EE369C Final Project: Accelerated Flip Angle Sequences

1. EE369C Final Project:Accelerated Flip Angle Sequences Jan 9, 2012 Jason Su

2. Overview • Originally wanted to further explore view sharing but instead pursued more formal optimization approach particularly because I was interested in applying SPIRiT • Tried to replicate the results of Velikina et. al. in “Accelerating Multi-Component Relaxometry in Steady State with an Application of Constrained Reconstruction in Parametric Dimension” ISMRM 2011:2740.

3. Goal • To accelerate variable flip angle relaxometry sequences like DESPOT1/2 or mcDESPOT by undersampling along the flip angle dimension • 3T, mcDESPOT, 1 mm^3 256x256x160 acquisition, 10 SPGRs and SSFPs with parallel imaging ~40min. • Exploit prior knowledge about the signal equation to regularize the reconstruction problem

4. The Problem • Velikina poses the problem as: • E is the encoding matrix including Fourier terms and coil sensititivies • m is the desired signal for all flip angles (FA, α) • y is the measured k-space data • Hybrid Huber-like norm to “promote sparsity and optimize SNR” • 1st term enforces data consistency, 2nd term smoothness in the signal curve

5. Regularization • To make the reconstruction problem more stable and allow greater undersampling, we use our prior knowledge that the signal curve is smooth • It is near zero for high angles in the 2nd derivative “space”

6. Velikina Results

7. Velikina Results

8. Data • 3T2, SPGR • 1:1:13 degrees • This is very different from Velikina, where up to 25 deg. was used, but a subset of 10 angles was taken • Note that the SPGR 2nd deriv. is only 0 for 15+ deg. • Nova 32ch head coil • 110x110x40 matrix • TR = 4.5ms

9. Alternate SPIRiT Problem • This requires knowledge of the coil sensitivities, instead I posed it as a regularized-SPIRiT problem: • x is the desired k-spaced data for all flips • G is the SPIRiT kernel • F-1 the inverse Fourier transform • Represents data consistency, self-consistency, and smoothness

10. Cartesian-based Acceleration Methods • Parallel imaging • SENSE – poses the reconstruction problem in the image domain • With coherent aliases, the problem can essentially become one of bookkeeping: keeping tracking of which pixels were folded onto a point then solving for the original pixels knowing the coil sensitivities • Optimal if coil sensitivities known • Limited to uniform undersampling • GRAPPA – frequency domain, over each coil • Uses a calibration region to learn how to interpolate samples with various configurations of surrounding collected data points • Limited to uniform undersampling

11. Cartesian-based Acceleration Methods • Parallel imaging • SPIRiT – optimization problem in the frequency domain over each coil • Adopts the idea of a calibration region from GRAPPA but only a single kernel interpolating from all surrounding points and coils • Key insight: applying the SPIRiT kernel, i.e. interpolating, on the reconstructed data should give back the same image: the result must be self-consistent • Enforce data and self-consistency for each coil image • Handles any sampling pattern (including non-cartesian) • Compressed Sensing – multiple domain solution • Exploits sparsity in natural images, typically in the Wavelet domain • Enforces data consistency and sparsity • Must have incoherent random undersampling to distinguish large Wavelet coefficients from background sampling artifacts

12. Solution • The solution to the alternate problem can be formulated as a Projection Over Convex Sets algorithm • Enforce each part of the problem in turn and iterate until convergence • Slow but simple to implement

13. Result

14. Result • Aggressive 5x random undersampling • Velikina used overall R=3.95 (R=3 for first and last 2 angles, R=5 else) • Slight signal gain in the center of the brain but no significant improvement • Computation time was about 1hr for one slice with 8 cores • Considered a compressed sensing variant as another approach

15. Compressed Sensing Problem • F is Fourier transform • Ψ is Wavelet transform • λ1 based on knowledge that image is about 85% sparse • λ2 set so that 2nd deriv. is about 25% sparse

16. Result

17. Result • Aggressive 5x random undersampling • Effectively no improvement at all with regularization • Solution converges within 5 minutes

18. Another Compressed Sensing Problem • Should we instead be forming a hybrid space and jointly enforcing sparsity in the Wavelet and 2nd derivative domains? • The sparsifying transform is now the Wavelet transform of the 2nd derivative images • This fails to converge!

19. Wavelet Coefficients

20. ΨΔ2 Coefficients

21. Conclusions • The Wavelet transform of the 2nd derivative images is not as sparse as the Wavelet transform alone • It is a poor sparsifying transform, explains why solution did not converge • Unable to reproduce the findings of Velikina, not sure 2nd deriv. is the correct thing to minimize • Only small for large angles well past the Ernst angle, which don’t need to be collected anyway but not sure what subset of angles they ultimately used

22. Ideas • Collect more angles? Pfile numbering problem • Linearize the signal curve first by dividing by the flip angle since sinα≈ α in this range • If perfectly linear, the 2nd deriv. would be 0 everywhere, there would only be content in the initial “position” and “velocity” frames • Led to strange behavior with negative values in the reconstruction • View sharing + SPIRiT?