“Fixels” (and “Dixels”)¶

Internally we have created a couple of new terms that we find invaluable when discussing diffusion MRI processing methods and statistics. We’d like to share these with our user base in the hope that others will gain advantages from using the same terminology, and also so that we all know what everyone else is talking about! Anyone using MRtrix3 to develop their own software may also see these terms scattered throughout the library code, so will need to know what they represent.

All MRtrix3 users should be familiar with the terms ‘pixel’ and ‘voxel’; these are abbreviations of “picture element” and “volume element”, corresponding to the smallest element within a 2D picture and 3D volume respectively. However in Diffusion MRI we also deal with orientation information within each image volume element; so we wanted terminology to allow us to convey the types of discrete elements that we deal with on a daily basis.

We have settled on the following two terms. The first of these, ‘fixel’, will appear frequently throughout the MRtrix3 documentation and in online discussions, and will therefore satisfy the requirements of the majority of users. The second, ‘dixel’, is typically reserved for internal technical discussion; however due to its occasional usage (and its inconsistent use in early presentations, see final note below), we are additionally providing its full definition here for interested readers.

‘Fixel’: Fibre bundle element¶

The term fixel refers to a specific fibre bundle within a specific voxel. Alternatively, consistently with the definitions of ‘pixel’ and ‘voxel’, it can be thought of as a “fibre bundle element”: the smallest discrete component of a fibre bundle. Each fixel is parameterized by the voxel in which it resides, the estimated mean orientation of the underlying fibres attributed to that bundle, a fibre density (or partial volume fraction), and potentially other metrics.

In reality, fixels have been used in the field of Diffusion MRI for a long time: multi-tensor fitting, ball-and-sticks, any diffusion model that is capable of fitting multiple anisotropic elements to each image voxel, can be considered as estimating fixels. However in the past, researchers have resorted either to lengthy descriptive labels in an attempt to express the nature of the data being manipulated, or have adopted existing terms, which can lead to confusion with the original sense of the terms. Furthermore, these labels are not applied inconsistently between publications; we hope that the term ‘fixel’, being unambiguous with other interpretations of “fibre bundle” or “fascicle” or other examples, will slowly become the standard term for describing these data.

Historically, in MRtrix we are accustomed to dealing with FODs that are continuous functions on the sphere, rather than having a discrete number of fibre directions in each voxel. However, if the FOD is segmented in any way (either through peak-finding as shown in this paper and implemented in the sh2peaks command, the segmentation algorithm described in the appendices of the SIFT NeuroImage paper and provided in the fod2fixel command, or more advanced methods), each discrete feature of a particular FOD can be labelled a ‘fixel’, as each represents a set of fibres within that voxel that form a coherent bundle in orientation space.

The term ‘fixel’ has now appeared in the literature with the publication of the statistical method, Connectivity-based Fixel Enhancement, as well as the more general framework of Fixel-Based Analysis, which together allow for the inference of group differences not just at the voxel level, but the fixel level; that is, if only one fibre bundle within a crossing-fibre voxel is affected in a cohort, we hope to both identify the bundle affected, and quantify the group effect that is specific to that bundle.

‘Dixel’: Directional Element¶

This term is used less frequently, and hence may not be relevant for all readers. If you have not seen it used before, you may in fact prefer to avoid the following text in order to keep things simple…

Imagine a single image voxel, the data for which is in fact a function on the sphere (i.e. varies with orientation). We now take samples of that function along a set of pre-defined directions on the unit sphere. Each of those samples is referred to as a ‘dixel’: a directional element within a specific voxel. Each dixel is described by the voxel in which it resides, the direction along which the relevant spherical function was sampled, and the intensity of the function in that direction.

Importantly, it is the combination of the voxel location and sampling direction that describe the dixel. If a different direction were used to sample the spherical function, this would be a different dixel with a different associated value; likewise, if the spherical function in an adjacent voxel were sampled along the same direction, that would also be a different dixel with a different associated value. Each dixel is a unique sample of a particular spatially-varying spherical function.

Most commonly, the term ‘dixel’ is used to refer to the situation where a set of directions on the unit sphere has been used to sample some function; for instance, sampling the amplitudes of a Fibre Orientation Distribution (FOD), which is otherwise a continuous function expressed in the Spherical Harmonic (SH) basis. However, by the definition of the term, ‘dixel’ could also be used to describe a single voxel within a particular image volume in a HARDI experiment; if the HARDI signal in a single voxel is considered to be discrete samples of the orientation dependence of the diffusion signal in that voxel, then each of those samples could be labelled a ‘dixel’.

Therefore, the fundamental disambiguation between ‘fixels’ and ‘dixels’ is as follows:

• A ‘dixel’ is typically assumed to represent a sample of a spherical function along some pre-determined direction, where that direction belongs to some dense basis set of equally-distributed unit directions that has been used to sample an otherwise continuous (hemi-)spherical function.
• ‘Fixel’, on the other hand, is used to describe a set of fibres within a voxel that are sufficiently similar in orientation that they are indistinguishable from one another, and therefore form a fibre ‘bundle’ within that voxel.

Some observations / contexts in which the term ‘dixel’ may be useful:

• The mrview “ODF overlay” tool is capable of loading “Dixel ODFs”. These can be either a set of direction-based samples on the sphere, or it can be used to directly visualise the diffusion signal within a particular b-value shell, since both of these cases correspond to a set of directions on the unit hemisphere, where each direction has associated with it an ‘intensity’ / ‘amplitude’.
• In the original Apparent Fibre Density (AFD) manuscript, the statistical analysis was performed by performing a t-test in each of 200 directions in each voxel, and then detecting connected clusters in position & orientation space. This can be thought of as “dixel-based cluster statistics”.
• In the FOD segmentation method provided in the fod2fixel command mentioned earlier, the algorithm first samples the amplitude of the FOD along a set of 1,281 directions, before identifying fixels based on accumulating these directions / samples. So this process can be thought of as converting the FOD from a continuous SH representation, to a dixel representation, then finally to a fixel representation.

Note

During the development of many of the aforementioned methods, a presentation was made at ISMRM demonstrating “Tractographic threshold-free cluster enhancement” (this is now referred to as “Connectivity-based Fixel Enhancement (CFE)”). During the presentation itself, the term ‘dixel’ was used to refer to a specific direction within a specific voxel; but a direction that corresponds to a particular fible bundle in that voxel. You may observe that this definition is in fact consistent with what we have labelled here as a ‘fixel’, rather than a ‘dixel’; this is because at the time when this presentation was made, these two terms had not yet been disambiguated. The definitions made within this documentation page are what will be used from now on by the MRtrix3 developers; and we hope by the wider community as well.