Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
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Liping Yu
, Wasim Jamshed
Abstract
Breast cancer diagnosis relies on breast ultrasound (BUS) and the early breast cancer screening saves lives. Computer-aided design (CAD) tools diagnose tumours via BUS tumour segmentation. Thus, breast cancer analysis automation may aid radiologists. Early detection of breast cancer might help the patients to survive and in context with this many approaches have been demonstrated by different researches, however, some of the works are weak in the segmentation of breast cancer images. to tackle these issues, this study propose a novel Hybrid Attendseg based gravitational clustering optimization (HA-GC) method which is utilized to segment breast cancer as normal malignant, and benign. For this we have taken the dataset known as breast ultrasound (BUS) images. This method constructively segments the breast cancer images. Prior to the segmentation, pre-processing is carried out which can be used to normalize the images incorporated with the removal of unwanted noises and format the images Optimization selects the best qualities. An experiment is conducted and compared the results with the parameters such as Dice coefficient, Jacquard, Precision, and Recall and attained over 90% and ensures the usage of present work in the segmentation of breast cancer images.
1 Introduction
Breast cancer is the most common malignancy in females and the leading cause of death among women worldwide [1]. In 2020, more than 7.8 million cases of breast cancer were detected worldwide, and 2.3 million women received a diagnosis. Worldwide, 685,000 individuals lost their lives to the illness [2]. Humans’ transition from a traditional to a modern lifestyle hastens the development of breast cancer. Through screening, breast cancer can be found early, which significantly lowers mortality and improves patient survival [3]. However, accurate and reliable diagnosis is necessary for early detection and localization. There are two types of tumours: benign and malignant. Malignant tumours are thought to be hazardous and are referred to as such, whereas benign tumours, which do not contain cancerous cells, are seen to be less toxic. The variables that show the level of risk in breast tumours include the tumour size, shape, location, behaviour, and growth rate.
Breast cancer may be detected and treated early with the use of a variety of imaging methods. Breast ultrasonography is often used in clinical practise for diagnosis [4]. Ultrasonography has lately been one of the most extensively used procedures for the clinical diagnosis of breast cancer due to its cheap cost, non-invasive nature, and absence of radioactivity. The most common abnormalities are masses and micro calcifications [5,6]. Micro calcifications, which are deposits of calcium, will show up as pinpoints of increased ultrasound signal. In order to make an accurate diagnosis of breast masses early on, image processing methods are essential due to the fact that breast masses and micro calcifications on an ultrasound seem like the backdrop. Considering the difficulty in finding and identifying breast masses. Researchers have developed a number of methods for localizing breast masses with high accuracy. If radiologists give ground truth indicators [7] that may be used to discriminate between real and false breast masses, we can evaluate the efficacy of these methods by comparing the acquired findings with the ground truth markers [8,9]. Because of the importance of accurately diagnosing and classifying breast cancer early, research has focused on creating an automated system to aid professional radiologists in this process. This justification underpins the following interventions aimed at improving breast cancer detection and classification.
A number of segmentation models based on convolutional neural networks (CNNs) have also been extensively researched. However, noise, stumpy contrast, unclear boundaries, various tumour sizes, and forms make it difficult to precisely split breast tumor cells on ultrasound pictures [10]. Additionally, the majority of traditional techniques only trained one network for tumour pixel segmentation [11]. Due to the high frequency of false-positive pixels in normal images that do not include tumours, this greatly lowers the segmentation performance during testing. In order to achieve precise tumour segmentation in breast ultrasound images, AttendSeg architecture with a gravitational clustering technique was developed in this research. Major contributions of the work are listed below,
This work is proposed for the segmentation of breast cancer images. to begin with the images are pre-processed to normalize the images and remove the unnecessary items from the images to focus the required area.
The proposed hybrid AttendSeg based Gravitational clustering is used to segment the breast cancer features from collected pre-processed images. the segmentation usually performed to classify the images as Benign, Malignant, and normal.
The remaining part of the article is arranged as; in Section 2 the relevant works are highlighted. The proposed work is elucidated in Section 3, and the simulation part is effectuated in Section 4, finally, the summarization is presented in Section 5.
2 Related works
A technique known as image segmentation extracts the area of interest in an image based on its pixel properties (gray, texture, etc.). The breast nodule is the area of attention on breast ultrasound pictures. To offer data as input for the subsequent classification operation, the sick nodule must be separated from the normal tissue region. Some of the most well-studied methods for ultrasonic image segmentation at the moment include the threshold and edge method, the area technique, the graph theory and clustering method, the energy functional method, and the neural network method [12].
Traditional image segmentation techniques, such as the threshold method, edge method, and area method, make it tough to immediately acquire the best segmentation results. General researchers will make significant changes and additions to the aforementioned algorithms. Liu et al. [13] proposed a framework for completely automated segmentation of the lesion area in breast ultrasound imaging by using morphological filtering and the Otsu-based adaptive thresholding (OBAT) technique to locate the region of interest (ROI) and determine the nodule shape. Wu et al. [14] worked on fetal ultrasound image quality assessment using deep convolution networks. The authors proposed a method for assessing the quality of fetal ultrasound images using deep convolutional networks. The paper presents experimental results demonstrating the effectiveness of their approach. Therefore, after initialising the nodule contour, Liu et al. [13] refined the Chan-Vese model to achieve precise nodule segmentation.
In recent years, a method for segmenting ultrasonic images using neural networks has been presented. Wu et al. [14] employed Convolutional Neural Networks (CNN) to detect the Region of Interest (ROI) in the fetal abdomen in the ultrasound picture, allowing them to screen high-quality ultrasound images for the measurement of fetal Abdominal Circumference (AC). After running the images through a second CNN, they rated them based on how well they described the stomach vesicle and umbilical vein, two crucial anatomical components. Ma et al. [15], as well as Ma et al. [16], arbitrarily separated the thyroid ultrasound picture into a large number of overlapping subgraphs. Using the pixel ratio between healthy tissue and the lesion location as the classification label, a Deep Convolutional Neural Network was trained (DCNN). With the use of trained deep convolutional neural networks (DCNNs) for subgraph-level classification, we were able to successfully segregate the lesion areas in thyroid ultrasound images. Wang and Jiao [17] used a Simplified Pulse-Coupled Neural Network equipped with morphological processing and fuzzy mutual information to extract the breast lesion area from the binary image with the highest fuzzy mutual information as the best choice criterion (SPCNN). Due to the nature of ultrasound imaging, breast ultrasound photographs still have issues like poor resolution, low contrast, and a lot of speckle noise despite the fact that deep learning has made enormous progress in the field of natural image processing.
When using the dataset driven deep learning technique [18] to the processing of breast ultrasound images, this presents some challenges. Using visual saliency as a training parameter to instruct the U-Net segmentation model to prioritize the highly significant spatial region features in the image, Vakanski et al. [19] were able to achieve promising experimental results on a small sample dataset consisting of 510 breast ultrasound pictures. In order to do a thorough ultrasound examination of the breast mass, Zhang et al. [20] suggested optimizing the convolution network AlexNet network with complete connection structure to capture the informational features of the image. To achieve picture information identification and segmentation, Ilesanmi et al. [21] enhanced the processing of breast ultrasound pictures based on cascaded convolution networks. Tang et al. [22] added the Transform Modal Ensemble Learning module to the nonlocal network of feature pyramids in order to build the model for breast ultrasound image-aided diagnosis and achieve autonomous image analysis. To implement breast abnormality picture analysis, Fang et al. [23] created the comprehensive convolution network model M-Net. Skin, mass, fibroglandular tissue, and fatty tissue were extracted from 3D breast ultrasound images using convolutional neural networks (CNNs) by Xu et al. [24]. Instead of using convolutional blocks, as is common practise in a standard U-Net, it has been suggested that SK blocks be utilized instead. The resulting U-Net kernel would be selective. Byra et al. [25] demonstrates that these SK blocks use both dilated and standard convolutions, and that the reflecting ratio of each output feature map may be calculated by applying attention coefficients.
In a separate investigation, Zhu et al. [26] integrated an ASP module into a U-Net-based core network. The first node of this combination network performs a full-feature-map convolution. Splitting the feature maps into four sections allows the second branch to perform four independent convolutions. These procedures guarantee that the global and local features of an input picture are used effectively. After that, you may combine these feature maps with the ones used in the last layer. Xue et al. [27] show that the BD module, to which each convolutional layer may be linked, can be used to locate the tumor’s borders. Using the binary picture that segmentation generates increases the contrast of the original image even more. For the detection phase, we use an anchor-free detection network to localize the tumour. Chen et al. [28] suggested the cascaded CNN technique, in which a C-Net is constructed from many networks. This model consists of three networks and is trained using deep supervision. The first model, UNet, generates and provides saliency maps. After being processed by an attention network and a refinement network, the segmentation result is sent to the output. The multistage method suggested by Cho et al. [29] uses ultrasound picture categorization and segmentation to isolate breast tumours. To determine if an ultrasound picture includes breast tumours, a breast tumour ensemble classification network (BTEC-Net) is developed. A residual feature selection UNet (RFS-UNet) is utilized just to segment pictures that were deemed aberrant by the BTEC-Net before moving on to the segmentation step. A completely automated diagnostic system is possible using the multistage segmentation technique.
Yu et al. [30] proposed a new U-Net-based method for improving the precision of tumour segmentation in breast ultrasound images. The next step was to integrate Res Path into the U-Net to help equalize the feature maps used by the encoder and the decoder. To combat the vanishing gradient issue and reduce feature information loss, a new connection, dense block from the input of the feature maps in the encoding to-decoding section, was introduced. For the concurrent segmentation and binary classification of breast ultrasound (BUS) images, Zhang et al. [31] published a multi-task learning (SHA-MTL) model based on soft and hard attention methods. A dense CNN encoder is linked to an up sampling decoder through two attention gated (AG) units using soft attention methods. However, there are downsides to using these methods. Because of the nature of deep networks, it’s possible that the picture segmentation model’s gradient may disappear throughout the recognition process, leading to less reliable results. Additionally, segmentation accuracy continues to decline when the depth model gets the image data owing to the independent phase interference induced by the substantial noise interference in the ultrasound picture. In order to overcome the limitations of the current depth network model, this research enhances the AttendSeg network and combines it with a hybrid gravitational clustering optimization technique to create an identification strategy for breast cancer ultrasound images.
3 Proposed methodology
Breast ultrasound pictures are first acquired in the suggested framework. Later, data augmentation techniques are employed to address the issue of limited data because deep neural networks require more data for their training and improved performance. Then, to identify breast cancer from breast ultrasound pictures, it is advised to utilize an AttendSeg, a low-precision, extremely compact deep NN architecture in combination with gravitational clustering-based optimization technique. Each stage in the approach is thoroughly explained in the subsections listed below:
3.1 Data pre-processing
Pre-processing is a vital step in the processing of medical images. Pre-processing breast US pictures aims to enhance image quality and remove extraneous information. Enhancement and despeckling are two pre-processing steps used in this study project. To enhance the quality of breast US images, a sigmoid filter is used. Let’s assume that the input breast image is U. The sigmoid filter can be mathematically represented as Eq. (1),
where, represents an input ultrasound image, U enh represents an enhanced image, min represents the lowest grey value, max represents the highest value, and a and A, respectively, denote the center and width.
3.2 Dataset augmentation
In this research, the BUS dataset was used for the validation technique. There are 780 pictures here, and on average, each one is 500 pixels wide and tall. Normal (133 shots), malignant (210 images), and benign (487 images) are the three primary groups represented in this collection, as shown in Figure 1 [10]. This whole dataset was divided into training and testing set with a 50/50 split. After then, the training pictures for each class included 56 healthy, 105 cancerous, and 105 benign images (243 images). Given the insufficiency of the dataset, an additional data augmentation phase is employed to train deep learning models. Three operations – a horizontal flip, a vertical flip, and a rotation of 90 degrees are performed to the initial ultrasound pictures in order to increase the variety of the original dataset. As many times as necessary for there to be 4,000 unique images in each class, the above operations are repeated. As a result of the enhancement procedure, the dataset now comprises 12,000. The sample images shown in Figure 1 are classified into Benign, Malignant, and Normal. Benign breast cancer is commonly the beginning stage of breast cancer and are not life threatening. Meanwhile Malignant is a tumor that occurs around the breast tissue, usually begins as a calcium deposit and grown as cancerous cell.

Sample ultrasound images of BUS dataset. (a) Benign. (b) Malignant. (c) Normal.
3.3 AttendSeg network
In order to develop low-precision deep neural networks tailored to breast tumours, this research makes use of two complimentary approaches. To begin, we will propose the idea of attention condensers as a unique self-attention mechanism for selective attention based on a fusion of local and cross-channel activation correlations learned by compacted embeddings. To automatically separate the tumours and strike the best balance between segmentation accuracy and network efficiency [32,33], a machine-driven design exploration approach is used that employs this novel attention condenser. In order to do image segmentation on low-power, high-capacity computing systems, researchers from DarwinAI and the University of Waterloo have created revolutionary neural network architecture.
Segmentation required the use of large, computationally expensive neural networks. This made it difficult to use these deep learning models offline, away from access to cloud servers. In this study, we make use of the AttendSeg neural network [34], which provides almost perfect segmentation while yet being compact enough to run on low-powered mobile devices. AttendSeg is a deep learning model that can segment data with almost the same accuracy as RefineNet but using fewer parameters [35,36]. The model requires less than two megabytes of memory, making it small enough to run on most edge devices. Our results demonstrate that low precision representation may be highly useful in such cases, and demonstrate that using AttenSeg does not limit the network’s generalizability. As can be seen in Figure 2, AttendSeg is a neural network that performs segmentation directly on the device itself.

Architecture of AttendSeg on-device segmentation neural network.
3.3.1 Attention condensers for segmentation
AttendSeg reduces the complexity of the model by using “attention condensers” to save space without compromising accuracy. There is a class of processes known as self-attention mechanisms that boosts the efficiency of neural networks by directing their attention to what’s most important. Self-attention techniques have shown to be quite useful. By using self-attention methods, transformers were able to extend their range. Models of deep learning, such as GPT-3, employ transformers and self-attention to construct long text strings that maintain coherence over time. used attentional principles to boost convolutional neural networks’ performance.
For the purpose of breast ultrasound picture segmentation, we used attention condensers, a highly efficient attention mechanism. Due to the fact that they allow for incredibly small deep neural network configurations, attention condensers are well suited for edge applications. As can be shown in Figure 3, the attention condensers are able to improve the functionality of convolutional neural networks with decreased memory consumption.

Attention condensers.
3.3.2 Machine-driven design of neural networks
Machine learning’s generative synthesis technique uses input parameters to generate new neural network designs. To find the best solution, researchers may give a machine learning model a problem space and let it learn from the data rather than manually trying out multiple setups and architectures. Humans provide the “Generative Synthesis” machine-driven design process (MDDP) [37] with an initial design prototype and desired operational requirements (such as size, accuracy, etc.), and then the MDDP takes over and learns from it to generate the best possible architecture design for the task at hand, taking into account the operational requirements, and the operational requirements. The revised deep model incorporates all of the information from the original model. The following parameters are used during the modified model’s training phase: The training used the stochastic gradient descent method for 300 iterations with the following parameters: a learning rate of 0.001, a mini batch size of 16, and a batch size of 16. The features were generated in the updated deep model’s Global Average Pooling (GAP) layer. The obtained attributes are then optimized using a tweaked optimization strategy [36,38].
3.4 Algorithm for gravitational clustering
A newly developed GCA algorithm based on studies of gravity research [39]. This algorithm is based on the gravitational search and balances the exploitation and exploration behaviours of the algorithm. Moreover, the searchability of the algorithm is high with less complexity and hence we selected the GCA for the below function. The actions listed below must be taken in order to implement GCA [40,41].
Step 1: Initializing the identity matrix M = I, which is the first step.
Step 2: Perform the following steps for each instance of P i . The first step is to identify a nearby point, P; j I that is closest to it; As a second step, we may calculate the total force F i acting on P i if d(P i ,P j ). Finally, under the assumptions M[i,j] = M[j,i] = 1 is equivalent to 1.
Step 3: The conditional step of calculating the transitive closure M* if (MI) is followed by merging mechanism execution based on similarity classes of M* and resuming at Step 2 if successful [42].
Step 4: Each new point P i that arises after merging two or more points is a new cluster in the cluster hierarchy if Step 2 is repeated. When this is the case, the height of any given point inside the cluster, h(P i ), is all that is needed to define it. Using Eqs. (2) and (3), and accordingly, we can determine η i for any value of P i and curr.
Step 5: Determine the new location of each point P i , P i + η curr F i and setting h(P i ) = h(P i ) + η curr
4 Experimental results
In the cluster hierarchy, any point P i that is the result of a merger of two or more points is treated as a new cluster if Step 2 is repeated. In this scenario, the height of any given point in the cluster, h(P i ), is all that is required. By using Eqs. (2) and (3), and correspondingly to determine I for each value of P i and curr:
4.1 Dataset
The breast ultrasound (BUS) images utilized in this research are from open-source databases [10]. Seven hundred eighty ultrasound scans from women aged 25 to 75 made up the BUSI dataset. We found 133 healthy photographs, 437 benign images, and 210 cancerous images. The tests to show the generalized performance divide the BUS dataset into training 80% and testing 20% subsets.
4.2 Evaluation metrics
To analyze the aforementioned technique, we use three objective and quantifiable indices for ultrasound picture segmentation: Dice, Intersection over Union (IoU), and Hausdorff Distance (HD). The Dice error assessment approach compares the area of the tumour to the area really present inside the segmented region [43]. The greater the value of the dice roll, the more effective the segmentation. In a way, IoU [44] may be thought of as being synonymous with the dice score index, which measures the degree to which a segmented area and a return on investment are comparable. The Hausdorff Distance (HD) [45] quantifies the greatest gap between the segmentation-generated border and the accurate partition of the real lesion area. The boundary error is less the lower the value. The difficulty of establishing a universal metric for evaluating various approaches to medical image processing is not new. Experts’ manual creation of ground truth introduces room for inaccuracy. Predicted values and ground truth values should be evaluated using a variety of quantitative metrics. As a result, it is difficult to judge how well a deep learning model is without some kind of performance indicator. Eq. (4) uses the Jaccard index [46,47], and [48] to calculate the degree of similarity between two pictures.
The Jaccard index between the true picture and the one that was generated is J(G, O). The suggested strategy improved the Jaccard index for segmentation accuracy. The overlap between segmented and ground truth pictures is often compared using DSCs. When comparing two sets of G and O, you may see how well they match up by averaging the sizes of the cross ovens [49]. The Dice Resemblance Coefficient is a useful metric for comparing the degree of visual similarity between two photographs (DSC).
where, D(G, O) is the DSC index of the ground truth and the original image. Hausdorff Distance (HD) is defined as
Where,
HD calculates the relative surface area change between the segmented and the hand-drawn contours. Maximum difference in surface area between segmented contour G and its equivalent manual contour R is computed using HD, whereas the average difference is calculated using d(x,C), where x and C are the coordinates of the two contours and d(x,C) is the Euclidean distance [50,51].
Segmentation performance with the comparison techniques is quantitatively evaluated in Table 1. For the comparative purpose we have taken the state-of-art works such as SegNet, PSPNet, DeepLabv3+, FC CNN, CNN, S2P-Net, and RFS-UNet. The Table 1 outlined the statistical analysis of Preciseness at the pixel level, Jaccard or Intersection over Union (IoU), Dice Coefficient, Hausdorff, and Processing Time. When compared to previous approaches, the DC value, IoU, accuracy, computing time, and Hausdroff of the AttendSeg gravitational clustering based Optimization Algorithm are all improved. This dataset includes manually annotated ground truth pictures that are used to evaluate the accuracy of the segmentation findings. The precision value of proposed approach is 98.957% which is higher than the other approaches such as SegNet, PSPNet, DeepLabv3+, FC CNN, CNN, S2P-Net, and RFS-UNet with 96.082, 96.223, 96.257, 96.6, 95.6, null, and 97.253. The null value indicates the work did not analyze the precision. Comparison of graphical plot for Segmentation performance is as shown in Figure 4.
Segmentation performance with the compared Algorithms on the BUS
Methods | Pixel Acc | IoU | DC | Hausdorff (mm) | Computational time (s) |
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SegNet | 96.082 | 60.642 | 68.844 | 61.34 | 74 |
PSPNet | 96.223 | 65.227 | 73.115 | 48.47 | 66 |
DeepLabv3+ | 96.257 | 64.879 | 72.254 | 47.27 | 44 |
FC CNN | 96.6 | 56.5 | 64.1 | 52.47 | 69 |
CNN | 95.6 | 63.56 | 70.9 | 31.87 | 33 |
S2P-Net | 93.56 | 76.39 | 84.7 | 47.27 | 48 |
RFS-UNet | 97.253 | 77.835 | 84.856 | 32.46 | 35 |
Proposed | 98.957 | 85.634 | 89.459 | 23.55 | 13 |

Comparison of graphical plot for segmentation performance.
The computational time of proposed work is lower than the other approaches with 13 s in which the method SegNet consumes more time of 74 s. Meanwhile, the Hausdorff of proposed approach is lower with the value of 23.55 mm and method SegNet with higher value of 61.34 mm. Moreover, the DC of proposed work is higher with 89.459 and other approaches SegNet, PSPNet, DeepLabv3+, FC CNN, CNN, S2P-Net, and RFS-UNet achieved the DC values of 68.844, 73.115, 72.254, 64.1, 70.9, 84.7, and 84.856 respectively. the Jaccard index or IoU of the proposed approach is 85.634 and the other approaches SegNet, PSPNet, DeepLabv3+, FC CNN, CNN, S2P-Net, and RFS-UNet are with the values, 60.642, 65.227, 64.879, 56.5, null, 76.39, and 77.835, respectively.
Figure 5 displays the final picture produced by the gradCAM segmentation process. Both the input ultrasound mask picture (Figure 5(a)) and the filtered prediction mask image (Figure 3(b)) are shown below. The gradCAM picture may be seen in Figure 3(c). In Figure 6, we see the final picture that the segmentation method produced using the anticipated and processed mask. The input ultrasonic mask picture is shown in Figure 5(a), and the predicted mask image is shown in Figure 6(b). The aberrant masses may be better identified with the use of the mask picture shown in Figure 3(c). The characteristics gleaned via a hybrid of AttendSeg and a clustering-based optimization technique inspired by gravitational fields.

(a) Orginal mask, (b) predicted mask, (c) GradCAM.

(a) Orginal mask, (b) predicted mask, (c) processed mask.
Finally, a segmented picture produced by the suggested approach is shown in Figure 7. Only a small selection of the photos in the BUS dataset are shown here. The malignant tumour segmented by the suggested approach appears as brown dots. This technique can distinguish between benign and malignant tumours in ultrasound pictures of the breast. As a result, this article’s suggested technique offers certain benefits over competing methods when applied to breast tumour segmentation, laying the groundwork for future clinical diagnostics. The consideration of the crucial elements in the preceding section leads us to the conclusion that the implementation of these elements in this algorithm yields superior results when applied to ultrasound breast pictures.

Segmented images of the proposed method.
In Figure 8 we can see the entropy loss of the suggested technique. Comparing this method to others, it is shown that the entropy loss is minimal. Our loss of approach train data is shown by the blue values in the graph, while the other values are test data. A demonstration of the suggested method’s precision is shown in Figure 9. In comparison to other methods, it is found that the accuracy is quite good.

Loss curve of the compared algorithms.

Accuracy of the proposed method.
5 Conclusion
The ability to accurately separate breast masses is critical for early detection of breast cancer. In a nutshell this work proposed an innovative approach known as hybrid Attend Segnetwork trained using a gravitational clustering based optimization technique for ultrasound images of breast tumours. Data augmentation was a utilized after an initial round of sigmoid filter pre-processing. The AttendSeg network model was used to isolate the regions of interest (ROIs) that contain breast masses. In addition, the ultrasound image segmentation model incorporates a hybrid attention mechanism network to better access the sample dataset’s global information characteristics and enhanced the analysis model’s image segmentation performance. The optimization approach uses gravitational clustering to extract the characteristics. Accurate identification and diagnosis of benign and malignant breast masses is made possible by the suggested breast mass segmentation approach based on the deep learning with optimization algorithm. The experimental findings demonstrate that the suggested strategy improves segmentation performance and gives radiologists trustworthy support in making medical diagnoses. Quantitative and qualitative analyses reveal that this technique outperforms the current gold standard for segmenting brain tumours in the treatment of breast cancer.
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Funding information: The authors state no funding involved.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: The data will made available on a reasonable request to the corresponding author.
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Articles in the same Issue
- Regular Articles
- Dynamic properties of the attachment oscillator arising in the nanophysics
- Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
- Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
- Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
- Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
- Analysis of the working mechanism and detection sensitivity of a flash detector
- Flat and bent branes with inner structure in two-field mimetic gravity
- Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
- Weighted survival functional entropy and its properties
- Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
- Study on the impulse mechanism of optical films formed by laser plasma shock waves
- Analysis of sweeping jet and film composite cooling using the decoupled model
- Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
- Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
- Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
- Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
- Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
- Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
- Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
- Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
- Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
- Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
- Controlling the physical field using the shape function technique
- Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
- Complex dynamics of a sub-quadratic Lorenz-like system
- Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
- Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
- Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
- A modified power-law model: Properties, estimation, and applications
- Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
- Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
- Predictability of machine learning framework in cross-section data
- Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
- Isomorphic shut form valuation for quantum field theory and biological population models
- Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
- Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
- Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
- A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
- Tangential electrostatic field at metal surfaces
- Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
- Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
- Influence of cationic surfactants on the growth of gypsum crystals
- Study on instability mechanism of KCl/PHPA drilling waste fluid
- Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
- A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
- Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
- Conserved vectors and solutions of the two-dimensional potential KP equation
- The reciprocal linear effect, a new optical effect of the Sagnac type
- Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
- The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
- Research on absolute ranging technology of resampling phase comparison method based on FMCW
- Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
- Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
- Fractional-order modeling: Analysis of foam drainage and Fisher's equations
- Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
- Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
- Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
- On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
- 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
- Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
- Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
- Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
- Does the Mott problem extend to Geiger counters?
- Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
- Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
- Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
- Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
- Study of fractional variable order COVID-19 environmental transformation model
- Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
- Influence of the regularization scheme in the QCD phase diagram in the PNJL model
- Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
- Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
- Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
- High-sensitivity on-chip temperature sensor based on cascaded microring resonators
- Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
- Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
- Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
- Numerical simulation and analysis of Airy's-type equation
- A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
- Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
- ΛCDM and the principle of equivalence
- Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
- HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
- Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
- Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
- Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
- A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
- Application of IoT network for marine wildlife surveillance
- Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
- Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
- Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
- Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
- Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
- Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
- Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
- Micropolar flow and heat transfer within a permeable channel using the successive linearization method
- Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
- Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
- Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
- Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
- Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
- Review Article
- Effects of stretching on phase transformation of PVDF and its copolymers: A review
- Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
- Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
- Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
- Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
- Special Issue on Advanced Energy Materials - Part II
- Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
- Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
- Adomian decomposition method for solution of fourteenth order boundary value problems
- New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
- On traveling wave solutions to Manakov model with variable coefficients
- Rational approximation for solving Fredholm integro-differential equations by new algorithm
- Special Issue on Predicting pattern alterations in nature - Part I
- Modeling the monkeypox infection using the Mittag–Leffler kernel
- Spectral analysis of variable-order multi-terms fractional differential equations
- Special Issue on Nanomaterial utilization and structural optimization - Part I
- Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations
Articles in the same Issue
- Regular Articles
- Dynamic properties of the attachment oscillator arising in the nanophysics
- Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
- Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
- Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
- Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
- Analysis of the working mechanism and detection sensitivity of a flash detector
- Flat and bent branes with inner structure in two-field mimetic gravity
- Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
- Weighted survival functional entropy and its properties
- Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
- Study on the impulse mechanism of optical films formed by laser plasma shock waves
- Analysis of sweeping jet and film composite cooling using the decoupled model
- Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
- Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
- Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
- Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
- Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
- Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
- Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
- Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
- Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
- Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
- Controlling the physical field using the shape function technique
- Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
- Complex dynamics of a sub-quadratic Lorenz-like system
- Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
- Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
- Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
- A modified power-law model: Properties, estimation, and applications
- Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
- Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
- Predictability of machine learning framework in cross-section data
- Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
- Isomorphic shut form valuation for quantum field theory and biological population models
- Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
- Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
- Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
- A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
- Tangential electrostatic field at metal surfaces
- Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
- Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
- Influence of cationic surfactants on the growth of gypsum crystals
- Study on instability mechanism of KCl/PHPA drilling waste fluid
- Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
- A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
- Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
- Conserved vectors and solutions of the two-dimensional potential KP equation
- The reciprocal linear effect, a new optical effect of the Sagnac type
- Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
- The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
- Research on absolute ranging technology of resampling phase comparison method based on FMCW
- Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
- Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
- Fractional-order modeling: Analysis of foam drainage and Fisher's equations
- Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
- Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
- Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
- On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
- 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
- Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
- Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
- Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
- Does the Mott problem extend to Geiger counters?
- Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
- Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
- Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
- Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
- Study of fractional variable order COVID-19 environmental transformation model
- Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
- Influence of the regularization scheme in the QCD phase diagram in the PNJL model
- Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
- Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
- Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
- High-sensitivity on-chip temperature sensor based on cascaded microring resonators
- Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
- Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
- Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
- Numerical simulation and analysis of Airy's-type equation
- A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
- Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
- ΛCDM and the principle of equivalence
- Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
- HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
- Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
- Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
- Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
- A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
- Application of IoT network for marine wildlife surveillance
- Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
- Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
- Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
- Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
- Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
- Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
- Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
- Micropolar flow and heat transfer within a permeable channel using the successive linearization method
- Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
- Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
- Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
- Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
- Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
- Review Article
- Effects of stretching on phase transformation of PVDF and its copolymers: A review
- Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
- Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
- Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
- Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
- Special Issue on Advanced Energy Materials - Part II
- Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
- Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
- Adomian decomposition method for solution of fourteenth order boundary value problems
- New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
- On traveling wave solutions to Manakov model with variable coefficients
- Rational approximation for solving Fredholm integro-differential equations by new algorithm
- Special Issue on Predicting pattern alterations in nature - Part I
- Modeling the monkeypox infection using the Mittag–Leffler kernel
- Spectral analysis of variable-order multi-terms fractional differential equations
- Special Issue on Nanomaterial utilization and structural optimization - Part I
- Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations