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Solving the diffusion equation for fiber tracking in the living human brain
... The directional information concerning fiber bundle orientation that diffusion imaging provides may be exploited to analyze cerebral anatomic connectivity noninvasively. A number of methods have been developed to infer connection between tissues in neighboring imaging voxels, or between brain regions (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). These methods fall into two broad groups: those that define a single route of connection for each start point (these will be referred to as "linear" approaches), and those that attempt to establish the spatially distributed degree of connection across the whole brain to a given start point or region (these will be referred to as "distributed" methods). ...
... A number of methods have been suggested that aim to generate distributed maps of the degree or probability of "connectivity" between brain regions (12,15,(23)(24)(25)(26)(27). In particular, attempts have been made to assess cerebral connectivity using models of the diffusion process, such as the diffusion ellipsoid, to propagate a simulated diffusion process, with the aim of establishing connectivity in a probabilistic sense using Monte Carlo processes (12,13,15). For example, a grid-based random walk process may be simulated (15), in which a particle is allowed to diffuse at a rate determined by a function of the magnitude of the diffusion coefficient along one of the possible inter-grid point (inter-voxel) directions chosen at random. ...
... This conclusion is reflected in the work of Koch et al (15), where and ADC profile-based profile is effectively modulated by an exponent a to make the PDF more focused. propagate a probabilistic connectivity measurement (12,13). This suffers from similar difficulties surrounding the dispersed nature of connectivity assignment, which is unlikely to reflect the underlying anatomy. ...
To establish a general methodology for quantifying streamline-based diffusion fiber tracking methods in terms of probability of connection between points and/or regions.
The commonly used streamline approach is adapted to exploit the uncertainty in the orientation of the principal direction of diffusion defined for each image voxel. Running the streamline process repeatedly using Monte Carlo methods to exploit this inherent uncertainty generates maps of connection probability. Uncertainty is defined by interpreting the shape of the diffusion orientation profile provided by the diffusion tensor in terms of the underlying microstructure.
Two candidates for describing the uncertainty in the diffusion tensor are proposed and maps of probability of connection to chosen start points or regions are generated in a number of major tracts.
The methods presented provide a generic framework for utilizing streamline methods to generate probabilistic maps of connectivity. J. Magn. Reson. Imaging 2003;18:242–254.
... The fiber tracking method presented in this paper exploits simulations of the diffusion process, through the solutions to the anisotropic diffusion equation. The idea of studying brain connectivity by way of simulating the anisotropic diffusion has been preliminarily explored in [22]–[24], and [25] further extends the diffusion equation to a diffusion-convection equation by adding a convection term. However, there is no (or at best limited) attempt at determining the route of fiber pathways and connectivity between anatomically or functionally defined regions in the brain using this method, although the concentration or flow field over the brain is calculated. ...
... The demonstration shows it is feasible to employ the diffusion-based tracking technique to noninvasively reconstruct white matter fiber tracts in the living human brain. Previous work, as shown in [22]–[24] , has attempted to perform fiber tracking by solving simplified variants of the general anisotropic diffusion equation (2). In the approach proposed in [23], a seed diffuses through the brain from a selected starting point by solving the full diffusion equation and the amount of concentration at some location is interpreted as a probability to reach that point. ...
... Both methods are completely different from our method to exploit the diffusion process, in which we construct successive three dimensional diffusion fronts to determine the fiber pathways by evaluating the distance and orientation from the computed fronts to their analogous diffusion seeds. In [24] , an anisotropic diffusion equation is solved, however, no effort has been made to show that the paths of fibers or the connections can be reconstructed between anatomical or functional regions of the brain. Regarding the use of diffusion tensor data for unveiling the organizational patterns of white matter structures, the diffusion-based tractography has several potential advantages. ...
A novel approach to noninvasively tracing brain white matter fiber tracts is presented using diffusion tensor magnetic resonance imaging (DT-MRI). This technique is based on successive anisotropic diffusion simulations over the human brain, which are utilized to construct three dimensional diffusion fronts. The fiber pathways are determined by evaluating the distance and orientation from the fronts to their corresponding diffusion seeds. Synthetic and real DT-MRI data are employed to demonstrate the tracking scheme. It is shown that the synthetic tracts are accurately replicated, and several major white matter fiber pathways can be reproduced noninvasively, with the tract branching being allowed. Since simulating the diffusion process, which is truly a physical phenomenon reflecting the underlying architecture of cerebral tissues, makes full use of the diffusion tensor data, including both the magnitude and orientation information, the proposed approach is expected to enhance robustness and reliability in white matter fiber reconstruction.
... The information in the DT-MRI data can be exploited to reconstruct the fiber pathways of the brain. A number of fiber tracking algorithms, or tractography, have been developed since the advent of DT-MRI [6,11,14,20,29,44,54,52]. In the following sections we will examine a few different fiber tracking techniques, explain their advantages, limitations, and potential applications, followed by a list of some available fiber tractography and visualization software packages. ...
... As the measured quantity in DT-MRI is for water diffusion, an intuitive way to gain insights from the diffusion tensor data is to carry out a direct simulation of water diffusion, which is anisotropic and governed by the diffusion equation, over the brain. The idea of studying brain connectivity by simulating the anisotropic diffusion has been preliminarily explored in [9,20,48], and [8] further extends the diffusion equation to a diffusion-convection equation by adding a convection term. However, in the above mentioned work based on this insight, there is no (or at best limited) attempt at determining the route of fiber pathways and connectivity between anatomical or functional regions in the brain, although the concentration or flow field over the brain is calculated and obtained. ...
Diffusion tensor magnetic resonance imaging (DT-MRI) is the first noninvasive in vivo imaging modality with the potential to generate fiber tract trajectories in soft fibrous tissues, such as the brain white matter. Several newly developed fiber reconstruction and tractography techniques based on DT-MRI are reviewed in this article, including streamline fiber tracking techniques, diffusion tensor deflection strategy, probabilistic Monte-Carlo method, fast marching tractography based on level set principles, and diffusion simulation-based tractography. We also discuss some potential applications of DT-MRI fiber tractography and list a few available tractography and visualization software packages.
... The directional information concerning fibre bundle orientation that DWI provides may be exploited to analyse cerebral anatomical connectivity non-invasively. A number of methods have been developed to infer connection between tissues in neighbouring imaging voxels, or between brain regions [4,[33][34][35][36][37][38][39][40][41][42][43]. These methods fall into two broad groups: those that define a single route of connection for each start point (these will be referred to as ''linear'' approaches), and those that attempt to establish the spatially distributed degree of connection across the whole brain to a given start point or region (these will be referred to as ''distributed'' methods). ...
... A number of methods have been suggested that aim to generate distributed maps of ''probability'' of inter-region connection or of ''connectivity'' between brain regions [39,42,[50][51][52][53][54]. For example, attempts have been made to assess cerebral connectivity by using models of the diffusion process, such as the diffusion ellipsoid, to propagate a simulated diffusion process, with the aim of establishing connectivity in a distributed manner using Monte Carlo methods [39,40,42]. This may be achieved using a gridbased simulated random walk process [42], in which a for two crossing fibres, generated with a two tensor model of diffusion. ...
Diffusion-weighted MR images provide information that is present in no other imaging modality. Whilst some of this information may be appreciated visually in diffusion weighted images, much of it may be extracted only with the aid of data post-processing. This review summarizes the methods available for interpreting diffusion weighted imaging (DWI) information using the diffusion tensor and other models of the DWI signal. This is followed by an overview of methods that allow the estimation of fibre tract orientation and that provide estimates of the routes and degree of anatomical cerebral white matter connectivity.
... In [3] a variety of these algorithms are described and reviewed. As the measured quantity in DT-MRI is water diffusion, an intuitive way to understand the diffusion data is to spread a virtual concentration peak of water [8], or to specify a starting point for tractography where a seed is diffused [4] . This approach makes use of the full information contained in the diffusion tensor and it is not dependent upon a point to point eigenvalue/eigenvector computation along a trajectory, thus in that sense hopefully is more robust. ...
We conduct simulations for the 3D unsteady state anisotropic diffusion process with DT-MRI data in the human brain by discretizing the governing diffusion equation on Cartesian grid and adopting a high performance differential–algebraic equation (DAE) solver, the parallel version of implicit differential–algebraic (IDA) solver, to tackle the resulting large scale system of DAEs. Parallel preconditioning techniques including sparse approximate inverse and banded-block-diagonal preconditioners are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of the two parallel preconditioners. The experimental results of the diffusion simulations on a parallel supercomputer show that the sparse approximate inverse preconditioning strategy, which is robust and efficient with good scalability, gives a much better overall performance than the banded-block-diagonal preconditioner.
... As DT-MRI is a fairly new field of research, many studies are yet to be made to compare the measured diffusion tensors to detailed tissue properties important for fiber path inference. However, in contrast to approaches such as solving the diffusion equation [6], it might be important to separate the physical phenomenon of water diffusion from the solution of the tracking problem through the use of a fiber model. In this way a priori knowledge about nerve fibers such as fiber stiffness could be taken into account [12]. ...
The development of Diffusion Tensor MRI has raised hopes in the neuro-science community for in vivo methods to track fiber paths in the white matter. A number of approaches have been presented, but there are still several essential problems that need to be solved. In this paper a novel fiber propagation model is proposed, based on stochastics and regularization, allowing paths origi- nating in one point to branch and return a probability distribution of possible paths. The proposed method utilizes the principles of a statistical Monte Carlo method called Sequential Importance Sampling and Resampling (SISR).
... However, the use of path curvature is not necessarily appropriate , as discussed above. Alternative approaches to generating connection " likelihood, " such as Monte-Carlo [31] and finite el- ement [32], [33] simulations of the diffusion process should be compared with our method and an anatomical gold standard (see for example [27]). The methodology presented here is related in its philosophy to graph search methods for establishing minimum cost paths between regions, as suggested in [34] for nondiffusion MR brain images. ...
... The physical laws that govern the dynamics of this substance are expressed as a set of partial differential equations (PDEs), which are solved numerically on the diffusion tensor field to produce a metric of connectivity. Early work in this area focused on developing methods modeled on the simple diffusion of heat or water through the image volume, whose dynamics is computed by solving the first order heat equation on the diffusion tensor field [41]- [43]. These methods have the advantages of level set and probabilistic methods and are based the same physical model used to construct the diffusion tensor field. ...
We introduce a fluid mechanics based tractography method for estimating the most likely connection paths between points in diffusion tensor imaging (DTI) volumes. We customize the Navier-Stokes equations to include information from the diffusion tensor and simulate an artificial fluid flow through the DTI image volume. We then estimate the most likely connection paths between points in the DTI volume using a metric derived from the fluid velocity vector field. We validate our algorithm using digital DTI phantoms based on a helical shape. Our method segmented the structure of the phantom with less distortion than was produced using implementations of heat-based partial differential equation (PDE) and streamline based methods. In addition, our method was able to successfully segment divergent and crossing fiber geometries, closely following the ideal path through a digital helical phantom in the presence of multiple crossing tracts. To assess the performance of our algorithm on anatomical data, we applied our method to DTI volumes from normal human subjects. Our method produced paths that were consistent with both known anatomy and directionally encoded color images of the DTI dataset.
... However, the use of path curvature is not necessarily appropriate , as discussed above. Alternative approaches to generating connection " likelihood, " such as Monte-Carlo [31] and finite el- ement [32], [33] simulations of the diffusion process should be compared with our method and an anatomical gold standard (see for example [27]). The methodology presented here is related in its philosophy to graph search methods for establishing minimum cost paths between regions, as suggested in [34] for nondiffusion MR brain images. ...
A method is presented for determining paths of anatomical connection between regions of the brain using magnetic resonance diffusion tensor information. Level set theory, applied using fast marching methods, is used to generate three-dimensional time of arrival maps, from which connection paths between brain regions may be identified. The method is demonstrated in the normal brain and it is shown that major white matter tracts may be elucidated and that multiple connections and tract branching are allowed. Maps of connectivity between brain regions are also determined. Four options are described for estimating the degree of connectivity between regions.
... This is accomplished by applying Fick's laws and solving the PDEs of diffusion. Gembris et al. [30] have solved these equations for the brain and simulated a peak concentration as it diffuses inside the brain. This method can be extended by introducing a convection term to enhance the anisotropy of the diffusion equation [31]. ...
Diffusion in structured tissue, such as white matter or muscle, is anisotropic. MR diffusion tensor imaging (DTI) measures anisotropy per pixel and provides the directional information relevant for MR tractography or fiber tracking in vivo. MR tractography is non-invasive, relatively fast, and can be repeated multiple times without destructing important tissue. Moreover, the combination with other MR images is relatively simple. In this paper, the basic principles of tractography are presented. Different tracking methods with varying degrees of complexity are introduced and their potential strengths and weaknesses are discussed. Clinical applications and different strategies for evaluating the fidelity of tracking results are reviewed.
... In [3] a variety of these algorithms are described and reviewed. As the measured quantity in DT-MRI is water diffusion, an intuitive way to understand the diffusion data is to spread a virtual concentration peak of water [8], or to specify a starting point for tractography where a seed is diffused [4]. This approach makes use of the full information contained in the diffusion tensor and it is not dependent upon a point to point eigenvalue/eigenvector computation along a trajectory, thus in that sense hopefully is more robust. ...
We conduct simulations for the 3D unsteady state anisotropic diffusion process in the human brain by discretizing the governing diffusion equation on Cartesian grid and adopting a high performance differential-algebraic equation (DAE) solver, the parallel version of implicit differential-algebraic (IDA) solver, to tackle the resulting large scale system of DAEs. Parallel preconditioning techniques including sparse approximate inverse and banded-block-diagonal preconditioners are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of the two parallel preconditioners.
Diffusion-Weighted Magnetic Resonance Imaging (DWI) enables the in-vivo visualization of fibrous tissues such as white matter in the brain. Diffusion-Tensor Imaging (DTI) specifically models the DWI diffusion measurements as a second order-tensor. The processing pipeline to visualize this data, from image acquisition to the final rendering, is rather complex. It involves a considerable amount of measurements, parameters and model assumptions, all of which generate uncertainties in the final result which typically are not shown to the analyst in the visualization. In recent years, there has been a considerable amount of work on the visualization of uncertainty in DWI, and specifically DTI. In this chapter, we primarily focus on DTI given its simplicity and applicability, however, several aspects presented are valid for DWI as a whole. We explore the various sources of uncertainties involved, approaches for modeling those uncertainties, and, finally, we survey different strategies to visually represent them. We also look at several related methods of uncertainty visualization that have been applied outside DTI and discuss how these techniques can be adopted to the DTI domain. We conclude our discussion with an overview of potential research directions.
Abstract We conduct simulations for the unsteady state anisotropic diffusion process in the human,brain by discretizing the governing diffusion equation on a face-centered cubic grid and adopting a high performance differential-algebraic equation solver, IDA, to deal with the resulting large-scale system of DAEs. Incomplete LU preconditioning techniques are used with the GMRES method,to accelerate the convergence,rate of the iterative solution. We then investigate and compare,the efficiency and effectiveness of a number of ILU preconditioners, and find out that the ILUT with a dual dropping strategy gives the best overall performance,when,it is provided with the optimum,choices of the fill-in parameter and the threshold dropping tolerance. © 2003 Elsevier B.V. All rights reserved. Keywords: Anisotropic diffusion; DT-MRI; FCC grid; Preconditioning
A family of methods aiming at the reconstruction of a putative fascicle map from any diffusion-weighted dataset is proposed. This fascicle map is defined as a trade-off between local information on voxel microstructure provided by diffusion data and a priori information on the low curvature of plausible fascicles. The optimal fascicle map is the minimum energy configuration of a simulated spin glass in which each spin represents a fascicle piece. This spin glass is embedded into a simulated magnetic external field that tends to align the spins along the more probable fiber orientations according to diffusion models. A model of spin interactions related to the curvature of the underlying fascicles introduces a low bending potential constraint. Hence, the optimal configuration is a trade-off between these two kind of forces acting on the spins. Experimental results are presented for the simplest spin glass model made up of compass needles located in the center of each voxel of a tensor based acquisition.
Diffusion-tensor magnetic resonance imaging (DT-MRI), also known simply as diffusion-tensor imaging (DTI), has rapidly evolved as a new in vivo approach to the investigation of white-matter abnormalities or tissue damages: quantifying the diffusivity of the water molecules in brain. The quantification of water diffusion in vivo is based on the characteristic movement of water molecules, which varies depending upon the tissue. For example, in pure liquids, such as cerebrospinal fluid, the motion of individual water molecules is random, meaning it has equal probability in all directions. However, the movement of water molecules within myelinated fibers is substantially restricted along the direction perpendicular to the orientation of the axons. Consequently, in white-matter fiber tracts, the principal direction of the water diffusion represents the direction of the fiber bundles. Thus, connecting points along the principal direction of the diffusion makes it possible to appreciate white-matter tracts within the brain. Such fiber-tracking schemes, often referred to collectively as fiber tractography, provide important information about the connectivity between brain regions. Therefore, DTI provides a quantitative assessment of the tissue-specific diffusivity and also provides information on anatomical connection. This chapter will focus on the basics and current advances of DTI for the quantification of white matter.
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