On the Use of Complementary Encoding Techniques to Improve MR Imaging

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On the Use of Complementary Encoding Techniques to Improve MR Imaging W. Scott Hoge shoge@bwh.harvard.edu Dept.

1 On the Use of Complementary Encoding Techniques to Improve MR Imaging W. Scott Hoge Dept. of Radiology, Brigham and Women s Hospital and Harvard Medical School, Boston, MA Graz, Austria Jul 2010 p.1/48

2 Overview Part I: Overview Fourier Encoding (MRI Foundation) Temporal Encoding (UNFOLD) Spatial Encoding (pmri: SENSE, GRAPPA) Review Strengths and Weaknesses Part II: Regularization in SENSE LSQR-Hybrid Algorithm for efficient solutions Accelerated partial-fourier imaging Part III: Useful Combinations GRAPPA+SENSE together Real-Time Self-Referenced pmri Cardiac CINE via UNFOLD/GRAPPA/SPACE RIP Part IV: EPI Artifact Correction Review of the problem A spatial + temporal encoding solution Graz, Austria Jul 2010 p.2/48

transform (data is acquired in k space domain) k space is sampled

3 Basics of MRI: Fourier Imaging MRI uses magnetic field gradients and RF signals to encode field-of-view F Encoding typically corresponds to a Fourier transform (data is acquired in k space domain) k space is sampled line-by-line Reduce number of lines Reduce acquisition time Graz, Austria Jul 2010 p.3/48

4 Fast MR Imaging: Keyhole Sampling One idea: acquire just the low frequency content of the image F Produces images with low resolution. Graz, Austria Jul 2010 p.4/48

Images can be recovered using complementary encoding.

5 Fast MR Imaging: Sub-sampling An Alternative: sub-sampling to increase imaging efficiency. F Sub-sampling in k-space produces spatial domain aliasing. Images can be recovered using complementary encoding. Temporal encoding, by varying the sampling pattern Spatial encoding, using multiple receiver coils Graz, Austria Jul 2010 p.5/48

6 Temporal Encoding A shift in the sampling grid alters the point spread function Odd Frames Even Frames k y k y k x k x odd frames (2n) 0 Point Spread Functions N/2 UNFOLD uses variations in sampling time to encode aliasing artifacts. [ 1 ] Madore, et al, Magn Reson Med, 1999; 42(5): even frames (2n+1) 0 N/2 Graz, Austria Jul 2010 p.6/48

7 UNFOLD Processing Alternating the sampling grid will cause artifact flicker With appropriate time domain filtering 0 10 Filter Response (in db) Simple interpolation Fermi filter with 95% passband Normalized Frequency the artifact can be removed. Graz, Austria Jul 2010 p.7/48

8 Spatial Encoding W 1 W 2 W W 3 Multiple receiver coils were first introduced to improve signal-to-noise. A secondary effect is that it provides a spatial domain encoding. Each receiver coil sees the FOV from a different perspective. This encoding complements traditional Fourier encoding. Graz, Austria Jul 2010 p.8/48

9 The Basic Parallel MRI (pmri) Problem W 1 W W 2 W 3 Sub-sampling combined with spatial domain encoding yields the basic parallel imaging problem. Each coil acquires an aliased image... weighted by the coil sensitivity... at each pixel that contributes s l (x,y) = W l (x,y)ρ(x,y) + W l (x,y + N/2)ρ(x,y + N/2) Graz, Austria Jul 2010 p.9/48

10 Cartesian SENSE Formulated Combining the equations for all coils, we have s 1 (x,y) s 2 (x,y) s 3 (x,y) s 4 (x,y) = W 1 (x,y) W 1 (x,y + N/2) W 2 (x,y) W 2 (x,y + N/2) W 3 (x,y) W 3 (x,y + N/2) W 4 (x,y) W 4 (x,y + N/2) [ ρ(x, y) ρ(x,y + N/2) ] or, in a compact matrix/vector form s = Pρ [ 2 ] Ra and Rim, et al, Magn Reson Med, 1993; 30(1): [ 3 ] Pruessmann, et al, Magn Reson Med, 1999; 42(5): Graz, Austria Jul 2010 p.10/48

11 Solving the SENSE Problem Recast as a minimization problem ˆρ = argmin{f(ρ) = Pρ s 2 2} ρ Adding regularization terms allows one to balance between noise reduction and artifact suppression. min ρ { s Pρ } All SENSE-related methods (including SPACE RIP, Generalized SENSE, Generalized GRAPPA) can be described this way. Graz, Austria Jul 2010 p.11/48

12 GRAPPA Coil 1 Coil l Coil L k y k x k x k x s (l) (j) = L m=1 3 b=0 2 c= 2 s m (c + k x, (2b 3) + k y ) n(m,b,c,l) GRAPPA generalizes so-called k-space methods: coil-by-coil reconstruction, flexible kernel size, multiple auto-calibration signal lines acquired. [ 4 ] Griswold, et al, MRM 2002, 47(6): Graz, Austria Jul 2010 p.12/48

13 GRAPPA Model While not explicit in the reconstruction model, GRAPPA does rely on spatial encoding via multiple coils. Consider the encoding model for the the full data in each coil [ ] s (a) [ ] l Ga s (u) = κ. G l u This gives two equations, acquired lines: s (a) l un-acquired lines: Eliminating κ yields s (u) l = G u (G as (a) l ). [ 5 ] Kholmovski and Samsonov, ISMRM 2005, pg s (u) l = G a κ = G u κ Graz, Austria Jul 2010 p.13/48

GRAPPA Visualized s (u) l = (G u G a)s (a) l The GRAPPA algorithm seeks to estimate a composite encoding matrix, (G u G a) to combine acquired coil data, and form unacquired

14 GRAPPA Visualized s (u) l = (G u G a)s (a) l The GRAPPA algorithm seeks to estimate a composite encoding matrix, (G u G a) to combine acquired coil data, and form unacquired coil data. s (a) l s (u) l GRAPPA is efficient (low number of parameters) because of the particular structure of the composite encoding matrix. Graz, Austria Jul 2010 p.14/48

15 Comparison of SENSE and GRAPPA SENSE GRAPPA Strengths - optimal in least-squares sense - easy to constrain the reconstruction (e.g. regularization) - does not require coil sensitivity estimates Weaknesses requires good coil sensitivity estimates sub-optimal solution in least-squares sense SENSE (and related methods) and GRAPPA use separate models to reconstruct an image. One can combine them and build on each of their strengths. Graz, Austria Jul 2010 p.15/48

16 Part II Regularization in SENSE Graz, Austria Jul 2010 p.16/48

17 SENSE with Damped Least Squares SENSE pmri problems can benefit from a single (scalar) regularization parameter { min Pρ s 2 ρ 2 + λ ρ 2 } 2 This problem can be solved quickly using LSQR-Hybrid solved in a projected space solutions for multiple λ can be found very efficiently best image selected automaticly via balance between error and solution norm. (e.g. L-curve, Discrepancy) [ 6 ] Hoge, et. al. Proc. ISBI 2006, pg Graz, Austria Jul 2010 p.17/48

18 LSQR Algorithm For the moment, assume that λ is fixed and let ρ 0 = 0. LSQR produces iterates ρ k such that ρ k = arg min ρ K k [ P λi ] ρ [ s 0 ]. The solution, ρ k, exists within a k-dimensional Krylov subspace: K k = span{p H s, (P H P)P H s,, (P H P) k 1 P H s}. Graz, Austria Jul 2010 p.18/48

19 LSQR-Hybrid Multiple regularized solutions can be found quickly because the Krylov subspace is independent of λ. That is, span{p H s, (P H P)P H s,, (P H P) k 1 P H s} describes the same space as span{p H s, (P H P + λ 2 I)P H s,, (P H P + λ 2 I) k 1 P H s}. Points on L-curve can be computed in O{N λ } flops. Recurrence relationships can update all ρ (λ) in O{NN λ } flops. k solutions [ 7 ] Kilmer and O Leary, SIAM J. Matrix Anal. Appl., 22(4): , Graz, Austria Jul 2010 p.19/48

20 LSQR-Hybrid Example Acceleration factor = 3, Variable density sub-sampling, 8-channel FSE data acquired on a 1.5T GE Signa EXCITE scanner. λ = 10 4 norm of solution, ρ (log scale) λ = a residual error, P ρ s (log scale) λ = λ = Graz, Austria Jul 2010 p.20/48

21 LSQR-Hybrid Example Acceleration factor = 3, Variable density sub-sampling, 8-channel FSE data acquired on a 1.5T GE Signa EXCITE scanner. λ = 10 4 norm of solution, ρ (log scale) λ = b residual error, P ρ s (log scale) λ = λ = Graz, Austria Jul 2010 p.20/48

22 LSQR-Hybrid Example Acceleration factor = 3, Variable density sub-sampling, 8-channel FSE data acquired on a 1.5T GE Signa EXCITE scanner. λ = 10 4 norm of solution, ρ (log scale) λ = c residual error, P ρ s (log scale) λ = λ = Graz, Austria Jul 2010 p.20/48

23 LSQR-Hybrid Example Acceleration factor = 3, Variable density sub-sampling, 8-channel FSE data acquired on a 1.5T GE Signa EXCITE scanner. λ = 10 4 norm of solution, ρ (log scale) λ = d residual error, P ρ s (log scale) λ = λ = Graz, Austria Jul 2010 p.20/48

24 LSQR-Hybrid Recon Time Number of λ solutions: LSQR recon time (sec): C-code implementation, 1 core of 2.16 GHz Centrino, 2GB RAM. 256 x 256 image, R = 3.2x, variable density subsampling Graz, Austria Jul 2010 p.21/48

25 Partial-Fourier Problem Formulation In partial-fourier imaging, only half-plane of k-space is acquired, and solution constrained to be nearly real. We use a two-parameter minimization, to constrain real and imaginary components separately. { min Pρ s 2 ρ 2 + λ re R{ρ} λ im I{ρ} 2 } 2 Downside: large regularization parameter search space Graz, Austria Jul 2010 p.22/48

26 Reformulating the partial-fourier problem To maintain scalar-lambda modification of Krylov subspace, we recast the 2-parameter partial-fourier problem as R{s} I{s} 0 0 = R{P } I{P } I{P } R{P } λ re [ I 0 0 λ im /λ re I hold the ratio c = λ im /λ re constant iterate over λ re repeat for multiple values of c. ] [ R{ρ} I{ρ} ] [9] Hoge, et. al., Proc ISBI 2007; Graz, Austria Jul 2010 p.23/48

27 LSQR-Hybrid with a regularization operator Define L = [ I 0 0 ci ], and c = λ im /λ re, and solve min ρ { Pρ s λ 2 re Lρ 2 2} A change of variables, y = Lρ, and ρ = L 1 y, gives min y { PL 1 y s λ 2 re y 2 2} Problem is now compatible with LSQR-Hybrid Constraint: L must be invertible. (e.g. non-zero λ s) Graz, Austria Jul 2010 p.24/48

28 Partial-Fourier pmri Results Highly efficient computations along lines of constant c λ re ) log10( log10( λ im ) Contour plot of residual error, Pρ s 2 2, Graz, Austria Jul 2010 p.25/48

2-D MRI with partial-fourier Results R=4x. (>2x k y acceleration + partial-fourier) 16 conjugate-symmetric lines (past k y = 0) 25 2.5 20 2.

29 2-D MRI with partial-fourier Results R=4x. (>2x k y acceleration + partial-fourier) 16 conjugate-symmetric lines (past k y = 0) λ LSQR λ LSQR + Homodyne Image ROI Image Error Graz, Austria Jul 2010 p.26/48

2-D MRI with partial-fourier Results R=4x. (>2x k y acceleration + partial-fourier) 8 conjugate-symmetric lines (past k y = 0) 4.0 20 3.5 3.

30 2-D MRI with partial-fourier Results R=4x. (>2x k y acceleration + partial-fourier) 8 conjugate-symmetric lines (past k y = 0) λ LSQR λ LSQR + Homodyne Image ROI Image Error Graz, Austria Jul 2010 p.26/48

31 2-D MRI with partial-fourier Results R=4x. (>2x k y acceleration + partial-fourier) 1 conjugate-symmetric line (past k y = 0) λ LSQR λ LSQR + Homodyne Image ROI Image Error Graz, Austria Jul 2010 p.26/48

3-D MRI with partial-fourier Results 20 50 50 40 100 100 60 80 150 150 100 120 200 200 140 160 250

at 3T, 8x total acceleration, 8 channel head coil unaccelerated acquisition time 28 min accelerated

32 3-D MRI with partial-fourier Results D FSE Data acquired at 3T, 8x total acceleration, 8 channel head coil unaccelerated acquisition time 28 min accelerated acquisition time 4 min nz = 3804 Graz, Austria Jul 2010 p.27/48

33 Part III Useful Combinations Graz, Austria Jul 2010 p.28/48

34 GRAPPA-enhanced self-referenced SENSE Strategy: use GRAPPA to improve coil sensitivity estimates in self-referenced SENSE methods. density 3x 2x 3x 1x 2x 3x density 3x N/2 0 +N/2 Acquisition pattern subdivided into three regions Use GRAPPA in light gray outer central region, estimate coil sensitivities using all gray regions, then use SENSE (LSQR-Hybrid) reconstruction on original data set. [10] Hoge and Brooks, Magn Reson Med, 2008; 60(2): Graz, Austria Jul 2010 p.29/48

35 GEYSER Signal Flow Diagram LSQR Hybrid GRAPPA Sensitivity Estimation Graz, Austria Jul 2010 p.30/48

a SPACE RIP (4 ACS lines) b GRAPPA alone (4 ACS

36 GEYSER + SPACE RIP 3.2x acceleration, exponentially weighted sampling distribution, 128-x-128 image size, 4 ACS lines, 8-channel cardiac coil, 3T scanner. a SPACE RIP (4 ACS lines) b GRAPPA alone (4 ACS lines) c GEYSER + SPACE RIP (20 ACS lines) Graz, Austria Jul 2010 p.31/48

GEYSER + LSQR-Hybrid in Real-Time MRI Short movie showing

4ms/1.9ms, 128x128 image size, 34 phase-lines acquired (3.

Image acquistion time: 163ms (6 frames/sec) Image

37 GEYSER + LSQR-Hybrid in Real-Time MRI Short movie showing real-time imaging capabilities Data acquired at 3T, TR/TE: 3.4ms/1.9ms, 128x128 image size, 34 phase-lines acquired (3.76x). Completely self-referenced. Image acquistion time: 163ms (6 frames/sec) Image reconstruction time: 100ms (8-core CPU, 4GB RAM) Credit: Renxin Chu Graz, Austria Jul 2010 p.32/48

38 UNFOLD Cardiac Imaging Example UNFOLD uses temporal variations in sampling to encode aliasing artifacts. Complements parallel imaging techniques. Separate temporal-frequency bands can be processed independently. k y f mid f lo t DC Ny [G] Madore, et al, MRM 1999; 42(5): Graz, Austria Jul 2010 p.33/48

39 UNFOLD Cardiac Imaging Images UNFOLD + SPACE RIP (lo) + SPACE RIP (mid) UNFOLD + GRAPPA (lo) + GRAPPA (mid) UNFOLD + GRAPPA (lo) + SPACE RIP (mid) Best results with GRAPPA and SPACE RIP combined [11] Chao, et al, Proc ISMRM, 2009; p Graz, Austria Jul 2010 p.34/48

40 Part IV Useful Combinations: Nyquist Ghost Correction Graz, Austria Jul 2010 p.35/48

41 Echo Planar Imaging EPI is a fast imaging method that acquires multiple k-space lines after each RF excitation. RF Excitation k y ky Gradient Readout Gradient k x Sampling Window Challenge: Data sampled on positive readout (+RO) and negative readout (-RO) gradients is often misaligned, introducing artifacts known as Nyquist ghosts. Graz, Austria Jul 2010 p.36/48

42 EPI Nyquist Ghost Correction Review Many methods have been proposed to correct them: use pre-scan data to estimate the shift (sensitive to scanner drift) acquire additional k-space lines w/ image data (extends echo train, yielding greater distortion) interleave data (loss of temporal resolution) self-referenced phase alignment of +RO and -RO data (not robust) use parallel imaging (the focus here) Graz, Austria Jul 2010 p.37/48

43 Ghost Correction via Temporal Encoding Temporal Encoding: alternate readout polarity on alternate frames Odd Frames Even Frames k y k y k x k x Ghost correction options: interleave adjacent data filter temporal series with UNFOLD Graz, Austria Jul 2010 p.38/48

Ghost Correction via Interleaves (PLACE)

and addition can remove any residual ghosts.

temporal variations 6 4 2 0.8 0.6 0.4 0.2 0.0 0.

44 Ghost Correction via Interleaves (PLACE) Interleaved data may be corrupted by scanner instability. a I n I n + ΨI p b (+) Phase-alignment and addition can remove any residual ghosts. (-) Signal cancellation, with loss of SNR and temporal variations [13] Xiang and Ye, Magn Reson Med, 2007; 57(4): Graz, Austria Jul 2010 p.39/48

45 Benefit to pmri Ghost-free data is critical for good pmri reconstructions Data: R=3.2x variable density EPI, 8-channel coil, 1.5T Recon: Double-pass GRAPPA (1st: 6 ACS; 2nd: 20 ACS) a b c Ghost Correction GRAPPA (a) pre-scan data self referenced (b) pre-scan data 20 ACS reference frame (c) interleaved data self referenced [14] Hoge, Tan, Kraft, ISMRM, 2009; p Graz, Austria Jul 2010 p.40/48

46 Ghost Elimination via Spatial/Temporal Enc. Temporal encoding: alternate polarity of readout gradients Spatial encoding: employ multiple channels and parallel MR imaging k(t 1) I p I n Ψ I p + ΨI n pmri Calibration I (t) p k(t) pmri Reconstruction pmri Reconstruction I (t) n Ψ I (t) p + Ψ I (t) n Graz, Austria Jul 2010 p.41/48

Comparison of Ghost Correction Methods Ahn & Cho static 250 15.272 1.

5 a Ahn & Cho dynamic inside ellipse 200 150 100 50 0 outside ellipse 4.

41 10 0.94 10 1.47 10 2.00 10 GSR (as percentage) 5.0 4.5 4.0 3.5 3.0 2.5 2.

5 1.0 20 40 60 80 100 120 Time point (t) b d PLACE c e EPI GESTE Using

47 Comparison of Ghost Correction Methods Ahn & Cho static a Ahn & Cho dynamic inside ellipse outside ellipse Real time PAGE GSR (as percentage) Ahn & Cho static Ahn & Cho dynamic Real time PAGE PLACE EPI GESTE Time point (t) b d PLACE c e EPI GESTE Using both temporal encoding and pmri, robust and exceptionally good Nyquist ghost correction can be obtained. Graz, Austria Jul 2010 p.42/48

48 EPI-GESTE summary Ghost Elimination via Spatial and Temporal Encoding: Parallel MRI calibration two temporally encoded frames are used interleaved data forms two images, +RO and -RO coherent summation of +RO and -RO image is ghost free ghost removal improves pmri coefficient calibration Image reconstruction recon +RO and -RO data separately w/ pmri coefficients adding images coherent cancels residual pmri artifacts Advantages no loss of temporal resolution self-referenced - no pre-scan data or longer ETL exceptionally good ghost suppression [15] Hoge, Tan, Kraft, Magn Reson Med, 2010; in press. Graz, Austria Jul 2010 p.43/48

49 GESTE + 3D EPI Data acquired as planes of kx-ky, at kz locations Standard Ghost Correction EPI GESTE Ghost Correction Credit: O. Afacan Graz, Austria Jul 2010 p.44/48

50 Measuring CBF with ASL MRI Arterial Spin Labeling (ASL) MRI for cerebral blood flow (CBF) measurement non-invasive and repeatable How does ASL work? wait Difference wait Spin Preparation Image Acquisition Credit: H. Tan, R. Kraft Graz, Austria Jul 2010 p.45/48

51 Measuring CBF with ASL MRI Arterial Spin Labeling (ASL) MRI for cerebral blood flow (CBF) measurement non-invasive and repeatable How does ASL work? wait Difference wait 1 average Spin Preparation Image Acquisition Credit: H. Tan, R. Kraft Graz, Austria Jul 2010 p.45/48

52 Measuring CBF with ASL MRI Arterial Spin Labeling (ASL) MRI for cerebral blood flow (CBF) measurement non-invasive and repeatable How does ASL work? wait Difference wait 20 averages Spin Preparation Image Acquisition Credit: H. Tan, R. Kraft Graz, Austria Jul 2010 p.45/48

53 Measuring CBF with ASL MRI Arterial Spin Labeling (ASL) MRI for cerebral blood flow (CBF) measurement non-invasive and repeatable How does ASL work? wait Difference wait 60 averages Spin Preparation Image Acquisition Credit: H. Tan, R. Kraft Graz, Austria Jul 2010 p.45/48

54 GESTE vs Other GC methods in ASL Real Time PAGE PLACE GESTE Fluctuations in ghost suppression can cause false brightening in the perfusion image Graz, Austria Jul 2010 p.46/48

55 Summary MR Imaging can benefit from spatial and temporal encoding. Spatial and Temporal encoding are complementary, and reduce the amount data of needed to create an image. Combining methods is often beneficial: artifacts visible in one method can be compensated with another. Leveraging complementary encoding to generate redundant data can be used to remove artifacts. also useful in: motion artifact correction, Graz, Austria Jul 2010 p.47/48

56 Acknowledgements Collaborators Dana H. Brooks (Northeastern) Misha E. Kilmer (Tufts) Robert A. Kraft (Wake-Forest) Bruno Madore (BWH) Istavan (Pisti) Morocz (BWH) Onur Afacan (Northeastern) Huan Tan (Wake-Forest) Renxin Chu (BWH) Zhikui Xiao (GE: ASL) Funding NC-IGT [NIH U41 RR A2 (Jolesz, PI)] Imaging Toolkit Functional Neuroimaging Lab [Silbersweig / Stern, PIs] Graz, Austria Jul 2010 p.48/48

MRI beyond Fourier Encoding: From array detection to higher-order field dynamics

MRI beyond Fourier Encoding: From array detection to higher-order field dynamics K. Pruessmann Institute for Biomedical Engineering ETH Zurich and University of Zurich Parallel MRI Signal sample: m γκ,