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Encyclopedia of Physical Science and Technology EN008M-395 June 29, 2001 15:52

Magnetic Resonance in Medicine 967

first principles, what those fields are as a function of time, ena within human tissues. In more general applications
much less their effect on the magnetization. In 1946 Felix the Bloch equations have been found to provide good pre-
Bloch cut the Gordian knot by lumping the total effect dictions for the behavior of liquid or liquidlike samples.
of all these internal processes into two parameters called In solids, however, the Bloch equations require substan-
the relaxation times. The effect of one of the parameters tial modification to give accurate results. The protons
called T 1 , the spin–lattice relaxation time, is to govern the in biological tissues behave, from a magnetic resonance
rate at which M z approaches it equilibrium value M 0 : standpoint, as though they were located in a liquid
environment.
dM z /dt = (M 0 − M z )/T 1 . (5)
In equilibrium, there is no transverse magnetization,
F. Relaxation Times
consequently the internal fields must act to reduce any M x
and M y that may be present. Bloch proposed quantifying The range of values taken by T 1 and T 2 in human tis-
this process by using a second relaxation time T 2 : sues is crucial to determining the practicality of MRI for
human tissues. If an unmagnetized sample (e.g., a human
dM x /dt = −M x /T 2 , dM y /dt = −M y /T 2 . (6)
patient) is placed in an uniform magnetic field, it is ini-
The reason that different relaxation times are needed to tially unmagnetized. The Bloch equations show that the
describe transverse and longitudinal relaxation is that the nuclear magnetization will gradually build up along the
strong external field biases the response of the spin system z direction and approach M 0 asymptotically according to
so strongly that these two magnetization components the exponential expression:
respond differently to the weak, internal fields. Analysis −t /T 1
M z (t) = M 0 (1 − e ). (8)
of the microscopic mechanisms responsible for relaxation
show that T 2 will always be shorter than, or at most equal Thus, M z will achieve 63.2% of its final value (M 0 ) ina
to, T 1 . time equal to T 1 , 86.5% of M 0 in 2T 1 , and so on. If it is
Bloch conjectured that the total motion of the magneti- desired to achieve 99% of the total possible magnetization
zation vector can be described as the superimposed effects it is necessary to wait for a time of 4.6T 1 . If T 1 is too long,
of the gyroscopic motion (driven by the externally applied a prohibitively long period can be required to achieve a
fields B 0 and B 1 ) and the relaxation processes (associated useful magnetization. The fact that Gorter failed to detect
with internally generated magnetic fields). This combina- nuclear magnetism in 1936 may have been the result of
tion leads to the final form for the Bloch equations in the an unfortunate choice of material, which had too long a
ˆ
stationary frame with unit vectors ˆ ı , ˆ , and k: value for T 1 . It should be noted that there are materials that
have T 1 values as long as hours or even days. Fortunately,
dM M x M y M 0 − M z
ˆ
= − ˆ ı − ˆ  + k +γ M ×[B 0 +B 1 (t)]. mobile protons in biological tissues have T 1 values of, at
dt T 2 T 2 T 1 most, a few seconds (Table II).
(7)
The only time a signal can be detected from the nuclear
The Bloch equations give a complete description of spins is when a transverse magnetization is present. This
the behavior of the magnetization within a body. Qualita- can be achieved by using a short burst of radio-frequency
tively, they express relatively simple ideas, the transverse (rf) energy, the B 1 field or rf pulse, at, or very near, the
◦
magnetization M x ˆ ı + M y ˆ  is constantly relaxing toward Larmor frequency. A 90 pulse will rotate a magnetization
zero while precessing rapidly at the Larmor frequency that is initially along the z axis into the transverse plane. If
γB 0 . The longitudinal magnetization M z is constantly re- the B 1 field is then turned off, the transverse magnetization

2
2
laxing toward its equilibrium value M 0 . If B 1 is not zero M t = M + M will precess at the Larmor frequency. Its
y
x
it is constantly rotating M about an axis parallel to B 1 in amplitude will decay according to the relation
the rotating frame. In practice, the solution to the Bloch −t /T 2
M t (t) = M 0 e cos ω 0 t . (9)
equations may be relatively complicated, particularly if
the frequency of B 1 (t) is not exactly equal to the Larmor The electric signal picked up during this time is called the
frequency. free induction decay or FID (Fig. 2). If T 2 is too short,
The relaxation times T 1 and T 2 provide only an em- the signal will decay away so rapidly that no useful in-
pirical treatment of the effects of the internal mag- formation can be extracted from the FID. Note that as
netic fields. In practice, they must be found by ex- soon as B 1 disturbs the longitudinal magnetization from
its equilibrium value M 0 , M z starts to rebuild according
periment rather than by calculation. Once T 1 and T 2
have been determined, experimental results indicate that to Eq. (8). Therefore, it is possible to put the spin sys-
the Bloch equations provide a completely satisfactory tem through a periodic excitation cycle using a series of
description of all nuclear magnetic resonance phenom- 90 pulses. Between pulses the longitudinal magnetization
◦