The impact of different estimator algorithms when obtaining effective diameter estimates from populations of scatterers of different sizes

Quantitative ultrasound using backscatter coefficients (BSCs) is a potentially powerful tool for estimating microstructural properties from tissues. However, most scattering models assume distributions of identical scatterers even though actual tissues exhibit different levels of spatial variations. It has been experimentally demonstrated that the use of single-size scattering models may yield effective scatterer diameter (ESD) estimates that are not physically meaningful, i.e., estimates that do not correspond to the size of actual physical structures present in the analysis medium. The objective of this study is to analyze three different estimator algorithms when estimating ESDs from media with populations of fluid-like spherical scatterers of different sizes. All estimator algorithms used a single-sized fluid sphere scattering model for ESD estimation. The estimators corresponded to minimizing (1) the variance of the ratio in decibels between the estimated BSCs and the scattering model, (2) the root mean square error (RMSE) between the estimated BSC and a scaled version of the scattering model, and (3) the RMSE between the estimated BSC and a linear transformation (allowing for an intercept) of the scattering model. In simulations, the inputs to the ESD estimators were obtained using portions of the theoretical BSC (neglecting multiple scattering and coherent scattering terms) with different center frequencies between 1 and 40 MHz and 100% fractional bandwidth corresponding to several scatterer size distributions ranging between 25 and 100 μm. In experiments, BSCs were estimated from a gelatin phantom with Sephadex spheres ranging in diameter from 70 to 130 μm and 5-, 7.5-, 10-, and 13-MHz focused transducers. In simulations, ESD estimates obtained with Estimator 1 were approximately inversely proportional to frequency and mostly independent of the underlying scatterer size distribution for sufficiently large analysis frequencies. Estimator 2 also converged to physically meaningless solutions for sufficiently large frequencies, but generally allowed convergence for higher frequencies than Estimator 1. Estimator 3, in contrast, produced meaningful estimates for all studied analysis frequencies and all simulated media. In experiments, Estimator 1 produced ESD estimates lower than 70 μm when using both the 10- and 13-MHz data. However, Estimators 2 and 3 produced ESD estimates between 80 and 100 μm from the same experimental BSCs. The results of this work demonstrate that ESD estimates are highly dependent on the algorithm used to produce them. Results from both simulations and experiments suggest that single-size scattering models may produce meaningful ESD estimates for moderately large ka factors with the use of a proper estimator algorithm.