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

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Gregory Barber
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1 MRI beyond Fourier Encoding: From array detection to higher-order field dynamics K. Pruessmann Institute for Biomedical Engineering ETH Zurich and University of Zurich

2 Parallel MRI Signal sample: m γκ, = ik e κ r s ( r γ ) ρ( r ) 3 d r gradient-driven Fourier encoding coil sensitivity signal density

3 Encoding Signal sample: ik r m = e γκ, κ s ( r γ ) ρ( r ) gradient-driven Fourier encoding coil sensitivity signal density

4 Encoding m γκ, = ik r e s (r) d(r ) γ dr κ 3 Discretisation m γκ, = κ ρ s (r ) ρ ik r e Encoding Matrix d(r ) γ ρ ρ m = E d

5 Reconstruction m = E d Encoding: Linear! I = F Decoding: m Reconstruction Matrix

6 Reconstruction Encoding: Decoding: m = E d I = F E d PSF SRF Depiction Matrix: F E =

7 Reconstruction Encoding: Decoding: m = E d I = F E d PSF SRF Depiction Matrix: F E =

8 Thermal noise Electrons Noise Voltage _ _ Ions Dipolar Molecules

9 Thermal noise Noise Voltage Statistics: Zero mean no autocorrelation Gaussian distribution white

10 Noise m = E d + η Noise Characteristics: Gaussian Zero mean ( ) Single coil: Variance ψ = Avg ηη

11 Noise m = E d + η Noise Characteristics: Gaussian Zero mean Multiple coils: Covariance Ψ, = Avg ( ηη ) γγ γ γ Ψ = γ γ Single channel noise variance

12 Noise m = E d + η Noise Characteristics: Gaussian Zero mean Multiple coils: Covariance Ψ, = Avg ( ηη ) γγ γ γ Ψ = γ γ Mutual noise correlation

13 Noise Structure of κ = 1 κ = 2 γ γ η κ Ψ= κ Ψ Ψ Ψ Ψ κ = N γ Ψ

14 Noise Propagation m = E d + η Reconstruction: I = FEd + F η Image noise Define Covariance Matrix of Image Noise: X = Avg (F ) (F ) ( η ) η ρρ, ρ ρ X = FΨ F H

15 Summary m = E + Encoding: d η Reconstruction: I = F m MR Signal Noise Data Encoding matrix E Ψ Image Depiction F E F Ψ F H

16 Summary m = E + Encoding: d η Reconstruction: I = F m MR Signal Noise Data Encoding matrix E Ψ Image Depiction F E F Ψ F H Identity Minimum

17 Reconstruction F E F Ψ F H Identity Minimum Enforce strictly Disregard any inverse of E e.g. Moore-Penrose: Other options: F = (E E) E H 1 H H 1 H F = (E XE) E X

18 Reconstruction F E F Ψ F H Identity Minimum Enforce strictly Minimize Moore-Penrose inverse with pre-whitening: F = (E Ψ E) E Ψ H 1 1 H 1 yields best SNR available with exact depiction

19 Reconstruction F E F Ψ F H Identity Minimum Minimize jointly: ( ) ( ) H H Tr FE Id Θ FE Id + Tr FΨ F Image domain inversion: Data domain inversion: F = (E Ψ E + Θ ) E Ψ H H 1 F = ΘE (EΘE +Ψ ) H H 1

20 Iterative Reconstruction Pseudoinverse: I = H 1 H (E E) E m Rewrite as: H E E I = E H m Solve by conjugate gradient algorithm: Residuum H E E CG H E Samples Expensive part when converged Image

21 FFT and Gridding E H K-Space K-Space E S* 1 FFT GRID GRID FFT S 1 + S* 2 FFT GRID GRID FFT S 2 S* N FFT GRID GRID FFT S N Reduces loop complexity from N 4 to N 2 logn!

22 K-Space S* 1 FFT GRID GRID FFT S 1 + S* 2 FFT GRID GRID FFT S 2 CG S* N FFT GRID GRID FFT S N 1 2 n Image Receive Channels Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med 2001;46(4):

23 Example Spiral R = 2.5 Residuum Initial Recon

24 Preconditioning Preconditioning = Modify equation such that right side is approximate solution Add density correction! Add intensity correction!

25 K-Space S* 1 FFT GRID D GRID FFT S 1 I S* 2 FFT GRID D GRID FFT S 2 + I CG S* N FFT GRID D GRID FFT S N I 1 2 n Image Receive Channels More advanced preconditioning: H. Eggers et al, Proc. ISMRM 2004

26 Preconditioning Initial Spiral, R = 2.5 Reconstruction

27 Sampling Patterns Initial Radial, R = 5.0 Reconstruction

28 Preconditioning Initial Random, R = 2.5 Reconstruction

29 High Reduction Factors Spiral, R = 3.0 Spiral, R = 4.0 Spiral, R = 5.0

30 High Reduction Factors R increases Conditioning deteriorates 1. Convergence slows down 2. Noise increases

31 Noise Propagation Cartesian, R = 1 Image Domain K-Space

32 Noise Propagation Cartesian, R = 6 Image Domain K-Space

33 Noise Propagation Spiral, R = 6 Image Domain K-Space

34 Noise Propagation Radial, R = 6 Image Domain K-Space

3+ liquid-state NMR sample glass capillary De Zanche et

35 Magnetic Field Monitoring NMR field probes for concurrent field mapping copper shield χ-tuned polymer Fe 3+ Dy 3+ Er 3+ liquid-state NMR sample glass capillary De Zanche et al, Magn Reson Med (2008) Barmet et al, Magn Reson Med (2008)

36 Probe Signal Spiral acquisition 2π π Probe signal Phase msec

37 Probe Signal Spiral acquisition 2π π Probe signal Phase Zoomed 0 2 msec

38 Signal Processing Spiral acquisition 100 π 0 Probe signal Phase Unwrapped -100 π 100 msec

39 Signal Processing probe position static frequency offset Phase of probe signal: ϕ i(t) = k(t) r i + φ B (t) + ωi t 0 actual k-space position global phase error (B 0 eddy current, drift)

40 Signal Processing probe position static frequency offset Phase of probe signal: ϕ i(t) = k(t) r i + φ B (t) + ωi t 0 Least squares fit actual k-space position global phase error (B 0 eddy current, drift)

41 Example: Spiral, 8 segments

42 Experiments Spiral, AQ = 30 ms, 25 µs gradient delay Nominal trajectory Monitor data

43 Magnetic Field Monitoring Spin-warp GE EPI Spiral

44 Segmented EPI Readout direction measured deviation from nominal (x40) Phase direction

45 Segmented EPI based on field probes nominal difference

46 Magnetic Field Monitoring Segmented gradient-echo EPI Reconstruction based on field probes only

47 Magnetic Field Monitoring Typical skewing, stretching in DTI scans Monitored k-space trajectory Reconstruction

48 Magnetic Field Monitoring Routine setup: - 16-channel 19 F field camera - 8-channel 1 H array

49 Higher-order field models Expand dynamic field into spherical harmonics N L 1 Br (,) t B () r c () t f () r + static l l l= 0 1 φ(,) r t ω () r t + k () t f () r N L static l l l= 0 Least-squares fit to probe phase data coefficient order basis function x 2 1 y 3 z 4 xy 5 zy 6 2 3z 2 - (x 2 + y 2 + z 2 ) 7 xz 8 x 2 - y 2 9 3x 2 y - y 3 10 xyz 11 y (5z 2 - (x 2 + y 2 + z 2 )) z 3-3z (x 2 + y 2 + z 2 ) 13 x (5z 2 - (x 2 + y 2 + z 2 )) 14 x 2 z y 2 z 15 x 3-3xy 2 ISMRM Honolulu - April 24,

50 Higher-order Reconstruction F = (E Ψ E + Θ ) E Ψ H H 1 E γκ ρ = s (r ) (, ), γ i ( r,t ) e φ ρ measured phase model Residuum H E E CG H E Samples matrix-vector multiplications when converged Image

51 Higher-Order Reconstruction Diffusion weighting causes: - eddy currents (all orders) - image distortion monitor higher-order field evolution incorporate in iterative reconstruction no diffusion weighting diffusion-weighted 1 st -order recon diffusion-weighted 3 rd -order recon

52 Higher-order reconstruction Diffusion tensor imaging w/o coregistration b 0 mean DW ADC 3rd vs. 1st order reconstruction: FA difference up to 10% FA FA

53 Beyond Fourier Encoding Non-Fourier encoding can boost encoding efficiency (e.g. parallel imaging) occurs inevitably with - imperfect hardware - subject-induced susceptibility effects Dynamic field measurements permit characterizing hardware and various field perturbations determine an accurate signal model enable image reconstruction from perturbed data Reconstruction maths are ready to handle general field evolutions continue to benefit from IT evolution a cheap alternative to expensive hardware optimization

Background II. Signal-to-Noise Ratio (SNR) Pulse Sequences Sampling and Trajectories Parallel Imaging. B.Hargreaves - RAD 229.

Background II. Signal-to-Noise Ratio (SNR) Pulse Sequences Sampling and Trajectories Parallel Imaging. B.Hargreaves - RAD 229. Background II Signal-to-Noise Ratio (SNR) Pulse Sequences Sampling and Trajectories Parallel Imaging 1 SNR: Signal-to-Noise Ratio Signal: Desired voltage in coil Noise: Thermal, electronic Noise Thermal