We describe a preliminary effort to model the growth and progression of glioblastoma multiforme, an aggressive form of primary brain cancer, in patients undergoing treatment for recurrence of tumor following initial surgery and chemoradiation. Two reaction-diffusion models are used: the Fisher-Kolmogorov equation and a 2-population model, developed by the authors, that divides the tumor into actively proliferating and quiescent (or necrotic) cells. The models are simulated on 3-dimensional brain geometries derived from magnetic resonance imaging (MRI) scans provided by the Barrow Neurological Institute. The study consists of 17 clinical time intervals across 10 patients that have been followed in detail, each of whom shows significant progression of tumor over a period of 1 to 3 months on sequential follow up scans. A Taguchi sampling design is implemented to estimate the variability of the predicted tumors to using 144 different choices of model parameters. In 9 cases, model parameters can be identified such that the simulated tumor, using both models, contains at least 40 percent of the volume of the observed tumor. We discuss some potential improvements that can be made to the parameterizations of the models and their initialization.
Keywords: Fisher-Kolmogorov model; ensemble prediction; glioblastoma multiforme; magnetic resonance imaging; parameter estimation; reaction-diffusion equations.