From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 197

idea that indicates that diffusion can be considered as a probe of the underlying
medium's microstructure. Isotropic diffusion can be understood as a special case of
anisotropic diffusion when D D I ,where I is the identity matrix. The idea of the

diffusion tensor is central to dMRI, since the fibrous quality of the cerebral white
matter also exhibits directional anisotropy.

=

Brownian Motion and Einstein's Random Walk Approach: Although Fick's
laws are concerned with the diffusion of molecules from regions of high con-
centration to regions of low concentration, they essentially describe the evolution
of the concentration gradient over time and space, and aren't concerned with the
movements of the molecules themselves. The molecular description of diffusion
emerged with Albert Einstein in 1905 when he related the molecular-kinetic theory
of heat to the observations made by Robert Brown in 1828. Brown had noted the
perpetual erratic motion of pollen grains suspended in water while observing them
under a microscope. This erratic movement came to be known by his name as
Brownian motion. When Einstein proposed [ 28 ] that due to the thermal kinetic
energy of molecules, particles suspended in a liquid large enough to be observed
under a microscope would exhibit random movements governed by the probabilistic
law he derived, his idea was quickly recognized to be the theoretical description
of Brownian motion. It turned out that the probabilistic law of Brownian motion
derived by Einstein also satisfied the diffusion equation. This provided the final link
and showed that diffusion was driven by the thermal kinetic energy of molecules
due to the ambient temperature, implying that diffusion, in the form of Brownian
motion also occurred in the absence of a temperature or a concentration gradient.
The special case of diffusion when the suspended particles belong to the liquid is
known as self diffusion .
To describe the erratic movement of a large number of particles undergo-
ing Brownian motion, Einstein adopted the probabilistic approach of a random
walk model [ 28 ]. He modelled diffusion using two Probability Density Functions
(PDF)s— f ( X ,t ) , the probability of finding a particle at the position X at a time t ,

and P (Δ X , Δ t ) , the transition probability or the probability of finding a particle

at a distance Δ X from its initial position after a time Δ t . Considering P (Δ X , Δ t )

symmetric, such that P (Δ X , Δ t )= P ( − Δ X , Δ t ) , Einstein proposed the relation

between f ( X ,t ) and P (Δ X , Δ t ) :

∞

f ( X ,t +Δ t )= f ( − Δ X ,t ) P (Δ X , Δ t ) d Δ X . (6.6)

X

−∞

He then showed that f , which can also be considered as the local particle

concentration, satisfies the diffusion equation:

( X ,t )

∂f (

X ,t

)

= D∇ 2 f (

X ,t ) , (6.7)

∂t

which introduces the diffusion coefficient D , showing that the random walk
approach can model diffusion. In the isotropic case discussed by Einstein, he further