We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Gene co-expression networks are increasingly used to explore the system-level functionality of genes. The network construction is conceptually straightforward: nodes represent genes and nodes are connected if the corresponding genes are significantly co-expressed across appropriately chosen tissue samples. In reality, it is tricky to define the connections between the nodes in such networks. An important question is whether it is biologically meaningful to encode gene co-expression using binary information (connected=1, unconnected=0). We describe a general framework for ;soft' thresholding that assigns a connection weight to each gene pair. This leads us to define the notion of a weighted gene co-expression network. For soft thresholding we propose several adjacency functions that convert the co-expression measure to a connection weight. For determining the parameters of the adjacency function, we propose a biologically motivated criterion (referred to as the scale-free topology criterion). We generalize the following important network concepts to the case of weighted networks. First, we introduce several node connectivity measures and provide empirical evidence that they can be important for predicting the biological significance of a gene. Second, we provide theoretical and empirical evidence that the ;weighted' topological overlap measure (used to define gene modules) leads to more cohesive modules than its ;unweighted' counterpart. Third, we generalize the clustering coefficient to weighted networks. Unlike the unweighted clustering coefficient, the weighted clustering coefficient is not inversely related to the connectivity. We provide a model that shows how an inverse relationship between clustering coefficient and connectivity arises from hard thresholding. We apply our methods to simulated data, a cancer microarray data set, and a yeast microarray data set.

A General Framework for Weighted Gene Co-Expression Network Analysis

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

/pdf/numerical-solution-of-saddle-point-problems-l583iob5c2.pdf

Numerical solution of saddle point problems

IEEE Transactions on Pattern Analysis and Machine Intelligence

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Chapter 1 introduces the most important terms and concepts of cardiovascular physiopathology ,while Chapter 2 illustrates the basic mathematical models for blood flow and biochemical transfer. The derivation of the equations that governs blood flow is covered in Chapter 3, while Chapter 4 is devoted to the treatment of medical images to obtain geometries suitable for numerical computations. Chapter 5 illustrates the important relationship between geometry and type of flow, focusing on the main characteristics of the different flow regimes encountered in the cardiovascular system. Mathematical models for blood rheology are discussed in Chapter 6. In Chapter 7 mathematical and numerical models of biochemical transport are explained in detail, with practical examples. The mathematical analysis of coupled models for fluid-structure interaction is addressed in Chapter 8, while Chapter 9 focuses on numerical methods for the mechanical coupling between blood flow and the vessel structure. Reduced models play an important role in cardiovascular modelling to enable the simulating of large parts of (or even th whole) vascular system. Their derivation is presented in Chapter 10. The intertwining of such models with more complex three dimensional ones is the foundation of the so called geometric multiscale approach, illustrated in detail in Chapter 11. Finally, Chapter 12 provides a set of well described and reproducible test cases and applications.

Cardiovascular Mathematics : Modeling and simulation of the circulatory system

In the cardiovascular system, morphology and functionality are closely related. Altered flow conditions play an important role in the development of arterial disease. In turn, these flow conditions are modified by arterial wall changes. A detailed understanding of the local hemodynamic environment, the influence of wall modifications on flow patterns and the long-term adaptations of the vascular wall after surgical procedures can have useful clinical applications, especially in view of reconstruction and revascularization operations. Some of these alterations are not well-understood, making it quite difficult to foresee short- and long-term evolution of the atherosclerotic disease and to plan an aggressive approach. Computational fluid dynamics (CFD) allows the carrying out of simulations at low cost and in completely controlled conditions. Three different issues are relevant to this subject and are addressed in this paper: (1) the definition of suitable mathematical models; (2) pre-processing of clinical data; and (3) the development of appropriate numerical techniques

Computational vascular fluid dynamics: problems, models and methods

In this paper we show how numerical solutions of human cardiovascular system may be devised by coupling models having different physical dimensions. One of the aspects of circulatory system is indeed its multiscale nature. Local flow features may have a global effect on circulation. For instance, a stenosis caused by an atherosclerotic plaque may change the overall characteristic of the vessels involved, and consequently have significant influence on the flow in the whole system.

Multiscale Modelling of the Circulatory System: a Preliminary Analysis

There is well-documented evidence that vascular geometry has a major impact in blood flow dynamics and consequently in the development of vascular diseases, like atherosclerosis and cerebral aneurysmal disease. The study of vascular geometry and the identification of geometric features associated with a specific pathological condition can therefore shed light into the mechanisms involved in the pathogenesis and progression of the disease. Although the development of medical imaging technologies is providing increasing amounts of data on the three-dimensional morphology of the in vivo vasculature, robust and objective tools for quantitative analysis of vascular geometry are still lacking. In this paper, we present a framework for the geometric analysis of vascular structures, in particular for the quantification of the geometric relationships between the elements of a vascular network based on the definition of centerlines. The framework is founded upon solid computational geometry criteria, which confer robustness of the analysis with respect to the high variability of in vivo vascular geometry. The techniques presented are readily available as part of the VMTK, an open source framework for image segmentation, geometric characterization, mesh generation and computational hemodynamics specifically developed for the analysis of vascular structures. As part of the Aneurisk project, we present the application of the present framework to the characterization of the geometric relationships between cerebral aneurysms and their parent vasculature.

A Framework for Geometric Analysis of Vascular Structures: Application to Cerebral Aneurysms

The purpose of the present article is to provide a theoretical analysis heterogeneous model coupling ordinary differential equations (ODEs) and partial differential equations (PDEs), providing a local-in-time existence result for the solution. The heterogeneous problem will actually be split into subproblems. The solution of the original problem will be regarded as the solution of a suitable fixed point problem, based on the successive solution of the subproblems. Moreover, the authors study the role of matching conditions between the two submodels for the numerical simulation.

Alessandro Veneziani

Papers

Cardiovascular Mathematics : Modeling and simulation of the circulatory system

Computational vascular fluid dynamics: problems, models and methods

Multiscale Modelling of the Circulatory System: a Preliminary Analysis

A Framework for Geometric Analysis of Vascular Structures: Application to Cerebral Aneurysms

Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations