Cellular Sociology: Parametrization of Spatial Relationships Based on Voronoi Diagram and Ulam Trees

At all levels (molecules, cells, organisms, ...), biological systems are made of structural and functional units. Those units are interacting with each other and, tend to use space in an optimal way with respect to their specific function and environmental constraints. Hence, cells can be defined as being the smallest structural and functional units capable of auto-reproduction. The rapid technological advances during the 20th century have made possible to describe the inner architecture of cells at different levels of organization. Our approach is an attempt to study cellular populations at the “sociological” level, i.e. at the level of their spatial organization and interrelationship in a given tissue. The topographies of cellular population have to be considered linked to morphogenesis, structural stabilities and functional state of a given tissue. This approach makes use of the relations that links form to disorder and is based on space partition constructed from a set of points defining the position of cells. This spatial partition consisting of a set of individual forms (Voronoi paving) associated to its dual (Delaunay triangulation) permits the calculation of Ulam trees which are descriptors of the local surrounding of a given cell and the search for characteristic features of the normal and pathological state.