Biomedical Engineering Reference
In-Depth Information
and another as a control. Thus, the ASL methods suffer from
poor temporal resolution, which is typically on the order of a few
seconds.
The formalism for ASL closely follows the one developed by
Kety for monitoring the kinetics of freely diffusible tracers
(55,
56)
. In ASL, the brain tissue longitudinal magnetization can be
described by the Bloch equations, modified to include the effects
of CBF:
M
b
−
M
b
(
t
)
dM
b
(
t
)
=
+
CBF
·
[
M
a
(
t
)
−
M
v
(
t
)
]
(13.1)
dt
T
1
b
where
M
b
is the brain tissue magnetization per gram of tissue,
M
b
is the equilibrium value of
M
b
,
T
1
b
is the longitudinal brain tissue
relaxation time constant, CBF is the cerebral blood flow expressed
in units of
ml blood
g tissue
s
,and
M
a
and
M
v
are the arterial and venous
blood magnetization per ml of blood, respectively. The above
equation describes brain tissue as a single-compartment that is
constantly receiving blood from the arterial side and losing blood
water on the venous side. Assuming water to be a freely-diffusible
tracer, the venous magnetization equals the brain magnetization
according to:
·
M
b
(
t
)
M
v
(
t
)
=
(13.2)
λ
where
is the brain-blood water partition coefficient, defined
as the ratio of the amount of water per gram of tissue and the
amount of water per ml of blood. In equilibrium, the amount of
water delivered by the arterial vasculature to the tissue compart-
ment must equal the amount of water leaving that compartment
on the venous side:
λ
M
b
λ
M
a
M
v
=
=
(13.3)
Therefore,
Equation (13.1)
can be rewritten as:
M
b
−
M
b
(
t
)
dM
b
(
t
)
CBF
λ
M
b
=
−
·
α (
)
2
t
(13.4)
dt
T
1
app
with the apparent longitudinal relaxation time for tissue water in
thepresenceofperfusion,
T
1
app
, and the degree of labeling effi-
ciency,
α
(
t
), defined as:
1
T
1
app
=
1
T
1
b
+
CBF
λ
M
a
−
M
a
(
t
)
α (
t
)
=
(13.5)
2
M
a
