An approach based on two‐dimensional graph theory for structural cluster detection and its histopathological application

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Abstract
An approach based on graph theory is described for detecting clusters of cells in tissue specimens (two-dimensional space). With a set of discrete basic elements (cell nuclei) having several measurable features (area, surface, main and minor axis of best-fitting ellipses) a graph is defined as having attributes associated with edges. Different minimum spanning trees (MSTs) can be constructed using different weight functions on the attributes (attributed MST). Analysis of the MST and of an attributed MST by use of a decomposition function allows detection of image areas with similar local properties. These clusters, which are then clusters of the tree, describe, for example, partial growth in different directions in a case of a human fibrosarcoma assuming that tumour cell nuclei are homogeneous with respect to their configuration and size. The model allows the separation of clusters of tumour cells growing in different directions and the approximation of the different growth angles. This decomposition also allows us to create new (higher) orders of structure (cluster tree).