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In this survey article, a variety of systems modeling tumor growth arediscussed. In accordance with the hallmarks of cancer, the described modelsincorporate the primary characteristics of cancer evolution. Specifically, wefocus on diffusive interface models and follow the phase-field approach thatdescribes the tumor as a collection of cells. Such systems are based on amultiphase approach that employs constitutive laws and balance laws forindividual constituents. In mathematical oncology, numerous biologicalphenomena are involved, including temporal and spatial nonlocal effects,complex nonlinearities, stochasticity, and mixed-dimensional couplings. Usingthe models, for instance, we can express angiogenesis and cell-to-matrixadhesion effects. Finally, we offer some methods for numerically approximatingthe models and show simulations of the tumor's evolution in response to variousbiological effects.

Tumor evolution models of phase-field type with nonlocal effects and angiogenesis