Modeling and simulation of blood flow problems

Medicine is one of the branches of scientific research in which numerical simulation can give a great support. In particular, a well mature branch like computational fluid dynamics (CFD) can provide a substantial help for the understanding of several pathologies of the cardiovascular system. The local behavior of blood flow may affect substantially the on-rise of such pathologies: a remarkable instance is atherosclerosis, which is unfortunately largely widespread in Western Countries. An accurate fluid dynamics analysis can better enlight these mechanisms, and CFD becomes necessary since physical data from in vivo'' measures are troublesome to obtain. Moreover, allowing completely controlled simulations, CFD can eventually provide a useful paradigm for clinical practice, for instance suggesting optimal strategy for a surgeric operation. par However, the study of biological systems is a difficult task not only from the numerical viewpoint, but also in the modeling phase. Indeed, different systems interact each other at mechanical or biochemical level. Therefore, making a hierarchy of simplifications in the models considered is almost always mandatory. These notes illustrate some specific features of the blood flow problem, in the case of large'' vessels, where the atherosclerotic plaques are localized. In particular, the interaction between blood and compliant vessel walls is of upmost interest. The purpose is to review shortly the existing literature on the field (limitedly to the most genuine mathematical aspects) and mention a first contribution from our side to the modeling of blood-wall interaction and its numerical simulation.

Modeling the fluid dynamics of the heart: from blood flows in idealized left ventricles to patient-specific aortic valve simulations

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

Efficient Solution of Fluid-Structure Interaction Problems in Computational Hemodynamics

Paolo Crosetto, Simone Deparis, Alfio Quarteroni

Cardiovascular diseases are the first cause of death in industrialized countries. It is now widely established that blood fluid dynamics and its interaction with the vascular wall play an essential role in the development and evolution of vascular pathologies. Mathematical modeling and numerical simulation are precious tools which may help to understand biomechanical phenomena in both physiological and pathological states and can lead to the development of new therapies or to assist physicians in clinical decision making and treatment. This paper reports on the simulation of blood flow and its interaction with the vascular wall using an aorta model. Then, to understand the complex flow patterns in pathological states, the simulation of a cerebral aneurysm is executed.

American Institute of Physics2010

A numerical study of isotropic and anisotropic constitutive models with relevance to healthy and unhealthy cerebral arterial tissues

Alfio Quarteroni, Adélia Sequeira, Paolo Tricerri

This paper presents an analysis of hyperelastic constitutive models for continuous bodies both from a modeling and numerical point of view. Contributions are made within the context of finite element numerical simulations. Numerical results with relevance to flows in the cardiovascular system are outlined in the case of a sophisticated fluid–structure interaction problem, in specific complex geometries of anatomically accurate cerebral arteries in diseased state. In this regard, the work carefully outlines the numerical validation of constitutive models for healthy and unhealthy cerebral arterial tissues by means of simulations of static inflation tests on an idealized specimen of anterior cerebral artery (ACA). The healthy tissue is described by means of isotropic and anisotropic models that, are fitted with respect to experimental data describing the mechanical behavior of the ACA; the numerical results are presented highlighting the most important numerical aspects influencing the correct and efficient simulation of the mechanics of continuous bodies such as, for instance, the arterial wall. We further consider numerical simulations of unhealthy conditions of the tissue by taking into account different levels of weakening of its mechanical properties. Taking the cerebral cardiovascular system as a challenging test problem, we focus on the study of the effects of the imposed mechanical levels of degradation on kinematic quantities of interest by simulating static inflation tests for the different models. This work does not aim to propose a new mathematical model for the mechanical damage occurring at the onset of cardiovascular diseases such as cerebral aneurysms. The modeling and numerical techniques presented may be applied to a wide range of problems, equally challenging to that of the cardiovascular system with complex structural models and fluid-structure coupling.