  The acid-mediated tumor invasion hypothesis proposes that altered glucose
metabolism exhibited by the vast majority of tumors leads to increased acid (H+
ion) production which subsequently facilitates tumor invasion [1-3]. The
reaction-diffusion model [2] that captures the key elements of the hypothesis
shows how the densities of normal cells, tumor cells, and excess H+ ions change
with time due to both chemical reactions between these three populations and
density-dependent diffusion by which they spread out in three-dimensional
space. Moreover, it proposes that each cell has an optimal pH for survival;
that is, if the local pH deviates from the optimal value in either an acidic or
alkaline direction, the cells begin to die, and that the death rate saturates
at some maximum value when the microenvironment is extremely acidic or
alkaline. We have previously studied in detail how the death-rate functions of
the normal and tumor populations depend upon the H+ ion density [4]. Here, we
extend previous work by investigating how the equilibrium densities (at which
the time rates of change of the cellular densities are equal to zero) reached
by the normal and tumor populations in three-dimensional space are affected by
the presence of the H+ ions, and we present detailed analytical and
computational techniques to analyze the dynamical stability of these
equilibrium densities. For a sample set of biological input parameters and
within the acid-mediation hypothesis, our model predicts the transformation to
a malignant behavior, as indicated by the presence of unstable sets of
equilibrium densities.
