# Velocity-Weighted Phase Contrast

Another straightforward case is coherent flow. This is equivalent to velocity- weighted imaging. Although a drift term was originally considered by Torrey [40], coherent flow effects are not traditionally considered alongside diffusion-weighted methods.

Coherent flow is described locally by a vector u, giving us

where the uM is the dyadic product of the two vectors. Substituting this into Eq. (15) yields the following solution

where Rz(9) is a rotation in the transverse plane by an angle 9. The angle of rotation is given by the dot product of the local flow and the q vector, manifesting as a phase shift without a change in magnitude.

The combination of coherent flow and diffusion is equivalent to an advection- diffusion equation where spins both flow and diffuse but the two processes are essentially uncoupled.

# Stretched-Exponentials and Space-Fractional Super-Diffusion

The physical interpretation of the stretched-exponential is sometimes a little mysterious. This section will show that they can be seen as modelling a particular form of transport process: the Levy walk. Particles executing Levy walks make many short steps with occasional much longer displacements. They are characterised by random walk-like processes with a power-law distribution of step lengths. i.e. step length ' follows

Just as a Brownian random walk leads to a Fickian flux, a similar calculation shows that a Levy walk leads to a flux with a fractional-order derivative (see Appendix for derivation). Fractional derivatives are derivatives of non-integer order. They have been extensively applied to transport theory. Whilst not providing a unique link to the microstructural details of the system, fractional approaches nonetheless provide access to systems in which diffusion-like transport it not well-described by Gaussian diffusion processes derived from the usual form of Fick’s law.

For mathematical simplicity, we will assume an isotropic process and treat diffusion in one dimension. In this case the flux is written as

where D is a constant and 0 < < 1 is the order of the derivative and step

distribution and щу is the Reisz fractional derivative [29]. The derivation of the fractional form of Fick’s law is given in the appendix.

The solution of the Bloch-Torrey equation with a fractional flux is

This form of spin transport is known as super-diffusion, so-called because the mean squared-displacement of particles increases faster than linear with time. The stretched exponential in this form is thus a model of super-diffusion.

Stretched exponentials have been employed by several authors [9, 12, 16], although the form chosen varies. The form derived here has the stretching exponent on the q terms only, although authors often apply the stretching exponent to both q and t. The model has been criticised on the grounds that the exponent is difficult to interpret physically, but the derivation here illustrates that it can be directly associated with the statistics of the underlying random walk: the fitted exponent is related to transport, not structure.