Topology across Scales on Heterogeneous Cell Data

TL;DR

This work develops a topology-centric framework for analyzing multiplexed spatial cell data by using 1-parameter persistent homology (PH) and novel visualisations to extract multiscale structure. A key contribution is persistence weighted death simplices (PWDS), which maps PH features back to the original tissue geometry to localise and interpret topological patterns, complemented by vectorisations including normalised Betti curves, elementary statistics, and persistence images with multiple weightings. The authors demonstrate that PWDS and these vectorisations can differentiate healthy vs diseased states in lupus spleen and COVID-19-affected lungs, identifying both large-scale architectural differences and small-scale cellular infiltrates, with endothelial patterns providing strong disease-stage separation. The approach emphasizes interpretability, stability, and compatibility with standard clustering, suggesting a practical pathway for integrating topological insights into spatial biology analyses and disease characterization. Overall, the paper shows that multiscale topological descriptors can uncover biologically meaningful patterns across tissue architectures and cell-type compositions, with potential to inform mechanistic understanding and diagnostic stratification.

Abstract

Multiplexed imaging allows multiple cell types to be simultaneously visualised in a single tissue sample, generating unprecedented amounts of spatially-resolved, biological data. In topological data analysis, persistent homology provides multiscale descriptors of ``shape" suitable for the analysis of such spatial data. Here we propose a novel visualisation of persistence homology (PH) and fine-tune vectorisations thereof (exploring the effect of different weightings for persistence images, a prominent vectorisation of PH). These approaches offer new biological interpretations and promising avenues for improving the analysis of complex spatial biological data especially in multiple cell type data. To illustrate our methods, we apply them to a lung data set from fatal cases of COVID-19 and a data set from lupus murine spleen.

Figures (6)

• Figure 1: Topological data analysis pipeline. Starting from point cloud data—e.g., cell centroid locations in tissue—we construct filtered simplicial complexes to approximate the underlying geometric structure. Persistent homology is then computed to extract topological features such as connected components, loops, and voids across multiple scales. These features are vectorised, and we explore weightings that emphasise structure at different scales. The resulting representations are used for clustering. We also introduce the persistence weighted death simplices (PWDS) visualisation to enhance interpretability of the topological features.
• Figure 2: Lupus murine spleen cell centroid data. A) Cell centroid spatial distribution of the 25 cell types in lupus murine spleen CODEX data set. The data set of consists of 3 healthy samples (BALBc-1, BALBc-2, BALBc-3), 3 samples in an early stage of the disease (MRL-4, MRL-5, MRL-6), 2 samples in an intermediate stage of the disease (MRL-7, MRL-8) and 1 sample in a late stage of the disease (MRL-9). B) Cell counts of the two types of pulp for each sample (entire slide) in tens of thousands of cells. Solid lines are guides for the eye. C) Point clouds corresponding to the main parts of the spleen, the red pulp and the white pulp, for each sample, classified by disease stage. The white pulp forms compartments surrounded by the red pulp.
• Figure 3: Examples of persistent homology. A) Example of the alpha filtration for a point cloud forming a loop. Corresponding persistence diagram with one persistent feature in degree 1 corresponding to the loop. B) Example of persistent homology for a point cloud consisting of four loops of various sizes and densities. For each loop, we indicate the corresponding feature in the persistence diagram. Less dense loops have larger birth values, bigger loops have larger death values. C) Steps of the computation of a persistence image with weight on persistence $w_1(b,p) = p$.
• Figure 4: Persistence weighted death simplices (PWDS) visualisation. A) Colouring and weighting of the persistence diagram. The colouring in the diagram distinguishes features based on their birth values, with red representing features formed by proximal points, blue for those formed by distal points, and a continuous gradation of colour for features in between. The intensity of the colour reflects the feature's persistence (death minus birth), with darker shades indicating more prominent features. B) Visualisation of persistence weighted death simplices (PWDS) for 5 loops with increasing noise with thresholds $b_{\text{prox}} = \langle P_{90} \rangle$ and $b_{\text{dist}}=\langle P_{98} \rangle$. Each triangle indicates the approximate location of a loop detected by persistence homology with alpha complexes, coloured as described above. The PWDS visualisation of the first loop shows only red triangles, indicating features formed by proximal points, with a prominent red triangle for the large loop and smaller red triangles for voids. As noise is added inside the loop, blue triangles appear for features formed by distal points, and the large loop to become less prominent. With increasing noise, the red triangles shrink and lighten, and eventually, only light-red triangles remain, indicating the loss of structure at the larger scale.
• Figure 5: Witness filtration. A) Example of the witness filtration for a point cloud forming a loop. Landmarks points, in black, are chosen randomly. The persistence diagram contains only one point in each degree, corresponding to a single connected component and the loop, both formed at $\epsilon = 0$.