Elsevier

NeuroImage

Volume 37, Issue 1, 1 August 2007, Pages 116-129
NeuroImage

A Bayesian framework for global tractography

https://doi.org/10.1016/j.neuroimage.2007.04.039Get rights and content

Abstract

We readdress the diffusion tractography problem in a global and probabilistic manner. Instead of tracking through local orientations, we parameterise the connexions between brain regions at a global level, and then infer on global and local parameters simultaneously in a Bayesian framework. This approach offers a number of important benefits. The global nature of the tractography reduces sensitivity to local noise and modelling errors. By constraining tractography to ensure a connexion is found, and then inferring on the exact location of the connexion, we increase the robustness of connectivity-based parcellations, allowing parcellations of connexions that were previously invisible to tractography. The Bayesian framework allows a direct comparison of the evidence for connecting and non-connecting models, to test whether the connexion is supported by the data. Crucially, by explicit parameterisation of the connexion between brain regions, we infer on a parameter that is shared with models of functional connectivity. This model is a first step toward the joint inference on functional and anatomical connectivity.

Introduction

Information about anatomical connexions in the human brain forms a crucial link for understanding brain function at a systems level (Passingham et al., 2002). The recent emergence of diffusion tractography has offered the potential to uncover this information at a level of detail that was previously impossible in the living human brain (Behrens et al., 2003b, Mori et al., 2005, Parker et al., 2005, Catani et al., 2005, Rushworth et al., 2006). Already, information from diffusion tractography is beginning to inform ideas about functional roles of cortical regions in humans (Croxson et al., 2005, Rushworth et al., 2006). However, such ideas are based only on the anatomical structure of the connexion patterns. A fundamental goal of systems neuroscience is to understand how different cortical regions interact with one another to produce observed behaviour. In order to answer this question, we need to know not only which connexions exist between different brain regions but also which of these connexions are recruited while subjects are solving specific problems. This is a problem that is tackled indirectly via functional connectivity analyses. Recent papers using fMRI (Friston et al., 2003) and M/EEG (David et al., 2006) have described techniques for analysing the passage of information through successive cortical regions. However, each of these techniques lacks information either in the spatial or temporal domain, and they would all benefit greatly from the connexional constraints that could be offered by diffusion tractography. A direct functional connexion between two regions is contingent on the existence of a direct anatomical connexion. This paper proposes a novel way to perform diffusion tractography, that makes it amenable to be combined naturally with functional connectivity analyses. It is a first step toward a symmetrical fusion between the two types of data.

To date, there have not been any attempts to merge functional data with diffusion tractography to infer on structural and functional connectivity, as there has not been an obvious approach for combining the two sets of data. Any attempt to merely constrain functional connectivity analyses with the output of tractography is hindered by inaccuracies in tractography data (Ramnani et al., 2004), particularly by the large number of connexions that are invisible to current tractography algorithms. This problem would be alleviated by solving the two problems simultaneously in a symmetric manner, with data from the functional study informing the diffusion tractography, and tractography connexions informing the functional connectivity analysis. Such an approach would improve the sensitivity and accuracy of both analyses and, potentially, would allow us to infer the anatomical connexions that are being recruited functionally during different phases of the task at hand.

Here, we take a first step towards such a solution. We readdress the tractography problem in a way that makes it amenable to symmetrical fusion with functional data. Instead of merely tracking through a local orientation field in either a deterministic (Conturo et al., 1999, Jones et al., 1999, Mori et al., 1999, Basser et al., 2000, Catani et al., 2002) or probabilistic (Behrens et al., 2003a, Hagmann et al., 2003, Parker and Alexander, 2005) manner, we parameterise the connexion between two regions at a global level [exactly as is done in functional connectivity experiments (Friston et al., 2003)]. By inferring on this new global model in a Bayesian framework, we can then compare the evidence for the models when the connexion is present or absent, allowing us to test whether the connexion is supported by the data. Again, this approach is deliberately borrowed from the world of functional connectivity (Penny et al., 2004), making it a simple conceptual step (although perhaps a challenging technical one!) to combine the two distinct sets of data.

This global approach to tractography immediately offers other important benefits. Unlike with local tractography approaches (Conturo et al., 1999, Jones et al., 1999, Mori et al., 1999, Basser et al., 2000, Behrens et al., 2003a, Hagmann et al., 2003, Parker and Alexander, 2005), small local regions of uncertainty within the image caused by noise or partial volume effects will not deflect pathways that are supported by the data along the rest of their length. We show that, if correct inference is performed, global connectivity information affects the local estimation of fibre orientation only at voxels that otherwise have high uncertainty. Lastly, in cases where there is a known connexion between two regions, acknowledging this connexion explicitly as part of the tractography process significantly increases the sensitivity and robustness of the tractography process. At first glance, this fact seems neither surprising nor interesting. If we tell our algorithm that a connexion exists, it is then better at finding it! However, there are two important implications. First, as described above, we can perform tractography twice, once enforcing a connexion and once enforcing its absence. The global information will improve the sensitivity of whichever is the correct model. We can then use Bayesian model comparison between the two models to test for a connexion in the data. Second, enforcing connexion will condition and constrain the tractography process even if the precise locations of the termination points are not given. We show an example where we enforce connexions from the internal capsule to the hand and face area of the primary motor cortex. Such connexions are notoriously difficult to trace using local tractography techniques. However, by enforcing the existence of the connexions, we can trace the connecting pathways, and infer directly on their locations within the internal capsule. This gives us a robust and natural method for performing connectivity-based parcellations.

Section snippets

Graphical description of the problem

The mechanism with which global information can be used to drive tractography can be understood through the Bayesian graphical model shown in Fig. 1a. The graph shows that the data Y are generated by the parameters of the local model (inside the dashed box). These parameters model the – local – diffusion properties (d, s0), fibre orientations (Θ, Φ), amount of anisotropy (f), and noise (Σ). After observing the data, the Bayesian theory allows us to infer on the posterior distribution of these

Inference on the model

We use an MCMC algorithm to infer the posterior distribution on all the parameters of the model. We use a Metropolis Hastings (MH) sampler for computing the posterior distribution of the parameters d, s0, f, Θ, Φ and F. Proposal distributions for these parameters are zero mean Gaussians with standard deviations tuned to give a jump acceptance of 0.5. The scale parameters Σ and Λ are excluded from the sampling because it is possible to integrate over their uncertainty analytically, and obtain a

Increasing local complexity – Multiple fibre partial volume model

We have shown how to add connectivity constraints to the partial volume-based probabilistic tractography. Until now, we used a simple partial volume model with one fibre direction per voxel. This model can be improved to account for more complexity in the data (Hosey et al., 2005, Behrens et al., 2007). In that case, Eq. (1) is written as:μij=s0j{(1l=1Lfjl)exp(bidj)+l=1Lfjlexp(bidjriTR(θjl,ϕjl)AR(θjl,ϕjl)Tri)}where L is the total number of fibre direction compartments in each voxel, and fjl

Decreasing prior knowledge – Inference on C?

An important, and still unsolved, issue in diffusion-based tractography is how to test whether two brain regions are connected, i.e. how to infer on the existence of a connexion. It was initially assumed that such a test would be possible with probabilistic tractography techniques. However, such techniques do not give probabilities on the existence of a connexion, but instead give probabilities on the trajectory of the strongest connexion. The method proposed in the present paper allows to

Results

Diffusion-weighted data were acquired by using echo planar imaging (72.2-mm-thick axial slices, matrix size 128 × 104, field of view 256 × 208 mm2, giving a voxel size of 2 × 2 × 2 mm). Diffusion weighting was isotropically distributed along 60 directions by using a b value of 1000 s/mm2. For each set of diffusion-weighted data, 5 volumes with no diffusion weighting were acquired at points throughout the acquisition. Three sets of diffusion-weighted data were acquired for subsequent averaging to

Summary

There are several advantages to using global information, such as priors on connexions among brain regions, to inform probabilistic tractography. This information helps to recover uncertainty in locations where the data are noisy. Moreover, this global model includes parameters such as the location of connexion extremities which are useful for connectivity-based segmentation. A global model allows us to formalise the problem of inferring on connexions in a Bayesian framework. Model comparison

Conclusion

Within the framework presented in this paper, it is possible to guide the probabilistic tractography by prior knowledge about regional connexions. This is a different way of doing tractography that gives equivalent results as probabilistic tractography for major anatomical tracts. It allows inference on local direction parameters which are guided by large-scale connexions when local data are highly uncertain. This modelling allows to address two important questions. First, it allows to perform

Acknowledgments

The authors would like to acknowledge funding from the Dr. Hadwen Trust for Humane Research (SJ), the UK Engineering and Physical Sciences Research Council (MWW) and the UK Medical Research Council (TEJB). We are also extremely grateful to Dr. Heidi Johansen-Berg and Dr. Matthew Rushworth for their valuable contribution to this work.

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