Need for Speed in MRI & Parallel Imaging I

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1 G Practical Magnetic Resonance Imaging II Sackler Institute of Biomedical Sciences New York University School of Medicine Need for Speed in MRI & Parallel Imaging I Ricardo Otazo, PhD ricardo.otazo@nyumc.org

2 Need for speed in MRI I feel the need for speed Top Gun movie The speed of conventional MRI (gradient encoding) is inherently limited Rapid, fast or accelerated MRI k-space undersampling One of the hottest topics in MRI research

3 Imaging speed in MRI Speed of k-space traversal Switching rate and amplitude of magnetic field gradients 2X faster!

4 Speed of conventional MRI is limited Slow 1. Amplitude R 2. Switching rate R 2 3. Dwell time / R R-faster What is the effect on the gradient amplifier? Power increases with R 3 What is the effect on SNR? SNR decreases with R It can also cause peripheral nerve stimulation

5 k-space undersampling Sample below the Nyquist rate Reduce the number of phase-encoding points Fully-sampled (R=1) 2X undersampled (R=2) k y k y k x k x How about the frequency-encoding dimension? Faster, but how do we reconstruct the image? Exploit data redundancy

6 Reconstruction of undersampled k-space data Partial Fourier imaging (already covered) Hermitian symmetry Parallel imaging Differences in coil sensitivities Compressed sensing Compressibility/sparsity of images To be continued

7 Parallel imaging Multiple receiver coils, different spatial sensitivities Coil 1 Coil 2 Coil 3 Coil Regular k-space undersampling k y More complicated image reconstruction Matrix inversion (linear) Sodickson DK, Manning WJ. Magn Reson Med. 1997; 38: Pruessmann KP et al. Magn Reson Med 1999; 42: k x

8 Multiple receiver coils First used to improve SNR Optimal combination (matched-filter) Sensitivity-weighted combination It requires knowledge of coil sensitivities ( ) m r = N c i= 1 c N i c i= 1 ( r) c i m ( r) i ( r) 2 m i (r): single coil images c i (r): coil sensitivities Roemer FB et al. Magn Reson Med. 1990; 16(2):

9 Multiple receiver coils First used to improve SNR Optimal combination (matched-filter) Sensitivity-weighted combination Unfiltered Filtered Coil #1 Coil #2 Coil #3 Coil #4 Sum

10 Multiple receiver coils First used to improve SNR Sum of squares Approximation to the optimal combination Images as coil sensitivities c i ( r) = m ( r) m( r) = m ( r) i N c i= 1 i 2

11 Multiple receiver coils 8-element circular array Complex sum Sum of squares Sensitivity-weighted sum

12 Gradient + coil-sensitivity encoding Image space k-space Gradient encoding only FT Only one k-space point at a time Gradient + Coil sensitivity t0 γ k( t0) = ( τ) dτ 2π G 0 FT Several k-space points at a time

13 Gradient + coil-sensitivity encoding Effective k-space oversampling Sodickson DK, McKenzie CA. Med Physics 2001; 28: Pruessmann K. NMR Biomed. 2006;19:

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15 Inversion of the full encoding matrix Direct inversion is not feasible due to matrix dimensionality NxN image N c coils R reduction factor Size of E N 2 N c x N 2 R Iterative methods Conjugate gradient, projection onto convex sets

16 Decoupling the encoding matrix Regular undersampling k-space Image space R=2 - Same k - Estimate k-space missing points from neighbors using the same kernel k-space methods SMASH, GRAPPA - Linear combination of R points - Unfold using coil sensitivities Image-space methods SENSE

17 SENSE: Unfolding aliased images Aliased images (R y = 2) a 1 a 2 ρ Fully-sampled image (R y = 1) Image reconstruction 2 coils 4 coils ( ) ˆ H H = ρ E Ψ E E Ψ a Encoding equation a E ρ = ) 2 ( ) 2 ( 2 ) ( 2 ) ( ) ( ) ( y y y y W r W r W r c r c W r c r c r a r a ρ ρ Classwork: Form the encoding equation for R=2 and 4 coils

18 Coil sensitivity estimation for SENSE Estimation of pure coil sensitivities (Pruessmann et al. MRM 1999). Separate low resolution image for each coil. REFERENCE COMBINATION RAW SENSITIVITY SMOOTHING EXTRAPOLATION = In vivo coil sensitivities (Sodickson et al. Med Phys 2001). Low resolution reference images as coil sensitivities. Post-multiply by reference combination after matrix inversion.

19 SENSE reconstruction examples Simulation of brain imaging acceleration 8-channel circular array coil R y = 3 R y = 4

20 SNR penalty in parallel imaging SNR acc = SNR no acc g-factor: noise amplification due to ill-conditioning of the encoding matrix g( r) = g R ( H 1 ) 1 H 1 E Ψ E E Ψ E ( r ) g-factor Why is it higher at the center? Coil sensitivities

21 How to reduce noise amplification? Use more coils Improve coil array design Regularization of the inverse reconstruction 2D acceleration instead of 1D acceleration for 3D imaging

22 Regularization of the inverse reconstruction Constrain the inverse problem to reduce noise amplification and control numerical instabilities Method 1: Tikhonov regularization Constrain the power of the solution { 2 } ( H λ λ ) H mˆ = min Em - s + m = E E + I E s m Lin FH et al, Magn Reson Med 2004; 51:

23 Regularization of the inverse reconstruction Method 2: Truncate, shift or weight the singular value decomposition (SVD) of the encoding matrix E = USV E H uv = VS U = σ 1 1 R * H k k k = 1 Full SVD Shifted SVD Truncated SVD k R=4

24 2D acceleration Vs. 1D acceleration 2D acceleration reduces g-factor R=4x1 R=2x2 Ohliger MA et al. MRM 2003;50:

25 Summary Speed of gradient encoding is limited Physical and physiological constraints on the gradient amplitude and switching rate Fast, rapid or accelerated MRI k-space undersampling Image reconstruction is more challenging, but more fun Exploit redundancies in the acquired data

26 Summary Parallel imaging Exploit additional encoding provided by multiple receiver coils with different sensitivities SNR penalty SENSE (image-domain) Unfolding images using coil sensitivities Matrix inversion GRAPPA and SPIRIT (k-space) In the following lectures Next lecture (Dan Sodickson) History and generalized parallel imaging