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In this talk, we study the persistent homology and related geometric properties of the evolution in time of a discrete-time stochastic process defined on the 2-dimensional regular square lattice. This process corresponds to a cell growth model called the Eden Growth Model (EGM). It can be described as follows: start with the cell square of the 2-dimensional regular square lattice of the plane that contains the origin; then make the cell structure grow by adding one cell at each time uniformly random to the perimeter. We give a characterization of the possible change in the rank of the first homology group of this process (the "number of holes"). Based on this result we have designed and implemented a new algorithm that computes the persistent homology associated to this stochastic process and that also keeps track of geometric features related to the homology. Also, we present obtained results of computational experiments performed with this algorithm, and we establish conjectures about the asymptotic behavior of the homology and other related geometric random variables. The EGM can be seen as a First Passage Percolation model after a proper time-scaling. This is the first time that tools and techniques from stochastic topology and topological data analysis are used to measure the evolution of the topology of the EGM and in general in FPP models.