Supertoroid-based Characterization of Cardiac Diffusion Tensor Fields

Introduction Characterization of biological tissue structure and organization using Diffusion Tensor (DT) Magnetic Resonance Imaging (MRI) relies on effective analysis and multidimensional data visualization methods. The toroid-based representation of the DT has been experimentally demonstrated to be less prone to visual ambiguity [1] and offers additional quantitative scalar maps to evaluate the diffusivity and the degree of anisotropy [2]. The purpose of this work is to introduce the supertoroids, which are the natural evolution of the toroid-based model to address the limitations of the superquadrics [3] and toroidal glyphs, by unifying the specific advantages of each representation. The methodology is applied on DT-MRI datasets of a normal and infarcted canine hearts. Results indicate that supertoroids enhance cardiac myofiber structure characterization compared to ellipsoidal, superquadrics and toroids. Methodology The supertoroid is a geometric primitive that incorporates the visual features conveyed by the increase in genus inherent to the toroids and a continuum that fully encodes the local eigensystem intrinsic to the superquadrics. Supertoroidal representation: The supertoroidal parametric equation is a function of the geometric shape metrics CL = λ1 −λ2 ( ) λ1 + λ2 + λ3 , CP = 2 λ2 − λ3 ( ) λ1 + λ2 + λ3 , and CS = 3λ3 λ1 + λ2 + λ3 [4] and is parameterized as follows: CS ≥ C P⇒I θ,φ ( )= cos1 θ CL +CP ( )+CS cos2 φ { },sin1 θ CL +CP ( )+CS cos2 φ { },sin2 φ ( ), CS < C P⇒I θ,φ ( )= cos1 θ CS + CL +CP ( )cos2 φ { },sin1 θ CS + CL +CP ( )cos2 φ { },sin2 φ ( ),