Quantitative ultrasound estimates from populations of scatterers with continuous size distributions: Effects of the size estimator algorithm

Quantitative ultrasonic techniques using backscatter coefficients (BSCs) may fail to produce physically meaningful estimates of effective scatterer diameter (ESD) when the analysis media contains scatterers of different sizes. In this work, three different estimator algorithms were used to produce estimates of ESD. The performance of the three estimators was compared over different frequency bands using simulations and experiments with physical phantoms. All estimators produced ESD estimates by comparing the estimated BSCs with a scattering model based on the backscattering cross section of a single spherical fluid scatterer. The first estimator consisted of minimizing the average square deviation of the logarithmically compressed ratio between the estimated BSCs and the scattering model. The second and third estimators consisted of minimizing the mean square error between the estimated BSCs and a linear transformation of the scattering model with and without considering an intercept, respectively. Simulations were conducted over several analysis bandwidths between 1 and 40 MHz from populations of scatterers with either a uniform size distribution or a distribution based on the inverse cubic of the size. Diameters of the distributions ranged between [25, 100], [25, 50], [50, 100], and [50, 75] m. Experimental results were obtained from two gelatin phantoms containing cross-linked dextran gel spheres ranging in diameter from 28 to 130 m and 70 to 130 m, respectively, and 5-, 7.5-, 10-, and 13-MHz focused transducers. Significant differences in the performances of the ESD estimator algorithms as a function of the analysis frequency were observed. Specifically, the third estimator exhibited potential to produce physically meaningful ESD estimates even for large ka values when using a single-size scattering model if sufficient analysis bandwidth was available.