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Cell Surface Tessellation: Model for Malignant Growth G. William Moore, MD, PhD, Raimond A. Struble, PhD, Lawrence A. Brown, MD, Grace F. Kao, MD, Grover M. Hutchins, MD. Departments of Pathology, Veterans Affairs Maryland Health Care System, University of Maryland Medical System, The Johns Hopkins Medical Institutions, Baltimore MD; Department of Mathematics, North Carolina State University, Raleigh, NC; and Department of Dermatology, George Washington University School of Medicine, Washington, DC.

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Cell Surface Tessellation: Abstract 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.

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Cuboidal or Columnar Epithelial Tumors 1. Common human malignancies. 2. Include: epithelial, mesothelial, endothelial tumors, in skin and mucus membrane. 3. Account for over twenty million new cases annually worldwide.

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Cuboidal or Columnar Epithelium 1. Benign: Cell surface with regular, repeated cell packing. Collection of equal cylinders resting side-by-side. 2. Malignant: Variably-sized cells, variable nuclear ploidy, disorganized surface, tendency to invade surrounding tissues.

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Mathematical Tessellation 1. Tiling: plane-filling arrangement of plane figures, or generalization to higher dimensions. 2. Mathematically: tiling is a collection of disjoint open sets, the closures of which cover the plane. 3. Tessellation: periodic tiling of the plane by polygons, or space by polyhedra. 4. Seen in many drawings by M. C. Escher.

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Tessellation

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Cross-section: Picket Fence

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En-face: Honeycomb

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En-face: Malignancy

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Nearly-Circular Cell Apices

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Cell Surface Tessellation Nearly-circular cell-apices. Each cell-pair has a unique TANGENT-LINE passing through a unique tangent-point. Each cell-triple has a unique CENTER- OPPOSITE-LINE drawn from center of cell to the opposite tangent-point. Cell-triple is BALANCED if and only if these six lines meet at a single intersection point.

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Tangent-line. Center-opposite-line.

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Balanced/Unbalanced Cell Triples

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Mutually Tangent Circle Theorem Tangent-lines and Center- opposite-lines intersect at a common point if and only if all three cell-radii are equal. Proof of If: High-school geometry. Circles, radius=1; all six points lie at coordinates: (0, 1/√3). Proof of Only-If: Advanced problem.

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Proof: If. For equal circles, radius=1: base = 2, edge = 2, height = √3, height-at-intersection = 1/√3, By Pythagorean Theorem.

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Proof: Only If. Construct points D, E, F.

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Proof: Only if. Point D.

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Only If, Part (i). Point D. There exists a unique point D at the intersection of center- opposite-tangent lines.

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Proof: Part (i). Point D. Ceva’s Theorem (1678): Products of alternating lengths on a triangle are equal, i.e., (Ab)(Ca)(Bc) = (aB)(cA)(bC). By construction, Ab=cA and Bc=aB. Thus Ca=cA and d=a.

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Proof: Only if. Point E.

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Only if, Part (ii). Point E. There exists a unique E at the intersection of tangent-lines

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Proof: Part (ii). Point E. Paired sets of congruent triangles, i.e., CaE = CbE, AcE = AbE, BaE = BcE.

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Proof: Only if. Point F.

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Only if, Part (iii). Point F. There exists a unique point F and internal circle radius r such that center-to-F minus r for an external circle equals the radius of the external circle.

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Proof: Part (iii). Point F. Form the maximal internal circle, tangent to the three external circles. Points A, B, and C pass through the center of the internal circle, F.

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Proof: Only If. Points D, E, F.

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Proof: Part (iv). Points D, E, F. Points D, E, F are coincident only for equilateral triangles.

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Points D, E, F are collinear.

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Summary: Mutually Tangent Circle Theorem Tangent-lines and center-opposite-lines intersect at a common point if and only if all three cell-radii are equal.

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Struble Triangle Theorem (i). There exists a unique interior point D, for which the three line segments emanating from the vertices and passing through D, intersect the edges of the triangle at three opposing points, a, b and c, satisfying length equalities Ab=Ac, Ba=Bc and Ca=Cb. (ii). There exists a unique interior point E, for which three line segments emanating from E to the points a, b and c are perpendicular to the edges of the triangle.

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Struble Triangle Theorem (iii). There exists a unique interior point F and positive number r, for which three line segments emanating from the vertices to F have lengths, when shortened by r, given by Ab, Bc and Ca. (iv). The interior points D, E and F are coincident only for equilateral triangles.

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Benign Cells have Equal Radii Benign cells have essentially equal radii. Premalignant and malignant cells do not have equal radii. Line-intersection property disappears in malignant degeneration. Common-intersection and equal-radii properties are mathematically equivalent.

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Mathematical Theories Can be used as alternatives to conventional models in pathology. Conventional model of cancer: invasion after tumor cells break through basement membrane. Alternative model of cancer: tumor proliferation as a property of cells, attempting to balance with neighboring cells.

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Possible Implications for Therapy Common-intersection and equal-radii properties equivalent. Processes are mathematically equivalent. Control one process, then you can control the other.

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Summary 1. Malignant transformation of cuboidal or columnar epithelium: variably-sized cells, variable nuclear ploidy, disorganized surface, tendency to invade surrounding tissue. 2. Cell surface: tessellation of nearly-circular cell-apices. 3. Cell-pair has tangent-line passing through tangent-point. 4. Cell-triple has line-segment from cell-center to opposite tangent-point.

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Summary 5. Cell-triple radii are equal if and only if six lines meet at one point. 6. Cell disorientation and radius-equality are geometric features of same process. 7. Therapy for one process might possibly control the other process. 8. Mathematical models can be used to propose alternatives to classical hypotheses in pathology.

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