A kinetic view of acid-mediated tumor invasion

According to the acid-mediated tumor invasion hypothesis, tumor-induced alteration of microenvironmental pH may provide a simple, yet complete mechanism for tumor invasion. The acid-mediation hypothesis analyzes the tumor growth and invasion process from a reaction-diffusion system perspective, where it incorporates the H⁺ ion concentration as a reaction factor and adds density-dependent diffusion parameters to the reaction terms, yielding independent reaction-diffusion equations for the normal, tumor, and acid populations. In this article, we apply the dynamical stability theory to the acid-mediation hypothesis. For reasonable biological parameters, we study the fixed points central to the model and their stability by calculating the eigenvalues of the Jacobian matrix of the partial differential equations that represent how these three populations evolve with time. For the case where a malignant behavior has not already taken place yet (the time rates of change of the densities of the three populations are equal to zero), our numerical results convey two different, yet possible configurations in three-dimensional space: stable and unstable dynamical equilibriums, and we discuss possible prospective trajectories for the normal and tumor populations starting from each configuration. Moreover, we discuss potential applications of our approach.