Resolving fiber crossing using advanced fast marching tractography based on diffusion tensor imaging
Introduction
One of the requirements for understanding brain function, both in physiological and pathological conditions, is the knowledge of the architectonical organization inside the brain's white matter. The technique of diffusion tensor imaging (DTI) (Basser et al., 1994, Pierpaoli et al., 1996) promises to deliver the required information in vivo in the human brain. DTI allows to measure and to describe the three-dimensional (3D) water diffusion properties across tissue by a local diffusion tensor on a voxel-by-voxel basis. For each voxel with a distinct fiber alignment, the tensor's principal eigenvector corresponds to the direction of the main diffusivity. All three orthogonal eigenvectors, weighted by the corresponding eigenvalues, span the diffusion ellipsoid which can be regarded as a 3D visualization of the diffusion distribution within a single voxel. However, it is important to mention that the diffusion tensor is a simplified model of the underlying real diffusion process in the tissue. The diffusion tensor in each voxel models a single compartment with Gaussian distributed water molecule displacements.
A further advancement of DTI is the 3D tracking of fibers, known as fiber tracking (for review articles, see Bammer et al., 2003, Mori and van Zijl, 2002, Mori et al., 2002a, Wieshmann et al., 2000). This technique allows to reconstruct cerebral fiber bundles in the white matter. Fiber tractography based on DTI provides a potential method for exploring the connectivity network of the brain. However, it is essential to be aware of the limitations of this procedure. Jones (2003) showed that each calculated principal diffusion eigenvector is associated with an uncertainty due to, e.g. partial volume effects. Note that the typical in-plane resolution is approximately 1.5 × 1.5 mm2, which is several orders of magnitude larger than the diameter of a single axon. Thus, due to DTI's voxel-averaged quantity, the principal eigenvector does not necessarily correspond to the main fiber direction, particularly when bundles intersect, branch or merge. A few groups (Barrick and Clark, 2004, Lazar and Alexander, 2005) observed tensor field singularities in such regions which result in an indistinct main diffusion direction. Consequently, misleading fiber pathways may be reconstructed if the tracking algorithm incorporates only the principal eigenvector for determining the propagation direction. This has been identified as one of the main problems for fiber tracking based on DTI data (Basser et al., 2000, Mori et al., 1999, Mori et al., 2002b, Watts et al., 2003, Xue et al., 1999). A further implication is that DTI cannot differentiate between fiber kissing and oblique fiber crossing or branching situations. Therefore, without additional prior knowledge, no tracking algorithm is able to resolve these situations correctly. Another problem stems from the image noise inherent to each MR acquisition procedure, deviating the directions of the principal eigenvectors (Lazar and Alexander, 2003, Lazar and Alexander, 2005, Tournier et al., 2002).
Different approaches have been proposed so far in order to solve these problems. One of the most promising for line propagation algorithms is the tensor deflection method (Lazar et al., 2003, Westin et al., 2002). It uses the entire information carried by the diffusion tensor to estimate the trajectory. When the diffusion distribution within the voxel of interest is planar, suggesting inhomogeneous fiber alignment, the propagation direction is determined by the superposition of the first and second eigenvector. Another approach uses a diffusion metric based on the diffusion ellipsoid's shape, differentiating between prolate, oblate and spherical fiber distributions (Weinstein et al., 1999, Westin et al., 2002, Zhang et al., 2004). Depending on that measure, the propagation direction is calculated by a weighted sum of all three diffusion eigenvectors.
In contrast to line propagation algorithms, the level set (Sethian, 1996b) and fast marching (FM) (Sethian, 1996a) techniques are based on evolving a 3D wave front through a volume of interest. Parker et al. proposed an FM algorithm (Parker et al., 2002b) adapted to DTI data. Its reliability was verified in the macaque brain (Parker et al., 2002a, Parker et al., 2002b). The accuracy of the FM algorithm was also investigated in vivo in human DTI data, where anterior callosal fibers, the optic radiation and the pyramidal tracts were reconstructed (Ciccarelli et al., 2003b). Additionally, a reproducibility study, investigating the same three pathways, was performed (Ciccarelli et al., 2003a).
The front evolution of FM tracking algorithms is controlled by a velocity function, where direction and speed depend on the diffusion tensor. The FM method can be visualized as the motion of a wave front emanating from a source. This concept permits trajectories to diverge and merge. Another benefit is the access to a voxel-based connectivity metric (Parker et al., 2002b). This allows to rank resulting connections of the voxels to the seed area. The connectivity metric can be used to color code the reconstructed fiber bundle corresponding to its likelihood or can serve as threshold criteria to abort fiber reconstruction.
Until now, all proposed FM algorithms based on low angular resolution DTI data consider solely the principal eigenvector. In regions where the principal eigenvector does not describe adequately the main fiber direction (e.g. in crossing situations), these algorithms may fail to reconstruct correct fiber pathways. Another intrinsic property of the FM method is its discrete front evolution, meaning that the possible propagation directions are limited by the number of neighboring voxels. This limitation causes discretization errors. Tournier et al. (2003) suggested an adaptive evolution grid to overcome this obstacle. However, their algorithm is not able to resolve fiber crossing.
In this study, an advanced implementation of FM is proposed. Combining the advantages of classical FM and of the tensor deflection approach, it is hypothesized that the proposed algorithm can resolve correct pathways in brain regions where fiber systems intersect. To this end, the algorithm takes into account the entire information contained in the diffusion tensor. Furthermore, it should allow to reduce the discretization error originating from the finite number of evolution directions.
Finally, the performance of the proposed algorithm is measured and compared with existing algorithms both in artificial diffusion data including fiber crossings and in vivo in well known anatomical structures of healthy volunteers.
Section snippets
The Fast Marching algorithm
The standard FM algorithm described in Parker et al. (2002b) starts from a manually defined seed area. From the start region, a wave front propagates through the 3D volume. During iteration, the voxels are subdivided into three groups: “front voxels” at the inner boarder of the spreading cloud, voxels in the narrow band (members of the adjacent outer shell) and the “outside voxels” (see Fig. 1). In each iteration step, the front expands from an origin front voxel at location r′ to a narrow band
Results
Fig. 6 shows a comparison of circular trajectories using the standard FM algorithm and the advanced FM algorithm incorporating additional front expansion directions. While the standard FM trajectories are of low quality due to discretization errors, the trajectories reconstructed using the advanced FM algorithm are in better agreement with the concentric arrangement of the principal eigenvectors. Compared to the standard FM result, the intrinsic discretization error is reduced.
The results
Discussion
An in vivo verification of fiber tracking algorithms faces serious challenges. The knowledge of the anatomy beyond the gross fiber pathways is absent, and the voxel resolution is restricted. A resolution and specificity comparable to the one of tracer and histological staining techniques would be required for a reliable validation. However, the strong invasive nature of these techniques prohibits an in vivo appliance. For these reasons, the properties of the presented technique were studied
Acknowledgments
The authors are grateful for the continuing support of Philips Medical Systems and the financial support by the Strategic Excellence Project Program (SEP) of the ETH Zurich.
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