Anterior mitral leaflet curvature in the beating ovine heart: a case study using videofluoroscopic markers and subdivision surfaces.

Abstract

The implantation of annuloplasty rings is a common surgical treatment targeted to re-establish mitral valve competence in patients with mitral regurgitation. It is hypothesized that annuloplasty ring implantation influences leaflet curvature, which in turn may considerably impair repair durability. This research is driven by the vision to design repair devices that optimize leaflet curvature to reduce valvular stress. In pursuit of this goal, the objective of this manuscript is to quantify leaflet curvature in ovine models with and without annuloplasty ring using in vivo animal data from videofluoroscopic marker analysis. We represent the surface of the anterior mitral leaflet based on 23 radiopaque markers using subdivision surfaces techniques. Quartic box-spline functions are applied to determine leaflet curvature on overlapping subdivision patches. We illustrate the virtual reconstruction of the leaflet surface for both interpolating and approximating algorithms. Different scalar-valued metrics are introduced to quantify leaflet curvature in the beating heart using the approximating subdivision scheme. To explore the impact of annuloplasty ring implantation, we analyze ring-induced curvature changes at characteristic instances throughout the cardiac cycle. The presented results demonstrate that the fully automated subdivision surface procedure can successfully reconstruct a smooth representation of the anterior mitral valve from a limited number of markers at a high temporal resolution of approximately 60 frames per minute.

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Biomech Model Mechanobiol (2010) 9:281–293
DOI 10.1007/s10237-009-0176-z
ORIGINAL PAPER
Anterior mitral leaflet curvature in the beating ovine heart: a case
study using videofluoroscopic markers and subdivision surfaces
S. Göktepe · W. Bothe · J.-P. E. Kvitting ·
J. C. Swanson · N. B. Ingels · D. C. Miller ·
Ellen Kuhl
Received: 21 February 2009 / Accepted: 13 October 2009 / Published online: 5 November 2009
© Springer-Verlag 2009
Abstract The implantation of annuloplasty rings is a
common surgical treatment targeted to re-establish mitral
valve competence in patients with mitral regurgitation. It is
hypothesized that annuloplasty ring implantation influences
leaflet curvature, which in turn may considerably impair
repair durability. This research is driven by the vision to
designrepairdevicesthatoptimizeleafletcurvaturetoreduce
valvular stress. In pursuit of this goal, the objective of this
manuscript is to quantify leaflet curvature in ovine models
with and without annuloplasty ring using in vivo animal data
S. Göktepe
Department of Mechanical Engineering,
Stanford University, Stanford, CA 94305, USA
e-mail: goktepe@stanford.edu
W. Bothe · J.-P. E. Kvitting · J. C. Swanson · N. B. Ingels ·
D. C. Miller
Department of Cardiothoracic Surgery, Stanford University,
Stanford, CA 94305, USA
e-mail: wbothe@stanford.edu
J.-P. E. Kvitting
e-mail: kvitting@stanford.edu
J. C. Swanson
e-mail: swansjul@stanford.edu
D. C. Miller
e-mail: dcm@stanford.edu
N. B. Ingels
Laboratory of Cardiovascular Physiology and Biophysics,
Palo Alto Medical Foundation, Palo Alto, CA 94304, USA
e-mail: ingels@stanford.edu
E. Kuhl (B )
Departments of Mechanical Engineering, Bioengineering,
and Cardiothoracic Surgery, Stanford University,
Stanford, CA 94305, USA
e-mail: kuhl@stanford.edu
from videofluoroscopic marker analysis. We represent the
surface of the anterior mitral leaflet based on 23 radiopaque
markers using subdivision surfaces techniques. Quartic box-
spline functions are applied to determine leaflet curvature
on overlapping subdivision patches. We illustrate the vir-
tual reconstruction of the leaflet surface for both interpolat-
ing and approximating algorithms. Different scalar-valued
metrics are introduced to quantify leaflet curvature in the
beating heart using the approximating subdivision scheme.
To explore the impact of annuloplasty ring implantation,
we analyze ring-induced curvature changes at characteristic
instancesthroughout thecardiaccycle. Thepresentedresults
demonstratethatthefullyautomatedsubdivisionsurfacepro-
cedure can successfully reconstruct a smooth representation
oftheanteriormitralvalvefromalimitednumberofmarkers
atahightemporal resolutionofapproximately 60framesper
minute.
Keywords
Annuloplasty ring · Curvature · Subdivision surfaces ·
Continuity
Mitral valve · Mitral regurgitation ·
1 Motivation
Mitral regurgitation is a progressive, valvular disorder that
affects approximately 4million people in the United States
with250,000newcasesoccurringeachyear.Annually,more
than 300,000 people worldwide, 44,000 in the United States
alone, undergo open heart surgery for mitral valve treatment
(Bornow et al. 2006; Cosgrove 2004). The mitral valve is a
bicuspid heart valve consisting of two leaflets, anterior and
posterior, surrounded by the mitral valve annulus. A normal
mitral valve allows unidirectional blood flow from the left
atrium into the left ventricle during diastole and prevents
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282S. Göktepe et al.
Fig. 1 Annuloplasty rings are a common surgical approach to repair a
leaking valve. The GeoForm®ring has a characteristic dogbone shape
with a significantly elevated posterior segment to bring the leaflets
closer together and prevent back flow
back flow during systole. In mitral regurgitation, the valve
fails to close properly and blood leaks backward with each
heartbeat, lowering pumping efficiency.
The most common surgical approach to repair a leaking
valve is to bring the leaflets closer together by implanting an
annuloplasty ring around the mitral valve annulus. Annulo-
plasty rings come in different shapes, oval versus dog bone
shaped, flat versus saddle-contoured, may be constructed
fromarigidorflexiblematerial,andareavailableinanassort-
ment of sizes. One example is the Edwards GeoForm®ring
illustrated in Fig. 1. Specifically designed to restore leaflet
coaptation and treat mitral regurgitation, it has a character-
istic dog bone shape to significantly reduce the valve sep-
tal-lateral dimensions (Bothe et al. 2009; Votta et al. 2007),
shown as the middle vertical dimension in the top view of
Fig. 1. Choosing the optimal ring type and size depends pri-
marily on the surgeon’s experience and personal preference.
It is hypothesized that both ring shape and stiffness consid-
erably influence leaflet and annular dynamics.
Elevated leaflet stresses have been postulated to cause
long-term repair failure. Unfortunately, it is impossible to
measure leaflet stresses in the beating heart. Continuum
mechanics-based descriptions of leaflet dynamics, solved
with finite element techniques, offer the potential to predict
stress profiles in vivo. These approaches, however, require
a precise understanding of the material properties, boundary
conditions, and forces of the entire mitral valve apparatus
(Krishnamurthy et al. 2008, 2009; Prot et al. 2007, 2009;
Votta et al. 2008). Early finite element-based stress anal-
yses indicate that the GeoForm®ring might significantly
reducemaximumprincipalleafletstresses(Vottaetal.2007).
The effect of annuloplasty rings on leaflet stresses has only
recently been quantified in vivo (Bothe et al. 2009), but the
complexinterplaybetweenringformandfunctionisnotfully
understood to date.
Rather than determining leaflet material properties, eval-
uating leaflet equilibrium, and then quantifying leaflet stress
(Krishnamurthy et al. 2009), we propose a more straightfor-
ward approach based on a direct analysis of leaflet kinemat-
ics (Karlsson et al. 1998; Sacks et al. 2006). We hypothesize
that mitral leaflet curvature immediately influences leaflet
stresses,andthushasadirectimpactonmitralvalvefunction
(Salgoetal.2002).Intheliterature,leafletcurvaturehasbeen
accessed using three-dimensional echocardiography (Ryan
et al. 2008b). The resulting images of the mitral leaflet were
manually segmented, manually merged, and approximated
with smooth splines. Based on a least squares fit, a param-
eterized surface representation was generated to determine
Gaussian curvature in ovine models with and without annu-
loplasty rings (Ryan et al. 2008a).
The present manuscript presents an alternative approach
to quantify leaflet curvature based on a high resolution vid-
eofluoroscopic marker analysis evaluated with a novel sub-
division surface algorithm. We illustrate the features of the
proposed approach in terms of two data sets, one with the
GeoForm®ring implanted and one with the ring released in
the same ovine heart. To define leaflet curvature in terms of
theacquiredmarkercoordinates,weadoptasubdivisionsur-
faceapproach.Theconceptofsubdivisionsurfaceswasintro-
ducedinthelate1970stoaddressthechallengeofgenerating
smooth free-form surfaces of arbitrary topologies (Catmull
and Clark 1978; Doo and Sabin 1978). Rather than assem-
bling individual patches of tensor product splines, subdivi-
sionsurfacealgorithmsgenerateasplinepatchasthelimitof
a repeated uniform knot intersection. Initially developed in
geometry and applied mathematics (Loop 1987; Reif 1995;
Reif and Schröder 2001; Zorin 2000), subdivision surface
algorithms are currently receiving a broad attention in com-
puter graphics (Umlauf 2000; Schröder 2002). Due to their
inherent C1-continuity, the structural mechanics community
has recently recognized subdivision surfaces as a new par-
adigm to characterize higher order derivatives in advanced
shell theories (Cirak et al. 2000; Cirak and Ortiz 2001; Cirak
et al. 2002). In this manuscript, we apply the generic idea
of subdivision, to create a parameterized surface representa-
tionoftheanteriormitralleaflet.Weutilizelocaloverlapping
patches of subdivision triangles to determine the global sec-
ond order curvature field with the help of quartic box splines
(Ciraketal.2000).InadditiontotheGaussiancurvaturefield
discussed in the literature (Ryan et al. 2008b), we also visu-
alize the mean and principal curvatures, and compare and
discuss the different curvature representations.
The manuscript is organized as follows: Sect. 2 briefly
summarizes our experimental technique to determine four-
dimensional leaflet marker coordinates. Section 3 compares
interpolating and approximating schemes and introduces the
concept of subdivision surfaces. In Sect. 4, we illustrate the
curvature computation and discuss different scalar-valued
curvature measures. The features of the proposed approach
aredemonstratedinSect.5bymeansofacharacteristicleaflet
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Anterior mitral leaflet curvature in the beating ovine heart283
Fig. 2 Intraoperative photograph showing some of the 23 tantalum
markers sewn to the mitral valve. Seven markers are sewn on the ante-
rior mitral annulus, seven on the mitral leaflet edge, and nine on the
leaflet belly
curvature analysis in an ovine model with and without annu-
loplastyring.Potentialapplicationsoftheproposedapproach
are discussed in Sect. 6.
2 Mitral leaflet coordinates
An adult Dorsett-hybrid male sheep was premedicated with
ketamine, anesthetized with sodium thiopental, intubated,
and mechanically ventilated with inhalational isoflurane.
A left thoracotomy was performed, and the heart was sus-
pended in a pericardial cradle. On cardiopulmonary bypass,
a total of 23 radiopaque tantalum markers were sewn to the
followingsites:sevenontheanteriormitralannulus,sevenon
themitralleafletedge,andnineontheleafletbelly,seeFig.2.
Toidentifymaximumleafletopening,anadditionmarkerwas
sewn on the central edge of the middle scallop of the pos-
terior mitral leaflet. Each tantalum marker weighed 3.2mg
and had an inner and outer diameter of 0.6 and 1.1mm,
respectively. After marker placement, a true-sized Geo-
Form®annuloplasty ring was implanted with a specifically
designed technique such that it could be released in the cath-
eterization laboratory while the heart is beating (Bothe et al.
2009). After ring implantation, the left atrium was closed,
the animal was weaned from cardiopulmonary bypass, and a
micromanometer transducer was placed in the left ventricle
throughtheleftatrium.Theanimalwasthentransferredtothe
experimental catheterization laboratory. It was placed in the
right lateral decubitus position for acquisition of data under
open-chestcondition,seeFig.3.Forleftventricularandatrial
pressure measurements, a micromanometer pressure trans-
ducer and a calibrated catheter were placed in the left ventri-
cleandleftatrium,respectively.Videofluoroscopicimagesat
Fig. 3 Data acquisition in the catheterization laboratory. The sheep is
imaged under open-chest conditions using biplane videofluoroscopy at
60 frames per second. Four-dimensional marker coordinates are gener-
ated by merging the time sequences from both cameras
60 frames per second were acquired of all radiopaque mark-
ers using a biplane videofluoroscopy system with the heart
in normal sinus rhythm and ventilation transiently arrested
at end expiration.
First, images were acquired with the ring attached to the
annulus.Thentheringwasreleased(Botheetal.2009).Ring
release was verified fluoroscopically and another data acqui-
sition was performed under baseline conditions to serve as
control. Marker coordinates from each of the biplane views
werethenmergedtoyieldthe3Dcoordinatesofeachmarker
centroidineachframeusingsemi-automatedimageprocess-
ing and digitization software developed in our laboratory
(Daughters et al. 1989; Niczyporuk and Miller 1991). Left
ventricular and atrial pressure were digitally recorded simul-
taneously during marker data acquisition and synchronized
with the marker coordinates. The previously described pro-
ceduregeneratesfour-dimensionalmarkercoordinatesforall
23 markers. In the following section, we compare different
techniques to represent the leaflet surface in terms of these
marker coordinates.
3 Surface representation
Subdivision surface algorithms area powerful tool tomathe-
maticallydefinearbitraryfree-formsurfacesbasedonagiven
setofcontrolpoints.Asubdivisionalgorithmconsistsoftwo
ingredients, a topological split algorithm defining how the
connectivity of the control mesh is refined, and a geometric
refinementalgorithmintroducingthenodalcoordinatesofthe
newrefinementlevel.Therefinementofacoarsemeshiscar-
ried out by quadrisecting triangular elements in the coarser
mesh.Duringquadrisection,eachedgeconnectingtwonodes
in the coarse mesh is divided into two edges through
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284S. Göktepe et al.
Interpolating scheme
control mesh1stsubdivision2ndsubdivision3rdsubdivision
Fig. 4 Interpolatingschemewithacontrolmeshbasedonthe23markercoordinates(lef t)andthreelevelsofsubdivision(right).Theinterpolating
algorithm generates surfaces which contain all control points but are not C1-continuous
Approximating scheme
control mesh1stsubdivision2ndsubdivision 3rdsubdivision
Fig. 5 Approximating scheme with a control mesh based on the 23 marker coordinates (lef t) and three levels of subdivision (right). The
approximating algorithm generates surfaces which are C1-continuous but only approximate the control points
insertionofnewlygeneratednodes.Dependingontheunder-
lyinggeometricrefinementalgorithm,surfacerepresentation
algorithms can be divided into two classes, interpolating and
approximating schemes. In what follows, we compare the
geometric refinement rules for these two schemes. Thereby,
we restrict ourselves to surfaces, which consist of triangular
faces only.
3.1 Interpolating schemes
Interpolating schemes preserve the nodal positions of the
coarser mesh upon refinement (Zorin 2000). This implies
that nodes of the original control mesh and all recursively
generated nodes will always lie exactly on the limit surface.
Accordingly, interpolating schemes are free of approxima-
tion errors; however, their limit surfaces do not have a well-
defined curvature. Figure 4 illustrates a typical sequence of
interpolating surfaces. The k + 1-th level of refinement is
generated by maintaining the coordinates of old nodes xk
the k-th level,
0of
xk+1
0
and by introducing new nodes xk+1
tions ofthenewly insertednodes arecalculated byaveraging
the weighted nodal positions of the unrefined mesh.
=1
8
Only the four nodes of two triangles that share the edge of
interest contribute the averaging Equation (2). The refine-
ment mask for the newly inserted nodes in Fig. 6 (right)
= xk
0
(1)
I
on its edges. The posi-
xk+1
I
?
3xk
0+ xk
I−1+ 3xk
I+ xk
I+1
?
(2)
demonstrates the distribution of the weighting factors 1/8
and 3/8 assigned to the four nodes (Loop 1987). Since the
subdivision process itself is linear, the resulting limit surface
is nothing but a linear combination of fundamental solutions
of the subdivision process.
3.2 Approximating schemes
Approximating schemes introduce an entirely new set of
nodal positions for each refinement step. Accordingly, nodes
of the original control mesh may no longer be part of
the limit surface. Approximating subdivision algorithms are
thus associated with an approximation error. In contrast to
interpolating schemes, approximating schemes can generate
C1-continuous limit surfaces with a well-defined curvature
providedthatelementswithirregularverticesarerecursively
refined. A typical approximating series of subdivision sur-
faces is illustrated in Fig. 5. The coordinates of the k + 1-th
refinement step are computed as weighted averages of the
nodal coordinates ofthe k-threfinement level. The oldnodes
xk
lowing formula,
0are assigned new coordinates xk+1
0
according to the fol-
xk+1
0
= [1 − vw] xk
0+ w
v
?
i=1
xk
I
(3)
while the newly generated nodes xk+1
edges of the k-th refinement are computed similarly to the
interpolating scheme.
?
I
associated with the
xk+1
I
=1
8
3xk
0+ xk
I−1+ 3xk
I+ xk
I+1
?
(4)
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Anterior mitral leaflet curvature in the beating ovine heart285
1
Existing node
New node
1
8
1
8
3
8
3
8
Fig. 6 Refinement mask for different algorithms. Existing nodes are
maintainedfortheinterpolatingschemeandrefinedfortheapproximat-
ingscheme(lef t).Newnodesarecalculatedbyaveragingtheweighted
nodal positions of the unrefined mesh for both schemes (right)
InEq.(3),w isaweightfunctionandv denotesthevalenceof
thenodeunderconsideration(Loop1987),i.e.,thenumberof
edges connected to it, see Fig. 6 (left). In a perfectly regular
triangular mesh, the valence of all inner nodes is v = 6, but
usually a few nodes of the original control mesh are irreg-
ular with a valence of v ?= 6. All newly generated nodes,
however, have a valence of v = 6 by construction.
Remark 1 (Specialtreatmentatsurfaceboundaries)Special
careneedstobetakenatkinematicboundarieswherethesub-
divisionalgorithmneedstobemodified.Inourexamples,we
use the method of artificial nodes that generates an artificial
mirror image of the original mesh along the boundary and
applies the standard subdivision rule as illustrated in Cirak
et al. (2000).
Remark 2 (Error of the approximation) While the interpo-
lating scheme always keeps the marker positions on the
subdivision surface, the approximating scheme inherently
introduces an approximation error. This approximation error
can be characterized through different error norms such as
the root mean square error norm ?e?L2or the average error
norm ?e?L1. In the following examples, we will explore the
average error norm
?e?L1=
1
nmrk
nmrk
?
i=1
???xk+1
0
− x1
0
???
(5)
which characterizes the average distance between the origi-
nallymeasuredmarkerpositions x1
on the subdivision surface of level k.
0andtheirrefinedposition
4 Curvature computation
The key objective of the approximating subdivision
algorithm outlined in Sect. 3.2 is the reconstruction of a suf-
ficiently smooth leaflet surface that allows for curvature ten-
sor field evaluation. At the end of the kth subdivision, our
T
Fig. 7 Regular patch with twelve nodes used for the curvature inter-
polation on triangle T
Fig. 8 Surface base vectors a1, a2, the surface normal a3at the inte-
gration point x of an element T on the surface S. The local curvilinear
coordinates θ1,θ2are assumed to be identical to the barycentric coor-
dinates of an element
database contains the nodal coordinates and the connectivity
of each triangle on the approximated surface. The curvature
fieldwithineachtriangle T iscomputedbasedonthecoordi-
nates of its own nodes and those of its immediate neighbor-
ing triangles, see Fig. 7. A regular patch consists of twelve
nodeswithnodalcoordinates xIfor I = 1,...,12.Thelocal
parametrizationofthelimitsurfacecanbeexpressedinterms
of quartic box-spline shape functions
x(θ1,θ2) =
12
?
I=1
NI(θ1,θ2) xI
(6)
where (θ1,θ2) are the barycentric coordinates of the master
triangle T = (θ1,θ2) with θ1,2∈ [0,1] and θ1+ θ2≤ 1,
see Fig. 8. For a regular twelve node patch, the box-spline
shape functions are given in Cirak et al. (2000). Irregular tri-
angles with nodal valences different from six require special
treatment. These irregular patches have to be locally refined
one-step further to obtain regular subpatches on which the
parametrization (6) can be applied.
Having the position vectors x at hand, we can determine
the covariant base vectors aαas
aα(θ1,θ2) =
12
?
I=1
NI,α(θ1,θ2)xI
for α = 1,2 (7)
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286S. Göktepe et al.
where NI,α:= ∂NI/∂θα. The normalized vector product of
base vectors yields the unit surface normal a3.
a3(θ1,θ2) :=
a1× a2
|a1× a2|
(8)
The key quantity describing the local curvature characteris-
tics of a surface is the second fundamental form (Kreyszig
1991),
B = Bαβaα⊗ aβ
with aα,β:= ∂aα/∂θβ. Due to the definition (8), the covari-
ant base vectors aαare orthogonal to the surface normal a3,
see Fig. 8. Hence, the identity Bαβ= aα,β· a3= −aα· a3,β
holds,andsodoestheequality Bαβdθαdθβ=−dx·da3.The
latter statement clearly illustrates the geometrical interpreta-
tion of the coefficients Bαβ. Recall that the contra-variant
base vectors aαare related to the covariant base vectors aβ
through the inverse of the local surface metric gαβ.
where Bαβ:= aα,β· a3
(9)
aα:= gαβaβ
The symmetric, positive definite surface metric gαβ, and its
inverse gαβare defined as follows.
(10)
gαβ:= aα· aβ
gαβ:= aα· aβ= (gαβ)−1
The surface metric gαβ, also referred to as the first funda-
mental form, is another key quantity required to evaluate
length measures. In light of the definitions given above, it
can readily be shown that contra-variant and covariant bases
are orthogonal aα·aβ= δβ
delta.
The principal curvatures κ at point x on the surface S,
Fig. 8, satisfy the principal value problem of the second fun-
damental form,
(11)
αwith δβ
αdenoting the Kronecker
B · n = κ g · n or Bαβnβ= κgαβnβ
where n are the corresponding contra-variant principal
directions. Non trivial solutions of (12) are obtained for
det(Bαβ− κgαβ) = 0 leading to the characteristic equation
of the principal curvatre problem.
(12)
κ2− IBκ + IIB= 0
Its coefficients IBand IIB
(13)
IB := tr(B) = Bαβgαβ= κ1+ κ2
IIB:= det(B) = det(Bαβ)/det(gαβ) = κ1κ2
are the first and the second principal invariants of the second
fundamental form B. Since B is symmetric, the principal
curvatures κ1,κ2arereal-valued andthecorresponding prin-
cipal directions n1,n2are orthogonal. The principal curva-
tures κ1,2can be determined as the roots of the characteristic
(14)
Equation (13).
κ1,2=1
Next to the principal curvatures κ1,2, the mean curva-
ture κmeanand the Gaussian curvature κgaussare commonly
employed in the literature to characterize a curvature dis-
tribution. In the forthcoming subsections, we introduce the
definitions of these curvature measures, illustrate their fea-
tures in terms of contour plot representations, and discuss
their physical interpretation. To this end, we evaluate the
second fundamental form B at the barycenter of each tri-
angle θ1= θ2= 1/3 using the isoparametric box-spline
parametrization (6).
2[IB±
?
I2
B− 4 IIB]
(15)
4.1 Mean curvature
The mean curvature is defined as the arithmetic mean of the
principal curvatures
κmean:=1
which is equivalent to κmean=1
(14)1.Asurfaceforwhichthemeancurvatureκmeanvanishes
at every point is called a minimal surface. It can be shown
that the minimal part of a surface with κmean= 0, bounded
byaclosedcurveΓ ,possessesthesmallestareacomparedto
any other part of the surface bounded by the same curve Γ ,
see (Kreyszig 1991, p. 244). The contours of κmeanin Fig. 9
illustrate the distribution of the mean curvature field over the
anterior leaflet at end diastole and at end systole for the third
subdivision level with 1,017 nodes.
2[κ1+ κ2]
(16)
2IB=1
2tr(B) according to
4.2 Gaussian curvature
Another widely used curvature measure is the Gaussian cur-
vature that is defined by the product of principal curvatures.
κgauss:= κ1κ2
It can also be expressed in terms of the second invariant
κgauss= IIB = det(B), see (14)2. The sign of the local
Gaussian curvature is used to classify the shape of a surface
intheneighborhoodofapoint x.Surfacepointswithpositive
κgauss> 0 are called elliptic, points with negative κgauss< 0
are called hyperbolic or saddle points, and points with van-
ishing κgauss= 0 are referred to as parabolic. The contours
of the Gaussian curvature κgaussover the leaflet at end dias-
tole and at end systole are depicted in Fig. 10. The chosen
color code clearly demonstrates the elliptic and hyperbolic
domains on the leaflet. Although the Gaussian curvature has
been used previously to illustrate the surface characteristics
of the leaflet (Ryan et al. 2008a,b), it does not readily reflect
theleafletcurvatureobservedonechocardiography.Atypical
(17)
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Anterior mitral leaflet curvature in the beating ovine heart287
..
Fig. 9 Mean curvature κmeanat end systole (top) and at end diastole
(bottom) for third subdivision with 1,017 nodes
..
Fig. 10 Gaussiancurvatureκgaussatendsystole(top)andatenddias-
tole (bottom) for third subdivision with 1,017 nodes
..
Fig. 11 Maximum principal curvature κmaxat end systole (top) and
at end diastole (bottom) for third subdivision with 1,017 nodes
example is an elliptic point with positive κgauss> 0, which
can be obtained either for κ1> 0 and κ2> 0 or for κ1< 0
and κ2 < 0. Accordingly, convex or concave areas can be
assigned the same color code.
4.3 Maximum principal curvature
To make curvature interpretation readily accessible and dis-
tinguish convex and concave areas, we propose to use the
maximum principal curvature κmax.
κmax:= max{κ1,κ2}
The maximum principal curvature contours over the leaflet
at end diastole and at end systole are depicted in Figs. 11 and
12. Compared to the Gaussian curvature contours in Fig. 10,
the κmaxcontours in Fig. 11 transparently distinguish con-
vex and concave regions in the elliptic domains of the leaflet
surface at end diastole. However, special attention should be
paid to the points where the maximum principal curvature is
positive. Positive κmax> 0 might represent both hyperbolic
and convex regions of the surface, and a clear distinction
can only be made by analyzing both the Gaussian and the
maximum principal curvature contours, see Figs. 10 and 11.
Another illustrative feature associated with the maximum
principal curvature κmaxis the maximum principal direction
n(κmax)accordingtotheprincipalvalueprobleminEq.(12).
(18)
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288S. Göktepe et al.
..
Fig. 12 Maximum principal curvature κmaxat end systole (top) and
at end diastole (bottom) for third subdivision with 1,017 nodes
Figure 13 illustrates the maximum principal directions, i.e.,
the directions with maximum curvature. These are orthogo-
nal to the directions of minimum curvature n(κmin), which,
in turn, might be correlated to the preferred microstructural
orientations of the leaflet, see, e.g., Kunzelman and Cochran
(1992), Itoh et al. (2009), for a discussion on leaflet anisot-
ropy, histological staining, and a conceptual model of the
leaflet microstructure, respectively.
4.4 Circumferential and radial curvatures
Apartfromthecurvaturemeasuresconsideredpreviously,we
now consider the projection of the second fundamental form
B on the circumferential and radial axes of the leaflet at end
diastole. For this purpose, we introduce the circumferential
κccand radial κrrcurvatures
κcc:= nc· B · nc
wherethevectors ncand nrdenotetheunitvectorsinthecir-
cumferentialandradialdirections,showninFig.14.Contour
plots of these curvature measures are depicted in Figs. 14
and 15, separately. Apparently, the contour plots in these
figures partially resemble the κmaxdistribution, Fig. 11, at
points where the principal maximum curvature directions,
Fig. 13, become nearly parallel to the circumferential and
and κrr:= nr· B · nr
(19)
Fig. 13 Maximum principal directions at end systole (top) and at end
diastole (bottom) for third subdivision with 1,017 nodes
.
.
Fig. 14 Circumferential curvature κccat end systole (top) and at end
diastole (bottom) for third subdivision with 1,017 nodes
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Anterior mitral leaflet curvature in the beating ovine heart289
..
Fig. 15 Radial curvature κrrat end systole (top) and at end diastole
(bottom) for third subdivision with 1,017 nodes
radialdirections.Overall,wefeelthatifweweretopickasin-
gle curvature contour plot, the maximum principal curvature
κmax, Fig. 11, may give the best picture of the leaflet shape.
In the following examples, for the sake of completeness, we
utilize the circumferential and radial curvature plots besides
themaximumprincipalcurvaturedistributiontocharacterize
ring-induced changes in leaflet kinematics.
5 Example: Ring-induced kinematic changes
In this section, we explore the impact of annuloplasty rings
on mitral leaflet kinematics. In particular, we compare the
leaflet surface area and curvature after the insertion of a
true-sized 28mm Edwards GeoForm®ring with the base-
line leaflet kinematics without ring. For the acquired data
sets of 23 control points at a resolution of 16.6ms, we per-
form the approximating subdivision up to third level and
calculate the total surface area and curvature over a repre-
sentative cardiac cycle for ring on and ring off, respectively.
We focus in particular on three representative configurations
of the mitral leaflet that correspond to maximum opening
(MO), end diastole (ED), and end systole (ES) as indicated
in the left ventricular pressure vs. time curve in Fig. 16.
Maximum opening is defined as the maximum distance
between the central anterior and posterior leaflet edge
800
20
40
60
80
100
0 100 200 300 400 500 600 700
0
Fig. 16 Evolutionofleftventricularpressure(LVP)duringarepresen-
tative cardiac cycle. Filled circles indicate stages of maximum opening
(MO), end diastole (ED), and end systole (ES)
Fig. 17 Evolution of approximation error ?e?L1during a representa-
tive cardiac cycle. Filled circles indicate stages of maximum opening
(MO), end diastole (ED), and end systole (ES)
markers. End diastole and end systole are identified as the
peak R-wave on the EKG and as the time frame preceding
the maximum negative pressure gradient, respectively.
5.1 Approximation error of subdivision surface scheme
Webeginouranalysisbyquantifyingtheapproximationerror
that provides information about the average distance of the
original marker positions from the final subdivision surface.
In Fig. 17, we illustrate the temporal evolution of the aver-
age approximation error ?e?L1according to Equation (5)
for the third level subdivision of a representative leaflet
without ring. The maximum average approximation error
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290S. Göktepe et al.
?e?L1= 0.0862 cm occurs in early stages of the car-
diac cycle when the valve is open. Given that the mean
circumferential dimension of an ovine mitral leaflet is
approximately 4cm, the average error of the approximat-
ing third subdivision surface is on the order of 2.3% with
respect to the mean circumferential dimension. The aver-
age approximation error ?e?L1is approximately on the
order of the digitalization error edig= 0.01 ± 0.03 cm of
the videofluoroscopic marker technique (Daughters et al.
1989). We thus conclude that the average error of the
approximating subdivision algorithm lies within a tolerable
range.
5.2 Effect of annuloplasty ring on leaflet surface area
The temporal evolution of the surface area of the anterior
mitralleafletwithringandwithoutringisdepictedinFig.18.
The kinematic constraint imposed by the implanted ring ini-
tiates a surface area reduction of approximately 0.25cm2.
The most significant reduction in surface area occurs at
end systole when the area ratio of the leaflet with ring to
the leaflet without ring is 93%. This ratio becomes 97% at
maximum opening and 98% at end diastole. The observed
ring-inducedareasurfacereductionmightbecomeevenmore
pronouncedinrealclinicalapplicationsinwhichdown-sized
rather than true-sized rings are applied to restore valvular
function.
Fig. 18 Evolution of anterior mitral leaflet surface area during a rep-
resentative cardiac cycle. Filled circles indicate stages of maximum
opening (MO), end diastole (ED), and end systole (ES)
5.3 Effect of annuloplasty ring on leaflet curvature
The impact of annuloplasty ring implantation on the max-
imum principal curvature and on the circumferential and
radial curvatures is illustrated in Figs. 19, 20 and 21, respec-
tively. The top row in each of these figures depicts the cur-
vature at maximum opening (left), end diastole (middle),
..
Fig. 19 Maximum principal leaflet curvature at maximum opening (lef t), end diastole (middle), and end systole (right) with ring (top row) and
without ring (bottom row) for third subdivision with 1,017 nodes
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Anterior mitral leaflet curvature in the beating ovine heart291
..
Fig. 20 Circumferentialleafletcurvatureatmaximumopening(lef t),
end diastole (middle), and end systole (right) with ring (top row) and
without ring (bottom row) for third subdivision with 1,017 nodes. The
circumferential direction nccorresponding to the end systolic and end
diastolic configurations is depicted in Fig. 14
..
Fig. 21 Radial leaflet curvature at maximum opening (lef t), end diastole (middle), and end systole (right) with ring (top row) and without ring
(bottom row) for third subdivision with 1,017 nodes. The radial direction nrcorresponding to the end systolic and end diastolic configurations is
depicted in Fig. 15
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292S. Göktepe et al.
and end systole (right) for the leaflet with ring. The bottom
rowsillustratethecorrespondingbaselinevaluesforthesame
leaflet without ring. The comparison of the different curva-
ture contours for the anterior leaflet with and without ring
suggests that the true-sized 28mm Edwards GeoForm®ring
has a small influence on the overall curvature distribution on
the anterior leaflet. The sequence of snapshots clearly doc-
uments the tremendous shape changes of the leaflet during
the cardiac cycle. The displayed stages of maximum open-
ing,enddiastole,andendsystolemimictheunderlyingblood
flow and chord anatomy. Maximum opening manifests itself
in a T-channel-shaped convex domain guiding blood flow
from the left atrium to left ventricle, shown in red in Fig. 19
(left). The slight increase in left ventricular pressure gen-
erates a slightly modified curvature pattern at end diastole.
This pressure-induced curvature change is more pronounced
for the leaflet without ring Figs. 19, 20 and 21 (bottom row,
middle) than for the leaflet with ring Figs. 19, 20 and 21 (top
row, middle). At end systole, the leaflet adapts to the signif-
icantly higher pressure by changing its curvature contours
from T-channel-shaped to funnel-shaped, see also Fig. 12.
The maximum curvature is now aligned with the circumfer-
ential direction rather than with the radial direction as it was
atenddiastole.Thischaracteristicfunnelshapeatendsystole
shown in red in Fig. 19 (right) is believed to guide the blood
flow underneath the leaflet, from the left ventricle to aorta.
Overall,weconcludethatring-inducedcurvaturechangesfor
this particular animal are less pronounced than surface area
changes.
6 Discussion
Annuloplasty ring implantation is a common surgical treat-
ment to repair a leaking valve. The shape and size of the ring
have been postulated to play a major role in long-term repair
durability.Accordingly,annuloplastyringscomeindifferent
sizes, shapes, and stiffnesses. To date, the appropriate ring
choice is mainly based on the surgeon’s experience and is
thus purely empirical. Although it is intuitive that ring size
andshapemightinfluencesleafletgeometrythereisnoquan-
titativeevidenceshowinghowleafletkinematicsareaffected
by ring implantation.
The goal of this study was to create a virtual test envi-
ronment to classify mitral annular rings based on their effect
on leaflet curvature. We acquired four-dimensional mitral
leaflet coordinates usingaspecificvideofluoroscopic marker
technique that allowed us to study leaflet dynamics with and
withoutringinthesameanimal.Sincecurvaturecalculations
require a non standard C1-continuous surface representation,
we explored smooth surface generation algorithms, which
had originally been developed for computer graphics appli-
cations. Among different surface representation schemes,
we chose a triangular approximating scheme capable of
generating smooth C1-continuous surfaces from a set of con-
trol points. On patches of neighboring triangles, we interpo-
lated the curvature tensor for each triangle and assembled
the elementwise curvature interpolation to the global cur-
vature field for the entire leaflet. From the resulting second
order curvature tensor field, we extracted and compared dif-
ferent scalar-valued curvature measures. We concluded that
the maximum principal curvature visualizes leaflet dynam-
ics most illustratively. To demonstrate the features of the
proposed approach, we compared maximum principal cur-
vatures in an ovine model with and without ring at three
characteristicpointsinthecardiaccycle.Fortheovinemodel
considered in this study, we found that the implantation of
a true-sized GeoForm®ring had rather small effects on cur-
vature profiles of the anterior leaflet of the particular animal
under investigation.
The proposed methodology offers many advantages with
respect to other classification schemes proposed in the lit-
erature: (i) in contrast to leaflet stress calculations, the
proposed leaflet curvature calculation uses exclusively raw
data. It does not imply assumptions about constitutive equa-
tions, boundary conditions, and forces acting on the mitral
valve apparatus. (ii) In contrast to echocardiography-based
data acquisition, the proposed marker technique is semi-
automated. It does not require manual sectioning or man-
ual merging of two-dimensional slices and is thus less likely
to generate errors. (iii) The proposed technique enables us to
studycurvaturewithandwithoutringinthesameanimaland
quantitatively compare curvature changes. (iv) The method
offers a high spatial and temporal resolution. This allows us
to answer where and when annuloplasty rings affect leaflet
curvature most, and how ring geometry could potentially be
improved. (v) The underlying smooth surface generations
follow well-defined schemes common in computer graph-
ics. They are easily reproducible and allow for a quantita-
tive assessment of the approximation error. (vi) The generic
determinationofthecurvaturetensorfieldallowsustoaccess
different curvature measures, e.g., mean curvature, Gaussian
curvature, and maximum principal curvature, and identify
the one that is most useful for the current application. The
disadvantage of the proposed approach is that it is inherently
invasive and based on ovine data. The suggested subdivision
surface and curvature calculation, however, is fairly generic
and can easily be adapted to control points generated from
any other data source, e.g., from echocardiography.
This manuscript has demonstrated the potential of com-
bined experimental/computational approaches to quantify
ring-induced curvature changes in the anterior mitral leaflet.
Curvature changes in the mitral leaflet have been associated
with surgical repair durability. The proposed algorithm is
currently being applied to quantify ring-induced leaflet cur-
vature changes in 60 sheep. These animals were divided in
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Anterior mitral leaflet curvature in the beating ovine heart293
to five groups of twelve animals each to compare five differ-
ent ring types. The proposed methodology could potentially
serve as a design tool for novel annuloplasty rings with a
more physiological shape. The overall goal of this project is
to predict curvature changes based on patient-specific mitral
valvegeometriesandidentifytheoptimalringshapeandsize
on a patient-specific individual basis. This must, of course,
bebasedonnoninvasiveimagemodalitiesthatdonotrequire
physical markers on the valve.
Acknowledgments
Lauren R. Davis for technical assistance, Maggie Brophy and Sigurd
Hartnett for careful marker image digitization, and George T. Daugh-
ters,andT.J.andW.S.Hartforextractionoffour-dimensionaldatafrom
markercoordinates.ThisworkwassupportedinpartbytheUSNational
Science Foundation grant EFRI-CBE 0735551 to Ellen Kuhl, by US
National Institutes of Health grants R01 HL29589 and R01 HL67025
to D. Craig Miller, by the Deutsche Herzstiftung, Frankfurt, Germany,
Research Grant S/06/07 to Wolfgang Bothe, by the U.S.- Norway Ful-
bright Foundation and the Swedish Heart-Lung Foundation to John-
Peder Escobar Kvitting, and by the Western States Affiliate American
Heart Association Fellowship to Julia C. Swanson.
We thank Paul A. Chang, Eleazar P. Briones, and
References
BotheW,KrishnamurthyG,ChangPA,SwansonJC,BrionesEP,Davis
LR, Itoh A, Ingels NB, Miller CD (2009) Effects of the Geoform®
annuloplasty ring on anterior mitral leaflet strains and stresses in
the normal ovine heart (submitted for publication)
Bornow RO, Carabello BA, Chatterjee K, deLeon AC, Faxon DP,
Freed MD, Gaasch WH, Lytle BW, Nischimura RA, O’Gara PT,
O’RourkeRA, Otto CM,
(2006) ACC/AHA 2006 Guidelines for the management of
patients with valvular heart disease. Circulation 114:E84–E231
Catmull E, Clark J (1978) Recursively generated B-spline surfaces on
arbitrary topological meshes. Comput Aided Des 10:350–355
Cirak F, Ortiz M, Schröder P (2000) Subdivision surfaces: a new para-
digm for thin-shell finite-element analysis. Int J Numer Methods
Eng 47:2039–2072
Cirak F, Ortiz M (2001) Fully C1-conforming subdivision elements for
finite deformation thin-shell analysis. Int J Numer Methods Eng
51:813–833
Cirak F, Scott MJ, Antonsson EK, Ortiz M, Schröder P (2002) Inte-
grated modeling, finite-element analysis, and engineering design
for thin-shell structures using subdivision. Comput Aided Des
34:137–148
Cosgrove D (2004) Review: view from North America’s cardiac sur-
geons. Eur J Cardiothorac Surg 26:S27–S31
Daughters G, Sanders W, Miller DC, Schwarzkopf A, Mead C,
Ingels NB (1989) A comparison of twoanalyticalsystems for 3-D
reconstruction from biplane videoradiograms. IEEE Comput Car-
diol 15:79–82
Doo D, Sabin M (1978) Behavior of recursive division surfaces near
extraordinary points. Comput Aided Des 10:356–360
Itoh A, Krishnamurthy G, Swanson J, Ennis D, Bothe W, Kuhl E,
Karlsson M, Davis L, Miller DC, Ingels NB (2009) Active stiff-
ening of mitral valve leaflets in the beating heart. Am J Physiol
Heart Circ Physiol 296:H1766–H1773
Karlsson MO, Glasson JR, Daughters GT, Komeda M, Foppiano LE,
Miller DC, Ingels NB (1998) Mitral valve opening in the ovine
heart. Am J Physiol Heart Circ Physiol 274:H552–H563
ShahPM, Shanewise JS
Kreyszig E (1991) Differential geometry. Dover, New York
Krishnamurthy G, Ennis DB, Itoh A, Bothe W, Swanson JC, Karlsson
M, Kuhl E, Miller DC, Ingels NB (2008) Material properties of
the ovine mitral valve anterior leaflet in vivo from inverse finite
element analysis. Am J Physiol Heart Circ Physiol 295:H1141–
H1149
Krishnamurthy G, Itoh A, Bothe W, Swanson J, Kuhl E, Karlsson M,
MillerDC,IngelsNB(2009)Stress–strainbehaviorofmitralvalve
leaflets in the beating ovine heart. J Biomech 42:1909–1916
Krishnamurthy G, Itoh A, Swanson J, Bothe W, Karlsson M, Kuhl E,
MillerDC,IngelsNB(2009)Regionalstiffeningofthemitralvalve
anteriorleafletinthebeatingovineheart.JBiomech.doi:10.1016/
j.jbiomech.2009.08.028
KunzelmanKS,CochranRP (1992) Stress/straincharacteristicsofpor-
cinemitralvalvetissue:parallelversusperpendicularcollagenori-
entation. J Card Surg 7:71–78
LoopC(1987)Smoothsubdivisionsurfacesbasedontriangles.Masters
Thesis, University of Utah, Department of Mathematics
Niczyporuk MA, Miller DC (1991) Automatic tracking and digitiza-
tion of multiple radiopaque myocardial markers. Comp Biomed
Res 24:129–142
ProtV,SkallerudB,HolzapfelGA (2007) Transverselyisotropicmem-
brane shells with application to mitral valve mechanics. Constitu-
tive modelling and finite element implementation. Int J Numer
Methods Eng 71:987–1008
ProtV,HaaverstadR,SkallerudB (2009) Finiteelementanalysisofthe
mitral apparatus: annulus shape effect and chordal force distribu-
tion. Biomech Model Mechanobiol 8:43–55
Reif U (1995) A unified approach to subdivision algorithms near
extraordinary points. Comput Aided Geom Des 12:153–174
Reif U, Schröder P (2001) Curvature integrability of subdivision sur-
faces. Adv Comput Math 14:157–174
RyanLP,JacksonBM,HamamotoH,EperjesiTJ,PlappertTJ,StJohn-
Sutton M, Gorman RC, Gorman JH III (2008) The influence of
annuloplasty ring geometry on mitral leaflet curvature. Ann Tho-
rac Surg 86:749–760
Ryan LP, Jackson BM, Eperjesi TJ, Plappert TJ, St John-Sutton M,
Gorman RC, Gormann JH III (2008) A methodology for assess-
ing human mitral leaflet curvature using real-time 3-dimensional
echocardiography. J Thorac Cardiovasc Surg 136:726–734
Sacks MA, Enomoto Y, Graybill JR, Merryman WD, Zeeshan A,
Yoganathan AP, Levy RJ, Gorman RC, Gorman JH III (2006)
In-vivo dynamic deformation of the mitral valve anterior leaflet.
Ann Thorac Surg 82:1369–1377
Salgo IS, Gorman JH III, Gorman RC, Jackson BM, Bowen FW,
Plappert T, St John Sutton MG, Edmunds LH (2002) Effect of
annular shape on leaflet curvature in reducing mitral leaflet stress.
Circulation 106:711–717
Schröder P (2002) Subdivision as a fundamental building block of
digital geometry processing algorithms. J Comput Appl Math
149:207–219
Umlauf G (2000) Analyzing the characteristic map of triangular subdi-
vision schemes. Constr Approx 16:145–155
VottaE,MaisanoF,BollingSF,AlfieriO,MontevecchiFM,RedaelliA
(2007) TheGeoformdisease-specificannuloplastysystem:afinite
element study. Ann Thorac Surg 84:92–101
Votta E, Caiani E, Veronesi F, Soncini M, Montevecchi FM, Redaelli A
(2008) Mitral valve finite-element modelling from ultrasound
data:apilotstudyforanewapproachtounderstandmitralfunction
and clinical scenarios. Phil Trans Royal Soc A 366:3411–3434
Zorin D (2000) A method for analysis of C1-continuity of subdivision
surfaces. SIAM J Numer Anal 37:1677–1708
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