Biomedical Engineering Reference
In-Depth Information
M
b
)
T
1
b
(
M
b
−
dM
b
dt
=−
−
k
−
1
M
b
−
k
2
M
b
+
k
1
M
a
+
k
−
2
M
c
(15.12b)
M
c
)
T
1
c
(
M
c
−
dM
c
dt
=−
−
k
−
2
M
c
+
k
2
M
b
(15.12c)
where
M
a
,
M
b
and
M
c
are the magnetizations of PCr,
-ATP and
Pi;
M
a
,
M
b
and
M
c
are the magnetizations at Boltzmann ther-
mal equilibrium;
k
1
and
k
−
2
are the pseudo first-order forward
rate constants involving ATP production through the CK reac-
tion and the ATP
ase
reaction, respectively;
k
−
1
and
k
2
are the
pseudo first-order reverse rate constants involving ATP utilization
through the CK reaction and the ATP
ase
reaction, respectively;
T
1
a
,
T
1
b
and
T
1
c
are the intrinsic spin-lattice relaxation times of
PCr,
γ
γ
-ATP and Pi, which are B
0
dependent. Four rate constants
involving the
PCr
Pi
exchange and their four associated
ATP metabolic fluxes can be determined by the following three-
step measurements with frequency-selective RF saturation on the
γ
↔
ATP
↔
-ATP (Step 1), Pi (Step 2) and PCr (Step 3) resonance peak,
respectively.
Step 1: Progressive saturation of
-ATP for determining
intrinsic spin-lattice relaxation times of PCr and Pi, forward
rate constants and fluxes
The first step of the
31
PMTmeasurementsistoapplya
frequency-selective RF pulse train for completely saturating the
γ
γ
-ATP resonance peak with varied saturation time (
t
)(i.e.,the
progressive saturation experiment commonly used in the CST
approach). For this case, the three-spin chemical exchange system
of
PCr
Pi
can be treated as two independent two-spin
chemical exchange systems (i.e.,
PCr
↔
ATP
↔
Pi
). Solv-
ing
Eqs. (15.12a)
and
(15.12c)
with the boundary condition of
M
b
=
↔
ATP
and
ATP
↔
0 results in
Eqs. (15.13a)
and
(15.13b)
(47)
,
M
a
k
1
α
a
e
−
α
a
t
1
α
a
T
1
a
M
a
(
t
)
=
+
(15.13a)
(
k
−
2
α
c
1
α
c
T
1
c
M
c
)
e
−
α
c
t
M
c
(
t
)
=
+
(15.13b)
with
k
1
;
k
−
2
+
.
1
T
app
1
a
1
T
1
a
1
T
app
1
c
1
T
1
c
α
=
=
+
α
=
=
a
c
Therefore, the parameters of apparent spin-lattice relaxation rates
(
α
a
and
α
c
) and intrinsic spin-lattice relaxation times (
T
1
a
and
