Comparison of Different Parallel Transport Methods for the Study of Deformations in 3D Cardiac Data

Abstract

Comparing the deformations of different beating hearts is a challenging operation. As in clinics the impaired condition is often recognized upon (local and global) deformation parameters, the particular nature of heart deformation during one beat can be compared among different individuals in the same ordination space more effectively if initial inter-individual form (shape + size) differences are filtered out. This is even more true if the shape of cardiac trajectory itself is under consideration. This need is satisfied by applying a geometric machinery named “parallel transport” in the field of differential geometry. In recent years several parallel transport methods have been applied to cardiological data acquired via echocardiography, CT scan or magnetic resonance. Concomitantly, some efforts were made for comparing different parallel transport algorithms applied to a variety of toy examples and real deformational data. Here we face the problem of comparing the heavily used LDDMM parallel transport with the recently proposed Riemannian “TPS space” in the context of the deformation of the right ventricle. Using local tensors diagnostics and global energy-based and shape distance-based parameters, we explored the maintenance of original deformations in transported data in four systo-diastolic deformations belonging to one healthy subject and three individuals affected by tetralogy of Fallot, atrial septal defect and pulmonary hypertension. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. We also do the same in a larger dataset relative to the left ventricle of 82 heathly subjects and 21 patients affected by hypertrophic cardiomyopathy. In particular, we contrasted the TPS space with classic LDDMM and a modified LDDMM able to manage spherical differences. Our results point toward a neat superiority of TPS space over classic LDDMM. The modified LDDMM performs similarly as it maintains better the chosen diagnostics.