Global Algorithms

The algorithms of this class use a radically different approach. In fact, whereas the line propagation algorithms use only local information (i.e. the data contained in a voxel and in those nearest to it), these employ the information in a global way by applying a mathematical function that reproduces the structural characteristics of the fibre tracts. For instance, the physical analogy used for the Fast Marching Technique [22 - 24] is that of an ink drop falling on adsorbent tissue. The stain extends faster along the direction of the tissue fibres than perpendicular to them. Assuming a vector field indicating the directions in which the ink spreads, a speed function for front propagation can be defined on the basis of the fibres' anisotropy value. This function reflects the fact that propagation is fastest along fibres and slowest perpendicular to them, and makes it possible to calculate the „shape" of the stain from any point at any given time. Its contours may be compared to the isobars of meteorological charts and, in the case of a vector field of DTI data, they represent a sort of map of the likelihood of connection starting from a given point. Using this technique, the course of the fibres coincides with the faster route, hence its name.

Another physical analogy, well known in the field of numerical simulations as the „travelling salesman problem", can help explain another class of methods. A travelling salesman needs to find the optimum route passing through all the towns where he will be calling. One solution is to define a function, e.g. petrol consumption or time, and find the route that minimizes it. Using DTI data, the function ensuring global energy minimization is related to paths along the direction of the field vectors, while those associated with greater energy expenditure are perpendicular to them [25]. Calculation of the value of the function for all possible trajectories makes it possible to identify the course that minimizes the energy function. However, methods like simulated annealing allow the solution to be found rapidly without calculating the energy for all the possible courses, while minimizing the effect of noise.

Halfway between line propagation and global algorithms are the Monte Carlo probabilistic methods [26-28]. With these techniques, thus named for their similarity to gambling, each time the tract is propagated from one voxel to the next, the various directions are given a probability value depending on the diffusion values measured. It is assumed that by repeating the line propagation a large number of times, the course that has been selected most often will correspond to the actual trajectory of the fibre.

Global methods have two main advantages. First, they can provide a semiquantitative estimate of the level of connectivity between two points or regions (Fig. 8.5). In fact, fibres like those shown in Fig. 8.4 provide a visual representation of the bundles, but not a „value" of the connection [29], for instance between two activation areas shown on functional MR. Secondly, they are less affected by the typical limitations of line propagation algorithms (addressed below). However, the level of calculation required to implement them is often close to the computational capacity of current processors, and they are still in an early phase of development compared with the more common line propagation algorithms.

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Fig. 8.4. Fibre reconstruction with a line propagation algorithm (left pyramidal tract in red), superimposed on axial T2-weighted images. Lower right corner: 3D reconstruction of the same tract overlaid on a coronal image

Fig. 8.5. Connectivity map generated using a Monte Carlo algorithm. Overlay on a fractional anisotropy map. Fractional anisot-ropy is derived from the diffusion tensor and represents white matter distribution. Colours represent the likelihood of connection with the seed point in the left lateral geniculate nucleus, according to an intensity scale

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