  We present and analyze new multi-species phase-field mathematical models of
tumor growth and ECM invasion. The local and nonlocal mathematical models
describe the evolution of volume fractions of tumor cells, viable cells
(proliferative and hypoxic cells), necrotic cells, and the evolution of MDE and
ECM, together with chemotaxis, haptotaxis, apoptosis, nutrient distribution,
and cell-to-matrix adhesion. We provide a rigorous proof of the existence of
solutions of the coupled system with gradient-based and adhesion-based
haptotaxis effects. In addition, we discuss finite element discretizations of
the model, and we present the results of numerical experiments designed to show
the relative importance and roles of various effects, including cell mobility,
proliferation, necrosis, hypoxia, and nutrient concentration on the generation
of MDEs and the degradation of the ECM.
