From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 196

first, concerning the spreading of energy in a solid medium, the second, concerning
the spreading of molecules from a region of high concentration to a region of low
concentration in fluids, and the third, concerning the random motion of molecules
and particles in fluids due to the ambient temperature—can all be described by
the same diffusion equation. However, while today their correspondence is widely
accepted, establishing this connection wasn't always an easily demonstrable task.

Fick's Laws of Diffusion: The phenomenological equations of diffusion were
proposed by Joseph Fourier in 1822 to describe the diffusion of heat in solids,
and then adapted by Adolf Fick in 1855 to describe the diffusion of molecules
in fluids in the presence of a concentration gradient [ 21 ]. Fick derived his “laws
of diffusion” from Fourier's laws by analogy, while attempting to describe the
experiments conducted by Thomas Graham in 1831 on the diffusion of gases.
These laws describe the molecular transfer or diffusion that takes place in a system
from regions of high concentration to regions of low concentration due to the
concentration gradient.
Fick's first law relates the rate of transfer of the diffusing substance per unit area,
or flux J , to the concentration gradient C causing the diffusion:

J = − D ∇ C , (6.2)

where D is the diffusion coefficient. Conservation of mass during the diffusion
process implies −∇ J = ∂ C /∂t . This leads to Fick's second law of diffusion:

∂ C
∂t =

∇ 2 C . D (6.3)

Fick's second law describes the change of the concentration field over time due
to the diffusion process. Equation ( 6.3 ), which relates the time derivative of the
concentration to the second order spatial derivative of the concentration is known as
the diffusion equation —it describes diffusion phenomenologically.
D being a scalar quantity in Eqs. ( 6.2 )and( 6.3 ) is an indication that diffusion is
equal in all directions. This is known as isotropic diffusion. However, certain media
such as crystals, textile fibers, etc. can be inherently anisotropic and can favour
diffusion in a certain spatial direction while hindering it in others. This results in
anisotropic diffusion, which is described by replacing the scalar diffusion coefficient
D by a generalized diffusion tensor D (

3 × 3 matrix) in Fick's laws [ 21 ]:

J = − D ∇ C , (6.4)

∂ C
∂t = ∇· ( D ∇ C ) .

(6.5)

Diagonalizing the diffusion tensor D into its eigenvalues and eigenvectors provides
a local orthogonal coordinate system that indicates the preferential diffusion
direction favoured by the anisotropy of the underlying material. This is the budding