Global Connectivity estimation: Theory

In the previous section we described techniques for estimating, at each voxel, probability distributions on every parameter in the chosen model of diffusion. In this section we use these local pdfs from the simple partial volume to infer on a model of global connectivity. The reason we choose this model is explained in detail in the previous section. We wish to maximize the chances that complex fiber structure will be represented by uncertainty in principal direction. We now require a model to take us from the local parameters in this model to parameters describing global connectivity. Note that, throughout the remainder of this paper, subscript $\vec{x}$ refers to ``every voxel in the brain''. Hence $(\theta,\phi)_{\vec{x}}$ refers to the complete set of principal diffusion directions.

Consider the case where the values of the local parameters are known with no uncertainty. What do they tell us about anatomical connectivity between voxels in the brain? In the case where our local model describes only a single fiber direction passing through the voxel, this global model can only take one form:

$$\mathcal{P}(\exists A \rightarrow B\vert(\theta,\phi)_{\vec{x}})=... ...eta,\phi)_{\vec{x}}\end{array}\\ &\\ 0 & \textrm{otherwise} \end{array} \right.$$ (20)

Where $\mathcal{P}(\exists A \rightarrow B\vert(\theta,\phi)_{\vec{x}})$ is the probability of a connection existing between points $A$ and $B$, given knowledge of local fiber direction.

In order to solve this equation we may simply start a connected path from a seed point, $A$, and follow local fiber direction until a stopping criterion is met. If $B$ lies on this path we may say that a connection exits between $A$ and $B$. This procedure is at the heart of all ``streamlining'' algorithms (e.g. [6,19,5]), which choose $(\theta,\phi)_\vec{x}$ to be the principal eigen-direction of the estimated diffusion tensor at each voxel.

However, in the case where there is uncertainty associated with $(\theta,\phi)_{\vec{x}}$ we would like to compute the probability of a connection existing given the data, $\vec{Y}_{\vec{x}}$, which is known. That is, we would like to compute $\mathcal{P}(\exists A \rightarrow B\vert\vec{Y}_{\vec{x}})$. In order to calculate this pdf we would have to perform the following integrations:

$$\mathcal{P}(\exists A \rightarrow B\vert Y)=$$

$$\int_0^{2\pi}\int_0^{\pi}\ldots\int_0^{2\pi}\int_0^{\pi}\mathcal{... ...ightarrow B\vert(\theta,\phi)_{x})\mathcal{P}((\theta,\phi)_{x_1}\vert Y)\ldots$$

$${P}((\theta,\phi)_{x_n}\vert Y)d\theta_{x_1}d\phi_{x_1}\ldots d\theta_{x_n} d\phi_{x_n}$$ (21)

That is, for each possible value of fiber direction at every voxel $(\theta,\phi)_\vec{x}$, we must incorporate the probability of connection given this $(\theta,\phi)_\vec{x}$, and also the probability of this $(\theta,\phi)_\vec{x}$ given the acquired MR data. This process is known as marginalization (see e.g. [20]).

It can be seen from equation 21, that $\mathcal{P}(\exists A \rightarrow B\vert Y)$ reduces to $\mathcal{P}(\exists A \rightarrow B\vert(\theta,\phi)_{\vec{x}})$ when the local pdfs on fiber direction $\mathcal{P}((\theta,\phi)_{\vec{x}})$ are delta functions . That is, when there is no uncertainty in the local fiber direction, equation 21 reduces to the streamlining (maximum likelihood) solution. However, when local fiber direction is uncertain, $\mathcal{P}(\exists A \rightarrow B\vert Y)$ will be non-zero for some $B$ not on the maximum likelihood streamlines. That is, the global connectivity pattern from $A$, will spread to incorporate the known uncertainty in local fiber direction.

However, even in the discrete data case, equation 21 represents a $v$ dimensional (where $v$ is the number of voxels in the brain) integral over distributions with no analytical representation (the local pdfs, generated with MCMC), and hence clearly cannot be solved analytically.

Fortunately, as we have seen in previous sections, even when explicit integration is unfeasible, it is often possible to compute integrals implicitly by drawing samples from the resulting distribution. In our case, in order to draw a sample from $\mathcal{P}(\exists A \rightarrow B\vert Y)$ we may draw a sample from the posterior pdf on fiber direction at each point in space and construct the streamline (henceforth referred to as a ``probabilistic streamline'') from $A$ given these directions. Computationally, this process is extremely cheap. Samples from the local pdfs at each voxel have already been generated, so to generate a single probabilistic streamline from seed point $A$, referring to the current ``front'' of the streamline as $\vec{z}$, it is sufficient simply to start $\vec{z}$ at $A$ and:

• Select a random sample, $(\theta,\phi)$ from $\mathcal{P}(\theta,\phi\vert\vec{Y})$ at $\vec{z}$.
• Move $\vec{z}$ a distance $s$ along $(\theta,\phi)$.
• Repeat until stopping criterion is met.

This probabilistic streamline is said to connect $A$ to all points $B$ along its path. By drawing many such samples, we may build the spatial pdf of $\mathcal{P}(\exists A \rightarrow B\vert Y)$ for all B. We may then discrete this distribution into voxels, by simply counting the number of probabilistic streamlines which pass through a voxel $B$, and dividing by the total number of probabilistic streamlines.