Quantifying immune memory in colorectal cancer using fractional-order differential equations.

Traditional mathematical models of tumor-immune dynamics employ integer-order ordinary differential equations that assume Markovian behavior, wherein the system's instantaneous rate of change depends only on its current state. However, biological processes such as T cell exhaustion and myeloid-derived suppressor cell (MDSC) accumulation exhibit memory effects, where past immunosuppressive exposures create persistent functional states through epigenetic modifications and microenvironmental feedback loops. We develop a fractional-order differential equation framework to quantify these non-Markovian dynamics in human colorectal cancer. By replacing integer-order derivatives with Caputo fractional derivatives of order α ∈ (0, 1] for T cell and MDSC compartments, we incorporate power-law memory kernels that capture temporal persistence of immunosuppressive states. Analyzing gene expression data from 498 patients in The Cancer Genome Atlas, we estimate patient-specific fractional orders from immune signatures and demonstrate substantial inter-patient heterogeneity. Lower fractional orders correlate strongly with T cell exhaustion markers and enable stratification into biologically distinct patient subgroups. Compared to standard integer-order models, the fractional framework achieves threefold higher variance explained in immune signatures. These findings establish fractional calculus as a powerful tool for modeling immune memory in colorectal cancer and suggest potential applications in precision immunotherapy.