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The complex and dynamic crosstalk between tumour and immune cells results intumours that can exhibit distinct qualitative behaviours - elimination,equilibrium, and escape - and intricate spatial patterns, yet share similarcell configurations in the early stages. We offer a topological approach toanalyse time series of spatial data of cell locations (including tumour cellsand macrophages) in order to predict malignant behaviour. We propose fourtopological vectorisations specialised to such cell data: persistence images ofVietoris-Rips and radial filtrations at static time points, and persistenceimages for zigzag filtrations and persistence vineyards varying in time. Todemonstrate the approach, synthetic data are generated from an agent-basedmodel with varying parameters. We compare the performance of topologicalsummaries in predicting - with logistic regression at various time steps -whether tumour niches surrounding blood vessels are present at the end of thesimulation, as a proxy for metastasis (i.e., tumour escape). We find that bothstatic and time-dependent methods accurately identify perivascular nicheformation, significantly earlier than simpler markers such as the number oftumour cells and the macrophage phenotype ratio. We find additionally thatdimension 0 persistence applied to macrophage data, representing multi-scaleclusters of the spatial arrangement of macrophages, performs best at thisclassification task at early time steps, prior to full tumour development, andperforms even better when time-dependent data are included; in contrast,topological measures capturing the shape of the tumour, such as tortuosity andpunctures in the cell arrangement, perform best at intermediate and laterstages. The logistic regression coefficients reveal detailed shape differencesbetween the classes.

Topological classification of tumour-immune interactions and dynamics