Diffusion model fitting and
tractography: A primer
Anastasia Yendiki
HMS/MGH/MIT Athinoula A. Martinos Center for
Biomedical Imaging
05/19/11
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White-matter imaging
•
•
•
•
From the National Institute on Aging
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Axons measure ~m
in width
They group together
in bundles that
traverse the white
matter
We cannot image
individual axons but
we can image
bundles with
diffusion MRI
Useful in studying
neurodegenerative
diseases, stroke,
aging,
development…
From Gray's Anatomy: IX. Neurology
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Diffusion in brain tissue
• Differentiate tissues based on the diffusion (random motion)
of water molecules within them
• Gray matter: Diffusion is unrestricted  isotropic
• White matter: Diffusion is restricted  anisotropic
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How to describe diffusion
• At every voxel we want to know:
 Is this in white matter?
 If yes, what pathway(s) is it part of?
 What is the orientation of diffusion?
 What is the magnitude of diffusion?
• A grayscale image cannot capture all this!
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Diffusion MRI
• Magnetic resonance imaging can
provide “diffusion encoding”
Diffusion
encoding in
direction g1
g2
g3
• Magnetic field strength is varied
by gradients in different
directions
• Image intensity is attenuated
depending on water diffusion in
each direction
g4
g5
• Compare with baseline images to
infer on diffusion process
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Why’n’how | Diffusion model fitting and tractography
g6
No
diffusion
encoding
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Need to know: Gradient directions
• True diffusion direction || Applied gradient direction
 Maximum attenuation
Diffusion-encoding gradient g
Diffusion detected
• True diffusion direction  Applied gradient direction
 No attenuation
Diffusion-encoding gradient g
Diffusion not detected
• To capture all diffusion directions well, gradient directions
should cover 3D space uniformly
Diffusion-encoding gradient g
Diffusion partly detected
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How many directions?
• Acquiring data with more gradient directions leads to:
+ More reliable estimation of diffusion measures
– Increased imaging time  Subject discomfort, more
susceptible to artifacts due to motion, respiration, etc.
• DTI:
– Six directions is the minimum
– Usually a few 10’s of directions
– Diminishing returns after a certain number [Jones, 2004]
• HARDI/DSI:
– Usually a few 100’s of directions
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Need to know: b-value
• 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

acquisition
G

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How high b-value?
• 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
diffusion measures in practice
• DTI: b ~ 1000 sec/mm2
• HARDI/DSI: b ~ 10,000 sec/mm2
• Data can be acquired at multiple b-values for trade-off
• Repeat acquisition and average to increase signal-to-noise ratio
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Looking at the data
A diffusion data set consists of:
• A set of non-diffusion-weighted a.k.a “baseline” a.k.a. “low-b”
images (b-value = 0)
• A set of diffusion-weighted (DW) images acquired with different
gradient directions g1, g2, … and b-value >0
• The diffusion-weighted images have lower intensity values
b=0
b1 , g 1
b2 , g 2
b3 , g 3
Baseline
image
Diffusionweighted
images
b4 , g 4
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b5, g5
b6 , g 6
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Data analysis steps
• Pre-process images
– FSL: eddy_correct, rotate_bvecs
• Fit a diffusion model at every voxel
– DTK: DSI, Q-ball, or DTI
– FSL: Ball-and-stick (bedpost) or DTI (dtifit)
• Compute measures of anisotropy/diffusivity
and compare them between populations
– Voxel-based, ROI-based, or tract-based statistical
analysis
• For tract-based: Reconstruct pathways
– DTK: Deterministic tractography using DSI, Qball, or DTI model
– FSL: Probabilistic tractography (probtrack) using
ball-and-stick model
DTK: www.trackvis.org, FSL: www.fmrib.ox.ac.uk/fsl
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Models of diffusion
Model
Software
Diffusion spectrum:
DTK (DSI option)
Full distribution of orientation
and magnitude
Orientation distribution
function (ODF):
DTK (Q-ball option)
No magnitude info
Ball-and-stick:
FSL (bedpost)
Orientation and magnitude for up
to N anisotropic compartments
(default N=2)
Tensor:
Single orientation and magnitude
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DTK (DTI option)
FSL (dtifit)
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A bit more about the tensor
• A tensor can be thought of as an ellipsoid
• It can be defined fully by:
– 3 eigenvectors e1, e2, e3 (orientations of ellipsoid axes)
– 3 eigenvalues 1 , 2, 3 (lengths of ellipsoid axes)
2 e2
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1 e1
3 e3
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Tensor: Physical interpretation
• Eigenvectors express diffusion direction
• Eigenvalues express diffusion magnitude
Isotropic diffusion:
1   2  3
1 e1
2 e2
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Anisotropic diffusion:
1 >> 2  3
2 e2
1 e1
3 e3
 3 e3
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Tensor: Summary measures
Faster
diffusion
Slower
diffusion
Anisotropic
diffusion
Isotropic
diffusion
FA(j)2 =
MD(j) = [1(j)+2(j)+3(j)]/3
• Fractional anisotropy (FA):
Variance of the 3 eigenvalues,
normalized so that 0 (FA) 1
3 [1(j)-MD(j)]2 + [2(j)-MD(j)]2 + [3(j)-MD(j)]2
2
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• Mean diffusivity (MD):
Mean of the 3 eigenvalues
1(j)2 + 2(j)2 + 3(j)2
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Tensor: More summary measures
• Axial diffusivity: Greatest eigenvalue
AD(j) = 1(j)
• Radial diffusivity: Average of 2 lesser eigenvalues
RD(j) = [2(j) + 3(j)]/2
• Inter-voxel coherence: Average angle b/w the major eigenvector
at some voxel and the major eigenvector at the voxels around it
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Tensor: Visualization
Image:
Tensor map:
An intensity value at each
voxel
A tensor at each voxel
Direction of eigenvector corresponding
to greatest eigenvalue
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Tensor: Visualization
Image:
Tensor map:
An intensity value at each
voxel
A tensor at each voxel
Direction of eigenvector corresponding
to greatest eigenvalue
Red: L-R, Green: A-P, Blue: I-S
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Tractography
• Use local diffusion orientation at each
voxel to determine pathway between
distant brain regions
?
• Local orientation comes from diffusion
model fit (tensor, ball-and-stick, etc.)
• Deterministic vs. probabilistic tractography:
– Deterministic assumes a single orientation at each voxel
– Probabilistic assumes a distribution of orientations
• Local vs. global tractography:
– Local fits the pathway to the data one step at a time
– Global fits the entire pathway at once
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Deterministic vs. probabilistic
• Deterministic methods give you an estimate of model
parameters
5
• Probabilistic methods give you the uncertainty (probability
distribution) of the estimate
5
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Deterministic vs. probabilistic
Sample 1
Sample 2
…
Deterministic tractography:
Probabilistic tractography:
One streamline per seed voxel
Multiple streamline samples per
seed voxel (drawn from probability
distribution)
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Deterministic vs. probabilistic
Deterministic tractography:
Probabilistic tractography:
One streamline per seed voxel
A probability distribution
(sum of all streamline samples from
all seed voxels)
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Local vs. global





Local tractography:
Fits pathway step-by-step, using
local diffusion orientation at
each step
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Global tractography:
Fits the entire pathway, using
diffusion orientation at all
voxels along pathway length
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Local tractography
• Best suited for exploratory
study of connections
• All connections from a seed
region, not constrained to a
specific target region
• How do we isolate a specific
white-matter pathway?
– Thresholding?
– Intermediate masks?
• Non-dominant connections
are hard to reconstruct
• Results are not symmetric between “seed” and “target” regions
• Sensitive to areas of high local uncertainty in orientation (e.g.,
pathaway crossings), errors propagate from those areas
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Global tractography





• Best suited for reconstruction
of known white-matter
pathways
• Constrained to connection of
two specific end regions
• Not sensitive to areas of high
local uncertainty in
orientation, integrates over
entire pathway
• Symmetric between “seed”
and “target” regions
• Need to search through a large solution space of all possible
connections between two regions:
– Computationally expensive
– Sensitive to initialization
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TRACULA
• TRActs Constrained by UnderLying Anatomy
• Automatic reconstruction of probabilistic
distributions of 18 major white-matter pathways
• No manual labeling of ROIs needed
• Use prior information on pathway anatomy
from training data:
– Manually labeled pathways in training
subjects
– FreeSurfer segmentations of same subjects
– Learn neighboring anatomical labels along
pathway
• Beta version available in FreeSurfer 5.1
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