Biological stoichiometry of tumor dynamics: Mathematical models and analysis

Abstract

Abstract. 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.

The P^* rule in the stochastic Holt-Lawton model of apparent competition

“…Volterra's mathematical derivation was for ordinary differential equation models where the per-capita growth rates of the competing species are linear functions of resource availability (see discussion in [13]). Since this work of Volterra, MathSciNet lists 279 publications on the "competitive exclusion principle" of which 19 appeared in Discrete and Continuous Dynamical Systems: Series B [21,29,40,51,22,4,56,53,27,39,38,19,55,7,1,26,2,25,50]. These 19 papers proved new principles of competitive exclusion for a diversity of situations including spatial chemostat models [21], within-host competition of multiple viral types [39], competing technologies [38], epidemiological models of competing disease strains [2], stoichiometric models of tumor growth [25], and discrete-time, size-structured chemostat models [50].…”

Tumor Modeling and Microwave Heating for Increasing Oxygen in Tumors during Photodynamic Therapy

“…In most occasions the early stage of cancer is exclusively recognized by a tissue biopsy. As a tumor grows, its capillary network grows simultaneously with it, in order to supply the comparatively larger requirement of nutrition and oxygen to the neoplasm [17]. The density of capillaries present in cancerous tissues, therefore, is comparatively higher than that present in healthy tissue.…”

Mathematical insights into neuroendocrine transdifferentiation of human prostate cancer cells

“…Others explored how different chemical resources would affect tumour growth, e.g. Kuang et al who proposed the KNE model [15], which applies ideas from ecological stoichiometry to tumour growth, and considers that tumour growth is limited by various physical restraints such as space, nutrients and vasculature. However, as the link between androgen, the male sex hormones testosterone and dihydrotestosterone, and prostate cancer was explored further [8,13,20], mathematical models began to focus on the response of PCa tumour growth to androgen concentration.…”