Spatial tessellation and the Morisita index

Spatial compartmentalisation was achieved automatically by partitioning DCIS tissue space using Voronoi tessellation, resulting in Voronoi polygons that contain one DCIS duct at the centre. The tessellation is a partition of space according to neighbourhood relations of a given set of points in the space. It has been suggested that Voronoi tessellation mimics the biological patterns present in the histological image and naturally emerged patterns³¹. This property combined with the ecological index aids to study the ecological characteristics of individual DCIS and is similarly used in histology studies to provide a spatial context of diverse cell types coexisting within the microenvironment³².

Let K be a set containing all coordinates of DCIS D and let (D_k)_k ∈ _K be the coordinates of a DCISk. A Voronoi region R_k generated by DCIS duct D_k contains all cells P that are not seeds and are closer to D_k than to any other seed D_j, j ≠ k. Let d(Q_i, Q_j) be the Euclidean distance function between two centroids of DCIS Q_i and Q_j then

Centroids from IM-Net segmentation were estimated and the resulting binary masks were mapped back to lower resolution (×1.25) of the image. The centroids were then used as a seed for calculating the Voronoi polygon. Because of these mathematical principles underlying Voronoi tessellation, lymphocytes within a polygon will be closer to its seed than any other seeds. This means that the closest DCIS duct for a lymphocyte is the one that ‘seeds’ the polygon containing this lymphocyte. Subsequently, lymphocytes and stromal cells within in situ microenvironments were reclassified as in situ lymphocytes and in situ stromal cells.

The Morisita–Horn similarity index is an ecological measure of community structure to quantify the extent of spatial colocalisation or overlap between two spatial variables. We have previously demonstrated its use in studying cancer-immune cell colocalisation in IDCs¹⁹. Here to measure colocalisation of DCIS and lymphocytes, the Morisita index was modified by restricting calculation in Voronoi polygons and estimating the density of epithelial cells and immune cells in the newly defined space. This space was further divided into Voronoi grids, following¹⁹. This is to provide more spatial points for calculation and to prevent a lack of power for samples with a low number of DCIS ducts. The number of immune cells and epithelial cells for each polygon i are denoted as nil and nie, based on Voronoi tessellation. Let pil and pie denote the fraction of immune cells and epithelial cells in polygon i, i.e. pil=nil∑inil,pie=nie∑inie. Morisita–Horn similarity index is henceforth referred to as DCIS immune colocalisation score throughout the manuscript and it is calculated as:

High DCIS immune colocalisation score indicates that TILs colocalise well with DCIS ducts within a sample, that is, spatial homogeneity; whereas, low DCIS immune colocalisation score could indicate TILs to only localise with part of the DCIS, i.e. high spatial variability.