non-cartesian parallel imaging based on the grappa method · • ill conditioned nature of weights...

Nicole Seiberlich

Workshop on Novel Reconstruction Strategies in NMR and MRI 2010Göttingen, Germany10 September 2010

Non-Cartesian Parallel Imaging

based on the GRAPPA Method

Non-Cartesian Parallel Imaging

Non-Cartesian Imaging

Efficient Coverage of K-Space

Tolerant of Undersampling

Acquisition of Center of k-Space

Parallel Imaging

Acceleration by removing phase encoding steps

Dedicated reconstruction

Efficiency of Non-Cartesian Trajectories

TR = 2.7 msPE lines = 128Time/Image = 355 ms

TR = 4.7 ms“PE” lines = 40Time/Image = 188 ms

This spiral is already 1.9x faster than Cartesian

Efficiency of Non-Cartesian Trajectories

TR = 2.7 msPE lines = 128Time/Image = 355 ms

TR = 2.7 ms“PE” lines = 200Time/Image = 540 ms

Hmm…how is this efficient?

Radial is forgiving to undersampling

200 proj

Ny: R=1 Cart: R=0.6

128 proj

Ny: R=1.6 Cart: R=1

100 proj

Ny: R=2 Cart: R=1.364 proj

Ny: R=3.1 Cart: R=2

50 proj

Ny: R=4 Cart: R=2.6

Parallel Imaging

Goal:• Acquire undersampled data to shorten scan• Use receiver coil sensitivity information to complement gradient

encoding

The Cartesian Case

SENSE1 GRAPPA2

[1] Pruessmann KP, et al. Magn Reson Med. 1999 Nov;42(5):952-62.[2] Griswold MA, et al. Magn Reson Med. 2002 Jun;47(6):1202-10.

These methods are used daily in clinical routine

How does GRAPPA work?

kernel

How does GRAPPA work?

6 source points and 4 coils = 24 source / target

4 coils = 4 target points

GRAPPA weight set [24 x 4]

[src ∙ NC x targ ∙ NC]

G∙srcˆtarg =

How can I get the GRAPPA weights?

Gtarg ∙ pinv(src) = ˆ ˆG∙srcˆtarg = ˆ ˆ

Undersampled Radial Trajectory

Undersampling Distance and Direction Changes

No regular undersampling pattern

Aliasing in all directionsAliasing with many pixels

What do we need for GRAPPA to work?

• GRAPPA• Requires regular undersampling• Patterns in k-space must be identifiable• Calibration data must also have these kernels

Non-Cartesian is a harder problem to tackle

Possible Approaches (and Outline)

• Radial GRAPPA

Dynamic imagingReal-Time Free-Breathing Cardiac ImagingBasics and Improvements to the method

• CASHCOW

Generalized GRAPPAMore Exotic look at GRAPPA WeightsNot yet ready for public consumption

Radial GRAPPA

and

Through-Time Non-Cartesian GRAPPA

Radial GRAPPA

Radial GRAPPA

Standard GRAPPA performed using approximation of identical kernels

Each segment calibrated / reconstructed separately

GRAPPAs for different trajectories

Cartesian Radial Spiral

PROPELLER Zig-Zag Rosette

Kernel of 2x3 and NC=1272 Weights

4 x1 (4) Segments = 3654 Equations

16 x 16 (256) Segments = 30 Equations



Trade off between not having enough equations and violating assumptions

18

Radial GRAPPA: Segment Size

Calibration Segment Size Affects Reco QualityR=7 Radial GRAPPA

Large segments

Geometry not Cartesian

R=7 Radial GRAPPASmall segments

Reco looks like calibration image

R=7 Radial Image (20 proj/128 base matrix)

Standard Radial GRAPPA fails at high acceleration factors due to segmentation

Can we calibrate radial GRAPPA without using segments?

Through-Time Radial GRAPPAFU

LLY

SA

MP

LED

time

Multiple Repetitions of Kernel Through Time

GRAPPA Weights

Through-Time Radial GRAPPAU

ND

ER

SA

MP

LED

GRAPPA Weights

Geometry-Specific Weights used for Reconstruction

Calibration Segment Size Affects Reco QualityR=7 Radial GRAPPA

Large segments

Geometry not Cartesian

R=7 Radial GRAPPASmall segments

Reco looks like calibration image

R=7 Through-TimeRadial GRAPPA

Many Repetitions of Pattern for CalibrationGeometry Conserved

• 1.5 T Siemens Espree

• 15 channel cardiac coil

• Radial bSSFP Sequence

• 30-50 Calibration Frames

• Free-breathing and not EKG Gated

• No view sharing or time-domain processing

Materials and Methods

Radial Through-Time GRAPPA

• Radial Trajectory

• Resolution =2 x 2 x 8 mm3

• 16 projection / image

• TR = 2.86 ms

• Temporal Resolution34.32 ms / image

Radial Through-Time GRAPPA


• Resolution =1.5 x 1.5 x 8 mm3


• TR = 3.1 ms

• Temporal Resolution31 ms / image




• TR = 2.7 ms


Radial Through-Time GRAPPA, PVCs

• bSSFP Spiral Sequence

• Variable Density

• 40 shots / 128 matrix

• TR = 4.8 ms

• Reconstruction based on through-time radial GRAPPA

Spiral Through-Time GRAPPA

• VD Spiral Trajectory


• 8 spiral arms / image

• TR = 4.78 ms



• VD Spiral Trajectory


• 4 spiral arms / image

• TR = 4.78 ms



Non-Cartesian GRAPPAs

• Rely on the approximation of same geometry through k-space

• Segmentation used to get enough patterns for calibration

Through-Time Non-Cartesian GRAPPA

• Geometry-specific weights yield better reconstructions

• High acceleration factors and frame rates (20 - 50 frames / s)

• Simple parallel imaging reconstruction

GROG / CASHCOW

Generalized GRAPPA

How do we calibrate this weight set?

GROG / GRAPPA Operator Concept

G G

G

G2

G0.5G-1

Jumps of arbitrary distances (with noise enhancement)

GROG allows freedom from standard shifts

Gy

Gx

Jumps of arbitrary direction and distance

DON’T FORGET!!

This is parallel imaging

Larger GRAPPA Operators

Gy

Gx

GRAPPA weights with size [NC ∙3 x NC∙3]

We can shift points aroundas long as the arrangement is the same

Can we make arbitrary operators?


Gxdx ˆ∙Gy

dy


Gxdx ˆ∙Gy

dy

Gxdx ˆ∙Gy

dy


Gxdx ˆ∙Gy

dy

Gxdx ˆ∙Gy

dy

Gxdx ˆ∙Gy

dy


Gxdx ˆ∙Gy

dy

Gxdx ˆ∙Gy

dyGline to arb

Gxdx ˆ∙Gy

dy

We can move from Cartesian points to arbitrary arrangement

Two Cartesian GRAPPA operators needed

ˆ= Garb to lineGline to arb

Moving from arbitrary points to grid

-1

CASHCOWCreation of Arbitrary Spatial Harmonics through

the Combination of Orthogonal Weightsets

Moving from arbitrary points to grid

CASHCOWCreation of Arbitrary Spatial Harmonics through

the Combination of Orthogonal Weightsets

• Generate weights for up/down and right/left shifts for a given configuration

• Use these weights to move from standard to arbitrary pattern

• Invert weights to move from arbitrary to standard pattern

How can we use CASHCOW?

Generation of Weight Set

Gcart_to_nc-1 =Gcart_to_ncˆ

Generation of Weight Set

Weight set to move from known points to unknown

Repeat for all Cartesian points

Gnc_to_cart

CASHCOW in Simulations

128 proj 64 proj 42 proj

32 proj 25 proj

CASHCOW in Simulations with Noise


32 proj 25 proj

Why did CASHCOW stop working?

GRAPPA operators are simply square matrices…

…often very ill-conditioned matrices

Typical condition number ~ 104

Crucial step in CASHCOW weights is an inversion

One solution Use regularization

CASHCOW with Noise + Regularization


32 proj 25 proj

CASHCOW with Noise + (more) Regularization


32 proj 25 proj

CASHCOW in vivo

144 proj 72 proj

48 proj

CASHCOW is not there yet….

But it demonstrates interesting properties of GRAPPA

• GRAPPA weights for arbitrary source and target points can be generated using Cartesian calibration data

• Ill conditioned nature of weights restricts CASHCOW

• Math + MRI Better solution for non-Cartesian parallel imaging

GRAPPA is a flexible tool for NC PI

Non-Cartesian GRAPPAs

• Standard Method uses geometrical approximationsSegmentation leads to errors in weights

• Through-time calibration removes the need for segmentsReal-time cardiac imagingFrame rates of 20 – 50 / sec using parallel imaging

GRAPPA is a flexible tool for NC PI

GROG / CASHCOW

• GRAPPA Operator ConceptWeights are manipulatable square matrices

• CASHCOWWeights for arbitrary configurations of points“Generalized” GRAPPAIll conditioned weights a problem – Regularization?

Acknowledgments

• Dr. Mark Griswold

• Dr. Jeff Duerk

• Dr. Felix Breuer

• Philipp Ehses

non-cartesian parallel imaging based on the grappa method · • ill conditioned nature of weights...

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