In this article, we propose a framework for modeling glioma growth and the subsequent mechanical impact on the surrounding brain tissue (mass-effect) in a medical imaging context. Glioma growth is modeled via nonlinear reaction-advection-diffusion, with a two-way coupling with the underlying tissue elastic deformation. Tumor bulk and infiltration and subsequent mass-effects are not regarded separately, but captured by the model itself in the course of its evolution, Our formulation is fully Eulerian and naturally allows for updating the tumor diffusion coefficient following structural displacements caused by tumor growth/infiltration. We show that model parameters can be estimated via optimization based on imaging data, using efficient solution algorithms on regular grids. We test the model and the automatic optimization framework on real brain tumor data sets, achieving significant improvement in landmark prediction compared to a simplified purely mechanical approach.