Issue:
Volume 1, Issue 2, December 2026
Pages:
80-91
Received:
11 July 2026
Accepted:
22 August 2026
Published:
3 November 2026
Abstract: Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate of change but do not explicitly represent this history dependence. In this work, a fractional-order tumor–immune–healthy tissue model is formulated using the Atangana–Baleanu Caputo (ABC) fractional derivative, whose nonsingular Mittag–Leffler kernel provides a nonlocal representation of biological memory. The model consists of three interacting populations: tumor cells, immune effector cells, and healthy tissue. Tumor proliferation is represented by logistic growth, immune-mediated tumor elimination and immune suppression are incorporated through nonlinear interaction terms, and healthy tissue regeneration is balanced against tumor-induced damage. The model is first analyzed mathematically. Under explicit regularity and parameter assumptions, local existence and uniqueness of solutions are established by a fixed-point argument. Positivity is obtained from the vector field on the boundary of the nonnegative orthant, and boundedness follows from comparison estimates. The fractional formulation is then converted into its equivalent Volterra integral equation and a history-weighted product-integration discretization with an implicit correction is implemented for numerical computation. Numerical illustrations are used to examine the effect of fractional order on tumor, immune, and healthy-tissue trajectories. The results indicate that decreasing the fractional order changes the transient response and introduces a stronger dependence on previous states, whereas the integer-order case is recovered at α=1. The study therefore provides a mathematically consistent framework for examining memory-dependent tumor dynamics and identifies directions for subsequent calibration with clinical or experimental data.
Abstract: Tumor progression is governed by nonlinear interactions among malignant cells, immune effector cells, and the surrounding healthy tissue. Such interactions are influenced by previous biological states, including accumulated immune stimulation, tissue damage, and cellular adaptation. Conventional integer-order models describe the instantaneous rate ...
Show More
