A space-time Caputo fractional order and modified homotopy perturbation method for evaluating the pathological response of tumor-immune cells

Tumors result from genetic mutations or environmental factors that prompt cells to divide uncontrollably. This study aims to examine the behavior of tumor-immune cell growth in the presence of chemotherapy drug diffusion at a Caputo fractional order. To accomplish this, we employed the modified homotopy perturbation method to solve a proposed system of nonlinear differential equations. We obtained the analytical solutions to study the spatiotemporal pathological response of tumor-immune cell growth. Our analysis also considered the impact of the Caputo-fractional order on the system's dynamics and compared the results with the classical integer-order scenario. Our findings demonstrated that the proposed method is an effective and precise technique for understanding the intricate interactions of tumor-immune cell growth. Additionally, we revealed that the Caputo-fractional order plays a significant role in the system's behavior and should not be overlooked in future analyses of such systems. In conclusion, this study holds important implications for cancer research by providing insights into the behavior of tumor-immune cell growth in the presence of time-fractional administration of chemotherapy drugs.