Data

Diffusion tensor magnetic resonance imaging (DT-MRI) [1] is a technique used to measure the anisotropic diffusion properties of the water molecules found within biological tissues as a function of the spatial position within the sample. Due to differing cell shape and cell membrane properties, the diffusion rates of the water molecules are different in different directions and locations. For instance, neural fibers are comprised mostly of bundles of long cylindrical cells that are filled with fluid and are bounded by less-water-permeable cell membranes. The average diffusion rate (at a spatial location) is fastest in the three-dimensional axis direction along the length of the neuron cells, since more of the water molecules are free to move in this direction. The average diffusion rate is slowest in the two transverse directions, where the cell membrane interferes, reducing and slowing down the movement of the water molecules. Other parts of the brain are primarily comprised of fluid without cell membranes, such as the ventricles. Here the average diffusion rate is larger and more uniform (almost the same in all directions). The diffusion properties can be represented with a symmetric second order tensor - $3$x$3$ matrix:

$${\mathbf T} = \left( \begin{array}{ccc} T^{xx} & T^{xy} &T^{xz} \\ T^{yx} & T^{yy} &T^{yz} \\ T^{zx} & T^{zy} &T^{zz} \\ \end{array}\right).$$ (1)

The 6 independent values (the tensor is symmetric) of the tensor elements vary continuously with spatial location. The 3-D local axis direction of the neuron fibers will correspond to the dominant eigenvector of the tensor. There should be one large eigenvalue, and two small eigenvalues. This can be seen from the physical interpretation of the diffusion tensor, which can be thought of as a vector-valued function whose input is the local 3-D concentration gradient and whose output is the 3-D directional vector flux² of the water molecules. The function is evaluated by multiplying the 3x3 matrix by the 3x1 concentration gradient, producing the 3x1 vector flux of the water molecules. Water will diffuse fastest in the direction along the axis of the neurons and slowest in the two transverse directions. For the ventricles, a dominant eigenvector should not exist: the three eigenvalues of the tensor should have roughly the same value. Water will diffuse roughly at the same speed in all directions. Hence, we can use the diffusion tensor to distinguish tissues with a primary diffusion axis from parts that do not. In this paper, the experimental dataset contains sampled values of the diffusion tensor on a regularly spaced grid of $121$x$88$x$60$ (cubic) voxels. We will denote these given tensor values as ${\mathbf T}_{ijk}^{\alpha \beta}$, where $\alpha$ and $\beta$ are the three dimensional tensor components $\{xx,xy,..,zz\}$, and $i,j,k$ are traditional integer indexes into the regular grid volume. Also, when no upper indexes are provided, the operations are assumed to be performed on the entire tensor component-wise ${\mathbf T}\equiv {\mathbf T}^{\alpha\beta}$, i.e., on each of the six independent values of the tensor.

 

 

 Figure 2: Sagittal and axial slices of anisotropy measure $c_\ell$ of the dataset. The lighter regions correspond to stronger anisotropy areas found in the white matter. See Eq. $(4)$.