There are two important, but separate issues here. The first, is that the parameters of real interest to the scientist are ones which relate directly to the underlying fiber structure, and not to the diffusion profile. These underlying parameters may have convincing markers within the fitted diffusion profile (for example anisotropy measures [37,2] from the diffusion tensor fit have been shown to be a marker for collinearity of fibers within a voxel), but any attempt to recreate the fiber structure from these profiles is essentially an educated guess. There has been no model proposed to predict how a specific structure or distribution of fiber directions within a voxel will reflect itself in the measured diffusion weighted NMR signal. The second issue is that, even when fitting a model of local diffusion, the resulting parameters have uncertainty associated with them. Factors such as noise in the NMR signal (both physical and physiological) and, crucially, the inadequacy of the proposed model, lead to this uncertainty which should be incorporated in any further processing (such as tractography schemes).
In this paper we have presented a method for the full treatment of this uncertainty. We have shown how, using Bayes' equation along with well established methods for its numerical solution, it is possible to form a complete representation of the uncertainty in the parameters in any generative model of diffusion, in the form of posterior probability density functions on these parameters. We have applied this Bayesian estimation technique to two simple local models of diffusion, the diffusion tensor model, and a simple partial volume model, with only a single anisotropically diffusing direction in the voxel. We have examined the results in these two cases, comparing the posterior distributions with empirical measurements of uncertainty.
We then consider uncertainty at a global level. We outline the theory behind moving from the pdfs on local PDD to an estimate on the probability distribution on global connectivity. When estimating global connectivity, we first have to choose between the available local models of diffusion. We have chosen to use a simple partial volume model. The reason for this choice is that, by choosing a model which allows for only a single fiber direction within a voxel, we maximize the chance that the effect of diverging or splitting fibers will be seen as uncertainty in the principal diffusion direction, and not as a change in the diffusion profile, as might be the case if the Diffusion Tensor model were chosen. However the similarity in uncertainty between the two models that we find in the empirical validation suggests that this decision is made largely for conceptual completeness, and that results would have been similar if the Diffusion Tensor model had been chosen.
The next stage is to define a model of global connectivity. The model we choose is identical to that used in streamlining algorithms (e.g. [4,19,6,5]). That is, given absolute knowledge of local fiber directions, connectivity is assumed between two points if, and only if, there exits a connected path between them through the data (see equation 20). The crucial difference between the probabilistic tractography proposed here, and the streamlining algorithms referenced above above can be seen in equation 21. Put simply, the result of this equation incorporates every possible fiber orientation at every voxel and the probability of each of these fiber directions given the acquired MR data. We simply allow for uncertainty in fiber direction when computing streamlines. The practicality of solving this equation is an algorithm similar in nature to others presented, along with this method, at ISMRM 2002 [38,39,40], effectively repeatedly sampling local pdfs to create streamlines, and regarding these streamlines as samples from a global pdf. A crucial difference between these methods and our method, is that we choose to compute the local pdfs in a rigorous fashion given the MR data . The methods referenced above all use heuristic experience-based relationships between the shape of the fitted diffusion tensor and the assumed pdf on local fiber orientation.
An important result of our procedure is that the recovered ``connectivity distributions'' are strictly probability distributions on the connected pathway through dominant fiber directions. That is, there is no explicit representation of splitting or diverging fibers in either the local or global model. We are strictly inferring on a single pathway leading from the seed point, and therefore in order to find, for example, splitting pathways, the effect of fiber divergence within a voxel must reveal itself as uncertainty in the PDD. It can be seen from the local results section that, at least in the cases presented here, this effect can be seen. Figure 4 (b) shows sensitivity to the splitting of fibers from the optic tract, into the superior collicular brachium, and the direct fibers of the optic radiations. Figure (d) also shows sensitivity to branching fibers. Descending fibers from the ventral lateral (motor) nucleus of the thalamus split into two distinct branches, as is to be expected from primate studies. The first heads down to brainstem, and the second into superior cerebellar cortex. However, because fiber divergence within a voxel is treated as uncertainty in principal diffusion direction, this sensitivity to diverging and branching fibers will be dependent on the experimental design; in general, the more information in the MR measurements, the lower the uncertainty in principal diffusion direction. Taking this effect to its logical extreme, if we were to gather an infinite number of MR measurements, there would be no uncertainty in principal fiber direction, and the marginal probability distribution on the dominant streamline would be infinitely narrow, i.e. the simple streamlining solution. Ideally we would like to infer, not on connectivity via a single connection, but on an anatomical distribution of connectivity. In order to do this we must allow for divergence, branching and crossing of fibers in our local model of diffusion. We propose one such model which will allow for inference on an underlying distribution of fiber orientations.
Probably the most important result in this paper is in Figure 4 (e,f,g). Here we seed every voxel in thalamus, and compute the respective connectivity distributions, recording the probability of connectivity to each of four cortical masks. There are two striking features in this figure. The correspondence of the connectivity-based thalamic segmentation between the left and right thalami (g) provides strong evidence for the robustness of the technique, even when seeding from deep gray matter areas. This is backed up by the marked similarity between the predicted cortical zones from primate data (f) and the connectivity based segmentation (g). This second feature also provides strong, albeit indirect, validation for the use of diffusion based tractography in any guise.
In summary, we have presented a technique for characterizing the uncertainty associated with parameter estimates in diffusion weighted MRI, and for propagating this uncertainty through the diffusion weighted data. This allows us to compute the probability distribution on the location of the dominant fiber pathway so that we may quantify our belief in the tractography results.