Traveling waves and free boundaries arising in tumor angiogenesis

In a paper appeared in 2021, a mathematical model was developed for tumor invasion with vessel cooption, including a fine analysis of angiogenesis driven by chemotaxis.Numerical solutions in spherical symmetry revealed that a traveling wave sets in. Results were in agreement with experimental data. In the present paper, we introduce some nontrivial changes inthe model and we further analyze the structure of the solutions as well as their dependence on some critical biological parameters, emphasizing for instance which are the most active zones where angiogenesis takes place. Moreover, we propose an alternative model characterized bythe presence of a free boundary (playing the role of the invasion front), showing that the new formulation (which is advantageous from the computational point of view) matches the results of the previous model for some biologically significant range of the parameters.