Since our continuous representation of the diffusion tensor, D( x) can furnish estimates of 2nd and higher derivatives of ϵ 1( x), it is prudent to use this information in a more robust and accurate numerical method to integrate these trajectories. While Euler's method is easy to explain and to implement, it is accurate only to 1st-order, and thus is susceptible to large accumulated errors and to numerical instabilities ( 19). This procedure can now be repeated starting at the new point, r(s 1) …, and can be iterated to predict the location of discrete points along the fiber trajectory, r(s). Thus, we can estimate r(s 1) from the values of r(s 0) and ϵ 1( r(s 0)). Previously, we proposed that a white matter fiber tract trajectory could be represented as a 3D space curve ( 3, 8, 11), i.e., a vector, r(s), parameterized by the arc length, s, of the trajectory. THEORY Evolution of Fiber Tract Trajectories The aims of this article are to 1) propose and describe a methodology to calculate continuous fiber-tract trajectories from discrete measured diffusion tensor MRI data 2) present a general framework for testing the fidelity and robustness of this (and of other) fiber tract following schemes 3) demonstrate that our method follows fiber tracts in the brain using in vivo DT-MRI data 4) elucidate artifacts and inherent limitations of fiber tract following schemes that employ DT-MRI data and 5) describe potential applications of DT-MRI fiber tractography. Second, the steps involved in implementing some of these more recent tract-following schemes have, to date, only been outlined schematically, making it difficult to reproduce them, and thus to compare their findings fairly with ours. First, there are many new biologically relevant findings presented here and methodological issues raised in this work, so that including additional material would make this article unnecessarily long. We do not attempt to compare and contrast our method or results with theirs. More recently, several groups have proposed tractography methods and have reported success in following fiber tracts, and even individual fascicles, over distances on a gross anatomical length scale ( 12- 15). Finally, a framework for following individual fiber tracts had to be developed, the underpinnings of which can be found in earlier works ( 3, 8, 11). This mathematical framework is described in ( 10). A methodology capable of generating a continuous, smooth representation of the measured DT-MRI data first had to be developed in order to ensure the reliability and robustness of DT-MRI fiber tractography. Just as in hydrodynamics, it is difficult to construct fluid streamlines accurately from discrete, noisy, velocity field data ( 9) here it is difficult to follow a white matter fiber trajectory using discrete, noisy, direction field data. Second, the macroscopic fiber-tract direction field, ϵ 1(x,y,z), is obtained from measured DT-MRI data that is discrete, coarsely sampled, noisy, and voxel-averaged ( 8). However, these problems have been ameliorated with the introduction of faster, more powerful gradients single-shot diffusion-weighted echo-planar imaging (DW-EPI) sequences ( 4) with higher SNR and reduced motion artifacts 5 as well as schemes to reduce eddy current artifacts ( 6), and B 0 distortion ( 7). However, until recently this end could not be realized primarily for technical and mathematical reasons: First, the resolution and quality of diffusion-weighted images (DWIs) in vivo was not adequate for this demanding application. Published 2000 Wiley-Liss, Inc.ĭiffusion tensor MRI (DT-MRI) ( 1) is the first noninvasive in vivo imaging modality with the potential to generate fiber-tract trajectories in soft fibrous tissues, such as nerves, muscles, ligaments, tendons, etc. Still, this method can provide quantitative information with which to visualize and study connectivity and continuity of neural pathways in the central and peripheral nervous systems in vivo, and holds promise for elucidating architectural features in other fibrous tissues and ordered media. Moreover, background noise in diffusion-weighted MRIs can cause a computed trajectory to hop from tract to tract. The method's reliability, however, degrades where the distribution of fiber tract directions is nonuniform. Corpus callosum and pyramidal tract trajectories were constructed and found to be consistent with known anatomy. This approach was validated using synthesized, noisy DT-MRI data. Then a Frenet equation, describing the evolution of a fiber tract, was solved. First, a continuous diffusion tensor field is constructed from this discrete, noisy, measured DT-MRI data. Fiber tract trajectories in coherently organized brain white matter pathways were computed from in vivo diffusion tensor magnetic resonance imaging (DT-MRI) data.
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