Digital Signal Processing Reference
In-Depth Information
Note that the phase component of (
2.42
) varies with
z
as
e
j
(
−
γ(
G
z
(
z
−
z
)τ
p
/
2
))
. To nullify
this, negative
z
-gradient
G
z
(along with the existent strong magnetic field
B
0
)is
applied. This is known as Refocussing gradient. Note that the external field (
2.32
)
and (
2.33
) are removed. When the resultant magnetic moment is having non-zero
−
α
and are kept with the strong magnetic field
−
(
G
z
(
z
−¯
z
)
+
B
0
)
for the time duration
τ
p
/
2, rotating transverse component (after
τ
p
/
2) is obtained and is given as follows.
e
j
(
−
γ(
G
z
(
z
−
z
)
+
B
0
)τ
p
/
2
+
φ)
e
j
(
−
γ(
−
G
z
(
z
−
z
)
+
B
0
)τ
p
/
2
)
2
M
0
sin
(α
τ
p
/
2
)
(2.43)
e
j
(
−
γ(
B
0
)τ
p
+
φ)
⇒
2
M
0
sin
(α
τ
p
/
2
)
(2.44)
Thus the resultant phase component is constant throughout the slice (not the function
of
z
). But note that there is still strong magnetic field
B
0
available in the
z
-axis. Hence
transverse magnetic moments along the slice are having the same phase, having
non-zero
value and are under the constant magnetic field
B
0
. Hence transverse
components of the magnetic field along the slice follows the equation as mentioned
below.
α
e
j
(
−
γ(
B
0
)τ
p
+
φ)
e
j
(
−
γ(
B
0
)
t
)
2
M
0
sin
(α
τ
p
/
2
)
(2.45)
τ
p
in the time scale
t
, where
t
In (
2.45
),
t
0 corresponds
to the middle of the sinc pulse applied. As described in the Sect.
2.4
, the resultant
transverse component (refer
2.46
) gradually decreases due to spin-spin interations.
This interactions start at the moment when there is non-zero
=
0 corresponds to
=
α
value. so at time
t
0 (middle of the RF pulse) itself, the transverse components decreses with time
constant
T
2
. If there is no refocussing gradient and other externally disturbing fields,
the disturbance in the transverse component is only due to spin-spin interaction,
which is completely described by the time constant
T
2
. For instance, after applying
refocussing gradient, the transverse component of the magnetic moment (including
the effect of spin-spin relaxation) is given as follows.
=
−
τ
p
T
2
(
−
t
T
2
(
e
j
(
−
γ(
B
0
)τ
p
+
φ)
e
j
(
−
γ(
B
0
)
t
)
e
(α
τ
p
/
2
)
2
M
0
sin
x
,
y
)
e
x
,
y
)
(2.46)
Note that
T
2
is the function of
due to different physical characteristics of the
tissues. The transverse component of the signal is sampled at some time instant
T
R
(read-out time instant) depends on the factor
T
2
(
(
x
,
y
)
x
,
y
)
and helps for
T
2
-MRI imaging
technique.
2.5.1 Summary of the Section
2.5
1. Apply positive
z
-gradient
G
z
to select the slice of the human body along the
z
-axis.
2. RF pulse
A
t
τ
Δ
v
γ
sinc
(Δ
vt
)
rect
(
p
)
is applied (Note that the duration is between
−
τ
p
/
τ
p
/
α
2 and
2) to obtain the identical
throughout the slice.
Search WWH ::
Custom Search