New insights into disease dynamics and treatment interventions with PID controller-based therapeutic strategies for pancreatic cancer

In this paper, we developed a mathematical model for pancreatic cancer progression using a system of nonlinear partial differential equations (PDEs) with time delays, capturing disease dynamics in the human body. The model represents six key cell populations involved in pancreatic cancer: cancer cells (C), pancreatic stellate cells (P), stromal cells (S), extracellular matrix-degrading enzymes (E), tumor-associated macrophages (N), and immunosuppressive cells (I). For biological feasibility, we established model existence and uniqueness via the method of continuity and Banach's contraction principle, with global stability verified through the Lyapunov method. Sensitivity analysis identified critical factors such as cancer cell division, stromal cell activation, and immune cell infiltration, as targets for effective treatment. Optimal control and PID strategies demonstrated potential in limiting cancer proliferation and reprogramming the tumor microenvironment, while simulations highlighted the need for timely and sustained interventions. The results emphasize the importance of early surgery and immunomodulation in maximizing treatment efficacy, offering new insights into personalized and adaptive approaches to improve patient outcomes in pancreatic cancer treatment.