Image Processing Reference
In-Depth Information
This is due to conservation of mass. Intuitively, it implies that, for example,
fluid flow outward from a point (divergence) should decrease the concentration
at that point while increasing the concentration at neighboring points. In the
steady state, the concentration does not change; consequently the steady-state
flux vector field is divergence free.
In diffusion MRI, magnetic field gradients are employed to sensitize the image
to diffusion in a particular direction. In each resulting diffusion-weighted image,
signal is lost wherever molecules diffuse in the direction of interest during
imaging. By repeating the process of diffusion weighting in multiple directions,
at each voxel a 3-D diffusion pattern can be estimated, which will reflect the
shape of the underlying anatomy.
In DT-MRI, the diffusion tensor field is calculated from a set of diffusion-
weighted images by solving the Stejskal-Tanner equation (Equation 13.4) [1].
This equation describes how the signal intensity at each voxel decreases in the
presence of diffusion:
T
ˆ
ˆ
SS
k
=
0
e
−
b g D g
k
(13.4)
k
Here,
S
is the image intensity at the voxel (measured with no diffusion gradient),
0
and
th diffusion-sensitizing
gradient. is a unit vector representing the direction of this diffusion-sensitizing
magnetic field gradient.
S
is the intensity measured after the application of the
k
k
ˆ
k
T
ˆ
ˆ
D
is the diffusion tensor, and the product
g Dg
represents
k
k
ˆ
the diffusion coefficient in direction
is LeBihan's factor describing
the pulse sequence, gradient strength, and physical constants. For rectangular gra-
dient pulses, the b-factor is defined by
. In addition,
b
k
b
=
γδ
22
(
∆ −
δ
)
|
g
|
2
,where
γ
is the proton
3
gyromagnetic ratio (42 MHz/Tesla),
|
g
|
the strength of the diffusion-sensitizing
gradient pulses,
δ
the duration of the diffusion gradient pulses, and
∆
the time
between diffusion gradient RF pulses [24]. For more information on the tensor
calculation process, see for example
Reference 24
.
13.3
ANISOTROPY AND MACROSTRUCTURAL
MEASURES
The geometric nature of the measured diffusion tensor within a voxel is a mean-
ingful measure of fiber tract organization. Factors affecting the shape of the
apparent diffusion tensor (shape of the diffusion ellipsoid) in the white matter
include the density of fibers, the degree of myelination, the average fiber diameter,
and the directional similarity of the fibers in the voxel. In addition, because MRI
methods obtain a macroscopic measure of a microscopic quantity (which neces-
sarily entails intravoxel averaging), the voxel dimensions influence the measured
diffusion tensor at any given location in the brain.
The advent of robust diffusion tensor imaging techniques has prompted the
development of quantitative measures for describing diffusion anisotropy. However,
to relate the measure of diffusion anisotropy to the structural geometry of the tissue,
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