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

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.

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

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” – 1 st term enforces data consistency, 2 nd term smoothness in the signal curve

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 2 nd derivative “space”

Velikina Results

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 2 nd deriv. is only 0 for 15+ deg. Nova 32ch head coil 110x110x40 matrix TR = 4.5ms

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

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

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

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

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 Result

Compressed Sensing Problem F is Fourier transform Ψ is Wavelet transform λ 1 based on knowledge that image is about 85% sparse λ 2 set so that 2 nd deriv. is about 25% sparse

Result

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

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

Wavelet Coefficients

ΨΔ 2 Coefficients

Conclusions The Wavelet transform of the 2 nd 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 2 nd 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

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 2 nd 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?