Biomedical Engineering Reference
In-Depth Information
called the equilibrium magnetization
M
z
which equals
M
0
. We refer to
M
z
as the longitudinal magnetization.
There is no transverse (
M
x
or
M
y
) magnetization here. It
is possible to change the net magnetization by exposing
the energy of a frequency equal to the energy difference
between the spin states. If enough energy is put into the
system, it is possible to saturate the spin system and
make
M
z
¼
0. The time constant that describes how
M
z
returns to its equilibrium value is called the spin lattice
relaxation time (
T
1
). The equation governing this be-
havior as a function of the time
t
after its displacement is
Table 6.1-2 Natural abundance of isotopes of interest to MRI
Element
Symbol
Natural abundance (%)
1
H
Hydrogen
99.985
2
H
Hydrogen
0.015
13
C
Carbon
1.11
14
N
Nitrogen
99.63
15
N
Nitrogen
0.37
23
Na
Sodium
100
M
z
¼ M
0
ð
1
e
t=T
1
Þ
(6.1.5)
31
P
Phosphorus
100
Therefore
T
1
is defined as the time required to change
the
Z
component of magnetization by a factor of
e.
If the net magnetization is placed along the
Z
axis, it
will gradually return to its equilibrium position along the
þZ
axis at a rate governed by
T
1
. The equation governing
this behavior as a function of the time
t
after its dis-
placement is
39
K
Potassium
93.1
43
Ca
Calcium
0.145
Spin packets
It is cumbersome to describeNMRon amicroscopic scale.
Amicroscope picture is more convenient. The first step in
developing the microscopic picture is to define the spin
packet. A spin packet is a group of spins experiencing the
same magnetic field strength. In this example, the spins
within each grid section represent a spin packet.
At any instant in time, the magnetic field due to the
spins in each spin packet can be represented by a mag-
netization vector. The size of each vector is proportional
to (N
þ
N
). The vector sum of the magnetization
vectors from all of the spin packets is the net magneti-
zation. To describe pulsed NMR, it is necessary to talk in
terms of the net magnetization. Adapting the conven-
tional NMR coordinate system, the external magnetic
field and the net magnetization vector at equilibrium are
both along the
z
axis.
M
z
¼ M
0
ð
1
2
e
t=T
1
Þ
(6.1.6)
The spin-lattice relaxation time (
T
1
) is the time to
reduce the difference between the longitudinal magne-
tization (
M
z
) and its equilibrium value by a factor of
e.
If the net magnetization is placed in the
XY
plane it
will rotate about the
Z
axis at a frequency equal to the
frequency of the photon that would cause a transition
between the two energy levels of the spin. This fre-
quency is called the Larmor frequency.
T
2
processes
In addition to the rotation, the net magnetization starts
to dephase because each of the spin packets making it up
experiences a slightly different magnetic field and rotates
at its own Larmor frequency. The longer the elapsed
time, the greater the phase difference. Here the net
magnetization vector is initially along
þY
. For this and all
dephasing examples you can think of this vector as the
overlap of several thinner vectors from the individual
spin packets.
The time constant that describes the return to equi-
librium of the transverse magnetization
M
xy
is called the
spin-spin relaxation time
T
2
.
T
1
processes
At equilibrium, the net magnetization vector lies along
the direction of the applied magnetic field
B
0
and is
Table 6.1-3 Biological elements of interest to MRI
Element
Biological abundance
Hydrogen (H)
0.63
Sodium (Na)
0.00041
M
xy
¼ M
xy
0
e
t=T
2
(6.1.7)
Phosphorus (P)
0.0024
T
2
is always less than or equal to
T
1
. The net magne-
tization in the
XY
plane goes to zero and then the longi-
tudinal magnetization grows in until we have
M
0
along
the
z
axis.
Any transverse magnetization behaves the same way.
The transverse component rotates about the direction of
Carbon (C)
0.094
Oxygen (O)
0.26
Calcium (Ca)
0.0022
Nitrogen (N)
0.015
