Tumour-induced angiogenesis, which is the mechanism by which a solid tumour promotes the growth of new blood vessels to sustain its own progression, is a biologically complex phenomenon that poses non-trivial challenges from a mathematical and computational standpoint. This talk focuses on a class of PDE systems designed to capture the key dynamics of this process, discussing both their analytical properties, such as the existence of solutions, conservation laws and positivity, and the construction of numerical schemes that are at the same time efficient and physically consistent. Particular attention is given to IMEX strategies and to the more recent use of Physics-Informed Neural Networks (PINNs) as a complementary tool for approximating solutions in regimes where classical methods become costly. Part of the presentation is devoted to the high-performance side of the problem: GPU-CUDA and Julia-CUDA implementations of the proposed methods have been tested on large-scale architectures, including the LEONARDO supercomputer at CINECA, yielding results that confirm both the numerical reliability and the computational scalability of the approach.