Advanced K-Space Based Parallel Imaging Methods
Mark A. Griswold
Universität Würzburg, Würzburg, Germany
mark@physik.uni-wuerzburg.de
Introduction:
Since the development of the NMR phased array in the late 1980s, multicoil arrays
have been designed to image almost every part of the human anatomy. These multicoil
arrays are primarily used for their increased signal-to-noise ratio (SNR) compared to
volume coils or large surface coils.
Recently several partially parallel acquisition (PPA) strategies have been proposed
which have the potential to revolutionize the field of fast MR imaging. These techniques
use spatial information contained in the component coils of an array to partially replace
spatial encoding which would normally be performed using gradients, thereby reducing
imaging time. In a typical PPA acquisition, only a fraction of the phase encoding lines are
acquired compared to the conventional acquisition. A specialized reconstruction is then
applied to the data to reconstruct the missing information, resulting in the full FOV image
in a fraction of the time.
Even though all of these techniques solve essentially the same set of linear imaging
equations, the various paths taken toward this inverse problem distinguishes the various
parallel imaging methods from each other. In this talk, we will focus on a specific set of
methods which have been developed to deal with specific problems in parallel imaging
which cannot be dealt with using the most simple methods. Specifically we cover
advanced methods to obtain coil sensitivity information with a specific focus on k-space
based autocalibrating methods. Next, we will cover various methods to increase imaging
efficiency which are based on using parallel imaging to encode more than one dimension
of the image. A specific focus will be the newly developed class of methods which
optimize multislice and volumetric acquisitions by modifying the appearance of the
aliasing artifacts through modified imaging sequences. Finally, we will discuss how nonCartesian (e.g. projection reconstruction and spiral trajectories) impact a parallel imaging
reconstruction.
Basic Imaging Equations:
Almost all parallel imaging methods solve the same basic set of imaging equations:
G  Ef
[1]
where the E matrix contains all of the encoding functions used in the imaging
experiment. In a normal completely gradient based acquisition, this matrix would be the
simple Fourier harmonics used in the acquisition. However, in parallel imaging, we
include the additional modulations provided by the imaging array. Specifically, the
encoding functions for a 2D image become:
E j  s j ( x, y )e
ik y y  ik x x
[2]
for a specific coil j. Once we have constructed the matrix of encoding functions, we only
have to invert the matrix to obtain the desired image.
However, real life application of this basic method is typically more difficult than it
appears. In the normal Cartesian imaging situation, this is almost entirely due to the
inherent difficulty of determining the coil sensitivity at each pixel in the image or at each
line in k-space. This information is contaminated by 1) noise, which degrades the coil
sensitivity information especially in regions of low spin density. Since the reconstruction
involves a matrix inverse, small errors in the coil sensitivity can lead to large errors in the
final image. 2) patient motion, which causes a misalignment between the undersampled
data and the coil sensitivity maps which again can cause large errors in the final image.
3) off-resonance, which can cause chemical shift errors as well as distortions in EPI.
4) Aliasing, which can in some cases lead to errors in the coil sensitivity maps, especially
when a low resolution acquisition or reconstruction is used. For these reasons, many
methods have been developed in the last few years to deal with these problems.
Coil Sensitivity Mapping in the Image Domain:
Pruessmann et al [1] proposed what
has become the standard method to deal
with noise in the coil sensitivity maps in
the SENSE method. The method is
based on a special acquisition designed
for coil sensitivity calibration which
collects information from both the array
and a coil with homogeneous sensitivity.
(See Figure 1.) Upon division of these
two images, a pure map of coil
sensitivity would be obtained in the
absence of noise. However, the presence
of noise (particularly in the body coil
image) can severely corrupt this simple
map. Since it can normally be assumed
that the coil sensitivity profiles are
relatively smooth, a local polynomial fit
is used to remove contribution due to
noise. Besides noise, there can be
regions of the image with no sensitivity
in either image, which leaves holes in
Figure 1: A) Raw array coil image B) Body coil image C) Raw
coil sensitivity map obtained by dividing A by B. D) Intensity
threshold E) Thresholded raw intensity map F) Final coil
sensitivity map after polynomial fitting and extrapolation.
the sensitivity map. This can lead to problems in later acquisitions if tissues move into
these areas (e.g. the lungs in cardiac exams). For this reason, data is extrapolated a few
pixels beyond the apparent borders of the object.
This method works very well in situations where there is enough time to acquire coil
sensitivity images of moderate resolution without patient motion, for example, in head
exams. However, this method can be sensitive to patient motion, particularly in
breathhold exams, and can produce serious errors whenever aliasing is present in the coil
sensitivity maps, since the assumption of smoothness is violated. The final difficulty in
this method is the requirement of an intensity threshold. Due to this threshold, all regions
which fall below the threshold which are also distant from the object are set to zero
intensity. This can lead to unexpected results, especially if tissues move into these
regions at a later time point.
In recent years, several groups have also proposed other methods for the
determination of the coil sensitivity maps. Walsh et al [2] initially proposed using an
adaptive matched filter for normal array combination to optimize the suppression of
background noise. This method has also been used for calculation of coil sensitivity maps
in parallel imaging [3-4]. The method is based on the calculation of the local signal and
noise covariance matrices at each pixel in the image. Walsh et al showed that the
eigenvector of these covariance matrices provides a nearly optimal estimate of the coil
sensitivity. The primary advantage of this method is that it works without a body coil
image. The method can also be used in many cases to form an intensity normalization for
the reconstructed image. Images are also produced with essentially normal background
appearance. On the down side, the method is still rather sensitive to aliasing the coil
images, and can be computationally slow. To counteract the slow calculation time, we
have implemented this method using a low resolution grid followed by interpolation.
While this works in many cases, large residual phase offsets from pixel to pixel resulting
from the eigenvector calculation can cause serious problems during interpolation if this is
not properly dealt with. Removing the phase of one coil from the phases of the others is
one solution to this problem.
Another approach to coil mapping is wavelet smoothing and denoising [5-6]. These
methods are relatively standard signal processing operations which can be performed
quickly. It is potentially possible to use these techniques without a body coil reference
image, although the performance is much improved with a body coil image. The method
can also work without an intensity threshold (when not using a body coil reference), so
that the method is more user independent. However, edges can still remain in the final
coil maps, which may require extrapolation, as in the polynomial fit method, in order to
avoid motion induced artifacts.
Autocalibration in K-Space and Image Domain Applications:
All of the techniques discussed above work in the image domain using a SENSE-like
reconstruction. Methods that operate in k-space have different coil mapping
requirements, and can therefore be optimized for different imaging situations. The first kspace method SMASH performed the reconstruction in k-space, but actually used coil
sensitivity maps in the image domain to determine the reconstruction parameters. For this
reason, pure SMASH shares the limitations of the coil mapping technique as in the
techniques above. The development of autocalibrated k-space methods (AUTOSMASH [7], VD-AUTOSMASH [8] and GRAPPA [9]) has removed many of these
limitations. In these
techniques, a small number
of extra lines are acquired
before, during or after the
acquisition of the
undersampled data. The
required reconstruction
parameters are then
determined directly in kFigure 2. (Left) Aliased mSENSE image with residual aliasing artifacts in the
space by fitting one or
lung. (Right) GRAPPA reconstruction from the same source data with no
several lines to other lines
artifacts.
in this calibration data set
using an equation such as:
N N b 1
S j (k y  mk y )    n( j, b, l , m) Sl (k y  bRk y )
[3]
l 1 b0
where Sj(ky) is the signal in coil j at line ky. In this case, Nb lines which are separated by
Rky are combined using the weights n(j,b,l,m) to form each line, corresponding to a
reduction factor R.
In this model, one reconstructs missing lines by first determining the weights to use in
the linear combination. Normally, a few additional lines are acquired at positions that
would normally be skipped. These data are then fit to the other normally acquired data
using Eq. 4 to determine the appropriate weights necessary for the reconstruction of
missing lines.
By fitting data to data, a pure coil sensitivity map is not needed, only the few lines of
extra data. No body coil image and no intensity thresholds are needed, thereby generating
normal appearing images even in the background. In addition, aliasing in the
reconstructed images is not a problem, thereby allowing slightly folded images to be
acquired without any problem with the reconstruction. Finally, patient motion is in
general not a problem when the extra data is acquired during the acquisition, since these
data will accurately track the coil positions as they move. While reconstructions of this
type are not guaranteed to be accurate (i.e. free from aliasing artifacts), they can be
accurate enough in practice to generate images without any visible artifacts in most cases
(e.g. Figure 2). In cases where this is not the case, any additional lines acquired in the
center of k-space for coil sensitivity mapping can be used in the final image
reconstruction to reduce any residual artifacts that may be present, as in refs [8, 9].
Volumetric Parallel Imaging
In general, all parallel imaging methods are limited by
the distribution of coil sensitivities at the various aliased
pixel locations. Ideally we would build coil arrays with
maximum possible intensity variations over the object,
since this would in general provide the best PPA
reconstruction. However, our ability to do this is limited by
electromagnetics, which limits the maximum variations a
given coil can have over a given distance, which in turn
limits our ability to encode in any one direction with
parallel imaging. The easiest solution therefore for very
high acceleration PPA is to use parallel imaging in more
than one direction simultaneously. In general, higher
acceleration factors can be achieved, approaching the
number of coil elements. In our lab, we have developed a
GRAPPA/SENSE hybrid which essentially uses SENSE to
encode one direction, while GRAPPA is used in the other.
However, the reconstruction is performed using a single 2D
GRAPPA reconstruction which unaliases both directions
simultaneously. This removes the need for accurate coil
sensitivity maps as well as intensity thresholds, etc. Figure
3 shows an example of this GRAPPA/SENSE hybrid used
with an eight element head array to achieve an acceleration
factor of 6 with excellent image quality.
Figure 3. A single partition from a
3D GRAPPA/SENSE hybrid
reconstruction (Acceleration
factor = 6.0)
Controlled Aliasing in Parallel Imaging (CAIPI)
Up to now, the all parallel MRI techniques have employed similar gradient fields and
identical pulse sequences to acquire k-space data. In this section we describe a new
approach in which the aliasing artifacts are controlled by modifying the data acquisition
period. As an example, aliasing artifacts in multislice imaging can be influenced by
shifting the individual slices by a fraction of the field of view with respect to other slices
using a POMP-type RF-phase cycle. In this case odd lines are acquired using a
specialized RF pulse which excites the two different slices with the same RF-phase (++).
On the other hand, the even lines are excited with a RF-phase difference of 180° (+-).
This acquisition scheme causes the appearance of one slice to be shifted with respect to
the other after Fourier transform.
An example is shown in Figure 4. In this case, two slices were excited which were
separated by only 5mm, resulting in essentially identical coil sensitivities in the two
slices. In the normal case (Figure 4a), the two slices appear exactly on top of one another,
while the slices appear shifted with respect to each other in the acquisition with CAIPI.
When these two folded slices are reconstructed with a conventional SENSE (Figure 4c),
the two slices cannot be reasonably separated from each other, resulting in a near total
SNR loss. However, since the two slices are shifted in the CAIPI acquisition, the aliased
areas of the two slices appear with different coil sensitivities and can therefore be
reconstructed with near perfect image quality. In this case, the CAIPI approach provides
a calculated SNR within a few percent of the ideal, while the conventional SENSE
approach leads to a 10 times increased geometry factor due to the essentially identical
coil sensitivity profiles.
A
Conv.:
C
CAIPI:
Slice 1
Slice 1
+
Slice 2
Slice 2
Slice 1
B
Conv.: R=2
D
CAIPI
Slice 1
Slice 1
Slice 2
Slice 2
Non-Cartesian Parallel Imaging
All of the methods discussed so far assume the simple aliasing pattern found with
normal mode of sampling on a Cartesian grid. The fact that this aliasing is simple allows
the simplification of the imaging equations substantially. For example, a normal SENSE
reconstruction with acceleration R and L coils requires only an inverse of an R x R matrix
in the pseudoinverse calculation. This is adequate to resolve the R pixels which are
aliased together in the undersampled acquisition.
However, this approach cannot be used with the more complicated aliasing patterns
found in non-Cartesian acquisitions, such as projection reconstruction or spiral. For
example, in accelerated spiral acquisitions, each pixel is aliased with entire rings! For
higher accelerations, multiple rings
of pixels are aliased together. This
clearly requires a more complex
method. In general, this requires
direct solution of the imaging
equations (Eq. 1), which requires the
inverse of very large matrices with
sizes on the order of the number of
the square of the number of pixels in
the image.
To date, the most used method
for solution of these large matrix
systems is the conjugate gradient
method [12]. This iterative method
gradually approaches the solution of
the inverse. While this method works
well in practice, the method is
computationally intensive, so that
most reconstructions can take several
minutes, compared to several
hundred milliseconds for a normal
SENSE reconstruction.
Figure 5. (Left) Undersampled PR images with 2x (top) and
For this reason, we have focused
4x undersampling (bottom)
on using the simple GRAPPA fitting
(Right) Direct GRAPPA reconstructions
concepts used above to derive fitting
relationships for simple nonCartesian trajectories, in particular, projection reconstruction trajectories. Figure 5 shows
two in vivo scans obtained with an 8 channel array and a real-time True-FISP acquisition.
The top, left image shows the 32 (projection) x 256 (read-out) normal reconstruction,
while the right shows the 2x GRAPPA reconstruction (64x256). The bottom row shows
an example from swallowing, however in this case the GRAPPA reconstruction is 3x
resulting in a 96x256 image. As can be seen, the aliasing artifacts are largely removed
using GRAPPA. Using this approach, all missing projections can be reconstructed in a
time comparable to several small GRAPPA reconstructions, each of which can be done in
much less than 1 sec using optimized code. However, further studies are needed to
determine if the accuracy of this method is good enough for practical clinical
implementation.
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