscillators are taken into account in a self-consistent way by a proper incorporation of the singular self-interaction terms. We show that in radiatively broadened media sufficiently close to the free-space resonance the real part of the index of refraction approaches the value $\ensuremath{-}2$ in the limit of $\ensuremath{\rho}{\ensuremath{\lambda}}^{3}⪢1$, where $\ensuremath{\rho}$ is the number density of scatterers. Since at the same time the imaginary part vanishes as $1∕\ensuremath{\rho}$, local field effects can have important consequences for realizing low-loss negative index materials.

A Polydisperse Sphere Model Describing the Propagation ofLight in Biological Tissue

A polydisperse sphere model with the complex refractive index isemployed to describe the propagation of light in biological tissue. Thescattering coefficient, absorption coefficient and scattering phasefunction are calculated. At the same time, the inverse problem onretrieving the particles size distribution, imaginary part of therefractive index and number density of scatterers is investigated. Theresult shows that the retrieval scheme together with the Chahine algorithmis effective in dealing with such an inverse problem. It is alsoclarified that a group of parameters including the scatteringcoefficient, absorption coefficient and phase function are associatedwith another group including the refractive index, particle sizedistribution and number density of scatterers, which is a problemdescribed in two different ways and the anisotropy factor is not anindependent variable, but is determined by the phase function.

Calculating the Jet Quenching Parameter

Models of medium-induced radiative parton energy loss account for the strong suppression of high-p(T) hadron spectra in square root of (S)NN=200 GeV Au-Au collisions at BNL RHIC in terms of a single "jet quenching parameter" q. We observe that q can be given a model-independent, nonperturbative, quantum field theoretic definition in terms of the short-distance behavior of a particular lightlike Wilson loop. We then use the anti-de Sitter/conformal-field-theory correspondence to obtain a strong-coupling calculation of q in hot N=4 supersymmetric QCD, finding q(SYM)=26.69 square root of alpha(SYM)N(c)T(3) in the limit in which both N(c) and 4pialpha(SYM)N(c) are large. Thus, at strong coupling q is not proportional to the entropy density s, or to some "number density of scatterers" since, unlike the number of degrees of freedom, q does not grow like N(c)(2).

Systematic Density Expansion of the Lyapunov Exponents for a Two-Dimensional Random Lorentz Gas

We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form A 0ñln ñ+B 0ñ, where A 0 and B 0 are known constants and ñ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order (ñ2), the positive Lyapunov exponent is of the form A 0ñln ñ+B 0ñ+A 1ñ2ln ñ +B 1ñ2. Explicit numerical values of the new constants A 1 and B 1 are obtained by means of a systematic analysis. This takes into account, up to O(ñ2), the effects of all possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.

Scatterer density estimation in the crust by seismic array processing

SUMMARY Both the number density of scatterers and scattering strength are parameters characterizing the inhomogeneity of the Earth’s crust. An attempt is made to estimate the number density of scatterers (n-value) using small-aperture seismic array data. Many estimates of scattering coefficients have been made previously; however, in a medium with a scattering strength, the wave energy scattered by a medium containing strong scatterers with a low number density is identical to that produced by a high density of weak scatterers. Knowledge of the number density besides the scattering strength supplies important information on stress concentration in the crust due to the strong scatterer reflecting strong inhomogeneity. Therefore, the number density of scatterers is estimated with array signal processing. A seismic array can decompose the approach direction of scattered waves arriving at the array. With array processing, it is not only possible to estimate the energy arriving at the array, but also the semblance value. Temporal variation of the energy depends strongly on the scattering coefficient. On the other hand, the difference in number density of the medium appears in the temporal variation of the semblance value. As a first-order approximation, the semblance value is inversely proportional to the n-value. For practical purposes, an array response should be considered for n-value estimation. Comparing simulated and observed semblances, number density and scattering strength can be estimated from the time sequence of semblances and stacking power. This method was applied to seismic array data from the aftershock area of the 2000 Western Tottori earthquake (Mw 6.8). A rough estimate of number density, compared to a simulation, is 3 × 10 −2 km −3 .

Frequency dependence of scattering from the cold‐temperate transition surface in a polythermal glacier

Radar surveys of the cold‐temperate transition in polythermal glaciers have been used to investigate the polythermal structure, as well as the stability of the temperature regime, in glaciers. The mapping is based on the detection of small water‐filled pockets present in the temperate ice beneath the cold surface layer. I compare radar profiles recorded at different frequencies on a polythermal glacier to show that the apparent depth to the cold‐temperate transition reflection at a center frequency of 155 MHz is significantly greater than that detected at frequencies above 345 MHz. This increase appears to be the result of a reduction of scattering efficiency from the uppermost part of the temperate ice, as the theory of electromagnetic scattering from small objects indicates a strong decrease in scattering efficiency from objects much smaller than the wavelength of the incident wave. Thus I interpret this apparent depth increase to indicate that the average radii of the water pockets of the cold‐temperate transition is on a subdecimeter scale. Spatial variation in the difference between cold‐temperate transition depths at 155 MHz and higher frequencies may arise from spatial variations in size or the number density of scatterers.

Examination of the Phenomena Concerning Optical Particle Measurements Using the Direct Finite-Difference Solution of Maxwell's Equation(Influence of Multiple Scattering and Three-Dimensionality of Scatterer on Diffraction)

At first, the accuracy of the scattering patterns predicted for typical examples by a direct finite-difference solution technique was examined by comparing them with the analytical solutions. Comparisons with regard to the scattering patters formed by a single cylinder, a pair of cylinders or a single true sphere showed good agreement. Based on this confirmation of accuracy of the direct finite-difference solution method, the influence of multiple-scattering and three-dimensionality of scatterer on the diffraction was discussed. Even if the volume fraction of particles is same, larger number density of scatterers leads to more remarkable distortion of scattering pattern and subsequent erroneous measurement of particle size distribution. When a cylindrical scatterer is long in the direction of optical axis, its scattering pattern is remarkably different from that formed by a true sphere of the same diameter. That is, the diffraction pattern is affected by the three-dimensionality of scatterer.

Statistical modeling of scattering from biological media

The statistics of the backscattered ultrasonic echo from tissue can provide information on its characteristics. Such information is useful in the classification of tissues in biomedicine. For example, some of the tissue properties may point to malignancies in certain lesions in liver, breast, or kidneys. The models employed in describing the backscattered echo are therefore very crucial to the success of these classification methods. These models must take into account the number density of scatterers, cross sections of scatterers, variation in cross sections of the scatterers, and any alignment (periodic, quasiperiodic, and purely random) of the scatterers. Parameters reflecting these features can be extracted from the backscattered echo using these models. They can be directly related to the properties of the tissue such as the presence of an abnormal growth, and further classification of the growth as benign and malignant. They may also be used to form parametric images to assist the clinicians in making a medical diagnosis. A number of models ranging from Rayleigh, Poisson, K-, Weibull, and Nakagami will be discussed along with the relevance of their parameters and utility of the parameters in biomedicine. Specific applications to classification of breast lesions in ultrasonic B-scans will be described. [Work supported by NIH-NCI No. 52823.]

Use of frequency diversity and Nakagami statistics in ultrasonic tissue characterization.

The Nakagami distribution was recently proposed as a generalized model for the envelope of the backscattered ultrasonic echo from tissue. The parameters of the Nakagami model were also shown to be useful in tissue characterization. This paper explores the possibility of enhancing the ability of these parameters for tissue characterization through the techniques of diversity and compounding. Frequency diversity has been used to create multiple versions of the envelope, which are then combined. This compounded envelope has been modeled, and its parameters have been analyzed. The ability of these new parameters to enhance tissue characterization is studied using computer simulation and experiments on tissue-mimicking phantoms. Results indicate that the use of frequency diversity and compounding may indeed improve the ability of the parameters of the Nakagami model to separate different number densities of scatterers. Therefore, it is suggested that such an approach may lead to better techniques in ultrasonic tissue characterization.

Intervening attenuation affects first-order statistical properties of ultrasound echo signals

Previous studies show that first-order statistical properties of ultrasound echo signals are related to the effective number of scatterers in the "resolution cell" of a pulse-echo ultrasound system. When the effective number of scatterers is large (~10 or more) this results in echo signals whose amplitude follows a Rayleigh distribution, with the RF echo signal obeying Gaussian statistics; deviation from Rayleigh or Gaussian statistics yields information on scatterer number densities. In this paper, the influence of the medium's attenuation on non-Gaussian properties of the echo signal is considered. Preferential attenuation of higher frequency components of a pulsed ultrasound beam effectively broadens the beam and increases the resolution cell size. Thus, the resultant non-Gaussian parameter for broad bandwidth excitation of the transducer depends not only on the scatterer number density but also on the attenuation in the medium. These effects can be reduced or eliminated by using narrow-band experiments.

Particle transport in a disordered medium: Numerical experiment

Using the multicenter Schrodinger equation for calculating the transmittance of a flat layer of randomly distributed point scattering centers through which a particle passes, we show that when the scattering length for one center is comparable to the particle wavelength λ or is larger, the Ioffe-Regel hypothesis holds. (According to this hypothesis, as the scatterer number density increases, the transmittance of the layer becomes constant, while the value of the particle’s effective mean free path remains of order λ.) When the scattering length is small compared to λ, the Ioffe-Regel hypothesis does not hold. As the scattering length decreases, the accuracy of the approximation of the effective scattering potential gradually increases, and, depending on the strength of the potential, particles may either tunnel or diffuse; the effective mean free path can be much smaller than λ.

Acoustic backscatter properties of the particle/bubble ultrasound contrast agent

Bubble-based suspensions with diameters in the 1–5 μm range have been developed for use as ultrasound contrast agents. Bubbles of these dimensions have resonance frequencies in the diagnostic ultrasonic range, thus improving their backscatter enhancement capabilities. The durability of these bubbles in the blood stream has been found to be limited, providing impetus for a number of approaches to further stabilize them. One of the approaches has been the development of micrometer-size porous particles or ‘nano-sponges’ with properties suitable for the entrapment and stabilization of gas bubbles. However, the complex morphology and surface chemistry involved in the production of this type of agent makes it unfeasible to directly measure the volume of the entrained gas. A model based on acoustic scattering principles is proposed which indicates that only a small volume fraction of gas should be necessary to significantly enhance the echogenicity of this type of particle-based contrast agent. In the model, the effective scattering cross-section is evaluated as a function of the volume fraction of gas contained in the overall scatterer and the overall scatterer diameter. Initially, the volume fraction of gas is considered as a discrete entity or single bubble. Using common mixture rules, it is then shown that the gas can be considered to be distributed throughout the particle and still arrive at a result that is similar to that for a single, discrete volume of gas. The main contribution to the increased scattering cross-section is due to the compressibility difference between gas and water. The backscatter coefficient is computed as the product of the resulting differential scattering cross-section and the scatterer number density. This approach facilitates comparison with known backscatter coefficients of biological targets such as liver and blood. Simple experimental results are presented for comparison with the model, and the implications relevant to clinical use are suggested.

Lasing in active, sub-mean-free path-sized systems with dense, random, weak scatterers

We report enhanced emission and gain narrowing in Rhodamine 590 perchlorate dye in an aqueous suspension of polystyrene microspheres. A systematic experimental study of the threshold condition for and the gain narrowing of the stimulated emission over a wide range of dye concentrations and scatterer number densities showed several interesting features, even though the transport mean free path far exceeded the system size. The conventional diffusive-reactive approximation to radiative transfer in an inhomogeneously illuminated random amplifying medium, which is valid for a transport mean-free path much smaller than the system size, is clearly inapplicable here. We propose a new probabilistic approach for the present case of dense, random, weak scatterers involving the otherwise rare and ignorable sub-mean-free-path scatterings, now made effective by the high gain in the medium, which is consistent with experimentally observed features.

Thermal Scattering Must Be Accounted for in the Determination of Adiabatic Compressibility

Sound velocity measurement is a widely used method for determining the compressibility of liquids, solutions, solutes, and macromolecules1-9 and for the determination of a wide range of thermodynamic parameters.10 Hitherto, the compressibility determined from sound velocity measurement has been interpreted as being adiabatic, and thermal scattering has been neglected. It is shown here that thermal scattering can be very significant and may be calculated by scattering theory. However, results indicate that the very large number density of scatterers in solutions, even at low mole fraction or volume fraction, causes additional scattering effects which are not yet included in scattering theory. It has previously been suggested that the heat flow associated with thermal scattering may produce a significant departure from adiabatic conditions.9 However, it is shown that propagation remains adiabatic in the presence of thermal scattering, under the usual experimental conditions.