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.
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 linesegment 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.
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.
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.
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.
Tessellation
Cross-section: Picket Fence
En-face: Honeycomb
En-face: Malignancy
Nearly-Circular Cell Apices
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 CENTEROPPOSITE-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.
Tangent-line.
Center-opposite-line.
Balanced/Unbalanced Cell Triples
Mutually Tangent Circle Theorem
Tangent-lines and Centeropposite-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.
Proof: If.
For equal circles, radius=1:
base = 2,
edge = 2,
height = √3,
height-at-intersection =
1/√3,
By Pythagorean Theorem.
Proof: Only If.
Construct points D, E, F.
Proof: Only if.
Point D.
Only If, Part (i).
Point D.
• There exists a unique
point D at the
intersection of centeropposite-tangent
lines.
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.
Proof: Only if.
Point E.
Only if, Part (ii).
Point E.
• There exists a unique
E at the intersection
of tangent-lines
Proof: Part (ii).
Point E.
• Paired sets of
congruent triangles,
i.e., CaE = CbE,
AcE = AbE,
BaE = BcE.
Proof: Only if.
Point F.
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.
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.
Proof: Only If.
Points D, E, F.
Proof: Part (iv).
Points D, E, F.
• Points D, E, F are
coincident only for
equilateral triangles.
Points D, E, F are collinear.
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.
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.
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.
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.
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.
Possible Implications for Therapy
• Common-intersection and equal-radii
properties equivalent.
• Processes are mathematically equivalent.
• Control one process, then you can control
the other.
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.
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.