Quantification of limitations in statistical analysis of ultrasound echo envelope amplitudes

Abstract

Ultrasound echo envelope statistics have been widely studied for quantitative tissue characterization. In ultrasound measurements, the size of the region-of-interest (ROI) is limited by several factors, such as the locality of the tissue characteristics and the depth dependence of the acoustic field of the ultrasound beam. In this case, the evaluated echo envelope statistics vary even when the envelope amplitudes follow the same population without any noise. In this study, the statistical variance of the moments caused by this finite number of samples was quantified based on the central limit theorem and the law of error propagation. The proposed principles were validated by random number simulation and used to quantify the statistical variance of Nakagami parameter estimation. Finally, the effective number of independent samples in an ultrasonic measurement was quantified based on the relationship between the ROI size and the ultrasound spatial resolution.

Appropriate indicator of modeling error for threshold-based model selection in statistics-based ultrasound tissue characterization

“…the number of samples in the region of interest (ROI)) is finite. As elucidated in our previous study, 37) the normalized moment calculated from samples with a finite data length varies around the theoretical moment…”

“…In an actual analysis, we generally normalize the echo envelope amplitude by its second moment. 37) The normalized theoretical moment is defined as:…”

“…In addition, the data length in an actual measurement is finite; therefore, the statistical properties vary even if the measured data follow exactly the same population without any noise. 37) Owing to this statistical variance, in some cases, model selection based on the minimization of modeling error causes the selection of an inappropriate model that has similar statistical properties as the appropriate model, as shown in Sect. 2.2.…”

“…This statistical variance comes from the central limit theorem; therefore, the amount of variance depends not on the number of samples but on the number of independent samples. 37) Therefore, an increase in the number of samples by up-sampling does not suppress this statistical variance.…”

“…For this purpose, we preliminarily examined the Mahalanobis distance (MD) of moments, which can normalize the statistical variance owing to the finite data length, as an indicator of modeling error, by conducting an ultrasonic simulation. 38) However, whether the MD of moments is an appropriate indicator for threshold-based model selection has not been examined. In addition, the MD of moments has not been compared with other indicators of modeling error.…”