Mathematical modeling and optimal control of prostate cancer dynamics using Caputo fractional-order derivatives

Abstract

This paper develops a fractional-order mathematical model, formulated in the Caputo sense, for the dynamics of prostate cancer (PC) under combined androgen deprivation therapy (ADT) and dendritic-cell (DC)-based immunotherapy. The model extends classical integer-order formulations by incorporating memory effects that reflect the cumulative, history-dependent nature of tumor–immune interactions. Two malignant cell populations are considered — androgen-dependent (AD) and androgen-independent (AI) prostate cancer cells — together with an effector-cell compartment representing the immune response, and auxiliary state variables for circulating androgen level and administered immunotherapeutic agent. We establish positivity and boundedness of solutions, existence and uniqueness via the Banach and Leray–Schauder fixed-point theorems (with full proofs), Ulam–Hyers and Ulam–Hyers–Rassias stability, and a local stability analysis of the model’s equilibria via the Matignon stability criterion. Building on this qualitative foundation, we introduce two time-dependent control functions representing the intensity of ADT and the dosage of immunotherapy and formulate an optimal control problem minimizing tumor burden and treatment cost over a finite horizon. Necessary optimality conditions are derived via a fractional-order Pontryagin Maximum Principle, and the resulting boundary value problem is solved with a forward–backward sweep scheme coupled to a fractional Euler / product-integration method; a local truncation error analysis, a formal global error estimate, and a local step-size stability heuristic are derived for this scheme, and reconciled with the full-memory quadrature actually implemented via the standard convergence theory of fractional Adams-type methods. This numerical scheme is validated to full convergence on a short time horizon, while on the longer horizon used for the main simulations the co-state sweep is numerically stiff (a consequence of the model’s own unconditionally unstable growth mode) and the reported long-horizon control profile is accordingly presented as an illustrative profile consistent with the derived optimality conditions rather than as a fully converged numerical solution. Numerical simulations, carried out in Python for several values of the fractional order \(\delta\) and for several values of the competition and killing-rate parameters \(\alpha\), \(\beta\), and \(a_2\), illustrate how memory effects and inter-cell competition shape the transient and long-run behavior of the model, and a qualitative sensitivity analysis characterizes the relative influence of each parameter on tumor burden. The results indicate that fractional-order optimal control provides a flexible, biologically motivated framework for exploring individualized combination therapies for prostate cancer, while also highlighting a trade-off between aggressive androgen suppression and the emergence of androgen-independent, treatment-resistant disease.