Effective Scatterer Diameter Estimates for Broad Scatterer Size Distributions

Abstract

Acoustic form factors have been used to model the frequency dependence of acoustic scattering in phantoms and tissues. This work demonstrates that a broad range of scatterer sizes, individually well represented by Faran theory or a Gaussian form factor is not accurately described by a single effective scatterer from either of these models. Contributions from a distribution of discrete scatterer sizes for two different form factor functions (Gaussian form factors and scattering functions from Faran’s theory) were calculated and linearly combined. Composite form factors created from Gaussian distributions of scatterer sizes centered at 50 µm with standard deviations of up to σ = 40µm were fit to each scattering model between 2 MHz and 12 MHz. Scatterer distributions were generated using one of two assumptions: the number density of the scatterer diameter distribution was Gaussian distributed, or the volume fraction of each scatterer diameter in the distribution was Gaussian distributed. Each simulated form factor was fit to a single diameter form factor model for Gaussian and exponential form factors. The mean squared error (MSE) between the composite simulated data and the best-fit single diameter model was smaller with an exponential form factor model, compared to a Gaussian model, for distributions with standard deviations larger than 30% of the centroid value. In addition, exponential models were shown to have better ability to distinguish between Faran scattering model-based distributions with varying center diameters than the Gaussian form factor model. The evidence suggests that when little is known about the scattering medium, an exponential scattering model provides a better first approximation to the scattering correlation function for a broad distribution of spherically symmetric scatterers than when a Gaussian form factor model is assumed.

A Geometric Model of Ultrasound Backscatter to Describe Microstructural Anisotropy of Tissue

“…15,31 A Gaussian model was used to estimate d e f f for the anisotropic phantom and the rectus femoris due to the narrow size distributions and more complex boundary conditions of the underlying acoustic impedance distribution. 32 This was accomplished using a least squares estimator. 33 While choosing a form factor model affects the accuracy of d e f f (a simple parameter that quantifies the slope of σ b ), all models assume spherical scatterer symmetry and isotropic scattering.…”

Quantitative Ultrasound Detects Smooth Muscle Activity at the Cervical Internal Os in Vitro

“…Parameterization was performed over a 4-to 9-MHz bandwidth and done for every time point, allowing for a direct comparison between force generation and backscatter parameter time series. It has been extensively reported that Gaussian correlation functions are well suited to describe scattering sources with continuously varying fluctuations in acoustic impedance or in the presence of narrow diameter distributions of discrete scatterers (Gerig et al 2003;Nordberg and Hall 2015;Zagzebski et al 2016). Acoustic form factors, functions proportional to the Fourier transform of the correlation function, may be estimated from BSCs directly , whereby deviation from f 4 Rayleigh scattering can be, in special cases (e.g., Bowman's capsule in the renal cortex), attributed to the finite size of the scatterer.…”

Limitations on estimation of effective scatterer diameters

“…Consequently, echographic contrast is primarily determined by regional variations in the concentrations and sizes of cell-based structures relative to the wavelength of sound, k 0 , near the pulse transmission frequency f 0 ¼ c=k 0 at the compressional sound speed c. For symmetric scatterers of radius a, the sensitivity of the echo signal to that reflector is indicated by the ratio of scatterer circumference to wavelength, 2pa=k 0 ¼ ka, where k is the wave number. Scatterer size can be measured with minimal prior information regarding reflector shape and orientation using spectral methods 4,8,10,[14][15][16] when the propagation medium is composed of randomly positioned monodisperse structures at pulse-echo frequencies near ka ¼ 1, 7,17 e.g., unclotted red blood cells 18 near 60 MHz. However, soft tissues are composed of polydisperse structures.…”

“…This method has been widely applied to measurements in monodisperse 10,14,16,[22][23][24] and polydisperse biological media. 15,17,25 Our study is an exploration of ESD measurements from simulated two-dimensional (2-D) scattering media and sound pulses. Initially, the scattering fields are weakly-reflecting, cell-sized disks with known sizes and random positional distributions.…”