1. NA-MIC National Alliance for Medical Image Computing http://na-mic.org Mathematical and physical foundations of DTI Anastasia Yendiki, Ph.D. Massachusetts General Hospital Harvard Medical School 13 th Annual Meeting of the Organization for Human Brain Mapping June 9th, 2007 Chicago, IL

2. Anastasia Yendiki Courtesy of Gordon Kindlmann Diffusion imaging Diffusion imaging: Image the major direction(s) of water diffusion at each voxel in the brain Clearly, direction can’t be described by a usual grayscale image

3. Anastasia Yendiki Tensors We express the notion of “direction” mathematically by a tensor D A tensor is a 3x3 symmetric, positive-definite matrix: D is symmetric 3x3  It has 6 unique elements It suffices to estimate the upper (lower) triangular part d 11 d 12 d 13 d 12 d 22 d 23 d 13 d 23 d 33 D =

4. Anastasia Yendiki Eigenvalues/vectors The matrix D is positive-definite  –It has 3 real, positive eigenvalues 1, 2, 3 > 0. –It has 3 orthogonal eigenvectors e 1, e 2, e 3. D = 1 e 1  e 1 ´ + 2 e 2  e 2 ´ + 3 e 3  e 3 ´ eigenvalue e 1x e 1y e 1z e 1 = eigenvector 1 e 1 2 e 2 3 e 3

5. Anastasia Yendiki Physical interpretation Eigenvectors express diffusion direction Eigenvalues express diffusion magnitude 1 e 1 2 e 2 3 e 3 1 e 1 2 e 2 3 e 3 Isotropic diffusion: 1  2  3 Anisotropic diffusion: 1 >> 2  3 One such ellipsoid at each voxel, expressing PDF of water molecule displacements at that voxel

6. Anastasia Yendiki Tensor maps Image: A scalar intensity value f j at each voxel j Tensor map: A tensor D j at each voxel j Courtesy of Gordon Kindlmann

7. Anastasia Yendiki Scalar diffusion measures Mean diffusivity (MD): Mean of the 3 eigenvalues Fractional anisotropy (FA): Variance of the 3 eigenvalues, normalized so that 0  (FA)  1 Faster diffusion Slower diffusion Anisotropic diffusion Isotropic diffusion MD(j) = [ 1 (j)+ 2 (j)+ 3 (j)]/3 [ 1 (j)-MD(j)] 2 + [ 2 (j)-MD(j)] 2 + [ 3 (j)-MD(j)] 2 FA(j) 2 = 1 (j) 2 + 2 (j) 2 + 3 (j) 2

8. Anastasia Yendiki MRI data acquisition Measure raw MR signal (frequency-domain samples of transverse magnetization) Reconstruct an image of transverse magnetization

9. Anastasia Yendiki DT-MRI data acquisition Must acquire at least 6 times as many MR signal measurements Need to reconstruct 6 times as many values  d 11 d 13 d 12 d 22 d 23 d 33

10. Anastasia Yendiki Spin-echo MRI Use a 180  pulse to refocus spins: 90  180  GyGy 90  180  acquisition Apply a field gradient Gy for location encoding Measure transverse magnetization at each location -- depends on tissue properties ( T 1,T 2 ) fast slow

11. Anastasia Yendiki Diffusion-weighted MRI For diffusion encoding, apply two gradient pulses: 90  180  GyGy GyGy Case 1: No spin diffusion 90  180  GyGyGyGy y = y 1, y 2 No displacement in y  No dephasing  No net signal change y = y 1, y 2 acquisition

12. Anastasia Yendiki Diffusion-weighted MRI 90  180  GyGy GyGy Case 2: Some spin diffusion 90  180  GyGyGyGy y = y 1, y 2 Displacement in y  Dephasing  Signal attenuation y = y 1 +  y 1, y 2 +  y 2 acquisition For diffusion encoding, apply two gradient pulses:

13. Anastasia Yendiki Diffusion tensor model f j b,g = f j 0 e -bg  D j  g where the D j the diffusion tensor at voxel j Design acquisition: –b the diffusion-weighting factor –g the diffusion-encoding gradient direction Reconstruct images from acquired data: –f j b,g image acquired with diffusion-weighting factor b and diffusion-encoding gradient direction g –f j 0 “baseline” image acquired without diffusion-weighting ( b=0 ) Estimate unknown diffusion tensor D j We need 6 or more different measurements of f j b,g (obtained with 6 or more non-colinear g ‘s)

14. Anastasia Yendiki Choice 1: Directions Diffusion direction || Applied gradient direction  Maximum signal attenuation Diffusion direction  Applied gradient direction  No signal attenuation To capture all diffusion directions well, gradient directions should cover 3D space uniformly Diffusion-encoding gradient g Displacement detected Diffusion-encoding gradient g Displacement not detected Diffusion-encoding gradient g Displacement partly detected

15. Anastasia Yendiki How many directions? Six diffusion-weighting directions are the minimum, but usually we acquire more Acquiring more directions leads to: +More reliable estimation of tensors –Increased imaging time  Subject discomfort, more susceptible to artifacts due to motion, respiration, etc. Typically diminishing returns beyond a certain number of directions [Jones, 2004] A typical acquisition with 10 repetitions of the baseline image + 60 diffusion directions lasts ~ 10min.

16. Anastasia Yendiki Choice 2: The b-value f j b,g = f j 0 e -bg  D j  g The b -value depends on acquisition parameters: b =  2 G 2  2 (  -  /3) –  the gyromagnetic ratio –G the strength of the diffusion-encoding gradient –  the duration of each diffusion-encoding pulse –  the interval b/w diffusion-encoding pulses 90  180  G acquisition  

17. Anastasia Yendiki How high b-value? f j b,g = f j 0 e -bg  D j  g Typical b -values for DTI ~ 1000 sec/mm 2 Increasing the b -value leads to: +Increased contrast b/w areas of higher and lower diffusivity in principle –Decreased signal-to-noise ratio  Less reliable estimation of tensors in practice Data can be acquired at multiple b -values for trade-off Repeat same acquisition several times and average to increase signal-to-noise ratio

18. Anastasia Yendiki Noise in DW images Due to signal attenuation by diffusion encoding, signal-to-noise ratio in DW images can be an order of magnitude lower than “baseline” image Eigendecomposition is sensitive to noise, may result in negative eigenvalues Baseline image DW images

19. Anastasia Yendiki Distortions in DW images The raw (k-space) data collected at the scanner are frequency-domain samples of the transverse magnetization In an ideal world, the inverse Fourier transform (IFT) would yield an image of the transverse magnetization Real k-space data diverge from the ideal model: –Magnetic field inhomogeneities –Shifts of the k-space trajectory due to eddy currents In the presence of such effects, taking the IFT of the k-space data yields distorted images

20. Anastasia Yendiki Field inhomogeneities Causes: – Scanner-dependent (imperfections of main magnetic field) – Subject-dependent (changes in magnetic susceptibility in tissue/air interfaces) Results: Signal loss in interface areas, geometric distortions Signal loss

21. Anastasia Yendiki Eddy currents Fast switching of diffusion- encoding gradients induces eddy currents in conducting components Eddy currents lead to residual gradients Residual gradients lead to shifts of the k-space trajectory The shifts are direction- dependent, i.e., different for each DW image Results: Geometric distortions From Chen et al., Correction for direction- dependent distortions in diffusion tensor imaging using matched magnetic field maps, NeuroImage, 2006. Error between images with eddy-current distortions and corrected images Gx on Gy on Gz on GxGy on GyGz on GxGz on

22. Anastasia Yendiki Distortion correction Images saved at the scanner have been reconstructed from k-space data via IFT, their phase has been discarded Post-process magnitude images (by warping) to reduce distortions: –Either register distorted images to an undistorted image [Haselgrove’96, Bastin’99, Horsfield’99, Andersson’02, Rohde’04, Ardekani’05, Mistry’06] –Or use side information on distortions from separate scans (field map, residual gradients) [Jezzard’98, Bastin’00, Chen’06; Bodammer’04, Shen’04]

23. Anastasia Yendiki Tensor estimation Estimate tensor from warped images: –Usually by least squares (implying Gaussian noise statistics) [Basser’94, Anderson’01, Papadakis’03, Jones’04, Chang’05, Koay’06] log( f j b,g / f j 0 ) = -bg  D j  g = -B  D j –Or accounting for Rician noise statistics [Fillard’06] Pre-smooth or post-smooth tensor map to reduce noise [Parker’02, McGraw’04, Ding’05; Chefd’hotel’04, Coulon’04, Arsigny’06]

24. Anastasia Yendiki Other models of diffusion Need a higher-order model to capture this: –A mixture of the usual (“rank-2”) tensors [Tuch’02] –A tensor of rank > 2 [Frank’02, Özarslan’03] –An orientation distribution function [Tuch’04] –A diffusion spectrum [Wedeen’05] More parameters to estimate at each voxel  More gradient directions needed (hence HARDI - high angular resolution diffusion imaging) The tensor is an imperfect model: What if more than one major diffusion direction in the same voxel, e.g., two fibers crossing?

25. Anastasia Yendiki Example: DTI vs. DSI From Wedeen et al., Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging, MRM, 2005