Parallel MRI
Ileana Pazos, Tomoyasu Mani, and Michel Bilello
1. General Overview of parallel MRI
Imaging speed is one of the most important considerations in clinical MRI. Advent of
parallel MRI accelerated image acquisition dramatically by extracting spatial information from
an array of surface coils instead of relying on increased gradient performance. Theoretical idea
of parallel MRI was conceived in late 1980’s but we had to wait until late 1990’s before the
practical implementation was carried out. The technique has been widely used in clinical
settings since 2000 and its improvement is still undergoing.
2. Spatial encoding and k-space
Three gradients are used to spatially encode the signals in MRI: slice selection (Gz),
frequency encoding (Gx), and phase encoding (Gy). As shown in Figure 1, the slice selections
are performed when we apply RF pulses and frequency encoding can only be performed during
the time echo is received. The phase encoding gradient is applied in between RF pulse and
receiving echo. The data is stored in data space (analogue k-space) and converted to k-space by
mathematical manipulation before Fourier transform (Twieg 1983 and Ljunggern 1983).
Figure 1 Pulse sequence diagram and spatial encoding gradients
The acquisition time depends on how quickly we can fill up the k-space. While
frequency encoding steps takes only milliseconds level gradient-phase encoding steps is time
consuming and it sometimes takes several minutes depending on the sequence. One way to
accelerate this phase encoding steps is to reduce the number of steps by the acceleration factor
R by increasing the distance of equidistantly sampled k-space lines (Dky). In image space, this
kind of undersampling the k-space yields in a reduces field of view (FOV) in phase encoding
direction associated with fold over artifacts in the coil images ad described in Figure 2.
Figure 2 (a) Conventional acquisition of fully sampled k-space, resulting in a full FOV image
after Fourier transform. (b) Undersampled acquisition (R=2), resulting in a reduced FOV
(FOV/2) with aliasing artifacts.
Using undersampling acquisition requires multiple coils to cover the entire FOV. To
provide full FOV we use array (surface) coils. One example is shown in the Figure 3 where fourcoil array is used to cover the spine. A very simple idea of parallel MRI is the following: acquire
reduced FOV covering a portion of the entire image by each coil and combine the images from
all the coils together to yield full-FOV image. Each element of an array coil is attached to a
separate independent receiver system. So for a given imaging experiment multiple images, one
for each coil element, can be reconstructed. Roemer explored how to combine these images
for optimal SNR (Roemer 1990). His study shows that to optimally reconstruct array coil images
we need to explicitly know the coil sensitivities at all pixel locations as shown in the following
equation.
Sopt=wiS1C1+wiS2C2+wiS3C3+…where wi=Ci2Cj2
3. Key Parameters in Parallel Imaging
3.1. Acceleration factor (R-factor)
As discussed in the previous section, image acquisition is accelerated by the
acceleration factor (R-factor). Theoretically imaging time can be reduced by R-factor equal to
the number of array coil elements. However decreased SNR and increased image nonuniformities limit acceleration factor well below the theoretical maximum R.
3.2. Geometry factor (g-factor)
The reason why we cannot achieve R-factor larger than 4 or 5 is because another
parameter called geometry factor comes into play. The g-facotr measures the level of noise
amplification which occurs as a result of the reconstruction process. It describes the ability
with the used coil configuration to separate pixels superimposed by aliasing. While 1 ≤ g ≤ 1.5
allows adequate image quality it turns out that g-factor is strongly dependent upon R-factor as
shown in the case of the six-coil array (Weiger 2005) and it is not practically possible to acquire
images with good SNR by using bigger R-factor than 4 or 5 (Figure 3).
Figure 3 The g-factor dependence on R-factor in the case of six-coil array.
Now the SNR for a pixel p of the reduced images can be described as the following equation
SNRpred=SNRpfullgpR
Since the acceleration is achieved at a cost in SNR, the acceleration by parallel imaging is
intrinsically limited by R-factor and g-factor.
3.3. Ideal Case
First, let’s look at the ideal case (Heidemann 2003). Suppose we have two coils, each with a
boxcar-like sensitivity profile C in phase encoding direction, by which each coil profile covers
exactly half of the FOV (Figure 4b). In this situation, the Coil 1 sees only the left side and the
Coil 2 sees only the right side of the whole object (brain). In a conventional acquisition
(assuming a homogeneous volume coil), the full FOV image would be acquired with N phaseencoding steps (Figure 4a). Using the same acquisition parameters with the two-coil array, the
resulting component coil images of the Coil 1 and the Coil 2 that would be acquired are shown
in c. It is now obvious that each component coil provides a full FOV image but contains only
spatial information from inside the individual sensitivity profiles. No signal can be detected
outside the individual sensitivity range. In order to speed up the imaging process, we now take
into account that each coil detects exactly one half of the FOV. Here we can reduce the FOV a
factor of two without aliasing, which can be achieved by acquiring only N/2 phase-encoding
steps over the reduced FOV. As a result, we simultaneously obtain two half-FOV images from
the two coils (Figure 4d). We then combine these two images to yield a full-FOV, full-resolution
image (Figure 4e). This image was obtained in half of the scan time with a reduction factor
R=2.
Figure 4 a-e A simplified parallel imaging procedure.
4. Reconstruction Algorithms
From the previously discussed ideal case reconstruction of the image ignores the realworld situation in which coil sensitivities extend over the FOV and overlaps. Firstly, Parallel
Imaging with Localized Sensitivities (PILS) will be discussed as the next simplified but non-ideal
case that foregoes calculated the coil sensitivity as long as it is smaller than the FOV.
Reconstruction algorithms that take coil sensitivities into account will be discussed with a focus
on Sensitivity encoding (SENSE), Simultaneous acquisition of spatial harmonics (SMASH), and
Generalized auto calibrating partially parallel acquisition (GRAPPA). The major theoretical
difference between different algorithms would be whether they are operated on in the image
or k-space domain and also if they are exact or approximations. These will be emphasized in
the following discussion.
4.1 Parallel Imaging with Localized Sensitivities (PILS)
In PILS, the receiver coils have spatially distinct localized sensitivities denoted Y c. With
an R value of two or greater, in other words, an accelerated pMRI acquisition reduces the FOV
in phase-encoding direction. This results in periodically repeating subimages of length Yi. If the
inequality Yc < Yi< Y holds, then subimages will be spatially resolved. Then the folding aliasing
that is seen could be removed by a simple threshold subtraction. Figure 5a is an image of a
brain processed with the PILS algorithm.
Four coils and there sensitivity profile is
shown in part a. The ful FOV from each
coil is shown in part b. With an R value of
2 there is a folding aliasing as seen in part
c. However this folding is spatially
resolved appear totally separated in the
full FOV whereas the position of the
correct subimages is lost. By knowing the
position of the coil the correct subimage is
selected and is boxed off in part d. In e,
the subimage is isolated by means of a
threshold value to remove the other
foldover artifacts and combined to form
the image f. There is some evidence of
aliasing in the final image since the values
of Yc and similar to Yi.
Figure 5 The PILS reconstruction method, steps a-f
M. A. Griswold et al., Magn. Reson. Med. 44, 602 (Oct, 2000).
M. Blaimer et al., Top Magn Reson Imaging 15, 223 (2004).
4.2 Sensitivity encoding (SENSE)
SENSE is the first reconstruction algorithm that is used for clinical applications and and
does not rely on ideal or simplified situation. It is an exact solution and the image is processed
in the image domain.
Figure 6, shows the
SENSE reconstruction
is done in the image
domain after Fourier
transforming the kspace data.
Figure 6- Schematic representation of SENSE reconstruction
When the under sampled k space data is Fourier transformed to the image domain, the
field of view (FOV) is reduced by the 1/R meaning that the same information is contained in a
smaller area which leads to fold over type aliasing artifacts.
In equation 1, the signal in one pixel at a certain location (x,y) received in the N’th
component coil image gN
Equation 1
for the case where R=2
 n ( x, y )
FOV   f ( x, y )
 g1 ( x, y )   s1 ( x, y )

s1 ( x, y 
) 
but could easily be
  1


2




 
 
generalized to a different 
  





 
accerlation factor. The g 
 





  s ( x, y ) s ( x, y  FOV ) 

FOV  
 f ( x, y 
)  n N ( x, y ) 
N
 g N ( x, y )  N
2  
2 
vector represents the complex coil image values including fold over artifacts, at the chosen
pixel. The matrix S denotes the sensitivities for each coil at the R=2. To generate the f vector,
the inverse of the S matrix is calculated by equation 2. This effectively combines the reference
data of coil sensitivities from a previously collected short low resolution scan and the subsampled target data.
Equation 2


f  ( Sˆ  Sˆ ) 1 Sˆ   g
Error! Reference source not found., below is an application of the SENSE reconstruction
for the short axis cardiac images. Each frame has different acceleration rates, specifically R=2, 3
and 4. The higher the R value the more folding aliasing artifacts present and the faster the
images are collected due to the higher frame per sec (fps) acquisition. Not qualitatively seen
here, but necessarily present, is the lower SNR value due to higher R values.
K. P. Pruessmann, M. Weiger, M. B. Scheidegger, P. Boesiger, Magnetic Resonance in Medicine
42, 952 (Nov, 1999).
Kellman, P. Parallel Imaging: The Basics. NHLBI, National Institutes of Health, (2004)
4.3
Simultaneous acquisition of spatial harmonics (SMASH)
SMASH is a reconstruction method that operates in the k-space. This is schematically
depicted in Figure 7. After the
algorithm it is Fourier transformed
to the image domain and
therefore there is no analog of the
fold over artifact that is present in
the SENSE images.
Figure 7 Schematic representation of SMASH reconstruction
Equation 4
The algorithm relies on
the linear combinations of
the estimated coil
Equation 5
sensitivities which directly generates
missing phase encoding steps. This
step is analogous to S matrix in
Equation 3
SENSE reconstruction obtained by a low resolution initial
scan.
The prior estimation of the individual coil sensitivities of
the receiver array is represented in equation 4 by the si.
The extra sampling of k space to calculate the
coil sensitivity is represented in equation 3.
The weighting factor ni from equation 3, is
approximated by a least square fitting to a
spatial harmonic
. This fitting for
order m=0 and m=1 is shown in Figure 8. By
fitting the coil sensitivities, equation 4 is fully
known and the accelerated intensity profile
can be Fourier transform to obtain a SMASH
Figure 8 Least square fit to spatial harmonic
image.
D. K. Sodickson, W. J. Manning, Magnetic Resonance in Medicine 38, 591 (Oct, 1997).
R. M. Heidemann, M. A. Griswold, A. Haase, P. M. Jakob, Magnetic Resonance in Medicine 45,
1066 (2001).
C. A. McKenzie, M. A. Ohliger, E. N. Yeh, M. D. Price, D. K. Sodickson, Magnetic Resonance
in Medicine 46, 619 (Sep, 2001).
Other review articles:
M. Blaimer et al., Top Magn Reson Imaging 15, 223 (2004, 2004).
D. J. Larkman, R. G. Nunes, Physics in Medicine and Biology 52, R15 (Apr, 2007).
4.5 GRAPPA



k-space method
Auto-calibration method (no explicit computation of sensitivity maps)
Approximate (interpolate missing k-space lines)
Compared to Auto-SMASH,
 Generates individual uncombined coil images before combination
 Data from multiple lines from all coils are used to fit an ACS line in single coil

Leads to better SNR and avoidance of phase cancellation
The additional (auto-calibration) k-space lines are fit to linear combinations of blocks of
acquired data, and the fitting coefficients are recorded. Next, the coefficients are used to
estimate the missing k-space data, by interpolation of acquired data.
In GRAPPA, the final image is constructed by using sum of squares to combine single coil images
together. Each coil contributes strongly in regions where its sensitivity is large and minimally in
regions where its sensitivity is low.
5. Application of pMRI
Virtually any sequence can accommodate pMRI. In addition to shortening the acquisition time,
pMRI can improve susceptibility artifacts seen at the bone-air interfaces on diffusion EPI
sequences.
Intrisincally high SNR imaging such as phase-contrast brain MRA can tolerate high acceleration
factors (e.g., 3) without significant apparent image degradation.
Not only pMRI can improve patient comfort by shortening time scan and hence prevent patient
motion, it also improve temporal resolution that can be critical to depict physiologic processes.
For example, a short acquisition on MRA (angiogram) can prevent getting venous
contamination on the images. Cardiac images can be performed within one breath hold. Also,
metastases that enhance only during the arterial phase (liver, breast) can be observed better
with the improved temporal resolution offered by pMRI.
6. Limitations of pMRI
SNR
As explained before, the SNR decreased with the square root of the acceleration factor r, and
directly with the geometry factor g. The g factor, which depends on the coil number and
geometry, is responsible for the noise amplification derived from reconstruction algorithms.
Although this is not stated explicitly in the literature, g probably depends on the conditioning of
the matrix to be inverted or pseudo-inverted. The process of matrix inversion is ubiquitous in all
reconstruction algorithms. Typically, there is a overdetermined system of linear equations to
solve, and the robustness of the solution is given by the conditioning of the (rectangular)
matrix, of which the pseudo-inverse is to be computed. It is probably the case that when the
matrix is well conditioned (ratio of largest to smallest non-zero singular values close to 1), the g
factor is close to 1, and increases with poor conditioning of the matrix.
Artifacts,
This primarily applies to algorithm where the coil sensitivities are explicitly measured (SENSE). If
the subject moved between the short acquisition for coil sensitivity measurement and the
actual sequence acquisition, the images will be degraded by artifacts. A similar artifact will be
present if the coils move between the two acquisitions.
Reference
Ljunggern S., Journal of Magnetic Resonance 54, 338-343 (1983)
Twieg D. B., Medical Physics 10, 610-621 (1983)
Heidemann R. M. et al., Eur. Radiol. 13, 2323-2337 (2003)
Weiger M. et al., Magnetic Resonance in Medicine 53, 177-185 (2005)
Roemer P. B., et al., Magnetic Resonance in Medicine 16, 192-225 (1990)