Estimating 3D Strain from 4D Cine-MRI and Echocardiography: In-Vivo Validation.

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Estimating 3D Strain from 4D Cine-MRI and
Echocardiography: In-Vivo Validation
Xenophon Papademetris1, Albert J. Sinusas23, Donald P. Dione2,
R. Todd Constable2and James S. Duncan13
1Departments of Electrical Engineering,2Diagnostic Radiology, and3Medicine,
Yale University New Haven, CT 06520-8042
papad@noodle.med.yale.edu
Abstract. The quantitative estimation of regional cardiac deformation
from 3D image sequences has important clinical implications for the as-
sessment of myocardial viability. The validation of such image-derived
estimates, however, is a non-trivial problem as it is very difficult to
obtain ground truth. In this work we present an approach to validat-
ing strain estimates derived from 3D cine-Magnetic Resonance (MR)
and 3D Echocardiography (3DE) images using our previously-developed
shape-based tracking algorithm. The images are segmented interactively
and then initial correspondence is established using a shape-tracking ap-
proach. A dense motion field is then estimated using a transversely linear
elastic model, which accounts for the fiber directions in the left ventricle.
The dense motion field is in turn used to calculate the deformation of
the heart wall in terms of strains. The strains obtained using our algo-
rithm are compared to strains estimated using implanted markers and
sonomicrometers, which are used as the gold standards. These prelimi-
nary studies show encouraging results.
1Introduction
It is the fundamental goal of many forms of cardiac imaging and image analysis
to measure the regional function of the left ventricle (LV) in an effort to isolate
the location and extent of ischemic or infarcted myocardium. In addition, the
current management of acute ischemic heart disease is directed at establishing
coronary reperfusion and, in turn, myocardial salvage. Regional function may
also be related to the degree of salvage achieved.
However, while there have been many methods proposed for the estimation of
3D left ventricular deformation (e.g. [1,6,7,16]), the validation of these results
is an extremely important and often neglected aspect of work in this area. Often
phantoms are used with known shapes and displacements usable as ground truth
(e.g. Kraitchman[8]). In Young [19] it was shown that away from the free surfaces
of the gel-phantom, a Rivlin-Mooney [9] analytic model accurately reproduced
the 2-D displacement of magnetic tags. This showed agreement between the
theory (model) and the image-derived displacements. However, in vivo measure-
ments of the beating heart usually present additional complexities. An alterna-
tive validation method is to use simulated images with known ground truth (e.g.

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Amini[1], Prince [14] and Haber [6]). One example[1] uses a kinematic model of
the left ventricular motion by Arts [2] within an MR tag image simulator [18]
to generate synthetic images with known displacements. In the shape-tracking
work of Shi[17], implanted markers are used as the gold standard. These markers
are physically implanted on the myocardium before the imaging. Here, algorithm
generated displacements are compared to the marker-displacements (these are
easily identifiable from the images). This technique has the disadvantage of com-
paring trajectories for a small number of points, however, it is done on real data
as opposed to simulations. However, in all previous efforts including those noted
above, to the best of our knowledge, there has been no validation of estimates
of in-vivo cardiac deformation (Shi et al[17] only validate displacements.)
In this paper we describe the experiments used to validate strains estimates
derived from our algorithm, which uses a transversely isotropic model for the left
ventricle and shape-based displacements [12]. We first briefly review the algo-
rithm used to estimate the deformation and then describe the methodology used
for the validation. We then present results for the validation of these deformation
estimates derived from Magnetic Resonance (MR) and 3D-Echocardiography
(3DE) image sequences.
2Our algorithm
2.1Image Acquisition
Canine MR-images: ECG-gated magnetic resonance imaging was performed on a
GE Signa 1.5 Tesla scanner. Axial images through the LV were obtained with the
gradient echo cine technique. The imaging parameters were: section thickness=5
mm, no intersection gap, 40 cm field of view, TE 13 msec, TR 28 msec, flip angle
30 degrees, flow compensation in the slice and read gradient directions, 256 x
128 matrix and 2 excitations. The resulting 3D image set consists of sixteen 2D
image slices per temporal frame, and sixteen temporal 3D frames per cardiac
cycle, with an in-plane resolution of 1.6mm and a slice thickness of 5mm.
3D Echocardiography (3DE): The 3DE images were acquired using an HP Sonos
5500 Ultrasound System with a 3D transducer (Transthoracic OmniPlane 21349A
(R5012)). The 3D-probe was placed at the apex of the left ventricle of an open-
chest dog using a small ultrasound gelpad (Aquaflex) as a standoff [12]. Each
acquisition consisted of 13–17 frames per cardiac cycle depending on the heart
rate. The angular slice spacing was 5 degrees resulting in 36 image slices for each
3D frame.
2.2Segmentation and Shape-Based Tracking
The endocardial and epicardial surfaces were extracted interactively[11] and then
sampled to 0.5 voxel resolution. Next, curvatures are calculated and the shape
based tracking algorithm applied to generate a set of initial matches and con-
fidence measures for all the points on the surface. Given a set of displacement

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vector measurements umand confidence measures cmwe model theses estimates
probabilistically by assuming that the noise in the individual measurements as
normally distributed with zero mean and a variance σ2equal to
sumptions result in a measurement probability of the form:
1
cm. These as-
p(um|u) =
1
√2πσ2e−(u−um)2
2σ2
(1)
2.3Modeling the myocardium
The left ventricular myocardium is modeled using a biomechanical model. We
use a transversely linear elastic model which allows us to incorporate information
about the preferential stiffness of the tissue along fiber directions from [5]. These
fiber directions are shown in figure 1. The model is described in terms of an
internal or strain energy function of the form:
W = ??C?
(2)
where ? is the strain and C is the 6 × 6 matrix containing the elastic constants
which define the material properties (see [12,13] for the details.)
Fig.1. Left: Fiber direction in the left ventricle as defined in Guccione[5]. Right:
Volumetric Model of the LV consisting of hexahedral elements.
The left ventricle of the heart is specifically modeled as a transversely elastic
material to account for the preferential stiffness in the fiber direction, using the
following material matrix C:

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C−1=



1
Ep
−νp
Ep
−νp
Ep
1
Ep
−νfp
Ef
−νfp
Ef
1
Ef
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
−νfpEf
Ep
0
0
0
−νfpEf
Ep
0
0
0
2(1+νp)
Ep
0
0
Gf
0
Gf



(3)
where Ef is the fiber stiffness, Ep is cross-fiber stiffness and νfp,νp are the
corresponding Poisson’s ratios and Gfis the shear modulus across fibers. (Gf≈
Ef/(2(1+νfp)). The fiber stiffness was set to be 3.5 times greater than the cross-
fiber stiffness [5]. The Poisson’s ratios were both set to 0.4 to model approximate
incompressibility. Using a Markov Random Field analogy, we can describe the
model probabilistically using an equivalent prior probability density function
p(u) of the Gibbs form:
p(u) = k1exp(−W(C,u))(4)
2.4Integrating Model and Data
Having defined both the data term (equation 1) and the model term (equation 4)
as probability density functions we naturally proceed to write the overall prob-
lem in a Bayesian estimation framework. Given a set of noisy input displacement
vectors um, the associated noise model p(um|u) (data term) and a prior prob-
ability density function p(u) (model term), find the best output displacements
ˆ u which maximize the posterior probability p(u|um). Using Bayes’ rule we can
write:
ˆ u =argmax
u
p(u|um) =argmax
u
?p(um|u)p(u)
p(um)
?
(5)
Taking logarithms in equation (5) and differentiating with respect to the dis-
placement field u results in a system of partial differential equations, which we
solve using the finite element method [3]. To achieve this, a volumetric model
of the LV is constructed using hexahedral elements as shown in figure 1. For
each frame between end-systole (ES) and end-diastole (ED), a two step problem
is posed: (i) solving equation (5) normally and (ii) adjusting the position of all
points on the endocardial and epicardial surfaces so they lie on the endocar-
dial and epicardial surfaces at the next frame using a modified nearest-neighbor
technique and solving equation (5) once more using this added constraint. This
ensures that there is a reduction in the bias in the strain estimates.
3Experimental Results
In this section we present validation of the image derived strains using implanted
markers and sonomicrometers as gold standards.

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3.1 Implanted Image-Opaque Markers:
Blood
Epicardium
Endocardium
Elastic String
Epicardial
Marker
Midwall
Marker
Endocardial
Marker
Fig.2. Implantation of Image-Opaque Markers. This figure shows the arrangement
of markers on the myocardium. First a small bullet-shaped copper bead attached to
an elastic string was inserted into the blood pool through a needle track. Then the
epicardial marker was sutured (stitched) to the myocardium and tied to the elastic
string. Finally, the midwall marker was inserted obliquely through a second needle
track to a position approximately half-way between the other two markers.
Methodology: To validate the image-derived strains, markers were implanted on
canine hearts as follows: First the canine heart was exposed through a thoraco-
tomy. Arrays of endocardial, midwall and epicardial pairs of markers were then
implanted as shown in figure 2. They were loosely tethered, combinations of
small copper beads (which show up dark in the MR images) at the endocardial
wall and the midwall region and small plastic capsules filled with a 200:1 mixture
of water to Gd-DTPA at the epicardial wall (which show up bright in the MR
images). Marker arrays were placed in two locations on the canine heart wall.
The location of each implanted marker is determined in each temporal frame
by first manually identifying all pixels which belong to the marker area (as the
marker ‘image’ extends to more than one voxel) and then computing the 3D
centroid of that cluster of points, weighted by the grey level1. figure 3 shows a
short-axis MR slice of the heart with the identified marker pixels shown in blue
(left). The marker centroids are shown on the right.
Results: The image-derived strains, estimated using the algorithm described in
section 2, were compared to strains derived from implanted markers2. In the case
of the markers the strains were computed as follows using only the epicardial and
endocardial markers. In each region of the LV that contained markers, groups of
either 6 or 8 markers (depending on the geometry) were connected to form wedge
1In the case of dark markers the image is first inverted.
2Note that in the estimation of the image-derived strains, we do not use any infor-
mation regarding the position of the implanted markers.

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Identified Point
Marker Centroids
Endo
Mid
Epi
Fig.3. Localization of implanted markers. Arrays consisting of 12 markers each were
placed at two positions on the left ventricle. In this figure, we show the portion of one
marker array as it intersected a short-axis MR image slice. A human observer identified
the pixels corresponding to each marker (left) and the marker positions (right) were
found by calculating centroids of these points.
R2 = 0.89
S.E.= 0.08
Fig.4. Algorithm-derived Strains vs. Implanted Marker-derived Strains. Left: Recon-
structed LV volume from cine-MRI at ED with marker positions noted as spheres.
Right: Scatter plot of principal strains derived from baseline and post-infarction cine-
MRI studies using algorithm vs. same strains derived from implanted marker clusters
at two positions in the LV wall for N = 4 dogs (There was a total of 12 usable extracted
marker arrays).

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or hexahedral elements. Given the known displacements, we then calculated
the strains between these markers. These strains were compared to the average
strains in the elements contained within each marker array. We used principal
strains, (these are the eigenvalues of the strain tensor[9]), as the marker arrays
where large and included elements where the cardiac-specific directions varied
widely.
Comparison results are shown in figure 4 for N = 4 dogs (2 acquisitions per
dog, one pre-occlusion and one post-occlusion). We observe a strong correlation
of the principal strain values (r2= 0.89).
3.2Sonomicrometers
Methodology: In the case of the 3DE images we validate the strain estimates
using implanted sonomicrometers (Sonometrics Corporation, London Ontario,
Canada.) The canine heart is again first exposed through a thoracotomy. With
the aid of an implantation device constructed in our laboratory, two crystal-
arrays each consisting of 12 crystals (3 sub-epicardial, ∼2.0 mm, 6 mid-wall and
3 sub-endocardial, ∼0.75 mm diameter) were placed in the heart wall. Finally,
to define a fixed coordinate space, 3 crystals attached to a plexi-glass frame were
secured in the pericardial space under the right ventricle.
Digital sonomicrometry employs the time of flight principal of ultrasound to
measure the distance between a transmitter and a receiver. A total of 27 crystals
are used in each study. The distances between all possible pairs of crystals are
recorded along with LV and aortic pressure at a sampling frequency of greater
than 125 Hz. There are a number of preprocessing steps involved in obtaining the
positions of the crystals over time from the crystal to crystal pair lengths. These
are described by Dione[4] (see also Ratcliffe[15].) The efficacy of this technique
was illustrated by additional work [10] that showed that the distances obtained
with sonomicrometers compared favorably (r = 0.992) with those obtained using
the more established technique of tracking implanted bead displacements using
biplane radiography.
Results: We compared our image-derived strains, estimated using the algorithm
described in section 2, to concurrently-estimated strains derived from sonomi-
crometers at several positions in the LV myocardium in the same dogs. The
sonomicrometers were visually located from the images and the two nearest sec-
tors [12] of algorithm-derived strains were selected for comparison purposes. The
comparison of the principal strain components in two separate regions for a set
of 3 studies showed a strong correlation (r2= 0.80). Here we again compare the
principal strains as in the last section. A scatter plot of algorithm-derived princi-
pal strains versus sonomicrometer-derived principal strains is shown in figure 5.
4 Conclusions
In this work, we describe methodology to validate 3D image-derived cardiac de-
formation estimates, and we use it to compare the output of our algorithm[12]

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R2 = 0.80
S.E.= 0.11
Arrays of
Sonomicrometers
Myocardium
Left−Ventricular
Blood−Pool
Fig.5. 3DE Algorithm-Derived Strains vs. Sonomicrometer-derived Strains. Left:
Placement of arrays of sonomicrometers in the Left Ventricular Wall (the crystals
are shown enlarged, they are barely visible in the images.) Right: Scatter plot of
principal strains derived from these 3DE studies using the algorithm vs. same strains
derived from sonomicrometer arrays (12 crystals in each cluster) at two positions in
the left ventricular wall. Note the high correlation between the two sets of strain values
(r2= 0.80).
to implanted markers and sonomicrometers. We demonstrate good correlation of
the image-derived estimates of strain, from both cine-MRI and 3D Echocardiog-
raphy, to strains estimated from implanted markers and sonomicrometers, used
as gold standards. At this point we only compare transmurally averaged principal
strains between end-diastole and end-systole. In the future we hope to extend
the validation to account for more time points and also to non-transmurally
averaged cardiac specific strains.
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