Active Cortex Tractography (ACT)

Abstract

Most existing diffusion tractography algorithms are affected by gyral bias, causing the termination of streamlines at gyral crowns instead of sulcal banks. We propose a tractography technique, called active cortex tractography (ACT), to overcome gyral bias by enabling fiber streamlines to curve naturally into the cortex. We show that the cortex can play an active role in cortical tractography by providing anatomical information to overcome orientation ambiguities as the streamlines enter the superficial white matter in gyral blades and approach the cortex. This is achieved by devising a direction scouting mechanism that takes into account the white matter surface normal vectors. The scouting mechanism allows probing of directions further in space to prepare the streamlines to turn at appropriate angles. The surface normal vectors guide the streamlines to turn into the cortex, perpendicular to the white-gray matter interface.

Methods

Deterministic tractography methods cite{Basser2000-iu,Tournier2012-hk,Mori1999-ct} typically reconstruct fiber streamlines by successively following local directions using the Euler forward method,

\(\mathbf{r}_{i+1} = \mathbf{r}_{i} +\rho \mathbf{t}_i,\)

where \(\mathbf{r}_{0}\) is a given seed point, and the streamline \(\mathcal{X}=[\mathbf{r}_{0}, \cdots , \mathbf{r}_{n} ]\) is estimated using a fixed step size \(\rho\). The forward direction \(\mathbf{t}_{i}\) is typically chosen from \(M\) directions \(\left\{ \mathbf{d}_1, \cdots , \mathbf{d}_M\right\}\) that correspond to the local maxima of the fiber orientation distribution function (FODF) cite{Jeurissen2019-qh} in the voxel containing \(\mathbf{r}_{i}\), minimizing the angular deviation with respect to the previous direction \(\mathbf{t}_{i-1}\):

\(\mathbf{t}_i = \underset{\mathbf{d}}{\text{argmin}} \arccos(\mathbf{t}_{i-1} \cdot \mathbf{d}), ~\mathbf{d} \in \left\{ \mathbf{d}_1, \cdots , \mathbf{d}_M\right\},~\text{s.t.}~\arccos(\mathbf{t}_{i-1} \cdot \mathbf{d}) \le \theta,\)

where the maximum allowed angular deviation is \(\theta\). FODFs are typically assumed to be antipodally symmetric; hence the configurations, such as fanning and bending, cannot be represented correctly.

We propose a tractography method to adaptively update the forward direction and the step size based on the asymmetric fiber orientation distribution (AFODF). We further incorporate surface prior to help the streamlines bend more naturally in gyral blades into the cortex.

Estimating sub-voxel fiber orientations in gyral blades is challenging due to the fanning and bending configurations. Asymmetric FODFs cite{wu2018multi,bastiani2017improved,Wu2020-ce} have been shown to improve tractography in gyral blades by explicitly considering sub-voxel orientation asymmetry and continuity across neighboring voxels. Let \(\mathbb{U}_{({\mathbf{r}}, {\mathbf{n}})}\) be the voxel subspace defined by the position vector \({\mathbf{r}}\) with the normal vector \({\mathbf{n}}\). Then, with an asymmetric FODF \(\mathcal{F}\), the forward direction \(\mathbf{t}_i\) can be determined in voxel subspace \(\mathbb{U}\) (see Fig.~ref{fig:space}, Left):

\(\mathbf{t}_i = \underset{\mathbf{t}^{*} \in {\mathbb{U}}}{\text{argmax}}\, \mathcal{F}_{\mathbf{r}_i}(\mathbf{t}^{*}), \quad {\mathbb{U}} := \mathbb{U}_{(\hat{\mathbf{r}}_i, \mathbf{r}_i - \hat{\mathbf{r}}_i)} \cap \mathbb{U}_{(\hat{\mathbf{r}}_i, {\mathbf{t}_{i-1}})},\)

where \(\mathbb{U}_{(\hat{\mathbf{r}}_i, \mathbf{r}_i - \hat{\mathbf{r}}_i)}\) is the voxel subspace containing \({\mathbf{r}}_i\) and \(\mathbb{U}_{(\hat{\mathbf{r}}_i, {\mathbf{t}_{i-1}})}\) is the voxel subspace defined by the previous direction \(\mathbf{t}_{i-1}\) and the current position \(\hat{\mathbf{r}}_i\).

Unlike conventional tractography algorithms in which the forward direction depends only on the fiber peaks at the current position, we adaptively update the forward direction via a scouting mechanism (see Fig.~ref{fig:space}, Right) based on the asymmetric FODF cite{wu2018multi,Wu2020-ce} as follows:

\(\tilde{\mathbf{t}}_i = {\delta}_i\mathbf{t}_i + {\delta}_{i+1}\mathbf{t}_{i+1},\)

where \({\delta}_i=\rho\mathcal{F}_{\mathbf{r}_{i}}(\mathbf{t}_{i})\). Then, the next position \(\mathbf{r}_{i+1}\) is updated with the adaptive step size:

\(\mathbf{r}_{i+1} = \mathbf{r}_{i} + {\delta}_i\tilde{\mathbf{t}}_i.\)

We harness the anatomical information provided by the cortex to guide the tracking of the streamline when it enters the gyral blade and into the cortex. This is achieved by updating the forward direction by

\(\tilde{\mathbf{t}}_i \leftarrow \tilde{\mathbf{t}}_i + \exp(-\|\mathbf{r}_i - {\mathbf{p}_i}\|) \mathbf{v}_i,\)

where \({\mathbf{p}_i}\) is a vertex on the WM surface with normal vector \(\mathbf{v}_i\), which is chosen such that

\(\{\mathbf{p}_i, \mathbf{v}_i\} = \underset{\mathbf{p},\mathbf{v}}{\text{argmin}} \exp(-\|\mathbf{r}_i - {\mathbf{p}}\|) \arccos(\mathbf{v} \cdot \tilde{\mathbf{t}}_i),\quad \mathbf{v} \in \mathbb{V},\)

and \(\mathbb{V}\) is a field of surface normals (see Fig.~ref{fig:surface}). The sign of the normal vector is flipped when the streamline leaves the cortex into superficial WM:

\(\mathbf{v}_i = \left\{{\begin{array}{cc}{\mathbf{v}_i,}&{~\text{if}~\arccos(\mathbf{v}_i \cdot \tilde{\mathbf{t}}_i) \leq \frac{\pi}{2},}\\{-\mathbf{v}_i,}&{~\text{if}~\arccos(\mathbf{v}_i\cdot \tilde{\mathbf{t}}_i) > \frac{\pi}{2}}.\end{array}}\right.\)

References

1. Wu, Y., Hong, Y., Ahmad, S. and Yap, P.T., 2021, September. Active Cortex Tractography. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 467-476). Springer, Cham.