A fluid-structure coupling model for haemodynamics simulation of normal aortic valve

2.1. Coupling approach of CFD and FEA codes

The weak coupling model assumes that the flow simulation would happen firstly and then the structure simulation follows based on variables from flow simulation. Flow simulation provides fluid variables such as pressure and stress distributing over leaflet surface. After applying above variables, structure simulation provides displacement (deformation), von-Mises stress etc., in return. The updated geometry would then be imported to conduct flow simulation.

FLUENT software starts computing first and sends data (pressure or shear stress) to ABAQUS software then computes one step and sends data (node moving positions or face deformation) back to FLUENT software for the next step. The data exchange is performed totally by utilizing commercial Computer Aided Design (CAD) packages.

The valve leaflets are assumed to suffer the transvalvular pressure only as mentioned above. Therefore, the whole deformation analysis of the valve during systole can be performed when transvalvular pressure is known. In previous study, this weak coupling approach has ever been used to investigate the mechanical behaviors of bio-prosthetic heart valves [2]. This study conducts the deformation analysis in ABAQUS software first by utilizing the pre-known $∆P$ and uses FLUENT software to study fluid dynamics of the flow regime (Fig. 1).

Fig. 1Quasi-steady model to solve FSI problem in the aortic valve during systole

The biomechanics of the valve during whole systolic phase is independently investigated in ABAQUS as described in the paradigm. With the utilization of pre-processing software such as Hypermesh, Solidworks and Gambit, the deformed geometry of the valve leaflet at each moment is then exported to FLUENT for fluid dynamics analysis. The final results which highlight fluid-structure interaction phenomena in the aortic valve during systole are proposed by combining deformation and fluid dynamic analysis in the Tecplot post-processing package.

2.2. Geometry development

The aortic valve consists of three highly flexible leaflets attached to the aortic sinus. There still have studies tried to present a comprehensive report about the dimensions of the valve although the aortic valve has been examined and represented in anatomy texts for centuries. This highlights the fact that the aortic valve geometry has not been completely understood. This may be the reason for various aortic valve models have been developed. According to the model of DeHart et al. [1], the aortic valve geometry (Fig. 2) is determined by four main parameters including the valve radius ($R_c$), the valve height ($H$), the angle $\alpha$ of the bottom surface and the free edge angle ($\mathrm{\Phi }$).

This study assumes that the valve base radius ($R_b$) is equal to the valve radius ($R_c$) and the angle $\mathrm{\Phi }$ is 0 degree to simplify the model.

Fig. 2Valve model [1]

Fig. 3Transvalvular pressure [2]

2.3. Boundary conditions

3.1. Opening and closing characteristics of normal aortic valve

Theoretically, the aortic valve opens during systole when the ventricle contracts and then closes during diastole as the ventricle relaxes and fills blood from the atrium. Systole lasts about one-third of the cardiac cycle which typically takes about 200 to 300 msec [8]. The systolic phase starts when the aortic valve opens. This study assumes that transvalvular pressure is the only load acting on the valve leaflet during a cardiac cycle. In fact, the aortic valve must suffer diastolic pressure which might result in excessive deformation of the valve leaflets during diastole. The natural valve in reality shows a complex fiber-reinforced composite texture to bear diastolic pressure. The simulation in current study will exclude the diastolic phase.

Fig. 4 presents a series of opening and closing behaviors of the aortic valve during systolic phase. The valve starts opening to create a stellular orifice (Fig. 4(b)) at the beginning of systolic phase. The transvalvular pressure is increasing and results in further opening of valve leaflets forming triangular shape of the valve orifice (Fig. 4(c)). The valve is in maximum opening level and the orifice approaches circular shape at 0.08 sec (Fig. 4(d)). The valve starts to close although the flow has not reached the peak at this moment. The blood flow from left ventricle reaches peak at 0.1 sec while the valve is closing already (Fig. 4(e)). The valve continues closing during remaining time of systolic phase and results in orifice form changing to small triangle (Fig. 4(f)) and small triangle to stellular shape (Fig. 4(g)). The valve completely closes at about 0.3 sec (Fig. 4(h)). The opening and closing mechanism of the aortic valve with orifice shape alterations during systolic phase has been described in the previous study [9]. Total opening and closing time of the simulation is about 0.3 sec which completely accords with the results of the theory prediction of Chandran et al. [8]. In other words, the simulation results with the proposed method are rather reliable.

Fig. 4Valve leaflet deformation during systole

a) 0 s

b) 0.02 s

c) 0.04 s

d) 0.08 s

e) 0.1 s

f) 0.17 s

g) 0.2 s

h) 0.3 s

The shape alterations during opening and closing of the aortic valve result in a cyclic stress pattern on the leaflets which is believed to be important in the analysis of fatigue behavior (Fig. (4)). At 0 sec, pressure acting on the valve is zero and therefore no stress found on the leaflets (Fig. 4(a)). The high von-Mises stress concentrates on the middle of the leaflets when the valve further opened (Fig. 4(b)) and the stress is more dominant on the aortic side than that on the ventricular side. The high von-Mises stress tends to a bidirectional spreading from the middle toward both free edges and valve bases from 0.04 sec to 0.08 sec. When the valve in fully opening situation (Fig. 4(d)), the stress almost uniformly distributes on whole surfaces of the leaflets. The stress on the valve begins decreasing from the free edges to the valve bases when the valve starts closing. When the valve is completely closed (Fig. 4(h)), the stress distributed on the whole valve is very small and returns the state nearly like that at 0 sec. This mechanical behavior of the aortic valve repeats cyclically during each cardiac cycle.

3.2. Haemodynamics of the normal aortic valve

Flow patterns through the aortic valve during systole are demonstrated in Fig. 5. At the beginning, the valve starts to open and forms the stellular orifice (Fig. 5(a)). At $t=$ 0.04 sec, the blood flow mainly concentrates in the central regime and creates a jet impingement through the orifice to the aortic endothelial linings since the orifice is smaller. At $t=$ 0.08 sec, the valve almost fully opens and blood flow ejection phenomena occurs (Fig. 5(c)). It should be noted that the aortic valve has already started closing at this moment. The flow ejection could still be observed at $t=$ 0.1 sec when the flow reaches the peak velocity and the valve is closing already (Fig. 5(d)). The valve closes rapidly and induces a decrease of blood flow during the remaining period from $t=$ 0.17 sec to 0.3 sec (Fig. 5(e)-(h)).

Fig. 5Flow patterns through the aortic valve

a) 0.02 s

b) 0.04 s

c) 0.08 s

d) 0.1 s

e) 0.17 s

f) 0.2 s

g) 0.26 s

h) 0.3 s

Pressure and velocity distribution on a middle plane of the valve is demonstrated in Fig. 6. It should be noted that during the deceleration period, especially for the moment $t=$ 0.17 sec (Fig. 6(e)), flow recirculation takes place in front of the leaflets free edges and induces vortex in the sinus cavity. The vortices are expected to create a transverse pressure difference along the leaflets and push them to close during the closing phase.