In this work, we report on kinetic Monte-Carlo calculations of resistive switching and the underlying growth dynamics of filaments in an electrochemical metallization device consisting of an Ag/TiO2/Pt sandwich-like thin film system. The developed model is not limited to (i) fast time scale dynamics and (ii) only one growth and dissolution cycle of metallic filaments. In particular, we present results from the simulation of consecutive cycles. We find that the numerical results are in excellent agreement with experimentally obtained data. Additionally we observe an unexpected filament growth mode that is in contradiction to the widely acknowledged picture of filament growth but consistent with recent experimental findings.