Bloch-Torrey partial differential equation

In diffusion MRI, a time-varying magnetic field gradient    is applied to the tissue to encode water diffusion. The resulting complex transverse water proton magnetization in the rotating frame satisfies the Bloch-Torrey PDE:

where

•  ,

•   where is the echo time at which the signal is measured,

•     is the gyromagnetic ratio of the water proton,

• is the imaginary unit,

• is the intrinsic diffusion tensor in ,

• is the -relaxation time in , and

• is the magnetization in .

The initial conditions are given by

where is the initial spin density in . The outer boundary conditions are given by

and the interface conditions by

where

• is the unit outward pointing normal vector of

• is a boundary relaxation coefficient for ,

•   is the permeability coefficient on ,

• and account for the spin density equilibrium between the two compartments, with either

  •   , in which case a uniform spin density across compartments is favored in the absence of a gradient, or

  •   and  , which ensures that the non-uniform initial spin density is preserved in the absence of a gradient [9].

The diffusion MRI signal is measured at echo time  . This signal is computed as the spatial integral of the final magnetization  :

Periodic boundary conditions

A sample of periodic tissue fills a box   , with the cells cut by its faces and their periodic images entering on the opposite faces (PeriodicBoxECS, VoronoiSetup). Because of the term  , the magnetization is not periodic, but the phase-shifted magnetization

is, where is the wave vector (see SpinDoctor.wavevector). It solves

in the compartments, with the interface conditions of (with replaced by  ), and periodic boundary conditions on the faces of the box. The finite element mesh is periodic (the opposite faces of the box have the same points), and the points on opposite faces are identified in each compartment (assemble_matrices(model; periodic = true)). For refocused sequences,   and the signal is the integral of .