Abstract: Tumor morphology provides important insight into growth and metastatic potential. We propose a topological feature obtained by persistent homology to characterize tumor progression in medical images, with particular emphasis on its association with time-to-event data. These features, which are invariant under scale-preserving transformations, capture complex and heterogeneous shape patterns. To incorporate this information, we develop a functional Cox proportional hazards model that represents these topological features in a functional space, utilizing them as functional predictors. This framework enables an interpretable assessment of how topological shape characteristics relate to survival risk.