Multivalued Geodesic Ray-Tracing for Computing Brain Connections Using Diffusion Tensor Imaging

N. Sepasian, J. H. M. ten Thije Boonkkamp, B. M. Ter Haar Romeny, A. Vilanova
2012 SIAM Journal of Imaging Sciences  
Diffusion tensor imaging (DTI) is a magnetic resonance technique used to explore anatomical fibrous structures, like brain white matter. Fiber-tracking methods use the diffusion tensor (DT) field to reconstruct the corresponding fibrous structure. A group of fiber-tracking methods trace geodesics on a Riemannian manifold whose metric is defined as a function of the DT. These methods are more robust to noise than more commonly used methods where just the main eigenvector of the DT is considered.
more » ... Until now, geodesic-based methods were not able to resolve all geodesics, since they solved the Eikonal equation, and therefore were not able to deal with multivalued solutions. Our algorithm computes multivalued solutions using an Euler-Lagrange form of the geodesic equations. The multivalued solutions become relevant in regions with sharp anisotropy and complex geometries, or when the first arrival time does not describe the geodesic close to the anatomical fibrous structure. In this paper, we compare our algorithm with the commonly used Hamilton-Jacobi (HJ) equation approach. We describe and analyze the characteristics of both methods. In the analysis we show that in cases where, e.g., U-shaped bundles appear, our algorithm can capture the underlying fiber structure, while other approaches will fail. A feasibility study with results for synthetic and real data is shown. SEPASIAN, TEN THIJE BOONKKAMP, TER HAAR ROMENY, VILANOVA turn in the trajectory, or when the fiber enters a region of low anisotropy where the main eigenvector cannot be clearly identified. The main eigenvector is not well defined, either due to noise or to the presence of crossing fibers. Furthermore, streamline methods are based on local characteristics and therefore sensitive to noise. A possible solution for resolving these limitations of classic fiber-tracking is to apply more global approaches such as geodesic-based algorithms [13, 27, 44, 32] . These techniques are based on the assumption that fibers follow the most efficient diffusion propagation paths. A Riemannian manifold is defined using as metric the inverse of the diffusion tensor. Paths on this manifold are shorter if the diffusion is stronger along that path. Therefore geodesics (i.e., shortest paths) on this manifold follow the most efficient diffusion paths. The geodesics are often computed from the stationary Hamilton-Jacobi (HJ) equation. Equidistant fronts are emanating from a certain initial point. The propagation speed of the fronts at each point in space is a local variable. Parker, Wheeler-Kingshott, and Barker [42] presented a similar approach where the local speed was based on a combination of the normal vector and the tensor dominant eigenvector. This approach is prone to incorrect propagation in anisotropic domains. In recent publications [27, 19, 20] , front propagation inside anisotropic domains has been considered, which is suitable for oriented domains. The propagation speed of the front is given by the diffusivity rate in the normal direction of the front; the fibers are extracted by back tracing along the characteristics of the front. One property of solving the HJ equation is that it gives only the single-valued viscosity solution corresponding to the minimizer of the length functional. It is also well known that the solution of the HJ equation can develop discontinuities in the gradient space, cusps. Cusps occur when the correct solution should become multivalued. HJ methods are not able to handle this situation. In this paper, we focus on developing an algorithm that can tackle this shortcoming. Recently, Sepasian et al. [49, 48] presented a ray-tracing algorithm for computing geodesics in anisotropic domains. They defined the metric as the inverse of the diffusion tensor [13, 27] . In contrast to other geodesic-based methods, this approach can capture multivalued geodesics connecting two given points by considering the geodesics as functions of position and direction. Moreover, it is based on the Euler-Lagrange (EL) equations, and therefore local changes in the geodesic can be taken into account. In [49] Sepasian et al. focused on the mathematical and numerical model for analytic and synthetic fields in two-dimensional domains. In a later paper [48] , Sepasian et al. presented the algorithm for the ray-tracing method in three dimensions with some examples of synthetic and brain data. The purpose of the current paper is to provide better insight into details of the ray-tracing algorithm. It presents the differences, advantages, and disadvantages compared to numerical solution of the HJ equation. In addition, we evaluate the algorithm for real brain DTI data, as well as for synthetic crossing fibers. It should be noted that the methods mentioned until now belong to the class of deterministic fiber-tracking methods; i.e., given the same input, these methods will always give the same result. In the case of geodesic-based methods, given two points in the domain, a finite number of geodesics paths (i.e., one in the case of HJ methods) will be found. Probabilistic fiber-tracking constitutes another class of methods, where the variation of the pathways due to model assumptions and/or noise is considered. A probabilistic distribution is built, and a random process generates many paths originating from one initial position [5, 41, 39, 10, 9, 51] .
doi:10.1137/110824395 fatcat:awnezmtkzjdvtgikdoc7qfihju