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Mathematical and physical foundations of DTI

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. Diffusion imaging.

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Mathematical and physical foundations of DTI

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  1. Mathematical and physical foundations of DTI Anastasia Yendiki, Ph.D. Massachusetts General Hospital Harvard Medical School 13th Annual Meeting of the Organization for Human Brain Mapping June 9th, 2007 Chicago, IL

  2. 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 Courtesy of Gordon Kindlmann

  3. Tensors • We express the notion of “direction” mathematically by a tensor D • A tensor is a 3x3 symmetric, positive-definite matrix: d11 d12 d13 d12 d22 d23 d13 d23 d33 D = • D is symmetric 3x3  It has 6 unique elements • It suffices to estimate the upper (lower) triangular part

  4. e1x e1y e1z e1 = Eigenvalues/vectors • The matrixD is positive-definite  • It has 3 real, positive eigenvalues 1, 2, 3> 0. • It has 3 orthogonal eigenvectors e1, e2, e3. D = 1 e1 e1´ + 2 e2 e2´ + 3 e3 e3´ eigenvalue eigenvector 1 e1 2 e2 3 e3

  5. Physical interpretation • Eigenvectors express diffusion direction • Eigenvalues express diffusion magnitude Isotropic diffusion: 1 2 3 • Anisotropic diffusion: • 1>> 2 3 1 e1 1 e1 2 e2 3 e3 2 e2 3 e3 • One such ellipsoid at each voxel, expressing PDF of water molecule displacements at that voxel

  6. Tensor maps Image: A scalar intensity valuefjat each voxelj Tensor map: A tensorDjat each voxelj Courtesy of Gordon Kindlmann

  7. Scalar diffusion measures Mean diffusivity (MD): Mean of the 3 eigenvalues Faster diffusion Slower diffusion MD(j) = [1(j)+2(j)+3(j)]/3 Fractional anisotropy (FA): Variance of the 3 eigenvalues, normalized so that 0 (FA) 1 Anisotropic diffusion Isotropic diffusion [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. MRI data acquisition Measure raw MR signal (frequency-domain samples of transverse magnetization) Reconstruct an image of transverse magnetization

  9. DT-MRI data acquisition Must acquire at least6 times as many MR signal measurements Need to reconstruct 6 times as many values  d11 d12 d13 d22 d23 d33

  10. Spin-echo MRI • Use a 180 pulse to refocus spins: acquisition Gy 90 180 • Apply a field gradient Gy for location encoding Measure transverse magnetization at each location -- depends on tissue properties (T1,T2) 90 180 slow fast slow fast

  11. Diffusion-weighted MRI • For diffusion encoding, apply two gradient pulses: acquisition Gy 90 180 Gy • Case 1: No spin diffusion y = y1, y2 y = y1, y2 Gy Gy 90 180 No displacement in y No dephasing  No net signal change

  12. Diffusion-weighted MRI • For diffusion encoding, apply two gradient pulses: acquisition Gy 90 180 Gy • Case 2: Some spin diffusion y = y1, y2 y = y1+y1, y2+y2 Gy Gy 90 180 Displacement in y Dephasing  Signal attenuation

  13. Diffusion tensor model • fjb,g = fj0 e-bgDjg where the Djthe diffusion tensor at voxel j • Design acquisition: • b the diffusion-weighting factor • g the diffusion-encoding gradient direction • Reconstruct images from acquired data: • fjb,gimage acquired with diffusion-weighting factor b and diffusion-encoding gradient direction g • fj0“baseline”image acquired without diffusion-weighting (b=0) • Estimate unknown diffusion tensor Dj • We need 6 or more different measurements of fjb,g (obtained with 6 or more non-colinear g‘s)

  14. 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. 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. Choice 2: The b-value • fjb,g = fj0 e-bgDjg • The b-value depends on acquisition parameters: b = 2G2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 acquisition 90  180 G 

  17. How high b-value? • fjb,g = fj0 e-bgDjg • Typical b-values for DTI ~ 1000 sec/mm2 • 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. 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. 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. Field inhomogeneities Signal loss • 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

  21. Eddy currents Error between images with eddy-current distortions and corrected images • 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 Gx on Gy on Gz on GxGy on GyGz on GxGz on From Chen et al., Correction for direction-dependent distortions in diffusion tensor imaging using matched magnetic field maps, NeuroImage, 2006.

  22. 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. 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( fjb,g / fj0) = -bgDjg = -BDj • 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. Other models of diffusion • The tensor is an imperfect model: What if more than one major diffusion direction in the same voxel, e.g., two fibers crossing? • 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)

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

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