2005 Scientific Session Abstracts
Infinite Papilloma: Model for Unbounded Tumor Growth
http://www.medparse.com/infnpapl.htm
G. William Moore, MD, PhD (George.Moore4@med.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
Context: Surface tumors in skin and mucus membrane are the most common human tumors, and typically arise as an upward, exophytic growth, or papilloma; or as a downward, endophytic growth, or acanthoma.
In benign growth, such as wound-healing, proliferation stops after the injured surface tissue has been replaced. In malignant growth, proliferation continues indefinitely.
Technology: This report proposes a mathematical model, using infinite products and Lebesgue integration, in which papillae/acanthi seek to fill a potential volume, above/below the normal tissue surface.
Design: For a potential tumor volume normalized to 1, one considers the n-product, Pn=(1-r1)x(1-r2)x...x(1-rn), where each ri represents the fraction of remaining volume removed by the ith papilla/acanthus. The infinite product, P, is the limit of the n-products as n approaches infinity.
Results: Mathematically, an infinite number of papillae fill the potential volume if and only if the infinite sum, S=r1+r2+... is DIVERGENT, an apparently paradoxical mathematical result. It is proposed that a convergent series corresponds to benign proliferation; whereas a divergent series corresponds to malignancy. Thus, a malignancy keeps trying to fill the potential volume; whereas benign proliferation is satisfied and ceases after the volume is partially filled.
Conclusion: The theory is completely general, because the exact functional form or values of the RI are not specified by the model. Mathematical models can be used to propose alternatives to conventional wisdom in pathology, and explore general properties.