Acquisition Parameters

  • \(\delta\): pulse duration
  • \(\Delta\): time separation between the two diffusion gradients
  • \(G\): gradient strength
  • \(\mathbf{n}\): gradient direction
  • TE: echo time
Diffusion-sensitizing gradients

A diffusion-sensitizing magnetic field gradient can be characterized by a so-called q-vector \(\mathbf{q}\) in q-space is defined as:

\[ \mathbf{q} = \frac{\gamma \delta}{2 \pi} \mathbf{g} = \frac{\gamma \delta G}{2\pi} \mathbf{n}, \]

where \(\gamma\) is the gyromagnetic ratio, \(\delta\) is the gradient pulse duration, \(\mathbf{g}\) is the diffusion-sensitizing gradient and \(G = \| \mathbf{g} \|\) is its norm. The direction of the q-vector is given by the unit vector \(\mathbf{n} = \mathbf{g} / \| \mathbf{g} \|\). The b-value is then defined as:

\[ b = \gamma^2 \delta^2 G^2 (\Delta - \delta/3), \] according to the Stejskal-Tanner equation (Stejskal and Tanner 1965), where \(\Delta\) is the diffusion time, which leads to:

\[ \mathbf{q} = \| \mathbf{q} \| \mathbf{n} = \frac{\gamma \delta G}{2 \pi} \mathbf{n} = \frac{1}{2 \pi} \sqrt{\frac{b}{\Delta - \delta / 3}} \mathbf{n}. \]

The application of a diffusion-sensitizing gradient leads to an MR signal decay \(E(\mathbf{q})\) with respect to the \(T_2\)-weighted signal in the absence of diffusion gradients.

Tissue Microstructure

Taken from

  • Axon diameter: it is of the order of \(1\mu m\)
  • Axonal density: both intra- and extra-axonal spaces exist
  • Number of axons: it depends on the voxel size
  • Orientation dispersion: axons are not perfectly aligned
Tissue Model