Complex blood flow patterns in an idealized left ventricle: A numerical study

In this paper, we study the blood flow dynamics in a three-dimensional (3D) idealized left ventricle of the human heart whose deformation is driven by muscle contraction and relaxation in coordination with the action of the mitral and aortic valves. We propose a simplified but realistic mathematical treatment of the valves function based on mixed time-varying boundary conditions (BCs) for the Navier-Stokes equations modeling the flow. These switchings in time BCs, from natural to essential and vice versa, model either the open or the closed configurations of the valves. At the numerical level, these BCs are enforced by means of the extended Nitsche's method (Tagliabue et al., Int. J. Numer. Methods Fluids, 2017). Numerical results for the 3D idealized left ventricle obtained by means of Isogeometric Analysis are presented, discussed in terms of both instantaneous and phase-averaged quantities of interest and validated against those available in the literature, both experimental and computational. The complex blood flow patterns are analysed to describe the characteristic fluid properties, to show the transitional nature of the flow, and to highlight its main features inside the left ventricle. The sensitivity of the intraventricular flow patterns to the mitral valve properties is also investigated. Published by AIP Publishing.

Numerical analysis of axisymmetric flows and methods for fluid-structure interaction arising in blood flow simulation

Simone Deparis

In this thesis we propose and analyze the numerical methods for the approximation of axisymmetric flows as well as algorithms suitable for the solution of fluid-structure interaction problems. Our investigation is aimed at, but are not restricted to, the simulation of the blood flow dynamics. The first part of this work deals with an axisymmetric fluid model based on three-dimensional incompressible Stokes or Navier–Stokes equations which are solved on a two-dimensional half-section of the domain under consideration. In particular we show optimal a priori error estimates for P1isoP2/P1 axisymmetric finite elements for the steady Stokes equations under the assumption that the domain and the data are axisymmetric and that the data have no angular component. Our analysis is carried out in the framework of weighted Sobolev spaces and takes advantage of a suitably defined Cl´ement type projection operator. We then introduce an axisymmetric formulation of the Navier–Stokes equations in moving domains and, starting from existing results in three-dimensions, we set up an Arbitrary Lagrangian–Eulerian (ALE) formulation and prove some stability results. In the second part, we deal with algorithms for the solution of fluid-structure interaction problems. We introduce the problem in a generic form where the fluid is described by means of incompressible Navier–Stokes equations and the structure by a viscoelastic model. We account for large deformations of the structure and we show how existing algorithms may be improved to reduce the computational time. In particular we show how to use transpiration boundary conditions to approximate the fluid-structure problem in a fixed point strategy. Moreover, in a quasi-Newton strategy we reduce the cost by replacing the Jacobian with inexact Jacobians stemming from reduced physical models for the problem at hand. To speed up the convergence of the Newton algorithm, we also define a dynamic preconditioner and an acceleration scheme which have been successfully tested in haemodynamics simulations in two and three dimensions.

EPFL2004

One dimensional and multiscale models for blood flow circulation

The aim of this work is to provide mathematically sound and computationally effective tools for the numerical simulation of the interaction between fluid and structures as occurring, for instance, in the simulation of the human cardiovascular system. This problem is global, in the sense that local changes can modify the solution far away. From the point of view of computing and modelling this calls for the use of multiscale methods, where simplified models are used to treat the global problem leaving to more accurate models the local description. Moreover it is characterised by the appearance of pressure waves inside the vessels. In large arteries the vessel wall dynamics can be described by a thin elastic membrane model (Navier equation) while the fluid motion can be represented by the Navier-Stokes equations for incompressible Newtonian fluids. Unfortunately, given the high levels of details furnished by this model, its computational complexity is dramatically high. Therefore reduced models have been developed. In particular, one-dimensional models, originally introduced by Euler, seem to be appropriate for the study in the time-space domain of pressure wave propagation induced by the interaction between the fluid and the vessel wall in the arterial tree. These reduced models are obtained after integrating the Navier-Stokes equations over a vessel section, supposed to be circular, and assuming an algebraic wall law to describe the relationship between pressure and wall deformation. They can be used in place of the more complex three dimensional fluid-structure models or in cooperation with them (multiscale approach). The first part of this work deals with one dimensional models. A reduced 1D model taken from literature is presented and analysed. Some extensions of the basic model, in particular with respect to vessel wall law (generalised string model) and more complex geometries (bifurcated and curved arteries), are also considered. Numerical schemes are proposed and some numerical results are presented. In the second part of this thesis we focus on a multiscale model. We consider a 55 arterial tree, described by the 1D model, coupled with lumped parameter models for heart and capillaries. In particular, specific attention has been devoted to the coupling between the left ventricle and the arterial system, whose physiopathological relevance is well known. This mathematical model gives good results in numerical tests and is able to describe the relevant features of the pressure wave propagation and reflections within the arterial system.

EPFL2004

One dimensional and multiscale models for blood flow circulation

EPFL2004

Numerical analysis of axisymmetric flows and methods for fluid-structure interaction arising in blood flow simulation

Simone Deparis

EPFL2004