Non-Cartesian Parallel Imaging based on the GRAPPA Method · • Ill conditioned nature of weights...
Transcript of 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
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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
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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
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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?
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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
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Parallel Imaging
Goal:• Acquire undersampled data to shorten scan• Use receiver coil sensitivity information to complement gradient
encoding
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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
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How does GRAPPA work?
kernel
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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 =
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How can I get the GRAPPA weights?
Gtarg ∙ pinv(src) = ˆ ˆG∙srcˆtarg = ˆ ˆ
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Undersampled Radial Trajectory
Undersampling Distance and Direction Changes
No regular undersampling pattern
Aliasing in all directionsAliasing with many pixels
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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
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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
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Radial GRAPPA
and
Through-Time Non-Cartesian GRAPPA
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Radial GRAPPA
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Radial GRAPPA
Standard GRAPPA performed using approximation of identical kernels
Each segment calibrated / reconstructed separately
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GRAPPAs for different trajectories
Cartesian Radial Spiral
PROPELLER Zig-Zag Rosette
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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
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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?
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Through-Time Radial GRAPPAFU
LLY
SA
MP
LED
time
Multiple Repetitions of Kernel Through Time
GRAPPA Weights
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Through-Time Radial GRAPPAU
ND
ER
SA
MP
LED
GRAPPA Weights
Geometry-Specific Weights used for Reconstruction
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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
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• 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
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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
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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
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• 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
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• bSSFP Spiral Sequence
• Variable Density
• 40 shots / 128 matrix
• TR = 4.8 ms
• Reconstruction based on through-time radial GRAPPA
Spiral Through-Time GRAPPA
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• 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
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• 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
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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
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GROG / CASHCOW
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Generalized GRAPPA
How do we calibrate this weight set?
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GROG / GRAPPA Operator Concept
G G
G
G2
G0.5G-1
Jumps of arbitrary distances (with noise enhancement)
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GROG allows freedom from standard shifts
Gy
Gx
Jumps of arbitrary direction and distance
DON’T FORGET!!
This is parallel imaging
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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
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Can we make arbitrary operators?
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Can we make arbitrary operators?
Gxdx ˆ∙Gy
dy
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Can we make arbitrary operators?
Gxdx ˆ∙Gy
dy
Gxdx ˆ∙Gy
dy
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Can we make arbitrary operators?
Gxdx ˆ∙Gy
dy
Gxdx ˆ∙Gy
dy
Gxdx ˆ∙Gy
dy
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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
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ˆ= Garb to lineGline to arb
Moving from arbitrary points to grid
-1
CASHCOWCreation of Arbitrary Spatial Harmonics through
the Combination of Orthogonal Weightsets
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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
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How can we use CASHCOW?
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Generation of Weight Set
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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
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CASHCOW in Simulations
128 proj 64 proj 42 proj
32 proj 25 proj
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CASHCOW in Simulations with Noise
128 proj 64 proj 42 proj
32 proj 25 proj
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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
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CASHCOW with Noise + Regularization
128 proj 64 proj 42 proj
32 proj 25 proj
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CASHCOW with Noise + (more) Regularization
128 proj 64 proj 42 proj
32 proj 25 proj
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CASHCOW in vivo
144 proj 72 proj
48 proj
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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
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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
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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?
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Acknowledgments
• Dr. Mark Griswold
• Dr. Jeff Duerk
• Dr. Felix Breuer
• Philipp Ehses