In this page you can compute any tensor metric from the eigenvalues of a diffusion tensor. Insert the values of the three eigenvalues in decreasing order, then click the HyperSpatioTensoral Button.

$1 \ge \lambda_1 \ge \lambda_2 \ge \lambda_3 \ge 0$

Lambda 1:

Lambda 2:

Lambda 3:

Metric Value
Mean Diffusivity
Fractional Anisotropy
Linearity
Planarity
Sphericity
Volume Ratio
Example of diffusion tensor in a crossing region.

### Mean Diffusivity

• Used in early stroke detection (it is sensitive to the presence of edema).
• High MD in CSF ($$9.6 \cdot 10^{-3} mm^2/s$$).
• Lower MD in GM and WM (between $$1.95\cdot 10^{-3} mm^2/s$$ and $$2.2\cdot10^{-3} mm^2/s$$).
• Water at 37° C has MD = $$9 \cdot 10^{-3} mm^2/s$$ .
• In ventricles or edema MD can be higher than in water due to water transportation.
$MD = \bar{\lambda} = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}$

### Fractional Anisotropy

• Measures how much of the observed diffusion is due to the anisotropy.
• A ball has $$FA=0$$.
• It is the distance between the tensor`s ellipsoid shape and a spherical isotropic tensor.
• Very often mistaken for a measure of WM integrity, but it is not.
$FA = \sqrt{\frac{3}{2}}\sqrt{\frac{\left(\lambda_1 - \bar{\lambda}\right)^2 + \left(\lambda_2 - \bar{\lambda}\right)^2 + \left(\lambda_3 - \bar{\lambda}\right)^2 }{\lambda_1^2 + \lambda_2^2 + \lambda_3^2}}$

### Linearity

• It measures how much the tensor is elongated along the main axis.
$C_L = \frac{\lambda_1 - \lambda_2}{\lambda_1}$

### Planarity

• It measures how much the tensor is close to a plane.
• If $$C_P$$ is high, the concept of principal fiber direction is not well defined.
$C_P = \frac{\lambda_2 - \lambda_3}{\lambda_1}$

### Sphericity

• It measures how much the tensor is close to being a sphere.
• If $$C_S$$ is high, the concept of principal fiber direction is not well defined.
• Notice that $$C_L + C_P + C_S = 1$$.
$C_S = \frac{\lambda_3}{\lambda_1}$

### Volume Ratio

• It is the ratio between the volume of the ellipsoid and that of a sphere with the same MD.
• It tells you how much the tensor is far from being a sphere.
$V_R = \frac{\lambda_1 \lambda_2 \lambda_3}{\bar{\lambda}^3}$