So far we have considered models with only a single, well-mixed continuum present in the sample. There is still a large class of models in the diffusion MR literature which we have not covered. To go beyond this we need to introduce the idea of multiple continua. The idea is that instead of a single Bloch-Torrey equation, we have a system of two or more. Here we have equations with their own relaxation process, their own fluxes, and potentially also exchange between them. A two- continuum system might be written as

where M_{;} is the magnetisation in compartment i, J_{;} and R_{;} describe the flux and relaxation processes in each one and E_{ik} defines the rate of exchange between compartments. Conservation of total number of spins also means we have that

The overall magnetisation of the system is the sum of the two components

where f defines the relative sizes of the compartments. This formulation can be readily extended to three or more compartments. This version of the system is quite general, but the solutions can be difficult to work with in practice. For example, there is a degeneracy between the different relaxation constants and the volume fractions which prevent unique fitting without fixing one or the other.

The simplest assumption is zero flux and zero exchange. This leads to a bi- or multi-exponential model of T1 and T2 decay, which is often applied in T2 imaging when more than one tissue type in present in a voxel. A more common approach in diffusion imaging, however, is to assume a common relaxation process in all compartments and to concentrate on flux.

Assuming zero exchange, (i.e. E is diagonal) we can immediately construct the two-tensor model in which we have a common relaxation process, separate Fickian fluxes and no exchange.

The solution of which is a superposition of two exponentials of tensors. This was a common model in the early days of HARDI methods, and also covers Behrens’ Ball and Stick model [8], in which we restrict the eigenvalues of the tensors such that one has all three equal and the other has only one non-zero.

The most general version of this model with a closed-form solution is two Fickian compartments with non-zero exchange. This is the Karger model [28], which models a pair of well-mixed fluids with a Poisson exchange process. The Karger model can

Table 2 Relaxation, flux, and exchange choices

R

J

E

Model

Compartmental

0

N

Multicomp. T_{2}

Uniform

Fickian

N

Multiexp./multitensor

Uniform

Fickian

Y

Karger

Uniform

Fractional

N

Multi-stretched exp.

be shown to be equivalent to a bi-exponential model with diffusivities transformed by the exchange process [15].

Assuming no exchange and the same relaxation terms in all compartments, multiple continua models allow descriptions of the diffusion signal to be built up additively from different components. Since we are free to choose any flux terms we like we can potentially describe multiple exponentials (or tensors), or potentially less common choices such as mixtures of stretched exponentials [17]. We summarise which choices lead to which models in Table 2.