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KING’S MEDICAL ENGINEERING CENTRE Philip Batchelor’s Work
in
MR-Reconstruction
Tobias Schaeffter
Division of Biomedical Engineering and Imaging Sciences
King’s College London
Wellcome Trust-EPSRC Medical Engineering Centre
Reconstruction
of
KING’S MEDICAL ENGINEERING CENTRE Undersampled Data
Magnetic Resonance in Medicine 54:1273–1280 (2005)
Matrix Description of General Motion Correction Applied
to Multishot Images
P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and
D. Larkman3
Motion of an object degrades MR images, as the acquisition is rection method could be used to spatially transform the
time-dependent, and thus k-space is inconsistently sampled. ghosted image by the transformation corresponding to a
This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot,
restrictive assumptions on the type of motions, for example, and repeat this operation for all shots (this is a version
that it is a translation or rotation, and use special properties of
of the method used in (1)). We could then rebuild an
k-space for these transformations. Such methods, however,
image by inverse Fourier transform. This method is in
cannot be generalized easily to nonrigid types of motions, and
even rotations in multiple shots can be a problem. Here, a general incorrect, as shown by the difference between
method is presented that can handle general nonrigid motion translations and rotations. Correcting translation repointwise phase changes in k-space. On the
models. A general matrix equation gives the corrupted image quires only
Magnetic Resonance in Medicine 63:1247–1257 (2010)
from the ideal object. Thus, inversion of this system allows us to other hand, correcting rotations requires knowledge of
get the ideal image from the corrupted one. This inversion is the data at neighboring k-space positions and these are
possible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier
the conjugate gradient method. A faster but empirical inversion rotation theorem, we would need to “synchronize”
is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions
ulated three-dimensional affine data and two-dimensional pul- such as nonrigid deformations cannot have a simple
sation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show
tion. All examples are
where
the object moves
2
2
2
2
Freddy
Odille,1* multishot
Sergio images
Uribe,
Philip
G. Batchelor,
Prieto,
Tobias
Schaeffter,
and
that itClaudia
is possible
to correct
complicated
motions,
inbetween shots. The results indicate that it is now possible to
1
David
Atkinson
correct for
nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical
many types of patient motion, although computation times re- description of the problems involved; the motion cormain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on
This
paper describes
an acquisition and reconstruction strategy reduce
gradient
interference
(4,6,7),
the magnetohydrodythe space
of images.
Thus,
inversion
of this matrix
Wiley-Liss,
Inc.
for cardiac cine MRI that does not require the use of electro- namic
effect
is difficult
to model
or correct,
especiallyisatof
correct
the motion’s
effects.
This approach
Key words: motion correction; ghosts; multishot; conjugate should
cardiogram or breath holding. The method has similarities with high
fields, asinterest,
it increases
the amplitude
static Bon
theoretical
but with
its practical
value of
depends
gradient; auto focus
0
self-gated techniques as information about cardiac and respi- field strength. The use of ECG also requires more patient
how easily we can find a solution of the linear system. It
ratory motion is derived from the imaging sequence itself; here,
Motion of an object can degrade MR images and imposes preparation
time.
Second,
holding
be a difficult
turns out that
with
somebreath
careful
linear can
algebra,
not only
by acquiring the center k-space line at the beginning of each
constraints on scan parameters that can in turn compro- task
are for
wemany
able to
invert or
in infants
a generalized
but alsodue
this
patients
and maysense,
be imperfect
segment of a balanced steady-state free precession sequence.
mise image
quality. The cause
the degradation
is that
can be over
donetime
efficiently
in practice.
For this we
organs drifting
(8). Another
issue associated
However,
the reconstruction
step isoffundamentally
different:
a toinversion
the acquisition
is time-dependent,
the Fourier
transuse breath
the LSQR
algorithm,
which
is a robust
implementaholding
is that the
physiology
is modified
comgeneralized
reconstruction
by inversionand
of coupled
systems
is with
form
of
the
image
seen
during
acquisition
changes
due
to
tion
of
the
conjugate
gradient
of
the
normal
equation
pared
to
the
“normal”
free-breathing
state.
Therefore,
it
is
used instead of conventional gating. By correcting for nonrigid
the deformation
of themotion,
object.generalized
This causes
inconsistencies
(see (4)).relevant to attempt to capture cardiac functional
cardiac
and respiratory
reconstruction
by clinically
Magnetic Resonance in Medicine 62:1331–1337 (2009)
inversion
of coupled
systems
all acquired data, information
in k-space
and hence
ghosts(GRICS)
in the uses
image.
This leaves
of finding what motion actually
in the
freequestion
breathing.
whereas
gatingmotion
rejects data
acquiredmethods
in certainmake
motionassumpstates.
Standard
correction
happened.
Different
costbeen
functions
have
been designed
Several strategies
have
proposed
in order
to addressto
The
method
relies
onofthe
processing
analysisthat
of the
tions
on the
type
motions,
for and
example,
it isk-a these
quantify
how
much anreal-time
image has
been corrupted
(1,tech5, 6).
issues,
including
imaging,
self-gated
space
central or
linea data:
local and
information
from a 32-channel
translation
rotation,
use formulas
on Fourier niques,
We explore
functions
conjunction
with
and different
combinedcost
cardiac
and in
respiratory
gating.
cardiac
coil is
in order
to automatically
extract
eigentransforms
toused
correct
the data
(1–3). We assume
here
that Real-time
our motion
correction.
Optimization
of
suchcardiac
cost functions
imaging
has
been
shown
to
allow
imagmodes of both cardiac and respiratory motion. In the GRICS
these data are acquired in shots. When the data posi- means repeating the matrix inversion iteratively. However,
framework, these eigenmodes are used as driving signals of ing during free breathing (9–11) but implies a comprotions at each shot are known, an empirical motion cor- inverting matrices repeatedly may be prohibitive even if it
a motion model. The motion model is defined piecewise, so mise between spatiotemporal resolution and signal-tois practicable
onItaisone-off
basis.
We therefore
also
invesratio (SNR).
possible
to combine
real-time
images
that each cardiac phase is reconstructed independently. Results noise
tigate
the useframes,
of the using
empirical
method
described
above.
from six healthy volunteers,
with various slice orientations, show
different
nonrigid
image
registration
in
1 C. Prieto,1 P.G. Batchelor,1 S.from
1 D. Atkinson,
2 H. Eggers,
3
1 Boubertakh,
R.
Uribe,
The
matrix
equationfor
allows
us to find
when
this
approxiimproved
image quality
comparedUniversity
to combined
and order
Medical Physics
& Bioengineering,
Collegerespiratory
London, London,
to compensate
respiratory
motion,
and
then
proUnited
Kingdom.
4 M.S.
5 R.S.2010.
1* show three-dimensional random
mation
is correct. We
cardiacSørensen,
gating. Magn
Reson Hansen,
Med 63:1247–1257,
© 20101 and
T.S.
Razavi,
T.
Schaeffter
duce
SNR-enhanced
cardiac
cine images (12). However,
2
DepartmentInc.
of Electrical Engineering, Pontifica Universidad Católica de
Wiley-Liss,
affine and pulsatile nonrigid motion corrections on simu-
Model-Based Reconstruction for Cardiac Cine MRI
Without ECG or Breath Holding
Liver Whole-Heart Imaging Using Undersampled Radial Phase
Encoding (RPE) and Iterative Sensitivity Encoding
(SENSE) Reconstruction
Chile, Chile.
this technique still requires the recording of an ECG sig-
lated data and an example of nonrigid correction of in vivo
Key
words: Artifacts; cardiac imaging; gating; motion correc3
Imaging Sciences, Imperial College London, London, United Kingdom.
nal during real-time scanning, and its extension to three
data (moving
tion;
navigators;
reconstruction
Whole-heart
isotropic
nonangulated cardiac magnetic reso- phology
of thelegs).
ventricle and great vessels. This removes
*Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengi- dimensions
remains challenging. Self-gated techniques can
nance
(CMR)
is becoming
an important
protocol
simplifying
neering,
University
College London,
Gower Street,
LondoninWCIE
6BT, UK, the need for time-consuming slice planning by a skilled
remove
need for either ECG (13,14) or breath hold
E-mail:
[email protected]
MRI,
since
it reduces
theheart,
need referred
of cumbersome
planning
of operator.the
The main problem of whole-heart acquisitions is
Dynamic
imaging
of the
to as cine
imaging,
or their combination (16). Self-gating relies on the
in part
as abstracts
at the 2nd International
on ParallelTHEORY
angulations.
However
the
acquisition
of Workshop
whole-heart
MRI (15),
rather
long acquisition times, particularly when acquiring
isPresented
theZürich,
reference
MRIMIUA,
method
forSept.
thetimes
study
of
cardiac
funcMRI,
2004, andto
2004.
of information about cardiac and/or respiratory
are
prohibitive
the London,
large fields
of view (FOVs) and the extraction
large
imaging
volumes
at high motion-corrupted
image resolutions.MR
Recently,
tion
patientsdue
with heart
standard
technique,
We need
a method
to handle
images
GrantinSponsor:
EPSRC;
Grant failure.
Numbers:The
GR/S30184,
AF/001381;
Grant motion
from
the imaging
data. This information was
shown
high
spatial resolution
required
for depicting
small
structures
Sponsor: Chilean
FONDECYT;
Grant Number:
based
Cartesian
scanning
available
on clinical
scanners,
is a 1030570.
segmented balanced fast volumetric
weretrospective
know acquisitions
the motion;
this
is aon
necessity
to correct
for
and vessels. To address this problem, we propose a three- towhen
allow
or prospective
synchronization
to
Received 14 March
2005;
revised 8sequence
June 2005; acquired
accepted 10during
June 2005
using
parallel
imaging
techniques
(2,3) were
introduced
steady-state
free
precession
susunknown
motions
an optimization
method.
Some
dimensional
(3D) acquisition scheme that combines Cartesian either the cardiac
orwith
respiratory
cycle. In particular,
the
DOI 10.1002/mrm.20656
for
this
purpose
(4).
However,
the
radio
frequency
(RF)
coil
pended
respiration
(1).
Images
from
each
cardiac
phase
are
MR techniques,
navigators,
to find
the
sampling
the9readout
direction
with InterScience
an undersampled
radial method
Published in
online
September
2005 in Wiley
(www.interscience.
in Buehrersuch
et al.as(16)
uses the allow
centralus
point
in the
used
in thosebut
studies
only modest
accelerareconstructed
by retrospective
gating,
usingundersampling
information array
motion
directly,
then allows
also require
an algorithm
to
wiley.com).
scheme
in the phase-encoding
plane.
Different
Magnetic
Resonance
in Medicine
–949 (2007)
k-space, which
can
be thought
of as a 57:939
zero-dimensional
from
electrocardiogram
(ECG) (2).with
In an
some
circumtion factors of 2 to 3 and thus the total scan time is still
patterns
were
investigated
in combination
iterative
sen- 1273
© 2005the
Wiley-Liss,
Inc.
navigator signal; the method in Uribe et al. (15) uses the
stances
however,
the method
may suffer
from
several issues
sitivity
encoding
(SENSE)
reconstruction
and
a 32-channel
car- rather long (up to 20 min). With the introduction of new
center
k-space
line
(CKL),
which
can
be
thought
of
as
a
associated
with
the use ofmaps
the ECG
breath hold.
First, MR scanner technology with up to 32 receive channels and
diac
coil. Noise
amplification
wereor
calculated
to compare
(1D) coils,
navigator
signal, producing
prothe
of distorted
the different
patterns
iterative
theperformance
ECG signal is
during
the using
MRI scan
dueSENSE
to the one-dimensional
corresponding receive
the signal-to-noise
ratioa(SNR)
the two-dimensional
or three-dimensional
image
reconstruction.
The radial effect
phase-encoding
(RPE) scheme and
was jection
magnetohydrodynamic
(3), to radiofrequency,
can be of
potentially
increased and
further acceleration
of
onto thebecomes
frequency-encoding
axis.studies
With all
these
implemented
a clinical
MR switching
scanner and(4,5).
tested
on phantoms
to magnetic on
field
gradient
Although
sig- content
these protocols
feasible. Some
have
althe self-gating
canmagnetic
be acquired
with
and
volunteers.
Thehave
proposed
method exhibits
better
nal healthy
processing
methods
been proposed
in order
to techniques,
ready addressed
whole-heartsignals
coronary
resonance
1
2
2
3
1
image
quality
even
for
high
acceleration
factors
(up
to
12)
in
minimal
distortion
of
steady-state
during
a massively
balanced
Claudia Prieto, Philip G. Batchelor, D.L.G. Hill,
Joseph
V. Hajnal,
Marcelo
Guarini,
angiography
(MRA)
in the
a single
breathhold
using
comparison to Cartesian acquisitions.
Magn Reson Med 62: steady-state free precession sequence. However, combined
1*
parallel imaging (5). Nevertheless, these breathholds are
and
Pablo2009.
Irarrazaval
1331–1337,
© 2009 Wiley-Liss, Inc.
cardiac and respiratory gating has several limitations: (i)
Reconstruction of Undersampled Dynamic Images by
Modeling the Motion of Object Elements
still quite long and the spatial resolution is relatively low.
Key words: whole-heart MRI; radial-Cartesian sampling; itera- it is relatively inefficient as data from undesired respiraIn order to image the complete heart with higher spatial
tive
SENSE
reconstruction;
parallel
32 channel
coil; tory phases are thrown away; (ii) if respiratory motion is
Dynamic
MRI
is restricted
due
to University
the imaging;
timeCollege
required
to obtain
1
missing adata
by exploiting
the high acquisition
spatiotemporal
resolution,
free-breathing
whole-heart
using
Centre for
Medical
Image Computing,
London,
London, the
radical
phase
enough
data
to encoding
reconstruct the image sequence. Several un- correlation
United Kingdom.
not
reproducible
from
breathing
cycle
to another,
the
of dynamic
sequences
or
from
prior
informaparallel
imaging
in
twoone
directions
was
proposed
(6). How2
dersampled
reconstruction
techniques
have been
proposed
to tion.
Division of Imaging
Sciences, King’s
College London,
London,
United Kingefficiency
decreases
further and
residual
artifacts
ever, although
a 32-channel
coil(iii)
array
was used,
the may
optiCardiac
resonance
(CMR)
has become
dom. themagnetic
reduce
acquisition
time. In most
ofimaging
these techniques
the a occur as motion within the acceptance window is not corTraditional
approaches
operate
on aindiscrete
k-t space
mal
phase encoding
directions
result
a moderate
accelclinically
useful
tool
in
noninvasive
imaging
of
cardiovasGrant
sponsor:
Engineering
and
Physical
Sciences
Research
Council;
Grant
nonacquired data are recovered by modeling the temporal inrected,
which
imposes
a tradeoff
between
quality
and
either
treatof
each
frame
separately
or image
consider
the
eration
factor
4.
number:diseases.
UK
EP/E001564.
cular
The
widespread
of cardiac
howformation
asEPSRC
varying
pixel
intensitiesuse
represented
in MRI,
time or
in
and
acquisition
efficiency
(16). Cartesian
*Correspondence to: Freddy Odille, CMIC, Malet Place Engineering Build- temporal
information
as time-varying
pixel intensities
repAlternatively
undersampled
acquisitions
for
temporal
Here
proposenature
a new of
approach
that
ever,
is frequencies.
hampered by
thewe
complex
the multiple
ing, University College London, Gower Street, London WC1E 6BT, United
In thisinwork,
propose
an alternative
strategy
that
time we
or MR
in
temporal
frequencies.
Therefore,
contrast-enhanced
angiography
(7) as well
as radial
recovers
missing
dataMR
through
a motion
estimation
of need
the resented
two-dimensional
(2D)
scanning
protocols
and the
Kingdom.the
E-mail:
[email protected]
is
more
efficient
than combined
cardiac-respiratory
selfacquisitions
for
whole-heart
MRI
were
proposed
to
reduce
each
pixel
is
considered
in
a
constant
position
over
time.
object
elements
(“obels,”
or
pieces
of
tissue)
of
the
image.
This
Received
9
June
2009;
revised
2
October
2009;
accepted
6
November
2009.
for highly individualized planning procedures. Wholegatingmethods
andtime
more
generally
applicable
real-time
imagmethod
assumes that an obel displacement through the se- These
(8,9).
In keyhole
these
acquisitions
the readout
diinclude
(8,9),than
reduced
encoding
DOI 10.1002/mrm.22312
heart
isotropic
nonangulated CMR is becoming an impor- the scan
ing,
without
requiring
the use2D
of or
either
theobtain
ECG or
breath
quence
has
lower
bandwidth
than(www.interscience.wiley.com).
fluctuations in pixel intensiPublished
online
in Wiley
InterScience
rection
is changed
in either
3D to
a (RIGR)
limited
imaging
with
generalized-series
reconstruction
tant protocol in simplifying MRI (1). Subsequent reformat- MR
ties
caused
by the
number
of projections
of the
imaging
volume.
particular
© 2010
Wiley-Liss,
Inc.motion, and thus it can be modeled with 1247
(10),
reduced
field of view
(rFOV)
(11),
hybridAtechnique
ting
of
any
slice
of
interest
can
be
obtained
from
the
3D
fewer parameters. Preliminary results show that this technique
asset
of the imaging
radial technique
is that the
point-spread funcdynamic
(12), unaliasing
by Fourier-encoding
volume
for reconstruct
qualitative (with
assessment
ofsquare
the complex
mor- for
can
effectively
root mean
(RMS) errors
tionoverlaps
(PSF) isusing
robustthe
with
respect dimension
to undersampling
(10).
the
temporal
(UNFOLD)
below 4%) cardiac images and joints with undersampling facAliased signal energy will appear only as slight streaking
tors of 8 and 4, respectively. Moreover, in the reconstruction (13), sensitivity encoding incorporating temporal filtering
artifact
and
thus
increased
pseudonoise,
whereas
under(TSENSE)
(14),
k-t
broad-use
linear
acquisition
speed-up
process
an
approximation
of
the
motion
vectors
is
obtained
for
1King’s College London, British Heart Foundation (BHF) Centre, Division of
sampling in a Cartesian acquisition will result in severe
Reconstruction
of
KING’S MEDICAL ENGINEERING CENTRE Compressed
Sensing
Undersampled Data
Magnetic Resonance in Medicine 54:1273–1280 (2005)
Matrix Description of General Motion Correction Applied
to Multishot Images
P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and
D. Larkman3
Motion of an object degrades MR images, as the acquisition is rection method could be used to spatially transform the
time-dependent, and thus k-space is inconsistently sampled. ghosted image by the transformation corresponding to a
This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot,
restrictive assumptions on the type of motions, for example, and repeat this operation for all shots (this is a version
that it is a translation or rotation, and use special properties of
of the method used in (1)). We could then rebuild an
k-space for these transformations. Such methods, however,
image by inverse Fourier transform. This method is in
cannot be generalized easily to nonrigid types of motions, and
even rotations in multiple shots can be a problem. Here, a general incorrect, as shown by the difference between
method is presented that can handle general nonrigid motion translations and rotations. Correcting translation repointwise phase changes in k-space. On the
models. A general matrix equation gives the corrupted image quires only
Magnetic Resonance in Medicine 63:1247–1257 (2010)
from the ideal object. Thus, inversion of this system allows us to other hand, correcting rotations requires knowledge of
get the ideal image from the corrupted one. This inversion is the data at neighboring k-space positions and these are
possible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier
the conjugate gradient method. A faster but empirical inversion rotation theorem, we would need to “synchronize”
is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions
ulated three-dimensional affine data and two-dimensional pul- such as nonrigid deformations cannot have a simple
sation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show
tion. All examples are
where
the object moves
2
2
2
2
Freddy
Odille,1* multishot
Sergio images
Uribe,
Philip
G. Batchelor,
Prieto,
Tobias
Schaeffter,
and
that itClaudia
is possible
to correct
complicated
motions,
inbetween shots. The results indicate that it is now possible to
1
David
Atkinson
correct for
nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical
many types of patient motion, although computation times re- description of the problems involved; the motion cormain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on
This
paper describes
an acquisition and reconstruction strategy reduce
gradient
interference
(4,6,7),
the magnetohydrodythe space
of images.
Thus,
inversion
of this matrix
Wiley-Liss,
Inc.
for cardiac cine MRI that does not require the use of electro- namic
effect
is difficult
to model
or correct,
especiallyisatof
correct
the motion’s
effects.
This approach
Key words: motion correction; ghosts; multishot; conjugate should
cardiogram or breath holding. The method has similarities with high
fields, asinterest,
it increases
the amplitude
static Bon
theoretical
but with
its practical
value of
depends
gradient; auto focus
0
self-gated techniques as information about cardiac and respi- field strength. The use of ECG also requires more patient
how easily we can find a solution of the linear system. It
ratory motion is derived from the imaging sequence itself; here,
Motion of an object can degrade MR images and imposes preparation
time.
Second,
holding
be a difficult
turns out that
with
somebreath
careful
linear can
algebra,
not only
by acquiring the center k-space line at the beginning of each
constraints on scan parameters that can in turn compro- task
are for
wemany
able to
invert or
in infants
a generalized
but alsodue
this
patients
and maysense,
be imperfect
segment of a balanced steady-state free precession sequence.
mise image
quality. The cause
the degradation
is that
can be over
donetime
efficiently
in practice.
For this we
organs drifting
(8). Another
issue associated
However,
the reconstruction
step isoffundamentally
different:
a toinversion
the acquisition
is time-dependent,
the Fourier
transuse breath
the LSQR
algorithm,
which
is a robust
implementaholding
is that the
physiology
is modified
comgeneralized
reconstruction
by inversionand
of coupled
systems
is with
form
of
the
image
seen
during
acquisition
changes
due
to
tion
of
the
conjugate
gradient
of
the
normal
equation
pared
to
the
“normal”
free-breathing
state.
Therefore,
it
is
used instead of conventional gating. By correcting for nonrigid
the deformation
of themotion,
object.generalized
This causes
inconsistencies
(see (4)).relevant to attempt to capture cardiac functional
cardiac
and respiratory
reconstruction
by clinically
Magnetic Resonance in Medicine 62:1331–1337 (2009)
inversion
of coupled
systems
all acquired data, information
in k-space
and hence
ghosts(GRICS)
in the uses
image.
This leaves
of finding what motion actually
in the
freequestion
breathing.
whereas
gatingmotion
rejects data
acquiredmethods
in certainmake
motionassumpstates.
Standard
correction
happened.
Different
costbeen
functions
have
been designed
Several strategies
have
proposed
in order
to addressto
The
method
relies
onofthe
processing
analysisthat
of the
tions
on the
type
motions,
for and
example,
it isk-a these
quantify
how
much anreal-time
image has
been corrupted
(1,tech5, 6).
issues,
including
imaging,
self-gated
space
central or
linea data:
local and
information
from a 32-channel
translation
rotation,
use formulas
on Fourier niques,
We explore
functions
conjunction
with
and different
combinedcost
cardiac
and in
respiratory
gating.
cardiac
coil is
in order
to automatically
extract
eigentransforms
toused
correct
the data
(1–3). We assume
here
that Real-time
our motion
correction.
Optimization
of
suchcardiac
cost functions
imaging
has
been
shown
to
allow
imagmodes of both cardiac and respiratory motion. In the GRICS
these data are acquired in shots. When the data posi- means repeating the matrix inversion iteratively. However,
framework, these eigenmodes are used as driving signals of ing during free breathing (9–11) but implies a comprotions at each shot are known, an empirical motion cor- inverting matrices repeatedly may be prohibitive even if it
a motion model. The motion model is defined piecewise, so mise between spatiotemporal resolution and signal-tois practicable
onItaisone-off
basis.
We therefore
also
invesratio (SNR).
possible
to combine
real-time
images
that each cardiac phase is reconstructed independently. Results noise
tigate
the useframes,
of the using
empirical
method
described
above.
from six healthy volunteers,
with various slice orientations, show
different
nonrigid
image
registration
in
1 C. Prieto,1 P.G. Batchelor,1 S.from
1 D. Atkinson,
2 H. Eggers,
3
1 Boubertakh,
R.
Uribe,
The
matrix
equationfor
allows
us to find
when
this
approxiimproved
image quality
comparedUniversity
to combined
and order
Medical Physics
& Bioengineering,
Collegerespiratory
London, London,
to compensate
respiratory
motion,
and
then
proUnited
Kingdom.
4 M.S.
5 R.S.2010.
1* show three-dimensional random
mation
is correct. We
cardiacSørensen,
gating. Magn
Reson Hansen,
Med 63:1247–1257,
© 20101 and
T.S.
Razavi,
T.
Schaeffter
duce
SNR-enhanced
cardiac
cine images (12). However,
2
DepartmentInc.
of Electrical Engineering, Pontifica Universidad Católica de
Wiley-Liss,
affine and pulsatile nonrigid motion corrections on simu-
Model-Based Reconstruction for Cardiac Cine MRI
Without ECG or Breath Holding
Liver Whole-Heart Imaging Using Undersampled Radial Phase
Encoding (RPE) and Iterative Sensitivity Encoding
(SENSE) Reconstruction
Chile, Chile.
this technique still requires the recording of an ECG sig-
lated data and an example of nonrigid correction of in vivo
Key
words: Artifacts; cardiac imaging; gating; motion correc3
Imaging Sciences, Imperial College London, London, United Kingdom.
nal during real-time scanning, and its extension to three
data (moving
tion;
navigators;
reconstruction
Whole-heart
isotropic
nonangulated cardiac magnetic reso- phology
of thelegs).
ventricle and great vessels. This removes
*Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengi- dimensions
remains challenging. Self-gated techniques can
nance
(CMR)
is becoming
an important
protocol
simplifying
neering,
University
College London,
Gower Street,
LondoninWCIE
6BT, UK, the need for time-consuming slice planning by a skilled
remove
need for either ECG (13,14) or breath hold
E-mail:
[email protected]
MRI,
since
it reduces
theheart,
need referred
of cumbersome
planning
of operator.the
The main problem of whole-heart acquisitions is
Dynamic
imaging
of the
to as cine
imaging,
or their combination (16). Self-gating relies on the
in part
as abstracts
at the 2nd International
on ParallelTHEORY
angulations.
However
the
acquisition
of Workshop
whole-heart
MRI (15),
rather
long acquisition times, particularly when acquiring
isPresented
theZürich,
reference
MRIMIUA,
method
forSept.
thetimes
study
of
cardiac
funcMRI,
2004, andto
2004.
of information about cardiac and/or respiratory
are
prohibitive
the London,
large fields
of view (FOVs) and the extraction
large
imaging
volumes
at high motion-corrupted
image resolutions.MR
Recently,
tion
patientsdue
with heart
standard
technique,
We need
a method
to handle
images
GrantinSponsor:
EPSRC;
Grant failure.
Numbers:The
GR/S30184,
AF/001381;
Grant motion
from
the imaging
data. This information was
shown
high
spatial resolution
required
for depicting
small
structures
Sponsor: Chilean
FONDECYT;
Grant Number:
based
Cartesian
scanning
available
on clinical
scanners,
is a 1030570.
segmented balanced fast volumetric
weretrospective
know acquisitions
the motion;
this
is aon
necessity
to correct
for
and vessels. To address this problem, we propose a three- towhen
allow
or prospective
synchronization
to
Received 14 March
2005;
revised 8sequence
June 2005; acquired
accepted 10during
June 2005
using
parallel
imaging
techniques
(2,3) were
introduced
steady-state
free
precession
susunknown
motions
an optimization
method.
Some
dimensional
(3D) acquisition scheme that combines Cartesian either the cardiac
orwith
respiratory
cycle. In particular,
the
DOI 10.1002/mrm.20656
for
this
purpose
(4).
However,
the
radio
frequency
(RF)
coil
pended
respiration
(1).
Images
from
each
cardiac
phase
are
MR techniques,
navigators,
to find
the
sampling
the9readout
direction
with InterScience
an undersampled
radial method
Published in
online
September
2005 in Wiley
(www.interscience.
in Buehrersuch
et al.as(16)
uses the allow
centralus
point
in the
used
in thosebut
studies
only modest
accelerareconstructed
by retrospective
gating,
usingundersampling
information array
motion
directly,
then allows
also require
an algorithm
to
wiley.com).
scheme
in the phase-encoding
plane.
Different
Magnetic
Resonance
in Medicine
–949 (2007)
k-space, which
can
be thought
of as a 57:939
zero-dimensional
from
electrocardiogram
(ECG) (2).with
In an
some
circumtion factors of 2 to 3 and thus the total scan time is still
patterns
were
investigated
in combination
iterative
sen- 1273
© 2005the
Wiley-Liss,
Inc.
navigator signal; the method in Uribe et al. (15) uses the
stances
however,
the method
may suffer
from
several issues
sitivity
encoding
(SENSE)
reconstruction
and
a 32-channel
car- rather long (up to 20 min). With the introduction of new
center
k-space
line
(CKL),
which
can
be
thought
of
as
a
associated
with
the use ofmaps
the ECG
breath hold.
First, MR scanner technology with up to 32 receive channels and
diac
coil. Noise
amplification
wereor
calculated
to compare
(1D) coils,
navigator
signal, producing
prothe
of distorted
the different
patterns
iterative
theperformance
ECG signal is
during
the using
MRI scan
dueSENSE
to the one-dimensional
corresponding receive
the signal-to-noise
ratioa(SNR)
the two-dimensional
or three-dimensional
image
reconstruction.
The radial effect
phase-encoding
(RPE) scheme and
was jection
magnetohydrodynamic
(3), to radiofrequency,
can be of
potentially
increased and
further acceleration
of
onto thebecomes
frequency-encoding
axis.studies
With all
these
implemented
a clinical
MR switching
scanner and(4,5).
tested
on phantoms
to magnetic on
field
gradient
Although
sig- content
these protocols
feasible. Some
have
althe self-gating
canmagnetic
be acquired
with
and
volunteers.
Thehave
proposed
method exhibits
better
nal healthy
processing
methods
been proposed
in order
to techniques,
ready addressed
whole-heartsignals
coronary
resonance
1
2
2
3
1
image
quality
even
for
high
acceleration
factors
(up
to
12)
in
minimal
distortion
of
steady-state
during
a massively
balanced
Claudia Prieto, Philip G. Batchelor, D.L.G. Hill,
Joseph
V. Hajnal,
Marcelo
Guarini,
angiography
(MRA)
in the
a single
breathhold
using
comparison to Cartesian acquisitions.
Magn Reson Med 62: steady-state free precession sequence. However, combined
1*
parallel imaging (5). Nevertheless, these breathholds are
and
Pablo2009.
Irarrazaval
1331–1337,
© 2009 Wiley-Liss, Inc.
cardiac and respiratory gating has several limitations: (i)
Reconstruction of Undersampled Dynamic Images by
Modeling the Motion of Object Elements
still quite long and the spatial resolution is relatively low.
Key words: whole-heart MRI; radial-Cartesian sampling; itera- it is relatively inefficient as data from undesired respiraIn order to image the complete heart with higher spatial
tive
SENSE
reconstruction;
parallel
32 channel
coil; tory phases are thrown away; (ii) if respiratory motion is
Dynamic
MRI
is restricted
due
to University
the imaging;
timeCollege
required
to obtain
1
missing adata
by exploiting
the high acquisition
spatiotemporal
resolution,
free-breathing
whole-heart
using
Centre for
Medical
Image Computing,
London,
London, the
radical
phase
enough
data
to encoding
reconstruct the image sequence. Several unUnited Kingdom.
not
reproducible
from
breathing
cycle
to another,
the
correlation
of dynamic
sequences
or
from
prior
informaparallel
imaging
in
twoone
directions
was
proposed
(6). How-
2
dersampled
reconstruction
techniques
have been
proposed
to tion.
Division of Imaging
Sciences, King’s
College London,
London,
United Kingefficiency
decreases
further and
residual
artifacts
ever, although
a 32-channel
coil(iii)
array
was used,
the may
optiCardiac
resonance
(CMR)
has become
dom. themagnetic
reduce
acquisition
time. In most
ofimaging
these techniques
the a occur as motion within the acceptance window is not corTraditional
approaches
operate
on aindiscrete
k-t space
mal
phase encoding
directions
result
a moderate
accelclinically
useful
tool
in
imaging
cardiovasGrant sponsor:
Engineering
andnoninvasive
Physical
Research
Council;
Grant
nonacquired
data
are
recovered
by Sciences
modeling
the of
temporal
inrected,
which
imposes
a tradeoff
between
image
quality
and
either
treat
each
frame
separately
or
consider
the
eration
factor
of
4.
number:
UK
EPSRC
EP/E001564.
cular diseases.
Thepixel
widespread
of cardiac
howformation
as varying
intensitiesuse
represented
in MRI,
time or
in
and
acquisition
efficiency
(16).
*Correspondence to: Freddy Odille, CMIC, Malet Place Engineering Build- temporal
information
as
time-varying
pixel
intensities
repAlternatively
undersampled
Cartesian
acquisitions
for
temporal
Here
proposenature
a new of
approach
that
ever,
is frequencies.
hampered by
thewe
complex
the multiple
ing, University College London, Gower Street, London WC1E 6BT, United
In thisinwork,
propose
an alternative
strategy
that
time we
or MR
in
temporal
frequencies.
Therefore,
contrast-enhanced
angiography
(7) as well
as radial
recovers
missing
dataMR
through
a motion
estimation
of need
the resented
two-dimensional
(2D)
scanning
protocols
and the
Kingdom.the
E-mail:
[email protected]
is
more
efficient
than combined
cardiac-respiratory
selfacquisitions
for
whole-heart
MRI
were
proposed
to
reduce
each
pixel
is
considered
in
a
constant
position
over
time.
object
elements
(“obels,”
or
pieces
of
tissue)
of
the
image.
This
Received
9
June
2009;
revised
2
October
2009;
accepted
6
November
2009.
for highly individualized planning procedures. Wholegatingmethods
andtime
more
generally
applicable
real-time
imagmethod
assumes that an obel displacement through the se- These
(8,9).
In keyhole
these
acquisitions
the readout
diinclude
(8,9),than
reduced
encoding
DOI 10.1002/mrm.22312
heart
isotropic
nonangulated CMR is becoming an impor- the scan
ing,
without
requiring
the use2D
of or
either
theobtain
ECG or
breath
quence
has
lower
bandwidth
than(www.interscience.wiley.com).
fluctuations in pixel intensiPublished
online
in Wiley
InterScience
rection
is changed
in either
3D to
a (RIGR)
limited
imaging
with
generalized-series
reconstruction
tant protocol in simplifying MRI (1). Subsequent reformat- MR
ties
caused
by the
number
of projections
of the
imaging
volume.
particular
© 2010
Wiley-Liss,
Inc.motion, and thus it can be modeled with 1247
(10),
reduced
field of view
(rFOV)
(11),
hybridAtechnique
ting
of
any
slice
of
interest
can
be
obtained
from
the
3D
fewer parameters. Preliminary results show that this technique
asset
of the imaging
radial technique
is that the
point-spread funcdynamic
(12), unaliasing
by Fourier-encoding
volume
for reconstruct
qualitative (with
assessment
ofsquare
the complex
mor- for
can
effectively
root mean
(RMS) errors
tionoverlaps
(PSF) isusing
robustthe
with
respect dimension
to undersampling
(10).
the
temporal
(UNFOLD)
below 4%) cardiac images and joints with undersampling facAliased signal energy will appear only as slight streaking
tors of 8 and 4, respectively. Moreover, in the reconstruction (13), sensitivity encoding incorporating temporal filtering
artifact
and
thus
increased
pseudonoise,
whereas
under(TSENSE)
(14),
k-t
broad-use
linear
acquisition
speed-up
process
an
approximation
of
the
motion
vectors
is
obtained
for
1King’s College London, British Heart Foundation (BHF) Centre, Division of
sampling in a Cartesian acquisition will result in severe
IOP PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 56 (2011) N99–N114
doi:10.1088/0031-9155/56/7/N02
NOTE
A computationally efficient OMP-based compressed
sensing reconstruction for dynamic MRI
M Usman1 , C Prieto1 , F Odille2 , D Atkinson2 , T Schaeffter1 and
P G Batchelor1
1
King’s College London, Division of Imaging Sciences and Biomedical Engineering, London,
UK
2 Centre for Medical Image Computing, University College London, London, UK
E-mail: [email protected]
Received 12 November 2010, in final form 7 February 2011
Published 2 March 2011
Online at stacks.iop.org/PMB/56/N99
FULL PAPERS
Magnetic Resonance in Medicine 000:000–000 (2011)
Abstract
Compressed sensing (CS) methods in MRI are computationally intensive.
Thus, designing novel CS algorithms that can perform faster reconstructions
is crucial for everyday applications. We propose a computationally efficient
k-t Group Sparse:
A Method
for Accelerating
orthogonal matching
pursuit (OMP)-based
reconstruction, specifically suited
to cardiac MR data. According to the energy distribution of a y–f space
Dynamic MRI
obtained from a sliding window reconstruction, we label the y–f space as static
or dynamic.
For staticand
y–fP.
space
images, a computationally efficient masked
M. Usman,* C. Prieto,
T. Schaeffter,
G. Batchelor
OMP reconstruction is performed, whereas for dynamic y–f space images,
standard
OMP reconstruction
used.called
The‘‘k-t
proposed
was tested
on a
Compressed sensing (CS)
is a data-reduction
technique that is
nique
sparse’’ method
(18) in which
the underhas been applied to speed
up the numerical
acquisition in phantom
MRI. How- and
sampled
data acquisition
was doneDepending
in the phase encodedynamic
two cardiac
MR datasets.
on the
ever, the use of this technique in dynamic MR applications
dimension (k-t space) by randomly skipping the
field
of maximum
view composition
of thetime
imaging
data,
compared
to
the
standard
OMP
has been limited in terms
of the
achievable reducphase encodes for each time frame. The sparsity was
tion factor. In general, noise-like artefacts and bad temporal fimethod, reconstruction speedupintroduced
factors ranging
from 1.5
2.5 areMR
achieved.
by transforming
the to
dynamic
data using
delity are visible in standard CS MRI reconstructions when
high reduction factors are used. To increase the maximum a wavelet transform and a Fourier transform along spatial
and temporal directions, respectively. Alternatively,
achievable reduction factor, additional or prior information can
be incorporated in the CS reconstruction. Here, a novel CS Gamper et al. (19) showed that a Fourier transform along
FULL PAPER
reconstruction method is proposed that exploits the structure the temporal dimension was sufficient to achieve sparMagnetic Resonance in Medicine 000:000–000 (2012)
within the sparse representation of a signal by enforcing the sity in x-f space, where x is spatial position and f is temsupport components to be in the form of groups. These groups poral frequency. For low reduction factors (up to 3-fold),
act like a constraint in the reconstruction. The information Gamper demonstrated that CS reconstructions exhibit
about the support region can be easily obtained from training
less error than k-t Broad-use Linear Acquisition Speeddata in dynamic MRI acquisitions. The proposed approach was
up Technique (BLAST) reconstructions for highly
tested in two-dimensional cardiac cine MRI with both downsampled and undersampled data. Results show that higher dynamic object features. For high reduction factors (>5),
high temporal frequency components with low amplitudes
acceleration factors (up to 9-fold), with improved spatial and
temporal quality, can be obtained with the proposed approach get interspersed with noise in the aliased x-f space and are
in comparison to the standard CS reconstructions. Magn not recovered in the CS reconstruction. This leads to noisy
C
V
2011 Wiley-Liss, Inc.
Reson Med 000:000–000, 2011.
reconstructions
with bad temporal fidelity.
Hence,
3
1
* Davidundersampling;
Atkinson,2group
Muhammad
Usman,1sensing;
Freddy
Odille,
Christoph
Kolbitsch,
improved
CS methods
for maintaining
both reasonable spaKey words: compressed
1
1,4at higher
sparsity; l1 minimization;1dynamic MRI
tial and temporal1quality of dynamic MR data
1. Introduction
In dynamic MRI, the motion of an object is measured by acquiring a series of images at a
Motion
Corrected Compressed Sensing for
high frame rate. However, the time resolution of dynamic MRI is limited by the number of
Free-Breathing
phase-encoding steps Dynamic
that are requiredCardiac
for each timeMRI
frame. For accelerated acquisition, the
data are typically undersampled along the phase encoding and/or time direction, exploiting the
correlations in k-space and/or time. Examples include UNFOLD (Madore et al 1999), Keyhole
Ghislain
Vaillant,
Tobias
G. Batchelor,
andThe
Claudia
Prieto in these
(Vanvaals
et al 1993),
k–t Schaeffter,
BLAST (TsaoPhilip
et al 2003),
among others.
reconstruction
reduction factors are very desirable. Several extensions
methods
a straightforward
linear
and hence isof very
fast. A SENSE
few years
back,
a
Dynamic
MRI uses
applications
usually require
high formulation
spatial and improvements
k-t-BLAST/k-t
and CS
techniand new
temporal
resolution.
speed
of MR can
Compressed
sensing
(CS) hasThe
beenacquisition
demonstrated
to‘compressed
accelerquessensing’
have
been(CS)
recently
proposed
(20–28).
be
introduced
in
the
MR image
reconstruction
due
data
reduction
technique
called
was
introduced
(Candes 2006,
images,
however, isbylimited
by physical
(e.g.,
gradient
ate
MRI acquisitions
reconstructing
sparse
images
of to One
approach
improve themovement
CS-based MR
reconstrucunwanted
or to
involuntary
during
acquisiDonoho
2006)
and
hasphysiological
beendata.
demonstrated
different
MR applications
(Lustig
et al 2007,
good
qualityand
from
highly
undersampled
Motion
strength
slew
rate)
and
(e.g., during
nerve intion.
tion can
be to exploit cardiac
the structure
the
MR images difin
In free-breathing
gated of
MR
acquisitions,
MR
scans
inconsistencies
in
resultstimulation)
constraints
(1).
thisdata,
issue,
sev- ferent
Jungcanetcause
al 2009,
JungToetaddress
alk-space
2007).
According
to the
CS theory,
perfect
reconstruction
of a
sparse
representation.
example,
thea sparse
represenk-space
profilesFor
belonging
to
specific
cardiac
ing
in reconstruction
strong motion artifacts
in the
reconstructed
images.
eral
techniques
have
been proposed
that phase
tation are
mayacquired
exhibit structure
in thebreathing
form of the
nonzeroor
at distinctive
positions
For
CSreconstruct
to be usefulMR
in these
applications,
motion
correction
can
images
of significant
quality
from and
coefficients
occurring
inPrinted
clusters.
dynamic from
cardiac
‘‘motion
states.’’
The combination
of profiles
the
0031-9155/11/070099+16$33.00
© 2011
Institute
of Physics
Engineering
in
Medicine
in theFor
UK
N99
techniques
needacquisitions.
to be combined
with
the
undersampled
reduced data
Following
Tsao
et al.’s classi- MR data, besides being sparse, the x-f space representareconstruction. Recently, joint motion correction and CS same cardiac phase but different respiratory motion
fication (2), these techniques can be classified into those tion tends to be in compact form (29), i.e., the support
approaches have been proposed to partially correct for states can result in inconsistencies in k-space, leading to
which exploit correlations in the k-space (3–8), in time elements
in x-f space
intensities
aboveInthe
noise
artifacts
in thehaving
reconstructed
images.
addition,
effects of motion. However, the main limitation of these motion
domain (9–13) or in both k-space and time domains levelunwanted
lie together
in few
Hence,
motion
can groups.
also reduce
the incorporating
sparsity level
approaches is that they can only correct for affine deforma- this
(2,14,15).
this
information
could
helprepresentation,
achieve high reduction
factions.
In this work, we propose a novel motion corrected CS of
MR
images in the
sparse
thus reducing
Recently,
new data reduction
titled
torsacceleration
in dynamic MRI.
framework
for afree-breathing
dynamic technique
cardiac MRI
that ‘‘comincor- the
factor achievable with CS reconstruction
pressed
sensing’’
(CS) correction
(16,17) has
been proposed
for
CS, the
‘‘block sparse’’
or
porates
a general
motion
formulation
directly into
(7).Recently,
Hence, toinbenefit
fromconcept
the highofacceleration
available
application
in MRI. According
to the
CScorrect
theory,for
perfect
‘‘group
signals
has been
introduced
in the signal
the
CS reconstruction.
This framework
can
arbi- from
CSsparse’’
methods
in these
applications,
additional
flexireconstruction
of a signal
is possible
sampling rates
trary
affine or nonrigid
motion
in the CSfrom
reconstructed
carprocessing
literature.
This refers
to exploiting
thewith
strucbility
is required
to combine
motion
correction
the
below
the Shannon–Nyquist
limit,benefiting
provided the
is ture of sparse signals that have support elements lying in
diac
images,
while simultaneously
fromsignal
highly
CS reconstruction.
accelerated
acquisition.
The transform
applicationdomain)
of this approach
sparse (inMR
itself
or in some
and the groups. Some work has been done on one-dimensional
Some approaches to combine CS reconstruction with
is measurement
demonstrated samples
both in simulations
andwith
in vivo
for 2D
are obtained
an data
incoherent
signals (such as speech) considering equal or unequal
correction techniques have been recently prorespiratory
CINE MRI,
basis. Forself-gated
dynamic free-breathing
MRI, Lustig etcardiac
al. proposed
a CS using
tech- motion
length groups (30–36). Specifically, these techniques
a golden angle radial acquisition. Results show that this posed (7,8). Jung et al. proposed a CS technique ‘‘k-t
make a partition of elements within the sparse represenFOCal
Underdetermined System Solver (FOCUSS)’’
approach
allows
for
the
reconstruction
of
respiratory
motion
King’s College London, Division of Imaging Sciences and Biomedical
tation into nonoverlapping groups and the group struccorrected
CINE Research
images Centre
with atsimilar
to (9–11) that incorporated a motion estimation procedure to
Engineering,cardiac
NIHR Biomedical
Guy’s andquality
St Thomas’
ture is enforced in the CS reconstruction via a ‘‘groupFoundation Trust,
London, UnitedMagn
Kingdom.
breath-held
acquisitions.
Reson Med 000:000–000, predict different cardiac phases from a fully sampled refsparse/block-sparse’’ formulation. Extensive simulations
C 2012 Wiley
*Correspondence
to:Periodicals,
Muhammad Inc.
Usman, Ph.D., Division of Imaging
V
2012.
erence
cardiac frame. The knowledge of motion between
have demonstrated that group sparse methods have an
Sciences, The Rayne Institute, 4th Floor, Lambeth Wing, St Thomas’ Hospital,
Key words: compressed sensing; undersampling; motion
Reconstruction
of
KING’S MEDICAL ENGINEERING CENTRE Compressed
Sensing
Undersampled Data
Pharmacokinetic
Modeling
Magnetic Resonance in Medicine 60:1524 –1530 (2008)
Magnetic Resonance in Medicine 54:1273–1280 (2005)
Matrix Description of General Motion Correction Applied
to Multishot Images
P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and
D. Larkman3
Motion of an object degrades MR images, as the acquisition is rection method could be used to spatially transform the
time-dependent, and thus k-space is inconsistently sampled. ghosted image by the transformation corresponding to a
This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot,
restrictive assumptions on the type of motions, for example, and repeat this operation for all shots (this is a version
that it is a translation or rotation, and use special properties of
of the method used in (1)). We could then rebuild an
k-space for these transformations. Such methods, however,
image by inverse Fourier transform. This method is in
cannot be generalized easily to nonrigid types of motions, and
even rotations in multiple shots can be a problem. Here, a general incorrect, as shown by the difference between
method is presented that can handle general nonrigid motion translations and rotations. Correcting translation repointwise phase changes in k-space. On the
models. A general matrix equation gives the corrupted image quires only
Magnetic Resonance in Medicine 63:1247–1257 (2010)
from the ideal object. Thus, inversion of this system allows us to other hand, correcting rotations requires knowledge of
get the ideal image from the corrupted one. This inversion is the data at neighboring k-space positions and these are
possible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier
the conjugate gradient method. A faster but empirical inversion rotation theorem, we would need to “synchronize”
is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions
ulated three-dimensional affine data and two-dimensional pul- such as nonrigid deformations cannot have a simple
sation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show
tion. All examples are
where
the object moves
2
2
2
2
Freddy
Odille,1* multishot
Sergio images
Uribe,
Philip
G. Batchelor,
Prieto,
Tobias
Schaeffter,
and
that itClaudia
is possible
to correct
complicated
motions,
inbetween shots. The results indicate that it is now possible to
1
David
Atkinson
correct for
nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical
many types of patient motion, although computation times re- description of the problems involved; the motion cormain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on
This
paper describes
an acquisition and reconstruction strategy reduce
gradient
interference
(4,6,7),
the magnetohydrodythe space
of images.
Thus,
inversion
of this matrix
Wiley-Liss,
Inc.
for cardiac cine MRI that does not require the use of electro- namic
effect
is difficult
to model
or correct,
especiallyisatof
correct
the motion’s
effects.
This approach
Key words: motion correction; ghosts; multishot; conjugate should
cardiogram or breath holding. The method has similarities with high
fields, asinterest,
it increases
the amplitude
static Bon
theoretical
but with
its practical
value of
depends
gradient; auto focus
0
self-gated techniques as information about cardiac and respi- field strength. The use of ECG also requires more patient
how easily we can find a solution of the linear system. It
ratory motion is derived from the imaging sequence itself; here,
Motion of an object can degrade MR images and imposes preparation
time.
Second,
holding
be a difficult
turns out that
with
somebreath
careful
linear can
algebra,
not only
by acquiring the center k-space line at the beginning of each
constraints on scan parameters that can in turn compro- task
are for
wemany
able to
invert or
in infants
a generalized
but alsodue
this
patients
and maysense,
be imperfect
segment of a balanced steady-state free precession sequence.
mise image
quality. The cause
the degradation
is that
can be over
donetime
efficiently
in practice.
For this we
organs drifting
(8). Another
issue associated
However,
the reconstruction
step isoffundamentally
different:
a toinversion
the acquisition
is time-dependent,
the Fourier
transuse breath
the LSQR
algorithm,
which
is a robust
implementaholding
is that the
physiology
is modified
comgeneralized
reconstruction
by inversionand
of coupled
systems
is with
form
of
the
image
seen
during
acquisition
changes
due
to
tion
of
the
conjugate
gradient
of
the
normal
equation
pared
to
the
“normal”
free-breathing
state.
Therefore,
it
is
used instead of conventional gating. By correcting for nonrigid
the deformation
of themotion,
object.generalized
This causes
inconsistencies
(see (4)).relevant to attempt to capture cardiac functional
cardiac
and respiratory
reconstruction
by clinically
Magnetic Resonance in Medicine 62:1331–1337 (2009)
inversion
of coupled
systems
all acquired data, information
in k-space
and hence
ghosts(GRICS)
in the uses
image.
This leaves
of finding what motion actually
in the
freequestion
breathing.
whereas
gatingmotion
rejects data
acquiredmethods
in certainmake
motionassumpstates.
Standard
correction
happened.
Different
costbeen
functions
have
been designed
Several strategies
have
proposed
in order
to addressto
The
method
relies
onofthe
processing
analysisthat
of the
tions
on the
type
motions,
for and
example,
it isk-a these
quantify
how
much anreal-time
image has
been corrupted
(1,tech5, 6).
issues,
including
imaging,
self-gated
space
central or
linea data:
local and
information
from a 32-channel
translation
rotation,
use formulas
on Fourier niques,
We explore
functions
conjunction
with
and different
combinedcost
cardiac
and in
respiratory
gating.
cardiac
coil is
in order
to automatically
extract
eigentransforms
toused
correct
the data
(1–3). We assume
here
that Real-time
our motion
correction.
Optimization
of
suchcardiac
cost functions
imaging
has
been
shown
to
allow
imagmodes of both cardiac and respiratory motion. In the GRICS
these data are acquired in shots. When the data posi- means repeating the matrix inversion iteratively. However,
framework, these eigenmodes are used as driving signals of ing during free breathing (9–11) but implies a comprotions at each shot are known, an empirical motion cor- inverting matrices repeatedly may be prohibitive even if it
a motion model. The motion model is defined piecewise, so mise between spatiotemporal resolution and signal-tois practicable
onItaisone-off
basis.
We therefore
also
invesratio (SNR).
possible
to combine
real-time
images
that each cardiac phase is reconstructed independently. Results noise
tigate
the useframes,
of the using
empirical
method
described
above.
from six healthy volunteers,
with various slice orientations, show
different
nonrigid
image
registration
in
1 C. Prieto,1 P.G. Batchelor,1 S.from
1 D. Atkinson,
2 H. Eggers,
3
1 Boubertakh,
R.
Uribe,
The
matrix
equationfor
allows
us to find
when
this
approxiimproved
image quality
comparedUniversity
to combined
and order
Medical Physics
& Bioengineering,
Collegerespiratory
London, London,
to compensate
respiratory
motion,
and
then
proUnited
Kingdom.
4 M.S.
5 R.S.2010.
1* show three-dimensional random
mation
is correct. We
cardiacSørensen,
gating. Magn
Reson Hansen,
Med 63:1247–1257,
© 20101 and
T.S.
Razavi,
T.
Schaeffter
duce
SNR-enhanced
cardiac
cine images (12). However,
2
DepartmentInc.
of Electrical Engineering, Pontifica Universidad Católica de
Wiley-Liss,
affine and pulsatile nonrigid motion corrections on simu-
Model-Based Reconstruction for Cardiac Cine MRI
Without ECG or Breath Holding
Liver Whole-Heart Imaging Using Undersampled Radial Phase
Encoding (RPE) and Iterative Sensitivity Encoding
(SENSE) Reconstruction
Chile, Chile.
this technique still requires the recording of an ECG sig-
lated data and an example of nonrigid correction of in vivo
Key
words: Artifacts; cardiac imaging; gating; motion correc3
Imaging Sciences, Imperial College London, London, United Kingdom.
nal during real-time scanning, and its extension to three
data (moving
tion;
navigators;
reconstruction
Whole-heart
isotropic
nonangulated cardiac magnetic reso- phology
of thelegs).
ventricle and great vessels. This removes
*Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengi- dimensions
remains challenging. Self-gated techniques can
nance
(CMR)
is becoming
an important
protocol
simplifying
neering,
University
College London,
Gower Street,
LondoninWCIE
6BT, UK, the need for time-consuming slice planning by a skilled
remove
need for either ECG (13,14) or breath hold
E-mail:
[email protected]
MRI,
since
it reduces
theheart,
need referred
of cumbersome
planning
of operator.the
The main problem of whole-heart acquisitions is
Dynamic
imaging
of the
to as cine
imaging,
or their combination (16). Self-gating relies on the
in part
as abstracts
at the 2nd International
on ParallelTHEORY
angulations.
However
the
acquisition
of Workshop
whole-heart
MRI (15),
rather
long acquisition times, particularly when acquiring
isPresented
theZürich,
reference
MRIMIUA,
method
forSept.
thetimes
study
of
cardiac
funcMRI,
2004, andto
2004.
of information about cardiac and/or respiratory
are
prohibitive
the London,
large fields
of view (FOVs) and the extraction
large
imaging
volumes
at high motion-corrupted
image resolutions.MR
Recently,
tion
patientsdue
with heart
standard
technique,
We need
a method
to handle
images
GrantinSponsor:
EPSRC;
Grant failure.
Numbers:The
GR/S30184,
AF/001381;
Grant motion
from
the imaging
data. This information was
shown
high
spatial resolution
required
for depicting
small
structures
Sponsor: Chilean
FONDECYT;
Grant Number:
based
Cartesian
scanning
available
on clinical
scanners,
is a 1030570.
segmented balanced fast volumetric
weretrospective
know acquisitions
the motion;
this
is aon
necessity
to correct
for
and vessels. To address this problem, we propose a three- towhen
allow
or prospective
synchronization
to
Received 14 March
2005;
revised 8sequence
June 2005; acquired
accepted 10during
June 2005
using
parallel
imaging
techniques
(2,3) were
introduced
steady-state
free
precession
susunknown
motions
an optimization
method.
Some
dimensional
(3D) acquisition scheme that combines Cartesian either the cardiac
orwith
respiratory
cycle. In particular,
the
DOI 10.1002/mrm.20656
for
this
purpose
(4).
However,
the
radio
frequency
(RF)
coil
pended
respiration
(1).
Images
from
each
cardiac
phase
are
MR techniques,
navigators,
to find
the
sampling
the9readout
direction
with InterScience
an undersampled
radial method
Published in
online
September
2005 in Wiley
(www.interscience.
in Buehrersuch
et al.as(16)
uses the allow
centralus
point
in the
used
in thosebut
studies
only modest
accelerareconstructed
by retrospective
gating,
usingundersampling
information array
motion
directly,
then allows
also require
an algorithm
to
wiley.com).
scheme
in the phase-encoding
plane.
Different
Magnetic
Resonance
in Medicine
–949 (2007)
k-space, which
can
be thought
of as a 57:939
zero-dimensional
from
electrocardiogram
(ECG) (2).with
In an
some
circumtion factors of 2 to 3 and thus the total scan time is still
patterns
were
investigated
in combination
iterative
sen- 1273
© 2005the
Wiley-Liss,
Inc.
navigator signal; the method in Uribe et al. (15) uses the
stances
however,
the method
may suffer
from
several issues
sitivity
encoding
(SENSE)
reconstruction
and
a 32-channel
car- rather long (up to 20 min). With the introduction of new
center
k-space
line
(CKL),
which
can
be
thought
of
as
a
associated
with
the use ofmaps
the ECG
breath hold.
First, MR scanner technology with up to 32 receive channels and
diac
coil. Noise
amplification
wereor
calculated
to compare
(1D) coils,
navigator
signal, producing
prothe
of distorted
the different
patterns
iterative
theperformance
ECG signal is
during
the using
MRI scan
dueSENSE
to the one-dimensional
corresponding receive
the signal-to-noise
ratioa(SNR)
the two-dimensional
or three-dimensional
image
reconstruction.
The radial effect
phase-encoding
(RPE) scheme and
was jection
magnetohydrodynamic
(3), to radiofrequency,
can be of
potentially
increased and
further acceleration
of
onto thebecomes
frequency-encoding
axis.studies
With all
these
implemented
a clinical
MR switching
scanner and(4,5).
tested
on phantoms
to magnetic on
field
gradient
Although
sig- content
these protocols
feasible. Some
have
althe self-gating
canmagnetic
be acquired
with
and
volunteers.
Thehave
proposed
method exhibits
better
nal healthy
processing
methods
been proposed
in order
to techniques,
ready addressed
whole-heartsignals
coronary
resonance
1
2
2
3
1
image
quality
even
for
high
acceleration
factors
(up
to
12)
in
minimal
distortion
of
steady-state
during
a massively
balanced
Claudia Prieto, Philip G. Batchelor, D.L.G. Hill,
Joseph
V. Hajnal,
Marcelo
Guarini,
angiography
(MRA)
in the
a single
breathhold
using
comparison to Cartesian acquisitions.
Magn Reson Med 62: steady-state free precession sequence. However, combined
1*
parallel imaging (5). Nevertheless, these breathholds are
and
Pablo2009.
Irarrazaval
1331–1337,
© 2009 Wiley-Liss, Inc.
cardiac and respiratory gating has several limitations: (i)
Reconstruction of Undersampled Dynamic Images by
Modeling the Motion of Object Elements
still quite long and the spatial resolution is relatively low.
Key words: whole-heart MRI; radial-Cartesian sampling; itera- it is relatively inefficient as data from undesired respiraIn order to image the complete heart with higher spatial
tive
SENSE
reconstruction;
parallel
32 channel
coil; tory phases are thrown away; (ii) if respiratory motion is
Dynamic
MRI
is restricted
due
to University
the imaging;
timeCollege
required
to obtain
1
missing adata
by exploiting
the high acquisition
spatiotemporal
resolution,
free-breathing
whole-heart
using
Centre for
Medical
Image Computing,
London,
London, the
radical
phase
enough
data
to encoding
reconstruct the image sequence. Several unUnited Kingdom.
not
reproducible
from
breathing
cycle
to another,
the
correlation
of dynamic
sequences
or
from
prior
informaparallel
imaging
in
twoone
directions
was
proposed
(6). How-
2
dersampled
reconstruction
techniques
have been
proposed
to tion.
Division of Imaging
Sciences, King’s
College London,
London,
United Kingefficiency
decreases
further and
residual
artifacts
ever, although
a 32-channel
coil(iii)
array
was used,
the may
optiCardiac
resonance
(CMR)
has become
dom. themagnetic
reduce
acquisition
time. In most
ofimaging
these techniques
the a occur as motion within the acceptance window is not corTraditional
approaches
operate
on aindiscrete
k-t space
mal
phase encoding
directions
result
a moderate
accelclinically
useful
tool
in
imaging
cardiovasGrant sponsor:
Engineering
andnoninvasive
Physical
Research
Council;
Grant
nonacquired
data
are
recovered
by Sciences
modeling
the of
temporal
inrected,
which
imposes
a tradeoff
between
image
quality
and
either
treat
each
frame
separately
or
consider
the
eration
factor
of
4.
number:
UK
EPSRC
EP/E001564.
cular diseases.
Thepixel
widespread
of cardiac
howformation
as varying
intensitiesuse
represented
in MRI,
time or
in
and
acquisition
efficiency
(16).
*Correspondence to: Freddy Odille, CMIC, Malet Place Engineering Build- temporal
information
as
time-varying
pixel
intensities
repAlternatively
undersampled
Cartesian
acquisitions
for
temporal
Here
proposenature
a new of
approach
that
ever,
is frequencies.
hampered by
thewe
complex
the multiple
ing, University College London, Gower Street, London WC1E 6BT, United
In thisinwork,
propose
an alternative
strategy
that
time we
or MR
in
temporal
frequencies.
Therefore,
contrast-enhanced
angiography
(7) as well
as radial
recovers
missing
dataMR
through
a motion
estimation
of need
the resented
two-dimensional
(2D)
scanning
protocols
and the
Kingdom.the
E-mail:
[email protected]
is
more
efficient
than combined
cardiac-respiratory
selfacquisitions
for
whole-heart
MRI
were
proposed
to
reduce
each
pixel
is
considered
in
a
constant
position
over
time.
object
elements
(“obels,”
or
pieces
of
tissue)
of
the
image.
This
Received
9
June
2009;
revised
2
October
2009;
accepted
6
November
2009.
for highly individualized planning procedures. Wholegatingmethods
andtime
more
generally
applicable
real-time
imagmethod
assumes that an obel displacement through the se- These
(8,9).
In keyhole
these
acquisitions
the readout
diinclude
(8,9),than
reduced
encoding
DOI 10.1002/mrm.22312
heart
isotropic
nonangulated CMR is becoming an impor- the scan
ing,
without
requiring
the use2D
of or
either
theobtain
ECG or
breath
quence
has
lower
bandwidth
than(www.interscience.wiley.com).
fluctuations in pixel intensiPublished
online
in Wiley
InterScience
rection
is changed
in either
3D to
a (RIGR)
limited
imaging
with
generalized-series
reconstruction
tant protocol in simplifying MRI (1). Subsequent reformat- MR
ties
caused
by the
number
of projections
of the
imaging
volume.
particular
© 2010
Wiley-Liss,
Inc.motion, and thus it can be modeled with 1247
(10),
reduced
field of view
(rFOV)
(11),
hybridAtechnique
ting
of
any
slice
of
interest
can
be
obtained
from
the
3D
fewer parameters. Preliminary results show that this technique
asset
of the imaging
radial technique
is that the
point-spread funcdynamic
(12), unaliasing
by Fourier-encoding
volume
for reconstruct
qualitative (with
assessment
ofsquare
the complex
mor- for
can
effectively
root mean
(RMS) errors
tionoverlaps
(PSF) isusing
robustthe
with
respect dimension
to undersampling
(10).
the
temporal
(UNFOLD)
below 4%) cardiac images and joints with undersampling facAliased signal energy will appear only as slight streaking
tors of 8 and 4, respectively. Moreover, in the reconstruction (13), sensitivity encoding incorporating temporal filtering
artifact
and
thus
increased
pseudonoise,
whereas
under(TSENSE)
(14),
k-t
broad-use
linear
acquisition
speed-up
process
an
approximation
of
the
motion
vectors
is
obtained
for
1King’s College London, British Heart Foundation (BHF) Centre, Division of
sampling in a Cartesian acquisition will result in severe
IOP PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 56 (2011) N99–N114
doi:10.1088/0031-9155/56/7/N02
NOTE
A computationally efficient OMP-based compressed
sensing reconstruction for dynamic MRI
M Usman1 , C Prieto1 , F Odille2 , D Atkinson2 , T Schaeffter1 and
P G Batchelor1
1
King’s College London, Division of Imaging Sciences and Biomedical Engineering, London,
UK
2 Centre for Medical Image Computing, University College London, London, UK
E-mail: [email protected]
Received 12 November 2010, in final form 7 February 2011
Published 2 March 2011
Online at stacks.iop.org/PMB/56/N99
FULL PAPERS
Magnetic Resonance in Medicine 000:000–000 (2011)
Abstract
Compressed sensing (CS) methods in MRI are computationally intensive.
Thus, designing novel CS algorithms that can perform faster reconstructions
is crucial for everyday applications. We propose a computationally efficient
k-t Group Sparse:
A Method
for Accelerating
orthogonal matching
pursuit (OMP)-based
reconstruction, specifically suited
to cardiac MR data. According to the energy distribution of a y–f space
Dynamic MRI
obtained from a sliding window reconstruction, we label the y–f space as static
or dynamic.
For staticand
y–fP.
space
images, a computationally efficient masked
M. Usman,* C. Prieto,
T. Schaeffter,
G. Batchelor
OMP reconstruction is performed, whereas for dynamic y–f space images,
standard
OMP reconstruction
used.called
The‘‘k-t
proposed
was tested
on a
Compressed sensing (CS)
is a data-reduction
technique that is
nique
sparse’’ method
(18) in which
the underhas been applied to speed
up the numerical
acquisition in phantom
MRI. How- and
sampled
data acquisition
was doneDepending
in the phase encodedynamic
two cardiac
MR datasets.
on the
ever, the use of this technique in dynamic MR applications
dimension (k-t space) by randomly skipping the
field
of maximum
view composition
of thetime
imaging
data,
compared
to
the
standard
OMP
has been limited in terms
of the
achievable reducphase encodes for each time frame. The sparsity was
tion factor. In general, noise-like artefacts and bad temporal fimethod, reconstruction speedupintroduced
factors ranging
from 1.5
2.5 areMR
achieved.
by transforming
the to
dynamic
data using
delity are visible in standard CS MRI reconstructions when
high reduction factors are used. To increase the maximum a wavelet transform and a Fourier transform along spatial
and temporal directions, respectively. Alternatively,
achievable reduction factor, additional or prior information can
be incorporated in the CS reconstruction. Here, a novel CS Gamper et al. (19) showed that a Fourier transform along
FULL PAPER
reconstruction method is proposed that exploits the structure the temporal dimension was sufficient to achieve sparMagnetic Resonance in Medicine 000:000–000 (2012)
within the sparse representation of a signal by enforcing the sity in x-f space, where x is spatial position and f is temsupport components to be in the form of groups. These groups poral frequency. For low reduction factors (up to 3-fold),
act like a constraint in the reconstruction. The information Gamper demonstrated that CS reconstructions exhibit
about the support region can be easily obtained from training
less error than k-t Broad-use Linear Acquisition Speeddata in dynamic MRI acquisitions. The proposed approach was
up Technique (BLAST) reconstructions for highly
tested in two-dimensional cardiac cine MRI with both downsampled and undersampled data. Results show that higher dynamic object features. For high reduction factors (>5),
high temporal frequency components with low amplitudes
acceleration factors (up to 9-fold), with improved spatial and
temporal quality, can be obtained with the proposed approach get interspersed with noise in the aliased x-f space and are
in comparison to the standard CS reconstructions. Magn not recovered in the CS reconstruction. This leads to noisy
C
V
2011 Wiley-Liss, Inc.
Reson Med 000:000–000, 2011.
reconstructions
with bad temporal fidelity.
Hence,
3
1
* Davidundersampling;
Atkinson,2group
Muhammad
Usman,1sensing;
Freddy
Odille,
Christoph
Kolbitsch,
improved
CS methods
for maintaining
both reasonable spaKey words: compressed
1
1,4at higher
sparsity; l1 minimization;1dynamic MRI
tial and temporal1quality of dynamic MR data
1. Introduction
In dynamic MRI, the motion of an object is measured by acquiring a series of images at a
Motion
Corrected Compressed Sensing for
high frame rate. However, the time resolution of dynamic MRI is limited by the number of
Free-Breathing
phase-encoding steps Dynamic
that are requiredCardiac
for each timeMRI
frame. For accelerated acquisition, the
data are typically undersampled along the phase encoding and/or time direction, exploiting the
correlations in k-space and/or time. Examples include UNFOLD (Madore et al 1999), Keyhole
Ghislain
Vaillant,
Tobias
G. Batchelor,
andThe
Claudia
Prieto in these
(Vanvaals
et al 1993),
k–t Schaeffter,
BLAST (TsaoPhilip
et al 2003),
among others.
reconstruction
reduction factors are very desirable. Several extensions
methods
a straightforward
linear
and hence isof very
fast. A SENSE
few years
back,
a
Dynamic
MRI uses
applications
usually require
high formulation
spatial and improvements
k-t-BLAST/k-t
and CS
techniand new
temporal
resolution.
speed
of MR can
Compressed
sensing
(CS) hasThe
beenacquisition
demonstrated
to‘compressed
accelerquessensing’
have
been(CS)
recently
proposed
(20–28).
be
introduced
in
the
MR image
reconstruction
due
data
reduction
technique
called
was
introduced
(Candes 2006,
images,
however, isbylimited
by physical
(e.g.,
gradient
ate
MRI acquisitions
reconstructing
sparse
images
of to One
approach
improve themovement
CS-based MR
reconstrucunwanted
or to
involuntary
during
acquisiDonoho
2006)
and
hasphysiological
beendata.
demonstrated
different
MR applications
(Lustig
et al 2007,
good
qualityand
from
highly
undersampled
Motion
strength
slew
rate)
and
(e.g., during
nerve intion.
tion can
be to exploit cardiac
the structure
the
MR images difin
In free-breathing
gated of
MR
acquisitions,
MR
scans
inconsistencies
in
resultstimulation)
constraints
(1).
thisdata,
issue,
sev- ferent
Jungcanetcause
al 2009,
JungToetaddress
alk-space
2007).
According
to the
CS theory,
perfect
reconstruction
of a
sparse
representation.
example,
thea sparse
represenk-space
profilesFor
belonging
to
specific
cardiac
ing
in reconstruction
strong motion artifacts
in the
reconstructed
images.
eral
techniques
have
been proposed
that phase
tation are
mayacquired
exhibit structure
in thebreathing
form of the
nonzeroor
at distinctive
positions
For
CSreconstruct
to be usefulMR
in these
applications,
motion
correction
can
images
of significant
quality
from and
coefficients
occurring
inPrinted
clusters.
dynamic from
cardiac
‘‘motion
states.’’
The combination
of profiles
the
0031-9155/11/070099+16$33.00
© 2011
Institute
of Physics
Engineering
in
Medicine
in theFor
UK
N99
techniques
needacquisitions.
to be combined
with
the
undersampled
reduced data
Following
Tsao
et al.’s classi- MR data, besides being sparse, the x-f space representareconstruction. Recently, joint motion correction and CS same cardiac phase but different respiratory motion
fication (2), these techniques can be classified into those tion tends to be in compact form (29), i.e., the support
approaches have been proposed to partially correct for states can result in inconsistencies in k-space, leading to
which exploit correlations in the k-space (3–8), in time elements
in x-f space
intensities
aboveInthe
noise
artifacts
in thehaving
reconstructed
images.
addition,
effects of motion. However, the main limitation of these motion
domain (9–13) or in both k-space and time domains levelunwanted
lie together
in few
Hence,
motion
can groups.
also reduce
the incorporating
sparsity level
approaches is that they can only correct for affine deforma- this
(2,14,15).
this
information
could
helprepresentation,
achieve high reduction
factions.
In this work, we propose a novel motion corrected CS of
MR
images in the
sparse
thus reducing
Recently,
new data reduction
titled
torsacceleration
in dynamic MRI.
framework
for afree-breathing
dynamic technique
cardiac MRI
that ‘‘comincor- the
factor achievable with CS reconstruction
pressed
sensing’’
(CS) correction
(16,17) has
been proposed
for
CS, the
‘‘block sparse’’
or
porates
a general
motion
formulation
directly into
(7).Recently,
Hence, toinbenefit
fromconcept
the highofacceleration
available
application
in MRI. According
to the
CScorrect
theory,for
perfect
‘‘group
signals
has been
introduced
in the signal
the
CS reconstruction.
This framework
can
arbi- from
CSsparse’’
methods
in these
applications,
additional
flexireconstruction
of a signal
is possible
sampling rates
trary
affine or nonrigid
motion
in the CSfrom
reconstructed
carprocessing
literature.
This refers
to exploiting
thewith
strucbility
is required
to combine
motion
correction
the
below
the Shannon–Nyquist
limit,benefiting
provided the
is ture of sparse signals that have support elements lying in
diac
images,
while simultaneously
fromsignal
highly
CS reconstruction.
accelerated
acquisition.
The transform
applicationdomain)
of this approach
sparse (inMR
itself
or in some
and the groups. Some work has been done on one-dimensional
Some approaches to combine CS reconstruction with
is measurement
demonstrated samples
both in simulations
andwith
in vivo
for 2D
are obtained
an data
incoherent
signals (such as speech) considering equal or unequal
correction techniques have been recently prorespiratory
CINE MRI,
basis. Forself-gated
dynamic free-breathing
MRI, Lustig etcardiac
al. proposed
a CS using
tech- motion
length groups (30–36). Specifically, these techniques
a golden angle radial acquisition. Results show that this posed (7,8). Jung et al. proposed a CS technique ‘‘k-t
make a partition of elements within the sparse represenFOCal
Underdetermined System Solver (FOCUSS)’’
approach
allows
for
the
reconstruction
of
respiratory
motion
King’s College London, Division of Imaging Sciences and Biomedical
tation into nonoverlapping groups and the group struccorrected
CINE Research
images Centre
with atsimilar
to (9–11) that incorporated a motion estimation procedure to
Engineering,cardiac
NIHR Biomedical
Guy’s andquality
St Thomas’
ture is enforced in the CS reconstruction via a ‘‘groupFoundation Trust,
London, UnitedMagn
Kingdom.
breath-held
acquisitions.
Reson Med 000:000–000, predict different cardiac phases from a fully sampled refsparse/block-sparse’’ formulation. Extensive simulations
C 2012 Wiley
*Correspondence
to:Periodicals,
Muhammad Inc.
Usman, Ph.D., Division of Imaging
V
2012.
erence
cardiac frame. The knowledge of motion between
have demonstrated that group sparse methods have an
Sciences, The Rayne Institute, 4th Floor, Lambeth Wing, St Thomas’ Hospital,
Key words: compressed sensing; undersampling; motion
Pharmacokinetic Modeling of Delayed Gadolinium
Enhancement in the Myocardium
Benjamin R. Knowles,1 Philip G. Batchelor,1 Victoria Parish,1 Matthew Ginks,1
Sven Plein,2 Reza Razavi,1 and Tobias Schaeffter1*
Delayed contrast-enhanced magnetic resonance imaging
(DCE-MRI) provides prognostic information by delineating regions of myocardial scar. The mechanism of this delayed enhancement in myocardial infarctions (MIs) is hypothesized to
result from altered kinetics and changes in the volumes of
distribution in the myocardium. Pharmacokinetic models with
two and three compartments were fitted to the concentrationtime curves of dynamic contrast-enhanced MRI data obtained
from five patients with known MI. Furthermore, the parameter
stability was investigated in simulations for the two different
models. The transfer constants and volumes of distribution
showed a good correlation with imaging findings on early and
delayed contrast-enhanced MRI. The two compartment model
showed higher parameter stability. The three compartment
model allows a more in-depth quantification of myocardial
scarring. These models have the potential to improve the diagnosis of myocardial pathologies involving scar, with differing
kinetics and volumes of distribution such as infarction or
cardiomyopathy. Magn Reson Med 60:1524 –1530, 2008.
© 2008 Wiley-Liss, Inc.
Key words: pharmacokinetics; delayed contrast enhancement;
viability; cardiac MRI; infarction
pothesized to result from alterations in wash-in/wash-out
kinetics (4) and volume of distribution (5). These parameters are likely to be different in areas with different tissue
characteristics. After an acute MI, MRI obtained early after
contrast agent administration often lack enhancement at
the region of microvascular obstruction (MVO), whereas
areas with fibrosis and necrosis are visible as areas of high
signal on images obtained at later time points. Such DCE is
not specific for MI and can be observed after cardiac interventions and in many other cardiac diseases, e.g., cardiomyopathy and myocarditis, making diagnosis sometimes
difficult. Recently, a combination of early and delayed
contrast-enhancement MRI has been proposed to differentiate between MI and myocarditis (6), indicating that the
pharmacokinetics of the contrast agent retention provides
additional information about the underlying pathology.
Here we propose the application of pharmacokinetic models for DCE MRI, thus providing a more quantitative approach to the diagnosis of myocardial pathologies. In particular, we have derived compartment models for delayed
enhancement that are similar to kinetic models used in
positron emission tomography (PET) (7).
Myocardial ischemia and myocardial infarction (MI), consequences of coronary artery disease (CAD), are the leading cause of death and highest medical care expense in the
THEORY
United States and Europe (1). The detection and evaluation of the myocardium damaged during ischemia is of Background and Perfusion Models
vital importance for the treatment and prognosis of pa- Pharmacokinetic modeling in MRI is concerned with modFULL
PAPER
tients with
ventricular
dysfunction. Other conditions such eling the time course of changes in the concentration of a
Magnetic Resonance in Medicine 000:000–000 (2012)
as different types of cardiomyopathy also lead to ventric- gadolinium-based contrast agent in a specific tissue of
ular failure. The use of magnetic resonance imaging (MRI) interest, (Ct(t)). There are various first-pass perfusion modfor diagnosis of different causes of ventricular failure and els in existence, such as those from Tofts and Kermode (8)
treatment monitoring is expanding. In particular, delayed or Larsson et al. (9), and model-independent techniques
contrast-enhanced (DCE) cardiac MRI, which was first de- such as in Jerosch-Herold et al. (10). The basis of model
scribed more than 10 years ago (2) is becoming the stan- derivation begins by assuming the tissue can be simplified
dard for the evaluation of the different patterns of myocar- into compartments, through which the passage of contrast
dial scar seen in MI and cardiomyopathies. Localization of agent can be modeled. These models assume two compartscar is performed by the administration of a gadolinium ments, consisting of a blood plasma volume and the extracontrast agent. Retention of contrast agent occurs in areas cellular extravascular space (EES), with respective fracof scarring or fibrosis, and these areas appear as an area of tional volumes of vp and ve, and respective concentrations
high signal intensity (3) due
Tissue uptake of contrast
1 to the T1 shortening effect1of of contrast agent Cp(t) and Ce(t).
* Amedeo
Chiribiri,
Niloufar
L. T. F. Hautvast,2 Masaki Ishida,1
the
contrast Zarinabad,
agent. The mechanism
of DCE
in MIs is hy-Gilion
agent occurs across a permeable barrier between the blood
1
1
Andreas Schuster,1 Zoran Cvetkovic,3 Philipplasma
G. Batchelor,
Eike
Nagel
volume and and
the EES.
How
easily contrast agents
can move between compartments is dependent on the
1King’s College London, Division of Imaging Sciences, London, United Kingparameter known as the transfer constant, Ktrans (11). This
The purpose of this study is to enable high spatial resolution parameter
dom.
techniques,
including Doppler
catheterization
andcomcorois dependent
on the permeability
of the
2voxel-wise
analysis
myocardial
perfusion
Academic Unitquantitative
of Cardiovascular
Medicine,ofUniversity
of Leeds,
Leeds Gen-in
nary sinus
thermo
areother
available
for measuring
partment
barrier
in alldilution,
conditions
than when
there is
eral
Infirmary,
Leeds, United Kingdom.
dynamic
contrast-enhanced
cardiovascular MR, in particular by
myocardial
blood
flow
in humans.
These methvery
low blood
flow
to (MBF)
the tissue.
Concentration-time
*Correspondence
to:
Tobias
Schaeffter,
King’s
College
London,
BHF
Centre,
finding the most favorable quantification algorithm in this context.
ods, which
variationsinofa indicator
dilution
methods,
curves
can bearemeasured
tissue from
dynamic
MR
Division of Imaging Science, NHR Biomedical Research Centre at Guy’s and
Four deconvolution algorithms—Fermi function modeling, deconSt. Thomas NHS Trust Foundation, London, United Kingdom SE1 7EH. Eare invasive and can only assess average perfusion of
volution
using B-spline basis, deconvolution using exponential ba- images, and then the model is fitted to these curves. Thus
mail:
[email protected]
whole coronary artery territories. Amongst noninvasive
sis, and22
autoregressive
moving
modeling
tested the shape of each curve will be dependent on the transfer
Received
January 2008; revised
26 average
June 2008;
accepted —were
7 July 2008.
imaging
techniques,
emission
tomography
between
the bloodpositron
volume and
the EES
(Ktrans), and(PET)
the
to calculate
voxel-wise perfusion estimates. The algorithms were rate
DOI
10.1002/mrm.21767
is currently
regarded
as each
a gold
standard forMost
the quantifivolume
size of
compartment.
imporPublished
online
Wiley InterScience
(www.interscience.wiley.com).
developed
on insynthetic
data and
validated against a true gold- respective
standard
using aInc.
hardware perfusion phantom. The accuracy of
©
2008 Wiley-Liss,
1524 cation of absolute MBF. However, this technique has
several drawbacks including low spatial resolution (makeach method was assessed for different levels of spatial averaging
and perfusion rate. Finally, voxel-wise analysis was used to genering it unsuitable for the detection of subtle subendocarate high resolution perfusion maps on real data acquired from five
dial perfusion defects), patient radiation exposure, and
patients with suspected coronary artery disease and two healthy high cost (2,3).
volunteers. On both synthetic and perfusion phantom data, the
Compared with PET, dynamic contrast-enhanced carB-spline method had the highest error in estimation of myocardial
diovascular magnetic resonance (DCE-CMR) imaging has
blood flow. The autoregressive moving average modeling and exseveral potential advantages: superior spatial resolution,
ponential methods gave accurate estimates of myocardial blood
absence of ionizing radiation, and availability of stable
flow. The Fermi model was the most robust method to noise. Both
simulations and maps in the patients and hardware phantom and inert contrast agents of low toxicity. Estimation of
MBF from DCE-CMR studies has been reported using a
showed that voxel-wise quantification of myocardium perfusion is
number of different analysis techniques including quanfeasible and can be used to detect abnormal regions. Magn
C 2012 Wiley Periodicals, Inc.
Reson Med 000:000–000, 2012. V
titative and semiquantitative methods (4–14).
Although favorable results with semiquantitative techKey words: myocardial perfusion; voxel-wise quantification;
accuracy; noise robustness
niques such as upslope analysis of the myocardial time–
intensity curve have been reported, these methods have
shown to underestimate the perfusion parameters
INTRODUCTION
(15,16). Moreover, semiquantitative analysis relies on a
Detection of myocardial ischemia is the key to the diag- ratio which introduce a bias on the data itself and the
nosis of coronary artery disease (1). Several invasive relationship between MBF and the semiquantitative
methods parameters such as the curve upslope is not as
clear-cut as the relationship between MBF and the
1
Division of Imaging Sciences and Biomedical Engineering, King’s College
impulse response amplitude which we get from quantitaLondon BHF Centre of Excellence, NIHR Biomedical Research Centre and
tive analysis (8,17), whereas using fully quantitative
Wellcome Trust and EPSRC Medical Engineering Centre at Guy’s and St.
analysis allows the absolute quantification of MBF in
Thomas’ NHS Foundation Trust, The Rayne Institute, St. Thomas’ Hospital,
London, United Kingdom.
units of ml/g/min and may permit more accurate and
2
Philips Healthcare, Imaging Systems–MR, Veenpluis 4-6, The Netherlands.
objective assessment of altered myocardial perfusion in
3
DivisionofEngineering,King’sCollegeLondon,Strand,London,UnitedKingdom.
patients with heart disease.
Grant sponsor: Wellcome Trust and the EPSRC; Grant number: WT
Quantitative methods can be further divided into two
088641/Z/09/Z; Grant sponsor: Department of Health via the National
Voxel-Wise Quantification of Myocardial Perfusion by
Cardiac Magnetic Resonance. Feasibility and Methods
Comparison
Magnetic Resonance in Medicine 54:1273–1280 (2005)
KING’S MEDICAL ENGINEERING CENTRE Matrix Description of General Motion Correction Applied
to Multishot Images
P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and
D. Larkman3
Motion of an object degrades MR images, as the acquisition is
time-dependent, and thus k-space is inconsistently sampled.
This causes ghosts. Current motion correction methods make
restrictive assumptions on the type of motions, for example,
that it is a translation or rotation, and use special properties of
k-space for these transformations. Such methods, however,
cannot be generalized easily to nonrigid types of motions, and
even rotations in multiple shots can be a problem. Here, a
method is presented that can handle general nonrigid motion
models. A general matrix equation gives the corrupted image
from the ideal object. Thus, inversion of this system allows us to
get the ideal image from the corrupted one. This inversion is
possible by efficient methods mixing Fourier transforms with
the conjugate gradient method. A faster but empirical inversion
is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves
between shots. The results indicate that it is now possible to
correct for nonrigid types of motion that are representative of
many types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005
Wiley-Liss, Inc.
Key words: motion correction; ghosts; multishot; conjugate
gradient; auto focus
Motion of an object can degrade MR images and imposes
rection method could be used to spatially transform the
ghosted image by the transformation corresponding to a
shot, pick the k-space lines corresponding to that shot,
and repeat this operation for all shots (this is a version
of the method used in (1)). We could then rebuild an
image by inverse Fourier transform. This method is in
general incorrect, as shown by the difference between
translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the
other hand, correcting rotations requires knowledge of
the data at neighboring k-space positions and these are
acquired at different times. Before applying the Fourier
rotation theorem, we would need to “synchronize”
neighboring values. Furthermore, complicated motions
such as nonrigid deformations cannot have a simple
description in Fourier space. Here, however, we show
that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical
description of the problems involved; the motion corruption is entirely described by a large matrix acting on
the space of images. Thus, inversion of this matrix
should correct the motion’s effects. This approach is of
theoretical interest, but its practical value depends on
how easily we can find a solution of the linear system. It
turns out that with some careful linear algebra, not only
KING’S MEDICAL ENGINEERING CENTRE Motion during Image Acquisition
Lung Liver KING’S MEDICAL ENGINEERING CENTRE Motion during Image Acquisition
Lung •  Blurring of moving structures
Liver KING’S MEDICAL ENGINEERING CENTRE Motion during Image Acquisition
Lung •  Blurring of moving structures
Liver •  Image Artifacts
KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating
1.  Cardiac Motion
2.  Breathing Motion
KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating
ECG
KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating
ECG
Respiration
accept
reject
KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating
ECG
Respiration
accept
reject
KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating Efficiency
How efficient is Cardiac MRI?
1.  Efficient:
acquiring during 50% of Time
2.  Doing OK:
acquiring during 20% of Time
3.  Inefficient:
acquiring during < 5% of Time
KING’S MEDICAL ENGINEERING CENTRE Answer:
ECG
Respiration
accept
reject
Cardiac Motion:
Respiration:
10%
50%
Overall:
5%
also
in other studies that fully overlapping blocks lead to a
958
better motion estimation. The appropriate search range has
to be adapted to the largest displacement that depends on
the speed of the motion. In the kinetic knee studies the
maximal displacement was 23 mm. Using an acquisition
time of 230 msec for each subset, the maximal detectable
speed was 0.1 m/sec.
KING’S MEDICAL ENGINEERING CENTRE Image Reconstruction With Motion Compensation
In Figure 2, the motion estimation technique has been
applied to the acquired sub-images as described in the
Motion Compensated Reconstruction
Motion
is estimated with respect to the
In previous
a first study, thesection.
feasibility of the
motion-compensated
FIG. 5. Mean squared difference (MSD), which was used as a quality
reconstruction was tested using static MR-images of the
measure for the hierarchical motion estimation. a: MSD for various
second subset used as a reference state of motion. However,
block sizes and different kinetic studies. b: MSD for different knee-joint at four different positions. These static images
Magnetic Resonance in Medicine 41:954–963 (1999)the ‘‘gold-standard’’ from which subsets representpixel-offsets between neighboring blocks. An offset of one pixel defined
the reference frame can be chosen arbitrarily, and motion
ing the various motion states were extracted. In these data,
indicates fully overlapping blocks.
motion
only between the
data subsets.
The data
cantakes
beplace
estimated
with
respect
to each of the four motion
Motion Compensated Projection Reconstruction
were subjected to the hierarchical motion compensated
states.
After
determining
all
displacement
fields d!i(x! ), the
three resolution levels was applied. The arrows represent reconstruction scheme described above, and the resulting
Tobias Schäffter,* Volker Rasche,
Ingwer C.for
Carlsen
theand
displacements
each block, indicating that the lower high resolution images were compared to the original
reconstruction
of the
high-resolution
Motion
in Image
spaceimage according to
limb is swinging to the left. The images demonstrate that images.
Figure
6
shows
the
results
of
the
hierarchical
MCthe continuity
the motion
field
is improved
if several
Over recent years, MRI has shown the capability
for real-time of After
the motion
information
has been
recorded during
the
Eq.
1 can
simply be modified to compensate for the motion:
applications. Although the acquisition timesresolution
of fast MRI methreconstruction
of one single motion state. The image in
levels are
used. it is used to reorder all acquired data
acquisition,
with
Motion Compensated MR in Image Space
ods have been reduced significantly, patient motion during a
respect
to the
different
motion
(7). However,
Figurethe6a has been reconstructed from the subsets of the
The sizes and the
overlaps
of the
blocks
of thestates
hierarchimagnetic resonance imaging (MRI) examination still causes
reconstruction
of images at
motion
is only motion states and shows severe image blur. For
different
cal
algorithm
were optimized
bydifferent
evaluating
thestates
artifacts in the image. In this paper, the effects
of matching
motion in MRI
R was reconstructed from 256
possible
as
a
post-processing
step
after
sufficient
data
have
comparison,
a referenceH
image
MCIt difference
using a radial acquisition scheme are examined.
is shown thatimages. In Figure 4 difference images with" "
MC
been
acquired.
Finally,
corrective
reconstruction
methods
motion can be estimated without the use of additional
out (Fig.measure4a) and with motion compensation for different static projections, which is given in Figure 6b. The image
i
are available, incorporating knowledge about the motion
ment, based on the acquired projections only.block
A new reconstrucin
Figure
6c
has
been
reconstructed
using
the
hierarchical
sizes (Fig. 4b,c)
are
shown
for
a
moving
knee.
Figure
tion technique is introduced that integrates a motion compensa- into the reconstruction process. In spin-warp MR-imaging
MC-reconstruction.
The artifacts in this image are signifi5 shows
theinMSD
for different
block sizes
andhas
overlaps.
transform
correction
been introduced
(8).
tion algorithm into the MR-reconstruction process,
resulting
a a generalized
cantly
reduced. However, the overall quality of the MCMSD were normalized
to correction
the MSD ofwithout
motion
significant reduction of blurring artifacts in The
the reconstructed
So far, only the
rigid body
movement
(9) and
images. The proposed method is applied tocompensation.
different kinds of In linear
(Fig. 6c) is worse than that
Figure
5a the MSD
for different
block
expansions
(10) have
been shown
to beimage
feasible.
" of the static image (Fig.
motion such as kinetic joint studies. Magn Reson
reconstruction
methods
have also
6b),develdue to imperfections of the motion estimation.
sizes Med
are41:954–
shown. Corrective
A minimum
in the MSD
of 9 pixels
is been
963, 1999. ! 1999 Wiley-Liss, Inc.
oped i.e.
for techniques
to projection
reconstruction
The MC-reconstruction was tested on MR-movies, i.e.
found for both cases,
a moving related
hand and
a moving
Key words: MRI; motion correction; projection-reconstruction
(11–13). To our knowledge, so far no corrective reconstruc-
I
(x! ) !
! BP(5p (u!
· (x! # d!i(x! )))6i)
u
! ! (cos ", sin ").
knee. Smaller blocks were found to give poorer results due
[4]
In this equation, a back-projection is applied to the filtered
projection p of the ith subset taking the displacement d!i(x! )
of each pixel with respect to one reference frame into
account, i.e. the back-projection is calculated at the position x! # d! (x! ). The motion compensated (MC) highresolution image I (x! ) represents the motion state with
respect to one reference frame. As described above, the reference frame can be chosen arbitrarily and for each subset a
high-resolution MC-image can be reconstructed using different
sets of displacement fields. Thus the MC images show different
motion states with high spatial resolution.
studies in which motion occurs continuously during the
tion technique can correct for complex motion, i.e. motion
to a less pronounced
maximum in the similarity measure, acquisition process. Although the assumed motion model
that cannot be modeled by a rigid body movement or an
Patient motion during magnetic resonance
imagingfor
(MRI)
whereas
largerexpansion
sizes themodel.
influence of elastic deforma- is not fully valid, the" MC-reconstruction significantly
data acquisition causes artifacts in the reconstructed
image
reduces
tions and rotationsWeofhave
the recently
blocks shown
is unfavorably
high.
the feasibility
of a new
correc- image blurring. Figure 7 shows a selection of
that obscure anatomical details. The main source of these
artifacts is macroscopic motion of organs or extremities, tive reconstruction technique for a projection based reconwhich is either caused by wanted movement as in kinetic struction (14). The method avoids the need for additional
studies, or by unwanted physiological motion due to flow, measurements, motion modeling or reordering of data.
respiration, peristalsis or cardiac pulsation. The appear- Motion is estimated by means of a block matching algoi
ance of these kinds of motion in the final MR-image rithm that can cope with even complex motion. The result
HR
strongly depends on the trajectory through k-space along of this algorithm is a motion field that represents the
MC
which the MR data are sampled. In spin-warp imaging, movement of each block by means of a motion vector.
motion results in ghost repetitions of the moving structures These estimated motion vectors are used during reconstrucin the phase encoding and blurring in the readout direction tion to either reduce artifacts in real-time MRI or to
(1). In spiral and radial acquisition schemes, motion results reconstruct different motion states from one single data set.
in blurring of the moving structures with superimposed The purpose of this paper is to demonstrate and discuss
radial streaking artifacts or spiral-like artifacts (2,3). Over results obtained from various regions of the human body
recent years, MRI has shown the capability for real-time and to outline the advantages and limitations of this
applications. Although fast imaging techniques are more approach.
immune to most types of macroscopic motion, for some
applications like kinetic MRI-studies the acquisition speed
of MRI is still too low in relation to motion.
METHODS
A number of strategies have been developed to degrade
This section introduces the new corrective reconstruction.
the effects of motion. A first strategy, which requires a high
A brief description of the data acquisition and reconstrucdegree of co-operation from the patient, is
to suppress
the of the knee joint. a: Reconstruction with motion using subsets of four different static positions. b: Reference image without
FIG.
6. MR-images
tion in projection reconstruction MRI is given. It is shown
motion using fixation devices or by means
breathcompensated reconstruction.
motion. of
c: Motion
that an interleaved MR-acquisition scheme allows the
holding (4). A second strategy is gating, where motion is
reconstruction of low-resolution images at different morecorded by navigator echoes (5) and only those data are
tion states during the acquisition of one high-resolution
accepted that correspond to one single motion state. This
image. The influence of motion on the image will be
strategy is very effective but has the severe disadvantage of
discussed for projection reconstruction MRI. It is shown
significantly increased scan time. A third strategy is to
that complex motion can be estimated by means of a block
reorder the sequence of profiles during acquisition. In
matching algorithm. A new reconstruction technique is
respiratory ordered phase encoding (ROPE) (6), for exintroduced that integrates estimated motion vectors. Fiample, the knowledge of the respiratory motion is used to
nally, an improved version of the motion-compensated
apply an acquisition scheme suited to spin-warp imaging.
reconstruction is given.
Reconstruction With a Hierarchical Motion Compensation
Philips Research Laboratories, Division Technical Systems, Hamburg, Ger-
According to Eq. 4, an MC-image is reconstructed using
displacement fields that are estimated from sub-images, i.e.
images reconstructed from one single data subset. The
accuracy of the motion estimation is thus limited by the
resolution of the sub-images. In the following, a reconstruc-
ing the da
resolution
3) Step
with a h
images ha
4) Mot
obtained
two levels
in step 1.
5) The
images is
resolution
the image
Due to
nique gen
can be co
ing the ac
METHOD
All meas
Philips G
Tesla resp
6000 syst
For signa
elements w
echo sequ
The hie
ent kinds
hierarchic
number o
each leve
the larges
the image
KING’S MEDICAL ENGINEERING CENTRE Motion Compensated MR in k-Space
Magnetic Resonance in Medicine 54:1273–1280 (2005)
Matrix Description of General Motion Correction Applied
to Multishot Images
P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and
D. Larkman3
Motion of an object degrades MR images, as the acquisition is
time-dependent, and thus k-space is inconsistently sampled.
This causes ghosts. Current motion correction methods make
restrictive assumptions on the type of motions, for example,
that it is a translation or rotation, and use special properties of
k-space for these transformations. Such methods, however,
cannot be generalized easily to nonrigid types of motions, and
even rotations in multiple shots can be a problem. Here, a
method is presented that can handle general nonrigid motion
models. A general matrix equation gives the corrupted image
from the ideal object. Thus, inversion of this system allows us to
get the ideal image from the corrupted one. This inversion is
possible by efficient methods mixing Fourier transforms with
the conjugate gradient method. A faster but empirical inversion
is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves
between shots. The results indicate that it is now possible to
correct for nonrigid types of motion that are representative of
many types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005
Wiley-Liss, Inc.
Key words: motion correction; ghosts; multishot; conjugate
gradient; auto focus
Motion of an object can degrade MR images and imposes
constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that
the acquisition is time-dependent, and the Fourier transform of the image seen during acquisition changes due to
the deformation of the object. This causes inconsistencies
in k-space and hence ghosts in the image.
Standard motion correction methods make assumptions on the type of motions, for example, that it is a
translation or a rotation, and use formulas on Fourier
transforms to correct the data (1–3). We assume here that
these data are acquired in shots. When the data positions at each shot are known, an empirical motion cor-
1
Medical Physics & Bioengineering, University College London, London,
United Kingdom.
2
Department of Electrical Engineering, Pontifica Universidad Católica de
Chile, Chile.
3
Imaging Sciences, Imperial College London, London, United Kingdom.
*Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengineering, University College London, Gower Street, London WCIE 6BT, UK,
E-mail: [email protected]
Presented in part as abstracts at the 2nd International Workshop on ParallelMRI, Zürich, 2004, and MIUA, London, Sept. 2004.
Grant Sponsor: EPSRC; Grant Numbers: GR/S30184, AF/001381; Grant
rection method could be used to spatially transform the
ghosted image by the transformation corresponding to a
shot, pick the k-space lines corresponding to that shot,
and repeat this operation for all shots (this is a version
of the method used in (1)). We could then rebuild an
image by inverse Fourier transform. This method is in
general incorrect, as shown by the difference between
translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the
other hand, correcting rotations requires knowledge of
the data at neighboring k-space positions and these are
acquired at different times. Before applying the Fourier
rotation theorem, we would need to “synchronize”
neighboring values. Furthermore, complicated motions
such as nonrigid deformations cannot have a simple
description in Fourier space. Here, however, we show
that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical
description of the problems involved; the motion corruption is entirely described by a large matrix acting on
the space of images. Thus, inversion of this matrix
should correct the motion’s effects. This approach is of
theoretical interest, but its practical value depends on
how easily we can find a solution of the linear system. It
turns out that with some careful linear algebra, not only
are we able to invert in a generalized sense, but also this
inversion can be done efficiently in practice. For this we
use the LSQR algorithm, which is a robust implementation of the conjugate gradient of the normal equation
(see (4)).
This leaves the question of finding what motion actually
happened. Different cost functions have been designed to
quantify how much an image has been corrupted (1, 5, 6).
We explore different cost functions in conjunction with
our motion correction. Optimization of such cost functions
means repeating the matrix inversion iteratively. However,
inverting matrices repeatedly may be prohibitive even if it
is practicable on a one-off basis. We therefore also investigate the use of the empirical method described above.
The matrix equation allows us to find when this approximation is correct. We show three-dimensional random
affine and pulsatile nonrigid motion corrections on simulated data and an example of nonrigid correction of in vivo
data (moving legs).
THEORY
We need a method to handle motion-corrupted MR images
78
spatial transformation. This matrix is to be distinguished
Motion
Ghosting
from the
inverse matrix of ut, which will not necessarily
exist.
But,
for an image sinduce
whose signal
in the FOV,
Spatial transformations
linear remains
image transforma!
u
u
s
"
s.
We
use
lowercase
letters
for
objects
to
t
t the images whose field of view (FOV) related
tions
on
is transimage For
space
and capital
letters for objects
to kformed.
an image
s, the transformed
imagerelated
is the image
space.
whose intensities at a pixel are the intensities of s at the
KING’S MEDICAL ENGINEERING CENTRE notation). Inversion of this matrix would recover an ob{uwhatever
3:
Spatially
transform
s0. corrupted view,
ts0}
ject’s
image
from
its motion
the
4:
Fourier transform.
{Futs0}
motion and time-sampling pattern in k-space. The matrix
5:
Extract the lines corresponding to shot t.
{AtFuts0}
is large, of size n1n2 % n1n2 for n1%n2 images,
but Algo6:
Set these lines in S.
{S " S # AtFuts0}
rithm 1 is an efficient implementation of multiplication by
7: end for
of Eq. (1)
wouldofbe
8:. A
Letk-space
s be theversion
inverse Fourier
transform
S. S "{s($
"t A
$ttFUHt)S
At0Fu"ts0}
!S0 where Ut " FutFH is the k-space representation of
motion
at time
t. form. From this algorithm, we obtain a
responding
matrix
Motion Compensated MR in k-Space
(inverse) transformed pixels (see (7) 3.3.2). The linear operations
on images are defined pixelwise.
Aliasing
Aliasing is an image space consequence of subsampling
Magnetic Resonance in Medicine 54:1273–1280 (2005)
Transformation
in the Fourier Matrices.
domain. As such, it can be represented by
Matrix Description
of General
Correction
Applied
a matrix. We
define a Motion
shot to be
a subsample
of k-space
The effect of a spatial transformation on images can be
to Multishot
andImages
make the assumption that all the corresponding
represented as a sum of linear image transformations and
Fourier
components
are
acquired
or in a
1
1
2
1 simultaneously,
P. G. Batchelor,
Atkinson,
P. Irarrazaval,
D. L. G. Hill,
J. Hajnal,3 and
can* D.
thus
be written
as matrix
multiplication.
In the exD. Larkman3 very short time in comparison to any processes causing
treme
case,
image transformation
on a single
change
toeach
the object.
In other words operates
shots correspond
to
Motion of an object
degrades MR
images, as the
acquisition
is rection
method
couldone
be usedlocation
to spatially transform
the
pixel,
moving
the
pixel
value
from
to
another.
time-dependent, and time
thus k-space
is inconsistently
sampled. ghosted
steps.
A partition
of image
k-space
in ns shots
leads
by the transformation
corresponding
to a to n s
This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot,
We
write uAt for
the matrix
acting
on
image space
that
restrictive assumptions
on the type of motions,
for example, and repeat
matrices
to the
corresponding
this operation
for all shots (this is shot
a versionposit that project
that it is a translation or rotation, and use special properties of
of deformation
the method used in (1)).
We could then rebuild
an
corresponds
to
a
spatial
dependent
on
time
t.
k-space for these
transformations.
Such
methods,
however,
tions and sum to theimage
identity
matrix
when
k-space
is
by inverse Fourier
transform.
This method
is in
cannot be generalized easily to nonrigid types of motions, and
!
general
incorrect,
as
shown
by
the
difference
between
If
the
spatial
transformation
can
be
inverted,
we
write
u
even rotations in multiple
shotscovered.
can be a problem.
Here, aaliasing matrices a are defined to be
t
fully
The
t
translation remethod is presented that can handle general nonrigid motion translations and rotations. Correcting
Hequation
quires only pointwise phase
changes inby
k-space.
Oninverse
the
for
the
matrix
of
the
transformation
induced
the
models. A general
matrix
gives
the corrupted
image
F
A
F,
where
Fourier
transformation
is
represented
by
t of this system allows us to other hand, correcting rotations requires knowledge of
from the ideal object. Thus, inversion
data atmatrix
neighboring
k-space
positions
and these are
spatial
transformation.
This
is
to(hermitian
be distinguished
get the ideal image
from
corrupted
inversion
is the
F the
and
FHone.isThisthe
conjugate
transpose
conjupossible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier
the conjugate gradient
method.
A faster
but empirical inversion
from
the
inverse
matrix
of
u
,
which
will
not
necessarily
rotation
theorem,
we
would
need
to
“synchronize”
gate) of F.
t
is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions
ulated three-dimensional
affine
data and
two-dimensional
pul- such
exist.
But,
for
an
image
s
whose
signal
remains
ina the
as
nonrigid
deformations
cannot
have
simple
For
rectangular
FOVs,
and
subsampling
along
oneFOV,
spasation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show
tion. All examplesu
are !
multishot
images
where
the
object
moves
u
s
"
s.
We
use
lowercase
letters
for
objects
related
to
tial
dimension,
the
computation
factors
along
dimensions,
that it is possible
to
correct
complicated
motions,
int
t
between shots. The results indicate that it is now possible to
correct for nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical
image
space
and
capital
letters
for
objects
related
to
kand
the
matrices
a
have
a
sparse
block
structure
corret description of the problems involved; the motion cormany types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on
sponding to the standard
ghosts.
the space
of images. Thus, inversion of this matrix
Wiley-Liss, Inc. space.
Key words: motion correction; ghosts; multishot; conjugate
gradient; auto focus
Motion of an object can degrade MR images and imposes
constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that
the acquisition is time-dependent, and the Fourier transform of the image seen during acquisition changes due to
the deformation of the object. This causes inconsistencies
in k-space and hence ghosts in the image.
Standard motion correction methods make assumptions on the type of motions, for example, that it is a
translation or a rotation, and use formulas on Fourier
transforms to correct the data (1–3). We assume here that
these data are acquired in shots. When the data positions at each shot are known, an empirical motion cor-
should correct the motion’s effects. This approach is of
theoretical interest, but its practical value depends on
how easily we can find a solution of the linear system. It
turns out that with some careful linear algebra, not only
are we able to invert in a generalized sense, but also this
inversion can be done efficiently in practice. For this we
use the LSQR algorithm, which is a robust implementation of the conjugate gradient of the normal equation
(see (4)).
This leaves the question of finding what motion actually t
happened. Different cost functions have been designed to
quantify how much an image has been corrupted (1, 5,t6).
We explore different cost functions in conjunction with
our motion correction. Optimization of such cost functions
means repeating the matrix inversion iteratively. However,
0 may be prohibitive even if it
inverting matrices repeatedly
is practicable on a one-off basis. We therefore also investigate the use of the empirical method described above.
The matrix equation allows us to find when this approximation is correct. We show three-dimensional random
affine and pulsatile nonrigid motion corrections on simulated data and an example of nonrigid
correction of in vivo
s
data (moving legs).
Motion during acquisition
Aliasing
We are now
to space
give the
exact formulaoffor
the effect of
Aliasing
is angoing
image
consequence
subsampling
acquisition
Suppose u deinany
the motion
Fourierduring
domain.
As such,init k-space.
can be represented
by
scribes
the
spatial
transformation
at
time
t,
and
A
is
the
a matrix. We define a shot to be a subsample of k-space
sampling
at this time.
observed
image s is
and
make of
thek-space
assumption
that The
all the
corresponding
related
to
the
ideal
object
image
s
by
the
sequence
Fourier components are acquired simultaneously, or inin
a
Algorithm
1.
The
expressions
in
curly
braces
are
the
corvery short time in comparison to any processes causing
Medical Physics & Bioengineering, University College London, London,
change to the object. In other words shots correspond to
United Kingdom.
Department of Electrical Engineering, Pontifica Universidad Católica de
Chile, Chile.
time steps. A partition of k-space in n shots leads to ns
Imaging Sciences, Imperial College London, London, United Kingdom.
*Correspondence to: matrices
P. G. Batchelor Room 2.20,
Physicsproject
& BioengiAMedical
that
to the corresponding shot posit London
neering, University College London, Gower Street,
WCIE 6BT, UK,
E-mail: [email protected]
tions
sum
to
the THEORY
identity matrix when k-space is
Presented in part as abstracts
at the and
2nd International
Workshop
on ParallelMRI, Zürich, 2004, and MIUA, London, Sept. 2004.
We need a matrices
method to handle motion-corrupted
MR imagesto be
Grant Sponsor: EPSRC;
Grant Numbers:
GR/S30184, AF/001381;
fully
covered.
The Grant
aliasing
a are defined
1
2
3
simple matrix description of the motion corruption: the
Empirical
Inversesimage is a superposition of aliased views
motion corrupted
of the
object in
different
positions,
namely,
The
empirical
motion
correction
method
mentioned in the
Motion inamounts
K-space
Introduction
to Algorithm 1 with the motions ut
!
!
n s !u1t . We write
replaced by their inverses
for the corresponding matrix, swhich
weat u
call
the
empirical
inverse.[1]
"
s
"
s
.
t
0
0
This algorithm is incorrect
in
the
sense
that
in
general
t"0
!
s " ! s0 & s0. The mathematical reason is that a sum
of
not the
inverse
of the sum.
physical
Theinverses
formulais here
is the
key insight
as it The
introduces
the
n s !1
reason
is
that
the
shot
lines
corresponding
to
t
were
indeed
matrix g ! $t"0 atut that entirely describes the effect of
transformed
a known
spatial
but only
motions on by
signals
acquired
intransformation,
the Fourier domain.
We
these
lines,
and
this
algorithm
applies
the
transformation
call this matrix the ghosting matrix (hence the choice of
to the whole image. It is easy to see that if the aliasing
notation). Inversion of this matrix would recover an obmatrices at and the spatial transformations matrices ut,
ject’s image from its motion corrupted view, whatever the
commute for all t,t', then ! inverts (assuming ut!uts0 "
motion and time-sampling pattern in k-space. The matrix
Inverse
sEmpirical
).
is0large, of size n1n2 % n1n2 for n1%n2 images, but Algorithm 1 is an efficient implementation of multiplication by
ns ! 1
ns ! 1
ns ! 1
. A k-space
version
of Eq. (1) would
be
S " ($t AtUt)S0 "
!
!
!
H
s
"
a
,u
a
u
s
"
a
u
a
ut s0
t
t
t
t
0
t'
t'
!S0 where Ut " FutF is the k-space trepresentation
of
t' " 0
t"0
t,t' " 0
motion at time t.
!" #
!" #
"
"
n s !1
at' at ut' !ut s0 !
Empirical"Inverses
t,t' " 0
"
"
ns ! 1
Batchelor
! et al.
t"0
at ut ut s0 "
"
ns ! 1
at s0 " s0.
t"0
The empirical motion correction method mentioned in the
FIG. 2. Nonrigid example (simulation),
Introduction
amounts to whose
Algorithm
1 with the motions
Spatial
transformations
transformation
matrices ut
which simulates a pulsation!in 16 shots. (a)!
replaced
by
their
inverses
u
.
We
write
for
the
commute
with
aliasing
matrices
are
then
special
in correthat
t (b) the correcThe motion-corrupted image;
sponding
matrix,
which
we
inverse.
they
allow
for
a
fast
exact
reconstruction.
This
means
that
$ call the empirical
tion by empirical inverse
; (c) the correcaliasing
first
and
transforming
the
Thistion
algorithm
isthen
incorrect
in (d)
thethemust
sense
that in to
general
by the
LSQR
algorithm;
gold-amount
!
!
s
s"
&
s
.
The
mathematical
reason
is
that
a
sum
standard0image.
0 (e) The deformation at each
of inverses
is
not
the
inverse
of
the
sum.
The
physical
of the 16 shots on a square checkerboard
reason
is that the shot lines corresponding to t were indeed
image.
transformed by a known spatial transformation, but only
these lines, and this algorithm applies the transformation
KING’S MEDICAL ENGINEERING CENTRE Motion Compensated Cardiac MRI
Magnetic Resonance in Medicine 54:1273–1280 (2005)
Knowledge [v.5.6]
Matrix Description of General Motion Correction Applied
http://cm.webofknowledge.com/viewCitationPrint.do
to Multishot Images
P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and
D. Larkman3
Print | Close
Motion of an object degrades MR images, as the acquisition is
time-dependent, and thus k-space is inconsistently sampled.
This causes ghosts. Current motion correction methods make
restrictive assumptions on the type of motions, for example,
that it is a translation or rotation, and use special properties of
k-space for these transformations. Such methods, however,
cannot be generalized easily to nonrigid types of motions, and
even rotations in multiple shots can be a problem. Here, a
method is presented that can handle general nonrigid motion
models. A general matrix equation gives the corrupted image
from the ideal object. Thus, inversion of this system allows us to
get the ideal image from the corrupted one. This inversion is
possible by efficient methods mixing Fourier transforms with
the conjugate gradient method. A faster but empirical inversion
is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves
between shots. The results indicate that it is now possible to
correct for nonrigid types of motion that are representative of
many types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005
Wiley-Liss, Inc.
Matrix description of general motion correction applied to multishot images
©2012
rection method could be used to spatially transform the
Tuesday, September 18 2012
ghosted image by the transformation corresponding to a
shot, pick the k-space lines corresponding to that shot,
and repeat this operation for all shots (this is a version
of the method used in (1)). We could then rebuild an
image by inverse Fourier transform. This method is in
general incorrect, as shown by the difference between
translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the
other hand, correcting rotations requires knowledge of
the data at neighboring k-space positions and these are
acquired at different times. Before applying the Fourier
rotation theorem, we would need to “synchronize”
neighboring values. Furthermore, complicated motions
such as nonrigid deformations cannot have a simple
description in Fourier space. Here, however, we show
that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical
description of the problems involved; the motion corruption is entirely described by a large matrix acting on
the space of images. Thus, inversion of this matrix
should correct the motion’s effects. This approach is of
theoretical interest, but its practical value depends on
how easily we can find a solution of the linear system. It
turns out that with some careful linear algebra, not only
are we able to invert in a generalized sense, but also this
inversion can be done efficiently in practice. For this we
use the LSQR algorithm, which is a robust implementation of the conjugate gradient of the normal equation
(see (4)).
This leaves the question of finding what motion actually
happened. Different cost functions have been designed to
quantify how much an image has been corrupted (1, 5, 6).
We explore different cost functions in conjunction with
our motion correction. Optimization of such cost functions
means repeating the matrix inversion iteratively. However,
inverting matrices repeatedly may be prohibitive even if it
is practicable on a one-off basis. We therefore also investigate the use of the empirical method described above.
The matrix equation allows us to find when this approximation is correct. We show three-dimensional random
affine and pulsatile nonrigid motion corrections on simulated data and an example of nonrigid correction of in vivo
Motion Compensated Cardiac MRI:
Motion of an object can degrade MR images and imposes
Michael
Schacht Hansen (NIH)
constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that
Johannes
Sebastian
Kozerke (ETH Zurich)
the acquisitionSchmidt
is time-dependent,…
and the
Fourier transform of the image seen during acquisition changes due to
the deformation of Usman
the object. This causes
Muhammad
…inconsistencies
Claudia Prieto (KCL)
in k-space and hence ghosts in the image.
Standard motion correction methods make assumpFreddy
Odille,Maria Filipovic, … Jacques Felblinger (Nancy)
tions on the type of motions, for example, that it is a
Key words: motion correction; ghosts; multishot; conjugate
gradient; auto focus
translation or a rotation, and use formulas on Fourier
transforms to correct the data (1–3). We assume here that
these data are acquired in shots. When the data positions at each shot are known, an empirical motion cor-
1
Medical Physics & Bioengineering, University College London, London,
United Kingdom.
2
Department of Electrical Engineering, Pontifica Universidad Católica de
Chile, Chile.
3
Imaging Sciences, Imperial College London, London, United Kingdom.