Image Processing Reference
In-Depth Information
which is equivalent to
Ct
Ct
()
()
...
()
Ct
Ct
(
)
0
...
0
0
Rt
Rt
()
()
...
()
VOI
1
AIF
1
1
(
)
Ct
(
)
...
VOI
2
AIF
2
AIF
1
2
=
CBF
⋅⋅
∆
t
⋅
⋅
(19.30)
...
..
.
...
...
Ct
Ct
(
)
Ct
(
)
...
Ct
(
)
Rt
N
VOI
N
AIF
N
AIF
N
−
1
AIF
1
In compact form:
C
=
CBF
⋅
∆
t
⋅
C
⋅
R
(19.31)
VOI
AIF
where
C
VOI
∈ℜ
Nx
1
,
C
AIF
∈ℜ
NxN
,
R
∈ℜ
Nx
1
, and
∆
t is the length of the equally
t
i−1
). Equation 19.32 is a standard matrix equation
that can be inverted to yield CBF·
R
if det(
C
AIF
) 0:
spaced sampling times (
∆
t
=
t
i
−
CBF
⋅
∆
t
⋅ = ⋅
RC C
−
1
(19.32)
AIF
VOI
This approach has been termed
raw deconvolution
in the literature [31]. Albeit
appealing in its simplicity, it is known to perform poorly being extremely sensitive
to noise. A widely used approach to solve Equation 19.18 that overcomes the
limitations of the raw deconvolution is singular value decomposition (SVD). This
was introduced as method to estimate R(t) by Ostergaard et al. [13,16]. The SVD
constructs matrices
V
,
W
, and
U
T
so that the inverse of C
AIF
can be written as
CVWU
AIF
−
1
=⋅
⋅
T
(19.33)
where
W
is a diagonal matrix, and
V
and
U
T
are orthogonal and transpose orthog-
onal matrices, respectively. Given this inverse matrix, CBF· R is found simply as
⋅ =
⋅
VWU C
∆
⋅
T
⋅
VOI
CBF
R
(19.34)
t
SVD has been shown to be a reliable technique for deconvolution because it
reduces the effect of noise on R(t) estimation. This is achieved by setting to zero
the elements in the diagonal matrix
W
obtained by SVD when they are smaller
than a threshold value given beforehand.
SVD represents the most used approach to quantify bolus-tracking MRI
data. However, in the last years its limitations have been pointed out [32-37].
In particular, it has been shown that CBF values obtained by SVD largely
depend on the threshold value selected to eliminate diagonal elements in
W
[32,33,35]. In addition, SVD introduces undesiderated oscillations and neg-
ative values in the reconstructed CBF · R(t), producing a nonphysiological
R(t). This is far from ideal because there are situations in which the actual
shape of the residue function, not just its maximum value, is of interest, i.e.,
when there is presence of bolus delay and dispersion and only an accurate
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