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.