DISCLAIMER.

ABSTRACT FOR SUBMISSION.

 Cell Surface Tessellation: Model for Malignant Growth.
 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].
                   http://www.medparse.com/celltess.htm 
                                       
 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.

NOTES.

2274.

(i) the three line-segments drawn from the centers to the opposing tangency-points (i.e., center-tangent-intersection) meet at a common point,

(ii) the three tangency-lines (i.e., tangent-line-intersection meet at a common point,

and (iii) these two meeting-points are coincident only if the three circles have equal radii. (Only then are they coincident with the center of the largest circle that can be embedded within the triple-spiked region which they form, as in Figure 2.)

2251.

2212.

2213.

2214.

2259.

4. Seen in many drawings by M. C. Escher.

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Cell Surface Tessellation.

9264.

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Cross-section: Picket Fence.

2286.


9929.

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En-face: Honeycomb.

2282.


9930.

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En-face: Malignancy.

2280.


9931.

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Nearly-Circular Cell Apices.

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Cell Surface Tessellation.

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.

2521.

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Balanced/Unbalanced Cell Triples.

2522.


2521.

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Mutually Tangent Circle Theorem.

2214.

Proof of Only-If: Advanced problem.

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Proof: If.

2214.

By Pythagorean Theorem.

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

2528.

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Proof: Only if. Point D.

2525.

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Only If, Part (i). Point D.

2525.

There exists a unique point D at the intersection of center-opposite-tangent lines.

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Proof: Part (i). Point D.

2540.

Thus Ca=cA and d=a.

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Proof: Only if. Point E.

2526.

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Only if, Part (ii). Point E.

2526.

There exists a unique E at the intersection of tangent-lines

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Proof: Part (ii). Point E.

2530.

Paired sets of congruent triangles, i.e., CaE = CbE, AcE = AbE, BaE = BcE.

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Proof: Only if. Point F.

2527.

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Only if, Part (iii). Point F.

2527.

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.

2531.

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.

2538.

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Proof: Part (iv). Points D, E, F.

2538.

Points D, E, F are coincident only for equilateral triangles.

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Points D, E, F are collinear.

2528.

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Summary: Mutually Tangent Circle Theorem.

2521.


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.

2531.

(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.

2531.

8. Mathematical models can be used to propose alternatives to classical hypotheses in pathology.

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Larry Brown has briefly vetted this idea, and he points out that actual cells on a surface are not quite circles, but rather, more-or-less-hexagons. If the Struble tangent-circle theorem only applies to circles, then it is, strictly speaking, irrelevant to cellular growth. I don't regard this as a big issue. It seems to me that you could find the area and center-of-gravity of each nearly-circular cell, and liken this to the circle of the same area and center-of-gravity. You might have to do a little more jostling to make everything fit together on a tiling or tessellation.

One of the amazing things about Struble's tangent-circle construction is that, even for wildly differing circle-sizes, the distance between the center-tangent-intersection and the tangent-line-intersection isn't very great. In fact, one needs to view the obtuse triangle case, to see any appreciable distance between the intersections. See:


2251.
This means that the amount of disorganization/imbalance evoked by malignant cells (if, indeed, this mechanism is appropriate) is not great, but is apparently sufficient for the next (and defining) step of malignancy, namely, tissue-invasion.

Furthermore, essentially all cells, even malignant cells, have a narrow size limit, from perhaps 9 nm minimum (small lymphocytes) to 200 nm maximum (malignant syncytiotrophoblasts, which might not even legitimately qualify as single cells). (Erythrocytes (7 nm) and platelets (3 nm) don't count as full-fledged cells, since they have no nuclei, and cannot reproduce.) Struble's manuscript shows that, even for a 20-fold difference of radii in a cell-triple, the center-tangent-intersection-point and the tangent-line-intersection-point are quite close.

I have used the terms tiling and tessellation somewhat sloppily. According to Mathworld on the internet, a TILING is a plane-filling arrangement of plane figures, or its generalization to higher dimensions. Topologically, a tiling is a collection of disjoint open sets, whose closure covers the plane. Wang's conjecture (1961) states that if a set of tiles tiles the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a TESSELLATION (Latin: tessella=tile).

1. Malignant tumors of surface cells, including tumors of ectodermal and endodermal origin, are the most common human malignancies, accounting for over twenty million new cases of malignant tumor annually worldwide.

2. A normal cell surface may be regarded as a collection of cylinders, packed side-by-side, where each cylinder is about 10µ in diameter and 20µ in height. Typically, the cells may be viewed over the surface (en face) or in cross-section ("picket fence").

Normal surface-cells, cross-section:
9929.


Normal endocervical cells, cross-section:
2286.


3. On a normal cell surface, viewed en face, the cells appear as approximately equi-radius circles, densely packed together in a honeycomb arrangement, over a nearly planar surface.

Normal surface-cells, en-face:
9930.


Normal endocervical surface-cells, en-face:
2282.


4. In malignant surface tumors, cells have a broader distribution of radii, less regular packing, and protrusions from the surface, either papillomatous (above-surface) or acanthomatous (below-surface).

Malignant surface cells, en-face, varying sizes:
9931.


Malignant surface cells, en-face, varying sizes:
2280.


5. We propose a mathematical model for a normal cell-surface tiling, or tessellation, in which contiguous cells sense their relationships to one another, and stabilize/balance their cell-radii, based upon intercellular relative positions of cells.

6. We propose that each cell LOOKS AT its immediate two, neighboring cells, to form a CELL TRIPLE, in a topological relationship. Each cell has a CENTER and a TANGENCY LOCUS with its two, neighboring cells:

2201.


In this illustration, there is a large cell (A, top), a medium cell (B, lower left), and a small cell (C, lower right), all viewed en face.

2202.


The centers of these cells are a (large), b (medium), and c (small), respectively:

2203.


Tangency-points (◅) are labeled TPab between the large and medium circle.......

2206.


Tangency-lines (——) are labeled TLab between the large and medium circle.......

2209.


7. The CELL NUCLEUS is located at the center of each cell, and the nucleus SENSES the TANGENCY LOCUS at its two neighboring cells.

8. Each TANGENCY-LOCUS is, itself, a TANGENCY-POINT; and projects out as a TANGENCY-PROJECTION-LINE. For each cell facing two other cells, there is a center-to-tangency-point line (CENTER-TANGENT-LINE) and a TANGENCY-PROJECTION-LINE.

9. Cells are IN BALANCE when all six lines (i.e., three center-tangent-lines and three tangency-projection-lines) intersect as a common point.

10. This formulation is analogous to the the so-called FRENCH FLAG PROBLEM in cellular biology. That is: if you begin with a single, vertical line of cells (i.e., the flagpole), and the cells divide and proliferate rightward, what chemical signal is sent between the cells that tell a line of cells to be BLUE, WHITE, or RED (in order from left to right, the colors of the French Flag, or tricoleur). Each cell can only send signals to its immediate neighbors, a sort of Markov/topological constraint.

11. Normal cells are in optimal balance, if and only if tangent-line-intersection equals center-tangent-intersection. Then their cell-radii are equal.

12. Struble Tangent Circle Theorem states that tangent-line-intersection equals center-tangent-intersection if and only if the cell-radii are equal.

13. Malignant surface cells are characterized by more size variation and less balanced packing. We suggest that these processes correspond to a loss of look-across-line and tangency-line coordination.

The U. S. National Cancer Institute (USNCI) of the U. S. National Institutes of Health (USNIH) estimates that there are one million new cases of skin cancer, the most common malignancy, annually in the USA:

http://www.cancer.gov/cancertopics/wyntk/skin

The next leading malignancies: prostate, breast, lung, colon, have on the order of 10⁵ new cases annually in the USA. The most common malignancies from these organs far and away/predominantly involve the ectodermal/endodermal embryonic cells. The USA has one-twentieth of the world's population, so, conservatively estimated, there are twenty million new malignancies annually worldwide. [Cancer is predominantly a disease of older persons, so that in areas of the world with short life-expectancies, the incidence of cancer may be less.]

A plane-filling arrangement of plane figures or its generalization to higher dimensions is called a TILING. Topologically, a tiling is a collection of disjoint open sets, the closures of which cover the plane.

Wang's conjecture (1961) states that if a set of tiles tiles the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a TESSELLATION.

See: http://mathworld.wolfram.com/Tessellation.html

French Flag Problem. "The pièce de résistance of the range, however, is a little genetic joke. Embryologists in the past spent a lot of time worrying about something they called the French flag problem. This was how simple chemical gradients in an embryo could result in complex body patterns. Their idealised version of this question was to ask how two signal chemicals, starting from opposite ends of an embryo, could result in a pattern that looked like the French flag (ie, three different coloured vertical stripes). The French flag problem was solved by the discovery of so-called "hox" genes that control embryonic development, and much of the early hox work was done in zebra fish. Dr Fril has used this knowledge to create a FrenchFlag fish, with a blue head, a white body and a red tail, and hopes to follow it soon with one depicting the stars and stripes. As a wag once put it "one man's fish is another man's poisson." See:
http://www.economist.com/diversions/displaystory.cfm?story_id=348179

Prove that if three cevians of a triangle are concurrent, then extensions of the three lines passing through their feet intersect extensions of the edges of the triangle along an external line. Conversely, if extensions of the edges of a triangle intersect at three points of an external line, then there exist concurrent cevians possessing feet which are collinear with those for intersections. (Appropriate versions hold interesections at infinity.) Upon defining a suitable topology for the space of external lines, show that this one-to-one association becomes topological. Show that whenever an interior (concurrent) point tends to a (non-vertex) edge-point, the corresponding exterior line tends to that edge-line. Show that any continuous internal curve leads to an external curve defined by a continuously-turning tangent. Indeed, all the geometry of interior points in a triangle is reflected by the geometry of exterior lines.

This problem is a bit of a project in synthetic geometry, with the show part involving angles and deviations from a fixed interior point (say, centroid), for example.


2549.


The one-to-one correspondence that you described to me last night, between points and topological neighborhoods within a triangle and lines and topological neighborhoods outside the triangle, if provable, would be quite a stunning result. It reminds me of Descartes's one-to-one correspondence between the points of Euclidean geometry and analytical geometry, which nearly doubled the number of theorems in 17th century mathematics, by making every theorem about Euclidean geometry a theorem about analytical geometry, and vice versa. (I say NEARLY doubled, because there must have been at least some common theorems of geometry and algebra known before Descartes.)

Dr. Larry Brown pointed out an associated passage in the New Testament, in which OUR LORD bestows the Keys to the Kingdom upon Saint Peter:

Matthew 16:18-19. And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

Let Euclidean geometry be GOD and analytical geometry be Saint Peter, and bind/loose properties correspond to theorems, and there you have it, the Biblical prophecy of analytical geometry, spoken by Jesus Himself. (Euclidean geometry predates Jesus.) This passage is considered very significant to Roman Catholics. It is engraved, in two-meter-high letters in Latin and Greek, on the western ceiling of the Sistine Chapel in the Vatican, and is believed by Roman Catholics to represent OUR LORD's commission to Saint Peter for the special status of the Pope as the Vicar of Christ.

Dr. Larry Brown does not claim to be the first person to point out this association between Descartes and Mt 16:18-19, but he knows of no prior source. Do you?

Dr. Jules Berman pointed out to me that the world's largest cell is an unfertilized ostrich egg, circa 10 cm. Compare this to bone marrow reticulocytes, circa 10 µ, giving a ratio of 100,000:1 for a pair of tangent cell-circles. The designation of an ostrich-egg as a single cell (just as the demotion of Pluto from planetary status) is scientifically somewhat arbitrary. The ostrich egg consists of a single cell nucleus, about 8 µ, at the periphery of a gigantic vacuole, consisting of egg-yolk and egg-white, or albumen. A vacuole is a bubble of non-living chemical within a cell, such as a fat-vacuole inside a fat-cell, or adipocyte. Furthermore, since an ostrich egg lives in isolation, with no surrounding cells, strictly speaking, it is not covered under our definition of cell tessellation, which requires a minimum of at least three tangent circles. Likewise, blood cells, which live as isolated boats in the fluid blood, are not strictly included under our definition of cell tessellation. Thus our theory makes no claims about hematogenous malignancies, such as leukemia.

http://www.newton.dep.anl.gov/askasci/mole00/mole00128.htm

The cell tessellation paper generated enough interest in Vancouver that I hope we can keep the idea alive for next year's meeting in Pittsburgh. Perhaps these considerations about tangent circles, triangles, and topological mappings, could evolve into further properties about surface cancer-cell relationships. An idea something like: the behavior of certain points in the triple-spike region (or equivalently, the corresponding triangle) influences the growth/cancer properties of the tessellation as-a-whole.

There is a small, obscure literature in this field, dating from the 1960s, which some National Cancer Institute researchers (especiallly Dr. Judith Prewitt) and even Dr. Mary B. Williams (when she was a graduate student at the University of London, in collaboration with the prominent tissue-culture researcher, Prof. Louis Wolpert) dabbled in, which I need to look into further.

Finally, it is interesting to note that the largest human cells include the human ovum and the syncytiotrophoblast, a cell formed by the embryo in a pregnant woman, whose job it is to dig into the lining of the uterus and seize a blood supply from the mother. More than one researcher has noted the similarity between pregnancy and invasive malignancy. In pregnancy, the embryo increases maternal heart output by 40%, and in the olden days of rheumatic heart disease before penicillin, would cause heart failure and sometimes death of the mother. Tumors may do the same sort of thing. The difference is: after nine months, the pregnancy is over.

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Wolpert L.
The French flag problem........ See:
http://www.economist.com/diversions/displaystory.cfm?story_id=348179

Ostrich egg has one cell:
http://www.newton.dep.anl.gov/askasci/mole00/mole00128.htm

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Last updated: 8/28/2006, by G. William Moore, MD, PhD.