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, = ik e κ r s ( r γ ) ρ( r ) 3 d r gradient-driven Fourier encoding coil sensitivity signal density
Encoding Signal sample: ik r m = e γκ, κ s ( r γ ) ρ( r ) gradient-driven Fourier encoding coil sensitivity signal density
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
Reconstruction m = E d Encoding: Linear! I = F Decoding: m Reconstruction Matrix
Reconstruction Encoding: Decoding: m = E d I = F E d PSF SRF Depiction Matrix: F E =
Reconstruction Encoding: Decoding: m = E d I = F E d PSF SRF Depiction Matrix: F E =
Thermal noise - - - - - - Electrons Noise Voltage _ + + + _ Ions Dipolar Molecules
Thermal noise Noise Voltage Statistics: Zero mean no autocorrelation Gaussian distribution white
Noise m = E d + η Noise Characteristics: Gaussian Zero mean ( ) Single coil: Variance ψ = Avg ηη
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
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
Noise Structure of κ = 1 κ = 2 γ γ η κ Ψ= κ Ψ Ψ Ψ Ψ κ = N γ Ψ
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
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
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
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
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
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 1 1 1 H 1 F = ΘE (EΘE +Ψ ) H H 1
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
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!
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):638-651.
Example Spiral R = 2.5 Residuum Initial Recon
Preconditioning Preconditioning = Modify equation such that right side is approximate solution Add density correction! Add intensity correction!
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
Preconditioning Initial Spiral, R = 2.5 Reconstruction
Sampling Patterns Initial Radial, R = 5.0 Reconstruction
Preconditioning Initial Random, R = 2.5 Reconstruction
High Reduction Factors Spiral, R = 3.0 Spiral, R = 4.0 Spiral, R = 5.0
High Reduction Factors R increases Conditioning deteriorates 1. Convergence slows down 2. Noise increases
Noise Propagation Cartesian, R = 1 Image Domain K-Space
Noise Propagation Cartesian, R = 6 Image Domain K-Space
Noise Propagation Spiral, R = 6 Image Domain K-Space
Noise Propagation Radial, R = 6 Image Domain K-Space
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)
Probe Signal Spiral acquisition 2π π Probe signal Phase 0 100 msec
Probe Signal Spiral acquisition 2π π Probe signal Phase Zoomed 0 2 msec
Signal Processing Spiral acquisition 100 π 0 Probe signal Phase Unwrapped -100 π 100 msec
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)
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)
Example: Spiral, 8 segments
Experiments Spiral, AQ = 30 ms, 25 µs gradient delay Nominal trajectory Monitor data
Magnetic Field Monitoring Spin-warp GE EPI Spiral
Segmented EPI Readout direction measured deviation from nominal (x40) Phase direction
Segmented EPI based on field probes nominal difference
Magnetic Field Monitoring Segmented gradient-echo EPI Reconstruction based on field probes only
Magnetic Field Monitoring Typical skewing, stretching in DTI scans Monitored k-space trajectory Reconstruction
Magnetic Field Monitoring Routine setup: - 16-channel 19 F field camera - 8-channel 1 H array
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 0 0 1 1 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 )) 12 3 5z 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, 2009 49
Higher-order Reconstruction F = (E Ψ E + Θ ) E Ψ H 1 1 1 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
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
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
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