Synopsis

A new class of diffusion MRI tractography processing methods is emerging, that involve the integration of ideas from existing "local" and "global" tractography algorithms in order to inherit the benefits of both. In this article, the basic premise behind these methods is explained, the history and details of these methods are provided, and the consequences and importance of these developments is contextualised.

Target audience

Outcomes / objectives

Purpose

Tractography methods for the reconstruction of white matter connectivity from diffusion MRI data are typically grouped into one of two classes (a third class, "geodesic", is not considered here):

• "Local" methods generate connection trajectories individually and independently, by propagating from some seed position in 3D space according to the local estimated fiber orientations^[1]. These methods are the most commonly used in the field. Their advantages include very good computational efficiency, and the ability to apply anatomical constraints during reconstruction to ensure that connections produced are biologically plausible^[2,3]. However a fundamental limitation of these approaches is that for the density of reconstructed connections is not necessarily reflective of the density of underlying axon fibers within that pathway^[4]; this places substantial restrictions on how these reconstructions can be used quantitatively.
• "Global" methods, conversely, aim to reconstruct all white matter connectivity in a concurrent manner, optimizing the whole tractogram while retaining consistency with the raw diffusion MRI data^[5-8]. This has the crucial advantage of reconstructing local reconstructed track densities that are consistent with the underlying biological fiber densities. However, these methods are computationally expensive, placing limitations on the achievable density of the reconstruction; furthermore, the anatomical plausibility of connection endpoints is not guaranteed, resulting in many connections terminating erroneously within the white matter (and hence not actually contributing to quantitative connectivity estimates).

Given the relative strengths and weaknesses of these two groups, research has been undertaken (and is ongoing) on methods that integrate concepts from these two classes, that aim to inherit the benefits of both approaches. Most of these methods involve taking a whole-brain fiber-tracking reconstruction generated using a "local" method, and use global image information to attribute quantitative characteristics to the tractogram. In this article, the basic mechanisms by which these methods operate is explained, and examples of such methods are provided.

Methods

A simple model is first described, that can be used to generate reconstructed image data from a tractogram, and hence enable comparison between a tractogram and the acquired image data.

Consider each streamline in a tractogram being represented as a cylinder (or "pipe") containing axon fibers, that follows a curved trajectory through 3D space. A principal parameter of such a cylinder would be the total cross-sectional area of the encapsulated fibers. By simple 3D geometry, multiplying this fiber cross-sectional area by the length of the cylinder yields the total spatial volume occupied by the fibers within that cylinder. Furthermore, if one is interested in the volume occupied by those fibers only within a specific region (an image voxel, for example), then a reasonable approximation is to multiply that fiber cross-sectional area by the cylinder's length within that region only.

Therefore, there is a simple way of generating a predicted fiber volume in each image voxel based on a reconstructed tractogram: For each streamline traversing that voxel, the volume contributed from that streamline is the product of the length of the streamline within that particular voxel, and some latent parameter that relates to the fiber cross-sectional area contributed by the streamline; these can simply be summed across all streamlines traversing that voxel to obtain the total fiber volume.

Now consider that in MRI, the intensity of the signal measured from any particular tissue type within an image voxel is proportional to the volume of that tissue within that particular image voxel. Based on this, the fiber volume reconstructed by the tractogram within each image voxel can be used to generate a predicted MR signal from the tractogram.

There are a few assumptions or limitations within this model:

• It does not consider higher-order fiber parameters, such as myelination or axon diameters;
• It does not consider other MR-related imaging parameters that may vary (e.g. diffusivity, T2 variations).
• It does not consider spatial intersections between those cylinders (e.g.^[9]); it only uses this construct to predict a total fiber volume in each voxel.

In addition to these, there are a number of confounds that must be considered when generating reconstructed images from a tractogram:

• In diffusion MRI, the image intensity is based on not only the volume occupied by the underlying fibers, but also the angle between the fiber orientation and the diffusion sensitization gradient. The local tangent direction of the streamlines must therefore be taken into account.
• In any particular image voxel, white matter fibers may not be the only constituent tissue type; other tissues with different properties, or cerebro-spinal fluid, may have contributed to the measured diffusion MRI signal.

The techniques proposed thus far generally adopt one of two approaches to address these confounds:

• Mapping to raw diffusion MRI data: A forward model is used to define how an individual streamline contributes toward the reconstructed diffusion MRI signal. The reconstructed signal is then generated through a spherical convolution between the streamlines orientations and this model. Optionally, models may also be defined for other tissues or fluid, and these may be optimized as part of the tractogram model.
• Mapping to a diffusion model: By first fitting a diffusion model (e.g. spherical deconvolution^[10]) to each image voxel, estimates of fiber bundle orientations and densities in each voxel are obtained. A tractogram can then be compared directly to these densities, by assigning streamlines to the appropriate fiber bundle elements based on their local orientations. Separation of contributions from different tissues or fluid is handled during the diffusion model fitting step.

Note however that these two approaches differ only by a spherical (de)convolution transformation.

Using one of these two techniques, it is possible to generate, from the tractogram, a reconstructed estimate of the data used to generate the tractogram itself (whether the raw diffusion MRI data, or the fitted diffusion model). Such a reconstruction can therefore be compared directly to the actual image data. If there is a substantial mismatch between the two, this is indicative of biases arising from the tractography reconstruction process: If the tractogram is an accurate representation of the underlying fiber configuration, then the MRI signal predicted from that tractogram should accurately mimic the actual acquired data.

For the methods described in this article, the overarching goal is to manipulate or perturb a whole-brain fiber-tracking reconstruction, in a manner that makes it more consistent with the diffusion image data. The algorithms by which this is achieved, and the forward models used for generating reconstructed image data from the tractogram, vary considerably between the methods proposed thus far; however this fundamental goal is consistent between them.

Results

The history of methods for integrating global image information into otherwise "local" tractography reconstructions (at least to the awareness of the author) can be summarized as follows:

• BlueMatter (2009)^[11]: The first method of its kind to be proposed. A very large database of "candidate streamlines" was defined (180 billion), from which a small subset (200,000) was stochastically selected that was found to provide consistency with the diffusion image data. Optimization required 9 days on a 2048-core BlueGene supercomputer.
• MicroTrack (2010)^[12]: Operating in a similar manner to the BlueMatter method, this technique additionally ascribes to each streamline one of a finite set of axon diameters, and penalizes spatial overlap between individual streamlines within a super-resolution grid. To the author's knowledge, it has not however been demonstrated for a whole-brain reconstruction.
• SIFT ("Spherical-deconvolution Informed Filtering of Tractograms") (2013)^[13]: This method proposed reduced complexity of both the model and optimization algorithm, in order to be more computationally feasible for application in group studies. Rather than fitting the tractogram directly to the image data, streamlines densities are instead compared to the fibre densities estimated by a diffusion model (e.g. spherical deconvolution^[10]); and rather than a stochastic global optimization, the algorithm instead iteratively removes streamlines from the reconstruction in a gradient descent-like fashion.
• COMMIT ("Convex Optimization Modeling for Micro-structure Informed Tractography") (2014)^[14] /
  LiFE ("Linear Fascicle Evaluation") (2014)^[15] /
  SIFT2^[16]:
  Developed approximately in parallel, these three methods all adopted a similar approach to the problem of fitting a pre-computed tractogram to the image data. Rather than explicitly selecting a subset of streamlines, as was performed in prior methods, these approaches instead calculate a unique (non-negative) weight for each streamline, which modulates the contribution of each streamline toward the reconstructed signal. In terms of the model described above, this corresponds to modulating the fiber cross-sectional area contributed by each streamline individually.

Alternative approaches have also been proposed, which combine aspects of local and global reconstruction techniques using altogether different strategies:

• Lemkaddem et al., 2014^[17]: Rather than incorporating global image information into a reconstruction generated using a "local" tractography method, this technique instead uses a "local" tractography method to initialize a "global" tractography optimization. The key advantage of this approach over other "global" methods is that it guarantees biological plausibility of the termination points for all reconstructed connections.
• "Dynamic seeding"^[16]: Typically when performing whole-brain fibre-tracking using a "local" method, streamline seeds are placed in some homogeneous fashion. In this framework, reconstructed streamlines densities are compared to fibre densities estimated from the diffusion model as tractography is being performed, and new streamline seeds are placed preferentially in those regions where the reconstruction density is insufficient.

Discussion

One of the most well-known limitations in applications of diffusion MRI tractography is that the streamline count between two regions cannot be used as a reliable marker of fiber connectivity strength^[4]. The reason for this limitation is that "local" tracking algorithms do not constrain the number of streamlines generated in any particular location in the brain to be consistent with the density of underlying fibers in that location. This is however precisely the inadequacy that this new class of methods aims to solve. By constraining the tractogram such that the streamlines density matches the density of underlying fibers throughout the image, the streamline count corresponding to a particular pathway can be used as a genuine proportional estimate of the fiber density within that pathway (taking into account other limitations of tractography reconstruction). This opens up the potential for applications of diffusion MRI tractography that would otherwise be considered non-robust due to this limitation.

This is particularly consequential for those using diffusion MRI to estimate the macroscopic structural connectome^[18,19]. For higher-order analyses of characteristics of this brain network to be sensible, it is necessary for the quantified connectivity between every pair of brain regions to be somehow proportional to the magnitude of direct communication (or "bandwidth") between those regions. The total fiber cross-sectional area connecting regions is possibly the most biologically relevant metric that can currently be extracted from a diffusion MRI experiment that is robust to crossing fibers^[20]. Moreover, it has been shown that the application of such methods prior to connectome construction has a significant and therefore non-negligible influence on network characteristics^[20,21].

An important aspect of the methods described here is that they cannot be applied in isolation to just a subset of brain connections. This is because a fundamental assumption intrinsic to the way in which these algorithms operate is that all of the fiber density present in a voxel must have a corresponding representation in the tractogram; presenting a subset of the tractogram to the algorithm would violate this assumption. It is however still possible to quantify the connectivity strength of just a particular pathway of interest: this is done by generating a whole-brain tractogram, applying a method such as those described, isolating those streamlines from the tractogram corresponding to the pathway of interest, and taking the sum of the fiber cross-sectional areas contributed by those streamlines.

Some of the methods described in this article have already been made freely available to the research public:

• COMMIT: https://github.com/daducci/COMMIT
• LiFE: https://francopestilli.github.io/life/
• SIFT, SIFT2, "Dynamic seeding": mrtrix.org

It is expected that research into more advanced methods will continue in this domain in the future, providing increasingly accurate reconstructions and quantitative measures for whole-brain structural connectivity analysis.

Conclusion

Acknowledgements

References

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