Abstract
Objectives:
Glioblastoma multiforme (GBM) is one of the most aggressive brain tumors, characterized by high heterogeneity and vascular proliferation. These features complicate its differentiation from normal brain tissue during surgical resection. Conventional ultrasound imaging provides limited quantitative insight into underlying tissue microstructure. This study investigates whether multiparametric quantitative ultrasound (QUS) features derived from envelope statistics can differentiate GBM from normal tissue. Specifically, the ability of the Homodyned-K (HK), Nakagami, and Burr distributions to capture microstructural differences is evaluated. The study also examines whether combining these features within a multiparametric framework and applying machine-learning-based classifications improves diagnostic performance.
Methods:
Ex vivo formalin-fixed human normal brain tissue and GBM samples (n = 20 per group) were embedded in agarose phantoms. Ultrasound data were acquired using a Verasonics Vantage 128 system with a 7.6 MHz linear array. Envelope data were analyzed within regions of interest (ROIs) selected within the imaging depth of field. HK, Nakagami, and Burr distributions were fitted to the envelope data to extract statistical parameters. These features were used for classification with logistic regression, support vector machines, random forests, and multilayer perceptron models. Model performance was assessed using accuracy, sensitivity, specificity, and area under the receiver operating characteristic curve.
Results:
Parameters derived from the HK, Nakagami, and Burr distributions were significantly different in GBM than in normal brain tissue, indicative of differences in tissue microstructure and scattering behavior. Among the individual statistical models, HK-derived features achieved the highest classification performance, with 90.0% accuracy. Multiparametric classification further improved performance, achieving 80.5% per-plane accuracy (AUC = 0.854) and 97.5% per-sample accuracy (AUC = 0.985). A reduced four-feature subset achieved comparable performance.
Conclusion:
These findings demonstrate that envelope statistics capture complementary aspects of GBM microstructure, enabling effective differentiation from normal brain tissue. The multiparametric framework improves characterization by integrating information across statistical models, while machine learning-based classification highlights its potential for tissue differentiation. These findings provide proof of concept for the use of multiparametric ultrasound envelope statistics in GBM tissue characterization. Further studies in freshly excised tissues and intraoperative settings are required to evaluate the translational potential of this approach.
1 Introduction
Glioblastoma multiforme (GBM) is a highly aggressive malignant brain tumor with a poor prognosis (). The global incidence of GBM ranges from 0.59 to 5 cases per 100,000 individuals and is rising (). The complex anisotropy and heterogeneity of brain tissue make complete resection of GBM challenging. Gross total resection may improve outcomes, but distinguishing tumor margins from adjacent normal tissue remains difficult.
Standard preoperative imaging techniques, such as Magnetic Resonance Imaging (MRI) and Computed Tomography (CT), are often inadequate for accurately assessing tumor margins intraoperatively. Surgical factors, such as bleeding and brain shift, prevent accurate assessment (; ). Effective intraoperative imaging tools for accurate tumor-margin identification remain limited, as fluorescence imaging has limited sensitivity and intraoperative MRI can be prohibitively expensive (). Conventional B-mode ultrasound provides useful anatomical information, but it offers limited insight into tissue differentiation through assessment of tissue microstructure (). Quantitative ultrasound (QUS) addresses this limitation by analyzing backscattered echo signals to derive parameters related to tissue scattering behavior. Statistical modeling of the echo envelope provides quantitative descriptors of tissue microstructure beyond conventional B-mode imaging (; ; ).
Several statistical models have been proposed to characterize the echo envelope. The Homodyned-K (HK) probability density function (PDF) accounts for both diffuse and coherent scattering components and provides parameters related to scattering clustering (α) and coherent-to-diffuse ratio (k) (; ). Previous studies have applied HK envelope statistics to capture variations in scatterer organization. These include characterization of tissues such as breast tumors (), lymph nodes (), and the detection and staging of early hepatic steatosis (; ).
The Nakagami PDF incorporates the amplitude, density, and the spatial organization of scatterers (; ). The Nakagami shape parameter (m) and scaling parameter (Ω) have been shown to correlate with spatial heterogeneity, the effective number of scatterers, and the backscattered energy of normal, benign, and malignant tissues (; ; ). Previous studies have applied Nakagami envelope statistics for tissue characterization, including breast lesion analysis, in which the ‘m’ parameter has been reported to be higher in malignant tumors than in benign lesions ().
The Burr PDF models the fractal and elongated characteristics of tissue microstructure by assuming that scatterers exhibit multi-scale, power-law behavior (). Within this framework, the shape parameter (b) reflects the density of fractal or elongated scattering structures, while the scale parameter (λ) represents the echo amplitude and system gain (). The Burr PDF has been used to describe ultrasound speckle statistics in soft biological tissues, including normal and fibrotic liver tissue and human placenta, demonstrating improved sensitivity in capturing tissue heterogeneity (; ).
Collectively, the HK, Nakagami, and Burr PDFs may provide complementary statistical descriptions of ultrasound backscatter, enabling characterization of tissue microstructure through variations in scatterer density, organization, and heterogeneity. Despite notable advancements in envelope statistics, their potential for characterizing brain tumors remains largely unexplored. The present study investigates whether parameters derived from the HK, Nakagami, and Burr PDFs can be used to effectively differentiate GBM from normal brain tissue ex vivo. In this work, envelope statistics were computed within defined regions of interest in ex vivo human brain tissue samples. Parameters obtained from all three models were compared between normal and GBM samples and combined within a multiparametric framework. In addition, the potential of these features for objective tissue classification using machine learning was investigated to evaluate the feasibility of improved tissue differentiation.
2 Materials and methods
2.1 Tissue sample preparation
Tissue samples of GBM were obtained from 20 individuals via surgical resection, with informed consent from the patient or a close relative (ethical clearance documents nos. NIMHANS/23rd (BS and NS DIV.)/2020 and NIMHANS/50th (BS and NS DIV.)/2024). The cohort included individuals aged 15–68 years, comprising 19 males and 1 female. The collected samples comprised tumor tissues from multiple anatomical regions, including the frontal, parietal, and temporal regions and related areas. Note that the sample distribution reflects availability at the time of collection and was not influenced by any selection based on age and gender.
Adjacent normal brain tissue was not collected during surgical excision to avoid neurological deficits. Therefore, normal brain tissues (n = 20 samples) were obtained from the Human Brain Tissue Repository (Brain Bank) at the National Institute of Mental Health and Neurosciences (NIMHANS). All samples were collected 24 h post-mortem. A corresponding set of control samples was included, derived from individuals with traumatic brain injury and collected 24 h post-mortem. The control cohort also included individuals across a broad age range and both sexes, with tissues obtained from anatomically comparable regions. All samples were collected with the informed consent of close relatives (ethical clearance documents nos. NIMHANS/IEC (BS and NS DIV.) 26th MEETING/2020–21 and NIMHANS/50th (BS and NS DIV.)/2024).
Both resected tumor and normal brain samples were carefully cut into blocks (2–6 mm thickness, 2–30 mm length and width), fixed in 10% neutral-buffered formalin, and subsequently characterized with ultrasound within 2 months of fixation. A fraction of tissue from each sample was processed for histopathological diagnosis. For this analysis, the tissues were sampled from different regions and processed in a tissue processor (through formalin, alcohol, xylene, and paraffin) for over 15 h. The processed tissue was embedded in paraffin wax, and Sections 3–4 µm thick were cut using a microtome for hematoxylin and eosin staining, followed by examination under light microscopy (Olympus Bx53 trinocular microscope, magnification lenses ∼1.25x, 4x, 10x, 20x, and 40x with DP23 camera equipped with the CellSens software). All histopathological slides were independently evaluated by two experienced neuropathologists with 10 and 25 years of experience, respectively.
Each brain tissue sample was embedded in an agarose phantom for ultrasound imaging according to the following protocol (ethical clearance document no. IITGN/IEC/2025–26/114). Agarose (2% w/v, J66369-22, Thermo Fisher Scientific, Mumbai, India) was dissolved in degassed, deionized (MilliQ) water by heating the solution in a 700 W microwave oven (NN-ST266B FDG, Panasonic, Madhya Pradesh, India). The agarose solution was degassed in an ultrasonic bath at 50 °C for 30 min to eliminate entrapped air bubbles, then poured into a mold (82 mm × 56 mm × 22 mm) until it reached half the mold’s volume and solidified at room temperature (25 °C) for 20 min. The brain tissue samples were placed gently on solidified agarose. The remaining agarose solution was cooled to 35 °C–40 °C and poured on the tissue sample. The agarose was then allowed to solidify at room temperature.
2.2 Ultrasound data acquisition and processing
2.2.1 Ultrasound data acquisition
A Verasonics Vantage 128 system (Kirkland, WA, United States) with a linear array transducer (L11-5v, Verasonics) was used to collect data at a center frequency of 7.6 MHz. A degassed, deionized water standoff was placed above the phantom to provide an acoustic propagation path (Figure 1A). The transducer was focused on the center of the tissue sample at an axial depth of 14 mm. The transducer was positioned to exclude multiple reflections from the water-phantom interface from the imaging field of view. A focused transmit imaging sequence with 32 active transmit elements and electronic shifting across the transducer aperture was employed. Beamformed in-phase and quadrature (IQ) data were acquired and reconstructed into radiofrequency (RF) signals using a 4× quadrature interpolation, in which interpolated I and Q components were phase-interleaved to recover the bandpass RF waveform. Time gain compensation (TGC) was not performed; the slider was set to the maximum value uniformly across the imaging depth. Brain tissues exhibit anisotropic microstructure, which may influence the ultrasound data depending on the orientation of the acoustic beam. The transducer was positioned normal to the macroscopic surface of each tissue specimen to ensure consistent acoustic coupling. The anatomical orientation of the excised tissues was not available; hence, the exact alignment of the acoustic beam relative to the microstructure could not be determined. Ultrasound RF data were obtained from five independent imaging planes for each tissue sample to account for tissue structural variability. The transducer was laterally translated by approximately 1 mm between acquisitions, which was greater than the lateral beamwidth (−6 dB) of 0.334 mm, ensuring that consecutive measurements were obtained from independent regions of the sample.
FIGURE 1
The reconstructed RF data were compensated for frequency-dependent attenuation across the ROI for further processing, as described in Section 2.2.3. Subsequent analyses were performed using Python (version 3.11.9) in Jupyter Notebook within the Visual Studio Code environment on a Windows-based computer equipped with a 64-bit 12th Gen Intel(R) Core(TM) processor (2.0 GHz) and 16.0 GB of RAM.
2.2.2 Acoustic attenuation measurements
The acoustic attenuation coefficients of the brain samples were measured using the broadband pulse-echo insertion-loss technique (; ). A single-element, polyvinylidene fluoride-focused transducer (Model TXZ_0056; Precision Acoustics, Dorchester, United Kingdom) was used. This transducer had a 20 MHz nominal center frequency, 40 mm focal depth, 2.5–35 MHz bandwidth (−6 dB), and 16 mm depth of focus (−6 dB), which was measured using a 0.2 mm needle hydrophone (Model 4,458; Precision Acoustics, Dorchester, United Kingdom).
A water stand-off was used for acoustic coupling, with a thin plastic wrap (∼20 μm) serving as an acoustic window to separate the coupling water from the tissue-containing medium, while allowing the ultrasonic beam to propagate through the agarose-embedded tissue and to reflect from the stainless-steel plate (Figure 1B, Supplementary Material, Section 2.1).
A broadband pulse was generated using a pulser/receiver (JSR DPR300; Imaginant Inc., Pittsford, New York, United States) at a 100 Hz pulse repetition frequency. Reflected RF signals from a stainless-steel plate were acquired (Supplementary Material, Section 1.1). The received signals were amplified by 40 dB, digitized at 2.5 G/s using a digital oscilloscope (MD03014; Tektronix Inc., Bangalore, India), and stored for analysis.
The frequency-dependent attenuation coefficients for agarose and tissue samples were computed from the spectral ratios of the reflected signals. A detailed description of the experimental setup and associated equations used for attenuation estimation is provided in the Supplementary Material, Section 1.1. The measured attenuation of the thin plastic wrap was 0.26 dB at 7.6 MHz (Supplementary Material, Section 2.1).
The thickness of the normal brain tissues ranged from 0.3 to 0.5 cm, while the GBM tissues varied from 0.15 to 0.4 cm. These measurements were derived from analyzing B-mode images of the samples. Attenuation measurements were repeated at five different locations within each sample and analyzed within the usable bandwidth of the imaging array (i.e., 5–11 MHz), which was used in subsequent experiments. As the sample thickness (0.15–0.5 cm) was well within the focal region of the transducer (1.6 cm depth of focus), diffraction effects along the propagation path were not considered. Due to limited sample thickness (0.2–0.35 cm), small lateral dimensions (0.2–1.3 cm), and reduced signal-to-noise ratio in a subset of samples, attenuation measurements could be performed for only 11 normal brain tissues and 11 GBM tissues. Group-specific average frequency-dependent attenuation parameters were calculated separately for the normal and GBM groups and subsequently used to compensate for attenuation in samples within their respective groups.
2.2.3 Acoustic attenuation compensation
The attenuation measurements were used to compensate for the frequency-dependent attenuation through the tissue sample prior to envelope statistics analysis. The frequency-dependent attenuation of the RF data was , where μtissue was the attenuation coefficient (converted to Np/cm) obtained from the insertion-loss measurements (Supplementary Material, Section 1.1), and x was the thickness (cm) of the sample (). To obtain attenuation-compensated RF data, the RF data were transformed into the frequency domain, multiplied by the factor , and transformed back to the time domain.
2.3 Envelope statistics
Envelope statistics were computed from the RF data by extracting the normalized echo envelope, A, using the Hilbert transform. The envelope distribution was modeled using the HK, Nakagami, and Burr PDFs within the predefined ROI.
2.3.1 Homodyned-K distribution
The HK PDF, denoted by pHK (A; ϵ, σ2, α), is given by Equation 1 (; ; ):where ϵ ≥ 0, σ2 > 0 andα>0, and denotes the zeroth-order Bessel function of the first kind. The parameter ϵ2 represents the coherent signal power, σ2 represents the diffuse signal power, α characterizes the scatterer clustering, and x is a dummy integration variable.
A derived k parameter is defined in Equation 2 ():which represents the ratio of coherent to diffuse signal components and describes the periodicity of scatterer distribution within the resolution cell.
2.3.2 Nakagami distribution
The Nakagami PDF, denoted by pN (A; m, Ω), is given by Equation 3 (; ; ):where m > 0, Ω > 0, and Γ is the gamma function. The parameter, m (shape parameter), represents the scattering conditions or effective scatterer density, while Ω (scale parameter) relates to the average energy of the backscattered envelope.
2.3.3 Burr distribution
The Burr PDF, denoted by pB (A; b, λ), is given by Equation 4 (; ):where b > 0 and λ > 0 The parameter b (shape parameter) is a power-law exponent that represents the fractal vessel network and is related to the number density of fractal and elongated scattering structures, while λ (scale parameter) relates to the echo amplitude and gain.
These models capture complementary scattering conditions ranging from coherent-diffuse mixtures (HK) to simplified regimes (Nakagami) and heavy-tailed distributions (Burr).
2.4 Parameter estimation
For all distributions, parameters were estimated using maximum likelihood estimation (MLE). Given a set of envelope amplitudes , the parameters of a distribution were obtained by maximizing the log-likelihood function as given in Equation 5:
For the HK distribution, parameter estimation was performed using numerical minimization of the negative log-likelihood due to the absence of a closed-form solution. The parameters (ϵ, σ2, α) were estimated using a constrained gradient-based optimization approach. For the Nakagami and Burr distributions, the parameters were estimated directly using MLE with appropriate constraints on the parameter space.
All estimations were performed independently for each region of interest for each sample plane. All analyses were implemented in Python using NumPy and SciPy.
2.4.1 Goodness-of-fit assessment
The quality of the distribution fitting was evaluated using the coefficient of determination (R2) and root mean square error (RMSE) between the empirical histogram and the fitted PDF. Additionally, the Kullback-Leibler (KL) divergence was computed to quantify the difference between the empirical and modeled distributions.
2.4.2 Statistical analysis
Power law regression analysis was performed on the attenuation measurements as a function of frequency, given as , wherein α0 and n are the power law constant (dB/cm/MHzn) and power law exponent, respectively. Outliers were identified using the interquartile range criterion (IQR) (25th-75th percentile), and values lying beyond 1.5×IQR were excluded prior to statistical analysis.
The Anderson-Darling test was used to assess the normality of the parameter distributions. Depending on the Anderson-Darling test result, either a two-sample t-test or the Wilcoxon rank-sum test (Mann-Whitney U test) was used to evaluate statistically significant differences in parameters between normal and GBM tissue samples. Effect sizes were quantified using Cohen’s d, computed as the difference between group means normalized by the pooled standard deviation. All statistical tests were performed at the 5% significance level using GraphPad Prism 9.0 (GraphPad Software, Boston, MA, United States) and MATLAB 2022a (MathWorks, Natick, MA, United States).
2.5 Classification analysis
To assess the discriminative potential of the estimated envelope statistics parameters for tissue characterization, supervised classification was performed to distinguish GBM from normal brain tissue (; ). Two levels of analysis were conducted: (i) per-sample classification, in which the estimated parameters were averaged across the five imaging planes for each tissue sample, yielding 40 observations (20 normal, 20 GBM), and (ii) per-plane classification, in which each imaging plane was treated as an independent observation, yielding 200 observations (100 per class). All classification analyses were implemented in Python (version 3.11) using scikit-learn, Optuna (), and matplotlib.
2.5.1 Feature extraction and standardization
Five feature subsets were constructed from the MLE-estimated distribution parameters. The first subset comprised six features: HK (α, k), Nakagami (m, Ω), and Burr (b, λ). The second subset comprised four features: HK (α, k), Nakagami m, and Burr b. This subset was obtained by excluding the scale parameters Ω and λ to emphasize features that are more directly linked to tissue microstructural organization. Additionally, to evaluate the contribution of each statistical distribution independently, three two-feature single-distribution sets were analyzed: HK (α, k), Nakagami (m, Ω), and Burr (b, λ). These single-distribution analyses were used as ablation comparisons against the multiparametric feature sets.
Prior to classification, all features were standardized using z-score normalization, wherein each feature value was transformed by subtracting the mean and dividing by the standard deviation of that feature computed exclusively from the training data within each cross-validation fold. This procedure ensures that the test data does not influence the scaling parameters, preventing information leakage.
2.5.2 Classification algorithms
Seven classification algorithms spanning linear, ensemble, and neural network approaches were evaluated.
2.5.2.1 Logistic regression
Logistic regression models the posterior probability of class membership using the logistic (sigmoid) function as given in Equation 6 ():where x is the feature vector, w is the learned weight vector, β0 is the intercept term, and denotes the class label (0 for normal, 1 for GBM). The model parameters were estimated by minimizing the negative log-likelihood with L1 or L2 regularization, controlled by the regularization parameter C. The decision boundary corresponds to the hyperplane , which is linear in the feature space.
2.5.2.2 Linear support vector machine
The linear SVM seeks a separating hyperplane that maximizes the margin between the two classes by solving the optimization problem as given by Equation 7 ():where C is a regularization parameter controlling the trade-off between the width of the margin and the number of misclassified training samples, and the second term (the hinge loss) penalizes samples that lie on the wrong side of the margin. Posterior class probabilities were obtained using Platt scaling, which fits a sigmoid function to the SVM decision function values.
2.5.2.3 Random forest
Random Forest is an ensemble method that constructs multiple decision trees, each trained on a bootstrap sample (a random subset of the training data drawn with replacement) (). At each node in a tree, the algorithm selects the best split from a random subset of features using the Gini impurity criterion as given by Equation 8:where pk is the proportion of training samples belonging to class k at a given node. Nodes are split to minimize Gini impurity, thereby creating regions that are increasingly homogeneous with respect to class labels. The final classification was determined by majority voting across all trees, and the posterior class probability was estimated as the fraction of trees voting for each class. The number of trees (50–500), the maximum tree depth (2–15), and the minimum samples per split and per leaf were treated as tunable hyperparameters. Feature importance was assessed using the mean decrease in impurity (MDI), which quantifies, for each feature, the total reduction in Gini impurity across all decision nodes in all trees of the ensemble where that feature was used for splitting (). Importance scores were averaged across cross-validation folds to obtain robust estimates, enabling identification of the envelope statistics parameters with the greatest discriminative power.
2.5.2.4 Multilayer perceptron
MLP classifiers were implemented as feedforward neural networks with ReLU or hyperbolic tangent activation functions and softmax output probabilities (). Four architectures were evaluated: one hidden layer with 64 or 128 neurons, and two hidden layers with either 64/32 or 128/64 neurons. Network weights were optimized using the Adam optimizer with cross-entropy loss. The L2 regularization parameter (10−5–10–1) and learning rate (10−4–10–1) were treated as tunable hyperparameters.
2.5.3 Cross-validation strategy
Cross-validation is a resampling technique used to evaluate classifier performance on unseen data by repeatedly partitioning the dataset into training and test subsets ().
For per-sample analysis, leave-one-out cross-validation (LOOCV) was employed. In each iteration, a single sample was held out as the test set, and the remaining n-1 = 39 samples were used for training. This process was repeated n = 40 times so that each sample served exactly once as the test case. LOOCV was selected because it maximizes the amount of training data in each fold, which is critical for reliable model estimation given the limited sample size.
For per-plane analysis, group K-fold cross-validation was employed, in which the data were partitioned into groups based on sample identity. All five imaging planes for a given tissue sample were assigned to the same fold, ensuring that the training and test sets did not contain planes from the same sample. This design prevents intra-sample data leakage, which would otherwise inflate performance estimates due to the spatial correlation among planes acquired from the same tissue specimen.
2.5.4 Hyperparameter optimization
Hyperparameter optimization was performed using Optuna, an automated framework that efficiently searches for the hyperparameter space using the Tree-structured Parzen Estimator (TPE). TPE is a sequential model-based optimization algorithm that models the probability distribution of hyperparameter configurations conditioned on their observed performance and preferentially samples configurations from regions of the search space that have yielded high performance in prior trials. For each classifier, 50 optimization trials were conducted, with the cross-validation accuracy (LOOCV for per-sample, group K-fold for per-plane) serving as the optimization objective. The configuration that achieved the highest cross-validation accuracy was selected as the final model for each classifier.
To further assess classifier generalizability, nested cross-validation was also performed. For each outer split, hyperparameter optimization using Optuna was performed only on the corresponding training data, and the held-out outer fold was then used for performance evaluation. Nested LOOCV was used for the per-sample analysis, whereas nested group K-fold cross-validation was used for the per-plane analysis to preserve sample-level grouping. This approach reduces the optimistic bias that can arise when hyperparameter selection and performance evaluation are performed on the same resampling splits.
2.5.5 Performance evaluation and visualization
Classification performance was evaluated using cross-validated predictions and metrics including accuracy, area under the receiver operating characteristic curve (AUROC), sensitivity, and specificity ().
To visualize class separability in the feature space, two-dimensional embeddings of the standardized feature vectors were generated using t-distributed stochastic neighbor embedding (t-SNE). t-SNE is a nonlinear dimensionality reduction technique that preserves the local neighborhood structure of the data points. The perplexity parameter, which controls the effective number of neighbors considered for each point, was set to n/3 (capped at 30), where n is the number of observations. Misclassified samples under LOOCV were highlighted with surrounding error rings. Receiver operating characteristic (ROC) curves were plotted for logistic regression and linear SVM to illustrate their discrimination performance across varying decision thresholds.
Posterior class probabilities were obtained from each classifier’s native probabilistic output: the sigmoid function for logistic regression, Platt scaling for the linear SVM, the fraction of trees voting for each class for random forest, and the softmax function for the MLP classifiers. These probability estimates provide per-prediction confidence scores, indicating the degree of certainty with which each sample is classified.
3 Results
3.1 Histopathology
Representative histopathology (hematoxylin and eosin-stained) images of normal and GBM samples are shown in Figure 2. Normal brain tissue with white matter containing predominantly neurons (Figure 2A), and grey matter containing glial cells and white matter fibers (Figure 2B) were compared with the GBM samples (Figures 2C,D). Tumor areas showed infiltrating high-grade glioma with varying proportions of infiltrating neoplastic astrocytes, necrosis, and vascular stromal components. These histopathology images were evaluated semi-quantitatively by visual grading in terms of the percentage of each component in the tumor samples (Table 1).
FIGURE 2
TABLE 1
| Sample | Tumor (%) | Necrosis (%) | Adjacent cortex (%) | Stromal component (%) |
|---|---|---|---|---|
| 1 | 60 | 40 | - | - |
| 2 | 60 | 20 | 20 | - |
| 3 | 60 | - | - | - |
| 4 | 30 | 20 | 30 | 20 |
| 5 | 30 | 20 | 50 | |
| 6 | 50 | 50 | - | - |
| 7 | 50 | 50 | - | - |
| 8 | 90 | 10 | - | - |
| 9 | 100 | 0 | - | - |
| 10 | 95 | 5 | - | - |
| 11 | 99 | 1 | - | - |
| 12 | 100 | - | - | - |
| 13 | - | 100 | - | - |
| 14 | 70 | 30 | - | - |
| 15 | 100 | - | - | - |
| 16 | 100 | - | - | - |
| 17 | 40 | 60 | - | - |
| 18 | 100 | - | - | - |
| 19 | 100 | - | - | - |
| 20 | - | 100 | - | - |
Histopathological information about GBM samples.
3.2 B-mode imaging
Representative B-mode ultrasound images of a normal brain tissue and GBM sample, along with the corresponding ROIs used for analysis, are shown in Figure 3. The normal brain tissue (Figure 3A) exhibits relatively homogeneous speckle patterns with consistent echotexture throughout the ROI. The GBM tissue (Figure 3B) shows increased echogenicity with spatial variations in scattering intensity.
FIGURE 3
3.3 Attenuation coefficient measurements
The frequency-dependent attenuation measurements of the normal and GBM tissue samples are shown in Figure 4. The power fit constants (α0) are 1.48 and 3.59 dB/cm/MHzn, and exponents (n) are 1.18 and 0.92 for normal tissue and GBM samples, respectively.
FIGURE 4
3.4 Envelope statistics
3.4.1 Distribution fitting
The histograms of echo envelope amplitudes from ROIs of a representative normal brain tissue and GBM sample are shown in Figure 5. The fitted distributions, including HK, Nakagami, and Burr distributions, follow the empirical histograms in both cases. For the normal brain tissue (Figure 5A), the distributions show a relatively narrower spread with lower variability. In contrast, the GBM sample (Figure 5B) exhibits a broader distribution with increased tailing, reflecting increased structural heterogeneity. The goodness-of-fit metrics (R2 and RMSE) indicate comparable fitting across distributions, suggesting that all three distributions capture the differences in tissue microstructure.
FIGURE 5
3.4.2 Parameter estimation
The HK, Nakagami, and Burr distribution parameters estimated from envelope amplitudes are shown in Figure 6. The HK parameters (α, k) (Figures 6A,B), Nakagami parameters (m, Ω) (Figures 6C,D), and Burr parameters (b, λ) (Figures 6E,F) were significantly different between normal tissue and GBM samples. Overall, the parameter distributions demonstrate clear separation between the two tissue types, with GBM samples exhibiting greater variability compared to normal tissues.
FIGURE 6
To assess the potential influence of attenuation compensation on the scale parameter, the parameter estimates were compared between before and after attenuation compensation. Paired t-test results indicated that the differences in the estimates of each scale parameter before and after attenuation compensation were not significant (p = 0.08 for the Nakagami Ω and p = 0.53 for the Burr λ).
In Table 2, the ranges of R2 and RMSE for the HK, Nakagami, and Burr distributions are shown. The goodness-of-fit metrics show comparable R2 and RMSE ranges across HK, Nakagami, and Burr distributions for both normal tissue and GBM samples, without variations observed between models.
TABLE 2
| Sample group | Distribution | R2 (min-max) | RMSE (min-max) |
|---|---|---|---|
| Normal | HK | (0.58–0.97) | (0.45–2.80) |
| Nakagami | (0.43–0.85) | (0.71–3.82) | |
| Burr | (0.80–0.99) | (0.23–0.84) | |
| GBM | HK | (0.57–0.98) | (0.31–1.73) |
| Nakagami | (0.60–0.92) | (0.48–2.68) | |
| Burr | (0.58–0.98) | (0.30–0.92) |
Goodness-of-fit metrics (R2 and RMSE) for HK, Nakagami, and Burr distributions fitted for the ROI.
The Kullback-Leibler (KL) divergence between the empirical distributions and fitted models for normal and GBM samples is shown in Figure 7. The KL divergence values show distinct trends across distributions between normal tissue and GBM samples, with lower values for Nakagami (Figure 7B) and higher values for Burr (Figure 7C) in GBM samples. HK (Figure 7A) did not show a statistically significant difference between the two tissue types.
FIGURE 7
In Table 3, the Cohen’s d parameter for HK, Nakagami, and Burr distribution parameter estimates are shown. Effect size magnitude was categorized as small (0.2≤|d|<0.5), medium (0.5≤|d|<0.8), and large (|d|≥0.8). The effect size analysis shows that most parameters exhibit medium to large differences between normal tissue and GBM samples.
TABLE 3
| Distribution | Parameter | Cohen’s d | Magnitude |
|---|---|---|---|
| HK | 1.077 | Large | |
| 0.699 | Medium | ||
| Nakagami | 1.124 | Large | |
| 1.277 | Large | ||
| Burr | 0.741 | Medium | |
| 1.047 | Large |
Cohen’s d parameter for HK (α, k), Nakagami (m, Ω), and Burr (b, λ) estimates.
3.5 Classification performance
In Table 4, the per-sample classification results using LOOCV are summarized for logistic regression, linear SVM, and the MLP (1 × 64) model. The complete classification results across all seven classifiers are shown in Supplementary Material, Section 2.2. For the six-feature set, linear SVM achieved the highest performance (97.5% accuracy, AUC = 0.985), and logistic regression and MLP (1 × 64) each reached 92.5% accuracy. For the four-feature set, linear SVM led with 95.0% accuracy (AUC = 0.953), followed by logistic regression (92.5%) and MLP (1 × 64) (90.0%).
TABLE 4
| Feature subset | Algorithm | Accuracy (%) | AUC | Sensitivity (%) | Specificity (%) |
|---|---|---|---|---|---|
| 6-feature | Logistic regression | 92.5 | 0.978 | 90.0 | 95.0 |
| Linear SVM | 97.5 | 0.985 | 100.0 | 95.0 | |
| MLP (1 × 64) | 92.5 | 0.978 | 90.0 | 95.0 | |
| 4-feature | Logistic regression | 92.5 | 0.963 | 95.0 | 90.0 |
| Linear SVM | 95.0 | 0.953 | 95.0 | 95.0 | |
| MLP (1 × 64) | 90.0 | 0.932 | 90.0 | 90.0 |
Per-sample classification performance (LOOCV, n = 40) for the six-feature and four-feature subsets.
In Table 5, the per-plane classification results from group K-fold cross-validation are summarized for the Optuna-tuned logistic regression, linear SVM, and MLP (1 × 64) classifiers. For the six-feature set, tuned logistic regression and MLP (1 × 64) achieved the highest accuracy of 80.5% (AUC = 0.854 and 0.845, respectively), with linear SVM at 76.5% (AUC = 0.858). For the four-feature set, tuned MLP (1 × 64) led with 79.5% accuracy (AUC = 0.837), followed by tuned logistic regression (79.0%, AUC = 0.854). Nested cross-validation results, together with a comparison against the corresponding tuned cross-validation estimates, are provided in Supplementary Material (Tables 5, 6). Nested cross-validation generally yielded lower performance estimates than the corresponding tuned cross-validation results, while preserving similar performance trends across feature subsets. In Table 6, the per-sample single-distribution ablation results are summarized for logistic regression, linear SVM, and MLP (1 × 64). Among the individual distributions, HK parameters achieved the highest performance, with linear SVM reaching 90.0% accuracy (AUC = 0.960). Nakagami-only and Burr-only features achieved lower accuracies (77.5% with MLP (1 × 64)), indicating that their parameters captured only part of the class-separating information compared with the multiparametric feature subsets.
TABLE 5
| Feature subset | Algorithm | Accuracy (%) | AUC | Sensitivity (%) | Specificity (%) |
|---|---|---|---|---|---|
| 6-feature | Logistic regression | 80.5 | 0.854 | 81.0 | 80.0 |
| Linear SVM | 76.5 | 0.858 | 81.0 | 72.0 | |
| MLP (1 × 64) | 80.5 | 0.845 | 80.0 | 81.0 | |
| 4-feature | Logistic regression | 79.0 | 0.854 | 82.0 | 76.0 |
| Linear SVM | 75.5 | 0.843 | 79.0 | 72.0 | |
| MLP (1 × 64) | 79.5 | 0.837 | 81.0 | 78.0 |
Per-plane classification performance (group K-fold, n = 200) using Optuna-tuned classifiers for the six-feature and four-feature subsets.
TABLE 6
| Feature set | Algorithm | Accuracy (%) | AUC | Sensitivity (%) | Specificity (%) |
|---|---|---|---|---|---|
| HK | Logistic regression | 85.0 | 0.953 | 85.0 | 85.0 |
| Linear SVM | 90.0 | 0.960 | 95.0 | 85.0 | |
| MLP (1 × 64) | 87.5 | 0.870 | 85.0 | 90.0 | |
| Nakagami | Logistic regression | 65.0 | 0.742 | 65.0 | 65.0 |
| Linear SVM | 72.5 | 0.775 | 80.0 | 65.0 | |
| MLP (1 × 64) | 77.5 | 0.855 | 75.0 | 80.0 | |
| Burr | Logistic regression | 70.0 | 0.450 | 95.0 | 45.0 |
| Linear SVM | 77.5 | 0.771 | 90.0 | 65.0 | |
| MLP (1 × 64) | 77.5 | 0.782 | 80.0 | 75.0 |
Per-sample classification performance (LOOCV, n = 40 samples) for single-distribution feature sets.
The t-SNE embeddings (Figures 8, 9) represent the separability between normal brain tissue and GBM samples in both six and four-feature spaces, respectively. The feature importance analysis using random forests (Figure 10) identifies the most discriminative parameters contributing to classification. The ROC curves (Figure 11) further demonstrate classification performance across different models and feature subsets, with the linear SVM achieving the highest AUC.
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
4 Discussion
This work reports that multiparametric ultrasound echo-envelope statistics derived from the HK, Nakagami, and Burr distributions can differentiate GBM from normal brain tissue in ex vivo tissue samples. The main contribution of this study is a combined framework for our multiparametric approach, integrating complementary information to improve the characterization of GBM microstructure.
4.1 Differentiation of GBM and normal brain tissue using HK, nakagami, and burr parameters
Ultrasound has played an evolving role in brain tumor diagnosis. Tanaka and Ito introduced ultrasound for brain tumor diagnosis in 1966 (). Subsequently, McGahan and colleagues demonstrated the utility of B-mode ultrasound in characterizing brain gliomas (). Then, Leroux et al. showed that B-mode ultrasound enhances tumor identification (). Strowitzkí and colleagues distinguished normal brain tissue, edema, and meningioma by comparing their frequency-dependent backscatter coefficients (). However, the utility of ultrasound echo-envelope statistics for differentiating GBM from normal brain tissue remains underexplored. Previous studies have applied these QUS models to capture variations in scatterer organization in different tissues. The HK distribution has been used to characterize breast tumors and lymph nodes and to stage hepatic steatosis (; ; ). Similarly, the Nakagami distribution has demonstrated its potential to differentiate malignant from normal breast tissues and to assess liver fibrosis in vivo (; ; ; ). The Burr distribution has also shown potential as a biomarker for tissue characterization (; ). Our work extends the potential application of echo envelope statistics for brain tumor characterization and provides a basis for future evaluation in intraoperative settings.
For the HK distribution, the parameter α was higher, whereas k was lower, in GBM than in normal brain tissue (Figures 6A,B). These findings suggest increased scatterer clustering together with a increased contribution of diffuse scattering in GBM. These findings align with histopathological observations, in which GBM regions are characterized by necrotic tumor areas, contributing to heterogeneous clustering, along with microvascular proliferation that causes diffuse scattering and intratumoral interfaces that introduce impedance mismatches.
For the Nakagami distribution, parameters m and Ω were higher in GBM samples than in normal brain tissues (Figures 6C,D). These differences represent an increased effective number density of scatterers and backscattered energy within GBM samples. These findings align with histopathological observations, in which GBM regions are characterized by high cellularity, large, pleomorphic tumor cells, and a high nuclear-to-cytoplasmic ratio.
For the Burr distribution, parameters b and λ were higher in GBM samples than in normal brain tissues (Figures 6E,F). These findings suggest that variations in tumor composition, including necrosis, adjacent cortex, and stromal components (Table 1), influence the density of fractal and elongated scattering structures, which could contribute to increased echo amplitudes in GBM samples. In addition to cellular contributions, extracellular matrix (ECM) remodeling also contributes to the observed scattering behavior (). In GBM, increased deposition of fibrillar collagens (types I and IV), fibronectin, and laminin leads to the formation of elongated, anisotropic scattering sites (; ), which can act as strong scatterers and are consistent with the heavy-tailed amplitude behavior captured by the Burr distribution.
The histopathological heterogeneity of the GBM samples contributed to the observed differences in the envelope-statistics parameters. The GBM samples included varying proportions of viable tumor, necrosis, adjacent cortex, and stromal components (Table 1). Consequently, the observed parameter differences reflect the combined scattering characteristics of these tissue constituents rather than those of viable tumor tissue alone.
Together, the consistent trends observed across all three envelope statistical models highlight complementary scattering mechanisms within GBM microstructure, supporting its differentiation from normal brain tissue. GBM exhibits structural heterogeneity, whereas healthy brain tissue typically exhibits structural homogeneity (; ). These distinct characteristics likely explain the differences in the statistical features of the backscattered ultrasound echoes from GBM tissue compared to those from normal tissue.
4.2 Model fitting and statistical analysis of envelope distributions
The goodness-of-fit analysis demonstrated comparable R2 and RMSE values across the HK, Nakagami, and Burr distributions for both normal brain tissue and GBM samples, indicating that all three distributions adequately represent the statistics of backscattered signals (Table 2).
However, differences in KL divergence demonstrate variations in the ability of these distributions to capture subtle distributional changes associated with tumor heterogeneity. The HK distribution demonstrated overlap in KL divergence across tissue types, suggesting that while it effectively models the overall scattering, it is less sensitive to differences in distribution shape between normal brain tissue and GBM samples (Figure 7A). The Nakagami distribution exhibited lower KL divergence in GBM samples than normal brain tissue, suggesting improved agreement with the altered scattering conditions in tumor regions (Figure 7B). In contrast, the Burr distribution showed higher KL divergence in GBM, indicating increased distributional complexity arising from multiscale and elongated scattering structures that are only partially captured by the distribution (Figure 7C).
The effect size analysis of the estimated distribution parameters further supports these observations, with parameters, including α, m, Ω, and λ, demonstrating large effect sizes, indicating strong differentiation between normal tissues and GBM samples, while the b and k parameters exhibited moderate effects (Table 3). These findings indicate that parameter-based features are more sensitive to underlying microstructural differences than distributional similarity measures.
Taken together, these results suggest that while all three models provide comparable global fits to the data, their parameter estimates capture distinct and complementary aspects of tissue microstructure. The differences observed in KL divergence further highlight variations in model sensitivity to distributional changes, supporting the use of a multiparametric framework to improve the differentiation of GBM samples from normal tissues.
4.3 Multiparametric classification analysis
The multiparametric classification results demonstrate that the extracted envelope statistics features provide consistent discriminative capability across classifiers. For the six-feature set, linear SVM achieved the highest per-sample accuracy of 97.5% (AUC = 0.985), with logistic regression and shallow MLP models also performing comparably (Table 4). The similar performance between the six-feature and reduced four-feature subsets (six-feature: 97.5%; four-feature: 95.0%) indicates that shape-related parameters (α, k, m, b) capture the most relevant microstructural information, while the scale parameters (Ω, λ) contribute comparatively less to classification in this dataset (Figures 8, 9) (Table 5).
To further understand the contribution of individual statistical models, the performance of multiparametric feature sets was compared with single-distribution feature sets (Tables 6, 7). Among the single-distribution sets, HK-based features achieved the highest per-sample accuracy (90.0%; AUC = 0.960) and per-plane accuracy (77.5%; AUC = 0.822). This finding suggests that the HK distribution captures key scattering characteristics associated with tissue microstructure, including scatterer clustering and the relative contributions of coherent and diffuse scattering components. However, the lower performance relative to the multiparametric subsets indicates that HK alone does not fully characterize the underlying tissue heterogeneity. In comparison, the Nakagami and Burr distributions capture additional microstructural properties related to the effective scatterer density, backscattered energy, and multiscale structural organization. The reduced performance for these models (per-sample: 77.5% and 80.0%, per-plane: 70.0% and 72.0%, respectively) confirms that each statistical model describes only part of the underlying scattering behavior. Classifier-dependent differences were observed for the single-distribution feature sets. For the Burr-derived features, logistic regression achieved an AUC of 0.45, whereas linear SVM and MLP achieved AUCs of 0.771 and 0.782, respectively (Table 6). This observation indicates that the classification performance of individual statistical models can vary across classifiers and further supports the use of multiparametric feature subsets for tissue characterization.
TABLE 7
| Feature set | Algorithm | Accuracy (%) | AUC | Sensitivity (%) | Specificity (%) |
|---|---|---|---|---|---|
| HK | Logistic regression | 72.5 | 0.821 | 73.0 | 72.0 |
| Linear SVM | 75.5 | 0.819 | 77.0 | 74.0 | |
| MLP (1 × 64) | 74.0 | 0.833 | 76.0 | 72.0 | |
| Nakagami | Logistic regression | 68.5 | 0.747 | 67.0 | 70.0 |
| Linear SVM | 68.5 | 0.748 | 67.0 | 70.0 | |
| MLP (1 × 64) | 70.0 | 0.756 | 70.0 | 70.0 | |
| Burr | Logistic regression | 67.0 | 0.744 | 61.0 | 73.0 |
| Linear SVM | 66.0 | 0.740 | 61.0 | 71.0 | |
| MLP (1 × 64) | 72.0 | 0.730 | 75.0 | 69.0 |
Per-plane classification performance (group K-fold, n = 200 planes) using Optuna-tuned classifiers for single-distribution feature sets.
In contrast, per-plane classification resulted in lower performance compared to per-sample analysis, reflecting increased intra-sample heterogeneity (Tables 5, 7). Individual imaging planes capture localized variations in tissue structure, thereby increasing classification variability. The improved performance of deeper models in the per-plane setting further suggests that larger and more variable datasets benefit from increased model complexity, whereas simpler models generalize more effectively in smaller datasets. Nevertheless, the ability to achieve reliable classification at the individual plane level indicates that even a single acquisition retains sufficient discriminative information, supporting future evaluation of these methods in intraoperative settings.
Feature importance analysis using random forests identified HK α and Nakagami Ω as the most discriminative parameters within the six-feature subset, while HK-derived parameters (α, k) dominated the reduced four-feature subset (Figure 10). These findings further support the importance of scatterer clustering, scattering behavior, and backscattered energy in differentiating GBM from normal brain tissue. Furthermore, the consistent performance trends observed across logistic regression, linear SVM, and shallow MLP classifiers suggest that classification performance is primarily driven by the discriminative capability of the extracted envelope statistics features rather than by model-specific complexity (Figure 11).
Notably, increasing model complexity did not consistently improve classification performance. For per-sample analysis, simpler models such as linear SVM and shallow MLP achieved optimal performance, whereas deeper architectures demonstrated reduced generalization. Similarly, for per-plane analysis, tuned logistic regression and shallow MLP architectures achieved the best overall performance, while deeper MLP models did not provide a consistent improvement. These findings suggest that the envelope-statistics-based features are largely separable using relatively simple decision boundaries, supporting the use of simpler and more interpretable models for tissue classification.
The nested cross-validation analysis further supports these observations. Although classification performance decreased relative to the corresponding tuned estimates, the overall performance trends across feature subsets remained similar. The highest per-sample accuracy decreased from 97.5% to 90.0% under nested cross-validation. This reduction is consistent with the optimistic bias expected when model selection and performance evaluation are performed on the same resampling splits. Nevertheless, multiparametric feature subsets continued to achieve the highest classification performance under nested cross-validation.
The nested cross-validation results also provide additional insight into the relative contribution of the different envelope-statistics models. HK-derived features achieved the highest performance among the individual distribution-based feature sets. Multiparametric feature subsets, however, maintained a performance advantage. This improvement was reduced under nested cross-validation, suggesting that a substantial proportion of the discriminative information was captured by the HK parameters. Taken together, these findings support the utility of combining complementary envelope-statistics features.
4.4 Limitations
One limitation of this study is the limited and imbalanced representation of samples from males and females within the dataset. The number of tissue samples from females was comparatively smaller, which may limit the generalizability of the findings. In addition, the classification analysis was limited by the sample size (n = 40 per sample, n = 200 per plane), which may limit the generalizability of the reported accuracies. Whole-ROI parameter averages were used as classification features, which do not capture the spatial heterogeneity within individual tissue samples. Future studies with larger and more balanced cohorts are required to further validate these observations.
Obtaining adjacent normal brain tissue from the same individual with GBM is challenging, limiting our ability to assess the effectiveness of this approach for determining tumor margins. Furthermore, the GBM and normal brain tissues were obtained from different sources. The GBM tissues were collected during surgical resection, whereas normal tissues were obtained from post-mortem brain bank specimens. Consequently, differences in post-mortem interval, tissue handling, fixation timing, age distribution, and sex balance represent potential confounding factors when interpreting the observed parameter differences and classification performance. Future in vivo studies are required to assess the efficacy of this approach for margin delineation. Larger and clinically matched cohorts are also required to further evaluate the influence of these factors.
This study utilized formalin-fixed tissue samples, which can alter acoustic properties relative to fresh tissue. Previous studies have shown that fixation can substantially modify attenuation due to protein cross-linking, loss of perfusion, and changes in water content. Kremkau et al. measured human brain tissue before and after formalin fixation at 2.25 MHz and reported attenuation changes ranging from −29% to +55%, while Hoffmeister et al. observed an increase of approximately 20% at 5 MHz in formalin-fixed bovine brain (; ). Although fixation alters tissue hardness, it preserves cellular architecture essential for histopathological diagnosis. Because both GBM and normal samples in this study were fixed under identical conditions, the relative trends in acoustic parameters are likely to remain comparable.
Pulse-echo measurements with a perfect reflector and a single-element transducer were used to compensate for attenuation. Here, attenuation compensation was performed using group-specific average attenuation coefficients, as sample-specific attenuation estimates could not be obtained for all tissues due to limited tissue dimensions and insufficient signal-to-noise ratio. Consequently, the applied compensation may not fully capture sample-to-sample variability in attenuation properties and their potential influence on the estimated envelope statistics parameters. However, for in vivo translation, attenuation can potentially be measured from backscattered data using spectral-shift methods (; ; ). Further, given that the probe is in close proximity to the region being assessed for tumor resection assessment, this application may be relatively robust to confounds of attenuation compared to extracorporeal applications. Nonetheless, it is imperative to conduct further investigations using freshly excised tissues or in vivo, with a larger sample size, to improve our understanding of the relationship between envelope statistical parameters and tissue-level microstructural changes.
The envelope statistics parameters reported may have some degree of system dependence due to a lack of compensation for receiver gain and beam diffraction. In the future, we envision using these compensation techniques to assess the impact of system-dependent factors and to facilitate quantitative parameter comparisons across ultrasound systems. Despite these limitations, this study provides proof of concept that GBM can be differentiated from normal brain tissues using envelope statistics. Our current findings lay the groundwork for subsequent in vivo studies, which will focus on the in-situ examination of brain tumors in patients using a specialized probe to assess the efficacy of these techniques in intraoperative tumor margin delineation.
4.5 Conclusion
The feasibility of using envelope statistics derived from HK, Nakagami, and Burr distributions to differentiate GBM samples from normal brain tissue was demonstrated in ex vivo samples. Significantly higher parameter values were observed in GBM compared to normal brain tissue, reflecting underlying variations in tissue microstructure and scattering behavior.
Multiparametric feature subsets achieved 97.5% per-sample accuracy and 80.5% per-plane accuracy, whereas HK-derived features alone achieved 90.0% per-sample accuracy. These findings indicate that the HK parameters contributed to tissue differentiation, while combining complementary envelope-statistics features provided additional information for tissue characterization. Although nested cross-validation yielded lower performance estimates than the corresponding tuned cross-validation analyses, the overall trends across feature subsets remained unchanged.
These methods could complement conventional B-mode ultrasound by providing quantitative information regarding tissue microstructure. Such information could assist in differentiating tumor and non-tumor tissues and potentially support tumor margin assessment. Further studies in freshly excised tissues and intraoperative settings are required to evaluate the clinical utility of this approach. Overall, these findings provide proof of concept for the use of multiparametric ultrasound envelope statistics in GBM tissue characterization.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.
Ethics statement
The studies involving humans were approved by National Institute of Mental Health and Neurosciences Institute Ethics Committee, and the Indian Institute of Technology Gandhinagar Institutional Ethics Committee. The studies were conducted in accordance with the local legislation and institutional requirements. Written informed consent for participation in this study was provided by the participants' legal guardians/next of kin.
Author contributions
JP: Conceptualization, Investigation, Writing – review and editing, Validation, Software, Writing – original draft, Formal Analysis, Visualization, Data curation, Methodology. MA: Writing – original draft, Writing – review and editing, Visualization, Formal Analysis, Software, Validation, Methodology, Investigation, Data curation. SR: Validation, Formal Analysis, Methodology, Writing – review and editing, Investigation, Data curation, Writing – original draft. VV: Data curation, Resources, Validation, Conceptualization, Writing – review and editing, Investigation, Writing – original draft. HP: Writing – review and editing, Conceptualization, Methodology, Writing – original draft. AM: Writing – original draft, Resources, Data curation, Validation, Writing – review and editing, Methodology. HS: Supervision, Writing – review and editing, Conceptualization, Writing – original draft, Visualization, Methodology. KM-S: Methodology, Supervision, Conceptualization, Writing – review and editing, Formal Analysis, Visualization, Writing – original draft, Project administration, Funding acquisition, Resources.
Funding
The author(s) declared that financial support was received for this work and/or its publication. This work was supported by the India Alliance, Department of Biotechnology (Grant No. IA/TSG/23/1/600493), Department of Science and Technology, Government of India (Grant No. DST/TDT/BDTD/08/2021), and the Gujarat State Biotechnology Mission, India (Grant No. GSBTM/JD (R&D)/610/20–21/344).
Acknowledgments
We thank Souritra Garai, Vishwas Trivedi, and Anushka Yadav for helpful discussions. JMP thanks the Ministry of Education, Government of India, for their support through the Prime Minister’s Research Fellowship.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/facou.2026.1882624/full#supplementary-material
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Summary
Keywords
brain tumors, envelope statistics, Homodyned-K distribution, Nakagami distribution, Burr distribution, multiparametric imaging, tissue characterization
Citation
Patil JM, Agarwal M, Rao S, Vazhayil V, Pandya HJ, Mahadevan A, Shekhar H and Mercado-Shekhar KP (2026) Ultrasound echo envelope statistics to differentiate glioblastoma multiforme from normal brain tissue ex vivo: a proof-of-concept study. Front. Acoust. 4:1882624. doi: 10.3389/facou.2026.1882624
Received
15 May 2026
Revised
19 June 2026
Accepted
22 June 2026
Published
11 August 2026
Volume
4 - 2026
Edited by
Chengzhi Shi, University of Michigan, United States
Reviewed by
Santiago Cepeda, Hospital Universitario Río Hortega, Spain
Hao Wu, The Second Affiliated Hospital of Xi’an Medical University, China
Updates
Copyright
© 2026 Patil, Agarwal, Rao, Vazhayil, Pandya, Mahadevan, Shekhar and Mercado-Shekhar.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Karla P. Mercado-Shekhar, karlamshekhar@iitgn.ac.in
Disclaimer
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