Introduction

Uncertainty and its representation have an important role to play in any situation where the goal is to infer useful information from noisy data. In Diffusion Weighted MRI (DWMRI) scientists attempt to infer information about, for example, diffusion anisotropy or underlying fiber tract direction, by fitting models of the diffusion and measurement processes to DWMRI data (e.g. [1,2]). In this scheme there is uncertainty caused both by the noise and artifacts present in any MR scan, but also by the incomplete modelling of the diffusion signal. That is, the true diffusion signal is more complicated than we choose to model. This additional complexity in the diffusion signal appears as residuals when we fit a simple model to the data, causing additional uncertainty in the model parameters. All of the uncertainty in these parameters may be represented in the form of probability density functions (pdfs). This paper is essentially divided into two parts, dealing separately with uncertainty at the local and global levels. In the first part, we describe a technique for estimating the pdfs on all parameters in any local model of diffusion. We will show results derived from two simple models of the diffusion process within a voxel: The diffusion tensor model which assumes a local 3D Gaussian diffusion profile, and a simple partial volume model of local diffusion, which assumes that a fraction of diffusion is along a single dominant direction, and that the remainder is isotropic. We will then make suggestions for the extension to more complete models of the diffusion process which are able to account for one, or more, distribution of fiber directions within the voxel. In all of these models, the use of Bayesian techniques allows for the application of prior constraints on parameters in the model where such constraints are sensible. For example, in the fitting of the diffusion tensor model, the eigenvalues of the diffusion tensor are constrained to be positive.

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 $A$ 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, $r_{final}-r_0$ of a particle with initial location $r_0$ in the voxel after a diffusion time $t_d$. There are crucial differences here, both conceptually and practically.