Statistical Characteristics of Flow Field through Open and Semi-Closed Bileaflet Mechanical Heart Valve

The formation of thrombi on the streamlined surface of the bileaflet mechanical heart valves is one of the main disadvantages of such valves. Thrombi block the valve leaflets and disrupt the cardiovascular system. Diagnosis of thrombosis of the bileaflet mechanical heart valves is relevant and requires the creation of effective diagnostic tools. Hydroacoustic registration of the heart noise is one of the methods for diagnosing the operation of a mechanical heart valve. The purpose of the research is to determine the statistical characteristics of the vortex and jet flow through the open and semi-closed bileaflet mechanical heart valve, to identify hydroacoustic differences and diagnostic signs to determine the operating conditions of the valve. Experimental studies were conducted in laboratory conditions on a model of the left atrium and left ventricle of the heart between which there was the bileaflet mechanical heart valve. Hydrodynamic noise was recorded by miniature pressure sensors, which were located downstream of the valve. The vortex and jet flow behind the prosthetic heart valve were non-linear, random processes and were analyzed using the methods of mathematical statistics and probability theory. The integral and spectral characteristics of the pressure field were obtained and the differences in the noise levels and their spectral components near the central and side jets for the open and semi-closed mitral valve were established. It was shown that hydroacoustic measurements could be an effective basis for developing diagnostic equipment for monitoring the bileaflet mechanical heart valve operation.


Introduction
The most important thing in human life is the smooth functioning of the cardiovascular system. Directed transfer of blood through arteries, veins, capillaries and other blood vessels, which is provided by the heart and valves, which are located between the atria and ventricles, allows a person to live. The heart is a biological muscle pump that supplies the body with the oxygenated blood and nutrients and takes away the blood saturated with carbon dioxide and waste metabolic products. The heart is divided by a partition into two cavities, which serve to collect the blood (left and right atria) and for muscle pumping of the blood (left and right ventricles). Between the atria and the ventricles are valves (mitral and tricuspid), which open when the ventricles are filled with the blood. From the ventricles, the blood through the aortic and pulmonary valves enters inside the aorta and pulmonary arteries, through which the blood enters into the lungs, where it is saturated with oxygen. These valves are opened when the ventricles are contracted and the mitral and 185 tricuspid valves are closed. Thus, the heart creates a unidirectional movement of the blood along the large (left side of the heart) and small (right side of the heart) circulation of the blood of the cardiovascular system and interferes with regurgitation or the reverse flow of the blood [1][2][3][4].
The cardiac cycle of the blood circulation in the human cardiovascular system consists of a diastole phase and a systole phase. When the left ventricle relaxes, the mitral valve (nature two-leaflets) is opened and the oxygenated blood in the lungs passes from the left atrium into the left ventricle through it. After the opening of the mitral valve, the blood velocity through the valve is increased and reaches its maximum in the diastole phase (wave E), and then it is decreased and flows with the small decreasing velocity (diastasis). Then the left atrium is contracted and in the blood flow again the velocity increase is observed (wave A). Typically, the maximum velocity in the wave E is higher than in the wave A. After this, the muscles of the left ventricle begin to contract (myocardial contraction) and the mitral valve is closed. When the pressure in the left ventricle exceeds the pressure in the aorta, then the aortic valve (nature three-leaflets) is opened. The blood flow in the systole phase is rushed inside the aorta and then inside other blood vessels of the circulatory system. Systole lasts about a third of the cardiac cycle. A similar cardiac cycle takes place in the small circulation of the blood of the cardiovascular system through the right atrium, the tricuspid valve, the right ventricle of the heart and the pulmonary valve [5][6][7][8].
Numerical and physical modeling of the features of the blood flow inside the cardiovascular system and the heart function has a long and rich history. Scientists all over the world pay considerable attention to this field of knowledge. This helps to reduce the number of diseases of the circulatory system and the entire cardiovascular system as a whole. New knowledge makes it possible to improve therapeutic and surgical methods of the treatment, to create effective artificial elements and designs of the replacement of damaged components of the cardiovascular system [1,9]. Experimental studies of the blood movement and the features of the vortex and jet flow near the heart valves are a complex task for in vivo and in vitro studies [10][11][12]. This is due to the spatial and temporal limitations of the measuring instruments, the opacity of the blood flow and its non-Newtonian properties, the need not to disturb the flow and the complexity, and sometimes the impossibility of direct measurements inside the heart. In recent years, laser Doppler velocity meters, ultrasound and tomography complexes, microPIV technique and other measuring instruments have been used in experiments [13][14][15][16][17]. But they have their advantages and disadvantages. Many new scientific results were obtained by numerical simulations using various methods and techniques [18][19][20][21][22][23][24] that require experimental confirmation and verification. Therefore, an urgent task of modern scientific hemodynamic and biomechanic studies is conducting experimental studies with high-precision and miniature measuring instruments [25][26][27][28]. These instruments must work in opaque non-Newtonian moving media and do not disturb the blood flow.
Heart diseases are one of the leading causes of death in people around the world. Pathologies of one or more valves lead to malfunction of the heart and the entire cardiovascular system. Most often, pathologies are recorded with the aortic valve, since it withstands the greatest pressure drops between the left ventricle and the main artery, which supplies the blood to the arterial system of the pulmonary circulation. Valve pathologies often require surgical treatment. More than 300,000 artificial heart valves are implanted annually in the world and among them about 70% are mechanical heart valves [12,21]. Bileaflet mechanical heart valves are the most common prosthetic valves. Typically, valve leaflets are made of pyrolytic carbon in the form of two lunate surfaces. The leaflets are attached to the annular surface of the valve base by means of hinge joints. Between the leaflets and the valve ring there is a technological gap of the order of 150 microns. It should be noted that this gap is one of the design flaws of the valve [29,30].
Bileaflet mechanical heart valves have increased durability compared to bioprostheses [1,17,31], but are subject to hemolysis, platelet activity, and thrombosis, thromboembolism. Therefore, for patients with such valves, life-long anticoagulation therapy is necessary [1,2,17,32,33]. These defects are caused by the non-physiological dynamics of the blood flow through the valve, the rigidity of the valve leaflets, the formation of intense vortex and jet flows, and the increased shear stresses [34][35][36]. Intensive backflows are formed in the gaps of the hinged joints of the leaflets and the base of the valve, which destroy the red blood cells and platelet shells. Here thrombi are formed, which are attached to the valve leaflets in the area of stagnant flows near the hinged joints [2,34,37,38]. Cavitation areas with microbubbles in the blood flow are formed, during the closing of the leaflets of the mechanical valves due to a sharp decrease in pressure. Cavitation destroys the blood structure and elements and cause erosion of the streamlined surfaces [34,[39][40][41]. The formation of the thrombi on the surface of the leaflets of the bileaflet mechanical heart valve leads to serious interferences with the operation of the valve leaflets. Thrombi interfere with the opening of the leaflets or the closing of them, which leads to the regurgitation of the blood flow, the stenosis and the changes to the geodynamics of the cardiovascular system and metabolism.
This seriously disrupts the human cardiovascular system and leads to death when two leaflets of the heart valve will be closed. In order to prevent such a situation, it is necessary to diagnose the operation of mechanical valves and take the necessary measures to reduce the formation of thrombi by both therapeutic and surgical methods. In this regard, there is an urgent need to develop and create effective methods and tools for diagnosing heart valves. Among 186 such methods it is offered to use hydroacoustic measurements of the noise for the operating conditions of the open and semi-closed bileaflet mechanical heart valve and on the basis of these measurements to reveal diagnostic signs that emphasize the urgency of carrying out such medical actions.
The purpose of this research is to determine the statistical characteristics of the vortex and jet flow through the open and semi-closed bileaflet mechanical heart valve, to identify hydroacoustic differences and diagnostic signs to determine the operating conditions of the valve.

Materials and Methods
When thrombi overlap the leaflet of the heart valves, their input sections are reduced. This leads to a significant increase in blood flow drag and increases flow velocity when fluid passes through the open valve leaflet. As a result, intense vortex and jet structures are separated from the valve leaflets, and the flow becomes unsteady and unstable with a high degree of turbulence. It is known that turbulent flows generate intense hydrodynamic noise, which has a sound and pseudo-sound nature [42][43][44]. Sound fluctuations of the velocity and pressure are associated with the compressibility of the medium, propagate into the environment with the sound velocity, which is determined by the elasticity of the medium, and obey the principle of superposition. Pseudo-sound fluctuations of the velocity and pressure are generated by vortex structures and jet flows, and these fluctuations are transferred with the velocity of movement of the vortex structures and jets. Pseudo-sound fluctuations in an unsteady turbulent flow are a nonlinear phenomenon and do not obey the principle of superposition. The intensity of the pseudo-sound fluctuations decays with distance from the source in proportion to the square of the distance.
Therefore, if to place the pressure fluctuation sensors near the jets that flow from the heart valve, they can record changes in hydrodynamic noise. This principle underlies the hydroacoustic diagnosis of the operation of the mitral valve. Naturally, pressure fluctuation sensors cannot be placed inside a person's heart. Therefore, heart noise sensors and accelerometers were located on the surface of the measuring bench in addition. Thus, hydrodynamic noises were recorded inside the model of the left ventricle, and noise and vibration fields were simultaneously recorded on the surface of the experimental bench. This makes it possible to determine the degree of transformation of hydrodynamic noise inside the bench into external noise and vibration on its surface. Differences in the intensity and spectral composition of the noise and vibration on the external surface of the heart model are the basis for creating a diagnostic complex of the operation conditions of the heart valve during thrombosis.
It is known that perturbations and instabilities in separating and jet flows, which are caused by the turbulence, are random processes [45,46]. Such processes are investigated using statistical analysis, which reflects the average and expected behavior of the characteristic properties and hydrodynamic parameters of such flows. In the formal representation of random processes, the vector components of velocity or vorticity, pressure and acceleration have a certain probability of accepting a specific value. A statistical description of such variables was carried out by obtaining integral and spectral characteristics. Four central statistical moments were determined in the studies, namely, the average values of the random variables, the probability density functions, variances and root-mean-square values of variables, skewness and kurtosis coefficients, and spectral characteristics of random variables. These functions and coefficients made it possible to determine the sources of pressure fluctuations and vibrations, their scales, directions and transfer velocities, intensities and spectral components [47,48]

Experimental Setup
Physical simulation of the flow and overlap of the leaflets of mechanical heart valves by thrombi was performed in the laboratory at the Technical University "Politecnico di Milano" (Italy). A bileaflet mechanical heart valve ( Figure 1) from Sorin Biomedica Cardio (Italy) with a diameter of d = 25 mm was installed in the position of the mitral valve between the model of the left atrium and the model of the left ventricle [49,50]. The studies were performed for the continual and pulsating flow of pure water and an aqueous solution of glycerol through an open and semi-closed valve. Behind the open bileaflet valve, the flow of fluid through the valve was divided into three jets: one narrow central jet between the leaflets and two "crescent" side jets between the leaflets and the annular base of the valve.
To solve this problem, miniature pressure, noise and acceleration sensors have been developed and manufactured. The pressure sensors were installed in a well-streamlined sensor's block, which consisted of five pressure fluctuation and absolute pressure sensors, which were mounted flush with the streamlined surface of this block and located at different distances downstream from the bileaflet valve (Figure 1b).

Figure 2. Schematic diagram (a) and photo (b) of the experimental stand
To conduct physical simulation of fluid flow through the bileaflet mechanical heart valve, an experimental stand was created [49][50][51], which is shown in Figure 2 The electrical signals of the sensors were amplified, filtered and fed through a 16-channel analog-to-digital converter to a personal computer, where the appropriate programs and algorithms processed and analyzed the experimental dates using the apparatus of probability theory and mathematical statistics [52]. Simultaneous multipoint recording of pressures, noise and vibrations allowed investigating the space-time characteristics of hydroacoustic and vibration parameters, to identify sources of noise and vibration [48,49,53]. The measurement error of the integral values of the velocity and pressure fields, and also vibration did not exceed 10% (95% reliability). The measurement error of the flow rate is no more than 3%. The measurement error of the spectral characteristics of the velocity, pressure fluctuations and accelerations is no more than 2 dB in the frequency range from 0.01 Hz to 2 kHz with a confidence probability of 0.95 or 2 . It is known [52] that the characteristic measure of the random variable (pressure fluctuations) is the probability () Pp  that the random variable ( p ) will take one value or another or falls into a given interval ( p  ). The first derivative of the distribution function of the random variable () pp  sets the velocity of change of the probability depending on the value of the pressure fluctuations. Usually, the function () pp  is estimated by calculating the probability that the instantaneous value of an individual implementation is in a narrow interval and is divided by the width of the interval according to the dependence:

Results and Discussion
The total area under the probability density graph is equal to unity, which indicates the reliability of the event. Therefore, the probability density is one of the main functions of describing the probable structure of the random process. It is also used to determine the normality of the process, identify non-linearity and analyze extreme values. The probability density functions of the pressure fluctuations near the side jet of the open and semi-closed valve, through which the steady and pulsating stream flows, are calculated according to dependence (3) and shown in Figure 5. Curve 1 in Figure 5a presents the results of measurements of the pressure fluctuations near the side jet of the pure water flow with flow rate of 15 l/min through the open valve. Curve 2 is obtained for the same research conditions, but for the semi-closed valve. Curves 3-6 were measured for the steady flow of the glycerol solution of different concentrations. Thus, curve 3 was measured near the side jet of the open valve, through which flowed the solution of 43% glycerol and 57% water. Curve 4 was measured for the semi-closed valve and 35% glycerol in the aqueous solution, curve 5 -38% glycerol and curve 6 -43% glycerol. Figure 5b shows the results of measurements of the hydrodynamic noise near the side jet of the pulsating flow of the pure water with a frequency ( f ) of 1 Hz or 60 beats per minute. Curves 1, 2 were measured for the open valve conditions, and curves 3, 4 were measured for the semi-closed valve conditions. Curve 1 was measured at the distance equal to the diameter ( d ) of the valve, curve 2 was measured at the distance x =1.2 d , curve 3 was measured at the distance x = d , and curve 4 was measured at the 190 distance x =1.2 d . It should be noted that the dependencies in Figure 5a are characteristic of random processes ("bellshaped" shape of the curve), and the dependencies in Figure 5b represent a set of the random process with the oscillatory determinate process [10,12]. This is not surprising, because in the pulsating flow through the open or semiclosed valve, harmonic oscillations with cardiac rhythm are superimposed on the field of stochastic pressure fluctuations of the jet and vortex flow. The pressure fluctuations near the side jet of the open valve, regardless of the viscosity of the fluid, have a low probability of high-amplitude pressure fluctuations. With increasing viscosity of the glycerol solution, the pressure fluctuations of small amplitude near the side jet of the steady flow through the semiclosed valve are significantly damped, which is clearly illustrated in Figure 5a.
The central statistical moments, the skewness coefficient (third moment) was determined: d . The results of the research showed that with the distance from the valve the difference of spectral levels in the hydrodynamic noise of the semi-closed valve and the open valve gradually decreases. The difference in spectra remains larger near the side jet and remains (2-3) times higher in the frequency range from 0.1 Hz to almost 100 Hz. It should also be noted that the hydrodynamic noise near the side jet of the semi-closed valve is higher (4)(5) times in the frequency range  Hz than near the same side jet, but the open valve (Figure 10a). Such a large difference in spectral levels in this frequency range can be used in hydroacoustic diagnostics of the operation of the bileaflet mechanical heart valve.
Thus, based on a statistical analysis of the results of experimental studies in laboratory conditions of the operation of the bileaflet mechanical heart valve at the position of the mitral valve, the hydroacoustic characteristics of the vortex and jet flow through the valve were obtained. The features of the hydrodynamic noise in the near wake of the valve were studied when modelling the conditions of thrombi on one of the valve leaflets. The integral and spectral characteristics of the pressure field were obtained inside the model of the left ventricle of the heart and in the atrium. The changes in the statistical moments of the pressure fluctuations depending on the operating conditions of the mitral valve for the flow of the pure water and the glycerol solution of the steady and pulsating flow are shown. The levels of the spectral components of the pressure fluctuation field were determined depending on the distance to the valve for various flow rates through the open and semi-closed valve. Differences in the levels of the hydrodynamic noise in the near wake of the open and semi-closed mitral valve were revealed. It is indicated that hydroacoustic diagnostics, especially in certain frequency ranges, can be an effective means of determining thrombosis on the bileaflet mechanical heart valve.

Conclusion
The results of the research showed that the statistical analysis of the experimental data of the researches of the hydroacoustic characteristics of the jet and vortex flow downstream from the bileaflet mechanical heart valve is an effective tool in diagnosing the operating conditions of the valve. It was found that the integral characteristics and especially the spectral characteristics of the pressure fields had significant differences in the operating conditions of Differences in hydrodynamic characteristics of the steady and pulsating flow through the open and semi-closed valve are revealed. The intensity of the pressure fluctuations near the side jet for the semi-closed valve is (3-4) times higher than for the open valve. It was found that the hydrodynamic noise of the semi-closed valve is several times higher than that noise of the open valve, which is especially evident in different frequency ranges of the pressure fluctuation spectrum depending on the operating conditions of the bileaflet mechanical heart valve. The research results show that the highest levels of the pressure fluctuations in the near wake of the valve are observed in the frequency range (5)(6)(7)(8)(9)(10)(11)(12) Hz. This frequency range of the hydrodynamic noise is due to the features of the vortex and jet flow, which is formed downstream from the open or semi-closed valve. In the frequency range from 30 Hz to 100 Hz, there is a significant difference between the hydrodynamic noise levels for the operating conditions of the open and semi-closed mitral valve. The hydrodynamic noise near the side jet of the semi-closed valve is higher (4-5) times in the frequency range (30-80) Hz than near the same side jet, but the open valve.

Data Availability
All data included in this study are available upon request by contact with the corresponding authors.