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Turner C.,Dalhousie University | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute
Seminars in Cancer Biology | Year: 2012

The last decade has witnessed significant advances in the application of mathematical and computational models to biological systems, especially to cancer biology. Here, we present stochastic and deterministic models describing tumour growth based on the cancer stem cell hypothesis, and discuss the application of these models to the epithelial-mesenchymal transition. In particular, we discuss how such quantitative approaches can be used to validate different possible scenarios that can lead to an increase in stem cell activity following induction of epithelial-mesenchymal transition, observed in recent experimental studies on human breast cancer and related cell lines. The utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells is discussed as well. © 2012 Elsevier Ltd.

Turner C.,University of Waterloo | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute
Journal of Theoretical Biology | Year: 2010

Under the cancer stem cell (CSC) hypothesis, sustained metastatic growth requires the dissemination of a CSC from the primary tumour followed by its re-establishment in a secondary site. The epithelial-mesenchymal transition (EMT), a differentiation process crucial to normal development, has been implicated in conferring metastatic ability on carcinomas. Balancing these two concepts has led researchers to investigate a possible link between EMT and the CSC phenotype-indeed, recent evidence indicates that, following induction of EMT in human breast cancer and related cell lines, stem cell activity increased, as judged by the presence of cells displaying the CD44high/CD24low phenotype and an increase in the ability of cells to form mammospheres. We mathematically investigate the nature of this increase in stem cell activity. A stochastic model is used when small number of cells are under consideration, namely in simulating the mammosphere assay, while a related continuous model is used to probe the dynamics of larger cell populations. Two scenarios of EMT-mediated CSC enrichment are considered. In the first, differentiated cells re-acquire a CSC phenotype-this model implicates fully mature cells as key subjects of de-differentiation and entails a delay period of several days before de-differentiation occurs. In the second, pre-existing CSCs experience accelerated division and increased proportion of self-renewing divisions; a lack of perfect CSC biomarkers and cell sorting techniques requires that this model be considered, further emphasizing the need for better characterization of the mammary (cancer) stem cell hierarchy. Additionally, we suggest the utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells. © 2010 Elsevier Ltd.

Hubard I.,National Autonomous University of Mexico | Orbanic A.,University of Ljubljana | Pellicer D.,National Autonomous University of Mexico | Pellicer D.,Fields Institute | Ivic Weiss A.,York University
Discrete and Computational Geometry | Year: 2012

We derive some general results on the symmetries of equivelar toroids and provide detailed analysis of the subgroup lattice structure of the dihedral group D 4 and of the octahedral group to complete classification by symmetry type of those in ranks 3 and 4. © 2012 Springer Science+Business Media, LLC.

Challal S.,York University | Lyaghfouri A.,York University | Lyaghfouri A.,Fields Institute | Rodrigues J.F.,University of Lisbon | And 2 more authors.
Interfaces and Free Boundaries | Year: 2014

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the p(x)-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth p(x) > 1, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for p(x)-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of C1 hypersurfaces: (i) by extending directly the finiteness of the (n - 1)-dimensional Hausdorff measure of the free boundary to the case of heterogeneous p-Laplacian type operators with constant p, 1 < p < ∞; (ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth p(x) > 1. © European Mathematical Society 2014.

Rose S.C.F.,Queens University | Rose S.C.F.,Fields Institute
Communications in Number Theory and Physics | Year: 2014

In this paper, we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms.

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