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Cell Surface Tessellation: Model for Malignant Growth

http://www.medparse.com/celltess.htm

G. William Moore, MD, PhD (George.Moore4@va.gov) [1,2,3];  Raimond A. Struble, PhD [4]; Lawrence A. Brown, MD [1,2];  Grace F. Kao, MD [1,5]; Grover M. Hutchins, MD [3].  Pathology and Laboratory Medicine Service, Veterans Affairs  Maryland Health Care System, Baltimore, MD [1];  Department of Pathology, University of Maryland Medical System,  Baltimore, MD [2]; Department of Pathology, The Johns Hopkins  Medical Institutions, Baltimore, MD [3]; Department of Mathematics,  North Carolina State University, Raleigh, NC [4]; and Department  of Dermatology, George Washington University School of Medicine,  Washington, DC [5]     
                                       
Context: Tumors of cuboidal or columnar epithelium are among the most  common human malignancies.  In benign cuboidal or columnar epithelium,  the cell surface exhibits a regular, repeated packing of cells,  resembling a collection of equal cylinders resting side-by-side.  Malignant transformation involves the apparently independent features of  variably-sized cells, variable nuclear ploidy, a disorganized surface,
 and tendency to invade surrounding tissues.
                               
Technology:  Mathematically, a TILING is a plane-filling arrangement of  plane figures, or its generalization to higher dimensions; a TESSELLATION  is a periodic tiling of the plane by polygons, or space by polyhedra.
                                      
Design: The cell surface is a tessellation of nearly-circular cell-apices.   Each cell-pair has a unique tangent-line passing through a unique  tangent-point; and each cell-triple has a unique line-segment drawn from the center of one cell to the opposite tangent-point.  A cell-triple is BALANCED if and only if these six lines meet at a single intersection point.
                    
Results: It is demonstrated that a cell-triple is balanced  if and only if all three cell-radii are equal.
                                               
Conclusion: Malignant surface cells are characterized by more size variation and less balanced packing. In this model, unequal cell size and cell disorientation are geometric features of the same underlying process.  Therapy for one process might possibly control the other process. Mathematical models can be used to propose alternatives to classical hypotheses in pathology, and explore general paradigms.