The present work compares the performance of two types of mechanical aortic valves under the same flow conditions, i.e. flow type (turbulent), boundary conditions, and fluid model (Newtonian/non-Newtonian). Our aim was also to determine the effect of Newtonian/non-Newtonian fluid flow assumption on blood flow downstream of the trileaflet valve. In the paper, we proposed our design of the mechanical TRI valve, which differs from those already presented in the literature by the shape of the leaflets (Figure 2). Usually, authors offer a design with thin leaflets which, in the closed position, form a dome-like construction (see, e.g. ). The inner curvature of the leaflets is an additional factor contributing to vortex formation in the blood flow. In addition, due to the light construction of the leaflets, they violently decelerate at the valve closure, which causes haemolysis by squeezing the blood cells. Studies on mechanical heart valves, including aortic valves, aim to decrease the risk of thrombosis, which requires anticoagulation treatment with various medicaments . Such research is commonly conducted by means of in silico modelling. An alternative, although more expensive, might be ex vivo modelling . Such a methodology was applied in the study of a novel trileaflet valve [15, 32]. The authors mounted the valve in a pulse duplicator that simulated the physiological system and studied, among others, clot formation. They found that the trileaflet valve causes only small and isolated deposits in the vicinity of the hinges. Platelet aggregation in the region of prosthesis hinges was also observed by Sari et al.  or Yun et al. . Chinese researchers designed a new bileaflet valve , which provides haemodynamic results similar to those obtained for the commonly used St. Jude valve. However, the valve design and function still do not prevent the use of anticoagulation therapy. The available in silico, ex vivo, and in vivo modelling approaches provide an understanding of the diseases involved and help clinicians to predict the patients reaction to the implanted valve. The in silico method we used allows one avoid medical interference (e.g., transoesophageal echocardiography ), which is troublesome to the patients’ and may result in medical complications.
Our numerical results seem to indicate that the proposed leaflets curvature in the TRI valve causes less turbulent blood flow. This is manifested by the occurrence of smaller vortices behind the valve (Figure 6). This is highly desirable as turbulent flow is one of the factors leading to haemolysis reaction. The vortices in the BIL valve during flow can be seen in Figure 6a, b.
Modelling blood flow through blood vessels, which form a branching structure, requires that the model of this structure must be truncated. Thus, a problem of proper boundary conditions at the distal ends of the vessels arises. To make the simulations more realistic, the smaller vessels beyond the truncation point must be substituted by boundary conditions. In our studies, we defined a combination of flow rate and pressure at the inlet and outlet, respectively. This approach in modelling the blood flow boundary conditions is commonly used [4, 6]. The choice of the defined boundary conditions is confirmed by obtaining blood flow velocity field values through the partially and fully open BIL valve (Figure 6b, c) corresponding to values for the natural trileaflet aortic valve . The instantaneous maximum velocity for the BIL valve corresponds to the moment of valve opening (Figure 6a). Compared to the BIL valve, there are significantly higher velocities in the TRI valve (maximum velocity value for BIL: 4,52 m/s, for TRI 5,74 m/s – non-Newtonian fluid and for TRI 5,89 m/s – Newtonian fluid). This is due to the curved shape of the TRI valve leaflets, which significantly affects the reduced flow field. Higher values of the flow velocity field for the case of a TRI valve compared to a BIL valve was also observed by Piatti et al. . The geometric orifice area is 318 mm2 for the BIL valve and 170 mm2 for the TRI valve. The velocity values for the Newtonian and non-Newtonian models are similar. However, the character of flow seems to be different for non-Newtonian and Newtonian fluid (Figure 6d-f and g-i). A closer analysis of Figure 6 shows that the definition of the blood flow as a non-Newtonian fluid seems to give more realistic results. We have used streamline techniques to visualize the flow and, in particular, its direction to make the analysis results more clear. The streamlines in Figure 6d-f show more clearly the peripheral flow than those in Figure 6g-i. TRI valve flow shows deceleration of the peripheral flow for non-Newtonian fluid during leaflet opening (Figure 6d). This reduction in flow velocity can have a negative effect on blood haemodynamics as it can lead to flow stagnation or cause haemolysis.
There is a common belief that in large vessels, blood can be modelled as a Newtonian fluid. However, such an assumption might be a too far-fetched simplification in certain situations, e.g. during a flow through a mechanical aortic valve. The blood flow through both BIL and TRI valves is highly inhomogeneous in space and time. This was also noticed and documented by De Vita et al., , who simulated blood flow through a bileaflet valve modelling blood as Newtonian and non-Newtonian fluid. They stated that the non-Newtonian fluid model should be assumed, mainly when blood cells damage is investigated. Although the quantitative results of haemolysis simulations can differ with a non-Newtonian model applied, such an approach seems to give a more realistic wall shear stress distribution than a Newtonian fluid model [8, 37].
One of the very significant parameters influencing the behaviour of blood cells during flow is shear stress. According to Ge et al.  shear stress must be above 150 Pa to cause haemolysis and above 10 Pa to cause platelet activation. A high value of shear stress in the ascending aorta for the BIL valve (i.e. 151.5 Pa, 126.88 Pa and 114.45 Pa for cusp position 40°, 20° and 0°, respectively) may indicate the possibility of haemolysis. The risk is high, but the duration of exposition would still need to be considered. Exceptionally high shear stress (151.5 Pa) occurs at the valve opening (40° - Figure 9a). This stress is because the flow runs close to the aortic wall (Figure 6a, Figure 7a). Furthermore, vortices occur during valve opening (Figure 6a, b), which further increase the impact of blood on the aortic wall. In the case of the BIL valve, a decrease of wall shear stress with the deceleration of flow can be noticed (compare Figure 6a-c and Figure 9a-c). This is due to the fact that the shear rate increases because the main streamlines of the flow are concentrated in the peripheral regions of the aorta, i.e. near the aorta wall, which changes the geometry of the flow drastically. As the blood is a shear-thinning fluid, which means that its viscosity decreases with a shear rate increase, a higher shear rate makes the blood less viscous, which causes lower wall shear stress. The wall shear stress for the TRI valve is much lower, i.e. 30.56 Pa, 49.64 Pa and 30.99 Pa for leaflet position 40°, 20° and 0°, respectively (non-Newtonian fluid) and 69.21 Pa, 47.90 Pa and 38.84 Pa for leaflet position 40°, 20° and 0°, respectively (Newtonian fluid) - see Figure 9. This is related to the central flow of blood (Figures 6, 7). Due to the more established flow and decreasing flow velocity field, haemolysis should also not occur further down the aorta. The highest shear stresses in the TRI valve occur at an opening angle of 20° (Figure 9e). Blood, in this case, flows through the gaps between the leaflets and the valve ring that form when the leaflet opens. Analysis of Figure 9g-i shows that for Newtonian fluid, the maximal wall shear stress occurs at the angle 40°, i.e. at the beginning of the valve opening. This is in accordance with Figure 7g, which present velocity distribution at the same leaflet position. The viscosity of a Newtonian fluid is constant, and the shear effects take place right after the beginning of the flow. In the case of non-Newtonian fluid, the viscosity changes with time. Therefore, we can observe the maximal wall shear stress at the mid-position of the leaflets (Figure 9e) when the flow rate is low.
The allowable stress value for the valve design is 32 MPa . The von Mises stress analysis indicates that the highest stresses occur at the hinges and the place of leaflet attachment (Table 2, Figure 5). The maximum stresses are comparable for both valves (Figure 5c, d). The moment of occurrence of the highest stresses, 0° for the BIL valve (5,64 MPa) and 40° for the TRI (5,66 MPa), is due to the highest velocity values near the hinges (Figure 7c, d). The low stress values in the centre of the TRI valve leaflets (0,02-0,08 MPa) are influenced by the adopted thickness of the leaflets. According to Figure 2, the TRI leaflets thickness value varies from 0.6 to 2.6 mm. For the BIL valve (leaflet thickness 0.4 mm), these values are in the range of 0.17-0.30 MPa. Figure 7 directly shows the central flow at the TRI valve. The results of our simulations indicate that the maximum stress values are much smaller than the allowable values. We, therefore, conclude that the construction of the valves will not fail. However, it should be noted that we considered 75 beats per minute. With an increase in heart rate, Nasif et al.  observed a significant increase in stress. Our stress values may be underestimated due to the lack of consideration of recirculating flow. Exceeding the allowable stresses can lead to malfunction and failure of the valve over a long period, so valve motion analysis, which will be performed in future research, is necessary. To avoid possible high stress values, a different way of fixing the leaflet has to be considered. This will also prevent the formation of gaps between the leaflet and the valve ring during valve opening. BIL valve flow is symmetrical (Figure 7a, b, c). It can therefore be concluded that the discs should not dislocate.
In our model, we did not consider the Valsalva sinuses. De Tulio et al. conducted numerical simulations of blood flow after a mechanical aortic valve and studied the influence of the aortic root geometry on blood behaviour in the region of sinuses . They considered three models, i.e. three sinuses, one sinus in the form of an axisymmetric bulb, and a simple aorta without sinuses. Their results indicate that the geometry of the aortic root affects only marginally the kinematic features of blood flow downstream of a mechanical valve. Only minor changes in velocities were observed. However, differences in dynamics of blood flow are resulting from the aortic root geometry are noticeable. The authors of  observed the formation of vortices in the region of the sinuses. Their numerical results show a presence of negative velocities that they interpreted as blood recirculation in the sinuses. However, researchers do not consider coronary arteries, which have their origins in the sinuses and may significantly affect fluid dynamics in the aortic root. In consequence, the wall boundary condition is imposed on the inner surfaces of the sinuses. This is a factor that benefits vortex formation in the region. In , the authors modelled the aortic root geometry with coronary arteries truncated at a close distance from the aorta and defined boundary conditions which “are limiting in that, in general, they do not accurately replicate vascular impedance of the downstream vasculature”. This means that they did not consider the inertia of the fluid of all the neglected parts of the vascular network, nor did they consider the compliance of the arteries. The primary characteristics of coronary flow were analysed by Querzoli et al. . They concluded that 75% of the flow in coronary arteries occurs during diastole. During systole, no distinct effects were observed, except for a secondary vortex region located at the inlet of the coronary vessel. The inclusion of coronary arteries in the model affects the delay and faster closure of the valve. Thus, it will be essential to model the coronary arteries when studying valve motion. In the present study, we decided to simulate fluid flow in a simplified aorta due to considerations for studying haemodynamic during valve opening in fixed positions.
Experimental studies on blood flow in the aorta showed its highly specific nature. Hansen et al., 2019 quantified the flow in ascending aorta by means of the vector flow imaging method . Earlier, the method was used in vivo on the heart during surgery to describe qualitatively and quantitatively the cardiac flow . The method makes it possible to measure the flow speed in two directions and proves a helical pattern of blood flow. Similar cardiac flow character was also visualised by means of magnetic resonance imaging . Secondary rotations in cardiac flow are due to the natural way the heart beats and the curvature of the aortic arch. The tangential components of the flow velocity shown in Figure 8 represent the flow in a specific cross-section. In particular, they represent the direction of the blood flow, which allows one to observe whether a spiral flow occurs during the flow. Our simulation results do not indicate spiral flow in the aorta. Figure 8 only shows the presence of small circulations in the ascending aorta. The lack of spiral flow may be due to the use of a simplified aortic root. According to , spiral flow correlates with an extension of the sinuses of Valsalva.
To better assess valve performance, clinicians have developed a parameter to determine the degree of stenosis: the effective orifice area (EOA), which is a measure of the effective valve opening during the forward flow phase. The highest EOA for the TRI valve corresponds to a valve opening angle of 20°. This is because the flow area at this position is the largest (170 mm2). The effective orifice area for the trileaflet valve indicates that the valve leaflets are too thick. The correct area should be approximately 1.5 mm2. Changing the curvature of the leaflet will increase the flow area and reduce the flow velocity field value. This will have a positive effect on decreasing the pressure gradient upstream and downstream of the valve.
In our simulations, we modelled turbulent blood flow . A similar assumption has been used by other researchers [4, 6, 7, 10, 11]. Our results indicate that the main turbulence occurs at the beginning of the leaflets opening and stabilises at the fully open position. In our opinion, this is due to the imposed boundary conditions. The analysis of blood flow through the valves at the 40° leaflets position does not take into account the fact that the blood is in constant move. The vortices visible in Figure 6a, b occur due to the fact that at the beginning, of the simulation the fluid after the valve was stationary.
The analyses confirm the validity of using three leaflets in the construction of the TRI valve. It is acknowledged that a TRI valve geometry causes more physiological closing compared to a bileaflet valve . The rate of the cups closing influences the stimulation of platelet activation. Moreover, slower closing velocity decreases cavitation intensity  – another phenomenon contributing to blood cells deterioration. According to , the minimization of cavitation is also affected by thicker leaflets and a small rotation radius. This highlights the desirability of using thicker leaflets in the construction.
Our subsequent study will consider the motion of the leaflets for a thorough comparative analysis and determine the adopted method's effect on haemodynamic results. The analysis will be performed on several cardiac cycles. This includes research on the influence of the inertia of the leaflets on the closing rate during the left ventricle diastole. Metal leaflets are expected to have a relatively high moment of inertia, resulting in far non-physiological blood flow. This factor can be reduced by applying different materials for the leaflets, such as polymers , polyurethanes, polytetrafluoroethylene, biodegradable elastomers, hydrogels  or biomaterials consisted of living tissues capable of active remodelling and self-repair .
Future research will focus on considering more real-world parameters of the blood. The Windkessel model will be used as a boundary condition. The influence of body fluids and other tissues on flow will also be taken into account. The aortic root will consider the sinuses of Valsalva and the coronary arteries. It will also be essential to show models characterising pathological states (restriction of leaflet motion in aortic stenosis).