Fractional spatiotemporal Hahnfeldt tumor model with convergence analysis and optimal control.

The mathematical modeling of tumor-immune interactions is pivotal for decoding the complex, multistage processes of cancer progression, yet classical integer-order models often fail to capture the hereditary properties and spatial heterogeneity intrinsic to malignant growth. This research presents a novel fractional spatiotemporal Hahnfeldt tumor model that integrates Caputo fractional derivatives with diffusion mechanisms to characterize memory dependent tumor vascular dynamics. Unlike standard formulations, this model explicitly accounts for the "biological memory" of the immune system and the "sub-diffusive" invasion patterns of tumor cells. We provide a rigorous mathematical analysis, establishing the non-negativity and boundedness of solutions, followed by a stability analysis of the Adams-Bashforth-Moulton predictor-corrector scheme. Numerical simulations reveal that lower fractional order inherently reproduce clinical phenomena such as tumor dormancy and delayed therapeutic response, which are mathematically inaccessible to integer-order counterparts. Furthermore, a fractional optimal control framework, solved via the Forward-Backward Sweep Method, demonstrates that adaptive chemo-immunotherapy significantly outperforms monotherapies by leveraging the system's memory to sustain remission. These findings offer a theoretically sound and clinically relevant computational tool for predicting long-term treatment outcomes in heterogeneous tissue environments.