Partial Parallel imaging Partial Parallel imaging (PPI) in MR for faster (PPI) in MR for faster

imagingimagingIMAIMA

Compressed SensingCompressed Sensing

June, 2007June, 2007

Acknowledgement: NIH Grants 5RO1CA092004 and 5P41RR008079, Pierre-Francois Van de Moortele, Gregor Adriany, Kamil Ugurbil

Our coilsOur coils

Open face coilOpen face coil 16 Channel 16 Channel “closed” coil“closed” coil

Intrinsically, surface coils offer a representation of signal as

Acquired k-space,

How to INTERPOLATE most stably to the

Non-acquired data.

we will see why it makes completely sense to think about interpolation

The sensitivities are complex valued

Field of View

Courtesy: Douglas Noll, University of Michigan

Undersampled images

Undersampled individual imagesUndersampled individual images

The linear system

The solution of the linear system gives rice to a spatially varying noise amplification. This is solely dependent on the sensitivities and is referred to as the geometry factor

11 1 " "unaliased H H reducedFOVI S S S y

The geometry factorThe geometry factorαα is the index of an aliased pixel, is the index of an aliased pixel, ββnn is the index of an unaliased is the index of an unaliased

pixel.pixel.

The reduced FOV is the RSOS of all the channels with a reduced FOV, only for illu.

Overall loss when using PPI is SNRred=SNRfull/(g sqrt(R))

Back to the equationBack to the equation

S indv. Channels. E encoding. p un-aliased image

S indv. Channels. E encoding. p un-aliased image (all in k-space)

K-space

Image space

Convolution operator

E is known, but we can make the formal separation of S, as follows:

Two matrix equations, two unknowns

E”acq” includes all of k-space for the sensitivities

SENSE/SMASH formalism (get one image)SENSE/SMASH formalism (get one image)

GRAPPA idea, get multiple images. The interpolation is essentially similar to Kriging

Courtesy: Yeh, et al, MRM Volume 53, Issue 6 , Pages 1383 - 1392

GRAPPAGRAPPA

Reconstructing the data for EACH coil

Courtesy: Griswold et al. MRM, 47(6):1202-1210 (2002)

Several reconstruction is found for EACH k-space point- Several reconstruction is found for EACH k-space point- due to the blocks. A weigthed average is used to compute due to the blocks. A weigthed average is used to compute just onejust one

GRAPPA formula to reconstruct signal in GRAPPA formula to reconstruct signal in one channelone channel

ACS (Auto-Calibration Signal) lines (no x)ACS (Auto-Calibration Signal) lines (no x)

where A represents the acceleration factor. Nb is the number of blocks used in the reconstruction, where a block is defined as a single acquired line and A-1 missing lines. 4-8 blocks are needed

,l

Temporal samplingTemporal sampling

PE

Interleaved/segmented (2)

Interleaved (2)

Works well for imaging of static objects.

For dynamic imaging, each image is not only undersampled, but also captures a different part of the “motion”/”change”. The acquisition is assumed faster than the motion

time½ k-space

½ k-space

PSF considerations (generally)PSF considerations (generally)

Let us start with imagingLet us start with imaging

PE or t

( )psf F

Standard PPI used to “unalias” the effects of the psf

UNFOLD (does not require UNFOLD (does not require multichannels)multichannels)

Specifically, alternate the sampling Specifically, alternate the sampling by a factor 2, such thatby a factor 2, such that

Remove aliasing by

…

½ k-space

Courtesy: Madore. MRM 48:493 (2002).

fMRI (UNFOLD)fMRI (UNFOLD)

FIG. 16. Results obtained for a single-trial fMRI experiment (4 spiral interleaves, 16 kz phase-encode values, axial images, matrix size128 3 128, TR 5 250 msec, TE 5 40 msec, 5 mm resolution along z, 24 cm FOV). Bilateral finger tapping was performed while a 2 sec audiocue was on, and then stopped for 12 sec. The acquisition time for a time frame (16 sec) is longer than a paradigm cycle (14 sec). UNFOLD isused to reduce the acquisition time by a factor 8, providing 7 frames per paradigm cycle. a: The acquired frames are corrupted by an 8-foldaliasing in the through slice direction. b: Temporal frequency spectrum for the highlighted image point in a. UNFOLD interleaves 8 spectra intothe same temporal bandwidth. Marks are placed on the axis at the locations of the DC, fundamental and harmonic frequencies for thenon-aliased material. Selecting only these frequencies, the aliasing seen in a is removed in c.

Remove aliased frequency by selective filtering

Courtesy: Madore. MRM 48:493 (2002).

Extend the concept of aliasingExtend the concept of aliasingLine in image

Unalias the support in x-f space, just like we unalias in x space with SENSE

Tsao et al, MRM 50: 1031-1042 (2003)

DATA challengeDATA challenge

1.1. Where is the support in x-f space?Where is the support in x-f space?

Used to define support in x-f space

“Equivalent” to a reference scan

Interleaved training set

Similar concepts hold for radial, where the center is the “prior”. This is used in speech imagingTsao et al, MRM 50: 1031-1042 (2003)

How do the methods compare?How do the methods compare?k-t SENSE, vs. Sliding Windowk-t SENSE, vs. Sliding Window

Consensus (in cardiac imaging) of:Consensus (in cardiac imaging) of:

Xu et al. MRM, 57:918-930 (2007)

What does the artifact meanWhat does the artifact mean

Xu et al. MRM, 57:918-930 (2007)

Looking at the temporal variation Looking at the temporal variation (in speech [radial]) (in speech [radial]) K-t SENSE Sliding window

ty

Comments/Conclusions

Michael S. Hansen. Workshop on Non-catesian MRI. 2007

FormallyFormally

With localized sensitivities (smooth in image space)

The mising information can be determined from the acquired data, if the coeeficients a(i,j,k) are known

SMASHSMASH Find weights nFind weights nkk

(m)(m)(x) [no x –readout dependence] such that we get (x) [no x –readout dependence] such that we get a new a new syntheticsynthetic sensitivity profile C sensitivity profile Cmm

compcomp

WE do parallel Imaging by finding ONE combined image (just like SENSE)

m is selected depending on how FAR the data must be interpolated. Only one line is used to advance the data

Generalised SMASHGeneralised SMASH

Find weights aFind weights akk(m)(m)(x) [(x) [withwith x – x –

readout dependence] such thatreadout dependence] such that

(x) ( , ) ( )q

m im kyj j

m p

C x y a x e

Express

( )( , ) ( )

( ) ( , )

y y

qik y i k m k ym

j y j jm p

qmj y

m p

s x k C e a x e

a x s x k m k

We use several phase-encoding lines to generate missing information. We use several phase-encoding lines to generate missing information.

For each readout point a new set of weights are comp.For each readout point a new set of weights are comp.

Two severe issuesTwo severe issues

The final image is that of a complex The final image is that of a complex sum image of the individual images. sum image of the individual images. Not optimal for SNRNot optimal for SNR

Total cancellation can occur with Total cancellation can occur with such complex sums.such complex sums.

Coils where phase-aligned PRIOR to Coils where phase-aligned PRIOR to reconstructionreconstruction

AUTO-SMASHAUTO-SMASH ACS (Auto-Calibration Signal) lines (no x), not ACS (Auto-Calibration Signal) lines (no x), not

fitting to a harmonic, but a “missing” PE-linefitting to a harmonic, but a “missing” PE-line