Evolutionary Dynamics of Cancer: Spatial and Heterogeneous Effects

Despite significant advances in the study of cancer and associated combination therapeutic treatments, cancer still remains one of the most common and complex often-terminal diseases. Acquisition of high-throughput experimental data from diverse cellular perspectives has thrown light on some of the regulatory mechanisms underlying the development of cancer. However, in general there is a lack of a general pattern and coherent model which can explain the development and evolution of the disease. To this end, evolutionary dynamics has been used, as a mathematical tool, in numerous studies to model various aspects of cancer over time periods. Our main focus in this thesis is on the use of stochastic and statistic methods to study cellular interactions within cancer tissues in order to understand the role of spatial structure, heterogeneity, and the microenvironment in cancer development. By constructing multi-cellular structures and using both analytic calculations and stochastic simulations, we have investigated the phenotypic hierarchy of stem cells within a heterogeneous system and in the presence of environmentally induced plasticity. Moreover, the effect of a random environment on the development of cancer has been explored in a general framework. As an important application of the multi-stage hierarchical model, the structure of the colonic/intestinal crypt has been taken into account to show the crucial role of these stem cells in the initiation and progression of colorectal/intestinal cancer. From an alternative viewpoint, we have envisaged the hierarchy of mutations as an evolutionary mechanism in the context of acute myeloid leukemia and carried out a statistical analysis of genetic data. Our findings in this thesis are general and most likely have many implications across a wide array of fields including different blood and solid cancers, bacterial growth, drug resistance and social networks. Moreover, the introduced methods and analyses should have important applications in diverse branches of evolution, ecology, and population genetics

Phenotypic heterogeneity in modeling cancer evolution

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and differentiated cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exact analytic results for the fixation probability of a mutant arising in each of the subpopulations. The analytic results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and differentiated cells turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. We discuss our model in the context of colorectal/intestinal cancer (at the epithelium). This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.Comment: 28 pages, 7 figures, 2 table

Multidimensional simple waves in fully relativistic fluids

A special version of multi--dimensional simple waves given in [G. Boillat, {\it J. Math. Phys.} {\bf 11}, 1482-3 (1970)] and [G.M. Webb, R. Ratkiewicz, M. Brio and G.P. Zank, {\it J. Plasma Phys.} {\bf 59}, 417-460 (1998)] is employed for fully relativistic fluid and plasma flows. Three essential modes: vortex, entropy and sound modes are derived where each of them is different from its nonrelativistic analogue. Vortex and entropy modes are formally solved in both the laboratory frame and the wave frame (co-moving with the wave front) while the sound mode is formally solved only in the wave frame at ultra-relativistic temperatures. In addition, the surface which is the boundary between the permitted and forbidden regions of the solution is introduced and determined. Finally a symmetry analysis is performed for the vortex mode equation up to both point and contact transformations. Fundamental invariants and a form of general solutions of point transformations along with some specific examples are also derived.Comment: 21 page