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

328
PRICE
Figure 10 A plot of the simulated echo attenuation for PFG diffusion measurements in a
cylinder with the cylinder oriented at three different polar angles with respect to the
gradient, i.e., 0 Ž . , 4 Ž . , and 2 Ž . versus calculated
using Eqs. 92 , 110 , and 112 ,
respectively. Also shown is the result of a power distribution
of polar angles Ž . versus calculated using Eqs. 110 and 119 .
The parameters used
in the simulation were the same, as far as possible, as those used for the sphere in Fig. 7, i.e.,
1 ms, D 5 1010 m2s1 , g 1Tm1 ,R8m, L 24 m, and 1H
2.6571 10 8 rad T 1 s 1 . The effects of the polar angle can be clearly seen on the
attenuation curves, and the curves go through three stages depending on Žthis
is most
obvious in the 2 case . , similar to the results for diffusion in a sphere Ž see Fig. 7 . . As
would be expected, since it is an average over all possible polar angles, the powder average
echo-attenuation curve is between the limits of the attenuation curves for 0 and
2. Because of the axial symmetry of the cylinder, it was unnecessary to average over
. However, the normalization factor in Eq. 119 was changed appropriately.
For example, it makes it difficult to compare the
diffusional characteristics of one sample to another.
A solution is to determine D itself by
measuring the diffusion coefficients in seven different
directions i.e.,
except for the case of
charged moieties, D is symmetric Ž 118. and so
there are only six independent elements .
However,
because of experimental imprecision it is
normal to perform a much larger number of
measurements and determine D statistically
Ž 114 . . As seen in Eq. 108 ,
the use of a single
gradient direction in a diffusion measurement
allows the diagonal elements of D to be probed.
The off-diagonal elements can be probed by applying
gradients along various oblique directions
Žconsider the pulse sequence shown in Fig. 2, but
with the possibility of gradients along all three
.
Cartesian directions . For example, if gradients
were applied along all three Cartesian directions,
then the echo attenuation would be described by
Ži.e., Eq. 105. Ž .
ln E b D b D b D
xx xx yy yy zz zz
Ž . Ž .
b b D b b D
xy yx xy xz zx xz
Ž .
b b D . 117
yz zy yz
As a further complication, the restricting geometries
may not all be uniformly aligned in the
same direction e.g., brain white matter Ž 119. or
even randomly aligned Že.g.,
a suspension of red
blood cells . . The apparent D is then an average
of the different orientations. Measuring diffusion
in three orthogonal directions so as to determine
the trace of the diffusion tensor has been pro-