Biomedical Engineering Reference
In-Depth Information
x 0 -axis and the corresponding component of
If the direction of the area is taken to be the
M
is used in Faraday's law, Eq. (16.51), for the cubic volume,
V voxel , of an image voxel
2
4
3
5
B 0 1 ð t Þ M xy
v ð t Þ¼ V voxel @
@ t
I
ð
16
:
69a
Þ
2
3
5 ¼ V voxel @
@ t
2
3
M xy
I
M xy
I
v ð t Þ¼ V voxel @
@ t
4
4
5
B 0 1 ð t Þ
B 0 1 ð t Þ
sin
y
0 1
0 1 and M xy
and Eq. (16.60f) for B
, and since the angle between B
From Eq. (16.67) for
M xy
ð t Þ
90 , the voltage becomes
is
y ¼
v ð t Þ¼ B
1
0
xy e j t j= T 2
e j o t
j o V
M
ð
16
:
69b
Þ
oxel
2
I
where
are
dropped. This detected voltage is called the free induction decay (FID), the real part of
which is shown in Figure 16.39.
The major reason the FID waveform is unsuitable for detection is that it is severely
affected by variations in the magnetic field that dominate the response over the signals of
the desired spin states. Also, the waveform weakens rapidly with the shorter time decay
constants. As shown in Figure 16.38, spin states experience different magnetic fields
because of their spatial location; therefore, the angular frequencies of their magnetization
vectors vary, some moving faster than others, according to Eq. (16.56).
A clever alternative is called the spin echo method. If a certain time,
V oxel is the cubic volume of an image voxel and the second-order terms in
dM
/
dt
T e /2, elapses so that
the spin vectors are severely but not totally dispersed, as shown in Figure 16.35c, a second
p
pulse with an amplitude twice that of the original
p
/2 pulse rotates the leading vectors back
to the
-axis, as shown in Figure 16.40a. The individual magnetization vectors rotate in an
opposite direction toward realignment. As the vectors draw into coherent phase, the
detected signal increases to an alignment peak at
þ y
and then decays, as shown in the spin
echo waveform in Figure 16.40b. The beauty of this approach is that the effects of magnetic
field variations are canceled out, resulting in a truer
T e
2 tissue response. Following a deriva-
tion similar to that of the FID waveform, we can obtain an expression for the spin echo:
T
0
xy e j t = T 2 j e j w t
v ð t Þ¼ jB
ð
=
2
I
Þo V
M
ð
16
:
70
Þ
1
oxel
Real(V)
t
FIGURE 16.39 Free induction decay waveform versus time. This waveform starts at
t ¼
0.
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