Fuzzy and fractional analysis of cancer tumor dynamics with depression effects on chemotherapy.

Cancer is a complex and heterogeneous condition marked by the unchecked growth and dissemination of abnormal cells, posing significant challenges in detection, treatment, and patient care. As one of the leading global causes of death, a deep understanding of its underlying biological mechanisms is essential for advancing therapeutic strategies and improving clinical outcomes. Mathematical modeling serves as a crucial tool in capturing the multifaceted dynamics of cancer initiation, progression, and response to interventions. By simulating critical aspects of cancer within a controlled computational framework, mathematical models enable scientists to explore innovative treatment strategies, investigate the disease's underlying biological dynamics, and identify novel therapeutic targets. This study presents a fuzzy-fractional differential model of tumor-immune interaction, incorporating tumor cells, immune effector cells, and the concentration of chemotherapy agents in the bloodstream, modeled using Caputo-type time-fractional derivatives. To better capture the uncertainty associated with patient-specific factors-particularly psychological impacts such as depression-triangular fuzzy numbers are integrated into the initial conditions, thereby enhancing the model's realism and predictive capability. The current model is addressed using the proposed modified He-Laplace-Carson algorithm for solution and analysis by creating multiple homotopies related to the perturbation method. The model's solution trajectories for chemotherapy levels, immune effector cells, and tumor cell populations are examined to evaluate the bidirectional interaction between immune response and tumor growth. Additionally, the simulation provides insight into the dynamic behavior of chemotherapy concentration over the duration of treatment, offering a clearer understanding of its therapeutic progression. An extensive graphical analysis is conducted by varying a range of parameters, including effect of depression, death rate of immune cells due to malignant cells attachment, maximum growth rate factors, and fuzzy parameters introduced in the cancer system. It was observed that as the fractional parameter ξ increased, all profiles rose, with effector cells showing a more notably faster growth than tumor cells. Furthermore, all fuzzy and non-fuzzy parameters generally showed a strong positive influence on effector cells, with tumor cell growth remaining comparatively subdued. The fractional parameter is analyzed under diverse conditions using 2D/3D visualizations and contour gradients, confirming the method's reliability in handling uncertainty and highlighting its adaptability to broader fuzzy-fractional systems. Such applications have the capability to significantly enhance the understanding of ongoing challenges faced in oncology.