Competition and natural selection in a mathematical model of cancer

Abstract

A malignant tumor is a dynamic amalgamation of various cell phenotypes, both cancerous (parenchyma) and healthy (stroma). These diverse cells compete over resources as well as cooperate to maintain tumor viability. Therefore, tumors are both an ecological community and an integrated tissue. An understanding of how natural selection operates in this unique ecological context should expose unappreciated vulnerabilities shared by all cancers. In this study I address natural selection's role in tumor evolution by developing and exploring a mathematical model of a heterogenous primary neoplasm. The model is a system of nonlinear ordinary differential equations tracking the mass of up to two different parenchyma cell types, the mass of vascular endothelial cells from which new tumor blood vessels are built and the total length of tumor microvessels. Results predict the possibility of a hypertumor-a focus of aggressively reproducing parenchyma cells that invade and destroy part or all of the tumor, perhaps before it becomes a clinical entity. If this phenomenon occurs, then we should see examples of tumors that develop an aggressive histology but are paradoxically prone to extinction. Neuroblastoma, a common childhood cancer, may sometimes fit this pattern. In addition, this model suggests that parenchyma cell diversity can be maintained by a tissue-like integration of cells specialized to provide different services.

Evolutionary biology of cancer

“…As tumors enlarge, and develop or recruit vascular tissue, they can become more heterogeneous both genetically and phenotypically [22], comprising a mix of cancerous and healthy cell types that cooperate as an integrated tissue but also compete for food and space. Nagy [23] developed an evolutionary ecological model that incorporated tumor heterogeneity and showed that interactions in such cellular communities could lead to competitive exclusion of cell lineages, in some situations giving rise to 'hypertumors' that exploit the developed vasculature to grow more quickly than do other cancer cell clones, but then die because they do not have a capacity to support further angiogenesis. He described how indirect evidence from the histology of some cancers supports the existence of hypertumors, and how the evolved balance between cooperation and competition in tumors has crucial clinical implications for optimizing cancer therapies.…”

Hypertumors in cancer can be caused by tumor phosphorus demand

“…To study the GRH's application to cancer in more detail, I modified an existing simple model of vascular tumor growth [6] to include P demand. Let P (t), x(t), y(t) and z(t) be intracellular [P], tumor parenchymal cell mass, vascular endothelial cell mass, and microvessel length density, respectively at time t. Assume interstitial [P] is a function of microvessel density v = z/x of the form P e = v/(a+v).…”

“…Phosphate leaks from cells at constant per unit cell mass rate l. Phosphate from dying cells is returned to the interstitial fluid. The per unit mass growth rate of tumor cells is assumed to be governed by an equation of the form (adapted from [6]),…”

“…From here one can develop an ODE model of vascular tumor growth along the lines of [6], resulting in a form identical with that of model (7) in [6] with the addition of equation (2) above. Biophysically, diffusion and phosphate transport occur on a much faster time scale than does cell proliferation and death.…”

“…where primes have been dropped, w = y/x, γ is basic maturation rate of vascular endothelial cells (VECs), δ is basic microvessel remodeling rate, and Ψ(v) represents growth rate of the VEC population (see [6] for details). Note that this model can be generalized to include multiple competing parenchyma strains as in [6] in a straightforward fashion.…”