Average Differential Scattering Cross Section Research Articles

Measurements of differential scattering cross section using a ring transducer.

A procedure for the measurement of intrinsic scattering object properties is presented and used to obtain illustrative results. The procedure is based on the measurement of the scattered acoustic field as a function of scattering angle and frequency. Measurements are normalized using analytically determined expressions for emitter and detector beams resulting from a combination of unfocused linear elements arranged in a circular configuration. The spatial effects of finite emitter pulse length and detector gate length are represented by a convolution formula valid for narrow-band transmitted signals and long receiver gates. The normalization includes correction for target absorption as well as measurement of the directly transmitted acoustic power in the free field and yields the average differential scattering cross section per unit volume. Under the Born approximation, this quantity is directly proportional to the spatial-frequency spectrum of the scattering medium inhomogeneities. Measured results are reported for two phantoms consisting of glass microspheres embedded in a weakly absorbing agar background medium. For the phantoms employed, scattering effects, rather than increased absorption, are shown to account for most of the difference in transmission loss between pure agar and agar with glass spheres. The measured differential scattering cross sections are compared with theoretical cross sections for distributions of glass spheres measured experimentally. The measured values show good relative agreement with theory for varying angle, frequency, and phantom properties. The results are interpreted in terms of wave space resolution and the potential for tissue characterization using similar fixed transducer configurations.

Quantitative modeling of the anisotropy of ultrasonic backscatter from canine myocardium

Reports extensions and new results of the First Time Domain Born approximation model used by Mottley and Miller (1982) to describe the anisotropy of ultrasonic backscatter measured in canine myocardium. The interaction of an ultrasonic plane wave impulse with a single cylindrical scatterer using time and frequency domain approaches is reviewed. Myocardial tissue is modeled as a suspension of aligned cylindrically shaped scatterers uniformly distributed in a homogeneous medium. The authors propose extensions to this model to deal with nonideal scatterer orientation, by introducing axial distribution functions and scatterer size distributions based on histology, modeled as a uniform distribution. The backscatter coefficient in the range 2.0-8.0 MHz is calculated. An algorithm to compute the average differential scattering cross section is presented. Ultrasonic elastic properties of myocardial tissue are discussed. Results of the anisotropy of the numerically computed backscatter parameters for model media having nominal mechanical and acoustic properties of canine myocardial tissue are presented and compared to available experimental data along with discussion of possible conclusions. >

Visualizations for studies of ultrasonic scattering and propagation in medical applications

Visualizations are used to show calculations of ultrasonic measurement system weight in scattering experiments, focal plane time history and point spread functions produced by beamforming algorithms, and scattered pressure and internal stress in a kidney stone model during lithotripsy. Knowledge of measurement system characteristics, especially the resolution, allows the design of scattering experiments for the determination of intrinsic parameters such as average differential scattering cross section or the power spectrum of medium variations. Depictions of focal plane time history and point spread functions that result from focusing algorithms in which wave front distortion is compensated, permit evaluation of algorithm performance. Simultaneous display of scattered pressure and internal stress in a kidney stone model following illumination by a plane-wave pulse elucidates the temporal buildup and reverberant decay of stress in a kidney stone during lithotripsy and shows scattering characteristics that may provide information about kidney stone disintegration. In each of these applications, visualization provides insight not readily obtained in other ways from the basic data.

An equation for acoustic propagation in an inhomogeneous medium with relaxation loss

An equation for acoustic propagation in an inhomogeneous medium with relaxation loss is systematically derived from the classical dynamic equations together with an equation of state for relaxation. The derivation assumes small acoustic perturbations but accommodates arbitrary spatial inhomogeneities in material compressibility, density, and parameters of relaxation. The linearized wave equation obtained for n relaxation mechanisms has order n + 2, is causal, and yields the expected dependence of attenuation on frequency. Exact analytic expressions valid at all frequencies are given for the spatially varying attenuation coefficient, as well as phase velocity. A Green's function is calculated for the equation. The results may be used to model scattering for image reconstruction and the determination of statistical properties, such as average differential scattering cross section.

Spectral power determinations of compressibility and density variations in model media and calf liver using ultrasound.

A model of scattering is used to relate average differential scattering cross section and power spectra of scattering medium variations. The model expresses the average differential scattering cross section as a sum of the power spectrum of medium compressibility variations, the power spectrum of density variations weighted by the square of the cosine of the scattering angle, and the cross-power spectrum of compressibility and density variations weighted by twice the cosine of the scattering angle. Known values of the average differential scattering cross section at a minimum of three different scattering angles and temporal frequencies corresponding to the same spatial frequency are used to calculate each of the three power spectra. Since noise and statistical fluctuations are present in actual measurements of average differential scattering cross section, the calculations of power spectra are obtained from an overdetermined set of equations to which a solution is found by using a singular value decomposition. Data derived from a model for scattering from a cloud of correlated particles are employed to show the influence of additive noise. Calculations are also made from measurements of scattering from three suspensions of particles that have a different average radius in each suspension but are similarly modeled by scattering from a cloud. Additionally, the calculations are applied to measurements of average differential scattering cross section of calf liver. The results show that determination of the power spectra of scattering medium variations can be made under practical conditions, and also imply that density variations contribute significantly to scattering by calf liver.

Estimates of wave front distortion from measurements of scattering by model random media and calf liver.

An expression based on a perturbation method is employed to estimate the correlation of path length difference in a plane normal to the direction of wave propagation from measurements of ultrasonic scattering by model random media and calf liver. The expression gives the correlation function of path length difference in terms of an integral of the correlation function of the medium variations or an equivalent integral of the power spectrum of medium variations, both for a scattering angle of zero degrees. Power spectra derived from measurements of average differential scattering cross section over a spatial-frequency window are used to fit analytic functions that extend over all spatial frequencies. The results for the windowed and unwindowed data yield correlation functions and corresponding power spectra that are used to estimate the correlation function of path length difference. The results suggest that the correlation length in calf liver is less than 100 microns and that a root-mean-square path length variation of about 20 microns results from propagation through a 100-mm calf liver path.

An ultrasonic scattering characterization of tissue

Average differential scattering cross section is obtained from measurements by a normalization that accounts for temporal waveforms and spatial apertures. A model is employed to express the average differential scattering cross section as a linear combination of the power spectrum of the medium compressibility variations, the power spectrum of density variations, and the cross power spectrum of compressibility and density variations. Since the scale factors depend on scattering angle and temporal frequency, known values of the average differential scattering section at different scattering angles and temporal frequencies corresponding to the same spatial frequency, which is considered to be the argument of the average differential scattering cross section and the power spectra, are used to calculate each of the three power spectra. Results are shown for model random media and calf liver. The results indicate that determinations of average differential scattering cross sections and the power spectra of scattering medium variations can be made under practical conditions and also imply that density variations contribute significantly to scattering by calf liver. [Work supported by NSF, NIH, NATO, and Industrial Associates.]

Ultrasonic scattering properties of three random media with implications for tissue characterization.

Random media with different structural properties were used to simulate some of the differences in liver morphology that may occur with disease. First, a reference medium consisting of glass spheres in agar was studied to verify the accuracy and precision of the data obtained with our equipment and processing procedures. Then, studies were conducted on a pair of media comprised of graphite particles in gelatin, one of the pair with twice as many particles as the other. Finally, studies were carried out on a set of media composed of Sephadex particles in water. Three samples were employed, each with a different size of Sephadex. The average differential scattering cross section per unit volume sigma sd (v) of each media was obtained as a function of scattering angle v and frequency. The measured sigma sd were compared with predictions based on models of scattering from the media. The agreement between the measured and predicted sigma sd of the glass sphere medium was excellent. The graphite medium with twice the number of particles as the other was observed to scatter twice as much power as the other. The shape of the angular scattering pattern measured from each size of Sephadex followed the prediction reasonably well. The largest size exhibited marked variations in the shape of sigma sd as a function of frequency, while the absolute magnitude of sigma sd of the smallest size grade was extremely sensitive to frequency. Our results suggest that the dependence of sigma sd (v) on scatterer number density and size, both in absolute magnitude and shape, can provide reliable information which may be useful in the diagnosis of some diseases.

Measurements of calf liver ultrasonic differential and total scattering cross sections

Ultrasonic scattering from calf liver has been measured and normalized to obtain the average differential scattering cross section per unit volume σ̄sd (ν) over scattering angles ν ranging in magnitude from 25 ° to 165 ° and frequencies between 3 and 7 MHz. The results indicate that, at each of the measured frequencies, calf liver is a weak scatterer with nearly one-half of the total scattered power directed between 25 ° and 45 °. The value of σ̄sd (165 °) , which is not expected to differ much from σ̄sd (180 °) , the backscattering coefficient, was found to be approximately 3.7 × 10−6f1.4 mm−1Ω−1, with f in MHz and Ω in steradians. By integrating σ̄sd over all the measured scattering angles to find the total scattering cross section σ̄st , the contribution of scattering loss to attenuation was found to be approximately 2% over the measured frequencies. Analysis of the shape of σ̄sd suggests that the correlation length of the liver was on the order of 1 mm. To assess the accuracy of the normalization method, experimental and theoretical results for σ̄st of a reference medium consisting of glass spheres in agar were compared and found to differ by less than 10%.

Normalization of ultrasonic scattering measurements to obtain average differential scattering cross sections for tissues.

An integral expression is derived for the normalization of ultrasonic scattering data to obtain an average differential scattering cross section per unit volume for tissue which is modeled as a random, fluidlike medium. The expression assumes narrowband signals and involves the incident beam, receiver beam pattern, and time gates. The beams and gates combine to form a window which limits the scattering volume. The derivation of the expression requires that the dimensions of the window be large compared to the correlation length of the scattering medium. Numerical values of the normalizing integral are given for cylindrical and beamlimited scattering volumes as a function of incident frequency and scattering angle under the assumptions of Gaussian beams and rectangular time gates. A set of curves is presented to relate the percent difference between the result for backscatter from a cylindrical scattering volume and from a beamlimited scattering volume which does not include the truncation effect of the cylinder boundary. Although similar in form to normalizations used by others, the integral in this paper is obtained from a derivation which treats physical parameters rigorously and provides a precise statement of conditions which are sufficient to obtain system-independent scattering data.

Omnidirectional Scattering of Acoustic Waves by Rough, Imperfectly Reflecting Surfaces

The omnidirectional scattering of acoustic waves by a rough, imperfectly reflecting surface is described using the method of physical optics. The expression for the far-zone intensity is derived and then specialized to the case of surfaces that are sufficiently rough to neglect the specularly reflected power. For such surfaces, the average differential scattering cross section σ0 is calculated and compared with the results of bistatic measurements made under conditions that satisfy the assumptions of the theory.

Omnidirectional Scattering of Acoustic Waves from Rough Surfaces of Known Statistics

The omnidirectional characteristics of the scattering of acoustic waves from statistically rough surfaces were experimentally investigated. The average differential scattering cross section was measured bistatically as a function of the angles of incidence and observation. Measurements for two surfaces of different roughnesses indicate the increase in the width of the scattering pattern's major lobe in the specular direction with increasing roughness. Scatter in directions other than near the specular (including the backscatter direction) is essentially independent of direction.

The Scattering of Sound in a Turbulent Medium

The scattering of a sound wave in a medium undergoing shear flow confined to a finite region is investigated under the assumption that the total velocity field is everywhere small compared to the velocity of sound. Formulas are obtained for the angular distribution and frequency distribution of the scattered wave in terms of the four-dimensional Fourier transform of the shear velocity field. Under typical conditions, the cross sections for the scattering of a plane wave of frequency ω by a shear flow of given scale and spatial structure are approximately of the form ω4M2, where M is a characteristic Mach number of the flow. The coupling between the shear and longitudinal velocity fields has a tensor character such the the scattering vanishes at 180 degrees and at 90 degrees. The spectrum of the scattered sound wave is very sharp in the forward direction and becomes broader at larger scattering angles. Explicit expressions for the cross sections are obtained for the case of scattering from a region of isotropic turbulence. When the frequencies of importance in the turbulence are small compared to the frequency of the incident sound wave, the average differential scattering cross section can be expressed directly in terms of the energy spectrum of the turbulence.

The Scattering of Sound in a Turbulent Medium

The scattering of a sound wave in a medium undergoing shear flow confined to a finite region is investigated under the assumption that the total velocity field is everywhere small compared to the velocity of sound. Formulas are obtained for the angular distribution and frequency distribution of the scattered wave in terms of the four-dimensional Fourier transform of the shear velocity field. The cross sections for the scattering of a plane wave of frequency ω by a shear flow of given scale and spatial structure go typically as ω4M2, where M is a characteristic Mach number of the flow. The coupling between the shear and longitudinal velocity fields has a tensor character such that the scattering vanishes at 180° and at 90°. The spectrum of the scattered sound wave is very sharp in the forward direction and becomes broader at larger scattering angles. Explicit expressions for the cross sections are obtained for the case of scattering from a region of isotropic turbulence. When the frequencies of importance in the turbulence are small compared to the frequency of the incident sound wave, the average differential scattering cross section can be expressed directly in terms of the energy spectrum of the turbulence.