Critical Patch Size of a Two-Population Reaction Diffusion Model Describing Brain Tumor Growth

The critical patch (KISS) size is the minimum habitat size needed for a population to survive in a region. Habitats larger than the critical patch size allow a population to persist, while smaller habitats lead to extinction. We perform a rigorous derivation of the critical patch size associated with a 2-population glioblastoma multiforme (GBM) model that divides the tumor cells into proliferating and quiescent/necrotic populations. We determine that the critical patch size of our model is consistent with that of the Fisher–Kolmogorov–Petrovsky–Piskunov equation, one of the first reaction-diffusion models proposed for GBM, and does not depend on parameters pertaining to the quiescent/necrotic population. The critical patch size may indicate that GBM tumors have a minimum size depending on the location in the brain. We also derive a theoretical relationship between the size of a GBM tumor at steady-state and its maximum cell density, which has potential applications for patient-specific parameter estimation based on magnetic resonance imaging data. Finally, we identify a positively invariant region for our model, which guarantees that solutions remain positive and bounded from above for all time.