A three-dimensional computational analysis of fluid–structure interaction in the aortic valve
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 bounded by Γf can be described by the well-known Navier–Stokes equation and continuity equation:where ρf denotes the density, t the time, is the velocity, the gradient operator with respect to the current configuration, pf the pressure, ηf the dynamic viscosity of the fluid and
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=ρfVr/ηf, 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
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