In
DTI 101 we described how diffusion tensor imaging estimates brain microstructure by modeling diffusion of water in the brain. We briefly discuss that this is done by using ellipsoid or ball shaped tensors. This leads to the question; what is a tensor?
"Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors." - University of Cambridge
In the scanner the diffusion tensor is measured by imaging the diffusion in individual gradient directions, like in the image below. If you are interested a more in depth mathematical explanation of this principle is provided in our post
here.
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| Example work out for 6 gradient directions, from diffusion image to tensor |
Once the water diffusion is measured there are a number of ways you can quantify the shape of the tensors in each voxel. In DTI there are 4 measures that are most commonly used; fractional anisotropy, mean diffusivity, axial diffusivity and radial diffusivity. These measure directly relate to the value of the three main eigenvalues of the tensor, indicated in the figure below with lambda 1, lambda 2 and lambda 3. What is an eigenvalue of a tensor? It is the value of the displacement/diffusion for each specific vector. Or, as defined by wikipedia:
"If a two-dimensional space is visualized as a rubber sheet, a linear map with two eigenvectors and associated eigenvalues λ1 and λ2 may be envisioned as stretching/compressing the sheet simultaneously along the two directions of the eigenvectors with the factors given by the eigenvalues." - Wikipedia
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| Example of the influence of the transformation (or in our case diffusion) on the eigenvalues |
There are roughly two archetypical examples of diffusion tensors. On the left you see the example where the tensor has roughly equal eigenvalues for each main vector, thus showing an isotropic diffusion profile. On the right an example of diffusion primarily in one direction, thus demonstrating an anisotropic diffusion profile.
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| Examples of tensor shapes |
Characterization of each eigenvalue in the tensor and combinations of them help constitute the main diffusion measures. Specifically, in axial diffusivity (AD) we only quantify the value of lambda 1. While in radial diffusivity (RD) we take the average of lambda 2 and lambda 3. Mean diffusivity (MD) provides an average of all three, lambda 1, lambda 2, and lambda 3, this measure is sometimes also referred to as trace (TR), which is the sum of lambda 1, lambda 2 and lambda 3. And finally fractional anisotropy (FA) provides the relative difference between the largest eigenvalue as compared to the others; it quantifies the fraction of diffusion that is anisotropic. The exact mathematical relations are shown below:
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Relation of diffusivity measures to the eigenvalues of the tensor
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When you plot the value of each a diffusivity measure in every voxel you can get scalar maps of the quantification of the local tensors. In the image below you see representative examples of each measure. When you look at the MD map it becomes apparent that the mean diffusivity measure is specifically sensitive to cerebral spinal fluid (CSF), which has high values of average diffusion. In voxels with much CSF mean diffusivity is high, and values are thus bright. While AD is only sensitive to diffusion in the longest eigenvalue. Here you see that highly organized structures like white matter pathways are brights. In addition large open cavities like ventricles have general high levels of diffusion which translates to high lambda 1 values. Then RD represents the two shortest eigenvalues and shows dark values in highly organized and dense structures like white matter pathways, intermediate values in gray matter, and high values in regions with CSF. Finally, FA plots the relative length of lambda 1, compared to lambda 2 and lambda 3. This leads to selective brightness in white matter, but not gray matter or CSF.
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| Examples of each diffusivity measure, FA, MD, AD and RD |
A guest post by Rodrigo Dennis Perea on the computational models for estimating the diffusion tensor from diffusion weighted images:
"When I first started working with DTI, I had a hard time understanding the derivation of the diffusion tensor from what was acquired on the scanner, the diffusion weighted image (DWI). Once I understood this process a little better I decided to create a brief review. I hope this will help you! It might be lacking information or it might be too implicit so please feel free to ask me anything if you need more detailed information."
First, it is worth noting that this example will highlight the computations in a single voxel (e.g. x=76, y=64, z=21). Essentially, a similar approach will be applied to every voxel in the entire MRI scan. The figure below illustrates our example where we have six DWI's, and the sample voxel is denoted by a small color coded square:
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| Diffusion weighted images |
The simplified diffusion matrix will look like this:
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| Diffusion matrix |
Which will translate into this tensor:
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| Tensor model |
Next the Stejskal-Tanner equation is of importance, this equation shows how there will be a reduction in the signal due to the application of a pulse gradient, this change will be proportional to the amount of diffusion:
[1]
It is important to notice that we will solve for tensor D given the other parameters from the MRI acquisition. Sk is the intensity at the "single voxel" when a specific gk gradient direction is applied (e.g. a 1x3 vector for the x,y,z direction). b0 is the no diffusion signal and S0 is the intensity value you get from the no-diffusion signal. So lets begin...
A random MRI-DWI sequence (lets say sample_001) is used as an example. Given the following gradient vectors with their specific intensity values at an specific voxel location in the brain, we will derive equation [1] to solve for the simplest case with only 7 volumes (1 3D volume with no diffusion and 6 3D volumes with pulse gradient directions).
A simple approach: The “H” approach: Six gradient directions (Sk) images “plus” a no-diffusion image (S0)
Given the following gradient direction pulses and intensities with a b-value of 800 s/mm2:
Given equation Stejskal-Tanner equation:
[1]
[2]
[3]
We can solve expand equation [1] in terms of D:
[4]
Where the subscript “i” denotes each gradient direction pulse ("i" in this case will go from 1 to 6). One approach to solve this system of equation is to use matrix algebra. Thus let express D as a six-element column vector, d:
[5]
With a six-element row matrix H containing a large M x 6 matrix, where M is the number of gradient directions (in this case M=6):
[6]
Lastly, let’s define a Y matrix for the left side of equation [4]:
[7]
Thus we can express equation [4] with the following 6 directions as:
[8]
With exactly six directions, there is an exact analytic solution and can be solved using standard methods such as the Cramer’s rule:
[9]
Using these equation, I was able to create functions in matlab. I also compared my accurate results with FSL dtifit, which gave me similar results.
I'll be more than happy to share my matlab functions and explain these in more detailed if needed.
The
development
of
diffusion
magnetic resonance imaging (dMRI)
enabled
the
research
of
white
matter
micro- and macro-structure in
vivo.
DMRI
measures
the
magnitude
and
orientation
of
water
diffusion.
This
is
done
in
multiple
directions
to
calculate
the
three
dimensional
representation
of
the
water
diffusion
profile.
Gray
matter
has
predominantly
isotropic (soccer ball shaped)
water
diffusion, while dense
white
matter
tracks
have
highly
anisotropic
(rugby ball shaped) diffusion
of
water
pointing
in
the
direction
of
the
fiber
bundle.
The
measure
most
commonly
used
to
characterize
directional
diffusion
is
fractional
anisotropy
(FA).
This
measure
gives
a
value
between
0
and
1
to
indicate
the
fraction
of
diffusion
that
is
in
the
longitudinal
direction
compared
to
the
proportion
of
diffusion
in
both
transverse
directions.
Other measures that can be used are axial diffusivity (AD), radial diffusivity (RD) and mean diffusivity (MD).
Voxel-Based Morphometry
There
are
two
main
methods
of
analyzing diffusion
images.
The
first
is
voxel‐based
analyses
(VBA)
,
which
is specifically suited for whole
brain
analysis.
It
is
a
voxel wise
method
to
statistically
compare
local
anisotropy
values
for
the
whole
brain
between
different
subjects. It has to be kept in mind that this method should correct for multiple comparisons.
One way to reduce the number or comparisons is to use an atlas based segmentation methods to selectively investigate white matter areas of interest.
Tract-Based Analysis
The
second
method
is
called
tract‐based
analysis.
It
uses
the
more
anisotropic
tensors
to
form
streamlines
of
tensors
leading
to
estimations
of
white
matter
fiber
tracts.
A
region
of
interest
is
used
as
seed
region
from
where
the
fibers
are
traced.
For
each
tract
mean
FA
values
can be
calculated.
These
values
per
tract
can
be
compared
across
groups
to
investigate
structural
connectivity.
VBA
and
fiber
tractography
are
two
methods
using
a
fairly
different
approach in dMRI.
In
VBA
the
whole
brain
is
investigated,
but
the
method
relies
heavily
on
effective
registration
between
subjects.
When
regions
of
abnormal
FA
values
do
not
map
onto
each
other
correctly
this
will
greatly
reduce
the
likelihood to
find
significant
results.
In
tract‐based
analyses
tracts can be delineated without relying on subject registration. Although specific
a
priori
regions
of
interest
or
specific
tracts
need to be
selected
for
comparison.
