From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 204

The easiest option for solving this is to use the least squares optimization d opt =
argmin d || || 2 , which translates to the Moore-Penrose pseudo-inverse X − Bd

solution:

B T B ) − 1 B T X . d =(

Due to its linear form which only involves matrix manipulations, this solution is
extremely rapid. However, it doesn't account for the signal noise or of the distortion
to the noise it introduces while taking the logarithms of the signal in the linearization
process. Due to DTI's popularity and maturity as a technique of probing tissue
microstructures, a number of sophisticated solutions exist for measuring D from the
dMRI signal. These range from Basser's original weighted least squares approach
[ 5 ] which accounts for the logarithmic distortion of the signal noise, to non-linear
optimization approaches that account for signal noise, spatial smoothing, and also
for constraining the DT to be positive definite [ 18 , 19 , 29 , 40 , 45 , 51 , 70 ].

Microstructure from DTI: The consistency between the phenomenological ap-
proach and the q-space formalism, under the NGP condition, implies that the prop-
agator describing the diffusion measured by DTI is the Gaussian PDF (Eq. 6.10 ).
This is an oriented Gaussian parameterized by the DT D , or its inverse. The
orientation of the PDF can be deduced from the eigen-decomposition of the DT.
The eigenvalues and eigenvectors of D form a local coordinate system that indicates
the preferential diffusion direction orienting the Gaussian PDF. In other words it
indicates the diffusion direction favoured by the microstructure of the medium.
This preferential orientation of the microstructure can be visually represented by
the ellipsoid represented by D whose implicit quadratic form is [ 6 ]:

X T D − 1 X
2

=1 . (6.22)

t

Since D is symmetric it can be diagonalized D

W T ΛW ,where W are its
orthonormal eigenvectors and Λ is a diagonal matrix whose diagonal elements
are its eigenvalues. The canonical form of the diffusion ellipsoid defined by D − 1
emerges in the coordinate frame of its eigenvectors:
x
√ 2

=

2 y

√ 2

2 z

√ 2

2

+ + =1 .

λ 1 t λ 2 t λ 3 t

To infer the microstructure of the cerebral white matter from DTI, the funda-
mental assumption is that the coherent fiber bundle structures formed by the axons
hinder the perpendicular diffusion of water molecules (spin bearing 1 H atoms) more
than the parallel diffusion. Therefore, the elongation and orientation of the DT are
good indicators of these coherent structures or fiber bundles locally. The eigenvector
corresponding to the largest eigenvalue, the major eigenvector, indicates the main
fiber direction, while the other eigenvectors and eigenvalues indicate diffusion
anisotropy in the perpendicular plane (Figs. 6.6 and 6.7 ).