# Topology across scales on heterogeneous cell data

## TL;DR

This paper introduces new ways to visualize and interpret complex spatial biological data using topology, helping to understand disease progression in tissues.

## Contribution

The novel visualization and vectorization methods for persistent homology improve the analysis of heterogeneous cell data.

## Key findings

- A new visualization method helps identify relevant features directly in biological data.
- The approach was successfully applied to tissue data from lupus and COVID-19, revealing cell patterns linked to disease progression.

## 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 persistent 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.

How cells are arranged within tissues is crucial to understand disease progression. Recent imaging technologies provide detailed spatial maps of tissues, creating large and complex data sets that can be challenging to interpret. Our work explores an avenue to quantify spatial data using ideas from topology, which is the mathematical field that describes shapes. Topological data analysis offers tools that capture the structure of complex data; an active area is visualising and interpreting the topological fingerprints in the original biological context. In this study, we adapt and extend topological methods to make the resulting insights more accessible. We introduce a simple visualisation that helps locate relevant features directly in the original data. By applying this method to tissue images from lupus murine spleen and COVID-19-infected human lungs, we show how it can highlight and quantify cell patterning that relates to disease progression. Our goal is to make these mathematical tools easier to use and understand, contributing to a growing set of interpretable methods for describing complex data.