A three-dimensional computational analysis of fluid–structure interaction in the aortic valve

doi:10.1016/S0021-9290(02)00244-0

Journal of Biomechanics

Volume 36, Issue 1, January 2003, Pages 103-112

https://doi.org/10.1016/S0021-9290(02)00244-0 Get rights and content

Abstract

Numerical analysis of the aortic valve has mainly been focused on the closing behaviour during the diastolic phase rather than the kinematic opening and closing behaviour during the systolic phase of the cardiac cycle. Moreover, the fluid–structure interaction in the aortic valve system is most frequently ignored in numerical modelling. The effect of this interaction on the valve's behaviour during systolic functioning is investigated. The large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood–valve interaction modelling. This has impeded numerical analyses of valves operating under physiological conditions. A numerical method, known as the Lagrange multiplier based fictitious domain method, is used to describe the large leaflet motion within the computational fluid domain. This method is applied to a three-dimensional finite element model of a stented aortic valve. The model provides both the mechanical behaviour of the valve and the blood flow through it. Results show that during systole the leaflets of the stented valve appear to be moving with the fluid in an essentially kinematical process governed by the fluid motion.

Introduction

Many numerical structural models have been developed that describe the behaviour of the aortic valve ignoring its interaction with the blood, e.g. see Black et al. (1991); Chandran et al. (1991); Krucinski et al. (1993); De Hart et al. (1998); Cacciola et al. (2000). The valve opening and closing during systole involves, however, a strong interaction between blood and the surrounding tissue. Several attempts have been made to analyse the valve behaviour using numerical fluid–structure interaction models, e.g. see Horsten (1990); Peskin and McQueen (1995); Makhijani et al. (1997). A detailed three-dimensional analysis of the valve kinematics, mechanics and fluid dynamics during the systolic phase has not been reported to date.

Modelling of such a fluid–structure interaction system is complicated due to the large motion of the thin leaflets through the computational fluid domain. The mathematical formulation of the equation of motion for a fluid is most conveniently described with respect to an Eulerian reference frame. However, this is incompatible with the Lagrangian formulation which is more appropriate to describe a structural phase. The arbitrary Lagrangian–Eulerian (ALE) method, first proposed by Donea et al. (1982), effectively combines the two different formulations and is frequently used in fluid–structure interaction analyses. Applied to the fluid phase, the ALE method requires a continuous adaptation of the fluid mesh without modification of the topology. Due to the large leaflet motions it is, however, difficult to adapt the fluid mesh in such a way that a proper mesh quality is maintained without changing the topology. Alternatively, remeshing of the fluid domain may be performed in conjunction with an ALE method, where remeshing is only performed if the mesh has degenerated too much. The change in topology during remeshing requires the use of interpolation techniques to recover state variables on the newly generated mesh. This not only introduces artificial diffusivity, but is also difficult and/or time-consuming to perform with sufficient robustness and accuracy for three-dimensional problems.

To resolve the limitations of these mesh update strategies we use a fictitious domain method to describe the interaction of the leaflets with the fluid. In this method, the different mathematical descriptions for the fluid and structure can be maintained, allowing convenient classical formulations for each of these subsystems. Moreover, the fluid mesh is not altered or interrupted by the presence of the immersed domain, and therefore preserves its original quality. Experimental validation of this method applied to a two-dimensional aortic valve model is demonstrated by De Hart et al. (2000). The application to a three-dimensional isotropic valve with rigid aortic root (mimicking a stented valve) and trileaflet symmetry is described in this paper. The model is used to study the effect of fluid–structure interaction on the valve behaviour for a reduced Reynolds number flow. We intend to address the importance of systolic functioning on the valve's (life-long) functionality. To this end, the influence of the fluid-structure interaction on the valve kinematics is investigated and the impact on the structural stress state and the associated fluid dynamical flow is analysed.

Section snippets

Methods

The blood flow is considered to be isothermal and incompressible. Assuming a Newtonian constitutive behaviour (Caro et al., 1978), the flow within the domain $Ω_{f}$ bounded by Γ_f can be described by the well-known Navier–Stokes equation and continuity equation: $ρ_{f} ∂ v →_{f} ∂ t + v →_{f} · ∇ → v →_{f} =− ∇ → p_{f} + ∇ → ·2η_{f} D_{f} in Ω_{f}, ∇ → · v →_{f} =0 in Ω_{f},$ where ρ_f denotes the density, t the time, $v →_{f}$ is the velocity, $∇ →$ the gradient operator with respect to the current configuration, p_f the pressure, η_f the dynamic viscosity of the fluid and $D_{f}$

Model properties

The aortic valve consists of three highly flexible leaflets, which are attached to the aortic root from one commissural point along a doubly curved line (aortic ring) towards a second commissural point, as illustrated in Fig. 3(a)–(c). Behind each leaflet the aortic root bulbs into a sinus cavity to form the beginning of the ascending aorta. Fig. 4 shows the relevant dimensions, which have frequently been used to describe the geometry of the valve. The values of these dimensions, based on the

Results

The valve kinematics is controlled by the surrounding fluid flow and the interaction of this flow with the leaflets, resulting in substantially different opening and closing configurations (Fig. 7). The opening behaviour is typical for stented valves (Cacciola, 1998) showing high curvatures of the free edge (Fig. 7(b)). The Reynolds number, defined as Re=ρ_fV_r/η_f, reaches a value of 900 at peak systolic mainstream velocity, i.e. V=300 (mm/s) at t=0.065 (s). However, the moment of complete opening

Discussion

A three-dimensional fictitious domain method is applied to model fluid–structure interaction in the aortic valve. The method is based on the imposition of kinematical constraints, using Lagrange multipliers, which represent the no-slip conditions along the fluid–structure interface. The implementation of this numerical technique is performed within the framework of the finite element method encoded in the SEPRAN software package (Segal, 2000). The essential feature of this approach is that

References (25)

M.M. Black et al.
A three-dimensional analysis of a bioprosthetic heart valve
Journal of Biomechanics
(1991)
G. Cacciola et al.
A three-dimensional mechanical analysis of a stentless fibre-reinforced aortic valve prosthesis
Journal of Biomechanics
(2000)
K.B. Chandran et al.
Stress distribution on the cusps of a polyurethane trileaflet heart valve prosthesis in the closed position
Journal of Biomechanics
(1991)
J. De Hart et al.
A two-dimensional fluid–structure interaction model of the aortic valve
Journal of Biomechanics
(2000)
J. Donea et al.
An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid–structure interactions
Computer Methods in Applied Mechanics and Engineering
(1982)
C. Farhat et al.
Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfacesmomentum and energy conservation, optimal discretization and application to aeroelastics
Computer Methods in Applied Mechanics and Engineering
(1998)
Z.B. Gao et al.
Bioprosthetic heart valve leaflet motion monitored by dual camera stereo photogrammetry
Journal of Biomechanics
(2000)
S. Krucinski et al.
Numerical simulation of leaflet flexure in bioprosthetic valves mounted on rigid and expansile stents
Journal of Biomechanics
(1993)
A.A. Van Steenhoven et al.
The effect of some hemodynamic factors on the behaviour of the aortic valve
Journal of Biomechanics
(1982)
F.P.T. Baaijens
A fictitious domain/mortar element method for fluid–structure interaction
International Journal for Numerical Methods in Fluids
(2001)

F. Brezzi

On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers

RAIRO—Operations Research

(1974)

Cacciola, G., 1998. Design, simulation and manufacturing of fiber reinforced polymer heart valves. Ph.D. Thesis,...

Cited by (328)

A mechanically consistent unified formulation for fluid-porous-structure-contact interaction
2024, Computer Methods in Applied Mechanics and Engineering
Fluid–structure interaction with contact poses profound mathematical and numerical challenges, particularly when considering realistic contact scenarios and the influence of surface roughness. Computationally, contact introduces challenges in altering the fluid domain topology and preserving stress balance. This work introduces a new mathematical framework for a unified continuum description of fluid-porous-structure-contact interaction (FPSCI), leveraging the Navier–Stokes–Brinkman (NSB) equations to incorporate porous effects within the surface asperities in the contact region. Our approach maintains mechanical consistency during contact, circumventing issues associated with contact models and complex interface coupling conditions, allowing for the modeling of tangential creeping flows due to surface roughness. The unified continuum and variational multiscale formulation ensure robustness by enabling stable and unified integration of fluid, porous, and solid sub-problems. Computational efficiency and ease of implementation – key advantages of our approach – are demonstrated by solving two benchmark problems of a falling ball and an idealized heart valve. This research has broad implications for fields reliant on accurate fluid–structure interactions and promising advancements in modeling and numerical simulation techniques.
An electromechanics-driven fluid dynamics model for the simulation of the whole human heart
2024, Journal of Computational Physics
We introduce a multiphysics and geometric multiscale computational model, suitable to describe the hemodynamics of the whole human heart, driven by a four-chamber electromechanical model. We first present a study on the calibration of the biophysically detailed RDQ20 active contraction model (Regazzoni et al., 2020) that is able to reproduce the physiological range of hemodynamic biomarkers. Then, we demonstrate that the ability of the force generation model to reproduce certain microscale mechanisms, such as the dependence of force on fiber shortening velocity, is crucial to capture the overall physiological mechanical and fluid dynamics macroscale behavior. This motivates the need for using multiscale models with high biophysical fidelity, even when the outputs of interest are relative to the macroscale. We show that the use of a high-fidelity electromechanical model, combined with a detailed calibration process, allows us to achieve a remarkable biophysical fidelity in terms of both mechanical and hemodynamic quantities. Indeed, our electromechanical-driven CFD simulations – carried out on an anatomically accurate geometry of the whole heart – provide results that match the cardiac physiology both qualitatively (in terms of flow patterns) and quantitatively (when comparing in silico results with biomarkers acquired in vivo). Moreover, we consider the pathological case of left bundle branch block, and we investigate the consequences that an electrical abnormality has on cardiac hemodynamics thanks to our multiphysics integrated model. The computational model that we propose can faithfully predict a delay and an increasing wall shear stress in the left ventricle in the pathological condition. The interaction of different physical processes in an integrated framework allows us to faithfully describe and model this pathology, by capturing and reproducing the intrinsic multiphysics nature of the human heart.
A novel mono-physics particle-based approach for the simulation of cardiovascular fluid-structure interaction problems
2024, Computer Methods and Programs in Biomedicine
Fluid-structure interaction (FSI) is required in the study of several cardiovascular engineering problems were the mutual interaction between the pulsatile blood flow and the tissue structures is essential to establish the biomechanics of the system. Traditional FSI methods are partitioned approaches where two independent solvers, one for the fluid and one for the structure, are asynchronously coupled. This process results into high computational costs. In this work, a new FSI scheme which avoids the coupling of different solvers is presented in the framework of the truly incompressible smoothed particle hydrodynamics (ISPH) method.
In the proposed FSI method, ISPH particles contribute to define both the fluid and structural domains and are solved together in a unified system. Solid particles, geometrically defined at the beginning of the simulation, are linked through spring bounds with elastic constant providing the material Young's modulus. At each iteration, internal elastic forces are calculated to restore the springs resting length. These forces are added in the predictor step of the fractional-step procedure used to solve the momentum and continuity equations for incompressible flows of all particles.
The method was validated with a benchmark test case consisting of a flexible beam immersed in a channel. Results showed good agreement with the system coupling approach of a well-established commercial software, ANSYS®, both in terms of fluid-dynamics and beam deformation. The approach was then applied to model a complex cardiovascular problem, consisting in the aortic valve operating function. The valve dynamics during opening and closing phases were compared qualitatively with literature results, demonstrating good consistency.
The method is computationally more efficient than traditional FSI strategies, and overcomes some of their main drawbacks, such as the impossibility of simulating the correct valve coaptation during the closing phase. Thanks to the incompressibility scheme, the proposed FSI method is appropriate to model biological soft tissues. The simplicity and flexibility of the approach also makes it suitable to be expanded for the modelling of thromboembolic phenomena.
Multiscale model for blood flow after a bileaflet artificial aortic valve implantation
2023, Computers in Biology and Medicine
Cardiovascular diseases are the leading cause of mortality in the world, mainly due to atherosclerosis and its consequences. The article presents the numerical model of the blood flow through artificial aortic valve. The overset mesh approach was applied to simulate the valve leaflets motion and to realize the moving mesh, in the aortic arch and the main branches of cardiovascular system. To capture the cardiac system’s response and the effect of vessel compliance on the outlet pressure, the lumped parameter model has been also included within the solution procedure. Three different turbulence modeling approaches were used and compared — the laminar, k- $ϵ$ and k- $ω$ model. The simulation results were also compared with the model excluding the moving valve geometry and the importance of the lumped parameter model for the outlet boundary condition was analyzed. Proposed numerical model and protocol was found as suitable for performing the virtual operations on the real patient vasculature geometry. The time-efficient turbulence model and overall solving procedure allows to support the clinicians in making decisions about the patient treatment and to predict the results of the future surgery.
A coupled SPH-PD model for fluid–structure interaction in an irregular channel flow considering the structural failure
2022, Computer Methods in Applied Mechanics and Engineering
Citation Excerpt :
Besides this, some grid-based methods, such as the immersed boundary (IB) method [29,30], naturally avoid the interface-tracking difficulty by distributing the nodal forces and interpolating the velocities between Eulerian and Lagrangian domains. Although grid-based methods have been successfully used in many FSI problems [31–36], some of their drawbacks still exist, such as the requirement for mesh updating and a lack of accuracy when dealing with large deformations and fractures in structures. Meshfree methods, on the other hand, overcome such difficulties by using arbitrarily distributed nodes (or particles) and integral interpolation for approximation.
Understanding fluid–structure interaction (FSI) is important because it dominates diverse natural phenomena and engineering problems. This paper presents an integrated particle model for FSI problems involving irregular channel flows and crack propagation in structures. The proposed model is implemented as follows: (1) we couple weakly compressible smoothed particle hydrodynamics (WCSPH) with bond-based peridynamics (BBPD) in a partitioned approach (this framework has a much simpler algorithm than the previously reported SPH-PD method); (2) we propose a novel periodic boundary conditions (PBCs) algorithm for modeling flows in non-regular channels; and (3) we incorporate crack propagation in structural responses under fluid dynamics, which was rarely considered in previous works. The new framework has been validated and illustrated to be effective and versatile in diverse FSI problems, including hydrostatic pressure-induced solid deformation, violent free-surface flows and channel flows interacting with flexible structures. Compared with conventional grid-based methods, this particle framework is more user-friendly, since no extra effort is required to update meshes, even when a discontinuity appears during the modeling process. The extendibility and potential of this framework is further demonstrated by the simulation of fluid-driven deformation and crack propagation in elastomers.
Fluid–Structure Interaction methods for the progressive anatomical and artificial aortic valve stenosis
2022, International Journal of Mechanical Sciences
Cardiovascular system diseases, as aortic valve stenosis, are the main cause of mortality and morbidity among patients. There is still a room for enhancement of the diagnostic and therapeutic procedures, which will lead to improvement of the treatment. One of the remedies are the computer tools to support the medical diagnoses and prostheses design. The development of a procedure for modeling the aortic valves: anatomical tricuspid valve and artificial bileaflet valve, still is a very challenging task. In presented work, the application of the novel, advanced moving mesh model, that consists of the coupling of the dynamic mesh smoothing and the overset mesh technique, to speed up the computation and improve the convergence and stability was shown. The real 2D and 3D vasculature and valve geometries were created based on the echocardiography images available in literature. The calculations of anatomical and artificial valve models were performed for the various severity of the atherosclerosis — not previously published for the bileaflet mechanical valve. The impact of calcification process onto natural and artificial aortic valves was assessed and compared.

View all citing articles on Scopus

View full text

A three-dimensional computational analysis of fluid–structure interaction in the aortic valve

Abstract

Introduction

Section snippets

Methods

Model properties

Results

Discussion

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Journal of Biomechanics

Journal of Biomechanics

Journal of Biomechanics

A fictitious domain/mortar element method for fluid–structure interaction

International Journal for Numerical Methods in Fluids

On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers

RAIRO—Operations Research