Two-dimensional color-mapping of turbulent shear stress distribution downstream of two aortic bioprosthetic valves in vitro.

Since artificial heart valve related complications such as thrombus formation, hemolysis and calcification are considered related to flow disturbances caused by the inserted valve, a thorough hemodynamic characterization of heart valve prostheses is essential. In a pulsatile flow model, fluid velocities were measured one diameter downstream of a Hancock Porcine (HAPO) and a Ionescu-Shiley Pericardial Standard (ISPS) aortic valve. Hot-film anemometry (HFA) was used for velocity measurements at 41 points in the cross-sectional area of the ascending aorta. Three-dimensional visualization of the velocity profiles, at 100 different instants during one mean pump cycle, was performed. Turbulence analysis was performed as a function of time by calculating the axial turbulence energy within 50 ms overlapping time windows during the systole. The turbulent shear stresses were estimated by using the correlation equation between Reynolds normal stress and turbulent (Reynolds) shear stress. The turbulent shear stress distribution was visualized by two-dimensional color-mapping at different instants during one mean pump cycle. Based on the velocity profiles and the turbulent shear stress distribution, a relative blood damage index (RBDI) was calculated. It has the feature of combining the magnitude and exposure time of the estimated shear stresses in one index, covering the entire cross-sectional area. The HAPO valve showed a skewed jet-type velocity profile with the highest velocities towards the left posterior aortic wall. The ISPS valve revealed a more parabolic-shaped velocity profile during systole. The turbulent shear stresses were highest in areas of high or rapidly changing velocity gradients. For the HAPO valve the maximum estimated turbulent shear stress was 194 N m-2 and for the ISPS valve 154 Nm-2. The RBDI was the same for the two valves. The turbulent shear stresses had magnitudes and exposure times that might cause endothelial damage and sublethal or lethal damage to blood corpuscules. The RBDI makes comparison between different heart valves easier and may prove important when making correlation with clinical observations.     

Comparison of the hemodynamic and thrombogenic performance of two bileaflet mechanical heart valves using a CFD/FSI model

The hemodynamic and the thrombogenic performance of two commercially available bileaflet mechanical heart valves (MHVs)--the ATS Open Pivot Valve (ATS) and the St. Jude Regent Valve (SJM), was compared using a state of the art computational fluid dynamics-fluid structure interaction (CFD-FSI) methodology. A transient simulation of the ATS and SJM valves was conducted in a three-dimensional model geometry of a straight conduit with sudden expansion distal the valves, including the valve housing and detailed hinge geometry. An aortic flow waveform (60 beats/min, cardiac output 4 l/min) was applied at the inlet. The FSI formulation utilized a fully implicit coupling procedure using a separate solver for the fluid problem (FLUENT) and for the structural problem. Valve leaflet excursion and pressure differences were calculated, as well as shear stress on the leaflets and accumulated shear stress on particles released during both forward and backward flow phases through the open and closed valve, respectively. In contrast to the SJM, the ATS valve opened to less than maximal opening angle. Nevertheless, maximal and mean pressure gradients and velocity patterns through the valve orifices were comparable. Platelet stress accumulation during forward flow indicated that no platelets experienced a stress accumulation higher than 35 dyne x s/cm2, the threshold for platelet activation (Hellums criterion). However, during the regurgitation flow phase, 0.81% of the platelets in the SJM valve experienced a stress accumulation higher than 35 dyne x s/cm2, compared with 0.63% for the ATS valve. The numerical results indicate that the designs of the ATS and SJM valves, which differ mostly in their hinge mechanism, lead to different potential for platelet activation, especially during the regurgitation phase. This numerical methodology can be used to assess the effects of design parameters on the flow induced thrombogenic potential of blood recirculating devices.     

Dynamic Behavior of Reciprocating Plunger Pump Discharge Valve Based on Fluid Structure Interaction and Experimental Analysis

The influence of spring stiffness and valve quality on the motion behaviors of reciprocating plunger pump discharge valves was investigated by fluid structure interaction (FSI) simulation and experimental analysis. The mathematical model of the discharge valve motion of a 2000-fracturing pump was developed and the discrete differential equations were solved according to FSI and results obtained by ANDINA software. Results indicate that spring stiffness influences the maximum lift, the opening resistance and shut-off lag angle, as well as the fluid velocity of the clearance, the impact stress and the volume efficiency of the pump valve in relation to the valve quality. An optimal spring stiffness parameter of 14.6 N/mm was obtained, and the volumetric efficiency of the pumping valve increased by 4‰ in comparison to results obtained with the original spring stiffness of 10.09N/mm. The experimental results indicated that the mathematical model and FSI method could provide an effective approach for the subsequent improvement of valve reliability, volumetric efficiency and lifespan.     

Velocity and turbulence measurements past mitral valve prostheses in a model left ventricle

Thrombogenesis and hemolysis have both been linked to the flow dynamics past heart valve prostheses. To learn more about the particular flow dynamics past mitral valve prostheses in the left ventricle under controlled experimental conditions, an in vitro study was performed. The experimental methods included velocity and turbulent shear stress measurements past caged-ball, tilting disc, bileaflet, and polyurethane trileaflet mitral valves in an acrylic rigid model of the left ventricle using laser Doppler anemometry. The results indicate that all four prosthetic heart valves studied create at least mildly disturbed flow fields. The effect of the left ventricular geometry on the flow development is to produce a stabilizing vortex which engulfs the entire left ventricular cavity, depending on the orientation of the valve. The measured turbulent shear stress magnitudes for all four valves did not exceed the reported value for hemolytic damage. However, the measured turbulent shear stresses were near or exceeded the critical shear stress reported in the literature for platelet lysis, a known precursor to thrombus formation.     

An in vitro comparative study of St. Jude Medical and Edwards-Duromedics bileaflet valves using laser anemometry

An in vitro comparative study of St. Jude (SJ) and Edwards-Duromedics (DM) Bileaflet valves was performed under steady and physiological pulsatile flow conditions in an axisymmetric chamber using Laser Doppler Anemometry (LDA). LDA measurements were conducted in two different orientations; in the first orientation, the LDA traverse was perpendicular and, in the second orientation, parallel to the tilt axis of the leaflets. The axial velocities were measured in both orientations at two different locations distal to the valves. The velocity profiles at peak systole show the presence of stronger vortex in the sinus region for flow past SJ valve in the first orientation compared to the DM valve. Velocity profile distal to the SJ valve in second orientation was relatively flat where as for the DM valve, a jet-like flow was present. The differences found in the velocity profiles between the two valves can be attributed to the differences in geometry with thicker leaflets, smaller angle of leaflets opening and the presence of the leaflet curvature for the DM valve. The results obtained in this study do not show any fluid dynamic advantages due to the curved leaflet geometry of the DM valve.     

The influence of the leaflets' curvature on the flow field in two bileaflet prosthetic heart valves

A successful mechanical prosthetic heart valve design is the bileaflet valve, which has been implanted for the first time more than 20 years ago. A key feature of bileaflet valves is the geometry of the two leaflets, which can be very important in determining the flow field. Laser Doppler anemometry (LDA) was used to perform an accurate study of the velocity and turbulence shear stress peak values (TSS(max)) fields at four distances from the valve plane. TSS(max) is a relevant parameter to assess the risk of hemolysis and platelet activation associated to the implantation of a prosthetic device, continuously interacting with blood. Two bileaflet valves were tested: the St. Jude HP and the Sorin Bicarbon, of the same nominal size (19mm). The former has flat leaflets, whereas the latter's leaflets have a cylindrical surface. A high regime (CO: 6l/min) was imposed, in order to test the two valves at maximum Reynolds number and consequent turbulence generation. The flat-leaflet design of the St. Jude generates a TSS field constant with distance; on the contrary, the Bicarbon's shear stress field undergoes an evident development, with an unexpected central peak at a distance comparable to the valve's dimensions (21mm). The two bileaflet valves tested, although very similar in design, behave very differently as for their turbulence properties. In particular, the concept of curved wake leads to conclude that the curvature of the leaflets' surface must be identified as an important parameter, which deserves careful attention in PHV design and development.     

In-vitro fluid dynamic characteristics of the abiomed trileaflet heart valve prosthesis

The need for better and longer lasting trileaflet valves has led to the design and development of the Abiomed polymeric trileaflet valve prosthesis. In-vitro fluid dynamic studies on sizes 25 and 21 mm valves in the aortic position indicate an overall improvement in performance compared to the Carpentier-Edwards and Ionescu-Shiley tissue valves in current clinical use. The pressure drop studies yielded effective orifice areas of 1.99 and 1.54 cm2, and performance indices of 0.41 and 0.45 for the Nos. 25 and 21 valves, respectively. Leaflet photography studies indicated that the two valve sizes had maximum opening areas of 225 and 145 mm2, respectively, at a normal resting cardiac output. Steady and pulsatile flow velocity measurements with a laser-Doppler anemometer (LDA) system indicate that the flow field downstream of the Abiomed valve is jetlike and turbulent. Maximum mean square axial velocity fluctuations of 55 and 83 cm/s, and turbulent shear stresses of 220 and 450 N/m2 were measured in the immediate vicinity of the nos. 25 and 21 valves, respectively. The Abiomed valves studied had been originally configured for use in valved conduits, and it is therefore our opinion that further improvements can be made to the valve and stent design, which would enhance its fluid dynamic performance.     

Estimation of turbulent shear stresses in pulsatile flow immediately downstream of two artificial aortic valves in vitro

Measuring turbulent shear stresses is of major importance in artificial heart valve evaluation. Bi- and unidirectional fluid velocity measurements enable calculation of Reynolds shear stress ( ) and Reynolds normal stress ( ). τ is important due to the relation to hemolysis and thrombus formation, but σ is the only obtainable parameter in vivo. Therefore, determination of a correlation factor between τ and σ is pertinent.

In a pulsatile flow model, laser Doppler (LDA) and hot-film (HFA) anemometry were used for simultaneous bi- and unidirectional fluid velocity measurements downstream of a Hall Kaster and a Hancock Porcine aortic valve. Velocities were registered in two flow field locations and at four cardiac outputs. The velocity signals were subjected to analog signal processing prior to digital turbulence analysis, as a basis for calculation of τ and σ.

A correlation factor of 0.5 with a correlation coefficient of 0.97 was found between the maximum Reynolds shear stress and Reynolds normal stress, implying . In vitro estimation of turbulent shear stresses downstream of artificial aortic valves, based on the axial velocity component alone, seems possible.     

Hydrodynamic in vitro comparison of a new artificial heart valve with the Bj?rk-Shiley valve (standard)

Measurements performed to compare a newly developed tilting disc valve with the Bj?rk-Shiley valve included velocity profiles downstream of the heart valves, valve-induced flow turbulence and pressure drop across the opened valves. The velocity profiles measured with pulsed Doppler ultrasound are similar, although they do not permit a quantitative comparison of the valves. The interpretation of the 90 degrees-component of Doppler signals as a measure of the turbulence permits a quantitative comparison without the need for extensive measurements. However, only large vortices are recorded, so that our turbulent shear stresses are lower than these reported in the literature. The pressure drop across the opened valve is a measure of the energy loss, and important parameters for the valve can be derived from it. The pressure drop is dependent on the test conditions, and is therefore not a characteristic constant of the valve. The transformation of the power law Q = C delta P beta into a relation between Re- and Eu-number gives a nondimensional similarity number that is characteristic for tilting disc valves. Its verification requires more investigations, involving variation of valve size and the viscosity of the test fluid.     

Three-dimensional visualization of stationary flow behind artificial heart valves]

The visualization and quantitative analysis of flow offers a possibility for the hydrodynamic characterization of artificial heart valves. Different types of valves can be compared if velocity profile and the turbulent shear stress caused by the prosthesis are known. The tracer technique was selected, since it permits visualization also of turbulent flow through the valve. With the aid of a simple optical device the three-dimensional flow pattern behind the valve is determinable. The main features of the method are: The regions of interest can easily be identified. Velocity profiles can be determined and shear stress and turbulence intensities estimated. The experimental setup is simple, calibration is not necessary, and it can be used for turbulent flows. The method can be used only with transparent fluids and vessels; measurements in blood are not possible. Because of the large number of measuring points required the method is very time-consuming. The use of an automatic picture analyzing system would make it possible to increase the number of pictures processed, and thus increase resolution. The velocity profile of a three-finger-valve, the TAD 29, was established at a distance of 20 mm from the ring, and compared with known profiles from the literature. The valve has an opening angle of 70 degrees. All typical regions for the flow of an artificial heart valve, such as jet, stagnation gone, backflow and turbulence were demonstrated.     

Two-component laser velocimeter measurements downstream of heart valve prostheses in pulsatile flow

Elevated turbulent shear stresses resulting from disturbed blood flow through prosthetic heart valves can cause damage to red blood cells and platelets. The purpose of this study was to measure the turbulent shear stresses occurring downstream of aortic prosthetic valves during in-vitro pulsatile flow. By matching the indices of refraction of the blood analog fluid and model aorta, correlated, simultaneous two-component laser velocimeter measurements of the axial and radial velocity components were made immediately downstream of two aortic prosthetic valves. Velocity data were ensemble averaged over 200 or more cycles for a 15-ms window opened at peak systolic flow. The systolic duration for cardiac flows of 8.4 L/min was 200 ms. Ensemble-averaged total shear stress levels of 2820 dynes/cm2 and 2070 dynes/cm2 were found downstream of a trileaflet valve and a tilting disk valve, respectively. These shear stress levels decreased with axial distance downstream much faster for the tilting disk valve than for the trileaflet valve.     

A detailed fluid mechanics study of tilting disk mechanical heart valve closure and the implications to blood damage

Hemolysis and thrombosis are among the most detrimental effects associated with mechanical heart valves. The strength and structure of the flows generated by the closure of mechanical heart valves can be correlated with the extent of blood damage. In this in vitro study, a tilting disk mechanical heart valve has been modified to measure the flow created within the valve housing during the closing phase. This is the first study to focus on the region just upstream of the mitral valve occluder during this part of the cardiac cycle, where cavitation is known to occur and blood damage is most severe. Closure of the tilting disk valve was studied in a "single shot" chamber driven by a pneumatic pump. Laser Doppler velocimetry was used to measure all three velocity components over a 30 ms period encompassing the initial valve impact and rebound. An acrylic window placed in the housing enabled us to make flow measurements as close as 200 microm away from the closed occluder. Velocity profiles reveal the development of an atrial vortex on the major orifice side of the valve shed off the tip of the leaflet. The vortex strength makes this region susceptible to cavitation. Mean and maximum axial velocities as high as 7 ms and 20 ms were recorded, respectively. At closure, peak wall shear rates of 80,000 s(-1) were calculated close to the valve tip. The region of the flow examined here has been identified as a likely location of hemolysis and thrombosis in tilting disk valves. The results of this first comprehensive study measuring the flow within the housing of a tilting disk valve may be helpful in minimizing the extent of blood damage through the combined efforts of experimental and computational fluid dynamics to improve mechanical heart valve designs.     

Pulsatile flow past aortic valve bioprostheses in a model human aorta

Pulsatile flow development past tissue valve prostheses in a model human aorta has been studied using qualitative flow visualization and quantitative laser-Doppler techniques. Experiments were conducted both in steady and physiological pulsatile flow situations and the measurements included the pressure drop across the valve, the instantaneous flow rate as well as the velocity profiles and turbulent stresses downstream to the valves. Our study shows that the velocity profiles with pericardial valves are closer to those measured past natural aortic valves. The porcine valves with a smaller valve opening area produce a narrower and stronger jet downstream from the valve with relatively larger turbulent axial stresses in the boundary of the jet. Our study suggests that the pericardial valves with turbulent stresses comparable to those of caged ball and tilting disc valves are preferable from a hemodynamic point of view.     

A numerical analysis of the backflow between the leaflets of a St Jude Medical cardiac valve prosthesis

Clinically significant, unexplained hemolysis has been reported to occur in St Jude Medical (SJM) cardiac valve prostheses. The leakage phase of backflow is identified as having the most hemolytic potential, when compared with the other phases of the cardiac cycle. A two-dimensional, laminar, constant fluid property finite element analysis (FEA) is used to calculate the peak shear stress obtained in the flow through the narrow slit formed by the closed leaflets of an idealized size 29 mm (tissue annulus diameter) SJM cardiac valve prosthesis, during the leakage phase of backflow. The flow geometry was such that the simple laminar flow theory (SLFT) could be used to model the flow through the region of peak shear stresses with reasonable accuracy. The SLFT predicts that the maximum shear stress depends upon the thickness of the clearance space and the average velocity through that space. These results suggest that the magnitude of the peak shear stress is of the order of 700-1000 Pa for a duration of the order of 0.5-0.4 ms for pressure drops across the valve of 150-300 torr. This suggests that hemolysis is possible for certain unfortunate combinations of clearances and pressure conditions. However, further research is needed before this flow phenomenon can be associated with the reported clinical hemolysis.     

Estimation of turbulent shear stresses in pulsatile flow immediately downstream of two artificial aortic valves in vitro

Measuring turbulent shear stresses is of major importance in artificial heart valve evaluation. Bi- and unidirectional fluid velocity measurements enable calculation of Reynolds shear stress ( ) and Reynolds normal stress ( ). τ is important due to the relation to hemolysis and thrombus formation, but σ is the only obtainable parameter in vivo. Therefore, determination of a correlation factor between τ and σ is pertinent.In a pulsatile flow model, laser Doppler (LDA) and hot-film (HFA) anemometry were used for simultaneous bi- and unidirectional fluid velocity measurements downstream of a Hall Kaster and a Hancock Porcine aortic valve. Velocities were registered in two flow field locations and at four cardiac outputs. The velocity signals were subjected to analog signal processing prior to digital turbulence analysis, as a basis for calculation of τ and σ.A correlation factor of 0.5 with a correlation coefficient of 0.97 was found between the maximum Reynolds shear stress and Reynolds normal stress, implying . In vitro estimation of turbulent shear stresses downstream of artificial aortic valves, based on the axial velocity component alone, seems possible.     

Three-dimensional visualization of velocity fields downstream of six mechanical aortic valves in a pulsatile flow model

Velocity fields downstream of 27 mm Bj?rk-Shiley Standard, Bj?rk-Shiley Convex-Concave, Bj?rk-Shiley Monostrut, Hall-Kaster (Medtronic-Hall), St. Jude Medical and Starr-Edwards Silastic Ball aortic valves were studied in a pulsatile mock circulation. Stroke volume was 70 cm3 and frequency 71 min-1 and 88 min-1. Fluid velocity was measured by a catheter mounted hot-film anemometer probe in a glycerol water mixture one and two diameters downstream of the aortic valve. Velocity fields were dynamically visualized by a three-dimensional technique and revealed qualitative independence of frequency. All profiles were flat in the acceleration phase of systole. From peak systole and throughout the systolic deceleration phase profiles characteristic of the individual valves appeared. The pivoting and tilting disc valves caused a skewed velocity profile with highest velocities downstream of the major orifice and lowest velocities downstream of the minor orifice. The differences between the three investigated Bj?rk-Shiley valves were remarkable. The St. Jude Medical valve generated velocity peaks downstream of the two major orifices and the central slit, and lower velocities in the hinge areas. A rather flat profile with central hollowing was seen downstream of the Starr-Edwards Ball valve. All velocity profiles were more or less dampened two diameters downstream.     

Turbulence downstream of subcoronary stentless and stented aortic valves

Regions of turbulence downstream of bioprosthetic heart valves may cause damage to blood components, vessel wall as well as to aortic valve leaflets. Stentless aortic heart valves are known to posses several hemodynamic benefits such as larger effective orifice areas, lower aortic transvalvular pressure difference and faster left ventricular mass regression compared with their stented counterpart. Whether this is reflected by diminished turbulence formation, remains to be shown. We implanted either stented pericardial valve prostheses (Mitroflow), stentless valve prostheses (Solo or Toronto SPV) in pigs or they preserved their native valves. Following surgery, blood velocity was measured in the cross sectional area downstream of the valves using 10MHz ultrasonic probes connected to a dedicated pulsed Doppler equipment. As a measure of turbulence, Reynolds normal stress (RNS) was calculated at two different blood pressures (baseline and 50% increase). We found no difference in maximum RNS measurements between any of the investigated valve groups. The native valve had significantly lower mean RNS values than the Mitroflow (p=0.004), Toronto SPV (p=0.008) and Solo valve (p=0.02). There were no statistically significant differences between the artificial valve groups (p=0.3). The mean RNS was significantly larger when increasing blood pressure (p=0.0006). We, thus, found no advantages for the stentless aortic valves compared with stented prosthesis in terms of lower maximum or mean RNS values. Native valves have a significantly lower mean RNS value than all investigated bioprostheses.     

Numerical simulation of cardiovascular dynamics with different types of VAD assistance

A variety of methods by which mechanical circulatory support (MCS) can be provided have been described. However, the haemodynamic benefits of the different methods have not been adequately quantified. The aim of this paper is to compare the haemodynamic effects of six forms of MCS by numerical simulation. Three types of ventricular assist device (VAD) are studied: positive displacement; impeller and a novel reciprocating-valve design. Similarly, three pumping modes are modelled: constant flow; counterpulsation and copulsation. The cardiovascular system is modelled using an approach developed previously, using the concentrated parameter method by considering flow resistance, vessel elasticity and inertial effects of blood in individual conduit segments. The dynamic modelling of displacement and impeller pumps is represented by VAD inlet/outlet flow-rate changes. The dynamics of the reciprocating-valve pump is modelled with a specified displacement profile. Results show that in each simulation, the physiological variables of mean arterial pressure and systemic flow are adequately maintained. Modulation of the impeller pump flow profile produces a small (5 mmHg) oscillatory component to arterial pressure, whereas the displacement and reciprocating-valve pumps generate substantial arterial pressure and flow pulsatility. The impeller pump requires the least power input, the reciprocating valve pump slightly more, and the displacement pump the most. The in parallel configuration of the impeller and displacement pump designs with respect to the left ventricle provides near complete unloading and can cause the aortic valve to remain closed throughout the entire cardiac cycle with the attendant risk of aortic valve leaflet fusion following prolonged support. The in series configuration of the reciprocating-valve pump avoids this shortcoming but activation must be carefully synchronized to the cardiac cycle to allow adequate coronary perfusion. The reciprocating-valve pump is associated with haemodynamic advantages and a favourable power consumption.     

Overview of SHOT hardware capabilities and range of cell biology experiment designs

Cell science hardware is currently available for lease or production by SHOT(R). The SHOT Avian Development Facility (ADF) is a single middeck locker experiment containing two carousels capable of rotating at variable g. Each carousel accommodates 18 40-ml vessels that can be adapted for the cultivation of avian or reptilian eggs, small invertebrates, zebra fish, small plants, seeds, spores, cells or tissues. Cellcult Cassettes contain a small (20-50 ml) rotating cylindrical culture vessel with feed media reservoir, pumps, waste bag and sample collection bags. Cultures can be rotated, perfused, fed, aerated and sampled automatically or on remote command. Fluid Processing Cassettes contain a flexible combination of feed, culture and fixative bags with valves and pumps programmable for various feeding and fixing protocols. Dynacult Cassettes contain a cylindrical bioreactor with differentially rotating walls, pumps and valves for feeding and sampling, and pH and oxygen measurement. All cassettes are doubly contained and are accommodated by a computerized, thermally controlled processing facility, the Advanced Space Experiment Processor, which processes three cassettes, or the Biotechnology Thermal Environment Carrier, which holds up to six cassettes.     

Comparison of hinge microflow fields of bileaflet mechanical heart valves implanted in different sinus shape and downstream geometry

The characterization of the bileaflet mechanical heart valves (BMHVs) hinge microflow fields is a crucial step in heart valve engineering. Earlier in vitro studies of BMHV hinge flow at the aorta position in idealized straight pipes have shown that the aortic sinus shapes and sizes may have a direct impact on hinge microflow fields. In this paper, we used a numerical study to look at how different aortic sinus shapes, the downstream aortic arch geometry, and the location of the hinge recess can influence the flow fields in the hinge regions. Two geometric models for sinus were investigated: a simplified axisymmetric sinus and an idealized three-sinus aortic root model, with two different downstream geometries: a straight pipe and a simplified curved aortic arch. The flow fields of a 29-mm St Jude Medical BMHV with its four hinges were investigated. The simulations were performed throughout the entire cardiac cycle. At peak systole, recirculating flows were observed in curved downsteam aortic arch unlike in straight downstream pipe. Highly complex three-dimensional leakage flow through the hinge gap was observed in the simulation results during early diastole with the highest velocity at 4.7 m/s, whose intensity decreased toward late diastole. Also, elevated wall shear stresses were observed in the ventricular regions of the hinge recess with the highest recorded at 1.65 kPa. Different flow patterns were observed between the hinge regions in straight pipe and curved aortic arch models. We compared the four hinge regions at peak systole in an aortic arch downstream model and found that each individual hinge did not vary much in terms of the leakage flow rate through the valves.