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Phipps C.,University of Waterloo | Molavian H.,University of Waterloo | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute for Research in Mathematical science
Journal of Theoretical Biology | Year: 2015

Cancer cells are notorious for their metabolic adaptations to hypoxic and acidic conditions, and especially for highly elevated glycolytic rates in tumor tissues. An end product of glycolysis is lactate, a molecule that cells can utilize instead of glucose to fuel respiration in the presence of oxygen. This could be beneficial to those cells that do not have sufficient oxygen as it conserves glucose for glycolysis. To better quantify this phenomenon we develop a diffusion-reaction mathematical model for nutrient concentrations in cancerous tissue surrounding a single cylindrical microvessel. We use our model to analyze the interdependence between cell populations[U+05F3] metabolic behaviors on a microscopic scale, specifically the emerging paradigm of metabolic symbiosis that exists between aerobic and glycolytic cells. The ATP turnover rates are calculated as a function of distance from the blood vessel, which exhibit a lactate-consuming population at intermediate distances from the vessel. We also consider the ramifications of the Warburg effect where cells utilize aerobic glycolysis along with this lactate-consuming respiration. We also investigate the effect of inhibiting metabolic pathways on cancer cells since insufficient ATP can trigger cell apoptosis. Effects that could be induced by metabolic inhibitors are analyzed by calculating the total ATP turnover in a unit tissue annulus in various parameter regimes that correspond to treatment conditions where specific metabolic pathways are knocked out. We conclude that therapies that target glycolysis, e.g. lactate dehydrogenase inhibitors or glycolytic enzyme inhibition, are the keys to successful metabolic repression. © 2014 Elsevier Ltd.

Phipps C.,University of Waterloo | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute for Research in Mathematical science
Computational and Mathematical Methods in Medicine | Year: 2011

We present a mathematical model for the concentrations of proangiogenic and antiangiogenic growth factors, and their resulting balance/imbalance, in host and tumor tissue. In addition to production, diffusion, and degradation of these angiogenic growth factors (AGFs), we include interstitial convection to study the locally destabilizing effects of interstitial fluid pressure (IFP) on the activity of these factors. The molecular sizes of representative AGFs and the outward flow of interstitial fluid in tumors suggest that convection is a significant mode of transport for these molecules. The results of our modeling approach suggest that changes in the physiological parameters that determine interstitial fluid pressure have as profound an impact on tumor angiogenesis as those parameters controlling production, diffusion, and degradation of AGFs. This model has predictive potential for determining the angiogenic behavior of solid tumors and the effects of cytotoxic and antiangiogenic therapies on tumor angiogenesis. Copyright © 2011 Colin Phipps and Mohammad Kohandel.

Schulze B.,Fields Institute for Research in Mathematical science | Whiteley W.,York University
Discrete and Computational Geometry | Year: 2012

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53-70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv: 0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561-598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric M d and the hyperbolic metric ℍ d. This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among E d, cones in E d+1, S d, M d, and ℍ d. We also consider the further extensions associated with the other Cayley-Klein geometries overlaid on the shared underlying projective geometry. © 2012 Springer Science+Business Media, LLC.

Nikolaev I.,Fields Institute for Research in Mathematical science
Finite Fields and their Applications | Year: 2014

We compute the number of points of projective variety V over a finite field in terms of invariants of the so-called Serre C*-algebra of V. © 2013 Elsevier Inc.

Kaveh K.,University of Waterloo | Kohandel M.,University of Waterloo | Sivaloganathan S.,University of Waterloo | Sivaloganathan S.,Fields Institute for Research in Mathematical science
Mathematical Biosciences | Year: 2016

The cancer stem cell hypothesis has evolved into one of the most important paradigms in cancer research. According to cancer stem cell hypothesis, somatic mutations in a subpopulation of cells can transform them into cancer stem cells with the unique potential of tumour initiation. Stem cells have the potential to produce lineages of non-stem cell populations (differentiated cells) via a ubiquitous hierarchal division scheme. Differentiation of a stem cell into (partially) differentiated cells can happen either symmetrically or asymmetrically. The selection dynamics of a mutant cancer stem cell should be investigated in the light of a stem cell proliferation hierarchy and presence of a non-stem cell population. By constructing a three-compartment Moran-type model composed of normal stem cells, mutant (cancer) stem cells and differentiated cells, we derive the replicator dynamics of stem cell frequencies where asymmetric differentiation and differentiated cell death rates are included in the model. We determine how these new factors change the conditions for a successful mutant invasion and discuss the variation on the steady state fraction of the population as different model parameters are changed. By including the phenotypic plasticity/dedifferentiation, in which a progenitor/differentiated cell can transform back into a cancer stem cell, we show that the effective fitness of mutant stem cells is not only determined by their proliferation and death rates but also according to their dedifferentiation potential. By numerically solving the model we derive the phase diagram of the advantageous and disadvantageous phases of cancer stem cells in the space of proliferation and dedifferentiation potentials. The result shows that at high enough dedifferentiation rates even a previously disadvantageous mutant can take over the population of normal stem cells. This observation has implications in different areas of cancer research including experimental observations that imply metastatic cancer stem cell types might have lower proliferation potential than other stem cell phenotypes while showing much more phenotypic plasticity and can undergo clonal expansion. © 2015 Elsevier Inc.

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