Biological stoichiometry of tumor dynamics: Mathematical models and analysis

Many lines of evidence lead to the conclusion that ribosomes, and therefore phosphorus, are potentially important commodities in cancer cells. Also, the population of cancer cells within a given tumor tends to be highly genetically and physiologically varied. Our objective here is to integrate these elements, namely natural selection driven by competition for resources, especially phosphorus, into mathematical models consisting of delay differential equations. These models track mass of healthy cells within a host organ, mass of parenchyma (cancer) cells of various types and the number of blood vessels within the tumor. In some of these models, we allow possible mechanisms that may reduce tumor phosphorous uptake or allow the total phosphorus in the organ to vary. Mathematical and numerical analyses of these models show that tumor population growth and ultimate size are more sensitive to total phosphorus amount than their growth rates are. In particular, our simulation results show that if an artificial mechanism (treatment) can cut the phosphorus uptake of tumor cells in half, then it may lead to a three quarter reduction in ultimate tumor size, indicating an excellent potential of such a treatment. Also, in general we find that tumors with a relatively high cell death rate are more susceptible to treatments that block phosphorus uptake by tumor cells. Similarly, tumors with a large phosphorus requirement and (or) low cell reproductive rates are also strongly affected by phosphorus limitation.