caps.diffusion_estimation.tensor_scalars.SecondOrderScalarParameters¶

SecondOrderScalarParameters¶

class caps.diffusion_estimation.tensor_scalars.SecondOrderScalarParameters[source]¶

Compute the Fractional Anisotropy [R3] (FA), Mean Diffusivity (MD) and the Westion Shapes coefficients [R3] (cl, cp, cs) of a second order tensor.

[+show/hide second order tensor scalars]

Second Order Tensor Scalar Parameters

At each voxel and from the six independant second order tensor parameters, an ellipsoid (or a peanut) can be ploted. Such a visualization is quite complex and hard to interpret. Therefore, a wide range of scalar measurements can be computed to reduce the six parameters of the diffusion tensor. The two most popular scalar maps are the fractional anisotropy (FA) map and the mean diffusivity (MD) map.

The MD map is calculated from the three eigenvalues $\lambda_1 \ge \lambda_2 \ge \lambda_3$ of the diffusion tensor as follows:

$md = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}$

The MD is invariant under tensor rotation and allows the characterization of the diffusivity of the medium: the greater the diffusion is restrain, the lower is the MD value (in the white matter for instance) and vice versa, the less the diffusion is restrain, the greater is the MD value (in the cerebrospinal fluid for instance).

The anisotropy can be characterized in several ways. It is interesting to access this parameter since it is directly related to the degree of tissue organization. The simplest way is to consider the ratio between the maximum and minimum eigenvalues $\lambda_1 / \lambda_3$ . When the ellipse is elongated this measure increases. Unfortunately, this measure is sensitive to measurement noise and do not take into account the second eigenvalue. Another way to characterize the anisotropy is based on the difference of the three eigenvalues, $(\lambda_1-\lambda_2)^2+(\lambda_1-\lambda_3)^2+(\lambda_2-\lambda_3)^2$ . This measure is zero for a sphere ( $\lambda_1=\lambda_2=\lambda_3$ ) and increases when the sphere is distorted. When this measure is normalized between 0 and 1 the FA is obtained:

$fa = \sqrt{\frac{1}{2}\frac{(\lambda_1-\lambda_2)^2+(\lambda_2- \lambda_3)^2+(\lambda_3-\lambda_1)^2}{\lambda_1^2+ \lambda_2^2+\lambda_3^2}}$

However, there are many other ways to represent the anisotropy of diffusion tensor. Westin and al. proposed to classify the diffusion tensor into three subcategorie:

Linear case ( $\lambda_1 \gg \lambda_2 \approx \lambda_3$ ): the diffusion is mainly in the direction of the eigenvector associated with the largest eigenvalue.

$\mathbf{D} \approx \lambda_1 \mathbf{D_l} = \lambda_1 \vec{e_1} \vec{e_1}^t$

Planar case ( $\lambda_1 \approx \lambda_2 \gg \lambda_3$ ): the diffusion is in the plane defined by the two eigenvectors corresponding to the two largest eigenvalues.

$\mathbf{D} \approx \lambda_1 \mathbf{D_p} = \lambda_1 (\vec{e_1} \vec{e_1}^t + \vec{e_2} \vec{e_2}^t)$

Spheric case ( $\lambda_1 \approx \lambda_2\approx \lambda_3$ ): the diffusion is isotropic.

$\mathbf{D} \approx \lambda_1 \mathbf{D_s} = \lambda_1 (\vec{e_1} \vec{e_1}^t + \vec{e_2} \vec{e_2}^t + \vec{e_3} \vec{e_3}^t)$

A diffusion tensor $\mathbf{D}$ can be represented by combination of these three cases:

$\textbf{D} &= \lambda_1\vec{e_1} \vec{e_1}^t + \lambda_2\vec{e_2} \vec{e_2}^t + \lambda_3\vec{e_3} \vec{e_3}^t \nonumber \\ &= (\lambda_1 - \lambda_2)\vec{e_1} \vec{e_1}^t + (\lambda_2 - \lambda_3)(\vec{e_1} \vec{e_1}^t + \vec{e_2} \vec{e_2}^t) + \lambda_3(\vec{e_1} \vec{e_1}^t+\vec{e_2} \vec{e_2}^t+ \vec{e_3} \vec{e_3}^t) \nonumber \\ &= (\lambda_1 - \lambda_2)\mathbf{D_l} + (\lambda_2 - \lambda_3)\mathbf{D_p} +\lambda_3 \mathbf{D_s}$

where $(\lambda_1 - \lambda_2)$ , $(\lambda_2 - \lambda_3)$ and $\lambda_3$ are the coordinates of $\mathbf{D}$ in the tensor basis $[\mathbf{D_l}, \mathbf{D_p}, \mathbf{D_s}]$ . This relationship between the eigenvalues of the diffusion tensor enables the classification of the diffusion tensor from its geometric shape. Using this new decomposition, it is possible to quantify the similarity to the linear, planar and spherical cases. The coefficients obtained are normalized as follows:

$c_l &= \frac{\lambda_1-\lambda_2}{\sqrt{\lambda_1^2+\lambda_2^2+ \lambda_3^2}} \nonumber \\ c_p &= \frac{2(\lambda_2-\lambda_3)}{\sqrt{\lambda_1^2+\lambda_2^2+ \lambda_3^2}} \nonumber \\ c_s &= \frac{3 \lambda_3}{\sqrt{\lambda_1^2+\lambda_2^2+ \lambda_3^2}}$

We have $c_l + c_p + c_s = 1$ . These three factors provide additional information characterizing the shape of a tensor. Finally, the anisotropy can be measured as a difference with the spherical case:

$c_a = c_l + c_p = 1 - c_s = 1 - \frac{3 \lambda_3}{\sqrt{\lambda_1^2+\lambda_2^2+ \lambda_3^2}}$

References

[R3]	(1, 2) C. Westin, S. Maier, H. Mamata, A. Nabavi, F. Jolesz and R. Kikinis : Processing and visualization of diffusion tensor MRI. Medical Image Analysis, 6(2):93-108, 2002.

Inputs¶

[Mandatory]

eigenvalues_file: a file name (mandatory)

a second order tensor eigen values

[Optional]

cl_basename: a string (optional)

the basename of the output linearity coefficients file

cp_basename: a string (optional)

the basename of the output planarity coefficients file

cs_basename: a string (optional)

the basename of the output sphericity coefficients file

fa_basename: a string (optional)

the basename of the output fa file

md_basename: a string (optional)

the basename of the output md file

output_directory: a directory name (optional)

the output directory where the tensor scalars will be written

Outputs¶

fractional_anisotropy_file: a file name

the name of the output fa file

linearity_file: a file name

the name of the fie that contains the linerity coefficients

mean_diffusivity_file: a file name

the name of the output md file

planarity_file: a file name

the name of the fie that contains the planarity coefficients

sphericity_file: a file name

the name of the fie that contains the sphericity coefficients