Fractional Order Temporal Dynamics of Tumor Immune Chemotherapy Interactions: A Reproducible Dimensionless Analysis and a Pharmacokinetic Caveat

Abstract

Fractional differential equations are increasingly used to represent temporal memory in tumour immune and pharmacokinetic models, but changing the derivative order without a consistent time scale can make comparisons difficult to interpret. We study a dimensionless, commensurate Caputo system for tumour, immune, normal cell, and drug states. We establish positivity, boundedness, the tumour invasion threshold, and local stability conditions. We then use a fractional Adams Bashforth Moulton predictor corrector method with an explicit mesh convergence study. Two conclusions correct potentially misleading interpretations of the model. First, the equilibrium locations are independent of the fractional order: decreasing the order changes transient time scales but does not by itself alter the model's asymptotic states. Second, after a unit bolus the drug equation has the exact solution C({tau})=E(-K{tau}). Its maximum is therefore the initial concentration, not a delayed peak. For 0<<1, the algebraic tail makes the infinite-horizon area under the concentration curve diverge. In the illustrative coupled simulations, the apparent late tumour reduction at small is consequently driven by persistent modelled drug exposure and must not be read as evidence of superior therapy. All parameters are dimensionless and illustrative; no patient data are fitted. Source code and generated tables are included to make every numerical statement reproducible.