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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