A Mathematical Model of Competition for Nutrients between Malignant and Secondary Tumors

Metastasis is a primary reason for cancer treatment failure. While a great deal of experimental work has been directed toward metastasis, much of the basic theory remains obscure. In particular, the question of how tumor cells survive to initiate secondary growth at a site distant from their origin despite competition for nutrients from cells in the primary tumor is still open. This thesis addresses this question using mathematical modeling. I developed a system of four ordinary differential equations that expresses the competition between a primary tumor and one of its metastatic offspring. In particular, the model addresses how a secondary tumor can survive with the primary tumor in an environment where nutrients are limited. The masses of both a primary and secondary tumor are modeled along with the dynamics of two nutrients: carbon and nitrogen. The parameters of the model are estimated from biological data whenever possible. The model predicts that one characteristic of importance with regards to tumor growth and competition is a given tumor’s optimal carbon to nitrogen (C:N) nutrient ratio. In general, the outcome of competition between a primary and secondary tumor can be predicted based on which tumor has a higher optimal C:N ratio, meaning the tumor can grow with less nitrogen per unit carbon. Comparisons of the results of the model are made to results of other models that consider nutrient dynamics, and experiments are suggested which could test the model’s predictions.