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Analytical Cellular Pathology
Volume 21 (2000), Issue 2, Pages 59-69

Caveats: Numerical Requirements in Graph Theory Based Quantitation of Tissue Architecture

J. Sudbø,1 R. Marcelpoil,2 and A. Reith1

1Section for Digital Pathology, The Norwegian Radium Hospital, Oslo, Norway
2University of Grenoble, Université Joseph Fourier, Grenoble, France

Accepted 30 October 2000

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Graph theory based methods represent one approach to an objective and reproducible structural analysis of tissue architecture. By these methods, neighborhood relations between a number of objects (e.g., cells) are explored and inherent to these methods are therefore certain requirements as to the number of objects to be included in the analysis. However, the question of how many objects are required to achieve reproducible values in repeated computations of proposed structural features, has previously not been adressed specifically.

After digitising HE stained slides and storing them as grey level images, cell nuclei were segmented and their geometrical centre of gravity were computed, serving as the basis for construction of the Voronoi diagram (VD) and its subgraphs. Variations in repeated computations of structural features derived from these graphs were related to the number of cell nuclei included in the analysis.

We demonstrate a large variation in the values of the structural features from one computation to another in one and the same section when only a limited number of cells (100–500) are included in the analysis. This variation decreased with increasing number of cells analyzed. The exact number of cells required to achieve reproducible values differ significantly between tissues, but not between separate cases of similar lesions. There are no significant differences between normal and malignantly changed tissues in oral mucosa with respect to how many cells must be included.

For graph theory based analysis of tissue architecture, care must be taken to include an adequate number of objects; for some of the structural features we have tested, more than 3000 cells.