Transient Study of Flow and Cavitation Inside a Bileaﬂet Mechanical Heart Valve

: A mechanical heart valve (MHV) is an e ﬀ ective device to cure heart disease, which has the advantage of long life and high reliability. Due to the hemodynamic characteristics of blood, mechanical heart valves can lead to potential complications such as hemolysis, which have damage to the blood elements and thrombosis. In this paper, ﬂowing features of the blood in the valve are analyzed and the cavitation mechanism in bileaﬂet mechanical heart valve (BMHV) is studied. Results show that the water hammer e ﬀ ect and the high-speed leakage ﬂow e ﬀ ect are the primary causes of the cavitation in the valve. Compared with the high-speed leakage ﬂow e ﬀ ect, the water hammer has a greater e ﬀ ect on the cavitation strength. The valve goes through four kinds of working condition within one heart beating period, including, fully opening stage, closing stage and fully closing stage. These four stages, respectively, make up 8.5%, 16.1%, 4.7% and 70.7% of the total period. The cavitation occurs on the fully closing stage. When the valve is in closing stage, the high pressure downstream of the valve lasts for about 20 ms and the high-speed leakage ﬂow lasts for about 200 ms. This study systematically analyzes the causes of cavitation emerged in the process of periodic motion, which proposes the method for characterizing the intensity of the cavitation, and can be referred to for the cavitation suppression of the BHMV and similar valves.


Introduction
The valvular heart disease is a significant health problem for more than 100 million people worldwide and has a high mortality rate [1]. According to previous studies [2], the incidence of these diseases would continue to increase, due to the increasing pathological burden of degenerative diseases and rheumatic fever. In the treatment of valvular heart disease, the bileaflet mechanical heart valve (BMHV), which has long service life and reliable performance, has a tremendous impact on the treatment of the valvular heart disease in the field of biomedicine, and has become the most common treatment device. The BHMV is usually implanted in the heart to replace the diseased heart valve. Although it is widely used in clinical studies, the BHMV still has defects and causes main complications including hemolysis, thrombocytopenic and thromboembolism. In addition, non-physiological geometry of the valve can cause abnormal velocity distribution of blood flow when the heart valve is implanted. quantify the movement of the leaflet by tracking the position of the leaflet tip during the whole closing process. In addition, the reverse leakage flow passing through the hinge gap is studied, to reduce damage to blood substances [29,30]. The leakage flow has a maximum speed of 4.7 m/s in the simplified axisymmetric sinus during the early diastolic phase. However, the flow field, with an effective spatial resolution of 167 µm and temporal resolution of 594 µs, is measured by two-component PIV. Higher speed leakage is observed t the hinge and the measured average speed can reach as high as 5.7 m/s [31].
To aid clinical decision making and the understanding of pathophysiology, the Lagrangian particle method can be used to improve insight into the transport mechanics of the downstream flow of the valve [32]. Based on this, the pulsating flow in BMHV is simulated by the smoothed particle hydrodynamic method [33]. The advantage of the smoothed particle hydrodynamic method is that it is a Lagrangian method, which can avoid the accuracy loss caused by the grid distortion when the deformation is extremely large. Results show that the non-physiological flow pattern and the main vertical structure have an important influence on the shear stress of blood components. The full three-dimensional geometry of the MHV with moving leaflet in a typical human heart cycle is simulated. The simulation can essentially reproduce the pressure distribution of the left ventricle and the aorta. From the simulation results, wall shear stress and vorticity can be derived to further understand the performance of the valve.
Cavitation is the process of the formation, development and collapse of vapor or gas cavities in the liquid or at the liquid-solid interface when the local pressure in the liquid decreases [34,35]. In early studies, researchers aimed at developing a physical model, supported by experimental observations, described the formation and growth of microbubbles seen in patients with mitral mechanical heart valve [36]. In order to fully-reveal the generation mechanism of microbubbles, an in vitro dynamic environment closest to the real situation was created. Results show that the vortex formation and microbubbles growth, which take place during valve closure, have far reaching clinical implications [37,38]. The violent collapse of microbubbles induces local erosion that may lead to structural damage and the microbubbles are caused by the cavitation [39], which inevitably leads to energy loss [40]. Although a large number of researchers have done a lot of work on the flow characteristics and structural improvement of MHV, few people pay attention to the cavitation effect. The collapse of the cavitation bubble produces high-speed leakage and shock waves, which damages the valve surface and blood components. Cavitation can increase the probability of thromboembolism. Therefore, it is of great significance to study the cavitation effect. In this work, a transient computational model is developed for calculating the flow and cavitation characteristics of mechanical heart valves and analyzing the mechanism of cavitation.

Geometric Model
The structure of the bileaflet mechanical heart valve, which is similar to the St. Jude Medical valve, is shown in Figure 1. The BMHV is mainly composed of valve housing and two leaflets. The two leaflets are connected with the valve housing through a hinge. The material density of the leaflet is 2.116 × 10 3 kg/m 3 , the inner diameter is 22.3 mm, the diameter of the upstream pipe is 25 mm, the length is 16.5 mm, and the length of the downstream pipe is 50 mm. The size of the valve is consistent with that in the literature [41].
The structure of the bileaflet mechanical heart valve, which is similar to the St. Jude Medical valve, is shown in Figure 1. The BMHV is mainly composed of valve housing and two leaflets. The two leaflets are connected with the valve housing through a hinge. The material density of the leaflet is 2.116 × 10 3 kg/m 3 , the inner diameter is 22.3 mm, the diameter of the upstream pipe is 25 mm, the length is 16.5 mm, and the length of the downstream pipe is 50 mm. The size of the valve is consistent with that in the literature [41].

Numerical Model
As illustrated in Figure 1, the flow channel is divided into three parts to generate mesh. Relatively coarse meshes are selected at the inlet and outlet parts, and relatively dense meshes are used at the vicinity of valve leaflet. The mesh re-generation method when the leaflets are moved is shown below.
When the difference between the latest calculation result and the last one is less than the set convergence criterion, it is considered that the current time step calculation converges and enters the next time step calculation. The detailed calculation process during the mesh updating process at each time step is shown in Figure 2. In order to reveal fully the flow characteristics of the blood in the MHV, the software FLUENT is used in this work. The inlet boundary condition is pressure inlet and the outlet boundary condition is pressure outlet. A no-slip boundary condition was imposed on the walls of the leaflets. The blood is considered as an incompressible Newtonian fluid with the density of 1.06 × 10 3 kg/m 3 and the dynamic viscosity of 3.5 × 10 −3 Pa·s. The pressure of saturated steam is 6.343 × 10 3 Pa, which is consistent with water at 37.5 • C. Flow is assumed to be laminar and the boundary conditions at heart rate of 70 bpm are loaded by means of the segmental application method.

Numerical Model
As illustrated in Figure 1, the flow channel is divided into three parts to generate mesh. Relatively coarse meshes are selected at the inlet and outlet parts, and relatively dense meshes are used at the vicinity of valve leaflet. The mesh re-generation method when the leaflets are moved is shown below.
When the difference between the latest calculation result and the last one is less than the set convergence criterion, it is considered that the current time step calculation converges and enters the next time step calculation. The detailed calculation process during the mesh updating process at each time step is shown in Figure 2. In order to reveal fully the flow characteristics of the blood in the MHV, the software FLUENT is used in this work. The inlet boundary condition is pressure inlet and the outlet boundary condition is pressure outlet. A no-slip boundary condition was imposed on the walls of the leaflets. The blood is considered as an incompressible Newtonian fluid with the density of 1.06 × 10 3 kg/m 3 and the dynamic viscosity of 3.5 × 10 −3 Pa·s. The pressure of saturated steam is 6.343 × 10 3 Pa, which is consistent with water at 37.5 °C. Flow is assumed to be laminar and the boundary conditions at heart rate of 70 bpm are loaded by means of the segmental application method. When the blood flows through the MHV, it follows the laws of mass conservation, momentum conservation and energy conservation. Equations are as follows: The equation of continuity is expressed as: When the blood flows through the MHV, it follows the laws of mass conservation, momentum conservation and energy conservation. Equations are as follows: The equation of continuity is expressed as: p is the pressure of the blood; t is the time; ρ is the density of the blood; i is the direction of the blood flow and i = 1, 2, 3.
The moment equation is expressed as: p is the pressure of the blood; g is the acceleration of gravity; τ ij is viscous stress tensor and can be expressed as follows: µ is dynamic viscosity.
The energy equation is expressed as: T is the temperature of the blood; P r is Prandtl number; C v is the specific heat at constant volume; C p is the specific heat at constant pressure.
The k − ε equation of the turbulence model in the turbulent viscosity coefficient method is selected to solve the problem of the turbulent flow in the valve. The transport equations of turbulent kinetic energy and turbulent dissipation rate in the standard k − ε turbulence model are as follows: k is the turbulent kinetic energy; ε is the turbulence dissipation; G k and G b are turbulent kinetic energy due to velocity gradient and buoyancy. C 1ε , C 2ε and C 3ε are model constants.
Checking for grid independence is performed by comparing three different kinds of mesh regimes, respectively with 0.7, 1.4 and 3.5 million cells. As illustrated in Figure 3, the velocity distributions on the symmetrical surface located downstream of the valve are compared when the leaflet is under the half-open stage. The velocity distribution has little difference for the cases with different grids. Therefore, meshing regime with 1.4 million grids is selected. In order to eliminate the influence of grids during the mesh updating process and initial boundary values on the results, four cardiac cycles are simulated exactly. Considering the time and accuracy of calculation, this work selects the most suitable cardiac cycle for analysis.
leaflet is under the half-open stage. The velocity distribution has little difference for the cases with different grids. Therefore, meshing regime with 1.4 million grids is selected. In order to eliminate the influence of grids during the mesh updating process and initial boundary values on the results, four cardiac cycles are simulated exactly. Considering the time and accuracy of calculation, this work selects the most suitable cardiac cycle for analysis.

Experimental Validation
In order to verify the accuracy of the numerical results, the opening time and closing time of the valve are compared with the data in the literature [41]. According to results obtained by Choi, the duration of opening stage is 66.2 ms, the duration of closing stage is 35.9 ms. In this paper, the duration of opening stage is 67.7 ms, the duration of closing stage is 37.4 ms. The relative error of the duration of opening stage is 2.3% and the relative error of the duration of closing stage is 4.2%. It can be found that the relative errors between the numerical results and reference values in the literature are very small, which verifies the accuracy of the numerical method. Figure 4 shows the motion of the leaflets. Initially, leaflets of the valve are in a closing stage. When the inlet pressure increases rapidly and is higher than the outlet pressure at about 0.155 s, leaflets begin to open. When the time reaches about 0.219 s, the leaflet reaches the fully opened stage. When the time reaches about 0.35 s, the leaflet starts to close because the outlet pressure is higher than the inlet pressure. These four stages, respectively, make up of 8.5%, 16.1%, 4.7% and 70.7% of the total period. It can be found that the closing time of the valve is shorter than opening time. It is caused by the difference between the inlet pressure and outlet pressure. The pressure difference is getting larger and larger and leads to the high-speed of valve leaflets during the closing process.

Flow Characteristics of the Bileaflet Mechanical Heart Valve
leaflets begin to open. When the time reaches about 0.219 s, the leaflet reaches the fully opened stage. When the time reaches about 0.35 s, the leaflet starts to close because the outlet pressure is higher than the inlet pressure. These four stages, respectively, make up of 8.5%, 16.1%, 4.7% and 70.7% of the total period. It can be found that the closing time of the valve is shorter than opening time. It is caused by the difference between the inlet pressure and outlet pressure. The pressure difference is getting larger and larger and leads to the high-speed of valve leaflets during the closing process.    From Figure 5, the flow inside the model likes three orifice flows, owing to the existence of the two leaflets. At the beginning of leaflets rotating, the velocity of orifice flows on both sides is higher than that in the middle orifice. As leaflets approach the fully opening stage, the fluid velocity in the middle orifice increases significantly and is larger than that in side orifices. In the meantime, the force acting on the leaflets decreases, so the movement speed of the leaflets decreases. When the outlet pressure is larger than the inlet pressure, the mechanical heart valve starts to close and the flow velocity in the blood vessel starts to decrease. When the leaflets are in the closing stage, the fluid From Figure 5, the flow inside the model likes three orifice flows, owing to the existence of the two leaflets. At the beginning of leaflets rotating, the velocity of orifice flows on both sides is higher than Appl. Sci. 2020, 10, 2548 8 of 13 that in the middle orifice. As leaflets approach the fully opening stage, the fluid velocity in the middle orifice increases significantly and is larger than that in side orifices. In the meantime, the force acting on the leaflets decreases, so the movement speed of the leaflets decreases. When the outlet pressure is larger than the inlet pressure, the mechanical heart valve starts to close and the flow velocity in the blood vessel starts to decrease. When the leaflets are in the closing stage, the fluid behind the valve is basically blocked by leaflets with a strong leakage flow between the leaflets and the valve housing. Figure 6 shows the pressure distribution on the middle surface at different times, where time f indicates the valve just reaches the fully closing stage. As can be seen from the Figure 6, during the opening and closing processes of the valve, the pressure is larger than the saturated vapor pressure of blood, indicating that there is no cavitation in the opening stage. When the leaflet just reaches the fully closing stage, as shown in moment f in Figure 6, the pressure has a dramatical variation at the tip of leaflets. The downstream pressure of the valve increases to near 1 MPa, which is about 10 times that of the normal pressure in the model, while the upstream pressure of the valve is significantly reduced to the saturated vapor pressure. The sharp increase of downstream pressure due to valve closing is a common phenomenon of water hammer in hydrodynamics. It indicates that the occurrence of water hammer may be the cause of cavitation in the bileaflet mechanical heart valve.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 Figure 6. The variation of pressure in the channel. Figure 7 shows the pressure distribution near the valve leaflet at four different times after the valve is closed. It can be found that after the valve closing, the downstream pressure of the valve leaflet back to normal level, which means the effect of water hammer decreases rapidly in a short time. However, the minimum upstream pressure of the valve is still the saturated vapor pressure, indicating that the occurrence of cavitation is also affected by other factors. From the pressure distribution near the leaflet at 230 microseconds after the valve closing, it can be found that the lowpressure area inside the valve decreases gradually and disappears eventually. In general, the cavitation occurs at the moment of valve closing, and the cavitation intensity increases firstly, then decreases gradually and disappears eventually.  Figure 7 shows the pressure distribution near the valve leaflet at four different times after the valve is closed. It can be found that after the valve closing, the downstream pressure of the valve leaflet back to normal level, which means the effect of water hammer decreases rapidly in a short time. However, the minimum upstream pressure of the valve is still the saturated vapor pressure, indicating that the occurrence of cavitation is also affected by other factors. From the pressure distribution near the leaflet at 230 microseconds after the valve closing, it can be found that the low-pressure area inside the valve decreases gradually and disappears eventually. In general, the cavitation occurs at the moment of valve closing, and the cavitation intensity increases firstly, then decreases gradually and disappears eventually.
time. However, the minimum upstream pressure of the valve is still the saturated vapor pressure, indicating that the occurrence of cavitation is also affected by other factors. From the pressure distribution near the leaflet at 230 microseconds after the valve closing, it can be found that the lowpressure area inside the valve decreases gradually and disappears eventually. In general, the cavitation occurs at the moment of valve closing, and the cavitation intensity increases firstly, then decreases gradually and disappears eventually.

Cavitation Characteristics of the Bileaflet Mechanical Heart Valve
To investigate the cavitation characteristics of the bileaflet mechanical heart valve, the minimum pressure in the blood vessel is monitored. Figure 8a shows the minimum pressure with time varying in the blood vessel at the moment of valve closing. It can be found that the time length of the saturated vapor pressure in the tube is about 210 microseconds after the moment of valve closing.

Cavitation Characteristics of the Bileaflet Mechanical Heart Valve
To investigate the cavitation characteristics of the bileaflet mechanical heart valve, the minimum pressure in the blood vessel is monitored. Figure 8a shows the minimum pressure with time varying in the blood vessel at the moment of valve closing. It can be found that the time length of the saturated vapor pressure in the tube is about 210 microseconds after the moment of valve closing. Results are consistent with the experimental results of Lee [42], which prove that numerical results are correct and effective. After valve closing, the volume distributions of the vapor phase at three different times are shown in Figure 8b. It can be found that cavitation mainly occurs on the Results are consistent with the experimental results of Lee [42], which prove that numerical results are correct and effective. After valve closing, the volume distributions of the vapor phase at three different times are shown in Figure 8b. It can be found that cavitation mainly occurs on the upstream of the valve leaflet, away from the rotating axis and in contact with the valve housing. The area where cavitation occurs decreases gradually and moves to the gap between the leaflet and the valve housing over time.
In order to find out the causes of the cavitation in the blood vessel after the valve is closed, the streamlines distribution near valve leaflets are analyzed. Figure 9 shows the streamlines distribution near valve leaflet at four different times after the valve closing. It can be found that in the initial stage of the valve closing, the main flow at downstream of leaflets is blocked but high-speed leakage flow can be found in the gap between leaflets and the valve housing. The maximum instantaneous velocity reaches 11 m/s, and most of the fluid flows directly are driven back to the inlet of the valve. Then, the flow passes through the gap and is driven back to the upstream surface of the leaflet by the larger pressure from the outlet, causing high-speed leakage flow in the blood vessel. Finally, the flow area and the maximum velocity of high-speed leakage flow decrease.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 Figure 9. The streamline distributions after valve closing. Figure 10 shows the pressure distribution near leaflets at different times after the valve closing, where t is the time of valve closing. High-pressure can be found downstream of the leaflet when the valve is closed, and then the upstream pressure and the downstream pressure of leaflets return to balance rapidly. However, the high-speed leakage flow at the gap reacts slower than the pressure field and lasts longer time. Based on the above, it can be inferred that the water hammer effect and the high-speed leakage flow effect are the causes of cavitation phenomenon and high-speed leakage flow plays a dominant role.  Figure 10 shows the pressure distribution near leaflets at different times after the valve closing, where t is the time of valve closing. High-pressure can be found downstream of the leaflet when the valve is closed, and then the upstream pressure and the downstream pressure of leaflets return to balance rapidly. However, the high-speed leakage flow at the gap reacts slower than the pressure field and lasts longer time. Based on the above, it can be inferred that the water hammer effect and the high-speed leakage flow effect are the causes of cavitation phenomenon and high-speed leakage flow plays a dominant role.
where t is the time of valve closing. High-pressure can be found downstream of the leaflet when the valve is closed, and then the upstream pressure and the downstream pressure of leaflets return to balance rapidly. However, the high-speed leakage flow at the gap reacts slower than the pressure field and lasts longer time. Based on the above, it can be inferred that the water hammer effect and the high-speed leakage flow effect are the causes of cavitation phenomenon and high-speed leakage flow plays a dominant role.

Conclusions
Based on the method of passive dynamic mesh, the transient analysis of the MHV is carried out. By monitoring the self-motion of the valve at the moment of the closing and the flow in the valve, the flow characteristics of the blood in the process of periodic motion are studied, and causes of cavitation emerged are analyzed systematically. The method of updating the grid in this work can avoid the failure of dynamic meshing updating, due to small moments of the inertia of leaflets. It is proven that this method is more stable in the process of numerical calculation. The valve goes through opening stage, fully opening stage, closing stage and fully closing stage in a cardiac cycle, and four stages, respectively, make up 8.5%, 16.1%, 4.7% and 70.7% of the total period, respectively. The cavitation of the bileaflet mechanical heart valve is mainly related to high-speed leakage flow through gap between leaflets and the valve housing, as well as the water hammer effect. The occurrence and

Conclusions
Based on the method of passive dynamic mesh, the transient analysis of the MHV is carried out. By monitoring the self-motion of the valve at the moment of the closing and the flow in the valve, the flow characteristics of the blood in the process of periodic motion are studied, and causes of cavitation emerged are analyzed systematically. The method of updating the grid in this work can avoid the failure of dynamic meshing updating, due to small moments of the inertia of leaflets. It is proven that this method is more stable in the process of numerical calculation. The valve goes through opening stage, fully opening stage, closing stage and fully closing stage in a cardiac cycle, and four stages, respectively, make up 8.5%, 16.1%, 4.7% and 70.7% of the total period, respectively. The cavitation of the bileaflet mechanical heart valve is mainly related to high-speed leakage flow through gap between leaflets and the valve housing, as well as the water hammer effect. The occurrence and intensity of cavitation at the sinus can be characterized by the highest velocity of leaflets rotating at the moment of valve closing. The higher the velocity of leaflets rotating at the moment of valve closing, the higher the possibility of cavitation occurring at the sinus. This study reveals the fundamental flow characteristics of the BMHV at different working conditions and can be referred to for the cavitation suppression of the BHMV and similar valves.