Fig. 7. DTI maps computed from data in Figs. 5 and 6. The images are (top-left ): T2-weighted "reference" (or b = 0) image from DTI data; (bottom-left): mean diffusivity (note similar contrast to T2-W image with CSF appearing hyperintense); (top-middle): fractional anisotropy (hyperintense in white matter); (bottom-middle) major eigenvector direction indicated by color (red = R/L, green = A/P, blue = S/I) weighted by the FA (note that specific tract groups can be readily identified). Conventional Tl-weighted and T2-weighted images (right column) at the same anatomical location are shown

(Szeszko et al. 2005), autism (Barnea-Goraly et al. 2004), HIV-AIDs (Pomara et al. 2001; Ragin et al. 2004), and Fragile X (Barnea-Goraly et al. 2003). In nearly all cases, diffusion anisotropy (e.g., fractional anisotropy - FA) is decreased and diffusivity increased in affected regions of diseased white matter relative to healthy controls, while the reverse is true for healthy white matter in development (FA increases, diffusivity decreases).

It is important to note that diffusion anisotropy does not describe the full tensor shape or distribution. This is because different eigenvalue combinations can generate the same values of FA (Alexander et al. 2000). So, for example, a decrease in FA may reflect a decreased major (largest) eigenvalue and/or increased medium/minor (smallest) eigenvalues. FA is likely to be adequate for many applications and appears to be quite sensitive to a broad spectrum of pathological conditions. However, changes simply indicate some difference exists in the tissue microstructure. Several recent studies have looked more directly at the diffusion eigenvalues to determine if they can provide more specific information about the microstructural differences. The results have suggested that the eigenvalue amplitudes or combinations of the eigenvalues (e.g., the radial diffusivity, Dr = (X2 + X3)/2) demonstrate specific relationships to white matter pathology. For example, the radial diffusivity appears to be specific to myelination in white matter (Song et al. 2005), whereas the axial diffusivity (Da = Ài) is more specific to axonal density, making it a good model of axonal degeneration (Song et al. 2002). Tensor shape can be fully described using a combination of spherical, linear and planar shape measures (Alexander et al. 2000; Westin et al. 2002), which may also be useful for understanding WM pathology. Consequently, it is important to consider alternative quantitative methods when trying to interpret DTI measurements.

Beyond the Diffusion Tensor

The diffusion tensor is a good model of the diffusion-weighted signal behavior for low levels of diffusion weighting (e.g., b < 1500s/mm2). However, the diffusion tensor model does not appear to be consistently accurate in describing the signal behavior for higher levels of diffusion-weighting (e.g., b > 2000 s/mm2). The problems with the simple diffusion tensor model arise from two sources -(1) apparent "fast" and "slow" diffusing components (Mulkern et al. 1999) that cause the signal decay with diffusion-weighting to appear bi-exponential; and (2) partial volume averaging (e.g.,Alexander et al. 2001a) between tissue groups with distinct diffusion tensor properties (e.g., crossing white matter (WM) tracts, averaging between WM and gray matter tissues). The fast and slow diffusion signals are likely to arise from local restriction effects from cellular membranes although some have hypothesized that these signals correspond to intra- and extra-cellular diffusion.

The effect of partial volume averaging causes ambiguities in the interpretation of diffusion tensor measurements. Whereas the diffusion anisotropy is generally assumed to be high in white matter, regions of crossing white matter tracts will have artifactually low diffusion anisotropy. Consequently, in regions with complex white matter organization, changes or differences in diffusion tensor measures may reflect changes in either the tissue microstructure or the partial volume averaging components. As the diffusion-weighting is increased, the profiles of apparent diffusivity reveal non-Gaussian diffusion behavior in voxels with partial volume averaging.

A growing number of strategies have been developed for measuring and interpreting complex diffusion behavior. The methods vary in their acquisition sampling and analysis approaches. For all of the approaches described here, increasing the maximum diffusion-weighting will improve the characterization of both the slow diffusion components and the partial volume effects, although the measurement SNR will be decreased.

Fast/Slow Diffusion Modeling: Diffusion-weighted measurements over a range of diffusion-weighting have been used to estimate apparent fast and slow components of both apparent diffusivities (BEDI: bi-exponential diffusion imaging) and diffusion tensors (MDTI: multiple diffusion tensor imaging) (Niendorf et al. 1996; Mulkern et al. 1999; Maier et al. 2004). In these cases, the measurements are fit to:

where Df and Ds are the fast and slow diffusion tensors, and k is the signal fraction from the fast compartment. For a fixed diffusion encoding direction, the signal decay appears bi-exponential with diffusion-weighting. Bi-exponential strategies are appropriate for the cases where there is no significant partial voluming expected and when the diffusion may be modeled using a combination of narrow and broad Gaussian distributions. As discussed earlier, partial volume effects (e.g., crossing WM fibers) will significantly complicate the interpretation of fast and slow diffusing components. In addition, the assignment of these components has been controversial.

High Angular Resolution Diffusion Imaging (HARDI): In order to better characterize the angular diffusion features associated with crossing white matter tracts, several diffusion encoding approaches have been developed that use a large number of encoding directions (Ne > 40 up to several hundred) at a fixed level of diffusion-weighting(Alexander et al. 2002; Frank 2002). Although HARDI studies have been reported with diffusion-weighting as low as b = 1000 s/mm2 (Alexander et al. 2002), the separation of tract components will be much better for higher diffusion-weighting. The original HARDI methods estimated the profiles of apparent diffusion coefficients and used spherical harmonic decomposition methods to estimate the complexity of the diffusion profiles (Alexander et al. 2002; Frank 2002).

Higher order spherical harmonic basis functions represent signal terms that may correspond to crossing white matter tracts in the voxel. Odd spherical harmonic orders do not correspond to meaningful diffusion measurements and are generally assumed to be noise and artifacts.

The HARDI 3D diffusion profiles may also be modeled using generalized diffusion tensor imaging (GDTI) (Ozarslan and Mareci 2003; Liu et al. 2004) which use higher order tensor statistics to model the ADC profile. The GDTI methods proposed by Liu et al. (2004) demonstrate the impressive ability to model asymmetrically bounded diffusion behavior, although the method requires the accurate measurement of the signal phase, which is nearly always discarded and may be difficult to obtain in practice. One problem with these approaches is that in the case of crossing white matter tracts, the directions of maximum ADC do not necessarily correspond to the fiber directions.

One approach to this problem is the q-ball imaging (QBI) solution described by Tuch (2004), which estimates the orientational distribution function (ODF) based upon the Funk-Radon Transform. According to this relationship, the ODF for a particular direction is equivalent to the circular integral about the equator perpendicular to the direction

This integral requires that the diffusivities be interpolated over the entire surface of the sphere. Whereas the peaks in the HARDI profile do not necessarily conform to the WM tract directions (see Fig. 8), the peaks in the ODF profiles do in fact correspond to the specific WM tract direction. Since the ODF is

Fig. 8. Example QBI orientational density function (ODF) map for region at the intersection of the corpus callosum, corona radiata and superior longitudinal fasciculus. Regions of crossing WM tracts are clearly observed

estimated by integrating several measurements together, the SNR of the ODF will be much higher than that of the ADC values in the original HARDI.

Diffusion Spectrum Imaging (DSI): The fast/slow diffusion modeling and HARDI approaches represent opposing approaches to complex diffusion characterization. The combination of high angular sampling at multiple levels of diffusion weighting may be used to provide information about both fast/slow diffusion and crossing WM tract orientations. The most basic approach for this application is diffusion spectrum imaging (DSI) (Wedeen et al. 2005) which uses diffusion-weighted samples on a Cartesian q-space lattice, where q = yG5 is the diffusion-weighting wave-vector analogous to wave-vector k used in k-space sampling for MR image acquisitions. An excellent discussion of q-space imaging is found in the text by Callaghan (1994). For a specified diffusion time, A, the probability distribution of diffusion displacements, P(R, A), is related to the distribution of sampled diffusion-weighted signals in q-space, E(q, A), through a Fourier Transform:

The derivations of q-space formalism assume that the widths of the diffusion-pulses, 5, are narrow relative to the pulse spacing, A, such that 5 << A. The maximum gradient amplitudes on current clinical MRI systems cause this assumption to be violated for diffusion spectrum imaging, since 5 ~ A. The effect of this will be to slightly, but consistently underestimate the diffusion displacements, but the overall distribution shape will be correct (Wedeen et al. 2005). Note that relationship of DSI (q-space) to diffusion tensor imaging is that P(R, A) is a multivariate Gaussian and the diffusion-weighting factor is b = |q|2 (A — 5/3) or b ~ |q|2A for small 5. The DSI approach yields empirical estimates of the distributions of diffusion displacements (e.g., model free), which are described using the standard definitions of Fourier sampling theory.

Since the distributions of diffusion displacements are model independent, the distributions may be challenging to quantify. Several features have been proposed including the zero-displacement probability, P(R = 0, A), which is higher in regions with more hindered or restricted diffusion; the mean squared displacement,

which is related to the diffusivity (see Fig. 9); the kurtosis of the diffusion distribution, which highlights regions of significant slow diffusion; and the orientational distribution function (ODF)(Wedeen et al. 2005):

Note that this definition of ODF (Eq (9)) for DSI is derived differently for DSI than it is for QBI (Tuch 2004).

While Cartesian sampling facilitates the straightforward FFT for estimation of the displacement densities, Cartesian sampling is not required. Recently, investigators have proposed non-Cartesian sampling strategies of q-space including sampling on concentric spherical shells of constant |q| (Assaf et al. 2004; Wu and Alexander 2005). Assaf et al. then applied a model (CHARMED) of slow and fast diffusing compartments to estimate what they deemed as hindered and restricted diffusion (Assaf et al. 2004). Wu and

Fig. 9. Example P(R = 0; A) and mean squared displacement maps from DSI study (Ne = 257; bmax = 9000 s/mm2)

Alexander (2005) demonstrated that the concentric q-space shell samples in hybrid diffusion imaging (HYDI) could be used for DTI, DSI and QBI in the same experiment.

Applications of High Diffusion-Weighting: The complexity and time required to perform advanced diffusion imaging methods with high diffusion-weighting has limited the number of clinical and research studies relative to the work in diffusion tensor imaging. The pathophysiologic significance of fast/slow diffusion measurements is unclear. Only one published study to date (Brugieres et al. 2004) has specifically examined the effects of pathology (ischemia) on the fast and slow diffusion components. Several small studies of hybrid DSI methods have shown promise in being sensitive to white matter changes associated with multiple sclerosis (Assaf et al. 2002a; Cohen and Assaf 2002), autoimmune neuritis (Assaf et al. 2002b), and vascular dementia (Assaf et al. 2002c). Clearly, more studies are necessary to justify longer imaging times than DTI. To date, none of these methods have been used to directly investigate the relationships to brain connectivity.

From Diffusion to Pathways: White Matter Tractography

In addition to providing information about the mean diffusivity and anisotropy, diffusion imaging methods can also yield novel information about the orientation of local anisotropic tissue features such as bundles of white matter fascicles. In diffusion tensor imaging, the direction of the major eigenvector, ei, is generally assumed to be parallel to the direction of white matter. This directional information can be visualized by breaking down the major eigenvector into x, y and z components, which can be represented using RGB colors - e.g., Red = e1x =Right/Left; Green= e1y =Anterior/Posterior; Blue= e1z =Inferior/Superior. Maps of WM tract direction can be generated by weighting the RGB color map by an anisotropy measure such as FA (Pajevic and Pierpaoli 1999). For many applications, the use of color labeling is useful for identifying specific WM tracts and visualizing their rough trajectories. An alternative strategy is white matter tractography (WMT), which uses the directional information from diffusion measurements to estimate the trajectories of the white matter pathways. WMT increases the specificity of WM pathway estimates and enables the 3D visualization of these trajectories, which may be challenging using cross-sectional RGB maps.

Deterministic Tractography Algorithms: Most WMT algorithms estimate trajectories from a set of "seed" points. Generally, WMT algorithms may be divided into two classes of algorithms - deterministic (e.g., streamline) and probabilistic (see below). Streamline algorithms are based upon the equation:

where r(T) is the path and vtraj is the vector field that defines the local path direction. Typically, streamline WMT algorithms use major eigenvector field to define the local trajectory directions vtraj = e1 at each step (Conturo et al. 1999; Mori et al. 1999; Basser et al. 2000) (see Fig. 10). Alternatively tensor deflection (TEND) vtraj = Dvin uses the entire diffusion tensor to define the local trajectory direction (Lazar et al. 2003). The integration of deterministic pathways may be performed using simple step-wise algorithms including FACT (Mori et al. 1999) and Euler (e.g., Ar = vtraj Ax) (Conturo et al. 1999) integration, or more continuous integration methods such as 2nd or 4th order Runge-Kutta (Basser et al. 2000), which enable more accurate estimates of curved tracts.

Deterministic Tractography Errors: WMT can be visually stunning (see Fig. 11). However, one significant limitation with WMT is that the errors in an estimated tract are generally unknown. Further, the visual aesthetic of WMT, which look like actual white matter patterns, can potentially instill a false sense of confidence in specific results. Unfortunately, there are many potential sources of error 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. Recent studies

Fig. 10. FA and ei color map depicting WM tract orientation. The principle concept of streamline WMT is depicted in a region of corpus callosum. The trajectory is started from a single seed point and the path estimated at discrete steps

Fig. 11. WMT (left) appears to be very similar to an actual white matter dissection (right) (Virtual Hospital). http://web.archive.org/web/20050407073533/ www.vh.org/adult/provider/anatomy/BrainAnatomy/BrainAnatomy.html

Fig. 11. WMT (left) appears to be very similar to an actual white matter dissection (right) (Virtual Hospital). http://web.archive.org/web/20050407073533/ www.vh.org/adult/provider/anatomy/BrainAnatomy/BrainAnatomy.html

have shown that the dispersion in tract estimates < Ax2 > 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 (AXj = h -j and SNR (Anderson 2001; Lazar et al. 2003)

where E is a factor related to the diffusion tensor encoding scheme and the diffusion tensor orientation, and j = 2, 3. Further, the tract dispersion is also affected by the local divergence of the tensor field (Lazar et al. 2003). Even in the complete absence of noise and image artifacts, most current deterministic methods cannot accurately map WM pathways in regions with crossing or converging fibers, which has led to the development of visualization tools to highlight these regions of uncertainty (Jones 2003; Jones et al. 2005c). An alternative approach, recently tested in visual cortex, is likely to be most applicable for mapping interhemispheric fibers. In this method, rather than placing seed voxels in regions of high coherence (e.g., splenium of the corpus callosum), the two hemispheres were seeded separately. Only those obtained tracts that overlapped in the corpus callosum were considered to be valid tracts (Dougherty et al. 2005). This method produced anatomically plausible results for projections from primary visual cortex, but the authors cautioned that many tracts were likely missed, due to the low specificity of WMT and the resolution of current DTI acquisition protocols. New diffusion imaging methods such as DSI and QBI described above are capable of resolving regions of white matter crossing and may ultimately improve WMT in regions of complex WM.

Probabilistic Tractography Algorithms: Although deterministic streamline algorithms are nice tools for visualizing WM patterns, they provide very little information about the reliability of specific results. They rely on accurate placement of seed and deflection point ROIs by the operator, and can vary as a function of ROI size and shape, making them susceptible to generating highly errant results arising from small errors at a single step. Probabilistic tractog-raphy algorithms can overcome some of these limitations. Most probabilistic WMT algorithms are based upon some sort of iterative Monte Carlo approach where multiple trajectories are generated from the seed points with random perturbations to the trajectory directions. Model based tractography algorithms include PICo (Probability Index of Connectivity (Parker et al. 2003), which uses a fast marching technique (Parker et al. 2002), RAVE (Random Vector (Lazar and Alexander 2002)) and ProbTrack (Behrens et al. 2003b). An alternative strategy is to acquire multiple DTI datasets and use bootstrap resampling to derive data-driven estimates of probabilistic tractography (e.g., BOOT-TRAC (Lazar and Alexander 2005) (see Fig. 12). The main difference between model and data-driven approaches is that the variance of the data driven approaches will include the effects of variance in the actual data (e.g., effects of physiologic and artifact noise), not just an idealized model. All of these algorithms create a distribution of tracts, which can be used to estimate the probability of connectivity for the tractography algorithm, which may be used as a surrogate measure of WMT confidence. Additionally, connection probability may be used to segment structures such as the thalamus (Behrens et al. 2003a), cerebral peduncles (Lazar and Alexander 2005), corpus callosum

Fig. 12. Probabilistic bootstrap tractography from a single seed point in the corpus callosum illustrating the tract dispersion associated with WMT at two planes above the seed point. The estimated tract density or probability is shown using a hot color scale. The dispersion increases with distance from the seed

Fig. 12. Probabilistic bootstrap tractography from a single seed point in the corpus callosum illustrating the tract dispersion associated with WMT at two planes above the seed point. The estimated tract density or probability is shown using a hot color scale. The dispersion increases with distance from the seed

(Ciccarelli et al. 2003a), and cortex (Rushworth et al. 2005) according to patterns of maximum connectivity.

Diffusion Imaging and Brain Connectivity: Issues and Considerations

To date, most studies using DTI have focused on analysis of scalar tensor data (anisotropy measures, diffusivity) and have been conducted at three levels of precision: whole-brain histograms; regions-of-interest, and single-voxel analyses. Early studies focused on analysis of whole-brain histograms (e.g., Cercignani et al. 2000; Rovaris et al. 2002), which identify non-specific, global changes in diffusion properties, and may be useful for laying the foundation for more focused analyses. More recently the focus has been on region-of-interest (ROI) and voxel-based analyses. Discussion is ongoing regarding the best methods for accomplishing each type of analysis. When using ROI analyses, it is important to consider the size of the ROI being used, as large ROIs may obscure interesting changes in diffusion measures, and there is a greater possibility that the underlying anatomy will not be consistent across observations. In addition to the usual requirement that the ROIs be placed by a well-trained operator, ROI analyses of DTI data may be may be more sensitive to placement bias in the presence of disease or atrophy. This is especially the case if FA maps are used to define the ROIs. Some have attempted to minimize this potential for bias by lowering the intensity threshold on the FA maps so that local variations in FA are no longer able to guide ROI placement (e.g.,Madden et al. 2004). For voxel-based analyses, the non-diffusion weighted images (b = 0) are often used to register subject data to a common space (Jones et al. 2002a), but this does not guarantee that the underlying fiber architecture (defined by FA or ei) is in register. This lack of correspondence is in part due to the high inter-subject variability of the smaller fiber bundles as well as tract characteristics such as their width, neither of which are evident on the b = 0 images. Inter-subject variability is clear when tracts or FA maps are transformed into stereotaxic space. In Fig. 13, optic radiations

Fig. 13. Optic radiation variability (n = 21). Maximum overlap was 70%. Similar variability would be present if FA maps had been transformed into stereotaxic space. (Reprinted from Ciccarelli et al. 2003b, with permission from Elsevier)

were first identified using probabilistic tractography for individual subjects in native image space. The individual subject data were then resampled into a standardized space, using the b = 0 images as the reference image (Ciccarelli et al. 2003b). Similar dispersion occurs if FA maps are resampled instead of tract probabilities (Jones et al. 2002a).

The large variability across subjects away from tract centers raises the possibility that when correlations of FA and some behavioural or functional measure are found at tissue interfaces, that they may arise simply from the increased variability in FA in these regions. Many published results of voxel-based assessment of group FA differences or FA correlations have identified significant effects in regions of more variable FA. These tend to be located at interfaces of white matter with gray matter or CSF (as seen on Ti-weighted images), or in regions of complex fibre architecture. An example of one such finding is shown in Fig. 14, where correlations of FA with performance on a working-memory task were strongest at tissue interfaces. Because of the error introduced by imperfect registration, residual noise from flow artifact and partial volume effects, as well as the application of smoothing filters (see below), most authors have interpreted such findings with caution. In fact, similar concerns prompted one group to abandon a preliminary voxel-based analysis for one using tractography to define ROIs in the corpus callosum (Kanaan et al. 2006).

Results seem to be more robust to these noise sources if mean tract FA is used rather than voxel-wise FA. An example is seen in recent work examining structure-function relations in the visual system (Toosy et al. 2004). In this study, dispersion was also seen in optic radiations, and it increased as more liberal thresholds were used to define connectivity (Fig. 15, left panel). However, since the regions of high overlap (red) dominated mean FA in the optic radiations, the magnitude of the correlation of FA with the BOLD response in visual cortex was not affected (Fig. 15, right panel).

In voxel-based analyses of functional MRI data, spatial smoothing filters are typically applied to bring the statistical properties of the data more in

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