Descripción del trabajo

In the following, we adapt the approach used by Friman et al in [46]. This approach al-lows us to employ a fully probabilistic framework exploiting distributions based on fibre dispersion while applying suitable priors to enforce smoothness on streamlines derived from the DW-MRI data. Friman’s approach accommodates only the uncertainty in the principle diffusion direction induced by noise, image artefacts and partial volume effects;

we instead incorporate underlying fibre dispersion directly in the ODF used to guide the tractography.

We may model a pathway as a train of vectors, as described in chapter 3, v1:n =

corresponding to the appropriate step pi( ˆv_i| ˆv_i−1, D) in sequential order and terminat-ing upon some appropriate criteria like exitterminat-ing a brain mask or a white matter mask.

p_i( ˆv_i| ˆv_i−1, D) is different for each step on the path and is dependent on the data at the relevant 3D location.

Hence the problem of building up the streamline v_1:n= { ˆv1, ..., ˆv_n} breaks down to an iterative process of sampling from the PDF p_i( ˆv_i| ˆv_i−1, D) at each step i. The one excep-tion is for the point i = 1 for which ˆv_i−1 is not known and ˆv₁ is sampled from p₁( ˆv₁|D).

To propagate the streamline through the image, starting from a seed, we choose a propagation direction v_i from a distribution formed from the product of the local ODF and a prior on the allowable deviation from the previous direction v_i−1:

P( ˆv_i| ˆv_i−1, D) =P( ˆv_i|D)P( ˆvi| ˆv_i−1)

P( ˆv_i) , (4.1)

This so far is similar to the approach given in [46]. In [46] the likelyhood P( ˆv_i|D) is calculated by modelling a cylindrical DT with the assumption of a Gaussian distribution of potential directions due to noise and comparing this with the underlying data. This is aimed at accounting for uncertainty in fibre direction due to noise, in contrast to our method, in which the model used is based on the distribution of fibre directions estimated in each voxel by the NODDI technique. Due to the complexity of sampling from the posterior distribution, Friman chooses to discretise the spherical posterior by evaluating it on 2562 unit vectors derived from the vertices of a 4-fold tessellation of an icosahedron, which is the chosen approach we use in the following.

The model used to describe the probability of an fibre existing in a voxel with ori-entation ˆv_i given the DW-MRI data D is the Bingham model described in section 2.4.7.

The Bingham model has a mean axisµ, a primary fanning axis µ1and two concentration parameters describing the degree of dispersion along the two orthogonal fanning axesκ1

andκ2. Therefore equation 4.1 can be rewritten:

P( ˆv_i| ˆv_i−1,µ,µ1,κ1,κ2) =P( ˆv_i|µ, µ1,κ1,κ2)P( ˆv_i| ˆv_i−1)

P( ˆv_i) , (4.2)

where P( ˆv_i|µ, µ1,κ1,κ2) is the Bingham distribution described above.

P( ˆv_i| ˆv_i−1) defines the prior used to impose smoothness on the propagated streamlines.

For this we use a distribution given by:

P( ˆv_i| ˆv_i−1) =

Sampling from this joint distribution allows exploration of the potential path directions in dispersive fibre regions while regularizing the curvature of the path. γ defines the strength of the curvature prior. Low values accommodate large degrees of deviation per streamline step, exploring more of the dispersion in each voxel, however, this also produces highly irregular streamlines. Higher values promote smooth, slowly curving pathways which correspond to known tract geometries.

We find the most appropriate value for γ using simulated data exhibiting dispersing structure, which is described in the next section, which models a region of highly dis-persive ODFs. Figure 4.1(a) shows that low values of γ such as 1, results in irregular streamlines. However, at a significantly higher value of γ = 50 (Figure 4.1(f)), such a strong prior on curvature can limit the potential trajectories of the streamlines, limiting full exploitation of the dispersive ODFs. Satisfactory results can be achieved for a range of intermediate values. For this demonstration of the algorithm we chooseγ = 24 (Figure 4.1(d)). γ may reasonably be tuned within this range for other applications if necessary.

Figure 4.2 shows the distibution of streamlines as they reach the top of the phantom at y = 6, which are binned into 4 separate bins per unit length on the x-axis, creating a total of 32 bins across the 8 units shown in Figure 4.4.γ is increased from 0 in intervals of 4.γ = 24 is chosen as it gives the distribution which most closely replicates the

distri-(a) γ = 1 (b) γ = 5

(c) γ = 15 (d) γ = 24

(e) γ = 35 (f)γ = 50

Figure 4.1: Tracking through a synthetic region of dispersion utilising different values of the constantγ in equation 3. In figure 4.1(a) γ = 1, in figure 4.1(d) γ = 24 and in figure 4.1(f) γ = 50. The blue lines represent the extremeties of the same phantom shown in figure 4.4.

(a) Binned streamlines for range of γ and for phantom streamlines.

(b) Binned streamlines forγ = 24 and phantom streamlines.

Figure 4.2: Streamlines binned according to x location at top of phantom ( y = 6) in Figure 4.4 for varying values ofγ, compared against streamlines extending from the same seed point in the phantom. γ = 24 gives the closest profile to the distribution of streamlines from the phantom. There are 4 bins per unit length, making a total of 32 bins.

at extreme values of γ = 0 and γ = 50 in in vivo data is shown in Figure 4.9. It can be observed that at γ = 0, what is effectively a random walk through the field of Bingham distributions does not create strong connections to the cortex, as the streamlines spread out too much before they reach the cortex, producing a lot of false positives. At the other extreme ofγ = 50, the curvature prior overwhelms the influence of the data, meaning the streamlines tend not to follow the data well, and they are focussed on a particular region of the cortex.

Sampling from the posterior distribution P( ˆv_i| ˆv_i−1,µ,µ1,κ1,κ2) is not trivial. Rejec-tion sampling can be used to sample from a standard Bingham distribuRejec-tion. However, to evaluate the full posterior, given by equation 4.2, a discrete approximation is required (Figure 4.3). Therefore, a set of vectors is defined which is very close to uniformly dis-tributed over the sphere using the vertices of a 4 fold tessellation of an icosahedron.

Equation 4.2 can then be evalutated for each of these vectors, and sampling from the resulting discrete distribution is then trivial.

Figure 4.3: Schematic illustration of discretely approximating posterior distribution in equation 4.2