Parallel Computation in Simulating Diffusion and Deformation in Human Brain - Parallel Computing for Bioinformatics and Computational Biology - page 123

tensor imaging reflects the directional organization of the underlying white matter
microstructure. DT-MRI characterizes the behavior on a voxel-by-voxel basis, and
for each voxel, the diffusion tensor yields the diffusion coefficient corresponding
to any direction in space [10]. The direction of the greatest diffusion can be deter-
mined by evaluating the diffusion tensor in each voxel, which is believed to point
along the dominant axis of the white matter fiber bundles traversing the voxel. Thus,
the panoramic view of the fastest diffusion direction can be generated to provide a
visualization of the white matter pathways and their orientations.
A number of fiber tracking algorithms have been developed since the appear-
ance of DT-MRI. In Refs. [11, 12] a variety of these algorithms are described and
reviewed. As the measured quantity in DT-MRI is water diffusion, an intuitive way
to gain insights from the diffusion tensor data is to carry out a virtual simulation of
water diffusion, which is anisotropic and governed by the diffusion equation, over
the brain. The white matter fiber bundles are assumed to proceed along the direc-
tion where the diffusion is the greatest. The idea of studying brain connectivity by
simulating the anisotropic diffusion has been explored in Refs. [13-15]. Batchelor
et al. [13] specify a starting point for tractography where a seed is diffused. A virtual
concentration peak of water is spread in Ref. [14]. In Ref. [15], successive virtual
anisotropic diffusion simulations are performed over the whole brain, which are uti-
lized to construct three-dimensional diffusion fronts and then the fiber pathways.
This technique of solving a diffusion equation makes use of the full information
contained in the diffusion tensor, and it is not dependent upon a point-to-point eigen-
value/eigenvector computation along a trajectory, thus in that sense may enhance
the robustness and reliability of fiber reconstruction algorithms. It is also intuitively
related to the underlying physical-chemical process in the central nervous system
[16, 17]. The diffusion process and related transport mechanism in the brain are
discussed in detail in Ref. [18].
Simply, anisotropic systems are those that exhibit a preferential flow direction, in
which the flowfield does not follow the concentration gradient directly, for thematerial
properties also affect diffusion. Therefore, the diffusion tensor, D , is introduced to
fully describe the molecular mobility along each direction and the correlation between
these directions. We have

D xx D xy D xz
D yx D yy D yz
D zx D zy D zz

,

D =

where the subscripts xx , xy , xz , and so on, denote the values of the individual coef-
ficients in the matrix that can be seen as the influence from directions in the input
(being the concentration) on the various directions in the output (being the flux). For
the brain system and other typical systems, the tensor is symmetric. Figure 5.1 shows
an axial slice of a diffusion tensor volume from the human brain.