Discrete Mereotopology for Spatial Reasoning in Automated Histological Image Analysis.

Discrete mereotopology (DM) is a first-order spatial logic that fuses together mereology (the theory of parthood relations) and topology to model discrete space. We show how a set of quasitopological functions defined within DM can be mapped to specific operators defined in mathematical morphology (MM) and easily implemented in scientific image processing programs. These functions provide the means to model topological properties of individual regions and spatial relations between them such as contact, overlap, and the relation of part to whole. DM not only extends the expressive power of image processing applications where mathematical morphology is used, but by functioning as a logic it also supplies the formal basis with which to prove the correctness of implemented algorithms as well as providing the computational basis to mechanically reason about segmented digital images using automated reasoning programs. In particular, we show how DM can supply a model-based and algorithmic context to the otherwise blind pixel-based image processing routines still dominating conventional imaging approaches. A number of worked examples drawn from the histological domain are given, including segmentation of cells in culture, identifying basal cell layers from stratified epithelia sections, and cell sorting in blood smears.

Morphologic characterization of cell neighborhoods in neoplastic and preneoplastic epithelium.

OBJECTIVE: To explore tissue organization based on the geometry of cell neighborhoods in histologic preparations. STUDY DESIGN: Local complexity of solid tissues was measured in images of discrete tissue compartments. Exclusive areas associated with cell nuclei (v-cells) were computed using a watershed transform of the nuclear staining intensity. Mathematical morphology was used to define neighborhood membership, distances and identify complete nested neighborhoods. Neighborhood complexity was estimated as the scaling of the number of neighbors relative to reference v-cells. RESULTS: The methodology applied to hematoxylin-eosin-stained sections from normal, dysplastic and neoplastic oral epithelium revealed that the scaling exponent, over a finite range of neighborhood levels, is nonunique and fractional. While scaling values overlapped across classes, the average was marginally higher in neoplastic than in dysplastic and normal epithelia. The best classificatory power of the exponent was 58% correct classification into 3 diagnostic classes (11 levels) and 83% between dysplastic and neoplastic classes (13 levels). CONCLUSION: V-cell architecture retains features of the original tissue classes and demonstrates an increase in tissue disorder in neoplasia. This methodology seems suitable for extracting information from tissues where identification of cell boundaries (and therefore segmentation into individual cells) is unfeasible.