Chemistry Reference
In-Depth Information
so that,
Dγ
2
t
0
t
G
(
t
)
dt
t
0
G
(
t
)
dt
dt
ln
S
(
t
)
=−
+
ln
S
(0)
(59)
0
This expression can be used to determine the effect on the signal intensity of
specific experimental design parameters for a gradient or spin echo experiments.
Thus Torrey [28] obtained the well-known Carr-Purcell [29] result (originally
obtained by them by considering the diffusion limit [13, 17, 18] of the discrete
time random walk of the phase) for the dephasing following the application of a
step gradient of magnitude
G
in a liquid characterized by a diffusion coefficient
D
, namely,
exp
3
Dγ
2
G
2
t
3
S
(
t
)
S
(0)
=
1
e
i
=
−
(60)
It follows that the microscopic diffusion coefficient of a particle in a fluid can
then be measured via the amplitude of the echo signal from nuclear spins subject
to an appropriate sequence of magnetic field pulses. We recall that the diffusion
coefficient is defined by the mean-square displacement of the Brownian particle
x
coordinate (ignoring the inertia), in an elapsed time interval
t
between observations,
x
)
2
x
(0)]
2
(
=
[
x
(
t
)
−
=
2
Dt
In order to further utilize Eqs. (54) or (59) we must first discuss diffusion sensitive
pulse sequences.
E.
Diffusion Sensitive Pulse Sequences
Magnetic field gradients are employed not only to spatially encode but also to
produce specific contrast within the MRI signal. In order to sensitize an imaging
sequence to molecular diffusion, a diffusion-weighting gradient must be included.
Diffusion weighting gradients can be incorporated in many pulse sequence, how-
ever, spin-echo sequences (Fig. 3a) are traditionally favored over gradient-echo
sequences (Fig. 3b)[31]. The diffusion-weighting gradient typically consists of
two lobes of equal area.
When applied to a gradient-echo sequence, the lobes have opposite polarity,
creating the so-called
bipolar
gradient. In the spin-echo sequence, the lobes have
the same polarity, and are placed symmetrically on either side of the 180
◦
refocus-
ing pulse. The refocusing rf pulse reverses the polarity of the second lobe in this
sequence. The purpose of the gradients is to magnetically label the spins carried by
the water molecules. The first gradient lobe causes a phase shift of the transverse
