The Human Epidermal Growth Factor Receptor 2 (HER2) is one of the aggressive subtypes of breast cancer. The HER2 status decides the requirement of breast cancer patients to receive HER2-targeted therapy. The HER2 testing involves combining Immunohistochemistry (IHC) screening, followed by fluorescence in situ hybridization (FISH) for cases where IHC results are equivocal. These tests may involve multiple trials, are time intensive, and tend to be more expensive for certain classes of people. Hematoxylin and Eosin (HE) staining is employed for visualizing general tissue morphology and is a routine, cost-effective method. In this study, we introduce a novel automated class-wise weighted average ensemble deep learning algorithm at the decision level. The proposed algorithm fuses three pre-trained deep-learning models at the decision level by assigning a weight to each class based on their performance of the model to classify the HE-stained breast histopathology images into multi-class HER2 statuses as HER2-0+, HER2-1+, HER2-2+, and HER2-3+. The class-wise weight allocation to the base classifiers is one of the key features of the proposed algorithm. The presented framework surpasses all the existing methods currently employed on the Breast Cancer Immunohistochemistry (BCI) dataset, establishing itself as a dependable approach for detecting HER2 status from HE-stained images. This study highlights the robustness of the proposed algorithm as well as the sufficient information encapsulated within HE-stained images for the effective detection of the HER2 protein present in breast cancer. Therefore, the proposed method possesses the potential to sideline the need for IHC laboratory tests, which hoard time and money.

In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step p-order method to order $p+3$. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples.

Vertebral fracture (VF) may induce spinal cord injury that can lead to serious consequences which eventually may paralyze the entire or some parts of the body depending on the location and severity of the injury. Diagnosis of VFs is crucial at the initial stage, which may be challenging because of the subtle features, noise, and homogeneity present in the computed tomography (CT) images. In this study, Wide ResNet-40, DenseNet-121, and EfficientNet-B7 are chosen, fine-tuned, and used as base models, and a Bayesian-based probabilistic ensemble learning method is proposed for fracture detection in cervical spine CT images. The proposed method considers the prediction’s uncertainty of the base models and combines the predictions obtained from them, to improve the overall performance significantly. This method assigns weights to the base learners, based on their performance and confidence about the prediction. To increase the robustness of the proposed model, custom data augmentation techniques are performed in the preprocessing step. This work utilizes 15,123 CT images from the RSNA-2022 C-spine fracture detection challenge and demonstrates superior performance compared to the individual base learners, and the other existing conventional ensemble methods. The proposed model also outperforms the best state-of-the-art (SOTA) model by 1.62%, 0.51%, and 1.29% in terms of accuracy, specificity, and sensitivity, respectively; furthermore, the AUC score of the best SOTA model is lagging by 5%. The overall accuracy, specificity, sensitivity, and F1-score of the proposed model are 94.62%, 93.51%, 95.29%, and 93.16%, respectively.