Cell Surface Tessellations in Malignant Growth: Morley's Theorem.
Grace F. Kao ; Pathology and Laboratory Medicine Service, VA MD Health Care System; Department of Dermatology, Univ. of MD Medical System; Dept. of Dermatology, George Washington Univ. School of Medicine; Grover M. Hutchins ; Department of Pathology, The Johns Hopkins Medical Institutions; Lawrence A. Brown ; Pathology and Laboratory Medicine Service, Veterans Affairs Maryland Health Care System; Department of Pathology, University of Maryland Medical System; Raimond A. Struble ; Department of Mathematics, North Carolina State University; G. William Moore ; Baltimore Veterans Affairs Medical Center;
Content:
Tumors of surface epithelium are among the most common human malignancies. In benign surface epithelium, the cell surface exhibits a regular, repeated packing of cells, or tessellation, resembling a collection of equal cylinders resting side-by-side. Malignant transformation involves variably-sized cells, a disorganized surface, and the tendency to invade surrounding tissues.
Technology:
Ordinary and synthetic geometry.
Design:
Mathematically, a tessellation is a periodic tiling of the plane by polygons, or space by polyhedra. An unbroken sheet of mucosal or epidermal cells, viewed en face, can be approximated as a collection of tangent circles. Any triple of tangent circles on a tessellation forms a triangle with vertices at the circle-centers. Morley's Theorem states that every tangent cell-triple has a unique internal Morley triangle, formed by trisecting the angles of primary triangle vertices. This Morley triangle may serve as a communication hub for cell-to-cell interaction.
Results:
It is demonstrated that the Morley triangle is maximal in an equilateral primary triangle.
Conclusion:
Malignant surface cells are characterized by more size variation and less balanced packing. In this model, unequal cell size and decreased Morley triangle ratio are geometric features of the same underlying process. Therapy for smaller Morley triangle ratio might possibly control the malignancy process. Mathematical models can be used to explore alternatives to classical hypotheses in pathology, and explore general paradigms.
