Cell Surface Tessellation: Model for Malignant Growth

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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, radius1 all six points lie at
  coordinates (0, 1/v3).
• Proof of Only-If Advanced problem.

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Proof If.

• For equal circles, radius1
• base 2, edge 2,
  height v3,
  height-at-intersection 1/v3,
• 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.

• Cevas Theorem (1678) Products of alternating
  lengths on a triangle are equal, i.e.,
  (Ab)(Ca)(Bc) (aB)(cA)(bC).
• By construction, AbcA and BcaB.
• Thus CacA and da.

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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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SummaryMutually 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 AbAc,
  BaBc and CaCb.
• (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.