non-cartesian parallel imaging based on the grappa method · • ill conditioned nature of weights...
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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
8 x 4 (32) Segments = 406 Equations
8 x 8 (64) Segments = 182 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
• Radial Trajectory
• Resolution =1.5 x 1.5 x 8 mm3
• 10 projection / image
• TR = 3.1 ms
• Temporal Resolution31 ms / image
• Radial Trajectory
• Resolution =2.3 x 2.3 x 8 mm3
• 16 projection / image
• TR = 2.7 ms
• Temporal Resolution44 ms / image
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
• Resolution =2.3 x 2.3 x 8 mm3
• 8 spiral arms / image
• TR = 4.78 ms
• Temporal Resolution38 ms / image
Spiral Through-Time GRAPPA
• VD Spiral Trajectory
• Resolution =2.3 x 2.3 x 8 mm3
• 4 spiral arms / image
• TR = 4.78 ms
• Temporal Resolution19 ms / image
Spiral Through-Time GRAPPA
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?
Can we make arbitrary operators?
Gxdx ˆ∙Gy
dy
Can we make arbitrary operators?
Gxdx ˆ∙Gy
dy
Gxdx ˆ∙Gy
dy
Can we make arbitrary operators?
Gxdx ˆ∙Gy
dy
Gxdx ˆ∙Gy
dy
Gxdx ˆ∙Gy
dy
Can we make arbitrary operators?
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
128 proj 64 proj 42 proj
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
128 proj 64 proj 42 proj
32 proj 25 proj
CASHCOW with Noise + (more) Regularization
128 proj 64 proj 42 proj
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
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