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
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. - PowerPoint PPT Presentation
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NA-MICNational 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
13th Annual Meeting of the Organization for Human Brain Mapping
June 9th, 2007
Chicago, IL
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
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
d11 d12 d13 d12 d22 d23
d13 d23 d33
D =
Anastasia Yendiki
Eigenvalues/vectors• The matrix D is positive-definite
– It has 3 real, positive eigenvalues 1, 2, 3 > 0.
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.
Anastasia Yendiki
Choice 2: The b-value
• fjb,g = fj
0 e-bgDjg
• The b-value depends on acquisition parameters:b = 2 G2 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
Anastasia Yendiki
How high b-value?• fj
b,g = fj0 e-bgDjg
• 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
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
Baselineimage
DWimages
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
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
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
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