Towards Image-Dependent Safety Hulls for Fiber Tracking

Introduction: The evaluation of fiber tracking algorithms is a challenging problem, given that software or hardware models are needed in order to have a ground truth to compare against (see for example [1-3]). In this work we introduce a novel global precision measure for tracked fibers, the "safety radius". We make use of a software model in order to systematically analyze the influence of image noise, fiber bundle diameter, number of seed points and tensor anisotropy on the safety radius. The latter is used to construct safety hulls, i.e. tubes that surround the tracked fibers and indicate their margin of error. Finally, we analyze the spatial distribution of tracked fibers. Methods: Our software model is given by a portion of a torus generated using a solid kernel as described in [4], which is used to compute a set of synthetic diffusion weighted images (DWI), see an example image shown in Fig. 1(a). The DW signal is computed according to the CHARMED model proposed in [5,6], where we restrict ourselves to the hindered model. We assume cylindrically symmetric tensors and for the fiber bundle we set the eigenvalues of the diffusion tensors parallel and perpendicular to the axonal fibers (denoted by and respectively) to values that are compatible with eigenvalues of tensors encountered in white matter, based on reports from [7,8]. For the background we use isotropic tensors. Partial volume effects are modeled by sampling the image at (0.1mm)3 and then linearly resampling it at (1mm)3. Image noise is simulated by adding Rice distributed noise to the DW images as in [9]. The DW images are used to compute the tensor valued images. The seed ROI used for fiber tracking is set as a circle located on a plane perpendicular to the fiber bundle, with the same diameter as the cross section of the fiber bundle (see Fig. 1(a)). For fiber tracking, we use the advection-diffusion based algorithm presented in [10]. To evaluate our fiber tracking results, we determine a so called safety radius . Given a cross section of the tracked fiber bundle, the safety radius is defined as the minimal radius that is needed so that if a circle with radius were placed around each fiber tracked inside the bundle, the aggregate of these circles would form a topological cover of the cross section of the modeled fiber bundle. In order to find , we first compute the Voronoi Diagram of the points inside the cross section of the fiber bundle; is then given by the maximal distance between one such point and the borders of the corresponding cell (see Fig. 1(b)). Finally, we are interested in the spatial distribution of the tracked fibers, which we analyze by looking at cross sections of the fiber bundle. We partition the cross section into six semi-annuli of equal area as shown in Fig. 1(c). At each cross section, we count how many fibers lie in each part. Figure 1: (a): DW image of a synthetic fiber bundle, gradient in x direction, with overlayed seed ROI shown in yellow. (b): Cross section of the tracked fiber bundle (location of fibers shown as dots) and the corresponding Voronoi Diagram. The length of the dashed black line segment corresponds to the safety radius. (c): Partitioned cross section of the tracked fiber bundle. Fibers with positive x-coordinate are on the interior of the circular fiber path. (d): A portion of a tracked fiber bundle. (e): Same portion of a tracked fiber bundle as in (d) but with a 3mm safety hull around each fiber.