  In this article, we present a multispecies reaction-advection-diffusion
partial differential equation (PDE) coupled with linear elasticity for modeling
tumor growth. The model aims to capture the phenomenological features of
glioblastoma multiforme observed in magnetic resonance imaging (MRI) scans.
These include enhancing and necrotic tumor structures, brain edema and the so
called "mass effect", that is, the deformation of brain tissue due to the
presence of the tumor. The multispecies model accounts for proliferating,
invasive and necrotic tumor cells as well as a simple model for nutrition
consumption and tumor-induced brain edema. The coupling of the model with
linear elasticity equations with variable coefficients allows us to capture the
mechanical deformations due to the tumor growth on surrounding tissues. We
present the overall formulation along with a novel operator-splitting scheme
with components that include linearly-implicit preconditioned elliptic solvers,
and semi-Lagrangian method for advection. Also, we present results showing
simulated MRI images which highlight the capability of our method to capture
the overall structure of glioblastomas in MRIs.
