A Mathematical Representation of the Wheatley Heart Valve

Starting from a hand-drawn contour plot, this note develops a set of intersecting and contiguous circles whose perimeter, upon extending appropriately to three dimensions, can be seen to be a natural mathematical representation of the Wheatley heart valve.


Introduction
The background and motivation for this study arise from the experience of one of the authors (DJW) as a clincal and academic cardiac surgeon with decades of experience of heart valve disease in different parts of the world and an international role through the European Association for Cardio-thoracic Surgery [1].He also has a longstanding research experience in the field of heart valve replacement surgery [2,3].Recent and ongoing work has culminated in a potential solution to the clinical problems associated with current prosthetic heart valves, particularly in the developing world [4].
The majority of heart valve disease presents in the developing world where rheumatic fever is still prevalent.Currently the mechanical valves available for surgical treatment of advanced irreparable valve disease require life-long anticoagulant or antiplatlet therapy to mitigate thrombo-embolic complications [5].Such therapy is often impractical or unavailable for those patients in the developing world who are vulnerable to heart valve disease.The alternative biological valves are less at risk of thrombo-embolic complications but have more limited durability, especially in young patients who are in the majority presenting with valve disease in developing countries [6,7].
The quest has been to design a prosthetic valve that would be sufficiently resistant to degradation in the body to offer the prospect of survival in excess of two decades or more, while at the same time not running an associated appreciable risk of thrombo-embolic complications attributable to abnormal flow conditions associated predominantly with mechanical valves [8].Currently available biostable polyurethanes show promise of durability in the human body [9].The novel design suggested by one of the authors (the Wheatley valve) is the subject of this paper [10][11][12].The design arises from experience with flexible-leaflet biological and synthetic valve prostheses.However, it differs from the usual design of flexible-leaflet valves [13] as it is intended to interact with known flow patterns within the ascending aorta [14].This interaction should facilitate leaflet opening and closing, as well as avoiding areas of abnormally high or low shear stress at the valve/blood interfaces and areas of poor washout of blood around the valve.The design is also intended to significantly reduce the stresses within the leaflet material that are inherent in the current designs modelled on the natural aortic valve.This is a further feature that offers a prospect of enhanced durability in comparison with many bioprosthetic and experimental synthetic, flexible-leaflet valves.
Producing a mathematical representation of the design is a useful first step before applying computational fluid dynamics to understand both the blood flow in the valve and the ascending aorta, and the internal leaflet stresses during valve function.This paper is therefore intended to characterize the shape of the Wheatley valve using simple mathematical functions.Of course, the shape could have been generated via splines, but the principal advantage of a mathematical representation is that it permits shape changes to be easily trialed.For example, there is no need for a linear increase in the vertical direction: any power of z may be employed, generating an infinite family.Furthermore, circles may be replaced by ellipses, thus generating an even greater number of potential valves, so that an optimal shape may be efficiently ascertained.Some illustrations are supplied.

The artificial aortic valve
The artificial valve is displayed in Figure 1.  to capture with simple mathematical functions.Indeed, close inspection of the valve and its associated contour lines suggests that, for each contour line, there exists six underlying symmetrically placed circles.These are displayed in Figure 2 with the appropriate arcs coloured red; the great circle is, for convenience and without any loss of generality, the unit circle.Note that the contours in Figure 1(a) are viewed from above the valve.The contour shown in Figure 2 is viewed from below and is therefore the mirror image.(1) Reverting to the X,Y -coordinate system this becomes Thus we see that the circle, provided by equation ( 2), passes through B(1, 0) and A(−a, 0); we wish it also to pass through P(− Thus the circle, given by equation ( 2), now takes the form Further, since the radius of the circle BP is 1 2 (1 − b), its equation is given by 4 Two further sets of circles We have established the equations of the two circles BPP and BP , given in Figure 3.However, from Figure 2 it is evident from symmetry that the two sets of circles B P P and PQ and B P P and P Q are exactly the same pairs of circles whose equations have already been developed: the only difference is that these pairs of circles have been rotated by 43 π and Fig. 3: A large circle and three smaller circles symmetrically placed around the origin Now, clockwise axis rotation from (X ,Y ) to (X,Y ) through 4  3 π is given by

A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e d
Fig. 4: Axes rotation through 4  3 π Thus equations ( 7) and ( 8) in the X,Y -coordinates become and 4.2 Circles B P P and P Q Similarly to the previous subsection, we note that the circles Q P P and P Q can be expressed in the X ,Y -coordinate system displayed in Figure 5: As we can see from Figure 5 we require to rotate from (X ,Y ) to (X,Y ) clockwise through 2 3 π.This is given by the transformation Thus equations ( 13) and ( 14) become

A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e d
and In summary, the six circles, depicted in Figure 2, are given by the equations ( 5), ( 6), ( 11), (12), and (17) , (18).
For equation ( 5) we write the parametrization directly as For equation ( 12) we have Solving the simultaneous equations yields In a similar way we obtain the parametrization for equations (11),( 18),( 17) and ( 6) as, respectively, Fig. 1(a): Contour plots of the Wheatley valve

A c c e
Fig. 1(b): The Wheatley Valve

1 4 )Fig. 2 :
Fig. 2: Three larger and three smaller circles within the unit circle: the red line indicates the required arcs

A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dFig. 5 :
Fig. 5: Axes rotation through 2 3 π

A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dFig. 6 :Fig. 7 :
Fig. 6: Mathematical representation of the Wheatley valve during systole, displaying the valve opening