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Topological data analysis (TDA) is an active field of mathematics forquantifying shape in complex data. Standard methods in TDA such as persistenthomology (PH) are typically focused on the analysis of data consisting of asingle entity (e.g., cells or molecular species). However, state-of-the-artdata collection techniques now generate exquisitely detailed multispecies data,prompting a need for methods that can examine and quantify the relations amongthem. Such heterogeneous data types arise in many contexts, ranging frombiomedical imaging, geospatial analysis, to species ecology. Here, we proposetwo methods for encoding spatial relations among different data types that arebased on Dowker complexes and Witness complexes. We apply the methods tosynthetic multispecies data of a tumor microenvironment and analyze topologicalfeatures that capture relations between different cell types, e.g., bloodvessels, macrophages, tumor cells, and necrotic cells. We demonstrate thatrelational topological features can extract biological insight, including thedominant immune cell phenotype (an important predictor of patient prognosis)and the parameter regimes of a data-generating model. The methods provide aquantitative perspective on the relational analysis of multispecies spatialdata, overcome the limits of traditional PH, and are readily computable.

Relational persistent homology for multispecies data with application to the tumor microenvironment