A dynamic quantum clustering approach to brain tumor segmentation

1 Introduction

Clustering is inherently an ill-posed problem, yet it plays a crucial role across numerous scientific and technological domains. Given a dataset, the primary objective is to identify potential structures by determining which data points are similar and can be grouped meaningfully. This step serves as a foundational stage prior to deeper analysis of the shared characteristics within each identified subset. Various unsupervised classification methods have been developed for this purpose, including k-means and fuzzy c-means clustering, both commonly used in image segmentation tasks (Wadhwa et al., 2019). More recently, Dynamic Quantum Clustering (DQC) has emerged as a powerful technique for exploring complex datasets, particularly in the contexts of Big Data analytics (Weinstein et al., 2013) and machine learning (Wittek, 2013), enabling the discovery of hidden relationships among data subsets (Weinstein and Horn, 2009; Horn and Gottlieb, 2001). Its high sensitivity in revealing the natural number of clusters has been successfully demonstrated in prior applications to medical imaging (Martín-Landrove et al., 2020; Torres-Hoyos et al., 2019). In the present study, we propose the full application of the DQC algorithm for the segmentation of medical images, aiming to leverage its strengths in detecting intrinsic data structure without the need for predefined class numbers.

2 Methods

2.1 Image selection

Images of high-grade gliomas were obtained from various publicly available datasets hosted by The Cancer Imaging Archive (TCIA) (Clark et al., 2013). Specifically, the TCGA-LGG collection (Pedano et al., 2016) and the REMBRANDT repository (Scarpace et al., 2015) were used for cases involving astrocytomas and oligodendrogliomas of grades II and III. For glioblastoma multiforme (GBM), images were sourced from both the TCGA-GBM collection (Scarpace et al., 2016) and the UPENN-GBM dataset (Bakas et al., 2021). In addition, benign brain tumor cases were extracted from the BraTS Meningioma Pre-operative Dataset, BraTS 2023 Challenge (LaBella et al., 2024). For the single-modality analysis, contrast-enhanced T1-weighted MRI images were selected across all collections. In the case of multi-modality analysis, only the UPENN-GBM dataset was used due to its comprehensive imaging protocol. In all cases, tumor lesions were pre-identified within the datasets and were segmented in such a way as to be clearly distinguished from surrounding anatomical structures prior to image processing.

2.2 Dynamic quantum clustering algorithm and image space

2.2.1 General

The DQC algorithm models the dataset as a collection of points in a high-dimensional space, where each point is associated with an intrinsic uncertainty, denoted by σ. The overall distribution of the data is represented by a Parzen estimator, φ, which is constructed as a sum of Gaussian kernels centered at each data point. This estimator serves as a probability density function and is interpreted as the ground-state wavefunction of a quantum system. Specifically, φ satisfies the time-independent Schrödinger equation, where the quantum potential is derived directly from the data distribution itself (Weinstein and Horn, 2009; Horn and Gottlieb, 2001). This quantum formalism allows the data points to evolve dynamically under the influence of the quantum potential, naturally revealing the underlying clustering structure without the need for prior assumptions about the number of clusters. Moreover, it is assumed that φ is the wave function associated to the ground state, that is, which leads to the evaluation of energy potential, It is common practice in medical image analysis to acquire multiple imaging modalities and combine them to improve the accuracy and diagnostic performance of image interpretation. The resulting dataset consists of the combined intensity information from all co-registered image modalities and can be represented by an n-dimensional joint histogram, where n denotes the number of modalities. In the present work, the dimensionality of the feature space is limited to the number of imaging modalities, since only image intensity values are considered. However, this is not an inherent limitation of the DQC framework. Additional image-derived features, such as quantitative parametric maps (e.g., apparent diffusion coefficient or pharmacokinetic parameter maps) and higher-order radiomic descriptors, can also be incorporated, thereby increasing the dimensionality of the feature space. Within the DQC framework, the Parzen estimator is obtained by convolving the n-dimensional joint histogram, h, with a Gaussian kernel, yielding where N is the total number of data points, corresponding to the number of bins in the n-dimensional histogram, x→ denotes an n-dimensional feature vector and within the scope of the present study its components are the intensity from each MRI modality, x→i represents the centers of the discrete histogram bins, and σ is the Gaussian kernel width that controls the degree of smoothing. Within the physical framework implied by Equation 1, the same value of σ is assumed for all imaging modalities under study. This operation yields a continuous approximation of the joint probability density function of the image intensities, which forms the basis for constructing the quantum potential in the DQC framework, as follows, The selection of σ² in Equation 4 is, in principle, arbitrary and generally not critical (Horn and Gottlieb, 2001), since the most representative minima of the quantum potential remain essentially unchanged over a broad range of σ² values.

-12σ2∇2φ+V(x→)φ=Eφ=0(1)

V(x→)=12σ2∇2φφ(2)

φ(x→)=∑l=1Nh(x→l)e-(x→-x→l)22σ2(3)

V(x→)=12σ4[1σ2∑i=1Nh(x→l)(x→-x→i)2e-(x→-x→i)22σ2∑i=1Nh(x→l)e-(x→-x→i)22σ2-n](4)

2.2.2 Dynamics

Using the Ehrenfest theorem, it is possible to derive the equation of motion for each image intensity vector, x→k in the dataset, treating each as a “particle” evolving under the influence of the quantum potential derived from the Parzen estimator, Equation 4. The motion of each point follows the classical expectation of quantum dynamics and is given by: where x→k(t) represents the trajectory of the k digital level over time, σ² represents the “mass” parameter and V(x→k(t)) is the quantum potential defined by Equation 2. To ensure convergence of the data points toward the potential energy minima, and thereby induce clustering, a dissipation term must be introduced. This modifies the system's dynamics into a second-order Langevin equation, which governs the motion of each data point under both the quantum potential and a damping force (Lafata et al., 2019). The equation takes the form: where γ is the damping coefficient. The damping term ensures that over time, the system loses kinetic energy and each point settles into a local minimum of the potential landscape. These equilibrium positions correspond to the clusters revealed by the data distribution. This approach effectively transforms the clustering problem into one of dissipative dynamics, where the stable configurations of the system naturally represent the underlying cluster structure of the data. The clustering process in the DQC framework requires the appropriate selection of several key parameters, including the Gaussian kernel width σ², which plays a role analogous to a mass in the dynamical system, the damping coefficient γ, and the integration time step used in the simulation of the particle dynamics. These parameters are not fixed universally but are estimated based on the scale and distribution of the digital intensity levels, and are thus adapted to the specific characteristics of each image dataset. A proper balance among these values ensures stable and rapid convergence of the data points to the minima of the quantum potential, effectively revealing the cluster structure. An illustrative example of the clustering of digital levels using this method is shown in Figure 1, where the dynamic evolution of the data leads to well-defined groupings corresponding to distinct tissue classes or tumor regions.

d2x→k(t)dt2=-∇V(x→k(t))σ2(5)

σ2d2x→kdt2=-∇V(x→k)-γdx→kdt(6)

2.3 Image segmentation

The general schematic of the proposed segmentation approach is presented in Figure 2. First, an initial three-dimensional region of interest (ROI) is extracted from the original image to reduce the matrix size encompassing the entire tumor lesion. The segmentation process, performed using the DQC algorithm, is subsequently divided into two main phases:

• Phase I involves a coarse, or gross, segmentation in which the original image is processed to identify and isolate the region of interest (ROI) by suppressing irrelevant anatomical structures and eliminating image artifacts. This step ensures that subsequent analyses focus exclusively on the relevant tissue or lesion areas, effectively constraining the digital intensity levels to those corresponding to the lesion and its immediate neighborhood.
• Phase II carries out a refined segmentation within the pre-selected ROI obteined in Phase I.

The structure and processing pipeline of each phase are as follows:

• Determination of the image histogram and Parzen estimator, φ(x→)
• Evaluation of the Schrödinger potential, V(x→)
• Determination of the acceleration, -∇V(x→k(t))σ2
• Dynamic A: the first application of the dynamic evolution is performed on the original image, using the quantum potential and corresponding acceleration field previously computed from the image histogram via the Parzen estimator. At the conclusion of this dynamic evolution, a new Parzen estimator is constructed based on the updated (clustered) image histogram. From this, a trap potential, denoted as V_trap, is derived. This trap potential reflects the stabilized configuration of the clustered data and reveals the number and approximate locations of dominant clusters. the number of distinct minima in V_trap serves as a data-driven estimate of the number of classes present in the image. This information is subsequently used to inform and initialize a conventional classification algorithm, such as k-means, which can then perform a final segmentation using the cluster centroids suggested by the quantum dynamics.
• Dynamic B: a second application of the dynamic evolution is performed on the original image, this time using the trap potential V_trap, along with the corresponding acceleration field computed from its gradient. By guiding the evolution of data points using this refined potential landscape, the algorithm enhances the segmentation resolution within the ROI, isolating subtle variations in intensity that may correspond to tumor substructures or heterogeneous tissue components.
• Following the dynamic clustering phase, both the original and the clustered images are subjected to classification using the k-means algorithm. The number of classes k is not predefined but rather determined by the number of local minima identified in the trap potential V_trap. The resulting classified images are then visually and analytically inspected by a qualified user to identify and select the classes of interest, typically corresponding to tumor regions, peritumoral edema, or other relevant anatomical features. This step introduces a semi-supervised component to the pipeline, where expert input refines the automatic classification output. A final segmentation mask is generated by combining the selected class labels from both the original and clustered images. This mask encapsulates the consensus segmentation, providing a comprehensive delineation of the regions of interest suitable for further analysis or clinical interpretation.
• The segmentation of the original image is performed based on the classification results obtained in the previous step. The resulting segmented image is then carefully reviewed to eliminate non-relevant anatomical structures and suppress potential image artifacts that may have been erroneously included in the classification. This stage requires the intervention of a qualified user, whose domain expertise ensures that only anatomically or clinically significant regions are retained. In the context of Phase I, this process yields a three-dimensional region of interest (ROI) that accurately encloses the suspected lesion or pathology. This ROI is then used as the input image for Phase II of the segmentation framework. By focusing subsequent analysis on this confined region, the dimensionality of the digital level space is significantly reduced, allowing for a more refined clustering and segmentation in the next phase. This targeted approach improves computational efficiency and enhances the precision of tumor delineation.

An illustrative example of the initial Schrödinger potential, the resulting clustered “trap” potential, and their corresponding acceleration fields is presented in Figure 3. These visualizations demonstrate the evolution of the potential landscape throughout the clustering process, highlighting how the dynamic quantum framework progressively shapes the data distribution to reveal underlying structural features. The differences between the initial and refined potentials reflect the convergence of data points toward stable regions, ultimately guiding the segmentation.

3 Results and discussion

3.1 One-dimensional analysis: single image modality

The proposed clustering procedure can be applied to any individual image modality, where the corresponding quantum potential is computed using Equation 4 with n = 1. This modality-specific application is particularly useful for parameter tuning, enabling the optimization of the algorithm's configuration before extending it to multi-modality image analysis. For the sake of simplicity and to provide a clear understanding of the clustering behavior, we focus the present analysis on contrast-enhanced T₁-weighted MRI images.

As previously discussed, DQCA requires careful selection of dynamic parameters, specifically the kernel width σ², the damping coefficient γ, and the simulation time interval. These parameters must be adapted to the properties of the image intensity space. Inappropriate parameter choices may lead to incomplete or unstable clustering, resulting in suboptimal classification performance. Critically, the number of clusters obtained is directly influenced by the number of minima in the computed trap potential V_trap, which in turn is sensitive to the value of σ² (Horn and Gottlieb, 2001).

As illustrated in Figure 4, increasing this parameter leads to a smoother potential landscape, effectively reducing the number of local minima. This reduction results in a lower number of detected clusters, thereby altering the final segmentation outcome. The example clearly demonstrates how tuning σ² controls the granularity of the segmentation, with noticeable impacts on the interpretation of the clustered image.

The selection of σ² is therefore a critical compromise in the dynamic quantum clustering framework. If σ² is set too small, the resulting potential landscape exhibits numerous local minima, leading to over-segmentation of the image. In such cases, the user is required to manually inspect and select from a larger number of classes, increasing the time, effort, and expertise needed for accurate segmentation. Conversely, if σ² is set too large, the Parzen estimator becomes overly smoothed, approaching a uni-modal distribution that effectively erases the structural details encoded in the image histogram. This results in under-segmentation, where distinct anatomical or pathological regions are merged into a single class, compromising the diagnostic utility of the segmentation. Thus, an optimal value of σ² must strike a balance between preserving meaningful image features and limiting the complexity of user-driven class selection.

In the present work, the parameter σ was selected to be proportional to the bin width, Δ_bin, of the image histogram and kept fixed throughout all experiments to ensure a consistent comparison with the k-means algorithm. Since Δ_bin depends on the dynamic range of digital intensity levels, this choice ensures that the scale of uncertainty used in the Parzen estimator remains consistent across different images, regardless of their specific intensity distributions. This strategy standardizes the smoothing behavior of the Parzen estimator and allows the dynamic quantum clustering algorithm to be applied uniformly to a variety of image datasets. The value of σ is thus defined as, where κ is a dimensionless proportionality constant, empirically selected to balance detail preservation and segmentation robustness. This definition facilitates reproducibility and simplifies parameter tuning for both single- and multi-modality image analysis.

σ=κΔbin(7)

The average behavior of the number of classes as a function of the parameter κ is presented in Figure 5, illustrating how the smoothness of the Parzen estimator influences the resulting segmentation complexity. As κ increases, a gradual reduction in the number of detected classes is observed, reflecting the smoothing effect of the estimator on the potential landscape.

In addition, the distribution of the number of classes obtained for different values of κ across the dataset is shown in Figure 6. This distribution highlights the sensitivity of the clustering process to the choice of κ demonstrating how small variations in this parameter can lead to significantly different segmentation outcomes.These results emphasize the importance of selecting an appropriate κ value to achieve a balance between over- and under-segmentation.

Analysis of Figure 5 reveals that the number of classes N_*C*, exhibits a power-law dependence on the parameter κ. This relationship can be expressed as: where λ is a positive exponent that characterizes the rate at which the number of classes decreases with increasing κ. The power-law behavior indicates a scale-invariant property in the segmentation process, suggesting that small increases in κ lead to significant reductions in segmentation granularity. Once σ² is appropriately selected, the Parzen estimator, potential, and acceleration are determined straightforwardly. The time interval, δt, and γ are then selected to provide a smooth pace of the dynamics as shown in the example of Figure 1. These parameters are, where f = 0.01 and a_*max* is the maximum absolute value of the acceleration matrix ã obtained from Equation 5 and with f₀ = 0.01.

NC~κ-λ(8)

δt=2fΔbinamax(9)

γ=f0δt(10)

The minima of the “trap” potential obtained through the DQC algorithm exhibit a close correspondence with the class centroids identified by the k-means algorithm, when the number of clusters is set equal to the number of potential minima. This relationship is illustrated in Figure 7, which was generated using an intermediate value of σ, corresponding to κ in the range 2 − 3. This result underscores the effectiveness of the quantum clustering framework in driving digital levels toward nearby stable points in the potential landscape. The dynamics inherently guide data points toward regions of maximal local data density, which align well with the centroids computed by traditional k-means clustering.

Importantly, this behavior implies that the trap potential minima serve as reliable and data-driven initial estimates for class centroids, effectively eliminating the need for additional centroid optimization strategies. In contrast to standard k-means approaches, which often require external optimization methods such as Discrete Particle Swarm Optimization (DPSO) (Kumar et al., 2022; Chen et al., 2019; Xiaoqiong and Zhang, 2020) to avoid suboptimal convergence, the proposed method naturally reveals the underlying class structure through the dynamics of the quantum potential. Thus, the dynamic quantum clustering approach offers a computationally efficient and theoretically grounded alternative for class initialization and segmentation in medical imaging.

A comparison between the segmentation results obtained using k-means and those from the DQC algorithm reveals a high degree of similarity, as illustrated in Figure 8. This consistency indicates that DQC, when properly parameterized, can serve as a standalone segmentation tool, potentially eliminating the need for subsequent k-means classification, provided that an expert review or an automated class selection criterion is incorporated. The ability of DQC to reproduce the key structures identified by k-means demonstrates its robustness in capturing the intrinsic clustering structure of the image intensity data through its dynamic evolution mechanism. This suggests that, following an initial expert-guided selection of relevant classes, DQC may be sufficient for automated segmentation, reducing algorithmic complexity while maintaining performance.

Nevertheless, it is important to note that the removal of irrelevant anatomical structures or image artifacts—whether due to normal anatomical variation, imaging conditions, or noise—still requires the intervention of a qualified user or the development of equivalent automated filtering strategies. As such, while DQC can simplify the segmentation workflow, some degree of manual curation or expert oversight remains necessary to ensure clinically relevant results.

Figure 9 presents an example of a detailed three-dimensional reconstruction of tumoral lesions, encompassing all tumor types analyzed in this study. This visualization highlights the capability of the proposed segmentation framework to accurately delineate and reconstruct complex tumor morphologies across a diverse set of brain tumor classes, including both high-grade and low-grade gliomas, as well as benign tumors.

The performance of a segmentation algorithm can be evaluated using the following indices: the Sørensen–Dice coefficient and the Jaccard index (Bertels et al., 2019; Carass et al., 2020; Eelbode et al., 2020; Müller et al., 2022), which are defined as follows, and where A and B represent different segmented datasets. Additionally, it is possible to define a distance between segmentations using the Jaccard index as,

Dice(A,B)=2|A⋂B||A|+|B|(11)

J(A,B)=|A⋂B||A⋃B|(12)

dJ(A,B)=1-J(A,B)(13)

Figure 10 compares the performance of two algorithms—Dynamic Quantum Clustering (DQC) and K-means clustering—using the BraTS Meningioma Pre-Operative Dataset from the BraTS 2023 Challenge (LaBella et al., 2024), which includes ground-truth segmentations provided for the challenge. For the construction of Figure 1, only segmentations corresponding to contrast-enhanced regions were extracted from the ground-truth data. The Dice and Jaccard indices were then calculated for both algorithms while progressively including an increasing number of classes in the segmentation. Specifically, the classes were ordered according to image intensity, with class 1 corresponding to the highest image intensity.

The analysis of Figure 10 and Table 1 indicates that, for segmentations involving up to four classes, the difference in the Dice similarity index is statistically significant (p < 0.05), with the K-means algorithm outperforming the Dynamic Quantum Clustering (DQC) method. Furthermore, increasing the number of classes beyond four leads to a reduction in the Dice similarity index for both algorithms. Notably, the obtained Dice values are comparable to the best results reported in the BraTS 2023 challenge for the segmentation of contrast-enhancing tumor regions (0.856) LaBella et al. (2024).

Table 1. Results of the ANOVA test comparing the Dice similarity index obtained from datasets segmented with the K-means and Dynamic Quantum Clustering (DQC) algorithms, evaluated as a function of the number of segmentation classes N_*C*.

The last column reports the Jaccard distance between the segmentations produced by both methods. Colored rows indicate similar Dice index distributions for both methods, as determined by the ANOVA test. Yellow highlights the number of classes corresponding to the maximum Dice index.

3.2 N-dimensional analysis: multimodality images

3.2.1 Direct approach: multimodality clustering

The application of the Dynamic Quantum Clustering (DQC) method to multimodal medical imaging arises naturally from its single-modality formulation, as governed by Equation 4. While the core principles of the algorithm remain unchanged, the dynamics in the multimodal context become more intricate due to the interaction between different imaging modalities—such as T₁-weighted, contrast-enhanced T₁-weighted, and T₂ FLAIR images. This complexity is illustrated in Figure 11, which depicts the elaborate trajectories followed by data points during the quantum evolution. A critical distinction in the multimodal case lies in the modality-dependent scaling of the “mass” parameter, denoted by σ², which modulates the quantum potential and thus shapes the clustering behavior. The influence of this parameter on the potential landscape and the resulting dynamics is further demonstrated in Figure 12.

Nevertheless, the clustering process in the multimodal setting introduces additional computational and methodological complexity. In particular, the number of potential minima—which corresponds to the number of identifiable clusters—remains undefined until a sufficient number of dynamic iterations has been performed. This requirement can lead to substantially increased computational time, particularly when combined with the challenge of appropriately defining the discretized Schrödinger equation and boundary conditions for the high-dimensional data space (or “potential box”).

Despite these complexities, partial clustering results can still be extracted from early stages of the dynamics, offering valuable insights into the underlying data structure. This is exemplified in Figure 13, where a representative multimodal clustering output demonstrates the feasibility and potential of the method even under constrained simulation conditions.

3.2.2 Alternative approach: independent modality clustering

Although the DQC algorithm can be directly applied to multimodal medical images, the analysis of the resulting clustered data points often entails extremely long processing times due to slow convergence. Additionally, the boundary conditions imposed on the potential “box” can introduce artifacts in the clustering process—for example, the artificial confinement of certain clusters near the boundaries of the potential domain. To address these limitations, an alternative approach involves performing the DQC algorithm independently on one-dimensional, single-modality images, and subsequently integrating the results to generate a multimodal clustering outcome. An illustrative example of this procedure is presented in Figure 14. For small values of the quantum parameter κ, the distribution of clustered points forms a dense “grid” that closely mirrors the original data distribution. As κ increases, the granularity of the clustering decreases, resulting in a sparser grid structure due to a reduction in the number of identified cluster classes. This trend is visually evident in the clustered images depicted in Figure 14, where the simplification of structure with increasing κ highlights the impact of this parameter on clustering resolution.

4 Final remarks

The one-dimensional (single-modality) implementation of Dynamic Quantum Clustering (DQC) provides a fast and reliable clustering approach, with the distinct advantage that the number of clusters is determined solely by the topology of the quantum potential. With an appropriate choice of the algorithm parameters, namely the “mass” (or σ²) and the simulation time interval—both of which are purely mathematical quantities—DQC achieves segmentation performance comparable to that of the k-means algorithm. In the present work, image intensity was used as the primary feature for segmentation because it is the most widely available attribute across medical imaging modalities. However, the DQC framework is equally applicable to other image-derived features, including higher-order radiomic descriptors or quantitative parametric maps that reflect the underlying physical properties of the tissue (e.g., apparent diffusion coefficient maps in diffusion MRI or pharmacokinetic parameter maps in dynamic contrast-enhanced MRI). The multidimensional (multimodality) extension of DQC enables the simultaneous integration of information from multiple co-registered imaging modalities. This can be implemented either by applying the algorithm directly to the complete multidimensional feature space, thereby fully exploiting the complementary information available across modalities, or by applying DQC independently to each modality and subsequently combining the resulting segmentations or feature representations. Although the former approach captures the full complexity of the multimodal dataset, it generally requires greater computational effort and exhibits slower convergence as the dimensionality of the feature space increases. The latter provides a computationally efficient alternative while preserving the flexibility to integrate complementary information from different imaging modalities. These complementary strategies offer promising directions for future research aimed at optimizing multimodal image segmentation and improving the computational efficiency of DQC in high-dimensional clinical imaging applications.

5 Conclusions

In the present work, an effective image segmentation method based on the Dynamic Quantum Clustering (DQC) algorithm was developed and applied to the analysis of medical images, with particular emphasis on brain tumor MRI. The algorithm parameters were optimized using single-modality data, specifically contrast-enhanced T₁-weighted images, providing a controlled framework for calibrating the algorithm and evaluating its segmentation performance. It is important to emphasize that the DQC algorithm operates exclusively on the mathematical representation of the input data and does not incorporate any explicit knowledge of the underlying physical, biological, or clinical meaning of the image features. Consequently, the resulting segmentation reflects similarities in the selected feature space rather than clinically meaningful tissue classes per se. As with many other unsupervised image segmentation methods, the interpretation of the segmentation results requires clinical expertise and should be performed in conjunction with a qualified medical professional.

For one-dimensional data spaces, corresponding to single-modality images, the proposed method demonstrated a high degree of consistency with conventional classification algorithms such as k-means. Furthermore, it offered a clear and interpretable clustering mechanism, driven by the dynamics of data evolution within the associated quantum potential landscape.

In the case of multimodality MRI data, the DQC algorithm extends naturally and preserves its theoretical foundation. However, the clustering process becomes significantly more complex, both computationally and in terms of interpretability. The increased dimensionality of the potential surface, along with the greater heterogeneity in data distributions across modalities, contributes to this complexity and hinders the identification of well-defined clusters. To address these challenges, an alternative strategy is proposed: applying the DQC algorithm independently to one-dimensional, single-modality images, followed by the integration of the resulting cluster structures to achieve a multimodality clustering representation.

Despite these difficulties, the method shows strong potential for integration in advanced medical image analysis pipelines, particularly when combined with expert guidance or automated class-selection criteria. These results support the feasibility of using DQC for robust, unsupervised segmentation tasks across both single- and multi-modality medical imaging settings.

Generative AI statement

The author(s) declared that Generative AI was not used in the creation of this manuscript.

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Footnotes

Edited by: Fatma Gargouri, INSERM U1127 Institut du Cerveau et de la Moelle épinière (ICM), France

Reviewed by: Stanislav Pekov, Moscow Institute of Physics and Technology, Russia

Gayathri N., Sri Eshwar College of Engineering, India