Sinisa Pajevic

In Vivo Fiber Tractography Using DT-MRI Data

Fiber tract trajectories in coherently organized brain white matter pathways were computed from in vivo diffusion tensor magnetic resonance imaging (DT‐MRI) data. First, a continuous diffusion tensor field is constructed from this discrete, noisy, measured DT‐MRI data. Then a Frenet equation, describing the evolution of a fiber tract, was solved. This approach was validated using synthesized, noisy DT‐MRI data. Corpus callosum and pyramidal tract trajectories were constructed and found to be consistent with known anatomy. The methods reliability, however, degrades where the distribution of fiber tract directions is nonuniform. Moreover, background noise in diffusion‐weighted MRIs can cause a computed trajectory to hop from tract to tract. Still, this method can provide quantitative information with which to visualize and study connectivity and continuity of neural pathways in the central and peripheral nervous systems in vivo, and holds promise for elucidating architectural features in other fibrous tissues and ordered media. Magn Reson Med 44:625–632, 2000. Published 2000 Wiley‐Liss, Inc.

Water diffusion changes in Wallerian degeneration and their dependence on white matter architecture.

This study investigates water diffusion changes in Wallerian degeneration. We measured indices derived from the diffusion tensor (DT) and T2-weighted signal intensities in the descending motor pathways of patients with small chronic lacunar infarcts of the posterior limb of the internal capsule on one side. We compared these measurements in the healthy and lesioned sides at different levels in the brainstem caudal to the primary lesion. We found that secondary white matter degeneration is revealed by a large reduction in diffusion anisotropy only in regions where fibers are arranged in isolated bundles of parallel fibers, such as in the cerebral peduncle. In regions where the degenerated pathway crosses other tracts, such as in the rostral pons, paradoxically there is almost no change in diffusion anisotropy, but a significant change in the measured orientation of fibers. The trace of the diffusion tensor is moderately increased in all affected regions. This allows one to differentiate secondary and primary fiber loss where the increase in trace is considerably higher. We show that DT-MRI is more sensitive than T2-weighted MRI in detecting Wallerian degeneration. Significant diffusion abnormalities are observed over the entire trajectory of the affected pathway in each patient. This finding suggests that mapping degenerated pathways noninvasively with DT-MRI is feasible. However, the interpretation of water diffusion data is complex and requires a priori information about anatomy and architecture of the pathway under investigation. In particular, our study shows that in regions where fibers cross, existing DT-MRI-based fiber tractography algorithms may lead to erroneous conclusion about brain connectivity.

Color Schemes to Represent the Orientation of Anisotropic Tissues From Diffusion Tensor Data: Application to White Matter Fiber Tract Mapping in the Human Brain

This paper investigates the use of color to represent the directional information contained in the diffusion tensor. Ideally, one wants to take into account both the properties of human color vision and of the given display hardware to produce a representation in which differences in the orientation of anisotropic structures are proportional to the perceived differences in color. It is argued here that such a goal cannot be achieved in general and therefore, empirical or heuristic schemes, which avoid some of the common artifacts of previously proposed approaches, are implemented. Directionally encoded color (DEC) maps of the human brain obtained using these schemes clearly show the main association, projection, and commissural white matter pathways. In the brainstem, motor and sensory pathways are easily identified and can be differentiated from the transverse pontine fibers and the cerebellar peduncles. DEC maps obtained from diffusion tensor imaging data provide a simple and effective way to visualize fiber direction, useful for investigating the structural anatomy of different organs. Magn Reson Med 42:526‐540, 1999. r 1999 Wiley-Liss, Inc.

Statistical artifacts in diffusion tensor MRI (DT-MRI) caused by background noise.

This work helps elucidate how background noise introduces statistical artifacts in the distribution of the sorted eigenvalues and eigenvectors in diffusion tensor MRI (DT‐MRI) data. Although it was known that sorting eigenvalues (principal diffusivities) by magnitude introduces a bias in their sample mean within a homogeneous region of interest (ROI), here it is shown that magnitude sorting also introduces a significant bias in the variance of the sample mean eigenvalues. New methods are presented to calculate the mean and variance of the eigenvectors of the diffusion tensor, based on a dyadic tensor representation of eigenvalue–eigenvector pairs. Based on their use it is shown that sorting eigenvalues by magnitude also introduces a bias in the mean and the variance of the sample eigenvectors (principal directions). This required the development of new methods to calculate the mean and variance of the eigenvectors of the diffusion tensor, based on a dyadic tensor representation of eigenvalue–eigenvector pairs. Moreover, a new approach is proposed to order these pairs within an ROI. To do this, a correspondence between each principal axis of the diffusion ellipsoid, an eigenvalue–eigenvector pair, and a dyadic tensor constructed from it is exploited. A measure of overlap between principal axes of diffusion ellipsoids in different voxels is defined that employs projections between these dyadic tensors. The optimal eigenvalue assignment within an ROI maximizes this overlap. Bias in the estimate of the mean and of the variance of the eigenvalues and of their corresponding eigenvectors is reduced in DT‐MRI experiments and in Monte Carlo simulations of such experiments. Improvement is most significant in isotropic regions, but some is also observed in anisotropic regions. This statistical framework should enhance our ability to characterize microstructure and architecture of healthy tissue, and help to assess its changes in development, disease, and degeneration. Mitigating these artifacts should also improve the characterization of diffusion anisotropy and the elucidation of fiber‐tract trajectories in the brain and in other fibrous tissues. Magn Reson Med 44:41–50, 2000. Published 2000 Wiley‐Liss, Inc.

Parametric and non-parametric statistical analysis of DT-MRI data.

In this work parametric and non-parametric statistical methods are proposed to analyze Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. A Multivariate Normal Distribution is proposed as a parametric statistical model of diffusion tensor data when magnitude MR images contain no artifacts other than Johnson noise. We test this model using Monte Carlo (MC) simulations of DT-MRI experiments. The non-parametric approach proposed here is an implementation of bootstrap methodology that we call the DT-MRI bootstrap. It is used to estimate an empirical probability distribution of experimental DT-MRI data, and to perform hypothesis tests on them. The DT-MRI bootstrap is also used to obtain various statistics of DT-MRI parameters within a single voxel, and within a region of interest (ROI); we also use the bootstrap to study the intrinsic variability of these parameters in the ROI, independent of background noise. We evaluate the DT-MRI bootstrap using MC simulations and apply it to DT-MRI data acquired on human brain in vivo, and on a phantom with uniform diffusion properties.

A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a statistical estimate of a symmetric, second-order diffusion tensor of water, D, in each voxel within an imaging volume. We propose a new normal distribution, p(D) /spl prop/ exp(-1/2 D : A : D), which describes the variability of D in an ideal DT-MRI experiment. The scalar invariant, D : A : D, is the contraction of a positive definite symmetric, fourth-order precision tensor, A, and D. A correspondence is established between D : A : D and the elastic strain energy density function in continuum mechanics-specifically between D and the second-order infinitesimal strain tensor, and between A and the fourth-order tensor of elastic coefficients. We show that A can be further classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When A is an isotropic fourth-order tensor, we derive an explicit analytic expression for p(D), and for the distribution of the three eigenvalues of D, p(/spl gamma//sub 1/, /spl gamma//sub 2/, /spl gamma//sub 3/), which are confirmed by Monte Carlo simulations. We show how A can be estimated from either real or synthetic DT-MRI data for any given experimental design. Here we propose a new criterion for an optimal experimental design: that A be an isotropic fourth-order tensor. This condition ensures that the statistical properties of D (and quantities derived from it) are rotationally invariant. We also investigate the degree of isotropy of several DT-MRI experimental designs. Finally, we show that the univariate and multivariate distributions are special cases of the more general tensor-variate normal distribution, and suggest how to generalize p(D) to treat normal random tensor variables that are of third- (or higher) order. We expect that this new distribution, p(D), should be useful in feature extraction; in developing a hypothesis testing framework for segmenting and classifying noisy, discrete tensor data; and in designing experiments to measure tensor quantities.

Noise characteristics of 3-D and 2-D PET images

The authors analyzed the noise characteristics of two-dimensional (2-D) and three-dimensional (3-D) images obtained from the GE Advance positron emission tomography (PET) scanner. Three phantoms were used: a uniform 20-cm phantom, a 3-D Hoffman brain phantom, and a chest phantom with heart and lung inserts. Using gated acquisition, the authors acquired 20 statistically equivalent scans of each phantom in 2-D and 3-D modes at several activity levels. From these data, they calculated pixel normalized standard deviations (NSDs), scaled to phantom mean, across the replicate scans, which allowed them to characterize the radial and axial distributions of pixel noise. The authors also performed sequential measurements of the phantoms in 2-D and 3-D modes to measure noise (from interpixel standard deviations) as a function of activity. To compensate for the difference in axial slice width between 2-D and 3-D images (due to the septa and reconstruction effects), they developed a smoothing kernel to apply to the 2-D data. After matching the resolution, the ratio of image-derived NSD values (NSD/sub 2D//NSD/sub 3D/)/sup 2/ averaged throughout the uniform phantom was in good agreement with the noise equivalent count (NEC) ratio (NEC/sub 3D//NEC/sub 2D/). By comparing different phantoms, the authors showed that the attenuation and emission distributions influence the spatial noise distribution. The estimates of pixel noise for 2-D and 3-D images produced here can be applied in the weighting of PET kinetic data and may be useful in the design of optimal dose and scanning requirements for PET studies. The accuracy of these phantom-based noise formulas should be validated for any given imaging situation, particularly in 3-D, if there is significant activity outside the scanner field of view.

Efficient Network Reconstruction from Dynamical Cascades Identifies Small-World Topology of Neuronal Avalanches

Cascading activity is commonly found in complex systems with directed interactions such as metabolic networks, neuronal networks, or disease spreading in social networks. Substantial insight into a systems organization can be obtained by reconstructing the underlying functional network architecture from the observed activity cascades. Here we focus on Bayesian approaches and reduce their computational demands by introducing the Iterative Bayesian (IB) and Posterior Weighted Averaging (PWA) methods. We introduce a special case of PWA, cast in nonparametric form, which we call the normalized count (NC) algorithm. NC efficiently reconstructs random and small-world functional network topologies and architectures from subcritical, critical, and supercritical cascading dynamics and yields significant improvements over commonly used correlation methods. With experimental data, NC identified a functional and structural small-world topology and its corresponding traffic in cortical networks with neuronal avalanche dynamics.

Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI

We propose a novel spectral decomposition of a 4th-order covariance tensor, @S. Just as the variability of vector (i.e., a 1st-order tensor)-valued random variable is characterized by a covariance matrix (i.e., a 2nd-order tensor), S, the variability of a 2nd-order tensor-valued random variable, D, is characterized by a 4th-order covariance tensor, @S. Accordingly, just as the spectral decomposition of S is a linear combination of its eigenvalues and the outer product of its corresponding (1st-order tensors) eigenvectors, the spectral decomposition of @S is a linear combination of its eigenvalues and the outer product of its corresponding 2nd-order eigentensors. Analogously, these eigenvalues and 2nd-order eigentensors can be used as features with which to represent and visualize variability in tensor-valued data. Here we suggest a framework to visualize the angular structure of @S, and then use it to assess and characterize the variability of synthetic diffusion tensor magnetic resonance imaging (DTI) data. The spectral decomposition suggests a hierarchy of symmetries with which to classify the statistical anisotropy inherent in tensor data. We also present maximum likelihood estimates of the sample mean and covariance tensors associated with D, and derive formulae for the expected value of the mean and variance of the projection of D along a particular direction (i.e., the apparent diffusion coefficient or ADC). These findings would be difficult, if not impossible, to glean if we treated 2nd-order tensor random variables as vector-valued random variables, which is conventionally done in multi-variate statistical analysis.

The organization of strong links in complex networks

Many complex systems reveal a small-world topology, which allows simultaneously local and global efficiency in the interaction between system constituents. Here, we report the results of a comprehensive study that investigates the relation between the clustering properties in such small-world systems and the strength of interactions between its constituents, quantified by the link weight. For brain, gene, social and language networks, we find a local integrative weight organization in which strong links preferentially occur between nodes with overlapping neighbourhoods; we relate this to global robustness of the clustering to removal of the weakest links. Furthermore, we identify local learning rules that establish integrative networks and improve network traffic in response to past traffic failures. Our findings identify a general organization for complex systems that strikes a balance between efficient local and global communication in their strong interactions, while allowing for robust, exploratory development of weak interactions.