Owing to its unrivaled invasiveness among primary tumors, glioblastoma multiforme (GBM) has become the subject of countless in vitro studies aimed at characterizing its growth and proliferation. While these studies have developed a greater understanding of the biochemistry underpinning GBM, they have done little, if anything, to answer questions about how the interactions of GBM with its surrounding collagen matrix shape its development. Important among such questions is: how prominent a role do haptotaxis and durotaxis, the directed growth of cells along adhesion and rigidity gradients, respectively, assume in GBM invasion?  Because simulation may readily address this concern, a multi-scale extended Q-Potts model based upon the Hamiltonian that follows was developed and its results analyzed using methods borrowed from topology and the study of glassy systems. Evidence indicates that the proposed model will not only be able to lay the above question to rest, but may be effortlessly adapted to elucidate other biological phenomena.