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Deterministic White Matter Tractography 

Deterministic White Matter Tractography
Chapter:
Deterministic White Matter Tractography
Author(s):

Andrew L. Alexander

DOI:
10.1093/med/9780195369779.003.0022
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Summary Box

  • The orientation information of the diffusion tensor may be used to reconstruct estimates of white matter pathways in the human brain. The methods for doing this are called white matter tractography.

  • A primary assumption of many tractography algorithms is that the direction of greatest diffusivity (the major eigenvector) is roughly parallel to the local white matter fiber bundle direction.

  • Different white matter tractography approaches include streamlines defined by the major eigenvector and tensor deflection defined by the full tensor. Tractography approaches vary in their methods for integration of the pathways, starting and stopping criteria, and constraints. The simplest tractography algorithms follow the major eigenvector direction at discrete locations in small, discrete steps.

  • The accuracy and precision of deterministic tractography algorithms are limited by image noise, artifacts, and crossing fibers. Models for estimating the amount of tract dispersion (variance) have been developed. In general, the tract error dispersion increases with distance and the inverse of the signal-to-noise ratio and the inverse of the diffusion anisotropy. Small errors in deterministic tractography can lead to very large errors.

  • Deterministic white matter tractography has several applications, including noninvasive visualization of white matter pathways, segmenting of specific tracts in the brain for image analysis, and relating white matter tract anatomy to brain tumors and lesions in patients who are candidates for neurosurgery.

The connections among brain regions are mainly contained within the white matter, which is comprised of myelinated axon bundles. The reader is referred to Chapter 1 by Catani for a detailed description of the anatomy of these pathways. Until recently, the mapping of these connections was performed using anatomical dissection techniques in cadavers and invasive chemical tracer studies in animal models (see the Chapter 3 by Axer for a review). The ability to noninvasively map the white matter connection patterns of the brain would be extremely valuable for delineating brain networks and examining the effects of normal changes (e.g., development and aging), disease, and injury of specific white matter pathways. Conventional structural imaging techniques like T1-, T2- and proton density–weighted imaging generally offer superb contrast between major tissue groups in the brain—gray matter (GM), white matter (WM), and cerebral spinal fluid (CSF), which is well suited for studying macrostructure but not specific white matter pathways. Diffusion-weighted imaging (DWI) techniques offer the ability to both probe tissue microstructure and map out the white matter connection patterns.

The diffusion of water in WM is greatest in the direction parallel to the fiber orientation. The anisotropic diffusion properties of biological tissues like white matter were first described nearly 20 years ago and provide a unique contrast mechanism (Chenevert et al., 1990; Moseley et al., 1990). Using DWI, it is possible to estimate not only the directional variance of the diffusion (e.g., the anisotropy) but also the local direction of greatest diffusion. The development of diffusion tensor imaging (DTI) provided a model basis for describing the diffusion anisotropy and orientation (Basser et al., 1994). This ability to map the orientation of WM fibers using DTI has been of great interest to both neuroimaging researchers and clinicians. The two main challenges associated with mapping the diffusion directions are 1) the visualization of the fiber direction and 2) the estimation of the WM fiber trajectories from the directional information derived from DTI. Tractography methods for estimating the fiber trajectories have recently been developed and appear promising for mapping the organization of major WM fiber pathways noninvasively on individual subjects. In this chapter, the basic principles of WM tractography are introduced and different algorithms are briefly discussed. The limitations of WM tractography are also described. Finally, example applications are discussed.

The Major Vector versus Fiber Orientation

Some of the early DWI studies (e.g., Moseley et al., 1990) noted that the signal in WM was more highly reduced when the diffusion-weighting gradients were applied in the direction parallel to regions of coherent WM fibers. In DTI the diffusion is characterized by the magnitude (trace), directional variance (anisotropy), and the orientation (eigenvectors) of the anisotropic diffusion. It is generally assumed that the direction of greatest diffusivity (the major eigenvector) is parallel to the direction of the WM fibers in a voxel. This is a reasonable assumption in relatively large, homogenous WM regions with parallel fiber organization. However, the parallel assumption is often violated in regions with complex fiber crossings and dispersion (see Chapter 27 by Alexander and Seunarine and Chapter 28 by Tournier). Despite this limitation, most WM tractography algorithms assume that the major eigenvector is tangent to the fiber bundle.

Visualizing Fiber Orientation

Several approaches have been implemented to visualize the directional organization and coherence of WM fiber tracts on the basis of DTI. Visualization techniques include vector maps and glyphs such as ellipsoids, cylinders, or superquadrics (Kindlmann and Westin, 2006; see Chapter 21 by Leemans in this book). The local coherence of the eigenvectors may be mapped by either using the dot-product sum of the eigenvectors or computing the scatter matrix in a neighborhood (e.g., Field et al., 2004; Wu et al. 2004). The most commonly used eigenvector visualization method is to map the direction of the major eigenvectors, e1, as a combination of RGB colors—e.g., red = e1x = right–left; green = e1y = anterior–posterior; blue = e1z = inferior–superior. Maps of WM tract directions may be generated by weighting the RGB color map intensity by an anisotropy measure such as fractional anisotropy (FA) (Pajevic and Pierpaoli, 1999). In most cases, color maps have been used to visualize only the major eigenvector organization; however, it is also possible to visualize the medium and minor eigenvectors using color mapping (Lazar et al. 2005). The three eigenvectors are assumed to be orthogonal. If any two eigenvectors are determined, the other eigenvector is already defined. In addition, the second and third eigenvectors are generally assumed to be normal to the local fiber direction except in areas of crossing WM fibers. Note that the minor (smallest) eigenvector is least likely to be affected by crossing fibers. For many applications such as presurgical WM mapping, the use of color labeling is useful for identifying specific WM tracts and visualizing their rough trajectories (Witwer et al., 2002; Jellison et al., 2004).

The most advanced visualization strategy is WM tractography (WMT), which uses the directional information from diffusion measurements to estimate the trajectories of the WM pathways. WMT increases the specificity of WM pathway estimates and enables the three-dimensional visualization of these trajectories, which may be challenging using cross-sectional RGB maps. Figure 22.1 illustrates several methods for mapping local tissue organization from DTI, including eigenvector color maps, ellipsoidal glyphs, major eigenvector maps, and tractography. WMT algorithms can be split into two major classes, deterministic and probabilistic tractography. Deterministic tractography is the focus of the remainder of this chapter, and probabilistic tractography is described in the following chapter.

Figure 22.1 DTI representations of DTI orientation information. A box centered on a region containing the genu of the corpus callosum and anterior limb of the internal capsule is expanded and cartoon examples of directional information are shown. The most common method for describing the orientation is the eigenvector color map (a). Ellipsoidal glyphs (b) and vector maps (c) have also been used to depict the local organization of the diffusion tensor field. Using the eigenvector information, the fiber trajectories may be estimated using tractography (d).

Figure 22.1
DTI representations of DTI orientation information. A box centered on a region containing the genu of the corpus callosum and anterior limb of the internal capsule is expanded and cartoon examples of directional information are shown. The most common method for describing the orientation is the eigenvector color map (a). Ellipsoidal glyphs (b) and vector maps (c) have also been used to depict the local organization of the diffusion tensor field. Using the eigenvector information, the fiber trajectories may be estimated using tractography (d).

Deterministic Tractography Algorithms

The general principle of deterministic tractography algorithms is to use the directional information described by the diffusion tensor. The most common directional assignment corresponds to the major eigenvector of the diffusion tensor. A deterministic tractogram is generated by starting from one or more “seed” locations, typically within white matter, and propagating the trajectories according to the tractography algorithm until the tracts are terminated (e.g., Figs. 22.1 and 22.2). Specific constraints may be placed on the tractography algorithm to determine whether or not the result is consistent with predicted connectivity patterns. The basic algorithms are described in the next sections.

Figure 22.2 Illustration of DTI tractography using different fiber integration methods. FACT (Mori et al., 1999) and Euler (EUL) are piecewise linear methods. FACT changes the direction at the interfaces between voxels, whereas Euler uses a fixed step size. The Runge-Katta (RK) method is also a stepwise integration method that yields a nonlinear and more accurate solution to curved trajectories. In general, the RK methods yield smoother solutions, although all approaches generate similar solutions. Smoother trajectories can also be generated with Euler using smaller step sizes.

Figure 22.2
Illustration of DTI tractography using different fiber integration methods. FACT (Mori et al., 1999) and Euler (EUL) are piecewise linear methods. FACT changes the direction at the interfaces between voxels, whereas Euler uses a fixed step size. The Runge-Katta (RK) method is also a stepwise integration method that yields a nonlinear and more accurate solution to curved trajectories. In general, the RK methods yield smoother solutions, although all approaches generate similar solutions. Smoother trajectories can also be generated with Euler using smaller step sizes.

Streamlines

The simplest approach for WM tractography is based on streamline algorithms operating on the vector field of major eigenvectors. Streamline algorithms were originally developed to visualize flow patterns in fluid mechanics. Streamline tractography algorithms were originally proposed by Basser (1998) for diffusion tensor imaging.

The streamlines are estimated by integrating the partial differential equation: [22.1]

r(τ)τ=νtraj(r(τ)).
where r (τ‎) is path, δτ‎ is the temporal “step,” and vtraj is the vector field that defines the tangent to local path direction. Most streamline algorithms for tractography use the major eigenvector to define the local trajectory directions, vtraj = e1, at each step (e.g., Conturo et al., 1999; Mori et al., 1999; Basser et al., 2000). Several approaches have been proposed for integrating the streamline pathway and are illustrated in Figure 22.2. The simplest approaches use linear step-wise algorithms such as FACT (Mori et al., 1999) or Euler integration (Conturo et al., 1999). These integration methods approximate Eq. [22.1] using a Taylor series expansion (Basser et al., 2000): [22.2]
r(τ1)~r(τ0)+αe1(r(τ0))
As long as a is sufficiently small relative to the curvature of the local tract, the estimated position r(τ‎1) at τ‎1 can be estimated from the initial position r(τ‎0) and the local eigenvector e1(r(τ‎0)). For most tractography studies, α‎ is usually fixed (e.g., Euler), although it can vary, as in the case of FACT. Another primary difference between FACT and Euler methods is the local estimation of e1. The FACT method uses the same e1 over the entire voxel, which is akin to nearest neighbor interpolation. Euler integration typically uses an interpolation of the DTI data to estimate the local diffusion tensor and major eigenvector.

Several approaches have been used to interpolate the local e1 direction, such as interpolating the raw diffusion-weighted image data (Conturo et al., 1999), interpolating the local diffusion tensors (most common approach), and fitting a continuous description of the tensor field (Basser et al., 2002). Higher-order integration methods with continuous derivatives such as second or fourth order Runge-Kutta (RK) have also been widely used, which enable more accurate estimates of curved tracts (Basser et al., 2000). Linear integration algorithms are slightly simpler and faster, although this is not a significant issue for most tractography applications with modern computers.

By using streamline methods, major WM tracts may be reconstructed from noninvasive DTI data in the living human brain. A beautiful example of the superior longitudinal fasciculus, which includes tracts involved with language function, is shown in Figure 22.3.

Figure 22.3 Streamlines tractography example of the superior longitudinal fasciculus. Apparent WM tracts originate in prefrontal and premotor cortex (Broca's area) and project back to Wernike's area before arching around the insula and projecting forward to the temporal lobe. Image courtesy of Derek Jones (adapted from Catani et al., 2002).

Figure 22.3
Streamlines tractography example of the superior longitudinal fasciculus. Apparent WM tracts originate in prefrontal and premotor cortex (Broca's area) and project back to Wernike's area before arching around the insula and projecting forward to the temporal lobe. Image courtesy of Derek Jones (adapted from Catani et al., 2002).

Tensor Deflection

The major eigenvector will work well for cases where the diffusion tensor is highly anisotropic and prolate (“cigar”) shaped. However, less anisotropic diffusion tensors will have a greater degree of ambiguity in the major eigenvector and may reflect regions of crossing WM fibers. In this case, use of the full diffusion tensor may yield more realistic tractography results. One approach, tensor deflection (TEND), defines the trajectory vector as the product of the diffusion tensor and the incoming vector direction (Lazar et al., 2003): [22.3]

vtraj=D.vin

This approach is used to propagate tracts through regions with low anisotropy and tends to penalize trajectories with high curvature. In regions of isotropic diffusion, the estimated tract direction will be unaltered for TEND, which is a reasonable guess of the direction as the major eigenvector is not well defined. The tensor deflection operation is illustrated in Figure 22.4. For crossing perpendicular fibers, TEND is able to pass through the region of intersection correctly (Lazar et al., 2003). However, for nonperpendicular crossing regions, the technique does not accurately follow the correct trajectory. Generally, the results for both conventional streamlines with the major eigenvector and tensor deflection are similar, although slightly smoother for TEND.

Figure 22.4 Illustration of the tensor deflection (TEND) operation, which uses the full diffusion tensor to estimate the trajectory direction. The deflected direction (red arrow) is a function of both the tensor anisotropy and orientation relative to the incoming vector (green arrow). The output deflection direction will be relatively undeviated when either the tensor is nearly perpendicular to the incoming vector direction or when the tensor is more isotropic.

Figure 22.4
Illustration of the tensor deflection (TEND) operation, which uses the full diffusion tensor to estimate the trajectory direction. The deflected direction (red arrow) is a function of both the tensor anisotropy and orientation relative to the incoming vector (green arrow). The output deflection direction will be relatively undeviated when either the tensor is nearly perpendicular to the incoming vector direction or when the tensor is more isotropic.

One variation of the TEND algorithm is called tensorlines (Weinstein et al., 1999; Lazar et al., 2003), which is a weighted combination of the major eigenvector (streamlines), tensor deflection (TEND), and an undeviated component: [22.4]

Vout=fe1+(1f)((1g)vin+gD.vin)

A comparison of tractography results with different algorithmic parameters is shown for corpus callosum and superior longitudinal fasciculus reconstructions in Figure 22.5. Although the patterns are similar, there are obvious differences in the estimated connection patterns. All of the results appear reasonable, so it is difficult to assess without any gold standard for comparison.

Figure 22.5 Comparison of results for different tractography algorithms in the corpus callosum (top row) and the superior longitudinal fasciculus (bottom row). The same seed region was used with each algorithm. It is clear that the results are influenced by the algorithm.

Figure 22.5
Comparison of results for different tractography algorithms in the corpus callosum (top row) and the superior longitudinal fasciculus (bottom row). The same seed region was used with each algorithm. It is clear that the results are influenced by the algorithm.

Seeding and Stopping Criteria

White matter tractography estimates are initialized at so-called seed locations. Two main strategies are used for seed placement. The first is to select a voxel or region of voxels where the tractograms will be initiated. The second approach is to select seeds over the entire brain. For both approaches, multiple seeds can be placed in a voxel with subvoxel locations. Subvoxel seed placements can occasionally generate multiple tract solutions, particularly in heterogeneous WM regions. However, it should be noted that multiple solutions do not indicate subvoxel resolution. The whole-brain seed methods generate nearly all possible pathways for the DTI data, though there is considerably higher redundancy in the pathways. Regional seed strategies are sometimes used when trying to extract a specific pathway or mapping tracts from a specific region. For example, seeds can be placed in the cerebral peduncles to map out the trajectories of the corona radiata. One potential issue with this approach is that regional seeding may lead to incomplete tract reconstructions. For this reason, whole-brain seeding is generally preferred.

Criteria must be defined to terminate tracts when they either leave the tissue region of interest or become unreliable, i.e., the estimate of the major eigenvector has a large variance. Common stopping criteria include the tract entering a region with low FA (e.g., FA 〈 0.2) or the trajectory bending more than a prescribed angle in a single step. The FA threshold is typically used to limit the results to regions where the major eigenvector is well defined, more specifically to the white matter. The obvious limitation is that FA can be quite low in regions of WM, including regions of the centrum semiovale where fiber tracts are crossing (e.g., see Fig. 22.6). This will cause some tracts to terminate prematurely. Another strategy would be to segment the WM from the DTI data or from a coregistered structural (i.e., T1-weighted) image. Thresholded FA maps and segmented WM maps are shown in Figure 22.6. Curvature criteria are based on the assumption that WM trajectories are smooth. In most regions, large angular deviations are not deemed to be realistic. Furthermore, in some cases tractography results can appear to bend back on themselves; an angular threshold can reduce these effects. Conversely, there are pathways such as the Meyer's loop portion of the optic pathway and the short subcortical U-fibers that have high curvature. A curvature stopping criterion in these cases may not be desirable, although some caution should be used in the interpretation.

Figure 22.6 Example FA threshold maps often used as stopping criteria for tractography studies. A white matter segmentation map using FA and mean diffusivity (MDC) is shown on the right. In this case, the FA threshold of 0.2 appears to do quite well at segmenting the white matter; however, there are white matter regions that are excluded because the FA is too low, for example, in the centrum semiovale (red arrow). Reducing the FA threshold to 0.1 corrected this problem, but much of the gray matter is also included.

Figure 22.6
Example FA threshold maps often used as stopping criteria for tractography studies. A white matter segmentation map using FA and mean diffusivity (MDC) is shown on the right. In this case, the FA threshold of 0.2 appears to do quite well at segmenting the white matter; however, there are white matter regions that are excluded because the FA is too low, for example, in the centrum semiovale (red arrow). Reducing the FA threshold to 0.1 corrected this problem, but much of the gray matter is also included.

Constraints

The goal of most WM tractography studies is to map out specific WM pathways. After the initial tractograms have been generated, regional constraints are often imposed to extract the pathways that meet the specific connection criteria. For example, the corticospinal tract connects the projection fibers from the cerebral peduncles through the posterior limb of the internal capsule to the subcortical WM of the motor cortex. The most common approach is to manually draw regions of interest that encompass the tract trajectory. Rules based on Boolean logic can be developed to select or exclude specific WM pathways (see Fig. 22.7). Several tractography studies describe approaches for selecting specific pathways (e.g., Conturo et al., 1999; Catani et al., 2002; Mori et al., 2002). Regions can be either inclusive or exclusive. For example, a pathway following one WM trajectory for a while (e.g., corticospinal tract) and then suddenly switching to another pathway (e.g., corpus callosum) violates known anatomy and the results should be excluded.

Figure 22.7 Illustration of tractography constraints using regions of interest.

Figure 22.7
Illustration of tractography constraints using regions of interest.

Inclusive regions should be specified near the ends of the main fiber trunk to minimize the amount of spurious branching. Other potential constraint regions include deep nuclei (basal ganglia, thalamus) or areas of cerebral cortex. Tract selection criteria for three WM pathways, the inferior longitudinal fasciculus (ILF), the inferior frontal-occipital fasciculus (IFO), and the uncinate fasciculus (UNC), are illustrated in Figure 22.8. Even within specific WM tracts it is possible to generate tract criteria to parcellate specific components of the tract (e.g., Catani et al., 2005; Huang et al., 2005).

Figure 22.8 Constrained tract-selection examples for the inferior longituindal fasciculus (ILF), inferior fronto-occipital fasciculus (IFO), and uncinate fasciculus (UNC) using manual regions of interest (yellow and blue outlined regions) selected in the occipital lobe (ILF and IFO), the frontal lobe (IFO and UNC), and the temporal lobe (ILF and UNC). The selected tracts are shown in red. Reproduced from (Mori et al., 2002) with permission from John Wiley & Sons, Inc.

Figure 22.8
Constrained tract-selection examples for the inferior longituindal fasciculus (ILF), inferior fronto-occipital fasciculus (IFO), and uncinate fasciculus (UNC) using manual regions of interest (yellow and blue outlined regions) selected in the occipital lobe (ILF and IFO), the frontal lobe (IFO and UNC), and the temporal lobe (ILF and UNC). The selected tracts are shown in red. Reproduced from (Mori et al., 2002) with permission from John Wiley & Sons, Inc.

Other Variations

It is very challenging to obtain accurate WM trajectories in regions of complex WM with multiple oblique tracts in a voxel. Although TEND can work better in some instances of crossing WM fibers, the solution is somewhat limited (Lazar et al., 2003). Another potential solution is to constrain the tract solutions. In one approach, G-TRACT (Cheng et al., 2006), the two tract termination regions are defined and a solution to connect these regions is sought. The G-TRACT approach enables one to evaluates multiple tract solutions and select the one with the lowest energy, which is a function of the distance and the coherence of the local tract direction and the major eigenvector. However, in regions with crossing WM fibers, the tract solution will evaluate potential routes that are oblique to the major eigenvector.

Other potential solutions to the tracking problem include finding a global tractography solution that fits the local DTI data. These solutions include spin-glass (Mangin et al., 2002) and Gibbs tracking algorithms (Kreher et al., 2008). Instead of performing a forward integration, these approaches attempt to estimate a global tractography solution that best fits the underlying DTI data. The Gibbs tracking algorithm (Kreher et al., 2008) uses an iterative optimization algorithm that 1) estimates a global tract connection pattern, 2) generates a corresponding synthetic DTI data set that is compared against the original DTI data, and 3) perturbs the tract connection pattern to minimize the difference between the measured and synthetic data. This process is iterated until an optimal solution is found. A comparison of corpus callosum reconstructions made with FACT, probabilistic tractography, and Gibbs tracking is shown in Figure 22.9. The Gibbs tracking method generates connections to more lateral brain regions compared with those generated with more conventional streamline algorithm methods. It is expected that the corpus callosum should connect both medial and lateral brain regions. However, the computational time for these global tractography methods is considerably longer than that of standard tractography approaches (Kreher et al., 2008).

Figure 22.9 Comparison of corpus callosum tractography results using deterministic FACT tractography (a, c), probabilistic tractography (PICo) (d), and a new Gibbs tracking algorithm (b, e). The Gibbs tracking method is used to find a global tractography solution and yields much more diffuse and lateral connection patterns. Note that most of the lateral connections are missing with both the conventional deterministic and probabilistic tractography methods. Reproduced from Kreher et al. (2008) with permission from John Wiley & Sons, Inc.

Figure 22.9
Comparison of corpus callosum tractography results using deterministic FACT tractography (a, c), probabilistic tractography (PICo) (d), and a new Gibbs tracking algorithm (b, e). The Gibbs tracking method is used to find a global tractography solution and yields much more diffuse and lateral connection patterns. Note that most of the lateral connections are missing with both the conventional deterministic and probabilistic tractography methods. Reproduced from Kreher et al. (2008) with permission from John Wiley & Sons, Inc.

Up to this point, the discussion has assumed a simple diffusion tensor model. More complex diffusion models (e.g., multiple tensor) and advanced diffusion image acquisition methods such as Q-ball imaging reconstruction and diffusion spectrum imaging (Tuch, 2004; Assaf and Basser, 2005; Wedeen et al., 2005; Wu and Alexander, 2007) can be used to estimate multiple fiber solutions within a voxel (see Chapters 27 and 28 for further discussion of these methods). Tractography algorithms have recently been implemented to estimate tract trajectories in complex WM (Chao et al., 2008; Wedeen et al., 2008).

Limitations and Errors

White matter tractography results are often visually stunning. However, a significant limitation of WMT is that the errors in an estimated tract are generally unknown. Further, the visual aesthetic of WMT, which looks like actual WM patterns, can instill a false sense of confidence in specific results. Unfortunately, there are many potential sources of errors that can confound WMT results. Very small perturbations in the image data (i.e., noise, distortion, ghosting, etc.) may lead to significant errors in a complex tensor field such as the brain. The main effects are summarized below.

Tract Dispersion

A model of the effects of image noise on the confidence in tractography data was originally described by Anderson (2001). Investigations of tract errors were also performed using tractography simulations in prescribed tensor fields. The main source of error in tractography results is tract dispersion, which describes the variance of the estimated trajectories. Since most streamline methods are based on a forward integration of the major eigenvector directions, the dispersion will increase with the distance from the seed location. Also, since the accuracy of the major eigenvector is affected by image noise, tract dispersion will increase as the signal-to-noise ratio (SNR) is decreased.

According to a perturbation theory model first described by Anderson (2001), the dispersion in tract estimates 〈Δ‎x2〉 from image noise is roughly proportional to the distance (N.w, where N is the number of voxels and w is voxel size) and inversely proportional to the squares of the eigenvalue differences (Δλ‎j = λ‎l−λ‎j) and SNR (Anderson, 2001; Lazar and Alexander, 2003): [22.5]

Δxj2=Nw2.E/(ΔλjSNR)2
where E is a factor related to the diffusion tensor encoding scheme and the diffusion tensor orientation, and j = 2,3 corresponding to the medium and minor eigenvalues. The eigenvalue contrast term reflects the relative dispersion associated with the tensor anisotropy. In general, the E term is smallest when the fiber direction is parallel to one of the encoding directions, although the directional dependence decreases as the number of encoding directions decrease (Lazar and Alexander, 2003; Jones, 2004).

Analytic plots of tractography dispersion versus the eigenvalue dispersion are shown in Figure 22.10. These plots show that even for a modest SNR (e.g., 20) and a large distance of 100 mm, the tractography error is only on the order of a voxel dimension (∼2 mm) or less. Figure 22.11 plots the tract dispersion versus distance for two voxel sizes, which have been adjusted for relative SNR values. The plots show that SNR is much more important than small voxels with respect to tractography estimates.

Figure 22.10 Modeled tract dispersion versus eigenvalue contrast (λ‎1 − λ‎2,3) at different levels of SNR. Other model parameters were six encoding directions, b = 1000 s/mm2, 2 mm isotropic voxels, and a distance of 100 mm.

Figure 22.10
Modeled tract dispersion versus eigenvalue contrast (λ‎1 − λ‎2,3) at different levels of SNR. Other model parameters were six encoding directions, b = 1000 s/mm2, 2 mm isotropic voxels, and a distance of 100 mm.

Figure 22.11 Modeled tract dispersion versus distance at two different levels of SNR and voxel size. Other model parameters were six encoding directions, b = 1000 s/mm2, and λ‎1 − λ‎2,3 = 1.

Figure 22.11
Modeled tract dispersion versus distance at two different levels of SNR and voxel size. Other model parameters were six encoding directions, b = 1000 s/mm2, and λ‎1 − λ‎2,3 = 1.

Recent studies by Lori et al. (2002) and Lazar and Alexander (2003) verified the dispersion model in both simulated tensor fields for several variations of WM tractography algorithms, including both streamlines and TEND. Fortunately, the predicted tractography dispersion is relatively small compared to the voxel dimensions and SNR levels used in most DTI studies. More recently, Lazar and Alexander (2005) validated the model in real human brain DTI data using bootstrap resampling methods (described in Chapter 23 by Parker, on probabilistic tractography). A comparison of the tract dispersion model with the measured bootstrap dispersion is shown in Figure 22.12. The model and measured tract dispersion showed moderately good agreement over a distance of roughly 40 mm from the seed point. To reduce the estimated tract dispersion, researchers have developed local smoothing or regularization methods (e.g., Mishra et al., 2006). Part of the motivation for probabilistic tractography, the topic of the next chapter, is to describe the uncertainty or dispersion of WM tractography.

Figure 22.12 Comparison of bootstrap tractography dispersion against modeled dispersion in human corpus callosum pathways (seed placed in midsagittal plane). Note the generally good agreement between model and measured errors out to roughly 40 mm in either direction from the center. The red and black curves are maximum and minimum dispersions in the axes perpendicular to the tracts. In the negative distance, the measured error is much higher, reflecting a branch in the tractography solution. Reprinted from Lazar and Alexander (2005) with permission from John Wiley & Sons, Inc.

Figure 22.12
Comparison of bootstrap tractography dispersion against modeled dispersion in human corpus callosum pathways (seed placed in midsagittal plane). Note the generally good agreement between model and measured errors out to roughly 40 mm in either direction from the center. The red and black curves are maximum and minimum dispersions in the axes perpendicular to the tracts. In the negative distance, the measured error is much higher, reflecting a branch in the tractography solution. Reprinted from Lazar and Alexander (2005) with permission from John Wiley & Sons, Inc.

Tract Deviation

While the tract dispersion describes the variance or uncertainty in tractography results, systematic errors in the tract position are described by the tract deviation. Tract-deviation errors were studied using simulations by Lazar and Alexander (2003). They observed that mean deviation errors were generally minimal for streamline algorithms, particularly for highly curved trajectories; however, mean displacement errors were much higher for TEND in curved trajectories. Predicted tract deviation errors are plotted in Figure 22.13 for FACT, as well as streamlines with both Euler and RK integration. Tract deviation is influenced by the step size. In general, the step size should be significantly smaller (an order of magnitude) than the local radius of curvature. For most tractography studies, a step size on the order of 0.5 mm or smaller is likely adequate.

Figure 22.13 Mean tract displacement error versus curvature at a distance of 50 mm. Both FACT and Runge-Kutta integration show minimal displacement errors; however, the errors with conventional streamlines and Euler integration are significant. The error decreases with smaller step sizes. Also note that the curved distance is much longer than that encountered in the human brain. Plots are derived from the model obtained in Lazar and Alexander (2003) with permission from Elsevier.

Figure 22.13
Mean tract displacement error versus curvature at a distance of 50 mm. Both FACT and Runge-Kutta integration show minimal displacement errors; however, the errors with conventional streamlines and Euler integration are significant. The error decreases with smaller step sizes. Also note that the curved distance is much longer than that encountered in the human brain. Plots are derived from the model obtained in Lazar and Alexander (2003) with permission from Elsevier.

Tract Divergence and Convergence

The dispersion model described in Eq. [22.5] appears to work well in regions with homogenous and parallel diffusion tensor fields. However, the tensor fields in the human brain are heterogeneous and are more difficult to directly model. The local heterogeneity of the diffusion tensor field is described by the divergence. When tracts are propagated in divergent tensor fields where the tracts are fanning outward, the dispersion will be increased. Conversely, when tracts move into regions with convergent tensor fields, where tracts are merging or funneling inward into narrower areas, the dispersion will be decreased. This effect was specifically examined in a tract-error simulation study (Lazar and Alexander, 2003); analytical models have not yet been developed.

Crossing Fibers

One of the major limitations of WM tractography based on DTI is the inaccuracy of the single-tensor model in regions of crossing WM fibers. Crossing WM tracts will cause the estimated major eigenvectors to not be coincident with the fiber orientations. The local error and ambiguity lead to significant challenges with respect to accurate tractography in regions with crossing fibers. One of the more common errors encountered is the poor accuracy in mapping the corticospinal tract (CST) connections between the brainstem and lateral motor cortex regions (e.g., see Fig. 22.14). Ideally, the lateral CST fibers should cross the centrum semiovale as they descend into the internal capsule. Many tractography studies fail to reveal these connections because they do not properly pass through the centrum semiovale, which also contains crossing fibers from the superior longitudinal fasciculus, corpus callosum, and short U-fibers. As described earlier, different strategies have been implemented to resolve crossing WM pathways. These include modeling of multiple tensors or fiber components or constrained algorithm (see section Other Variations, above). These approaches may better enable the mapping of crossing WM tracts, including more lateral fibers of the CST.

Figure 22.14 Streamline tractography estimate of the corticospinal tract. Tracts were readily identified to the medial primary motor cortex region, but not the lateral motor cortex areas. The fibers to the lateral motor cortex were not able to cross through the centrum semiovale toward the internal capsule and brainstem.

Figure 22.14
Streamline tractography estimate of the corticospinal tract. Tracts were readily identified to the medial primary motor cortex region, but not the lateral motor cortex areas. The fibers to the lateral motor cortex were not able to cross through the centrum semiovale toward the internal capsule and brainstem.

DTI Artifacts

Irrespective of recent improvements to tractography algorithms, one of the main issues is that the tractography results are aesthetically appealing and appear like virtual WM tracts. Tractography is most effective in large, simple tracts, with minimal fiber crossing and high-quality DTI data, for example, in the trunk of the corpus callosum. However, the accuracy of DTI measurements is highly sensitive to a wide spectrum of artifact sources including acquisition noise, physiological noise, scanner stability, head motion, image distortions, and partial-volume averaging. Since most tractography algorithms are based on forward integration methods, errors at any step may lead to significant errors “downstream.” Yet, the appearance of smooth erroneous trajectories can often appear realistic.

An illustrative extreme case is shown in Figure 22.15. In this example, a research subject was scanned using DTI. Initial examination of tractography results in the corpus callosum revealed highly abnormal connection patterns, which at first glance yielded an impression of highly impaired connectivity in the corpus ca llosum. However, closer inspection of the raw DTI data revealed significant artifacts and noise in the measurements (see FA map shown in Fig. 22.15). The data were obtained using a suboptimal DTI acquisition protocol on a 1.5T scanner. The subject was subsequently scanned 1 year later with a more optimal DTI protocol on a 3T scanner and the tractography in the corpus callosum appeared more consistent with tractography results in other subjects.

Figure 22.15 Tractography estimates of the corpus callosum in the same individual collected using different MRI scanners and pulse sequence protocols. The corpus callosum reconstruction on the case on the left showed highly disorganized connection patterns, whereas the reconstruction on the right is much more consistent with typical white matter connectivity patterns. Closer inspection of the raw DTI data (e.g., FA maps in lower sections) revealed significant artifacts and poor image quality for the tractography study on the left.

Figure 22.15
Tractography estimates of the corpus callosum in the same individual collected using different MRI scanners and pulse sequence protocols. The corpus callosum reconstruction on the case on the left showed highly disorganized connection patterns, whereas the reconstruction on the right is much more consistent with typical white matter connectivity patterns. Closer inspection of the raw DTI data (e.g., FA maps in lower sections) revealed significant artifacts and poor image quality for the tractography study on the left.

Correspondence to Known White Matter Anatomy

Despite the known limitations in tractography algorithms, many studies have demonstrated that WM tractography can reliably generate tract estimates that are consistent with known WM anatomy. Careful and detailed tractography studies have resulted in virtual anatomic atlases of WM connections in the human brain (e.g., Catani et al., 2002, 2008; Wakana et al., 2004; Lawes et al., 2008). Recent tractography work in animal models also offers the ability to perform comparative validation with invasive tracers (Dauguet et al., 2007). The results from WM tractography yield data that rival ex vivo anatomic dissection or invasive tracer methods for mapping WM connections. The reader is referred to Chapter 26 by Lawes and Clark for more details on validation of tractography results. The other obvious advantage of DTI tractography is that it can be applied to large cohorts of living human subjects noninvasively.

Applications

White matter tractography has several potential applications. First, it offers the unique ability to noninvasively visualize the organization of specific WM pathways in individual subjects. This can be used to create stunning visual depictions of WM organization for neuroanatomic studies.

Another application is that it may be used to segment specific WM pathways or portions of WM pathways. An example of WM segmentation using tractography is shown in Figure 22.16. This use enables performance of tract-specific measurements, such as tract volume and cross-sectional dimensions, and computation of the statistics of quantitative measurements within the pathways, such as the mean diffusivity and the fractional anisotropy. Several studies have used WMT to perform measurements in specific WM pathways, for example, frontotemporal connections in schizophrenia (Jones et al., 2005), pyramidal tract development in preterm newborns (Partridge et al., 2005), and the pyramidal tracts and corpus callosum in multiple sclerosis (Vaithianathar et al., 2002; Lin et al., 2005). Although these techniques are gaining popularity because they are tract specific, there are many related issues that have not been fully addressed. The first is how to handle aberrant WM pathways. Often tracts that have undesirable sections (e.g., false branching) are completely culled, including the sections that may be correct. Also, the averaging of measures over the entire tractography volume may be complicated by differences between the measures in the trunk and those in the branches. Since most tractography studies use an anisotropy stopping threshold, this could bias the overall results. Further, the peripheral branches of the tracts will usually have much different DTI values than those in the trunk. These issues for DTI tractography analyses have not been adequately addressed.

Figure 22.16 Example segmentation of major white matter pathways using tractography derived from DTI data. The top image shows a coronal section. The middle row is two axial planes. The bottom row displays the estimated tractography pathways in a sagittal view. Images courtesy of M. Lazar.

Figure 22.16
Example segmentation of major white matter pathways using tractography derived from DTI data. The top image shows a coronal section. The middle row is two axial planes. The bottom row displays the estimated tractography pathways in a sagittal view. Images courtesy of M. Lazar.

A third application of WM tractography is to visualize specific WM patterns relative to pathology, including brain tumors (see Chapter 36 by Clark and Byrnes), multiple sclerosis lesions, and vascular malformations. The increased specificity of WM trajectories may ultimately be useful for planning surgeries (Holodny et al., 2001) as well as following the patterns of brain reorganization after surgery (Lazar et al., 2006). An example of tractography both before and after surgical tumor resection is shown in Figure 22.17. This example nicely depicts the potential utility of tractography to visualize the trajectories of WM tracts relative to pathology. However, it should be noted that WMT reconstructions still need further validation before advocating its use as a tool for surgical guidance on a widespread basis. Indeed, one recent study demonstrated that their WMT method underestimated the dimensions of the specific tract of interest, leading to an incorrect clinical interpretation and subsequent surgical complications (Kinoshita et al., 2005). A deeper discussion of this issue is provided in Chapter 36. Other studies have started to examine the relationship between specific WM tracts affected by multiple sclerosis lesions and specific clinical impairments (Lin et al., 2005).

Figure 22.17 Comparison of corticospinal tract reconstructions both before and after surgical resection of tumor mass between the cerebral peduncles and internal capsule. The tractography results nicely depict the tracts splayed around the tumor before surgery and the preservation of the pathways after surgery.

Figure 22.17
Comparison of corticospinal tract reconstructions both before and after surgical resection of tumor mass between the cerebral peduncles and internal capsule. The tractography results nicely depict the tracts splayed around the tumor before surgery and the preservation of the pathways after surgery.

Summary and Outlook

Since publication of the original WM tractography studies (Conturo et al., 1999; Jones et al., 1999; Mori et al., 1999; Basser et al., 2000), the neuroscience research community has exhibited great excitement about and interest in the ability to noninvasively map WM connections in the brain. This approach has revolutionized our ability to investigate the role of WM connections in the brain as they relate to brain function, behavior, and disease. However, it is important to consider the limitations of DTI measurements and tractography when interpreting the virtual connection maps.

A wide variety of software packages for tractography estimation are now available through commercial image analysis software, academic research software, and the MRI scanner hardware vendors. Improvements to these packages are constantly being developed that will improve the accuracy and reliability of estimated WM connections. These software developments should include improvements to DTI estimation accuracy (e.g., tensor field regularization [Wang et al., 2004; Arsigny et al., 2006; Lu et al., 2006] and robust artifact correction [Chang et al., 2005]), DTI artifact and outlier detection, tractography methods to better resolve crossing WM fibers, constrained tractography methods, and estimates of tractogram confidence (e.g., Lazar and Alexander, 2003). The latter piece is largely addressed using probabilistic tractography methods, which is the focus of Chapter 23.

Acknowledgments

I would like to thank Mariana Lazar, Aaron Field, Janet Lainhart, Derek Jones, Susumu Mori, Valerij K. and B. Kreher for generously providing figures for this chapter. I am also grateful to Derek Jones, John Ollinger, and Yu-Chien Wu for making suggestions for the chapter. Part of this work was supported by NIH grants MH62015 and MH80826.

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