Abstract: Tumor shape and size have been used as important markers for cancer diagnosis and treatment. This paper proposes a topological feature computed by persistent homology to characterize tumor progression from digital pathology and radiology images and examines its effect on the time-to-event data. The proposed topological features are invariant to scale-preserving transformation and can summarize various tumor shape patterns. The topological features are represented in functional space and used as functional predictors in a functional Cox proportional hazards model. The proposed model enables interpretable inference about the association between topological shape features and survival risks. Two case studies are conducted using lung cancer pathology and brain tumor radiology images. The results show that the topological features predict survival prognosis after adjusting clinical variables, and the predicted high-risk groups have significantly worse survival outcomes than the low-risk groups (p-values<0.005 for both studies). Also, the topological shape features found to be positively associated with survival hazards are irregular and heterogeneous shape patterns, which are known to be related to tumor progression. Our study provides a new perspective for understanding tumor shape and patient prognosis.