Biomedical Engineering Reference
In-Depth Information
From Eq. (16.57), the excess population can be written as
ð
n
þ
n
Þ=
n
þ
¼
1
n
=
n
þ
¼
1
exp
ðD
E
=
KT
Þ
ð
16
:
58
Þ
Substituting the necessary values,
10
26
10
23
ð
n
þ
n
Þ=
n
þ
¼
1
exp
½
1
:
325
J
=ð
1
:
3805
J
=
K
Þ
300 K
10
6
This is a small number based on the material under examination being entirely of one type.
In reality, certain types of isotopes occur more commonly than others. The ratio of one type of
isotope to the total number available in percent is called the natural isotropic abundance and is
listed in Table 16.2. A third relevant question is how many of these isotopes occur in the human
body? The third factor is sensitivity relative to the hydrogen isotope or the equivalent number of
nuclei in a field, also listed in Table 16.2. Fortunately, hydrogen is plentiful in the body, especially
in fat and water.
¼
3
:
2
EXAMPLE PROBLEM 16.12
Compare the isotopes
1
H,
13
C, and
31
P for imaging. Use Table 16.2.
Solution
Note that
1
H is 99.98 percent naturally abundant.
13
C is not, and occurs at only 1.11 percent, so
this isotope is not suitable. On the other hand,
31
P is abundant but is difficult to detect due to its
low sensitivity. Because the body is 60 percent water by weight, it is not surprising that
1
H, with
its high sensitivity and abundance, is usually used for MR imaging.
16.3.4 Precession
To excite hydrogen dipoles into a number of spin states for imaging, an external mag-
netic field can be applied. The natural inclination for the spin magnetic dipoles to align
along the
z
-axis makes detection by a coil difficult if the coil is placed perpendicular to
the
-axis as in the fourth case from Section 16.3.2. It is necessary to find a means to bring
the dipoles down into the
x
plane so they can be detected by a coil in a manner analogous
to case 4. A force is required to push the dipoles into a precession, a downward spiraling
orbit. This mechanism is similar to the action of gravity on a spinning top, which is initially
vertical and eventually is tilted by the force of gravity into a precessing orbit and finally
into a final horizontal position (Figure 16.33).
At first, the net magnetization vector,
x
-
y
M
0
, is aligned with the static magnetic field along
the
-axis as shown in Figure 16.33a. The application of a time-varying magnetic field, B
1
,
along the
z
x
-axis at the Larmor frequency causes the magnetization to precess at an angle
f
-axis at this frequency, as shown in Figure 16.33b.
To clarify what happens to
M
0
, next, it is worth introducing a reference frame notation to
simplify the description. This frame will have coordinates described by a prime notation,
about the
z
