Abstract This study introduces a novel single-step hybrid block method with four intra-step points that attains six-order accuracy, ensures A-stability, consistency, and provides an efficient, accurate, and computationally economical tool for solving ordinary differential equations. The scheme incorporates intra-step points, which provide richer information within each integration step and significantly improve both precision and stability. When function values are not naturally defined at the chosen nodes, suitable interpolation techniques are introduced to approximate the missing terms without compromising accuracy. A detailed theoretical framework is established, including the analysis of convergence behavior and the derivation of local truncation error expressions. The stability of the method is further examined by identifying its stability regions and proving zero-stability under practical constraints on the step size. These theoretical guarantees ensure that the scheme is not only accurate but also reliable for long-time numerical integration. To complement the analysis, a series of comprehensive numerical experiments are conducted on benchmark problems frequently used in literature. The experimental results consistently demonstrate the superiority of the proposed method over existing approaches in terms of accuracy, efficiency, and overall robustness.

Abstract Colorectal cancer (CRC) remains a major global health burden with high mortality in advanced stages, highlighting the urgent need for more effective and safer therapies. Aberrant β-catenin stabilization and nuclear accumulation promote oncogenic transcriptional programs and remains an attractive yet challenging therapeutic target. Here, an in silico screen of 25 naturally derived compounds was performed against β-catenin (PDB: 1JDH) using AutoDock Vina 1.2.5. Ligands and receptors were prepared in PyMOL 3.0 and AutoDockTools 1.5.7. Blind docking was conducted using a whole-protein search space centered at x = −2.857, y = 9.859, z = 40.811 with a box size of approximately 107 × 59 × 121 Å in triplicate runs. Binding poses and interaction patterns were visualized in PyMOL 3.0 and BIOVIA Discovery Studio 2024 (3D and 2D interaction maps). Nine compounds achieved predicted binding affinities (ΔG_bind) of ≤ −7.0 kcal/mol, led by silibinin (−9.9 kcal/mol), followed by quercetin (-7.8 kcal/mol), luteolin (-7.7 kcal/mol), ellagic acid (-7.5 kcal/mol), garcinol (-7.5 kcal/mol), betulinic acid (-7.4 kcal/mol), ursolic acid (-7.4 kcal/mol), derricin (-7.0 kcal/mol) , and epigallocatechingallate (EGCG) (-7.0 kcal/mol). Silibinin showed a consistent predicted pose with multiple hydrogen-bond and pi-alkyl hydrophobic contacts within a putative pocket. Drug-likeness analysis using Lipinski’s Rule of Five indicated that most of the top hit ligands complied with criteria for molecular weight, hydrogen bond donors/acceptors, and lipophilicity, suggesting favorable oral bioavailability, while EGCG exceeded the recommended limits for hydrogen bond donors and acceptors, and garcinol surpassed the molecular weight and cLogP thresholds. Additionally, ADMET predictions highlighted potential concerns for quercetin due to a predicted mutagenic/tumorigenic risk. Overall, by applying a curated ligand set under a single standardized docking–ADMET workflow, this study reports novel screening outputs, including docking scores, predicted binding poses, and residue-level interaction profiles, together with an ADMET-informed prioritization. Based on these in silico results, silibinin emerged as the leading scaffold for prioritized experimental validation.

Abstract This study investigates the development and clinical applications of Large Deformation Diffeomorphic Metric Mapping (LDDMM) in medical image registration. Through systematic comparison between conventional optimization-based methods and contemporary deep learning techniques, we evaluate their respective performance in registration accuracy, computational efficiency, and clinical utility. Our methodology encompasses a thorough examination of both mathematical foundations and neural network implementations in LDDMM. Results demonstrate that traditional approaches maintain superior precision for complex anatomical variations via rigorous variational optimization, whereas deep learning methods achieve substantial computational acceleration (reducing processing time from hours to seconds) through learned deformation patterns. Critical analysis reveals important trade-offs: while deep learning offers remarkable speed improvements, traditional methods preserve accuracy advantages in specialized clinical scenarios. We identify key challenges including computational complexity, implementation difficulties, and domain adaptation limitations, while proposing hybrid architecture and transfer learning as potential solutions. The study concludes that integrating the mathematical robustness of conventional LDDMM with the computational efficiency of deep learning presents the most viable path forward. Such synergistic approaches promise to advance medical image analysis pipelines and promote wider clinical implementation of sophisticated registration technologies.