Modeling melanoma-immune interactions with a physiological delay incorporating dendritic cell vaccines and anti-PD-1...

Malignant melanoma is an aggressive skin cancer with limited responsiveness to traditional therapies. Notably, the combination of dendritic cell (DC) vaccines with anti-programmed cell death protein 1 (anti-PD-1) therapy has shown stronger clinical potential than conventional approaches. Understanding the tumor-immune interplay is essential for optimizing melanoma immunotherapy strategies. In this paper, we formulate a melanoma-specific tumor-immune interaction model of tumor cells (TCs), DCs, and effector CD8⁺ T cells (ECs). A key threshold value is identified to characterize tumor growth. Using this threshold, we determine the conditions for tumor-free and tumorous equilibria, consistent with cancer immunoediting theory. Furthermore, bifurcation analysis indicates that the model exhibits oscillatory behavior under certain conditions. Sensitivity and parameter heterogeneity analyses reveal that tumor burden is mainly regulated by the intrinsic tumor growth and immune activation rate. Moreover, to better reflect physiological realism, we extend the model to a time-delayed system by incorporating a constant delay for DC-to-EC activation. Analytical and numerical results demonstrate a supercritical Hopf bifurcation at a critical delay τ₀ ≈ 4.68 days, leading to stable periodic solutions. Finally, an optimal control framework is proposed to design DC vaccines and anti-PD-1 injection protocols. Compared with the constant dosing strategy, optimal control achieves enhanced tumor suppression for the same total treatment intensity. This work elucidates the dynamical mechanisms of melanoma-immune interactions and establishes a theoretical foundation for personalized combination immunotherapies.