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We study growth of solid tumors in a partial differential equation modelintroduced by Hillen et al for the interaction between tumor cells (TCs) andcancer stem cells (CSCs). We find that invasion into the cancer-free state maybe separated into two regimes, depending on the death rate of tumor cells. Inthe first, staged invasion regime, invasion into the cancer-free state is leadby tumor cells, which are then subsequently invaded at a slower speed by cancerstem cells. In the second, TC extinction regime, cancer stem cells directlyinvade the cancer-free state. Relying on recent results establishing frontselection propagation under marginal stability assumptions, we use geometricsingular perturbation theory to establish existence and selection properties offront solutions which describe both the primary and secondary invasionprocesses. With rigorous predictions for the invasion speeds, we are then ableto heuristically predict how the total cancer mass as a function of timedepends on the TC death rate, finding in some situations a tumor invasionparadox, in which increasing the TC death rate leads to an increase in thetotal cancer mass. Our methods give a general approach for verifying lineardeterminacy of spreading speeds of invasion fronts in systems with fast-slowstructure.

Growth of cancer stem cell driven tumors: staged invasion, linear determinacy, and the tumor invasion paradox