This section describes models which take a slightly different approach to that considered so far. The continuity and Bloch-Torrey approach is fundamental to diffusion MRI, but can be formulated under slightly different assumptions than those so far. Here we consider two further cases.
Random Permeable Barriers
This model assumes a single, effective medium-level description of a population of diffusing spins in a compartment containing randomly oriented permeable barriers . This description is on a length scale long enough that the contributions of the barriers can be treated as an spatial average, and that the central limit theorem applies locally, but still much shorter than the size of a typical scan voxel. This means that care is required during the derivation of he Bloch-Torrey equations to step upwards in scale from the microscopic, disordered regime to the intermediate, ensemble average. The authors employ a renormalisation group approach, and show that diffusion at this scale is described by a diffusivity with a power-law time dependence, specifically
where Dis the asymptotic, long time diffusivity, A is a constant, and в is an exponent defining the time dependence.
This approach is powerful, identifying different universality classes in temporal scaling which contain information about environmental disorder. The exponent value differentiates between disorder classes. в = 0.5 corresponding to long-range order and в = | for the short-range order case of dimension d. The short time limit is also related to the surface area to volume ratio experienced by spins and has been used to estimate surface area to volume ratio in tissue .
The approach requires measurements over a wide range of diffusion times to infer critical exponents, and is often used in conjunction with oscillating gradient acquisitions, which provide access to shorter diffusion times than the more common PGSE. Fitting is typically performed in the frequency domain. This model has been applied to imaging in brain  and muscle .