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Publication# Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs

Abstract

For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions are becoming common practice. In this paper we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional (0D) model, describing the global characteristics of the circulatory system, and a one-dimensional (1D) model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODEs and the 1D hyperbolic system of PDEs); then we use fixed-point techniques. The abstract result is then applied to the original blood flow case under very realistic hypotheses on the data.

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Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic function

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Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.

Simone Deparis, Elena Faggiano, Marco Fedele, Davide Forti, Alfio Quarteroni, Anna Tagliabue

The computational fluid dynamics of the heart represents a challenging task both in terms of mathematical and numerical modeling; this is mainly due to the pulsatile nature of the blood flow, to its complex interaction with the valves, and, more in general, to the reciprocal action of the components responsible for the heart functioning. Even if one focuses on the study of the left ventricle, the blood flow patterns result to be significantly dependent on the mechanical contraction and relaxation of the muscle, on the conformation of the chamber, and on the interaction with the valves, which define a complex fluid-structure interaction problem. In this respect, also the blood flow in the aorta, and hence in the downstream circulation, is strongly affected by the aortic valve, whose behavior should be suitably mathematically and numerically modeled. In this work, we firstly focus on the study of the fluid dynamics inside the left ventricle in idealized configurations, for which we propose a mathematical model based on the Navier-Stokes equations endowed with mixed, time-dependent boundary conditions, which allow a simplified treatment of the aortic and mitral valves’ behavior. In this idealized setting, we perform numerical simulations which highlight the role and influence of modeling the valves to study and characterize the blood flow patterns inside the ventricle, as well as other parameters of clinical relevance. In addition, we consider a reduced, patient-specific fluid-structure-interaction model for the simulation of the blood flow through the aortic valve. Specifically, we propose an efficient coupled model which represents the valve dynamics by means of a zero-dimensional (0D) equation with the opening angle as primitive variable, while the blood flow by means of the full 3D Navier-Stokes equations. In this coupled model, the valve’s leaflets, which are reconstructed from MRI data of the open and closed configurations for a specific patient, influence the Navier-Stokes equations by means of resistive immersed surfaces, whose position depends on the opening angle of the valve. Moreover, the dynamics of the valve described by the 0D model is dependent on the velocity and pressure variables, specifically on the pressure jump and the flow rate through the valve itself. We per form patient-specific numerical simulations of the aortic valve based on this reduced 3D-0D model, for which we highlight its ability to correctly capture the fluid dynamics indicators expected for the patient.

2015