Supertoroid-based Fusion of Cardiac Dt-mri with Molecular and Physiological Information

INTRODUCTION The supertoroid-based representation is an evolution of the toroidal model that enhances the three-dimensional perception of myocardial tissue structure and organization using DT-MRI [1]. The supertoroidal model, as a superset of genus-0 and genus-1 shapes, improves the effectiveness of the diffusion tensor characterization of myocardial architecture post myocardial infarction (MI) [1]. However, successful characterization of tissue properties relies on relating structural information with physiological or molecular information obtained from different sources. Because the supertoroidal model offers the ability to incorporate two additional parameters, γ1 and γ 2 , we are able to fuse standard structural DT-MRI information with parameters from other modalities. Changes in myocardial structure and function are dependent on alterations in the extracellular matrix (ECM), which is modulated by activation of matrix metalloproteinases (MMPs). The purpose of this work is to employ the supertoroids to combine the regional structural information derived from cardiac DT-MRI, with regional activation of MMPs, and associated changes in myocardial perfusion and cellular viability in porcine heart at 2 weeks post-MI. METHODOLOGY The supertoroid is a geometric primitive that incorporates the visual features conveyed by the increase in genus and a continuum that fully encodes the local eigensystem. 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 [2] and is parameterized as follows: CS ≥ C P⇒I θ,φ ( )= cos 1−Cp ( ) γ1 θ CL + CP ( )+ CS cos 1−CS ( ) γ2 φ { },sin 1−Cp ( )1 θ CL + CP ( )+ CS cos 1−CS ( )2 φ { },sin 1−CS ( )2 φ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ,