A Bloch-Torrey Equation for Diffusion in a Deforming Media Page: 3 of 19
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in an appropriate basis where the tensor is diagonal
3
D = A2zd2d, RART,
2=1
where A is the diagonal matrix diag(Al, A2, A3) with
A1 > A2 > A3. R (di, d2, d3) is the rotation matrix
which transforms the initial Cartesian basis into the
local basis of the ellipsoid. This visualization is shown
in Fig. 1
A3 e3
A2 e2
Figure 1: Visualization of the diffusion tensor as an ellipsoid
with the corresponding basis vector.
However, this formalism associated with the tensor
of diffusion reaches some limits as soon as the mate-
rial exhibits a more complex configuration. If more
than one direction is promoted, as in the case of a
fiber crossing in the biological tissue such as with the
white matter in the brain [40], the diffusion tensor is
unable to solve the problem accurately. In this case,
the principal direction of diffusion will be aligned with
the average of the two crossing directions leading to
a large error of interpretation as illustrated in Fig. 2.
This problem is especially limiting in the case of real
studies because of the partial volume effect. If large
voxels are taken in account during the acquisition, the
diffusion tensor has a high probability for averaging
the directions that follow different paths inside this
voxel. In order to have a more accurate model, higher
order tensors can be used (q-ball) [41 43]. However,
these methods still require a longer acquisition time
and a strong gradient system [44] for a good signal
to noise ratio (SNR). Moreover, for the purpose of
the muscular fibers of the heart, the bundles are sup-
posed to follow the same orientation in a local neigh-
boring and therefore the direction of diffusion varies
smoothly. For this reason, the tensor notation will be
kept for the purpose of the current study.
2.2 The Diffusion Equation
Performing a numerical or mathematical study of a
macroscopic structure taking into account every mi-.Fiber Crossing
I
Figure 2: Example of fiber crossing. The red tube repre-
sents an example of fiber. The blue circle shows the intersec-
tion of the fiber crossing. With the tensor approximation, the
principal direction of diffusion is the average between the two
directions of the fiber.
croscopic molecule is not feasible. The studied ele-
ment, such as water, or spin orientation in the case
of the MR signal, is studied as a continuous func-
tion. The macroscopic behavior is therefore known
by equations that are linked to the microscopic struc-
ture via the diffusion tensor.
A scalar density of the diffusing material is intro-
duced using the notation 0, which represents the wa-
ter density or the spin orientation in one direction.
This density depends on the position x and on the
time t. Denoting j the density flux, Fick's first law
states that this flux is given by the product of the dif-
fusion coefficient or tensor with the spatial variation
of the density (the diffusion occurs from the highest
concentration to the lowest):j -D V .
(21
Therefore j is a vector oriented through the local
lowest concentration at each position. The diffusion
equation can be derived from the continuity equation.
If no sink or sources of diffusing material exist, the
time rate of change of the density equals the opposite
of the divergence of the density flux:
And using Eq. 2 to express the flux,V (DV p) .3
In the case of a constant isotropic diffusion D D I,
where I is the 3 x 3 identity matrix and D is the scalar
coefficient of diffusion, Equation 3 simplifies then to
the Laplace equation:
= D V - (V 0)3
(3)
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Rohmer, Damien & Gullberg, Grant T. A Bloch-Torrey Equation for Diffusion in a Deforming Media, report, December 29, 2006; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc890298/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.