By modifying and calibrating an active vertex model to experiments, we have simulated numerically a confluent cellular monolayer spreading on an empty space and the collision of two monolayers of different cells in an antagonistic migration assay. Cells are subject to inertial forces and to active forces that try to align their velocities with those of neighboring ones. In agreement with experiments in the literature, the spreading test exhibits formation of fingers in the moving interfaces, there appear swirls in the velocity field, and the polar order parameter and the correlation and swirl lengths increase with time. Numerical simulations show that cells inside the tissue have smaller area than those at the interface, which has been observed in recent experiments. In the antagonistic migration assay, a population of fluidlike Ras cells invades a population of wild type solidlike cells having shape parameters above and below the geometric critical value, respectively. Cell mixing or segregation depends on the junction tensions between different cells. We reproduce the experimentally observed antagonistic migration assays by assuming that a fraction of cells favor mixing, the others segregation, and that these cells are randomly distributed in space. To characterize and compare the structure of interfaces between cell types or of interfaces of spreading cellular monolayers in an automatic manner, we apply topological data analysis to experimental data and to results of our numerical simulations. We use time series of data generated by numerical simulations to automatically group, track and classify the advancing interfaces of cellular aggregates by means of bottleneck or Wasserstein distances of persistent homologies. These techniques of topological data analysis are scalable and could be used in studies involving large amounts of data. Besides applications to wound healing and metastatic cancer, these studies are relevant for tissue engineering, biological effects of materials, tissue and organ regeneration. Author summary Confluent motion of cells in tissues plays a crucial role in wound healing, tissue repair, development, morphogenesis and in numerous pathological processes such as tumor invasion and metastatic cancer. For such complex processes, controlled experiments help clarifying the roles of chemical, mechanical and biological cues. Among them, spreading of cellular tissues on an empty space and antagonistic migration assays between cancerous and normal cells are quite revealing. The interfaces between confluent cellular aggregates uncover properties thereof when a combination of modeling, numerical simulation and data analysis is used. Here we have modified an active vertex model with a dynamics that includes inertia, friction and active forces that tend to align cells based on interaction with its immediate neighborhood. Selecting appropriately junction tensions among cells and using the SAMoS software, we have succeed in simulating assays of cellular tissue spreading on an empty space and the invasion of healthy tissue by cancerous one. We have introduced topological data analysis to characterize, track and compare in an automatic manner the interfaces of the tissue both in numerical simulations and from experimental data of normal and Ras modified precancerous Human Embryonic Kidney cells. We find good agreement when normal cells are solidlike and modified cells are liquidlike according to their shape parameters. In addition, cell variability means that a fraction of randomly distributed cells favor mixing, the others segregation. Topological data analysis techniques are scalable and could be used in studies involving large amounts of data. Besides applications to wound healing and metastatic cancer, these studies are relevant in ascertaining how the biophysical features of materials may affect tissue and organ regeneration.