The convolution methods assume the form of the fibre response kernel, but nothing about the FOD. Other approaches make different and stronger assumptions, however, leading to a parametric representation. Given an assumption for the form of a particular compartment, a continuous version can be constructed by assuming a distribution over one or more parameters. The result is a distributed compartment with distribution parameters which can be fitted to the data.

A simple example of this approach is Jbabdi’s model of distributed diffusivities [22]. Here we assume that rather than a single free diffusion compartment, there is a continuum of non-exchanging compartments with diffusivities given by a gamma distribution. The resulting model has a closed form, with the shape and scale parameters of the gamma distribution taking the place of the single diffusivity in the single compartment version. This is similar to an argument made by Bennett [9] in relation to the stretched exponential, although the final form is quite different.

A related assumption is made in the CHARMED model [5], and other similar approaches such as AxCalibre [6] and ActiveAx [2], in which compartments are constructed from cylinders with gamma-distributed radii. In this case care must be taken to weight the cylinders’ contributions by volume fraction in the distributed compartment: larger cylinders take up more space than smaller ones and hence each one has a larger volume fraction and a larger contribution to the signal.

Another common variant of this approach is to assume a distribution of orientations on some directed geometry. Models such as VERDICT [37] assume a uniform distribution of orientation in cylinders or sticks. Between the limiting cases of completely parallel orientation and uniform orientation distribution lies orientation distributions with some finite width distribution. NODDI [45] makes use of a compartment in which the orientations of a continuum of sticks is described by a Watson distribution, which provides directional analogues to a mean and standard deviation. NODDI interprets the standard deviation as an estimate of fibre dispersion in the voxel.

The above makes it clear that NODDI is a combination of continuous and noncontinuous compartments without exchange, making strong assumptions on the form of each of them. In the current context, it is worth emphasising that it is therefore a solution of a quite artificial set of Bloch-Torrey equations. It also fixes intrinsic diffusivity and requires it to be equal across all compartments. Recent work has pointed out that this set of assumptions is flawed [26], and that the resulting model can lead to misleading conclusions [34], and well as highly biased parameter estimates [23].

Both NODDI and VERDICT combine continuous compartments with other compartments without distributions, such as spherical shells or tensors. They also restrict the parameters describing different compartments to have the same values. Table 3 summarises compartment choices in different microstructure models.

One difficulty with constructing compartment-based models is the sheer number of possible combinations to choose from. Ten to twelve possible compartments with various different constraints lead to hundreds of potential two compartment

Compartments

Model

Tensor, gamma-cylinders, ball

CHARMED [5]

Ball, astrosticks, sphere

VERDICT [37]

Tensor, cylinder, dot

ActiveAx [2]

Tensor, Watson-sticks, dot

NODDI [45]

Table 3 Compartment combinations in microstructure models models and thousands of possible three compartment models. Comparing these combinations is a significant and time-consuming undertaking, and although such comparisons have been undertaken [13, 37], strictly a model selection step is necessary in each individual application. Making a priori choices about tissue geometry also requires prior information, for example from histology, and intuition about how to relate underlying biological complexity to the very simple geometries for which explicit expressions for compartments are possible.

Another important caveat is packing. Multi-compartment models make no explicit assumption about how geometric compartments are arranged in space. Packing is captured indirectly via volume fractions and a tortuosity assumption in the extra cellular space. This can be important since these approximations may be more or less valid in different volume fraction regimes.