Abstract
The homodyned K distribution (HK) can generally describe the ultrasound backscatter envelope statistics distribution with parameters that have specific physical meaning. However, creating robust and reliable HK parameter estimates remains a crucial concern. The maximum likelihood estimator (MLE) usually yields a small variance and bias in parameter estimation. Thus, two recent studies have attempted to use MLE for parameter estimation of HK distribution. However, some of the statements in these studies are not fully justified and they may hinder the application of parameter estimation of HK distribution based on MLE. In this study, we propose a new parameter estimator for the HK distribution based on the MLE (i.e., MLE1), which overcomes the disadvantages of conventional MLE of HK distribution. The MLE1 was compared with other estimators, such as XU estimator (an estimation method based on the first moment of the intensity and tow log-moments) and ANN estimator (an estimation method based on artificial neural networks). We showed that the estimations of parameters α and k are the best overall (in terms of the relative bias, normalized standard deviation, and relative root mean squared errors) using the proposed MLE1 compared with the others based on the simulated data when the sample size was N = 1000. Moreover, we assessed the usefulness of the proposed MLE1 when the number of scatterers per resolution cell was high (i.e., α up to 80) and when the sample size was small (i.e., N = 100), and we found a satisfactory result. Tests on simulated ultrasound images based on Field II were performed and the results confirmed that the proposed MLE1 is feasible and reliable for the parameter estimation from the ultrasonic envelope signal. Therefore, the proposed MLE1 can accurately estimate the HK parameters with lower uncertainty, which presents a potential practical value for further ultrasonic applications.
Keywords
Introduction
Biopsies often serve as the gold standard for diagnosing certain diseases. However, a biopsy is associated with invasive procedures, sampling errors, and time and cost constraints. It is unnecessary particularly in the case of misdiagnosis or ambiguity.1 -3 Medical imaging, particularly ultrasound imaging, has emerged as a noninvasive alternative to biopsies. Compared to other medical imaging modalities, ultrasound imaging incurs a lower cost, is user-friendly, and uses non-ionizing radiation. Therefore, it has gained widespread acceptance and has been widely used in routine diagnostics.2,4,5 Ultrasound echo signals are generated by a process in which a transducer transmits ultrasound waves into a tissue being examined and receives the backscattered signal, which is caused by the interaction between the tissue and the incident waves. 4 Conventional B-mode ultrasound images are constructed from beam-formed radiofrequency (RF) echo signals after they have been subjected to scan conversion (resampling onto a rectangular grid), logarithmic compression, and other proprietary nonlinear mappings built into the ultrasound machine. 6 These processes are necessary to make raw ultrasound data readable to the human eye. However, they inevitably lose some useful properties that are theoretically derived from the RF signal. In addition, conventional B-mode images lack functional information and suffer from the drawbacks of being qualitative and operator-dependent.2,5 -10 Therefore, quantitative ultrasound (QUS) techniques, a complement to conventional B-mode ultrasound, have been proposed. In general, QUS refers to any method that extracts quantitative tissue information from ultrasound backscattered signals for ultrasound tissue characterization. Moreover, QUS techniques consist of spectral-based parameterization of ultrasound signals, flow estimation through Doppler, tissue elastography techniques, shear wave imaging, and envelope statistics.4,11 -18
Ultrasound backscatter envelope statistics, both statistical model-based and non-model-based, is an important group of QUS techniques. 2 It extracts statistical information related to tissue microstructures from the envelope of backscattered RF signals and has been applied to characterize different types of tissue. In particular, statistical model-based ultrasound envelope statistics parametric imaging techniques, in contrast to non-model-based techniques, such as ultrasound kurtosis imaging and ultrasound entropy imaging, have received broad interest in the past three decades.2,19 -23 Various statistical models have been proposed to estimate the information regarding sub-resolution material properties.4,24 Among them, the Rayleigh distribution is the most basic and provides the least amount of information. The Rayleigh distribution arises when numerous nearly identical and randomly located scatterers contribute to the echo signal. 11 The K distribution is another model that is considerably more general than the Rayleigh distribution. It can provide information on the scatter density, variation in the scattering amplitudes within the range cell, and mean scattering amplitude from the two parameters of the K distribution, namely, the effective number of scatterers and the scaling parameter.25 -27 The K distribution approaches the Rayleigh distribution under conditions of high values of the effective number of scatterers.4,11,24 The K distribution, although better than the Rayleigh distribution, is not general enough to account for the existence of periodic scatterers. This flaw can be addressed by adopting the Nakagami distribution or homodyned K (HK) distribution.25,26,28 Nakagami distribution, a good approximation of the HK distribution, has been the most frequently adopted model in tissue characterization because of its simplicity, even though the HK distribution is more versatile.1,4,26 In recent years, however, comparative studies have reported that the HK parameter, α, is more appropriate than the Nakagami parameter, m, to describe media with a high number of scatterers per resolution cell.4,7 In addition, the parameters of the HK distribution contain more information regarding the tissue microstructure than those of the Nakagami distribution.1,4 Therefore, with the development of effective parameter estimators, the HK model has been used in ultrasound parametric imaging for various medical applications, such as classification of breast lesions, temperature monitoring during focused ultrasound treatment, and hepatic steatosis evaluation.2,7,21,29 -31
To estimate the parameters of the statistical distribution models, methods based on maximum likelihood estimator (MLE), moments, or artificial neural networks (ANNs) are commonly adopted. For the HK distribution, Dutt and Greenleaf 32 first proposed a method based on the moments of the intensity (i.e., MI estimator). A method based on the first three moments of the amplitude was used to estimate the HK distribution in 1995 (i.e., MA estimator), but a closed-form expression of the amplitude moments could not be found until an explicit expression of an arbitrary moment of the amplitude was given by Hruska and Oelze. 34 Moreover, improved estimators for the HK distribution, which are based on the integer or fractional moments of envelope signals and ANNs, have been proposed. They include (i) the estimator based on the signal-to-noise (SNR) of fractional order moments of the amplitude 33 ; (ii) the level-curve estimator based on the SNR, skewness, and kurtosis of the fractional order moments of the amplitude, namely the RSK estimator 34 ; (iii) the estimator based on the mean intensity and two log-moments, namely the XU estimator 35 ; and (iv) the estimator based on an ANN, namely the ANN estimator. 36
MLE usually has desirable asymptotic properties.37,38 Additionally, comparative studies have presented parameter estimators based on MLE or maximum a posteriori estimation that have a small normalized mean squared error in the case of the Rice, K, or Nakagami distribution. 4 Thus, as a classical parameter estimation method, the MLE has been used to estimate the parameters of the HK distribution. Hu et al. 39 first used MLE to estimate parameters of HK distribution to assess the physical meanings of parameters and goodness of fit of HK distribution for ultrasound envelope data from scatterer distributions with wide-varying spatial organizations. They used a high-order global adaptive Gauss-Kronrod quadrature method to calculate the probability density function (PDF) of the HK distribution, and then used the Newton-Raphson method to numerically solve the log-likelihood equation. Moreover, Wang et al. 40 proposed a similar MLE for parameter estimation of HK distribution but adapted a cloud adaptive particle swarm optimization (CAPSO) to calculate the PDF of the HK distribution. However, these studies have significant limitations, and some of the statements are misleading. We will further discuss them in detail in Section 2.5.
MLE can be used to develop an accurate and stable parameter estimation method for the HK distribution, however conventional MLE is flawed. Therefore, there is an urgent need to propose an improved MLE which can overcome the shortcomings of conventional MLE and facilitate the application of parameter estimator for HK distribution based on MLE in ultrasound tissue characterization. In addition, the sample size is an important factor affecting the performance of the HK distribution parameter estimator, so we explored the dependence of the proposed MLE1 performance on the sample size. Furthermore, inspired by the work in Cristea et al., 7 we further explored the usefulness of the proposed MLE1 for characterization of both sparse and concentrated media. The work of this study is two folds. First, we proposed a novel parameter estimation method of the HK distribution based on the MLE (MLE1) after analyzing the deficiencies of conventional MLE of HK distribution. Second, we evaluated the feasibility and performance of the proposed MLE1 through simulation experiments based on independent and identically distributed (i.i.d.) samples from the HK distribution and ultrasound images.
The remainder of this paper is organized as follows. In Section 2, the theoretical background of parameter estimation of the HK distribution is presented. In Section 3, the proposed estimation method is explained, and computer simulations for sample sets of the HK distribution and ultrasound images are described. The simulation results and discussions are presented in Section 4, and the conclusions are provided in Section 5.
Theoretical Background
Homodyned K Distribution
The PDF of the HK distribution is significantly complicated and does not have a closed form. In 1987, the expression of HK distribution was introduced by Jakeman and Tough 27 in terms of an improper integral. It (two-dimensional) is defined as follows:
where A is the amplitude of the backscattered ultrasound envelope,
where
The PDF of HK distribution can also be defined in a compound form. 27
where
RSK Estimator
The RSK estimator estimates the parameters α and k of the HK distribution based on the SNR, skewness S, and kurtosis K from the backscattered ultrasound envelope. It can be defined as33,34
where A is the amplitude of the backscattered ultrasound envelope, v is the power of the amplitude, and
XU Estimator
The estimation method of the HK distribution based on the mean intensity and X- and U-statistics can be presented as follows 35 :
where X and U are defined as
The expressions of
where
ANN Estimator
Recently, a parameter estimator of the HK distribution based on an ANN was proposed, which has a much faster computing speed and a smaller normalized standard deviation for parameters α and k compared with the RSK and XU estimators.
36
The architecture of the ANN is shown in Figure 1, which is a four-layer feed-forward backpropagation (BP) network. It consists of an input layer, two hidden layers, and an output layer. By using equations (7) and (9), SNR R, skewness S, kurtosis K, and X- and U-statistics can be extracted from the i.i.d. envelope signal samples of the HK distribution with respect to different combinations of α and k parameters, and can be input into the corresponding neurons in the input layer. The numbers of neurons in the two hidden layers were 30 and 10, respectively. The output layer is made up of two neurons, which correspond to the

Architecture of the feed-forward artificial neural network (ANN) for the proposed ANN estimator of homodyned K model: α and k parameters in Zhou et al. 36 The ANN consists of an input layers, two hidden layers, and an output layer. Each layer contains several neurons.
Conventional MLE for Parameter Estimation of HK Distribution
The conventional MLE for parameter estimation of HK distribution mainly consists of two parts. A numerical integration method is used to calculate the PDF of the HK distribution, and then a numerical calculation method is used to find the parameters that maximize the log-likelihood equation.
First, the authors in Hu et al. 39 adapted a numerical integration function quadgk in the MATLAB platform to calculate the PDF of the HK distribution. Whereas the authors in Wang et al. 40 considered that significant oscillation is produced by the Bessel function contained in the PDF of HK distribution, therefore they used an advanced CAPSO algorithm to calculate the PDF of the HK distribution. However, such a statement is not fully justified. As mentioned above, the PDF of the HK distribution has two expressions. In fact, for different PDF expressions of HK distribution, the properties of their integrands are different. It can be seen from Figure 2 that for equation (1), its integrand oscillates and does not converge when α is small. In this case, the numerical integration method cannot be used to calculate the integral. Moreover, when the value of α is large, its integrand converges quickly and the function curve is very steep. Therefore, if equation (1) is used to calculate the PDF of the HK distribution, the performance of the numerical integration method will have a great influence on the calculation result. On the contrary, the integrand of equation (4) is convergent, and the function curve is flat. We can conclude that it is more reasonable to use equation (4) instead of equation (1) to calculate the PDF of the HK distribution. Thus, the oscillation may appear in Hu et al. 39 which uses equation (1) rather than in Wang et al. 40 which uses equation (4). Moreover, the performance of the numerical integration method has slight effect on the calculation results when adapting equation (4) and we will further verify it through simulation experiments.

The first row: the function curve of the integrand of equation (1) under the corresponding parameters. The second row: the function curve of the integrand of equation (4) under the corresponding parameters.
Second, both the article Hu et al.
39
and Wang et al.
40
use the Newton-Raphson method to find the critical point that maximizes the log-likelihood equation. And the authors believe that sole solution can be obtained based on the Newton-Raphson algorithm to solve the log-likelihood equation numerically.
40
However, this statement is true unless the log-likelihood equation has only one critical point in the parameter domain because Newton-Raphson method can only find one root near the initial value. Recall that the Maximum likelihood estimation can be defined as a point
Generation of Independent and Identically Distributed Samples From Homodyned K Distributions
To evaluate the estimators, Monte Carlo simulations are usually performed with i.i.d. samples from the HK distribution, which are generated with known parameters. The i.i.d. samples can be obtained as33,34
where
Moreover, if we restrict the average intensity to 1, we can obtain the following expression by using equations (2), (3), and (11). 36
Materials and Methods
In this section, first, we proposed a novel parameter estimator for HK distribution based on the MLE, using a numerical integration method to approximate the PDF of the HK distribution and a non-gradient global optimization algorithm to find a critical point that maximizes the log-likelihood function in the given parameter domain. Second, we carried out a series of simulation experiments based on the i.i.d. samples of HK distribution with known parameters. Specifically, (1) we adopted the method in Destrempes et al.
35
to test the proposed MLE1 through Monte Carlo computer simulation in the domain
Proposed MLE
The MLE is a generalized statistical method to estimate parameters, and it is defined as a critical point that maximizes the log-likelihood function 38 :
where
The flow charts of the proposed MLE1 for estimating the parameters of the HK distribution are shown in Figures 3 to 5. First, we obtained the i.i.d. HK distribution samples Ai to be fitted, and initialized the values of the parameters to be estimated for their upper and lower boundaries, respectively. In addition, the termination condition (i.e., MaxIterations and MeshTolerance) for this function needed to be specified, and the MeshTolerance was set at 0.001 in this study. Next, we used the pattern search algorithm to repeatedly update and iterate over the estimated parameters to minimize

Flow chart for MLE1.

Flow chart for computing log-likelihood equations using trapezoidal numerical integration method.

Flow chart for computing log-likelihood equations using numerical integration method based on CAPSO.
The following aspects are worth discussing with regard to the proposed MLE1 method. First, one may consider how to select the initial values of the parameters to be estimated and their upper and lower boundaries. In general, based on values estimated from ultrasound data, the range of parameter α is between 1 and 80, and between 0 and 1 for parameter k.
7
Additionally, according to property 1 discussed in the previous section, we can limit the value range of the HK distribution samples. For example, if we multiply all the samples by
Second, the upper limit of numerical integration must be chosen. According to the PDF of the HK distribution, it is known that the upper limit is supposed to be set to positive infinity in theory, but this is unrealistic in practice. Meanwhile, a larger upper limit corresponds to a large computational cost, which is time-consuming. Fortunately, this challenge can be dealt with well by using property 1. Based on the aforementioned example, when we limit the maximum value of all sample values to 10, the range of parameters ɛ and σ can be 0 to 10, and the range for parameter α is 0 to 80. In this parameter domain, we found that setting the upper limit of integration to 200 is sufficient to calculate the PDF well by comparing the PDF curve calculated by the numerical integration method with the histogram envelope of samples generated using equation (11) under the same parameters.
Finally, as the PDF of the HK distribution contains a Bessel function, it may lead to a very large value. Thus, there is a problem with the range of parameters to be estimated with respect to numerical considerations. Considering that the maximum positive floating point number of the MATLAB platform is
Simulations of Independent and Identically Distributed Samples From Homodyned K Distributions
To evaluate the performance of the proposed MLE1, we tested it through several Monte Carlo computer simulations by generating sets of i.i.d. samples of the HK distribution with known parameters. We adopted the same approach as in Destrempes et al.
35
to compare the reported experimental results. Sets of samples were generated in the domains
Previous simulation studies mostly focused on low numbers of scatterers per resolution cell, while it is meaningful to evaluate scattering properties from concentrated media. Inspired by the work of Cristea et al.,
7
we assessed the usefulness of the proposed MLE1 for estimating concentrated media using i.i.d. samples from the HK distribution. Sets of samples were generated in the domains
Recently, an ANN estimator based on an ANN has been proposed, which inherits some advantages of RSK and XU estimators by using five feature parameters from ultrasound backscatter envelope signals as its input. As shown in Zhou et al., 36 compared to the RSK estimator and XU estimator, the ANN estimator performs better in terms of NSD and RRMSE of the parameters to be estimated and the calculation speed. However, the disadvantage of an ANN is that its accuracy of parameter estimation is greatly affected by the sample size N. Within a certain range, the larger the size of the samples, the better is the performance of parameter estimation. As the ultrasonic backscatter envelope signal is generally not stationary and ergodic, its statistical characteristics may vary with the actual measurement position. A larger sample size corresponds to a larger actual measurement size, so using a larger sample size will make quantitative ultrasound images have lower resolution and weaker specificity. In general, we hope that the sample size of each estimation is as small as possible while ensuring that the sample size is statistically significant, and the parameter estimation results have acceptable accuracy.
To evaluate the impact of the sample size on the ANN estimator, we followed the method in Zhou et al.,
36
to estimate the parameter of the HK distribution when the sample sizes were 100, 1000, 5000, and 10,000 in the domains of
Estimation Based on Simulated Ultrasound Images
Experimental setup
To verify the performance of the proposed MLE1 in the ultrasonic envelope signal, the ultrasound simulation software Field II was used to simulate a series of ultrasound images. Field II software is an ultrasound simulation software based on the MATLAB platform, which generates ultrasound echo signals from the scatterer phantoms using the linear acoustics theory. A linear array transducer with a focal length of 20 mm was used, and it was excited with a Hanning windowed sinusoidal pulse. The number of physical elements was 512 and 64 for the active elements. The width and height of the element were 0.77 and 5 mm, respectively. Its center and sampling frequency frequencies were 10 and 100 MHz, respectively. The sound speed and wavelength were 1540 m/s and
We generated a three-dimensional computational phantom with dimensions of 20 mm (axial), 20 mm (lateral), and 2 mm (elevation). The center of the volume was located at the geometric focus of the transducer. The resolution cell was considered to have an ellipsoidal shape with diameters equal to the resolutions in each dimension: 0.154 mm (axial), 0.64 mm (lateral), and 0.64 mm (elevation) which was obtained by measuring the width of the envelope of the Point-Spread Function at −6 dB (Full Width at Half Maximum). This configuration leads to a resolution cell with a volume of 0.2642 mm
3
, based on an ellipsoid model. Thus, within the computational phantom, the average number of scatterers per unit volume (i.e., the density of scatterers) of Ns per resolution cell can be formulated as
Generation of simulation phantom and parameters estimation
To perform the simulations, a set of phantoms with various scatterer densities was scanned to generate the simulated ultrasonic echo signals. The signals containing only the diffuse component were simulated by placing randomly located scatterers in the phantom volume at spatial positions distributed according to a uniform distribution. The amplitudes of all scatterers were distributed according to a normal distribution of mean 0 and variance 1. Theoretically, the coherent component is related to the strength of the specular reflection or the periodic organization of the scatterers. Hence, periodically located scatterers along the transducer axis were used to simulate the ultrasonic echo signal containing a coherent component. The coordinates in the plane perpendicular to the transducer axis were randomly placed according to a uniform distribution, whereas their coordinates along that axis were separated by a constant distance,
Sequences of ultrasound images were simulated by randomly located scatterer phantoms with the number of scatterers per resolution cell varying from 1 to 10, which only contain diffuse components. A total of 60 simulated ultrasonic images were generated for each scatterer density value. To simulate phantoms that can generate ultrasonic echo signals that contain not only diffuse components but also coherent components, a fixed density of five randomly located scatterers per resolution cell was considered, and the scatterer density for periodic scatterers was varied from 1 to 5 scatterers per resolution cell with a step of 1. A total of 60 images were simulated for each scatterer density value.
For each simulated sequence, the simulated ultrasonic envelope signals were estimated using the MLE1, RSK, XU, and ANN. The estimated parameter values of α and 1/(k + 1) were averaged over 60 images within a region of interest (ROI) . The ROI was a rectangular area centered at the geometric focus of the simulation phantom, and the size of sample N was 2000. The parameter α is called the effective number of scatterers and can be expressed as
Results and Discussion
Computer Simulation of Independent and Identically Distributed Samples From HK Distributions
The total absolute RRMSE, RB, and NSD obtained from the parameters that were estimated by MI, MA, RSK, XU, and the proposed MLE1 in the domains of
Main Parameters and Statistics Discussed in This Paper.
Improvements in Bias and Variance of Estimators. The Sample Size is N = 1000. The Parameter Domain is

Relative bias (RB), normalized standard deviation (SD), and relative root mean squared error (RRMSE) of the parameter estimates based on the MLE1. The sample size is N = 1000. The numerical integration method adopts CAPSO.

Relative bias (RB), normalized standard deviation (SD), and relative root mean squared error (RRMSE) of the parameter estimates based on the MLE1. The sample size is N = 1000. The numerical integration method adopts trapezoidal numerical integration method.
Figure 8 shows the biases and standard deviations for the estimated parameters using the proposed MLE1 in the domains of

Standard deviations and biases for parameter α and k as a function of the true values of α and k for sample size (N): 1000. Each value was obtained by averaging 100 estimations.
Table 3 shows the mean RB, NSD, and RRMSE of the parameter estimated by the ANN estimator in the domains of
Total Absolute Value of RB, NSD, and RRMSE Between the Parameter Estimated by the ANN Estimator or MLE1, and the Reference Standard of α and k for Different Sample Size, N, Based on 100 Estimates of Computer Simulations. The Parameter Domain is

Relative bias (RB), normalized standard deviation (SD), and relative root mean squared error (RRMSE) of the parameter estimates based on the MLE1. The sample size, N, is 100. The numerical integration method adopts CAPSO.
Estimation Based on Simulated Ultrasound Images
Examples of simulated B-mode images corresponding to different concentrated simulated phantoms, described in Section 3.3, are illustrated in Figure 10, where log-compression and downsampling were applied to the simulated ultrasonic echo signal for visualization. Figure 11 shows the variation curve of the theoretical parameter values and the corresponding estimated parameter values obtained from the ultrasonic echo signal containing only the diffuse component using different estimators. As can be seen from this figure, the scatterer clustering parameter α increased with the growth of the average number of randomly located scatterers per resolution cell, and the results of parameter estimation using MLE1 are closer to corresponding theoretical values than that using other estimators. Furthermore, examples of the histogram of the ultrasonic echo envelope signals that contain both diffuse and coherent components and the probability density function curve of the HK distribution plotted by the parameters estimated by using the MLE1 from the corresponding envelope signals are shown in Figure 12. It can be observed that the curve drawn by the estimated parameters fits the histogram envelope well. The mean RB, NSD, and RRMSE for α were .1176, .0596, and .1357, respectively, and for k were .2847, .1686, and .3678, respectively.

Examples of simulated B-mode images with Ns randomly located scatterers per resolution cell and Nc periodic scatterers per resolution cell. Top left: Ns = 1, Nc = 0; Top middle: Ns = 5, Nc = 0; Top right: Ns = 10, Nc = 0; Bottom left: Ns = 5, Nc = 1; Bottom middle: Ns = 5, Nc = 3; Bottom right: Ns = 5, Nc = 5; A log-compression was applied to the echo envelope solely for visualization purposes.

Value of the scatterer clustering parameter, α (left), and of parameter 1/(k + 1) (right) as a function of the average number of randomly located scatterers per resolution cell. No coherent component was included in these simulations. The estimated values are averaged over 60 samples, and the sample size, N, is 2000.

Examples of the statistical histogram of different the ultrasonic echo envelope signals that contain both diffuse and coherent components and the probability density function curve of the HK distribution plotted with the parameters estimated by the MLE1 from the corresponding envelope signals. Ns is the number of randomly located scatterers per resolution cell. Nc is the number of periodic scatterers per resolution cell. L denotes the value of the log-likelihood equation under the corresponding estimation parameters (σ, ɛ, α).
Note that the theoretical value of k is calculated as
Discussion
In this paper, we systematically investigated the application of MLE in parameter estimation of HK distribution and proposed a new MLE for the HK distribution on the basis of overcoming the mistakes or deficiencies in the conventional MLE. We conclude that it is more reasonable to use equation (4) to calculate the PDF of HK distribution, and the performance of numerical integration method has no obvious influence on the accuracy of parameter estimation. One can choose specific numerical integration methods based on actual demand. Moreover, the results obtained from the simulation experiments using simulated i.i.d. samples from the HK distribution and simulated ultrasonic echo signals confirmed the feasibility of using the proposed MLE1 to obtain accurate parameter estimation for the HK distribution. In addition, compared to using the XU estimator, the proposed MLE1 had a smaller bias and variance in the parameter estimation of concentrated media. This advantage will greatly contribute to the diagnosis of diseases when the tissue has a high number of scatterers per resolution cell based on statistical QUS technology, such as the classification of cancerous lymph nodes. Furthermore, there was a weak dependency between the performance of the proposed MLE1 and the sample size, and a relatively accurate parameter estimation can be obtained even when the sample size is small. This is a considerable advantage of the proposed MLE1 over the latest XU estimator and ANN estimator because the hypothesis that speckles over a wide area of tissues scanned in clinical disease detection that have the same statistical properties may fail. Although there are several methods to increase the number of uncorrelated samples, as mentioned in Cristea et al., 7 the parameter estimation method based on a small sample size is more popular because it can effectively improve the resolution and specificity of quantitative ultrasound images and thus enhance the accuracy of disease diagnosis. Moreover, as shown in Figure 11, in the simulation experiments based on ultrasonic echo signals, the proposed MLE1 yields a more reliable and stable HK parameter estimation compared to other estimators. In conclusion, compared with other estimators, the MLE1 proposed in this paper is more accurate and versatile. It can characterize in a wide range of number density of scatterers and is less dependent on sample size. Therefore, it will further improve the accuracy and applicability of the statistical QUS technology based on HK distribution.
It is well-known that MLE has the disadvantage of being more time-consuming than the methods of moments and ANNs. In the case of the first experiment detailed in Section 3.2, one estimation took an average of 4s (using the traditional trapezoidal numerical integration method) and 8s (using the numerical integration method based on CAPSO). The computational load of the MLE1 is determined by the selection of the numerical integration method, global optimization algorithm, and sample size. In addition, the accuracy of parameter estimation using the proposed MLE1 is significantly affected by the global optimization algorithm. It is worth mentioning that the selection of the initial values and upper and lower boundaries of the parameters to be estimated in the pattern search global optimization algorithm used in this study plays an important role in the performance of the MLE1. Therefore, we can use some HK distribution parameter estimators with fast operation speed, such as XU and ANN estimators, to first estimate the HK distribution parameters, and then set the estimated parameter value as the initial value, and set the upper and lower boundaries of the parameters to be estimated near the initial value. This operation may make the MLE1 faster and more accurate. Moreover, the pattern search algorithm used in this study was determined by comparing several common gradient-free global optimization algorithms, such as the particle swarm optimization algorithm and genetic algorithm, under a trade-off between the calculation cost and parameter estimation accuracy. In future work, new and effective numerical integration methods and global optimization algorithms can be used to increase the accuracy and efficiency of the MLE1.
Note that the scatterers used in the field II ultrasound simulation are treated as dimensionless points with a position and amplitude. This hypothesis is inappropriate for biological tissues, as described in Section 5.4 of Destrempes et al. 35 Furthermore, the validation of the proposed MLE1 in this study is based on simulation data, which is a limitation of our study. Therefore, in future work, more clinical experiments can be conducted to further validate the performance of the proposed MLE1. Moreover, this paper only discusses the usefulness of the proposed MLE1 and XU estimator for the characterization of both sparse and concentrated media, more comprehensive comparative experiments involving more estimators should be performed to further evaluate the performance of each estimator.
Conclusions
In this study, we proposed an effective parameter estimation method of the HK distribution based on the MLE after analyzing the deficiencies of conventional MLE of HK distribution. The simulation experiment results confirmed the feasibility of the proposed MLE1 and indicated that the proposed MLE1 is more accurate and robust than other estimators when estimating parameters α and k of the HK distribution. Thus, we can conclude that the proposed MLE1 has potential practical value for ultrasonic parameter imaging in tissue characterization.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Grants from the Yunnan Fundamental Research Projects (No. 202101AS070031), and University Key Lab of Electronic Information Processing of High Altitude Medicine, Yunnan Province.
