BSC validation using tissue-mimicking phantoms
To validate the BSC-based QUS processing pipeline, BSC estimation was performed using an agar-based tissue-mimicking phantom containing 60 µm diameter borosilicate glass beads as the sample phantom and an agar-based phantom containing 18 µm diameter borosilicate glass beads as the reference phantom. These phantoms were selected because their scatterer size, material properties, and acoustic properties had been independently characterized, enabling comparison between experimentally estimated BSC values and theoretical BSC predictions. The structural and acoustic properties of both phantoms were known a priori. The attenuation properties of the 18 µm reference phantom consisted of an attenuation coefficient of 0.028 dB/cm/MHz1.862 and exponent of 1.862, as reported in Rohrbach et. al.8. The 60 µm phantom had an attenuation coefficient of 0.34 dB/cm/MHz and an attenuation intercept of −0.8 dB/cm over the 3.5–6 MHz processing bandwidth.
BSC estimation was performed using the reference phantom method, as described in Step 4.3, for both PW and FW acquisition RF data with the focal depth fixed at 20 mm, as shown in Figure 5A–D. The theoretical BSC of the 60 µm phantom was computed using Faran scattering theory with the following borosilicate glass bead acoustic and mechanical properties: speed of sound of 5640 m/s, mass density of 2.5 g/cm3, and Poisson’s ratio of 0.2. Analysis was performed within a 30 mm × 15 mm ROI subdivided into 5 mm × 3 mm sliding sub-ROIs, as shown in Figure 5A–D. Figure 5E shows the estimated BSC curves for the 60 µm phantom averaged over the entire ROI using a single frame for both PW and FW acquisition modes, together with the theoretical BSC. At the mid-band frequency of 4.75 MHz, the estimated BSC using PW acquisition was −0.43 ± 1.60 dB/m/sr, whereas FW acquisition yielded a mean BSC of −0.80 ± 1.26 dB/m/sr. Figure 5F shows the estimated BSC curves averaged over 20 frames for both acquisition modes. Averaging over 20 frames reduced the variance of the estimated BSC curves and improved agreement with the theoretical BSC. At 4.75 MHz, the estimated mean BSC using PW acquisition increased to −0.25 ± 0.52 dB/m/sr, whereas FW acquisition yielded a mean BSC of −0.01 ± 0.51 dB/m/sr. The theoretical BSC at 4.75 MHz calculated using Faran scattering theory was -0.15 dB/m/sr. The close agreement between the estimated and theoretical BSC values at the mid-band frequency under both single-frame and 20-frame averaging conditions demonstrates that the pipeline generated BSC estimates consistent with the theoretical predictions.

Figure 5. Validation of backscatter coefficient (BSC) estimation using tissue-mimicking phantoms with known scatterer sizes. (A–D) Plane-wave (PW) and focused-wave (FW) B-mode images of the reference and sample phantoms showing the processing region of interest (ROI) and sliding sub-regions of interest (sub-ROIs). (E,F) Estimated BSC curves obtained using single-frame averaging and 20-frame averaging compared with theoretical BSC values calculated using Faran scattering theory. Shaded regions represent ±1 standard deviation (SD). Please click here to view a larger version of this figure.
To further evaluate the BSC-based parameter estimation, Gaussian scattering model fitting and linear fitting were applied to the estimated BSC curves for both acquisition modes to obtain the ESD, EAC, SS, I0, and MBF. The estimated parameters, averaged over the entire ROI, are summarized in Table 1 for both single-frame and 20-frame averaging. The expected model parameters, obtained by applying the same fitting procedures to the theoretical Faran BSC curve, are presented in the final row of Table 1.
| Acquisition | Number of frames | ESD
(μm) | EAC
(dB/m³) | SS
(dB/sr/m/MHz) | I0
(dB/sr/m) | MBF
(dB/sr/m) |
| FW | 1 | 74.6 ± 28.6 | 100 ± 7.20 | 2.83 ± 0.57 | −14.0 ± 2.92 | −0.49 ± 0.58 |
| FW | 20 | 84.3 ± 10.9 | 97.2 ± 3.07 | 2.68 ± 0.26 | −13.1 ± 1.28 | −0.3 ± 0.28 |
| PW | 1 | 77.6 ± 33.6 | 100 ± 13.1 | 2.74 ± 0.78 | −13.9 ± 3.69 | −0.86 ± 0.51 |
| PW | 20 | 85.8 ± 10.4 | 96.5 ± 2.79 | 2.65 ± 0.24 | −13.1 ± 1.12 | −0.52 ± 0.30 |
| Expected values | — | 77.8 | 98.5 | 2.9 | −14.5 | −0.66 |
Table 1: Mean ± standard deviation of backscatter coefficient (BSC)-based quantitative ultrasound (QUS) parameters estimated for the 60 µm tissue-mimicking phantom using plane-wave (PW) and focused-wave (FW) acquisition modes. Mean values were calculated over a 30 mm × 15 mm region of interest (ROI) using 5 mm × 3 mm sliding sub-regions of interest (sub-ROIs), as illustrated in Figure 5. Expected values were calculated from the theoretical backscatter coefficient using Faran scattering theory.
For single-frame estimation, corresponding to the near real-time operating condition of the pipeline, the estimated ESD was 74.6 ± 28.6 µm and 77.6 ± 33.6 µm for FW and PW acquisition modes, respectively, compared with the expected value of 77.8 µm. The estimated MBF was −0.49 ± 0.58 and −0.86 ± 0.51 dB/m/sr for FW and PW acquisition modes, respectively, compared with the expected value of −0.66 dB/m/sr. The relatively large standard deviations observed under single-frame conditions reflect the sensitivity of model-based BSC fitting to fluctuations in the estimated power spectra caused by measurement noise and speckle variability between sub-ROIs. Despite this increased variability, the mean estimated parameters remained close to the expected theoretical values for both acquisition modes. As shown in Table 1, averaging over 20 frames substantially reduced parameter variability across all reported parameters, with the standard deviations of ESD, EAC, and I0 reduced by more than 50%, further improving agreement with the expected values. These findings demonstrate that the pipeline generated consistent BSC-based parameter estimates under near real-time single-frame conditions, while temporal averaging further reduced estimation variability.
Envelope statistics validation
Envelope statistics parameters were estimated for the 60 µm tissue-mimicking phantom using a 30 mm × 35 mm ROI with 5 mm × 3 mm sub-ROIs and a single acquired frame for both PW and FW acquisition modes (focal depth = 25 mm), representing the near real-time operating condition of the pipeline. Figure 6 shows the spatial maps of the H-K structure and clustering parameters, the Nakagami shape and scale parameters, and the Rayleigh SNR. For PW and FW acquisition modes, the mean Nakagami shape parameter was 1.00 ± 0.08 and 1.00 ± 0.09, respectively. The corresponding Rayleigh SNR values were 1.91 ± 0.11 and 1.90 ± 0.11. These values are consistent with a fully developed speckle pattern expected from a dense random scatterer distribution, for which the Nakagami shape parameter approaches 116 and the Rayleigh envelope SNR approaches 1.9137.

Figure 6. Envelope statistics parameter maps estimated for the 60 µm tissue-mimicking phantom. Quantitative ultrasound parameter maps obtained using (top) plane-wave (PW) acquisition and (bottom) focused-wave (FW) acquisition with a focal depth of 25 mm. Maps include the Homodyned-K structure parameter, Homodyned-K clustering parameter, Nakagami shape parameter, Nakagami scale parameter, and Rayleigh signal-to-noise ratio (SNR). Please click here to view a larger version of this figure.
The mean Nakagami scale parameter was 5.51 ± 0.15 and 6.07 ± 0.41 for PW and FW acquisition modes, respectively, reflecting the local backscattered signal energy within the ROI. Differences in the estimated scale parameter between acquisition modes are expected because of differences in the transmitted acoustic field, beamforming characteristics, and local signal intensity. For PW and FW acquisition modes, the mean H-K structure parameter was 0.35 ± 0.29 and 0.35 ± 0.32, respectively, whereas the mean H-K clustering parameter was 34.8 ± 33.2 and 25.6 ± 30.3, respectively. These large clustering parameter values, together with their large standard deviations, indicate unstable estimates for this phantom. As described in Protocol Steps 4.5.7–4.5.10, H-K parameter estimation relies on an iterative fitting procedure without a closed-form solution. When the fitting algorithm fails to converge, the estimated parameter values saturate and the fitting procedure terminates. Estimation failure is likely to occur for this phantom because the high scatterer number density and random spatial distribution of the glass beads produce scattering conditions approaching a fully developed Rayleigh regime, in which the H-K model has reduced sensitivity. In contrast, H-K parameter estimation has been reported to perform more reliably in biological tissue, where scattering conditions are more favorable for stable convergence38. Overall, the Nakagami and Rayleigh SNR parameters obtained using both PW and FW acquisition modes were in close agreement, demonstrating consistent parameter estimation across the two acquisition configurations for these envelope statistics-based parameters.
ATS phantom validation of BSC-based mid-band fit and effective acoustic concentration parameters
To validate the MBF and EAC parameters, QUS processing was performed on an ATS tissue-mimicking phantom containing contrast targets with known relative backscatter levels using PW acquisition mode. A processing sub-ROI size of 4.1 mm (lateral) × 3 mm (axial) was used to ensure that each sub-ROI remained within the boundaries of an individual contrast target. Figure 7A shows the ATS phantom layout with the annotated contrast targets corresponding to relative backscatter levels of +3 dB, −3 dB, and −6 dB with respect to the surrounding background. The MBF parameter was estimated within each contrast target, yielding values of 4.60 dB, −1.37 dB, and −4.58 dB for the +3 dB, −3 dB, and −6 dB targets, respectively, as shown in Figure 7B. The corresponding relative differences in MBF between adjacent contrast targets were −5.97 dB and −3.21 dB, which closely matched the expected differences of −6 dB and −3 dB specified for the phantom. Similarly, the EAC parameter, obtained by Gaussian scattering model fitting of the estimated BSC, yielded values of 91.2 dB, 85.2 dB, and 82.1 dB for the +3 dB, −3 dB, and −6 dB targets, respectively, as shown in Figure 7C. The corresponding relative differences between adjacent targets were −6.05 dB and −3.09 dB, again closely matching the expected contrast differences. These results demonstrate that both MBF and EAC accurately capture relative differences in acoustic backscatter strength and support the ability of the pipeline to quantify acoustic contrast.

Figure 7. Quantitative ultrasound (QUS) parameter mapping of an ATS tissue-mimicking phantom. (A) Phantom layout highlighting the contrast targets. (B) Mid-band fit (MBF) parameter map. (C) Effective acoustic concentration (EAC) parameter map demonstrating differentiation of the +3 dB, −3 dB, and −6 dB contrast inclusions. Please click here to view a larger version of this figure.
QUS parameter maps of biological tissue
To demonstrate the applicability of the pipeline to biological tissue, the QUS imaging pipeline was applied to an ex vivo chicken breast sample. Figure 8 shows the estimated QUS parameter maps, including BSC-based parameters (ESD, EAC, MBF, SS, and I0), attenuation-based parameters (attenuation coefficient and attenuation intercept), and envelope statistics-based parameters (Nakagami shape and scale parameters and H-K structure and clustering parameters), overlaid on the corresponding B-mode images. Parameter estimation was performed over a 30 mm (lateral) × 27 mm (axial) ROI subdivided into 5 mm × 3 mm sliding sub-ROIs.

Figure 8. Quantitative ultrasound (QUS) parameter maps estimated from ex vivo chicken breast tissue. Envelope statistics-, attenuation-, and backscatter coefficient (BSC)-based parameter maps are displayed as color overlays on the corresponding B-mode images. The displayed parameters include the Homodyned-K structure parameter, Homodyned-K clustering parameter, Nakagami shape parameter, Nakagami scale parameter, Rayleigh signal-to-noise ratio (SNR), spectral slope (SS), spectral intercept (I0), mid-band fit (MBF), effective scatterer diameter (ESD), effective acoustic concentration (EAC), attenuation coefficient (AttenAlpha), attenuation intercept (AttenIntercept), and attenuation at the mid-band (AttenMB). Please click here to view a larger version of this figure.
The envelope statistics results showed a mean Nakagami shape parameter of 0.56 ± 0.15 and a Rayleigh SNR of 1.19 ± 0.28, indicating a deviation from fully developed Rayleigh speckle consistent with pre-Rayleigh scattering behavior expected from structurally heterogeneous biological tissue. The mean Nakagami scale parameter was 5.05 ± 0.78, reflecting the local backscattered signal energy within the ROI. The mean H-K structure parameter was 0.10 ± 0.05, and the mean H-K clustering parameter was 1.02 ± 2.33. The large spatial variability observed in the H-K clustering parameter likely reflects tissue heterogeneity.
The attenuation coefficient was estimated to be 0.56 ± 0.29 dB/cm/MHz, with an attenuation intercept of 1.56 ± 2.05 dB/cm. The relatively large variability in the attenuation intercept is expected because it is obtained by extrapolating the attenuation fit to 0 MHz, outside the measured processing bandwidth, making it sensitive to small variations in the estimated attenuation curve. In contrast, the attenuation within the processing bandwidth remained relatively stable, yielding a mean attenuation of 4.22 ± 0.99 dB/cm at the mid-band frequency of 4.75 MHz. Because Coila et al.39 reported attenuation using a zero-intercept linear fit, comparison was performed using attenuation at the common mid-band frequency rather than the fitted slope and intercept parameters. Coila et al.39 reported an attenuation coefficient of 0.83 ± 0.43 dB/cm/MHz for chicken breast tissue, corresponding to an attenuation of 3.94 dB/cm at 4.75 MHz. The estimated mid-band attenuation of 4.22 ± 0.99 dB/cm obtained in the present study is in good agreement with this value, with the observed differences likely attributable to variations in tissue preparation and estimation methodology.
The estimated attenuation coefficient and attenuation intercept were subsequently used as inputs to the reference phantom method for BSC estimation. The BSC-based linear fit parameters were estimated as follows: SS = 1.49 ± 1.04 dB/sr/m/MHz, I0 = 17.4 ± 7.87 dB/sr/m, and MBF = 24.4 ± 3.90 dB/sr/m. Gaussian scattering model fitting yielded an ESD of 121 ± 28.4 µm and an EAC of 115 ± 3.88 dB/m3.
These representative results demonstrate that the pipeline can generate a comprehensive set of QUS parameter maps and quantitative tissue descriptors from biological tissue using the integrated spectral-based, attenuation-based, and envelope statistics-based processing methods.
Performance metrics
Table 2 summarizes the processing frame rate for different QUS parameter configurations using both PW and FW acquisition modes (focal depth = 25 mm). The reported processing frame rates represent the end-to-end processing pipeline following beamformed RF acquisition, including data transfer, loading of the beamformed RF data, QUS computation, and display of the selected parameter map. Performance was evaluated over a 30 mm × 30 mm ROI with a scan depth of 55 mm. All measurements were obtained using a laptop equipped with an Intel Core Ultra 7 255H processor (20 cores, approximately 2.4 GHz) and 32 GB of RAM. Overall, the processing frame rate decreased as additional QUS parameters were selected for computation, consistent with the increased computational demand of each additional processing step.
| Configuration | Parameters Computed | FPS (PW) | FPS (FW) |
| Envelope statistics only (Nakagami + Rayleigh SNR) | Nakagami shape, Nakagami scale, Rayleigh SNR | 2.0 | 1.8 |
| Envelope statistics only (all, including Homodyned-K) | Nakagami shape, Nakagami scale, Rayleigh SNR, Homodyned-K structure, Homodyned-K clustering | 1.5 | 1.5 |
| Backscatter coefficient (BSC)-based parameters only | Effective scatterer diameter (ESD), effective acoustic concentration (EAC), mid-band fit (MBF), spectral slope (SS), spectral intercept (I0) | 0.8 | 0.8 |
| Attenuation-based parameters only | Attenuation coefficient (AttenAlpha), attenuation intercept (AttenIntercept), attenuation at the mid-band (AttenMB) | 0.8 | 0.8 |
| Envelope statistics (excluding Homodyned-K) + BSC-based parameters | Nakagami shape, Nakagami scale, Rayleigh SNR, ESD, EAC, MBF, SS, I0 | 0.7 | 0.7 |
| All parameters (excluding Homodyned-K) | Nakagami shape, Nakagami scale, Rayleigh SNR, ESD, EAC, MBF, SS, I0, AttenAlpha, AttenIntercept, AttenMB | 0.4 | 0.4 |
| All parameters | Nakagami shape, Nakagami scale, Rayleigh SNR, Homodyned-K structure, Homodyned-K clustering, ESD, EAC, MBF, SS, I0, AttenAlpha, AttenIntercept, AttenMB | 0.4 | 0.4 |
Table 2: Processing frame rate (frames per second [FPS]) for different quantitative ultrasound (QUS) parameter configurations over an imaging depth of 10–40 mm using 5 mm × 3 mm sliding sub-regions of interest (sub-ROIs). Processing performance is reported for plane-wave (PW) and focused-wave (FW) acquisition modes using different combinations of envelope statistics-, backscatter coefficient (BSC)-based, and attenuation-based QUS parameters.
As summarized in Table 2, envelope statistics-based processing alone, consisting of Nakagami fitting and Rayleigh SNR estimation, yielded the highest frame rates of 2.0 frames/s and 1.8 frames/s for PW and FW acquisition modes, respectively. Adding H-K parameter estimation reduced the processing frame rate to 1.5 frames/s for both acquisition modes, reflecting the higher computational cost of iterative H-K fitting. Independently computing either BSC-based parameters or attenuation-based parameters resulted in frame rates of approximately 0.8 frames/s for both PW and FW acquisition modes. Combining envelope statistics-based parameters (excluding H-K) with BSC-based parameters further reduced the processing frame rate to 0.7 frames/s.
Computing all implemented parameters except H-K yielded a processing frame rate of 0.4 frames/s. When H-K fitting was included together with all other parameters, a frame rate of approximately 0.4 frames/s was achieved for the tissue-mimicking phantoms evaluated in this study. The processing time required for H-K fitting depended on the imaging medium. Media exhibiting nearly homogeneous scattering or air-like scattering properties required additional iterations for convergence of the iterative H-K fitting algorithm, resulting in frame rates below 0.4 frames/s.
These results demonstrate the trade-off between processing speed and the number of QUS parameters computed. For near real-time imaging applications, combining envelope statistics-based parameters (excluding H-K) with BSC-based parameters provided a practical balance between parameter diversity and computational performance, achieving a processing frame rate of approximately 0.7 frames/s. H-K parameter estimation is therefore better suited for offline analysis or applications in which higher temporal resolution is not required.