Declaration of Conflict of Interest or Relationship

1. Declaration of Conflict of Interest or Relationship David Atkinson: I have no conflicts of interest to disclose with regard to the subject matter of this presentation.

2. Image Reconstruction: Motion Correction David AtkinsonD.Atkinson@ucl.ac.uk Centre for Medical Image Computing, University College London with thanks to Freddy Odille, Mark White, David Larkman, Tim Nielsen, Murat Askoy, Johannes Schmidt.

3. Problem: Slow Phase Encoding • Acquisition slower than physiological motion. • motion artefacts. • Phase encode FOV just large enough to prevent wrap around. • minimises acquisition time, • Nyquist: k-space varies rapidly making interpolation difficult.

4. Motion and K-Space Fourier Transform image k-space acquired in time The sum in the Fourier Transform means that motion at any time can affect every pixel.

5. K-Space Corrections for Affine Motion Image Motion Translation (rigid shift) Rotation Expansion General affine K-Space Effect Phase ramp Rotation (same angle) Contraction Affine transform [Guy Shechter PhD Thesis]

6. Rotation Example Time Example rotation mid-way through scan. Ghosting in phase encode direction.

7. Interpolation, Gridding and Missing Data FFT requires regularly spaced samples. Rapid variations of k-space make interpolation difficult. K-space missing in some regions.

8. Prospective Motion Correction Motion determined during scan & plane updated using gradients. • Prevents pie-slice missing data. • Removes need for interpolation. • Prevents through-slice loss of data. • Can instigate re-acquisition. • Reduces reliance on post-processing. • Introduces relative motion of coil sensitivities, distortions & field maps. • Difficult to accurately measure tissue motion in 3D. • Gradient update can only compensate for affine motion.

9. Non-Rigid Motion • Most physiological motion is non-rigid. • No direct correction in k-space or using gradients. • A flexible approach is to solve a matrix equation based on the forward model of the acquisition and motion.

10. Forward Model and Matrix Solution Measured data “Encoding” matrix with motion, coil sensitivities etc Artefact-free Image Least squares solution: Conjugate gradient techniques such as LSQR.

11. coil sensitivity FFT Measured k-space for shot sample shot = motion k i The Forward Model as Image Operations motion-free patient Image transformation at current shot Multiplication of image by coil sensitivity map Fast Fourier Transform to k-space Selection of acquired k-space for current shot

12. Shots spin echo 1 readout = 1 shot single-shot EPI multi-shot

13. coil sensitivity FFT Measured k-space for shot sample shot = motion k i Forward Model as Matrix-Vector Operations motion-free patient *

14. Converting Image Operations to Matrices • The trial motion-free image is converted to a column vector. motion-free patient image n n n2

15. Expressing Motion Transform as a Matrix FFT Measured image = coil sample motion k i ? = • Matrix acts on pixels, not coordinates. • One pixel rigid shift – shifted diagonal. • Half pixel rigid shift – diagonal band, width depends on interpolation kernel. • Shuffling (non-rigid) motion - permutation matrix.

16. Converting Image Operations to Matrices • Pixel-wise image multiplication of coil sensitivities becomes a diagonal matrix. • FFT can be performed by matrix multiplication. • Sampling is just selection from k-space vector. patient FFT Measured image = coil sample motion k i =

17. Stack Data From All Shots, Averages and Coils *

18. Conjugate Gradient Solution • Efficient: does not require E to be computed or stored. • User must supply functions to return result of matrix-vector products • We know the correspondence between matrix-vector multiplications and image operations, hence we can code the functions.

19. The Complex Transpose EH H H H H motion FFT • Reverse the order of matrix operations and take Hermitian transpose. • Sampling matrix is real and diagonal hence unchanged by complex transpose. • FFT changes to iFFT. • Coil sensitivity matrix is diagonal, hence take complex conjugate of elements. • Motion matrix ... coil sample

20. Complex transpose of motion matrix Options: • Approximate by the inverse motion transform. • Approximate the inverse transform by negating displacements. • Compute exactly by assembling the sparse matrix (if not too large and sparse). • Perform explicitly using for-loops and accumulating the results in an array.

21. Example Applications of Solving Matrix Eqn averaged cine ‘sensors’ from central k-space lines input to coupled solver for motion model and artefact-free image. multi-shot DWI artefact free image example phase correction

22. Summary: Forward Model Method • Efficient Conjugate Gradient solution. • Incorporates physics of acquisition including parallel imaging. • Copes with missing data or shot rejection. • Interpolates in the (more benign) image domain. • Can include other artefact causes e.g. phase errors in multi-shot DWI, flow artefacts, coil motion, contrast uptake. • Can be combined with prospective acquisition. • Often regularised by terminating iterations. • Requires knowledge of motion.

23. Alternative Iterative Reconstruction - • Fourier-transform each interleave. • Initialize image: I=0 measurementdata weightwiththe coilsensitivities combine updates fold unfold coil 1 rotate +shift uncorrected motion compensated rotate +shift coil 2 coil 3 + [Nielsen et al. #3048]

24. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models.

25. Estimating Motion ECG, respiratory bellows, optical tracking, ultrasound (#3961), spirometer (#1553), accelerometer (#1550). • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models.

26. Estimating Motion pencil beam navigator, central k-space lines, orbital navigators, rapid, low resolution images, FID navigators. • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models.

27. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. repeated acq near k-space centre, PROPELLER, radial & spiral acquisitions, spiral projection imaging,

28. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. Predict and compare k-space lines. Detect and minimise artefact source to make multiple coil images consistent.

29. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. Find model parameters to minimise cost function e.g. image entropy, coil consistency.

30. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. Link a model to scan-time signal. Solve for motion model and image in a coupled system (GRICS).

31. Example combined prospective and retrospective methods at ISMRM 2010 Motion corrected 2xSENSE 20 mm window 2x SENSE 20 mm window 5 mm gatingwindow 34.7 min 9 min 9 min Retrospective correction: motion model from low res images, LSQR solution, 6 iterations. Implicit SENSE allows undersampling [Schmidt et al #492]

32. Example combined prospective and retrospective methods at ISMRM 2010 no correction prospective optical tracking additional entropy-based autofocus [Aksoy et al #499]

33. Outlook • Reconstruction times, 3D and memory still challenging. • Expect intelligent use of prior knowledge: sparsity, motion models, atlases etc. • Optimum solution target dependent. Power in combined acquisition and reconstruction methods.

34. www.ucl.ac.uk/cmic

35. Physiological Motion can be useful • Functional information • cardiac wall motion, bowel motility. • Elastography. • Randomising acquisition for compressed sensing reconstruction.