A Study on the Effects of Depth-dependent Power Loss on Speckle Statistics Estimation

Abstract

Characterization of the interference patterns observed in B-mode images (i.e. speckle statistics) is a valuable tool in tissue characterization. However, changes in echo amplitudes unrelated to speckle, including power loss due to attenuation and diffraction, can bias these metrics, undermining their utility. Tissue with high attenuation such as the uterine cervix are especially affected. The purpose of this study is to demonstrate and quantify the effects of attenuation and diffraction on speckle statistics and to propose methods of compensation. Analysis was performed on simulated diffuse-scattering phantoms of varying attenuation with simulated transducers at 9 MHz and 5 MHz center frequency. Application in the in vivo macaque cervix using a clinical scanner is also presented. Parameters of the Nakagami and homodyned K distribution were calculated in parameter estimation regions (PERs) of varying size within simulations and experiments. Changes in speckle statistics parameters with respect to PER size and depth were compared with and without two different compensation schemes. It is shown that compensation for attenuation and diffraction is necessary to produce speckle statistics estimates that do not depend on the attenuation of the medium or the PER size. Reducing the dependence on these factors connects speckle statistics estimates more closely with the microstructure of the probed medium.

Introduction

The estimation of speckle statistics is a sub-field of quantitative ultrasound which seeks to characterize tissue microstructure by analyzing the speckle patterns ubiquitous in diagnostic ultrasound images. Because it is an interference pattern arising from microstructure, speckle can reveal valuable information which would not be easily extracted in qualitative ultrasound imaging [1, 2, 3]. Successful implementation of speckle statistics estimation for tissue characterization has been shown in liver [4, 5, 6], breast [7, 8, 9], vasculature [10, 11, 12], and various other tissue characterization applications [13, 14, 15, 16].

The prevailing paradigm in first order speckle statistics is that ultrasonic speckle is the result of a sum of individual phasors which represent reflected pulses detected by the signal receiver when a waveform interacts with a discrete, weakly scattering structure. Scatterer amplitude and position are assumed to be independent of one another, and each scatterer is considered independent from other scatterers [3]. Without direct observation of these microscopic components, each envelope amplitude sample can be considered as a statistical process occurring within the “resolution cell,” defined here as the finite volumetric region encompassing signal intensities within 6 dB of the peak value for a point scatterer placed within it [2]. Statistical models such as the homodyned K [17] or Nakagami [18] distributions are used to describe the macroscopic texture that emerges in the received signal envelope. The $\alpha$ parameter of the homodyned K distribution can be interpreted as the “scatterer clustering parameter” related to the number density of scattering structures, and the derived $k$ parameter relates to the ratio of coherent to diffuse scattering which arises from organized and spatially random scatterer structure, respectively [3]. The $m$ parameter of the Nakagami model is affected by both the number density of scatterers and the presence of coherent scattering [18].

Speckle statistics analysis relies on the magnitude of envelope signal samples. The phasor model does not account for power loss from attenuation and diffraction. Noting this, authors in the speckle statistics literature of biomedical ultrasound have applied power loss compensation in a wide variety of ways. Dutt and Greenleaf [17] found that the use of exponential time gain compensation function decreased variance in parameter estimation for the homodyned K distribution. Hao et al. [19] demonstrated that increasing attenuation in simulations from 0 to 2 dB cm⁻¹ MHz⁻¹ causes increasing bias in estimated homodyned K parameters. They proposed an attenuation compensation method in which local attenuation was estimated and then applied inversely to the respective section of acquired RF data. Cristea et al. [20] note that attenuation compensation is “essential” for envelope statistics estimates, since high attenuation causes a broadening of the envelope distribution. The authors applied point attenuation compensation to the spectra of acquired RF signals and transformed them back to the time domain for parameter estimation.

The application of speckle statistics in a wide number of tissues has motivated clinical interest, with initial commercial implementation in hepatic steatosis assessment [21, 22, 23]. Additionally, many clinical ultrasound devices allow the user to extract raw data for research purposes, opening the possibility for the neglect of crucial steps. Thus it is increasingly important to track and report these sources of measurement error. While previous studies have applied attenuation compensation in specific circumstances, there is little agreement on best practices, and many speckle statistics investigations do not address it at all. To our knowledge there is no systematic study of the effects of depth-dependent power loss on speckle statistics estimates. Our investigation seeks to remedy this shortfall by increasingly applying attenuation to simulated phantoms and exploring the resulting speckle statistics estimates at different parameter estimation region sizes and depths. The next section describes the creation of the ultrasound speckle simulations with varying degrees of applied attenuation that were analyzed. Simulations were used because they allow a well-controlled, systematic variation of physical properties. To provide evidence of generality, acquisitions of a homogeneous phantom and in vivo macaque cervix on a clinical scanner were also explored. Then, two methods of compensation for attenuation and diffraction were explored. Lastly, speckle parameters were estimated and summarized. It was found that depth-dependent power loss significantly affects speckle statistics parameter estimation, but that this estimation error can be compensated using measured and theoretical attenuation estimates.

Methods

Simulations

Simulations of ultrasonic speckle with and without applied attenuation were created in Field II [24, 25]. Point scatterers in Field II maintain the assumptions of weak scattering and scatterer independence while allowing the observation of constructive and destructive interference of reflected pulses essential to speckle statistics analysis. A simulated 64-element transducer with a 9 MHz central frequency was focused at 5 mm with an f-number of 2. These parameters were chosen due to similarity to those used for transvaginal imaging of the human cervix [26], a highly-attenuating tissue which presents challenges for quantitative ultrasound [27, 28]. Another 64-element 5 MHz transducer was simulated with the same focal distance and an f-number of 1. Frequency-dependent attenuation was set to 0 dB cm⁻¹ MHz⁻¹, 0.5 dB cm⁻¹ MHz⁻¹, (e.g. liver, cardiac muscle, fat [29]), 1 dB cm⁻¹ MHz⁻¹ (e.g. skeletal muscle, kidney [29]), or 2 dB cm⁻¹ MHz⁻¹ (e.g. cervix tissue [27, 30], connective tissue [29]). Element pitch was selected at appropriately small lengths to avoid grating lobes, at 100 and 200 μm respectively, and element height was 2 mm. The virtual transducer was excited by a two-cycle sine function, with an impulse response modulated by a Hann window. Thirty independent frames of RF data were generated for each transducer and attenuation value. Signal envelopes were obtained by calculating the absolute value of the Hilbert transform on each scan line.

A unique diffuse-scattering virtual phantom was created for each simulated frame by randomly generating uniformly-distributed 3-dimensional spatial coordinates for the positions of point scatterers. The total number of scatterers in the phantom was chosen so that there were ten scatterers on average in each theoretical resolution cell, which is typically used as the “rule-of-thumb” to produce fully-developed speckle [31, 32]. The axial, lateral, and elevational resolution lengths ($d_{\mathit{ax}}$, $d_{\mathit{lat}}$, and $d_{\mathit{el}}$ respectively) were calculated following the theoretical full-width at half-maximum for a Gaussian pulse emitted from an array transducer,

$$d_{\mathit{ax}}=\frac{1}{2}{N}_c\lambda$$ (1)

$$d_{\mathit{lat}}=1.2\lambda F∕D$$ (2)

$$d_{\mathit{el}}=1.2\lambda F∕h$$ (3)

where $\lambda$ is the carrier wavelength, $N_c$ is the number of cycles of the pulse, $F$ is the focal distance, $D$ is the length of active aperture, and $h$ is element height [33, 2]. The resolution cell volume was modeled as an ellipsoid with its three axes defined by the equations above. The size of the virtual phantom was two resolution lengths elevationally (0.51 mm for 9 MHz, and 0.92 mm for 5 MHz), 6 cm laterally, and 3 cm axially.

In addition to these homogeneous simulations, another virtual phantom with a diffuse scattering background of 10 scatterers per resolution cell and a 1 cm cylindrical sparse scattering inclusion with 1 scatterer per resolution cell was simulated. The inclusion was positioned in the center of the phantom at a depth of 2 cm, and a 9 MHz simulated transducer focused at 2 cm with an f-number of 2 was used. An attenuation of 0.5 dB cm⁻¹ MHz⁻¹ was applied. Thirty independent frames of RF data were generated. Another set of thirty frames without the low scatterer density inclusion using the same simulated transducer settings and applied attenuation were also produced for power loss compensation.

Experimental Validation

To demonstrate applicability to clinical ultrasound, in vivo scans in non-human primates were also investigated. A 4-weeks pregnant rhesus macaque was scanned using Siemens ACUSON Sequoia clinical ultrasound system (Siemens Healthineers, Issaquah, WA, USA). An 18H6 prototype transrectal probe operated at 8 MHz was placed against the anterior rectal wall to obtain longitudinal views of the cervix. Eight images of the cervix within a similar field-of-view but non-identical imaging plane were obtained. Access to raw IQ data was provided through a research agreement with Siemens Healthineers. This macaque study was approved by the Institutional Animal Care and Use Committee at UW-Madison under project number G006499-A03.

To enable correction for depth-dependent power loss from system diffraction, a phantom was also scanned using the same parameters as the in vivo acquisitions. The phantom was manufactured in-house and produces a homogeneous speckle pattern from diffuse scattering. The attenuation coefficient and speed of sound have previously been measured at 0.5 dB cm⁻¹ MHz⁻¹ and 1560 m/s, respectively, using a through-transmission method [34]. Thirty independent frames of IQ data were obtained and used for analysis.

Power Loss Compensation

Power loss compensation was performed using one of two methods. The first method, “compensation-by-average,” relies only on acquired data and does not estimate attenuation or incorporate knowledge of attenuation value. The average envelope A-line $\overline{A(z)}$ for a given attenuation value was computed over the acquired A-line envelopes from all data frames. When compensation-by-average is applied, each measured A-line $A_{\mathit{Raw}}(z)$ is divided by the normalized average A-line to create the compensation-by-average signal, $A_{\mathit{CBA}}(z)$, before calculating speckle statistics:

$$A_{\mathit{CBA}}(z)=\frac{A_{\mathit{Raw}}(z)}{\overline{A(z)}∕{\max }_z(\overline{A(z)})}$$ (4)

The same steps were performed in the lateral direction using the mean lateral profile. Although this method is unlikely to be practical in tissue due to heterogeneity and lack of sufficient independent data, the results can provide an upper limit to the best possible compensation.

The second method, “compensation-by-theory,” requires a mixture of acquired data and a knowledge of the theoretical power loss that should occur due to attenuation. Diffraction effects of the simulated system were measured via A-line and lateral profile averaging in the non-attenuated data set. Each frame in the attenuated data sets were divided by the normalized average non-attenuated lateral and axial profiles. Then, attenuation was compensated for by multiplying each A-line by a time-gain compensation factor:

$$\exp \left(\frac{2za{f}_c(z)}{8.686}\right)$$ (5)

where $a$ is the known attenuation (dB cm⁻¹ MHz⁻¹), $z$ is the signal depth (cm), and $f_c$ is the depth-dependent center frequency (MHz).

In order to calculate the depth-dependent compensation, the depth-dependent center frequency was measured from the simulated data. A multi-taper power spectral estimator [35] was applied to the simulated RF signals in 4 mm x 4 mm power spectral estimation regions (PSERs) which overlapped by 85%. The frequency of the spectral peak was plotted as a function of PSER center depth, and a least-squares regression was performed to obtain a linear function of peak frequency with respect to depth.

Because the attenuation was known and applied homogeneously to the simulations, this compensation could be performed with perfect a priori knowledge. However, attenuation estimation in biological tissue is complex and prone to error in estimation. In order to demonstrate the degree of accuracy needed to sufficiently compensate for attenuation effects on speckle statistics, an artificial fractional error $\gamma$ was applied to the attenuation $a$ using a multiplier of ($(1+\gamma )$). The fractional error was given values of 0, 0.25, 0.5, 0.75 or ± 1.

Envelope Statistics Estimation

The Nakagami and homodyned K distributions are the most commonly used models for speckle statistics tissue characterization, and are therefore of interest for this work.

The homodyned K distribution has three parameters: $\alpha$, $\sigma$, and $ϵ$ [17]. The $\alpha$ parameter is also known as the “clustering parameter” and is related to the local scatterer number density. A derived parameter $k$ is often used when characterizing tissue using the homodyned K distribution, defined as the ratio of coherent to diffuse scattering, $ϵ∕\sigma$. The $\alpha$ and $k$ parameters were estimated using the neural network architecture proposed by Zhou et al. [36] and later improved by Gao et al. [37]. The network for this study closely follows this architecture and was trained on 500 independent, identically-distributed samples of the homodyned K distribution each for values of $\log \alpha$ ranging from −1 to 2 in steps of 0.01 and values of $k$ ranging from 0 to 2 in steps of 0.01.

The Nakagami distribution has two parameters, $m$ and $\Omega$, but only $m$ is related to scatterer properties. $\Omega$ is related to total signal energy, and is thus operator dependent and not typically used for speckle statistics analysis. The use of only one parameter results in some ambiguity between signal coherence and scatterer number density. The Nakagami $m$ parameter can differentiate between large numbers of randomly-positioned scatterers (i.e. diffuse scattering) where $m=1$, sparse or clustered scattering where $m<1$, and spatially periodic scatterers where $m>1$ [18]. Parameter estimation was performed using a maximum likelihood estimator in which maximum likelihood is solved through a binary search algorithm [3].

In the homogeneous simulations, the region of interest (ROI) in which parameters were estimated spanned the lateral extent of the simulated frame. The axial extent of the ROI spanned the depths of 3.6 mm to 11.3 mm. Before parameter estimation, each frame of envelope data was compensated for power loss using one of the above mentioned methods, or power loss compensation was neglected for comparison. The ROI in each frame of data was then subdivided into smaller, overlapping parameter estimation regions (PERs) for speckle statistics analysis. Speckle statistics were estimated from the envelope data contained within each PER, and the average estimate amongst PERs was taken as the parameter estimate for that frame. The parameter estimate presented for each transducer and attenuation value is the average of parameter estimates for all frames in the appropriate simulated or acquired data set.

When selecting a PER size, there is a trade off between estimate stability and resolution [38]; however, the effects of power loss will be more prominent at larger PER sizes. To probe these effects, PER size was varied between an axial extent of 1 mm and 6 mm. Lateral PER size was kept to a constant 2.5 mm for the 9 MHz data and 4 mm for the 5 MHz data, which were chosen to roughly match the lengths of the respective simulated transducers. Choosing a large lateral PER size was done to reduce estimate variance as much as possible. PERs were allowed to overlap by 50% in homogeneous simulations, and 90% to create parametric images. A schematic of the division of diffuse scattering simulations into an ROI and PERs is provided in Figure 1a.

Figure 1:

a.) The region of interest (ROI) for the diffuse scattering phantom simulations spans nearly the entire lateral extent of the image and the axial range of 3.6 11.3 mm. The ROI is divided into parameter estimation regions (PERs) which reside entirely within the ROI. PERs are swept axially through the ROI with a step size of the axial PER extent, so that they overlap by 50%. b.) B-mode image of one frame of the macaque cervix acquisition, showing the PER that was used for speckle statistics estimation within a delineation of the cervix boundaries. The PER spans 9.8-14.1 mm axially and −6.6-2.2 mm laterally in all eight acquired frames.

An investigation of speckle statistics estimates with and without compensation for power loss was also performed while varying the depth of PERs within the region of interest. In this experiment, the axial PER extent was kept to a constant length of 3 mm and PERs were only estimated at a single selected axial depth. The mean of each speckle statistics estimate amongst all simulated frames was calculated at depths of 5.6, 7.1, 8.6, and 10.1 mm, with the PER centered axially at each depth.

Speckle statistics were additionally estimated within the experimental phantom. As in the simulations, the lateral extent of each PER was kept to a constant 4 mm, and the axial extent was varied between 1 mm and 6 mm. Compensation by the average phantom scan line was applied, and the resulting estimates were compared to estimates made with no compensation.

Due to the tissue heterogeneity of the macaque cervix, only one PER in a relatively homogeneous speckle region of the anterior cervix was used. An image of the PER overlaid on a B-mode image of the cervix is shown in Figure 1b. The experimental phantom scans were used to compensate for power losses due to diffraction. To isolate power loss due to system diffraction, each scan line of the phantom acquisitions was multiplied by the depth-dependent factor in Equation 5 using the nominal attenuation coefficient of 0.5 dB cm⁻¹ MHz⁻¹. Then, the mean normalized attenuation-free phantom scan line was used to remove depth-dependent diffraction effects on signal power from the in vivo acquisitions, using Equation 4. Finally, to compensate for tissue attenuation in the cervix, each in vivo acquisition scan line was multiplied by the depth-dependent factor in Equation 5 using a literature-informed [27] attenuation coefficient of 1.5 dB cm⁻¹ MHz⁻¹.

Statistical Analysis

A Kruskal Wallis test was used to detect the presence of significant differences in speckle statistics parameter estimates due to the power loss effects of attenuation and diffraction. Parameter estimates from each PER were grouped by the applied attenuation of each simulation, and the statistical significance of the differences in speckle statistics parameter were tested at each PER size. In the case of the in vivo experiment, the statistical significance of differences in speckle statistics estimates before versus after compensation was probed. The null hypothesis in all cases is that all estimates come from the same distribution, since they have equivalent underlying scattering properties.

Results

Simulations

Figure 2a illustrates an intermediate step in speckle statistics estimation, demonstrating that depth-dependent power loss can change the observed envelope statistics. The envelope amplitude histogram of a single 9 MHz simulated data frame from a 1 dB cm⁻¹ MHz⁻¹ diffuse scattering phantom simulation is shown, extracted from the 5 mm to 6 mm region (blue) and the 5 mm to 8 mm region (orange). The histogram skews noticeably to lower amplitudes when lengthening the PER. Since speckle statistics estimates provide information about histogram shape, they will change with axial PER length even when underlying scatterer conditions are the same. Figure 2b demonstrates that it is possible to remove these biases by compensating for depth-dependent power losses, in this case with “compensation-by-average.” After correction is applied, the shapes of the histograms of a 1 mm and a 3 mm PER are similar.

Figure 2:

a.) When power loss as a function of depth is not considered, changing the axial extent of the PER from 1 mm to 3 mm introduces changes in the shape of the envelope histogram. b.) When compensating for power losses with compensation-by-average, the PER length has less impact on the observed statistics.

Nakagami maximum likelihood estimates and homodyned K neural network estimates as a function of axial PER length without any compensation for power loss are provided in Figure 3a-c for the 9 MHz diffuse scattering simulations. A clear negative trend can be observed, and even in the case where no attenuation is applied a downward trend exists due to beam diffraction. The Kruskal Wallis p-value did not exceed 0.001 at any PER length for Nakagami or homodyned K parameters; that is, the attenuation applied in these simulations causes statistically significant changes in parameter estimates. Figure 3d-f reveals similar results for the 5 MHz simulations. The negative trend is consistent, although to a lesser degree due to the decreased effects of attenuation at lower probe frequencies. The only case in which the null hypothesis is confirmed is the homodyned K $k$ estimate at 1 mm PER extent (p = 0.222). In all other cases, the p-value did not exceed 0.016, similarly revealing that power loss within PERs causes statistically significant changes in speckle statistics estimation.

Figure 3:

Speckle statistics estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter in 9 MHz simulations and d.)-f.) in the 5 MHz simulations as a function of axial PER span. No power loss compensation has been applied. A clear downward trend can be observed in all parameters, with severity correlating with applied attenuation in each set of simulations.

When power loss is accounted for by the “compensation-by-average” method, trends with PER length become negligible, as Figures 4a-c and 4d-f demonstrate for the 9 MHz and 5 MHz simulations respectively. After compensation, the null hypothesis was confirmed for every parameter estimate at all PER sizes, with p-value not less than 0.141 for the 9 MHz case and 0.074 for the 5 MHz case. Intuitively, this is because the compensation is exactly matched with the data, and this represents the best-case scenario for compensation. Unfortunately, the decrease in estimate variance observed when compensation was not applied is no longer as pronounced. It may be that the improvement in variance seen in the uncompensated data was somewhat “forced” due to the leftward skew of the amplitude histogram; indeed, Nakagami and homodyned K estimate variance are expected to decrease as parameter value decreases [39, 20].

Figure 4:

Speckle statistics estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter in 9 MHz simulations and d.)-f.) in the 5 MHz simulations as a function of axial PER span. “Compensation-By-Average” has been applied to envelope frames.

“Compensation-by-theory” also removes the negative trends due to power loss. This method represents the best outcome that may be achieved practically, where it is unlikely that there would be enough independent homogeneous data to perform compensation by averaging. The 9 MHz case is demonstrated in Figure 5a-c, and the 5 MHz case is presented in Figure 5d-f, with no artificial error applied to the known attenuation value. Nakagami and $\alpha$ parameter estimates at 1 cm had the lowest p-value at 0.034 in the 9 MHz case; all others were greater than 0.07, confirming the null hypothesis. As in the “compensation-by-average” results, the decrease in estimate variance is somewhat lost.

Figure 5:

Speckle statistics estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter in 9 MHz simulations and d.)-f.) in the 5 MHz simulations as a function of axial PER span. “Compensation-By-Theory” has been applied to envelope frames with no artificial error.

Figures 6 and 7 provides speckle statistics estimates in the 9 MHz simulations as a function of PER center depth while maintaining a PER length of 3 mm, before and after compensation-by-theory. These results are relevant for a clinical context in which the depth of a selected region of interest may vary over time or in different patients. A noticeable downward trend in speckle statistics is observed in the absence of compensation for all levels of applied attenuation, suggesting that diffraction is the driver of estimate bias when PER length is kept constant. The applied compensation for power loss appears sufficient to mitigate this bias.

Figure 6:

Speckle statistics estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter in 9 MHz simulations as a function of the depth of the center of each PER. The axial PER span was kept constant at 3 mm. No compensation has been applied.

Figure 7:

Speckle statistics estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter in 9 MHz simulations as a function of the depth of the center of each PER. The axial PER span was kept constant at 3 mm. “Compensation-By-Theory” has been applied to envelope frames with no artificial error.

To show how this analysis is affected by error in the supposed attenuation value, Figure 8a-c provides speckle statistics estimates in the 9 MHz simulations as a function of PER size, with the attenuation set to 1 dB cm⁻¹ MHz⁻¹. Figure 8d-f presents results of the same analysis in the 5 MHz case. A range of artificial error values was applied to the attenuation value used for compensation. The case in which the applied error $\gamma$ is equal to −1 represents a compensation that accounts only for diffraction effects, since the exponent of Equation 5 would equate to zero. Interestingly, an applied error as high as 25% does not appear to distort parameter estimates. Higher errors in attenuation value, however, neutralize attempts at power loss compensation to an increasing extent.

Figure 8:

Speckle statistics estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter in 9 MHz simulations and d.)-f.) in the 5 MHz simulations with 1dB cm⁻¹ MHz⁻¹ attenuation as a function of axial PER span. “Compensation-By-Theory” has been applied to envelope frames with varying degrees of artificial error.

Homodyned K $\alpha$ parametric images of the simulated inclusion phantom with and without compensation are provided in Figure 9. Figure 9a presents the case of no compensation for power losses, and Figure 9b presents the parametric image after compensation by the average frame without the low scatterer density inclusion. A ring of low $\alpha$ values surrounding the inclusion are caused by PERs encompassing two regions of different scattering properties, and are an artifact. As can be observed in the images and in the mean axial profiles presented in Figure 9c, the compensation improves the regional uniformity of $\alpha$ estimates in a way that better represents the two distinct scattering regions of the simulated phantom.

Figure 9:

Parametric images of homodyned K $\alpha$ estimates in the inclusion phantom experiment a.) without any compensation for power losses and b.) with compensation using the average of simulated homogeneous frames. c.) The mean axial profile of both $\alpha$ images shows the effects of the compensation as a function of depth.

Experimental Validation

Speckle statistics estimates in the experimental phantom demonstrate that compensation for attenuation and diffraction has consequences in physical ultrasound measurements as well. Figure 10 presents speckle statistics estimates as a function of axial PER extent before and after compensation by the average phantom scan line. A clear negative trend with PER length is observed in all parameters, which is mostly removed through compensation in the case of the Nakagami parameter and homodyned K $\alpha$. The negative trend in the $k$ parameter remains, although to a lesser degree and with a positive offset after power loss compensation.

Figure 10:

Estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter amongst the thirty total experimental phantom frames are shown before and after ‘Compensation-By-Average” frame. Error bars represent standard error from the mean.

Finally, the changes in speckle statistics parameters in in vivo acquisitions before and after power loss compensation are shown in Figure 11. Although the true underlying scatterer conditions are unknown in this case - therefore resulting parameters cannot be compared to expected values - it is shown that power loss compensation does cause a statistically significant positive difference in speckle statistics estimation. The p-values between the compensated and uncompensated estimates are 0.012 in the Nakagami parameter and 0.021 in the homodyned K $\alpha$ parameter. The change in the $k$ parameter was not significant with a p-value of 0.294.

Figure 11:

a.) Estimates of the a.) Nakagami parameter, b.) homodyned K $\alpha$ parameter, and c.) homodyned K $k$ parameter amongst the eight total in vivo frames are presented before and after ‘Compensation-By-Theory.”

Discussion

The goal of this investigation was to quantify the effects of unaddressed power loss within the parameter estimation region on speckle statistics estimates. It was demonstrated that in several simulations with tissue-like attenuation conditions, statistically different speckle statistics estimates were computed depending on attenuation value even though underlying scatterer conditions were held constant. A trend with PER size was observed, as effects of power loss increased as more data was included in the axial dimension. Additionally, a trend with PER center depth was observed in simulations, which has implications in practical use where the region of interest may vary in location over time and between individuals. Methods for compensation were proposed, and it was demonstrated that these methods successfully reversed the effects of power loss and made speckle estimates PER size-, depth-, and attenuation value-independent. Similar investigation in phantom and in vivo acquisitions on a clinical scanner demonstrate the potential clinical impact of these effects and their compensation.

The fact that depth-dependent power losses alone can cause statistically significant changes in speckle statistics estimates raises doubts about reported values in tissue if it is unclear whether depth-dependent power loss was factored into analysis. For example, Nakagami estimates in human breast tissue have been observed to be lower in malignant carcinoma than benign fibroadenoma, with mean values of m = 0.58 and m = 0.7, respectively [40]. However, the attenuation coefficient in carcinomas has been measured to be higher than that of fibroadenomas [41], with values of 1.16 and 0.94 dB cm⁻¹ MHz⁻¹, respectively. Effects of increased attenuation and differing scatterer conditions may both contribute to the lower Nakagami values in carcinomas. Likewise, homodyned K parameters measured in human skin have been found to increase with degree of lymphedema: $\alpha$ between 1.5 and 2.4, and $k$ between 0.51 and 0.60 in the hypodermis [42]. This increase in speckle statistics estimates also occurs alongside a decrease in the attenuation coefficient from 1.09 dB cm⁻¹ MHz⁻¹ to 0.93 dB cm⁻¹ MHz⁻¹, which may partially contribute to the observed change.

When both scatterer configuration and tissue attenuation contribute to measured speckle statistics, it is difficult to separate the two and gain a clear understanding of microstructural change. Without accounting for power loss as a potential source of estimation error, our understanding of each disease’s effect on tissue is confounded. To produce more robust estimates, compensation for power losses due to attenuation and diffraction should be performed before speckle statistics parameters are estimated. Additional systematic exploration of the effects of these power losses on speckle statistics estimates in various tissues and pathological states is beyond the scope of this work but would be valuable additions to the literature.

The use of simulations alone allowed for flexibility in this study, and isolates the effects of diffraction and attenuation in an environment where the underlying scattering physics can be held constant. In the simulation component of this study, the diffraction effects were measured in a simulated phantom with no attenuation. As this is not feasible in reality, the attenuation of the phantom would need to be known and compensated for before using it to account for diffraction in tissue, as was done for the in vivo component of this study. Another feasible option is to use beamforming methods that reduce the effects of diffraction on the spatial variation in brightness of the final reconstructed image, such as multiple focusing, plane-wave compounding [43], or synthetic aperture techniques [2]. As discussed by other authors [44, 45, 46], such beamforming methods also produce resolution cells that are more uniform with depth, so that speckle statistics in turn do not vary with depth. Because diffuse scattering was expected in all simulations and experiments, these effects were not probed in this study.

Summary and conclusions

The extent to which power loss caused by attenuation and acoustic beam diffraction degrades first-order speckle statistics estimates was measured in virtual phantoms of varying acoustic attenuation, as well as in physical phantoms and in vivo acquisitions. It was demonstrated that compensation for these phenomena is necessary to produce estimates that do not depend on the size and depth of the estimation region. These effects should be considered when using speckle statistics for tissue characterization since they cause statistically significant differences in estimates unrelated to scatterer number density and spatial configuration.

Data Availability Statement

The research data from this article is available and can be accessed upon request.