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This paper describes two three-dimensional (3D) analytical techniques based on 3D mathematical morphology that have been found useful in quantifying the 3D spatial distribution of S-phase cells in a tubular tumor of the human breast. One technique is based on determining the normalized radial distribution of the S-phase cells with respect to the central axis of the tumor. The other technique is a novel extension of the polyhedra of Voronoi to quantify the distribution. The Voronoi polyhedron of a given S-phase cell nucleus is that polyhedron of minimal volume defined by planes all of which are perpendicular bisectors of the vectors extending from the given cell to all other S-phase cells in the tumor. Methods are demonstrated for generating these polyhedra and for histogramming their volumes. An illustration is given of using the histogram to sort the S-phase cells according to their 3D positional relationships. Displays showing the sorted cells in 3D and their associated Voronoi polyhedra are provided.

Ongoing research on the differentiation between small-cell and large-cell follicular lymphomas ascertained that image analysis by three-dimensional morphologic transforms permits subtyping of follicular lymphomas with greater accuracy and reproducibility than can be accomplished visually by expert pathologists. The present studies improved the use and interpretation of the minimum tracking algorithm and identical to-filters (binary ranking filters). These filters operate on the voxels that constitute the three-dimensional space encodings of gray-level images and are implemented by the Triakis software system.

Non-Hodgkin's lymphomas with a follicular pattern are subdivided on the basis of morphologic criteria into four subtypes. Each subtype has a distinctive natural history, response to therapy and survival, and precise histologic diagnosis is essential for optimal treatment. However, recent studies by a panel of seven expert hematopathologists showed that there was a 20% to 40% likelihood of disagreement in subtyping non-Hodgkin's follicular lymphomas; thus, the likelihood of a patient receiving inappropriate treatment is high. To resolve some of the problems relating to subjective morphologic diagnosis, we have continued to investigate the possibility of using computerized image analysis to perform automated subtyping of follicular lymphomas. In the portion of the study reported herein, a total of 37 cases were selected from the set subtyped by the panel of expert hematopathologists, and digitized images of slides from each case were processed by computer. The computer-generated subtypes were compared with those arrived at by the panel. For the cases analyzed, the computer subtyping was at least as good as, and in some instances superior to, that of most of the expert panelists.