Download Left Ventricular Dynamic Geometry in the Intact and Open Chest Dog

573
Left Ventricular Dynamic Geometry in the Intact and
Open Chest Dog
Keith R. Walley, Maleah Grover, Gilbert L. Raff, J. William Benge, Blake Hannaford, and
Stanton A. Glantz
From the Cardiovascular Research Institute and Department of Medicine, University of California, San Francisco, California
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SUMMARY. No approach to describing the heart's dynamic geometry has been widely adopted,
probably because all require questionable assumptions of chamber shape, symmetry, or placement
of the measuring devices. In other words, these approaches require assumptions about shape to
reach conclusions about shape. We present an analysis that avoids such assumptions and provides
an objective description of how the left ventricle deforms and rotates during the cardiac cycle. We
only assume that the deformation of the left ventricular cavity is homogeneous, and explicitly
validate this assumption. Our analysis yields the following new information about the contracting
left ventricle: three principal directions of deformation and the relative length change along these
directions: the axis and angle of rotation, and relative volume. All these changes are referenced to
the ventricle's configuration at end-diastole. We instrumented 13 dogs with tantalum screws without
opening their chests. During systole, the three principal directions of deformation essentially are
aligned along apex-base, anterior-posterior, and septum-free wall directions. There is little length
change in the apex-base direction. The anterior and septal principal directions do not remain fixed
with respect to the heart's anatomy during systole. During isovolumic relaxation and early filling,
systolic shape changes are reversed. During slow filling, only small shape changes occur. Opening
the pleura or performing a sternotomy and pericardiectomy m,akes the heart change orientation
within the chest, but does not alter the magnitude of shortening, relative to the left ventricle's enddiastolic configuration. (Ore Res 50: 573-589, 1982)
THE heart ejects blood by deforming. The complex
three-dimensional deformation that occurs during
contraction and relaxation has forced investigators to
make major simplifying assumptions to interpret their
data, and these assumptions can themselves introduce
subtle biases (Sandier and Alderman, 1974). It is
common to assume a shape or symmetry (for example, an elliptical chamber) to study ventricular shape
changes. Measuring devices fixed anatomically, such
as radiopaque markers or ultrasonic crystals, are assumed to bear a fixed and physically meaningful
relationship to the deforming ventricle (for example,
when the markers are assumed to lie on the major
and minor axes of an assumed elliptical chamber).
The results of angiographic, echocardiographic, and
other methods based on imaging depend on the orientation of the observer relative to the heart; furthermore, this orientation changes as the heart beats. In
addition to these problems associated with interpreting the observations, thoracotomy and pericardiectomy necessary to implant measuring devices might
distort ventricular geometry. Finally, few methods
provide information concerning the dynamic geometry of the entire ventricle. Most methods measure
only one or two chamber dimensions or a single
geometric variable such as volume. We present a new
analysis which avoids any assumptions about chamber shape, chamber symmetry, or placement of radiopaque markers. This analysis yields an objective decription of the pattern of deformation and rotation of
the ventricular cavity. Interventions such as volume
loading, opening the pleura, or resecting the pericardium do not significantly affect normal left ventricular
dynamic geometry during systole.
Methods
Experimental Preparation
The surgical method has already been described (Davis
et al., 1980; Raff et al., 1981). With the aid of a fluoroscope,
we implanted seven to 15 tantalum screws ( 1 X 2 mm
helices of tantalum wire) in the left ventricular endocardium
of 13 healthy mongrel dogs with a Medi-tech steerable
catheter introduced via a carotid artery. The chest wall,
pleura, and pericardium remained intact. We always placed
one to three (usually two) screws in the aortic valve ring,
one at approximately the left ventricular apex, and at least
four spaced approximately evenly about the left ventricular
equator. In some dogs, we implanted additional screws to
define the ventricular cavity more completely.
Protocol
Four to 6 weeks after screw implantation, we performed
the following experiment: we premedicated the dogs with
5 mg/kg morphine injected subcutaneously and anesthetized them with 70 mg/kg a-chloralose injected intravenously. Every hour, we gave the dogs an additional 15 mg
morphine. This anesthetic combination maintained normal
heart rates and a physiological response to volume loading
(Afors et al., 1971; Vatner and Boetticher, 1978). We measured left ventricular pressure, aortic pressure, and right
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ventricular pressure with Millar catheter-tip pressure transducers.
A set of data consisted of an analog recording of simultaneous pressures and ECG for approximately 15 seconds.
This 15-second interval included several seconds of synchronized biplane X-ray cineradiographs filmed at 30
frames/sec. We selected this sampling rate because Rankin
et al. (1976) reported that dimensional changes can be
adequately represented by the first five harmonics of the
dimension change. Since the heart rates in our dogs were of
the order of 1-2 Hz, our 30 Hz sampling rate seemed more
than adequate. The analog data were digitized every 5 msec
(Horowitz and Glantz, 1979) and the three-dimensional
positions of each of the tantalum screws were computed
every 33.3 msec (V30 sec) after projecting the biplane films
onto a Talos digitizing tablet and manually indicating the
positions of the marker shadows. We located digitizing
errors by checking the resulting screw image coordinate for
discontinuities. All suspect frames were then redigitized.
The resulting data were employed to compute the screws'
three-dimensional coordinates, using the equations of Davis
et al. (1980).
We collected a set of data during a control condition and
after volume loading with a 1:1 mixture of saline and blood.
We recorded data at end-diastolic pressures approximately
5, 10, 15, and 20 mm Hg above control (approximately
every 15 minutes).
We recorded the next set of data after inserting bilateral
chest tubes to release pleural pressure; we refer to this as
"open pleura." The dog was again given volume, and
additional data sets were recorded.
Similarly, we collected data after a median sternotomy
and pericardiectomy, before and after volume loading; we
refer to this as "open pericardium."
Circulation Research/Vol. 50, No. 4, April 1982
FIGURE 1. X-rays of 1-cm slices parallel to the plane of the mitral
valve, showing locations of endocardial tantalum screws. The apical
and aortic screws defined one axis of the coordinate system fixed
in the heart. The free wall screw defined a second axis perpendicular to the first, and the third axis completed a righthand coordinate
system. Principal directions of deformation are expressed in this
coordinate system. Note that the free wall screw is anterior of the
free wall papillary muscle. (There are also four screws in the right
ventricle; we do not include them in the analysis.)
Volume Validation
In three experiments, we removed the hearts and inserted
a balloon to check our computed volumes against actual left
ventricular volumes (Rankin et al., 1976; Suga and Sagawa,
1979). We opened the left atrium, excised mitral valvular
tissue, cut the chordae tendineae, and then sewed a Lucite
disk with an attached thin-walled latex balloon into the
mitral annulus and tied off the aortic annulus. We inserted
a small catheter transmurally to evacuate any fluid or air
that might have prevented the balloon from conforming to
the endocardia! surface. We filled the balloon with a dilute
radiopaque dye solution (1:10 Renografin 76: water) and
took biplane x-rays over a range of balloon volumes from
20 to 80 ml. These data provided simultaneous measurements of screw positions and ventriculograms at known
volumes. These data were obtained within an hour after the
dogs were killed.
We then fixed the hearts in formalin and cut them into
1-cm slices along places perpendicular to the long axis of
the heart to precisely locate the screws. Figure 1 consists of
x-rays of these slices, showing the screws implanted in the
left ventricular endocardium, as well as screws in the right
ventricle (the latter are not included in the present analysis).
We now develop the concepts and computational
methods needed to convert the data on the threedimensional movement of all the tantalum screws into
objective information about how the left ventricle
deforms (changes shape and size) and rotates in the
chest as it beats. More precisely, we will identify the
three mutually perpendicular directions along which
the left ventricle deforms during the cardiac cycle and
by how much (as a fraction of some control value).
The ventricle expands or shrinks along each of these
three directions. These so-called principal directions
of the deformation in general do not pass through
pairs of tantalum screws and may change relative to
the heart's anatomy during the cardiac cycle. We will
also identify the axis of rotation and the amount the
heart rotates about this axis during the cardiac cycle.
Statistical Methods
We tested the null hypothesis that opening the pleura or
pericardium did not alter the variables of interest by performing the two-way analysis of variance with a mixed
effects model. When this procedure detected a significant
difference, we performed multiple comparisons using the
Student-Neumann-Keuls method with a = 0.05.
The Assumption of Homogeneous Deformation
In contrast to other approaches to describing the
left ventricle's dynamic geometry, we make no assumptions about the shape or symmetry of the ventricular lumen. We only assume that the ventricular
cavity deforms homogeneously. Since the screws out-
Analytical Approach
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Walley et a/./Left Ventricular Dynamic Geometry
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line the ventricular cavity, this assumption need only
apply to the deformation of the blood-filled lumen.
We present experimental evidence to justify this assumption.
When a space deforms homogeneously, any part of
the space deforms in the same manner as the whole
space deforms. For example, parallel lines change
length and orientation but remain parallel, similar
triangles change shape and orientation but remain
similar, and ellipsoids change orientation and eccentricity but remain ellipsoids. The assumption of homogeneous deformation has been implicit in all previous studies which measured two or three diameters
of the left ventricle and assumed they were the major
and minor axes of an assumed ellipsoidal ventricle. In
our analysis, we avoid the need for specific shape
assumptions and the exact anatomical location of the
principal axes of the deformation; we retain the much
less restrictive assumption that the ventricular cavity
deforms homogeneously.
Nomenclature
To simplify the equations, we use matrix notation
to describe the screw positions and the transformations of these positions during the cardiac cycle. We
record the three-dimensional coordinates of all n
screws in a single 3 X n screw position matrix, X.
X =
Xi
X2
X3
Xn
yi
y2
y3
Zi
Z2
Z3
yn
ZnJ
(1)
The ith column of X gives the x, y, and z coordinates
of the ith screw. Since the positions of the screws
change with time, the matrix X changes with time;
X(t) is the matrix of three-dimensional screw positions
at time t.
Since the ventricular cavity is assumed to deform
homogeneously, the change in endocardial marker
positions between two times, ti and t2 (for example,
between end-diastole and end-systole) can be represented by a linear transformation, T, that depends on
ti and t2 (Meier et al., 1980a):
X(t 2 )=T(t,,t 2 )X(t 1 )
(2)
T(ti, t2) is the 3 X 3 matrix that describes how the
heart deforms and rotates between times ti and t2.
We always reference T to end-diastole, so T describes
how the heart deforms and rotates relative to its enddiastolic shape and orientation. Therefore, we can
simplify our notation and rewrite Equation 2:
X(t) = T(t)XED
(3)
where t is the time after end-diastole and X E D is the
screw position matrix at end-diastole.
In most cases, XED contains the screw coordinates
at the end-diastole immediately before the beat begins. Therefore, T (and the resulting dilation and
rotation) describes how the left ventricle changes with
respect to its configuration at end-diastole. In some
cases, we take XED to be the screw coordinates for
end-diastole with the chest intact, even when analyzing beats after opening the pleura or pericardium. In
these cases T describes the dilation and rotation due
to opening the pleura or pericardium with respect to
the left ventricle's size, shape, and orientation before
performing any of these interventions.
T represents dilation and rotation but not translation. Therefore, translation of the heart as it beats
must be subtracted from X(t) before computing T.
Incomplete removal of the translational component of
ventricular motion introduces errors in the estimate
of T because Equation 3 does not account for translation of the whole heart. If the ventricular cavity
deformed exactly homogeneously, any point which
moved with the heart could be used as the origin of
a moving coordinate system to subtract out the translation. However, any inhomogeneity of deformation
would result in the choice of origin influencing the
analysis. To test the magnitude of this potential difficulty, we performed three separate analyses with the
origin at the aortic valve, at the centroid of the screws,
and at the apex of the heart. The measures of deformation at end-systole obtained in these three analyses
differed by less than 2%, less than the measurement
error. No particular origin was identified as providing
a consistently better fit to the data. In other words,
the deformation is close enough to being homogeneous that one can choose any origin that translates with
the heart to describe the screw locations without
affecting the results of the analysis. We placed the
origin at the apex when computing the results presented in this paper.
Estimating the Transformation from Data
Rather than being given the initial screw positions
at end-diastole and the matrix T to estimate the screw
positions at some later time (say, end-diastole), we
wish to use the observed screw positions at two
different times to estimate T. We use a least squares
best fitting procedure analogous to linear regression
to estimate T. We estimate T with the transformation
that minimizes the sum of squares of the distances
between the measured final screw positions at time t
and the positions of the screw predicted from Equation 3. Appendix A shows that
f (t) =
(4)
is the best estimate of T(t) given that the ventricular
cavity may not deform exactly homogeneously and
that the data always contain some random measurement errors.
Since we film the heart at 30 frames/sec, there is
rarely a frame that corresponds exactly to end-diastole
(defined by examining the ECG R-wave peak and the
left ventricular pressure recording). We estimate XED
by interpolating linearly in time between the two cine
frames that span the time of end-diastole. Since the
aortic valve ring does not deform in the same way as
the left ventricular cavity, we only include one aortic
576
Circulation Research/Vol. 50, No. 4, April 1982
change) in three independent perpendicular directions
and a rigid-body rotation about some axis (Fig. 2). In
our application to the heart, this dilation represents
contraction of the ventricle during systole and expansion of the ventricle during diastole (Meier et al.,
1980a, 1980b). The Polar Decomposition Theorem
states that any linear transformation, T, can be
uniquely decomposed into a dilation represented by
the matrix D and a rotation represented by the matrix
R, where
T = RD.
(5)
Dilation. The dilation can be described by reporting
the three principal directions along which the ventricular cavity is changing size and shape, together with
the fractional change in length along each principal
direction. Appendix B shows that the dilation matrix
D equals
(TTT) 1/2
(6a)
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We estimate D with
D = (fT f) 1/2
(6b)
The three principal directions of dilation or contraction of the deforming ventricle are given by the
three mutually perpendicular eigenvectors of D.
To visualize what principal directions are, consider
stretching the rubber band shown in Figure 3. Lines
I and II change length but not direction when the
rubber band is stretched. These two lines are parallel
FIGURE 2. The polar decomposition theorem states that any hoto the principal directions of the deformation; they
mogeneous deformation can be decomposed into a pure dilation,
are the directions that do not change orientation when
D (panel A to B), and a pure rotation without shape change, R
the rubber band is deformed. This situation contrasts
(panel B to C). The three mutually perpendicular principal direcwith
line III (or a line in any other direction) which
tions of deformation (eigenvectors) are in directions given by the
changes not only in length but apparent orientation
vectors ei, ej, and d, and the associated eigenvalues are given by
A i, A 2 and A 3, respectively. The rotation 6 is about an axis in the when the rubber band is stretched.
—ei direction. (To simplify the illustration, the axis of rotation
Each principal direction (eigenvector) has associcoincides with one of the principal directions of dilation; this is not
ated with it a number equal to the fractional change
the case in general.) (Adapted with permission from an illustration
in length along that axis as the heart deforms. These
provided by George Meier.)
numbers are called eigenvalues of D. We denote these
three eigenvalues Ai, A2, and A3. Since we reference
screw in the computation of T. Which aortic screw
all computations to end diastole, Ai, A2/ and A3 are
one uses does not affect the results.
1 at end-diastole. Therefore, as the left ventricular
Decomposition of T into a Dilation and Rotation
cavity shrinks along principal axis i, Ai decreases. If
Any homogeneous deformation of space can be
the heart expands beyond end-diastolic length along
principal axis i, Ai will exceed 1. For example, the
uniquely decomposed into a dilation (size or shape
FIGURE 3. Stretching a rubber band with the lines I, 11, and III drawn on it reveals that lines parallel to and perpendicular to the direction of
stretch (principal directions of deformation) do not change orientation during the stretch. However, lines not oriented along the principal
directions of deformation, such as line III, change orientation.
577
Walley et ai./Left Ventricular Dynamic Geometry
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rubber band in Figure 3 lengthens to 150% of its
original length along line I (principal direction I), so
the associated eigenvalue is 1.50 and the rubber band
shortens along line II (principal direction II) to 80% of
its original length, so the associated eigenvalue is 0.80.
Figure 3 illustrates a two-dimensional deformation, so
has two eigenvectors and associated eigenvalues; the
left ventricle deforms in three dimensions, so it has
three principal directions and associated eigenvalues.
In short, the eigenvalues quantify the amount of
shortening or lengthening along each of the principal
directions of deformation.
So long as the eigenvalues are different, the eigenvectors are uniquely defined. However, when two of
the eigenvalues are identical, the deformation is symmetrical in the plane defined by the two associated
eigenvectors, so all directions lying in that plane are
principal directions (eigenvectors), and any pair of
perpendicular lines in that plane can be principal
directions. (This fact may be significant in light of the
variability of two of the principal directions at endsystole, when the associated eigenvalues are similar.)
Likewise, any three mutually perpendicular directions
are principal directions at or near end-diastole, when
all three eigenvalues are, by definition, equal to 1.
Rotation. The rotation matrix R defines the axis
and angle of rotation of the entire heart as it beats.
This is a pure rotation in which there is no deformation of the heart. From Equation 5,
The other two eigenvalues give the rotation angle, 0
(Appendix B). 6 is zero at end-diastole and positive
for righthand rotations about the axis of rotation.
Volume Measurement
Since the eigenvalues specify the fractional
shortening or lengthening in three mutually perpendicular directions, multiplying the three eigenvalues
together gives a value which should be directly proportional to ventricular volume. Since all eigenvalues
equal 1 at end-diastole, the product A1A2A3 equals the
fraction of the end-diastolic volume remaining in the
ventricle, in theory 1 minus the ejection fraction. The
determinant of the dilation matrix, D, equals the
product of its eigenvalues, A1A2A3, and the determinant
of the rotation matrix R is 1. Therefore,
VE
= | R D | = | R | | D | = A1A2A3
(8)
where VE is the relative volume, which we denote the
eigenvolume. No assumptions about the ventricular
shape, symmetry, or placement of markers have been
made in deriving this volume.
Results
(7a)
Hemodynamics
Table 1 shows that left ventricular end-diastolic
and peak systolic pressure and dp/dtmax did not
change significantly (P > 0.25) with opening the
pleura or pericardium.
R = TD- i
(7b)
One eigenvalue of R is always 1, and the associated
eigenvector is the axis about which the heart rotates.
Volume Validation
Figure 4 shows the linear relationship between the
eigenvolume, VE, and the abolute volume in the intraventricular balloon, VA, in the three excised hearts.
The eigenvolumes have been referenced to the control
=
Tn-'
We estimate R with
TABLE 1
Hemodynamic Data
End-diastolic pressure
(mm Hg)
Open
Dog
109
111
116
124
126
127
129
206
213
215
216
217
218
Mean
SD
dp/dt™,
(mm Hg/sec)
Maximum systolic pressure
(mm Hg)
Open
Closed
chest
Open
pleura
pericardium
Closed
chest
Open
pleura
Open
pericardium
Closed
chest
3.9
6.6
4.6
4.4
3.5
18.0
174
149
185
115
181
159
3200
2000
3100
2100
3300
2500
2.3
8.1
7.7
8.1
1.9
2.3
15.0
16.0
2.0
2.9
2.1
6.2
3.0
6.2
8.2
3.0
5.0
3.0
4.3
11.3
10.7
129
192
129
171
120
113
98
200
162
156
129
207
127
151
178
164
139
168
145
174
114
138
153
177
2600
3300
4100
3900
2300
3800
2900
3200
2800
3200
3200
3700
2200
3900
3600
2900
2400
3400
3500
3400
1800
3000
4300
2600
5.8
9.4
5.8
116
142
136
1800
2100
2500
5.9
4.0
6.9
4.3
6.8
4.8
145
35
163
42
153
21
3000
3000
3000
710
640
690
2.1
9.0
10.6
periOpen
pleura cardium
Circulation Research/Vol. 50, No. 4, April 1982
578
beat in the intact dog at the start of each experiment.
The deviations between the observed points and the
regression line are of the same order as the 2-4 ml
uncertainty in the balloon volume measurements and
the 0.05 uncertainty in the eigenvolumes. The regression line relating VE to VA could be used as a calibration curve for all data obtained in the experiment.
In theory, this regression line should pass through
the origin, but it had a positive intercept for all three
dogs. This positive intercept means that when the
balloon volume is zero, there is still a residual volume
defined by the screws. It also means that 1 — VE does
not precisely equal the ejection fraction.
To estimate the residual volume, let VR denote the
residual volume and VAI denote the balloon volume
corresponding to an eigenvolume of 1, Then,
Eigenvolume,
VE
i.2r
217
l.O
0.6
25
35
45
55
Balloon Volume, V4 (ml)
VE = m VA + b
(9)
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where m and b are the slope and intercept of the
regression line. Eigenvolume is related to the balloon
and residual volume according to
Eigenvolume,
VE
l.2r
(.0,
218
Substitute from Equation 10 into Equation 9, set VA
equal to zero, and solve VR.
1.0
VR
0.8
Vc --0O9 VA t .50
r--.93O
0.6
B
25
35
45
55
Bolloon Volume, V4 (ml)
65
Eigenvolume,
VE
i.2r
301
1.0
=
(11)
1-b
Using the values for b (0.45, 0.50, 0.65) and VAi (66,
51, 55 ml) in Figure 4 yields estimates of residual
volumes of 54, 51, and 94 ml, respectively.
This residual volume is an artifact that probably
arises from a combination of three factors. First, our
analysis assumes the markers are at the endocardial
surface, but they are actually 2-4 mm below the
surface. A 4-mm shell outside a spherical cavity holding 60 ml has a volume of 35 ml. Second, this residual
volume includes the volume of the papillary muscles
and other intraventricular structures. Third, the balloon probably does not completely fill the ventricular
cavity. These three factors taken together explain the
first two residual volumes (54 and 51 ml), but probably will not explain the entire 94-ml residual volume.
We cannot explain this discrepancy.
We can use Equation 9 to derive a relationship
between the true volume ejection fraction and the
eigenvolume ejection fraction. By definition, the actual volume ejection fraction is
EF A = ( V A D -
VA.S)/VAD
(12)
where the subscripts D and S refer to end-diastole
and end-systole, respectively. From Equation 9,
0.6<25
35
45
55
Bolloon Volume, V4 (ml)
VA = (VE - b)/m.
65
FIGURE 4. Eigenvolume, VE, correlates well with actual ventricular
volume, VA, measured by an intraventricular balloon in all three
postmortem hearts we studied. The scatter of points about the
regression line is less than the uncertainty of our measurements.
(13)
Substitute from Equation 13 into Equation 12 to obtain
EFA =
(VE S
- b)/m - (V E D - b)/m
- b)/m
(14)
579
W'alley et al./ Left Ventricular Dynamic Geometry
_
VED
-
VES
_
VED
VED-b
-
VES
VEn
VE,,
V E D -b
(15)
u
the eigenvolume ejection
I)
fraction and, by definition, VED = 1, so
EF A =
1-b
EFE
(16)
Since we found values of b of 0.45, 0.50, and 0.65, the
actual volume ejection fraction is approximately twice
the eigenvolume ejection fraction.
Prediction of Three-Dimensional Screw Positions
during Systole
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To test whether or not the homogeneity assumption
led to realistic predictions of screw coordinates, we
compared the observed end-systolic screw positions
with those predicted from the transformation
). End-systole is defined to be the time minimum
dp/dt; this time coincides with aortic valve closure
(Abel, 1981; Raff and Gantz, 1981). This procedure is
analogous to plotting a regression line with the raw
data to observe the size and distribution of the residuals. If the linear model is a good one, the differences
between the regression line and points will be small
and randomly distributed about the line. In our analysis, if the assumption of a homogeneous deformation
is valid, the differences between the predicted and
observed coordinates will be small and not depend
on location in the heart. Figure 5 shows a comparison
of observed end-diastolic (outside) and predicted and
observed end-systolic (inside) screw positions. The
outside (end-diastolic) screws have been connected to
make it easier to interpret the figure. Notice that the
end-systolic (inside) points deform considerably from
the end-diastolic (outside) points and that the predicted and observed end-systolic points agree well.
The predicted points lie on both sides of the observed
points (as expected from a best-fitting procedure)
which suggests that volume measurements will be
accurate despite small inhomogeneities and random
measurement errors.
To quantify the closeness of the fit, we defined a
measure of fit analogous to a standard correlation
coefficient,
(17)
where SSres is the sum of squared deviations (residuals) between the regression line and the data points,
and SStot is the sum of squared deviations of the
observed points about the mean value (Glantz, 1981).
By analogy, we define SSres to be the sum of squared
distances (in three dimensions) between the predicted
and observed screw positions at end-systole and SStot
to be the sum of squared distances between the
observed screw positions at end-diastole and endsystole.
ANTERIOR
FREE
WALL
SEPTUM
APEX
FIGURE 5. Predicted and actual screw positions observed in two perpendicular views. Roman numbers indicate end-diastolic screw positions
observed in two perpendicular views. Italic numbers indicate end-systolic screw positions. Dots indicate end-systolic screw positions
predicted by homogeneous deformation from the end-diastolic screw positions. The outside screws in each view are connected by lines.
Solid lines connect measured screw positions, whereas dashed lines connect predicted screw positions. The differences between the observed
and predicted systolic screw positions are small compared to the differences between diastolic and systolic screw positions, and of the same
order as the size of the screws (2 mmj. r = 0.95 for this beat; Table I indicates that this is relatively poor agreement between predicted and
observed screw coordinates, compared to the other beats we analyzed.
580
Circulation Research/Voi. 50, No. 4, April 1982
TABLE 2
Correlations between Predicted and Observed ThreeDimensional Screw Screw Positions at End-Systole
Minimum
25 Percentile
Median
75 Percentile
Maximum
Number of beats
Closed
chest
Open
pleura
Open
pericardium
0.838
0.954
0.968
0.986
0.996
0.794
0.937
0.967
0.982
0.994
0.728
0.959
0.978
0.987
0.995
48
39
33
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Table 2 summarizes the values of r for the 120
beats we analyzed, r exceeded 0.90 in 111 (93%) of
the beats and exceeded 0.95 in 90 (75%) of the beats.
Similar high values of r were observed throughout the
entire cardiac cycle. These high values indicate that
the differences between the predicted and observed
end-systolic screw positions were small compared to
the difference between measured end-diastolic and
end-systolic screw positions. That is, the errors introduced by assuming homogeneous deformation, as
well as random measurement errors, are small compared to the actual deformation.
Principal Directions and Shortening (Eigenvectors
and Eigenvalues of D)
To visualize the three mutually perpendicular principal directions of dilation, imagine placing a globe
around the heart with the north pole at the midpoint
of the aortic valve, the south pole at the apex, and 0°
longitude (the Greenwich meridian) passing through
the screw implanted in the left ventricular free wall
(Fig. 6). [The free wall screw tended to be located
anterior of the free wall insertion of the papillary
muscle (cf. Fig. 1)] Thus, —90° longitude is anterior,
+90° longitude is posterior, and +180° or —180°
longitude is on the septum (the international date
line). Figure 7A shows an Aitoff equal area projection
of the globe in Figure 6, together with the three
mutually perpendicular directions of the dilation for
a typical systole in a closed-chest dog. Figure 7B
summarizes the location of the principal directions
for all 13 dogs. Qualitative examination of the locations of the principal directions in the heart did not
reveal any systematic changes with opening the pleura
or pericardium, and Figure 7B was constucted by
pooling information for all analyzed beats.
One of the principal directions always lines up
generally along the long axis of the heart (Fig. 7B).
This principal direction is generally the most stable
of the three principal directions during systole.
Figure 8 shows the eigenvalues for the same beat
illustrated in Figure 7A, and Table 3 summarizes the
end-systolic values of the eigenvalues for all dogs
with the chest intact, with the pleura open, and with
the pericardium open. (Table 3 contains the mean
value for all beats for those cases in which we ana-
lyzed multiple beats.) These maneuvers did not produce significantly different values of any of these
eigenvalues (P > 0.25).
These figures and this table show that there is little
shortening along the long axis principal direction,
given by ALONG- The mean end-systolic value was
0.95, indicating only a 5% shortening.
The other two eigenvectors complete the orthogonal set and therefore lie in approximately the equatorial plane of the ventricle. One of the principal
directions tends to lie in an anterior-posterior (AP)
direction and the other lies at right angles, in approximately a septum-free wall (SF) direction. The meann
end-systolic values, 0.84 for AAP and 0.86 for ASF, of
these two equatorial eigenvalues were not significantly different (P > 0.25). The similarity in the endsystolic values of AAP and ASF may explain the variability of the associated eigenvectors. These two eigenvalues are, however, significantly smaller (P < 0.0005)
than the corresponding values of ALONG, indicating
that most of the heart's shape and volume change
occurs along directions parallel to the equatorial
plane.
Whereas the 30 frame/sec sampling rate appears
to have been fast enough to compute the eigenvalues
accurately, subsequent experiments using a 60 frame/
sec sampling rate (unpublished observations) revealed
that it was not fast enough to track precisely the
motion of the eigenvectors. Therefore, we have cho^
sen to display the region of systolic motion for each'
of the principal directions in each of the dogs in
Bose
Apex
FIGURE 6. Heart enclosed, in the unit sphere used to describe the
orientation and motion of the three mutually perpendicular principal directions of dilation. The north pole is toward the base, the
south pole toward the apex and 0° longitude along the free wall
Walley et ai./Left Ventricular Dynamic Geometry
581
Base
Apex
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116
|
"T297T
/
217
—"124
/
217
i;
/
(
. —
2y
-
•
—
*
129
/
218
/
FIGURE 7. A: Aitoff equal-area projection of
three principal directions of dilation on a
globe fixed in the heart during the same systole in a closed-chest dog. Meridians are
drawn every 30° of latitude and longitude.
Each point represents the results from one
cine frame, so the points are 1/30 sec apart in
time. B and C: Regions enclosing principal
directions of dilation during systole for all 13
dogs.
—
-—
y/~
/
O9
^ >
V V
B
—*
~77T——__
IS
—
— in
/
/
206
^i_
206
2«
\
2I5\
\
127
/
Figure 7, B and C, without making any firm conclusions about the nature of the motion. Resolving that
question will have to wait for new data recorded at 60
frame/sec.
Since R quantifies the rotation of the entire ventricle, the change in principal axis orientation described
here indicates that shortening of the ventricular cavity
is occurring in different directions at different times.
/
These measurements demonstrate that the ventricle
does not necessarily contract from end-diastole to
end-systole along an anatomically fixed set of directions. In the equatorial plane, shortening occurring
early in systole may be in directions that are quite
different from the directions late in systole.
During isovolumic relaxation and rapid diastolic
filling, the apex-base principal direction remained
Circulation Research/Voi. 50, No. 4, April 1982
582
e
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LV dp/dt
(mm Hg /s)
J -40
LV
Pressure
(mmHq)
-I0 1 Time (s)
FIGURE 8. Eigenvalues, eigenvolume, and rotation for the same beat shown in Figure 7A. At the onset of systole, X SF decreases, which
indicates shortening along the septum-free wall principal direction. Later in systole, X Ar decreases faster than X SF, reaching about the same
minimum value. \ LONG decreases significantly less than A.SF and XAI-, indicating that apex-base shortening is less than cross-sectional
shortening. During slow diastolic filling, all directions lengthen slowly and evenly, so no principal directions of deformation are defined. The
entire left ventricular cavity rotates slightly during atrial systole, then rotates more in the reverse direction during ventricular ejection. The
vertical lines indicate the times of left ventricular end-diastole and dp/dt„,,„.
relatively stable and the equatorial principal directions tended to return to their end-diastolic orientations.
By the end of rapid filling, the ALONG had increased
to 0.99 (mean), ASF to 0.96, and \Ap to 0.96 of their
diastolic values. Thus, most of the shape change that
occurred during systole is reversed by the end of
rapid filling.
During late diastolic filling, the eigenvectors have
no fixed orientations and change considerably from
frame to frame. The lengthening along the three
eigenvectors is small (less than 5-10%) and occurs
steadily until atrial contraction. The erratic orientation
of the eigenvectors may reflect, in part, the fact that
the ventricular cavity at this time is small in shape
and expands uniformly to the reference end-diastolic
shape so the numerical algorithm cannot reliably
estimate the eigenvectors. In sum, during late diastolic
filling, only small ventricular cavity shape changes
occur. Since the lengthening that occurs has no fixed
orientation, these changes are consistent with concentric expansion described by Rankin et al. (1976).
Volume loading did not produce consistent changes
in the basic pattern of deformation.
Volume
Figure 8 demonstrates the well-known phases of
the cardiac cycle: isovolumic contraction, ejection,
isovolumic relaxation, early rapid filling, slow filling,
and atrial contraction. Table 3 shows that the mean
end-systolic eigenvolume is 0.70, corresponding to a
30% eigenvolume ejection fraction, or approximately
60-70% actual volume ejection fraction. The data
consistently revealed a small (less than 4%) eigenvolume increase at the time of aortic valve closure in all
13 dogs. Data published by others (Hinds et al., 1969;
Mitchell et al., 1969; Rankin et al., 1976; Yellin et al.,
1980) shows similar results. Reflux or bulging as the
valve closes probably accounts for this small volume
change.
Volume loading always increased end-diastolic eigenvolume computed with respect to the end-diastolic
screw positions during the closed chest baseline experimental condition. Increases in equatorial eigenvalues, not the apex-base eigenvalue, accounted for
this increase. Opening the pleura or opening the
pericardium did not significantly change end-diastolic
or end-systolic eigenvolume, computed with respect
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0.85
O.c)2
0.85
0.83
0.05
216
217
218
Mean
SO
0.89
0.11
0.74
0.84
0.85
0.91
0.80
0.76
0.93
0.84
0.92
0.89
0.99
1.17
0.87
0.83
0.88
0.80
0.93
0.81
0.85
0.89
0.76
0.84
0.79
0.94
0.78
0.85
0.06
Open
peric
Open
pleura
pcric = pericardium.
' Bv definition.
213
0.86
0.74
127
215
0.85
126
0.92
0.83
124
0.84
0.75
0.84
116
129
0.86
206
0.77
111
chest
Int.ict
109
DOR
ASK
0.82
0.06
0.82
0.77
0.76
0.78
0.88
0.87
0.78
0.83
0.70
0.84
0.90
0.79
0.93
Intact
chest
0.86
0.06
0.94
0.81
0.85
0.85
0.81
0.89
0.93
0.86
0.74
0.78
0.84
0.89
0.94
0.83
0.05
0.84
0.81
0.78
0.89
0.87
0.88
0.78
0.91
0.83
0.78
0.78
0.88
Open Open
pleura peric
AI.ONI;
0.94
0.05
0.97
0.04
0.93
0.95
1.01
0.92
0.93
1.02
0.98
0.96
0.95
1.03
0.98
1.01
0.95
0.95
0.05
0.98
0.93
0.98
0.98
0.95
1.02
0.90
0.93
0.99
0.99
0.90
0.85
Open
peric
End-systole
Intact Open
chest pleura
0.94
0.95
0.98
0.88
0.91
1.00
1.00
0.93
0.91
1.00
0.99
0.87
0.90
TABIX 3
0,65
0.07
0.59
0.63
0.62
0.57
0.68
0.64
0.72
0.65
0.48
0.71
0.75
0.64
0.71
0.70
0.08
0.75
0.64
0.76
0.63
0.70
0.74
0.78
0.73
0.54
0.68
0.65
0.85
0.68
0.70
0.08
0.62
0.63
0.65
0.80
0.66
0.68
0.65
0.70
0.75
0.68
0.70
0.88
Intact Open Open
chest pleura peric
VK
4.7
2.2
4.1
0.8
4.8
4.2
4.8
2.7
4.1
7.4
5.8
3.1
5.7
5.2
9.9
4.2
2.3
2.2
6.9
3.0
2.2
8.1
3.6
3.4
6.0
7.6
1.5
4.9
4.3
1.5
5.0
2.9
3.8
5.1
7.3
3.3
1.5
4.4
7.5
7.5
4.5
4.5
0.0
10.8
Intact Open Open
chest pleura peric
6
6.9
2.4
4.1
4.3
5.3
5.2
8.7
4.0
9.4
9.7
7.8
5.8
5.7
8.3
11.1
6.9
2.3
5.6
8.2
8.6
2.7
9.0
3.8
5.6
7.8
11.0
6.0
6.3
9.2
5.8
Intact Open
chest pleura
0m.,.
7.1
2.7
4.8
7.2
7.5
4.4
4.4
6.1
7.6
0
0
0
0
0
0
0
0
0
0
0
0
8.1
0
5.8
0
0*
18.6
10.9
18.0
13.5
15.2
15.3
13.2
24.6
43.8
21.6
6.7
5.3
28.3
6.3
28.8
31.4
19.0
38.7
11.0
27.2
36.1
23.8
12.1
19.5
22.1
33.2
30.9
38.1
38.8
Intact Open Open
chest pleura peric
12.5
4.9
11.9
Open
peric
9
0
1.00
1.00*
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Intact
chest
0.96
0.10
1.00
1.10
0.80
1.03
0.91
0.91
1.16
0.85
0.95
0,84
0.92
0.97
1.04
Open
pleura
VK
0.93
0.26
1.00
1.14
1.34
0.81
1.00
1.20
1.12
0.91
0.50
0.66
0.77
0.64
Open
peric
0.65
0.07
0.59
0.63
0.62
0.57
0.68
0.64
0.72
0.65
0.48
0.71
0.75
0.64
0.71
Intact
chest
0.67
0.10
0.75
0.70
0.61
0.65
0.64
0.67
0.90
0.62
0.51
0.57
0.60
0.82
0.70
0.64
0.14
0.63
0.75
0.89
0.65
0.66
0.82
0.73
0.65
0.40
0.42
0.54
0.56
Open Open
pleura peric
VK
End-syst ole
Referenced to intact chest end-diasti)le
End-distole
Eigenvalues, Eigenvolumes, and Rotation Angles Referenced to End-Diastole of Current Beat
Circulation Research/VoJ. 50, No. 4, April 1982
584
to the end-diastolic screw positions with the chest
intact at the beginning of the experiment.
Rotation
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FIGURE 9. Definition of the spherical coordinate system fixed in
the dog used to describe the axis about which the left ventricle
rotates. The dog's head points toward the north pole, its chest
toward 0° longitude.
The rotation matrix R specifies the axis and angle
of the rigid-body rotation of the entire left ventricular
cavity as it beats. As with the principal directions of
deformation, we will describe the orientation of the
axis of rotation in a spherical coordinate system.
However, unlike the deformation, which we described
in a coordinate system that rotated with the heart, we
must describe the axis and angle of rotation in a fixed
coordinate system that does not rotate with the heart.
Figure 9 shows this coordinate system, which is fixed
in the dog.
With the chest closed, the axis of rotation did not
systematically change with respect to the dog during
the cardiac cycle. Figure 10A shows the axis of rotation during systole for the beat depicted in Figures
7A and 8. Figure 10B shows the general orientation
for the axis of rotation during systole for all the dogs.
Figure 8 shows the pattern of rotation with the
chest intact. Recall that, by defintion, 0 = 0 at enddiastole. Very little rotation occurs during diastole.
Early in systole, there is a small left-hand rotation in
Head
Toil
FIGURE 10. (A) Orientation for the axis of
rotation during systole for the beats shown in
Figures 7A and 8. (B) Orientation for axis of
rotation during systole for all dogs with their
chests intact.
HEAD
SPINE
TAIL
Walley ef ai./Left Ventricular Dynamic Geometry
585
I2r
o
o
T3
eg
o
DC
0-
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0.6
0.8
1.0
1.2
Ekjenvolume
FIGURE 11. The rotation of the heart in the chest generally follows
ejection, as can be seen from this plot of rotation angle vs. eigenvolume for the beat shown in Figures 7A, 8, and 10A.
most dogs that coincides with the a-wave, followed
by a larger right-hand rotation (to a mean maximum
of 7.0, then falls to a mean of 4.6° at end-systole)
(Table 3), that generally follows ejection (Fig. 11).
Isovolumic relaxation is accompanied by a rotation
back to the diastolic baseline.
Opening the pleura and opening the pericardium
did not consistently change the magnitude of rotation
during the beat (P > 0.25) (Table 3).
By contrast, opening of both the pleura and pericardium produces significant (P < 0.0005) changes in
the orientation of the heart in the dog's chest. (Recall
that the dog is lying on its back). We quantified this
change by computing the rotation axis and angle with
respect to the closed chest end-diastolic screw positions rather than the end-diastolic screw positions of
the current beat. Table 3 shows that, at end-diastole,
opening the pleura caused the heart to rotate a mean
of 18.6° from its end-diastolic orientation with the
chest closed. Opening the pericardium caused a further significant rotation to a mean of 31.4° at the enddiastole. Figure 12 shows the orientation of the axes
of rotation describing how the heart changes its enddiastolic orientation upon opening the pleura or pericardium.
OPEN PLEURA Axis of Rotation
HEAD
'SPINE
TAIL
FIGURE 12. Orientation of axis about which
the left ventricle rotates from end-diastole
with the chest intact and end-diastole after the
pleura (A), and the pericardium (B) have been
opened.
OPEN PERICARDIUM Axis of Rotation
HEAD
SPINE
TAIL
Circulation Research/Vo^. 50, No. 4, April 1982
586
Discussion
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Implanted markers that move with the myocardium, such as tantalum screws, provide data on how
the heart is deforming as it contracts and relaxes
(Sandier and Alderman, 1974). The lack of an integrated, theoretically sound approach to interpreting
these data, however, has prevented investigators from
extracting much of the physiologically meaningful
information from them. In particular, investigators
have typically made simplifying assumptions about
chamber shape, chamber symmetry, and location of
the markers with respect to the principal directions of
deformation when making statements about left ventricular dynamic geometry. The analysis presented
here avoids these restrictive assumptions. We only
need the assumption that the left ventricular cavity
deforms homogeneously. This assumption is implicit
in most earlier studies.
Unlike most assumptions made in the process of
describing the left ventricle's dynamic geometry, we
could explicitly test the validity of the homogeneity
assumption. Most important, there is close agreement
between the predicted and observed screw positions,
with no systematic errors (Table 1). The facts that
eigenvolume varies linearly with the cavity volume
(Fig. 4) and that our results do not depend on the
point used to remove the translational component of
the heart's movement in the chest add more support
to the validity of assuming that the left ventricular
cavity deforms homogeneously.
Our analysis used all the markers to estimate the
transformation T—and so the dilation D and rotation
R—throughout the cardiac cycle. We found not only
that the principal directions of dilation and axis of
rotation did not lie along any line connecting two of
the screws, but also that the orientation of these
principal directions may change with respect to the
ventricle's anatomy during the cardiac cycle. An anatomically fixed direction will reflect different contributions of the three different principal directions of
deformations at different times (analogous to Line III
in Fig. 3).
We attempted to distribute the screws evenly
around the ventricular cavity so that inhomogeneities
in ventricular motion would not prevent obtaining
globally representative result. For example, if all the
screws had been implanted at the apex, then any
difference in deformation of the apex with respect to
the rest of the ventricle would have been overemphasized. Apart from this consideration, the consequence of the theoretical derivation is that endocardial
markers need not be placed in perpendicular planes
or other fixed configurations. In fact, T, in theory,
does not require the precise placement of screws at
any specific anatomical sites. By repeating the analysis
with different sets of markers, we tested this assumption in several dogs that had many tantalum screws in
place. The differences in the results were less than the
measurement error.
With one important exception, the screws included
in the analysis did not affect the results. If one includes both of the aortic screws in the computation of
T, the result is a marked deterioration in the quality
of the fit between the observed and predicted marker
positions, together with a decrease in the amount of
shortening observed in each of the principal directions. The result occurs because the aortic valve ring
does not deform homogeneously with the rest of the
cavity. This result is itself indirect evidence in favor
of the assumption that the cavity deforms homogeneously, since if one introduces a significant inhomogeneity (e.g., the aortic valve ring), it produces a
noticeable deterioration in the fit.
Opening the chest and opening the pericardium
affect ventricular function (Rushmer, 1954a; Leshkin
et al., 1972; Glantz et al., 1978; Stokland et al., 1980)
and, therefore, possibly affect ventricular dynamic
geometry. Even in chronic studies, Rushmer (1954b)
has noted the formation of large pleural and pericardial adhesions which may alter dynamic geometry.
By avoiding constraints of precise placement of wall
markers, surgical manipulation was reduced to inserting a steerable catheter through a carotid artery.
Therefore, we could obtain measurements without
opening the chest and we could quantitate the effects
of opening the pleura and the pericardium. The resulting data revealed that these manipulations led the
heart to change significantly its orientation in the
chest, but not end-diastolic or end-systolic eigenvolume or the pattern of contraction.
Stokland et al. (1980) observed similarly modest
changes in chamber geometry with the pericardium
open; they concluded that the major acute effect of
opening the pericardium occurred on the pressures,
not on the dimension. Our failure to find a change in
pressure may have been due to limitations in the
pressure amplifiers, the time delay between taking
the open pleura and open pericardium data, or differences in the dogs' volume status due to the slow
infusion that was maintained during the experiment.
The transformation T can be used to estimate the
change in any ventricular length, area, or shape simply
by multiplying T by the matrix containing the points
defining that length, area or shape. This information,
together with the relative volumes (eigenvolumes),
could provide a tool to help clarify the significance of
the results of common experimental methods (such as
measurement of chords or segment lengths using
pulse transit time ultrasonic crystals) or clinically
useful techniques (such as angiography or echocardiography).
Our measurements, being more fundamental than
those obtained with other techniques, offer a firmer
basis for earlier notions of cardiac geometry. There is
a relatively stable long axis and perpendicular to this
axis, the relative deformations (quantified by the eigenvalues AAP and XSF are larger and similar.
We implanted the screws endocardially, outlining
the left ventricular cavity. Therefore, this study describes chamber dynamic geometry. The approach
outlined here could be modified to study mid-wall or
Walley et ai./Left Ventricular Dynamic Geometry
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epicardial dynamic geometry, depending on screw
placement. Inhomogeneities of segmental differences
in wall motion of normal and pathological hearts can
be studied using the same analysis. One would simply
examine smaller segments of the wall with this linear
analysis, analogous to fitting a curved line with small
straight line segments. (T can be computed from as
few as three screws.) As the region of the ventricular
cavity which is assumed to deform homogeneously
gets smaller, the assumption of homogeneous deformation gets even more accurate. In fact, our work was
motivated by the use, by Meier et al. (1980a, 1980b),
of a similar analysis to study the deformation and
rotation of very small regions of the right ventricular
myocardium. They showed that the principal directions of deformation appear to line up with local fiber
orientation and that rotations appear consistent with
the combined effects of myocardial fiber orientation
and depolarization pattern.
Our results, together with previous experimental
and theoretical work, suggest that, during systole, the
shape and rotation of the deforming ventricle depends
on the pattern of electrical activity (Hotta, 1967), and
the fiber orientation of the myocardium (Streeter and
Ross, 1980). During slow filling, the principal directions of deformation are ill-defined, and the rates of
change of all the eigenvalues are similar. This pattern
of deformation suggests a symmetric expansion,
which, in turn, suggests that the late diastolic ventricle
is reflecting the relatively isotropic elastic (or viscoelastic) properties of the myocardium. This fact may
explain why simple models of late diastole based on
the seemingly unrealistic assumption of a spherical
cavity comprised of homogeneous and isotropic myocardium still agree with experimental data (Glantz
and Kernoff, 1975).
This paper presents the first measurement of the
heart's rotation in the chest that did not require any
assumptions about the orientation of the axis of rotation. The results are similar to those obtained by
Mirro et al. (1979), who tracked a papillary muscle by
means of echocardiography. They also appear quantitatively and qualitatively similar to predictions of
torsion about the left ventricle's long axis predicted
by Arts et al. (1980) on the basis of the helical fiber
orientation of the myocardium. While we may be
measuring this torsion, there are two reasons for
doubting that this is the case. First, by assuming
homogeneous deformation we by definition exclude
torsion. Second, the axis of rotation, r, does not align
in any consistent way with the principal direction
associated with ALONGThe description of ventricular dynamic geometry as
outlined here yields important new information for
three reasons. First, the description is more objective
than earlier approaches since it does not rely on shape
assumptions, on orientation of the measuring device,
or on precise marker placement. The directions of
deformation and corresponding amounts of shortening are determined whether or not wall markers are
implanted at specific sites or in specific planes. Sec-
587
ond, although the use of linear transformations to
describe deformations of the left ventricle is new, it
is based in a large body of knowledge concerning
deformation of materials and spaces. The analysis and
results in this paper provide a step toward applying
methods of classical physics to obtain a better understanding of cardiac function. Third, and most important, the description of dynamic geometry follows
from a truly three-dimensional analysis of shape of
the entire ventricular cavity, rather than a one- or
two-dimensional analysis based on one or two measured parameters.
Although the approach described here applies well
to the normal heart, there are several important questions that need to be answered to determine the
generality of the results. Is the assumption of homogeneous cavity deformation tenable in the presence
of inhomogeneous local deformation, such as occurs
when part of the left ventricle is ischemic or infarcted?
What errors are introduced by torsion of the heart
about its long axis (Arts et al., 1980)? How would
abnormal patterns of depolarization affect the results?
Nevertheless, this approach has yielded a description
of how the normal heart contracts under various
physiological conditions that is more complete than
was possible with existing techniques. These results
can be simply interpreted in terms of cardiac electrophysiology and structure; they can, as well, provide
a standard against which to compare other experimental and theoretical approaches.
Appendix A
Least Squares Estimate of T(t) from Screw Coordinate
Matrices
Let XED and X(t) be the 3xn matrices of screw coordinate
matrices observed at end-diastole and time t. Then, from
Equation 3, let the 3x3 matrix T(t) represent a homogeneous
deformation (linear transformation) that predicts screw positions at time t from the screw coordinates at end-diastole
according to
X(t) = T(t)XED.
(A.I)
Let A be the 3xn matrix of deviations between the predicted
and observed screw coordinates
A = X(t) - X(t) = T(t)XED - X(t).
(A.2)
Therefore, the sum of the squared deviations (distances)
between the predicted and observed screw positions is
Q = 22apjp = trace AAT.
(A.3)
We now find the transformation f(t) which minimizes Q.
A is a minimum when all elements of the 3xn matrix
dQ/dT equal zero.
For simplicity, let X(t) = X and T(t) = T; then Equation
A.2 becomes
(A.4)
A = TXED - X
and
A'=X E D T'-X'
T
(A.5)
T
T
AA = (TXED - X)(Xl D T - X )
(A.6)
Circulation Research/Vo/. 50, No. 4, April 1982
588
AA T = T X E D X I D T T - XX^ D T T
T
- TX ED X + XX
AA
T
= T ( X E D X E D ) T T - (XXi D )T T
T T
- [(XXED)T ]
(A.7)
T
+ XX
(A.8)
T
Q will therefore be given by the sum of the traces of each
term in Equation A.8. Because the second and third terms
are transposes of each other, their traces are identical and
Q = trace [T(X E DXED)T T - 2(XXED)T T + XXT].
(A.9)
To compute dQ/dT we use the facts that if C and T are
square matrices
d (trace CTT)
dT
This routine returns the eigenvectors ordered according
to the magnitudes of the associated eigenvalues and with
arbitrary sign. It is necessary to compare each E matrix with
that computed from previous frames and permute the comumns of E to produce results in which a given eigenvalue
and eigenvector correspond to the same principal axis
throughout the cardiac cycle. We do this in two steps. First,
we ensure that the three columns of E define a righthand
system by computing | E |. If this determinant is negative
(indicating a lefthand system), we reverse the sign of the
last column of E. Second, we permute the columns of E to
maximize the trace of EEL T where EL represents a weighted
sum of the last three matrices of eigenvalues.
EL = 0.5 E-i + 0.3E-2 + 0.2E-3
_
(A. 10)
and if C is symmetric,
d (trace TCT T )
= 2TC.
dT
(A.11)
(E-i, E_2, and E_3 are initialized to the identity matrix at
end-diastole.) This procedure minimizes the three-dimensional angular change between the three orthogonal unit
vectors given by E and EL.
To find the axis and angle of rotatdom from R, first
calculate R from
Downloaded from http://circres.ahajournals.org/ by guest on May 5, 2017
Since XXED is a square (3x3) matrix and XED XED is a
R = TD1
symmetric matrix
dT
+ 0.
— 2T(XEDXED) — 2
f = XXEDCXEDX^D]"
1
(B.8)
(A.13)
The axis of rotation is the eigenvector of R associated
with eigenvalue 1. We compute the eigenvalues and eigenvectors of R using a norm reducing Jacobi type method
(Eberlein and Boothroyd, 1971) and identify the eigenvector
whose eigenvalue is closest to 1 as the axis of rotation, r.
We adjust the sign of t to make r •ft,positive, where
(A.14)
ft = 0.5r_i + 0.3r_2 + O.2r_3
(A. 12)
The best estimate, T, is the value that makes dQ/dT = 0:
0 = 2f (XEDXID) - 2 (XXED)
(B.7)
which is Equation 4.
(B.9)
as in Equation B.7.
The remaining two eigenvalues form a complex conjugate1
pair that describe the angle of rotation. Rather than use this
information to compute the rotation angle directly, it is
computationally simpler to multiply R by
Appendix B
Polar Decomposition of T
Given
ri
T=RD,
(B.I)
TT = DTRT
(B.2)
hence,
(B.10)
nr2 + r3
Lnr3 which represents a 90% righthand rotation about r
[ T , then make use of the fact that
and
trace R = 2 cos 6 + 1
T
T
T
T T = D R RD.
(B.3)
However, R represents a pure rotation (an orthogonal transformation), so R"1 = RT, and Equation B.3 becomes
T T T = D T D.
trace PR = 2 cos(0 + 90°) + 1
D is a symmetric matrix, so D = D, and
(B.12)
and
(B.4)
T
DD = D 2 = T T T.
(B.ll)
- 1 trace PR"1
0 = cos
90°.
(B.13)
(B.5)
2
Since D is symmetric, D is symmetric.
D 2 can be diagonalized to the matrix A2 by computing a
similarity transformation given by the matrix E.
A2 = E'D'E.
(B.6)
2
E also diagonalizes D, so the eigenvectors of D (and D ) are
the columns of E. The eigenvalues of D are just the square
root of the eigenvalues of D 2 , which are also the three
diagonal elements of A2.
We compute the eigenvalues and eigenvectors of D 2
using subroutine EIGEN in the Digital Equipment Corporation Scientific Subroutines Package which uses the Jacobi
method as adapted by Von Neumann (Ralston and Wilf,
1962).
We thank William Parmley, Julien Hoffman, David Bristow,
John Tyberg, Robert Willett, Bruce Brundage, Jonathan Melvin,
George Meier, and Harold Sandier for their useful suggestions
during our work and their thoughtful criticism of the manuscript.
We thank Referee 2 for his thorough review and suggesting the
derivation in Appendix A. We thank Donald Holmes for helping
digitize the films. Dean Forbes for getting the results into our
computer, Gordon Dower and David Berghofer for suggesting the
Aitoff equal-area projection of the globe and providing software to
draw the pictures, Ed Stokes for writing usable and accurate
programs to do the analysis. We thank Jim Stoughton and Rich
Sievers for technical assistance. We thank Larry Wood for making
it possible for Keith Walley to visit the CVRI, Mary Hurtado for
typing the manuscript, and Mary Helen Briscoe and Rene Meier
for preparing the illustrations.
Walley et al./Left Ventricular Dynamic Geometry
Funds for the support of this study have been allocated by the
NASA-Ames Research Center, Moffet Field, California, under
Interchange No. NCA2-OR665-905, by NIH Research Grant HL25869, Program Project Grant HL-06285, Training Grant HL-01792,
and by the San Francisco Division of the University of California
Academic Senate. Dr. Glantz holds an NIH Research Career Development Award.
Address for reprints: Stanton A. Glantz, Ph.D., Associate Professor of Medicine, University of California, San Francisco, Division of Cardiology, Room 1186M, San Francisco, California 94143.
Received November 20, 1980; accepted publication January 14,
1982.
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Circ Res. 1982;50:573-589
doi: 10.1161/01.RES.50.4.573
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