Evolution of cancer stem cell lineage involving feedback regulation.

Tumor emergence and progression is a complex phenomenon that assumes special molecular and cellular interactions. The hierarchical structuring and communication via feedback signaling of different cell types, which are categorized as the stem, progenitor, and differentiated cells in dependence of their maturity level, plays an important role. Under healthy conditions, these cells build a dynamical system that is responsible for facilitating the homeostatic regulation of the tissue. Generally, in this hierarchical setting, stem and progenitor cells are yet likely to undergo a mutation, when a cell divides into two daughter cells. This may lead to the development of abnormal characteristics, i.e. mutation in the cell, yielding an unrestrained number of cells. Therefore, the regulation of a stem cell's proliferation and differentiation rate is crucial for maintaining the balance in the overall cell population. In this paper, a maturity based mathematical model with feedback regulation is formulated for healthy and mutated cell lineages. It is given in the form of coupled ordinary and partial differential equations. The focus is laid on the dynamical effects resulting from acquiring a mutation in the hierarchical structure of stem, progenitor and fully differentiated cells. Additionally, the effects of nonlinear feedback regulation from mature cells into both stem and progenitor cell populations have been inspected. The steady-state solutions of the model are derived analytically. Numerical simulations and results based on a finite volume scheme underpin various expected behavioral patterns of the homeostatic regulation and cancer evolution. For instance, it has been found that the mutated cells can experience significant growth even with a single somatic mutation, but under homeostatic regulation acquire a steady-state and thus, ensuing healthy cell population to either a steady-state or a lower cell concentration. Furthermore, the model behavior has been validated with different experimentally measured tumor values from the literature