Towards inference of human brain connectivity from MR diffusion tensor data
Introduction
During the last decade, the development of magnetic resonance imaging (MRI) has led to the design of numerous segmentation methods dedicated to brain structures. For instance, cortex, white matter and basal ganglia can be extracted from standard T1-weighted MR images (Mangin et al., 1995, Poupon et al., 1998b. While such structures can be identified from a priori knowledge on MR signal and simple anatomical considerations, further brain parcellations are much more difficult to perform from macroscopic images.
However, discussions concerning the nature of psychological phenomena and their neurobiological bases often make reference to networks of brain areas. For instance, the thalamus is made up of numerous elementary nuclei; the cortical sheet can be divided in areas according to various maps related to architectural features. Moreover, information about forward and backward connections between these elementary areas has been inferred from invasive anatomical techniques for several animal species (Young et al., 1995). While a number of objections can be raised against these oversimplified architectural models, they have provided invaluable reference systems for neuroscience studies. Indeed, these parcellations and their connectivity are considered to be reproducible between individuals of the same species and share important similarities across species.
Unfortunately, most of the architectural information underlying these reference systems cannot be accessed in the living human brain. Therefore, the neuroimaging community has designed its own reference system in a very different way. This system, inspired from Talairach proportional system for surgery planning (Talairach et al., 1967), simply relies on 3D coordinates indicating a location in a template brain. Each new brain is endowed with this coordinate system through spatial normalization, namely a 3D deformation matching as far as possible the new brain macroscopic anatomy with that of the template. While spatial normalization is often required to compare functional images across individuals and across experiments (Fox et al., 1985), no simple link between the proportional system and usual architectural parcellations can be provided apart from a statistical one (Roland and Zilles, 1994).
Although impressive refinements of the normalization scheme have been achieved by the image analysis community during the last years (Miller et al., 1993, Thirion, 1998), most of these developments are bound to drift towards pure morphing approaches without consistent architectural justification. Indeed, considering the absence of a gold standard, more attention should be given to architectural value of the different features used to drive the deformation processes. In fact nobody really knows today how brains should be matched. Furthermore, nobody knows to which extent matching two different brains with a continuous deformation makes sense from a neuroscience point of view.
A first approach to increasing the role of brain architecture in spatial normalization procedures consists in imposing the perfect matching of a few well-known cortical folds (Thompson et al., 1996; Lohman and Von Cramon, 1998). Indeed, the largest cortical fissures are endowed with clear architectural value. Unfortunately, extending this approach to a higher number of folds requires better understanding of the inter-individual variability of the folding patterns (Régis et al., 1995, Lohman and Von Cramon, 1998). Furthermore, the putative architectural value of secondary folds has still to be proven.
While an increasing number of studies are dedicated to the cortex folding patterns, few groups try to analyse the complex shape of white matter (Mangin et al., 1996, Naf et al., 1996). Indeed, standard imaging modalities give little information on the underlying bundle entanglement. In this paper, we describe an emerging MR technique which may radically modify the situation. Indeed, this technique called diffusion tensor imaging (DTI), gives access to the macroscopic organization of brain white matter (Le Bihan, 1995). The basic principle stems from the orientational information provided by the phenomenon of water diffusion anisotropy in white matter. Diffusion tensor imaging characterises the diffusional behaviour of water in tissue on a voxel by voxel basis. For each voxel, the diffusion tensor yields the diffusion coefficient corresponding to any direction in space (Basser et al., 1994b). Given that one may ascribe diffusion anisotropy in white matter to a greater hindrance or restriction to diffusion across fiber axes than along them, the direction corresponding to the highest diffusion coefficient may be considered to point along a putative fiber bundle traversing the voxel. Thus, maps of the highest diffusion direction can be produced which provide a striking visualization of the white matter pathways and their orientation.
This paper addresses the reconstruction of fiber trajectories from such direction maps. The general aim is the mapping of structural connectivity, that is, the possibility to assert which cortical areas or basal ganglia are connected by fascicles embedded in white matter bundles. This would allow the neuroimaging community to bring its methodology closer to the ones used in standard neuroscience. Furthermore, this would allow the improvement of current models of the human cortex connectivity. Indeed, up to now, tract tracing methodologies dedicated to the human brain have been restricted to post mortem methods, which are not competitive compared with invasive methods used for animals (Young et al., 1995).
The possibility to track the putative trajectories of some fascicles, namely small fiber bundles, has been convincingly proven by several recent studies (Poupon et al., 1998a, Mori et al., 1999, Conturo et al., 1999). However, the robustness of methods which simply consist in following the direction of highest diffusion in a step by step fashion may be discussed with regard to low spatial resolution and artefacts inherent to in vivo MR diffusion data. In this paper, we address the ill-posed nature of the problem which in our opinion calls for a regularization framework.
The paper is organized as follows. In the first section, the various stages of the process leading to the computation of tensor images from a sequence of diffusion-weighted data are briefly delineated. The second section outlines a simple fascicle tracking algorithm dealing with potential junctions between fascicles. This algorithm converts any direction map extracted from diffusion data into a tracking map. The third section highlights the ill-posed nature of the tracking problem which calls for a regularization framework. This need lead us to design Markovian models aimed at modeling the geometry of white matter which is compared to the geometry of spaghetti plates. Then, one of these models is used as a priori knowledge to infer a regularized direction map from raw tensor diffusion data. In a final section, we study the tracking algorithm behaviour according to the regularization weight and describe some tracking results obtained in a series of normal volunteers.
Section snippets
Diffusion tensor images
In brain tissues, molecules are endowed with a Brownian motion macroscopically leading to a diffusion process (Le Bihan, 1995). In the case of isotropic liquids, the probability that a molecule covers distance r during time t follows a Gaussian law with variance 6ct where c is the diffusion coefficient that characterizes molecule mobility. In anisotropic environments, mobility is different along each direction of space. Hence, diffusion is a tridimensional process which can be modeled by a
Computation of the tracking map
This section outlines a method which transforms any direction map inferred from diffusion data into a tracking map. A direction map is an image of unitary vectors which indicate the putative local fiber directions. Usually, such direction maps are simply made up of the tensor eigenvector associated with the highest eigenvalue. In the following, we will propose a method to construct regularized direction maps according to a priori knowledge on white matter geometry. A tracking map endows each
Ill-posed nature of the tracking problem
The tracking map inferred from the direction map relies on the definition of links towards best forward and backward neighbors. These links which stem from the hypothesis of slow direction variations of actual white matter fascicles are clearly dependent on errors in the direction map. Indeed a small error on can switch the links to different neighbors. Furthermore, some errors can lead to ‘pathological’ voxels turning out to be a dead end for the tracking process (empty or )
Results
All the results described in the following stem from deterministic minimization of U(D) (cf. Eq. (11)). The state space of the random variables has been descretized in 162 uniformly distributed directions. An ICM like algorithm is used to get the minimum the nearest to the direction map (Besag, 1986). The computation time is about 1 hour on a conventional workstation. Further work is required to assess the interest of using stochastic minimization.
Discussion and conclusion
Mapping the living human brain connectivity with clinical scanners and reasonable acquisition times is a difficult challenge. Indeed, fascicle tracking is a mathematically ill-posed problem especially sensitive to noise in the direction map inferred from the DTI data. For instance, each error in the direction map can lead to a random fork in the tracking process without underlying anatomical meaning. Furthermore, the typical spatial resolution of echo planar images results in partial volume
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Toward global tractography
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