Biomedical Engineering Reference
In-Depth Information
high temporal resolution, or in measurement of vascular transit-
time maps associated with diagnosis of cerebrovascular diseases, it
would be essential to determine CBF dynamically. One efficient
way to accomplish dynamic measurements of CBF with ASL is to
impose a systematic and periodic variation of the degree of label-
ing
α
(
t
). According to
Equation (13.4)
, periodic variations in
α
(
t
) would cause periodic variations in the tissue magnetization,
and a temporal analysis of the tissue response would yield dynamic
quantification of
T
1
b
,
and CBF. This transient analysis has been
named dynamic ASL (DASL)
(53, 65)
.
The dynamic analysis can be introduced by simply allowing
the degree of labeling
τ
(
t
)in
Equation (13.5)
to be a time-
dependent, periodic function. In such case,
Equation (13.5)
can
be integrated characterizing the time evolution after steady-state
has been reached of a system subject to a specific time-varying
labeling function, and is given by
(65)
:
α
e
−
(
t
−
τ )
T
1
app
CBF
λ
M
b
(
t
)
M
b
e
−
τ
/
T
1
a
M
b
(
t
)
=
−
2
α
⊗
(13.8)
where
denotes the convolution product. From
Equation
(13.8)
, it is important to note that independent of the labeling
function, the tissue response has a time shift introduced by the
transit time
⊗
.
In theory, the labeling function may be arbitrarily defined as
long as the arterial spins can be inverted accordingly, so that defi-
nition of the labeling function should be determined by the exper-
imental goals. For example, the use of a periodic square labeling
function of period 2
τ
defined as follows:
e
−
τ/
T
1
a
α
0
·
,
0
<
t
≤
α
(
t
)
=
(13.9)
0
,
<
t
≤
2
will
produce
a
periodic
tissue
magnetization
response,
given by
(65)
:
1
α
0
e
−
τ
/
T
1
a
CBF
λ
M
b
M
b
(
τ<
t
≤
+
τ
)
=
−
2
T
1
app
1
e
−
T
1
app
e
−
(
t
−
τ )
T
1
app
e
T
1
app
⎛
e
−
(
t
−
τ )
T
1
app
⎞
⎤
1
−
⎝
⎠
⎦
×
+
−
e
−
T
1
app
−
1
2
T
1
app
α
0
e
−
τ
/
T
1
a
CBF
λ
M
b
M
b
(
+
τ<
t
≤
2
+
τ
)
=
−
1
e
−
T
1
app
e
−
T
1
app
e
T
1
app
⎛
⎛
e
−
T
1
app
⎞
⎠
e
−
(
t
−
−
τ )
T
1
app
⎞
⎤
−
1
⎝
⎝
⎠
⎦
×
+
−
e
−
T
1
app
−
(13.10)
