Infinite Papilloma: Model for Unbounded Tumor Growth

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Infinite Papilloma Model for Unbounded Tumor
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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Infinite Papilloma Abstract

• Context Surface tumors in skin and mucus
  membrane most common human tumors. Tumors
  typically arise as exophytic growth, or
  papilloma or as endophytic growth, or acanthoma.
  In benign growth, proliferation stops after
  injured tissue is replaced. Malignant growth,
  proliferation continues indefinitely.
• Technology Mathematical model infinite products
  and Lebesgue integration. Papillae/acanthi seek
  to fill a potential volume.
• Design For potential tumor volume normalized to
  1, one considers the n-product, Pn(1-r1)x(1-r2)x
  ...x(1-rn), where each rn represents the fraction
  of remaining volume removed by the nth papilla.
  P limit of the n-products as n approaches
  infinity.
• Results Infinite number of papillae fill the
  volume if and only if the infinite sum,
  Sr1r2... Is divergent. Convergent series
  corresponds to benign proliferation. Divergent
  series corresponds to malignancy.
• Conclusion Theory is completely general. Exact
  form or values of rn not specified by model.
  Mathematical models can suggest alternatives to
  conventional models in pathology.

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Surface Tumors

• 1. Most common human tumors.
• 2. Include epithelial, mesothelial, endothelial
  tumors, in skin and mucus membrane,
• 3. Account for over twenty million new cases per
  year worldwide.

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Surface Tumors arise.

• 1. As exophytic growth (papilloma),
• 2. As endophytic growth (acanthoma).
• 3. In benign growth, e.g., wound-healing, growth
  stops after injured tissue is replaced.
• 4. In malignant growth, growth continues
  indefinitely.

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Papilloma/acanthoma
Endophytic growth acanthoma
Exophytic growth papilloma
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Mathematical model

• 1. Papillae (or acanthi) seek to fill potential
  volume BILLBOARD, above/below tissue surface,
  possibly infinite.
• 2. Height/depth of tumor is the TOWER.
  
• 3. Proposal there is TERMINAL VOLUME, at which
  tumor-cells sense that they should no longer
  continue to divide.

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Mathematical model

• 1. Normalize terminal tumor volume to 1.
• 2. At the nth step in papilloma-growth, let the
  nth papilla occupy rn of the volume remaining.
• 3. Unfilled terminal volume remaining after step
  n is Pn (1-r1)x(1-r2)x ... x(1-rn).
• 4. Final unfilled volume, P, is Pn as n
  approaches infinity.

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Model r1 r2 r3 ½
P1 ½ 0.5
P2 ½ x ½ ¼ 0.25
P4 ½ x ½ x ½ x ½ 1/16 0.06
P3 ½ x ½ x ½ 1/8 0.13
P0.
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Model r1 r2 r3 ¼
P1 ¾ 0.75
P2 ¾ x ¾ 9/16 0.56
P3 ¾ x ¾ x ¾ 27/64 0.42
P4 ¾ x ¾ x ¾ x ¾ 81/256 0.32
P0.
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Model r11/2, r21/4, r31/8,
P1 ½ 0.5
P2 ½ x ¾ 3/8 0.38
P3 ½ x ¾ x 7/8 21/64 0.33
P4½ x ¾ x 7/8 x 15/16 315/10240.31
P 0.28878811.
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Struble Filling Theorem

• Terminal volume FILLS if and only if
  corresponding infinite series,
  S r1 r2 r3 ... DIVERGES.
• 2. Example HARMONIC SERIES, namely 1/2 1/3
  1/4 ..., DIVERGES, i.e., 1/2 1/3 1/4
  ... infinity. P0.
• 3. Example GEOMETRIC SERIES, namely 1/2 1/4
  1/8 ..., CONVERGES, i.e., 1/2 1/4 1/8 ...
  lt infinity. P0.289gt0.

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Infinite Products and Calculus

• 1. Differential calculus examines the slope of a
  particular curve, or mathematical function, at a
  given point.
• 2. Integral calculus determines the area/volume
  under a particular function.
• 3. The traditional method for determining
  integrals is Riemann integration.

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Riemann Integration
In Riemann integration, one calculates the sum of
vertical bars under the curve.
Riemann integration fails for extremely irregular
functions, as shown above.
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Lebesgue Integration

• 1. In the early twentieth century, French
  mathematician Henri Lebesgue, proposed a system
  for adding up HORIZONTAL SLABS.
• 2. Lebesgue integration handles certain
  statistical and probability distributions, as
  well as image analysis and Fourier series.
• 3. Mikusinski proposed an integration model
  involving rectangular bricks.
• 4. Struble proposed an integration model
  involving arbitrary areas.
• 5. Papillae or acanthi filling a terminal volume
  is mathematically equivalent to Lebesgue
  integration.

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Divergence malignancy

• Proposal CONVERGENT SERIES corresponds to BENIGN
  GROWTH.
• 2. DIVERGENT SERIES corresponds to MALIGNANCY.
• 3. Malignancy keeps trying to fill the potential
  volume forever.
• 4. Benign proliferation ceases after the volume
  is partially filled.

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Geometric vs Harmonic Papilloma
Geometric/pedunculated
Harmonic/sessile
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Infinite Papilloma
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Mathematical Theories.

• Can be used as alternatives to conventional
  models in pathology.
• 2. Conventional model of cancer invasion after
  surface-tumor breaks through the basement
  membrane.
• 3. Alternative model of cancer tumor
  proliferation as a property of cells, attempting
  to fill a potential volume.

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Implications for Diagnosis and Therapy.

• 1. Is there a geometric property of malignant
  tumors that could be used in diagnosis?
• 2. Theoretically, a space-filling infinite
  product can be converted into a non-space-filling
  infinite product, merely by expanding the
  terminal tumor volume.
• 3. Is there some way by which one could trick the
  tumor cells into sensing that the terminal tumor
  volume has expanded?
• 4. This trick might convert a malignant tumor
  into a benign tumor.

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Summary.

• Surface tumors in skin and mucus membrane arise
  as a papilloma or acanthoma. In malignant growth,
  proliferation continues indefinitely.
• 2. For potential tumor volume normalized to 1,
  one considers the n-product, Pn(1-r1)x(1-r2)x
  ...x(1-rn), where each rn represents the fraction
  of remaining volume removed by the nth papilla.
• 3. Infinite number of papillae fills the volume
  if and only if the infinite sum, Sr1r2... is
  divergent.
• 4. Convergent series corresponds to benign
  proliferation. Divergent series corresponds to
  malignancy.
• 5. Theory is completely general. Mathematical
  models can suggest alternatives to conventional
  models in pathology.