Quantitative ultrasound moment-based double Nakagami distribution method

Abstract

Background: Ultrasound imaging is a valuable diagnostic tool, but quantifying tissue characteristics can be challenging. While models like the Nakagami distribution help characterize tissue microstructure based on envelope statistics, they may not fully capture the complexity of tissues with multiple scatterer types.

Purpose: This study aims to develop and validate an enhanced version of the Double Nakagami Distribution (DND) model using moment equations for quantitative ultrasound imaging. We seek to establish its theoretical foundation and demonstrate its effectiveness through numerical simulations and experimental results.

Methods: Five versions of the DND estimation model were developed to compute the five associated model parameters. Using the method of moments, the estimators directly computed 5, 4, or 3 DND parameters with any remaining parameters derived from statistical relationships. After selecting the initial solution for the DND methods, Monte Carlo simulations were employed to generate random combinations of Nakagami parameters within two-scatterer media. For experimental validation, four phantoms with different mixtures of nylon and acrylic scatterers were used. Ex vivo validations were conducted using radio-frequency data from four excised fatty rat livers, each exhibiting low and high concentrations of fat droplets. The median and interquartile range of error values from numerical simulations were analyzed, and the Kruskal-Wallis test was used to assess statistical differences, with post hoc Dunn tests with Bonferroni correction for pairwise comparisons. Effect sizes were calculated using Cohen's d to quantify improvements in fitting performance.

Results: The DND estimation model with three parameter estimations demonstrated the least computation time (p < 0.05) and was identified as the most robust of the proposed DND models for further assessments. In simulations with 10⁶ independents, identically distributed random data points, the errors of all five DND parameters remained below 5%. Our results indicated that increasing the mode ratio of the two scatterers' probability density function histograms enhanced the model's performance. In in vitro phantoms, the DND method estimated the scatterer mixture ratios with errors of less than 6%. Additionally, the DND estimation model exhibited lower Kullback-Leibler divergence (KLD) values compared to the Single Nakagami Distribution (p < 0.0001), indicating that DND provided a superior fit. The effect sizes were consistently large (d > 0.8), further supporting the improved performance of DND.

Conclusions: A DND estimation model of envelope statistics with estimations of three parameters was the most robust method regarding computation speed, KLD values, and accuracy.

Keywords: Kullback–Leibler divergence; Monte–Carlo; Nakagami distribution; double Nakagami; ex vivo fatty liver; in vitro phantom; methods of moments; quantitative ultrasound; simulations.

FIGURE 1

Flowchart of initial solution finding.

FIGURE 2

Initial solution for NLSE. The two NDs are represented by red and blue dashed lines, while the cut-off line for clusters 1 and 2 is indicated by a grey dashed line.

FIGURE 3

Probability density of normalized echo amplitude and PDF models of single and DND in different MRs.

FIGURE 4

Boxplot of the computation time of all methods in Monte-Carlo simulation with 10⁶ points and 100 realizations. *p < 0.05, ***p < 0.001, ****p < 0.0001.

FIGURE 5

(a) Errors of estimated parameters in different MR ranges for Monte-Carlo simulations with 10⁶ points, and (b) error of parameter estimates for increasing PDF pixel count. 300 random selections of MR within [0.1, 3).

FIGURE 6

PDFs of the phantoms. The red dashed line corresponds to the single ND fit, while the blue dashed line represents the DND fit. Green and black dotted lines illustrate the individual scatterer fits within the DND model.

FIGURE 7

PDFs of the rat livers. The red dashed line corresponds to the single ND fit, while the blue dashed line represents the DND fit. Green and black dotted lines illustrate the individual scatterer fits within the DND model.