Dynamic modelling of prosthetic chorded mitral valves using the immersed boundary method

Current artificial heart valves either have limited lifespan or require the recipient to be on permanent anticoagulation therapy. In this paper, effort is made to assess a newly developed bileaflet valve prosthesis made of synthetic flexible leaflet materials, whose geometry and material properties are based on those of the native mitral valve, with a view to providing superior options for mitral valve replacement. Computational analysis is employed to evaluate the geometric and material design of the valve, by investigation of its mechanical behaviour and unsteady flow characteristics. The immersed boundary (IB) method is used for the dynamic modelling of the large deformation of the valve leaflets and the fluid-structure interactions. The IB simulation is first validated for the aortic prosthesis subjected to a hydrostatic loading. The predicted displacement fields by IB are compared with those obtained using ANSYS, as well as with experimental measurements. Good quantitative agreement is obtained. Moreover, known failure regions of aortic prostheses are identified. The dynamic behaviour of the valve designs is then simulated under four physiological pulsatile flows. Experimental pressure gradients for opening and closure of the valves are in good agreement with IB predictions for all flow rates for both aortic and mitral designs. Importantly, the simulations predicted improved physiological haemodynamics for the novel mitral design. Limitation of the current IB model is also discussed. We conclude that the IB model can be developed to be an extremely effective dynamic simulation tool to aid prosthesis design.     

A two-dimensional fluid-structure interaction model of the aortic valve [correction of value

Failure of synthetic heart valves is usually caused by tearing and calcification of the leaflets. Leaflet fiber-reinforcement increases the durability of these valves by unloading the delicate parts of the leaflets, maintaining their physiological functioning. The interaction of the valve with the surrounding fluid is essential when analyzing its functioning. However, the large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood-valve interaction modeling. This has, so far, obstructed numerical analyses of valves operating under physiological conditions. A two-dimensional fluid-structure interaction model is presented, which allows the Reynolds number to be within the physiological range, using a fictitious domain method based on Lagrange multipliers to couple the two phases. The extension to the three-dimensional case is straightforward. The model has been validated experimentally using laser Doppler anemometry for measuring the fluid flow and digitized high-speed video recordings to visualize the leaflet motion in corresponding geometries. Results show that both the fluid and leaflet behaviour are well predicted for different leaflet thicknesses.     

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.     

A Model for Fluid Drainage by the Lymphatic System

This study investigates the fluid flow through tissues where lymphatic drainage occurs. Lymphatic drainage requires the use of two valve systems, primary and secondary. Primary valves are located in the initial lymphatics. Overlapping endothelial cells around the circumferential lining of lymphatic capillaries are presumed to act as a unidirectional valve system. Secondary valves are located in the lumen of the collecting lymphatics and act as another unidirectional valve system; these are well studied in contrast to primary valves. We propose a model for the drainage of fluid by the lymphatic system that includes the primary valve system. The analysis in this work incorporates the mechanics of the primary lymphatic valves as well as the fluid flow through the interstitium and that through the walls of the blood capillaries. The model predicts a piecewise linear relation between the drainage flux and the pressure difference between the blood and lymphatic capillaries. The model describes a permeable membrane around a blood capillary, an elastic primary lymphatic valve and the interstitium lying between the two.     

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.     

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.     

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.     

The central axis prosthetic cardiac valve: an in vitro study of pressure drop assessment under steady-state flow conditions

In this work, a new mechanical prosthetic heart valve, the central axis valve, is presented. This new prosthesis has been tested in vitro, and compared with four other common prosthetic cardiac valves (Starr-Edwards 6120, Bjork-Shiley monostrut, Medtronic-Hall, and St Jude Medical valves). All valves studied have the same orifice diameter of 22 mm. The prostheses were installed inside a transparent mitral test chamber, which enables pressure drop measurement to be made under steady-state flow conditions using a blood analogue fluid. Pressure drop loss is one important factor affecting the overall performance of a prosthetic heart valve. Steady-state flow tests are essential to predict certain flow characteristics and pressure gradient loss before more complicated, expensive, and difficult-to-interpret pulsatile flow tests are conducted. All experiments were performed in vitro and at steady volumetric flow rates of 10 to 30 l/min. The Starr-Edwards SE 6120 showed the highest values for pressure drop. The St Jude Medical valve offers the minimum resistance to flow. The central axis valve comes second to the Starr-Edwards valve for this type of measurement. The new valve is promising. A complete valve evaluation programme, covering initial conceptional design through to clinical use, is in progress. Materials for the fabrication of the new valve are also under consideration.     

A mechanical analysis of the closed Hancock heart valve prosthesis

In order to obtain mechanical specifications for the design of an artificial leaflet valve prosthesis, a geometrically non-linear numerical model is developed of a closed Hancock leaflet valve prosthesis. In this model, the fibre reinforcement of the leaflet and the viscoelastic properties of frame and leaflets are incorporated. The calculations are primarily restricted to 1/6 part of the valve and a time varying pressure load is applied. The calculations are verified experimentally by measuring the commissure displacements and leaflet centre displacement of a Hancock valve. The numerically obtained commissure displacements are found to be linearly dependent on the pressure load, and the slope of the curves is hardly dependent on loading type and loading velocity. Experimentally a difference is found between the three commissure displacements, which is also predicted numerically using a simplified asymmetric total valve model. Besides, experimentally a clear dependency of commissure displacements on frame size is found. For the leaflet centre displacement, a qualitative agreement exists between numerical prediction and experimental result, although the numerical predicted values are systematically higher. The numerically obtained stress distributions revealed that the maximum von Mises intensity in the membranes occurs in the vicinity of the commissure in the free leaflet area (0.2 N mm-2). Wrinkling of the membranes may occur in the coaptation area near the leaflet suspension. The maximum fibre stress is found near the aortic ring in the fibres which form the boundaries of the coaptation area (0.64 N mm-2). These locations seem to correlate with some common regions of tissue valve failure.     

A fully coupled fluid-structure interaction model of the secondary lymphatic valve

Abstract

The secondary lymphatic valve is a bi-leaflet structure frequent throughout collecting vessels that serves to prevent retrograde flow of lymph. Despite its vital function in lymph flow and apparent importance in disease development, the lymphatic valve and its associated fluid dynamics have been largely understudied. The goal of this work was to construct a physiologically relevant computational model of an idealized rat mesenteric lymphatic valve using fully coupled fluid-structure interactions to investigate the relationship between three-dimensional flow patterns and stress/deformation within the valve leaflets. The minimum valve resistance to flow, which has been shown to be an important parameter in effective lymphatic pumping, was computed as 268?g/mm⁴?s. Hysteretic behavior of the lymphatic valve was confirmed by comparing resistance values for a given transvalvular pressure drop during opening and closing. Furthermore, eddy structures were present within the sinus adjacent to the valve leaflets in what appear to be areas of vortical flow; the eddy structures were characterized by non-zero velocity values (up to ～4?mm/s) in response to an applied unsteady transvalvular pressure. These modeling capabilities present a useful platform for investigating the complex interplay between soft tissue motion and fluid dynamics of lymphatic valves and contribute to the breadth of knowledge regarding the importance of biomechanics in lymphatic system function.     

Particle Image Velocimetry Study of Pulsatile Flow in Bi-leaflet Mechanical Heart Valves with Image Compensation Method

Particle Image Velocimetry (PIV) is an important technique in studying blood flow in heart valves. Previous PIV studies of flow around prosthetic heart valves had different research concentrations, and thus never provided the physical flow field pictures in a complete heart cycle, which compromised their pertinence for a better understanding of the valvular mechanism. In this study, a digital PIV (DPIV) investigation was carried out with improved accuracy, to analyse the pulsatile flow field around the bi-leaflet mechanical heart valve (MHV) in a complete heart cycle. For this purpose a pulsatile flow test rig was constructed to provide the necessary in vitro test environment, and the flow field around a St. Jude size 29 bi-leaflet MHV and a similar MHV model were studied under a simulated physiological pressure waveform with flow rate of 5.2 l/min and pulse rate at 72 beats/min. A phase-locking method was applied to gate the dynamic process of valve leaflet motions. A special image-processing program was applied to eliminate optical distortion caused by the difference in refractive indexes between the blood analogue fluid and the test section. Results clearly showed that, due to the presence of the two leaflets, the valvular flow conduit was partitioned into three flow channels. In the opening process, flow in the two side channels was first to develop under the presence of the forward pressure gradient. The flow in the central channel was developed much later at about the mid-stage of the opening process. Forward flows in all three channels were observed at the late stage of the opening process. At the early closing process, a backward flow developed first in the central channel. Under the influence of the reverse pressure gradient, the flow in the central channel first appeared to be disturbed, which was then transformed into backward flow. The backward flow in the central channel was found to be the main driving factor for the leaflet rotation in the valve closing process. After the valve was fully closed, local flow activities in the proximity of the valve region persisted for a certain time before slowly dying out. In both the valve opening and closing processes, maximum velocity always appeared near the leaflet trailing edges. The flow field features revealed in the present paper improved our understanding of valve motion mechanism under physiological conditions, and this knowledge is very helpful in designing the new generation of MHVs.     

Simulation of a chain of collapsible contracting lymphangions with progressive valve closure

The aim of this investigation was to achieve the first step toward a comprehensive model of the lymphatic system. A numerical model has been constructed of a lymphatic vessel, consisting of a short series chain of contractile segments (lymphangions) and of intersegmental valves. The changing diameter of a segment governs the difference between the flows through inlet and outlet valves and is itself governed by a balance between transmural pressure and passive and active wall properties. The compliance of segments is maximal at intermediate diameters and decreases when the segments are subject to greatly positive or negative transmural pressure. Fluid flow is the result of time-varying active contraction causing diameter to reduce and is limited by segmental viscous and valvular resistance. The valves effect a smooth transition from low forward-flow resistance to high backflow resistance. Contraction occurs sequentially in successive lymphangions in the forward-flow direction. The behavior of chains of one to five lymphangions was investigated by means of pump function curves, with variation of valve opening parameters, maximum contractility, lymphangion size gradation, number of lymphangions, and phase delay between adjacent lymphangion contractions. The model was reasonably robust numerically, with mean flow-rate generally reducing as adverse pressure was increased. Sequential contraction was found to be much more efficient than synchronized contraction. At the highest adverse pressures, pumping failed by one of two mechanisms, depending on parameter settings: either mean leakback flow exceeded forward pumping or contraction failed to open the lymphangion outlet valve. Maximum pressure and maximum flow-rate were both sensitive to the contractile state; maximum pressure was also determined by the number of lymphangions in series. Maximum flow-rate was highly sensitive to the transmural pressure experienced by the most upstream lymphangions, suggesting that many feeding lymphatics would be needed to supply one downstream lymphangion chain pumping at optimal transmural pressure.     

Near valve flows and potential blood damage during closure of a bileaflet mechanical heart valve

Blood damage and thrombosis are major complications that are commonly seen in patients with implanted mechanical heart valves. For this in vitro study, we isolated the closing phase of a bileaflet mechanical heart valve to study near valve fluid velocities and stresses. By manipulating the valve housing, we gained optical access to a previously inaccessible region of the flow. Laser Doppler velocimetry and particle image velocimetry were used to characterize the flow regime and help to identify the key design characteristics responsible for high shear and rotational flow. Impact of the closing mechanical leaflet with its rigid housing produced the highest fluid stresses observed during the cardiac cycle. Mean velocities as high as 2.4 m/s were observed at the initial valve impact. The velocities measured at the leaflet tip resulted in sustained shear rates in the range of 1500-3500 s(-1), with peak values on the order of 11,000-23,000 s(-1). Using velocity maps, we identified regurgitation zones near the valve tip and through the central orifice of the valve. Entrained flow from the transvalvular jets and flow shed off the leaflet tip during closure combined to generate a dominant vortex posterior to both leaflets after each valve closing cycle. The strength of the peripheral vortex peaked within 2 ms of the initial impact of the leaflet with the housing and rapidly dissipated thereafter, whereas the vortex near the central orifice continued to grow during the rebound phase of the valve. Rebound of the leaflets played a secondary role in sustaining closure-induced vortices.     

Replacement of the Aortic and Mitral Valves Using the Starr-Edwards Ball-Valve Prosthesis: A Report of 50 Cases

The aortic and mitral valves were replaced in 50 patients at the University of Alberta Hospital using the Starr-Edwards ball-valve prosthesis. The basis of the selection of 20 patients for isolated aortic valve replacement and 27 for mitral valve replacement using this type of prosthesis is presented, and the techniques of insertion of the aortic and mitral valve are described in detail. Of the 27 patients in whom the mitral valve was replaced by the Starr-Edwards prosthesis six died within 30 days of surgery and two after discharge from hospital at two and a half and four months, respectively. Left atrial thrombosis was the cause of death in four of these patients. In 20 patients in whom the aortic valve was replaced, four died in hospital and two died more than 30 days after returning home. Three of these six patients died from bleeding—the result of the use of anticoagulants. The difficulty in assessing whether or not anticoagulants are needed following replacement by a Starr-Edwards prosthesis is considered. It is felt, in our present state of knowledge, that anticoagulants should be used following mitral valve replacement but are probably not essential following replacement of the aortic valve. Two patients survived replacement of both aortic and mitral valves and have been followed up 18 months and seven months, respectively.     

In vitro validation of finite-element model of AAA hemodynamics incorporating realistic outlet boundary conditions

The purpose of this study is to validate numerical simulations of flow and pressure in an abdominal aortic aneurysm (AAA) using phase-contrast magnetic resonance imaging (PCMRI) and an in vitro phantom under physiological flow and pressure conditions. We constructed a two-outlet physical flow phantom based on patient imaging data of an AAA and developed a physical Windkessel model to use as outlet boundary conditions. We then acquired PCMRI data in the phantom while it operated under conditions mimicking a resting and a light exercise physiological state. Next, we performed in silico numerical simulations and compared experimentally measured velocities, flows, and pressures in the in vitro phantom to those computed in the in silico simulations. There was a high degree of agreement in all of the pressure and flow waveform shapes and magnitudes between the experimental measurements and simulated results. The average pressures and flow split difference between experiment and simulation were all within 2%. Velocity patterns showed good agreement between experimental measurements and simulated results, especially in the case of whole-cycle averaged comparisons. We demonstrated methods to perform in vitro phantom experiments with physiological flows and pressures, showing good agreement between numerically simulated and experimentally measured velocity fields and pressure waveforms in a complex patient-specific AAA geometry.     

The ideal hydrodynamic form of the concavo‐convex productide brachiopod shell

Shiino, Y & Suzuki, Y. 2011: The ideal hydrodynamic form of the concavo‐convex productide brachiopod shell. Lethaia, Vol. 44, pp. 329–343. Water‐flume experiments were performed to determine whether the concavo‐convex Permian brachiopod Waagenoconcha imperfecta was hydrodynamically adapted for feeding. The generation of passive currents inside the valves was observed experimentally. The use of four transparent, hollow polyhedron models, each differing in a single morphological feature, permitted observation of the currents inside the valves and allowed evaluation of the hydrodynamic significance of the ears and the prominent geniculated trail. Regardless of the direction of ambient flow, only the approximate‐imitation model generated a stable flow pattern consisting of inhalation from the ear gapes and exhalation from the anterior trail gape; models lacking or with small changes in these morphological features failed to generate stable flow patterns. The stable flow pattern was probably maintained by a pressure difference between the posterior lower ear gapes (maximum pressure) and the anterior trail gape (minimum pressure). Notably, bilaterally rotating internal currents formed parallel to the brachial ridges; such flow patterns would facilitate the capture of food particles by the animal via tentacles on its lophophore, which is most likely were located on the brachial ridges. Our results demonstrate that the immobile brachiopod W. imperfecta, an animal incapable of widely opening its valves, probably fed on the passive internal currents generated by its shell form. This unique valve morphology appears to be perfectly adapted from a hydrodynamic point of view. □Biomechanics, ecomorphology, evolution, morphological disparity, Productidina, suspension feeder.     

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.     

Numerical simulation of steady turbulent flow through trileaflet aortic heart valves—II. Results on five models

Turbulent flow simulations are run for five aortic trileaflet valve geometries, ranging from a valve leaflet orifice area of 1.1 cm² (Model A1—very stenotic) to 5.0 cm² (Model A5—natural valve). The simulated data compares well with experimental measurements made downstream of various aortic trileaflet valves by Woo (PhD Thesis, 1984). The location and approximate width and length of recirculation regions are correctly predicted. The less stenotic valve models reattach at the end of the aortic sinus region, 1.1 diameters downstream of the valve. The central jet exiting the less stenotic valve models is not significantly different from fully developed flow, and therefore recovers very quickly downstream of the reattachment point. The more stenotic valves disturb the flow to a greater degree, generating recirculation regions large enough to escape the sinuses and reattach further downstream. Peak turbulent shear stress values downstream of the aortic valve models which approximated prosthetic valves are 125 and 300 N m⁻², very near experimental observations of 150 to 350 N m⁻². The predicted Reynolds stress profiles also present the correct shape, a double peak profile, with the location of the peak occuring at the location of maximum velocity gradient, which occurs near the recirculation region. The pressure drop across model A2 (leaflet orifice area 1.6 cm²) is 20 mmHg at 1.6 diameters downstream. This compares well with values ranging from 19.5 to 26.2 mmHg for valves of similar orifice areas. The pressure drop decreases with decreasing valve stenosis, to a negligible value across the least stenotic valve model. Based on the good agreement between experimental measurements of velocity, shear stress and pressure drop, compared to the simulated data, the model has the potential to be a valuable tool in the analysis of heart valve designs.     

Fluid dynamic models as a guide to determine prosthetic heart valve diameter

The diameter of prosthetic heart valves is usually chosen according to the anatomic annulus size as determined during open-heart surgery. Therefore, this approach does not take into account the dimensional changes induced by heart pathology and surgical procedures. In addition, current practice fails to consider the variations of heart dimensions due to hemodynamic improvement following valve replacement. Here we suggest a method to determine the appropriate prosthesis diameter according to the hemodynamic features of the patient, to its kind of activity, and to the type of prosthesis. Assuming that the pressure drop across a valve can be calculated as delta p = apv 2/2, and considering the variation of blood flow with time and its change induced by frequency, it is possible to obtain the relationship between pressure drop and prosthetic valve diameter. The results obtained with this analytical method have been plotted on diagrams which allow the graphical determination of the proper valve diameter.