Fluid-Structure Interaction and Biomedical Applications

In this chapter we deal with some specific existence and numerical results applied to a 2D/1D fluid–structure coupled model, for an incompressible fluid and a thin elastic structure. We will try to underline some of the mathematical and numerical difficulties that one may face when studying this kind of problems such as the geometrical nonlinearities or the added mass effect. In particular we will point out the link between the strategies of proof of weak or strong solutions and the possible algorithms to discretize these type of coupled problems.

Fluid–structure interaction (FSI) problems arise in many applications. They include multi-physics problems in engineering such as aeroelasticity and propeller turbines, as well as biofluidic application such as self-propulsion organisms, fluid–cell interactions, and the interaction between blood flow and cardiovascular tissue. A comprehensive study of these problems remains to be a challenge due to their strong nonlinearity and multi-physics nature. To make things worse, in many biological applications the structure is composed of several layers, each with different mechanical characteristics. This is, for example, the case with arterial walls, which are composed of three main layers: the intima, media, and adventitia, separated by thin elastic laminae. A stable and efficient FSI solver that simulates the interaction between an incompressible, viscous fluid and a multi-layered structure would be an indispensable tool for the computational studies of solutions.

The multi-physics nature of this class of problems suggests the use of partitioned, modular algorithms based on an operator splitting approach that would separate the different physics in the problem. This chapter presents such a scheme, which can be used not only in computations, but also to prove existence of weak solutions to this class of problems. Particular attention will be payed to multi-physics FSI problems involving structures consisting of multiple layers.

It is well known that elastic solids, when subjected to a time-periodic load of frequency ω, may respond with a drastic increase of the magnitude of basic kinematic and dynamic quantities, such as displacement, velocity and energy, whenever ω is near to one of the “proper frequencies” of the solid. This phenomenon is briefly described as resonance. Objective of our analysis is to investigate whether the interaction of an elastic solid with a dissipative agent can affect and possibly prevent the occurrence of resonance. We shall study this problem in a broad class of dynamical systems that we call partially dissipative, and whose dynamics is governed by strongly continuous semigroups of contractions. For such systems we will provide sharp necessary and sufficient conditions for the occurrence of resonance. Afterward, we shall furnish a number of applications to physically relevant problems including thermo- and magneto-elasticity, as well as several liquid–structure interaction models.

In this paper, we review recent results devoted to the interactions between a collection of rigid bodies \((\mathcal{B}_{i})_{i=1,\ldots,n}\) and a surrounding viscous fluid \(\mathcal{L}\), the whole system filling a container \(\Omega \). We assume that the motion of \(\mathcal{L}\) (resp. the rigid bodies \(\mathcal{B}_{i}\)) is governed by the incompressible Navier Stokes equations (resp. Newton laws), and that velocities and stress tensors are continuous at the fluid/body interfaces. Our concern is the well-posedness of the associated Cauchy problem, with a specific eye towards the handling of contact between bodies or between one body and the container boundary.

Recently, the numerical solution of FSI problems has become important also in biomechanics, among others in voice modelling. The numerical analysis of this case is very complicated: Human voice is created by passage of air flow between vocal folds, where the constriction formed by the vocal folds induces acceleration of the flow and vocal fold oscillations, which generates the sound. The modelling of such a complex phenomenon encounters many difficulties as it is a result of coupling complex fluid dynamics and structural behavior. We focus on mathematical and numerical modelling of nonlinear coupled problems of fluid–structure interactions (FSI). The main attention is paid to the mathematical description of a relevant problem and to the description of the applied numerical methods. The mathematical description consists of the elasticity equations describing the motion of an elastic structure, and the air flow modelled by the Navier–Stokes equations. Both models are coupled via interface conditions.

The solution of dynamic elasticity equations is realized with the aid of conforming finite elements or the elastic structure motion is modelled by a simplified model of vibrating rigid body. Both compressible and incompressible fluid model is considered. The approximation of flow in moving domains is treated with the aid of the arbitrary Lagrangian–Eulerian method. The incompressible Navier–Stokes equations are approximated by the stabilized finite element method. The compressible Navier–Stokes equations are discretized by the discontinuous Galerkin finite element method. The time discretization based on a semi-implicit linearized scheme is described and the solution of the coupled problem of FSI is realized by a coupling algorithm.

Numerical methods for incompressible fluid dynamics have recently received a strong impulse from the applications to the cardiovascular system. In particular, fluid–structure interaction methods have been extensively investigated in view of an accurate and possibly fast simulation of blood flow in arteries and veins. This has been strongly motivated by the progressive interest in using numerical tools not only for understanding the general physiology and pathology of the vascular system. The opportunity offered by medical images properly preprocessed and elaborated to simulate blood flow in real patients highlighted the potential impact of scientific computing on the clinical practice. Therefore, in silico experiments are currently extensively used in bioengineering for completing (and sometimes driving) more traditional in vivo and in vitro investigations. Parallel to the development of numerical models, the need for quantitative analysis for diagnostic purposes has strongly stimulated the design of new methods and instruments for measurements and imaging. Thanks to these developments, a huge amount of data is nowadays available. Data Assimilation is the accurate merging of measures (including images) and numerical simulations for a mathematically sound integration of different sources of information. The outcome of this process includes both the patient-specific measures and the general principles underlying the development of mathematical models. In this way, simulations are adapted to the availability of individual data and are therefore supposed to be more reliable; measures are correspondingly filtered by the mathematical models assumed to describe the underlying phenomena, resulting in a (hopefully) significant reduction of the noise.

This chapter provides an introduction to methods for data assimilation, mostly developed in fields like meteorology, applied to computational hemodynamics. We focus mainly on two of them: methods based on stochastic arguments (Kalman filtering) and variational methods. We also address some examples that have been approached with different techniques, in particular the estimation of vascular compliance from displacement measures.

This chapter presents an overview and introduction to blood coagulation models. The historical exposure of the development of classical coagulation modeling theories is followed by a basic overview of blood coagulation biochemistry. The recent developments of cell-based models are explained in detail to demonstrate the current shift from the classical cascade/waterfall models. This phenomenological overview is followed by a survey of available mathematical concepts used to describe the blood coagulation process at various spatial scales including some of the related biophysical phenomena. A comprehensive survey of basic literature is provided for each of these topics.