Higher-Order Tensors and Differential Topology in Diffusion MRI Modeling and Visualization

Dissertation, University of Bonn, Mai 2019
 

Abstract

Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a noninvasive method for creating three-dimensional scans of the human brain. It originated mostly in the 1970s and started its use in clinical applications in the 1980s. Due to its low risk and relatively high image quality it proved to be an indispensable tool for studying medical conditions as well as for general scientific research. For example, it allows to map fiber bundles, the major neuronal pathways through the brain. But all evaluation of scanned data depends on mathematical signal models that describe the raw signal output and map it to biologically more meaningful values. And here we find the most potential for improvement. In this thesis we first present a new multi-tensor kurtosis signal model for DW-MRI. That means it can detect multiple overlapping fiber bundles and map them to a set of tensors. Compared to other already widely used multi-tensor models, we also add higher order kurtosis terms to each fiber. This gives a more detailed quantification of fibers. These additional values can also be estimated by the Diffusion Kurtosis Imaging (DKI) method, but we show that these values are drastically affected by fiber crossings in DKI, whereas our model handles them as intrinsic properties of fiber bundles. This reduces the effects of fiber crossings and allows a more direct examination of fibers. Next, we take a closer look at spherical deconvolution. It can be seen as a generalization of multi-fiber signal models to a continuous distribution of fiber directions. To this approach we introduce a novel mathematical constraint. We show, that state-of-the-art methods for estimating the fiber distribution become more robust and gain accuracy when enforcing our constraint. Additionally, in the context of our own deconvolution scheme, it is algebraically equivalent to enforcing that the signal can be decomposed into fibers. This means, tractography and other methods that depend on identifying a discrete set of fiber directions greatly benefit from our constraint. Our third major contribution to DW-MRI deals with macroscopic structures of fiber bundle geometry. In recent years the question emerged, whether or not, crossing bundles form two-dimensional surfaces inside the brain. Although not completely obvious, there is a mathematical obstacle coming from differential topology, that prevents general tangential planes spanned by fiber directions at each point to be connected into consistent surfaces. Research into how well this constraint is fulfilled in our brain is hindered by the high precision and complexity needed by previous evaluation methods. This is why we present a drastically simpler method that negates the need for precisely finding fiber directions and instead only depends on the simple diffusion tensor method (DTI). We then use our new method to explore and improve streamsurface visualization.

Download: https://nbn-resolving.org/urn:nbn:de:hbz:5n-54149

Bibtex

@PHDTHESIS{ankele2019phd,
    author = {Ankele, Michael},
     title = {Higher-Order Tensors and Differential Topology in Diffusion MRI Modeling and Visualization},
      type = {Dissertation},
      year = {2019},
     month = may,
    school = {University of Bonn},
  abstract = {Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a noninvasive method for creating
              three-dimensional scans of the human brain. It originated mostly in the 1970s and started its use in
              clinical applications in the 1980s. Due to its low risk and relatively high image quality it proved
              to be an indispensable tool for studying medical conditions as well as for general scientific
              research. For example, it allows to map fiber bundles, the major neuronal pathways through the
              brain. But all evaluation of scanned data depends on mathematical signal models that describe the
              raw signal output and map it to biologically more meaningful values. And here we find the most
              potential for improvement.
              In this thesis we first present a new multi-tensor kurtosis signal model for DW-MRI. That means it
              can detect multiple overlapping fiber bundles and map them to a set of tensors. Compared to other
              already widely used multi-tensor models, we also add higher order kurtosis terms to each fiber. This
              gives a more detailed quantification of fibers. These additional values can also be estimated by the
              Diffusion Kurtosis Imaging (DKI) method, but we show that these values are drastically affected by
              fiber crossings in DKI, whereas our model handles them as intrinsic properties of fiber bundles.
              This reduces the effects of fiber crossings and allows a more direct examination of fibers.
              Next, we take a closer look at spherical deconvolution. It can be seen as a generalization of
              multi-fiber signal models to a continuous distribution of fiber directions. To this approach we
              introduce a novel mathematical constraint. We show, that state-of-the-art methods for estimating the
              fiber distribution become more robust and gain accuracy when enforcing our constraint. Additionally,
              in the context of our own deconvolution scheme, it is algebraically equivalent to enforcing that the
              signal can be decomposed into fibers. This means, tractography and other methods that depend on
              identifying a discrete set of fiber directions greatly benefit from our constraint.
              Our third major contribution to DW-MRI deals with macroscopic structures of fiber bundle geometry.
              In recent years the question emerged, whether or not, crossing bundles form two-dimensional surfaces
              inside the brain. Although not completely obvious, there is a mathematical obstacle coming from
              differential topology, that prevents general tangential planes spanned by fiber directions at each
              point to be connected into consistent surfaces. Research into how well this constraint is fulfilled
              in our brain is hindered by the high precision and complexity needed by previous evaluation methods.
              This is why we present a drastically simpler method that negates the need for precisely finding
              fiber directions and instead only depends on the simple diffusion tensor method (DTI). We then use
              our new method to explore and improve streamsurface visualization.},
       url = {https://nbn-resolving.org/urn:nbn:de:hbz:5n-54149}
}