Laboratory for Topology and Neuroscience, EPFL, Station 8, Lausanne, 1015, Switzerland.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, UK.
Bull Math Biol. 2024 Sep 17;86(11):128. doi: 10.1007/s11538-024-01353-6.
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.
拓扑数据分析（TDA）是一门用于对复杂数据中的形状进行量化的数学活跃领域。TDA 中的标准方法，如持久同调（PH），通常专注于分析由单个实体（例如，细胞或分子物种）组成的数据。然而，最先进的数据采集技术现在生成了极其详细的多物种数据，这促使我们需要能够检查和量化它们之间关系的方法。这种异质数据类型出现在许多背景下，从生物医学成像、地理空间分析到物种生态学。在这里，我们提出了两种基于 Dowker 复形和 Witness 复形的方法，用于编码不同数据类型之间的空间关系。我们将这些方法应用于肿瘤微环境的合成多物种数据，并分析了捕获不同细胞类型之间关系的拓扑特征，例如血管、巨噬细胞、肿瘤细胞和坏死细胞。我们证明了关系拓扑特征可以提取生物学见解，包括主导免疫细胞表型（患者预后的重要预测因子）和数据生成模型的参数范围。这些方法为多物种空间数据的关系分析提供了定量视角，克服了传统 PH 的限制，并且易于计算。
