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

Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation