A mathematical model for pattern formation of glioma cells outside the tumor spheroid core

Glioblastoma is the most common and the most aggressive type of brain cancer. The median survival time from the time of diagnosis is approximately one year. Invasion of glioma cells from the core tumor into the surrounding brain tissue is a major reason for treatment failure: these migrating cells are not eliminated in surgical resection and cause tumor recurrence. Variations are seen in number of invading cells, and in the extent and patterns of migration. Cells can migrate diffusely and can also be seen as clusters of cells distinct from the main tumor mass. This kind of clustering is also evident in vitro using 3D spheroid models of glioma invasion. This has been reported for U87 cells stably expressing the constitutively active EGFRVIII mutant receptor, often seen expressed in glioblastoma. In this case the cells migrate as clusters rather than as single cells migrating in a radial pattern seen in control wild type U87 cells. Several models have been suggested to explain the different modes of migration, but none of them, so far, has explored the important role of cell-cell adhesion. The present paper develops a mathematical model which includes the role of adhesion and provides an explanation for the various patterns of cell migration. It is shown that, depending on adhesion, haptotactic, and chemotactic parameters, the migration patterns exhibit a gradual shift from branching to dispersion, as has been reported experimentally.