Comparative Analyses of Blood Flow Through Mechanical Trileaet and Bileaet Aortic Valves

The paper describes one of many issues concerning the human circulatory system. The simulation of blood ow through an articial aortic heart valve using the nite element method (FEM) is the main subject of the paper. The studies aim to verify the performance of mechanical aortic valves of two types, i.e. bileaet (BIL) and trileaet (TRI) valves. The blood was modelled as Newtonian and non-Newtonian. Although the design of our TRI valve is preliminary and needs to be optimised, our results highlight some advances of such a valve geometry. This is manifested mainly by a central blood jet, contributing to more physiological blood ow and decreasing the risk of haemolysis. The central ow minimises the risk of leaet dislocation. In addition, lower stresses extend the durability of the valve. However, the TRI valve geometry has also disadvantages, for instance, the occurrence of small peripheral streams or relatively low effective orice area. The valves' performance was assessed by means of the reduced stress in the valves, the shear stress in the aortic wall, ow velocity eld, and the effective orice area. The maximum von Mises stress for the BIL valve leaets is 0.3 MPa, and for the TRI valve: 0.06 MPa. The maximum ow velocity for the BIL valve is 4.52 m/s for 40° and for the TRI valve is 5.74 m/s. Higher shear stress is present in the BIL (151.5 Pa) than for the TRI valve (49.64 Pa).


Introduction
Arti cial heart valves, often referred to as mechanical valves, have been implanted since the 1950s. Every year, around 300 000 arti cial heart valves are clinically applied [1]. By the highest systemic loads, the most frequently replaced valve is the aortic valve.
The function of the aortic valve is to regulate the blood ow constantly. Any malfunction of the valve can lead to the creation of coagulated masses of blood, which ows through the cardiovascular system can cause undesirable obstruction of vessels of small cross-sections. The bilea et (BIL) mechanical aortic valve has been studied for years. Many aspects related to its functioning were investigated in vitro or in silico, such as structural strength [2], damage of platelets [3], ow patterns and shear stress distribution [4], the valve overall hydrodynamic performance, the valve closing sounds or the cavitation phenomenon.
Despite many studies, there is still a problem with thrombus formation, leading consequently to anticoagulation therapy, in patients with an implanted mechanical heart valve. It has been reported that an unphysiological uid ow pattern determines thrombosis formation. Therefore, we started to work on designing a mechanical trilea et heart valve that would correspond to the construction of the natural aortic valve and allow for central blood ow. Unlike other researchers, we included the ventricle in the simulation and adopted the non-Newtonian nature of blood ow to represent the physiological ow of the blood stream.
Recent numerical simulations of blood ow through mechanical valves have shown various approaches in modelling the haemodynamic effects. The haemodynamic consequences of blood ow across a mechanical valve were studied in silico with different ow patterns (laminar or turbulent), blood was de ned as a Newtonian or non-Newtonian uid, and various boundary conditions were assumed. Abbas et al. [5] investigated how the tilting angle of BIL implantation in uences blood ow rate and shear stress distribution in the lea ets. They modelled blood as a non-Newtonian uid and assumed the ow to be laminar. A physiological blood velocity pro le was set at the inlet of the numerical model to simulate the ow. On the other hand, Kuan et al. [6] investigated hinge micro ow elds of bilea et mechanical heart valves, modelled blood as a Newtonian uid. They prescribed the velocity and pressure boundary conditions at the inlet and outlet by using the given wavefront. A similar approach related to boundary conditions was presented in [7], where the authors presented the effects of pannus formation on blood ow through a bilea et heart valve and the dynamical performance of BIL during the whole systolic phase, respectively. Also, in [8], the velocity pro le was set at the inlet. However, the authors implemented the three-element Windkessel model to predict pressure at outlets. The numerical model presented in [9] assumed pressure boundary conditions at the inlet and outlet of the blood vessel under consideration.
The authors developed an integrated uid-structure interaction (FSI) model using smoothed particle hydrodynamics for the uid domain coupled to a nonlinear nite element formulation. The FSI model was also applied in [10], where the effect of nonlinear lea et material properties on aortic valve dynamics was investigated. When the ow is fully developed at the out ow, the boundary conditions can be of the Neumann type, i.e. a zero-gradient pressure or a zero-gradient velocity. Such an approach is presented in [11], where the authors showed the results of blood ow simulations and described trends of ow elds and shear stress distribution in the aorta wall. Table 1 summarises brie y the review of blood ow modelling through the bilea et mechanical aortic valve. There are relatively few reported studies of blood ow simulation through a trilea et (TRI) mechanical aorta valve. Claiborne et al. [12] conducted an optimisation process of a trilea et valve simulating blood as a Newtonian uid in a turbulent ow. They imposed velocity boundary conditions at the inlet and zero pressure at the outlet. They aimed to optimise the lea et pro le to obtain the lowest pressure gradient in forwarding ow, lowest max velocity in forwarding ow, and the highest effective ori ce area. The ndings they achieved allowed them to advance valves closer to clinical viability. A polymeric valve was also a subject of interest of other studies [13]. The authors analysed the haemodynamic and thrombogenic performance of the trilea et valve using simulation of blood ow as a Newtonian uid. They de ned the boundary conditions in terms of energy sources using the Mie-Grüneisen equation of state relating pressure and internal energy per unit volume. However, they do not provide the Grüneisen parameters for blood, they used parameters adequate for water. Recently, several studies related to how a TRI valve affects blood ow have been developed. Bruecker and Li [14] introduced an in-vitro pulseduplicator generating early helical ow in the valve plane and experimentally investigated the in uence of that ow on uid behaviour after bi-and trilea et valves in the ascending aorta. They hypothesised that physiological right-handed helix in the ascending aorta might partly be maintained by early swirl in the ventricle out ow tract entering the aortic arch. They concluded that the TRI valve better conserved the helical ow than the BIL valve. Schaller et al. [15] implanted four novel mechanical prostheses of the aortic valve in sheep. The mechanical valves consisted of three lea ets made of poly-ether-ether-ketone.
The housing was manufactured from medical-grade titanium-aluminium-vanadium alloy (TiAl6V4). Their research aimed to ameliorate the mechanical valve's haemodynamic performance and reduce anticoagulation treatment. The in-vivo preliminary results were auspicious because the valves induced excellent haemodynamic and are characterised by a shallow risk of thrombotic events.
The primary aim of the present study is to compare the BIL and TRI aortic valves' performance during uniform blood ow model and boundary conditions. To complete the task, we performed numerical simulations of the ow through the TRI valve. The review of the literature shows that the blood ow simulations are conducted under various assumptions. The available comparative studies pertaining BILs and TRIs show superiority of TRI valves over BILs [16,17]. In the cited research, however, the authors simulated blood as a Newtonian uid and investigated a polymeric TRI valve [16] or a metallic one with thin lea ets [17]. The authors of the latter paper stated that their ndings might help to improve the development of the TRIs further. Our research follows that conclusion and introduces a design of a TRI valve and compares its haemodynamic performance with a BIL valve. The results of the simulations may contribute to the determination of new design parameters for trilea et aortic valves, which will improve their cardiologic performance and ensure proper haemodynamic parameters. The TRI design presented in the paper is the rst conceptual approach that we wanted to verify.
The secondary aim of our study is to determine the effect of Newtonian/non-Newtonian uid ow assumption on blood ow directly behind the trilea et valve, as that concerning the bilea et valve is very well known [18].
In the paper, 3D ow patterns and velocity pro les of blood ow in the aorta behind the two types of mechanical valves are presented. Achieving more real behaviour of blood stream during cardiac cycling is possible by including the ventricle in the numerical model. By direct comparison of the numerical results, we could assess the performance of the valves from the haemodynamic point of view.

Geometrical modelling
The geometrical model of the whole system consisted of the left ventricle and a fragment of the aorta. The model was based on the data available on the GrabCAD platform (https://grab cad.com/), which enables users to download geometric models, including models of anatomical parts of the human body. We used SolidWorks 2019 CAD software to create the 3D models of the anatomical components of our system and those of the valves ( Figure 1).
In our study, we considered two types of mechanical valves, i.e. bilea et valve and trilea et valve. It has to be noted that although the models are based on the known designs of bi-and trilea et valves, our models of the valves differ from them mainly in the construction of the annular ring. The internal diameter of the outlet of both prostheses is 21 mm, the external diameter of the ring is 27 mm, the pro le height is 16,5 mm. Additional, for the TRI valve, we proposed a new lea et curvature ( Figure 2). The shape of the lea ets in uences the pattern of velocity elds, which determine the haemodynamic quality of the valve. The design of the valve prostheses enables unidirectional blood ow. The valve outlet surface is tapered. It is modelled in a way to reduce ow turbulence. In the presented model of a BIL valve, the ring has additional covers to protect the lea ets in the closed position. The lea ets of both valves are embedded in blind holes, allowing them to rotate freely between 0° (fully open) and 60° (fully closed). In the open position, the plane of the lea ets of a BIL valve forms an angle of 90° to the surface of the outlet. The presented models are simpli ed. In fact, the ring's rim is lined with material, attached through titanium fastening rings, which enables the valve to be implanted in the outlet of the aorta.
The internal diameter of the ascending aorta was assumed to be 27 mm and its length 40 mm. The valve ring inside the aortic outlet was projected, which allowed us to de ne the constraints determining the valve position in the system.

Boundary conditions
Two different numerical methods are widely used in the study of ow through heart valves. One approach neglects the lea et motion, focusing on xed lea ets with constant or pulsatile ow [19,20]. The other method is to simulate the movement of the lea ets. In our research, we decided to analyse the blood ow at three positions of the valves' lea ets de ned using angle with respect to the vertical plane (passing through the symmetry axis of the valve) 40°, 20° and 0° (fully opened valve). A scheme of the complete opening of the valve is shown in Figure 3.
A representation of pulsatile ow was obtained by measuring blood ow velocity [21][22][23]. The correlation of the ow velocity and the time opening of the lea ets was determined utilising the Doppler ultrasound examination in an adult human being [21]. A 6th-degree polynomial interpolation (Equation (1)) was used to describe the ow mathematically. In the time span from 0.36 s to 0.38 s a linear function was adopted (Equation (2)). The ow velocity at the time that followed until the end of the cycle was described using Equation (3). The length of one cycle was assumed to be 0.8 s. Heart rate was 75 beats per minute.
It was assumed that the valve opens at 0.04 s of the ow cycle, and the process of the valve opening lasts 0.04 s [24]. The valves are, thus, fully opened after 0.08 s of the ow cycle. Regarding this, the values of the ow velocity at the considered lea et positions were estimated utilising Equation (1).
The ow velocity values were determined at the inlet, which was de ned at the entrance to the left ventricle. A constant pressure of 14 kPa was described at the aortic outlet, corresponding to the average pressure of the systolic and diastolic phases in a healthy human [25]. Boundary conditions are summarised in Figure 4.
The pulsatile nature of the blood and the geometry of the natural system means that vortices can form behind the aortic valve. To re ect physiological ow as accurately as possible, we have assumed turbulent blood ow and utilized the k-ε model to de ne its turbulent characteristics.
There is a general opinion that blood can be modelled as Newtonian in the case of large arteries. In our work, we investigate the shear effect, for which the uid model adopted can be of great importance. We, therefore, considered two blood models (Newtonian and non-Newtonian) during blood ow through the TRI valve. Based on the results, we determined how the adopted blood model affects the results. Blood ow through the BIL valve was modelled as non-Newtonian.
By de ning blood as incompressible Newtonian uid we assumed the density ρ = 1060 kg/m 3 and viscosity µ = 0.0035 kg/(m·s) [26]. For the non-Newtonian blood ow, we de ned the Carreau model. We have assumed the following parametric values: relaxation time constant λ = 3.313 s, zero shear rate limit µ 0 = 0.056 Pa•s, in nite shear rate limit µ∞=0.0035 Pa•s and power low index n=0.3568 [27].
The ventricle and aorta materials were assumed to be isotropic and elastic medium with the Young modulus E a = 1 MPa and Poisson's ratio ν = 0.49. The lea ets were also modelled as an isotropic elastic solid material with the Young modulus El = 2884 MPa, ν l = 0.39, and the ring material was modelled as titanium alloy Ti-6Al-4V, whose characteristic mechanical parameters are: E r = 1.07•10 5 MPa, ν r = 0.3.

Numerical modelling of blood ow
Blood ow through a valve is a complex phenomenon. It includes both the movement of the uid and the movement of the valve lea ets. The dynamics of blood circulation were determined using ANSYS 2020 R2 software. Three modules for blood ow analysis were used. The Mechanical module allowed us to determine the stress in the valve as well as in soft tissues. The Fluent module was used to de ne the ow parameters. The System Coupling module provided data exchange. Such a solution made it possible to perform FSI analyses, which made it possible to determine ow characteristics and the in uence of structural response on cyclic uid movement.
Referring to the previously presented boundary conditions and turbulent blood ow, the k-ε ow model was adopted. Taking into account the possible deformation of the mesh during the calculation, the option "Smoothing" was selected, allowing for smoothing of the mesh and the "Remeshing" option, allowing for regeneration of the too much distorted mesh. For the spatial discretisation of the momentum and turbulence equations within the liquid zone, a "Second Order Upwind" was used. To determine the stress distribution across the entire system, the lea et rotation constraints and the contact between the lea et guiding elements and blind holes were de ned. The friction coe cient was assumed to be equal 0.1. In order to determine stress distribution, the FSI bonds were determined. This allowed us to take into account the contribution of the uid on the stress distribution in the valves and aorta.
A pressure-based solver was used for the calculations. The model mesh was made of 150 000 nite elements. The mesh densi cation was performed for the valve lea ets, valve ring and ascending aorta (these areas participated in the FSI analysis). The mesh density mentioned above is optimal. We have veri ed that the number of the nite elements, which de ne our model, higher than the mentioned value 150 000 in uence the results, i.e. values of stress, by approx. 1%. The solid imitating blood was created with 3 million nite elements. The number of blood nite elements was also veri ed in terms of the accuracy of the results. The settings of the simulations were de ned such that the data exchange between the Fluent module and Transient Structural was allowed. Each step was recalculated ve times (maximum iteration = 5), preventing sudden stress value jumps and solver errors. The selection of a good quality mesh and an appropriate time step allowed for stable computations.

Results
Determination of stress distribution in the lea ets and streamlines of the blood ow velocity eld in the aorta was our main interest. Figure 5 shows the stress distributions in the BIL and TRI valve lea ets in the positions considered. The mounting of the lea ets in the valve ring was also analysed. These stresses are a consequence of the interaction of owing blood with the lea ets. The results of the study are summarised in Table 2. The results obtained give some insight into the strength of the lea et attachment structure in annular rings. Figure 6 depicts the ow of blood in the aorta for the same lea ets' positions. In addition, the ow velocity eld was shown in Figure 7 to verify the symmetry of the ow. Secondary rotations in the ascending aorta were also analysed. In Figure 8, tangential components of the blood velocity for fully opened valves are presented. Figure 9 shows shear stress in the ascending aorta wall induced by the ow in the considered positions of the lea ets, i.e. 0°, 20° and 40°.
To evaluate the performance of the two types of mechanical valves, we also calculated the effective ori ce area E OA , representing the cross-sectional area of the jet issuing from the valve at the point of its where: Q is the root mean square of forwarding ow in mL/s, Δp is the mean pressure difference across the valve in mmHg, and ρ is the density of blood in g/cm 3 . The number 51.6 is the gravitational acceleration constant. We assumed the blood density to be 1.06 g/cm 3 . The mean pressure difference Δp, as well as the blood ow, were calculated in the numerical simulations of blood ow.
The maximum effective ori ce area is 1.53 cm 2 for a BIL valve for an opening angle 0° and for a TRI valve 0.49 cm 2 (opening angle -20°).

Discussion
The present work compares the performance of two types of mechanical aortic valves under the same ow conditions, i.e. ow type (turbulent), boundary conditions, and uid model (Newtonian/non-Newtonian). Our aim was also to determine the effect of Newtonian/non-Newtonian uid ow assumption on blood ow downstream of the trilea et 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 lea ets ( Figure 2). Usually, authors offer a design with thin lea ets which, in the closed position, form a dome-like construction (see, e.g. [29]). The inner curvature of the lea ets is an additional factor contributing to vortex formation in the blood ow. In addition, due to the light construction of the lea ets, 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 [30]. Such research is commonly conducted by means of in silico modelling. An alternative, although more expensive, might be ex vivo modelling [31]. Such a methodology was applied in the study of a novel trilea et 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 trilea et 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. [33] or Yun et al. [3]. Chinese researchers designed a new bilea et valve [34], 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 [33]), which is troublesome to the patients' and may result in medical complications.

√
Our numerical results seem to indicate that the proposed lea ets curvature in the TRI valve causes less turbulent blood ow. This is manifested by the occurrence of smaller vortices behind the valve (Figure 6). This is highly desirable as turbulent ow is one of the factors leading to haemolysis reaction. The vortices in the BIL valve during ow can be seen in Figure 6a, b.
Modelling blood ow 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 de ned a combination of ow rate and pressure at the inlet and outlet, respectively. This approach in modelling the blood ow boundary conditions is commonly used [4,6]. The choice of the de ned boundary conditions is con rmed by obtaining blood ow velocity eld values through the partially and fully open BIL valve (Figure 6b, c) corresponding to values for the natural trilea et aortic valve [35]. The instantaneous maximum velocity for the BIL valve corresponds to the moment of valve opening (Figure 6a). Compared to the BIL valve, there are signi cantly higher velocities in the TRI valve (maximum velocity value for BIL: 4,52 m/s, for TRI 5,74 m/s -non-Newtonian uid and for TRI 5,89 m/s -Newtonian uid). This is due to the curved shape of the TRI valve lea ets, which signi cantly affects the reduced ow eld. Higher values of the ow velocity eld for the case of a TRI valve compared to a BIL valve was also observed by Piatti et al. [13].
The geometric ori ce area is 318 mm 2 for the BIL valve and 170 mm 2 for the TRI valve. The velocity values for the Newtonian and non-Newtonian models are similar. However, the character of ow seems to be different for non-Newtonian and Newtonian uid (Figure 6d-f and g-i). A closer analysis of Figure 6 shows that the de nition of the blood ow as a non-Newtonian uid seems to give more realistic results. We have used streamline techniques to visualize the ow and, in particular, its direction to make the analysis results more clear. The streamlines in Figure 6d-f show more clearly the peripheral ow than those in Figure 6g-i. TRI valve ow shows deceleration of the peripheral ow for non-Newtonian uid during lea et opening (Figure 6d). This reduction in ow velocity can have a negative effect on blood haemodynamics as it can lead to ow stagnation or cause haemolysis.
There is a common belief that in large vessels, blood can be modelled as a Newtonian uid. However, such an assumption might be a too far-fetched simpli cation in certain situations, e.g. during a ow through a mechanical aortic valve. The blood ow through both BIL and TRI valves is highly inhomogeneous in space and time. This was also noticed and documented by De Vita et al., [36], who simulated blood ow through a bilea et valve modelling blood as Newtonian and non-Newtonian uid. They stated that the non-Newtonian uid 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 uid model [8,37].
One of the very signi cant parameters in uencing the behaviour of blood cells during ow is shear stress. According to Ge et al.
[38] 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 ow 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 ow 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 ow are concentrated in the peripheral regions of the aorta, i.e. near the aorta wall, which changes the geometry of the ow drastically. As the blood is a shear-thinning uid, 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 Figure  9. This is related to the central ow of blood (Figures 6, 7). Due to the more established ow and decreasing ow velocity eld, 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, ows through the gaps between the lea ets and the valve ring that form when the lea et opens. Analysis of Figure 9g-i shows that for Newtonian uid, 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 lea et position. The viscosity of a Newtonian uid is constant, and the shear effects take place right after the beginning of the ow. In the case of non-Newtonian uid, the viscosity changes with time. Therefore, we can observe the maximal wall shear stress at the mid-position of the lea ets ( Figure   9e) when the ow rate is low.
The allowable stress value for the valve design is 32 MPa [39]. The von Mises stress analysis indicates that the highest stresses occur at the hinges and the place of lea et 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 lea ets (0,02-0,08 MPa) are in uenced by the adopted thickness of the lea ets. According to Figure 2 Figure 7 directly shows the central ow 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. [39] observed a signi cant increase in stress. Our stress values may be underestimated due to the lack of consideration of recirculating ow. 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 xing the lea et has to be considered. This will also prevent the formation of gaps between the lea et and the valve ring during valve opening. BIL valve ow 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 ow after a mechanical aortic valve and studied the in uence of the aortic root geometry on blood behaviour in the region of sinuses [40]. 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 ow downstream of a mechanical valve. Only minor changes in velocities were observed. However, differences in dynamics of blood ow are resulting from the aortic root geometry are noticeable. The authors of [18] 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 signi cantly affect uid 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 bene ts vortex formation in the region. In [40], the authors modelled the aortic root geometry with coronary arteries truncated at a close distance from the aorta and de ned 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 uid of all the neglected parts of the vascular network, nor did they consider the compliance of the arteries.
The primary characteristics of coronary ow were analysed by Querzoli et al. [41]. They concluded that 75% of the ow 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 uid ow in a simpli ed aorta due to considerations for studying haemodynamic during valve opening in xed positions.
Experimental studies on blood ow in the aorta showed its highly speci c nature. Hansen et al., 2019 quanti ed the ow in ascending aorta by means of the vector ow imaging method [42]. Earlier, the method was used in vivo on the heart during surgery to describe qualitatively and quantitatively the cardiac ow [43]. The method makes it possible to measure the ow speed in two directions and proves a helical pattern of blood ow. Similar cardiac ow character was also visualised by means of magnetic resonance imaging [44]. Secondary rotations in cardiac ow are due to the natural way the heart beats and the curvature of the aortic arch. The tangential components of the ow velocity shown in Figure 8 represent the ow in a speci c cross-section. In particular, they represent the direction of the blood ow, which allows one to observe whether a spiral ow occurs during the ow. Our simulation results do not indicate spiral ow in the aorta. Figure 8 only shows the presence of small circulations in the ascending aorta. The lack of spiral ow may be due to the use of a simpli ed aortic root. According to [45], spiral ow 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 ori ce area (E OA ), which is a measure of the effective valve opening during the forward ow phase. The highest E OA for the TRI valve corresponds to a valve opening angle of 20°. This is because the ow area at this position is the largest (170 mm 2 ). The effective ori ce area for the trilea et valve indicates that the valve lea ets are too thick. The correct area should be approximately 1.5 mm 2 . Changing the curvature of the lea et will increase the ow area and reduce the ow velocity eld 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 ow [46]. 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 lea ets opening and stabilises at the fully open position. In our opinion, this is due to the imposed boundary conditions. The analysis of blood ow through the valves at the 40° lea ets 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 uid after the valve was stationary.
The analyses con rm the validity of using three lea ets in the construction of the TRI valve. It is acknowledged that a TRI valve geometry causes more physiological closing compared to a bilea et valve [47]. The rate of the cups closing in uences the stimulation of platelet activation. Moreover, slower closing velocity decreases cavitation intensity [48] -another phenomenon contributing to blood cells deterioration. According to [47], the minimization of cavitation is also affected by thicker lea ets and a small rotation radius. This highlights the desirability of using thicker lea ets in the construction.
Our subsequent study will consider the motion of the lea ets 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 in uence of the inertia of the lea ets on the closing rate during the left ventricle diastole. Metal lea ets are expected to have a relatively high moment of inertia, resulting in far non-physiological blood ow. This factor can be reduced by applying different materials for the lea ets, such as polymers [13], polyurethanes, polytetra uoroethylene, biodegradable elastomers, hydrogels [49] or biomaterials consisted of living tissues capable of active remodelling and self-repair [50].
Future research will focus on considering more real-world parameters of the blood. The Windkessel model will be used as a boundary condition. The in uence of body uids and other tissues on ow 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 lea et motion in aortic stenosis).

Conclusions
Although the design of our TRI valve is preliminary and needs to be optimised, our results highlight some advances of such a valve geometry. This is manifested mainly by a central blood jet, contributing to more physiological blood ow and decreasing the risk of haemolysis (maximum shear stress for the BIL valve is 151.5 Pa, for the TRI valve 49.64 Pa) and, therefore, avoiding anticoagulation therapy in transplant patients. This will increase the possibility of implanting mechanical aortic valves in more patients. The central ow minimises the risk of lea et dislocation. In addition, lower stresses extend the durability of the valve (maximum von Mises stress for BIL valve lea ets is 0.3 MPa and for the TRI valve 0.06 MPa).
Another feature of the TRI valve is that it ensures similar blood ow regardless of the implantation angle. This is not the case for the BIL valve, which causes different ow patterns under various implantation angular orientations. Our numerical results indicate that the proposed lea et curvature in the TRI valve results in less turbulent blood ow. This is a positive aspect of the TRI valve lea et design, as vortices increase the risk of haemolysis.
The analyses also point to construction elements that should be improved. For instance, the parameter E OA should be increased. Reducing the curvature of the lea et will increase the ow area and reduce the ow velocity eld value. This will have a positive effect on decreasing the pressure gradient upstream and downstream of the valve.
The analyses con rm the validity of using three lea ets in constructing the TRI valve and indicate the advisability of further optimisation of the construction.

Declarations
On behalf of all authors, the corresponding author states that there is no con ict of interest.