Zahra Aminzare

Applied Mathematics

Ph.D.
3.32

Publications

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    Zahra Aminzare, Eduardo D Sontag
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    Zahra Aminzare, Eduardo D. Sontag
    01/2015; 1(2). DOI:10.1109/TNSE.2015.2395075
  • Zahra Aminzare, Eduardo D. Sontag
    CDC 2014; 12/2014
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    ABSTRACT: Practically, all chemotherapeutic agents lead to drug resistance. Clinically, it is a challenge to determine whether resistance arises prior to, or as a result of, cancer therapy. Further, a number of different intracellular and microenvironmental factors have been correlated with the emergence of drug resistance. With the goal of better understanding drug resistance and its connection with the tumor microenvironment, we have developed a hybrid discrete-continuous mathematical model. In this model, cancer cells described through a particle-spring approach respond to dynamically changing oxygen and DNA damaging drug concentrations described through partial differential equations. We thoroughly explored the behavior of our self-calibrated model under the following common conditions: a fixed layout of the vasculature, an identical initial configuration of cancer cells, the same mechanism of drug action, and one mechanism of cellular response to the drug. We considered one set of simulations in which drug resistance existed prior to the start of treatment, and another set in which drug resistance is acquired in response to treatment. This allows us to compare how both kinds of resistance influence the spatial and temporal dynamics of the developing tumor, and its clonal diversity. We show that both pre-existing and acquired resistance can give rise to three biologically distinct parameter regimes: successful tumor eradication, reduced effectiveness of drug during the course of treatment (resistance), and complete treatment failure.
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    Zahra Aminzare, Eduardo D. Sontag
    CDC 2014; 12/2014
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    ABSTRACT: We present conditions that guarantee spatial uniformity of the solutions of reaction-diffusion partial differential equations. These equations are of central importance to several diverse application fields concerned with pattern formation. The conditions make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators on the given spatial domain. We present analogous conditions that apply to the solutions of diffusively-coupled networks of ordinary differential equations. We derive numerical tests making use of linear matrix inequalities that are useful in certifying these conditions. We discuss examples relevant to enzymatic cell sig-naling and biological oscillators. From a systems biology perspective, the paper's main contributions are unified verifiable relaxed conditions that guarantee spatial uniformity of biological processes. Keywords Reaction-diffusion systems · Turing phenomenon · Diffusive instabili-ties · Compartmental systems · Contraction methods for stability · Matrix measures The authors Zahra Aminzare and Yusef Shafi contributed equally.
    A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations, Edited by V. Kulkarni, G.-B. Stan, and K. Raman, 01/2014: pages 73-101; Springer-Verlag.
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    Zahra Aminzare, Eduardo D. Sontag
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    ABSTRACT: In this note, we present a condition which guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction-diffusion PDE with Neumann boundary conditions in one dimension, using the Jacobian matrix of the reaction term and the first Dirichlet eigenvalue of the Laplacian operator on the given spatial domain. We also derive an analog of this PDE result for the synchronization of a network of identical ODE models coupled by diffusion terms.
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    Zahra Aminzare, Eduardo D Sontag
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    ABSTRACT: This paper proves that ordinary differential equation systems that are contractive with respect to L-p norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well.
    Nonlinear Analysis 05/2013; 83:31-49. DOI:10.1016/j.na.2013.01.001 · 1.61 Impact Factor
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    Zahra Aminzare, Eduardo D. Sontag
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    ABSTRACT: This note works out an advection-diffusion approximation to the density of a population of E. coli bacteria undergoing chemotaxis in a one-dimensional space. Simulations show the high quality of predictions under a shallow-gradient regime.
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    ABSTRACT: We present conditions that guarantee spatial uniformity in diffusively-coupled systems. Diffusive coupling is a ubiquitous form of local interaction, arising in diverse areas including multiagent coordination and pattern formation in biochemical networks. The conditions we derive make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators, and generalize and unify existing theory about asymptotic convergence of trajectories of reaction-diffusion partial differential equations as well as compartmental ordinary differential equations. We present numerical tests making use of linear matrix inequalities that may be used to certify these conditions. We discuss an example pertaining to electromechanical oscillators. The paper's main contributions are unified verifiable relaxed conditions that guarantee synchrony.
    American Control Conference (ACC), 2013; 01/2013
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    Zahra Aminzare
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    ABSTRACT: In [1], we showed contractivity of reaction-diffusion PDE: \frac{\partial u}{\partial t}({\omega},t) = F(u({\omega},t)) + D\Delta u({\omega},t) with Neumann boundary condition, provided \mu_{p,Q}(J_F (u)) < 0 (uniformly on u), for some 1 \leq p \leq \infty and some positive, diagonal matrix Q, where J_F is the Jacobian matrix of F. This note extends the result for Q weighted L_2 norms, where Q is a positive, symmetric (not merely diagonal) matrix and Q^2D+DQ^2>0.

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