This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells $K$ (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate $K$ of cancerous cells depend on the cells concentration.

Marfan syndrome (MS) is a genetic disorder often associated with the development of aortic aneurysms, leading to severe vascular complications. The progression of this condition is intricately linked to hemodynamic factors such as wall shear stress (WSS) and von Mises stress, as abnormal distributions can contribute to thrombus formation, endothelial damage, and the worsening of aneurysmal conditions. In this study, six vascular models were analyzed: four representing diseased aortas with Marfan syndrome aneurysms and two healthy aortic models for comparison. The models were sourced from Vascular Model Repository, and computational fluid dynamics (CFD) simulations were conducted using a Newtonian fluid model and the shear stress transport (SST) k–ω turbulent transitional model to evaluate WSS and von Mises stress. Fluid–structure interaction was employed to incorporate vessel wall interaction, and pulsatile inlet velocity profiles were used to simulate physiological blood flow, capturing time-dependent hemodynamic variations. The results revealed significant differences between healthy and diseased aortic models. In healthy models, WSS was uniformly distributed, with values consistently below 40 Pa, reflecting stable vascular conditions. Conversely, the diseased models exhibited highly non-uniform WSS distributions, with notably lower values in aneurysmal regions, contributing to thrombus formation, with elevated WSS in areas like the carotid and subclavian arteries due to geometric and hemodynamic complexities. The von Mises stress analysis identified regions of heightened rupture risk, particularly on the superior side of case MS1, where both von Mises stress and WSS reached their highest values among all cases. Physics-informed neural networks demonstrated strong agreement with CFD results while significantly reducing computational cost, highlighting their potential for real-time clinical applications. These findings underscore the critical role of hemodynamic factors in aneurysm progression and rupture risk, offering valuable insights for optimizing diagnostic and therapeutic strategies in vascular diseases.

An aneurysm's rupture is commonly associated with its maximum diameter, yet biomechanical studies emphasize the significant influence of mechanical wall shear stress (WSS) in this process. This study focuses on two models of aortic aneurysms: abdominal aortic aneurysm and descending thoracic aortic aneurysm. Five cases, comprising two for model 1 and three for model 2, are examined to explore both axisymmetric and asymmetric shapes, as patient geometry may manifest as either fusiform (axisymmetric) or saccular (asymmetric), while maintaining a consistent aneurysm diameter and adjusting the bulge shape factor to induce asymmetry. Hemodynamic factors, including WSS and wall shear stress gradient, are computed to evaluate thrombus formation and rupture risk within the aneurysms. Our results indicate the presence of recirculation zones in both the medial and transverse planes, generating vortices within the aneurysm. These vortices are more prominent in asymmetric cases compared to axisymmetric cases, leading to increased blood residence time within the aneurysm and a higher likelihood of thrombus formation. Thrombus formation can further impede blood flow, heightening the risk of embolism or ischemic events. Rupture occurs when the WSS surpasses tissue strength; thus, if the tissue strength of all aneurysms is same, our findings suggest that rupture risk varies according to asymmetry. In the transverse direction, our results demonstrate that in model 1, case 1 exhibits uniform WSS on both sides, while in case 2, WSS is higher at the posterior sides of the aneurysm sac. Conversely, in model 2, WSS is higher at the anterior side of the aneurysm. In the medial direction of the aneurysm, WSS is highest for case 5, followed by case 3, case 4, case 2, and case 1, respectively, indicating elevated WSS when the anterior bulge dominates over the posterior bulge for each model. Overall, a higher rupture risk is observed in model 2 compared to model 1 due to increased mechanical stresses.

This paper proposes a mathematical modeling approach to examine the two-dimensional flow stagnates at over a heated stretchable sheet in a porous medium influenced by nonlinear thermal radiation, variable viscosity, and MHD. This study’s main purpose is to examine how thermal radiation and varying viscosity affect fluid flow motion. Additionally, we consider the convective boundary conditions and incorporate the gyrotactic microorganisms equation, which describes microorganism behavior in response to fluid flow. The partial differential equations (PDEs) that represent the conservation equations for mass, momentum, energy, and microorganisms are then converted into a system of coupled ordinary differential equations (ODEs) through the inclusion of nonsimilarity variables. Using MATLAB’s built-in solver bvp4c, the resulting ODEs are numerically solved. The model’s complexity is assessed by plotting two-dimensional graphics of the solution profiles at various physical parameter values. The physical parameters considered in this study include skin friction coefficient, local Nusselt number, local Sherwood number, and density of motile microorganisms. These parameters measure, respectively, the roughness of the sheet, the transformation rate of heat, the rate at which mass is transferred to it, and the rate at which microorganisms are transferred to it. Our study shows that, depending on the magnetic parameter M, the presence of a porous medium causes a significant increase in fluid velocity, ranging from about 25% to 45%. Furthermore, with an increase in the Prandtl number , we have seen a notable improvement of about 6% in fluid thermal conductivity. Additionally, our latest findings are in good agreement with published research for particular values. This study provides valuable insights into the behavior of fluid flow under various physical conditions and can be useful in designing and optimizing industrial processes.