From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 198

showed that the diffusion coefficient is directly proportional to the variance of the

particle displacement (Δ ) 2 =2Δ

tD , where without loss of generality x is any
chosen spatial direction. A similar development in the anisotropic case along the
lines proposed by Einstein results in the anisotropic diffusion equation and connects
the diffusion tensor to the covariance tensor:
∂f

x

X ,t

( )

= ∇· ( D ∇ f ( X ,t )) , (6.8)

∂t

(Δ

) 2 Δ

⎛ ⎞

x x y Δ x z

Δ Δ

(Δ

) 2 Δ

1
2Δ

⎝ ⎠ .

D := Δ y x y y z (6.9)

Δ Δ

(Δ

) 2

t

Δ z x Δ z y z

Δ Δ

Finally Einstein also derived that under the initial condition f

X , δ

( 0) = ( ) ,which

X

corresponds to free diffusion, the local particle concentration f

X ,t

( ) is a Gaussian

tD . This, however, implies that the Green's
function of the diffusion equation, or the transition probability is also a Gaussian in
the case of free diffusion:

function with the derived variance 2

.

X T D − 1 Δ

1

− Δ

X

P (Δ X , Δ t )=

| 1 / 2 exp

(6.10)

(4 π Δ t ) 3 / 2 | D 4Δ t

In the anisotropic case it is an oriented Gaussian parameterized by the covariance
tensor.

6.3.3 The Stejskal-Tanner PGSE Experiment

After Hahn who first noticed the effects of diffusion in NMR in his spin echo
experiment [ 31 ], Carr and Purcell measured the diffusion coefficient for the first
time from NMR. In their modification to Hahn's experiment they employed a
temporally constant magnetic gradient field and modelled the diffusion of spin
bearing particles with discrete jumps [ 14 ]. However, the continuous description was
formulated by Torrey in 1956. He modified the phenomenological Bloch equations
by adding to it Fick's diffusion equation (Eq. 6.3 )[ 64 ]. This came to be known as
the Bloch-Torrey equation for describing the net magnetization vector M (without
flow):

⎛ ⎞ ⎛ ⎞

− T 2 00
0 − T 2 0
00 − T 1

0
0 M 0
T 1

∂ M
∂t =

⎝ ⎠ ⎝ ⎠ +

∇ 2 M ,

γ M × B + M + D (6.11)

where Fick's law is employed to describe the self diffusion of the net magnetization.
About a decade later, in 1965, Stejskal and Tanner designed the pulsed gradient
spin echo (PGSE) experiment by modifying Hahn's spin echo experiment with two
identical magnetic gradients around the 180 ◦ RF pulse to encode the transverse