Errors in ultrasonic scatterer size estimates due to phase and amplitude aberration

Current ultrasonic scatterer size estimation methods assume that acoustic propagation is free of distortion due to large-scale variations in medium attenuation and sound speed. However, it has been demonstrated that under certain conditions in medical applications, medium inhomogeneities can cause significant field aberrations that lead to B-mode image artifacts. These same aberrations may be responsible for errors in size estimates and parametric images of scatterer size. This work derives theoretical expressions for the error in backscatter coefficient and size estimates as a function of statistical parameters that quantify phase and amplitude aberration, assuming a Gaussian spatial autocorrelation function. Results exhibit agreement with simulations for the limited region of parameter space considered. For large values of aberration decorrelation lengths relative to aberration standard deviations, phase aberration errors appear to be minimal, while amplitude aberration errors remain significant. Implications of the results for accurate backscatter and size estimation are discussed. In particular, backscatter filters are suggested as a method for error correction. Limitations of the theory are also addressed. The approach, approximations, and assumptions used in the derivation are most appropriate when the aberrating structures are relatively large, and the region containing the inhomogeneities is offset from the insonifying transducer.     

Theoretical and phantom based investigation of the impact of sound speed and backscatter variations on attenuation slope estimation

Quantitative ultrasound features such as the attenuation slope, sound speed and scatterer size, have been utilized to evaluate pathological variations in soft tissues such as the liver and breast. However, the impact of variations in the sound speed and backscatter due to underlying fat content or fibrotic changes, on the attenuation slope has not been addressed. Both numerical and acoustically uniform tissue-mimicking experimental phantoms are used to demonstrate the impact of sound speed variations on attenuation slope using clinical real-time ultrasound scanners equipped with linear array transducers. Radiofrequency data at center frequencies of 4 and 5 MHz are acquired for the experimental and numerical phantoms respectively. Numerical phantom sound speeds between 1480 and 1600 m/s in increments of 20 m/s for attenuation coefficients of 0.3, 0.4, 0.5, 0.6, and 0.7 dB/cm/MHz are simulated. Variations in the attenuation slope when the backscatter intensity of the sample is equal, 3 dB higher, and 3 dB lower than the reference is also evaluated. The sound speed for the experimental tissue-mimicking phantoms were 1500, 1540, 1560 and 1580 m/s respectively, with an attenuation coefficient of 0.5 dB/cm/MHz. Radiofrequency data is processed using three different attenuation estimation algorithms, i.e. the reference phantom, centroid downshift, and a hybrid method. In both numerical and experimental phantoms our results indicate a bias in attenuation slope estimates when the reference phantom sound speed is higher (overestimation) or lower (underestimation) than that of the sample. This bias is introduced via a small spectral shift in the normalized power spectra of the reference and sample with different sound speeds. The hybrid method provides the best estimation performance, especially for sample attenuation coefficient values lower than that of the reference phantom. The performance of all the methods deteriorates when the attenuation coefficient of the reference phantom is lower than that of the sample. In addition, the hybrid method is the least sensitive to sample backscatter intensity variations.     

Spectral estimation for characterization of acoustic aberration

Spectral estimation based on acoustic backscatter from a motionless stochastic medium is described for characterization of aberration in ultrasonic imaging. The underlying assumptions for the estimation are: The correlation length of the medium is short compared to the length of the transmitted acoustic pulse, an isoplanatic region of sufficient size exists around the focal point, and the backscatter can be modeled as an ergodic stochastic process. The motivation for this work is ultrasonic imaging with aberration correction. Measurements were performed using a two-dimensional array system with 80 x 80 transducer elements and an element pitch of 0.6 mm. The f number for the measurements was 1.2 and the center frequency was 3.0 MHz with a 53% bandwidth. Relative phase of aberration was extracted from estimated cross spectra using a robust least-mean-square-error method based on an orthogonal expansion of the phase differences of neighboring wave forms as a function of frequency. Estimates of cross-spectrum phase from measurements of random scattering through a tissue-mimicking aberrator have confidence bands approximately +/- 5 degrees wide. Both phase and magnitude are in good agreement with a reference characterization obtained from a point scatterer.     

Improved scatterer size estimation using backscatter coefficient measurements with coded excitation and pulse compression

Scatterer size estimates from ultrasonic backscatter coefficient measurements have been used to differentiate diseased tissue from normal. A low echo signal-to-noise ratio (eSNR) leads to increased bias and variance in scatterer size estimates. One way to improve the eSNR is to use coded excitation (CE). The normalized backscatter coefficient was measured from three tissue-mimicking phantoms by using CE and conventional pulsing (CP) techniques. The three phantoms contained randomly spaced glass beads with median diameters of 30, 45, and 82 mum, respectively. Measurements were made with two weakly focused, single-element transducers (f(0)=5 MHz and f(0)=10 MHz). For CE, a linear frequency modulated chirp with a time bandwidth product of 40 was used and pulse compression was accomplished by the use of a Wiener filter. Preliminary results indicated that improved estimation bias versus penetration depth was obtained by using CE compared to CP. The depth of penetration, where the accuracy of scatterer diameter estimates (absolute divergence <25%) were obtained with the 10 MHz transducer, was increased up to 50% by using CE versus CP techniques. In addition, for a majority of the phantoms, the increase in eSNR from CE resulted in a modest reduction in estimate variance versus depth of penetration.     

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.     

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%.     

A stochastic discrete model of tissue and its ultrasonic backscattering coefficient

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Angular scatter ultrasound imaging of wavelength scale targets

A bistatic ultrasound imaging system is demonstrated that uses two 32-element linear phased array transducers oriented at an angle of 40 degrees to one another. The system simultaneously acquires and displays in real time one conventional backscatter image and one "angular scatter" image formed using side-scattered echoes from the same B-mode sector region. Experiments are presented that show differences in the magnitudes of backscatter and angular scatter signals acquired from three nylon monofilaments with diameters less than one wavelength and from soft tissue structures in vivo. The relative magnitudes of angular scatter signals from the monofilaments are qualitatively consistent with a theoretical analysis of acoustic scattering from elastic cylinders. Larger tissue features are more clearly defined in angular scatter images. This result is attributed to the orientation of specularly reflecting surfaces and the expected influence of scattering angle on the system's sensitivity to different scatterer spacings.     

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.     

Spectral integral representations of monostatic backscattering from three-dimensional distributions of sediment volume inhomogeneities

A theory is developed for generating short time, monostatic reverberation realizations caused by three-dimensionally distributed volume inhomogeneities in stratified media. A wave number integral approach to treating the propagation to and from the scatterers, combined with a two-dimensional spectral representation of the azimuthally averaged scatterer realizations and a novel numerical implementation, combine to yield an efficient, high fidelity reverberation simulator for predicting monostatic backscatter from horizontally stratified sediments.     

Influence of the precision of spectral backscatter measurements on the estimation of scatterers size in cancellous bone

The goal of this study is to propose a model for the ultrasonic frequency-dependent backscatter coefficient in femoral cancellous bone. This model has been developed with success to predict backscatter in human calcaneal bone [Jenson, Ultr. Med. Biol. 2003]. A weak scattering model is used and the backscatter coefficient is expressed in terms of a Gaussian autocorrelation function of the medium. The backscatter coefficient is computed and comparison is made with experimental data for 37 specimens and for frequency ranging from 0.4 to 1.2 MHz. An excellent agreement between experimental data and predictions is found for both the magnitude and the frequency-dependence of the backscatter coefficient. Then, a nonlinear regression is performed for each specimen, and the mean trabecular thickness is estimated. Experimental data and theoretical predictions are averaged over the 37 specimens. We also find a close agreement between theoretical predictions obtained using the Gaussian autocorrelation function (scatterer size=134+/-15 microm) and the mean trabecular thickness (Tb.Th=132+/-12 microm) derived from the analysis of bone 3-D micro-architecture using high-resolution micro-tomography. However, the correlation between individual experimental and estimated Tb.Th values is moderate (R(2)=0.44). The performance of the estimator are limited mainly by two factors: interference noise due to random positioning of the scatterers and attenuation. We show that the fundamental limitation of our estimator due to the speckle noise is around 5 microm for trabecular thickness estimation. This limitation is lower than the observed biological variability which is around 30 microm and should not be a limiting factor for individual prediction. A second limitation is the tremendous attenuation encountered in highly scattering media such as cancellous bone, which results in highly damped backscatter signals. The compensation for attenuation is difficult to perform, and it may be a critical point that limits the precision of the estimator.     

High-frequency ultrasound scattering from microspheres and single cells

Assessing the proportion of biological cells in a volume of interest undergoing structural changes, such as cell death, using high-frequency ultrasound (20-100 MHz), requires the development of a theoretical model of scattering by any arbitrary cell ensemble. A prerequisite to building such a model is to know the scattering by a single cell in different states. In this paper, a simple model for the high-frequency acoustic scattering by one cell is proposed. A method for deducing the backscatter transfer function from a single, subresolution scatterer is also devised. Using this method, experimental measurements of backscatter from homogeneous, subresolution polystyrene microspheres and single, viable eukaryotic cells, acquired across a broad, continuous range of frequencies were compared with elastic scattering theory and the proposed cell scattering model, respectively. The resonant features observed in the backscatter transfer function of microspheres were found to correspond accurately to theoretical predictions. Using the spacing of the major spectral peaks in the transfer functions obtained experimentally, it is possible to predict microsphere diameters with less than 4% error. Such good agreement was not seen between the cell model and the measured backscatter from cells. Possible reasons for this discrepancy are discussed.     

Anisotropy of the ultrasonic backscatter of myocardial tissue: I. Theory and measurements in vitro

This research addresses the variations in the ultrasonic backscatter from specimens consisting of a suspension of approximately aligned cylindrical scatterers in a fluid medium as a function of the angle of propagation in the sample. Predictions of the angular dependence of backscatter based on the time-domain Born approximation described by Rose and Richardson [J. H. Rose and J. M. Richardson, J. Nondestr. Eval. 3, 45-53 (1982)] were compared with experimental measurements of the backscatter from both tissue-mimicking phantoms consisting of graphite fibers suspended in gelatin and from canine myocardial tissue. The angular dependence of the backscatter was predicted and measured to be maximum for propagation perpendicular to the cylinder axes and minimum for propagation parallel to the axes. Maximum to minimum (i.e., perpendicular to parallel) changes in the integrated backscatter were predicted to be between 5 and 10 dB in the phantom. The corresponding quantity measured in both the phantom and in canine myocardial tissue was approximately 6 dB.     

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.     

Identification of human myocardial infarction in vitro based on the frequency dependence of ultrasonic backscatter.

It has been reported previously that acute and mature myocardial infarction in dogs can be differentiated in vitro and in vivo by ultrasonic tissue characterization based on measurement of the frequency dependence of ultrasonic backscatter. To characterize human infarction with an index of the frequency dependence of backscatter that could be obtained in patients, cylindrical biopsy specimens from 7 normal regions and 12 regions of infarction of 6 fixed, explanted human hearts in 2-deg steps around their entire circumference with a 5-MHz broadband transducer were insonified. One to six consecutive transmural levels were studied for each specimen. The dependence of apparent (uncompensated for attenuation or beam width) backscatter, /B(f)/2, on frequency (f) was computed from spectral analyses of radio-frequency data as /B(f)/2 = afn, where from theoretical considerations the magnitude of n decreases as scatterer size increases. Apparent integrated backscatter was computed as the average of /B(f)/2 from 3 to 7 MHz. The average value for n for normal tissue (0.9 +/- 0.1) exceeded that for tissue from regions of infarction (0.6 +/- 0.1; p less than 0.05). Infarct manifested a significant decrease of n from epicardial to endocardial levels (epi----mid----endo: 0.9----0.7----0.2; p less than 0.05) whereas normal tissue manifested similar values for n at each transmural level (0.8----1.1----0.9; p = NS). Average integrated backscatter across all transmural levels for infarct was significantly greater than for normal tissue (-48.3 +/- 0.5 vs -53.4 +/- 0.4 dB, infarct versus normal; p less than 0.05). The presence of fibrosis was associated with smaller values of n and greater integrated backscatter.(ABSTRACT TRUNCATED AT 250 WORDS)     

Eigenfunction analysis of stochastic backscatter for characterization of acoustic aberration in medical ultrasound imaging

Presented here is a characterization of aberration in medical ultrasound imaging. The characterization is optimal in the sense of maximizing the expected energy in a modified beamformer output of the received acoustic backscatter. Aberration correction based on this characterization takes the form of an aberration correction filter. The situation considered is frequently found in applications when imaging organs through a body wall: aberration is introduced in a layer close to the transducer, and acoustic backscatter from a scattering region behind the body wall is measured at the transducer surface. The scattering region consists of scatterers randomly distributed with very short correlation length compared to the acoustic wavelength of the transmit pulse. The scatterer distribution is therefore assumed to be delta correlated. This paper shows how maximizing the expected energy in a modified beamformer output signal naturally leads to eigenfunctions of a Fredholm integral operator, where the associated kernel function is a spatial correlation function of the received stochastic signal. Aberration characterization and aberration correction are presented for simulated data constructed to mimic aberration introduced by the abdominal wall. The results compare well with what is obtainable using data from a simulated point source.     

Ultrasound speckle analysis based on the K distribution

The departure of speckle magnitude from Rayleigh statistics was applied to examine insonated phantoms with both low and high concentrations of scatterers. A mathematical model, the K distribution of Jakeman, was used to characterize non-Rayleigh statistics. This model contains a parameter, alpha, which characterizes the clustering of the scattering sites in a medium. It is shown from phantom experiments that alpha is linearly proportional to the log-scaled scatterer concentration in a range from about 1 to 30 scatterers per sample volume.     

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.     

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.     

大空间生物组织透射的Monte Carlo模拟方法的加速

乳腺等类型生物组织的透射Monte Carlo模拟(简称MC模拟)耗时很长,这主要是由于被模拟的组织较厚和Monte Carlo模拟自身统计特性引起的所需光子数量多造成的。针对光源与接收面在同一直线上,采用光纤接收透射光的模型进行分析,通过考虑生物组织自身的光学特性,进行了边界约束和有效后向散射次数的限制来缩短仿真时间,经过多次实验验证,以两次散射所到位置为分界,并充分利用组织光学参数计算约束边界,计算相对比较简单,同时考虑到入射与出射的实际情况进行微扩,从而得到的约束边界。对光学参数相同但厚度不同的组织进行仿真,分析进入同一位置光纤的光子数所发生后向散射次数,发现后向散射次数随着组织厚度与散射系数的增大而增大,随着吸收系数与各向异性因子的增大而减小,通过限定后向散射次数来节省时间。仿真结果表明,在传统计算机上,该方法能够显著缩短MC模拟的运算时间50%左右,尤其适用于较厚且边界较为复杂的高散射物质的MC模拟。该提速方法主要应用于光源与接收面处于同一直线,且生物组织较厚,散射系数较大的情况,能够有效节约仿真所需时间,有利于分析组织透射成像。