Topological classification of tumour-immune interactions and dynamics.

The complex and dynamic crosstalk between tumour and immune cells results in tumours that can exhibit distinct qualitative behaviours-elimination, equilibrium, and escape-and intricate spatial patterns, yet share similar cell configurations in the early stages. We offer a topological approach to analyse time series of spatial data of cell locations (including tumour cells and macrophages) in order to predict malignant behaviour. We propose four topological vectorisations specialised to such cell data: persistence images of Vietoris-Rips and radial filtrations at static time points, and persistence images for zigzag filtrations and persistence vineyards varying in time. To demonstrate the approach, synthetic data are generated from an agent-based model with varying parameters. We compare the performance of topological summaries in predicting-with logistic regression at various time steps-whether tumour niches surrounding blood vessels are present at the end of the simulation, as a proxy for metastasis (i.e., tumour escape). We find that both static and time-dependent methods accurately identify perivascular niche formation, significantly earlier than simpler markers such as the number of tumour cells and the macrophage phenotype ratio. We find additionally that dimension 0 persistence applied to macrophage data, representing multi-scale clusters of the spatial arrangement of macrophages, performs best at this classification task at early time steps, prior to full tumour development, and performs even better when time-dependent data are included; in contrast, topological measures capturing the shape of the tumour, such as tortuosity and punctures in the cell arrangement, perform best at intermediate and later stages. We analyse the logistic regression coefficients for each method to identify detailed shape differences between the classes.