Biomedical Engineering Reference
In-Depth Information
In the presence of a radio frequency field (
3.26
), to rotate the magnetic moment
vectors, the effective field (
3.28
)
2
, must be extended to
B
eff
:
¼
B
0
þ
1
c
X
þ
B
1
ð
3
:
29
Þ
where B
0
þ
c
X is referred to as the external field B
ext
:
Using the L
AMOR
-Frequency (
3.9
), of the magnetic dipole moment, existent
when excited by a single magnetic field B
0
;
as well as the angular velocity of the
clockwise rotating reference frame X
¼
xe
u
e
r
¼
xe
z
and the precessional
angular frequency x
1
¼
cB
1
(
3.26
) about the rf-field axis as a result of the
B
1
-field, the effective magnetic field defined in (
3.29
) can be written as
B
eff
¼
1
c
ð
x
L
x
Þ
e
z
þ
x
1
e
r
:
ð
3
:
30
Þ
c
Equation (
3.28
)
1
, thus yields
d
G
l
dt
¼
l
½ð
x
L
x
Þ
e
z
þ
x
1
e
r
:
ð
3
:
31
Þ
If the radial frequency of irradiation x
;
i.e. of the rotating reference frame
matches the precessional frequency of the magnetic dipole moments x
L
the first
term in brackets of (
3.31
) vanishes (i.e. the external field B
ext
vanishes from the
rotating frame of reference) and there is only a precession about the rotating
e
r
-axis with the precessional frequency x
1
:
In terms of frequency of oscillation of
the circular polarized field B
1
;
the B
1
-field is thus most efficiently synchronized
to rotate the magnetic dipole moments if its frequency x is equivalent to
x
L
(resonance condition).
The B
LOCH
-Equations derived in
Sect. 3.1.3
are valid for a static external
magnetic field parallel to the z-direction. In case of a combined static field and an
rf-field, the resulting motion of the magnetic dipole vectors and magnetization,
respectively, derives as follows.
With respect to the rotating frame, the (effective) magnetic field is given by
(
3.30
). The projection of the (effective) magnetic field vector B
eff
onto the rotating
frame axis yields the magnitude B
1
along the r-axis, zero along the u-axis and B
ext
or
ð
x
L
x
Þ=
c along the z-axis. The projections of the net magnetization vector
M onto the rotating frame axis are given by M
r
;
M
u
;
and M
z
:
M
jj
¼
M
z
e
z
and
M
?
¼
M
r
e
r
þ
M
u
e
u
:
ð
3
:
32
Þ
Expanding the B
LOCH
-Equation (
3.12
), to the effective magnetic field with
respect to the rotating reference frame one obtains regarding (
3.9
), (
3.17
), (
3.30
)
and (
3.32
)
