Using alpha shapes to quantify spatial heterogeneity in virus-mediated syncytia formation

Some viruses have the ability to cause neighboring cells to fuse into a cingle multi-mucleated cell known as a syncytium. Syncytia formation is often studied through the use of cell-cell fusion assays that use fluorescent dyes to watch the formation of syncytia over time. In order to develop accurate mathematical models of the syncytia formation process, we need to quantify the spatial heterogeneity created by syncytiia. Our method uses topological theories including Delaunay Tessellations and Voronoi Graphs to represent cell-cell interaction probability. By applying a filtration to the Delaunay Tessellation of our system, we can obtain unique Alpha Shapes that have cell-cell interactions removed. The filtration finds the alpha shapes where there are no more connections crossing syncytia. This defines an alpha number that characterizes the ``clumpiness'' of the system. Alpha numbers can then be used to characterize spatial differences in syncytia formation for different viruses or under different environmental conditions.