Asian Journal of Pure and Applied Mathematics

Mathematical models of cancer dynamics typically assume integer-order derivatives, which neglect memory effects. This study examines the influence of temporal memory on a phenomenological tumour-immune-chemotherapy model by replacing integer-order derivatives with Caputo fractional derivatives of order \(\alpha\) between 0.5 and 1.0. The fractional system represents tumour density, immune cell density, normal cell density, and drug concentration. Numerical solutions were obtained using the fractional Adams-Bashforth-Moulton predictor-corrector method, with convergence verification and 90% uncertainty intervals generated from 500 parametric bootstrap runs. The simulations indicate that lower \(\alpha\) values, representing stronger memory, delay tumour growth while producing a higher final tumour burden. For \(\alpha\) = 0.5, the time to 50% of maximum tumour density was 47.3 days compared with 21.4 days for \(\alpha\) = 1.0, whereas the corresponding final burdens were 0.82 and 0.53. Stronger memory also produced later, higher, and more sustained immune responses. Fractional pharmacokinetics generated greater drug exposure, with AUC increasing from 12.8 at \(\alpha\) = 1.0 to 28.4 at \(\alpha\) = 0.5. The reported therapeutic index increased as \(\alpha\) decreased, and the fractional formulation had the lowest AIC and BIC values among the compared model variants.

These findings are entirely model-based and depend on the stated parameterisation and assumptions. The results therefore support further investigation of fractional-order formulations as theoretical tools for studying memory effects in cancer-immune and chemotherapy dynamics, while requiring independent experimental and clinical validation before any clinical interpretation.