In this talk, we present a mathematical model describing the evolution of tumor density under treatment while accounting for a delayed effect. After a medical motivation, we introduce a conceptual scheme linking the observed biological mechanisms to the choice of model variables. The resulting system is formulated as a fractional differential inclusion with Caputo derivative and two maximal monotone operators, representing respectively the intrinsic tumor dynamics and the treatment-induced response. This modeling approach provides a rigorous framework to incorporate complex biological phenomena and delayed therapeutic effects.

The main analytical challenges arise from the multivalued nonlinear structure and the memory property of the fractional term. We employ advanced tools from functional analysis, in particular monotone operator theory and regularization techniques, to establish an existence result for the solution. This solution u(t) exhibits a medically consistent temporal behavior and reflects key biological insights observed in clinical studies. The proposed framework allows us to interpret the “delayed” effect observed in certain therapies and may be used to predict tumor evolution, adjust dosing frequency, optimize treatment strategies, or analyze the effectiveness of novel therapeutic interventions.

We conclude with several perspectives for future work: detailed stability analysis, fractional optimal control, extensive numerical simulations, and systematic comparison with clinical data, highlighting the potential practical impact of the model in guiding personalized treatment.