Mathematical modeling and optimal control of lung cancer dynamics: a fractional-order approach enhanced by artificial intelligence

Abstract

Lung cancer is one of the leading causes of mortality worldwide, representing a major challenge for modern healthcare systems due to its rapid progression and complex biological mechanisms. In this work, we propose a mathematical model based on fractional differential equations in the sense of Caputo to describe the dynamics of lung cancer while incorporating memory effects inherent to biological systems. The proposed model considers the interactions between normal cells, tumor cells, immune response, therapeutic treatment, and a tumor biomarker. A rigorous mathematical analysis is carried out to establish the fundamental properties of the system, including existence, uniqueness, positivity, and boundedness of solutions, ensuring the biological consistency of the model. Furthermore, an optimal control problem is formulated to determine an effective therapeutic strategy that minimizes tumor growth while reducing treatment-related costs. Pontryagin’s Maximum Principle is applied to characterize the optimal control, leading to an analytical expression of the optimal therapeutic protocol. Numerical simulations are performed to illustrate the behavior of the system. The results show that the application of a therapeutic control significantly reduces tumor growth compared to the uncontrolled case. In addition, the influence of the fractional order is investigated, highlighting the importance of memory effects in tumor evolution. Finally, artificial intelligence techniques are used to estimate the model parameters from simulated data, improving prediction accuracy and model calibration.