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The dynamic 3D shape of a cell acts as a signal of its physiological state, reflecting the interplay of environmental stimuli and intra- and extra-cellular processes. However, there is little quantitative understanding of cell shape determination in 3D, largely due to the lack of data-driven methods that analyse 3D cell shape dynamics. To address this, we have developed MorphoSense, an interpretable, variable-length multivariate time series classification (TSC) pipeline based on multiple instance learning (MIL). We use this pipeline to classify 3D cell shape dynamics of perturbed cancer cells and learn hallmark 3D shape changes associated with clinically relevant and shape-modulating small molecule treatments. To show the generalisability across datasets, we apply our pipeline to classify migrating T-cells in collagen matrices and assess interpretability on a synthetic dataset. Across datasets, our pipeline offers increased predictive performance and higher-quality interpretations. To our knowledge, our work is the first to utilise MIL for multivariate, variable-length TSC, focusing on interpretable 3D morphodynamic profiling of biological cells.