  Topological data analysis (TDA) is an active field of mathematics for
quantifying shape in complex data. Standard methods in TDA such as persistent
homology (PH) are typically focused on the analysis of data consisting of a
single entity (e.g., cells or molecular species). However, state-of-the-art
data collection techniques now generate exquisitely detailed multispecies data,
prompting a need for methods that can examine and quantify the relations among
them. Such heterogeneous data types arise in many contexts, ranging from
biomedical imaging, geospatial analysis, to species ecology. Here, we propose
two methods for encoding spatial relations among different data types that are
based on Dowker complexes and Witness complexes. We apply the methods to
synthetic multispecies data of a tumor microenvironment and analyze topological
features that capture relations between different cell types, e.g., blood
vessels, macrophages, tumor cells, and necrotic cells. We demonstrate that
relational topological features can extract biological insight, including the
dominant immune cell phenotype (an important predictor of patient prognosis)
and the parameter regimes of a data-generating model. The methods provide a
quantitative perspective on the relational analysis of multispecies spatial
data, overcome the limits of traditional PH, and are readily computable.
