Pulsed-field gradient nuclear magnetic resonance as a tool for ...

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cluding the presence of a background gradient,
Ž .
we obtain 45, 86
ž / 0 1 2 1
SŽ 0. ž
21 2 2
Ž . 2 2 Ž . 2 2 2
S 2 exp exp g D 2 3 g D Ž 3.
2 T2 T1 3 g 0term
g0 term
/
2
2 2 2 Ž . 2
gg D t t t t 2Ž .
0 1 2 1 2 1 2 1
3
where t21t 1.
The same simplifications
can be applied as in the case of Eq. 34 .
By
comparing Eqs. 34 and 36 ,
and by realizing
that the duration in the Stejskal and Tanner
sequence is generally much longer than 1 in the
STE sequence, it can be seen that the last term in
Eq. 36 Ži.e., 2Ž . .
1 2 1 is much smaller than
Ž 2 the corresponding term in Eq. 34 i.e., 2 . ;
thus, the effect of the cross-term g g 0 is smaller
for the STE sequence Ž 86 . . As an aside, we note
that the effects of background gradients can, of
course, also be included with the shaped gradient
pulse versions of the Stejskal and Tanner sequences
given in Modulated Gradients, or similarly
with the STE sequence.
In the case of nonuniform gradients, when the
² 2 : 12
equality g g 0 is not met, the interpretation
of Eq. 34 Žor 36. becomes very difficult
Ž 70, 87 . . If the distribution of g 0 is symmetric
about g0 0 and not too large, a series expansion
can be used to correct for the background
gradients Ž 81 . . Perhaps counterintuitively, the
measured diffusion in the presence of internal
gradients is often found to be lower than the
actual diffusion coefficient Ž 87, 88 . . The reason
for this is the following. The measured diffusion
is in essence an ensemble average, and the internal
gradients will weight this distribution at the
time of signal acquisition, since the degree of
dephasing caused by the internal gradients is a
function of the diffusivity. The faster diffusing
spins will be more attenuated, and consequently it
is the more slowly diffusing spins that contribute
most to the echo signal Ž 87 . . This is analogous to
the effect found for spins diffusing in a restricted
geometry having an absorbing wall Ž 89 . . Since the
attenuation due to the background gradients may
be indistinguishable from the attenuation due to
the applied gradient, the effects of background
gradients can be mistaken for restricted diffu-
Ž .
sion 75 .
gg cross-terms
0 36
The basis of most sequences for the removal of
the g 0 term is to add additional pulses to the
PFG sequence to refocus the dephasing effects of
g 0 in a way analogous to the CPMG sequence
Ž 90 . . Clearly, such sequences must be designed
with an odd number of pulses between the
gradient pulses, since an even number of pulses
would simply result in the effects of the second
gradient pulse adding to the dephasing effects of
the first gradient pulse Ž 90 . . However, removal of
the g g 0 cross-term is more problematic. As
noted above, one solution to the background gradient
problem is to use applied gradients that are
much larger than the background gradients. When
this is not possible, more sophisticated pulse sequences
must be used. In 1980, Karlicek and
Lowe Ž 91. proposed the use of alternating Žbi-
polar. pulsed-field gradients in a modified
CarrPurcell sequence Fig. 17Ž A. to eliminate
the contribution of the g g 0 cross-term, since the
number of positive g intervals equals the number
of negative g intervals. The attenuation due to
diffusion for the Karlicek and Lowe sequence can
be calculated using the theory developed in the
first article Žsee
Part 1, The Macroscopic Approach.
to be Ž 91.
ž
2
2 2 3
3
0
EŽ g,2n. exp D ng Ž n1.
3
2
Ž .
2
1 2
g ž 2 /
ž / /
Ž . 1 2
37 2
Ž n1.
where the integer n, 1, and 2 are defined in Fig.
17Ž A . . Systematic errors due to the cross-term
can also be eliminated in a CarrPurcell se-