Multi-valued geodesic tractography for diffusion weighted imaging

Diffusion-Weighted Imaging (DWI) is a Magnetic Resonance(MR) technique that measures water diffusion characteristics in tissue for a given direction. The diffusion profile in a specific location can be obtained by combining the DWI measurements of different directions. The diffusion profile gives information about the underlying fibrous structure, e.g., in human brain white matter, based on the assumption that water molecules are moving less freely perpendicularly to the fibrous structure. From the DW-MRI measurements often a positive definite second-order tensor is defined, the so-called diffusion tensor (DT). Neuroscientists have begun using diffusion tensor images (DTI) to study a host of various disorders and neurodegenerative diseases including Parkinson, Alzheimer and Huntington. The techniques for reconstructing the fiber tracts based on diffusion profiles are known as tractography or fiber tracking. There are several ways to extract fibers from the raw diffusion data. In this thesis, we explain and apply geodesic-based tractography techniques specifically, where the assumption is that fibers follow the most efficient diffusion propagation paths. A Riemannian manifold is defined using as metric the inverse of the diffusion tensor. A shortest path in this manifold is one with the strongest diffusion along this 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 equation (HJ). One characteristic 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, i.e., cusps. Cusps occur when the correct solution should become multi-valued. HJ methods are not able to handle this situation. To solve this, we developed a multi-valued solution algorithm for geodesic tractography in a metric space defined by given by diffusion tensor imaging data. The algorithm can capture all possible geodesics arriving at a single voxel instead of only computing the first arrival. Our algorithm gives the possibility of applying different cost functions in a fast post-processing. Moreover, the algorithm can be used for capturing possible multi-path connections between two points. In this thesis, we first focus on the mathematical and numerical model for analytic and synthetic fields in twodimensional domains. Later, we present the algorithm in three-dimensions with examples of synthetic and brain data. Despite the simplicity of the DTI model, the tractography techniques using DT are shown to be very promising to reveal the structure of brain white matter. However, DTI assumes that each voxel contains fibers with only one main orientation and it is known that brain white matter has multiple fiber orientations, which can be arbitrary many in arbitrary directions. Recently, High Angular Resolution Diffusion Imaging (HARDI) acquisition and its modeling techniques have been developed to overcome this limitation. As a next contribution we propose an extension of the multi-valued geodesic algorithm to HARDI data. First we introduce the mathematical model for more complex geometries using Finsler geometry. Next, we propose, justify and exploit the numerical methods for computing the multi-valued solution of these equations

Multi-valued geodesic tractography for diffusion weighted imaging

Diffusion-Weighted Imaging (DWI) is a Magnetic Resonance(MR) technique that measures water diffusion characteristics in tissue for a given direction. The diffusion profile in a specific location can be obtained by combining the DWI measurements of different directions. The diffusion profile gives information about the underlying fibrous structure, e.g., in human brain white matter, based on the assumption that water molecules are moving less freely perpendicularly to the fibrous structure. From the DW-MRI measurements often a positive definite second-order tensor is defined, the so-called diffusion tensor (DT). Neuroscientists have begun using diffusion tensor images (DTI) to study a host of various disorders and neurodegenerative diseases including Parkinson, Alzheimer and Huntington. The techniques for reconstructing the fiber tracts based on diffusion profiles are known as tractography or fiber tracking. There are several ways to extract fibers from the raw diffusion data. In this thesis, we explain and apply geodesic-based tractography techniques specifically, where the assumption is that fibers follow the most efficient diffusion propagation paths. A Riemannian manifold is defined using as metric the inverse of the diffusion tensor. A shortest path in this manifold is one with the strongest diffusion along this 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 equation (HJ). One characteristic 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, i.e., cusps. Cusps occur when the correct solution should become multi-valued. HJ methods are not able to handle this situation. To solve this, we developed a multi-valued solution algorithm for geodesic tractography in a metric space defined by given by diffusion tensor imaging data. The algorithm can capture all possible geodesics arriving at a single voxel instead of only computing the first arrival. Our algorithm gives the possibility of applying different cost functions in a fast post-processing. Moreover, the algorithm can be used for capturing possible multi-path connections between two points. In this thesis, we first focus on the mathematical and numerical model for analytic and synthetic fields in twodimensional domains. Later, we present the algorithm in three-dimensions with examples of synthetic and brain data. Despite the simplicity of the DTI model, the tractography techniques using DT are shown to be very promising to reveal the structure of brain white matter. However, DTI assumes that each voxel contains fibers with only one main orientation and it is known that brain white matter has multiple fiber orientations, which can be arbitrary many in arbitrary directions. Recently, High Angular Resolution Diffusion Imaging (HARDI) acquisition and its modeling techniques have been developed to overcome this limitation. As a next contribution we propose an extension of the multi-valued geodesic algorithm to HARDI data. First we introduce the mathematical model for more complex geometries using Finsler geometry. Next, we propose, justify and exploit the numerical methods for computing the multi-valued solution of these equations

A ray tracing method for geodesic based tractography in diffusion tensor images

We present multi-valued solution algorithm for geodesic-based fiber tracking in a tensor-warped space given by diffusion tensor imaging data. This technique is based on solving ordinary differential equations describing geodesics by a ray tracing algorithm. The algorithm can capture all possible geodesics connecting two given points instead of a single geodesic captured by Hamilton-Jacobi based algorithms. Once the geodesics have been computed, using suitable connectivity measures, we can choose among all solutions the most likely connection pathways which correspond best to the underlying real fibrous structures. In comparison with other approaches, our algorithm gives the possibility of applying different cost functions in a fast post-processing. Moreover, the algorithm can be used for capturing possible multi-path connections between two points that can happen when, e.g., pathologies are presented. Synthetic second order diffusion tensor data in a two dimensional space are employed to illustrate the potential applications of the algorithm to fiber tracking

An innovative geodesic based multi-valued fiber-tracking algorithm for diffusion tensor imaging

We propose a new geodesic based algorithm for fiber tracking in diffusion tensor imaging data. Our algorithm computes the multi-valued solutions from the Euler-Lagrange form of the geodesic equations. Compared to other geodesic based approaches, multi-valued solutions at each grid point are computed rather than just computing the viscosity solution. This allows us to compute fibers in a region with sharp orientation, or when the correct physical solution is not the fiber computed from the first arrival time. Compared to the classical stream-line approach, our method is less sensitive to noise, since the complete tensor is used. We also compare our algorithm with the Hamilton-Jacobi equation (HJ) based approach. We show that in the cases where U-shaped bundles appear, our algorithm can capture the underlying fiber structure while other approaches may fail. The results for synthetic and real data are shown for both methods

Multi-valued geodesic based fiber tracking for diffusion tensor imaging

In this paper, we propose a new geodesic based algorithm for diffusion tensor fiber tracking. This technique is based on computing multi-valued solutions from the Euler-Lagrange form of the geodesic equations. Compared to other geodesic based approaches, multi-valued solutions at each grid point have been considered other than just computing the viscosity solution. This allows us to compute fibers in a region with sharp orientation, or when the correct physical solution is not the fiber computed from the first arrival time. Compared to the classical stream-line approach, our approach is less sensitive to noise, since the complete tensor is used. We also compare our algorithm with the PDE approach, using the Hamilton-Jacobi equation.We show that in the cases where the U-shaped bundles appear, our algorithm can capture the underlying fiber structure while other approaches may fail. The results for a realistic synthetic data field is shown for both methods

Multivalued geodesic ray-tracing for computing brain connections using diffusion tensor imaging

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. 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

Multi-valued geodesic based fiber tracking for diffusion tensor imaging

In this paper, we propose a new geodesic based algorithm for diffusion tensor fiber tracking. This technique is based on computing multi-valued solutions from the Euler-Lagrange form of the geodesic equations. Compared to other geodesic based approaches, multi-valued solutions at each grid point have been considered other than just computing the viscosity solution. This allows us to compute fibers in a region with sharp orientation, or when the correct physical solution is not the fiber computed from the first arrival time. Compared to the classical stream-line approach, our approach is less sensitive to noise, since the complete tensor is used. We also compare our algorithm with the PDE approach, using the Hamilton-Jacobi equation.We show that in the cases where the U-shaped bundles appear, our algorithm can capture the underlying fiber structure while other approaches may fail. The results for a realistic synthetic data field is shown for both methods