Increasing the Complexity - A Distribution of Fibers?

In the partial volume model presented above, only a single fibre orientation is modeled in each voxel. In fact, there will be a distribution, $H(\theta,\phi)$, of fibre orientations in the voxel. In order to estimate this distribution we must build a model which, given this distribution, could predict the Diffusion Weighted MR measurements.

Such a model clearly requires some assumptions. We start by assuming that each subvoxel has only one fibre direction through it and that the MR signal is the sum of the signal from arbitrarily small subvoxels, and that the signal from each subvoxel behaves as described by Equation 12. (Note that this is a strong assumption to make, but it is explicit in the model. Any other model of the local diffusion characteristics of a single fibre orientation may be used as a replacement.)

$$\vec{\mu}_{total}=\sum_{j\in sub-voxels}\vec{\mu}_j$$ (16)

where $\mu_{total}$ is the vector of MR signal from the voxel at each gradient direction and strength, and $\vec{\mu}_j$ is the same vector for each sub-voxel.

If we now consider, instead of the individual sub-voxels, the set $\Theta\Phi$ of major directions ( $\theta,\phi$) in these subvoxels (note the discretization of $\Theta\Phi$) , then Equation 16 is identically equivalent to (see equation 12):

$$\mu_i=\sum_{(\theta,\phi)\in\Theta\Phi}\left(\sum_{j\in V_{\theta\phi}}\frac{S_{0_j}}{N}\left[(1-f_j)\exp{(-b_id_j)}+\right.\right.$$

$$\left.\vphantom{\sum_{(\theta,\phi)\in\Theta\Phi}\left(\sum_{j\in... ...}_i^T\vec{R}_{\theta\phi}\vec{A}\vec{R}_{\theta\phi}^T\vec{r}_i)}\right]\right)$$ (17)

where $V_{\theta\phi}$ is the set of all voxels whose principal fibre direction is $(\theta,\phi)$ and $N$ is the number of subvoxels. This equation, although fearsome at first sight, is actually very straight forward. The first part of the argument to the summation (on the top line) represents the signal due to all of the isotropic compartments, and the second part represents the signal due to all of the fibre compartments. If we now further assume that $S_0$ (the signal with no diffusion gradients applied) and $d$ (the diffusivity) are constant across the voxel, then the inner summation (over voxels which have the same principal direction) may be replaced by a constant for the isotropic compartment, and in the anisotropic compartments, by the distribution function $H(\theta,\phi)$ defined earlier. With a little more manipulation and by letting the sub-voxel size tend to zero, it is easy to arrive at:

$$\frac{\mu_i}{S_0}=(1-f)\exp{(-b_id)}+$$

$$f\int_0^{2\pi}\int_0^{\pi}\mathcal{H}(\theta,\phi)\exp{(-b_id\vec... ...}_{\theta\phi}\vec{A}\vec{R}_{\theta\phi}\vec{r}_i)}\sin(\theta)d\theta d\phi .$$ (18)

where $1-f$ is now the proportion of the whole voxel showing isotropic diffusion.Note that the integral is over $\sin(\theta)d\theta d\phi$ in order to maintain elemental area over the sphere. Finally, if we write the gradient direction $\vec{r}_i$ in spherical polar coordinates $\vec{r}_i=[\begin{array}{ccc} \sin{\alpha_i}\cos{\beta_i}& \sin{\alpha_i}\sin{\beta_i}& \cos{\alpha_i} \end{array}],$ and define $\gamma_i$ as the angle between gradient direction, $(\alpha_i,\beta_i)$, and fibre direction $(\theta_i,\phi_i)$, then the exponent inside the integral reduces dramatically. We may now write:

$$\frac{\mu_i(\alpha_i,\beta_i)}{S_0}=(1-f)\exp{(-b_id)}+$$

$$f\int_0^{2\pi}\int_0^{\pi}\mathcal{H}(\theta,\phi)\exp{}[-b_id\cos^2\gamma_i]\sin(\theta)d\theta d\phi .$$ (19)

This equation reveals a great deal about the diffusion measurement process. The real ``signal'' of interest is $\mathcal{H}(\theta,\phi)$, the distribution of fibers within the voxel. When we measure the diffusion profile of this signal, we are measuring a version of this signal which is smoothed in angular space, with a kernel, predicted by this model, of $\exp{(-bd\cos^2{\gamma})}$. We would like to deconvolve the effect of the measurement process from the signal. However, we leave the details of this estimation process, and validation thereof, as future work.