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Networks & Heterogeneous Media

2011 , Volume 6 , Issue 4

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Multiscale model of tumor-derived capillary-like network formation
Marco Scianna and Luca Munaron
2011, 6(4): 597-624 doi: 10.3934/nhm.2011.6.597 +[Abstract](53) +[PDF](2009.1KB)
Solid tumors recruit and form blood vessels, used for maintenance and growth as well as for formation and spread of metastases. Vascularization is therefore a pivotal switch in cancer malignancy: an accurate analysis of its driving processes is a big issue for the development of treatments. In vitro experiments have demonstrated that cultured tumor-derived endothelial cells (TECs) are able to organize in a connected network, which mimics an in vivo capillary-plexus. The process, called tubulogenesis, is promoted by the activity of soluble peptides (such as VEGFs), as well as by the following intracellular calcium signals. We here propose a multilevel approach, reproducing selected features of the experimental system: it incorporates a continuous model of microscopic VEGF-induced events in a discrete mesoscopic Cellular Potts Model (CPM). The two components are interfaced, producing a multiscale framework characterized by a constant flux of information from finer to coarser levels. The simulation results, in agreement with experimental analysis, allow to identify the key mechanisms of network formation. In particular, we provide evidence that the nascent pattern is characterized by precise topological properties, regulated by the initial cell density in conjunction with the degree of the chemotactic response and the directional persistence of cell migration.
Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model
Tong Li and Kun Zhao
2011, 6(4): 625-646 doi: 10.3934/nhm.2011.6.625 +[Abstract](62) +[PDF](457.2KB)
This paper is concerned with an initial-boundary value problem on bounded domains for a one dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that $L^\infty$ entropy weak solutions exist globally in time when the initial data are large, rough and contains vacuum states. Furthermore, based on entropy principle and the theory of divergence measure field, it is shown that any $L^\infty$ entropy weak solution converges to a constant equilibrium state exponentially fast as time goes to infinity. The physiological relevance of the theoretical results obtained in this paper is demonstrated.
A model for biological dynamic networks
Alessia Marigo and Benedetto Piccoli
2011, 6(4): 647-663 doi: 10.3934/nhm.2011.6.647 +[Abstract](69) +[PDF](493.2KB)
The main aim of this paper is to introduce a mathematical framework to study stochastically evolving networks. More precisely, we provide a common language and suitable tools to study systematically the probability distribution of topological characteristics, which, in turn, play a key role in applications, especially for biological networks. The latter is possible via suitable definition of a random network process and new results for graph isomorphism, which, under suitable generic assumptions, can be stated in terms of the graph walk matrix and computed in polynomial time.
Modeling and analysis of pooled stepped chutes
Graziano Guerra, Michael Herty and Francesca Marcellini
2011, 6(4): 665-679 doi: 10.3934/nhm.2011.6.665 +[Abstract](75) +[PDF](989.8KB)
We consider a mathematical model describing pooled stepped chutes where the transport in each pooled step is described by the shallow-water equations. Such systems can be found for example at large dams in order to release overflowing water. We analyze the mathematical conditions coupling the flows between different chutes taken from the engineering literature. For the case of two canals divided by a weir, we present the solution to the Riemann problem for any initial data in the subcritical region, moreover we give a well-posedness result. We finally report on some numerical experiments.
Shock formation in a traffic flow model with Arrhenius look-ahead dynamics
Dong Li and Tong Li
2011, 6(4): 681-694 doi: 10.3934/nhm.2011.6.681 +[Abstract](83) +[PDF](396.8KB)
We consider a nonlocal traffic flow model with Arrhenius look-ahead dynamics. We provide a complete local theory and give the blowup alternative of solutions to the conservation law with a nonlocal flux. We show that the finite time blowup of solutions must occur at the level of the first order derivative of the solution. Furthermore, we prove that finite time singularities do occur for several types of physical initial data by analyzing the solutions on different characteristic lines. These results are new and are consistent with the blowups observed in previous numerical simulations on the nonlocal traffic flow model [6].
Time-continuous production networks with random breakdowns
Simone Göttlich, Stephan Martin and Thorsten Sickenberger
2011, 6(4): 695-714 doi: 10.3934/nhm.2011.6.695 +[Abstract](89) +[PDF](1052.1KB)
Our main objective is the modelling and simulation of complex production networks originally introduced in [15, 16] with random breakdowns of individual processors. Similar to [10], the breakdowns of processors are exponentially distributed. The resulting network model consists of coupled system of partial and ordinary differential equations with Markovian switching and its solution is a stochastic process. We show our model to fit into the framework of piecewise deterministic processes, which allows for a deterministic interpretation of dynamics between a multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately tracing stochastic events. Numerical results are presented for three exemplary networks, including a comparison with the long-chain model proposed in [10].
Ginzburg-Landau model with small pinning domains
Mickaël Dos Santos and Oleksandr Misiats
2011, 6(4): 715-753 doi: 10.3934/nhm.2011.6.715 +[Abstract](55) +[PDF](857.3KB)
We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\epsilon\to0$; here, $\epsilon$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\partial \Omega$, with topological degree ${\rm deg}_{\partial \Omega} (g) = d >0$. Our main result is that, for small $\epsilon$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to $1$. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.
The needle problem approach to non-periodic homogenization
Ben Schweizer and Marco Veneroni
2011, 6(4): 755-781 doi: 10.3934/nhm.2011.6.755 +[Abstract](64) +[PDF](545.3KB)
We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation $-\nabla\cdot (a^\epsilon\nabla u^\epsilon) = f$. On the coefficients $a^\epsilon$ we assume that solutions $u^\epsilon$ of homogeneous $\epsilon$-problems on simplices with average slope $\xi\in \mathbb{R}^n$ have the property that flux-averages $f a^\epsilon\nabla u^\epsilon\in \mathbb{R}^n$ converge, for $\epsilon\to 0$, to some limit $a^\star(\xi)$, independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an $\epsilon$-problem in each simplex and are affine on faces.
Erratum and addendum to "Feedback stabilization of a coupled string-beam system" by K. Ammari, M. Jellouli and M. Mehrenberger; N. H. M: 4 (2009), 19--34
Kaïs Ammari, Mohamed Jellouli and Michel Mehrenberger
2011, 6(4): 783-784 doi: 10.3934/nhm.2011.6.783 +[Abstract](56) +[PDF](216.2KB)

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