Biological Frequency Combs: A Case of Breast Cancer Dynamics

Abstract We explore the emergence of frequency-comb structure in a fractional-order model of breast-cancer dynamics. The model couples tumor cells, cytotoxic T lymphocytes (CTLs), and chimeric antigen receptor (CAR-T) cells through tumor growth, immune activation, saturating cytotoxic killing, cellular decay, and CAR-T-cell therapy. At the nominal parameter values, the three populations undergo relaxation oscillations with a common period, characterized by the frequency comb spectra consisting of frequencies separated by equal intervals. Within the comb regime, the frequency comb spacing increases with the CTL and CAR-T-cell decay rates (d and ζ), the CTL-mediated tumor-cell deactivation rate β, and the CTL-to-CAR-T-cell conversion rate σ. The comb spacing decreases as CTL-dependent CAR-T-cell activation μ, tumor-induced CAR-T-cell deactivation b, or the tumor-cell and CTL carrying capacities (k1 and k2) increase. Changes in the tumor-induced CTL deactivation rate ξ and CTL growth rate γ2 have little effect on the spacing. The comb disappears at sufficiently large β, b, or ζ, while it only appears when μ is increased above a threshold. Ultimately, the comb represents an interesting spectral fingerprint associated with the tumor–immune dynamics.