Image Processing Reference
In-Depth Information
1.6.3
T
HE
B
LOCH
E
QUATIONS
From the preceding description of the observed behavior of
M
(
t
) the following
equation can be postulated for the complete description of this motion:
dt
dt
M
()
=
γ
MB
()
t
×
()
t
−
Rt
{ ()
MM
−
}
(1.22)
0
where
B
(
t
) is composed of the static field and the RF field i.e.,
B
(
t
)
=
B
0
+
B
1
(
t
)
and R is the relaxation matrix:
&
12 0 0
0120
0
/
T
)
(
(
(
+
+
+
R
=
/
T
(1.23)
0
1
/
T
1
'
*
and the vector
M
0
[0, 0, M
0
].
This is the set of equations used when constructing models of MRI. For most
applications, these equations are transformed into the rotating frame of reference.
=
1.6.3.1
Rotating Frame of Reference
Usually, MR experiments involve the application of a sequence of RF pulses.
Viewed from fixed coordinates (stationary frame), a description of the motion of
the magnetization results can be complicated and difficult to visualize, especially
when two or more RF pulses are applied. Hence, we consider the motion of the
magnetization from the point of view of an observer rotating about an axis parallel
with
B
0
, in synchronism with the precessing nuclear magnetic moments: this is
the so-called
rotating frame of reference
.
A rotating frame is a coordinate system whose transverse plane is rotating
clockwise at an angular frequency
ω
. To distinguish it from the conventional sta-
tionary frame, we use x
′
, y
′
, and z
′
to denote the three orthogonal axes of this frame,
and correspondingly,
i
as their unit directional vectors. Mathematically,
this frame is related to the stationary frame by the following transformations:
′
,
j
′
, and
k
′
"
"
i
′ =
cos
()
ω
t
i
−
sin
()
ω
t
j
j
′ =
sin
()
ω
t
i
+
cos
()
ω
t
j
(1.24)
k
′ =
k
1.7
MULTIPLE RF PULSES
Up to now, we have described single-RF-pulse effects and subsequent detection
of FID (one-pulse experiments). Now we consider the effects of multiple pulses
on the magnetization.
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