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Volume 1, 2006

Networks & Heterogeneous Media

2006 , Volume 1 , Issue 4

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Special issue from the launching meeting of networks and heterogeneous media
Benedetto Piccoli
2006, 1(4): i-ii doi: 10.3934/nhm.2006.1.4i +[Abstract](27) +[PDF](32.8KB)
The launching meeting of Networks and Heterogeneous Media took place on June 21-23 2006 in Maiori (Salerno, Italy). The meeting was sponsored by the American Institute of Mathematical Sciences, the Istituto per le Applicazioni del Calcolo of Roma, and the DIIMA of University of Salerno. For more information please click the "Full Text" above.
The impact of cell crowding and active cell movement on vascular tumour growth
Russell Betteridge , Markus R. Owen , H.M. Byrne , Tomás Alarcón and  Philip K. Maini
2006, 1(4): 515-535 doi: 10.3934/nhm.2006.1.515 +[Abstract](43) +[PDF](665.5KB)
A multiscale model for vascular tumour growth is presented which includes systems of ordinary differential equations for the cell cycle and regulation of apoptosis in individual cells, coupled to partial differential equations for the spatio-temporal dynamics of nutrient and key signalling chemicals. Furthermore, these subcellular and tissue layers are incorporated into a cellular automaton framework for cancerous and normal tissue with an embedded vascular network. The model is the extension of previous work and includes novel features such as cell movement and contact inhibition. We present a detailed simulation study of the effects of these additions on the invasive behaviour of tumour cells and the tumour's response to chemotherapy. In particular, we find that cell movement alone increases the rate of tumour growth and expansion, but that increasing the tumour cell carrying capacity leads to the formation of less invasive dense hypoxic tumours containing fewer tumour cells. However, when an increased carrying capacity is combined with significant tumour cell movement, the tumour grows and spreads more rapidly, accompanied by large spatio-temporal fluctuations in hypoxia, and hence in the number of quiescent cells. Since, in the model, hypoxic/quiescent cells produce VEGF which stimulates vascular adaptation, such fluctuations can dramatically affect drug delivery and the degree of success of chemotherapy.
Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit
Hiroshi Matano , Ken-Ichi Nakamura and  Bendong Lou
2006, 1(4): 537-568 doi: 10.3934/nhm.2006.1.537 +[Abstract](36) +[PDF](401.5KB)
We study a curvature-dependent motion of plane curves in a two-dimensional cylinder with periodically undulating boundary. The law of motion is given by $V=\kappa + A$, where $V$ is the normal velocity of the curve, $\kappa$ is the curvature, and $A$ is a positive constant. We first establish a necessary and sufficient condition for the existence of periodic traveling waves, then we study how the average speed of the periodic traveling wave depends on the geometry of the domain boundary. More specifically, we consider the homogenization problem as the period of the boundary undulation, denoted by $\epsilon$, tends to zero, and determine the homogenization limit of the average speed of periodic traveling waves. Quite surprisingly, this homogenized speed depends only on the maximum opening angle of the domain boundary and no other geometrical features are relevant. Our analysis also shows that, for any small $\epsilon>0$, the average speed of the traveling wave is smaller than $A$, the speed of the planar front. This implies that boundary undulation always lowers the speed of traveling waves, at least when the bumps are small enough.
The many facets of internet topology and traffic
D. Alderson , H. Chang , M. Roughan , S. Uhlig and  W. Willinger
2006, 1(4): 569-600 doi: 10.3934/nhm.2006.1.569 +[Abstract](42) +[PDF](609.5KB)
The Internet's layered architecture and organizational structure give rise to a number of different topologies, with the lower layers defining more physical and the higher layers more virtual/logical types of connectivity structures. These structures are very different, and successful Internet topology modeling requires annotating the nodes and edges of the corresponding graphs with information that reflects their network-intrinsic meaning. These structures also give rise to different representations of the traffic that traverses the heterogeneous Internet, and a traffic matrix is a compact and succinct description of the traffic exchanges between the nodes in a given connectivity structure. In this paper, we summarize recent advances in Internet research related to (i) inferring and modeling the router-level topologies of individual service providers (i.e., the physical connectivity structure of an ISP, where nodes are routers/switches and links represent physical connections), (ii) estimating the intra-AS traffic matrix when the AS's router-level topology and routing configuration are known, (iii) inferring and modeling the Internet's AS-level topology, and (iv) estimating the inter-AS traffic matrix. We will also discuss recent work on Internet connectivity structures that arise at the higher layers in the TCP/IP protocol stack and are more virtual and dynamic; e.g., overlay networks like the WWW graph, where nodes are web pages and edges represent existing hyperlinks, or P2P networks like Gnutella, where nodes represent peers and two peers are connected if they have an active network connection.
On the variational theory of traffic flow: well-posedness, duality and applications
Carlos F. Daganzo
2006, 1(4): 601-619 doi: 10.3934/nhm.2006.1.601 +[Abstract](125) +[PDF](337.5KB)
This paper describes some simplifications allowed by the variational theory of traffic flow(VT). It presents general conditions guaranteeing that the solution of a VT problem with bottlenecks exists, is unique and makes physical sense; i.e., that the problem is well-posed. The requirements for well-posedness are mild and met by practical applications. They are consistent with narrower results available for kinematic wave or Hamilton-Jacobi theories. The paper also describes some duality ideas relevant to these theories. Duality and VT are used to establish the equivalence of eight traffic models. Finally, the paper discusses how its ideas can be used to model networks of multi-lane traffic streams.
Exogenous control of vascular network formation in vitro: a mathematical model
V. Lanza , D. Ambrosi and  L. Preziosi
2006, 1(4): 621-637 doi: 10.3934/nhm.2006.1.621 +[Abstract](37) +[PDF](470.4KB)
The reconstitution of a proper and functional vascular network is a major issue in tissue engineering and regeneration. The limited success of current technologies may be related to the difficulties to build a vascular tree with correct geometric ratios for nutrient delivery. The present paper develops a mathematical model suggesting how an anisotropic vascular network can be built in vitro by using exogenous chemoattractant and chemorepellent. The formation of the network is strongly related to the nonlinear characteristics of the model.
Hyperbolic-elliptic models for well-reservoir flow
Steinar Evje and  Kenneth H. Karlsen
2006, 1(4): 639-673 doi: 10.3934/nhm.2006.1.639 +[Abstract](27) +[PDF](577.0KB)
We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This $2 \times 2$ system of conservation laws is coupled to the integral equation through natural coupling conditions accounting for the flow between well and surrounding reservoir. By imposing simplifying assumptions we obtain various hyperbolic-parabolic and hyperbolic-elliptic systems. In particular, by assuming that the fluid is incompressible we obtain a hyperbolic-elliptic system for which we present existence and uniqueness results. Numerical examples demonstrate formation of steep gradients resulting from a balance between a local nonlinear convective term and a non-local diffusive term. This balance is governed by various well, reservoir, and fluid parameters involved in the non-local diffusion term, and reflects the interaction between well and reservoir.
Optimal control for continuous supply network models
Claus Kirchner , Michael Herty , Simone Göttlich and  Axel Klar
2006, 1(4): 675-688 doi: 10.3934/nhm.2006.1.675 +[Abstract](46) +[PDF](314.5KB)
We consider a supply network where the flow of parts can be controlled at the vertices of the network. Based on a coarse grid discretization provided in [6] we derive discrete adjoint equations which are subsequently validated by the continuous adjoint calculus. Moreover, we present numerical results concerning the quality of approximations and computing times of the presented approaches.
On the control volume finite element methods and their applications to multiphase flow
Zhangxin Chen
2006, 1(4): 689-706 doi: 10.3934/nhm.2006.1.689 +[Abstract](35) +[PDF](260.2KB)
In this paper we systematically review the control volume finite element (CVFE) methods for numerical solutions of second-order partial differential equations. Their relationships to the finite difference and standard (Galerkin) finite element methods are considered. Through their relationship to the finite differences, upstream weighted CVFE methods and the conditions on positive transmissibilities (positive flux linkages) are studied. Through their relationship to the standard finite elements, error estimates for the CVFE are obtained. These estimates are comparable to those for the standard finite element methods using piecewise linear elements. Finally, an application to multiphase flows in porous media is presented.

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