MBI Publications

MBI Publications for Duan Chen (7)

  • D. Chen and G. Wei
    Quantum dynamics in continuum models for proton transport-Generalized correlatio
    Journal of Chemical PhysicsVol. 136 No. 134109 (2012)

    Abstract

  • D. Chen, J. Roda, C. Marsh, T. Eubank and A. Friedman
    Hypoxia Inducible Factors-mediated inhibition of cancer by GM-CSF: A mathematical model
    (2012) (Under Review)

    Abstract

  • D. Chen and A. Friedman
    Analysis of a two-phase free boundary problem for a parabolic-hyperbolic system: an application to tumor growth.
    (2012) (Submitted)

    Abstract

  • D. Chen, J. Roda, C. Marsh, T. Eubank and A. Friedman
    Hypoxia inducible factors mediated-inhibition of cancer by GM-CSF: A mathematical model
    Bulletin of Mathematical Biology (2012)

    Abstract

    Under hypoxia, tumor cells, and tumor-associated macrophages produce VEGF (vascular endothelial growth factor), a signaling molecule that induces angiogenesis. The same macrophages, when treated with GM-CSF (granulocyte/macrophage colony-stimulating factor), produce sVEGFR-1 (soluble VEGF receptor-1), a soluble protein that binds with VEGF and inactivates its function. The production of VEGF by macrophages is regulated by HIF-1α (hypoxia inducible factor-1α), and the production of sVEGFR-1 is mediated by HIF-2α. Recent experiments measured the effect of inhibiting tumor growth by GM-CSF treatment in mice with HIF-1α-de?cient or HIF-2α-de?cient macrophages. In the present paper, we represent these experiments by a mathematical model based on a system of partial differential equations. We show that the model simulations agree with the above experiments. The model can then be used to suggest strategies for inhibiting tumor growth. For example, the model qualitatively predicts the extent to which GM-CSF treatment in combination with a small molecule inhibitor that stabilizes HIF-2α will reduce tumor volume and angiogenesis.
  • D. Chen and A. Friedman
    A two-phase free boundary problem with discontinuous velocity: Application to tumor model
    Journal of Mathematical Analysis and Applications (2012)

    Abstract

    We consider a two-phase free boundary problem consisting of a hyperbolic equation for w and a parabolic equation for u, where w and u represent, respectively, densities of cells and cytokines in a simpli?ed tumor growth model. The tumor region Ω(t) is enclosed by the free boundary Γ(t), and the exterior of the tumor, D(t), consists of a healthy normal tissue. Due to cancer cells proliferation, the convective velocity ~v of cells is discontinuous across the free boundary; the motion of the free boundary Γ(t) is determined by ~v. We prove the existence and uniqueness of a solution to this system in the radially symmetric case for a small time interval 0 t T, and apply the analysis to the full tumor growth model.
  • L. Hu, D. Chen and G. Wei
    High-order fractional partial differential equation for molecular surface construction
    Molecular based Mathematical Biology (2012)

    Abstract

    Fractional derivative or fractional calculus plays a signifcant role in theoretical modeling of scienti?c and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
  • L. Hu and D. Chen
    High-order fractional partial differential equation transform for molecular surface construction.
    Molecular based mathematical biologyVol. 1 (2013)

    Abstract

    Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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