Andrea Fuster

A Novel Riemannian Metric for Geodesic Tractography in DTI

One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor . Such a metric is used for white matter tractography and connectivity analysis. We propose a modified metric tensor given by the adjugate rather than the inverse diffusion tensor. Tractography experiments on real brain diffusion data show improvement in the vicinity of isotropic diffusion regions compared to results for inverse (sharpened) diffusion tensors.

Higher-Order Tensors in Diffusion Imaging

Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathematical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging. This survey gives a careful introduction to the foundations of higher-order tensor algebra, and explains how some concepts from linear algebra generalize to the higher-order case. From the application side, it reviews a variety of distinct higher-order tensor models that arise in the context of diffusion imaging, such as higher-order diffusion tensors, q-ball or fiber Orientation Distribution Functions (ODFs), and fourth-order covariance and kurtosis tensors. By bridging the gap between mathematical foundations and application, it provides an introduction that is suitable for practitioners and applied mathematicians alike, and propels the field by stimulating further exchange between the two.

Sheet Probability Index (SPI): Characterizing the geometrical organization of the white matter with diffusion MRI

The question whether our brain pathways adhere to a geometric grid structure has been a popular topic of debate in the diffusion imaging and neuroscience societies. Wedeen et al. (2012a, b) proposed that the brains white matter is organized like parallel sheets of interwoven pathways. Catani et al. (2012) concluded that this grid pattern is most likely an artifact, resulting from methodological biases that cause the tractography pathways to cross in orthogonal angles. To date, ambiguities in the mathematical conditions for a sheet structure to exist (e.g. its relation to orthogonal angles) combined with the lack of extensive quantitative evidence have prevented wide acceptance of the hypothesis. In this work, we formalize the relevant terminology and recapitulate the condition for a sheet structure to exist. Note that this condition is not related to the presence or absence of orthogonal crossing fibers, and that sheet structure is defined formally as a surface formed by two sets of interwoven pathways intersecting at arbitrary angles within the surface. To quantify the existence of sheet structure, we present a novel framework to compute the sheet probability index (SPI), which reflects the presence of sheet structure in discrete orientation data (e.g. fiber peaks derived from diffusion MRI). With simulation experiments we investigate the effect of spatial resolution, curvature of the fiber pathways, and measurement noise on the ability to detect sheet structure. In real diffusion MRI data experiments we can identify various regions where the data supports sheet structure (high SPI values), but also areas where the data does not support sheet structure (low SPI values) or where no reliable conclusion can be drawn. Several areas with high SPI values were found to be consistent across subjects, across multiple data sets obtained with different scanners, resolutions, and degrees of diffusion weighting, and across various modeling techniques. Under the strong assumption that the diffusion MRI peaks reflect true axons, our results would therefore indicate that pathways do not form sheet structures at every crossing fiber region but instead at well-defined locations in the brain. With this framework, sheet structure location, extent, and orientation could potentially serve as new structural features of brain tissue. The proposed method can be extended to quantify sheet structure in directional data obtained with techniques other than diffusion MRI, which is essential for further validation.

A Riemannian Scalar Measure for Diffusion Tensor Images

We study a well-known scalar quantity in differential geometry, the Ricci scalar, in the context of Diffusion Tensor Imaging (DTI). We explore the relation between the Ricci scalar and the two most popular scalar measures in DTI: Mean Diffusivity and Fractional Anisotropy. We discuss results of computing the Ricci scalar on synthetic as well as real DTI data.

Finsler pp-waves

In this work we present Finsler gravitational waves. These are a Finslerian version of the well-known pp-waves, generalizing the very special relativity line element. Our Finsler pp-waves are an exact solution of Finslerian Einstein’s equations in vacuum and describe gravitational waves propagating in an anisotropic background.

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor. Such a metric is used for geodesic tractography and connectivity analysis in white matter. We propose a metric tensor given by the adjugate rather than the previously proposed inverse diffusion tensor. The adjugate metric can also be employed in the sharpening framework. Tractography experiments on synthetic and real brain diffusion data show improvement for high-curvature tracts and in the vicinity of isotropic diffusion regions relative to most results for inverse (sharpened) diffusion tensors, and especially on real data. In addition, adjugate tensors are shown to be more robust to noise.

Myocardial deformation from local frequency estimation in tagging MRI

We consider a new method to analyse deformation of the myocardial wall from tagging magnetic resonance images. The method exploits the fact that a regular pattern of stripe tags induces a time-dependent frequency covector field tightly coupled to the myocardial tissue and not affected by tag fading. The corresponding local frequency can be disambiguated with the help of the Gabor transform. The transformation of the tagging frequency covector field is governed by the deformation tensor field. Reversely, the deformation (and strain) tensor field can be retrieved from local frequency estimates given at least n (independent) tagging sequences, where n denotes spatial dimension. For the sake of illustration we consider the conventional case n=2. Moreover, we make use of an overdetermined system by exploiting 4 instead of 2 tagging directions, which contributes to the robustness of the results. The method does not require explicit knowledge of material motion or tag line extraction. Displacement estimations are compared to HARP.

Quantifying the brain's sheet structure with normalized convolution

HighlightsA new method is proposed to quantify the extent of sheet structure in the brain.Clustering and computing derivatives of diffusion MRI fiber directions is required.Normalized convolution is adopted to calculate these derivatives.The reliability of the method is demonstrated with simulations and experimental data.The method is more robust than a recent method that is based on tractography. ABSTRACT The hypothesis that brain pathways form 2D sheet‐like structures layered in 3D as “pages of a book” has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of “path neighborhoods” reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications of the sheet structure hypothesis for our understanding of brain structure and development, it is still considered controversial by many for lack of quantitative analysis. A means to quantify sheet structure is therefore necessary to reliably investigate its occurrence in the brain. Previous work has proposed the Lie bracket as a quantitative indicator of sheet structure, which could be computed by reconstructing path neighborhoods from the peak orientations of dMRI orientation density functions. Robust estimation of the Lie bracket, however, is challenging due to high noise levels and missing peak orientations. We propose a novel method to estimate the Lie bracket that does not involve the reconstruction of path neighborhoods with tractography. This method requires the computation of derivatives of the fiber peak orientations, for which we adopt an approach called normalized convolution. With simulations and experimental data we show that the new approach is more robust with respect to missing peaks and noise. We also demonstrate that the method is able to quantify to what extent sheet structure is supported for dMRI data of different species, acquired with different scanners, diffusion weightings, dMRI sampling schemes, and spatial resolutions. The proposed method can also be used with directional data derived from other techniques than dMRI, which will facilitate further validation of the existence of sheet structure. Graphical abstract Figure. No Caption available.

3D winding number: theory and application to medical imaging

We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological number, which is the generalization to three dimensions of the 2D winding number. We illustrate our method by considering three different biomedical applications, namely, detection and counting of ovarian follicles and neuronal cells and estimation of cardiac motion from tagged MR images. Qualitative and quantitative evaluation emphasizes the reliability of the results.

Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

We consider Riemann-Finsler geometry as a potentially powerful mathematical framework in the context of diffusion weighted magnetic resonance imaging. We explain its basic features in heuristic terms, but also provide mathematical details that are essential for practical applications, such as tractography and voxel-based classification. We stipulate a connection between the (dual) Finsler function and signal attenuation observed in the MRI scanner, which directly generalizes Stejskal-Tanner’s solution of the Bloch-Torrey equations and the diffusion tensor imaging (DTI) model inspired by this. The proposed model can therefore be regarded as an extension of DTI. Technically, reconstruction of the (dual) Finsler function from diffusion weighted measurements is a fairly straightforward generalization of the DTI case. The extension of the Riemann differential geometric paradigm for DTI analysis is, however, nontrivial.