consideration of the factors discussed above results in the identification of certain design goals which are achieved by the prosthetic heart valve of the present invention . first , the prosthetic heart valve must have enough material in the leaflet for wide opening and low closing , but more than this amount increases the energy barrier to opening . to ensure that there is sufficient , but not an excess of material , a draping analysis discussed in more detail below is used . second , to ensure sufficient material for wide opening and low closing , the valve can only be manufactured in a partially open position : ( a ) by deforming the stent posts outwards during manufacture ; ( b ) by introducing multiple curves in the leaflet free edge ( but see below ); ( c ) by making the closed position asymmetric ; and ( d ) combinations of the above . third , if there is enough material for low closing and wide opening , the energy barrier to opening may be high enough to prevent opening of all leaflets at low flow . the energy barrier can be minimized by : ( a ) introducing multiple curves in the leaflet ; ( b ) making the leaflet asymmetric ; and combinations of the above . fourth , open commissures are needed for blood handling and closed commissures are needed for regurgitation , so the valve should have partially open commissures . in particular the included angle between adjacent leaflet free edges at the valve commissures ( for example see angle α of the symmetric leaflets shown in fig1 ) should be in the range of 10 - 55 °, preferably in the range 25 - 55 °. as discussed above , the use of multiple curves in the leaflet helps assure wide opening and more complete closure of the valve and to minimize the energy barrier to opening of the valve . however , the introduction of multiple curves of more than 1 . 5 wavelengths to the leaflet can be a disadvantage . while there may be sufficient material in the leaflet to allow full opening , in order for this to happen , the bends in the leaflet must straighten out completely . the energy available to do this arises only from the pressure gradient across the open valve , which decreases as the leaflets becomes more open , i . e ., as the valve orifice area increases . this energy is relatively small ( the more successful the valve design the smaller it becomes ), and does not provide enough energy to remove leaflet curves of more than 1 . 5 wavelengths given the stiffness of the materials available for valve manufacture . the result is they do not straighten out and the valve does not open fully . a draping analysis is used as a first approximation to full finite element analysis to determine if the starting shape of a membrane is such that it will take on a desired final shape when placed in its final position . from a durability standpoint the focus is on the closed position , and the desired shape of the leaflet in its closed position is defined . draping analysis allows the leaflet to be reformed in a partially open position . draping analysis assumes that very low energy deformation is possible ( in reality any form of deformation requires energy ). in order for this to occur the bending stiffness of the leaflet / membrane must be small , each element of the membrane should be free to deform relative to its neighbour , and each element should be free to change shape , i . e ., the shear modulus of the material is assumed to be very low . in applying the draping analysis , it is assumed that the leaflet can be moved readily from an original defined closed position to a new position in which it is manufactured . when the valve is actually cycled , it is assumed that the leaflet when closing will move from the manufactured position to the originally defined closed position . this allows the closed position to be optimised from a stress distribution aspect , and the manufactured position to be optimised from the point of view of reducing the energy barrier to opening . both symmetric and asymmetric shapes of the leaflet can allow incorporation of sufficient material in the leaflet free edge to allow full opening . fig1 is a diagrammatic view comparing the shape of symmetric ( solid line ) and asymmetric ( dashed line ) leaflets and also showing the commissure area 12 where the leaflets connect to the frame . an advantage of the asymmetric shape is that a region of higher radius of curvature 14 is produced than is achieved with a symmetric curve having a lower radius of curvature 16 . this region can buckle more readily and thereby the energy barrier to opening is reduced . an asymmetric leaflet also reduces the energy barrier through producing unstable buckling in the leaflet . during opening symmetric leaflets buckle symmetrically i . e ., the leaflet buckles are generally mirrored about the centerline of the leaflet thus balancing the bending energies about this centerline . in the asymmetric valve the region of higher radius buckles readily , and because these bending energies are not balanced about the center line , this buckle proceeds to roll through the leaflet producing a sail - like motion producing a low energy path to open . an additional feature of the asymmetric valve is that the open position is also slightly asymmetric , as a result of which it offers a somewhat helical flow path , and this can be matched to the natural helical sense of the aorta . suggested benefits of this helical flow path include reduction of shear stress non - uniformity at the wall , and consequent reduction of platelet activation . first and second embodiments of the valve prosthesis will be described with reference to the accompanying drawings . fig2 is a perspective view of a heart valve prosthesis made in accordance with the present invention . the valve 10 comprises a stent or frame 1 and attached leaflets 2 a , 2 b , and 2 c . the leaflets are joined to the frame at scallops 5 a , 5 b , and 5 c . between each scallop is post 8 , the most down - stream part of which is known as a stent tip 6 . leaflets 2 a , 2 b , and 2 c have free edges 3 a , 3 b , and 3 c , respectively . the areas between the leaflets at the stent tips 6 form commissures 4 . the following describes a particular way of designing a first embodiment of a valve of the present invention . other different design methodology could be utilized to design a valve having the structural features of the valve disclosed herein . five computational steps are involved in this particular method : ( 1 ) define the scallop geometry ( the scallop , 5 , is the intersection of the leaflet , 2 , with the frame , 1 ); ( 2 ) geometrically define a valve leaflet in the closed position c ; ( 3 ) map and compute the distribution of area across the leaflet in the closed position ; ( 4 ) rebuild the leaflet in a partially open position p ; and ( 5 ) match the computed leaflet area distribution in the partially open or molded position p to the defined leaflet in the closed position c . this ensures that when an increasing closing pressure is applied to the leaflets , they eventually assume a shape which is equivalent to that defined in closed position c . this approach allows the closed shape of the leaflets in position c to be optimised for durability while the leaflets shaped in the molded partially open shape p can be optimised for hemodynamics . this allows the use of stiffer leaflet materials for valves which have good hemodynamics . an xyz co - ordinate system is defined as shown in fig2 , with the z axis in the flow direction of blood flowing through the valve . the leaflets are mounted on the frame , the shape of which results from the intersection of the aforementioned leaflet shape and a 3 - dimensional geometry that can be cylindrical , conical or spherical in nature . a scallop shape is defined through intersecting the surface enclosed by the following equations with a cylinder of radius r ( where r is the internal radius of the valve ): x ell = e so − e sj .√{ square root over ( 1 −( z / e sn ) 2 )} h sj = e so − e sj √{ square root over (( 1 −( z / e sn ) 2 ))}− h so x hyp = h so + h sj √{ square root over (( 1 −( y / h sn ) 2 ))} the shape of the scallop can be varied using the constants e s0 , e sj , h s0 , ƒ ( z ). the definition of parameters used in these and the other equations herein are contained in table 4 . the shape of the leaflet under back pressure ( i . e ., in the closed position c ) can be approximated mathematically using elliptical or hyperbolic co - ordinates , or a combination of the above in an xyz co - ordinate system where xy is the plane of the valve perpendicular to the blood flow and z is the direction parallel to the blood flow . the parameters are chosen to define approximately the shape of the leaflet under back pressure so as to allow convenient leaflet re - opening and minimize the effect of the stress component which acts in the direction parallel to the blood flow , whilst also producing an effective seal under back pressure . the closed leaflet geometry in closed position c is chosen to minimize stress concentrations in the leaflet particularly prone to occur at the valve commissures . the specifications for this shape include : ( 1 ) inclusion of sufficient material to allow a large open - leaflet orifice ; ( 2 ) arrangement of this material to minimize redundancy ( excess material in the free edge , 3 ) and twisting in the centre of the free edge , 3 ; and ( 3 ) arrangement of this material to ensure the free edge , 3 , is under low stress i . e ., compelling the frame and leaflet belly to sustain the back - pressure . fig3 is a partial sectional view ( using the section 3 - 3 shown in fig2 ) showing only the intended position of the leaflet in the closed position . the shape of this intended position is represented by the function x closed ( z ). this function can be used to arrange the shape of the leaflet in the closed position c to meet the aforementioned specification . the curve is defined using the following equation and manipulated using the constants e cj , e co , z co and the functions e cn ( z ) and x t ( z ). x closed ⁡ ( z ) = - [ e cj · ( 1 - ( z - z e ⁢ ⁢ o e c ⁢ ⁢ n ⁡ ( z ) ) 2 ) ] 0 . 5 + e co - x t ⁡ ( z ) where e cn is a function changing linearly with z and x t ( z ) is a function changing nonlinearly with z thus the scallop shape and the function x closed ( z ) are used to form the prominent boundaries for the closed leaflet in the closed position c . the remaining part of the leaflet is formed using contours s ( x , y ) n sweeping from the scallop to the closed leaflet belly function x closed ( z ), where n is an infinite number of contours , two of which are shown in fig4 b . the length of the leaflet ( or contours s ( x , y ) n ) in the circumferential direction ( xy ) is calculated and repeated in the radial direction ( z ) yielding a function l ( z ) which is used later in the definition of the geometry in the partially open position p . the area contained between respective contours is also computed yielding a function k ( z ) which is also used in the definition of the geometry in position p . the area contained between contours is approximated using the process of triangulation as shown in fig4 b . this entire process can be shortened by reducing the number of contours used to represent the surface ( 100 & lt ; n & lt ; 200 ). the aforementioned processes essentially define the leaflet shape and can be manipulated to optimise for durability . in order to optimise for hemodynamics , the same leaflet is molded in a position p which is intermediate in terms of valve opening . this entails molding large radius curves into the leaflet which then serve to reduce the energy required to buckle the leaflet from the closed to the open position . the large radius curves can be arranged in many different ways . some of these are outlined herein . the leaflet may be molded on a dipping former as shown in fig1 . preferably the former is tapered with an included angle θ so that the end 29 has a diameter which is greater than the end 22 . ( this ensures apposition of the frame and former during manufacture .) in this case , the scallop shape , defined earlier , is redefined to lie on a tapered geometry ( as opposed to the cylindrical geometry used in the definition of the closed leaflet shape ). this is achieved by moving each point on the scallop radially , and in the same movement , rotation of each point about an x - y plane coincident with the bottom of the scallop , until each point lies on the tapered geometry . the geometry of the leaflet shape can be defined as a trigonometric arrangement ( or other mathematical function ) preferably sinusoidal in nature in the xy plane , comprising one or more waves , and having anchoring points on the frame . thus the valve leaflets are defined by combining at least two mathematical functions to produce composite waves , and by using these waves to enclose the leaflet surface with the aforementioned scallop . one such possible manifestation is a composite curve consisting of an underlying low frequency sinusoidal wave upon which a second higher frequency sinusoidal wave is superimposed . a third wave having a frequency different from the first and second waves could also be superimposed over the resulting composite wave . this ensures a wider angle between adjacent leaflets in the region of the commissures when the valve is fully open thus ensuring good wash - out of this region . the composite curve , and the resulting leaflet , can be either symmetric or asymmetric about a plane parallel to the blood flow direction and bisecting a line drawn between two stent tips such as , for leaflet 2 a , the section along line 3 - 3 of fig2 . the asymmetry can be effected either by combining a symmetric underlying curve with an asymmetric superimposed curve or vice versa . the following describes the use of a symmetric underlying function with an asymmetric superimposed function , but the use of an asymmetric underlying function will be obvious to one skilled in the art . the underlying function is defined in the xy plane and connects the leaflet attachment points to the scallop at a given height from the base of the valve . this underlying function shown in fig5 , can be trigonometric , elliptical , hyperbolic , parabolic , circular , or other smooth analytic function or could be a table of values . using sine functions , one possible underlying wave is shown in fig5 and is defined using the following equation . x u = x ( n , 0 ) + a u · sin ⁡ [ ( 0 . 5 ⁢ π y ( n , 0 ) ) · ( y - y ( n , 0 ) ) ] the superimposed wave is defined in the xy plane , and connects the attachment points of the leaflet to the scallop at a given height above the base of the valve . the superimposed wave is of higher frequency than the underlying wave , and can be trigonometric , elliptic , hyperbolic , parabolic , circular , or other smooth analytic function , or a table of values . using sine functions , one possible symmetric leaflet design is formed when the underlying wave is combined with a superimposed wave formed using the following equation . x s = - a s · b s ⁡ ( y ) · sin ⁡ [ ( 1 . 5 · π y ( n , 0 ) ) · ( y - y ( n , 0 ) ) ] a s can be varied across the leaflet to produce varying wave amplitude across the leaflet , for example lower amplitude at the commissures than in the leaflet centre . b s can be varied to adjust the length of the wave . the superimposed wave is shown in fig6 . the composite wave formed by combining the underlying wave ( fig5 ) with the superimposed wave ( fig6 ) is shown in fig7 . using sine functions , one possible asymmetric leaflet design is formed when the underlying wave ( fig5 ) is combined with a superimposed wave formed using the following equation . x s = - a s · b s ⁡ ( y ) · sin ⁡ [ ( π y ( n , 0 ) ) · ( y - y ( n , 0 ) ) ] 0 y ( n , 0 ) x s = 0 . 5 · a s ⁢ b s ⁡ ( y ) · sin ⁡ [ ( 2 . 0 · π y ( n , 0 ) ) · y ] ( - y ( n , 0 ) ) 0 a s can be varied across the leaflet to produce varying wave amplitude across the leaflet , for example lower amplitude at the commissures than in the leaflet centre . b s ( y ) can be varied to adjust the length of the wave . the superimposed wave is shown in fig8 . the resulting asymmetric composite wave is shown in fig9 . the composite wave w ( x c , y c ) n is created by offsetting the superimposed wave normal to the surface of the underlying wave ( fig7 ). while the general shape of the leaflet in position p has been determined using the composite wave , at this stage it is not specified in any particular position . in order to specify the position of p , the shape of the partially open leaflet position can be defined as x open ( z ). this is shown as reference numeral 7 in fig1 . x open ⁡ ( z ) = - [ e oj · ( 1 - ( z - z oo e on ) 2 ) ] 0 . 5 + e oo in order to manipulate the composite wave to produce the belly shape x open ( z ) the respective amplitudes of the individual sine waves can be varied from the free edge to the leaflet base . for example , the degree of ‘ openness ’ of the leaflet in position p can be varied throughout the leaflet . the composite wave is thus defined to produce the molded “ buckle ” in the leaflet , and x open ( z ) is used to define the geometry of the leaflet at position p . at this stage it may bear no relation to the closed leaflet shape in position c . in order to match the area distribution of both leaflet positions , ( thus producing essentially the same leaflet in different positions ) the composite wave length is iterated to match the length of the relevant leaflet contour in position c . thus the amplitude and frequency of the individual waves can be varied in such a manner as to balance between : ( a ) producing a resultant wave the length of which is equal to the relevant value in the length function l ( z ) thus approximating the required closed shape when back pressure is applied , and ( b ) allowing efficient orifice washout and ready leaflet opening . also the area contained between the contours in the open leaflet is measured using the same process of triangulation as in the closed position c , and is iterated until it matches with the area contained between relevant contours in position c ( denoted k ( z )) ( through tilting the contours in p relative to each other ). thus the composite waves ( p ( x , y ) n ) pertaining to the contour n and length l ( z ) can be tilted at an angle to the xy plane about attachment points x ( n , 0 ). y ( n 0 ) and x ( n , 0 ), − y ( n , 0 ) until the correct area is contained between p ( x , y ) n and p ( x , y ) n - 1 ( see fig1 & amp ; 11 ). this process identifies the values of b s , a u and the contour tilt angle to be used in constructing the mold for the valve leaflet . as long as the constants such as b s and a u , and the tilt angle of the contours relative to the xy plane , are known , the surface of the leaflet in its molded position can be visualised , enclosed and machined in a conventional manner . as a result of this fitting process the composite wave retains the same basic form but changes in detail from the top of the leaflet to the bottom of the leaflet . a composite wave can be defined in the leaflet surface as the intersection of the leaflet surface with a plane normal to the z axis . this composite wave will have the same general form as the composite wave used in the leaflet design but will differ from it in detail as a result of the tilting process described above . in summary therefore one possible method of designing the leaflet of the first embodiment of the present invention is in the following way : ( 1 ) define a scallop shape ; ( 2 ) define a shape approximating the shape of the closed leaflet using elliptical , hyperbolic , parabolic or circular functions , smooth analytical functions or table of values ; ( 3 ) compute the functions l ( z ) and k ( z ), which define the length of the leaflet in the xy plane along the z axis and the area distribution of the leaflet along the z axis ; ( 4 ) use one or more associated sine waves to generate a geometry which is partially - open , which pertains to a leaflet position which is between the two extreme conditions of normal valve function , i . e ., leaflet open and leaflet closed ; ( 5 ) vary the frequency and amplitude of the sinewaves to fit to the length function l ( z ) and the angle at which the contour is tilted to the xy plane to fit to the area function k ( z ); and ( 6 ) the respective amplitudes of the individual sine waves can be varied from the free edge to leaflet base , for example the degree of ‘ openness ’ of the leaflet can be varied throughout the leaflet . examples 1 and 2 set forth hereafter are examples of how the invention of the first embodiment can be put into practice . using the scallop constants in table 1 , the constants required to produce an example of a symmetric leaflet valve ( example 1 , fig1 ) and an example of an asymmetric leaflet valve ( example 2 , fig1 ) are given in table 2 and table 3 respectively . these constants are used in conjunction with the aforementioned equations to define the leaflet geometry . with one leaflet described using the aforementioned equations , the remaining two leaflets are generated by rotating the geometry about the z axis through 120 ° and then through 240 °. these leaflet shapes are inserted as the leaflet forming surfaces of the dipping mold ( otherwise known as a dipping former ), which then forms a 3 - dimensional dipping mold . the composite wave described in the aforementioned equations , therefore substantially defines the former surface which produces the inner leaflet surface . as seen in fig1 the dipping mold 20 is slightly tapered so that the end 29 has a diameter which is greater than the end 22 , and has a first end 22 having an outside diameter slightly smaller than the inside diameter of the frame . the former includes at least two and preferably three leaflet forming surfaces 24 which are defined by scalloped edges 26 and flats 28 . sharp edges in the manufacturing former and on the frame are radiused to help reduce stress concentrations in the finished valve . during the dip molding process the frame is inserted over end 22 of the former so that the scallops 5 and stent posts 8 of the frame align with the scalloped edges 26 and flats 28 of the former . the leaflet forming surfaces 24 are configured to form leaflets during the molding process which have the geometry described herein . this mold can be manufactured by various methods , such as , machining , electrical discharge machining , injection molding . in order that blood flow is not disturbed , a high surface finish on the dipping mold is essential . for the frame there are preferably three posts with leaflets hung on the frame between the posts . a crown - like frame or stent , 1 , is manufactured with a scallop geometry , which matches the dipping mold scallop . the frame scallop is offset radially by 0 . 1 mm to allow for the entire frame to be coated with a thin layer of leaflet material to aid adhesion of the leaflets . leaflets may be added to the frame by a dip - molding process , using a dipping former machined or molded to create the multiple sinewave form . the material of preference should be a semi - rigid fatigue - and creep - resistant frame material such as polyetheretherketone ( peek ), high modulus polyurethane , titanium , reinforced polyurethane , or polyacetal ( delrin ) produced by machining or injection - molding etc . alternatively , a relatively low modulus polymer may be used , which may be fibre - reinforced , to more closely mimic the aortic wall . the frame can be machined or injection molded , and is manufactured preferably from peek or polyacetal ( delrin ). the frame is treated by exposure to a gas plasma or other methods to raise its surface energy above 64 mn / m ( millinewtons / meter ). then the frame is dipped in a polyurethane solution ( preferably elast - eon ™ manufactured by aortech biomaterials pty , sydney australia ) in order to apply a coating of approximately 0 . 1 mm thick . having dried the frame with applied coating in an oven overnight , it is placed on the dipping former and aligned with the former scallops . the combination of frame and three dimensional dipping mold is then dipped into polyurethane solution , which forms a coating of solution on frame and mold . this coating flows slowly over the entire mold surface ensuring a smooth coating . the new coating on the frame and dipping mold solvates the initial frame coating thus ensuring a good bond between leaflet and frame . the dipping mold with polyurethane covering is dried in an oven until all the solvent has been removed . one or more dips may be used to achieve a leaflet with a mean thickness between 40 μm and 500 μm . the shape of the former , and the viscosity and solvent interactive properties of the polyurethane solution , control the leaflet thickness and the distribution of thickness over the leaflet . a dipping process does not allow precise control of leaflet thickness and its variation across a leaflet . in particular , surfaces that are convex on the dipping former result in reduced leaflet thickness when compared with surfaces that are concave . additionally the region of the leaflet adjacent to the frame essentially provides a very small concave radius which traps further polymer solution and this results in thickening of these regions . the shape of the former is substantially defined by the composite wave . radiusing and polishing of the former can both contribute to some variation of the shape . the shape of the inner surface of the leaflets will closely replicate the shape of the former . the shape of the outer surface of the leaflets will be similar to the shape of the inner surface but variations will result from the processing properties of the polymer solution and details of the dipping process used to produce the valve . the leaflet may be formed from polyurethanes having a young &# 39 ; s modulus less than 100 mpa , preferably in the range 5 to 50 mpa . the valve is next removed from the dipping mold . the stent posts , which had been deflected by the taper on the former , now recover their original position . the shape of the leaflets changes slightly as a result of the movement of the stent posts . at this stage the dipping mold and frame is covered with an excess of polyurethane due to the drain - off of the polymer onto the region of the mold known as the drain - off area 30 . leaflet free edges may be trimmed of excess material using a sharp blade rotated around the opened leaflets or using laser - cutting technology . an alternate valve manufacturing method is injection molding . a mold is constructed with a cavity which allows the valve frame to be inserted in the mold . the cavity is also designed with the leaflet geometry , as defined above , as the inner leaflet surface . a desired thickness distribution is defined for the leaflet and the outer leaflet surface of the mold is constructed by adding the leaflet thickness normally to the inner leaflet surface . the leaflet may be of uniform thickness throughout , in the range 40 to 500 microns , preferably 50 to 200 microns , more preferably 80 to 150 microns . the leaflet may be thickened towards its attachment to the frame . alternatively the thickness of the leaflet , along a cross - section defined by the intersection of a plane perpendicular to the blood flow axis and the leaflet , can change gradually and substantially continuously from a first end of the cross - section ( i . e ., first edge of the leaflet ) to a second end of the cross - section ( i . e ., second edge of the leaflet ) in such a way that the mean thickness of the first half of the leaflet is different from the mean thickness of the second half of the leaflet . this mold is inserted in a conventional injection molding machine , the frame is inserted in the mold and the machine injects molten polymer into the cavity to form the leaflets and bond them to the frame . the polymer solidifies on cooling and the mold is opened to allow the complete valve to be removed . the leaflets may also be formed using a reaction - molding process ( rim ) whereby the polymer is synthesized during the leaflet forming . a mold is constructed as described above . this mold is inserted in a reaction - injection molding machine , the frame is inserted in the mold and the machine injects a reactive mixture into the cavity . the polymer is produced by the reaction in the cavity to form the leaflets and bond them to the frame . when the reaction is complete , the mold is opened to allow the complete valve to be removed . yet a further option is to compression mold a valve initially dipped . this approach allows the leaflet thickness or thickness distribution to be adjusted from that initially produced . by varying the thickness of the leaflets the dynamics of the valve opening and closing can be modified . for example , the thickness of the leaflet along a cross - section defined by the intersection of a plane perpendicular to the blood flow axis and the leaflet can be varied so that the thickness changes gradually and substantially continuously from a first end of the cross - section ( i . e ., first edge of the leaflet ) to a second end of the cross - section ( i . e ., second edge of the leaflet ) in such a way that the mean thickness of the first half of the leaflet is different from the mean thickness of the second half of the leaflet . this will result in the thinner half of the leaflet opening first and creating a sail - like opening motion along the free edge of the leaflet . leaflet shape resulting from conventional injection molding , reaction injection molding or compression molding , is substantially defined by the composite wave described above . it will differ in detail for many of the same reasons identified for dip molding . the valves of the present invention are manufactured in the neutral position or close to it and are therefore substantially free of bending stresses in this position . as a result when the leaflet is moved to its closed position the total bending energy at the leaflet center free edge and at the commissures is reduced compared to a valve made according to u . s . pat . no . 5 , 376 , 113 ( jansen et al .). the valves of the present invention may be used in any required position within the heart to control blood flow in one direction , or to control flow within any type of cardiac assist device . the following examples 1 and 2 use the same scallop geometry described using the constants set forth in table 1 : while the examples described herein relate to one valve size , the same method can be used to produce valves from a wide range of sizes . this can be carried out by modifying the constants used in the equations , by resealing the bounding curves such as x closed ( z ) and computing and iterating in the normal fashion or by rescaling the leaflet . the parameters described in the preceding sections are assigned the values set forth in table 2 and are used to manufacture a symmetric valve . the included angle between adjacent leaflet free edges at the valve commissure for this valve is approximately 50 °. the parameters described in the preceding sections are assigned the values set forth in table 3 and are used to manufacture an asymmetric valve . the included angle between adjacent leaflet free edges at the valve commissure for this valve is approximately 48 °. table 4 definition of parameters r internal radius of valve scallop ( fig2 ) x ell , h sj , h sn , x hyp are used to define a surface which , when intersected with a cylinder , scribe a function which forms the scallop for one leaflet . this method for creating a scallop is described in mackay et al ., biomaterials 17 1996 , although an added variable f ( z ) is used for added versatility . x ell scribes an ellipse in the radial direction . x hyp scribes a hyperbola in the circumferential direction . e so ellipse x - axis offset e sj major axis of the ellipse e sn minor axis of the ellipse h sj major axis of the hyperbola h sn minor axis of the hyperbola h so hyperbola x - axis offset f ( z ) creates a varying relationship between h sn and h sj closed leaflet geometry c ( fig3 & amp ; 4 ) x closed ( z ) is defined as an ellipse ( with a minor axis e cn ( z ) which changes with z ) in the xz axis in the plane defined in fig2 by cutting plane 3 - 3 . it is defined using the following constants and functions . z co closed ellipse z - axis offset e cn ( z ) closed ellipse minor axis which changes with z e co closed ellipse x - axis offset e cj closed ellipse major axis x t ( z ) offset function which serves to increase the amount of material in the belly molded position p p is enclosed by a number ( n ) of contours p ( x , y ) n which run from one side of the scallop to the other . the underlying function x u is used in defining both symmetric and asymmetric leaflets . x u is simply an ellipse ( or other such function ) running in a plane from one side of the scallop to the other . the points on the scallop are designated x ( n , 0 ) , y ( n , 0 ) where n refers to the contour number ( see fig5 , 9 , 11b ). y variable in plane from y ( n , 0 ) to − y ( n , 0 ) a u a u is the amplitude of the underlying wave a s ( y ) a s is a function which biases the wave amplitude in a defined way , e . g . the amplitude of the wave can be increased near the commissure if so desired . b s b s is the amplitude of the superimposed wave composite curve ( fig7 & amp ; 9 ) x c x coordinate for defining the composite curve . this is derived using x u and x s y c y coordinate for defining the composite curve . this is derived using x u and x s open leaflet position ( fig1 ) x open ( z ) is defined as an ellipse in the xz axis in the plane defined in fig2 by cutting plane 3 - 3 . the contours defined in composite curve are married to the open leaflet position x open ( z ) to produce the molded leaflet p . it is defined using the following constants . e oj open ellipse major axis z oo open ellipse z - axis offset e oo open ellipse x - axis offset e on open ellipse minor axis θ former taper angle the following describes another particular way of designing a second embodiment of a valve of the present invention . other different design methodology could be utilized to design a valve having the structural features of the valve disclosed herein . five computational steps are involved in this particular method : ( 1 ) define the scallop geometry ( the scallop , 5 , is the intersection of the leaflet , 2 , with the frame , 1 ); ( 2 ) define a contour length function l ( z ) and use this function to define a valve leaflet in the closed position c and optimize the stress distribution on the valve . the stress distribution can be confirmed using finite element analysis ( fea ). thus the resulting stress distribution results from the length function l ( z ) and fea is used to confirm the optimal l ( z ); ( 3 ) rebuild the leaflet in a partially open position p ; and ( 4 ) match , using contour lengths , the computed leaflet area distribution in the partially open or molded position p to the defined leaflet in the closed position c . this ensures that when an increasing closing pressure is applied to the leaflets , they eventually assume a shape which is equivalent to that defined in closed position c . this approach allows the closed shape of the leaflets in position c to be optimised for durability while the leaflets shaped in the molded partially open shape p can be optimised for hemodynamics . this allows the use of stiffer leaflet materials for valves which have good hemodynamics . an xyz co - ordinate system is defined as shown in fig2 , with the z axis in the flow direction of blood flowing through the valve . the leaflets are mounted on the frame , the shape of which results from the intersection of the aforementioned leaflet shape and a 3 - dimensional geometry that can be cylindrical , conical or spherical in nature . the leaflets are mounted on the frame , the shape of which results from the intersection of the aforementioned leaflet shape and a 3 - dimensional geometry that can be cylindrical , conical or spherical in nature . a scallop shape is defined through cutting a cylinder of radius r ( where r is the internal radius of the valve ) with a plane at an inclined angle . the angle of the cutting plane is dictated by the desired height of the leaflet and the desired distance between the leaflets at the commissures . the closed leaflet geometry in closed position c is chosen to minimize stress concentrations in the leaflet particularly prone to occur at the valve commissures . the specifications for this shape include : ( 1 ) inclusion of sufficient material to allow a large open - leaflet orifice ; ( 2 ) arrangement of this material to minimize redundancy ( excess material in the free edge , 3 ) and twisting in the centre of the free edge , 3 ; and ( 3 ) arrangement of this material to ensure the free edge , 3 , is under low stress i . e ., compelling the frame and leaflet belly to sustain the back - pressure . the closed leaflet geometry is formed using contours s ( x , y ) n sweeping from attachment points on one side of the scallop to the congruent attachment point on the opposite side of the scallop , where n is an infinite number of contours , two of which are shown in fig4 b . the geometry of the contours s ( x , y ) n can be simple circular arcs or a collection of circular arcs and tangential lines ; the length of each contour is defined by l ( z ). hence the geometry is defined and modified using the length function l ( z ). thus the scallop shape and the contours s ( x , y ) n are used to form the prominent boundaries for the closed leaflet in the closed position c . this process can be shortened by reducing the number of contours used to represent the surface ( 5 & lt ; n & lt ; 200 ). for design iteration , the ease with which the leaflet shape can be changed can be improved by reducing the number of contours to a minimum ( i . e ., n = 5 ), although the smoothness of the resulting leaflet could be compromised to some extent . upon optimising the function l ( z ) for stress distribution , the number of contours defining the leaflet can be increased to improve the smoothness of the resulting leaflet ( 100 & lt ; n & lt ; 200 ). the function l ( z ) is used later in the definition of the geometry in the partially open position p . the aforementioned processes essentially define the leaflet shape and can be manipulated to optimise for durability . in order to optimise for hemodynamics , the same leaflet is molded in a position p which is intermediate in terms of valve opening . this entails molding large radius curves into the leaflet which then serve to reduce the energy required to buckle the leaflet from the closed to the open position . the large radius curves can be arranged in many different ways . some of these are outlined herein . as previously described with respect to the first embodiment the leaflet may be molded on a dipping former as shown in fig1 . however , in this embodiment to aid removal of the valve from the former and reduce manufacturing stresses in the leaflet the former is preferably not tapered . the geometry of the leaflet shape can be defined as a circular and trigonometric arrangement ( or other mathematical function ) preferably circular and sinusoidal in nature in the xy plane , comprising one or more waves , and having anchoring points on the frame . thus the valve leaflets are defined by combining at least two mathematical functions to produce composite waves , and by using these waves to enclose the leaflet surface with the aforementioned scallop . one such possible manifestation is a composite curve consisting of an underlying circular arc or wave upon which a second higher frequency sinusoidal wave is superimposed . a third wave having a frequency different from the first and second waves could also be superimposed over the resulting composite wave . this ensures a wider angle between adjacent leaflets in the region of the commissures when the valve is fully open thus ensuring good wash - out of this region . the composite curve , and the resulting leaflet , can be either symmetric or asymmetric about a plane parallel to the blood flow direction and bisecting a line drawn between two stent tips such as , for leaflet 2 a , the section along line 3 - 3 of fig2 . the asymmetry can be effected either by combining a symmetric underlying curve with an asymmetric superimposed curve or vice versa , or by utilising a changing wave amplitude across the leaflet . the following describes the use of a symmetric underlying function with an asymmetric superimposed function , but the use of an asymmetric underlying function will be obvious to one skilled in the art . the underlying function is defined in the xy plane and connects the leaflet attachment points to the scallop at a given height from the base of the valve . this underlying function shown in fig1 , can be trigonometric , elliptical , hyperbolic , parabolic , circular , or other smooth analytic function or could be a table of values . the superimposed wave is defined in the xy plane , and connects the attachment points of the leaflet to the scallop at a given height above the base of the valve . the superimposed wave is of higher frequency than the underlying wave , and can be trigonometric , elliptic , hyperbolic , parabolic , circular , or other smooth analytic function , or a table of values . one possible asymmetric leaflet design is formed when the underlying wave formed using a circular arc is combined with a superimposed wave formed using the following equation . x s = - a s · b s ⁡ ( y ) · sin ⁡ [ ( 1 . 5 · π y ( n , 0 ) ) · ( y - y ( n , 0 ) ) ] a circular arc is defined by its cord length , 2y ( n , o ) , and amplitude , a u , as shown in fig1 . a s can be varied across the leaflet to produce varying wave amplitude across the leaflet , for example lower amplitude in one commissure than the opposite commissure . b s can be varied to adjust the length of the wave . the superimposed wave is shown in fig1 . the composite wave formed by combining the underlying wave ( fig1 ) with the superimposed wave ( fig1 ) is shown in fig1 . the composite wave w ( x c , y c ) n is created by offsetting the superimposed wave normal to the surface of the underlying wave ( fig1 ). positive γ is defined as the direction of the normal to the underlying wave relative to the x - axis . when y is positive , the composite curve is created by offsetting in the direction positive γ and where y is negative the composite curve is created by offsetting in the direction negative γ ( the offset direction is shown by arrows for a positive y point and a negative y point in fig1 . while the general shape of the leaflet in position p has been determined using the composite wave , at this stage it is not specified in any particular position . in order to specify the position of p , the shape of the partially open leaflet position can be defined using the ratio of the amplitude of the circular arc a u to the amplitude of the sinusoidal wave b s . a large ratio results in a leaflet which is substantially closed and vice versa . in this example the ratio changes from 10 at the base of the leaflet to 4 at the free edge of the leaflet . the result of this is a leaflet which effectively is more open at the free edge than at the base of the leaflet . in this way , the degree of ‘ openness ’ of the leaflet in position p can be varied throughout the leaflet . the composite wave is thus defined to produce the molded “ buckle ” in the leaflet , and the amplitude ratio is used to define the geometry of the leaflet at position p . at this stage it may bear no relation to the closed leaflet shape in position c . in order to match the area distribution of both leaflet positions , ( thus producing essentially the same leaflet in different positions ) the composite wave length is iterated to match the length of the relevant leaflet contour in position c . thus the amplitude and frequency of the individual waves can be varied in such a manner as to balance between : ( a ) producing a resultant wave the length of which is equal to the relevant value in the length function l ( z ) thus approximating the required closed shape when back pressure is applied , and ( b ) allowing efficient orifice washout and ready leaflet opening . this process identifies the values of a u and b s to be used in constructing the mold for the valve leaflet . as long as the constants such as a u and b s are known , the surface of the leaflet in its molded position can be visualised , enclosed and machined in a conventional manner . as a result of this fitting process the composite wave retains the same basic form but changes in detail from the top of the leaflet to the bottom of the leaflet . a composite wave can be defined in the leaflet surface as the intersection of the leaflet surface with a plane normal to the z axis . in summary therefore one possible method of designing the leaflet of the second embodiment of the present invention is in the following way : ( 1 ) define a scallop shape ; ( 2 ) define a shape representing the closed leaflet using a contour length function l ( z ); ( 3 ) use circular arcs and sine waves to generate a geometry which is partially - open , which pertains to a leaflet position which is between the two extreme conditions of normal valve function , i . e ., leaflet open and leaflet closed ; ( 5 ) vary the amplitude of the arcs and the sinewaves to fit to the length function l ( z ); and ( 6 ) the respective amplitudes of the circular arcs and sine waves can be varied from the free edge to leaflet base , for example the degree of ‘ openness ’ of the leaflet can be varied throughout the leaflet . example 3 set forth hereafter is an example of how the invention of the second embodiment can be put into practice . using the scallop constants in table 5 , the constants required to produce an example of an asymmetric leaflet valve are given in table 6 . these constants are used in conjunction with the aforementioned equations to define the leaflet geometry . with one leaflet described using the aforementioned equations , the remaining two leaflets are generated by rotating the geometry about the z axis through 120 ° and then through 240 °. these leaflet shapes are inserted as the areas of the dipping mold ( otherwise known as a dipping former ), which form the majority of the leaflet forming surfaces , and which then forms a 3 - dimensional dipping mold . the composite wave described in the aforementioned equations , therefore substantially defines the former surface which produces the inner leaflet surface . a drain - off area 30 is also created on the former to encourage smooth run - off of polymer solution . the drain - off region 30 is defined by extruding the leaflet free edge away from the leaflet and parallel to the flow direction of the valve for a distance of approximately 10 mm . the transition from leaflet forming surface of the dipping mold 24 to the drain - off surface of the dipping mold 30 is radiused with a radius greater than 1 mm and preferably greater than 2 mm to eliminate discontinuities in the leaflet . the details of the manufacture of the valve of the second embodiment are similar to those previously described with respect to the valve of the first embodiment until the valve is removed from the dipping mold . since the former used in making the valve of the second embodiment is not tapered the stent posts are not deflected by the former and do not move or change the leaflet shape when the valve is removed from the mold . at this stage the dipping mold and frame is covered with an excess of polyurethane due to the drain - off of the polymer onto the region of the mold known as the drain - off area 30 . to maintain the integrity of the frame coating , the leaflet is trimmed above the stent tips at a distance of between 0 . 025 to 5 mm preferably 0 . 5 mm to 1 . 5 mm from the stent tip . thus part of the surface of the leaflet is formed on the drain - off region 30 which is substantially defined using the composite wave w ( x c , y c ) 0 . leaflet free edges may be trimmed of excess material using a sharp blade rotated around the opened leaflets or using laser - cutting technology or other similar technology . the valve of the second embodiment may be used in any required position within the heart to control blood flow in one direction , or to control flow within any type of cardiac assist device . the following example 3 uses the same scallop geometry described using the constants set forth in table 5 : while the example 3 described herein relates to one valve size , the same method can be used to produce valves from a wide range of sizes . this can be carried out by modifying the constants used in the equations , and computing and iterating in the normal fashion or by rescaling the leaflet . the parameters described in the preceding sections are assigned the values set forth in table 6 and are used to manufacture an asymmetric valve according to the second embodiment . the included angle between adjacent leaflet free edges at the valve commissure for this valve is approximately 30 °. table 7 definition of parameters r internal radius of valve scallop ( fig2 ) the scallop is defined using a simple straight line , defined using a slope and intersection , to cut with a cylinder . closed leaflet geometry c l ( z ) is used to modify the inherent geometry of the leaflet . circular arcs and straight lines can be used to enclose the surface defined using l ( z ). molded position p p is enclosed by a number ( n ) of contours w ( x , y ) n which run from one side of the scallop to the other . the underlying function is used in defining both symmetric and asymmetric leaflets . running in a plane from one side of the scallop to the other . the points on the scallop are designated x ( n , 0 ) , y ( n , 0 ) where n refers to the contour number ( see fig1 , 16 , 17 , 18 ). y variable in plane from y ( n , 0 ) to − y ( n , 0 ) a u a u is the amplitude of the underlying wave a s ( y ) a s is a function which biases the wave amplitude in a defined way , e . g . the amplitude of the wave can be varied from commissure to commissure to produce asymmetry in the leaflet . b s b s is the amplitude of the superimposed wave composite curve ( fig1 ) x c x coordinate for defining the composite curve . y c y coordinate for defining the composite curve . open leaflet position ( fig1 ) the open leaflet position is defined using a ratio which determines the degree of “ openness ” of the leaflet . θ former taper angle
a cardiac valve prosthesis having a frame and two or more leaflets attached to the frame . the leaflets are attached to the frame between posts , with a free edge which can seal the leaflets together when the valve is closed under back pressure . the leaflets are created in a mathematically defined shape allowing good wash - out of the whole leaflet orifice , including the area close to the frame posts , thereby relieving the problem of thrombus deposition under clinical implant conditions .