From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 200

experiment, therefore, makes it possible to measure the diffusion coefficient D a
regular spin-echo experiment in NMR.
In the same year, Stejskal further considered diffusion in anisotropic media by
employing the anisotropic Fick's law (Eq. 6.5 ) instead of the isotropic law (Eq. 6.3 ),
in the Bloch-Torrey equation, which introduces the diffusion tensor D [ 61 ]. He
was able to derive the modified Stejskal-Tanner equation incorporating the diffusion
tensor in anisotropic media [ 61 ]:

γ 2 δ 2 g 2 Δ

g T Dg

S 0 exp − b g T Dg .

δ
3

S S 0 exp − −

= = (6.13)

However, Stejskal in this seminal paper fell short of providing a method for
measuring the diffusion tensor from NMR, which could have preempted diffusion
tensor imaging by almost three decades. But he did lay the foundations of the
q-space formalism with the “pulsed” gradient assumption.

6.3.4 Narrow Gradient Pulse PGSE: q-Space Formalism

Diffusion in the PGSE experiment can also be modelled from a probabilistic
or random-walk model driven by the thermal kinetic energy of the spin bearing
particles. The PGSE experiment (Fig. 6.4 ) spatially encodes or labels the transverse
phase of the spins using the first gradient, which results in a deliberate dephasing of
the transverse magnetization. The purpose of the second gradient after the 180 ◦
RF pulse is to undo the effects of the first gradient and rephase the transverse
magnetization. However, if the spins diffuse away from their position between the
two gradients, then the transverse magnetization isn't entirely rephased after the
second gradient, resulting in a loss of the transverse signal. This can be described
by using a random-walk approach for the spin bearing particles.
Under the assumption δ Δ , which is known as the narrow gradient pulse

(NGP) condition, which implies that the spins are static during the application of
the diffusion encoding gradients G

, the dephasing accrued by a spin in the initial
position r 0 during the first gradient is [ 44 ] φ 1 =

( t )

γ δ
0

G t

) ·

r 0 dt γδ G · r 0 , when

( =

G g g . Similarly the dephasing accrued by the spin, now in the position

r due to diffusion, during the second gradient is φ 2 = γ Δ + δ
Δ

t G

( )= =

G ( t ) · r dt = γδ G · r .
Since the second gradient is applied after the 180 ◦ RF pulse, which flips the spins
around, the net phase shift accrued by a spin is φ = φ 2 − φ 1 = γδ G · ( r − r 0 )

.
Of course, if the spins hadn't diffused and had remained static during the period Δ
(between the gradients), then the net phase shift would have cancelled out. In other
words the amount of net phase shift is proportional to the diffused distance .

The NGP condition δ Δ can also be interpreted in the way Stejskal proposed
it δ → 0

( r − r 0 )

, with δ G finite. Although in practice the NGP condition can never be
achieved, it provides a powerful insight into the process of measuring diffusion
from NMR.