Chemistry Reference
In-Depth Information
2
δ
2
3
2
→
2
δ
=
G
dt
2
2
kTγ
2
mβ
α
2
δ
15
δ
2
t
1
+
Gδ
2
−
8
δt
1
+
3
δ
2
t
2
t
1
)
2
−
α
G
2
δ
15
δ
2
t
1
+
dt
1
(
α
1
Gδ
2
×
−
8
δt
1
+
−
1)
(
t
2
−
2
δ
(178)
Thus,
2
2
(
t
)
2
Dγ
2
G
2
δ
2
+
α
8
(5
α
2
))
=
(3
+
α
)(12
+
α
(24
+
9
α
+
(179)
+
α
)
which yields Eq.(72) in the normal diffusion limit
α
→
1.
F.
Conclusion
We have shown that the method outlined in Section III. (viz. [1]), the application
of the Langevin equation to the dephasing of the magnetization due to diffusion,
may also be extended to anomalous diffusion in a transparent fashion in order to
provide a possible microscopic justification for the use of stretched exponentials
to describe the dephasing in tissue. Specifically, we suggest that the anomalous
diffusion may ultimately have its origin in memory effects giving rise to fractional
Brownian motion. This process naturally introduces the new fitting parameter,
α
,
related to the Hurst index [3] indicating the role played by fractional dynamics in
the time for the complex diffusion, which is observed in human neuronal tissue.
In normal diffusion,
α
1, so that we regain the classical expressions of Carr and
Purcell, and so on, once more.
=
V.
MAGNETIC RESONANCE IMAGING METHODS
It is a well-established fact that water in neuronal tissue does not diffuse freely.
In MRI, the application of diffusion weighting gradients in an imaging sequence,
causes a loss in signal intensity. As the degree of diffusion weighting is increased
during a series of image acquisitions, the signal intensity decreases with a monoex-
ponential decay for freely diffusing water, while the decay in neuronal tissue can be
described empirically with a biexponential model. This deviation from free diffu-
sion arises from the complexity of the cellular environment, with factors including
molecular crowding and transient binding events thought to play important roles.
Diffusion MRI is a noninvasive technique, and as such we cannot expect to
determine a 'correct' diffusion coefficient in
in vivo
experiments, hence we cannot
hope to determine which mathematical model is the most suitable. As we have seen,
many empirical expressions have been proposed that fit the diffusion weighted MRI
