A Mathematical Model with Fractional-Order Dynamics for the Combined Treatment of Metastatic Colorectal Cancer.

Colorectal cancer (CRC) poses a significant global public health challenge as its prevalence continues to increase. This study delves into the intricate dynamics of metastatic CRC by utilizing a novel fractional-order model. Through a comprehensive analysis of existing literature and clinical data, we introduce an innovative fractional-order model tailored to metastatic CRC. This model encapsulates key factors like tumor growth, immune response, and therapeutic interventions. Our findings underscore the crucial role of epithelial cells (E) within the tumor microenvironment, emphasizing their significance due to their high sensitivity. Additionally, the influence of adenomatous polyps (P) is noteworthy, given their potential to transform into cancerous entities. Remarkably, the concentration of inflammatory cells (C) significantly impacts the model, shedding light on the pivotal role of the immune response. We implement a clever fusion of feedback control and Proportional–Integral–Derivative (PID) controllers, enabling dynamic self-adjustment and precision. Our results indicate that achieving high efficacy in treating metastatic colorectal cancer necessitates a synergistic and simultaneous application of immunotherapy and surgery. Also, surgical intervention should exert a slightly higher intensity than immunotherapy, even when administered concurrently. The PID control strategy offers a promising method for fine-tuning concentrations over time. These investigations enable a thorough exploration of how fractional-order parameters influence tumor progression, interactions with the immune system, and treatment outcomes. We conduct sensitivity analysis to pinpoint crucial model parameters governing the system’s dynamic behavior. The insights gleaned from our study hold significant promise for advancing treatment protocols and personalized medicine tailored to CRC patients.