ABSTRACT We introduce a novel unifying approach to jointly estimate the motion and the dynamic images in first pass cardiac perfusion MR imaging. We formulate the recovery as an energy minimization scheme using a unified objective function that combines data consistency, spatial smoothness, motion and contrast dynamics penalties. We introduce a variable splitting strategy to simplify the objective function into multiple sub problems, which are solved using simple algorithms. These sub-problems are solved in an iterative manner using efficient continuation strategies. Preliminary validation using a numerical phantom and in-vivo perfusion data demonstrate the utility of the proposed scheme in recovering the perfusion images from considerably under-sampled data

Christophe Chefd&apos

Li Zhang

Mariappan Nadar

Mathews Jacob

Sajan G Lingala

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UNIFIED RECONSTRUCTION AND MOTION ESTIMATION IN CARDIAC PERFUSION MRI
Sajan G Lingala!, Mariappan Nadar†, Christophe Chefd’hotel†, Li Zhang† and Mathews Jacob!
!Department of Biomedical Engineering, University of Rochester, NY, USA
†Siemens Corporate Research, Princeton, NJ, USA
ABSTRACT
We introduce a novel unifying approach to jointly estimate
the motion and the dynamic images in first pass cardiac per-
fusion MR imaging. We formulate the recovery as an energy
minimization scheme using a unified objective function that
combines data consistency, spatial smoothness, motion and
contrast dynamics penalties. We introduce a variable splitting
strategy to simplify the objective function into multiple sub
problems, which are solved using simple algorithms. These
sub-problems are solved in an iterative manner using efficient
continuation strategies. Preliminary validation using a numer-
ical phantom and in-vivo perfusion data demonstrate the util-
ity of the proposed scheme in recovering the perfusion images
from considerably under-sampled data.
1. INTRODUCTION
Myocardial first-pass perfusion schemes track the contrast
changes resulting from the passage of a contrast agent through
the heart. Since it enables the quantitative evaluation of both
ischemic and non-ischemic heart disease, perfusion imag-
ing is a key component of most clinical cardiac MRI ex-
ams. The standard method is to collect the k-space data of
each timeframe during the diastole phase of each heartbeat,
where the cardiac motion is minimal. The risk of peripheral
nerve stimulation and hardware limitations often restricts the
spatio-temporal resolution and spatial coverage, resulting in
dark-rim artifacts, underestimation of the transmural extent of
defects, inability to visualize all heart regions and inaccurate
fitting of the kinetic model.
Conventional k ! t space acceleration techniques, which
are originally designed for breath-held acquisitions, exploit
the sparsity of the signal in the x ! f space. Since the data’s
sparsity in the x ! f space is disturbed due to non-periodic
respiratory motion and temporal contrast variations, the util-
ity of these schemes are limited in perfusion imaging. To
overcome these problems, Pederson et. al proposed to unify
the reconstruction of the images and the motion compensa-
tion into a single algorithm [1]. They represented the contrast
variations using a parametric perfusion model, while motion
was modeled as a modulation of a 2-D displacement field,
This work is supported by NSF award CCF-0844812.
which is estimated from two images acquired at end inspira-
tion and end expiration. The fewer degrees of freedom in this
model may be restrictive in practical perfusion imaging ap-
plications. Jung et. al, have extended their k ! t FOCUSS
scheme with motion estimation and compensation for cardiac
cine MRI [2]. This scheme approximate the dynamic images
as the deformation of fully sampled reference frames, col-
lected before and after the dynamic acquisition. The residu-
als are then reconstructed from under-sampled k-space data
using k ! t FOCUSS. Unlike cine MRI, the contrast of the
dynamic images are significantly different from the reference
images. Hence, the subtraction of the deformed reference im-
age may not generate sparse residuals. Moreover, more com-
plex mutual information similarity measures may be needed
for the registration. Fessler recently introduced an elegant en-
ergy minimization framework to reconstruct a static image of
a moving organ from its measurements [3]. The formulation
of the problem as a unified energy minimization scheme en-
ables the appreciation of the tradeoffs in the modeling. How-
ever, this scheme is not designed to recover image time series
with dynamic contrast variations.
We introduce a novel energy minimization formulation
for the joint estimation of the deformation and the images
in myocardial perfusion imaging. We model the respiratory
motion as an elastic deformation, whose parameters are es-
timated from the data. We assume the contrast variations
due to bolus passage to be smooth in time, once the respi-
ratory motion is removed. This model is considerably less
constrained than the parametric scheme used in [1]. Since
we do not model the dynamic frames as deformations of pre-
contrast reference images, our approach is robust to contrast
variations due to bolus passage, in comparison to [2]. Our
motion estimation scheme estimates the deformation by reg-
istering the dynamic data to a reference dataset that is free of
respiratory motion, which is derived from the measurements
themselves. We introduce a variable splitting frame work with
continuation to minimize the energy function, and thus derive
the deformation and the dynamic images. The validation of
the proposed scheme using numerical phantom and in-vivo
data demonstrate the feasibility in accurately recovering per-
fusion MRI data from considerably under-sampled data.
2. THE OBJECTIVE FUNCTION
The temporal contrast variations of any voxel in perfusion
MRI is reasonably smooth if the subject is holding his breath.
In this context, a simple approach to recover the dynamic
imaging data from under-sampled measurements is to exploit
the spatio-temporal smoothness of the signal. However, the
temporal smoothness is significantly disturbed in the presence
of respiratory motion. Simple temporal smoothing will aver-
age out the contrasts from different spatial regions and hence
result in artifacts and degraded performance. In this context,
we propose to simultaneously recover the motion D and the
dynamic images f(x, t) from under-sampled data b(k, t) us-
ing the following energy minimization scheme:
{f,D}! = argmin
f,D
"A(f)! b"2! "# $
data consistency
+! "(Df)t"2! "# $
temporal regln.
+µ "#f"!1! "# $
spatial regln.
.
(1)
Here, A is the Fourier sampling operator, D is the deforma-
tion model (Df is the motion compensated version of f ), #f
and ()t denotes the spatial gradient and first order temporal
derivative respectively. Note that the temporal regulariza-
tion term penalizes the roughness of the motion compensated
dataset Df , rather than f . This approach ensures that the re-
sulting temporal smoothing does not result in artifacts. The
rapid variations in f , induced by respiratory motion will be
captured by D, which is also estimated during the joint es-
timation scheme. This approach is very different from [2],
where they estimate the deformation between a reference im-
age and the dynamic images. We model the deformation D as
a diffeomorphism [4, 5], which is considerably more flexible
than the constrained model used in [1].
3. THE OPTIMIZATION ALGORITHM
We reformulate the unconstrained problem in (1) to a con-
strained one in (2) by introducing auxiliary variables g and
w:
arg min
f,D,g,w
"A(f)! b"2 + ! "(g)t"2 + µ " w"!1 ,
s.t. Df = g and #f = w (2)
The main motivation for this approach is that the constrained
scheme is easier to solve than (1). We then relax the equality
constraints and penalize their violations by quadratic func-
tions as:
arg min
f,D,g,w
"A(f)! b"2 + ! "(g)t"2 +
"1
2
"Df ! g"2 +
µ "w"!1 +
"2
2
"#f ! w"2 (3)
When the penalty parameters "1 and "2 approach $, the
solution of (3) tends to that of (2) and hence (1). The cost in
(3) has to be now minimized with four variables f,D, g and
w. Although, this appears complicated, we solve for each of
the variables in an alternating fashion similar to [3], assum-
ing the others to be fixed. Thus, the algorithm involves the
following four steps:
Derivation of g: With f,D and w fixed, the minimization of
(3) with respect to g is a quadratic smoothing problem, which
is solved exactly in the temporal Fourier domain:
F(g) = F(Df)
(1 + 2"#1#
2)
(4)
where F is the temporal Fourier operator and # the temporal
frequency. Thus, g is a low pass filtered version of the de-
formed object Df . For a fixed !, low value of "1 would have
a stringent cut off passing only components of extremely low
frequency and vice-versa.
Derivation of w: To solve this problem, we adapt the multi-
dimensional shrinkage solution derived by [6] to yield,
w =
f
#f
%
#f ! µ
"2
&
+
(5)
where (.)+ is the shrinkage operator given by,
(.)+ =
'
. if . % 0
0 else (6)
Derivation of f : We collect the terms in (3) with f and
rewrite the minimization problem as
argmin
f
"A (f)! b"2 + "1
2
"f !D"1g"2 + "2
2
"#f ! w"2
Here, we assume the Jacobian of the transformation D to be
approximately equal to one. Minimizing the above expres-
sion, we get
A!A(f)+ "1
2
f+
"2
2
#2f = AT b+ "1
2
D"1g+ "2
2
#Tw (7)
As shown by the authors in [7], the above problem can be
efficiently solved in the spatial Fourier domain if the operator
A obtains the samples on a Cartesian grid.
Derivation of the deformation D: Assuming all the vari-
ables in (3) to be known, we solve for the deformation as
D! = argmin
D
"Df ! g"2 (8)
This is a registration problem, where the dynamic scene
f(x, t) is registered frame by frame with a reference scene
g(x, t). Note that the reference series is derived from the
measurements itself as shown in (4); we do not require addi-
tional high resolution reference frames. The temporal profiles
of the reference dataset g is significantly more smooth com-
pared to f . This approach enables us to decouple the effects
(a) Ideal: perfusion and mo-
tion
(b) Ideal: only perfusion (c) Direct IFFT (d) Estimated f (e) Estimated Df
Fig. 1. Performance demonstration of the proposed unifying frame work on PINCAT data: (top row): A spatial frame, (bottom
row): image time series profile through the line in (a). The reconstructions in f (d) is devoid of artifacts (which are present in
the zero filled direct IFFT reconstruction (c)) and is close to the ideal data (a). The motion is corrected in Df and is close to the
ideal motion free data (b) except for small residual inaccuracies
of smooth perfusion induced contrast changes and the more
rapid changes resulting from respiratory motion.
We use a fast flexible deformation algorithm by [4, 5] in
our experiments. It uses a Gaussian based regularizer which
penalizes irregular deformations. The degree of flexibility of
the deformation could be controlled by tuning the width of
the regularizer. Specifically, it estimates the deformation such
that the vector field that gives for each pixel on the reference
its corresponding location on the target image is smooth. It
uses a template propagation method to estimate large defor-
mations by the composition of small displacements. A multi-
scale approach is used to reduce the non convex problem to
multiple simpler problems that can be solved efficiently. The
implementation of the algorithm was highly optimized for
speed and practical usage.
For a fixed "1 and "2, we solve the above four individual
sub problems in an alternating manner until convergence is
met. Note that one requires a high value of "1 and "2 for the
constraints in (2) to hold; however at high values of "1 and "2,
the problem becomes ill conditioned and would result in poor
convergence properties. On the other hand, choosing a low
value for "1 and "2, would ensure fast convergence but with
the accuracy compromised. We incorporate a continuation
strategy that has been used in related works [7] that uses the
variable splitting frame work. Specifically, we start with low
values of "1 and "2, to solve the modified cost in (3). Once
the stopping criterion is met, we then increment "1 and "2 and
again solve (3). This is repeated until the constraints in (2)
are satisfied. This continuation strategy improves the overall
convergence rate significantly while maintaining the desired
accuracy levels.
4. RESULTS AND DISCUSSIONS
We perform our experiments on the Physiologically Im-
proved Non uniform CArdiac Torso (PINCAT) phantom [8]
and in-vivo perfusion data. We focus on a single slice of
the phantom (128x128), which has the cross section of the
heart and consider free breathing dynamic contrast enhanced
images with a temporal resolution of one heart-beat (&1
sec) with 70 frames. The in-vivo data was acquired with
a FLASH-saturation recovery (TR/TE=2.5/1 ms, saturation
recovery time=100 ms) sequence on a Siemens 3T scanner at
the University of Utah in accordance with the institute’s re-
view board. A single short axis slice of size phase/frequency
encodes/time frame:(90x190x70) of the in-vivo data was con-
sidered. The subject was not able to hold his breath during
the entire imaging duration and hence the data had resid-
ual breathing motion. We use an equiangular pseudo-radial
trajectory to perform the sampling in the k! t space. The tra-
jectory is rotated by a certain angle in each temporal frame.
We consider an under-sampling factor of approximately 5
which corresponds to 25 radial spokes/frame for the PINCAT
data and 20 in the in-vivo data set.
In figure 1, we show the validations on the PINCAT data.
It is seen that the reconstructed dataset f closely matches the
fully-sampled data. This is in sharp contrast to the standard
zero filled IFFT reconstruction. The performance of the mo-
tion correction task can be evaluated by comparing the im-
age time profiles of f and Df . During the iterations, we
used a strategy to initially correct for simpler deformations
and as the algorithm converge, we correct for more complex
deformations. This is very similar to the coarse to fine correc-
tion strategies popularly used in many registration algorithms.
Specifically, in the initial iterations, as the floating dynamic
scene, f is noisy due to the streak artifacts, we use a highly
rigid model, which is robust to noise but less accurate to ob-
(a) Fully sampled (b) Direct IFFT (c) f (d) Df
Fig. 2. Performance demonstration on in-vivo data: (top row):
A spatial frame, (bottom row): Image time profile through
the cross section marked by the line in (a). The estimated
quality of f is close to that of the fully sampled data, resolving
the artifacts in the zero filled direct IFFT reconstruction. The
motion (ripples in the first three columns) has been corrected
in Df (last column)
tain a decent first guess estimate of the deformations. Note
that these artifacts are filtered out in the reference scene, g (4).
As the iterations proceed, the floating data gets more cleaner,
hence we incorporate a more flexible deformation algorithm
thats more accurate to update the estimate of D.
Figure 2 shows the verification of the algorithm on in-
vivo data. Similar to the PINCAT results, we observe good
correlation of f with the fully sampled data, in contrast to
the direct IFFT reconstructions. To estimate the motion, we
followed the same strategy of adaptively tuning the flexibility
of the deformation algorithm through the iterates.
This work presented the formulation of the unifying re-
construction and motion estimation frame work and prelim-
inary results at moderate under-sampling factors. A more
comprehensive analysis of the utility of the proposed frame
work at higher under-sampling factors in comparison with re-
construction schemes without motion compensation is a study
which we plan to investigate in the future. Our current model
cannot account for through plane deformations. To address
this, we plan to investigate the use of 3D registration algo-
rithms in the future by extending our frame work to 3D ac-
quisitions.
5. CONCLUSION
We introduced a novel framework to unify reconstruction and
motion estimation steps in first pass cardiac perfusion MRI.
The unifying framework had a global objective function with
penalties on the spatial smoothness, motion and the contrast
dynamics. The global objective was solved by a variable
splitting strategy with continuation, where we decompose the
complex cost to simpler sub problems. We had sub problems
that took well defined forms: low pass filtering of the de-
formed object, TV shrinkage, analytical Fourier replacement
and a l2 minimizing problem that could be dealt by a standard
registration tool. Adaptive tuning strategies to update the es-
timates of the deformations at different points of iterations
were used. Experiments and initial findings on the PINCAT
phantom data and in-vivo data demonstrated the usefulness
of the proposed scheme. Further validations on multiple in-
vivo data sets are required to thoroughly study the robustness,
efficiency and accuracy of the developed scheme.
6. ACKNOWLEDGEMENT
We thank Dr. Edward Dibella of University of Utah for pro-
viding the in-vivo data used in this study.
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Unified reconstruction and motion estimation in cardiac perfusion MRI

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Unified reconstruction and motion estimation in cardiac perfusion MRI

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