Exact Response to Perturbations and Dynamical Indicators in a Tumor-Growth Model

Abstract

We formulate a three-dimensional tumor–immune model obtained by adding to an established tumor–host–effector system two phenomenological perturbation terms, whose signs are motivated by the reported tumor-promoting and effector-suppressing influence of regulatory B cells; their coefficients are dynamical control parameters and are not calibrated to biological rates. The perturbed and unperturbed dynamics are compared using phase portraits, Lyapunov exponents, parameter sensitivity, and multifractal indicators. Because trajectory-wise prediction is intrinsically limited in chaotic regimes and initial cell populations are only partially known, we also extend exact response theory, originally developed in nonequilibrium molecular dynamics, to the relevant non-compact positive set of populations values. Admissible initial densities are introduced to control boundary terms and ensure the validity of the exact response identity for finite-time ensemble averages. The resulting formulation quantifies both instantaneous and cumulative changes in tumor, host, and effector-cell observables under finite parameter perturbations, without requiring a small-perturbation approximation. Numerical tests verify the response identity against direct Monte Carlo propagation and assess sampling and time-discretization errors. For the parameter ranges studied, the perturbations can move the model between periodic and chaotic regimes, while ensemble averages remain considerably smoother than individual trajectories, and are capable of significant predictions. Long-time averaged responses are consistent across three admissible initial ensembles, although instantaneous dispersion can remain substantial. The analysis therefore separates attractor geometry from finite-time statistical response and provides a mathematically controlled framework for uncertainty-aware comparison of tumor-growth models, especially relevant in the potentially most interesting transients. The biological interpretation remains qualitative: The perturbation coefficients are not clinically calibrated rates, and any correspondence with regulatory B-cell biology requires independent experimental validation.