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Publication# On the coupling of 1D and 3D Diffusion-Reaction Equations: Application to Tissue Pefusion Problems

Abstract

In this paper we consider the coupling between two diffusion-reaction problems, one taking place in a three-dimensional domain Ω, the other in a one-dimensional subdomain Λ. This coupled problem is the simplest model of fluid flow in a three-dimensional porous medium featuring fractures that can be described by one-dimensional manifolds. In particular this model can provide the basis for a multiscale analysis of blood flow through tissues, in which the capillary network is represented as a porous matrix, while the major blood vessels are described by thin tubular structures embedded into it: in this case, the model allows the computation of the 3D and 1D blood pressures respectively in the tissue and in the vessels. The mathematical analysis of the problem requires non-standard tools, since the mass conservation condition at the interface between the porous medium and the one-dimensional manifold has to be taken into account by means of a measure term in the 3D equation. In particular, the 3D solution is singular on Λ. In this work, suitable weighted Sobolev spaces are introduced to handle this singularity: the well-posedness of the coupled problem is established in the proposed functional setting. An advantage of such an approach is that it provides a Hilbertian framework which may be used for the convergence analysis of finite element approximation schemes. The investigation of the numerical approximation will be the subject of a forthcoming work.

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Alfio Quarteroni, Christian Vergara

This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of '90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous problems featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions. Our review starts with the introduction of the stand-alone problems, namely the 3D fluid-structure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called "defective problems" naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works-from different research groups-which addressed the geometric multiscale approach in the past years. A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled. (C) 2016 Elsevier B.V. All rights reserved.

Isogeometric analysis (IGA) is a computational methodology recently developed to numerically approximate Partial Differential Equation (PDEs). It is based on the isogeometric paradigm, for which the same basis functions used to represent the geometry are then used to approximate the unknown solution of the PDEs. In the case in which Non-Uniform Rational B-Splines (NURBS) are used as basis functions, their mathematical properties lead to appreciable benefits for the numerical approximation of PDEs, especially for high order PDEs in the standard Galerkin formulation. In this framework, we propose an a priori error estimate, extending existing results limited to second order PDEs. The improvements in both accuracy and efficiency of IGA compared to Finite Element Analysis (FEA), encourage the use of this methodology in the haemodynamic applications. In fact, the simulation of blood flow in arteries requires the numerical approximation of Fluid-Structure Interaction (FSI) problems. In order to account for the deformability of the vessel, the Navier-Stokes equations representing the blood flows, are coupled with structural models describing the mechanical response of the arterial wall. However, the FSI models are complex from both the mathematical and the numerical points of view, leading to high computational costs during the simulations. With the aim of reducing the complexity of the problem and the computational costs of the simulations, reduced FSI models can be considered. A first simplification, based on the assumption of a thin arterial wall structure, consists in considering shell models to describe the mechanical properties of the arterial walls. Moreover, by means of the additional kinematic condition (continuity of velocities) and dynamic condition (balance of contact forces), the structural problem can be rewritten as generalized boundary condition for the fluid problem. This results in a generalized Navier-Stokes problem which can be expressed only in terms of the primitive variables of the fluid equations (velocity and pressure) and in a fixed computational domain. As a consequence, the computational costs of the numerical simulations are significantly reduced. On the other side, the generalized boundary conditions associated to the reduced FSI model could involve high order derivatives, which need to be suitably approximated. With this respect, IGA allows an accurate, straightforward and efficient numerical approximation of the generalized Navier-Stokes equations characterizing the reduced FSI problem. In this work we consider the numerical approximation of reduced FSI models by means of IGA, for which we discuss the numerical results obtained in Haemodynamic applications.

2012,

This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of ’90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous problems featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions. Our review starts with the introduction of the stand-alone problems, namely the 3D fluidstructure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called “defective problems” naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works - from different research groups - which addressed the geometric multiscale approach in the past years. A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled.