Three-dimensional Myocardial Deformations - Semantic Scholar
Cardiac Walter C. O’Dell, Elias A. Zerhouni,
PhD MD
Christopher C. Moore, BSE Elliot R. McVeigh, PhD
William
#{149}
C. Hunter,
#{149}
three-dideformaof parallel(MR)
M
Model deformations were reconstructed with a 3D tracking error of less than 0.3 mm. Error between estimated and observed onedimensional displacements along the tags in 10 human subjects was 0.00 mm ± 0.36 (mean ± standard deviation). Robustness to noise in the tag displacement data was demonstrated by using a Monte Carlo simulation. RESULTS:
CONCLUSION: The combination of rapidly acquired parallel-tagged MR images and field-fitting analysis is a valuable tool in cardiac mechanics research and in the clinical assessment of cardiac mechanical function.
breath-hold
netic resonance cardium, MR.
Heart, (MR), 511.1214
function, physics,
51.91 511.1214
imaging
tools
search
for
Mag#{149} Myo-
#{149}
of the
of this
the
1995;
195:829-835
was
at many
of the
in both The
neobjec-
to measure
non-
three-dimensional
field within time points
deformation
wall
study
wall settings.
study
invasively
(8,9) are pnomis-
heart
clinical
(3D)
the heart in the heart
cycle.
Myocandiab tags are regions where the magnetization has been perturbed before imaging and that, therefore, produce a signal intensity difference relative to that of nomtagged regions for a time proportional to Ti. Because the
tags
the the
magnetization deformation
reflects tissue
result
from
perturbations
of
of the tissue itself, of the tags accurately
the motion of the underlying (10-12). Special techniques
are
needed to reconstruct the 3D motion of the heart from MR image planes that are fixed in space, because different
sections
of tissue
are
sampled
achieved
tiom from sets
by combining
both
of tagged
Previously
longheart
and
short-axis into
for the presented
infonma-
images
a umi-
3D displacemotion
Figure
1.
MR images
(7.0/2.3,
15#{176} flip angle)
of an in vivo human heart of a healthy 30year-old male volunteer obtained with the parallel-tagging and -imaging protocol. The progression (from left to right) through three phases in the cardiac cycle is shown: early,
middle,
and
late systole.
Two
axis images and one long-axis shown; each displays tag lines ent set of mutually orthogonal
cardiac
short-
image are from a differtag planes.
at
different times. The movement of the heart through short-axis image planes, known as cardiac through-plane motion, is typically 10 mm at the base of the left ventricle (13). (This was confirmed with the analysis of 10 healthy human subjects discussed in this antide.) Correction for this is crucial, even for two-dimensional analysis of wall deformation (14). This correction
fled expression ment field. Radiology
(MR) imagand fast,
nominvasive
and
tive
was
terms:
resonance tagging (1-7)
ing ing
Deformations: Field Fitting
AGNETIC
function
MATERIALS AND METHODS: Displacement information in the direction normal to the undeformed tag planes was obtained at points along tag lines. Three independent sets of one-dimensional displacement data were used to fit an analytical series expression to describe 3D displacement as a function of deformed position. The technique was demonstrated with computer-generated models of the deformed left ventricle with data from healthy human volunteers.
Index
PhD
#{149}
Three-dimensional Myocardial Calculation with Displacement to Tagged MR Images’ PURPOSE: To reconstruct mensional (3D) myocardial lions from orthogonal sets tagged magnetic resonance images.
Radiology
recon-
I From the Departments of Biomedical Engineering (W.G.O., C.C.M., W.C.H., E.R.M.) and Radiology (E.A.Z.), Medical Imaging Laboratory, 407 Traylor Bldg. 720 Rutland Aye, The Johns Hopkins University School of Medicine, Baltimore, MD 21205. Received October 31, 1994; revision requested December 8; revision received January 30, 1995; accepted February 6. Supported in part by grants HL45090 and HL45683 from the National Institutes of Health, by a fellowship from the Merck Sharp and Dohme Corp (C.C.M.), and by a Whitaker Foundation Biomedical Engineering Research Grant. Address reprint requests to E.R.M. RSNA, 1995
struction schemes for MR imaging tag data require identifiable points within the images such as intersections be-
tween
tags
(11,12),
intersections
tweem tags and myocandiab (14), or points along striped (i5).
Analyses
intersection valuable
with
data, information
intervening
portions
such
becontours tag lines
sparse
tag
however, neglect contained in the
of the
tag lines.
Accordingly, the descriptions of deformation that result suffer from poor spatial resolution. These methods also require images with high spatial neso-
lution
both
in the
frequency
and
phase directions, which results in acquisition times of longer than one breath hold and thus limits their clinicab applicability.
Abbreviations: RMSD standard
=
3D = three-dimensional, root-mean-square deviation, deviation of the error.
SDE
=
In this study, we developed a method for reconstructing the 3D deformation field of the left ventricle from tagged MR images with use of position and displacement infonmation along the entire length of each tag. This method relies on accurately defined tag profiles (16) rather than on poorly defined heart contours (17). This
approach
eliminates
the
need
of this
agation
method.
properties
by means
of a Monte
with human deformation.
MATERIALS
A typical
human
consisted
Field
to the readout diat eight to 12 time
coordinate tive human
define
were
Fitting
parameter
processed by means of software package (18) to
along
positions
was
In general terms, field fitting is a technique for estimation of the value of some parameter throughout a particular region of interest, given discrete samples of that
comeCartesian
axes. Portions of a representaimage set are shown in Figure
1. The images a semiautomated
for
point
in and around
that region.
the tag lines and
around the endoeardial and epicardiab heart contours. A typical tag and contour data set for a healthy human after image processing is shown in Figure 2. For a tag separation in the reference state of 6.0 mm, approximately 12 tags were produced in the myocardium in each set. At a point separation along each tag of 1 mm, this produced more than 4,600 raw data
prop-
simulation
and
Sets set of an in vivo
of three
sets of
multiphasie images: one in the cardiac bong-axis view and two in the cardiac short-axis view. The short-axis image sets consisted of stacks of six or seven contiguous parallel sections and the long-axis image set consisted of six sections prescribed radially around the cardiac long axis with
a. Figure
b.
2. (a) Typical 3D short-axis tag data set for x-displacement in an in vivo heart of a healthy volunteer. Image shows the appearance of the tags and the contours near end systole on seven short-axis image planes. (b) Typical 3D contour data set and the estimated prolate spheroidal centroid (upper gray circle) and apical focal point (lower black circle).
Tag
at t=t
Surface
Image
xLLY
3. Figures
3, 4.
ten detection tag
surface
(3) Depiction
of tag points within
the
heart
of a short-axis
at 1-mm wall
Radiology
#{149}
image
intervals. with
an automated tag detection algorithm. figure at the upper left shows the region
830
In
displacement field fitting, the parameter of interest is the 3D displacement vector, and the samples are the values of one-dimensional displacement measured at points on tags in the deformed heart wall. Although field-fitting is generally applicable to any motion-detection method, it has been applied here to the analysis of three independent one-dimensional sets of displacement measurements from parallel-tagged MR images. This type of data is depicted
METHODS
Data 3D tag data
ented perpendicularly rection was obtained
other
tested
geometry
AND
heart
then
Carlo
cardiac
Parallel-tagged
Noise
were
points, of which every used in the fitting.
frames throughout systole. The three sets of tag planes sponded to three orthogonal
accurate simultaneous measurement of displacement in two dimensions and permits computation of the 3D deformation gradient tensor at any point in the heart wall. The method was tested on a cornputer-generated model of a prolate spheroid that undergoes deformations that simulate those measured in the beating heart. Finally, paralleltagged, breath-hold cine data sets from 10 human hearts were analyzed by means
an angular separation of 30#{176}. For each image set, a stack of parallel tag planes ori-
four
at some
(4) Deformation vertical
represents of the heart
image
time
4. initial
after
tagging.
of a tag plane planes.
The
the one-dimensional depicted.
enlarged
Image
Inset
depicts
into a tag surface dots
displacement
on
these
an
enlarged
view
in 3D. The bold curves
associated
are
with
the
of one
curves tag
points
Plane
Plane
2
deformed
tag
line
are the intersections generated
the nth tag point.
4
by
af-
of the
means
The rectangle
June
of
in the
1995
in Figure image
3, which
and
illustrates
a deformed
a short-axis
set
late
The key concept of the method is shown in Figure 4. In three dimensions, each observed tag line represents the tersection
the
of a deformed
tag
two-dimensional
surface initially tag by
four
within parallel
plane,
the
lines between heart
For
of an depicts
a is cut
along tag
point
placement the
and
coefficients the
of digital
sampling
reconstruct
position
data
field.
on
and
Fourier
Here,
nectionab
displacement
to solve for an the displacement tion, independently
compute
data
directions.
displacement determined,
the
to
the
entire each set
was
Once
eliminates
a p
=
(In
evaluated ix,,, was
the large-scale stretches and
fitted
=
italic
italic;
by
using
for
efficiently
spheroidal
series
surements
of displacement
prolate
displacements
spheroidal
x
x
fcosh +
jy
=
point
heart,
the
bowed
by a probate
spheroidal
residual
displacement
fields.
shows
first-order
the
system;
prolate
longitudinal
coordinate,
means
coordi-
z was moThis
andfis related
the focal
sin (4) cos (0) IIX (0) l4
fsinh
(X) sin
(4)
(0) 0,
(X) sin
x
=
=
sin
(4k) sin
(5)
(0) iX
fsinh
(X) cos (4i) sin
(0) 114
+
fsinh
(A) sin
(4) cos
(0) 110,
liz
which as
can
=
fsinh
()
cos
(It)
fcosh
(6)
IIX
(A) sin (di) 1k,
where
Sx
be
expressed
bx
in vector
=
(7)
length. coordinates
notation
I BA, BA
[x,by,1z], J is the Jacobian matrix transformation.
circumfer-
=
(8) [1iX,,10],
=
for
this
and
coordinate
For a, b, and
of
y
de-
(4i) cos
4 is the
to Cartesian
directional
(X) cos
coordinate 0 is the
and
are related
fol5
component,
infini-
and
fit of the
spheroidal
A is the radial
the
The
+
the
Figure
onto
axes.
following
(It)
fcosh
for gen-
fit was
the measured
in Cartesian
in the
Cartesian
meaCarte-
fsinh
-
de-
from in the
coordinates
of the
=
density param-
projected
spheroidal
by means rivatives:
data
of the prolate
directions, were
-
to
this
expansions
prolate
describe
numtagging
direction.
tesimal
expected
were
,
typical
for
coordinate
fitting method)
deformations
and
50 free-fitting
scalars,
to a power series in prolate spheroi-
curvilinear
per
local
value
the
appropriate generated
all tag equations coefficients,
singular
(a least-squares
and
displacements
=
value, the
fitting
order, a,
For
font;
and
radial
fitting
coefficients,
sian coordinate p
de-
x displacement By considering
dal coordinates-To
were genstate.
bulk shears.
bold
most
(20) eters
be written are
angular
To fit for the coefficients
x
and
Legendre
N is the
sequentially.
was
un-
form
vectors
displacements and p vectors points, a series of simultaneous was created and the unknown
by
to remove and linear
the can
bold
and the measured.
by using
used tions
4),
and
plain italic.) For each sampled tag point (x,,,y,,,z,), each term in the vector p was
was computed
power
article,
a lowercase
are
an initial step, a first-order series expansion in x, y, and
fit
[a,al,a,,a1]
=
uppercase
angle,
notes-As
this
a
this
with
tensors,
ential
in Cartesian
a,, took
where
These
to a power series
(x,y,z)
associated
function,
and imaging geometry and the typical image signal-to-noise characteristics, we determined that a fit with N = 1 and L = 4
expression
in x, x(x,y,z)(Fig
a1x + a2Y + a3z. In vector notation,
system. time frame
Fitting
fo-
+
Fitting
three
bered
Cartesian The
of position
coefficients,
vector field in an any coordinate The displacement field at each tag planes reference
problem.
P71 is the
order, L is the the unknown
sphe-
singularity
the
(19). Independent series expressions the y and z displacement fields were erated and fitted analogously.
the
polynomial
which
a mathematical this
where
or apical
used
of any
independently
centroid
wall;
pro-
fixed
deforma-
the myocardium,
heart
composition
in the deformed heart wall. At higher complexity, displacement data from multiple directions can be used simultaneously to solve for the full displacement
the time at which the erated as the universal
the
bulk
a, were
field expresthey were used
3D displacement
the
the
left unidi-
interpolation function for field in that one direeof the motion in the
Cartesian
independent sions were
that
analysis
the case,
create
within
of the
by aligning
axis of the prolate system at each time
cause
would
noted
to use
described
throughout In the simplest
may
efficiency
large
cus to intersect
[1,x,y,zl.
one-dimensional
this expression
In addition,
x
for
expression is analogous
frame.
a0
field dis-
to solve
This
a continuous,
displacement ventricle.
other
displacement the tag point
of a series
describes
signal.
of the by using
and long coordinate
known
the deformed tag surface, the component of its displacement vector that was normal to the tag plane (ix, in Fig 4) and its position were measured. Reconstruction was performed
centroid roidal
as a function
tag
and epicar-
each
the
fitting
for the displacement
In each
marked
the endocardial
contours.
tag
that
increased
spheroidal
tions
The
wall
planes.
were
inwith
plane.
heart
image
tag points
dial
surface
image
results from the deformation flat tag plane. The figure
surface
greatly
of tags.
fsinh
(1) sin
fsinh
(X) sin
() cos (4) sin
c, defined as the vectors of coefficients for the 1IX, 64, and
unknown
(0),
( )
o
(0),
(2)
vector
expressions,
generating
and
respectively,
of 50 series
terms
function,
and
p, the
derived
M
a
=
from
the
b
p,
.
0 = c p (the a vector for the Cartesian fit and the a vector here are different). and
z =fcosh
For each
prolate
spheroidal
displacement field, prolate spheroidal
created
by using
analogous
,0
(4).
(X)cos
spherical
1a1P”uI
expansion (X4,0)
with was
function harmonic
cos
(mO), (mO),
>
series:
0
0,
case
(4)
in which
the
by the three
tag normal vectors) are aligned with the prolate
4X
in in
the
(as defined
dinate (1)-(3), ments
[cos
sin .
For
coordinate
a series coordinates
a generating
to the
(3)
Cartesian
axes
orthogonal centered spheroidal
sets of and coor-
system as described in Equations the measured Cartesian displacecan be expressed as fcosh
(X) sin
(4) cos (0) (a
.
p)
+
fsinh
(X) cos
(4) cos (0) (b . p)
-
fsinh
(X) sin
(4)
sin
(0)
(c
.
p)
(9)
y
Focal
Point
Figure 5. Prolate spheroidal coordinate tern. Surfaces of constant X are ellipsoids surfaces 4i
of constant
are hyperboloids.
Vf1IIYThD
1Q
sysand
longitudinal coordinate const = constant.
#{149} KTiiriIior
021
x
a(Jp)+
=
+
y
b(Joip) (Jo’p)
C
u
=
(10)
Px’
a(110p)+b(111p)
=
c (J12p)
+
(11)
a
and
Xz
a
=
(J2p)
+
where .
. .
now and
,C49]
JolP49,J0Po,
were
a .
defined
method
.
b
.
.
(J2IP)
,a4g,b0,
p .
(12) ,b49,C0,
. .
J())P49,J01P0
vectors
analogously.
of solution
series
.
[Joopo ,J02P49]. The
Px . .
[a0,
=
+
(J22p)= a
C
p
Similar
and p. to the
of the Cartesian
coefficients
above,
power
each measured
displacement value was collected into a single row vector, .x, and the corresponding p vectors were inserted as columns in a matrix P. The unknown coefficients, a, were then solved for by using the equation x = a P. In this way, all the unknown coefficients can be solved for simultaneously by using all available tag data and singular value decomposition. Although, for finite deformations, the relations between the displacements
and
in x, y, and
0 are not
least-squares
z and
those
exact,
the simultaneous
fitting
will
always
,
in X, result
for
each
time
frame
in the
data set. Because the heart geometry changes throughout the cardiac cycle, a new prolate spheroidal coordinate system was
calculated
for
every
time
frame
on
the
basis of the measured epicardial contour point data. This was done first by estimating the three coordinates of the centroid
and
the
focus
three
coordinates
(point
on
the
apex, at a distance the centroid) and squares
and
optimal
apical
cardiab Within
focus
were
until
then
X value
at this
for the
given
contour loop, the
toward
and
the
of the
dinate centroid that Equation for
the
set of epi-
by the normal
of myocardial
state at a material point at any given imaging time
is fully
described
mation
gradient
FdX,
=
point
means
defor-
in the
deformed
state
(21). F
from
calculated
the
and
X
state
displacement
gradient tensor Vu, which was computed numerically by taking partial derivatives of the displacement expressions with respect to the three coordinate directions. Because the terms in the series expansion were computed on the basis of the coordinates of the tag points in their deformed state, Vu described the motion from the deformed state back to the undeformed
when
the tags were created. Vu requires F = (I
I
is the
Thus, Analytical
an
identity
tensor.
To
The
transposition.
coordi-
spheroidal
coor-
neces-
of the new spheroiwith those defined
vectors
to the undeformed
through
space.
A mesh
points
regularly
spaced
The
cardiab
the
ated
by
the
aries
between
achieved
basal
and
as a function nates.
Axel
image endocardial
section
model
(5), which by
contour
constructed.
coordi-
described
by Young
was
on 3D strains
based of cine
coordinates
of the initial deformation 0
apical
ti
and
radiography
(X,,0)
coordinates parameters =
0
+
+
Cfsinh
of
as a function
(A,,O) and the were as follows:
Bfeos
A +
(4)
(A),
Dfeos
+
=
Efsin
+
(13)
(CF) (t)
sinh
(A),
(14)
and
was
Vol
epipoints
was
deformative
means
first
the left ventricle
defor-
spheroidal
axisymmetnically
deformed
0 was
locations
a known
computer-gener-
model
bongitudimost
with
the anterolateral free wall of the left yentnicbe with implanted metallic beads (22). The governing equations that describe the
seeheart.
at the
method
a realistic, to prolate
measured
sections and radially by and epicardial contours. of the contour bound-
by fitting
the first time late spheroidal
and
and
imaging of the
constrained
most
short-axis image the endocardiab The interpolation
,
an
cylinder
of 96 material
locations
were
Case
deformation
nates
wall in the
in X,
the
field,
We adapted
and 3D displacewere evaluated as they moved
basis of the and the shape
material-point
time frame
as
the heart
first imaging time frame ments and deformations over time at these points
selected on tion locations
from sets of im-
ages with arbitrary tag normal vectors, long as three dimensions were spanned. Computing strains at material points-A material point is defined as an infinitesi-
within
test
mation
For both the mathematical and experimental models, a set of material points
was defined
Test
-
and
focal
spheroid
The
of its
in the reference
was
sets for the prolate
tensor, F, according where x are the coordinates
are its coordinates
mitted
Radiology
by
tissue.
tag data
with added Cartesian motions were modeled for in each view. The angles of tilt are due to simulated
mechanical in the heart
nally
#{149}
modes planes
mal volume
tag planes. For the transformation to the new coordinate system x’, defined with the rotation matrix R such that x’ = R . x, Equation (8) becomes Bx = R ‘J1IX. This modification was used throughout the derivation. In addition, this formalism per-
832
(b) long-axis
adjusted
axes system
3D reconstruction
deformation and 10 tag error.
and
Lagrangian finite strain tensor, E, can then be computed from E = y2(FTF - I), where the superscript T denotes matrix
radial
transformation
the
(a) short-axis
Physiologic image sections prescription
where
and apical focus required (8) be modified to account
coordinate
to align dal coordinate sary
prolate
Simulated
to compute F from inverse operation:
nate A were derived that produced a prolate spheroid with the smallest fitting error to the measured left ventricle contour points. Variation
model. seven image
state
points. six coordi-
of centroid
coordinates
the
centroid
independently
a combination
point
axis,
of a focal length from by computing a least-
left ventricle an iteration
nates
of the apical
long
b. 6.
to dx of the
in
the optimal fit of the given displacement data to the series expansion. The fitting procedure was performed independently
a. Figure
(1.)
of
frame to a fourth-order procoordinate expansion for X of the two angular coordi-
=
4/3-rf
=
[Vol
Vol
+
where
tor, B
A is the is the
axial
torsion
cosh (A)
-
(X) sinh2 Vol
(X)
(‘endo)]
(15)
(‘endo)’
rotation around
at the the
equa-
central
June 1995
Bulk motions and linear deformations that incorporate ellipticalization (the change in cross-sectional shape of the left ventricle
from
a circle
at end
diastole
computed tribution trajectories
to an
ellipse at end systole) and shears in x, y. and z were then superimposed on the axisymmetric modes in the manner of Arts et
al (23). The magnitudes
of these
additional
modes were chosen to provide the overall model with approximately physiologic rigid body motions and to reproduce those deformations observed by Arts et al (23). These deformation values are listed
in Table
2. Image
prescription
error,
which
occurs when the central axis of the imaging volume is not coincident with the long
axis of the left ventricle,
was also simu-
bated with representative set about the x axis and
values of 5#{176} offof 8#{176} offset about
the y axis. Seven
were tion
image
sections
simulated between
and
for each adjacent
10 tag planes
image
set. Separa-
tag points
along
a
given tag was fixed at 2 mm. Throughout the entire 3D data set, this produced 2,400 tag points. deformation model
Human
Data
The prolate spheroidal is shown in Figure 6.
imaging
Sets
sequence
with
parallel
left side of the chest. The imaging parameters
per Figure (7.0/2.3,
7.
Shortand 15#{176} flip angle)
long-axis MR images of a human heart ob-
tamed in a healthy, 21-year-old male teer at 228 msec into systole overlaid point fit.
locations
predicted
the
C
axial
stretch,
is the
transmural
E is the
nab shear,fis
twist,
coordinate
volume
D is the
Monte
within
of the pro-
system,
a prolate
Vol(X)
spheroidal
shell of constant X, and endocardial measurement. in the radial component, to a volume-conserving
“endo” denotes Deformation A., was limited (incompressible) mode by setting a = 1. The deformed short-axis endocar-
contraction
initial
and equatorial
I sinh(X,fld0), realistic
radii,fsinh(Aefldo)
were
heart
chosen
geometry
ejection fraction of spheroidal deformation listed in Table 1.
Volume
195
Number
#{149}
(35
msec
time
one signal views resolution).
was
Carlo
tuned
to achieve
a 180#{176}
Simulations
and
to approximate and
resulted
in an
50%. The prolate parameters are
3
A representative human data set was fitted; the reconstructed displacement field and heart geometry were then used to generate a physiologic noise-free paral-
beb tag data
set. The effect
tag point position on the point tracking predictions
by using
a Monte
One-dimensional with 0.25-1.00-mm
Carlo
were added to the sets, and the resultant trajectories terial points mean-square
of uncertainty
in
final materialwas evaluated
simulation
Gaussian standard .x, y,
the
formation
were
prolate
model:
evalu-
spheroidal
tracking
de-
and
fitting.
The 3D tracking performance was evaluated by comparing the estimated material-point trajectories denived
from
the
tiom
to the
exact
from tions
field-fitting
neconstruc-
solution
the deformation (13)-(15). The
computed
model of Equamean absolute
value and the standard deviation the tracking error were computed the set of 96 material points. The
of for fit-
ting performance computing the
was standard
evaluated deviation
by
the
around
a mean
error
(SDE;
of
error
at
The
SDE
model
Deformation of the
tag data
Model
fit to the
points
2,400
was
0.095
mm.
This is on the order of the expected uncertainty in the determination of tag point displacements for typical in vivo data sets (16,17). This implies that the fitting error for a human data set will be dominated by the uncentainty in the tag point displacement
data
rather
than
by error
in recom-
structiom of the displacement field at the measurement locations. The 3D tracking error for the set of 96 material points was 0.28 mm ± 0.i6
(mean ± standard deviation). The relatively large value of the 3D tracking error ( 0.3 mm) compared to the SDE of the fit ( 0.1 mm) suggests
transmural-longitudi-
the focal length
spheroidal
is the
dial
frame
with
Mathematical Results
see-
The tagging pulse consisted of five nonseleetive radio-frequency pulses with relative amplitudes of 0.7, 0.9, 1.0, 0.9, and 0.7 separated by spatial modulation of magnetization (2) encoding gradients to achieve a tag spacing of 6 mm. The tagging tip angle flip angle.
axis,
late
from
volun-
with tag 3D field
movie
ated
of the displacement
performance
with the
tion were a repetition time of 7.0 msec and an echo time of 2.3 msec (7.0/2.3, fractional echo), a 15#{176} flip angle, 110 phase encoding steps, I .25 x 2.9 x 7 mm voxels, acquired, and five phase-encoded
measures
ments were made, whereas the tracking performance reflects both the quality of the fitting and the interpolatiom of the displacement field between the measurement locations.
tags
for each
Two field-fitting
variations in the displacement field the locations where the measure-
was used (8). Eighteen acquisitions were performed in each subject during a breath hold. Volunteers underwent imaging at
end expiration in the supine position a flexible surface coil wrapped around
RESULTS
of 0) between the estimated and the measured one-dimensional displacements at each tag point. Thus, the fitting performance reflects the ability of the reconstruction to account for the
Ten healthy human volunteers gave informed consent and underwent imaging with a i.5-T scanner (Sigma; GE Medical Systems, Milwaukee, Wis). A cine breath-
hold
for each point from the 3D disof the cloud of estimated point around the expected location.
(19).
noise profiles deviation and z data
3D material-point
were computed for the 96 mafor 100 trials. A single rootdeviation (RMSD) value was
that further improvements to the meconstruction cam be made by increasing the tag density and number of fitting parameters. The corresponding
circumferential material points a mange
strain were
of -0.25
errors 0.006
to 0.06,
the
at the ±
nab strain errors were 0.0003 for a range of -0.23 to -0.08, and rnidwall radial strain errors were 0.017
± 0.026
for
a mange
96
0.012 for bongitudi± 0.0070
the
of rnidwall
Radiology
833
#{149}
DISCUSSION
radial strains of 0.18-0.52. The high accuracy of the strain estimations, even where 3D tracking errors were _
0.3 mm,
is due
to the
correlation
of
tracking errors between neighboring myocandial regions. These errors and standard deviations are well below the threshold necessary to detect motion abnormalities in the isehemic heart wall (24).
In Vivo
Human
Heart
Results
The SDE of the fit (a mean error of 0) to the tag data at 228 msec after the electrocardiograph R wave in the 10 subjects was 0.36 mm ± 0.06 (mean SDE ± standard deviation SDE). This reflects both the error in recomstmuetion of the displacement field, as seen in the fit results for the mathematical model, and the uncertainty in the determination of the tag point locations. The accuracy of the fitting can be illustrated by the ability to reconstruct the positions of deformed tags. The fitted 3D displacement field was used to generate points that started from the reference state tag plane locations and moved into the image plane at 228 msec into systole. Figure 7 shows representative shortand long-axis human cardiac images obtained at 228 msee into systole and demonstrates the agreement between the predicted locations of the tag points and the actual tags on the MR images.
Monte
Carlo
Noise
Analysis
The precision of the fitting algonithm was tested by using a Monte Carlo simulation as described in Monte Carlo Simulations in Materials and Methods. The RMSD of the cornputed material-point position increased linearly as a function of input noise level as shown in Figure 8. It was determined that the scatter of points around the expected value was isotropic. The RMSDs at the endocamdial and epicandial material points were equal and were greaten than the RMSD at the midwall. At an input noise standard deviation of 0.5 mm, the midwall RMSD tracking error was 0.077 mm ± 0.015, and the endocandial and epicardial RMSD was 0.126 mm ± 0.030. The RMSD computation with a subset that consisted of 50 trials produced results that were not sigmificantly different, which suggests that 100 trials were sufficient to obtain a stable convergence of the estimate in tracking precision.
834
Radiology
#{149}
The combination of parallel-tag MR imaging and displacement field fitting is am accurate and robust method for reconstruction of the 3D deformation field throughout the entire left yentricle. Because of its reliance on parallel-tag data, the displacement fieldfitting method is inherently less susceptible to tag- and contoun-detectiom errors than were previous MR imaging reconstruction schemes, because only the center line of any tag need be determined during image analysis. The incorporation of a greater number of points along each tag line increases the sampling density throughout the heart compared with the sampling density obtained with techniques that use only those points located at intersections betweem tags or between tags and myocardial contours. The fitting of displacement fields cam also be realized by using other methods, which include energy minimizatiom techniques (5,25). In this approach, the fits can be computed mumenicalby for a predefimed set of material points (nodes), and intenpobatiom between these points is aceomplished by means of linear on finite element interpolation functions. Although field fitting by solving for series coefficients has some mathematical similarities to energy minimization and other methods, the series method has several advantages. First, it is independent of material strain energy models. Second, the global fit provides the ability to express the 3D deformation at any myocandial point with a relatively small number of parametems, independently of a pmedefined set of material points. The field-fitting approach is immediately suitable for other methods of 3D strain analysis in which motions in multiple directions are independently measured. For example, three sets of velocity-encoded images (26) could be used to solve for a set of expressions that describe the velocity field throughout a prescribed region of interest in a manner analogous to that described for tag-based displacement fields. This would have the advantage of smooth interpolation of the vebocity field. Grid tag data sets cam also be analyzed by interpreting the grid as two independent sets of parallel tags. We have demonstrated the application of displacement field fitting by using am analytical series expression for the analysis of tagged MR imaging cardiac data. The important features of this method are that it allows all
E E
.
0.3
.
Mid Wall Endo, Epi Walls
0) C .
0 Ca
0.2
I0 C,) 0
0 C/)
a:
-
o.o4 0.00
Figure
Standard
8.
0.25
Deviation
RMSD
0.50
of 1 D Input
of the computed
0.75
Noise
1.00
(mm)
material-
point position as a function of input noise level on a human heart geometry and deformation field. The distribution of tracked points formed an isotropic cloud around the noise-free reference trajectory. The RMSD of the distribution about the endocardial and
epicardial material points were the same and were larger than those about midwall material points. The error bars correspond to the standard Monte
deviation Carlo trials.
in the
ID
=
RMSD for the one-dimensional.
100
the available tag data to be considered and not just points at tag-tag on tagcontour intersections, it allows the determination of the local 3D deformatiom gradient tensor at any point in the heart wall, it has predictable noise propagation characteristics, and it is suitable for the analysis of both gridand parallel-tagged MR data. This last property is important in light of mecent advances in cardiac breath-hold imaging that are virtually free of rnotion artifact and in which only parallel tags are produced. The cascade of a first-order Cartesian basis set fit and a geometrically more appropriate prolate spheroidal fit enable reconstruction of materialpoint trajectories to within 0.3 mm for a physiologic deformation model. At bate systole in 10 healthy human subjects, tag displacement data was meconstructed with an average error of 0.00 mm ± 0.36, in which the standard deviation of the error approximately matched the expected uncentaimty in the determination of the in vivo tag point displacements. This suggests that all the displacement information contained in the tag data has been accounted for in the necomstruction. The combination of rapidly acquired parallel-tagged MR images and field-fitting analysis is a valuable tool for cardiac mechanics research and for the clinical assessment of cardiac mechanical function. U Acknowledgments: acknowledge Andrew thoughtful discussions ivieri, MD, for assistance human heart data.
The authors
gratefully
Douglas, PhD, for and Carlos Lugo-Olin acquisition of the
June
1995
10.
Zerhouni EA, Parish DM, Rogers WJ, Yang A, Shapiro EP. Human heart: tagging with MR imaging-a method for noninvasive assessment of myocardial motion. Radiology 1988; 169:59-63. Axel L, Dougherty L. MR imaging of mo-
Moore CC, Reeder SB, MeVeigh ER. Tagged MR imaging in a deforming phantom: photographic validation. Radiology 1994; 190:765-769.
11.
Young
tion with spatial modulation of magnetization. Radiology 1989; 171:841-845.
12.
References 1.
2.
3.
Mosher
TJ, Smith
MB.
A DANTE
tagging
sequence for the evaluation of translational sample motion. Magn Reson Med 1990; 15:334-339.
4.
5.
AA, Axel
motion
L.
with
spatial
of the heart modulation
250. 7.
8.
9.
Fischer SE, MeKinnon Boesiger P. Improved
GC, Maier SE, myocardial tagging
contrast. 200.
Med
Magn
Reson
1993;
Noninvasive
measurement of transmural gradients in myocardial strain with magnetic resonance imaging. Radiology 1991; 180:677-683. Rogers WJ, Shapiro EP, WeissJL, et al. Quantification of and correction for left
22.
Moore
CC, O’Dell EA.
WG, McVeigh
Calculation
ER, Zer-
15.
16.
Moore houni
CC, EA.
23.
thickness
for the
by M.
measurement
IEEE Trans
Med Imaging
of the
heart
in a single
breath
17.
24.
25.
of motion 1994;
A, Guttman
18.
strain
Guttman
Imaging
Cambridge
3
MA,
Prince
1994;
by MR tag-
1994; 29:427-433. JL, MeVeigh
ER.
26.
W, Douglas
transleft yen-
A, of the of the left ventricle by a kineJ Biomech 1992; 25:1119-1127.
R, Corday
E.
A, Muijtjens
Description
matic model. Lugo-Olivieri CH, Moore CC, Poon EGC, Lima JAC MCVeigh ER, Zerhouni EA. Temporal evolution of three dimensional deformation in the isehemic human left ventricle: assessment by MR tagging. In: Proceedings of the Society of Magnetic Resonance. Berkeley, Calif: Society of Magnetic Resonance, 1994; 1482. Denney TS Jr. Prince JL. 3D displacement field reconstruction on an irregular domain from planar tagged cardiac MR images. In: Proceedings of the IEEE Workshop on and
Articulate
Motion,
Austin,
Texas 1994. Los Alamitos, Calif: IEEE Computer Society, 1994. Pelc NJ. Myocardial motion analysis with phase contrast one MRI. In: Book of abstracts: Society of Magnetic Resonance in Medicine 1991. Berkeley, Calif: Society of Magnetic Resonance in Medicine, 1991; 17.
in tagged MR IEEE Trans
13:74-88.
Press W, Flannery B, Teukolsky 5, Vetterling W. Numerical recipes in C: the art of scientific
Number
calculation
Radiol
tnicle. Arts T, Hunter
Non-rigid
ER, Zer-
myocardial
Med
#{149}
McVeigh
Impact of semiautomated verimage segmentation errors on
Tag and contour detection images of the left ventricle. 19.
195
MA,
houni EA. sus manual ging. Invest
hold
sequence.
Bazille
analysis of three-dimensional finite deformation in canine J Biomech 1991; 24:539-548.
deformation
A, Zertagging to
measure three-dimensional endocardial and epicardial deformation throughout the canine left ventricle during isehemia (abstr). JMRI 1993; 3(P):124. Atalar E, MeVeigh ER. Optimum tag
nance, 1994; 1483. Malvern LE. Introduction to the mechanics of a continuous medium. Englewood Cliffs, NJ: Prentice Hall, 1969. MeCulloch AD, Omens JH. Nonhomoge-
Reneman
of three-dimen-
MeVeigh ER, Mebazaa Use of striped radial
O’Dell WG, Moore CC, MeVeigh ER. Optimization of displacement field fitting for 3D deformation analysis. In: Proceedings of the Society of Magnetic Resonance. Berkeley, Calif: Society of Magnetic Reso-
neous mural
13:152-160.
with a segmented turboFLASH Radiology 1991; 178:357-360.
Volume
EA.
21.
sional left ventricular strains from bi-planar tagged MR images. JMRI 1992; 2:165-175.
30:191-
MeVeigh ER, Atalar E. Cardiac tagging with breath hold cine MRI. Magn Reson Med 1992; 28:318-327. Atkinson DJ, Edelman RR. Cineangiography
L, Parenteau
of tagging with MR imagmaterial deformation. Radi-
ology 1993; 188:101-108. McVeigh ER, Zerhouni
houni
wall: of mag-
netization-a model-based approach. Radiology 1992; 185:241-247. O’Dell WG, SchoenigerJS, Blackband SJ, McVeigh ER. A modified quadrapole gradient set for use in high resolution MRI tagging. Magn Reson Med 1994; 32:246-
L, Dougherty
ventricular systolic long-axis shortening by magnetic resonance tissue tagging and slice isolation. Circulation 1991; 84:721-731.
14.
Three-dimensional
and deformation
estimation
6.
13.
Bolster BD, MeVeigh ER, Zerhouni EA. Myoeardial tagging in polar coordinates with striped tags. Radiology 1990; 177:769772. Yount
AA, Axel
CS. Validation ing to estimate
20.
computing.
University
Cambridge,
Press,
England:
1988.
Radiology
835
#{149}
PhD MD
Christopher C. Moore, BSE Elliot R. McVeigh, PhD
William
#{149}
C. Hunter,
#{149}
three-dideformaof parallel(MR)
M
Model deformations were reconstructed with a 3D tracking error of less than 0.3 mm. Error between estimated and observed onedimensional displacements along the tags in 10 human subjects was 0.00 mm ± 0.36 (mean ± standard deviation). Robustness to noise in the tag displacement data was demonstrated by using a Monte Carlo simulation. RESULTS:
CONCLUSION: The combination of rapidly acquired parallel-tagged MR images and field-fitting analysis is a valuable tool in cardiac mechanics research and in the clinical assessment of cardiac mechanical function.
breath-hold
netic resonance cardium, MR.
Heart, (MR), 511.1214
function, physics,
51.91 511.1214
imaging
tools
search
for
Mag#{149} Myo-
#{149}
of the
of this
the
1995;
195:829-835
was
at many
of the
in both The
neobjec-
to measure
non-
three-dimensional
field within time points
deformation
wall
study
wall settings.
study
invasively
(8,9) are pnomis-
heart
clinical
(3D)
the heart in the heart
cycle.
Myocandiab tags are regions where the magnetization has been perturbed before imaging and that, therefore, produce a signal intensity difference relative to that of nomtagged regions for a time proportional to Ti. Because the
tags
the the
magnetization deformation
reflects tissue
result
from
perturbations
of
of the tissue itself, of the tags accurately
the motion of the underlying (10-12). Special techniques
are
needed to reconstruct the 3D motion of the heart from MR image planes that are fixed in space, because different
sections
of tissue
are
sampled
achieved
tiom from sets
by combining
both
of tagged
Previously
longheart
and
short-axis into
for the presented
infonma-
images
a umi-
3D displacemotion
Figure
1.
MR images
(7.0/2.3,
15#{176} flip angle)
of an in vivo human heart of a healthy 30year-old male volunteer obtained with the parallel-tagging and -imaging protocol. The progression (from left to right) through three phases in the cardiac cycle is shown: early,
middle,
and
late systole.
Two
axis images and one long-axis shown; each displays tag lines ent set of mutually orthogonal
cardiac
short-
image are from a differtag planes.
at
different times. The movement of the heart through short-axis image planes, known as cardiac through-plane motion, is typically 10 mm at the base of the left ventricle (13). (This was confirmed with the analysis of 10 healthy human subjects discussed in this antide.) Correction for this is crucial, even for two-dimensional analysis of wall deformation (14). This correction
fled expression ment field. Radiology
(MR) imagand fast,
nominvasive
and
tive
was
terms:
resonance tagging (1-7)
ing ing
Deformations: Field Fitting
AGNETIC
function
MATERIALS AND METHODS: Displacement information in the direction normal to the undeformed tag planes was obtained at points along tag lines. Three independent sets of one-dimensional displacement data were used to fit an analytical series expression to describe 3D displacement as a function of deformed position. The technique was demonstrated with computer-generated models of the deformed left ventricle with data from healthy human volunteers.
Index
PhD
#{149}
Three-dimensional Myocardial Calculation with Displacement to Tagged MR Images’ PURPOSE: To reconstruct mensional (3D) myocardial lions from orthogonal sets tagged magnetic resonance images.
Radiology
recon-
I From the Departments of Biomedical Engineering (W.G.O., C.C.M., W.C.H., E.R.M.) and Radiology (E.A.Z.), Medical Imaging Laboratory, 407 Traylor Bldg. 720 Rutland Aye, The Johns Hopkins University School of Medicine, Baltimore, MD 21205. Received October 31, 1994; revision requested December 8; revision received January 30, 1995; accepted February 6. Supported in part by grants HL45090 and HL45683 from the National Institutes of Health, by a fellowship from the Merck Sharp and Dohme Corp (C.C.M.), and by a Whitaker Foundation Biomedical Engineering Research Grant. Address reprint requests to E.R.M. RSNA, 1995
struction schemes for MR imaging tag data require identifiable points within the images such as intersections be-
tween
tags
(11,12),
intersections
tweem tags and myocandiab (14), or points along striped (i5).
Analyses
intersection valuable
with
data, information
intervening
portions
such
becontours tag lines
sparse
tag
however, neglect contained in the
of the
tag lines.
Accordingly, the descriptions of deformation that result suffer from poor spatial resolution. These methods also require images with high spatial neso-
lution
both
in the
frequency
and
phase directions, which results in acquisition times of longer than one breath hold and thus limits their clinicab applicability.
Abbreviations: RMSD standard
=
3D = three-dimensional, root-mean-square deviation, deviation of the error.
SDE
=
In this study, we developed a method for reconstructing the 3D deformation field of the left ventricle from tagged MR images with use of position and displacement infonmation along the entire length of each tag. This method relies on accurately defined tag profiles (16) rather than on poorly defined heart contours (17). This
approach
eliminates
the
need
of this
agation
method.
properties
by means
of a Monte
with human deformation.
MATERIALS
A typical
human
consisted
Field
to the readout diat eight to 12 time
coordinate tive human
define
were
Fitting
parameter
processed by means of software package (18) to
along
positions
was
In general terms, field fitting is a technique for estimation of the value of some parameter throughout a particular region of interest, given discrete samples of that
comeCartesian
axes. Portions of a representaimage set are shown in Figure
1. The images a semiautomated
for
point
in and around
that region.
the tag lines and
around the endoeardial and epicardiab heart contours. A typical tag and contour data set for a healthy human after image processing is shown in Figure 2. For a tag separation in the reference state of 6.0 mm, approximately 12 tags were produced in the myocardium in each set. At a point separation along each tag of 1 mm, this produced more than 4,600 raw data
prop-
simulation
and
Sets set of an in vivo
of three
sets of
multiphasie images: one in the cardiac bong-axis view and two in the cardiac short-axis view. The short-axis image sets consisted of stacks of six or seven contiguous parallel sections and the long-axis image set consisted of six sections prescribed radially around the cardiac long axis with
a. Figure
b.
2. (a) Typical 3D short-axis tag data set for x-displacement in an in vivo heart of a healthy volunteer. Image shows the appearance of the tags and the contours near end systole on seven short-axis image planes. (b) Typical 3D contour data set and the estimated prolate spheroidal centroid (upper gray circle) and apical focal point (lower black circle).
Tag
at t=t
Surface
Image
xLLY
3. Figures
3, 4.
ten detection tag
surface
(3) Depiction
of tag points within
the
heart
of a short-axis
at 1-mm wall
Radiology
#{149}
image
intervals. with
an automated tag detection algorithm. figure at the upper left shows the region
830
In
displacement field fitting, the parameter of interest is the 3D displacement vector, and the samples are the values of one-dimensional displacement measured at points on tags in the deformed heart wall. Although field-fitting is generally applicable to any motion-detection method, it has been applied here to the analysis of three independent one-dimensional sets of displacement measurements from parallel-tagged MR images. This type of data is depicted
METHODS
Data 3D tag data
ented perpendicularly rection was obtained
other
tested
geometry
AND
heart
then
Carlo
cardiac
Parallel-tagged
Noise
were
points, of which every used in the fitting.
frames throughout systole. The three sets of tag planes sponded to three orthogonal
accurate simultaneous measurement of displacement in two dimensions and permits computation of the 3D deformation gradient tensor at any point in the heart wall. The method was tested on a cornputer-generated model of a prolate spheroid that undergoes deformations that simulate those measured in the beating heart. Finally, paralleltagged, breath-hold cine data sets from 10 human hearts were analyzed by means
an angular separation of 30#{176}. For each image set, a stack of parallel tag planes ori-
four
at some
(4) Deformation vertical
represents of the heart
image
time
4. initial
after
tagging.
of a tag plane planes.
The
the one-dimensional depicted.
enlarged
Image
Inset
depicts
into a tag surface dots
displacement
on
these
an
enlarged
view
in 3D. The bold curves
associated
are
with
the
of one
curves tag
points
Plane
Plane
2
deformed
tag
line
are the intersections generated
the nth tag point.
4
by
af-
of the
means
The rectangle
June
of
in the
1995
in Figure image
3, which
and
illustrates
a deformed
a short-axis
set
late
The key concept of the method is shown in Figure 4. In three dimensions, each observed tag line represents the tersection
the
of a deformed
tag
two-dimensional
surface initially tag by
four
within parallel
plane,
the
lines between heart
For
of an depicts
a is cut
along tag
point
placement the
and
coefficients the
of digital
sampling
reconstruct
position
data
field.
on
and
Fourier
Here,
nectionab
displacement
to solve for an the displacement tion, independently
compute
data
directions.
displacement determined,
the
to
the
entire each set
was
Once
eliminates
a p
=
(In
evaluated ix,,, was
the large-scale stretches and
fitted
=
italic
italic;
by
using
for
efficiently
spheroidal
series
surements
of displacement
prolate
displacements
spheroidal
x
x
fcosh +
jy
=
point
heart,
the
bowed
by a probate
spheroidal
residual
displacement
fields.
shows
first-order
the
system;
prolate
longitudinal
coordinate,
means
coordi-
z was moThis
andfis related
the focal
sin (4) cos (0) IIX (0) l4
fsinh
(X) sin
(4)
(0) 0,
(X) sin
x
=
=
sin
(4k) sin
(5)
(0) iX
fsinh
(X) cos (4i) sin
(0) 114
+
fsinh
(A) sin
(4) cos
(0) 110,
liz
which as
can
=
fsinh
()
cos
(It)
fcosh
(6)
IIX
(A) sin (di) 1k,
where
Sx
be
expressed
bx
in vector
=
(7)
length. coordinates
notation
I BA, BA
[x,by,1z], J is the Jacobian matrix transformation.
circumfer-
=
(8) [1iX,,10],
=
for
this
and
coordinate
For a, b, and
of
y
de-
(4i) cos
4 is the
to Cartesian
directional
(X) cos
coordinate 0 is the
and
are related
fol5
component,
infini-
and
fit of the
spheroidal
A is the radial
the
The
+
the
Figure
onto
axes.
following
(It)
fcosh
for gen-
fit was
the measured
in Cartesian
in the
Cartesian
meaCarte-
fsinh
-
de-
from in the
coordinates
of the
=
density param-
projected
spheroidal
by means rivatives:
data
of the prolate
directions, were
-
to
this
expansions
prolate
describe
numtagging
direction.
tesimal
expected
were
,
typical
for
coordinate
fitting method)
deformations
and
50 free-fitting
scalars,
to a power series in prolate spheroi-
curvilinear
per
local
value
the
appropriate generated
all tag equations coefficients,
singular
(a least-squares
and
displacements
=
value, the
fitting
order, a,
For
font;
and
radial
fitting
coefficients,
sian coordinate p
de-
x displacement By considering
dal coordinates-To
were genstate.
bulk shears.
bold
most
(20) eters
be written are
angular
To fit for the coefficients
x
and
Legendre
N is the
sequentially.
was
un-
form
vectors
displacements and p vectors points, a series of simultaneous was created and the unknown
by
to remove and linear
the can
bold
and the measured.
by using
used tions
4),
and
plain italic.) For each sampled tag point (x,,,y,,,z,), each term in the vector p was
was computed
power
article,
a lowercase
are
an initial step, a first-order series expansion in x, y, and
fit
[a,al,a,,a1]
=
uppercase
angle,
notes-As
this
a
this
with
tensors,
ential
in Cartesian
a,, took
where
These
to a power series
(x,y,z)
associated
function,
and imaging geometry and the typical image signal-to-noise characteristics, we determined that a fit with N = 1 and L = 4
expression
in x, x(x,y,z)(Fig
a1x + a2Y + a3z. In vector notation,
system. time frame
Fitting
fo-
+
Fitting
three
bered
Cartesian The
of position
coefficients,
vector field in an any coordinate The displacement field at each tag planes reference
problem.
P71 is the
order, L is the the unknown
sphe-
singularity
the
(19). Independent series expressions the y and z displacement fields were erated and fitted analogously.
the
polynomial
which
a mathematical this
where
or apical
used
of any
independently
centroid
wall;
pro-
fixed
deforma-
the myocardium,
heart
composition
in the deformed heart wall. At higher complexity, displacement data from multiple directions can be used simultaneously to solve for the full displacement
the time at which the erated as the universal
the
bulk
a, were
field expresthey were used
3D displacement
the
the
left unidi-
interpolation function for field in that one direeof the motion in the
Cartesian
independent sions were
that
analysis
the case,
create
within
of the
by aligning
axis of the prolate system at each time
cause
would
noted
to use
described
throughout In the simplest
may
efficiency
large
cus to intersect
[1,x,y,zl.
one-dimensional
this expression
In addition,
x
for
expression is analogous
frame.
a0
field dis-
to solve
This
a continuous,
displacement ventricle.
other
displacement the tag point
of a series
describes
signal.
of the by using
and long coordinate
known
the deformed tag surface, the component of its displacement vector that was normal to the tag plane (ix, in Fig 4) and its position were measured. Reconstruction was performed
centroid roidal
as a function
tag
and epicar-
each
the
fitting
for the displacement
In each
marked
the endocardial
contours.
tag
that
increased
spheroidal
tions
The
wall
planes.
were
inwith
plane.
heart
image
tag points
dial
surface
image
results from the deformation flat tag plane. The figure
surface
greatly
of tags.
fsinh
(1) sin
fsinh
(X) sin
() cos (4) sin
c, defined as the vectors of coefficients for the 1IX, 64, and
unknown
(0),
( )
o
(0),
(2)
vector
expressions,
generating
and
respectively,
of 50 series
terms
function,
and
p, the
derived
M
a
=
from
the
b
p,
.
0 = c p (the a vector for the Cartesian fit and the a vector here are different). and
z =fcosh
For each
prolate
spheroidal
displacement field, prolate spheroidal
created
by using
analogous
,0
(4).
(X)cos
spherical
1a1P”uI
expansion (X4,0)
with was
function harmonic
cos
(mO), (mO),
>
series:
0
0,
case
(4)
in which
the
by the three
tag normal vectors) are aligned with the prolate
4X
in in
the
(as defined
dinate (1)-(3), ments
[cos
sin .
For
coordinate
a series coordinates
a generating
to the
(3)
Cartesian
axes
orthogonal centered spheroidal
sets of and coor-
system as described in Equations the measured Cartesian displacecan be expressed as fcosh
(X) sin
(4) cos (0) (a
.
p)
+
fsinh
(X) cos
(4) cos (0) (b . p)
-
fsinh
(X) sin
(4)
sin
(0)
(c
.
p)
(9)
y
Focal
Point
Figure 5. Prolate spheroidal coordinate tern. Surfaces of constant X are ellipsoids surfaces 4i
of constant
are hyperboloids.
Vf1IIYThD
1Q
sysand
longitudinal coordinate const = constant.
#{149} KTiiriIior
021
x
a(Jp)+
=
+
y
b(Joip) (Jo’p)
C
u
=
(10)
Px’
a(110p)+b(111p)
=
c (J12p)
+
(11)
a
and
Xz
a
=
(J2p)
+
where .
. .
now and
,C49]
JolP49,J0Po,
were
a .
defined
method
.
b
.
.
(J2IP)
,a4g,b0,
p .
(12) ,b49,C0,
. .
J())P49,J01P0
vectors
analogously.
of solution
series
.
[Joopo ,J02P49]. The
Px . .
[a0,
=
+
(J22p)= a
C
p
Similar
and p. to the
of the Cartesian
coefficients
above,
power
each measured
displacement value was collected into a single row vector, .x, and the corresponding p vectors were inserted as columns in a matrix P. The unknown coefficients, a, were then solved for by using the equation x = a P. In this way, all the unknown coefficients can be solved for simultaneously by using all available tag data and singular value decomposition. Although, for finite deformations, the relations between the displacements
and
in x, y, and
0 are not
least-squares
z and
those
exact,
the simultaneous
fitting
will
always
,
in X, result
for
each
time
frame
in the
data set. Because the heart geometry changes throughout the cardiac cycle, a new prolate spheroidal coordinate system was
calculated
for
every
time
frame
on
the
basis of the measured epicardial contour point data. This was done first by estimating the three coordinates of the centroid
and
the
focus
three
coordinates
(point
on
the
apex, at a distance the centroid) and squares
and
optimal
apical
cardiab Within
focus
were
until
then
X value
at this
for the
given
contour loop, the
toward
and
the
of the
dinate centroid that Equation for
the
set of epi-
by the normal
of myocardial
state at a material point at any given imaging time
is fully
described
mation
gradient
FdX,
=
point
means
defor-
in the
deformed
state
(21). F
from
calculated
the
and
X
state
displacement
gradient tensor Vu, which was computed numerically by taking partial derivatives of the displacement expressions with respect to the three coordinate directions. Because the terms in the series expansion were computed on the basis of the coordinates of the tag points in their deformed state, Vu described the motion from the deformed state back to the undeformed
when
the tags were created. Vu requires F = (I
I
is the
Thus, Analytical
an
identity
tensor.
To
The
transposition.
coordi-
spheroidal
coor-
neces-
of the new spheroiwith those defined
vectors
to the undeformed
through
space.
A mesh
points
regularly
spaced
The
cardiab
the
ated
by
the
aries
between
achieved
basal
and
as a function nates.
Axel
image endocardial
section
model
(5), which by
contour
constructed.
coordi-
described
by Young
was
on 3D strains
based of cine
coordinates
of the initial deformation 0
apical
ti
and
radiography
(X,,0)
coordinates parameters =
0
+
+
Cfsinh
of
as a function
(A,,O) and the were as follows:
Bfeos
A +
(4)
(A),
Dfeos
+
=
Efsin
+
(13)
(CF) (t)
sinh
(A),
(14)
and
was
Vol
epipoints
was
deformative
means
first
the left ventricle
defor-
spheroidal
axisymmetnically
deformed
0 was
locations
a known
computer-gener-
model
bongitudimost
with
the anterolateral free wall of the left yentnicbe with implanted metallic beads (22). The governing equations that describe the
seeheart.
at the
method
a realistic, to prolate
measured
sections and radially by and epicardial contours. of the contour bound-
by fitting
the first time late spheroidal
and
and
imaging of the
constrained
most
short-axis image the endocardiab The interpolation
,
an
cylinder
of 96 material
locations
were
Case
deformation
nates
wall in the
in X,
the
field,
We adapted
and 3D displacewere evaluated as they moved
basis of the and the shape
material-point
time frame
as
the heart
first imaging time frame ments and deformations over time at these points
selected on tion locations
from sets of im-
ages with arbitrary tag normal vectors, long as three dimensions were spanned. Computing strains at material points-A material point is defined as an infinitesi-
within
test
mation
For both the mathematical and experimental models, a set of material points
was defined
Test
-
and
focal
spheroid
The
of its
in the reference
was
sets for the prolate
tensor, F, according where x are the coordinates
are its coordinates
mitted
Radiology
by
tissue.
tag data
with added Cartesian motions were modeled for in each view. The angles of tilt are due to simulated
mechanical in the heart
nally
#{149}
modes planes
mal volume
tag planes. For the transformation to the new coordinate system x’, defined with the rotation matrix R such that x’ = R . x, Equation (8) becomes Bx = R ‘J1IX. This modification was used throughout the derivation. In addition, this formalism per-
832
(b) long-axis
adjusted
axes system
3D reconstruction
deformation and 10 tag error.
and
Lagrangian finite strain tensor, E, can then be computed from E = y2(FTF - I), where the superscript T denotes matrix
radial
transformation
the
(a) short-axis
Physiologic image sections prescription
where
and apical focus required (8) be modified to account
coordinate
to align dal coordinate sary
prolate
Simulated
to compute F from inverse operation:
nate A were derived that produced a prolate spheroid with the smallest fitting error to the measured left ventricle contour points. Variation
model. seven image
state
points. six coordi-
of centroid
coordinates
the
centroid
independently
a combination
point
axis,
of a focal length from by computing a least-
left ventricle an iteration
nates
of the apical
long
b. 6.
to dx of the
in
the optimal fit of the given displacement data to the series expansion. The fitting procedure was performed independently
a. Figure
(1.)
of
frame to a fourth-order procoordinate expansion for X of the two angular coordi-
=
4/3-rf
=
[Vol
Vol
+
where
tor, B
A is the is the
axial
torsion
cosh (A)
-
(X) sinh2 Vol
(X)
(‘endo)]
(15)
(‘endo)’
rotation around
at the the
equa-
central
June 1995
Bulk motions and linear deformations that incorporate ellipticalization (the change in cross-sectional shape of the left ventricle
from
a circle
at end
diastole
computed tribution trajectories
to an
ellipse at end systole) and shears in x, y. and z were then superimposed on the axisymmetric modes in the manner of Arts et
al (23). The magnitudes
of these
additional
modes were chosen to provide the overall model with approximately physiologic rigid body motions and to reproduce those deformations observed by Arts et al (23). These deformation values are listed
in Table
2. Image
prescription
error,
which
occurs when the central axis of the imaging volume is not coincident with the long
axis of the left ventricle,
was also simu-
bated with representative set about the x axis and
values of 5#{176} offof 8#{176} offset about
the y axis. Seven
were tion
image
sections
simulated between
and
for each adjacent
10 tag planes
image
set. Separa-
tag points
along
a
given tag was fixed at 2 mm. Throughout the entire 3D data set, this produced 2,400 tag points. deformation model
Human
Data
The prolate spheroidal is shown in Figure 6.
imaging
Sets
sequence
with
parallel
left side of the chest. The imaging parameters
per Figure (7.0/2.3,
7.
Shortand 15#{176} flip angle)
long-axis MR images of a human heart ob-
tamed in a healthy, 21-year-old male teer at 228 msec into systole overlaid point fit.
locations
predicted
the
C
axial
stretch,
is the
transmural
E is the
nab shear,fis
twist,
coordinate
volume
D is the
Monte
within
of the pro-
system,
a prolate
Vol(X)
spheroidal
shell of constant X, and endocardial measurement. in the radial component, to a volume-conserving
“endo” denotes Deformation A., was limited (incompressible) mode by setting a = 1. The deformed short-axis endocar-
contraction
initial
and equatorial
I sinh(X,fld0), realistic
radii,fsinh(Aefldo)
were
heart
chosen
geometry
ejection fraction of spheroidal deformation listed in Table 1.
Volume
195
Number
#{149}
(35
msec
time
one signal views resolution).
was
Carlo
tuned
to achieve
a 180#{176}
Simulations
and
to approximate and
resulted
in an
50%. The prolate parameters are
3
A representative human data set was fitted; the reconstructed displacement field and heart geometry were then used to generate a physiologic noise-free paral-
beb tag data
set. The effect
tag point position on the point tracking predictions
by using
a Monte
One-dimensional with 0.25-1.00-mm
Carlo
were added to the sets, and the resultant trajectories terial points mean-square
of uncertainty
in
final materialwas evaluated
simulation
Gaussian standard .x, y,
the
formation
were
prolate
model:
evalu-
spheroidal
tracking
de-
and
fitting.
The 3D tracking performance was evaluated by comparing the estimated material-point trajectories denived
from
the
tiom
to the
exact
from tions
field-fitting
neconstruc-
solution
the deformation (13)-(15). The
computed
model of Equamean absolute
value and the standard deviation the tracking error were computed the set of 96 material points. The
of for fit-
ting performance computing the
was standard
evaluated deviation
by
the
around
a mean
error
(SDE;
of
error
at
The
SDE
model
Deformation of the
tag data
Model
fit to the
points
2,400
was
0.095
mm.
This is on the order of the expected uncertainty in the determination of tag point displacements for typical in vivo data sets (16,17). This implies that the fitting error for a human data set will be dominated by the uncentainty in the tag point displacement
data
rather
than
by error
in recom-
structiom of the displacement field at the measurement locations. The 3D tracking error for the set of 96 material points was 0.28 mm ± 0.i6
(mean ± standard deviation). The relatively large value of the 3D tracking error ( 0.3 mm) compared to the SDE of the fit ( 0.1 mm) suggests
transmural-longitudi-
the focal length
spheroidal
is the
dial
frame
with
Mathematical Results
see-
The tagging pulse consisted of five nonseleetive radio-frequency pulses with relative amplitudes of 0.7, 0.9, 1.0, 0.9, and 0.7 separated by spatial modulation of magnetization (2) encoding gradients to achieve a tag spacing of 6 mm. The tagging tip angle flip angle.
axis,
late
from
volun-
with tag 3D field
movie
ated
of the displacement
performance
with the
tion were a repetition time of 7.0 msec and an echo time of 2.3 msec (7.0/2.3, fractional echo), a 15#{176} flip angle, 110 phase encoding steps, I .25 x 2.9 x 7 mm voxels, acquired, and five phase-encoded
measures
ments were made, whereas the tracking performance reflects both the quality of the fitting and the interpolatiom of the displacement field between the measurement locations.
tags
for each
Two field-fitting
variations in the displacement field the locations where the measure-
was used (8). Eighteen acquisitions were performed in each subject during a breath hold. Volunteers underwent imaging at
end expiration in the supine position a flexible surface coil wrapped around
RESULTS
of 0) between the estimated and the measured one-dimensional displacements at each tag point. Thus, the fitting performance reflects the ability of the reconstruction to account for the
Ten healthy human volunteers gave informed consent and underwent imaging with a i.5-T scanner (Sigma; GE Medical Systems, Milwaukee, Wis). A cine breath-
hold
for each point from the 3D disof the cloud of estimated point around the expected location.
(19).
noise profiles deviation and z data
3D material-point
were computed for the 96 mafor 100 trials. A single rootdeviation (RMSD) value was
that further improvements to the meconstruction cam be made by increasing the tag density and number of fitting parameters. The corresponding
circumferential material points a mange
strain were
of -0.25
errors 0.006
to 0.06,
the
at the ±
nab strain errors were 0.0003 for a range of -0.23 to -0.08, and rnidwall radial strain errors were 0.017
± 0.026
for
a mange
96
0.012 for bongitudi± 0.0070
the
of rnidwall
Radiology
833
#{149}
DISCUSSION
radial strains of 0.18-0.52. The high accuracy of the strain estimations, even where 3D tracking errors were _
0.3 mm,
is due
to the
correlation
of
tracking errors between neighboring myocandial regions. These errors and standard deviations are well below the threshold necessary to detect motion abnormalities in the isehemic heart wall (24).
In Vivo
Human
Heart
Results
The SDE of the fit (a mean error of 0) to the tag data at 228 msec after the electrocardiograph R wave in the 10 subjects was 0.36 mm ± 0.06 (mean SDE ± standard deviation SDE). This reflects both the error in recomstmuetion of the displacement field, as seen in the fit results for the mathematical model, and the uncertainty in the determination of the tag point locations. The accuracy of the fitting can be illustrated by the ability to reconstruct the positions of deformed tags. The fitted 3D displacement field was used to generate points that started from the reference state tag plane locations and moved into the image plane at 228 msec into systole. Figure 7 shows representative shortand long-axis human cardiac images obtained at 228 msee into systole and demonstrates the agreement between the predicted locations of the tag points and the actual tags on the MR images.
Monte
Carlo
Noise
Analysis
The precision of the fitting algonithm was tested by using a Monte Carlo simulation as described in Monte Carlo Simulations in Materials and Methods. The RMSD of the cornputed material-point position increased linearly as a function of input noise level as shown in Figure 8. It was determined that the scatter of points around the expected value was isotropic. The RMSDs at the endocamdial and epicandial material points were equal and were greaten than the RMSD at the midwall. At an input noise standard deviation of 0.5 mm, the midwall RMSD tracking error was 0.077 mm ± 0.015, and the endocandial and epicardial RMSD was 0.126 mm ± 0.030. The RMSD computation with a subset that consisted of 50 trials produced results that were not sigmificantly different, which suggests that 100 trials were sufficient to obtain a stable convergence of the estimate in tracking precision.
834
Radiology
#{149}
The combination of parallel-tag MR imaging and displacement field fitting is am accurate and robust method for reconstruction of the 3D deformation field throughout the entire left yentricle. Because of its reliance on parallel-tag data, the displacement fieldfitting method is inherently less susceptible to tag- and contoun-detectiom errors than were previous MR imaging reconstruction schemes, because only the center line of any tag need be determined during image analysis. The incorporation of a greater number of points along each tag line increases the sampling density throughout the heart compared with the sampling density obtained with techniques that use only those points located at intersections betweem tags or between tags and myocardial contours. The fitting of displacement fields cam also be realized by using other methods, which include energy minimizatiom techniques (5,25). In this approach, the fits can be computed mumenicalby for a predefimed set of material points (nodes), and intenpobatiom between these points is aceomplished by means of linear on finite element interpolation functions. Although field fitting by solving for series coefficients has some mathematical similarities to energy minimization and other methods, the series method has several advantages. First, it is independent of material strain energy models. Second, the global fit provides the ability to express the 3D deformation at any myocandial point with a relatively small number of parametems, independently of a pmedefined set of material points. The field-fitting approach is immediately suitable for other methods of 3D strain analysis in which motions in multiple directions are independently measured. For example, three sets of velocity-encoded images (26) could be used to solve for a set of expressions that describe the velocity field throughout a prescribed region of interest in a manner analogous to that described for tag-based displacement fields. This would have the advantage of smooth interpolation of the vebocity field. Grid tag data sets cam also be analyzed by interpreting the grid as two independent sets of parallel tags. We have demonstrated the application of displacement field fitting by using am analytical series expression for the analysis of tagged MR imaging cardiac data. The important features of this method are that it allows all
E E
.
0.3
.
Mid Wall Endo, Epi Walls
0) C .
0 Ca
0.2
I0 C,) 0
0 C/)
a:
-
o.o4 0.00
Figure
Standard
8.
0.25
Deviation
RMSD
0.50
of 1 D Input
of the computed
0.75
Noise
1.00
(mm)
material-
point position as a function of input noise level on a human heart geometry and deformation field. The distribution of tracked points formed an isotropic cloud around the noise-free reference trajectory. The RMSD of the distribution about the endocardial and
epicardial material points were the same and were larger than those about midwall material points. The error bars correspond to the standard Monte
deviation Carlo trials.
in the
ID
=
RMSD for the one-dimensional.
100
the available tag data to be considered and not just points at tag-tag on tagcontour intersections, it allows the determination of the local 3D deformatiom gradient tensor at any point in the heart wall, it has predictable noise propagation characteristics, and it is suitable for the analysis of both gridand parallel-tagged MR data. This last property is important in light of mecent advances in cardiac breath-hold imaging that are virtually free of rnotion artifact and in which only parallel tags are produced. The cascade of a first-order Cartesian basis set fit and a geometrically more appropriate prolate spheroidal fit enable reconstruction of materialpoint trajectories to within 0.3 mm for a physiologic deformation model. At bate systole in 10 healthy human subjects, tag displacement data was meconstructed with an average error of 0.00 mm ± 0.36, in which the standard deviation of the error approximately matched the expected uncentaimty in the determination of the in vivo tag point displacements. This suggests that all the displacement information contained in the tag data has been accounted for in the necomstruction. The combination of rapidly acquired parallel-tagged MR images and field-fitting analysis is a valuable tool for cardiac mechanics research and for the clinical assessment of cardiac mechanical function. U Acknowledgments: acknowledge Andrew thoughtful discussions ivieri, MD, for assistance human heart data.
The authors
gratefully
Douglas, PhD, for and Carlos Lugo-Olin acquisition of the
June
1995
10.
Zerhouni EA, Parish DM, Rogers WJ, Yang A, Shapiro EP. Human heart: tagging with MR imaging-a method for noninvasive assessment of myocardial motion. Radiology 1988; 169:59-63. Axel L, Dougherty L. MR imaging of mo-
Moore CC, Reeder SB, MeVeigh ER. Tagged MR imaging in a deforming phantom: photographic validation. Radiology 1994; 190:765-769.
11.
Young
tion with spatial modulation of magnetization. Radiology 1989; 171:841-845.
12.
References 1.
2.
3.
Mosher
TJ, Smith
MB.
A DANTE
tagging
sequence for the evaluation of translational sample motion. Magn Reson Med 1990; 15:334-339.
4.
5.
AA, Axel
motion
L.
with
spatial
of the heart modulation
250. 7.
8.
9.
Fischer SE, MeKinnon Boesiger P. Improved
GC, Maier SE, myocardial tagging
contrast. 200.
Med
Magn
Reson
1993;
Noninvasive
measurement of transmural gradients in myocardial strain with magnetic resonance imaging. Radiology 1991; 180:677-683. Rogers WJ, Shapiro EP, WeissJL, et al. Quantification of and correction for left
22.
Moore
CC, O’Dell EA.
WG, McVeigh
Calculation
ER, Zer-
15.
16.
Moore houni
CC, EA.
23.
thickness
for the
by M.
measurement
IEEE Trans
Med Imaging
of the
heart
in a single
breath
17.
24.
25.
of motion 1994;
A, Guttman
18.
strain
Guttman
Imaging
Cambridge
3
MA,
Prince
1994;
by MR tag-
1994; 29:427-433. JL, MeVeigh
ER.
26.
W, Douglas
transleft yen-
A, of the of the left ventricle by a kineJ Biomech 1992; 25:1119-1127.
R, Corday
E.
A, Muijtjens
Description
matic model. Lugo-Olivieri CH, Moore CC, Poon EGC, Lima JAC MCVeigh ER, Zerhouni EA. Temporal evolution of three dimensional deformation in the isehemic human left ventricle: assessment by MR tagging. In: Proceedings of the Society of Magnetic Resonance. Berkeley, Calif: Society of Magnetic Resonance, 1994; 1482. Denney TS Jr. Prince JL. 3D displacement field reconstruction on an irregular domain from planar tagged cardiac MR images. In: Proceedings of the IEEE Workshop on and
Articulate
Motion,
Austin,
Texas 1994. Los Alamitos, Calif: IEEE Computer Society, 1994. Pelc NJ. Myocardial motion analysis with phase contrast one MRI. In: Book of abstracts: Society of Magnetic Resonance in Medicine 1991. Berkeley, Calif: Society of Magnetic Resonance in Medicine, 1991; 17.
in tagged MR IEEE Trans
13:74-88.
Press W, Flannery B, Teukolsky 5, Vetterling W. Numerical recipes in C: the art of scientific
Number
calculation
Radiol
tnicle. Arts T, Hunter
Non-rigid
ER, Zer-
myocardial
Med
#{149}
McVeigh
Impact of semiautomated verimage segmentation errors on
Tag and contour detection images of the left ventricle. 19.
195
MA,
houni EA. sus manual ging. Invest
hold
sequence.
Bazille
analysis of three-dimensional finite deformation in canine J Biomech 1991; 24:539-548.
deformation
A, Zertagging to
measure three-dimensional endocardial and epicardial deformation throughout the canine left ventricle during isehemia (abstr). JMRI 1993; 3(P):124. Atalar E, MeVeigh ER. Optimum tag
nance, 1994; 1483. Malvern LE. Introduction to the mechanics of a continuous medium. Englewood Cliffs, NJ: Prentice Hall, 1969. MeCulloch AD, Omens JH. Nonhomoge-
Reneman
of three-dimen-
MeVeigh ER, Mebazaa Use of striped radial
O’Dell WG, Moore CC, MeVeigh ER. Optimization of displacement field fitting for 3D deformation analysis. In: Proceedings of the Society of Magnetic Resonance. Berkeley, Calif: Society of Magnetic Reso-
neous mural
13:152-160.
with a segmented turboFLASH Radiology 1991; 178:357-360.
Volume
EA.
21.
sional left ventricular strains from bi-planar tagged MR images. JMRI 1992; 2:165-175.
30:191-
MeVeigh ER, Atalar E. Cardiac tagging with breath hold cine MRI. Magn Reson Med 1992; 28:318-327. Atkinson DJ, Edelman RR. Cineangiography
L, Parenteau
of tagging with MR imagmaterial deformation. Radi-
ology 1993; 188:101-108. McVeigh ER, Zerhouni
houni
wall: of mag-
netization-a model-based approach. Radiology 1992; 185:241-247. O’Dell WG, SchoenigerJS, Blackband SJ, McVeigh ER. A modified quadrapole gradient set for use in high resolution MRI tagging. Magn Reson Med 1994; 32:246-
L, Dougherty
ventricular systolic long-axis shortening by magnetic resonance tissue tagging and slice isolation. Circulation 1991; 84:721-731.
14.
Three-dimensional
and deformation
estimation
6.
13.
Bolster BD, MeVeigh ER, Zerhouni EA. Myoeardial tagging in polar coordinates with striped tags. Radiology 1990; 177:769772. Yount
AA, Axel
CS. Validation ing to estimate
20.
computing.
University
Cambridge,
Press,
England:
1988.
Radiology
835
#{149}
