Peskin-1975

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The Dgerential Geometry of Heart Valve? 
by CHARLES S. PESKIN 
Courant Institute of Mathmatical Sciences 
.New York University, New York City, .New York 
ABSTRACT : Heart valves when closed consist of membranes under ten&m bearing a pressure 
load. Along the linee where the lea$eta meet, ape&al support &u&urea under tension are 
needed (ajibroua ridge for the arterial valvea and a network of torch for the atrioventricular 
valves). The equilibrium of thae ~up-port~ requirea that the valve leaflets form cuapa along 
each line of closure. In the aortk and pulmonary valves there is evidence that the stress in 
the leaflet is borne by a single famgly of load-bearing fibers covering the surface. Under this 
hypothesis, we derive a differential equation for the leaflet and show that the Jibera are 
geodetic lines under constant tension. During the large motions of valve opening and closure 
certain constraints are imposed on the surface metric by the presence of (almost) inextensible 
Jibera in the surface. The severity of the constraint.9 dependa on the number of independent 
familiea of fibers. 
I. Introduction 
Like soap bubbles, suspension bridges and other structures under 
tension (l), the valves of the human heart invite analysis. Since the valve 
leaflets are essentially curved surfaces, the logical tool for studying their 
form is differential geometry (2). Moreover, the pressure load on the closed 
valve is everywhere normal to the surface and constant in magnitude. This 
suggests that the surface normal vector will play an important role in the 
geometry of heart valves, as it does generally in the geometry of surfaces 
embedded in three-dimensional space. 
We begin by demonstrating that the main feature of heart valve anatomy, 
the fact that the leaflets form cusps along their line of closure, follows directly 
from the equations of equilibrium of the closed valve. This is an easy con- 
sequence of a theorem (3) that the edge cables supporting a membrane under 
tension are asymptotic lines of the membrane surface. Next, we propose the 
hypothesis that the tension in the closed valve leaflet is supported by a single 
family of load-bearing fibers. We derive a differential equation for the fibers 
(and hence for the surface) from the conditions of equilibrium, which lead 
to the conclusion that each fiber is under constant tension and lies along a 
geodetic line of the surface. (Here the fact that the load is always normal to 
the surface plays a crucial role.) Finally, we discuss briefly the question of 
valve leaflet motion formulated as follows: Assuming that the load-bearing 
fibers are inextensible, is there any constraint on the kind of surface that the 
valve leaflet can cover during its motion ‘1 We shall show that the answer to 
* This work was supported in part by U.S. Public Health Service Grant 5T5GM1674 
from the N&ional Institutes of Health, Institute of General Medical Sciences. 
335 
Charles X. Peskin 
this question is essentially different if the number of families of load-bearing 
fibers is 1, as assumed above, 2, or 3 or more. 
The reader who is unfamiliar with cardiac anatomy will find an excellent 
presentation of 
Netter (4). 
the subject in the detailed and beautiful illustrations of 
II. Geometrical Properties of the Line of Closure 
The leaflets of the closed valve are membranes under tension. They touch 
each other along curves which are called the lines of closure. These curves are 
near the free edge of the leaflets ; for purposes of discussion we shall assume 
that they are the free edge. It is the aim of this section to show that the 
leaflets not only touch along these curves but are tangent there. 
We begin by discussing the equilibrium of the edge of a membrane under 
tension acoording to the method of (3). If the membrane stress does not 
vanish at the edge it must be balanced by some sort of support such as a 
cable under tension. Let the edge curve be given by x(s), where s measures 
distance. Then 
dx 
T = ds 
is the unit vector tangent to the edge. Let G(s) with dimensions of force be 
the tension in the edge cable, and let f(s)ds be the force exerted by the 
membrane on the arc ds, so that f(s) with dimensions of force per unit length 
lies in the tangent plane to the membrane at each point. The equations of 
equilibrium are 
-&G(s)T)+f(s) = 0 
or 
$f~+Gz+f(s) = 0. 
Let L be the vector in the membrane surface normal to T. Then we can 
resolve f(s) into jr(s) +r +fi(s) L. Then (3) becomes 
[g+f&)] ~+Gg+f,(s)L = o. 
Now 7-L = 0, and, since 1~1 = 1, ~.dT/ds = 0. Therefore (4) can be separated 
into two parts as follows : 
dG 
z+&(s) = 09 
Gz+fi(s)L = 0. 
336 Journal of The Franklin Institute 
The Differential Geometry of Heart Valves 
Equation (6) shows that the vectors L and dr/ds lie along the same line 
(in opposite directions). Now dr/ds is the principal curvature vector to the 
edge cable, and L is a vector parallel to the membrane surface. It follows 
at once that the edge cable forms an asymptotic line of the surface, i.e. a 
curve whose osculating plane (the plane at each point which most nearly 
contains the curve) is tangent to the surface. This result is proved in (3). 
(a) (b) 
FIG. 1. The line of closure of two valve leaflets. (a) The osculating plane of the line of 
closure contains the vectors 7, the unit tangent to the line of closure; dr/ds, the 
curvature vector ; L, the unit normal to the line of closure lying parallel to the surface ; 
and f, the force per unit length exerted by the membrane on the line of closure. (b) A 
plane normal to the line of closure cuts the valve leaflets along curves which form a 
cusp at 23. The tangent to both curves is the curvature vector dr/ds. 
We now remark that when two valve leaflets touch along a curve at their 
edges, the curve which constitutes the line of closure is common to both 
leaflets. Moreover, the two leaflets must be separately in equilibrium because 
tension cannot be transmitted across the line of closure from one leaflet to 
the other. By the arguments given above, therefore, both leaflets are tangent 
to the osculating plane of the line of closure at each of its points. Consequently 
they are tangent to each other. 
If a plane is constructed normal to the line of closure, each leaflet intersects 
this plane along a curve. What we have shown is that these curves meet at 
the line of closure to form a cusp (Fi g. 1). This feature of heartvalve anatomy 
is so fundamental that the leaflets are often referred to as the “cusps” of the 
valve. 
Before concluding this section we note two further consequences of 
Eqs. (5) and (6). From (5) the tension G is constant along the line of closure 
if there is no tangential stress, fi(s) = 0. From (6) and the fact that L and 
dr/ds are along the same line 
G g =f2(s). I I (7) 
Let Y-~ = IdrIds 1, so that r is the radius of curvature of the line of closure. 
Then 
f = f,(s) 
vol. 297,No. 5,iway 1974 337 
Charles S. Peskin 
which shows that the normal tension in the leaflet at the edge is related to 
the tension in the line of closure through the radius of curvature of the line 
of closure. 
As a practical matter, supporting the tension G in the line of closure poses 
a problem for the valve leaflet. Since G has dimensions of force this problem 
is of a different order of magnitude from that of supporting the stress in the 
interior of the leaflet, as the latter has dimensions of force per unit length. 
To see this more clearly, let p be the pressure, R a typical radius of curvature 
of the leaflet, 6 the thickness of the leaflet, r a typical radius of curvature of 
the line of closure and 62 the cross-sectional area of the “line” of closure. 
Then the true material stress (force per unit area) in the interior of the 
leaflet is of order pR/6, while the material stress in the line of closure is of 
order (pR/S) (r/S) = pRr/P. Two ways to go about solving this problem are 
to thicken the leaflet alongthe line of closure, or to break up the line of 
closure into a sequence of short, sharply curved arches. Both of these 
solutions are actually used : the first in the aortic valve where a fibrous ridge 
supports each leaflet along the back of the line of closure and the second in 
the mitral valve where each leaflet ends in a chain of short arches supported 
at their junctions by a network of cords. 
ZZZ. The Hypothesis of a Single Family of Load-bearing Fibers 
It is well known that the valve leaflets are mainly supported by a layer of 
fibrous tissue, the “fibrosa” (5). We shall assume in this section that the 
tibrosa is organized in such a way that at each point of the leaflet there is a 
definite fiber orientation which can be described by a unit vector field t 
parallel to the leaflet. To our knowledge, no map of fiber orientation has yet 
been made for a heart valve leaflet, but measurements of stress-strain 
relations for adult human aortic and pulmonary valves (6) show that the 
valve tissue is markedly anisotropic. For example, at 10 per cent elongation 
the ratio of radial to circumferential stress is about 0.1 for the aortic valve 
and O-025 for the pulmonary valve. This strongly suggests that the valve is 
stiffened by a family of relatively inextensible fibers oriented primarily in 
the circumferential direction. In the mitral and tricuspid valves, by contrast, 
the stress-strain curves are nearly identical in the radial and circumferential 
directions (6). It appears, then, that the hypothesis of this section is correct 
for the aortic and pulmonary valves but not for the mitral and tricuspid 
valves. We may speculate that for the latter two valves the fibrosa is com- 
posed of several layers of fibers oriented in different directions. 
If the stress in the surface is supported by a one-parameter family of fibers 
covering the surface, then it is appropriate to use the fibers as parameter 
curves (Fig. 2). That is, we seek the equation of the surface in the form 
X(U, V) where the lines v = constant are the load-bearing fibers and where the 
parameter u measures arc length along these fibers. As the initial line u = 0, 
we choose the line of closure as discussed in the previous section. Along this 
338 Journal of The Franklin Institute 
The DiJerential Geometry of Heart Valves 
u=o 
FIG. 2. Coordinate system when load-bearing fibers are used as prtrameter curves 
v = constant. The parameter u meaares arc length along the fibers. At equilibrium 
the fibers are geodesics. 
line, we let v measure arc length, but in general v cannot measure arc length 
because the distance between fibers changes as they fan out over the surface. 
The unit vector t which characterizes the directions of the fibers is given by 
The vector (t x ax/i%) dudv points normal to the surface and has as its 
magnitude the area of the patch (du, dv). Consequently the force exerted on 
the patch by the pressure is 
Let tTdv be the force transmitted by the fibers across an arc dv along 
u = constant. There is no force transmitted across the lines v = constant 
because no fibers cross such lines. Then the force exerted on the patch 
(du, dv) by the fibers in the surface is 
; (tT) du dv. 
Therefore, the equation of equilibrium for the surface is 
Taking the dot product of both sides with t and noting that 
t.at=_ 
au fT$ti2= 0, 
t* tx; = (txt).$= 0 ( 1 
we find 
aT 0 au= 9 
T = T(v) 
(11) 
(12) 
Vol. 297, NO. 5, May 1974 339 
Charles 8. Peskin 
and hence 
T(u)& tx; =O. 
( 1 
(13) 
The two Eqs. (9) and (13) can be written as a first-order system as follows : 
(14) 
From Eqs. (ll)-(la), we conclude that the tension is constant along 
each of the load-bearing fibers and that the fiber itself is a geodetic line in 
the surface, since such lines may be characterized by the fact that their 
curvature vector points normal to the surface (7). 
Initial conditions which suffice to determine the form of a leaflet are the 
line of closure x(0, v), the initial fiber directions t(O, v) and the tension function 
T(v). These are not completely arbitrary. It follows from the discussion of 
the previous section that t lies in the osculating plane to the curve x(O,v). 
Moreover, once x(0, v) and t(O,v) have been chosen the function T(v) is 
determined up to constant of proportionality. To see this, eliminate G 
between Eqs. (5) and (8) to obtain 
(15) 
In this case s = v and f = Tt, so that 
fi = Tcos8, 
fi = Tsine, 
where 0 is the angle between the fiber and the line of closure. Therefore, (15) 
becomes 
-$vTsin8)+TcosB = 0, 
where r and 0 are given functions of v. This has the solution 
rT sin 0 
r0 To sin 8, (17) 
In the special case 0 = ~12 we have simply rT = constant. 
The significance of the two possible signs in Eqs. (14) is as follows. On the 
two leaflets which meet along the line of closure u = 0, the pressure has the 
same magnitude but points in the opposite direction. The two signs give US 
the two leaflets which meet in the line of closure. Consider the special case 
where these two leaflets have fibers with the same initial directions. Then both 
leaflets obey (14) with identical initial conditions but different signs. If the 
line of closure is a plane curve, this means that the leaflets will be mirror 
340 Journal of The Franklin Institute 
The Differential Geometry of Heart Valves 
images of each other. In the general case, however, the line of closure need 
not be a plane curve, and the relationship between the leaflets which meet 
along it need not be so simple. 
The nature of Eqs. (14) may be somewhat clarified by reducing them to a 
system of ordinary differential equations in a case where axial symmetry 
applies. Let the initial curve be the unit circle 
x = cosv, IJ = sinv 
with radial initial fiber directions. The resulting surface will be symmetrical 
about the z axis, and it suffices to determine the form of the fiber v = 0, 
which lies in the x, x plane. In these circumstances (14) reduces to 
ae 
&=k- ' x 
I 
(18) 
ax 
au ' 
sin 8, 
where R = T/p and 0 is the angle between t and the z axis. As the fibers fan 
out, their density goes down in proportion to their distance from the x axis, 
but their curvature goes up accordingly, and the normal pressure that they 
can sustain remains the same. The same principle applies in the general case, 
but the fibers are not necessarily plane curves because of the torsion of the 
line of closure. 
IV. Remarks on the Deformations during Leaflet Motion 
In the preceding sections we have only considered the closed valve. Here 
we approach the more difficult problem of leaflet motion, a full resolution of 
which involves the fluid dynamics of the blood. Even the kinematics of 
leaflet motion involves an interesting problem which we may pose as follows. 
Let xa(u, v) be the surface of a valve leaflet when the valve is closed. When 
the leaflet moves it is deformed into a surface x(u,v) where the point (u,v) 
in the parameter plane corresponds to the same material point in both cases. 
Find the class of possible surfaces x into which x0 can be deformed under the 
following restrictions. First, a given boundary curve is held fixed in space. 
This is the line where the valve leaflet attaches to the tissues that support it. 
It forms only part of the border of the leaflet, the remainder being the free 
edge. Second, we require that the length of each arc along a load-bearing 
fiber be fixed during the deformation. This is an idealization based on the 
idea that the stresses during leaflet motion are so much smaller than the 
stresses when the valve is closed that these fibers will hold their lengths 
relatively constant during the motion. Finally, we assume that the load- 
bearing fibers do not slip through the material of the leaflet, so that each 
fiber is represented by a fixed curve in the U, v plane. 
Vol . 297 ,No . 5,May 1974 341 
17 
Charles S. Pet&in 
The problem then divides itself into two parts: 
(I) What restrictions are imposed on the surface metric by these conditions Z 
(II) Whut restrictions are imposed on the shape of the surface in space by the 
surface metric T 
Only the first of these problems will be discussed here in any detail. 
By the surface metric we mean the differential form 
ds2 = Edu2+2Bdudv+Gdv2 (19) 
which determines the length of a differential arc when the displacements 
du, dw in the parameter plane are given. 
If there is only one family of load-bearing fibers then it is reasonable to 
introduce parameter curves like those of the last section so that the fibers 
are given by w = constant with u measuring arc length. In this case the metric 
has E = 1, but P and G are arbitrary. Since such parameters can be intro- 
duced along an arbitrary family of curves in an arbitrary surface, there is no 
restriction on the shape of the surface in space. 
If there are two families of load-bearing fibers, we can let u = constant be 
the members of one family and v = constant be the members of the other. 
In this case neither u nor v measures arc length in general, and the metric 
has E and G as given functions of (u, v) with F arbitrary. In other words, the 
parameter curves form a net of inextensible fibers with fixed intersections, 
but the angle between the fibers can be varied at will. Such a net can be 
considered as a generalization of the Tchebychef net (8) because the 
Tchebychef net has E = G = 1 with P arbitrary. In Ref. (8) it is shown that 
a Tchebychef net cannot smoothly cover a surface whose integrated Gaussian 
curvature is too large. This suggests that the deformations with two families 
of load-bearing fibers are not completely unrestricted. 
When the leaflet is covered by three inextensible families of fibers, we can 
use two of them as parameter curves. This fixes E and G as functions of (u,v). 
The third family of fibers will then have the form u(r,s),v(r,s) where r 
identifies the fiber and s measures length along it. During the deformations 
of the surface these functions remain fixed because of the no-slip condition 
discussed above. Taking differentials along such a fiber we have 
1= E@2+2F(z) @) +G($)2. (20) 
Now E and G are known at each point (u, v), and du/ds, dv/ds are also known 
as functions of (r, s) and hence of (u, w). Therefore we can solve for 
F = 1 - E(du/ds)z- G(dv/ds)2 
Z(du/ds) (dv/ds) . (21) 
F becomes undetermined only when du,ids or dv/ds = 0, which would imply 
that the third family of fibers was parallel to one of the first two. 
If this exception is excluded then we see that for three families of fibers 
(or more) the surface metric is completely determined. This means that the 
class of surfaces onto which x0 can be deformed is the class of surfaces 
342 Journal of The Franklin Institute 
The Differential Geometry of Heart Valves 
isometric to x,,. That is, the deformation is required to leave the lengths of 
all’curves in the surface invariant. 
From the functional point of view such a constraint has two attractive 
features. First it guarantees a minimum of wear on the material. Second, it 
provides the most control over the leaflet motion that is possible from within 
the leaflet. This may be important for the rapid and accurate approximation 
of the leaflets during valve closure. 
V. Conclusions 
Where valve leaflets touch, they form cusps. The tangent plane to both 
leaflets is the osculating plane of the line of closure. 
If the load in the closed valve is borne by a single family by fibers covering 
the surface, then these fibers are geodesics and the tension is constant along 
each fiber. The distribution of tension in the surface is determined up to a 
constant of proportionality by the geometrical character of the line of 
closure. 
If the concept of load-bearing fibers is valid, then the changes in length 
of these fibers during the motion of the leaflet should be small. This imposes 
(approximately) certain constraints on the surface metric. The metric is 
completely determined when the number of families of load-bearing fibers is 
three or more. 
Our effort has been to show how the local fiber structure will have a 
profound influence on the geometry of the heart valve and on the geometry 
of its motions. Hopefully this will make further experimental investigations 
of the local mechanics of the tissue seem worthwhile. 
Acknowledgement 
Helpful discussions with Dr. Edward L. Yellin and Ted Feit are gratefully 
acknowledged. 
References 
(1) F. Otto, “Tensile Structures”, The M.I.T., Cambridge Press, 1967. 
(2) J. J. Stoker, “Differential Geometry”, Wiley, New York, 1969. 
(3) F. Otto, op. cit., Vol. 1, pp. 183-184. 
(4) F. H. Netter, “Heart”, The Ciba Collection of Medical Illustrations, Oiba, Vol. 5, 
pp. S-12, 1969. 
(5) L. Gross and M. A. Kugel, “Topographic anatomy and histology of valves in the 
human heart”, Am. J. Path., Vol. 7, pp. 445-474, 1931. 
(6) H. Yamada, “Strength of Biological Materials”, p. 111, Williams & Wilkins, 
Baltimore, 1970. 
(7) J. J. Stoker, op. cit., p. 165. 
(8) J. J. Stoker, op. cit., pp. 199-200. 
VoL297 ,No . 5,May 1974 343