This Article Full Text Full Text (PDF) CME: Take the course for this article: Breast Cancer: New Treatments and Improved Therapies Supplemental DVD eLetters: Submit a response to this article Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services E-mail this article link to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Reprints/Permissions Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Norton, L. Search for Related Content PubMed PubMed Citation Articles by Norton, L.

Conceptual and Practical Implications of Breast Tissue Geometry: Toward a More Effective, Less Toxic Therapy

Memorial Sloan-Kettering Cancer Center, New York, New York, USA

Correspondence: Correspondence: Larry Norton, M.D., Memorial Sloan-Kettering Cancer Center, 1275 York Avenue, New York, New York 10021-6007, USA. Telephone: 212-639-5319; Fax: 212-717-3743; e-mail: nortonl{at}mskcc.org

Mathematics provides greater understanding of the complex process of tumorigenesis.

Based on the Gompertzian phenomenon and the Norton-Simon hypothesis, enhanced cell kill can be obtained through a greater chemotherapy dose rate. Results from the 1995 Bonadonna et al. study and the CALGB/Intergroup C9741 study demonstrated that patients in the dose-dense arms had significantly longer disease-free survival and overall survival.

Because of the demonstrated applicability of Gompertzian kinetics, attention has been turned to the etiology of the Gompertzian curve. Breast tumor dimensions, as with all tissue dimensions in biology, can be calculated by fractals. A less cell-dense tissue usually has a lower fractal dimension than a tissue with more cells (i.e., a higher cell density is usually due to a higher fractal dimension). Density is the number of cells divided by the tissue volume. When allowed to grow, the density of a tissue with a lower fractal dimension drops quickly. However, a tumor, since it has a higher fractal mass dimension, maintains a high density as it grows bigger, resulting in a more rapid growth rate and a larger final size. Fractal dimensions of infiltrating ductal adenocarcinomas of the breast are high (i.e., 2.98), which results in a very dense tissue compared with normal breast tissue (with a fractal dimension of about 2.25). As expected, the higher fractal dimension results in a high rate of growth.

The reason for this high fractal dimension is that breast cancer can be considered as a conglomerate of many small Gompertzian tumors, each of which has a high cell density and hence ratio of mitosis to apoptosis. In mathematical terms, each component of the conglomerate can be considered a small metastasis in itself. Thus, the primary tumor is composed of multiple self-metastases that form around a seed from the tumor to itself. Conventional thinking is that cancers metastasize because they are large, but in fact it may be that they are large because they are self-metastatic.

Many genes are associated with the biology of metastasis; these include: A) obligatory cancer genes (most of which regulate mitosis and mitotic rate); B) genes relating to self-metastasis and growth of tumors at local sites, conferring the ability to invade and grow with high cell density; and C) genes that relate to the ability of the cancer to metastasize to distant areas. Additionally, fibroblasts may send out abnormal growth signals causing abnormal breast tissue growth. Consequently, we are not only dealing with abnormal cancer cells, but also with the tissue that surrounds them, or the microenvironment, that is, the "Smith-Bissell" model.

These new insights may lead us to change the thrust of our attack from genes involved in mitosis to those involved in metastasis, including metastasis to self, and to use and further improve dose-dense regimens.

Key Words. Breast cancer • Growth kinetics • Geometry • Mathematics • Gompertzian • Dose dense • Self-metastasis • Self-seeding • Mitosis • Apoptosis

This article has been cited by other articles:

				R. Rao, L. Feng, H. M. Kuerer, S. E. Singletary, I. Bedrosian, K. K. Hunt, M. I. Ross, G. N. Hortobagyi, B. W. Feig, F. C. Ames, et al. Timing of Surgical Intervention for the Intact Primary in Stage IV Breast Cancer Patients Ann. Surg. Oncol., June 1, 2008; 15(6): 1696 - 1702. [Abstract] [Full Text] [PDF]

				C. J. Twelves Defining the Optimal Schedule of Drug Administration: Art or Science? J. Clin. Oncol., April 10, 2008; 26(11): 1781 - 1782. [Full Text] [PDF]

				T. A. Traina, M. Theodoulou, K. Feigin, S. Patil, K. L. Tan, C. Edwards, U. Dugan, L. Norton, and C. Hudis Phase I Study of a Novel Capecitabine Schedule Based on the Norton-Simon Mathematical Model in Patients With Metastatic Breast Cancer J. Clin. Oncol., April 10, 2008; 26(11): 1797 - 1802. [Abstract] [Full Text] [PDF]

				L.-P. P. Tran and J. L. Grabinski Chemotherapy for Early-Stage Breast Cancer: A Paradigm in Flux Journal of Pharmacy Practice, February 1, 2008; 21(1): 46 - 56. [Abstract] [PDF]

				D. Dingli and F. Michor Successful Therapy Must Eradicate Cancer Stem Cells Stem Cells, December 1, 2006; 24(12): 2603 - 2610. [Abstract] [Full Text] [PDF]

				S. M. Tolaney, A. H. Partridge, R. G. Sheib, H. J. Burstein, and E. P. Winer Pneumocystis Carinii Pneumonia During Dose-Dense Chemotherapy for Breast Cancer J. Clin. Oncol., November 20, 2006; 24(33): 5330 - 5331. [Full Text] [PDF]

				P. S. Yan, C. Venkataramu, A. Ibrahim, J. C. Liu, R. Z. Shen, N. M. Diaz, B. Centeno, F. Weber, Y.-W. Leu, C. L. Shapiro, et al. Mapping Geographic Zones of Cancer Risk with Epigenetic Biomarkers in Normal Breast Tissue. Clin. Cancer Res., November 15, 2006; 12(22): 6626 - 6636. [Abstract] [Full Text] [PDF]

				D. P. Ryan Nonsurgical Approaches to Colorectal Cancer Oncologist, October 1, 2006; 11(9): 999 - 1002. [Abstract] [Full Text] [PDF]

				B. A. Chabner and M. J. Murphy Jr. Breast Cancer: A Tale of Two Centuries: With Implications for Understanding Cancer Metastasis and Cancer Stem Cell Biology Oncologist, June 1, 2005; 10(6): 369 - 369. [Full Text] [PDF]