The distributions on parameters in a diffusion model are of great significance when making inference on the basis of these parameters. Inference may be at a group level; for example there have been studies showing reduced anisotropy in groups of Multiple Sclerosis patients, in comparison with groups of normal subjects (e.g. [3]). However, inference may also be within a single subject. There have been many recent papers (e.g. [4,5,6]) describing techniques for using parameters from a diffusion tensor fit to follow major white matter pathways in the brain. However, none of these techniques attempt to quantify the uncertainty in the resulting white matter connections. The output of these algorithms is a set of nodes describing the maximum likelihood pathway through the DTI data, with no measure of confidence on the location of this pathway. The lack of this information makes interpretation of the output pathways difficult, and also makes it hard to devise strategies for tracing reliably in uncertain areas. For both of these reasons, streamlining algorithms to date have chosen not to trace pathways through areas of low diffusion anisotropy
(e.g. [5,7]). Diffusion anisotropy tends to be low in areas of high uncertainty in fiber direction (although the converse is not necessarily true [8]), and therefore, by tracing fibers only when anisotropy is high, streamlining algorithms have tended to generate pathways which (if they had been calculated) would have had narrow confidence bounds on them. This knowledge means that reconstructed pathways are often interpretable as major fiber tracts in the brain [9], but places limits on areas where it is possible to create them. In the second part of this manuscript, we give the mathematical formulation for deriving spatial PDF on connectivity between point and every other point in the data field given the local pdfs. This PDF is an explicit representation of the confidence regions for pathways in the data. We go on to present a sampling technique to generate this PDF in a computationally efficient manner, and describe and discuss technical details, such as data interpolation, required in any fiber tracing algorithm. We present resulting connectivity PDFs from seed voxels in the thalamus, a deep gray matter structure with relatively low diffusion anisotropy. We show that connectivity distributions estimated from diffusion imaging data in human correspond well with predictions from sacrificial tracer studies in primate. Further results from this study appear with detailed discussion and interpretation in [8].
An important point to note is that, throughout this paper, the estimated probability distributions are pdfs on parameters in a model. This is to be contrasted with the Gaussian distribution described by the diffusion tensor fit [10], and with more recent work (e.g. [11]) which have attempted to recreate the diffusion spectrum as a probability distribution on the displacement, of a particle with initial location in the voxel after a diffusion time . There are crucial differences here, both conceptually and practically.