Nicole Seiberlich Workshop on Novel Reconstruction Strategies in NMR and MRI 2010 Göttingen, Germany 10 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 ms PE lines = 128 Time/Image = 355 ms TR = 4.7 ms “PE” lines = 40 Time/Image = 188 ms This spiral is already 1.9x faster than Cartesian

Efficiency of Non-Cartesian Trajectories TR = 2.7 ms PE lines = 128 Time/Image = 355 ms TR = 2.7 ms “PE” lines = 200 Time/Image = 540 ms Hmm…how is this efficient?

Radial is forgiving to undersampling 200 proj Ny: R=1Cart: R= proj Ny: R=1.6Cart: R=1 100 proj Ny: R=2Cart: R= proj Ny: R=3.1Cart: R=2 50 proj Ny: R=4Cart: R=2.6

Parallel Imaging Goal: Acquire undersampled data to shorten scan Use receiver coil sensitivity information to complement gradient encoding

The Cartesian Case SENSE 1 GRAPPA 2 [1] Pruessmann KP, et al. Magn Reson Med Nov;42(5): [2] Griswold MA, et al. Magn Reson Med Jun;47(6): 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∙src ˆ targ =

How can I get the GRAPPA weights? G ˆ targ ∙ pinv(src) = ˆˆ G∙src ˆ targ = ˆˆ

Undersampled Radial Trajectory Undersampling Distance and Direction Changes No regular undersampling pattern Aliasing in all directions Aliasing 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 imaging Real-Time Free-Breathing Cardiac Imaging Basics and Improvements to the method CASHCOW Generalized GRAPPA More Exotic look at GRAPPA Weights Not yet ready for public consumption

Radial GRAPPA and Through-Time Non-Cartesian GRAPPA

Radial GRAPPA

Standard GRAPPA performed using approximation of identical kernels Each segment calibrated / reconstructed separately

GRAPPAs for different trajectories CartesianRadialSpiral PROPELLERZig-ZagRosette

Kernel of 2x3 and NC=12 72 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 Quality R=7 Radial GRAPPA Large segments Geometry not Cartesian R=7 Radial GRAPPA Small 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 GRAPPA FULLY SAMPLED time 2x3 GRAPPA Kernel Exact Geometry Multiple Repetitions of Kernel Through Time GRAPPA Weights

Through-Time Radial GRAPPA UNDERSAMPLED GRAPPA Weights Geometry-Specific Weights used for Reconstruction

Calibration Segment Size Affects Reco Quality R=7 Radial GRAPPA Large segments Geometry not Cartesian R=7 Radial GRAPPA Small segments Reco looks like calibration image R=7 Radial Image (20 proj/128 base matrix) R=7 Through-Time Radial GRAPPA Many Repetitions of Pattern for Calibration Geometry Conserved

1.5 T Siemens Espree 15 channel cardiac coil Radial bSSFP Sequence 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 mm 3 16 projection / image TR = 2.86 ms Temporal Resolution ms / image

Radial Through-Time GRAPPA Radial Trajectory Resolution = 1.5 x 1.5 x 8 mm 3 10 projection / image TR = 3.1 ms Temporal Resolution 31 ms / image

Radial Trajectory Resolution = 2.3 x 2.3 x 8 mm 3 16 projection / image TR = 2.7 ms Temporal Resolution 44 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 mm 3 8 spiral arms / image TR = 4.78 ms Temporal Resolution 38 ms / image Spiral Through-Time GRAPPA

VD Spiral Trajectory Resolution = 2.3 x 2.3 x 8 mm 3 4 spiral arms / image TR = 4.78 ms Temporal Resolution 19 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 ( frames / s) Simple parallel imaging reconstruction

GROG / CASHCOW

Generalized GRAPPA How do we calibrate this weight set?

GROG / GRAPPA Operator Concept G ˆ = matrix of size [NC x NC] G ˆ G ˆ ˆ G2G2 ˆ G 0.5 ˆ G -1 Jumps of arbitrary distances (with noise enhancement)

GROG allows freedom from standard shifts ˆ GyGy ˆ GxGx Jumps of arbitrary direction and distance DON’T FORGET!! This is parallel imaging

Larger GRAPPA Operators ˆ GyGy ˆ GxGx GRAPPA weights with size [NC∙3 x NC∙3] We can shift points around as long as the arrangement is the same

Can we make arbitrary operators?

ˆ G x dx ˆ ∙G y dy

Can we make arbitrary operators? ˆ G x dx ˆ ∙G y dy ˆ G x dx ˆ ∙G y dy

Can we make arbitrary operators? ˆ G x dx ˆ ∙G y dy ˆ G x dx ˆ ∙G y dy ˆ G x dx ˆ ∙G y dy

Can we make arbitrary operators? ˆ G x dx ˆ ∙G y dy ˆ G x dx ˆ ∙G y dy ˆ G line to arb ˆ G x dx ˆ ∙G y dy We can move from Cartesian points to arbitrary arrangement Two Cartesian GRAPPA operators needed

ˆ = G arb to line ˆ G line to arb Moving from arbitrary points to grid CASHCOW Creation of Arbitrary Spatial Harmonics through the Combination of Orthogonal Weightsets

Moving from arbitrary points to grid CASHCOW Creation 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

G cart_to_nc -1 = ˆ G cart_to_nc ˆ Generation of Weight Set Weight set to move from known points to unknown Repeat for all Cartesian points G nc_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 ~ 10 4 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 approximations Segmentation leads to errors in weights Through-time calibration removes the need for segments Real-time cardiac imaging Frame rates of 20 – 50 / sec using parallel imaging

GRAPPA is a flexible tool for NC PI GROG / CASHCOW GRAPPA Operator Concept Weights are manipulatable square matrices CASHCOW Weights for arbitrary configurations of points “Generalized” GRAPPA Ill conditioned weights a problem – Regularization?

Acknowledgments Dr. Mark Griswold Dr. Jeff Duerk Dr. Felix Breuer Philipp Ehses