From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 201

The complex signal generated by individual spins with a net phase shift φ is
exp( )

is the averaged net signal from the spin ensemble, or it is the expected value of the
complex signal given the probability of spins starting at r 0 and diffusing to r in the
time Δ . This probability is the product of the probabilities f

iφ )=exp[ iγδ G · ( r − r 0 )] [ 13 , 44 ]. However, the spin echo signal E ( G ,Δ

( r 0 , 0) , of finding a spin

initially at r 0 ,and P ( r | r 0 ,Δ ) , of a single spin starting at r 0 and diffusing to r in

time Δ . The product f introduces the random-walk model for the

spin bearing particles diffusing from r 0 to r ,and[ 13 ]:

( r 0 , 0) P ( r | r 0 ,Δ )

E ( G ,Δ )= f ( r 0 , 0) exp[ iγδ G · ( r − r 0 )] P ( r | r 0 ,Δ ) d r d r 0 . (6.14)

This indicates that in the absence of diffusion encoding gradients E ( 0 ,t )=1 . In

practice E is obtained by dividing the echo signal amplitude from a PGSE

experiment with diffusion gradients by the echo signal amplitude from a Hahn spin
echo experiment without gradients E

( G ,Δ )

/S 0 .
This leads to the q-space formalism by defining a reciprocal space q where [ 13 ]

( G ,Δ )= S ( G )

q := γδ G / 2 π. Inserting q in Eq. ( 6.14 ) gives the q-space signal:

E ( q ,Δ )= f ( r 0 , 0) exp[ i 2 π q · ( r − r 0 )] P ( r | r 0 ,Δ ) d r d r 0 . (6.15)

to be translationally invariant or
that the movement of a spin is independent of the movements of the other spins
and also of its own position and movements in the past—as in a random-walk,
implies that P

Assuming the transition probability P ( r | r 0 ,Δ )

, which is the diffusion propagator. Also since
in a random-walk the movements of all the particles are independent and identical,
and since the complex signal and the diffusion propagator for a spin only depend
on the spin displacement

( r | r 0 ,Δ )= P (Δ r ,Δ )

, it is useful to consider the ensemble
average propagator (EAP), which describes the average probability of any spin in
the ensemble diffusing by

Δ r =( r − r 0 )

Δ r during the time Δ t [ 13 ]:

P (Δ r , Δ t )= P (Δ r , Δ t ) f ( r 0 , 0) d r 0 . (6.16)

Combining Eqs. ( 6.15 )and( 6.16 ) gives the main result of the q-space formalism
[ 13 ]:

E ( q ,t )= P (Δ r ,t )exp( i 2 π q · Δ r ) d Δ r , (6.17)

which establishes an inverse Fourier transform relationship between the EAP,
henceforth denoted P

.
This Fourier relationship between the ensemble average diffusion propagator
and the diffusion NMR signal ushers in the paradigm change that diffusion can be
viewed more than just an intrinsic property, but also as a probe of the microstructure

( r ) , and the normalized echo signal, henceforth denoted E ( q )