Download Left ventricular regional wall curvedness and wall stress in patients

Am J Physiol Heart Circ Physiol 296: H573–H584, 2009.
First published January 2, 2009; doi:10.1152/ajpheart.00525.2008.
Innovative Methodology
TRANSLATIONAL PHYSIOLOGY
Left ventricular regional wall curvedness and wall stress in patients with
ischemic dilated cardiomyopathy
Liang Zhong,1 Yi Su,2 Si-Yong Yeo,2 Ru-San Tan,1 Dhanjoo N. Ghista,3 and Ghassan Kassab4,5,6
1
Deparment of Cardiology, National Heart Centre, 2Institute of High Performance Computing, Agency for Science,
Technology, and Research, and 3Parkway Health, Singapore; and Departments of 4Biomedical Engineering, 5Surgery,
and 6Cellular and Integrative Physiology, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana
Submitted 17 May 2008; accepted in final form 9 December 2008
three-dimensional modeling; left ventricle; curvature; cine magnetic
resonance imaging; human heart
ISCHEMIC DILATED CARDIOMYOPATHY (IDCM) is a degenerative
disease of the myocardial tissue accompanied by left ventricular (LV) remodeling (34). LV remodeling is a multistep process
that involves acute dilation of the infarcted area, increase of LV
volume, lengthening of the LV perimeter, and decrease of LV
curvature (10, 42). Natural history studies show that progressive
LV remodeling is directly related to future deterioration of LV
performance and a poor clinical course (20).
Address for reprint requests and other correspondence: G. S. Kassab, Dept.
of Biomedical Engineering, Indiana Univ. Purdue Univ. Indianapolis, Indianapolis, IN 46202 (e-mail: [email protected]).
http://www.ajpheart.org
The elevation of LV wall stress in IDCM is associated with
morphological changes in the myocardium that may cause
regional hypokinesis (20 –22). LV wall stress is in part determined by the local curvature of the ventricular wall; i.e.,
decreased curvature will increase wall stress (23). In addition
to increasing LV size, IDCM can alter myocardial properties and
normal LV shape curvature. The border zone will have a higher
stress, which makes it more susceptible to ischemia and infarction
and may accelerate the remodeling process (44). Therefore, therapeutic approaches for IDCM include LV size reduction to disrupt
the downward-spiral cycle of heart failure (12).
The geometry of the LV in IDCM is complicated, and a
three-dimensional (3-D) approach is necessary to characterize
the ventricular mechanics of contractility and regional shape
changes throughout the cardiac cycle. Many studies have
shown that MRI is a better and more comprehensive approach
to quantify 3-D ventricular structure and function (5, 28) than
echocardiography (17), ventriculography (16, 48), angiography
(40), and indicator-dilution (11) methods.
End-systolic wall stress is governed by the systolic pressure
of the LV and a geometric factor of the shape of the LV (19,
23, 47). Several mathematical models have been used to
calculate wall stress (47). Some approaches are limited by
geometric assumptions and are only valid for spherical or
ellipsoidal shapes and, hence, only allow calculation of global
wall stress. Other approaches are applicable to any arbitrary
shape of the LV (23) but have some errors in the measurement
of the thickness of the LV wall and of the radius of curvature
(6). The technical limitations of these methods do not permit
precise regional measurements of 3-D wall curvature.
In this study, we aim to 1) assess the regional variations of
LV shape in 3-D (i.e., in terms of surface curvedness), 2) assess
the 3-D regional variations of wall stress (by incorporation of
wall curvature), and 3) determine the relationship between
peak systolic wall stress and the regional extent of myocardial
infarct.
Glossary
IDCM
d␴*/dtmax
EDVI
Ischemic dilated cardiomyopathy
LV contractility index [1.5 ⫻ (dV/dtmax)/Vm]
Indexed end-diastolic volume
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0363-6135/09 $8.00 Copyright © 2009 the American Physiological Society
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Zhong L, Su Y, Yeo S, Tan R, Ghista DN, Kassab G. Left
ventricular regional wall curvedness and wall stress in patients
with ischemic dilated cardiomyopathy. Am J Physiol Heart Circ
Physiol 296: H573–H584, 2009. First published January 2, 2009;
doi:10.1152/ajpheart.00525.2008.—Geometric remodeling of the left
ventricle (LV) after myocardial infarction is associated with changes
in myocardial wall stress. The objective of this study was to determine
the regional curvatures and wall stress based on three-dimensional
(3-D) reconstructions of the LV using MRI. Ten patients with ischemic dilated cardiomyopathy (IDCM) and 10 normal subjects underwent MRI scan. The IDCM patients also underwent delayed gadolinium-enhancement imaging to delineate the extent of myocardial
infarct. Regional curvedness, local radii of curvature, and wall thickness were calculated. The percent curvedness change between end
diastole and end systole was also calculated. In normal heart, a shortand long-axis two-dimensional analysis showed a 41 ⫾ 11% and 45 ⫾
12% increase of the mean of peak systolic wall stress between basal
and apical sections, respectively. However, 3-D analysis showed no
significant difference in peak systolic wall stress from basal and apical
sections (P ⫽ 0.298, ANOVA). LV shape differed between IDCM
patients and normal subjects in several ways: LV shape was more
spherical (sphericity index ⫽ 0.62 ⫾ 0.08 vs. 0.52 ⫾ 0.06, P ⬍ 0.05),
curvedness at end diastole (mean for 16 segments ⫽ 0.034 ⫾ 0.0056
vs. 0.040 ⫾ 0.0071 mm⫺1, P ⬍ 0.001) and end systole (mean for 16
segments ⫽ 0.037 ⫾ 0.0068 vs. 0.067 ⫾ 0.020 mm⫺1, P ⬍ 0.001)
was affected by infarction, and peak systolic wall stress was significantly increased at each segment in IDCM patients. The 3-D quantification of regional wall stress by cardiac MRI provides more precise
evaluation of cardiac mechanics. Identification of regional curvedness
and wall stresses helps delineate the mechanisms of LV remodeling in
IDCM and may help guide therapeutic LV restoration.
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Indexed end-systolic volume
Heart rate
Sphericity index
Curvedness
Curvedness at end diastole
Curvedness at end systole
Percent curvedness change
Wall thickness
Wall radius
Wall stress
Wall thickening
End-diastolic wall thickness
End-systolic wall thickness
2-D radius in the short-axis plane
2-D wall thickness in the short-axis plane
Wall stress in the short-axis plane
Wall thickening in the short-axis plane
2-D radius in the long-axis plane
2-D wall thickness in the long-axis plane
Wall stress in the long-axis plane
Wall thickening in the long-axis plane
3-D radius accounting for 3-D curvature
3-D wall thickness accounting for 3-D curvature
3-D wall stress
3-D wall thickening
METHODS
Subjects. Ten normal subjects and 10 patients with IDCM participated
in the study. All subjects underwent diagnostic MRI scan. None of the
normal subjects had 1) significant valvular or congenital cardiac disease,
2) history of myocardial infarction, 3) coronary artery lesions, or
4) abnormal LV pressure, end-diastolic volume, or ejection fraction. All
subjects were recruited without consideration of gender or ethnicity and
gave informed consent. The study was approved by the Human Subjects
Review Committee of the National Heart Centre, Singapore.
MRI scans. MRI scanning was performed using steady-state free
precession cine gradient echo sequences. Subjects were imaged on a
1.5-T scanner (Avanto, Siemens Medical Solutions, Erlangen, Germany). Some preliminary short-axis acquisitions were used to locate
the plane passing through the mitral and aortic valves. This allowed
the acquisition of the oblique long-axis plane of the LV, which is
orthogonal to the short-axis plane and passes through the mitral valve,
apex, and aortic valve. In addition, the corresponding vertical longaxis planes (connecting the short-axis plane images) were acquired.
TrueFISP (fast imaging with steady-state precession) magnetic resonance pulse sequence with segmented k-space and retrospective electrocardiographic gating was used to acquire a parallel stack of twodimensional (2-D) cine images of the LV in the short-axis plane, from
LV base to apex (8-mm interslice thickness, no interslice gap). The
field of view was typically 320 mm with in-plane spatial resolution of
⬍1.5 mm. Each slice was acquired in a single breath hold, with 25
temporal phases per heart cycle. Figure 1 depicts the magnetic
resonance short- and long-axis plane or image during end diastole and
end systole. The average duration of entire image acquisition period
was 30 min. The short- and long-axis views (or planes) derived from
the MRI were utilized to carry out 3-D LV reconstruction (at end
diastole and end systole) using customized software (see Data processing and LV geometry reconstruction).
Delayed gadolinium-enhancement imaging. For evaluation of myocardial viability, 10 to 15 contiguous short-axis views were acquired
with 2-D MRI. Standard extracellular magnetic resonance contrast
agents were injected (0.2 mmol gadolinium/kg iv, gadoterate dimeglumine; Schering, Berlin, Germany). All images were acquired
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Fig. 1. Sample segmented TrueFISP 2-dimensional (2-D) cine MRIs of
short-axis slices (A) acquired at the base (top), middle (middle), and apex
(bottom) and long-axis slices (B) of left ventricle (LV) in end-diastolic (left)
and end-systolic (right) phases.
during end-expiratory breath holds at 5–15 min after injection. T1
weighting was achieved with an inversion-recovery fast low-angle
shot (IR-FLASH) pulse sequence. Typical parameters were as follows: TE ⫽ 2 ms, TR ⫽ 6 ms, and voxel size ⫽ 1 ⫻ 1 ⫻ 6 mm, with
300-ms inversion delay and k-space data segmented over four cardiac
cycles (32 lines per cycle) and data acquired every cardiac cycle.
Images were acquired at middiastole during breath hold (8 s).
Using commercially available image analysis software (Syngo 3D,
Siemens Medical Solution), one author (R.-S. Tan, with more than 10
years of experience in cardiac magnetic imaging) designated the regions
of interest in the viable myocardium (dark) and in the nonviable myocardium (highly enhanced) for each image frame in the short-axis cine
DE series. The thickness of the hyperenhancement was planimetered on
all short-axis images from base to apex. Each 3-D reconstructed LV
model was divided into 16 segments (see Fig. 4, Tables 2– 4). The
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ESVI
HR
SI
C
CED
CES
⌬C
T
R
WS
WS
EDT
EST
2DSR
2DST
2DSWS
2DSWT
2DLR
2DLT
2DLWS
2DLWT
3DR
3DT
3DWS
3DWT
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lated using the following formula: 1.5 ⫻ (dV/dtmax)/Vm, where Vm is
myocardial volume at the end-diastolic phase, as previously reported
(49). Global LV shape was assessed by calculation of the sphericity
index (SI) in diastole using the following formula: SI ⫽ AP/BA,
where BA was measured in the four-chamber view from the apex to
the midpoint of the mitral valve and AP was measured as the axis that
perpendicularly intersects the midpoint of the long axis. A small SI
value implies an ellipsoidal LV, whereas values approaching 1 suggest a more spherical LV.
Computation of 3-D surface shape descriptors. Software developed
in-house was used to reconstruct the LV endocardial meshes at end
diastole and end systole as well as to calculate the surface shape
descriptors, expressed in terms of local normal curvature and curvedness. The formulations for these descriptors are shown in APPENDIX A.
To evaluate the shape at a particular point on the mesh, a local
surface geometry was fitted to the region. The normal curvature at that
point is then calculated as
␬共␭兲 ⫽
L ⫹ 2M␭ ⫹ N␭ 2
E ⫹ 2F␭ ⫹ G␭ 2
(1)
where ␭ ⫽ dv/du, such that u and v are the parameters of the
underlying geometry, and {E, F, G} and {L, M, N} are components of
the first and second fundamental forms, respectively.
Fig. 2. Three-dimensional (3-D) reconstruction of LV endocardial surface at end diastole and end systole for an
ischemic dilated cardiomyopathy (IDCM) patient (A) and a
normal subject (B).
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myocardial contrast in each of these segments was quantified in terms
of scar scores. A scar extent score was assigned to each region as
follows: 0 ⫽ no scarring, 1 ⫽ 1–25% scarring, 2 ⫽ 26 –50% scarring,
3 ⫽ 51–75% scarring, 4 ⫽ 76 –99% scarring, and 5 ⫽ 100% scarring.
Data processing and LV geometry reconstruction. The MRI data
were processed using a semiautomatic technique provided in the
CMRtools suite (CVIS, London, UK). Short- and long-axis images
were displayed simultaneously, such that segmentation in the two
planes proceeded interactively to reduce registration errors (35). For
each phase, every control point on the endocardium was constrained
to lie on the intersection of the short- and long-axis views.
The construction of long-axis planes (orthogonal to the short-axis
planes oriented at regular angular intervals) enabled the fitting of a
series of B-spline curves to represent the contours of the endocardial
surface. This allowed the addition or manipulation of control points to
obtain the desired boundary locations. The papillary muscles and
trabeculae were included in the chamber volume to obtain smooth
endocardial contours suitable for shape analysis. The 3-D reconstructions of a typical IDCM and normal LV during end-diastolic and
end-systolic phases are shown in Fig. 2.
From the reconstructed LV, the chamber volume, wall mass, stroke
volumes, and EF were determined. A six-order polynomial function
was then used to curve fit the volume-time data to determine the
volume rate (dV/dt). The contractility index (d␴*/dtmax) was calcu-
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LEFT VENTRICULAR SHAPE
The extreme values ␬1 and ␬2 of ␬(␭) are the maximum and
minimum principal curvatures, respectively, and they are obtained
from the roots of the equation
det
冋
册
L ⫺ ␬E M ⫺ ␬F
⫽0
M ⫺ ␬F N ⫺ ␬G
A shape descriptor is characterized in terms of the curvedness value
(C), as presented by Koenderink and Van Doorn (25). It is a measure
of the degree of regional curvature and is defined as
(2)
The corresponding directions of ␬1 and ␬2 are the principal directions and are orthogonal to each other. Figure 3 presents typical
principal curvature on the endocardial wall of the LV in IDCM and
normal hearts.
C⫽
冑
␬ 12 ⫹ ␬ 22
2
(3)
The value of C indicates the magnitude of the curvedness at a point,
which is a measure of the extent to which a region deviates from
flatness. A normal LV will exhibit a larger value of C.
The percent curvedness change (⌬C) between the end-diastolic and
end-systolic phase is defined as
⌬C ⫽ 共C ES ⫺ CED兲/CES ⫻ 100%
(4)
WT ⫽
EST ⫺ EDT
EDT
(5)
where EST and EDT are wall thickness at end systole and end
diastole, respectively.
Regional peak systolic wall stress. The wall stress is obtained by
the equilibrium of forces due to stresses in the wall and blood pressure
acting on the wall. Following Grossman et al. (19), the regional peak
systolic wall stress (WS) was determined from the inner radius of
curvature (R) and T at end systole by
WS ⫽ 0.133 ⫻ SP ⫻
R
冉
2T ⫻ 1 ⫹
Fig. 3. Principal curvature analysis on LV endocardial wall. Arrowheads
represent directions of maximum (left) and minimum (right) principal curvature of the endocardial surface of IDCM patients (A) and normal subjects (B).
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冊
T
2R
(6)
where SP is the peak systolic ventricular blood pressure (in mmHg).
In this study, SP was assessed from the systolic noninvasive blood
pressure (37a); a conversion factor of 0.133 was used to express the
final results in 1,000 N/m2. WS was calculated using R and T values
derived with the 3-D curvature method described above. WS was
determined using Eq. 6 and as illustrated in Fig. B1 in the circumferential and longitudinal (or meridional) regions in 2-D as 2DSWS and
2DLWS and for a 3-D element with radius that is the inverse of
curvature (computed from Eq. 3).
To determine regional LV properties of curvedness, wall stress, and
wall thickening, the LV was divided into a 16-segment model (8) from
apex to base (Fig. 4).
Statistical analysis. LV volume, function, SI, and curvedness data
were compared between IDCM patients and normal subjects using a
t-test. 2-D short-axis plane, long-axis plane, and 3-D evaluations of
wall stress were compared using ANOVA. If there was a significant
interaction (P ⬍ 0.05) between multiple measurements, selected
pairwise comparisons were examined further. The difference in
curvedness, wall stress, and wall thickening among various zones was
assessed by ANOVA. Statistical significance for comparison of regional LV systolic wall stress between IDCM patients and normal
subjects was determined using a two-tailed Student’s t-test. A similar
analysis was made for wall thickening calculated in the 2-D short-axis
plane, 2-D long-axis plane, and direction of the 3-D curvature.
Bivariate correlation was performed between curvedness difference,
wall stress, and wall thickening and extent of myocardial infarction
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where CED and CES represent curvedness at the end-diastolic and
end-systolic phase, respectively. Positive values of ⌬C indicate LV
wall regions of increasing curvature during systole; negative values of
⌬C indicate wall regions of decreasing curvature.
Regional wall thickening. In addition to the evaluation of curvature
measures on each point of the surface mesh, the LV radius (R) and
wall thickness (T) were deduced from the 3-D geometry of the LV.
The steps of the computation of wall radius-to-curvature and thickness
are summarized in APPENDIX B. The wall thickening (WT) is expressed
at each segment by the following formula
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using Spearman’s correlation coefficient. Values are means ⫾ SD,
and significance was defined as P ⬍ 0.05.
RESULTS
Global LV function. The hemodynamic and volumetric parameters of the subjects are summarized in Table 1. The
cardiac contractility index d␴*/dtmax and LV ejection fraction
(LVEF) were significantly lower in IDCM patients than normal
subjects. In addition, LV end-diastolic and end-systolic volumes were greater in IDCM patients than normal subjects. By
visual inspection, the LV has a broader apex in IDCM patients
than normal subjects. The increase in dilated volume in the LV
with IDCM was accompanied by a corresponding increase in
sphericity. Consequently, the LV was significantly more spherical in IDCM patients than normal subjects.
Variation of curvedness, peak systolic wall stress, and wall
thickening from base to apex in normal subjects. Calculated
regional values for curvedness at end diastole and end systole in
normal subjects and IDCM patients are summarized in Table 2. In
general, normal hearts demonstrated the following regional
differences: curvedness is highest at the apex and higher in the
inferior regions than in the lateral region (among the 4 circumferential zones), especially at end diastole. In IDCM patients,
curvedness was highest at the apex, whereas there was no
significant difference among the six circumferential zones.
Similar to normal subjects, the gradient from base to apex was
significant (P ⬍ 0.001, ANOVA).
Wall thickness and radius of the cavity were measured in the
short-axis plane, long-axis plane (perpendicular to the short
axis), and 3-D surface. Wall stress calculated with these data is
shown in Fig. 5, A–C. Peak systolic wall stress in the short-axis
plane (Fig. 5A) and long-axis plane (Fig. 5B) revealed a
significant difference from base to apex (P ⬍ 0.0001,
ANOVA). The short- and long-axis wall stress showed 41 ⫾
11% and 45 ⫾ 12% increase of peak systolic wall stress
between basal and apical sections, respectively. When wall
thickness and radius of the cavity were calculated in the 3-D
space, the variation of wall stress (3DWS) from base to apex
was no longer observed (Fig. 5C). The difference between
2DSWS and 3DWS, 2DLWS, and 3DWS was reduced more at
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Table 1. Characteristics of normal subjects
and IDCM patients
Normal (n ⫽ 10)
IDCM (n ⫽ 10)
P
39⫾17
67⫾15
169⫾8
73⫾12
122⫾17
70⫾9
3.3⫾0.4
73⫾10
26⫾6
65⫾5
0.52⫾0.06
5.7⫾1.3
52⫾9
71⫾16
164⫾8
70⫾9
113⫾12
81⫾18
2.3⫾0.4
144⫾27
114⫾32
22⫾9
0.62⫾0.08
2.4⫾0.9
0.05
0.57
0.18
0.54
0.19
0.10
⬍0.001
⬍0.001
⬍0.001
⬍0.001
⬍0.05
⬍0.001
Age, yr
Weight, kg
Height, cm
Diastolic pressure, mmHg
Systolic pressure, mmHg
HR, beats/min
CI, ml/m2
EDVI, ml/m2
ESVI, ml/m2
EF, %
SI
d␴*/dtmax, s⫺1
Values are means ⫾ SD. IDCM, ischemic dilated cardiomyopathy; HR,
heart rate; CI, cardiac index; EDVI, end-diastolic volume index; ESVI,
end-systolic volume index; EF, ejection fraction; SI, sphericity index; d␴*/
dtmax, cardiac contractility index (⫽1.5 ⫻ (dV/dtmax)/Vm, where dV/dtmax is
maximum volume rate and Vm is myocardial volume).
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Fig. 4. Standardized myocardial segmentation (see Tables 2– 4 for segment
identification).
the base. The 3DWS values tend to be highest in the anterior
region and lowest in the inferior region (Table 3).
Wall thickening values determined in the short-axis plane,
long-axis plane, and 3-D surface are shown in Fig. 5, D–F.
Wall thickening did not differ significantly from base to apex.
The comparison between 3DWT and a 2-D assessment of wall
thickening (i.e., 2DSWT and 2DLWT) did not reveal significant differences at the basal and midzone, anterior, septal, and
lateral regions.
Curvedness, radius-to-thickness ratio, peak systolic wall
stress, and wall thickening in IDCM patients and normal
subjects. Regional variations of LV curvature are highlighted
by curvedness values from base to apex in normal subjects and
IDCM patients in Table 2. Significant differences in enddiastolic curvedness (CED) were noted in all regions, except
the base and anterior. Also, significant differences in endsystolic curvedness (CES) and ⌬C were noted in all regions
between normal subjects and IDCM patients. Significant
differences in end-diastolic radius-to-thickness ratio (R/
TED) were noted in 9 of 16 segments. Also, significant
differences in end-systolic radius-to-thickness ratio (R/TES)
were noted in all regions between normal subjects and
IDCM patients (Table 4).
In IDCM patients, 3DWS was significantly increased and
3DWT was decreased compared with normal subjects (Fig. 6A).
There is a gradient of mean wall stress from base to apex with
3DWS. Also, 3DWS was highest at the apex in IDCM patients
and 3DWT was significantly decreased in all regions compared
with normal subjects (Fig. 6B). In addition, 3DWT is smallest
at the inferior segments and highest at the anterior segments.
There is also significant variation among the four circumferential regions (P ⬍ 0.001, ANOVA) and from base to apex
(P ⬍ 0.001, ANOVA).
Analysis of segment scar extent in IDCM patients. The
distribution of mean values of wall stress and wall thickening
with the extent of segment scar is shown in Fig. 7 for a total of
160 segments from the IDCM patients. As expected, there was
a positive correlation between the extent of myocardial infarct
and peak systolic wall stress (3DWS; r ⫽ 0.652, P ⬍ 0.0001;
Fig. 7A). A negative correlation between the extent of myo-
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Table 2. LV regional curvedness analysis in normal subjects and IDCM patients
Normal (n ⫽ 10)
Segment
Basal anterior
Basal anterior septal
Basal inferior septal
Basal inferior
Basal inferior lateral
Basal anterior lateral
Middle anterior
Middle anterior septal
Middle inferior septal
Middle inferior
Middle inferior lateral
Middle anterior lateral
Apical anterior
Apical septal
Apical inferior
Apical lateral
0.040⫾0.0098
0.037⫾0.0058
0.030⫾0.0036
0.038⫾0.0053
0.035⫾0.0040
0.036⫾0.0063
0.034⫾0.0035
0.043⫾0.0057
0.037⫾0.0059
0.043⫾0.0044
0.037⫾0.0025
0.033⫾0.0029
0.043⫾0.0061
0.052⫾0.0092
0.056⫾0.0077
0.048⫾0.0091
CES, mm
IDCM (n ⫽ 10)
⌬C, %
⫺1
0.054⫾0.0064
0.058⫾0.010
0.047⫾0.0062
0.056⫾0.0079
0.055⫾0.0096
0.050⫾0.0065
0.057⫾0.010
0.070⫾0.016
0.056⫾0.0096
0.060⫾0.010
0.063⫾0.011
0.052⫾0.011
0.083⫾0.016
0.10⫾0.024
0.10⫾0.030
0.10⫾0.026
25⫾21
33⫾13
36⫾7
31⫾10
36⫾8
27⫾16
39⫾10
38⫾8
34⫾7
28⫾10
39⫾10
34⫾11
47⫾9
47⫾12
43⫾14
50⫾13
CED, mm
⫺1
0.031⫾0.0073*
0.032⫾0.0064*
0.030⫾0.0084
0.032⫾0.0074*
0.031⫾0.0070
0.031⫾0.0057
0.029⫾0.0054*
0.031⫾0.0042‡
0.031⫾0.0044*
0.033⫾0.0042‡
0.032⫾0.0052*
0.030⫾0.0078
0.043⫾0.017
0.040⫾0.010†
0.047⫾0.0045†
0.045⫾0.0093
CES, mm⫺1
⌬C, %
0.036⫾0.0079‡
0.034⫾0.0063‡
0.033⫾0.0096†
0.036⫾0.011‡
0.033⫾0.0079†
0.035⫾0.0062‡
0.032⫾0.0056‡
0.033⫾0.0050‡
0.031⫾0.0048‡
0.034⫾0.0056‡
0.034⫾0.0059‡
0.030⫾0.0042‡
0.049⫾0.016‡
0.042⫾0.012‡
0.050⫾0.0087‡
0.051⫾0.014‡
12⫾13
8⫾7‡
8⫾17‡
8⫾17†
5⫾15‡
8⫾16*
6⫾12‡
6⫾8‡
⫺1⫾8‡
2⫾7‡
4⫾13‡
⫺1⫾16‡
11⫾12‡
6⫾8‡
3⫾16‡
10⫾14‡
Values are means ⫾ SD. LV, left ventricular; CED, curvedness in end-diastole; CES, curvedness in end-systole; ⌬C, curvedness change between end diastole
and end systole. *P ⬍ 0.05; †P ⬍ 0.01; ‡P ⬍ 0.001 vs. normal.
cardial infarct and wall thickening (3DWT) can be seen (r ⫽
⫺0.622, P ⬍ 0.0001; Fig. 7B).
DISCUSSION
This study investigated the regional 3-D shape variations (e.g.,
in terms of curvedness) and wall stress in the LV. Accordingly,
we developed methods to determine and map the local curvatures,
local radius, and wall thickness in a 3-D model of the reconstructed LV. This approach yields new insights and demonstrates
the potential of using 3-D regional analysis to provide details
associated with local mechanics that are unattainable with 2-D
analysis or simplified geometric modeling.
Fig. 5. Variation of LV systolic wall stress and
wall thickening from base to apex in normal
subjects. A: wall stress in short-axis direction
(2DSWS). B: wall stress in long-axis direction
(2DLWS). C: 3-D wall stress (3DWS). D: wall
thickening in short-axis direction (2DSWT).
E: wall thickening in long-axis direction
(2DLWT). F: 3-D wall thickening (3DWT).
Values are means ⫾ SD. P, 2-tailed significance of paired differences; NS, not significant.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
CED, mm
⫺1
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Table 3. LV end-systolic wall stress that takes into account curvature in normal heart
P
Segment
Basal anterior
Basal anterior septal
Basal inferior septal
Basal inferior
Basal inferior lateral
Basal anterior lateral
Middle anterior
Middle anterior septal
Middle inferior septal
Middle inferior
Middle inferior lateral
Middle anterior lateral
Apical anterior
Apical septal
Apical inferior
Apical lateral
2DLWS,
⫻1,000 N/m2
2DSWS,
⫻1,000 N/m2
3DWS vs.
2DSWS
3DWS vs. 2DLWS
2DSWS vs.
2DLWS
10.9⫾3.92
10.1⫾3.96
12.4⫾3.21
10.3⫾4.07
10.4⫾4.69
12.5⫾5.50
13.7⫾4.52
10.2⫾3.85
11.9⫾4.69
11.1⫾5.77
12.6⫾9.63
17.3⫾11.12
12.5⫾6.02
8.50⫾4.23
10.0⫾6.40
10.7⫾7.40
8.19⫾3.55
7.99⫾4.38
12.1⫾5.56
8.00⫾2.67
7.80⫾3.46
9.50⫾4.68
9.10⫾3.88
7.41⫾4.02
10.5⫾6.60
8.35⫾3.91
8.00⫾5.89
12.4⫾9.20
6.27⫾3.38
6.07⫾3.79
7.16⫾4.13
5.87⫾3.14
10.1⫾4.99
8.81⫾2.62
7.54⫾2.81
8.08⫾3.26
8.73⫾3.83
9.39⫾5.80
10.0⫾3.93
9.85⫾3.33
7.41⫾3.44
8.05⫾5.74
10.8⫾9.49
10.2⫾7.31
7.15⫾3.49
5.53⫾2.54
5.08⫾3.88
10.7⫾7.40
0.01
0.006
NS
0.009
0.003
⬍0.001
⬍0.0001
0.024
NS
0.039
0.021
⬍0.0001
⬍0.0001
NS
NS
0.045
0.030
NS
⬍0.0001
0.03
0.021
0.003
⬍0.0001
NS
⬍0.0001
⬍0.0001
⬍0.0001
0.006
⬍0.0001
0.015
0.003
0.006
NS
NS
NS
NS
NS
NS
NS
0.048
NS
NS
NS
NS
NS
NS
NS
NS
Values are means ⫾ SD. 3DWS, 3-dimensional (3-D) wall stress; 2DLWS, wall stress in long-axis plane; 2DSWS, wall stress in short-axis plane; NS, not
significant.
Normal hearts. The radius of curvature of the LV is usually
estimated from the short- and long-axis diameters (2-D analysis). These parameters, however, do not necessarily correspond
to the radii of curvature at a particular point, especially when
local pathology is present (45). In the present study, we have
compared the difference between 3-D and 2-D analysis (i.e.,
short axis and long axis). The wall stress values derived from
short- and long-axis methods tend to vary from base to apex,
with the smallest wall stress value at the apex. 3DWS is fairly
uniform, however, from base to apex (P ⫽ 0.298, ANOVA)
among the four circumferential regions (P ⬎ 0.05, ANOVA).
Also, 3DWS is ⬃10% lower at the apex and 20% lower at the
inferior region than elsewhere. These modest variations in wall
stress are in agreement with the relatively uniform radius-tothickness ratio from base to apex (Table 4). Previous studies of
wall stress at the apex are controversial, with some predicting
high values (36) and others predicting low values (3, 28).
ICDM. LV remodeling in IDCM is a multistep process that
has been investigated in numerous studies (9, 34, 39). The loss
of contractile function following coronary occlusion is accompanied by acute dilatation of the infarction area, increase of LV
volume, lengthening of the LV perimeter, and blunting of the
normal curvature. Geometrically, the increase of LV volume
and sphericity result in a corresponding increase of the local
radii of curvature. This is also consistent with the increase of
SI, which is the short axis-to-long axis ratio. The development
of IDCM is accompanied by a decrease of LV function (e.g.,
LVEF, d␴*/dtmax, and wall thickening), on the one hand, and
a progressive increase of LV wall stress, on the other. In the
present study, we find that wall stress increases in each region,
which may in turn cause decreased function (wall thickening).
The increase of global stress has been shown to be a measure
of afterload following infarction (47). However, information
regarding regional distribution of wall stress is lacking. Since
Table 4. LV radius-to-thickness ratio that takes into account 3-D curvature in normal subjects and IDCM patients
Normal (n ⫽ 10)
Segment
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Basal anterior
Basal anterior septal
Basal inferior septal
Basal inferior
Basal inferior lateral
Basal anterior lateral
Middle anterior
Middle anterior septal
Middle inferior septal
Middle inferior
Middle inferior lateral
Middle anterior lateral
Apical anterior
Apical septal
Apical inferior
Apical lateral
IDCM (n ⫽ 10)
R/TED
R/TES
R/TED
R/TES
5.04⫾1.35
4.22⫾1.26
4.59⫾1.06
3.90⫾0.60
4.95⫾1.19
5.65⫾1.83
6.98⫾3.17
4.77⫾1.20
4.74⫾1.25
4.18⫾0.83
5.15⫾1.00
6.26⫾1.87
5.86⫾1.67
4.55⫾0.83
4.78⫾1.15
5.67⫾2.42
1.86⫾0.65
1.66⫾0.58
2.01⫾0.53
1.72⫾0.61
1.84⫾0.80
2.15⫾0.96
2.10⫾0.70
1.65⫾0.55
1.94⫾0.69
1.83⫾0.84
2.03⫾1.34
2.57⫾1.53
1.97⫾0.86
1.41⫾0.59
1.59⫾0.89
1.72⫾1.01
5.64⫾1.63
5.62⫾1.43*
6.13⫾2.06*
5.00⫾1.07*
5.30⫾1.30
5.31⫾1.38
8.79⫾2.16*
8.04⫾1.56‡
6.18⫾1.51*
5.27⫾1.10*
5.56⫾1.44*
7.06⫾2.49
7.51⫾3.27
7.59⫾2.64†
5.74⫾1.24
6.10⫾2.03
4.35⫾1.03‡
4.64⫾1.29‡
4.84⫾1.62‡
3.98⫾1.14‡
4.42⫾1.72‡
4.50⫾1.60‡
7.02⫾1.77‡
6.55⫾1.57‡
5.72⫾1.37‡
4.84⫾1.33‡
5.26⫾1.96‡
6.10⫾1.84‡
6.23⫾2.65‡
6.37⫾2.22‡
4.90⫾1.45‡
5.17⫾2.02‡
Values are means ⫾ SD. R/TED, end-diastolic radius-to-thickness ratio; R/TES, end-systolic radius-to-thickness ratio. *P ⬍ 0.05; †P ⬍ 0.01; ‡P ⬍ 0.001 vs.
normal.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
3DWS,
⫻1,000 N/m2
Innovative Methodology
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Fig. 6. Variation of LV end-systolic wall stress (3DWS) in normal subjects
and IDCM patients. A: 3DWS assessment from 3-D curvature. B: 3DWT
assessment from 3-D curvature. Values are means ⫾ SD. *Significantly
different from normal.
infarction results in regional inhomogeneity of wall stress and
the regional load may vary between different zones of an LV,
the response to the load should be examined regionally.
Measurement of regional LV curvature. The importance of
shape characteristics of ventricular chambers has been appreciated for many centuries. Much of the previous work, however, has been of a qualitative nature or based on assumptions
regarding ideal geometry of the ventricle. Furthermore, many
earlier studies are based on 2-D models of short and/or long
axis from ventriculography (4, 20, 30, 31, 33), MRI (3), and
echocardiography (29).
The present study draws on the strength of 3-D geometric
methodology, which uses an analytic approach to extract local
differential properties and curvature information by means of
local surface fitting (14, 18, 24, 50). This approach is proven to
be robust and produces accurate results, as shown by Garimella
and Swartz (15) and Cazals and Pouget (8). The approach is
also quantitative and provides specific regional information,
and it can be extended to the characterization of 3-D LV wall
stress.
AJP-Heart Circ Physiol • VOL
Fig. 7. Distribution of peak systolic wall stress (3DWS, A) and wall thickening
(3DWT, B) based on different segment scar extent in IDCM patients. Values
are means ⫾ SD.
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The choice of the descriptor used to quantify the variation of
the surface shape is important. The common shape-related
differential properties of surfaces are the Gaussian curvature
(K ⫽ k1k2) and the mean curvature [H ⫽ (k1 ⫹ k2)/2]. The
Gaussian curvature is considered the most widely used index of
surface shape, and it depends only on the intrinsic geometry of
the surface. However, it does not intuitively describe the extent
of surface bending. For example, in Fig. 8, the Gaussian
curvature K at the parabolic line on a toroidal surface is zero,
even though it is actually curved. On the other hand, the mean
curvature H can also be misleading. For example, at the saddle
point illustrated in Fig. 8, H is zero, even though the surface is
curved. Koenderink and van Doorn (25) also showed that K is
not very indicative of local shape, and they introduced a more
significant measure of local shape known as curvedness (C).
The curvedness value (C) describes the magnitude of the
curvature at a surface point, i.e., a measure of degree of
curvature at a point. Since we are interested in the magnitude
or extent of curvature locally, we chose C as the curvature
metric for analyzing LV regional shape. Our results show that
surface curvedness (CES) of the LV of an IDCM patient is
significantly lower than CES of the normal LV. This is explicitly demonstrated by the lower systolic function (i.e., LVEF
and wall thickening).
Limitations of the study. The peak systolic blood pressure
only provides a global value of stress for the whole ventricle.
Innovative Methodology
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LEFT VENTRICULAR SHAPE
Fig. 8. Schematics of difference between curvedness, Gaussian curvature, and mean curvature.
AJP-Heart Circ Physiol • VOL
patients with IDCM compared with normal subjects. This
decrease in wall surface curvature may be an important prognostic factor of LV remodeling.
APPENDIX A
Computation of 3-D shape descriptors. We compute LV surface
curvatures via an analytic approach using a local surface patch-fitting
method.
Local surface patch fitting. In the vicinity of a point x, the surface
can be approximated by an osculating paraboloid that may be represented by a quadratic polynomial with parameters du and dv. This
polynomial can be considered a Taylor expansion at x of the surface
by omitting those after the quadratic term
x共u ⫹ du, v ⫹ dv兲 ⫽ x00 ⫹ xudu ⫹ xvdv ⫹ 共xuudu2 ⫹ 2xuvdudv
⫹ xvvdv2兲/2
(A1)
The coefficients x00, xu, and xv are the zero, first, and second
derivatives of x with respect to u and v at the surface point x00 ⫽ x(u,
v), respectively. With these six derivatives, the curvatures of the
surface can be easily computed. Considering the function given by
z ⫽ f(x, y), a second-order polynomial of the form
z ⫽ c 1 ⫹ c 2 p ⫹ c 3 q ⫹ c 4 p 2 ⫹ c 5 pq ⫹ c 6 q 2
(A2)
is used to fit the approximating surface patch. In this equation, p ⫽
x ⫺ x0 and q ⫽ y ⫺ y0, where x0 and y0 are the x and y coordinates
of the center point x0 of the surface patch under consideration. The six
coefficients of the paraboloid are then obtained by least-squares
solution of an overdetermined system of linear equations
冦冧 冤
z⬘1
1
z⬘2
1
z⬘3 ⫽ 1
⯗
⯗
z⬘n
1
p1
p2
p3
⯗
pn
q1
q2
q3
⯗
qn
p 12
p 22
p 32
⯗
p n2
p 1q 1
p 2q 2
p 3q 3
⯗
p nq n
q 12
q 22
q 32
⯗
q n2
冥冦 冧
c1
c2
c3
c4
c5
c6
(A3)
where n is the number of points in the neighborhood of a selected
surface point.
It is desired that c1–c6 are acquired, such that the divergence of the
fitted paraboloid and the data points are minimized, which is similar
to minimizing the differences between zn (measured value of z) and z⬘n
(calculated value of z). The minimization problem at this point is then
expressed as
N
E共z兲 ⫽
兺 G 共z
n
N
⬘
n
⫺ z n兲 ⫽
n⫽1
2
兺 G 关z共p , q 兲 ⫺ z 兴
n
n
n
n
2
(A4)
n⫽1
where N represents the number of points within the fitted paraboloid
and G is a distance weighting function
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Consequently, regional variations of the wall stress reported
here only account for geometric factors; they do not integrate
the local changes in pressure.
The evaluation of the wall surface curvature depends on factors
such as image resolution and the surface reconstruction process.
The spacing between short-axis image slices for cardiac MRI in
clinical practice is typically ⬃5–10 mm. Therefore, a considerable
amount of interpolation between image slices has been used, and
this may affect the accuracy of the curvature evaluation. The
accuracy of the surface curvature also depends on the mesh
quality and sampling density of the LV models, since the
differential properties of each surface point are computed using
additional information from the neighboring vertices. Because
of the resolution of the MRIs used and a more intricate shape
of the apical region, segmentation of LV contours is often more
difficult in the apical region than in the middle and basal
regions. This could affect the evaluation of the regional
curvedness at the apical region, especially at end systole. An
additional limitation is the absence of an age-matched control
group.
Several parameters, such as wall thickening and regional
circumferential or radius strain, have been developed to quantify regional myocardial function. Since exclusion of trabeculation at end systole is impossible because the trabeculae are
compacted with the full thickness of myocardium, the wall
thickening may be overestimated. Furthermore, calculation of
wall thickening will be affected by the axis rotation during the
cardiac cycle. Regional strains were not calculated in the
present study, however, and cannot be compared.
Traditional techniques of regional ventricular function parameters (i.e., wall thickening and regional circumferential or
radius strain) depend on assumptions about coordinates, reference, and the uniformity of ventricular contraction. Our
method of shape analysis was developed to allow the quantification of regional curvedness and is independent of some of
those assumptions.
Conclusions. A framework has been developed for the
analysis of LV regional curvature, wall stress, and wall thickening for normal subjects and patients with IDCM. The methodology relies on magnetic resonance anatomic data and local
surface-fitting techniques to interrogate LV geometry at end
diastole and end systole. The proposed 3-D approach is found
to be better than the 2-D approaches for precise evaluation of
the regional wall stress. A decrease of end-systolic curvedness
and an increase of peak systolic wall stress are observed in
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Fig. B1. A: LV geometry. B: schematics of 2DSR (cavity radius) and 2DST (wall thickness) in short-axis
plane. C: schematics of 2DLR and 2DLT in long-axis
plane. D: schematics of 3DR and 3DT accounting for
3-D curvature.
2
2
i ⫽ 1, 2, 3,. . .N
(A5)
where f and d are arbitrary constants, which can be adjusted accordingly. The error function E will have a minimum when
⳵E
⫽0
⳵c i
i ⫽ 1, 2, 3, 4, 5, 6
(A6)
thus yielding the six linear equations required to compute c1–c6.
The first coefficient c1 is the z value of the center point x0 on the
fitted surface patch, while c2–c6 can be interpreted as the first and
second derivatives of x with respect to p and q at x0. By acquiring
these coefficients, we can then determine the shape descriptor, which
is based on the curvedness (C) presented by Koenderink and Van
Doorn (25) as
冑
px ⫹ qy ⫹ rz ⫽ 0
(B1)
If we include the quadric equation in the plane equation, we obtain
px ⫹ qy ⫹ r共ax 2 ⫹ bxy ⫹ cy 2 ⫹ dx ⫹ ey兲 ⫽ 0
arx 2 ⫹ brxy ⫹ cry 2 ⫹ 共dr ⫹ p兲x ⫹ 共er ⫹ q兲y ⫽ 0
␬ 12 ⫹ ␬ 22
C⫽
2
(A7)
where ␬1 and ␬2 are the maximum and minimum principal curvatures
at the vertex of the surface, the values of which are obtained from the
roots of
冋
find the curvature of the intersection curve, as well as the vector
defining the direction, to measure the wall thickness. The wall radius
is then the inverse of the curvature. The direction vector will be used
to perform a ray-triangle intersection with the epicardial surface mesh
to determine the wall thickness.
The calculation is carried out in the local frame of the quadric
surface obtained during the patch fitting. As the intersection plane
passes through the origin of the quadric surface, the implicit form in
the local frame is given by
册
L ⫺ kE M ⫺ kF
det
⫽0
M ⫺ kF N ⫺ kG
(A8)
where {E, F, G} and {L, M, N} are components of the first and second
fundamental forms, respectively.
The parametric curve that describes the intersection between the
quadric and the plane is then given by
冋册 冋
(B3)
where ␣ is the curve parameter. A suitable parameterization is to take
␣ ⫽ x. On the basis of differential geometry, the curvature ␬ of r at
the origin is
␬⫽
兩ṙ共0兲 ⫻ r̈共0兲兩
兩ṙ共0兲兩3
(B4)
where ṙ and r̈ are the first and second derivatives of r with respect to
the curve parameter, respectively.
To calculate these derivatives, we evaluate the y component of r.
The derivative of Eq. B2 with respect to x gives
Fig. B2. Normal (n̂) at endocardial surface.
AJP-Heart Circ Physiol • VOL
册
x
x
y
r共␣兲 ⫽ y ⫽
z
ax2 ⫹ bxy ⫹ cy2 ⫹ dx ⫹ ey
APPENDIX B
Computation of radius and wall thickness using the short- and
long-axis approach and the 3-D approach. To enable the calculations
of 2DSR and 2DLR, we need to determine the properties of the
intersection curve between the quadric surface and the intersecting
plane, be it the short- or long-axis plane (Fig. B1). The objective is to
(B2)
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2
G i ⫽ f 䡠 e ⫺共p i ⫹q i 兲/d
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2arx ⫹ bry ⫹ brx
dy
dy
dy
⫹ 2cry ⫹ 共dr ⫹ p兲 ⫹ 共er ⫹ q兲 ⫽ 0
dx
dx
dx
(B5)
dy ⫺2arx ⫺ bry ⫺ dr ⫺ p
⫽
dx
brx ⫹ 2cry ⫹ er ⫹ q
(B6)
Differentiating Eq. B5, again with respect to x, yields
2ar ⫹ 2br
冉 冊
dy
d 2y
dy
⫹ brx 2 ⫹ 2cr
dx
dx
dx
2
⫹ 2cry
d 2y
d 2y
⫽0
2 ⫹ 共er ⫹ q兲
dx
dx 2
冉冊
2
dz
dy
dy
dy
⫽ 2ax ⫹ by ⫹ bx ⫹ 2cy ⫹ d ⫹ e
dx
dx
dx
dx
(B8)
If we differentiate Eq. B8 with respect to x, we obtain
冉 冊
2
⫹ 共bx ⫹ 2cy ⫹ e兲
d 2y
dx 2
(B9)
At ␣ ⫽ 0, x ⫽ y ⫽ 0. Using the results in Eqs. B6 and B8, we obtain
冤冥冤冥冤
dx
dx
1
d␣
dx
⫺2arx ⫺ bry ⫺ dr ⫺ p
dy
dy
brx ⫹ 2cry ⫹ er ⫹ q
ṙ共␣兲 ⫽
⫽
⫽
d␣
dx
dy
dz
dz
2ax ⫹ by ⫹ 共bx ⫹ 2cy ⫹ e兲 ⫹ d
dx
d␣
dx
ṙ共␣ ⫽ 0兲 ⫽
冥
冋 册
(B10)
1
m
d ⫹ em
where m ⫽ 共⫺dr ⫺ p兲/共er ⫹ q兲.
Furthermore, using the results in Eqs. B7 and B9, we obtain
冤冥冤冥
冤 冉冊
d 2x
d 2x
2
d␣
dx 2
2
d y
d 2y
r̈共␣兲 ⫽
⫽
d␣ 2
dx 2
d 2z
d 2z
d␣ 2
dx 2
冥
0
2
dy
dy
⫺2ar ⫺ 2br ⫺ 2cr
dx
dx
⫽
brx ⫹ 2cry ⫹ er ⫹ q
2
dy
dy
d 2y
2a ⫹ 2b ⫹ 2c
⫹ 共bx ⫹ 2cy ⫹ e兲 2
dx
dx
dx
r̈共␣ ⫽ 0兲 ⫽
冋
0
n
2a ⫹ 2bm ⫹ 2cm 2 ⫹ en
where n ⫽
冉冊
册
冋册
⫺d
⫺e
冑d 2 ⫹ e2 ⫹ 1 1
1
(B12)
GRANTS
Next, we evaluate the z component of r by differentiating the quadric
equation z ⫽ ax2 ⫹ bxy ⫹ cy2 ⫹ dx ⫹ ey with respect to x
dy
dy
d 2z
⫹ 2c
2 ⫽ 2a ⫹ 2b
dx
dx
dx
n̂ ⫽
(B7)
(B11)
d 2y ⫺2r 共a ⫹ bm ⫹ cm2兲
⫽
dx2
er ⫹ q
Using Eqs. B10 and B11, we can then use Eq. B4 to determine
the curvature ␬ of the intersection curve r. The tangent of the curve
t̂ at the origin was calculated as follows: t̂ ⫽ ṙ(0)/兩ṙ(0)兩. The
AJP-Heart Circ Physiol • VOL
This work was supported in part by a research grant from the National Heart
Centre Singapore and National Heart, Lung, and Blood Institute Grant HL84529.
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dy
dy
⫺2ar ⫺ 2br ⫺ 2cr
dx
dx
d 2y
⫽
dx 2
brx ⫹ 2cry ⫹ er ⫹ q
direction of the wall thickness was then taken as the cross-product
of the plane normal n̂ and t̂ (in the global coordinate space), as
illustrated in Fig. B2.
To calculate the wall thickness, a ray was defined with origin at the
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of the quadric surface at the origin and is given by
Innovative Methodology
H584
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