| In this thesis we develop a stochastic mathematical model, to study the dynamics of growing cell clusters where there is a progenitor-progeny relationship between the cells, regulated by feedback. One cell type mimics the combined effects of stem cell and progenitor cells while the other characterizes terminally differentiated cells. We first give a microscopic (stochastic) derivation of the model. Then using averaging and a mean field approximation (MFA), we derive a macroscopic (deterministic) description of the model. In this thesis we focus on understanding how noise, motilities of cells and feedback mechanisms in a lineage affect the development of an invasive tumor. Moreover we investigate how the interaction of motilities and feedback lead to heterogeneous patterning of cells in a tumor. Under what conditions do patterns of cells emerge spontaneously at the tumor front and in the tumor interior?;Our results indicate that instability develops as a consequence of the interaction between feedback and local stochastic fluctuations. Front instability is not observed in the mean-field model. Our analysis shows that front instability depends on spatial heterogeneity in cell types at the front that arises from strong negative feedback on the self renewal probabilities of the stem/progenitor cells from terminally differentiated cells. This is consistent with experimental observations. Moreover we show cell motilities ratio play an important role in front patterning and invasiveness. A Turing instability analysis on the MFA model reveals that pattern formation in the tumor bulk can be achieved using positive feedback on self renewal probability of stem cell.;The front speed is mainly controlled by the self-renewal capacity of the stem cells, the cell motilities, the proliferation and death rates. These are also critical parameters in mean field models. These results have important consequences on the progression of locally invasive tumors and their treatment. |
