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Molzahn D.K.,Argonne National Laboratory | Mehta D.,University of Notre Dame | Mehta D.,University of Adelaide | Niemerg M.,Fields Institute for Research in Mathematical science
Proceedings of the American Control Conference | Year: 2016

The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the high-voltage solution to the power flow equations, which is of primary interest, there are potentially many low-voltage solutions that are useful for certain analyses. This paper addresses the number of solutions to the power flow equations. There exist upper bounds on the number of power flow solutions; however, there is only limited work regarding bounds that are functions of network topology. This paper empirically explores the relationship between the network topology, as characterized by the maximal cliques, and the number of power flow solutions. To facilitate this analysis, we use a numerical polynomial homotopy continuation approach that is guaranteed to find all complex solutions to the power flow equations. The number of solutions obtained from this approach upper bounds the number of real solutions. Testing with many small networks informs the development of upper bounds that are functions of the network topology. Initial results include empirically derived expressions for the maximum number of solutions for certain classes of network topologies. © 2016 American Automatic Control Council (AACC).

Schulze B.,Fields Institute for Research in Mathematical science | Sljoka A.,York University | Whiteley W.,York University
AIP Conference Proceedings | Year: 2011

How a protein functions depends both on having basic stable forms (tertiary structure) and having some residual flexibility supported within that structure. The modeling of protein flexibility and rigidity in terms imported from physics and engineering has been developed through the theory of generically rigid frameworks and a fast combinatorial algorithm is available in programs such as FIRST. Recent theoretical work on rigidity of frameworks has modified this generic analysis to include the basic symmetry and to predict additional motions. In particular, a framework which would normally count to be combinatorially minimally rigid in generic realizations has been shown to become flexible when realized with 2-fold rotational symmetry in 3-space. Protein dimers, formed by two copies of a protein are a good case study for impact of this added flexibility due to 2-fold rotational symmetry, as dimers generally self-assemble with a 2-fold rotational axis, for reasons of minimal energy. We describe an algorithm for predicting possible additional flexibility and consider the question: What is the significance of this for the behavior of dimers, such as tryptophan repressor? © 2011 American Institute of Physics.

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.

Manem V.S.K.,University of Waterloo | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute for Research in Mathematical science | Hodgson D.C.,University of Toronto | And 3 more authors.
International Journal of Radiation Biology | Year: 2015

Purpose: Numerous studies have implicated elevated second cancer risks as a result of radiation therapy. Our aim in this paper was to contribute to an understanding of the effects of radiation quality on second cancer risks. In particular, we developed a biologically motivated model to study the effects of linear energy transfer (LET) of charged particles (including protons, alpha particles and heavy ions Carbon and Neon) on the risk of second cancer. Materials and methods: A widely used approach to estimate the risk uses the so-called initiation-inactivation-repopulation model. Based on the available experimental data for the LET dependence of radiobiological parameters and mutation rate, we generalized this formulation to include the effects of radiation quality. We evaluated the secondary cancer risks for protons in the clinical range of LET, i.e., around 410 (KeV/μm), which lies in the plateau region of the Bragg peak. Results: For protons, at a fixed radiation dose, we showed that the increase in second cancer risks correlated directly with increasing values of LET to a certain point, and then decreased. Interestingly, we obtained a higher risk for proton LET of 10 KeV/μm compared to the lower LET of 4 KeV/μm in the low dose region. In the case of heavy ions, the risk was higher for Carbon ions than Neon ions (even though they have almost the same LET). We also compared protons and alpha particles with the same LET, and it was interesting to note that the second cancer risks were higher for protons compared to alpha particles in the low-dose region. Conclusion: Overall, this study demonstrated the importance of including LET dependence in the estimation of second cancer risk. Our theoretical risk predictions were noticeably high; however, the biological end points should be tested experimentally for multiple treatment fields and to improve theoretical predictions. © 2015 Informa UK, Ltd.

Pandey A.,Harvard University | Kulkarni A.,Harvard University | Roy B.,Harvard University | Goldman A.,Harvard University | And 9 more authors.
Cancer Research | Year: 2014

Nanomedicines that preferentially deploy cytotoxic agents to tumors and molecular targeted therapeutics that inhibit specific aberrant oncogenic drivers are emerging as the new paradigm for the management of cancer. While combination therapies are a mainstay of cancer chemotherapy, few studies have addressed the combination of nanomedicines and molecular targeted therapeutics. Furthermore, limited knowledge exists on the impact of sequencing of such therapeutics and nanomedicines on the antitumor outcome. Here, we engineered a supramolecular cis-platinum nanoparticle, which induced apoptosis in breast cancer cells but also elicited prosurvival signaling via an EGF receptor/phosphoinositide 3-kinase (PI3K) pathway. A combination of mathematical modeling and in vitro and in vivo validation using a pharmacologic inhibitor of PI3K, PI828, demonstrate that administration of PI828 following treatment with the supramolecular cis-platinum nanoparticle results in enhanced antitumor efficacy in breast cancer as compared with when the sequence is reversed or when the two treatments are administered simultaneously. This study addresses, for the first time, the impact of drug sequencing in the case of a combination of a nanomedicine and a targeted therapeutic. Furthermore, our results indicate that a rational combination of cis-platinum nanoparticles and a PI3K-targeted therapeutic can emerge as a potential therapy for breast cancer. © 2013 AACR.

Kaveh K.,University of Waterloo | Manem V.S.K.,University of Waterloo | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute for Research in Mathematical science | And 2 more authors.
Radiation and Environmental Biophysics | Year: 2015

Although the survival rate of cancer patients has significantly increased due to advances in anti-cancer therapeutics, one of the major side effects of these therapies, particularly radiotherapy, is the potential manifestation of radiation-induced secondary malignancies. In this work, a novel evolutionary stochastic model is introduced that couples short-term formalism (during radiotherapy) and long-term formalism (post-treatment). This framework is used to estimate the risks of second cancer as a function of spontaneous background and radiation-induced mutation rates of normal and pre-malignant cells. By fitting the model to available clinical data for spontaneous background risk together with data of Hodgkin’s lymphoma survivors (for various organs), the second cancer mutation rate is estimated. The model predicts a significant increase in mutation rate for some cancer types, which may be a sign of genomic instability. Finally, it is shown that the model results are in agreement with the measured results for excess relative risk (ERR) as a function of exposure age and that the model predicts a negative correlation of ERR with increase in attained age. This novel approach can be used to analyze several radiotherapy protocols in current clinical practice and to forecast the second cancer risks over time for individual patients. © 2014, Springer-Verlag Berlin Heidelberg.

Powathil G.,University of Waterloo | Powathil G.,Fields Institute for Research in Mathematical science | Kohandel M.,University of Waterloo | Kohandel M.,Fields Institute for Research in Mathematical science | And 3 more authors.
Computational and Mathematical Methods in Medicine | Year: 2012

Tumor oxygenation status is considered one of the important prognostic markers in cancer since it strongly influences the response of cancer cells to various treatments; in particular, to radiation therapy. Thus, a proper and accurate assessment of tumor oxygen distribution before the treatment may highly affect the outcome of the treatment. The heterogeneous nature of tumor hypoxia, mainly influenced by the complex tumor microenvironment, often makes its quantification very difficult. The usual methods used to measure tumor hypoxia are biomarkers and the polarographic needle electrode. Although these techniques may provide an acceptable assessment of hypoxia, they are invasive and may not always give a spatial distribution of hypoxia, which is very useful for treatment planning. An alternative method to quantify the tumor hypoxia is to use theoretical simulations with the knowledge of tumor vasculature. The purpose of this paper is to model tumor hypoxia using a known spatial distribution of tumor vasculature obtained from image data, to analyze the accuracy of polarographic needle electrode measurements in quantifying hypoxia, to quantify the optimum number of measurements required to satisfactorily evaluate the tumor oxygenation status, and to study the effects of hypoxia on radiation response. Our results indicate that the model successfully generated an accurate oxygenation map for tumor cross-sections with known vascular distribution. The method developed here provides a way to estimate tumor hypoxia and provides guidance in planning accurate and effective therapeutic strategies and invasive estimation techniques. Our results agree with the previous findings that the needle electrode technique gives a good estimate of tumor hypoxia if the sampling is done in a uniform way with 5-6 tracks of 20-30 measurements each. Moreover, the analysis indicates that the accurate measurement of oxygen profile can be very useful in determining right radiation doses to the patients. Copyright © 2012 Gibin Powathil et al.

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.

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.

Batkam C.J.,Fields Institute for Research in Mathematical science
Mathematical Methods in the Applied Sciences | Year: 2016

The aim of this note is to investigate the existence of signed and sign-changing solutions to the Kirchhoff type problem {-(a+b∫Ω|∇u|2Δu=∫Ω1/p|u|p)2/p|u|p-2uinΩu=0on∂Ω, where Ω is a bounded smooth domain in RN(N = 1,2,3), a,b > 0 and 2 < p < 2∗, with 2∗=+∞ if N = 1,2 and 2∗=6 if N = 3. Using variational methods, we show that (0.1) possesses three solutions of mountain pass type (one positive, one negative and one sign-changing) and infinitely many high-energy sign-changing solutions. Copyright © 2015 John Wiley & Sons, Ltd.

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