From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 199

Fig. 6.4 The pulsed gradient spin echo (PGSE) sequence. Two identical gradients are applied
around the 180 ◦ RF pulse of Hahn's spin echo experiment. This encodes the transverse phase of
the diffusing spin bearing particles. It then becomes easier to measure the decay of the signal due
to diffusion

phase of the diffusing spin bearing particles [ 61 , 62 ](Fig. 6.4 ). This made it easier
to measure the decay in the transverse signal due to diffusion, and from there the
diffusion coefficient. The PGSE experiment established the field of dNMR.
In the PGSE experiment the first gradient G of duration δ spatially encodes the
phase of the individual spins (by dephasing them by an amount dependent on their
position), and the effects of this gradient are undone by the second identical gradient
after the 180 ◦ RF pulse which flips the spins around (implying an effect G

from the second gradient). This results in a complete recovery of the signal since the
magnitude of transverse magnetization vector M xy depends on the phase coherence
of the individual spins. However, if the individual spins move due to diffusion
during the period Δ , between the two pulsed gradients, then the effects of the
second gradient isn't the exact opposite of the first gradient (

−

G ) that was used
to encode their phases. This leads to a partial phase incoherence—resulting in a
reduced transverse magnetization M xy , implying a loss in the spin echo signal.
Since the signal decay is related to the rate of diffusion or the diffusion coefficient,
measuring the signal decay makes it possible to measure the diffusion coefficient.
Stejskal and Tanner provided the mathematical solution to the Bloch-Torrey
differential equation for their PGSE experiment, which became the corner stone
equation for dNMR as the Stejskal-Tanner equation for the signal:

−

γ 2 δ 2 g 2 Δ

D

δ
3

S = S 0 exp − − = S 0 exp( − bD ) , (6.12)

where S is the magnitude of the signal decay due to diffusion, S 0 is the magnitude of
the signal in the absence of a diffusion encoding gradient, Δ is the time between the
two gradients, δ is the application time of each gradient, and b

− 3
is the b-value. The modifications introduced by Stejskal and Tanner in the PGSE

γ 2 δ 2 g 2 Δ

=