Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model

Objectives: Glioblastomas are aggressive primary brain cancers that are characterized by extensive infiltration into the brain and are highly resistant to treatment. Through mathematical modelling, we model the process of invasion and predict the relative importance of mechanisms contributing to malignant invasion. Clinically, we predict patterns of tumour recurrence following various modes of therapeutic intervention. Materials and methods: Our mathematical model uses a realistic three-dimensional brain geometry and considers migrating and proliferating cells as separate classes. Several mechanisms for infiltrative migration are considered. Methods are developed for simulating surgical resection, radiotherapy and chemotherapy. Results: The model provides clinically realistic predictions of tumour growth and recurrence following therapeutic intervention. Specific results include (i) invasiveness is governed largely by the ability of glioblastoma cells to degrade and migrate through the extracellular matrix and the ability of single migrating cells to form colonies; (ii) tumours originating deeper in the brain generally grow more quickly than those of superficial origin; (iii) upon surgery, the margins and geometry of resection significantly determine the extent and pattern of postoperative recurrence; (iv) radiotherapy works synergistically with greater resection margins to reduce recurrence; (v) simulations in both two- and three-dimensional geometries give qualitatively similar results; and (vi) in an actual clinical case comprising several surgical interventions, the model provides good qualitative agreement between the simulated and observed course of the disease. Conclusions: The model provides a useful initial framework by which biological mechanisms of invasion and efficacy of potential treatment regimens may be assessed.