Biomedical Engineering Reference
In-Depth Information
Fig. 6.5
Diffusion weighted images of the brain acquired along different gradient encoding
directions showing different contrasts
gradients in addition to the pure diffusion encoding gradients [
5
,
6
]. They formulated
the PGSE echo signal to be:
S
=
S
0
exp(
−tr
(
bD
))
,
(6.18)
where
tr
(
A
)
represents the trace of the matrix
A
. This simplifies to Stejskal's
formulation (Eq.
6.13
)
S
=
S
0
exp
−b
g
T
Dg
in the absence of the imaging
gradients, or under the consideration that the imaging gradients are small compared
to the diffusion encoding gradients, which is mostly true. Otherwise, the
b
-matrix
has to be computed from the dynamics of the imaging and the diffusion encoding
gradients.
DTI Estimation:
D
is a covariance tensor, therefore, it is symmetric and positive
definite. This implies that there are six unknowns to be estimated from the
DTI signal in Eq. (
6.18
). Therefore, at least six DWIs, acquired along linearly
independent and non-coplanar gradient directions, and a non diffusion weighted
or Hahn spin echo (
S
0
) image is required to measure the six unknown coefficients
of
D
. The linearized version of Eq. (
6.18
) provides the simplest scheme for doing
this [
5
,
6
]:
S
S
0
ln
=
−
b
ij
D
ij
.
(6.19)
In practice, often more than six DWIs are used to account for acquisition noise. In
the case of
N
DWIs, the linearized equation for the signal is written in a matrix form:
⎡
⎤
⎡
⎤
⎡
⎤
D
11
D
12
D
13
D
22
D
23
D
33
b
11
2
b
12
2
b
13
b
22
2
b
23
b
33
−
ln(
S
1
/S
0
)
⎣
⎦
⎣
⎦
⎣
⎦
b
11
2
b
12
2
b
13
b
22
2
b
23
b
33
−
ln(
S
2
/S
0
)
b
11
2
b
12
2
b
13
b
22
2
b
23
b
33
−
ln(
S
3
/S
0
)
.
,
=
(6.20)
.
.
.
.
.
.
b
11
2
b
12
2
b
13
b
22
2
b
23
b
33
−
ln(
S
N
/S
0
)
X
=
Bd
.
(6.21)
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