  We study growth of solid tumors in a partial differential equation model
introduced by Hillen et al for the interaction between tumor cells (TCs) and
cancer stem cells (CSCs). We find that invasion into the cancer-free state may
be separated into two regimes, depending on the death rate of tumor cells. In
the first, staged invasion regime, invasion into the cancer-free state is lead
by tumor cells, which are then subsequently invaded at a slower speed by cancer
stem cells. In the second, TC extinction regime, cancer stem cells directly
invade the cancer-free state. Relying on recent results establishing front
selection propagation under marginal stability assumptions, we use geometric
singular perturbation theory to establish existence and selection properties of
front solutions which describe both the primary and secondary invasion
processes. With rigorous predictions for the invasion speeds, we are then able
to heuristically predict how the total cancer mass as a function of time
depends on the TC death rate, finding in some situations a tumor invasion
paradox, in which increasing the TC death rate leads to an increase in the
total cancer mass. Our methods give a general approach for verifying linear
determinacy of spreading speeds of invasion fronts in systems with fast-slow
structure.
