## Ultrastructure

DTI provides information on fibre bundles, not on individual axon branches. In addition, the diffusion tensor is unable to model adequately voxels containing more than one axon population with different directions [37]. For instance, if the relationship among the three eigenvalues of the diffusion tensor is of the type A; = A2 > A3, the FA may still be sufficiently high as to

Fig. 8.6. The crossing of two fibre bundles may result in a spherical diffusion tensor, erroneously indicating an absence of fibres fail to halt a line propagation algorithm, even in the absence of a major direction with an eigenvalue greater than the other two. In this case, the plane defined by the two major eigenvectors contains several more or less equivalent directions, leading to error. This problem stems from the nature of the diffusion tensor itself, which being a mere second-order approximation of the diffusion process, cannot adequately represent complex situations like the one described. The tensorline technique partially obviates this problem by selecting, among the directions of the plane defined by the two main eigenvectors, the one minimizing the curvature of the trajectory according to the original direction [33].

These methods do not address the possibility that a dominant direction is not identified in a voxel due to crossing bundles giving rise to different directions. The inability to resolve a single direction within each voxel is a significant general limitation of DTI. In fact, in the millimetre scale of the MR voxel, voxels typically exhibit a number of fibre orientations. Common situations of intra-voxel orientational heterogeneity may be due to the intersection of different white matter bundles, or to the complex architecture of subcortical or junctional fibres. In the presence of fascicles with multiple directions within the same voxel, for example due to crossing or divergence, DTI will estimate the prevalent direction, which does not necessarily correspond to any actual direction. For instance, if in a voxel a vertical bundle branches off a horizontal bundle, DTI will show the presence of a single direction corresponding to their diagonal, thus failing to represent either.

The standard diffusion tensor reconstruction technique using DTI data cannot resolve this problem. Indeed, even using several different diffusion-weighting directions to reconstruct the tensor, its mathematical nature prevents it from identifying the different directions when, for instance, two or three different bundles cross. The inability of the DTI technique to resolve fibres with multiple directions derives from the assumption of the Gaussian nature of the tensor model, because a Gaussian function has a single directional peak, preventing the recognition of multidirectional diffusion by the tensor model.

Methods capable of using all the diffusion information are HARDI techniques [38-41], which measure diffusion in several directions with an equal distribution in the three-dimensional space but do not calculate the diffusion tensor. These approaches to the resolution ofmultiple fibre directions in voxels are based on much more complex models of diffusion in nerve tissue.

The most promising method is currently an approach that is independent of a priori models, does not require the prediction of a single diffusion direction, and can therefore recognize crossing or branching structures. The methods that measure diffusion independently of a model are called q-space techniques, i.e. q-space imaging, diffusion spectrum and q-ball imaging. q-Space imaging directly measures the microscopic diffusion function without making a priori assumptions. The diffusion function, P(r), describes the likelihood of a water molecule undergoing a displacement r over a diffusion time t. q-Space imaging is based on the Fourier relationship between the diffusion function P(r) and the MR signal S(yDI), where D and I indicate the duration and intensity of diffusion-sensitizing gradients and y is the gyromagnetic ratio. The Fourier relationship P(R) = F[S(yDI)] enables direct reconstruction of the diffusion function through Fourier transform of the diffusion signal.

The degree of anisotropy can also be estimated directly using HARDI methods without calculating the tensor. In this case an anisotropy index, the spherical diffusion variance, is used instead of conventional FA. Given that, unlike the tensor, this method is not based on an a priori physical model of diffusion, it has the advantage of not losing any information contained in the data. HARDI and its future developments appear to be very promising, and the evolution of tractography is likely to be based on it even though it requires specialized acquisition sequences, generally longer acquisition times than tensor methods, more complex processing algorithms and, save for q-ball imaging, very powerful gradients [39].

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