Model for breast cancer diversity and spatial heterogeneity

We present and analyze a growth model of an avascular tumor that considers the basic biological principles of proliferation, motility, death, and genetic mutations of the cell. From an analysis of genomic data and considering the results of a regulatory network analysis we identify two sets of genes---a set of sixteen and six genes---that are believed to play an important role in the evolution of breast cancer. Considering that cancer cells shape the tissue microenvironment and niches to their competitive advantage, the model assumes that cancer and normal cells compete for essential nutrients and that the rate of mutations is affected by nutrients availability. To this end, we propose a coupling between the transport of nutrients and gene mutations dynamics. Gene mutation dynamics are modeled as a Yule-Furry Markovian process, while transport of nutrients is described with a system of reaction-diffusion equations. For each representative tumor we calculate its diversity, represented by the Shannon index, and its spatial heterogeneity, measured by its fractal dimension. These quantities are important in the clinical diagnosis of tumor malignancy. A tumor malignancy diagram, obtained by plotting diversity versus fractal dimension, is calculated for different values of a parameter $\ensuremath{\beta}$, that modulate proliferation rate. It is found that, when $\ensuremath{\beta}l1$, tumors show greater diversity and more spatial heterogeneity as compared with $\ensuremath{\beta}g1$. More importantly, it is found that the results and conclusions are similar when we use the six-gene set versus the sixteen-gene set.