  In this work we analyse a PDE-ODE problem modelling the evolution of a
Glioblastoma, which includes chemotaxis term directed to vasculature. First, we
obtain some a priori estimates for the (possible) solutions of the model. In
particular, under some conditions on the parameters, we obtain that the system
does not develop blow-up at finite time. In addition, we design a fully
discrete finite element scheme for the model which preserves some pointwise
estimates of the continuous problem. Later, we make an adimensional study in
order to reduce the number of parameters. Finally, we detect the main
parameters determining different width of the ring formed by proliferative and
necrotic cells and different regular/irregular behaviour of the tumor surface.
