We believe that the medical community has not yet fully benefited from what dif-fusion imaging offers and this is in part due to the fact that well-established robust quantitative methods for analysis of diffusion MRI data are largely missing. This thesis proposes a framework to enable the quantitative analysis and in particular to answer the following questions:
• How to obtain the point correspondence between the trajectories efficiently?
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Figure 1-3: An example of tract-oriented quantitative analysis on a population and for the splenium parts of the corpus collasum. (a) Sagittal view of the trajectories from all cases, registered into the atlas space, and colored with the local fractional anisotropy. (b) Sagittal view of the trajectories in (a) that are clustered into the upper and lower parts of the splenium. (c) Axial view of the trajectories shown in (b). (d) Average fractional anisotropy along the tract arc length for the normal (blue) and schizophrenia (red) cases, where a drop in FA is seen for diseased cases. (e) A plot of the corresponding p-values which indicates that a significant drop in the FA is only observed in the mid-portion of each side of the splenium.
• How to define the similarity between 3-D trajectories?
• How to incorporate anatomical information in clustering of the trajectories?
• How to perform tract-oriented analysis over a population?
The answer to the last question is the ultimate goal of this thesis. It is worth mentioning again that being able to perform such analysis is valuable since it enables us to study the local changes of quantitative parameters along the fibers. This is especially interesting for studying temporal changes of fiber tracts during brain de-velopment and it also opens new possibilities to compare normal and pathological subjects.
As an answer to the first question mentioned above, we propose a novel approach for calculating the point correspondence between the trajectories by building a dis-tance map and a Voronoi diagram on the same space. This provides a computationally efficient alternative to curve matching algorithms when dealing with a large number of curves (feature vectors) when performing population analysis. This enables us to use the point correspondence in the clustering step. Since the number of trajectories is quite high, other approaches sacrifice the accuracy of determining point correspon-dence either by using a simple similarity measure or by using information from a limited number of trajectory points.
Having the knowledge of point correspondences, we define a similarity measure in which spatial information is used explicitly while shape similarity is included implic-itly through penalty terms for missed and repeated point matches.
Theoretical models are also developed to include anatomical information from an atlas of fiber tracts. We describe two levels for including the anatomical prior. In the first approach, the priors are fixed and are given by an anatomical atlas. The second approach uses a Dirichlet distribution to control the influence of the atlas. To the best of our knowledge these are the first implementations in which anatomical priors are used in a mathematically principled framework. By using an atlas, the correspondence between clusters in different subjects is automatically known.
To ensure robustness, quantitative analysis is performed probabilistically using the membership weights (probabilities) assigned to each trajectory in the clustering step. In the mixture model clustering implemented in this work, we develop the idea of building the mixture model based on the distance between the trajectories and the cluster centers to deal with the variable length of the trajectories. The output of clus-tering is the probabilistic assignment of trajectories to anatomically-known bundles of fiber tracts and the point correspondences between trajectories to the mean tra-jectory of each cluster. This information is used to compute proper statistics on any attribute feature vector, such as scalar diffusion measures, e.g., fractional anisotropy and mean diffusivity, defined along the trajectories. In population studies, the tra-jectories are first transformed to a common space, clustered, and the quantitative parameters of interest are reported along the tracts.
As will be shown in Chapter 6, the proposed tract-oriented quantitative analysis addresses the following clinical questions:
• To what extent and where on a white matter fiber tract the diffusion parameters change between two time points during the brain development?
• Is there diffusion asymmetry between the two brain hemispheres and how are such asymmetries affected by diseases?
• Is there any localized group difference in a given fiber tract between normal and diseased populations?
As a byproduct of the proposed clustering method, a spatial model of the fiber bundles represented by the mean trajectory and its spatial variation is also obtained.
This is shown in Figure 1-4 in which the abstract models of five fiber bundles are visualized by their spatial mean and iso-surfaces corresponding to the mean plus three standard deviations (3σ) of the 3-D coordinates calculated along the cluster center. Such an abstract spatial model for fiber bundles could be used for neurosurgery applications. It enables one to easily visualize the extent of the fiber tracts adjacent to the brain lesions to help minimize the damage to the bundles when removing
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Figure 1-4: (a) Trajectories of 5 different clusters: splenium (yellow), corticospinal (red), corticobulbar (green), middle cerebellar peduncle (blue), and genu (magenta).
(b) A model representation of the bundles as the mean trajectory and the isosurfaces corresponding to spatial variation of the clusters.
the lesion. Conventional DT-MRI-based approaches usually visualize a map of the fractional anisotropy [50] or bundle segmentations based on the FA map. However, such approaches are not accurate at fiber crossings where FA is low. An alternative approach is to visualize the fiber trajectories [44], but rendering such a large dataset can be confusing and not be efficient in real-time applications.