Journal of Healthcare Engineering

Volume 2017 (2017), Article ID 8154780, 14 pages

https://doi.org/10.1155/2017/8154780

## Assessment of Homodyned K Distribution Modeling Ultrasonic Speckles from Scatterers with Varying Spatial Organizations

^{1}Department of Electronic Engineering, Yunnan University, Kunming City, Yunnan Province, China^{2}Cardiovascular Department, The Second Affiliated Hospital of Kunming Medical University, Kunming City, Yunnan Province, China

Correspondence should be addressed to Yufeng Zhang

Received 6 March 2017; Revised 9 June 2017; Accepted 22 June 2017; Published 5 September 2017

Academic Editor: Maria Lindén

Copyright © 2017 Xiao Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

*Objective*. This paper presents an assessment of physical meanings of parameter and goodness of fit for homodyned K (HK) distribution modeling ultrasonic speckles from scatterer distributions with wide-varying spatial organizations. *Methods*. A set of 3D scatterer phantoms based on gamma distributions is built to be implemented from the clustered to random to uniform scatterer distributions continuously. The model parameters are obtained by maximum likelihood estimation (MLE) from statistical histograms of the ultrasonic envelope data and then compared with those by the optimally fitting models chosen from three single distributions. Results show that the parameters of the HK distribution still present their respective physical meanings of independent contributions in the scatterer distributions. Moreover, the HK distribution presents better goodness of fit with a maximum relative MLE difference of 6.23% for random or clustered scatterers with a well-organized periodic structure. Experiments based on ultrasonic envelope data from common carotid arterial B-mode images of human subjects validate the modeling performance of HK distribution. *Conclusion*. We conclude that the HK model for ultrasonic speckles is a better choice for characterizing tissue with a wide variety of spatial organizations, especially the emphasis on the goodness of fit for the tissue in practical applications.

#### 1. Introduction

The ultrasonic imaging has many advantages over other techniques due to utilizing nonionizing radiation, scanning in real time, and distinguishing soft tissues with high sensitivity and resolution [1, 2]. The speckle, which manifests the granular structure in the ultrasound images, is caused by diffuse scattering of the ultrasound, and the background texture of the speckle is connected with the tissue microstructure. Therefore, the ultrasound imaging shows good potential for diagnosing diseases by statistical analysis of the speckle properties in the images to extract corresponding distribution parameters [1, 2]. Two kinds of statistic models including single distributions [3–5], such as the K distribution (K), Rayleigh distribution (RA), Rician distribution (RI), and Nakagami distribution, as well as compound distributions [6–9], such as the homodyned K distribution (HK), generalized K distribution, Rician inverse Gaussian distribution (RiIG), Nakagami-generalized inverse Gaussian distribution (NGIGD), have been investigated for analyzing the statistical properties of the ultrasonic-echoed envelope data. As commonly used models, the single distributions have been widely employed since the 1980s [3–5]. This kind of method is used as the histological descriptors with a one-to-one relationship between the distribution type and tissue characterization. According to the results from the researches, the K distribution corresponds to the tissue with low density of scatterers and without a deterministic component; the Rayleigh distribution refers to the tissue with high density of scatterers and a without deterministic component; the Rician distribution represents tissue with high density of scatterers and a deterministic component. On the contrary, the compound distributions model the tissue speckle pattern in images through modulating the parameters to represent the scatterer clustering degree or effective density, diffuse signal power, and coherent signal component [6–9]. The quantitative measurements, such as log-likelihood cross-validation or Kullback-Leibler distance, are used to verify the model performance.

As a generalized compound distribution, the HK model has drawn more attention over the other compound versions because its parameters present respective physical meanings from independent contributions in the scatterer distributions. In order to investigate the parameter meaning of the HK distribution, Prager et al. [10] described a method to estimate the ratio of the mean to the standard deviation and the skewness for the statistical model of the HK distribution based on arbitrary powers of the simulating ultrasound echo envelope signals. The parameters of the HK distribution were also estimated by the moments of the distribution. As a unifying point of view, Destrempes and Cloutier [11] compared the HK distribution and other statistical models based on theoretical computation for the modulated distribution, modulating distribution, and modulated parameters on the mean and the signal-to-noise ratio of the signal intensity. The authors conclude that the HK distribution is the only model that the parameters have their physical meanings in certain cases, even though the other distributions may better fit ultrasound signals. In addition, the authors suggest that the goodness of fit for HK distribution should be further assessed by the simulation or clinical test. Destrempes et al. [12] presented a new estimation method for the parameters of HK distribution by the mean intensity and two Log-moments. Then, they made a comparison between this method and the methods based on the first three moments of the intensity, the amplitude, or the signal-to-noise ratio (SNR), skewness, and kurtosis of two fractional orders of amplitude. The results indicate that this estimation method is the best. However, the method of moments for parameter estimation is deficient because the solutions of the equations based on the even moments are not always real or positive. The selection criteria for a set of parameters are various and nonunique, and the computational complexity for the high order moments is also a problem [13]. Moreover, the distribution based on the moment method may not be the optimum one for fitting ultrasound signals. As an important aspect in practice, applications of the best fitting ultrasound signals using statistical models involve tissue segment [14], speckle reduction [15], modeling for localizing a thin surgical tool [16], ultrasound kidney images [17], carotid artery plaque assessment [18], or classification of breast lesions [19]. It is necessary to assess the parameter meanings and goodness of fit of HK distribution for ultrasound echo signals under an optimum condition.

The objective of this paper is to assess the physical meanings of parameter and goodness of fit of HK distribution for ultrasonic envelope data from scatterer distributions with wide-varying spatial organizations by using maximum likelihood estimation (MLE) criteria. A 3D scatterer phantom based on gamma distributions is built to be implemented from the clustered to random to uniform scatterer distributions continuously. The model parameters and maximum likelihood estimation are obtained by MLE from statistical histograms of the ultrasonic envelope data. In order to evaluate the parameter meanings and goodness of fit, the mean and standard deviation (MSD) of these estimated values based on 30 simulation realizations are compared with those based on the optimally fitting models chosen from commonly used three single distributions, that is, the K, Rayleigh, and Rician (OKRR) distributions. Experiments based on ultrasonic envelope images from common carotid arteries (CCA) of 30 human subjects validate the simulation results of HK distribution for tissues with varying scatterer spatial organizations.

#### 2. Methods

##### 2.1. The Speckle Models

###### 2.1.1. Three Single Distributions

*(i)**Rayleigh Distribution*. The Rayleigh distribution [20] arises with a large number of scatterers in the effective resolution cell. The scattering structure is too fine to be resolved and fully forms a speckle pattern in ultrasonic B-mode images. The Rayleigh distribution for ultrasonic envelope amplitude is defined by
where represents the variance of scatterer strength. This distribution is a classical statistical model that assumes many fine randomly distributed scattering sites in the space without any well-organized periodic structure.*(ii)**Rician Distribution*. The Rician distribution [21] describes the analogous textures as the Rayleigh distribution, but the difference is the existence of the coherent signal echoed from the well-organized periodic scatterer structure to the diffuse signal from randomly distributed scatterers. The Rician distribution is expressed as
where and , respectively, are the variance and mean in scatterer strength. is the modified Bessel function of the first kind and order zero. The special case is that the Rician distribution becomes the Rayleigh model with a small value of or Gaussian model for .*(iii)**K Distribution*. Another commonly used model for ultrasonic envelope data is called the K distribution [22], which may describe the signals from the structures with a small number of scatterers in the effective resolution cell. The probability density function for the envelope amplitude can be written as
where is the modified Bessel function of the second kind and order . is the gamma function, and (where is the second moment of ). For the case of , this model turns into the Rayleigh distribution.

###### 2.1.2. The Homodyned K Distribution

The HK distribution [10], as a more universal statistical model, is used to describe the signals from the structure filling of variable density scatterers with or without well-organized periodic components. The HK distribution models the ultrasonic envelope amplitude by where , , and , respectively, denote the coherent component, diffuse component, and scatterer clustering degree in the signal. is the Bessel function of the first kind with order 0. The model shades into the K distribution with , the Rayleigh distribution with and , and the Rician distribution with . The parameters of the HK distribution have their own physical meanings of the independent contributions from clustered, random, and regular components in the scatterer distributions.

##### 2.2. Ultrasound-Echoed Data Simulation

In order to objectively and fully assess the HK distribution performance, it is required to synthesize a varied ultrasonic data source with the scatterer distributions, whose density and spatial organization can be tuned along the continuum from clustering to random to regular. In a present study, a 3D simulation for the ultrasonic envelope images is performed by the Field II software on the MATLAB platform. This library achieves this target by setting the scatterer phantom geometry, density, strength, and organization as well as the probe and ultrasonic scanning parameters in relevant functions.

A generalized Poisson process is used to setup the 3D scatterer distribution by a given scatterer number and phantom dimension as well as shape and scale parameters of the gamma distribution for the scatterer space. A one-dimensional scatterer model proposed by Cramblitt and Parker [23] is given by
where and are position and strength of the *i*th scatterer, respectively. A Poisson process is considered to define the distance between two continuous points [24]. In this case, the space of scatterers is the gamma distribution to generalize this Poisson process with the shape parameter and scale parameter as
where , , and . The mean and variance of the space are and , respectively. Therefore, the scatterer distribution could be characterized by the density parameter and shape parameter . For , the scatterer distribution is clustering with high space variance; with , the Poisson process with gamma distribution turns into exponential distribution, and the space is random; for , the scatterers are distributed evenly in the space with low space variance. In other words, is set to equality. Figure 1(a) demonstrates the one-dimensional scatterer positions with different values of shape parameter under a certain density condition (determined by ). The shape parameter is set as 0.01, 0.1, 1, 10, and 100, while the scale parameter changes with the by with 50 scatterers. In this figure, the clustering degree of scatterers is the highest for . With the value of the shape parameter increasing, the clustering degree is decreasing, and the scatterer positions are randomly distributed with , while the distribution tends to evenly spread as . Therefore, with the shape parameter increasing, the scatterer distribution is changing from clustered to random to regular continuously. By the given varied mean distance (density) and shape parameter, the scatterer distance distribution could be smoothly changed from irregularity to regularity, which makes this scatterer model agilely and continuously adjustable.