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microenvironment. The concept of a distribution of diffusion coefficients is a very
attractive one, given the averaged nature of the signal from each image voxel,
which by necessity must contain a wide range of cell types and structures. While
the so-called “heterogeneity index”,
α
, appears insensitive to applied diffusion
gradient direction [55], a more comprehensive diffusion measurement, the
D
DC
could facilitate improved fiber tracking in DTI [56].
Hall et al. [56] developed the stretched exponential model by introducing the
hypothesis that the stretch parameter is related to the fractal nature of the movement
of the diffusing spins, and therefore the complexity of the tissue environment.
They related their stretching parameter to the fractal dimension and thus linked
this model to the concept of anomalous diffusion.
5. An Anomalous Diffusion Model
One of the first experiments to measure anomalous diffusion in tissue using mag-
netic resonance methods was that by Ozarslan et al. [57]. Q-space spectroscopy
data was obtained from human red blood cell ghosts, tissue from a human grade-4
astrocytoma and a human erythrocyte ghost model. Using a simple model that de-
scribes anomalous diffusion in disordered media, they found that water diffusion
in the human tissue samples was anomalous, (i.e., that the mean-square displace-
ments varied more slowly than linearly with time).
Another group, Magin et al. [38], developed the idea of applying fractional
diffusion in a phenomenological manner to the diffusion decay observed with DWI.
Beginning with the Bloch-Torrey equation [Eq.(55)] [28], by simply replacing
∂M
2
M
by fractional time and space derivatives, respectively, (cf. Eq. 8
of their article), they transformed it into a fractional order form. Hence, they derived
two functions, one which is fractional in space, and the other fractional in time.
Their expression for diffusion decay due to fractional-order dynamics in space for
the Stejskal-Tanner experiment is their Eq. (16c)
/∂t
and
⊥
⊥
S
0
exp
Dμ
2(
β
−
1)
γG
z
δ
2
β
1
δ
2
β
−
1
S
(
t
)
=
−
−
(82)
2
β
+
Here,
μ
is a fractional-order space constant required to preserve units and
β
is the order of the fractional-spatial derivative and constrained to the values
(1
/
2
<β
1) (Fig. 14).
Experimental data was acquired by the authors in phantoms, human cartilage
in
vitro
, and human brain
in vivo
. The diffusion coefficients calculated by fitting with
the anomalous diffusion model above, were almost identical to those produced by
fitting with the monoexponential function,
S
≤
bD
]. The novel informa-
tion was therefore provided by the other parameters,
β
and
μ
. As expected when
the diffusion decay deviated from monoexponential behavior, the values of
β
were
found to be
<
1.
=
S
0
exp[
−
