From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 207

the geometric forms of the measured EAPs, or their characteristics. Such EAPs
clearly represented non-free diffusion and were non-Gaussian. DSI was based on
the Fourier transform relationship between the signal and the EAP described in
Eq. ( 6.17 ). Although in practice it couldn't satisfy the NGP condition required by the
q-space formalism, DSI essentially imaged the q-space densely and reconstructed
the EAP via a fast Fourier transform (FFT). In fact, δ Δ in the DSI experiments

conducted in [ 67 , 71 , 72 ]. Nonetheless, the angular results produced by such DSI
experiments clearly spoke in favour of q-space imaging—as seen in [ 58 ].
DSI, however, considered the modulus Fourier transform:

≈

P ( r )= | E ( q ) | exp ( − i 2 π q · r ) d q ,

in place of the true Fourier transform, justifying that in the case of pure diffusion
the modulus Fourier transform is equal to the true Fourier transform. This was done
to counter biological motion such as cardiac pulsation, which tended to contaminate
the phase of the signal.
Initially visualizing iso-surfaces of the estimated EAP [ 71 ], DSI techniques later
introduced the orientation distribution function (ODF), to emphasize the angular
results, which indicated underlying fibers. Two ODFs were used, namely [ 72 ]
(Fig. 6.8 ):

r 2 dr, u Ψ SA ( u )= P ( r u ) = r / | r | , (6.23)

R +

and the one introduced by Tuch [ 67 ]:

Ψ T ( u )= P ( r u ) dr, u = r / | r | . (6.24)

R +

However, DSI had severe acquisition setbacks. To correctly estimate the EAP,
the q-space had to be densely sampled, and also at very high b-values. The DSI
sampling scheme was a Cartesian grid inside a sphere in q-space, where both
the radius of the sphere—the maximum b-value, and the number of grid-points—
the number of acquisitions played important roles in determining the accuracy of the
estimated EAP. For example in [ 71 , 72 ], the maximum b-value was in the range of
20,000 s/mm 2 , and the number of acquisitions were more than 500. In comparison,
DTI acquisitions are done for b-values of 1,000 s/mm 2 , and only require a minimum
of 6 acquisitions. These demanding requirements played unfavourably for DSI,
since its clinical viability was near impossible.
However, DSI was the proof of concept for q-space imaging, which quickly
became the popular approach for dMRI, and gave rise to a plethora of techniques
for estimating complex EAPs or their characteristics like the ODF. These q-space
techniques were developed to overcome the acquisition limitations of DSI.