Biomedical Engineering Reference
In-Depth Information
3.1.3 B
LOCH
-Equation and Static Field Solutions
If, in addition to the external magnetic field, other magnetic fields are superposed,
the magnetization M changes, and due to spin-spin-interaction and spin-lattice-
interaction, the magnetization approaches a relaxed or equilibrium steady state, not
instantaneously, but within a definite time. This empirical process is described by
introducing specific relaxation terms in (
3.11
) containing distinct times to reach
the relaxed state. Equation (
3.11
) is modified accordingly.
The augmented form of (
3.11
), the so-called B
LOCH
-Equation, thus reads (
3.12
).
It was introduced by Felix B
LOCH
in 1946 and is referred to as the B
LOCH
-Equation.
Felix B
LOCH
(*1905-1983) was (together with E. M. P
URCELL
) awarded the N
OBEL
Price in physics in 1952 for his contribution to nuclear magnetic measurements,
which provided the underlying principles of MR-imaging)
M
¼
cM
B
0
1
T
1
ð
M
jj
M
0
Þ
1
T
2
M
?
:
ð
3
:
12
Þ
Dividing the macroscopic net magnetization M into portions parallel and
orthogonal to the vector field B
0
leads to
M
¼
M
?
þ
M
jj
ð
3
:
13
Þ
with the following properties
M
?
B
0
¼
0
M
jj
B
0
¼
0
:
ð
3
:
14
Þ
Thus, (
3.12
) can be written as
M
?
þ
M
jj
¼
cM
?
B
0
1
T
1
ð
M
jj
M
0
Þ
1
T
2
M
?
:
ð
3
:
15
Þ
After separation, and using relation (
3.14
)
2
,(
3.15
) leads to the following
coupled vector-valued linear first-order system of differential equations for both
portions, M
?
and M
jj
;
M
?
¼
cM
?
B
0
1
T
2
M
?
ð
3
:
16
Þ
M
jj
¼
1
T
1
ð
M
jj
M
0
Þ:
To
solve
(
3.16
),
it
is
convenient
to
represent
all
vectors
in
Cartesian
coordinates:
M
?
¼
M
x
e
x
þ
M
y
e
y
;
M
jj
¼
M
z
e
z
;
M
0
¼
M
0
e
z
;
B
0
¼
B
0
e
z
: ð
3
:
17
Þ
Thus, together with (
3.17
), all terms in (
3.16
) read
