Complex dynamics of tumors: modeling an emerging brain tumor system with coupled reaction–diffusion equations

One of the hallmarks of malignant brain tumors is their extensive tissue invasion, which represents a major obstacle for effective treatment. In this paper we specifically model the invasive behavior of such tumors viewed as complex dynamic biosystems. Based on the spatio-temporal patterns seen in an experimental setting for multicellular brain tumor spheroids we propose an invasion-guiding, dynamical profile of heterotype and homotype attractor substances. We present a novel theoretical and numerical framework for a mathematical tumor model composed of a set of coupled reaction–diffusion equations describing chemotactic and haptotactic cell behavior. In particular, our continuum model simulates tumor cell motility guided by the principle of least resistance, most permission and highest attraction. Preliminary numerical results indicate that the computational algorithm is capable of reproducing patterns similar to the experimentally observed behavior.