Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images

Abstract

This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process Key words: DT-MRI; tractography; tensor; invariant; anisotropyThe utility of magnetic resonance imaging (MRI) to characterize biologic tissue is amplified by analysis and visualization methods that help researchers to see and understand the structures within the data. Tensor-valued imaging is an increasingly important source of information about tissue structure and dynamics, such as with diffusion tensor MRI (DT-MRI), and strain tensors derived from displacement encoded MRI. An effective and established way of describing structure in tensor fields is through a function that maps tensors to more readily understood scalar metrics. The medical imaging literature provides a variety of such metrics (e.g., bulk mean diffusivity, fractional anisotropy).This paper uses a combination of mathematics and visualizations to generate a rigorous and intuitive exposition of tensor shape, characterized by sets of orthogonal tensor invariants. Each invariant set decomposes tensor shape with an orthogonal basis. When invariants have application-specific significance (as they do in DT-MRI imaging), orthogonality is a useful property of an invariant set, because it isolates the measurement of variation in one physiologic property from variations in another. These sets of orthogonal tensor invariants are then used to create informative visualizations of tensor field structure. The result is the establishment of two sets of orthogonal tensor invariants that incorporate the established use of fractional anisotropy (FA) and apparent diffusion coefficient (ADC).Diffusion tensors are symmetric and thus can be decomposed into an eigensystem of three real eigenvalues and three mutually orthogonal eigenvectors. We adopt the terminology that tensor "shape" refers to those degrees of freedom in tensor values (components of the matrix representation of a tensor) spanned by changes in the eigenvalues, while keeping eigenvectors fixed. "Orientation," on the other hand, refers to the complementary degrees of freedom associated with changes in the eigenvectors, while keeping the eigenvalues fixed. Defining shape and orientation in terms of the tensor eigensystem coincides with the standard visuali...

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