TDA approaches for spatial biology problems.

Spatial biology studies how cells and tissues are arranged and interact in their natural 2D/3D environment. At single-cell resolution, data can be modeled as chromatic (labeled) point clouds, where each point represents a cell and its color denotes cell type. The challenge is capturing not only the overall geometry of the cloud, but also relationships between labeled subsets, which standard persistent homology often misses. Recent TDA tools such as Dowker persistence or mix-up barcodes, address the analysis of pairs of labels. For multiple labels, the chromatic alpha filtration generalizes the classical alpha filtration; inclusion maps between labeled datasets induce maps between filtrations and derive a set of persistence diagrams encoding spatial relations. However, computing chromatic alpha values can be costly, motivating the use of chromatic Delaunay–Čech filtration as a cheaper alternative (Natarajan et al.). Using Morse-theoretic techniques, these two filtrations are linked by simplicial collapses, extending a Bauer–Edelsbrunner result to the chromatic setting. Within this framework, we describe some spatial biology problems of interest: (1) quantifying cancer–immune interactions in multiplex-imaged tumors, (2) probing stem-cell differentiation controlled by spatial signaling such as BMP, and (3) analyzing evolving 3D tissue organization during embryonic development using simulations like EmbryoMaker.

Maria Jose Jimenez is funded by the Spanish government, MCIU, under grant PID2024-155867NB-I00.